LMFDB - Embedded newform 1045.2.a.i.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 28x^{4} - 17x^{3} - 28x^{2} + 6x + 8 $ x^8 - 2x^7 - 9x^6 + 12x^5 + 28x^4 - 17x^3 - 28x^2 + 6*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.06639$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0663929 q^{2} -0.255194 q^{3} -1.99559 q^{4} +1.00000 q^{5} -0.0169431 q^{6} +2.56056 q^{7} -0.265279 q^{8} -2.93488 q^{9} +O(q^{10})$	\(q+0.0663929 q^{2} -0.255194 q^{3} -1.99559 q^{4} +1.00000 q^{5} -0.0169431 q^{6} +2.56056 q^{7} -0.265279 q^{8} -2.93488 q^{9} +0.0663929 q^{10} +1.00000 q^{11} +0.509263 q^{12} -6.37708 q^{13} +0.170003 q^{14} -0.255194 q^{15} +3.97357 q^{16} -0.997292 q^{17} -0.194855 q^{18} -1.00000 q^{19} -1.99559 q^{20} -0.653439 q^{21} +0.0663929 q^{22} +4.24483 q^{23} +0.0676976 q^{24} +1.00000 q^{25} -0.423393 q^{26} +1.51454 q^{27} -5.10983 q^{28} -10.4615 q^{29} -0.0169431 q^{30} +0.460672 q^{31} +0.794375 q^{32} -0.255194 q^{33} -0.0662131 q^{34} +2.56056 q^{35} +5.85682 q^{36} -11.7533 q^{37} -0.0663929 q^{38} +1.62739 q^{39} -0.265279 q^{40} -3.65137 q^{41} -0.0433837 q^{42} +0.845851 q^{43} -1.99559 q^{44} -2.93488 q^{45} +0.281827 q^{46} +9.94088 q^{47} -1.01403 q^{48} -0.443552 q^{49} +0.0663929 q^{50} +0.254503 q^{51} +12.7260 q^{52} +0.244293 q^{53} +0.100555 q^{54} +1.00000 q^{55} -0.679262 q^{56} +0.255194 q^{57} -0.694572 q^{58} -12.7989 q^{59} +0.509263 q^{60} -7.48745 q^{61} +0.0305854 q^{62} -7.51491 q^{63} -7.89440 q^{64} -6.37708 q^{65} -0.0169431 q^{66} -12.8572 q^{67} +1.99019 q^{68} -1.08326 q^{69} +0.170003 q^{70} -3.59925 q^{71} +0.778561 q^{72} -1.87705 q^{73} -0.780337 q^{74} -0.255194 q^{75} +1.99559 q^{76} +2.56056 q^{77} +0.108047 q^{78} +13.1277 q^{79} +3.97357 q^{80} +8.41813 q^{81} -0.242425 q^{82} -13.7505 q^{83} +1.30400 q^{84} -0.997292 q^{85} +0.0561585 q^{86} +2.66972 q^{87} -0.265279 q^{88} +3.61316 q^{89} -0.194855 q^{90} -16.3289 q^{91} -8.47095 q^{92} -0.117561 q^{93} +0.660004 q^{94} -1.00000 q^{95} -0.202720 q^{96} +1.69508 q^{97} -0.0294487 q^{98} -2.93488 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) $ 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 17 q^{13} + 12 q^{14} - 7 q^{15} + 18 q^{16} - 9 q^{17} - 2 q^{18} - 8 q^{19} + 10 q^{20} + q^{21} - 6 q^{22} - 8 q^{23} + q^{24} + 8 q^{25} + 10 q^{26} - 34 q^{27} - 22 q^{28} - 3 q^{29} - q^{31} - 37 q^{32} - 7 q^{33} - 8 q^{34} - 11 q^{35} + 30 q^{36} - 17 q^{37} + 6 q^{38} + 14 q^{39} - 18 q^{40} - 5 q^{41} + 15 q^{42} - 21 q^{43} + 10 q^{44} + 11 q^{45} - 2 q^{46} - 8 q^{47} + 10 q^{48} + 19 q^{49} - 6 q^{50} - 16 q^{51} + 9 q^{52} - 19 q^{53} - 3 q^{54} + 8 q^{55} + 24 q^{56} + 7 q^{57} + 37 q^{58} - 33 q^{59} - 7 q^{60} - q^{61} - 42 q^{62} - 20 q^{63} + 48 q^{64} - 17 q^{65} - 18 q^{67} - 37 q^{68} + 16 q^{69} + 12 q^{70} - 18 q^{71} + 13 q^{72} - 18 q^{73} + 15 q^{74} - 7 q^{75} - 10 q^{76} - 11 q^{77} - 51 q^{78} - 5 q^{79} + 18 q^{80} + 32 q^{81} + 12 q^{82} - 33 q^{83} - 51 q^{84} - 9 q^{85} - 16 q^{86} - 26 q^{87} - 18 q^{88} - 20 q^{89} - 2 q^{90} + 6 q^{91} - 3 q^{92} + 18 q^{93} + 30 q^{94} - 8 q^{95} + 21 q^{96} - 69 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 17 * q^13 + 12 * q^14 - 7 * q^15 + 18 * q^16 - 9 * q^17 - 2 * q^18 - 8 * q^19 + 10 * q^20 + q^21 - 6 * q^22 - 8 * q^23 + q^24 + 8 * q^25 + 10 * q^26 - 34 * q^27 - 22 * q^28 - 3 * q^29 - q^31 - 37 * q^32 - 7 * q^33 - 8 * q^34 - 11 * q^35 + 30 * q^36 - 17 * q^37 + 6 * q^38 + 14 * q^39 - 18 * q^40 - 5 * q^41 + 15 * q^42 - 21 * q^43 + 10 * q^44 + 11 * q^45 - 2 * q^46 - 8 * q^47 + 10 * q^48 + 19 * q^49 - 6 * q^50 - 16 * q^51 + 9 * q^52 - 19 * q^53 - 3 * q^54 + 8 * q^55 + 24 * q^56 + 7 * q^57 + 37 * q^58 - 33 * q^59 - 7 * q^60 - q^61 - 42 * q^62 - 20 * q^63 + 48 * q^64 - 17 * q^65 - 18 * q^67 - 37 * q^68 + 16 * q^69 + 12 * q^70 - 18 * q^71 + 13 * q^72 - 18 * q^73 + 15 * q^74 - 7 * q^75 - 10 * q^76 - 11 * q^77 - 51 * q^78 - 5 * q^79 + 18 * q^80 + 32 * q^81 + 12 * q^82 - 33 * q^83 - 51 * q^84 - 9 * q^85 - 16 * q^86 - 26 * q^87 - 18 * q^88 - 20 * q^89 - 2 * q^90 + 6 * q^91 - 3 * q^92 + 18 * q^93 + 30 * q^94 - 8 * q^95 + 21 * q^96 - 69 * q^98 + 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.0663929	0.0469469	0.0234734	−	0.999724i	$-0.492527\pi$
$2$	0.0663929	0.0469469	0.0234734	+	0.999724i	$0.492527\pi$
$3$	−0.255194	−0.147336	−0.0736682	−	0.997283i	$-0.523471\pi$
$3$	−0.255194	−0.147336	−0.0736682	+	0.997283i	$0.523471\pi$
$4$	−1.99559	−0.997796
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.0169431	−0.00691698
$7$	2.56056	0.967799	0.483900	−	0.875124i	$-0.339220\pi$
$7$	2.56056	0.967799	0.483900	+	0.875124i	$0.339220\pi$
$8$	−0.265279	−0.0937903
$9$	−2.93488	−0.978292
$10$	0.0663929	0.0209953
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.509263	0.147012
$13$	−6.37708	−1.76868	−0.884342	−	0.466840i	$-0.845392\pi$
$13$	−6.37708	−1.76868	−0.884342	+	0.466840i	$0.845392\pi$
$14$	0.170003	0.0454351
$15$	−0.255194	−0.0658908
$16$	3.97357	0.993393
$17$	−0.997292	−0.241879	−0.120939	−	0.992660i	$-0.538591\pi$
$17$	−0.997292	−0.241879	−0.120939	+	0.992660i	$0.538591\pi$
$18$	−0.194855	−0.0459278
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.99559	−0.446228
$21$	−0.653439	−0.142592
$22$	0.0663929	0.0141550
$23$	4.24483	0.885109	0.442554	−	0.896742i	$-0.354072\pi$
$23$	4.24483	0.885109	0.442554	+	0.896742i	$0.354072\pi$
$24$	0.0676976	0.0138187
$25$	1.00000	0.200000
$26$	−0.423393	−0.0830342
$27$	1.51454	0.291474
$28$	−5.10983	−0.965666
$29$	−10.4615	−1.94266	−0.971329	−	0.237739i	$-0.923594\pi$
$29$	−10.4615	−1.94266	−0.971329	+	0.237739i	$0.923594\pi$
$30$	−0.0169431	−0.00309337
$31$	0.460672	0.0827392	0.0413696	−	0.999144i	$-0.486828\pi$
$31$	0.460672	0.0827392	0.0413696	+	0.999144i	$0.486828\pi$
$32$	0.794375	0.140427
$33$	−0.255194	−0.0444236
$34$	−0.0662131	−0.0113555
$35$	2.56056	0.432813
$36$	5.85682	0.976136
$37$	−11.7533	−1.93223	−0.966117	−	0.258104i	$-0.916902\pi$
$37$	−11.7533	−1.93223	−0.966117	+	0.258104i	$0.916902\pi$
$38$	−0.0663929	−0.0107704
$39$	1.62739	0.260591
$40$	−0.265279	−0.0419443
$41$	−3.65137	−0.570249	−0.285124	−	0.958491i	$-0.592035\pi$
$41$	−3.65137	−0.570249	−0.285124	+	0.958491i	$0.592035\pi$
$42$	−0.0433837	−0.00669425
$43$	0.845851	0.128991	0.0644955	−	0.997918i	$-0.479456\pi$
$43$	0.845851	0.128991	0.0644955	+	0.997918i	$0.479456\pi$
$44$	−1.99559	−0.300847
$45$	−2.93488	−0.437505
$46$	0.281827	0.0415531
$47$	9.94088	1.45003	0.725013	−	0.688735i	$-0.241834\pi$
$47$	9.94088	1.45003	0.725013	+	0.688735i	$0.241834\pi$
$48$	−1.01403	−0.146363
$49$	−0.443552	−0.0633646
$50$	0.0663929	0.00938937
$51$	0.254503	0.0356375
$52$	12.7260	1.76479
$53$	0.244293	0.0335562	0.0167781	−	0.999859i	$-0.494659\pi$
$53$	0.244293	0.0335562	0.0167781	+	0.999859i	$0.494659\pi$
$54$	0.100555	0.0136838
$55$	1.00000	0.134840
$56$	−0.679262	−0.0907702
$57$	0.255194	0.0338013
$58$	−0.694572	−0.0912017
$59$	−12.7989	−1.66627	−0.833137	−	0.553067i	$-0.813457\pi$
$59$	−12.7989	−1.66627	−0.833137	+	0.553067i	$0.813457\pi$
$60$	0.509263	0.0657456
$61$	−7.48745	−0.958670	−0.479335	−	0.877632i	$-0.659122\pi$
$61$	−7.48745	−0.958670	−0.479335	+	0.877632i	$0.659122\pi$
$62$	0.0305854	0.00388435
$63$	−7.51491	−0.946790
$64$	−7.89440	−0.986800
$65$	−6.37708	−0.790979
$66$	−0.0169431	−0.00208555
$67$	−12.8572	−1.57075	−0.785377	−	0.619018i	$-0.787531\pi$
$67$	−12.8572	−1.57075	−0.785377	+	0.619018i	$0.787531\pi$
$68$	1.99019	0.241346
$69$	−1.08326	−0.130409
$70$	0.170003	0.0203192
$71$	−3.59925	−0.427153	−0.213576	−	0.976926i	$-0.568511\pi$
$71$	−3.59925	−0.427153	−0.213576	+	0.976926i	$0.568511\pi$
$72$	0.778561	0.0917543
$73$	−1.87705	−0.219692	−0.109846	−	0.993949i	$-0.535036\pi$
$73$	−1.87705	−0.219692	−0.109846	+	0.993949i	$0.535036\pi$
$74$	−0.780337	−0.0907124
$75$	−0.255194	−0.0294673
$76$	1.99559	0.228910
$77$	2.56056	0.291802
$78$	0.108047	0.0122339
$79$	13.1277	1.47698	0.738490	−	0.674264i	$-0.235539\pi$
$79$	13.1277	1.47698	0.738490	+	0.674264i	$0.235539\pi$
$80$	3.97357	0.444259
$81$	8.41813	0.935347
$82$	−0.242425	−0.0267714
$83$	−13.7505	−1.50932	−0.754659	−	0.656118i	$-0.772197\pi$
$83$	−13.7505	−1.50932	−0.754659	+	0.656118i	$0.772197\pi$
$84$	1.30400	0.142278
$85$	−0.997292	−0.108172
$86$	0.0561585	0.00605573
$87$	2.66972	0.286224
$88$	−0.265279	−0.0282788
$89$	3.61316	0.382994	0.191497	−	0.981493i	$-0.438666\pi$
$89$	3.61316	0.382994	0.191497	+	0.981493i	$0.438666\pi$
$90$	−0.194855	−0.0205395
$91$	−16.3289	−1.71173
$92$	−8.47095	−0.883158
$93$	−0.117561	−0.0121905
$94$	0.660004	0.0680742
$95$	−1.00000	−0.102598
$96$	−0.202720	−0.0206900
$97$	1.69508	0.172109	0.0860545	−	0.996290i	$-0.472574\pi$
$97$	1.69508	0.172109	0.0860545	+	0.996290i	$0.472574\pi$
$98$	−0.0294487	−0.00297477
$99$	−2.93488	−0.294966
$100$	−1.99559	−0.199559
$101$	−11.0884	−1.10334	−0.551669	−	0.834063i	$-0.686009\pi$
$101$	−11.0884	−1.10334	−0.551669	+	0.834063i	$0.686009\pi$
$102$	0.0168972	0.00167307
$103$	−12.2645	−1.20845	−0.604226	−	0.796813i	$-0.706518\pi$
$103$	−12.2645	−1.20845	−0.604226	+	0.796813i	$0.706518\pi$
$104$	1.69170	0.165885
$105$	−0.653439	−0.0637691
$106$	0.0162193	0.00157536
$107$	3.72084	0.359707	0.179854	−	0.983693i	$-0.442438\pi$
$107$	3.72084	0.359707	0.179854	+	0.983693i	$0.442438\pi$
$108$	−3.02241	−0.290832
$109$	11.1518	1.06814	0.534072	−	0.845439i	$-0.320661\pi$
$109$	11.1518	1.06814	0.534072	+	0.845439i	$0.320661\pi$
$110$	0.0663929	0.00633032
$111$	2.99938	0.284688
$112$	10.1746	0.961405
$113$	6.22765	0.585848	0.292924	−	0.956136i	$-0.405372\pi$
$113$	6.22765	0.585848	0.292924	+	0.956136i	$0.405372\pi$
$114$	0.0169431	0.00158686
$115$	4.24483	0.395833
$116$	20.8770	1.93838
$117$	18.7159	1.73029
$118$	−0.849755	−0.0782263
$119$	−2.55362	−0.234090
$120$	0.0676976	0.00617992
$121$	1.00000	0.0909091
$122$	−0.497114	−0.0450066
$123$	0.931808	0.0840183
$124$	−0.919314	−0.0825569
$125$	1.00000	0.0894427
$126$	−0.498937	−0.0444488
$127$	−10.4223	−0.924832	−0.462416	−	0.886663i	$-0.653017\pi$
$127$	−10.4223	−0.924832	−0.462416	+	0.886663i	$0.653017\pi$
$128$	−2.11288	−0.186754
$129$	−0.215856	−0.0190051
$130$	−0.423393	−0.0371340
$131$	7.90946	0.691053	0.345526	−	0.938409i	$-0.387700\pi$
$131$	7.90946	0.691053	0.345526	+	0.938409i	$0.387700\pi$
$132$	0.509263	0.0443257
$133$	−2.56056	−0.222028
$134$	−0.853625	−0.0737419
$135$	1.51454	0.130351
$136$	0.264561	0.0226859
$137$	9.25621	0.790812	0.395406	−	0.918506i	$-0.370604\pi$
$137$	9.25621	0.790812	0.395406	+	0.918506i	$0.370604\pi$
$138$	−0.0719205	−0.00612228
$139$	18.2599	1.54878	0.774391	−	0.632707i	$-0.218056\pi$
$139$	18.2599	1.54878	0.774391	+	0.632707i	$0.218056\pi$
$140$	−5.10983	−0.431859
$141$	−2.53685	−0.213642
$142$	−0.238965	−0.0200535
$143$	−6.37708	−0.533278
$144$	−11.6619	−0.971828
$145$	−10.4615	−0.868783
$146$	−0.124623	−0.0103139
$147$	0.113192	0.00933590
$148$	23.4548	1.92798
$149$	−4.03498	−0.330558	−0.165279	−	0.986247i	$-0.552852\pi$
$149$	−4.03498	−0.330558	−0.165279	+	0.986247i	$0.552852\pi$
$150$	−0.0169431	−0.00138340
$151$	13.6255	1.10883	0.554413	−	0.832242i	$-0.312943\pi$
$151$	13.6255	1.10883	0.554413	+	0.832242i	$0.312943\pi$
$152$	0.265279	0.0215170
$153$	2.92693	0.236628
$154$	0.170003	0.0136992
$155$	0.460672	0.0370021
$156$	−3.24761	−0.260017
$157$	−2.37388	−0.189456	−0.0947280	−	0.995503i	$-0.530198\pi$
$157$	−2.37388	−0.189456	−0.0947280	+	0.995503i	$0.530198\pi$
$158$	0.871586	0.0693396
$159$	−0.0623421	−0.00494405
$160$	0.794375	0.0628008
$161$	10.8691	0.856607
$162$	0.558904	0.0439116
$163$	−14.8206	−1.16084	−0.580421	−	0.814316i	$-0.697112\pi$
$163$	−14.8206	−1.16084	−0.580421	+	0.814316i	$0.697112\pi$
$164$	7.28665	0.568992
$165$	−0.255194	−0.0198668
$166$	−0.912938	−0.0708577
$167$	−7.66518	−0.593149	−0.296575	−	0.955010i	$-0.595844\pi$
$167$	−7.66518	−0.593149	−0.296575	+	0.955010i	$0.595844\pi$
$168$	0.173343	0.0133737
$169$	27.6671	2.12824
$170$	−0.0662131	−0.00507831
$171$	2.93488	0.224436
$172$	−1.68797	−0.128707
$173$	19.8851	1.51184	0.755919	−	0.654665i	$-0.227190\pi$
$173$	19.8851	1.51184	0.755919	+	0.654665i	$0.227190\pi$
$174$	0.177251	0.0134373
$175$	2.56056	0.193560
$176$	3.97357	0.299519
$177$	3.26620	0.245503
$178$	0.239888	0.0179804
$179$	12.1637	0.909159	0.454579	−	0.890706i	$-0.349790\pi$
$179$	12.1637	0.909159	0.454579	+	0.890706i	$0.349790\pi$
$180$	5.85682	0.436541
$181$	12.5315	0.931456	0.465728	−	0.884928i	$-0.345793\pi$
$181$	12.5315	0.931456	0.465728	+	0.884928i	$0.345793\pi$
$182$	−1.08412	−0.0803604
$183$	1.91075	0.141247
$184$	−1.12606	−0.0830146
$185$	−11.7533	−0.864121
$186$	−0.00780521	−0.000572305 0
$187$	−0.997292	−0.0729292
$188$	−19.8379	−1.44683
$189$	3.87808	0.282089
$190$	−0.0663929	−0.00481665
$191$	24.9196	1.80312	0.901559	−	0.432656i	$-0.142424\pi$
$191$	24.9196	1.80312	0.901559	+	0.432656i	$0.142424\pi$
$192$	2.01460	0.145392
$193$	−3.57666	−0.257454	−0.128727	−	0.991680i	$-0.541089\pi$
$193$	−3.57666	−0.257454	−0.128727	+	0.991680i	$0.541089\pi$
$194$	0.112541	0.00807998
$195$	1.62739	0.116540
$196$	0.885149	0.0632249
$197$	−15.6847	−1.11749	−0.558744	−	0.829340i	$-0.688716\pi$
$197$	−15.6847	−1.11749	−0.558744	+	0.829340i	$0.688716\pi$
$198$	−0.194855	−0.0138477
$199$	23.0476	1.63380	0.816902	−	0.576776i	$-0.195690\pi$
$199$	23.0476	1.63380	0.816902	+	0.576776i	$0.195690\pi$
$200$	−0.265279	−0.0187581
$201$	3.28107	0.231429
$202$	−0.736191	−0.0517982
$203$	−26.7873	−1.88010
$204$	−0.507884	−0.0355590
$205$	−3.65137	−0.255023
$206$	−0.814273	−0.0567331
$207$	−12.4581	−0.865895
$208$	−25.3398	−1.75700
$209$	−1.00000	−0.0691714
$210$	−0.0433837	−0.00299376
$211$	6.54118	0.450313	0.225157	−	0.974323i	$-0.427711\pi$
$211$	6.54118	0.450313	0.225157	+	0.974323i	$0.427711\pi$
$212$	−0.487509	−0.0334822
$213$	0.918507	0.0629351
$214$	0.247037	0.0168871
$215$	0.845851	0.0576866
$216$	−0.401777	−0.0273375
$217$	1.17958	0.0800750
$218$	0.740397	0.0501460
$219$	0.479013	0.0323687
$220$	−1.99559	−0.134543
$221$	6.35981	0.427807
$222$	0.199137	0.0133652
$223$	5.13331	0.343752	0.171876	−	0.985119i	$-0.445017\pi$
$223$	5.13331	0.343752	0.171876	+	0.985119i	$0.445017\pi$
$224$	2.03404	0.135905
$225$	−2.93488	−0.195658
$226$	0.413472	0.0275038
$227$	5.69498	0.377989	0.188994	−	0.981978i	$-0.439477\pi$
$227$	5.69498	0.377989	0.188994	+	0.981978i	$0.439477\pi$
$228$	−0.509263	−0.0337268
$229$	0.804113	0.0531373	0.0265686	−	0.999647i	$-0.491542\pi$
$229$	0.804113	0.0531373	0.0265686	+	0.999647i	$0.491542\pi$
$230$	0.281827	0.0185831
$231$	−0.653439	−0.0429931
$232$	2.77522	0.182202
$233$	−21.5122	−1.40931	−0.704656	−	0.709549i	$-0.748899\pi$
$233$	−21.5122	−1.40931	−0.704656	+	0.709549i	$0.748899\pi$
$234$	1.24261	0.0812316
$235$	9.94088	0.648472
$236$	25.5414	1.66260
$237$	−3.35011	−0.217613
$238$	−0.169542	−0.0109898
$239$	−17.9352	−1.16013	−0.580064	−	0.814571i	$-0.696973\pi$
$239$	−17.9352	−1.16013	−0.580064	+	0.814571i	$0.696973\pi$
$240$	−1.01403	−0.0654555
$241$	−26.2994	−1.69409	−0.847045	−	0.531521i	$-0.821621\pi$
$241$	−26.2994	−1.69409	−0.847045	+	0.531521i	$0.821621\pi$
$242$	0.0663929	0.00426790
$243$	−6.69189	−0.429285
$244$	14.9419	0.956557
$245$	−0.443552	−0.0283375
$246$	0.0618655	0.00394440
$247$	6.37708	0.405764
$248$	−0.122207	−0.00776013
$249$	3.50905	0.222377
$250$	0.0663929	0.00419906
$251$	−11.3237	−0.714743	−0.357371	−	0.933962i	$-0.616327\pi$
$251$	−11.3237	−0.714743	−0.357371	+	0.933962i	$0.616327\pi$
$252$	14.9967	0.944704
$253$	4.24483	0.266870
$254$	−0.691969	−0.0434180
$255$	0.254503	0.0159376
$256$	15.6485	0.978033
$257$	−2.49470	−0.155615	−0.0778076	−	0.996968i	$-0.524792\pi$
$257$	−2.49470	−0.155615	−0.0778076	+	0.996968i	$0.524792\pi$
$258$	−0.0143313	−0.000892229 0
$259$	−30.0950	−1.87001
$260$	12.7260	0.789236
$261$	30.7033	1.90049
$262$	0.525132	0.0324428
$263$	−4.40937	−0.271893	−0.135947	−	0.990716i	$-0.543408\pi$
$263$	−4.40937	−0.271893	−0.135947	+	0.990716i	$0.543408\pi$
$264$	0.0676976	0.00416650
$265$	0.244293	0.0150068
$266$	−0.170003	−0.0104235
$267$	−0.922057	−0.0564290
$268$	25.6577	1.56729
$269$	0.564118	0.0343949	0.0171974	−	0.999852i	$-0.494526\pi$
$269$	0.564118	0.0343949	0.0171974	+	0.999852i	$0.494526\pi$
$270$	0.100555	0.00611958
$271$	−22.2776	−1.35327	−0.676634	−	0.736319i	$-0.736562\pi$
$271$	−22.2776	−1.35327	−0.676634	+	0.736319i	$0.736562\pi$
$272$	−3.96281	−0.240281
$273$	4.16703	0.252200
$274$	0.614547	0.0371261
$275$	1.00000	0.0603023
$276$	2.16174	0.130121
$277$	20.2596	1.21728	0.608641	−	0.793446i	$-0.291715\pi$
$277$	20.2596	1.21728	0.608641	+	0.793446i	$0.291715\pi$
$278$	1.21233	0.0727105
$279$	−1.35202	−0.0809431
$280$	−0.679262	−0.0405936
$281$	5.60402	0.334308	0.167154	−	0.985931i	$-0.446542\pi$
$281$	5.60402	0.334308	0.167154	+	0.985931i	$0.446542\pi$
$282$	−0.168429	−0.0100298
$283$	11.9527	0.710513	0.355257	−	0.934769i	$-0.384393\pi$
$283$	11.9527	0.710513	0.355257	+	0.934769i	$0.384393\pi$
$284$	7.18264	0.426211
$285$	0.255194	0.0151164
$286$	−0.423393	−0.0250357
$287$	−9.34955	−0.551886
$288$	−2.33139	−0.137379
$289$	−16.0054	−0.941495
$290$	−0.694572	−0.0407867
$291$	−0.432574	−0.0253579
$292$	3.74583	0.219208
$293$	−14.8636	−0.868343	−0.434172	−	0.900830i	$-0.642959\pi$
$293$	−14.8636	−0.868343	−0.434172	+	0.900830i	$0.642959\pi$
$294$	0.00751513	0.000438291 0
$295$	−12.7989	−0.745180
$296$	3.11791	0.181225
$297$	1.51454	0.0878828
$298$	−0.267894	−0.0155187
$299$	−27.0696	−1.56548
$300$	0.509263	0.0294023
$301$	2.16585	0.124837
$302$	0.904636	0.0520559
$303$	2.82969	0.162562
$304$	−3.97357	−0.227900
$305$	−7.48745	−0.428730
$306$	0.194327	0.0111090
$307$	7.37424	0.420870	0.210435	−	0.977608i	$-0.432512\pi$
$307$	7.37424	0.420870	0.210435	+	0.977608i	$0.432512\pi$
$308$	−5.10983	−0.291159
$309$	3.12981	0.178049
$310$	0.0305854	0.00173713
$311$	13.2381	0.750667	0.375333	−	0.926890i	$-0.377528\pi$
$311$	13.2381	0.750667	0.375333	+	0.926890i	$0.377528\pi$
$312$	−0.431713	−0.0244409
$313$	22.3406	1.26277	0.631384	−	0.775470i	$-0.282487\pi$
$313$	22.3406	1.26277	0.631384	+	0.775470i	$0.282487\pi$
$314$	−0.157609	−0.00889437
$315$	−7.51491	−0.423417
$316$	−26.1975	−1.47373
$317$	−28.0306	−1.57436	−0.787179	−	0.616725i	$-0.788459\pi$
$317$	−28.0306	−1.57436	−0.787179	+	0.616725i	$0.788459\pi$
$318$	−0.00413907	−0.000232108 0
$319$	−10.4615	−0.585733
$320$	−7.89440	−0.441310
$321$	−0.949536	−0.0529979
$322$	0.721633	0.0402150
$323$	0.997292	0.0554908
$324$	−16.7991	−0.933286
$325$	−6.37708	−0.353737
$326$	−0.983986	−0.0544979
$327$	−2.84586	−0.157376
$328$	0.968632	0.0534838
$329$	25.4542	1.40333
$330$	−0.0169431	−0.000932685 0
$331$	20.1669	1.10847	0.554235	−	0.832360i	$-0.313011\pi$
$331$	20.1669	1.10847	0.554235	+	0.832360i	$0.313011\pi$
$332$	27.4405	1.50599
$333$	34.4945	1.89029
$334$	−0.508913	−0.0278465
$335$	−12.8572	−0.702462
$336$	−2.59648	−0.141650
$337$	2.94865	0.160623	0.0803115	−	0.996770i	$-0.474408\pi$
$337$	2.94865	0.160623	0.0803115	+	0.996770i	$0.474408\pi$
$338$	1.83690	0.0999143
$339$	−1.58926	−0.0863168
$340$	1.99019	0.107933
$341$	0.460672	0.0249468
$342$	0.194855	0.0105365
$343$	−19.0596	−1.02912
$344$	−0.224387	−0.0120981
$345$	−1.08326	−0.0583205
$346$	1.32023	0.0709761
$347$	−5.15823	−0.276908	−0.138454	−	0.990369i	$-0.544213\pi$
$347$	−5.15823	−0.276908	−0.138454	+	0.990369i	$0.544213\pi$
$348$	−5.32767	−0.285593
$349$	11.7199	0.627352	0.313676	−	0.949530i	$-0.398439\pi$
$349$	11.7199	0.627352	0.313676	+	0.949530i	$0.398439\pi$
$350$	0.170003	0.00908703
$351$	−9.65837	−0.515526
$352$	0.794375	0.0423403
$353$	−20.0730	−1.06838	−0.534189	−	0.845365i	$-0.679383\pi$
$353$	−20.0730	−1.06838	−0.534189	+	0.845365i	$0.679383\pi$
$354$	0.216852	0.0115256
$355$	−3.59925	−0.191028
$356$	−7.21039	−0.382150
$357$	0.651669	0.0344900
$358$	0.807584	0.0426821
$359$	−15.9410	−0.841336	−0.420668	−	0.907215i	$-0.638204\pi$
$359$	−15.9410	−0.841336	−0.420668	+	0.907215i	$0.638204\pi$
$360$	0.778561	0.0410338
$361$	1.00000	0.0526316
$362$	0.832000	0.0437289
$363$	−0.255194	−0.0133942
$364$	32.5858	1.70796
$365$	−1.87705	−0.0982495
$366$	0.126860	0.00663110
$367$	−4.25756	−0.222243	−0.111122	−	0.993807i	$-0.535444\pi$
$367$	−4.25756	−0.222243	−0.111122	+	0.993807i	$0.535444\pi$
$368$	16.8671	0.879260
$369$	10.7163	0.557870
$370$	−0.780337	−0.0405678
$371$	0.625525	0.0324757
$372$	0.234603	0.0121636
$373$	16.2272	0.840215	0.420107	−	0.907474i	$-0.361992\pi$
$373$	16.2272	0.840215	0.420107	+	0.907474i	$0.361992\pi$
$374$	−0.0662131	−0.00342380
$375$	−0.255194	−0.0131782
$376$	−2.63711	−0.135998
$377$	66.7140	3.43595
$378$	0.257477	0.0132432
$379$	27.9614	1.43628	0.718140	−	0.695898i	$-0.244994\pi$
$379$	27.9614	1.43628	0.718140	+	0.695898i	$0.244994\pi$
$380$	1.99559	0.102372
$381$	2.65972	0.136261
$382$	1.65448	0.0846508
$383$	26.5606	1.35718	0.678592	−	0.734515i	$-0.262590\pi$
$383$	26.5606	1.35718	0.678592	+	0.734515i	$0.262590\pi$
$384$	0.539195	0.0275157
$385$	2.56056	0.130498
$386$	−0.237465	−0.0120866
$387$	−2.48247	−0.126191
$388$	−3.38268	−0.171730
$389$	16.5015	0.836659	0.418329	−	0.908295i	$-0.362616\pi$
$389$	16.5015	0.836659	0.418329	+	0.908295i	$0.362616\pi$
$390$	0.108047	0.00547119
$391$	−4.23334	−0.214089
$392$	0.117665	0.00594298
$393$	−2.01845	−0.101817
$394$	−1.04135	−0.0524625
$395$	13.1277	0.660526
$396$	5.85682	0.294316
$397$	−25.0026	−1.25484	−0.627422	−	0.778679i	$-0.715890\pi$
$397$	−25.0026	−1.25484	−0.627422	+	0.778679i	$0.715890\pi$
$398$	1.53020	0.0767020
$399$	0.653439	0.0327128
$400$	3.97357	0.198679
$401$	−25.3814	−1.26749	−0.633744	−	0.773543i	$-0.718483\pi$
$401$	−25.3814	−1.26749	−0.633744	+	0.773543i	$0.718483\pi$
$402$	0.217840	0.0108649
$403$	−2.93774	−0.146339
$404$	22.1279	1.10091
$405$	8.41813	0.418300
$406$	−1.77849	−0.0882650
$407$	−11.7533	−0.582591
$408$	−0.0675143	−0.00334245
$409$	−2.31939	−0.114686	−0.0573432	−	0.998355i	$-0.518263\pi$
$409$	−2.31939	−0.114686	−0.0573432	+	0.998355i	$0.518263\pi$
$410$	−0.242425	−0.0119725
$411$	−2.36213	−0.116515
$412$	24.4748	1.20579
$413$	−32.7723	−1.61262
$414$	−0.827126	−0.0406510
$415$	−13.7505	−0.674987
$416$	−5.06579	−0.248371
$417$	−4.65981	−0.228192
$418$	−0.0663929	−0.00324738
$419$	9.68668	0.473226	0.236613	−	0.971604i	$-0.423963\pi$
$419$	9.68668	0.473226	0.236613	+	0.971604i	$0.423963\pi$
$420$	1.30400	0.0636285
$421$	−9.68352	−0.471946	−0.235973	−	0.971760i	$-0.575828\pi$
$421$	−9.68352	−0.471946	−0.235973	+	0.971760i	$0.575828\pi$
$422$	0.434288	0.0211408
$423$	−29.1753	−1.41855
$424$	−0.0648057	−0.00314724
$425$	−0.997292	−0.0483758
$426$	0.0609824	0.00295461
$427$	−19.1720	−0.927800
$428$	−7.42528	−0.358914
$429$	1.62739	0.0785712
$430$	0.0561585	0.00270820
$431$	0.0789944	0.00380503	0.00190251	−	0.999998i	$-0.499394\pi$
$431$	0.0789944	0.00380503	0.00190251	+	0.999998i	$0.499394\pi$
$432$	6.01815	0.289548
$433$	37.4851	1.80142	0.900711	−	0.434420i	$-0.143047\pi$
$433$	37.4851	1.80142	0.900711	+	0.434420i	$0.143047\pi$
$434$	0.0783156	0.00375927
$435$	2.66972	0.128003
$436$	−22.2543	−1.06579
$437$	−4.24483	−0.203058
$438$	0.0318030	0.00151961
$439$	−17.2262	−0.822162	−0.411081	−	0.911599i	$-0.634849\pi$
$439$	−17.2262	−0.822162	−0.411081	+	0.911599i	$0.634849\pi$
$440$	−0.265279	−0.0126467
$441$	1.30177	0.0619891
$442$	0.422246	0.0200842
$443$	−38.3728	−1.82315	−0.911573	−	0.411138i	$-0.865132\pi$
$443$	−38.3728	−1.82315	−0.911573	+	0.411138i	$0.865132\pi$
$444$	−5.98553	−0.284061
$445$	3.61316	0.171280
$446$	0.340815	0.0161381
$447$	1.02970	0.0487032
$448$	−20.2141	−0.955025
$449$	25.0230	1.18091	0.590454	−	0.807071i	$-0.298949\pi$
$449$	25.0230	1.18091	0.590454	+	0.807071i	$0.298949\pi$
$450$	−0.194855	−0.00918555
$451$	−3.65137	−0.171936
$452$	−12.4279	−0.584557
$453$	−3.47714	−0.163370
$454$	0.378106	0.0177454
$455$	−16.3289	−0.765509
$456$	−0.0676976	−0.00317023
$457$	7.67694	0.359112	0.179556	−	0.983748i	$-0.442534\pi$
$457$	7.67694	0.359112	0.179556	+	0.983748i	$0.442534\pi$
$458$	0.0533874	0.00249463
$459$	−1.51044	−0.0705015
$460$	−8.47095	−0.394960
$461$	3.61276	0.168263	0.0841316	−	0.996455i	$-0.473188\pi$
$461$	3.61276	0.168263	0.0841316	+	0.996455i	$0.473188\pi$
$462$	−0.0433837	−0.00201839
$463$	6.14146	0.285418	0.142709	−	0.989765i	$-0.454419\pi$
$463$	6.14146	0.285418	0.142709	+	0.989765i	$0.454419\pi$
$464$	−41.5697	−1.92982
$465$	−0.117561	−0.00545175
$466$	−1.42826	−0.0661628
$467$	−23.5950	−1.09185	−0.545923	−	0.837835i	$-0.683821\pi$
$467$	−23.5950	−1.09185	−0.545923	+	0.837835i	$0.683821\pi$
$468$	−37.3494	−1.72648
$469$	−32.9215	−1.52017
$470$	0.660004	0.0304437
$471$	0.605799	0.0279137
$472$	3.39528	0.156280
$473$	0.845851	0.0388923
$474$	−0.222423	−0.0102162
$475$	−1.00000	−0.0458831
$476$	5.09599	0.233574
$477$	−0.716969	−0.0328278
$478$	−1.19077	−0.0544644
$479$	−17.1964	−0.785722	−0.392861	−	0.919598i	$-0.628515\pi$
$479$	−17.1964	−0.785722	−0.392861	+	0.919598i	$0.628515\pi$
$480$	−0.202720	−0.00925285
$481$	74.9519	3.41751
$482$	−1.74609	−0.0795322
$483$	−2.77374	−0.126209
$484$	−1.99559	−0.0907087
$485$	1.69508	0.0769695
$486$	−0.444294	−0.0201536
$487$	−11.3122	−0.512606	−0.256303	−	0.966597i	$-0.582504\pi$
$487$	−11.3122	−0.512606	−0.256303	+	0.966597i	$0.582504\pi$
$488$	1.98626	0.0899140
$489$	3.78214	0.171034
$490$	−0.0294487	−0.00133036
$491$	−5.99437	−0.270522	−0.135261	−	0.990810i	$-0.543187\pi$
$491$	−5.99437	−0.270522	−0.135261	+	0.990810i	$0.543187\pi$
$492$	−1.85951	−0.0838332
$493$	10.4332	0.469888
$494$	0.423393	0.0190493
$495$	−2.93488	−0.131913
$496$	1.83051	0.0821925
$497$	−9.21609	−0.413398
$498$	0.232976	0.0104399
$499$	−9.66851	−0.432822	−0.216411	−	0.976302i	$-0.569435\pi$
$499$	−9.66851	−0.432822	−0.216411	+	0.976302i	$0.569435\pi$
$500$	−1.99559	−0.0892456
$501$	1.95611	0.0873924
$502$	−0.751810	−0.0335549
$503$	−24.0691	−1.07319	−0.536595	−	0.843840i	$-0.680290\pi$
$503$	−24.0691	−1.07319	−0.536595	+	0.843840i	$0.680290\pi$
$504$	1.99355	0.0887997
$505$	−11.0884	−0.493427
$506$	0.281827	0.0125287
$507$	−7.06049	−0.313567
$508$	20.7987	0.922794
$509$	−24.7303	−1.09615	−0.548075	−	0.836429i	$-0.684639\pi$
$509$	−24.7303	−1.09615	−0.548075	+	0.836429i	$0.684639\pi$
$510$	0.0168972	0.000748220 0
$511$	−4.80630	−0.212618
$512$	5.26471	0.232670
$513$	−1.51454	−0.0668688
$514$	−0.165630	−0.00730565
$515$	−12.2645	−0.540436
$516$	0.430761	0.0189632
$517$	9.94088	0.437199
$518$	−1.99810	−0.0877914
$519$	−5.07456	−0.222749
$520$	1.69170	0.0741862
$521$	−20.9083	−0.916008	−0.458004	−	0.888950i	$-0.651436\pi$
$521$	−20.9083	−0.916008	−0.458004	+	0.888950i	$0.651436\pi$
$522$	2.03848	0.0892219
$523$	16.8518	0.736879	0.368439	−	0.929652i	$-0.379892\pi$
$523$	16.8518	0.736879	0.368439	+	0.929652i	$0.379892\pi$
$524$	−15.7841	−0.689530
$525$	−0.653439	−0.0285184
$526$	−0.292751	−0.0127645
$527$	−0.459425	−0.0200129
$528$	−1.01403	−0.0441301
$529$	−4.98141	−0.216583
$530$	0.0162193	0.000704522 0
$531$	37.5632	1.63010
$532$	5.10983	0.221539
$533$	23.2851	1.00859
$534$	−0.0612180	−0.00264916
$535$	3.72084	0.160866
$536$	3.41074	0.147321
$537$	−3.10411	−0.133952
$538$	0.0374534	0.00161473
$539$	−0.443552	−0.0191051
$540$	−3.02241	−0.130064
$541$	−27.9122	−1.20004	−0.600020	−	0.799985i	$-0.704841\pi$
$541$	−27.9122	−1.20004	−0.600020	+	0.799985i	$0.704841\pi$
$542$	−1.47908	−0.0635317
$543$	−3.19795	−0.137237
$544$	−0.792224	−0.0339663
$545$	11.1518	0.477689
$546$	0.276661	0.0118400
$547$	−7.36255	−0.314800	−0.157400	−	0.987535i	$-0.550311\pi$
$547$	−7.36255	−0.314800	−0.157400	+	0.987535i	$0.550311\pi$
$548$	−18.4716	−0.789069
$549$	21.9748	0.937860
$550$	0.0663929	0.00283100
$551$	10.4615	0.445676
$552$	0.287365	0.0122311
$553$	33.6142	1.42942
$554$	1.34509	0.0571476
$555$	2.99938	0.127316
$556$	−36.4393	−1.54537
$557$	−46.6423	−1.97630	−0.988149	−	0.153500i	$-0.950945\pi$
$557$	−46.6423	−1.97630	−0.988149	+	0.153500i	$0.950945\pi$
$558$	−0.0897643	−0.00380003
$559$	−5.39406	−0.228144
$560$	10.1746	0.429953
$561$	0.254503	0.0107451
$562$	0.372067	0.0156947
$563$	−20.6573	−0.870602	−0.435301	−	0.900285i	$-0.643358\pi$
$563$	−20.6573	−0.870602	−0.435301	+	0.900285i	$0.643358\pi$
$564$	5.06252	0.213171
$565$	6.22765	0.261999
$566$	0.793573	0.0333564
$567$	21.5551	0.905228
$568$	0.954806	0.0400628
$569$	−19.6820	−0.825112	−0.412556	−	0.910932i	$-0.635364\pi$
$569$	−19.6820	−0.825112	−0.412556	+	0.910932i	$0.635364\pi$
$570$	0.0169431	0.000709667 0
$571$	38.7259	1.62063	0.810313	−	0.585997i	$-0.199297\pi$
$571$	38.7259	1.62063	0.810313	+	0.585997i	$0.199297\pi$
$572$	12.7260	0.532103
$573$	−6.35933	−0.265665
$574$	−0.620744	−0.0259093
$575$	4.24483	0.177022
$576$	23.1691	0.965379
$577$	−20.6720	−0.860587	−0.430294	−	0.902689i	$-0.641590\pi$
$577$	−20.6720	−0.860587	−0.430294	+	0.902689i	$0.641590\pi$
$578$	−1.06265	−0.0442002
$579$	0.912742	0.0379323
$580$	20.8770	0.866868
$581$	−35.2090	−1.46072
$582$	−0.0287198	−0.00119047
$583$	0.244293	0.0101176
$584$	0.497943	0.0206050
$585$	18.7159	0.773809
$586$	−0.986840	−0.0407660
$587$	25.0952	1.03579	0.517894	−	0.855445i	$-0.326716\pi$
$587$	25.0952	1.03579	0.517894	+	0.855445i	$0.326716\pi$
$588$	−0.225885	−0.00931533
$589$	−0.460672	−0.0189817
$590$	−0.849755	−0.0349839
$591$	4.00264	0.164646
$592$	−46.7027	−1.91947
$593$	−33.4798	−1.37485	−0.687424	−	0.726256i	$-0.741259\pi$
$593$	−33.4798	−1.37485	−0.687424	+	0.726256i	$0.741259\pi$
$594$	0.100555	0.00412582
$595$	−2.55362	−0.104688
$596$	8.05216	0.329830
$597$	−5.88162	−0.240719
$598$	−1.79723	−0.0734942
$599$	−5.45944	−0.223067	−0.111533	−	0.993761i	$-0.535576\pi$
$599$	−5.45944	−0.223067	−0.111533	+	0.993761i	$0.535576\pi$
$600$	0.0676976	0.00276374
$601$	18.8988	0.770896	0.385448	−	0.922730i	$-0.374047\pi$
$601$	18.8988	0.770896	0.385448	+	0.922730i	$0.374047\pi$
$602$	0.143797	0.00586073
$603$	37.7342	1.53666
$604$	−27.1909	−1.10638
$605$	1.00000	0.0406558
$606$	0.187872	0.00763176
$607$	32.5947	1.32298	0.661490	−	0.749954i	$-0.269925\pi$
$607$	32.5947	1.32298	0.661490	+	0.749954i	$0.269925\pi$
$608$	−0.794375	−0.0322162
$609$	6.83597	0.277007
$610$	−0.497114	−0.0201276
$611$	−63.3938	−2.56464
$612$	−5.84095	−0.236107
$613$	−10.8158	−0.436845	−0.218423	−	0.975854i	$-0.570091\pi$
$613$	−10.8158	−0.436845	−0.218423	+	0.975854i	$0.570091\pi$
$614$	0.489597	0.0197585
$615$	0.931808	0.0375741
$616$	−0.679262	−0.0273682
$617$	31.3112	1.26054	0.630272	−	0.776375i	$-0.282944\pi$
$617$	31.3112	1.26054	0.630272	+	0.776375i	$0.282944\pi$
$618$	0.207797	0.00835884
$619$	21.1120	0.848563	0.424282	−	0.905530i	$-0.360527\pi$
$619$	21.1120	0.848563	0.424282	+	0.905530i	$0.360527\pi$
$620$	−0.919314	−0.0369206
$621$	6.42899	0.257986
$622$	0.878919	0.0352415
$623$	9.25170	0.370662
$624$	6.46656	0.258870
$625$	1.00000	0.0400000
$626$	1.48326	0.0592830
$627$	0.255194	0.0101915
$628$	4.73729	0.189038
$629$	11.7215	0.467367
$630$	−0.498937	−0.0198781
$631$	3.92468	0.156239	0.0781195	−	0.996944i	$-0.475108\pi$
$631$	3.92468	0.156239	0.0781195	+	0.996944i	$0.475108\pi$
$632$	−3.48250	−0.138526
$633$	−1.66927	−0.0663475
$634$	−1.86104	−0.0739112
$635$	−10.4223	−0.413598
$636$	0.124409	0.00493315
$637$	2.82857	0.112072
$638$	−0.694572	−0.0274984
$639$	10.5634	0.417880
$640$	−2.11288	−0.0835190
$641$	−31.3797	−1.23942	−0.619712	−	0.784829i	$-0.712750\pi$
$641$	−31.3797	−1.23942	−0.619712	+	0.784829i	$0.712750\pi$
$642$	−0.0630425	−0.00248809
$643$	−34.1180	−1.34548	−0.672741	−	0.739878i	$-0.734883\pi$
$643$	−34.1180	−1.34548	−0.672741	+	0.739878i	$0.734883\pi$
$644$	−21.6903	−0.854719
$645$	−0.215856	−0.00849933
$646$	0.0662131	0.00260512
$647$	−15.7468	−0.619072	−0.309536	−	0.950888i	$-0.600174\pi$
$647$	−15.7468	−0.619072	−0.309536	+	0.950888i	$0.600174\pi$
$648$	−2.23315	−0.0877265
$649$	−12.7989	−0.502400
$650$	−0.423393	−0.0166068
$651$	−0.301021	−0.0117979
$652$	29.5760	1.15828
$653$	9.67304	0.378535	0.189268	−	0.981926i	$-0.439389\pi$
$653$	9.67304	0.378535	0.189268	+	0.981926i	$0.439389\pi$
$654$	−0.188945	−0.00738833
$655$	7.90946	0.309048
$656$	−14.5090	−0.566481
$657$	5.50892	0.214923
$658$	1.68998	0.0658822
$659$	12.7974	0.498518	0.249259	−	0.968437i	$-0.419813\pi$
$659$	12.7974	0.498518	0.249259	+	0.968437i	$0.419813\pi$
$660$	0.509263	0.0198230
$661$	42.1039	1.63765	0.818826	−	0.574042i	$-0.194625\pi$
$661$	42.1039	1.63765	0.818826	+	0.574042i	$0.194625\pi$
$662$	1.33894	0.0520392
$663$	−1.62299	−0.0630315
$664$	3.64773	0.141559
$665$	−2.56056	−0.0992941
$666$	2.29019	0.0887432
$667$	−44.4074	−1.71946
$668$	15.2966	0.591842
$669$	−1.30999	−0.0506471
$670$	−0.853625	−0.0329784
$671$	−7.48745	−0.289050
$672$	−0.519075	−0.0200238
$673$	−25.9451	−1.00011	−0.500056	−	0.865993i	$-0.666687\pi$
$673$	−25.9451	−1.00011	−0.500056	+	0.865993i	$0.666687\pi$
$674$	0.195769	0.00754075
$675$	1.51454	0.0582949
$676$	−55.2123	−2.12355
$677$	−36.2924	−1.39483	−0.697415	−	0.716668i	$-0.745666\pi$
$677$	−36.2924	−1.39483	−0.697415	+	0.716668i	$0.745666\pi$
$678$	−0.105516	−0.00405230
$679$	4.34034	0.166567
$680$	0.264561	0.0101454
$681$	−1.45332	−0.0556915
$682$	0.0305854	0.00117117
$683$	−9.68713	−0.370668	−0.185334	−	0.982676i	$-0.559337\pi$
$683$	−9.68713	−0.370668	−0.185334	+	0.982676i	$0.559337\pi$
$684$	−5.85682	−0.223941
$685$	9.25621	0.353662
$686$	−1.26542	−0.0483141
$687$	−0.205205	−0.00782905
$688$	3.36105	0.128139
$689$	−1.55787	−0.0593503
$690$	−0.0719205	−0.00273797
$691$	22.0462	0.838677	0.419339	−	0.907830i	$-0.362262\pi$
$691$	22.0462	0.838677	0.419339	+	0.907830i	$0.362262\pi$
$692$	−39.6826	−1.50851
$693$	−7.51491	−0.285468
$694$	−0.342470	−0.0130000
$695$	18.2599	0.692636
$696$	−0.708221	−0.0268450
$697$	3.64149	0.137931
$698$	0.778118	0.0294522
$699$	5.48979	0.207643
$700$	−5.10983	−0.193133
$701$	33.8317	1.27780	0.638902	−	0.769288i	$-0.279389\pi$
$701$	33.8317	1.27780	0.638902	+	0.769288i	$0.279389\pi$
$702$	−0.641247	−0.0242023
$703$	11.7533	0.443285
$704$	−7.89440	−0.297531
$705$	−2.53685	−0.0955434
$706$	−1.33271	−0.0501570
$707$	−28.3925	−1.06781
$708$	−6.51800	−0.244961
$709$	41.1885	1.54687	0.773433	−	0.633878i	$-0.218538\pi$
$709$	41.1885	1.54687	0.773433	+	0.633878i	$0.218538\pi$
$710$	−0.238965	−0.00896819
$711$	−38.5282	−1.44492
$712$	−0.958495	−0.0359211
$713$	1.95548	0.0732332
$714$	0.0432662	0.00161920
$715$	−6.37708	−0.238489
$716$	−24.2738	−0.907155
$717$	4.57694	0.170929
$718$	−1.05837	−0.0394981
$719$	12.1123	0.451711	0.225855	−	0.974161i	$-0.427482\pi$
$719$	12.1123	0.451711	0.225855	+	0.974161i	$0.427482\pi$
$720$	−11.6619	−0.434615
$721$	−31.4038	−1.16954
$722$	0.0663929	0.00247089
$723$	6.71144	0.249601
$724$	−25.0077	−0.929403
$725$	−10.4615	−0.388532
$726$	−0.0169431	−0.000628816 0
$727$	−16.8354	−0.624389	−0.312195	−	0.950018i	$-0.601064\pi$
$727$	−16.8354	−0.624389	−0.312195	+	0.950018i	$0.601064\pi$
$728$	4.33171	0.160544
$729$	−23.5466	−0.872098
$730$	−0.124623	−0.00461250
$731$	−0.843561	−0.0312002
$732$	−3.81308	−0.140936
$733$	−0.203260	−0.00750759	−0.00375379	−	0.999993i	$-0.501195\pi$
$733$	−0.203260	−0.00750759	−0.00375379	+	0.999993i	$0.501195\pi$
$734$	−0.282672	−0.0104336
$735$	0.113192	0.00417514
$736$	3.37199	0.124293
$737$	−12.8572	−0.473600
$738$	0.711488	0.0261902
$739$	46.2418	1.70103	0.850516	−	0.525949i	$-0.176290\pi$
$739$	46.2418	1.70103	0.850516	+	0.525949i	$0.176290\pi$
$740$	23.4548	0.862217
$741$	−1.62739	−0.0597837
$742$	0.0415305	0.00152463
$743$	−38.6255	−1.41703	−0.708517	−	0.705694i	$-0.750635\pi$
$743$	−38.6255	−1.41703	−0.708517	+	0.705694i	$0.750635\pi$
$744$	0.0311864	0.00114335
$745$	−4.03498	−0.147830
$746$	1.07737	0.0394454
$747$	40.3561	1.47655
$748$	1.99019	0.0727685
$749$	9.52742	0.348124
$750$	−0.0169431	−0.000618673 0
$751$	−49.1929	−1.79507	−0.897537	−	0.440938i	$-0.854646\pi$
$751$	−49.1929	−1.79507	−0.897537	+	0.440938i	$0.854646\pi$
$752$	39.5008	1.44045
$753$	2.88973	0.105308
$754$	4.42934	0.161307
$755$	13.6255	0.495882
$756$	−7.73906	−0.281467
$757$	7.56374	0.274909	0.137454	−	0.990508i	$-0.456108\pi$
$757$	7.56374	0.274909	0.137454	+	0.990508i	$0.456108\pi$
$758$	1.85644	0.0674289
$759$	−1.08326	−0.0393197
$760$	0.265279	0.00962268
$761$	32.5350	1.17939	0.589697	−	0.807624i	$-0.299247\pi$
$761$	32.5350	1.17939	0.589697	+	0.807624i	$0.299247\pi$
$762$	0.176586	0.00639705
$763$	28.5547	1.03375
$764$	−49.7293	−1.79914
$765$	2.92693	0.105823
$766$	1.76344	0.0637156
$767$	81.6195	2.94711
$768$	−3.99341	−0.144100
$769$	18.2500	0.658112	0.329056	−	0.944310i	$-0.393270\pi$
$769$	18.2500	0.658112	0.329056	+	0.944310i	$0.393270\pi$
$770$	0.170003	0.00612647
$771$	0.636633	0.0229278
$772$	7.13756	0.256886
$773$	−24.5139	−0.881703	−0.440852	−	0.897580i	$-0.645323\pi$
$773$	−24.5139	−0.881703	−0.440852	+	0.897580i	$0.645323\pi$
$774$	−0.164818	−0.00592427
$775$	0.460672	0.0165478
$776$	−0.449668	−0.0161422
$777$	7.68007	0.275521
$778$	1.09558	0.0392785
$779$	3.65137	0.130824
$780$	−3.24761	−0.116283
$781$	−3.59925	−0.128791
$782$	−0.281064	−0.0100508
$783$	−15.8445	−0.566235
$784$	−1.76249	−0.0629459
$785$	−2.37388	−0.0847273
$786$	−0.134011	−0.00478000
$787$	−10.1527	−0.361905	−0.180953	−	0.983492i	$-0.557918\pi$
$787$	−10.1527	−0.361905	−0.180953	+	0.983492i	$0.557918\pi$
$788$	31.3002	1.11502
$789$	1.12524	0.0400598
$790$	0.871586	0.0310096
$791$	15.9463	0.566984
$792$	0.778561	0.0276650
$793$	47.7481	1.69558
$794$	−1.66000	−0.0589110
$795$	−0.0623421	−0.00221104
$796$	−45.9937	−1.63020
$797$	28.7406	1.01804	0.509021	−	0.860754i	$-0.330007\pi$
$797$	28.7406	1.01804	0.509021	+	0.860754i	$0.330007\pi$
$798$	0.0433837	0.00153577
$799$	−9.91396	−0.350731
$800$	0.794375	0.0280854
$801$	−10.6042	−0.374680
$802$	−1.68515	−0.0595046
$803$	−1.87705	−0.0662398
$804$	−6.54768	−0.230919
$805$	10.8691	0.383086
$806$	−0.195045	−0.00687018
$807$	−0.143959	−0.00506761
$808$	2.94152	0.103482
$809$	−12.8392	−0.451402	−0.225701	−	0.974197i	$-0.572467\pi$
$809$	−12.8392	−0.451402	−0.225701	+	0.974197i	$0.572467\pi$
$810$	0.558904	0.0196379
$811$	31.9835	1.12309	0.561546	−	0.827445i	$-0.310207\pi$
$811$	31.9835	1.12309	0.561546	+	0.827445i	$0.310207\pi$
$812$	53.4566	1.87596
$813$	5.68511	0.199386
$814$	−0.780337	−0.0273508
$815$	−14.8206	−0.519144
$816$	1.01129	0.0354021
$817$	−0.845851	−0.0295926
$818$	−0.153991	−0.00538417
$819$	47.9232	1.67457
$820$	7.28665	0.254461
$821$	−6.12398	−0.213728	−0.106864	−	0.994274i	$-0.534081\pi$
$821$	−6.12398	−0.213728	−0.106864	+	0.994274i	$0.534081\pi$
$822$	−0.156829	−0.00547003
$823$	40.9700	1.42813	0.714063	−	0.700082i	$-0.246853\pi$
$823$	40.9700	1.42813	0.714063	+	0.700082i	$0.246853\pi$
$824$	3.25350	0.113341
$825$	−0.255194	−0.00888471
$826$	−2.17585	−0.0757074
$827$	−20.9834	−0.729663	−0.364832	−	0.931074i	$-0.618873\pi$
$827$	−20.9834	−0.729663	−0.364832	+	0.931074i	$0.618873\pi$
$828$	24.8612	0.863986
$829$	−23.3772	−0.811925	−0.405962	−	0.913890i	$-0.633064\pi$
$829$	−23.3772	−0.811925	−0.405962	+	0.913890i	$0.633064\pi$
$830$	−0.912938	−0.0316885
$831$	−5.17013	−0.179350
$832$	50.3432	1.74534
$833$	0.442351	0.0153266
$834$	−0.309378	−0.0107129
$835$	−7.66518	−0.265264
$836$	1.99559	0.0690190
$837$	0.697709	0.0241164
$838$	0.643127	0.0222165
$839$	22.6825	0.783087	0.391543	−	0.920160i	$-0.371941\pi$
$839$	22.6825	0.783087	0.391543	+	0.920160i	$0.371941\pi$
$840$	0.173343	0.00598092
$841$	80.4437	2.77392
$842$	−0.642917	−0.0221564
$843$	−1.43011	−0.0492557
$844$	−13.0535	−0.449321
$845$	27.6671	0.951778
$846$	−1.93703	−0.0665965
$847$	2.56056	0.0879818
$848$	0.970715	0.0333345
$849$	−3.05025	−0.104684
$850$	−0.0662131	−0.00227109
$851$	−49.8909	−1.71024
$852$	−1.83297	−0.0627964
$853$	39.6950	1.35913	0.679565	−	0.733615i	$-0.262168\pi$
$853$	39.6950	1.35913	0.679565	+	0.733615i	$0.262168\pi$
$854$	−1.27289	−0.0435573
$855$	2.93488	0.100371
$856$	−0.987061	−0.0337370
$857$	−42.3803	−1.44768	−0.723842	−	0.689966i	$-0.757626\pi$
$857$	−42.3803	−1.44768	−0.723842	+	0.689966i	$0.757626\pi$
$858$	0.108047	0.00368867
$859$	−16.1630	−0.551474	−0.275737	−	0.961233i	$-0.588922\pi$
$859$	−16.1630	−0.551474	−0.275737	+	0.961233i	$0.588922\pi$
$860$	−1.68797	−0.0575594
$861$	2.38595	0.0813129
$862$	0.00524467	0.000178634 0
$863$	15.4142	0.524704	0.262352	−	0.964972i	$-0.415502\pi$
$863$	15.4142	0.524704	0.262352	+	0.964972i	$0.415502\pi$
$864$	1.20312	0.0409308
$865$	19.8851	0.676115
$866$	2.48875	0.0845711
$867$	4.08448	0.138716
$868$	−2.35396	−0.0798985
$869$	13.1277	0.445326
$870$	0.177251	0.00600935
$871$	81.9912	2.77816
$872$	−2.95833	−0.100182
$873$	−4.97484	−0.168373
$874$	−0.281827	−0.00953293
$875$	2.56056	0.0865626
$876$	−0.955914	−0.0322973
$877$	14.4051	0.486427	0.243213	−	0.969973i	$-0.421798\pi$
$877$	14.4051	0.486427	0.243213	+	0.969973i	$0.421798\pi$
$878$	−1.14370	−0.0385979
$879$	3.79311	0.127938
$880$	3.97357	0.133949
$881$	−39.6149	−1.33466	−0.667330	−	0.744762i	$-0.732563\pi$
$881$	−39.6149	−1.33466	−0.667330	+	0.744762i	$0.732563\pi$
$882$	0.0864283	0.00291019
$883$	−8.84444	−0.297639	−0.148819	−	0.988864i	$-0.547547\pi$
$883$	−8.84444	−0.297639	−0.148819	+	0.988864i	$0.547547\pi$
$884$	−12.6916	−0.426864
$885$	3.26620	0.109792
$886$	−2.54768	−0.0855910
$887$	−30.5801	−1.02678	−0.513389	−	0.858156i	$-0.671610\pi$
$887$	−30.5801	−1.02678	−0.513389	+	0.858156i	$0.671610\pi$
$888$	−0.795672	−0.0267010
$889$	−26.6870	−0.895052
$890$	0.239888	0.00804107
$891$	8.41813	0.282018
$892$	−10.2440	−0.342994
$893$	−9.94088	−0.332659
$894$	0.0683649	0.00228646
$895$	12.1637	0.406588
$896$	−5.41015	−0.180741
$897$	6.90801	0.230652
$898$	1.66135	0.0554400
$899$	−4.81934	−0.160734
$900$	5.85682	0.195227
$901$	−0.243631	−0.00811653
$902$	−0.242425	−0.00807188
$903$	−0.552712	−0.0183931
$904$	−1.65207	−0.0549469
$905$	12.5315	0.416560
$906$	−0.230858	−0.00766973
$907$	20.7675	0.689575	0.344787	−	0.938681i	$-0.387951\pi$
$907$	20.7675	0.689575	0.344787	+	0.938681i	$0.387951\pi$
$908$	−11.3649	−0.377156
$909$	32.5431	1.07939
$910$	−1.08412	−0.0359383
$911$	−0.520574	−0.0172474	−0.00862370	−	0.999963i	$-0.502745\pi$
$911$	−0.520574	−0.0172474	−0.00862370	+	0.999963i	$0.502745\pi$
$912$	1.01403	0.0335779
$913$	−13.7505	−0.455076
$914$	0.509694	0.0168592
$915$	1.91075	0.0631676
$916$	−1.60468	−0.0530201
$917$	20.2526	0.668801
$918$	−0.100283	−0.00330982
$919$	−14.4946	−0.478132	−0.239066	−	0.971003i	$-0.576841\pi$
$919$	−14.4946	−0.478132	−0.239066	+	0.971003i	$0.576841\pi$
$920$	−1.12606	−0.0371252
$921$	−1.88186	−0.0620094
$922$	0.239862	0.00789943
$923$	22.9527	0.755498
$924$	1.30400	0.0428983
$925$	−11.7533	−0.386447
$926$	0.407750	0.0133995
$927$	35.9946	1.18222
$928$	−8.31038	−0.272802
$929$	−52.5793	−1.72507	−0.862535	−	0.505998i	$-0.831124\pi$
$929$	−52.5793	−1.72507	−0.862535	+	0.505998i	$0.831124\pi$
$930$	−0.00780521	−0.000255943 0
$931$	0.443552	0.0145368
$932$	42.9296	1.40621
$933$	−3.37830	−0.110600
$934$	−1.56654	−0.0512588
$935$	−0.997292	−0.0326149
$936$	−4.96494	−0.162284
$937$	−21.7312	−0.709928	−0.354964	−	0.934880i	$-0.615507\pi$
$937$	−21.7312	−0.709928	−0.354964	+	0.934880i	$0.615507\pi$
$938$	−2.18575	−0.0713674
$939$	−5.70120	−0.186052
$940$	−19.8379	−0.647042
$941$	−9.98432	−0.325480	−0.162740	−	0.986669i	$-0.552033\pi$
$941$	−9.98432	−0.325480	−0.162740	+	0.986669i	$0.552033\pi$
$942$	0.0402208	0.00131046
$943$	−15.4995	−0.504732
$944$	−50.8573	−1.65526
$945$	3.87808	0.126154
$946$	0.0561585	0.00182587
$947$	34.3766	1.11709	0.558544	−	0.829475i	$-0.311360\pi$
$947$	34.3766	1.11709	0.558544	+	0.829475i	$0.311360\pi$
$948$	6.68545	0.217133
$949$	11.9701	0.388566
$950$	−0.0663929	−0.00215407
$951$	7.15325	0.231960
$952$	0.677422	0.0219554
$953$	59.0115	1.91157	0.955786	−	0.294064i	$-0.0950079\pi$
$953$	59.0115	1.91157	0.955786	+	0.294064i	$0.0950079\pi$
$954$	−0.0476017	−0.00154116
$955$	24.9196	0.806379
$956$	35.7912	1.15757
$957$	2.66972	0.0862998
$958$	−1.14172	−0.0368872
$959$	23.7011	0.765347
$960$	2.01460	0.0650211
$961$	−30.7878	−0.993154
$962$	4.97627	0.160441
$963$	−10.9202	−0.351899
$964$	52.4828	1.69036
$965$	−3.57666	−0.115137
$966$	−0.184156	−0.00592514
$967$	−61.6249	−1.98172	−0.990862	−	0.134882i	$-0.956934\pi$
$967$	−61.6249	−1.98172	−0.990862	+	0.134882i	$0.956934\pi$
$968$	−0.265279	−0.00852639
$969$	−0.254503	−0.00817581
$970$	0.112541	0.00361348
$971$	−29.7990	−0.956295	−0.478147	−	0.878280i	$-0.658691\pi$
$971$	−29.7990	−0.956295	−0.478147	+	0.878280i	$0.658691\pi$
$972$	13.3543	0.428339
$973$	46.7554	1.49891
$974$	−0.751051	−0.0240652
$975$	1.62739	0.0521183
$976$	−29.7519	−0.952336
$977$	−6.12178	−0.195853	−0.0979266	−	0.995194i	$-0.531221\pi$
$977$	−6.12178	−0.195853	−0.0979266	+	0.995194i	$0.531221\pi$
$978$	0.251107	0.00802952
$979$	3.61316	0.115477
$980$	0.885149	0.0282750
$981$	−32.7290	−1.04496
$982$	−0.397983	−0.0127002
$983$	36.6301	1.16832	0.584160	−	0.811639i	$-0.301424\pi$
$983$	36.6301	1.16832	0.584160	+	0.811639i	$0.301424\pi$
$984$	−0.247189	−0.00788010
$985$	−15.6847	−0.499756
$986$	0.692691	0.0220598
$987$	−6.49576	−0.206762
$988$	−12.7260	−0.404869
$989$	3.59050	0.114171
$990$	−0.194855	−0.00619290
$991$	−14.5788	−0.463110	−0.231555	−	0.972822i	$-0.574381\pi$
$991$	−14.5788	−0.463110	−0.231555	+	0.972822i	$0.574381\pi$
$992$	0.365947	0.0116188
$993$	−5.14646	−0.163318
$994$	−0.611883	−0.0194077
$995$	23.0476	0.730660
$996$	−7.00264	−0.221887
$997$	−35.5504	−1.12589	−0.562947	−	0.826493i	$-0.690332\pi$
$997$	−35.5504	−1.12589	−0.562947	+	0.826493i	$0.690332\pi$
$998$	−0.641920	−0.0203196
$999$	−17.8009	−0.563197

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.i.1.6	✓	8
3.2	odd	2		9405.2.a.bf.1.3		8
5.4	even	2		5225.2.a.o.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.i.1.6	✓	8	1.1	even	1	trivial
5225.2.a.o.1.3		8	5.4	even	2
9405.2.a.bf.1.3		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0663929 q^{2} -0.255194 q^{3} -1.99559 q^{4} +1.00000 q^{5} -0.0169431 q^{6} +2.56056 q^{7} -0.265279 q^{8} -2.93488 q^{9} +O(q^{10})\)	\(q+0.0663929 q^{2} -0.255194 q^{3} -1.99559 q^{4} +1.00000 q^{5} -0.0169431 q^{6} +2.56056 q^{7} -0.265279 q^{8} -2.93488 q^{9} +0.0663929 q^{10} +1.00000 q^{11} +0.509263 q^{12} -6.37708 q^{13} +0.170003 q^{14} -0.255194 q^{15} +3.97357 q^{16} -0.997292 q^{17} -0.194855 q^{18} -1.00000 q^{19} -1.99559 q^{20} -0.653439 q^{21} +0.0663929 q^{22} +4.24483 q^{23} +0.0676976 q^{24} +1.00000 q^{25} -0.423393 q^{26} +1.51454 q^{27} -5.10983 q^{28} -10.4615 q^{29} -0.0169431 q^{30} +0.460672 q^{31} +0.794375 q^{32} -0.255194 q^{33} -0.0662131 q^{34} +2.56056 q^{35} +5.85682 q^{36} -11.7533 q^{37} -0.0663929 q^{38} +1.62739 q^{39} -0.265279 q^{40} -3.65137 q^{41} -0.0433837 q^{42} +0.845851 q^{43} -1.99559 q^{44} -2.93488 q^{45} +0.281827 q^{46} +9.94088 q^{47} -1.01403 q^{48} -0.443552 q^{49} +0.0663929 q^{50} +0.254503 q^{51} +12.7260 q^{52} +0.244293 q^{53} +0.100555 q^{54} +1.00000 q^{55} -0.679262 q^{56} +0.255194 q^{57} -0.694572 q^{58} -12.7989 q^{59} +0.509263 q^{60} -7.48745 q^{61} +0.0305854 q^{62} -7.51491 q^{63} -7.89440 q^{64} -6.37708 q^{65} -0.0169431 q^{66} -12.8572 q^{67} +1.99019 q^{68} -1.08326 q^{69} +0.170003 q^{70} -3.59925 q^{71} +0.778561 q^{72} -1.87705 q^{73} -0.780337 q^{74} -0.255194 q^{75} +1.99559 q^{76} +2.56056 q^{77} +0.108047 q^{78} +13.1277 q^{79} +3.97357 q^{80} +8.41813 q^{81} -0.242425 q^{82} -13.7505 q^{83} +1.30400 q^{84} -0.997292 q^{85} +0.0561585 q^{86} +2.66972 q^{87} -0.265279 q^{88} +3.61316 q^{89} -0.194855 q^{90} -16.3289 q^{91} -8.47095 q^{92} -0.117561 q^{93} +0.660004 q^{94} -1.00000 q^{95} -0.202720 q^{96} +1.69508 q^{97} -0.0294487 q^{98} -2.93488 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9}+O(q^{10}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9	\( 8 q - 6 q^{2} - 7 q^{3} + 10 q^{4} + 8 q^{5} - 11 q^{7} - 18 q^{8} + 11 q^{9} - 6 q^{10} + 8 q^{11} - 7 q^{12} - 17 q^{13} + 12 q^{14} - 7 q^{15} + 18 q^{16} - 9 q^{17} - 2 q^{18} - 8 q^{19} + 10 q^{20} + q^{21} - 6 q^{22} - 8 q^{23} + q^{24} + 8 q^{25} + 10 q^{26} - 34 q^{27} - 22 q^{28} - 3 q^{29} - q^{31} - 37 q^{32} - 7 q^{33} - 8 q^{34} - 11 q^{35} + 30 q^{36} - 17 q^{37} + 6 q^{38} + 14 q^{39} - 18 q^{40} - 5 q^{41} + 15 q^{42} - 21 q^{43} + 10 q^{44} + 11 q^{45} - 2 q^{46} - 8 q^{47} + 10 q^{48} + 19 q^{49} - 6 q^{50} - 16 q^{51} + 9 q^{52} - 19 q^{53} - 3 q^{54} + 8 q^{55} + 24 q^{56} + 7 q^{57} + 37 q^{58} - 33 q^{59} - 7 q^{60} - q^{61} - 42 q^{62} - 20 q^{63} + 48 q^{64} - 17 q^{65} - 18 q^{67} - 37 q^{68} + 16 q^{69} + 12 q^{70} - 18 q^{71} + 13 q^{72} - 18 q^{73} + 15 q^{74} - 7 q^{75} - 10 q^{76} - 11 q^{77} - 51 q^{78} - 5 q^{79} + 18 q^{80} + 32 q^{81} + 12 q^{82} - 33 q^{83} - 51 q^{84} - 9 q^{85} - 16 q^{86} - 26 q^{87} - 18 q^{88} - 20 q^{89} - 2 q^{90} + 6 q^{91} - 3 q^{92} + 18 q^{93} + 30 q^{94} - 8 q^{95} + 21 q^{96} - 69 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q - 6 * q^2 - 7 * q^3 + 10 * q^4 + 8 * q^5 - 11 * q^7 - 18 * q^8 + 11 * q^9 - 6 * q^10 + 8 * q^11 - 7 * q^12 - 17 * q^13 + 12 * q^14 - 7 * q^15 + 18 * q^16 - 9 * q^17 - 2 * q^18 - 8 * q^19 + 10 * q^20 + q^21 - 6 * q^22 - 8 * q^23 + q^24 + 8 * q^25 + 10 * q^26 - 34 * q^27 - 22 * q^28 - 3 * q^29 - q^31 - 37 * q^32 - 7 * q^33 - 8 * q^34 - 11 * q^35 + 30 * q^36 - 17 * q^37 + 6 * q^38 + 14 * q^39 - 18 * q^40 - 5 * q^41 + 15 * q^42 - 21 * q^43 + 10 * q^44 + 11 * q^45 - 2 * q^46 - 8 * q^47 + 10 * q^48 + 19 * q^49 - 6 * q^50 - 16 * q^51 + 9 * q^52 - 19 * q^53 - 3 * q^54 + 8 * q^55 + 24 * q^56 + 7 * q^57 + 37 * q^58 - 33 * q^59 - 7 * q^60 - q^61 - 42 * q^62 - 20 * q^63 + 48 * q^64 - 17 * q^65 - 18 * q^67 - 37 * q^68 + 16 * q^69 + 12 * q^70 - 18 * q^71 + 13 * q^72 - 18 * q^73 + 15 * q^74 - 7 * q^75 - 10 * q^76 - 11 * q^77 - 51 * q^78 - 5 * q^79 + 18 * q^80 + 32 * q^81 + 12 * q^82 - 33 * q^83 - 51 * q^84 - 9 * q^85 - 16 * q^86 - 26 * q^87 - 18 * q^88 - 20 * q^89 - 2 * q^90 + 6 * q^91 - 3 * q^92 + 18 * q^93 + 30 * q^94 - 8 * q^95 + 21 * q^96 - 69 * q^98 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more