LMFDB - Embedded newform 4029.2.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4029,2,Mod(1,4029)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4029, 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("4029.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4029 = 3 \cdot 17 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4029.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:	$32.1717269744$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	4029.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.88694 q^{2} +1.00000 q^{3} +1.56054 q^{4} -0.846288 q^{5} -1.88694 q^{6} +3.01609 q^{7} +0.829236 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.88694 q^{2} +1.00000 q^{3} +1.56054 q^{4} -0.846288 q^{5} -1.88694 q^{6} +3.01609 q^{7} +0.829236 q^{8} +1.00000 q^{9} +1.59689 q^{10} -0.920932 q^{11} +1.56054 q^{12} -3.85540 q^{13} -5.69118 q^{14} -0.846288 q^{15} -4.68580 q^{16} +1.00000 q^{17} -1.88694 q^{18} +1.27900 q^{19} -1.32067 q^{20} +3.01609 q^{21} +1.73774 q^{22} -0.718821 q^{23} +0.829236 q^{24} -4.28380 q^{25} +7.27490 q^{26} +1.00000 q^{27} +4.70673 q^{28} +4.88556 q^{29} +1.59689 q^{30} -5.70052 q^{31} +7.18334 q^{32} -0.920932 q^{33} -1.88694 q^{34} -2.55248 q^{35} +1.56054 q^{36} +2.37427 q^{37} -2.41340 q^{38} -3.85540 q^{39} -0.701773 q^{40} +10.5478 q^{41} -5.69118 q^{42} -4.21626 q^{43} -1.43715 q^{44} -0.846288 q^{45} +1.35637 q^{46} +5.34685 q^{47} -4.68580 q^{48} +2.09681 q^{49} +8.08326 q^{50} +1.00000 q^{51} -6.01650 q^{52} -4.27035 q^{53} -1.88694 q^{54} +0.779374 q^{55} +2.50105 q^{56} +1.27900 q^{57} -9.21876 q^{58} +0.996670 q^{59} -1.32067 q^{60} -2.60886 q^{61} +10.7565 q^{62} +3.01609 q^{63} -4.18293 q^{64} +3.26278 q^{65} +1.73774 q^{66} +10.0505 q^{67} +1.56054 q^{68} -0.718821 q^{69} +4.81638 q^{70} +13.2458 q^{71} +0.829236 q^{72} +9.80450 q^{73} -4.48011 q^{74} -4.28380 q^{75} +1.99593 q^{76} -2.77762 q^{77} +7.27490 q^{78} +1.00000 q^{79} +3.96553 q^{80} +1.00000 q^{81} -19.9030 q^{82} +14.1014 q^{83} +4.70673 q^{84} -0.846288 q^{85} +7.95583 q^{86} +4.88556 q^{87} -0.763670 q^{88} +3.60832 q^{89} +1.59689 q^{90} -11.6282 q^{91} -1.12175 q^{92} -5.70052 q^{93} -10.0892 q^{94} -1.08240 q^{95} +7.18334 q^{96} +14.0403 q^{97} -3.95656 q^{98} -0.920932 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9}+O(q^{10}) $ 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9} + 5 q^{10} + 26 q^{11} + 34 q^{12} + 7 q^{13} + 19 q^{14} + 11 q^{15} + 40 q^{16} + 31 q^{17} + 4 q^{18} + 32 q^{19} + 23 q^{20} + 4 q^{21} + 2 q^{22} + 29 q^{23} + 12 q^{24} + 32 q^{25} + 13 q^{26} + 31 q^{27} - 13 q^{28} + 25 q^{29} + 5 q^{30} + 22 q^{31} + 28 q^{32} + 26 q^{33} + 4 q^{34} + 20 q^{35} + 34 q^{36} - 4 q^{37} + 19 q^{38} + 7 q^{39} - 3 q^{40} + 33 q^{41} + 19 q^{42} + 6 q^{43} + 30 q^{44} + 11 q^{45} - 11 q^{46} + 23 q^{47} + 40 q^{48} + 31 q^{49} + 6 q^{50} + 31 q^{51} - 7 q^{52} + 12 q^{53} + 4 q^{54} + 40 q^{56} + 32 q^{57} + 9 q^{58} + 27 q^{59} + 23 q^{60} - 4 q^{61} + 25 q^{62} + 4 q^{63} + 10 q^{64} + 54 q^{65} + 2 q^{66} + 34 q^{68} + 29 q^{69} - 59 q^{70} + 35 q^{71} + 12 q^{72} + 5 q^{73} + 48 q^{74} + 32 q^{75} + 32 q^{76} + 42 q^{77} + 13 q^{78} + 31 q^{79} + 24 q^{80} + 31 q^{81} + 5 q^{82} + 67 q^{83} - 13 q^{84} + 11 q^{85} - 20 q^{86} + 25 q^{87} - 7 q^{88} + 22 q^{89} + 5 q^{90} + 16 q^{91} + 57 q^{92} + 22 q^{93} + 45 q^{94} + 73 q^{95} + 28 q^{96} - 13 q^{97} - 19 q^{98} + 26 q^{99}+O(q^{100}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9 + 5 * q^10 + 26 * q^11 + 34 * q^12 + 7 * q^13 + 19 * q^14 + 11 * q^15 + 40 * q^16 + 31 * q^17 + 4 * q^18 + 32 * q^19 + 23 * q^20 + 4 * q^21 + 2 * q^22 + 29 * q^23 + 12 * q^24 + 32 * q^25 + 13 * q^26 + 31 * q^27 - 13 * q^28 + 25 * q^29 + 5 * q^30 + 22 * q^31 + 28 * q^32 + 26 * q^33 + 4 * q^34 + 20 * q^35 + 34 * q^36 - 4 * q^37 + 19 * q^38 + 7 * q^39 - 3 * q^40 + 33 * q^41 + 19 * q^42 + 6 * q^43 + 30 * q^44 + 11 * q^45 - 11 * q^46 + 23 * q^47 + 40 * q^48 + 31 * q^49 + 6 * q^50 + 31 * q^51 - 7 * q^52 + 12 * q^53 + 4 * q^54 + 40 * q^56 + 32 * q^57 + 9 * q^58 + 27 * q^59 + 23 * q^60 - 4 * q^61 + 25 * q^62 + 4 * q^63 + 10 * q^64 + 54 * q^65 + 2 * q^66 + 34 * q^68 + 29 * q^69 - 59 * q^70 + 35 * q^71 + 12 * q^72 + 5 * q^73 + 48 * q^74 + 32 * q^75 + 32 * q^76 + 42 * q^77 + 13 * q^78 + 31 * q^79 + 24 * q^80 + 31 * q^81 + 5 * q^82 + 67 * q^83 - 13 * q^84 + 11 * q^85 - 20 * q^86 + 25 * q^87 - 7 * q^88 + 22 * q^89 + 5 * q^90 + 16 * q^91 + 57 * q^92 + 22 * q^93 + 45 * q^94 + 73 * q^95 + 28 * q^96 - 13 * q^97 - 19 * q^98 + 26 * 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 $

$2$	−1.88694	−1.33427	−0.667134	−	0.744938i	$-0.732479\pi$
$2$	−1.88694	−1.33427	−0.667134	+	0.744938i	$0.732479\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.56054	0.780270
$5$	−0.846288	−0.378472	−0.189236	−	0.981932i	$-0.560601\pi$
$5$	−0.846288	−0.378472	−0.189236	+	0.981932i	$0.560601\pi$
$6$	−1.88694	−0.770340
$7$	3.01609	1.13998	0.569988	−	0.821653i	$-0.306948\pi$
$7$	3.01609	1.13998	0.569988	+	0.821653i	$0.306948\pi$
$8$	0.829236	0.293179
$9$	1.00000	0.333333
$10$	1.59689	0.504982
$11$	−0.920932	−0.277672	−0.138836	−	0.990315i	$-0.544336\pi$
$11$	−0.920932	−0.277672	−0.138836	+	0.990315i	$0.544336\pi$
$12$	1.56054	0.450489
$13$	−3.85540	−1.06930	−0.534648	−	0.845075i	$-0.679556\pi$
$13$	−3.85540	−1.06930	−0.534648	+	0.845075i	$0.679556\pi$
$14$	−5.69118	−1.52103
$15$	−0.846288	−0.218511
$16$	−4.68580	−1.17145
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.88694	−0.444756
$19$	1.27900	0.293423	0.146711	−	0.989179i	$-0.453131\pi$
$19$	1.27900	0.293423	0.146711	+	0.989179i	$0.453131\pi$
$20$	−1.32067	−0.295310
$21$	3.01609	0.658165
$22$	1.73774	0.370488
$23$	−0.718821	−0.149885	−0.0749423	−	0.997188i	$-0.523877\pi$
$23$	−0.718821	−0.149885	−0.0749423	+	0.997188i	$0.523877\pi$
$24$	0.829236	0.169267
$25$	−4.28380	−0.856759
$26$	7.27490	1.42673
$27$	1.00000	0.192450
$28$	4.70673	0.889488
$29$	4.88556	0.907226	0.453613	−	0.891199i	$-0.350135\pi$
$29$	4.88556	0.907226	0.453613	+	0.891199i	$0.350135\pi$
$30$	1.59689	0.291552
$31$	−5.70052	−1.02384	−0.511922	−	0.859032i	$-0.671066\pi$
$31$	−5.70052	−1.02384	−0.511922	+	0.859032i	$0.671066\pi$
$32$	7.18334	1.26985
$33$	−0.920932	−0.160314
$34$	−1.88694	−0.323607
$35$	−2.55248	−0.431448
$36$	1.56054	0.260090
$37$	2.37427	0.390328	0.195164	−	0.980771i	$-0.437476\pi$
$37$	2.37427	0.390328	0.195164	+	0.980771i	$0.437476\pi$
$38$	−2.41340	−0.391505
$39$	−3.85540	−0.617358
$40$	−0.701773	−0.110960
$41$	10.5478	1.64729	0.823644	−	0.567107i	$-0.191937\pi$
$41$	10.5478	1.64729	0.823644	+	0.567107i	$0.191937\pi$
$42$	−5.69118	−0.878169
$43$	−4.21626	−0.642974	−0.321487	−	0.946914i	$-0.604183\pi$
$43$	−4.21626	−0.642974	−0.321487	+	0.946914i	$0.604183\pi$
$44$	−1.43715	−0.216659
$45$	−0.846288	−0.126157
$46$	1.35637	0.199986
$47$	5.34685	0.779918	0.389959	−	0.920832i	$-0.372489\pi$
$47$	5.34685	0.779918	0.389959	+	0.920832i	$0.372489\pi$
$48$	−4.68580	−0.676336
$49$	2.09681	0.299545
$50$	8.08326	1.14315
$51$	1.00000	0.140028
$52$	−6.01650	−0.834339
$53$	−4.27035	−0.586577	−0.293289	−	0.956024i	$-0.594750\pi$
$53$	−4.27035	−0.586577	−0.293289	+	0.956024i	$0.594750\pi$
$54$	−1.88694	−0.256780
$55$	0.779374	0.105091
$56$	2.50105	0.334217
$57$	1.27900	0.169408
$58$	−9.21876	−1.21048
$59$	0.996670	0.129755	0.0648777	−	0.997893i	$-0.479334\pi$
$59$	0.996670	0.129755	0.0648777	+	0.997893i	$0.479334\pi$
$60$	−1.32067	−0.170497
$61$	−2.60886	−0.334031	−0.167015	−	0.985954i	$-0.553413\pi$
$61$	−2.60886	−0.334031	−0.167015	+	0.985954i	$0.553413\pi$
$62$	10.7565	1.36608
$63$	3.01609	0.379992
$64$	−4.18293	−0.522866
$65$	3.26278	0.404698
$66$	1.73774	0.213901
$67$	10.0505	1.22786	0.613931	−	0.789360i	$-0.289587\pi$
$67$	10.0505	1.22786	0.613931	+	0.789360i	$0.289587\pi$
$68$	1.56054	0.189243
$69$	−0.718821	−0.0865359
$70$	4.81638	0.575668
$71$	13.2458	1.57199	0.785996	−	0.618231i	$-0.212151\pi$
$71$	13.2458	1.57199	0.785996	+	0.618231i	$0.212151\pi$
$72$	0.829236	0.0977264
$73$	9.80450	1.14753	0.573765	−	0.819020i	$-0.305482\pi$
$73$	9.80450	1.14753	0.573765	+	0.819020i	$0.305482\pi$
$74$	−4.48011	−0.520802
$75$	−4.28380	−0.494650
$76$	1.99593	0.228949
$77$	−2.77762	−0.316539
$78$	7.27490	0.823721
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	3.96553	0.443360
$81$	1.00000	0.111111
$82$	−19.9030	−2.19792
$83$	14.1014	1.54783	0.773913	−	0.633292i	$-0.218297\pi$
$83$	14.1014	1.54783	0.773913	+	0.633292i	$0.218297\pi$
$84$	4.70673	0.513546
$85$	−0.846288	−0.0917928
$86$	7.95583	0.857899
$87$	4.88556	0.523787
$88$	−0.763670	−0.0814075
$89$	3.60832	0.382481	0.191240	−	0.981543i	$-0.438749\pi$
$89$	3.60832	0.382481	0.191240	+	0.981543i	$0.438749\pi$
$90$	1.59689	0.168327
$91$	−11.6282	−1.21897
$92$	−1.12175	−0.116950
$93$	−5.70052	−0.591117
$94$	−10.0892	−1.04062
$95$	−1.08240	−0.111052
$96$	7.18334	0.733146
$97$	14.0403	1.42558	0.712788	−	0.701380i	$-0.247432\pi$
$97$	14.0403	1.42558	0.712788	+	0.701380i	$0.247432\pi$
$98$	−3.95656	−0.399673
$99$	−0.920932	−0.0925572
$100$	−6.68503	−0.668503
$101$	−0.778050	−0.0774188	−0.0387094	−	0.999251i	$-0.512325\pi$
$101$	−0.778050	−0.0774188	−0.0387094	+	0.999251i	$0.512325\pi$
$102$	−1.88694	−0.186835
$103$	−0.390840	−0.0385106	−0.0192553	−	0.999815i	$-0.506130\pi$
$103$	−0.390840	−0.0385106	−0.0192553	+	0.999815i	$0.506130\pi$
$104$	−3.19704	−0.313495
$105$	−2.55248	−0.249097
$106$	8.05789	0.782651
$107$	2.33288	0.225528	0.112764	−	0.993622i	$-0.464030\pi$
$107$	2.33288	0.225528	0.112764	+	0.993622i	$0.464030\pi$
$108$	1.56054	0.150163
$109$	−17.8437	−1.70912	−0.854558	−	0.519356i	$-0.826172\pi$
$109$	−17.8437	−1.70912	−0.854558	+	0.519356i	$0.826172\pi$
$110$	−1.47063	−0.140219
$111$	2.37427	0.225356
$112$	−14.1328	−1.33542
$113$	−18.2810	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$113$	−18.2810	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$114$	−2.41340	−0.226035
$115$	0.608330	0.0567270
$116$	7.62411	0.707881
$117$	−3.85540	−0.356432
$118$	−1.88066	−0.173128
$119$	3.01609	0.276485
$120$	−0.701773	−0.0640628
$121$	−10.1519	−0.922899
$122$	4.92277	0.445686
$123$	10.5478	0.951062
$124$	−8.89589	−0.798875
$125$	7.85677	0.702731
$126$	−5.69118	−0.507011
$127$	−12.4803	−1.10745	−0.553725	−	0.832700i	$-0.686794\pi$
$127$	−12.4803	−1.10745	−0.553725	+	0.832700i	$0.686794\pi$
$128$	−6.47374	−0.572203
$129$	−4.21626	−0.371221
$130$	−6.15666	−0.539975
$131$	−2.16959	−0.189558	−0.0947791	−	0.995498i	$-0.530214\pi$
$131$	−2.16959	−0.189558	−0.0947791	+	0.995498i	$0.530214\pi$
$132$	−1.43715	−0.125088
$133$	3.85758	0.334495
$134$	−18.9646	−1.63830
$135$	−0.846288	−0.0728369
$136$	0.829236	0.0711064
$137$	17.0324	1.45517	0.727587	−	0.686015i	$-0.240642\pi$
$137$	17.0324	1.45517	0.727587	+	0.686015i	$0.240642\pi$
$138$	1.35637	0.115462
$139$	3.52916	0.299340	0.149670	−	0.988736i	$-0.452179\pi$
$139$	3.52916	0.299340	0.149670	+	0.988736i	$0.452179\pi$
$140$	−3.98325	−0.336646
$141$	5.34685	0.450286
$142$	−24.9941	−2.09746
$143$	3.55056	0.296913
$144$	−4.68580	−0.390483
$145$	−4.13459	−0.343359
$146$	−18.5005	−1.53111
$147$	2.09681	0.172942
$148$	3.70514	0.304561
$149$	10.4477	0.855912	0.427956	−	0.903800i	$-0.359234\pi$
$149$	10.4477	0.855912	0.427956	+	0.903800i	$0.359234\pi$
$150$	8.08326	0.659996
$151$	20.4933	1.66772	0.833862	−	0.551972i	$-0.186125\pi$
$151$	20.4933	1.66772	0.833862	+	0.551972i	$0.186125\pi$
$152$	1.06059	0.0860255
$153$	1.00000	0.0808452
$154$	5.24119	0.422347
$155$	4.82429	0.387496
$156$	−6.01650	−0.481706
$157$	−5.04399	−0.402554	−0.201277	−	0.979534i	$-0.564509\pi$
$157$	−5.04399	−0.402554	−0.201277	+	0.979534i	$0.564509\pi$
$158$	−1.88694	−0.150117
$159$	−4.27035	−0.338661
$160$	−6.07917	−0.480601
$161$	−2.16803	−0.170865
$162$	−1.88694	−0.148252
$163$	19.3163	1.51297	0.756486	−	0.654010i	$-0.226915\pi$
$163$	19.3163	1.51297	0.756486	+	0.654010i	$0.226915\pi$
$164$	16.4602	1.28533
$165$	0.779374	0.0606742
$166$	−26.6084	−2.06521
$167$	−22.6179	−1.75023	−0.875113	−	0.483919i	$-0.839213\pi$
$167$	−22.6179	−1.75023	−0.875113	+	0.483919i	$0.839213\pi$
$168$	2.50105	0.192960
$169$	1.86410	0.143392
$170$	1.59689	0.122476
$171$	1.27900	0.0978076
$172$	−6.57964	−0.501693
$173$	−5.37195	−0.408422	−0.204211	−	0.978927i	$-0.565463\pi$
$173$	−5.37195	−0.408422	−0.204211	+	0.978927i	$0.565463\pi$
$174$	−9.21876	−0.698872
$175$	−12.9203	−0.976685
$176$	4.31530	0.325278
$177$	0.996670	0.0749143
$178$	−6.80867	−0.510332
$179$	17.5000	1.30801	0.654006	−	0.756490i	$-0.273087\pi$
$179$	17.5000	1.30801	0.654006	+	0.756490i	$0.273087\pi$
$180$	−1.32067	−0.0984366
$181$	25.4752	1.89356	0.946778	−	0.321889i	$-0.104318\pi$
$181$	25.4752	1.89356	0.946778	+	0.321889i	$0.104318\pi$
$182$	21.9418	1.62643
$183$	−2.60886	−0.192853
$184$	−0.596072	−0.0439430
$185$	−2.00932	−0.147728
$186$	10.7565	0.788708
$187$	−0.920932	−0.0673452
$188$	8.34397	0.608546
$189$	3.01609	0.219388
$190$	2.04243	0.148173
$191$	−4.97527	−0.359998	−0.179999	−	0.983667i	$-0.557609\pi$
$191$	−4.97527	−0.359998	−0.179999	+	0.983667i	$0.557609\pi$
$192$	−4.18293	−0.301877
$193$	8.95833	0.644835	0.322417	−	0.946598i	$-0.395505\pi$
$193$	8.95833	0.644835	0.322417	+	0.946598i	$0.395505\pi$
$194$	−26.4932	−1.90210
$195$	3.26278	0.233652
$196$	3.27216	0.233726
$197$	12.5153	0.891678	0.445839	−	0.895113i	$-0.352905\pi$
$197$	12.5153	0.891678	0.445839	+	0.895113i	$0.352905\pi$
$198$	1.73774	0.123496
$199$	−25.9024	−1.83617	−0.918086	−	0.396381i	$-0.870266\pi$
$199$	−25.9024	−1.83617	−0.918086	+	0.396381i	$0.870266\pi$
$200$	−3.55228	−0.251184
$201$	10.0505	0.708906
$202$	1.46813	0.103297
$203$	14.7353	1.03422
$204$	1.56054	0.109260
$205$	−8.92647	−0.623452
$206$	0.737491	0.0513835
$207$	−0.718821	−0.0499615
$208$	18.0656	1.25262
$209$	−1.17787	−0.0814752
$210$	4.81638	0.332362
$211$	12.2946	0.846397	0.423199	−	0.906037i	$-0.360907\pi$
$211$	12.2946	0.846397	0.423199	+	0.906037i	$0.360907\pi$
$212$	−6.66404	−0.457688
$213$	13.2458	0.907590
$214$	−4.40201	−0.300915
$215$	3.56817	0.243347
$216$	0.829236	0.0564224
$217$	−17.1933	−1.16716
$218$	33.6700	2.28042
$219$	9.80450	0.662527
$220$	1.21624	0.0819991
$221$	−3.85540	−0.259342
$222$	−4.48011	−0.300685
$223$	0.593490	0.0397430	0.0198715	−	0.999803i	$-0.493674\pi$
$223$	0.593490	0.0397430	0.0198715	+	0.999803i	$0.493674\pi$
$224$	21.6656	1.44759
$225$	−4.28380	−0.285586
$226$	34.4952	2.29459
$227$	−18.5422	−1.23069	−0.615344	−	0.788259i	$-0.710983\pi$
$227$	−18.5422	−1.23069	−0.615344	+	0.788259i	$0.710983\pi$
$228$	1.99593	0.132184
$229$	7.01948	0.463860	0.231930	−	0.972732i	$-0.425496\pi$
$229$	7.01948	0.463860	0.231930	+	0.972732i	$0.425496\pi$
$230$	−1.14788	−0.0756891
$231$	−2.77762	−0.182754
$232$	4.05128	0.265980
$233$	−17.8632	−1.17026	−0.585130	−	0.810940i	$-0.698956\pi$
$233$	−17.8632	−1.17026	−0.585130	+	0.810940i	$0.698956\pi$
$234$	7.27490	0.475575
$235$	−4.52497	−0.295177
$236$	1.55534	0.101244
$237$	1.00000	0.0649570
$238$	−5.69118	−0.368905
$239$	−6.45389	−0.417467	−0.208734	−	0.977973i	$-0.566934\pi$
$239$	−6.45389	−0.417467	−0.208734	+	0.977973i	$0.566934\pi$
$240$	3.96553	0.255974
$241$	−23.6366	−1.52257	−0.761285	−	0.648418i	$-0.775431\pi$
$241$	−23.6366	−1.52257	−0.761285	+	0.648418i	$0.775431\pi$
$242$	19.1560	1.23139
$243$	1.00000	0.0641500
$244$	−4.07123	−0.260634
$245$	−1.77451	−0.113369
$246$	−19.9030	−1.26897
$247$	−4.93106	−0.313756
$248$	−4.72708	−0.300170
$249$	14.1014	0.893637
$250$	−14.8252	−0.937630
$251$	15.7470	0.993942	0.496971	−	0.867767i	$-0.334445\pi$
$251$	15.7470	0.993942	0.496971	+	0.867767i	$0.334445\pi$
$252$	4.70673	0.296496
$253$	0.661986	0.0416187
$254$	23.5496	1.47763
$255$	−0.846288	−0.0529966
$256$	20.5814	1.28634
$257$	16.7472	1.04466	0.522331	−	0.852743i	$-0.325063\pi$
$257$	16.7472	1.04466	0.522331	+	0.852743i	$0.325063\pi$
$258$	7.95583	0.495308
$259$	7.16102	0.444964
$260$	5.09169	0.315773
$261$	4.88556	0.302409
$262$	4.09389	0.252921
$263$	−9.30111	−0.573531	−0.286766	−	0.958001i	$-0.592580\pi$
$263$	−9.30111	−0.573531	−0.286766	+	0.958001i	$0.592580\pi$
$264$	−0.763670	−0.0470007
$265$	3.61394	0.222003
$266$	−7.27903	−0.446306
$267$	3.60832	0.220825
$268$	15.6842	0.958063
$269$	19.0596	1.16208	0.581041	−	0.813874i	$-0.302646\pi$
$269$	19.0596	1.16208	0.581041	+	0.813874i	$0.302646\pi$
$270$	1.59689	0.0971839
$271$	29.3396	1.78225	0.891127	−	0.453754i	$-0.149915\pi$
$271$	29.3396	1.78225	0.891127	+	0.453754i	$0.149915\pi$
$272$	−4.68580	−0.284118
$273$	−11.6282	−0.703773
$274$	−32.1391	−1.94159
$275$	3.94509	0.237898
$276$	−1.12175	−0.0675213
$277$	31.4076	1.88710	0.943550	−	0.331231i	$-0.107464\pi$
$277$	31.4076	1.88710	0.943550	+	0.331231i	$0.107464\pi$
$278$	−6.65931	−0.399399
$279$	−5.70052	−0.341281
$280$	−2.11661	−0.126492
$281$	17.1452	1.02279	0.511397	−	0.859344i	$-0.329128\pi$
$281$	17.1452	1.02279	0.511397	+	0.859344i	$0.329128\pi$
$282$	−10.0892	−0.600802
$283$	−10.0764	−0.598981	−0.299490	−	0.954099i	$-0.596817\pi$
$283$	−10.0764	−0.598981	−0.299490	+	0.954099i	$0.596817\pi$
$284$	20.6707	1.22658
$285$	−1.08240	−0.0641160
$286$	−6.69969	−0.396161
$287$	31.8131	1.87787
$288$	7.18334	0.423282
$289$	1.00000	0.0588235
$290$	7.80173	0.458133
$291$	14.0403	0.823056
$292$	15.3003	0.895383
$293$	−15.1049	−0.882438	−0.441219	−	0.897399i	$-0.645454\pi$
$293$	−15.1049	−0.882438	−0.441219	+	0.897399i	$0.645454\pi$
$294$	−3.95656	−0.230751
$295$	−0.843470	−0.0491087
$296$	1.96883	0.114436
$297$	−0.920932	−0.0534379
$298$	−19.7142	−1.14202
$299$	2.77134	0.160271
$300$	−6.68503	−0.385960
$301$	−12.7166	−0.732975
$302$	−38.6697	−2.22519
$303$	−0.778050	−0.0446978
$304$	−5.99314	−0.343730
$305$	2.20785	0.126421
$306$	−1.88694	−0.107869
$307$	−5.28167	−0.301441	−0.150720	−	0.988576i	$-0.548159\pi$
$307$	−5.28167	−0.301441	−0.150720	+	0.988576i	$0.548159\pi$
$308$	−4.33458	−0.246986
$309$	−0.390840	−0.0222341
$310$	−9.10313	−0.517023
$311$	−19.8699	−1.12672	−0.563359	−	0.826213i	$-0.690491\pi$
$311$	−19.8699	−1.12672	−0.563359	+	0.826213i	$0.690491\pi$
$312$	−3.19704	−0.180997
$313$	17.3863	0.982734	0.491367	−	0.870953i	$-0.336497\pi$
$313$	17.3863	0.982734	0.491367	+	0.870953i	$0.336497\pi$
$314$	9.51770	0.537115
$315$	−2.55248	−0.143816
$316$	1.56054	0.0877872
$317$	30.2204	1.69735	0.848674	−	0.528916i	$-0.177401\pi$
$317$	30.2204	1.69735	0.848674	+	0.528916i	$0.177401\pi$
$318$	8.05789	0.451864
$319$	−4.49927	−0.251911
$320$	3.53997	0.197890
$321$	2.33288	0.130209
$322$	4.09094	0.227979
$323$	1.27900	0.0711655
$324$	1.56054	0.0866966
$325$	16.5157	0.916129
$326$	−36.4487	−2.01871
$327$	−17.8437	−0.986759
$328$	8.74661	0.482951
$329$	16.1266	0.889088
$330$	−1.47063	−0.0809556
$331$	−24.6484	−1.35480	−0.677400	−	0.735615i	$-0.736893\pi$
$331$	−24.6484	−1.35480	−0.677400	+	0.735615i	$0.736893\pi$
$332$	22.0057	1.20772
$333$	2.37427	0.130109
$334$	42.6786	2.33527
$335$	−8.50560	−0.464711
$336$	−14.1328	−0.771007
$337$	2.38693	0.130024	0.0650121	−	0.997884i	$-0.479291\pi$
$337$	2.38693	0.130024	0.0650121	+	0.997884i	$0.479291\pi$
$338$	−3.51745	−0.191324
$339$	−18.2810	−0.992889
$340$	−1.32067	−0.0716231
$341$	5.24980	0.284292
$342$	−2.41340	−0.130502
$343$	−14.7885	−0.798502
$344$	−3.49628	−0.188507
$345$	0.608330	0.0327514
$346$	10.1365	0.544944
$347$	12.7112	0.682372	0.341186	−	0.939996i	$-0.389171\pi$
$347$	12.7112	0.682372	0.341186	+	0.939996i	$0.389171\pi$
$348$	7.62411	0.408695
$349$	−13.8757	−0.742747	−0.371374	−	0.928484i	$-0.621113\pi$
$349$	−13.8757	−0.742747	−0.371374	+	0.928484i	$0.621113\pi$
$350$	24.3799	1.30316
$351$	−3.85540	−0.205786
$352$	−6.61537	−0.352600
$353$	4.35714	0.231907	0.115954	−	0.993255i	$-0.463008\pi$
$353$	4.35714	0.231907	0.115954	+	0.993255i	$0.463008\pi$
$354$	−1.88066	−0.0999558
$355$	−11.2098	−0.594954
$356$	5.63092	0.298438
$357$	3.01609	0.159629
$358$	−33.0214	−1.74524
$359$	−1.47835	−0.0780242	−0.0390121	−	0.999239i	$-0.512421\pi$
$359$	−1.47835	−0.0780242	−0.0390121	+	0.999239i	$0.512421\pi$
$360$	−0.701773	−0.0369867
$361$	−17.3642	−0.913903
$362$	−48.0701	−2.52651
$363$	−10.1519	−0.532836
$364$	−18.1463	−0.951126
$365$	−8.29743	−0.434308
$366$	4.92277	0.257317
$367$	1.22044	0.0637064	0.0318532	−	0.999493i	$-0.489859\pi$
$367$	1.22044	0.0637064	0.0318532	+	0.999493i	$0.489859\pi$
$368$	3.36825	0.175582
$369$	10.5478	0.549096
$370$	3.79146	0.197109
$371$	−12.8798	−0.668684
$372$	−8.89589	−0.461230
$373$	30.4212	1.57515	0.787574	−	0.616220i	$-0.211337\pi$
$373$	30.4212	1.57515	0.787574	+	0.616220i	$0.211337\pi$
$374$	1.73774	0.0898566
$375$	7.85677	0.405722
$376$	4.43380	0.228656
$377$	−18.8358	−0.970093
$378$	−5.69118	−0.292723
$379$	−29.8919	−1.53545	−0.767723	−	0.640782i	$-0.778610\pi$
$379$	−29.8919	−1.53545	−0.767723	+	0.640782i	$0.778610\pi$
$380$	−1.68913	−0.0866507
$381$	−12.4803	−0.639386
$382$	9.38803	0.480334
$383$	32.0422	1.63728	0.818640	−	0.574307i	$-0.194728\pi$
$383$	32.0422	1.63728	0.818640	+	0.574307i	$0.194728\pi$
$384$	−6.47374	−0.330362
$385$	2.35066	0.119801
$386$	−16.9038	−0.860382
$387$	−4.21626	−0.214325
$388$	21.9104	1.11233
$389$	35.9615	1.82332	0.911659	−	0.410947i	$-0.134802\pi$
$389$	35.9615	1.82332	0.911659	+	0.410947i	$0.134802\pi$
$390$	−6.15666	−0.311755
$391$	−0.718821	−0.0363523
$392$	1.73875	0.0878203
$393$	−2.16959	−0.109441
$394$	−23.6156	−1.18974
$395$	−0.846288	−0.0425814
$396$	−1.43715	−0.0722195
$397$	31.2960	1.57070	0.785351	−	0.619051i	$-0.212483\pi$
$397$	31.2960	1.57070	0.785351	+	0.619051i	$0.212483\pi$
$398$	48.8763	2.44995
$399$	3.85758	0.193121
$400$	20.0730	1.00365
$401$	−8.29049	−0.414007	−0.207004	−	0.978340i	$-0.566371\pi$
$401$	−8.29049	−0.414007	−0.207004	+	0.978340i	$0.566371\pi$
$402$	−18.9646	−0.945870
$403$	21.9778	1.09479
$404$	−1.21418	−0.0604076
$405$	−0.846288	−0.0420524
$406$	−27.8046	−1.37992
$407$	−2.18654	−0.108383
$408$	0.829236	0.0410533
$409$	−10.0389	−0.496390	−0.248195	−	0.968710i	$-0.579837\pi$
$409$	−10.0389	−0.496390	−0.248195	+	0.968710i	$0.579837\pi$
$410$	16.8437	0.831851
$411$	17.0324	0.840146
$412$	−0.609921	−0.0300487
$413$	3.00605	0.147918
$414$	1.35637	0.0666620
$415$	−11.9338	−0.585808
$416$	−27.6946	−1.35784
$417$	3.52916	0.172824
$418$	2.22257	0.108710
$419$	17.1659	0.838611	0.419305	−	0.907845i	$-0.362274\pi$
$419$	17.1659	0.838611	0.419305	+	0.907845i	$0.362274\pi$
$420$	−3.98325	−0.194363
$421$	−7.71328	−0.375922	−0.187961	−	0.982176i	$-0.560188\pi$
$421$	−7.71328	−0.375922	−0.187961	+	0.982176i	$0.560188\pi$
$422$	−23.1992	−1.12932
$423$	5.34685	0.259973
$424$	−3.54113	−0.171972
$425$	−4.28380	−0.207795
$426$	−24.9941	−1.21097
$427$	−7.86857	−0.380787
$428$	3.64056	0.175973
$429$	3.55056	0.171423
$430$	−6.73293	−0.324691
$431$	21.7270	1.04655	0.523275	−	0.852164i	$-0.324710\pi$
$431$	21.7270	1.04655	0.523275	+	0.852164i	$0.324710\pi$
$432$	−4.68580	−0.225445
$433$	28.5630	1.37265	0.686326	−	0.727294i	$-0.259222\pi$
$433$	28.5630	1.37265	0.686326	+	0.727294i	$0.259222\pi$
$434$	32.4427	1.55730
$435$	−4.13459	−0.198239
$436$	−27.8458	−1.33357
$437$	−0.919373	−0.0439796
$438$	−18.5005	−0.883988
$439$	−16.0063	−0.763940	−0.381970	−	0.924175i	$-0.624754\pi$
$439$	−16.0063	−0.763940	−0.381970	+	0.924175i	$0.624754\pi$
$440$	0.646285	0.0308104
$441$	2.09681	0.0998482
$442$	7.27490	0.346032
$443$	15.3898	0.731191	0.365596	−	0.930774i	$-0.380865\pi$
$443$	15.3898	0.731191	0.365596	+	0.930774i	$0.380865\pi$
$444$	3.70514	0.175838
$445$	−3.05368	−0.144758
$446$	−1.11988	−0.0530278
$447$	10.4477	0.494161
$448$	−12.6161	−0.596055
$449$	−15.7432	−0.742966	−0.371483	−	0.928440i	$-0.621151\pi$
$449$	−15.7432	−0.742966	−0.371483	+	0.928440i	$0.621151\pi$
$450$	8.08326	0.381049
$451$	−9.71380	−0.457405
$452$	−28.5283	−1.34186
$453$	20.4933	0.962861
$454$	34.9880	1.64207
$455$	9.84084	0.461346
$456$	1.06059	0.0496668
$457$	34.9860	1.63658	0.818289	−	0.574807i	$-0.194923\pi$
$457$	34.9860	1.63658	0.818289	+	0.574807i	$0.194923\pi$
$458$	−13.2453	−0.618914
$459$	1.00000	0.0466760
$460$	0.949322	0.0442624
$461$	−28.0810	−1.30786	−0.653931	−	0.756554i	$-0.726882\pi$
$461$	−28.0810	−1.30786	−0.653931	+	0.756554i	$0.726882\pi$
$462$	5.24119	0.243842
$463$	−31.6142	−1.46923	−0.734617	−	0.678482i	$-0.762638\pi$
$463$	−31.6142	−1.46923	−0.734617	+	0.678482i	$0.762638\pi$
$464$	−22.8927	−1.06277
$465$	4.82429	0.223721
$466$	33.7068	1.56144
$467$	−10.2872	−0.476037	−0.238018	−	0.971261i	$-0.576498\pi$
$467$	−10.2872	−0.476037	−0.238018	+	0.971261i	$0.576498\pi$
$468$	−6.01650	−0.278113
$469$	30.3132	1.39973
$470$	8.53835	0.393845
$471$	−5.04399	−0.232415
$472$	0.826475	0.0380416
$473$	3.88289	0.178536
$474$	−1.88694	−0.0866700
$475$	−5.47898	−0.251393
$476$	4.70673	0.215733
$477$	−4.27035	−0.195526
$478$	12.1781	0.557013
$479$	−8.35097	−0.381566	−0.190783	−	0.981632i	$-0.561103\pi$
$479$	−8.35097	−0.381566	−0.190783	+	0.981632i	$0.561103\pi$
$480$	−6.07917	−0.277475
$481$	−9.15377	−0.417376
$482$	44.6009	2.03152
$483$	−2.16803	−0.0986488
$484$	−15.8424	−0.720110
$485$	−11.8821	−0.539540
$486$	−1.88694	−0.0855933
$487$	−24.0747	−1.09093	−0.545465	−	0.838134i	$-0.683647\pi$
$487$	−24.0747	−1.09093	−0.545465	+	0.838134i	$0.683647\pi$
$488$	−2.16336	−0.0979309
$489$	19.3163	0.873515
$490$	3.34839	0.151265
$491$	19.0373	0.859142	0.429571	−	0.903033i	$-0.358665\pi$
$491$	19.0373	0.859142	0.429571	+	0.903033i	$0.358665\pi$
$492$	16.4602	0.742085
$493$	4.88556	0.220035
$494$	9.30461	0.418634
$495$	0.779374	0.0350303
$496$	26.7115	1.19938
$497$	39.9507	1.79203
$498$	−26.6084	−1.19235
$499$	−12.3272	−0.551840	−0.275920	−	0.961181i	$-0.588983\pi$
$499$	−12.3272	−0.551840	−0.275920	+	0.961181i	$0.588983\pi$
$500$	12.2608	0.548319
$501$	−22.6179	−1.01049
$502$	−29.7136	−1.32619
$503$	6.74107	0.300569	0.150285	−	0.988643i	$-0.451981\pi$
$503$	6.74107	0.300569	0.150285	+	0.988643i	$0.451981\pi$
$504$	2.50105	0.111406
$505$	0.658454	0.0293008
$506$	−1.24913	−0.0555304
$507$	1.86410	0.0827877
$508$	−19.4760	−0.864109
$509$	25.2221	1.11795	0.558975	−	0.829184i	$-0.311195\pi$
$509$	25.2221	1.11795	0.558975	+	0.829184i	$0.311195\pi$
$510$	1.59689	0.0707117
$511$	29.5713	1.30816
$512$	−25.8884	−1.14412
$513$	1.27900	0.0564693
$514$	−31.6009	−1.39386
$515$	0.330763	0.0145752
$516$	−6.57964	−0.289653
$517$	−4.92409	−0.216561
$518$	−13.5124	−0.593701
$519$	−5.37195	−0.235802
$520$	2.70561	0.118649
$521$	6.19871	0.271570	0.135785	−	0.990738i	$-0.456644\pi$
$521$	6.19871	0.271570	0.135785	+	0.990738i	$0.456644\pi$
$522$	−9.21876	−0.403494
$523$	−12.1289	−0.530361	−0.265181	−	0.964199i	$-0.585432\pi$
$523$	−12.1289	−0.530361	−0.265181	+	0.964199i	$0.585432\pi$
$524$	−3.38573	−0.147906
$525$	−12.9203	−0.563889
$526$	17.5506	0.765244
$527$	−5.70052	−0.248319
$528$	4.31530	0.187799
$529$	−22.4833	−0.977535
$530$	−6.81929	−0.296211
$531$	0.996670	0.0432518
$532$	6.01991	0.260996
$533$	−40.6659	−1.76144
$534$	−6.80867	−0.294640
$535$	−1.97429	−0.0853561
$536$	8.33422	0.359983
$537$	17.5000	0.755181
$538$	−35.9642	−1.55053
$539$	−1.93102	−0.0831750
$540$	−1.32067	−0.0568324
$541$	37.1875	1.59881	0.799407	−	0.600789i	$-0.205147\pi$
$541$	37.1875	1.59881	0.799407	+	0.600789i	$0.205147\pi$
$542$	−55.3621	−2.37800
$543$	25.4752	1.09324
$544$	7.18334	0.307983
$545$	15.1009	0.646852
$546$	21.9418	0.939021
$547$	42.8321	1.83137	0.915685	−	0.401897i	$-0.131649\pi$
$547$	42.8321	1.83137	0.915685	+	0.401897i	$0.131649\pi$
$548$	26.5797	1.13543
$549$	−2.60886	−0.111344
$550$	−7.44414	−0.317419
$551$	6.24864	0.266201
$552$	−0.596072	−0.0253705
$553$	3.01609	0.128257
$554$	−59.2642	−2.51790
$555$	−2.00932	−0.0852908
$556$	5.50740	0.233566
$557$	2.89913	0.122840	0.0614201	−	0.998112i	$-0.480437\pi$
$557$	2.89913	0.122840	0.0614201	+	0.998112i	$0.480437\pi$
$558$	10.7565	0.455361
$559$	16.2554	0.687529
$560$	11.9604	0.505420
$561$	−0.920932	−0.0388818
$562$	−32.3519	−1.36468
$563$	13.9220	0.586741	0.293370	−	0.955999i	$-0.405223\pi$
$563$	13.9220	0.586741	0.293370	+	0.955999i	$0.405223\pi$
$564$	8.34397	0.351344
$565$	15.4710	0.650871
$566$	19.0136	0.799201
$567$	3.01609	0.126664
$568$	10.9839	0.460875
$569$	−5.20319	−0.218129	−0.109065	−	0.994035i	$-0.534786\pi$
$569$	−5.20319	−0.218129	−0.109065	+	0.994035i	$0.534786\pi$
$570$	2.04243	0.0855479
$571$	−19.7899	−0.828180	−0.414090	−	0.910236i	$-0.635900\pi$
$571$	−19.7899	−0.828180	−0.414090	+	0.910236i	$0.635900\pi$
$572$	5.54079	0.231672
$573$	−4.97527	−0.207845
$574$	−60.0294	−2.50558
$575$	3.07928	0.128415
$576$	−4.18293	−0.174289
$577$	14.0552	0.585124	0.292562	−	0.956247i	$-0.405492\pi$
$577$	14.0552	0.585124	0.292562	+	0.956247i	$0.405492\pi$
$578$	−1.88694	−0.0784863
$579$	8.95833	0.372296
$580$	−6.45219	−0.267913
$581$	42.5310	1.76448
$582$	−26.4932	−1.09818
$583$	3.93270	0.162876
$584$	8.13025	0.336432
$585$	3.26278	0.134899
$586$	28.5020	1.17741
$587$	30.3566	1.25295	0.626475	−	0.779442i	$-0.284497\pi$
$587$	30.3566	1.25295	0.626475	+	0.779442i	$0.284497\pi$
$588$	3.27216	0.134942
$589$	−7.29098	−0.300419
$590$	1.59158	0.0655242
$591$	12.5153	0.514811
$592$	−11.1254	−0.457249
$593$	−14.4046	−0.591527	−0.295763	−	0.955261i	$-0.595574\pi$
$593$	−14.4046	−0.591527	−0.295763	+	0.955261i	$0.595574\pi$
$594$	1.73774	0.0713005
$595$	−2.55248	−0.104642
$596$	16.3041	0.667842
$597$	−25.9024	−1.06011
$598$	−5.22935	−0.213844
$599$	44.1246	1.80288	0.901442	−	0.432900i	$-0.142510\pi$
$599$	44.1246	1.80288	0.901442	+	0.432900i	$0.142510\pi$
$600$	−3.55228	−0.145021
$601$	−12.9118	−0.526685	−0.263342	−	0.964702i	$-0.584825\pi$
$601$	−12.9118	−0.526685	−0.263342	+	0.964702i	$0.584825\pi$
$602$	23.9955	0.977985
$603$	10.0505	0.409287
$604$	31.9807	1.30127
$605$	8.59142	0.349291
$606$	1.46813	0.0596388
$607$	7.79738	0.316486	0.158243	−	0.987400i	$-0.449417\pi$
$607$	7.79738	0.316486	0.158243	+	0.987400i	$0.449417\pi$
$608$	9.18750	0.372602
$609$	14.7353	0.597105
$610$	−4.16608	−0.168680
$611$	−20.6142	−0.833963
$612$	1.56054	0.0630811
$613$	38.3292	1.54810	0.774050	−	0.633124i	$-0.218228\pi$
$613$	38.3292	1.54810	0.774050	+	0.633124i	$0.218228\pi$
$614$	9.96619	0.402203
$615$	−8.92647	−0.359950
$616$	−2.30330	−0.0928026
$617$	36.2737	1.46032	0.730162	−	0.683274i	$-0.239444\pi$
$617$	36.2737	1.46032	0.730162	+	0.683274i	$0.239444\pi$
$618$	0.737491	0.0296663
$619$	34.8928	1.40246	0.701229	−	0.712936i	$-0.252635\pi$
$619$	34.8928	1.40246	0.701229	+	0.712936i	$0.252635\pi$
$620$	7.52849	0.302351
$621$	−0.718821	−0.0288453
$622$	37.4932	1.50334
$623$	10.8830	0.436019
$624$	18.0656	0.723203
$625$	14.7699	0.590796
$626$	−32.8070	−1.31123
$627$	−1.17787	−0.0470397
$628$	−7.87134	−0.314101
$629$	2.37427	0.0946684
$630$	4.81638	0.191889
$631$	36.1616	1.43957	0.719786	−	0.694196i	$-0.244240\pi$
$631$	36.1616	1.43957	0.719786	+	0.694196i	$0.244240\pi$
$632$	0.829236	0.0329852
$633$	12.2946	0.488668
$634$	−57.0241	−2.26472
$635$	10.5619	0.419138
$636$	−6.66404	−0.264247
$637$	−8.08405	−0.320302
$638$	8.48985	0.336116
$639$	13.2458	0.523997
$640$	5.47865	0.216563
$641$	−40.8595	−1.61385	−0.806926	−	0.590652i	$-0.798871\pi$
$641$	−40.8595	−1.61385	−0.806926	+	0.590652i	$0.798871\pi$
$642$	−4.40201	−0.173734
$643$	−10.3584	−0.408494	−0.204247	−	0.978919i	$-0.565475\pi$
$643$	−10.3584	−0.408494	−0.204247	+	0.978919i	$0.565475\pi$
$644$	−3.38330	−0.133321
$645$	3.56817	0.140497
$646$	−2.41340	−0.0949538
$647$	−43.4012	−1.70628	−0.853139	−	0.521683i	$-0.825304\pi$
$647$	−43.4012	−1.70628	−0.853139	+	0.521683i	$0.825304\pi$
$648$	0.829236	0.0325755
$649$	−0.917866	−0.0360294
$650$	−31.1642	−1.22236
$651$	−17.1933	−0.673859
$652$	30.1439	1.18053
$653$	−20.2481	−0.792370	−0.396185	−	0.918171i	$-0.629666\pi$
$653$	−20.2481	−0.792370	−0.396185	+	0.918171i	$0.629666\pi$
$654$	33.6700	1.31660
$655$	1.83610	0.0717424
$656$	−49.4248	−1.92971
$657$	9.80450	0.382510
$658$	−30.4299	−1.18628
$659$	2.17029	0.0845425	0.0422713	−	0.999106i	$-0.486541\pi$
$659$	2.17029	0.0845425	0.0422713	+	0.999106i	$0.486541\pi$
$660$	1.21624	0.0473422
$661$	−43.3422	−1.68582	−0.842908	−	0.538057i	$-0.819158\pi$
$661$	−43.3422	−1.68582	−0.842908	+	0.538057i	$0.819158\pi$
$662$	46.5101	1.80766
$663$	−3.85540	−0.149731
$664$	11.6934	0.453790
$665$	−3.26463	−0.126597
$666$	−4.48011	−0.173601
$667$	−3.51185	−0.135979
$668$	−35.2961	−1.36565
$669$	0.593490	0.0229456
$670$	16.0495	0.620048
$671$	2.40259	0.0927508
$672$	21.6656	0.835769
$673$	32.2301	1.24238	0.621190	−	0.783660i	$-0.286650\pi$
$673$	32.2301	1.24238	0.621190	+	0.783660i	$0.286650\pi$
$674$	−4.50399	−0.173487
$675$	−4.28380	−0.164883
$676$	2.90900	0.111885
$677$	−36.3410	−1.39670	−0.698348	−	0.715758i	$-0.746081\pi$
$677$	−36.3410	−1.39670	−0.698348	+	0.715758i	$0.746081\pi$
$678$	34.4952	1.32478
$679$	42.3468	1.62512
$680$	−0.701773	−0.0269118
$681$	−18.5422	−0.710538
$682$	−9.90605	−0.379322
$683$	−27.6294	−1.05721	−0.528605	−	0.848868i	$-0.677285\pi$
$683$	−27.6294	−1.05721	−0.528605	+	0.848868i	$0.677285\pi$
$684$	1.99593	0.0763163
$685$	−14.4143	−0.550742
$686$	27.9049	1.06542
$687$	7.01948	0.267810
$688$	19.7566	0.753211
$689$	16.4639	0.627225
$690$	−1.14788	−0.0436991
$691$	8.03647	0.305722	0.152861	−	0.988248i	$-0.451151\pi$
$691$	8.03647	0.305722	0.152861	+	0.988248i	$0.451151\pi$
$692$	−8.38314	−0.318679
$693$	−2.77762	−0.105513
$694$	−23.9852	−0.910467
$695$	−2.98669	−0.113292
$696$	4.05128	0.153564
$697$	10.5478	0.399526
$698$	26.1825	0.991023
$699$	−17.8632	−0.675649
$700$	−20.1627	−0.762077
$701$	28.6491	1.08206	0.541031	−	0.841003i	$-0.318034\pi$
$701$	28.6491	1.08206	0.541031	+	0.841003i	$0.318034\pi$
$702$	7.27490	0.274574
$703$	3.03670	0.114531
$704$	3.85220	0.145185
$705$	−4.52497	−0.170420
$706$	−8.22166	−0.309426
$707$	−2.34667	−0.0882556
$708$	1.55534	0.0584534
$709$	13.2281	0.496793	0.248397	−	0.968658i	$-0.420096\pi$
$709$	13.2281	0.496793	0.248397	+	0.968658i	$0.420096\pi$
$710$	21.1522	0.793828
$711$	1.00000	0.0375029
$712$	2.99215	0.112135
$713$	4.09766	0.153458
$714$	−5.69118	−0.212987
$715$	−3.00480	−0.112373
$716$	27.3094	1.02060
$717$	−6.45389	−0.241025
$718$	2.78955	0.104105
$719$	−1.77194	−0.0660821	−0.0330410	−	0.999454i	$-0.510519\pi$
$719$	−1.77194	−0.0660821	−0.0330410	+	0.999454i	$0.510519\pi$
$720$	3.96553	0.147787
$721$	−1.17881	−0.0439012
$722$	32.7651	1.21939
$723$	−23.6366	−0.879056
$724$	39.7550	1.47748
$725$	−20.9288	−0.777274
$726$	19.1560	0.710945
$727$	−38.2746	−1.41953	−0.709764	−	0.704440i	$-0.751198\pi$
$727$	−38.2746	−1.41953	−0.709764	+	0.704440i	$0.751198\pi$
$728$	−9.64256	−0.357377
$729$	1.00000	0.0370370
$730$	15.6568	0.579482
$731$	−4.21626	−0.155944
$732$	−4.07123	−0.150477
$733$	15.2987	0.565069	0.282535	−	0.959257i	$-0.408825\pi$
$733$	15.2987	0.565069	0.282535	+	0.959257i	$0.408825\pi$
$734$	−2.30289	−0.0850014
$735$	−1.77451	−0.0654537
$736$	−5.16354	−0.190330
$737$	−9.25581	−0.340942
$738$	−19.9030	−0.732641
$739$	−14.4646	−0.532088	−0.266044	−	0.963961i	$-0.585717\pi$
$739$	−14.4646	−0.532088	−0.266044	+	0.963961i	$0.585717\pi$
$740$	−3.13562	−0.115268
$741$	−4.93106	−0.181147
$742$	24.3033	0.892203
$743$	−17.9639	−0.659032	−0.329516	−	0.944150i	$-0.606886\pi$
$743$	−17.9639	−0.659032	−0.329516	+	0.944150i	$0.606886\pi$
$744$	−4.72708	−0.173303
$745$	−8.84180	−0.323938
$746$	−57.4029	−2.10167
$747$	14.1014	0.515942
$748$	−1.43715	−0.0525474
$749$	7.03620	0.257097
$750$	−14.8252	−0.541341
$751$	−10.3650	−0.378223	−0.189111	−	0.981956i	$-0.560561\pi$
$751$	−10.3650	−0.378223	−0.189111	+	0.981956i	$0.560561\pi$
$752$	−25.0542	−0.913634
$753$	15.7470	0.573853
$754$	35.5420	1.29436
$755$	−17.3433	−0.631186
$756$	4.70673	0.171182
$757$	−27.2481	−0.990347	−0.495174	−	0.868794i	$-0.664896\pi$
$757$	−27.2481	−0.990347	−0.495174	+	0.868794i	$0.664896\pi$
$758$	56.4043	2.04870
$759$	0.661986	0.0240286
$760$	−0.897568	−0.0325582
$761$	−26.0036	−0.942630	−0.471315	−	0.881965i	$-0.656221\pi$
$761$	−26.0036	−0.942630	−0.471315	+	0.881965i	$0.656221\pi$
$762$	23.5496	0.853112
$763$	−53.8182	−1.94835
$764$	−7.76410	−0.280895
$765$	−0.846288	−0.0305976
$766$	−60.4617	−2.18457
$767$	−3.84256	−0.138747
$768$	20.5814	0.742668
$769$	24.9618	0.900146	0.450073	−	0.892992i	$-0.351398\pi$
$769$	24.9618	0.900146	0.450073	+	0.892992i	$0.351398\pi$
$770$	−4.43556	−0.159846
$771$	16.7472	0.603136
$772$	13.9798	0.503145
$773$	−35.7594	−1.28617	−0.643087	−	0.765793i	$-0.722347\pi$
$773$	−35.7594	−1.28617	−0.643087	+	0.765793i	$0.722347\pi$
$774$	7.95583	0.285966
$775$	24.4199	0.877188
$776$	11.6427	0.417949
$777$	7.16102	0.256900
$778$	−67.8571	−2.43279
$779$	13.4906	0.483352
$780$	5.09169	0.182312
$781$	−12.1985	−0.436497
$782$	1.35637	0.0485038
$783$	4.88556	0.174596
$784$	−9.82524	−0.350901
$785$	4.26867	0.152355
$786$	4.09389	0.146024
$787$	−50.1097	−1.78622	−0.893109	−	0.449840i	$-0.851481\pi$
$787$	−50.1097	−1.78622	−0.893109	+	0.449840i	$0.851481\pi$
$788$	19.5306	0.695749
$789$	−9.30111	−0.331128
$790$	1.59689	0.0568149
$791$	−55.1373	−1.96046
$792$	−0.763670	−0.0271358
$793$	10.0582	0.357178
$794$	−59.0536	−2.09574
$795$	3.61394	0.128173
$796$	−40.4217	−1.43271
$797$	−25.3853	−0.899194	−0.449597	−	0.893232i	$-0.648432\pi$
$797$	−25.3853	−0.899194	−0.449597	+	0.893232i	$0.648432\pi$
$798$	−7.27903	−0.257675
$799$	5.34685	0.189158
$800$	−30.7720	−1.08795
$801$	3.60832	0.127494
$802$	15.6436	0.552396
$803$	−9.02928	−0.318636
$804$	15.6842	0.553138
$805$	1.83478	0.0646675
$806$	−41.4708	−1.46075
$807$	19.0596	0.670928
$808$	−0.645187	−0.0226976
$809$	13.9436	0.490232	0.245116	−	0.969494i	$-0.421174\pi$
$809$	13.9436	0.490232	0.245116	+	0.969494i	$0.421174\pi$
$810$	1.59689	0.0561091
$811$	6.74545	0.236865	0.118432	−	0.992962i	$-0.462213\pi$
$811$	6.74545	0.236865	0.118432	+	0.992962i	$0.462213\pi$
$812$	22.9950	0.806967
$813$	29.3396	1.02898
$814$	4.12587	0.144612
$815$	−16.3472	−0.572617
$816$	−4.68580	−0.164036
$817$	−5.39260	−0.188663
$818$	18.9427	0.662316
$819$	−11.6282	−0.406324
$820$	−13.9301	−0.486460
$821$	−12.7394	−0.444609	−0.222305	−	0.974977i	$-0.571358\pi$
$821$	−12.7394	−0.444609	−0.222305	+	0.974977i	$0.571358\pi$
$822$	−32.1391	−1.12098
$823$	−49.2113	−1.71540	−0.857699	−	0.514151i	$-0.828107\pi$
$823$	−49.2113	−1.71540	−0.857699	+	0.514151i	$0.828107\pi$
$824$	−0.324099	−0.0112905
$825$	3.94509	0.137350
$826$	−5.67223	−0.197362
$827$	51.9279	1.80571	0.902855	−	0.429946i	$-0.141467\pi$
$827$	51.9279	1.80571	0.902855	+	0.429946i	$0.141467\pi$
$828$	−1.12175	−0.0389835
$829$	−23.6474	−0.821310	−0.410655	−	0.911791i	$-0.634700\pi$
$829$	−23.6474	−0.821310	−0.410655	+	0.911791i	$0.634700\pi$
$830$	22.5184	0.781624
$831$	31.4076	1.08952
$832$	16.1269	0.559099
$833$	2.09681	0.0726503
$834$	−6.65931	−0.230593
$835$	19.1413	0.662410
$836$	−1.83812	−0.0635726
$837$	−5.70052	−0.197039
$838$	−32.3911	−1.11893
$839$	54.2295	1.87221	0.936105	−	0.351721i	$-0.114403\pi$
$839$	54.2295	1.87221	0.936105	+	0.351721i	$0.114403\pi$
$840$	−2.11661	−0.0730300
$841$	−5.13128	−0.176941
$842$	14.5545	0.501581
$843$	17.1452	0.590511
$844$	19.1863	0.660418
$845$	−1.57757	−0.0542700
$846$	−10.0892	−0.346873
$847$	−30.6190	−1.05208
$848$	20.0100	0.687146
$849$	−10.0764	−0.345822
$850$	8.08326	0.277254
$851$	−1.70668	−0.0585041
$852$	20.6707	0.708165
$853$	−32.2549	−1.10439	−0.552193	−	0.833716i	$-0.686209\pi$
$853$	−32.2549	−1.10439	−0.552193	+	0.833716i	$0.686209\pi$
$854$	14.8475	0.508072
$855$	−1.08240	−0.0370174
$856$	1.93451	0.0661203
$857$	−15.2326	−0.520337	−0.260168	−	0.965563i	$-0.583778\pi$
$857$	−15.2326	−0.520337	−0.260168	+	0.965563i	$0.583778\pi$
$858$	−6.69969	−0.228724
$859$	−9.13714	−0.311755	−0.155878	−	0.987776i	$-0.549821\pi$
$859$	−9.13714	−0.311755	−0.155878	+	0.987776i	$0.549821\pi$
$860$	5.56827	0.189877
$861$	31.8131	1.08419
$862$	−40.9975	−1.39638
$863$	24.4648	0.832793	0.416397	−	0.909183i	$-0.363293\pi$
$863$	24.4648	0.832793	0.416397	+	0.909183i	$0.363293\pi$
$864$	7.18334	0.244382
$865$	4.54622	0.154576
$866$	−53.8967	−1.83148
$867$	1.00000	0.0339618
$868$	−26.8308	−0.910698
$869$	−0.920932	−0.0312405
$870$	7.80173	0.264503
$871$	−38.7486	−1.31295
$872$	−14.7966	−0.501077
$873$	14.0403	0.475192
$874$	1.73480	0.0586805
$875$	23.6967	0.801096
$876$	15.3003	0.516950
$877$	−50.8356	−1.71660	−0.858298	−	0.513152i	$-0.828478\pi$
$877$	−50.8356	−1.71660	−0.858298	+	0.513152i	$0.828478\pi$
$878$	30.2030	1.01930
$879$	−15.1049	−0.509476
$880$	−3.65199	−0.123108
$881$	42.0380	1.41630	0.708149	−	0.706063i	$-0.249531\pi$
$881$	42.0380	1.41630	0.708149	+	0.706063i	$0.249531\pi$
$882$	−3.95656	−0.133224
$883$	−19.5189	−0.656863	−0.328432	−	0.944528i	$-0.606520\pi$
$883$	−19.5189	−0.656863	−0.328432	+	0.944528i	$0.606520\pi$
$884$	−6.01650	−0.202357
$885$	−0.843470	−0.0283529
$886$	−29.0396	−0.975605
$887$	4.03208	0.135384	0.0676920	−	0.997706i	$-0.478436\pi$
$887$	4.03208	0.135384	0.0676920	+	0.997706i	$0.478436\pi$
$888$	1.96883	0.0660697
$889$	−37.6418	−1.26247
$890$	5.76210	0.193146
$891$	−0.920932	−0.0308524
$892$	0.926164	0.0310103
$893$	6.83862	0.228846
$894$	−19.7142	−0.659343
$895$	−14.8100	−0.495045
$896$	−19.5254	−0.652298
$897$	2.77134	0.0925324
$898$	29.7064	0.991315
$899$	−27.8503	−0.928858
$900$	−6.68503	−0.222834
$901$	−4.27035	−0.142266
$902$	18.3294	0.610301
$903$	−12.7166	−0.423183
$904$	−15.1593	−0.504190
$905$	−21.5593	−0.716657
$906$	−38.6697	−1.28471
$907$	15.0180	0.498664	0.249332	−	0.968418i	$-0.419789\pi$
$907$	15.0180	0.498664	0.249332	+	0.968418i	$0.419789\pi$
$908$	−28.9358	−0.960269
$909$	−0.778050	−0.0258063
$910$	−18.5691	−0.615559
$911$	−19.2566	−0.638001	−0.319001	−	0.947755i	$-0.603347\pi$
$911$	−19.2566	−0.638001	−0.319001	+	0.947755i	$0.603347\pi$
$912$	−5.99314	−0.198453
$913$	−12.9864	−0.429787
$914$	−66.0165	−2.18363
$915$	2.20785	0.0729893
$916$	10.9542	0.361936
$917$	−6.54369	−0.216092
$918$	−1.88694	−0.0622783
$919$	42.4425	1.40005	0.700024	−	0.714119i	$-0.253173\pi$
$919$	42.4425	1.40005	0.700024	+	0.714119i	$0.253173\pi$
$920$	0.504449	0.0166312
$921$	−5.28167	−0.174037
$922$	52.9871	1.74504
$923$	−51.0680	−1.68092
$924$	−4.33458	−0.142597
$925$	−10.1709	−0.334417
$926$	59.6540	1.96035
$927$	−0.390840	−0.0128369
$928$	35.0947	1.15204
$929$	−16.1756	−0.530705	−0.265352	−	0.964151i	$-0.585488\pi$
$929$	−16.1756	−0.530705	−0.265352	+	0.964151i	$0.585488\pi$
$930$	−9.10313	−0.298504
$931$	2.68183	0.0878933
$932$	−27.8763	−0.913117
$933$	−19.8699	−0.650510
$934$	19.4114	0.635160
$935$	0.779374	0.0254883
$936$	−3.19704	−0.104498
$937$	−41.0799	−1.34202	−0.671011	−	0.741448i	$-0.734140\pi$
$937$	−41.0799	−1.34202	−0.671011	+	0.741448i	$0.734140\pi$
$938$	−57.1991	−1.86762
$939$	17.3863	0.567382
$940$	−7.06140	−0.230317
$941$	−19.5457	−0.637171	−0.318585	−	0.947894i	$-0.603208\pi$
$941$	−19.5457	−0.637171	−0.318585	+	0.947894i	$0.603208\pi$
$942$	9.51770	0.310103
$943$	−7.58198	−0.246903
$944$	−4.67019	−0.152002
$945$	−2.55248	−0.0830323
$946$	−7.32678	−0.238214
$947$	−31.5035	−1.02372	−0.511862	−	0.859067i	$-0.671044\pi$
$947$	−31.5035	−1.02372	−0.511862	+	0.859067i	$0.671044\pi$
$948$	1.56054	0.0506840
$949$	−37.8003	−1.22705
$950$	10.3385	0.335425
$951$	30.2204	0.979965
$952$	2.50105	0.0810596
$953$	−38.6397	−1.25166	−0.625831	−	0.779958i	$-0.715240\pi$
$953$	−38.6397	−1.25166	−0.625831	+	0.779958i	$0.715240\pi$
$954$	8.05789	0.260884
$955$	4.21051	0.136249
$956$	−10.0715	−0.325737
$957$	−4.49927	−0.145441
$958$	15.7578	0.509111
$959$	51.3712	1.65886
$960$	3.53997	0.114252
$961$	1.49598	0.0482575
$962$	17.2726	0.556891
$963$	2.33288	0.0751762
$964$	−36.8859	−1.18801
$965$	−7.58133	−0.244052
$966$	4.09094	0.131624
$967$	−38.4574	−1.23671	−0.618353	−	0.785900i	$-0.712200\pi$
$967$	−38.4574	−1.23671	−0.618353	+	0.785900i	$0.712200\pi$
$968$	−8.41831	−0.270575
$969$	1.27900	0.0410874
$970$	22.4209	0.719890
$971$	15.7133	0.504265	0.252132	−	0.967693i	$-0.418868\pi$
$971$	15.7133	0.504265	0.252132	+	0.967693i	$0.418868\pi$
$972$	1.56054	0.0500543
$973$	10.6443	0.341240
$974$	45.4275	1.45559
$975$	16.5157	0.528927
$976$	12.2246	0.391300
$977$	−55.2388	−1.76724	−0.883622	−	0.468201i	$-0.844902\pi$
$977$	−55.2388	−1.76724	−0.883622	+	0.468201i	$0.844902\pi$
$978$	−36.4487	−1.16550
$979$	−3.32301	−0.106204
$980$	−2.76919	−0.0884585
$981$	−17.8437	−0.569705
$982$	−35.9223	−1.14633
$983$	−52.6483	−1.67922	−0.839609	−	0.543191i	$-0.817216\pi$
$983$	−52.6483	−1.67922	−0.839609	+	0.543191i	$0.817216\pi$
$984$	8.74661	0.278832
$985$	−10.5915	−0.337475
$986$	−9.21876	−0.293585
$987$	16.1266	0.513315
$988$	−7.69511	−0.244814
$989$	3.03074	0.0963719
$990$	−1.47063	−0.0467397
$991$	58.3156	1.85246	0.926228	−	0.376963i	$-0.123032\pi$
$991$	58.3156	1.85246	0.926228	+	0.376963i	$0.123032\pi$
$992$	−40.9488	−1.30013
$993$	−24.6484	−0.782194
$994$	−75.3845	−2.39105
$995$	21.9209	0.694939
$996$	22.0057	0.697278
$997$	10.1087	0.320147	0.160074	−	0.987105i	$-0.448827\pi$
$997$	10.1087	0.320147	0.160074	+	0.987105i	$0.448827\pi$
$998$	23.2606	0.736302
$999$	2.37427	0.0751186

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.k.1.6	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.k.1.6	✓	31	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4029 = 3 \cdot 17 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4029.a (trivial)

\(f(q)\)	\(=\)	\(q-1.88694 q^{2} +1.00000 q^{3} +1.56054 q^{4} -0.846288 q^{5} -1.88694 q^{6} +3.01609 q^{7} +0.829236 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.88694 q^{2} +1.00000 q^{3} +1.56054 q^{4} -0.846288 q^{5} -1.88694 q^{6} +3.01609 q^{7} +0.829236 q^{8} +1.00000 q^{9} +1.59689 q^{10} -0.920932 q^{11} +1.56054 q^{12} -3.85540 q^{13} -5.69118 q^{14} -0.846288 q^{15} -4.68580 q^{16} +1.00000 q^{17} -1.88694 q^{18} +1.27900 q^{19} -1.32067 q^{20} +3.01609 q^{21} +1.73774 q^{22} -0.718821 q^{23} +0.829236 q^{24} -4.28380 q^{25} +7.27490 q^{26} +1.00000 q^{27} +4.70673 q^{28} +4.88556 q^{29} +1.59689 q^{30} -5.70052 q^{31} +7.18334 q^{32} -0.920932 q^{33} -1.88694 q^{34} -2.55248 q^{35} +1.56054 q^{36} +2.37427 q^{37} -2.41340 q^{38} -3.85540 q^{39} -0.701773 q^{40} +10.5478 q^{41} -5.69118 q^{42} -4.21626 q^{43} -1.43715 q^{44} -0.846288 q^{45} +1.35637 q^{46} +5.34685 q^{47} -4.68580 q^{48} +2.09681 q^{49} +8.08326 q^{50} +1.00000 q^{51} -6.01650 q^{52} -4.27035 q^{53} -1.88694 q^{54} +0.779374 q^{55} +2.50105 q^{56} +1.27900 q^{57} -9.21876 q^{58} +0.996670 q^{59} -1.32067 q^{60} -2.60886 q^{61} +10.7565 q^{62} +3.01609 q^{63} -4.18293 q^{64} +3.26278 q^{65} +1.73774 q^{66} +10.0505 q^{67} +1.56054 q^{68} -0.718821 q^{69} +4.81638 q^{70} +13.2458 q^{71} +0.829236 q^{72} +9.80450 q^{73} -4.48011 q^{74} -4.28380 q^{75} +1.99593 q^{76} -2.77762 q^{77} +7.27490 q^{78} +1.00000 q^{79} +3.96553 q^{80} +1.00000 q^{81} -19.9030 q^{82} +14.1014 q^{83} +4.70673 q^{84} -0.846288 q^{85} +7.95583 q^{86} +4.88556 q^{87} -0.763670 q^{88} +3.60832 q^{89} +1.59689 q^{90} -11.6282 q^{91} -1.12175 q^{92} -5.70052 q^{93} -10.0892 q^{94} -1.08240 q^{95} +7.18334 q^{96} +14.0403 q^{97} -3.95656 q^{98} -0.920932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9}+O(q^{10}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9} + 5 q^{10} + 26 q^{11} + 34 q^{12} + 7 q^{13} + 19 q^{14} + 11 q^{15} + 40 q^{16} + 31 q^{17} + 4 q^{18} + 32 q^{19} + 23 q^{20} + 4 q^{21} + 2 q^{22} + 29 q^{23} + 12 q^{24} + 32 q^{25} + 13 q^{26} + 31 q^{27} - 13 q^{28} + 25 q^{29} + 5 q^{30} + 22 q^{31} + 28 q^{32} + 26 q^{33} + 4 q^{34} + 20 q^{35} + 34 q^{36} - 4 q^{37} + 19 q^{38} + 7 q^{39} - 3 q^{40} + 33 q^{41} + 19 q^{42} + 6 q^{43} + 30 q^{44} + 11 q^{45} - 11 q^{46} + 23 q^{47} + 40 q^{48} + 31 q^{49} + 6 q^{50} + 31 q^{51} - 7 q^{52} + 12 q^{53} + 4 q^{54} + 40 q^{56} + 32 q^{57} + 9 q^{58} + 27 q^{59} + 23 q^{60} - 4 q^{61} + 25 q^{62} + 4 q^{63} + 10 q^{64} + 54 q^{65} + 2 q^{66} + 34 q^{68} + 29 q^{69} - 59 q^{70} + 35 q^{71} + 12 q^{72} + 5 q^{73} + 48 q^{74} + 32 q^{75} + 32 q^{76} + 42 q^{77} + 13 q^{78} + 31 q^{79} + 24 q^{80} + 31 q^{81} + 5 q^{82} + 67 q^{83} - 13 q^{84} + 11 q^{85} - 20 q^{86} + 25 q^{87} - 7 q^{88} + 22 q^{89} + 5 q^{90} + 16 q^{91} + 57 q^{92} + 22 q^{93} + 45 q^{94} + 73 q^{95} + 28 q^{96} - 13 q^{97} - 19 q^{98} + 26 q^{99}+O(q^{100}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9 + 5 * q^10 + 26 * q^11 + 34 * q^12 + 7 * q^13 + 19 * q^14 + 11 * q^15 + 40 * q^16 + 31 * q^17 + 4 * q^18 + 32 * q^19 + 23 * q^20 + 4 * q^21 + 2 * q^22 + 29 * q^23 + 12 * q^24 + 32 * q^25 + 13 * q^26 + 31 * q^27 - 13 * q^28 + 25 * q^29 + 5 * q^30 + 22 * q^31 + 28 * q^32 + 26 * q^33 + 4 * q^34 + 20 * q^35 + 34 * q^36 - 4 * q^37 + 19 * q^38 + 7 * q^39 - 3 * q^40 + 33 * q^41 + 19 * q^42 + 6 * q^43 + 30 * q^44 + 11 * q^45 - 11 * q^46 + 23 * q^47 + 40 * q^48 + 31 * q^49 + 6 * q^50 + 31 * q^51 - 7 * q^52 + 12 * q^53 + 4 * q^54 + 40 * q^56 + 32 * q^57 + 9 * q^58 + 27 * q^59 + 23 * q^60 - 4 * q^61 + 25 * q^62 + 4 * q^63 + 10 * q^64 + 54 * q^65 + 2 * q^66 + 34 * q^68 + 29 * q^69 - 59 * q^70 + 35 * q^71 + 12 * q^72 + 5 * q^73 + 48 * q^74 + 32 * q^75 + 32 * q^76 + 42 * q^77 + 13 * q^78 + 31 * q^79 + 24 * q^80 + 31 * q^81 + 5 * q^82 + 67 * q^83 - 13 * q^84 + 11 * q^85 - 20 * q^86 + 25 * q^87 - 7 * q^88 + 22 * q^89 + 5 * q^90 + 16 * q^91 + 57 * q^92 + 22 * q^93 + 45 * q^94 + 73 * q^95 + 28 * q^96 - 13 * q^97 - 19 * q^98 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more