LMFDB - Embedded newform 8007.2.a.e.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$46$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.938469 q^{2} +1.00000 q^{3} -1.11928 q^{4} -3.16176 q^{5} -0.938469 q^{6} +3.19339 q^{7} +2.92734 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.938469 q^{2} +1.00000 q^{3} -1.11928 q^{4} -3.16176 q^{5} -0.938469 q^{6} +3.19339 q^{7} +2.92734 q^{8} +1.00000 q^{9} +2.96721 q^{10} -3.89690 q^{11} -1.11928 q^{12} +3.99640 q^{13} -2.99690 q^{14} -3.16176 q^{15} -0.508669 q^{16} -1.00000 q^{17} -0.938469 q^{18} -2.61070 q^{19} +3.53888 q^{20} +3.19339 q^{21} +3.65712 q^{22} -2.63739 q^{23} +2.92734 q^{24} +4.99672 q^{25} -3.75050 q^{26} +1.00000 q^{27} -3.57429 q^{28} -8.17474 q^{29} +2.96721 q^{30} +9.36114 q^{31} -5.37732 q^{32} -3.89690 q^{33} +0.938469 q^{34} -10.0967 q^{35} -1.11928 q^{36} +6.64774 q^{37} +2.45006 q^{38} +3.99640 q^{39} -9.25556 q^{40} +0.367288 q^{41} -2.99690 q^{42} +5.19918 q^{43} +4.36171 q^{44} -3.16176 q^{45} +2.47511 q^{46} -8.99268 q^{47} -0.508669 q^{48} +3.19775 q^{49} -4.68927 q^{50} -1.00000 q^{51} -4.47307 q^{52} +10.5866 q^{53} -0.938469 q^{54} +12.3211 q^{55} +9.34815 q^{56} -2.61070 q^{57} +7.67174 q^{58} +5.79433 q^{59} +3.53888 q^{60} -8.59133 q^{61} -8.78514 q^{62} +3.19339 q^{63} +6.06378 q^{64} -12.6357 q^{65} +3.65712 q^{66} -13.3867 q^{67} +1.11928 q^{68} -2.63739 q^{69} +9.47547 q^{70} -11.1243 q^{71} +2.92734 q^{72} -8.17971 q^{73} -6.23870 q^{74} +4.99672 q^{75} +2.92210 q^{76} -12.4443 q^{77} -3.75050 q^{78} +6.91010 q^{79} +1.60829 q^{80} +1.00000 q^{81} -0.344688 q^{82} +4.30590 q^{83} -3.57429 q^{84} +3.16176 q^{85} -4.87927 q^{86} -8.17474 q^{87} -11.4076 q^{88} -0.653746 q^{89} +2.96721 q^{90} +12.7621 q^{91} +2.95197 q^{92} +9.36114 q^{93} +8.43935 q^{94} +8.25441 q^{95} -5.37732 q^{96} +17.7265 q^{97} -3.00099 q^{98} -3.89690 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) $ 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9 - 10 * q^10 - 25 * q^11 + 43 * q^12 - 8 * q^13 - 28 * q^14 - 19 * q^15 + 33 * q^16 - 46 * q^17 - 5 * q^18 - 2 * q^19 - 56 * q^20 + q^21 - 19 * q^22 - 64 * q^23 - 18 * q^24 + 11 * q^25 - 13 * q^26 + 46 * q^27 - 38 * q^28 - 51 * q^29 - 10 * q^30 - 19 * q^31 - 61 * q^32 - 25 * q^33 + 5 * q^34 - 39 * q^35 + 43 * q^36 - 46 * q^37 - 48 * q^38 - 8 * q^39 - 10 * q^40 - 53 * q^41 - 28 * q^42 - 33 * q^43 - 62 * q^44 - 19 * q^45 + 2 * q^46 - 45 * q^47 + 33 * q^48 + 21 * q^49 - 60 * q^50 - 46 * q^51 - 63 * q^52 - 47 * q^53 - 5 * q^54 + 5 * q^55 - 82 * q^56 - 2 * q^57 - 21 * q^58 - 65 * q^59 - 56 * q^60 - 37 * q^61 - 46 * q^62 + q^63 + 74 * q^64 - 85 * q^65 - 19 * q^66 - 52 * q^67 - 43 * q^68 - 64 * q^69 - 20 * q^70 - 48 * q^71 - 18 * q^72 - 39 * q^73 - 16 * q^74 + 11 * q^75 + 42 * q^76 - 78 * q^77 - 13 * q^78 - 26 * q^79 - 78 * q^80 + 46 * q^81 + 3 * q^82 - 47 * q^83 - 38 * q^84 + 19 * q^85 - 6 * q^86 - 51 * q^87 - 58 * q^88 - 58 * q^89 - 10 * q^90 - 43 * q^91 - 68 * q^92 - 19 * q^93 - 78 * q^95 - 61 * q^96 - 44 * q^97 - 4 * q^98 - 25 * 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 $

$2$	−0.938469	−0.663598	−0.331799	−	0.943350i	$-0.607656\pi$
$2$	−0.938469	−0.663598	−0.331799	+	0.943350i	$0.607656\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.11928	−0.559638
$5$	−3.16176	−1.41398	−0.706991	−	0.707223i	$-0.749948\pi$
$5$	−3.16176	−1.41398	−0.706991	+	0.707223i	$0.749948\pi$
$6$	−0.938469	−0.383128
$7$	3.19339	1.20699	0.603494	−	0.797367i	$-0.293775\pi$
$7$	3.19339	1.20699	0.603494	+	0.797367i	$0.293775\pi$
$8$	2.92734	1.03497
$9$	1.00000	0.333333
$10$	2.96721	0.938315
$11$	−3.89690	−1.17496	−0.587480	−	0.809239i	$-0.699880\pi$
$11$	−3.89690	−1.17496	−0.587480	+	0.809239i	$0.699880\pi$
$12$	−1.11928	−0.323107
$13$	3.99640	1.10840	0.554201	−	0.832383i	$-0.313024\pi$
$13$	3.99640	1.10840	0.554201	+	0.832383i	$0.313024\pi$
$14$	−2.99690	−0.800955
$15$	−3.16176	−0.816363
$16$	−0.508669	−0.127167
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.938469	−0.221199
$19$	−2.61070	−0.598936	−0.299468	−	0.954106i	$-0.596809\pi$
$19$	−2.61070	−0.598936	−0.299468	+	0.954106i	$0.596809\pi$
$20$	3.53888	0.791318
$21$	3.19339	0.696855
$22$	3.65712	0.779701
$23$	−2.63739	−0.549934	−0.274967	−	0.961454i	$-0.588667\pi$
$23$	−2.63739	−0.549934	−0.274967	+	0.961454i	$0.588667\pi$
$24$	2.92734	0.597542
$25$	4.99672	0.999344
$26$	−3.75050	−0.735533
$27$	1.00000	0.192450
$28$	−3.57429	−0.675477
$29$	−8.17474	−1.51801	−0.759005	−	0.651084i	$-0.774314\pi$
$29$	−8.17474	−1.51801	−0.759005	+	0.651084i	$0.774314\pi$
$30$	2.96721	0.541736
$31$	9.36114	1.68131	0.840656	−	0.541570i	$-0.182170\pi$
$31$	9.36114	1.68131	0.840656	+	0.541570i	$0.182170\pi$
$32$	−5.37732	−0.950584
$33$	−3.89690	−0.678364
$34$	0.938469	0.160946
$35$	−10.0967	−1.70666
$36$	−1.11928	−0.186546
$37$	6.64774	1.09288	0.546441	−	0.837498i	$-0.315982\pi$
$37$	6.64774	1.09288	0.546441	+	0.837498i	$0.315982\pi$
$38$	2.45006	0.397453
$39$	3.99640	0.639936
$40$	−9.25556	−1.46343
$41$	0.367288	0.0573607	0.0286804	−	0.999589i	$-0.490870\pi$
$41$	0.367288	0.0573607	0.0286804	+	0.999589i	$0.490870\pi$
$42$	−2.99690	−0.462432
$43$	5.19918	0.792868	0.396434	−	0.918063i	$-0.370248\pi$
$43$	5.19918	0.792868	0.396434	+	0.918063i	$0.370248\pi$
$44$	4.36171	0.657552
$45$	−3.16176	−0.471327
$46$	2.47511	0.364935
$47$	−8.99268	−1.31172	−0.655859	−	0.754884i	$-0.727693\pi$
$47$	−8.99268	−1.31172	−0.655859	+	0.754884i	$0.727693\pi$
$48$	−0.508669	−0.0734200
$49$	3.19775	0.456821
$50$	−4.68927	−0.663162
$51$	−1.00000	−0.140028
$52$	−4.47307	−0.620304
$53$	10.5866	1.45418	0.727090	−	0.686543i	$-0.240872\pi$
$53$	10.5866	1.45418	0.727090	+	0.686543i	$0.240872\pi$
$54$	−0.938469	−0.127709
$55$	12.3211	1.66137
$56$	9.34815	1.24920
$57$	−2.61070	−0.345796
$58$	7.67174	1.00735
$59$	5.79433	0.754358	0.377179	−	0.926140i	$-0.376894\pi$
$59$	5.79433	0.754358	0.377179	+	0.926140i	$0.376894\pi$
$60$	3.53888	0.456868
$61$	−8.59133	−1.10001	−0.550004	−	0.835162i	$-0.685374\pi$
$61$	−8.59133	−1.10001	−0.550004	+	0.835162i	$0.685374\pi$
$62$	−8.78514	−1.11571
$63$	3.19339	0.402330
$64$	6.06378	0.757973
$65$	−12.6357	−1.56726
$66$	3.65712	0.450161
$67$	−13.3867	−1.63545	−0.817723	−	0.575612i	$-0.804764\pi$
$67$	−13.3867	−1.63545	−0.817723	+	0.575612i	$0.804764\pi$
$68$	1.11928	0.135732
$69$	−2.63739	−0.317504
$70$	9.47547	1.13254
$71$	−11.1243	−1.32021	−0.660103	−	0.751175i	$-0.729487\pi$
$71$	−11.1243	−1.32021	−0.660103	+	0.751175i	$0.729487\pi$
$72$	2.92734	0.344991
$73$	−8.17971	−0.957362	−0.478681	−	0.877989i	$-0.658885\pi$
$73$	−8.17971	−0.957362	−0.478681	+	0.877989i	$0.658885\pi$
$74$	−6.23870	−0.725234
$75$	4.99672	0.576972
$76$	2.92210	0.335187
$77$	−12.4443	−1.41816
$78$	−3.75050	−0.424660
$79$	6.91010	0.777447	0.388724	−	0.921354i	$-0.372916\pi$
$79$	6.91010	0.777447	0.388724	+	0.921354i	$0.372916\pi$
$80$	1.60829	0.179812
$81$	1.00000	0.111111
$82$	−0.344688	−0.0380645
$83$	4.30590	0.472634	0.236317	−	0.971676i	$-0.424060\pi$
$83$	4.30590	0.472634	0.236317	+	0.971676i	$0.424060\pi$
$84$	−3.57429	−0.389987
$85$	3.16176	0.342941
$86$	−4.87927	−0.526145
$87$	−8.17474	−0.876424
$88$	−11.4076	−1.21605
$89$	−0.653746	−0.0692970	−0.0346485	−	0.999400i	$-0.511031\pi$
$89$	−0.653746	−0.0692970	−0.0346485	+	0.999400i	$0.511031\pi$
$90$	2.96721	0.312772
$91$	12.7621	1.33783
$92$	2.95197	0.307764
$93$	9.36114	0.970706
$94$	8.43935	0.870453
$95$	8.25441	0.846885
$96$	−5.37732	−0.548820
$97$	17.7265	1.79985	0.899926	−	0.436043i	$-0.143620\pi$
$97$	17.7265	1.79985	0.899926	+	0.436043i	$0.143620\pi$
$98$	−3.00099	−0.303146
$99$	−3.89690	−0.391653
$100$	−5.59271	−0.559271
$101$	4.42409	0.440214	0.220107	−	0.975476i	$-0.429359\pi$
$101$	4.42409	0.440214	0.220107	+	0.975476i	$0.429359\pi$
$102$	0.938469	0.0929223
$103$	−4.43823	−0.437312	−0.218656	−	0.975802i	$-0.570167\pi$
$103$	−4.43823	−0.437312	−0.218656	+	0.975802i	$0.570167\pi$
$104$	11.6988	1.14717
$105$	−10.0967	−0.985340
$106$	−9.93518	−0.964990
$107$	−7.01299	−0.677971	−0.338986	−	0.940792i	$-0.610084\pi$
$107$	−7.01299	−0.677971	−0.338986	+	0.940792i	$0.610084\pi$
$108$	−1.11928	−0.107702
$109$	−2.69617	−0.258246	−0.129123	−	0.991629i	$-0.541216\pi$
$109$	−2.69617	−0.258246	−0.129123	+	0.991629i	$0.541216\pi$
$110$	−11.5629	−1.10248
$111$	6.64774	0.630976
$112$	−1.62438	−0.153489
$113$	3.47775	0.327159	0.163579	−	0.986530i	$-0.447696\pi$
$113$	3.47775	0.327159	0.163579	+	0.986530i	$0.447696\pi$
$114$	2.45006	0.229469
$115$	8.33879	0.777596
$116$	9.14979	0.849537
$117$	3.99640	0.369467
$118$	−5.43780	−0.500590
$119$	−3.19339	−0.292738
$120$	−9.25556	−0.844913
$121$	4.18585	0.380532
$122$	8.06270	0.729962
$123$	0.367288	0.0331172
$124$	−10.4777	−0.940926
$125$	0.0103688	0.000927417 0
$126$	−2.99690	−0.266985
$127$	−15.2780	−1.35571	−0.677853	−	0.735197i	$-0.737090\pi$
$127$	−15.2780	−1.35571	−0.677853	+	0.735197i	$0.737090\pi$
$128$	5.06396	0.447595
$129$	5.19918	0.457762
$130$	11.8582	1.04003
$131$	13.0166	1.13727	0.568633	−	0.822591i	$-0.307472\pi$
$131$	13.0166	1.13727	0.568633	+	0.822591i	$0.307472\pi$
$132$	4.36171	0.379638
$133$	−8.33699	−0.722909
$134$	12.5630	1.08528
$135$	−3.16176	−0.272121
$136$	−2.92734	−0.251018
$137$	−8.48559	−0.724973	−0.362486	−	0.931989i	$-0.618072\pi$
$137$	−8.48559	−0.724973	−0.362486	+	0.931989i	$0.618072\pi$
$138$	2.47511	0.210695
$139$	−16.1044	−1.36596	−0.682980	−	0.730437i	$-0.739316\pi$
$139$	−16.1044	−1.36596	−0.682980	+	0.730437i	$0.739316\pi$
$140$	11.3010	0.955112
$141$	−8.99268	−0.757320
$142$	10.4398	0.876086
$143$	−15.5736	−1.30233
$144$	−0.508669	−0.0423891
$145$	25.8466	2.14644
$146$	7.67640	0.635303
$147$	3.19775	0.263746
$148$	−7.44066	−0.611618
$149$	−7.77400	−0.636871	−0.318436	−	0.947945i	$-0.603157\pi$
$149$	−7.77400	−0.636871	−0.318436	+	0.947945i	$0.603157\pi$
$150$	−4.68927	−0.382877
$151$	−6.12281	−0.498267	−0.249134	−	0.968469i	$-0.580146\pi$
$151$	−6.12281	−0.498267	−0.249134	+	0.968469i	$0.580146\pi$
$152$	−7.64242	−0.619882
$153$	−1.00000	−0.0808452
$154$	11.6786	0.941090
$155$	−29.5977	−2.37734
$156$	−4.47307	−0.358133
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	−6.48492	−0.515912
$159$	10.5866	0.839571
$160$	17.0018	1.34411
$161$	−8.42222	−0.663764
$162$	−0.938469	−0.0737331
$163$	18.3261	1.43541	0.717705	−	0.696347i	$-0.245192\pi$
$163$	18.3261	1.43541	0.717705	+	0.696347i	$0.245192\pi$
$164$	−0.411097	−0.0321013
$165$	12.3211	0.959194
$166$	−4.04095	−0.313639
$167$	7.02150	0.543340	0.271670	−	0.962390i	$-0.412424\pi$
$167$	7.02150	0.543340	0.271670	+	0.962390i	$0.412424\pi$
$168$	9.34815	0.721226
$169$	2.97121	0.228555
$170$	−2.96721	−0.227575
$171$	−2.61070	−0.199645
$172$	−5.81932	−0.443719
$173$	7.26500	0.552348	0.276174	−	0.961108i	$-0.410933\pi$
$173$	7.26500	0.552348	0.276174	+	0.961108i	$0.410933\pi$
$174$	7.67174	0.581593
$175$	15.9565	1.20620
$176$	1.98223	0.149416
$177$	5.79433	0.435529
$178$	0.613521	0.0459853
$179$	23.1191	1.72800	0.864001	−	0.503490i	$-0.167951\pi$
$179$	23.1191	1.72800	0.864001	+	0.503490i	$0.167951\pi$
$180$	3.53888	0.263773
$181$	−9.53173	−0.708488	−0.354244	−	0.935153i	$-0.615262\pi$
$181$	−9.53173	−0.708488	−0.354244	+	0.935153i	$0.615262\pi$
$182$	−11.9768	−0.887780
$183$	−8.59133	−0.635089
$184$	−7.72054	−0.569166
$185$	−21.0186	−1.54531
$186$	−8.78514	−0.644158
$187$	3.89690	0.284970
$188$	10.0653	0.734087
$189$	3.19339	0.232285
$190$	−7.74651	−0.561991
$191$	21.7974	1.57720	0.788601	−	0.614906i	$-0.210806\pi$
$191$	21.7974	1.57720	0.788601	+	0.614906i	$0.210806\pi$
$192$	6.06378	0.437616
$193$	3.66453	0.263779	0.131889	−	0.991264i	$-0.457896\pi$
$193$	3.66453	0.263779	0.131889	+	0.991264i	$0.457896\pi$
$194$	−16.6358	−1.19438
$195$	−12.6357	−0.904858
$196$	−3.57916	−0.255655
$197$	11.3572	0.809168	0.404584	−	0.914501i	$-0.367416\pi$
$197$	11.3572	0.809168	0.404584	+	0.914501i	$0.367416\pi$
$198$	3.65712	0.259900
$199$	−6.77838	−0.480506	−0.240253	−	0.970710i	$-0.577230\pi$
$199$	−6.77838	−0.480506	−0.240253	+	0.970710i	$0.577230\pi$
$200$	14.6271	1.03429
$201$	−13.3867	−0.944225
$202$	−4.15187	−0.292125
$203$	−26.1051	−1.83222
$204$	1.11928	0.0783650
$205$	−1.16128	−0.0811070
$206$	4.16514	0.290199
$207$	−2.63739	−0.183311
$208$	−2.03284	−0.140952
$209$	10.1736	0.703726
$210$	9.47547	0.653870
$211$	−0.321094	−0.0221050	−0.0110525	−	0.999939i	$-0.503518\pi$
$211$	−0.321094	−0.0221050	−0.0110525	+	0.999939i	$0.503518\pi$
$212$	−11.8493	−0.813814
$213$	−11.1243	−0.762221
$214$	6.58147	0.449900
$215$	−16.4386	−1.12110
$216$	2.92734	0.199181
$217$	29.8938	2.02932
$218$	2.53027	0.171371
$219$	−8.17971	−0.552733
$220$	−13.7907	−0.929767
$221$	−3.99640	−0.268827
$222$	−6.23870	−0.418714
$223$	5.06897	0.339444	0.169722	−	0.985492i	$-0.445713\pi$
$223$	5.06897	0.339444	0.169722	+	0.985492i	$0.445713\pi$
$224$	−17.1719	−1.14734
$225$	4.99672	0.333115
$226$	−3.26376	−0.217102
$227$	−12.3705	−0.821061	−0.410530	−	0.911847i	$-0.634656\pi$
$227$	−12.3705	−0.821061	−0.410530	+	0.911847i	$0.634656\pi$
$228$	2.92210	0.193521
$229$	4.56787	0.301853	0.150927	−	0.988545i	$-0.451774\pi$
$229$	4.56787	0.301853	0.150927	+	0.988545i	$0.451774\pi$
$230$	−7.82569	−0.516011
$231$	−12.4443	−0.818777
$232$	−23.9303	−1.57110
$233$	−26.2130	−1.71727	−0.858636	−	0.512585i	$-0.828688\pi$
$233$	−26.2130	−1.71727	−0.858636	+	0.512585i	$0.828688\pi$
$234$	−3.75050	−0.245178
$235$	28.4327	1.85474
$236$	−6.48546	−0.422167
$237$	6.91010	0.448859
$238$	2.99690	0.194260
$239$	14.5206	0.939262	0.469631	−	0.882863i	$-0.344387\pi$
$239$	14.5206	0.939262	0.469631	+	0.882863i	$0.344387\pi$
$240$	1.60829	0.103815
$241$	−17.0645	−1.09922	−0.549611	−	0.835421i	$-0.685224\pi$
$241$	−17.0645	−1.09922	−0.549611	+	0.835421i	$0.685224\pi$
$242$	−3.92829	−0.252520
$243$	1.00000	0.0641500
$244$	9.61607	0.615606
$245$	−10.1105	−0.645937
$246$	−0.344688	−0.0219765
$247$	−10.4334	−0.663862
$248$	27.4033	1.74011
$249$	4.30590	0.272875
$250$	−0.00973083	−0.000615432 0
$251$	12.1258	0.765375	0.382687	−	0.923878i	$-0.374999\pi$
$251$	12.1258	0.765375	0.382687	+	0.923878i	$0.374999\pi$
$252$	−3.57429	−0.225159
$253$	10.2776	0.646150
$254$	14.3380	0.899644
$255$	3.16176	0.197997
$256$	−16.8799	−1.05500
$257$	−4.73637	−0.295446	−0.147723	−	0.989029i	$-0.547194\pi$
$257$	−4.73637	−0.295446	−0.147723	+	0.989029i	$0.547194\pi$
$258$	−4.87927	−0.303770
$259$	21.2288	1.31910
$260$	14.1428	0.877098
$261$	−8.17474	−0.506004
$262$	−12.2157	−0.754687
$263$	0.820477	0.0505928	0.0252964	−	0.999680i	$-0.491947\pi$
$263$	0.820477	0.0505928	0.0252964	+	0.999680i	$0.491947\pi$
$264$	−11.4076	−0.702088
$265$	−33.4722	−2.05618
$266$	7.82401	0.479721
$267$	−0.653746	−0.0400086
$268$	14.9834	0.915258
$269$	−21.8863	−1.33443	−0.667214	−	0.744866i	$-0.732514\pi$
$269$	−21.8863	−1.33443	−0.667214	+	0.744866i	$0.732514\pi$
$270$	2.96721	0.180579
$271$	−30.4449	−1.84940	−0.924698	−	0.380703i	$-0.875682\pi$
$271$	−30.4449	−1.84940	−0.924698	+	0.380703i	$0.875682\pi$
$272$	0.508669	0.0308426
$273$	12.7621	0.772396
$274$	7.96346	0.481090
$275$	−19.4717	−1.17419
$276$	2.95197	0.177687
$277$	−14.7706	−0.887481	−0.443741	−	0.896155i	$-0.646349\pi$
$277$	−14.7706	−0.887481	−0.443741	+	0.896155i	$0.646349\pi$
$278$	15.1135	0.906448
$279$	9.36114	0.560437
$280$	−29.5566	−1.76635
$281$	4.52998	0.270236	0.135118	−	0.990830i	$-0.456859\pi$
$281$	4.52998	0.270236	0.135118	+	0.990830i	$0.456859\pi$
$282$	8.43935	0.502556
$283$	23.1198	1.37433	0.687164	−	0.726503i	$-0.258856\pi$
$283$	23.1198	1.37433	0.687164	+	0.726503i	$0.258856\pi$
$284$	12.4511	0.738838
$285$	8.25441	0.488949
$286$	14.6153	0.864222
$287$	1.17289	0.0692338
$288$	−5.37732	−0.316861
$289$	1.00000	0.0588235
$290$	−24.2562	−1.42437
$291$	17.7265	1.03914
$292$	9.15535	0.535776
$293$	10.0891	0.589413	0.294706	−	0.955588i	$-0.404778\pi$
$293$	10.0891	0.589413	0.294706	+	0.955588i	$0.404778\pi$
$294$	−3.00099	−0.175021
$295$	−18.3203	−1.06665
$296$	19.4602	1.13110
$297$	−3.89690	−0.226121
$298$	7.29566	0.422626
$299$	−10.5401	−0.609547
$300$	−5.59271	−0.322895
$301$	16.6030	0.956982
$302$	5.74607	0.330649
$303$	4.42409	0.254157
$304$	1.32798	0.0761650
$305$	27.1637	1.55539
$306$	0.938469	0.0536487
$307$	−1.13614	−0.0648432	−0.0324216	−	0.999474i	$-0.510322\pi$
$307$	−1.13614	−0.0648432	−0.0324216	+	0.999474i	$0.510322\pi$
$308$	13.9286	0.793658
$309$	−4.43823	−0.252482
$310$	27.7765	1.57760
$311$	−16.6346	−0.943262	−0.471631	−	0.881796i	$-0.656335\pi$
$311$	−16.6346	−0.943262	−0.471631	+	0.881796i	$0.656335\pi$
$312$	11.6988	0.662316
$313$	−21.6848	−1.22570	−0.612849	−	0.790200i	$-0.709977\pi$
$313$	−21.6848	−1.22570	−0.612849	+	0.790200i	$0.709977\pi$
$314$	−0.938469	−0.0529609
$315$	−10.0967	−0.568887
$316$	−7.73431	−0.435089
$317$	23.3728	1.31275	0.656373	−	0.754436i	$-0.272090\pi$
$317$	23.3728	1.31275	0.656373	+	0.754436i	$0.272090\pi$
$318$	−9.93518	−0.557137
$319$	31.8562	1.78360
$320$	−19.1722	−1.07176
$321$	−7.01299	−0.391427
$322$	7.90399	0.440472
$323$	2.61070	0.145263
$324$	−1.11928	−0.0621820
$325$	19.9689	1.10767
$326$	−17.1985	−0.952535
$327$	−2.69617	−0.149098
$328$	1.07518	0.0593668
$329$	−28.7172	−1.58323
$330$	−11.5629	−0.636519
$331$	−31.4738	−1.72996	−0.864978	−	0.501809i	$-0.832668\pi$
$331$	−31.4738	−1.72996	−0.864978	+	0.501809i	$0.832668\pi$
$332$	−4.81949	−0.264504
$333$	6.64774	0.364294
$334$	−6.58946	−0.360559
$335$	42.3255	2.31249
$336$	−1.62438	−0.0886171
$337$	28.1301	1.53234	0.766172	−	0.642636i	$-0.222159\pi$
$337$	28.1301	1.53234	0.766172	+	0.642636i	$0.222159\pi$
$338$	−2.78839	−0.151668
$339$	3.47775	0.188885
$340$	−3.53888	−0.191923
$341$	−36.4795	−1.97547
$342$	2.45006	0.132484
$343$	−12.1421	−0.655610
$344$	15.2198	0.820596
$345$	8.33879	0.448945
$346$	−6.81798	−0.366537
$347$	3.22192	0.172962	0.0864810	−	0.996254i	$-0.472438\pi$
$347$	3.22192	0.172962	0.0864810	+	0.996254i	$0.472438\pi$
$348$	9.14979	0.490480
$349$	−14.1179	−0.755714	−0.377857	−	0.925864i	$-0.623339\pi$
$349$	−14.1179	−0.755714	−0.377857	+	0.925864i	$0.623339\pi$
$350$	−14.9747	−0.800430
$351$	3.99640	0.213312
$352$	20.9549	1.11690
$353$	−2.75038	−0.146388	−0.0731941	−	0.997318i	$-0.523319\pi$
$353$	−2.75038	−0.146388	−0.0731941	+	0.997318i	$0.523319\pi$
$354$	−5.43780	−0.289016
$355$	35.1722	1.86675
$356$	0.731723	0.0387812
$357$	−3.19339	−0.169012
$358$	−21.6966	−1.14670
$359$	−20.4598	−1.07983	−0.539913	−	0.841721i	$-0.681543\pi$
$359$	−20.4598	−1.07983	−0.539913	+	0.841721i	$0.681543\pi$
$360$	−9.25556	−0.487811
$361$	−12.1842	−0.641276
$362$	8.94523	0.470151
$363$	4.18585	0.219700
$364$	−14.2843	−0.748700
$365$	25.8623	1.35369
$366$	8.06270	0.421444
$367$	−5.34969	−0.279252	−0.139626	−	0.990204i	$-0.544590\pi$
$367$	−5.34969	−0.279252	−0.139626	+	0.990204i	$0.544590\pi$
$368$	1.34156	0.0699335
$369$	0.367288	0.0191202
$370$	19.7253	1.02547
$371$	33.8071	1.75518
$372$	−10.4777	−0.543244
$373$	11.3104	0.585632	0.292816	−	0.956169i	$-0.405408\pi$
$373$	11.3104	0.585632	0.292816	+	0.956169i	$0.405408\pi$
$374$	−3.65712	−0.189105
$375$	0.0103688	0.000535444 0
$376$	−26.3247	−1.35759
$377$	−32.6695	−1.68257
$378$	−2.99690	−0.154144
$379$	21.2526	1.09167	0.545836	−	0.837892i	$-0.316212\pi$
$379$	21.2526	1.09167	0.545836	+	0.837892i	$0.316212\pi$
$380$	−9.23896	−0.473949
$381$	−15.2780	−0.782718
$382$	−20.4561	−1.04663
$383$	−7.00071	−0.357720	−0.178860	−	0.983875i	$-0.557241\pi$
$383$	−7.00071	−0.357720	−0.178860	+	0.983875i	$0.557241\pi$
$384$	5.06396	0.258419
$385$	39.3460	2.00526
$386$	−3.43905	−0.175043
$387$	5.19918	0.264289
$388$	−19.8408	−1.00727
$389$	−32.5726	−1.65150	−0.825748	−	0.564039i	$-0.809247\pi$
$389$	−32.5726	−1.65150	−0.825748	+	0.564039i	$0.809247\pi$
$390$	11.8582	0.600462
$391$	2.63739	0.133378
$392$	9.36091	0.472797
$393$	13.0166	0.656601
$394$	−10.6584	−0.536962
$395$	−21.8481	−1.09930
$396$	4.36171	0.219184
$397$	−32.0993	−1.61102	−0.805508	−	0.592585i	$-0.798108\pi$
$397$	−32.0993	−1.61102	−0.805508	+	0.592585i	$0.798108\pi$
$398$	6.36130	0.318863
$399$	−8.33699	−0.417372
$400$	−2.54168	−0.127084
$401$	−33.7957	−1.68768	−0.843839	−	0.536596i	$-0.819710\pi$
$401$	−33.7957	−1.68768	−0.843839	+	0.536596i	$0.819710\pi$
$402$	12.5630	0.626586
$403$	37.4109	1.86357
$404$	−4.95178	−0.246360
$405$	−3.16176	−0.157109
$406$	24.4989	1.21586
$407$	−25.9056	−1.28409
$408$	−2.92734	−0.144925
$409$	−21.7914	−1.07751	−0.538757	−	0.842461i	$-0.681106\pi$
$409$	−21.7914	−1.07751	−0.538757	+	0.842461i	$0.681106\pi$
$410$	1.08982	0.0538224
$411$	−8.48559	−0.418563
$412$	4.96760	0.244736
$413$	18.5036	0.910501
$414$	2.47511	0.121645
$415$	−13.6142	−0.668295
$416$	−21.4899	−1.05363
$417$	−16.1044	−0.788637
$418$	−9.54765	−0.466991
$419$	−35.1932	−1.71930	−0.859650	−	0.510883i	$-0.829319\pi$
$419$	−35.1932	−1.71930	−0.859650	+	0.510883i	$0.829319\pi$
$420$	11.3010	0.551434
$421$	−26.1770	−1.27579	−0.637895	−	0.770124i	$-0.720194\pi$
$421$	−26.1770	−1.27579	−0.637895	+	0.770124i	$0.720194\pi$
$422$	0.301337	0.0146689
$423$	−8.99268	−0.437239
$424$	30.9906	1.50504
$425$	−4.99672	−0.242377
$426$	10.4398	0.505808
$427$	−27.4355	−1.32770
$428$	7.84947	0.379418
$429$	−15.5736	−0.751900
$430$	15.4271	0.743960
$431$	7.91517	0.381260	0.190630	−	0.981662i	$-0.438947\pi$
$431$	7.91517	0.381260	0.190630	+	0.981662i	$0.438947\pi$
$432$	−0.508669	−0.0244733
$433$	−19.3800	−0.931345	−0.465673	−	0.884957i	$-0.654188\pi$
$433$	−19.3800	−0.931345	−0.465673	+	0.884957i	$0.654188\pi$
$434$	−28.0544	−1.34665
$435$	25.8466	1.23925
$436$	3.01775	0.144524
$437$	6.88544	0.329375
$438$	7.67640	0.366793
$439$	−16.6192	−0.793190	−0.396595	−	0.917994i	$-0.629808\pi$
$439$	−16.6192	−0.793190	−0.396595	+	0.917994i	$0.629808\pi$
$440$	36.0680	1.71947
$441$	3.19775	0.152274
$442$	3.75050	0.178393
$443$	−26.8218	−1.27434	−0.637171	−	0.770723i	$-0.719895\pi$
$443$	−26.8218	−1.27434	−0.637171	+	0.770723i	$0.719895\pi$
$444$	−7.44066	−0.353118
$445$	2.06699	0.0979846
$446$	−4.75707	−0.225254
$447$	−7.77400	−0.367698
$448$	19.3640	0.914865
$449$	−28.1842	−1.33010	−0.665048	−	0.746800i	$-0.731589\pi$
$449$	−28.1842	−1.33010	−0.665048	+	0.746800i	$0.731589\pi$
$450$	−4.68927	−0.221054
$451$	−1.43129	−0.0673966
$452$	−3.89256	−0.183091
$453$	−6.12281	−0.287675
$454$	11.6094	0.544854
$455$	−40.3506	−1.89166
$456$	−7.64242	−0.357889
$457$	−31.2171	−1.46027	−0.730136	−	0.683302i	$-0.760543\pi$
$457$	−31.2171	−1.46027	−0.730136	+	0.683302i	$0.760543\pi$
$458$	−4.28680	−0.200309
$459$	−1.00000	−0.0466760
$460$	−9.33341	−0.435172
$461$	−11.1046	−0.517194	−0.258597	−	0.965985i	$-0.583260\pi$
$461$	−11.1046	−0.517194	−0.258597	+	0.965985i	$0.583260\pi$
$462$	11.6786	0.543339
$463$	−31.2253	−1.45116	−0.725581	−	0.688137i	$-0.758429\pi$
$463$	−31.2253	−1.45116	−0.725581	+	0.688137i	$0.758429\pi$
$464$	4.15823	0.193041
$465$	−29.5977	−1.37256
$466$	24.6001	1.13958
$467$	−5.84983	−0.270698	−0.135349	−	0.990798i	$-0.543216\pi$
$467$	−5.84983	−0.270698	−0.135349	+	0.990798i	$0.543216\pi$
$468$	−4.47307	−0.206768
$469$	−42.7490	−1.97396
$470$	−26.6832	−1.23080
$471$	1.00000	0.0460776
$472$	16.9620	0.780740
$473$	−20.2607	−0.931588
$474$	−6.48492	−0.297862
$475$	−13.0449	−0.598543
$476$	3.57429	0.163827
$477$	10.5866	0.484726
$478$	−13.6272	−0.623292
$479$	−11.6305	−0.531412	−0.265706	−	0.964054i	$-0.585605\pi$
$479$	−11.6305	−0.531412	−0.265706	+	0.964054i	$0.585605\pi$
$480$	17.0018	0.776022
$481$	26.5670	1.21135
$482$	16.0145	0.729441
$483$	−8.42222	−0.383224
$484$	−4.68512	−0.212960
$485$	−56.0469	−2.54496
$486$	−0.938469	−0.0425698
$487$	−13.6357	−0.617894	−0.308947	−	0.951079i	$-0.599976\pi$
$487$	−13.6357	−0.617894	−0.308947	+	0.951079i	$0.599976\pi$
$488$	−25.1498	−1.13848
$489$	18.3261	0.828735
$490$	9.48840	0.428642
$491$	17.2416	0.778101	0.389050	−	0.921216i	$-0.372803\pi$
$491$	17.2416	0.778101	0.389050	+	0.921216i	$0.372803\pi$
$492$	−0.411097	−0.0185337
$493$	8.17474	0.368172
$494$	9.79143	0.440537
$495$	12.3211	0.553791
$496$	−4.76172	−0.213808
$497$	−35.5241	−1.59347
$498$	−4.04095	−0.181079
$499$	9.47865	0.424323	0.212161	−	0.977235i	$-0.431950\pi$
$499$	9.47865	0.424323	0.212161	+	0.977235i	$0.431950\pi$
$500$	−0.0116056	−0.000519018 0
$501$	7.02150	0.313697
$502$	−11.3797	−0.507901
$503$	20.6318	0.919928	0.459964	−	0.887938i	$-0.347862\pi$
$503$	20.6318	0.919928	0.459964	+	0.887938i	$0.347862\pi$
$504$	9.34815	0.416400
$505$	−13.9879	−0.622454
$506$	−9.64525	−0.428784
$507$	2.97121	0.131956
$508$	17.1003	0.758705
$509$	−19.1511	−0.848858	−0.424429	−	0.905461i	$-0.639525\pi$
$509$	−19.1511	−0.848858	−0.424429	+	0.905461i	$0.639525\pi$
$510$	−2.96721	−0.131390
$511$	−26.1210	−1.15553
$512$	5.71337	0.252498
$513$	−2.61070	−0.115265
$514$	4.44493	0.196058
$515$	14.0326	0.618351
$516$	−5.81932	−0.256181
$517$	35.0436	1.54122
$518$	−19.9226	−0.875349
$519$	7.26500	0.318898
$520$	−36.9889	−1.62207
$521$	1.14721	0.0502601	0.0251301	−	0.999684i	$-0.492000\pi$
$521$	1.14721	0.0502601	0.0251301	+	0.999684i	$0.492000\pi$
$522$	7.67174	0.335783
$523$	−3.21216	−0.140458	−0.0702289	−	0.997531i	$-0.522373\pi$
$523$	−3.21216	−0.140458	−0.0702289	+	0.997531i	$0.522373\pi$
$524$	−14.5692	−0.636457
$525$	15.9565	0.696398
$526$	−0.769992	−0.0335732
$527$	−9.36114	−0.407778
$528$	1.98223	0.0862656
$529$	−16.0442	−0.697573
$530$	31.4126	1.36448
$531$	5.79433	0.251453
$532$	9.33140	0.404567
$533$	1.46783	0.0635788
$534$	0.613521	0.0265496
$535$	22.1734	0.958639
$536$	−39.1875	−1.69264
$537$	23.1191	0.997663
$538$	20.5396	0.885524
$539$	−12.4613	−0.536747
$540$	3.53888	0.152289
$541$	−22.0075	−0.946176	−0.473088	−	0.881015i	$-0.656861\pi$
$541$	−22.0075	−0.946176	−0.473088	+	0.881015i	$0.656861\pi$
$542$	28.5716	1.22725
$543$	−9.53173	−0.409046
$544$	5.37732	0.230551
$545$	8.52463	0.365155
$546$	−11.9768	−0.512560
$547$	−13.7768	−0.589053	−0.294526	−	0.955643i	$-0.595162\pi$
$547$	−13.7768	−0.589053	−0.294526	+	0.955643i	$0.595162\pi$
$548$	9.49771	0.405722
$549$	−8.59133	−0.366669
$550$	18.2736	0.779190
$551$	21.3418	0.909191
$552$	−7.72054	−0.328608
$553$	22.0667	0.938370
$554$	13.8618	0.588930
$555$	−21.0186	−0.892188
$556$	18.0253	0.764443
$557$	−2.29854	−0.0973923	−0.0486961	−	0.998814i	$-0.515507\pi$
$557$	−2.29854	−0.0973923	−0.0486961	+	0.998814i	$0.515507\pi$
$558$	−8.78514	−0.371905
$559$	20.7780	0.878816
$560$	5.13589	0.217031
$561$	3.89690	0.164527
$562$	−4.25125	−0.179328
$563$	−5.41074	−0.228035	−0.114018	−	0.993479i	$-0.536372\pi$
$563$	−5.41074	−0.228035	−0.114018	+	0.993479i	$0.536372\pi$
$564$	10.0653	0.423825
$565$	−10.9958	−0.462597
$566$	−21.6972	−0.912000
$567$	3.19339	0.134110
$568$	−32.5645	−1.36638
$569$	41.5767	1.74299	0.871493	−	0.490408i	$-0.163152\pi$
$569$	41.5767	1.74299	0.871493	+	0.490408i	$0.163152\pi$
$570$	−7.74651	−0.324465
$571$	−26.5040	−1.10916	−0.554579	−	0.832131i	$-0.687121\pi$
$571$	−26.5040	−1.10916	−0.554579	+	0.832131i	$0.687121\pi$
$572$	17.4311	0.728832
$573$	21.7974	0.910598
$574$	−1.10072	−0.0459434
$575$	−13.1783	−0.549573
$576$	6.06378	0.252658
$577$	40.9331	1.70407	0.852033	−	0.523488i	$-0.175370\pi$
$577$	40.9331	1.70407	0.852033	+	0.523488i	$0.175370\pi$
$578$	−0.938469	−0.0390352
$579$	3.66453	0.152293
$580$	−28.9294	−1.20123
$581$	13.7504	0.570464
$582$	−16.6358	−0.689574
$583$	−41.2549	−1.70860
$584$	−23.9448	−0.990843
$585$	−12.6357	−0.522420
$586$	−9.46832	−0.391133
$587$	−2.78441	−0.114925	−0.0574625	−	0.998348i	$-0.518301\pi$
$587$	−2.78441	−0.114925	−0.0574625	+	0.998348i	$0.518301\pi$
$588$	−3.57916	−0.147602
$589$	−24.4392	−1.00700
$590$	17.1930	0.707825
$591$	11.3572	0.467173
$592$	−3.38150	−0.138979
$593$	13.5205	0.555222	0.277611	−	0.960694i	$-0.410457\pi$
$593$	13.5205	0.555222	0.277611	+	0.960694i	$0.410457\pi$
$594$	3.65712	0.150054
$595$	10.0967	0.413926
$596$	8.70125	0.356417
$597$	−6.77838	−0.277421
$598$	9.89152	0.404494
$599$	13.9623	0.570483	0.285241	−	0.958456i	$-0.407926\pi$
$599$	13.9623	0.570483	0.285241	+	0.958456i	$0.407926\pi$
$600$	14.6271	0.597150
$601$	−35.0318	−1.42898	−0.714488	−	0.699647i	$-0.753341\pi$
$601$	−35.0318	−1.42898	−0.714488	+	0.699647i	$0.753341\pi$
$602$	−15.5814	−0.635051
$603$	−13.3867	−0.545149
$604$	6.85311	0.278849
$605$	−13.2346	−0.538065
$606$	−4.15187	−0.168658
$607$	−7.49108	−0.304054	−0.152027	−	0.988376i	$-0.548580\pi$
$607$	−7.49108	−0.304054	−0.152027	+	0.988376i	$0.548580\pi$
$608$	14.0386	0.569339
$609$	−26.1051	−1.05783
$610$	−25.4923	−1.03215
$611$	−35.9384	−1.45391
$612$	1.11928	0.0452441
$613$	45.0264	1.81860	0.909301	−	0.416140i	$-0.136617\pi$
$613$	45.0264	1.81860	0.909301	+	0.416140i	$0.136617\pi$
$614$	1.06624	0.0430298
$615$	−1.16128	−0.0468272
$616$	−36.4288	−1.46776
$617$	−27.3417	−1.10074	−0.550368	−	0.834922i	$-0.685513\pi$
$617$	−27.3417	−1.10074	−0.550368	+	0.834922i	$0.685513\pi$
$618$	4.16514	0.167546
$619$	−0.00114521	−4.60299e−5 0	−2.30149e−5	−	1.00000i	$-0.500007\pi$
$619$	−0.00114521	−4.60299e−5 0	−2.30149e−5		1.00000i	$0.500007\pi$
$620$	33.1280	1.33045
$621$	−2.63739	−0.105835
$622$	15.6111	0.625946
$623$	−2.08767	−0.0836406
$624$	−2.03284	−0.0813789
$625$	−25.0164	−1.00066
$626$	20.3505	0.813371
$627$	10.1736	0.406296
$628$	−1.11928	−0.0446640
$629$	−6.64774	−0.265063
$630$	9.47547	0.377512
$631$	−19.2656	−0.766949	−0.383475	−	0.923551i	$-0.625273\pi$
$631$	−19.2656	−0.766949	−0.383475	+	0.923551i	$0.625273\pi$
$632$	20.2282	0.804636
$633$	−0.321094	−0.0127623
$634$	−21.9346	−0.871136
$635$	48.3055	1.91694
$636$	−11.8493	−0.469856
$637$	12.7795	0.506342
$638$	−29.8960	−1.18359
$639$	−11.1243	−0.440069
$640$	−16.0110	−0.632892
$641$	30.9125	1.22097	0.610486	−	0.792027i	$-0.290974\pi$
$641$	30.9125	1.22097	0.610486	+	0.792027i	$0.290974\pi$
$642$	6.58147	0.259750
$643$	−8.79402	−0.346802	−0.173401	−	0.984851i	$-0.555476\pi$
$643$	−8.79402	−0.346802	−0.173401	+	0.984851i	$0.555476\pi$
$644$	9.42678	0.371467
$645$	−16.4386	−0.647268
$646$	−2.45006	−0.0963964
$647$	7.62021	0.299581	0.149791	−	0.988718i	$-0.452140\pi$
$647$	7.62021	0.299581	0.149791	+	0.988718i	$0.452140\pi$
$648$	2.92734	0.114997
$649$	−22.5800	−0.886341
$650$	−18.7402	−0.735051
$651$	29.8938	1.17163
$652$	−20.5120	−0.803311
$653$	−25.8874	−1.01305	−0.506526	−	0.862225i	$-0.669070\pi$
$653$	−25.8874	−1.01305	−0.506526	+	0.862225i	$0.669070\pi$
$654$	2.53027	0.0989413
$655$	−41.1554	−1.60807
$656$	−0.186828	−0.00729441
$657$	−8.17971	−0.319121
$658$	26.9502	1.05063
$659$	−31.3890	−1.22274	−0.611371	−	0.791344i	$-0.709382\pi$
$659$	−31.3890	−1.22274	−0.611371	+	0.791344i	$0.709382\pi$
$660$	−13.7907	−0.536801
$661$	23.3594	0.908576	0.454288	−	0.890855i	$-0.349894\pi$
$661$	23.3594	0.908576	0.454288	+	0.890855i	$0.349894\pi$
$662$	29.5372	1.14800
$663$	−3.99640	−0.155207
$664$	12.6048	0.489163
$665$	26.3596	1.02218
$666$	−6.23870	−0.241745
$667$	21.5600	0.834805
$668$	−7.85899	−0.304074
$669$	5.06897	0.195978
$670$	−39.7212	−1.53456
$671$	33.4796	1.29246
$672$	−17.1719	−0.662420
$673$	−34.9821	−1.34846	−0.674231	−	0.738520i	$-0.735525\pi$
$673$	−34.9821	−1.34846	−0.674231	+	0.738520i	$0.735525\pi$
$674$	−26.3992	−1.01686
$675$	4.99672	0.192324
$676$	−3.32561	−0.127908
$677$	−6.96778	−0.267794	−0.133897	−	0.990995i	$-0.542749\pi$
$677$	−6.96778	−0.267794	−0.133897	+	0.990995i	$0.542749\pi$
$678$	−3.26376	−0.125344
$679$	56.6076	2.17240
$680$	9.25556	0.354934
$681$	−12.3705	−0.474040
$682$	34.2348	1.31092
$683$	24.1975	0.925890	0.462945	−	0.886387i	$-0.346793\pi$
$683$	24.1975	0.925890	0.462945	+	0.886387i	$0.346793\pi$
$684$	2.92210	0.111729
$685$	26.8294	1.02510
$686$	11.3950	0.435062
$687$	4.56787	0.174275
$688$	−2.64466	−0.100827
$689$	42.3082	1.61182
$690$	−7.82569	−0.297919
$691$	−32.2655	−1.22744	−0.613719	−	0.789525i	$-0.710327\pi$
$691$	−32.2655	−1.22744	−0.613719	+	0.789525i	$0.710327\pi$
$692$	−8.13154	−0.309115
$693$	−12.4443	−0.472721
$694$	−3.02368	−0.114777
$695$	50.9183	1.93144
$696$	−23.9303	−0.907074
$697$	−0.367288	−0.0139120
$698$	13.2492	0.501490
$699$	−26.2130	−0.991468
$700$	−17.8597	−0.675034
$701$	−7.90448	−0.298548	−0.149274	−	0.988796i	$-0.547694\pi$
$701$	−7.90448	−0.298548	−0.149274	+	0.988796i	$0.547694\pi$
$702$	−3.75050	−0.141553
$703$	−17.3553	−0.654566
$704$	−23.6300	−0.890588
$705$	28.4327	1.07084
$706$	2.58115	0.0971428
$707$	14.1279	0.531333
$708$	−6.48546	−0.243738
$709$	−19.3028	−0.724931	−0.362466	−	0.931997i	$-0.618065\pi$
$709$	−19.3028	−0.724931	−0.362466	+	0.931997i	$0.618065\pi$
$710$	−33.0080	−1.23877
$711$	6.91010	0.259149
$712$	−1.91374	−0.0717204
$713$	−24.6890	−0.924610
$714$	2.99690	0.112156
$715$	49.2399	1.84147
$716$	−25.8767	−0.967056
$717$	14.5206	0.542283
$718$	19.2009	0.716570
$719$	−43.6511	−1.62791	−0.813957	−	0.580926i	$-0.802691\pi$
$719$	−43.6511	−1.62791	−0.813957	+	0.580926i	$0.802691\pi$
$720$	1.60829	0.0599374
$721$	−14.1730	−0.527830
$722$	11.4345	0.425549
$723$	−17.0645	−0.634636
$724$	10.6686	0.396497
$725$	−40.8469	−1.51702
$726$	−3.92829	−0.145792
$727$	10.6095	0.393484	0.196742	−	0.980455i	$-0.436964\pi$
$727$	10.6095	0.393484	0.196742	+	0.980455i	$0.436964\pi$
$728$	37.3590	1.38462
$729$	1.00000	0.0370370
$730$	−24.2709	−0.898307
$731$	−5.19918	−0.192299
$732$	9.61607	0.355420
$733$	−14.4560	−0.533943	−0.266972	−	0.963704i	$-0.586023\pi$
$733$	−14.4560	−0.533943	−0.266972	+	0.963704i	$0.586023\pi$
$734$	5.02052	0.185311
$735$	−10.1105	−0.372932
$736$	14.1821	0.522758
$737$	52.1667	1.92158
$738$	−0.344688	−0.0126882
$739$	25.0656	0.922051	0.461026	−	0.887387i	$-0.347482\pi$
$739$	25.0656	0.922051	0.461026	+	0.887387i	$0.347482\pi$
$740$	23.5256	0.864817
$741$	−10.4334	−0.383281
$742$	−31.7269	−1.16473
$743$	39.2069	1.43836	0.719180	−	0.694824i	$-0.244518\pi$
$743$	39.2069	1.43836	0.719180	+	0.694824i	$0.244518\pi$
$744$	27.4033	1.00465
$745$	24.5795	0.900524
$746$	−10.6145	−0.388624
$747$	4.30590	0.157545
$748$	−4.36171	−0.159480
$749$	−22.3952	−0.818303
$750$	−0.00973083	−0.000355320 0
$751$	3.99017	0.145603	0.0728016	−	0.997346i	$-0.476806\pi$
$751$	3.99017	0.145603	0.0728016	+	0.997346i	$0.476806\pi$
$752$	4.57430	0.166807
$753$	12.1258	0.441889
$754$	30.6593	1.11655
$755$	19.3589	0.704541
$756$	−3.57429	−0.129996
$757$	11.8841	0.431936	0.215968	−	0.976400i	$-0.430709\pi$
$757$	11.8841	0.431936	0.215968	+	0.976400i	$0.430709\pi$
$758$	−19.9449	−0.724431
$759$	10.2776	0.373055
$760$	24.1635	0.876502
$761$	14.8864	0.539633	0.269816	−	0.962912i	$-0.413037\pi$
$761$	14.8864	0.539633	0.269816	+	0.962912i	$0.413037\pi$
$762$	14.3380	0.519410
$763$	−8.60991	−0.311700
$764$	−24.3973	−0.882662
$765$	3.16176	0.114314
$766$	6.56995	0.237382
$767$	23.1565	0.836132
$768$	−16.8799	−0.609102
$769$	49.8262	1.79678	0.898389	−	0.439201i	$-0.144738\pi$
$769$	49.8262	1.79678	0.898389	+	0.439201i	$0.144738\pi$
$770$	−36.9250	−1.33068
$771$	−4.73637	−0.170576
$772$	−4.10162	−0.147621
$773$	21.2647	0.764838	0.382419	−	0.923989i	$-0.375091\pi$
$773$	21.2647	0.764838	0.382419	+	0.923989i	$0.375091\pi$
$774$	−4.87927	−0.175382
$775$	46.7750	1.68021
$776$	51.8915	1.86280
$777$	21.2288	0.761580
$778$	30.5684	1.09593
$779$	−0.958879	−0.0343554
$780$	14.1428	0.506393
$781$	43.3501	1.55119
$782$	−2.47511	−0.0885097
$783$	−8.17474	−0.292141
$784$	−1.62660	−0.0580927
$785$	−3.16176	−0.112848
$786$	−12.2157	−0.435719
$787$	16.9550	0.604380	0.302190	−	0.953248i	$-0.402282\pi$
$787$	16.9550	0.604380	0.302190	+	0.953248i	$0.402282\pi$
$788$	−12.7119	−0.452841
$789$	0.820477	0.0292097
$790$	20.5037	0.729490
$791$	11.1058	0.394877
$792$	−11.4076	−0.405350
$793$	−34.3344	−1.21925
$794$	30.1242	1.06907
$795$	−33.4722	−1.18714
$796$	7.58688	0.268910
$797$	47.3156	1.67600	0.838001	−	0.545668i	$-0.183724\pi$
$797$	47.3156	1.67600	0.838001	+	0.545668i	$0.183724\pi$
$798$	7.82401	0.276967
$799$	8.99268	0.318138
$800$	−26.8690	−0.949961
$801$	−0.653746	−0.0230990
$802$	31.7162	1.11994
$803$	31.8755	1.12486
$804$	14.9834	0.528424
$805$	26.6290	0.938549
$806$	−35.1089	−1.23666
$807$	−21.8863	−0.770433
$808$	12.9508	0.455609
$809$	−3.49688	−0.122944	−0.0614718	−	0.998109i	$-0.519579\pi$
$809$	−3.49688	−0.122944	−0.0614718	+	0.998109i	$0.519579\pi$
$810$	2.96721	0.104257
$811$	36.6303	1.28626	0.643132	−	0.765755i	$-0.277635\pi$
$811$	36.6303	1.28626	0.643132	+	0.765755i	$0.277635\pi$
$812$	29.2189	1.02538
$813$	−30.4449	−1.06775
$814$	24.3116	0.852121
$815$	−57.9427	−2.02964
$816$	0.508669	0.0178070
$817$	−13.5735	−0.474877
$818$	20.4505	0.715036
$819$	12.7621	0.445943
$820$	1.29979	0.0453906
$821$	−31.6148	−1.10337	−0.551683	−	0.834054i	$-0.686014\pi$
$821$	−31.6148	−1.10337	−0.551683	+	0.834054i	$0.686014\pi$
$822$	7.96346	0.277758
$823$	−26.5119	−0.924146	−0.462073	−	0.886842i	$-0.652894\pi$
$823$	−26.5119	−0.924146	−0.462073	+	0.886842i	$0.652894\pi$
$824$	−12.9922	−0.452605
$825$	−19.4717	−0.677919
$826$	−17.3650	−0.604207
$827$	39.6677	1.37938	0.689690	−	0.724105i	$-0.257747\pi$
$827$	39.6677	1.37938	0.689690	+	0.724105i	$0.257747\pi$
$828$	2.95197	0.102588
$829$	36.5725	1.27022	0.635108	−	0.772423i	$-0.280955\pi$
$829$	36.5725	1.27022	0.635108	+	0.772423i	$0.280955\pi$
$830$	12.7765	0.443479
$831$	−14.7706	−0.512387
$832$	24.2333	0.840139
$833$	−3.19775	−0.110795
$834$	15.1135	0.523338
$835$	−22.2003	−0.768272
$836$	−11.3871	−0.393832
$837$	9.36114	0.323569
$838$	33.0277	1.14092
$839$	−46.3732	−1.60098	−0.800490	−	0.599346i	$-0.795427\pi$
$839$	−46.3732	−1.60098	−0.800490	+	0.599346i	$0.795427\pi$
$840$	−29.5566	−1.01980
$841$	37.8263	1.30436
$842$	24.5663	0.846611
$843$	4.52998	0.156021
$844$	0.359393	0.0123708
$845$	−9.39426	−0.323172
$846$	8.43935	0.290151
$847$	13.3671	0.459297
$848$	−5.38507	−0.184924
$849$	23.1198	0.793468
$850$	4.68927	0.160841
$851$	−17.5327	−0.601012
$852$	12.4511	0.426568
$853$	42.1101	1.44182	0.720911	−	0.693027i	$-0.243723\pi$
$853$	42.1101	1.44182	0.720911	+	0.693027i	$0.243723\pi$
$854$	25.7473	0.881056
$855$	8.25441	0.282295
$856$	−20.5294	−0.701681
$857$	13.0924	0.447229	0.223615	−	0.974678i	$-0.428214\pi$
$857$	13.0924	0.447229	0.223615	+	0.974678i	$0.428214\pi$
$858$	14.6153	0.498959
$859$	−44.8238	−1.52937	−0.764684	−	0.644406i	$-0.777105\pi$
$859$	−44.8238	−1.52937	−0.764684	+	0.644406i	$0.777105\pi$
$860$	18.3993	0.627410
$861$	1.17289	0.0399721
$862$	−7.42814	−0.253004
$863$	5.07199	0.172653	0.0863263	−	0.996267i	$-0.472487\pi$
$863$	5.07199	0.172653	0.0863263	+	0.996267i	$0.472487\pi$
$864$	−5.37732	−0.182940
$865$	−22.9702	−0.781010
$866$	18.1876	0.618039
$867$	1.00000	0.0339618
$868$	−33.4594	−1.13569
$869$	−26.9280	−0.913470
$870$	−24.2562	−0.822362
$871$	−53.4986	−1.81273
$872$	−7.89261	−0.267277
$873$	17.7265	0.599951
$874$	−6.46177	−0.218573
$875$	0.0331118	0.00111938
$876$	9.15535	0.309331
$877$	39.2280	1.32463	0.662317	−	0.749223i	$-0.269573\pi$
$877$	39.2280	1.32463	0.662317	+	0.749223i	$0.269573\pi$
$878$	15.5966	0.526359
$879$	10.0891	0.340298
$880$	−6.26734	−0.211272
$881$	27.9967	0.943232	0.471616	−	0.881804i	$-0.343671\pi$
$881$	27.9967	0.943232	0.471616	+	0.881804i	$0.343671\pi$
$882$	−3.00099	−0.101049
$883$	50.0321	1.68371	0.841857	−	0.539701i	$-0.181463\pi$
$883$	50.0321	1.68371	0.841857	+	0.539701i	$0.181463\pi$
$884$	4.47307	0.150446
$885$	−18.3203	−0.615830
$886$	25.1714	0.845650
$887$	−52.7111	−1.76986	−0.884932	−	0.465720i	$-0.845796\pi$
$887$	−52.7111	−1.76986	−0.884932	+	0.465720i	$0.845796\pi$
$888$	19.4602	0.653042
$889$	−48.7888	−1.63632
$890$	−1.93980	−0.0650224
$891$	−3.89690	−0.130551
$892$	−5.67358	−0.189966
$893$	23.4772	0.785635
$894$	7.29566	0.244003
$895$	−73.0970	−2.44336
$896$	16.1712	0.540242
$897$	−10.5401	−0.351922
$898$	26.4500	0.882649
$899$	−76.5249	−2.55225
$900$	−5.59271	−0.186424
$901$	−10.5866	−0.352690
$902$	1.34322	0.0447242
$903$	16.6030	0.552514
$904$	10.1806	0.338600
$905$	30.1370	1.00179
$906$	5.74607	0.190900
$907$	−31.8539	−1.05769	−0.528846	−	0.848718i	$-0.677375\pi$
$907$	−31.8539	−1.05769	−0.528846	+	0.848718i	$0.677375\pi$
$908$	13.8460	0.459497
$909$	4.42409	0.146738
$910$	37.8678	1.25530
$911$	−7.88328	−0.261185	−0.130592	−	0.991436i	$-0.541688\pi$
$911$	−7.88328	−0.261185	−0.130592	+	0.991436i	$0.541688\pi$
$912$	1.32798	0.0439739
$913$	−16.7797	−0.555326
$914$	29.2962	0.969034
$915$	27.1637	0.898005
$916$	−5.11271	−0.168929
$917$	41.5671	1.37267
$918$	0.938469	0.0309741
$919$	8.67506	0.286164	0.143082	−	0.989711i	$-0.454299\pi$
$919$	8.67506	0.286164	0.143082	+	0.989711i	$0.454299\pi$
$920$	24.4105	0.804790
$921$	−1.13614	−0.0374372
$922$	10.4214	0.343209
$923$	−44.4570	−1.46332
$924$	13.9286	0.458219
$925$	33.2169	1.09216
$926$	29.3040	0.962988
$927$	−4.43823	−0.145771
$928$	43.9582	1.44300
$929$	0.0643093	0.00210992	0.00105496	−	0.999999i	$-0.499664\pi$
$929$	0.0643093	0.00210992	0.00105496	+	0.999999i	$0.499664\pi$
$930$	27.7765	0.910828
$931$	−8.34837	−0.273607
$932$	29.3396	0.961051
$933$	−16.6346	−0.544592
$934$	5.48989	0.179635
$935$	−12.3211	−0.402942
$936$	11.6988	0.382388
$937$	19.3821	0.633185	0.316593	−	0.948562i	$-0.397461\pi$
$937$	19.3821	0.633185	0.316593	+	0.948562i	$0.397461\pi$
$938$	40.1186	1.30992
$939$	−21.6848	−0.707657
$940$	−31.8240	−1.03799
$941$	25.3346	0.825885	0.412942	−	0.910757i	$-0.364501\pi$
$941$	25.3346	0.825885	0.412942	+	0.910757i	$0.364501\pi$
$942$	−0.938469	−0.0305770
$943$	−0.968681	−0.0315446
$944$	−2.94740	−0.0959296
$945$	−10.0967	−0.328447
$946$	19.0140	0.618200
$947$	54.7938	1.78056	0.890280	−	0.455413i	$-0.150508\pi$
$947$	54.7938	1.78056	0.890280	+	0.455413i	$0.150508\pi$
$948$	−7.73431	−0.251199
$949$	−32.6894	−1.06114
$950$	12.2423	0.397192
$951$	23.3728	0.757915
$952$	−9.34815	−0.302975
$953$	46.8870	1.51882	0.759410	−	0.650612i	$-0.225488\pi$
$953$	46.8870	1.51882	0.759410	+	0.650612i	$0.225488\pi$
$954$	−9.93518	−0.321663
$955$	−68.9180	−2.23013
$956$	−16.2526	−0.525647
$957$	31.8562	1.02976
$958$	10.9149	0.352644
$959$	−27.0978	−0.875034
$960$	−19.1722	−0.618781
$961$	56.6310	1.82681
$962$	−24.9323	−0.803851
$963$	−7.01299	−0.225990
$964$	19.0999	0.615167
$965$	−11.5864	−0.372978
$966$	7.90399	0.254307
$967$	19.3935	0.623652	0.311826	−	0.950139i	$-0.399059\pi$
$967$	19.3935	0.623652	0.311826	+	0.950139i	$0.399059\pi$
$968$	12.2534	0.393840
$969$	2.61070	0.0838678
$970$	52.5983	1.68883
$971$	−11.5844	−0.371761	−0.185881	−	0.982572i	$-0.559514\pi$
$971$	−11.5844	−0.371761	−0.185881	+	0.982572i	$0.559514\pi$
$972$	−1.11928	−0.0359008
$973$	−51.4277	−1.64870
$974$	12.7967	0.410033
$975$	19.9689	0.639516
$976$	4.37014	0.139885
$977$	28.4649	0.910674	0.455337	−	0.890319i	$-0.349519\pi$
$977$	28.4649	0.910674	0.455337	+	0.890319i	$0.349519\pi$
$978$	−17.1985	−0.549947
$979$	2.54759	0.0814212
$980$	11.3165	0.361491
$981$	−2.69617	−0.0860820
$982$	−16.1807	−0.516346
$983$	−35.7667	−1.14078	−0.570390	−	0.821374i	$-0.693208\pi$
$983$	−35.7667	−1.14078	−0.570390	+	0.821374i	$0.693208\pi$
$984$	1.07518	0.0342754
$985$	−35.9088	−1.14415
$986$	−7.67174	−0.244318
$987$	−28.7172	−0.914077
$988$	11.6779	0.371522
$989$	−13.7123	−0.436025
$990$	−11.5629	−0.367494
$991$	1.18549	0.0376582	0.0188291	−	0.999823i	$-0.494006\pi$
$991$	1.18549	0.0376582	0.0188291	+	0.999823i	$0.494006\pi$
$992$	−50.3378	−1.59823
$993$	−31.4738	−0.998791
$994$	33.3383	1.05743
$995$	21.4316	0.679427
$996$	−4.81949	−0.152711
$997$	−9.92711	−0.314395	−0.157197	−	0.987567i	$-0.550246\pi$
$997$	−9.92711	−0.314395	−0.157197	+	0.987567i	$0.550246\pi$
$998$	−8.89542	−0.281580
$999$	6.64774	0.210325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.e.1.18	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.e.1.18	✓	46	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-0.938469 q^{2} +1.00000 q^{3} -1.11928 q^{4} -3.16176 q^{5} -0.938469 q^{6} +3.19339 q^{7} +2.92734 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.938469 q^{2} +1.00000 q^{3} -1.11928 q^{4} -3.16176 q^{5} -0.938469 q^{6} +3.19339 q^{7} +2.92734 q^{8} +1.00000 q^{9} +2.96721 q^{10} -3.89690 q^{11} -1.11928 q^{12} +3.99640 q^{13} -2.99690 q^{14} -3.16176 q^{15} -0.508669 q^{16} -1.00000 q^{17} -0.938469 q^{18} -2.61070 q^{19} +3.53888 q^{20} +3.19339 q^{21} +3.65712 q^{22} -2.63739 q^{23} +2.92734 q^{24} +4.99672 q^{25} -3.75050 q^{26} +1.00000 q^{27} -3.57429 q^{28} -8.17474 q^{29} +2.96721 q^{30} +9.36114 q^{31} -5.37732 q^{32} -3.89690 q^{33} +0.938469 q^{34} -10.0967 q^{35} -1.11928 q^{36} +6.64774 q^{37} +2.45006 q^{38} +3.99640 q^{39} -9.25556 q^{40} +0.367288 q^{41} -2.99690 q^{42} +5.19918 q^{43} +4.36171 q^{44} -3.16176 q^{45} +2.47511 q^{46} -8.99268 q^{47} -0.508669 q^{48} +3.19775 q^{49} -4.68927 q^{50} -1.00000 q^{51} -4.47307 q^{52} +10.5866 q^{53} -0.938469 q^{54} +12.3211 q^{55} +9.34815 q^{56} -2.61070 q^{57} +7.67174 q^{58} +5.79433 q^{59} +3.53888 q^{60} -8.59133 q^{61} -8.78514 q^{62} +3.19339 q^{63} +6.06378 q^{64} -12.6357 q^{65} +3.65712 q^{66} -13.3867 q^{67} +1.11928 q^{68} -2.63739 q^{69} +9.47547 q^{70} -11.1243 q^{71} +2.92734 q^{72} -8.17971 q^{73} -6.23870 q^{74} +4.99672 q^{75} +2.92210 q^{76} -12.4443 q^{77} -3.75050 q^{78} +6.91010 q^{79} +1.60829 q^{80} +1.00000 q^{81} -0.344688 q^{82} +4.30590 q^{83} -3.57429 q^{84} +3.16176 q^{85} -4.87927 q^{86} -8.17474 q^{87} -11.4076 q^{88} -0.653746 q^{89} +2.96721 q^{90} +12.7621 q^{91} +2.95197 q^{92} +9.36114 q^{93} +8.43935 q^{94} +8.25441 q^{95} -5.37732 q^{96} +17.7265 q^{97} -3.00099 q^{98} -3.89690 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9 - 10 * q^10 - 25 * q^11 + 43 * q^12 - 8 * q^13 - 28 * q^14 - 19 * q^15 + 33 * q^16 - 46 * q^17 - 5 * q^18 - 2 * q^19 - 56 * q^20 + q^21 - 19 * q^22 - 64 * q^23 - 18 * q^24 + 11 * q^25 - 13 * q^26 + 46 * q^27 - 38 * q^28 - 51 * q^29 - 10 * q^30 - 19 * q^31 - 61 * q^32 - 25 * q^33 + 5 * q^34 - 39 * q^35 + 43 * q^36 - 46 * q^37 - 48 * q^38 - 8 * q^39 - 10 * q^40 - 53 * q^41 - 28 * q^42 - 33 * q^43 - 62 * q^44 - 19 * q^45 + 2 * q^46 - 45 * q^47 + 33 * q^48 + 21 * q^49 - 60 * q^50 - 46 * q^51 - 63 * q^52 - 47 * q^53 - 5 * q^54 + 5 * q^55 - 82 * q^56 - 2 * q^57 - 21 * q^58 - 65 * q^59 - 56 * q^60 - 37 * q^61 - 46 * q^62 + q^63 + 74 * q^64 - 85 * q^65 - 19 * q^66 - 52 * q^67 - 43 * q^68 - 64 * q^69 - 20 * q^70 - 48 * q^71 - 18 * q^72 - 39 * q^73 - 16 * q^74 + 11 * q^75 + 42 * q^76 - 78 * q^77 - 13 * q^78 - 26 * q^79 - 78 * q^80 + 46 * q^81 + 3 * q^82 - 47 * q^83 - 38 * q^84 + 19 * q^85 - 6 * q^86 - 51 * q^87 - 58 * q^88 - 58 * q^89 - 10 * q^90 - 43 * q^91 - 68 * q^92 - 19 * q^93 - 78 * q^95 - 61 * q^96 - 44 * q^97 - 4 * q^98 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more