LMFDB - Embedded newform 8043.2.a.n.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.83203 q^{2} +1.00000 q^{3} +1.35635 q^{4} +1.36968 q^{5} -1.83203 q^{6} +1.00000 q^{7} +1.17919 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.83203 q^{2} +1.00000 q^{3} +1.35635 q^{4} +1.36968 q^{5} -1.83203 q^{6} +1.00000 q^{7} +1.17919 q^{8} +1.00000 q^{9} -2.50930 q^{10} -5.37273 q^{11} +1.35635 q^{12} -1.02371 q^{13} -1.83203 q^{14} +1.36968 q^{15} -4.87301 q^{16} +5.32646 q^{17} -1.83203 q^{18} -1.77346 q^{19} +1.85776 q^{20} +1.00000 q^{21} +9.84303 q^{22} -1.40182 q^{23} +1.17919 q^{24} -3.12398 q^{25} +1.87548 q^{26} +1.00000 q^{27} +1.35635 q^{28} +4.08916 q^{29} -2.50930 q^{30} +4.59327 q^{31} +6.56915 q^{32} -5.37273 q^{33} -9.75826 q^{34} +1.36968 q^{35} +1.35635 q^{36} -8.65651 q^{37} +3.24904 q^{38} -1.02371 q^{39} +1.61511 q^{40} -10.0319 q^{41} -1.83203 q^{42} -3.93042 q^{43} -7.28731 q^{44} +1.36968 q^{45} +2.56818 q^{46} +12.4813 q^{47} -4.87301 q^{48} +1.00000 q^{49} +5.72325 q^{50} +5.32646 q^{51} -1.38851 q^{52} -3.44770 q^{53} -1.83203 q^{54} -7.35891 q^{55} +1.17919 q^{56} -1.77346 q^{57} -7.49149 q^{58} -12.1040 q^{59} +1.85776 q^{60} +7.33415 q^{61} -8.41503 q^{62} +1.00000 q^{63} -2.28889 q^{64} -1.40216 q^{65} +9.84303 q^{66} +10.8300 q^{67} +7.22455 q^{68} -1.40182 q^{69} -2.50930 q^{70} -9.83352 q^{71} +1.17919 q^{72} +10.0412 q^{73} +15.8590 q^{74} -3.12398 q^{75} -2.40543 q^{76} -5.37273 q^{77} +1.87548 q^{78} +2.35585 q^{79} -6.67446 q^{80} +1.00000 q^{81} +18.3788 q^{82} +4.42317 q^{83} +1.35635 q^{84} +7.29553 q^{85} +7.20067 q^{86} +4.08916 q^{87} -6.33546 q^{88} +7.35599 q^{89} -2.50930 q^{90} -1.02371 q^{91} -1.90136 q^{92} +4.59327 q^{93} -22.8661 q^{94} -2.42907 q^{95} +6.56915 q^{96} -14.9568 q^{97} -1.83203 q^{98} -5.37273 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) $ 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9 - 14 * q^10 - 29 * q^11 + 33 * q^12 - 40 * q^13 - 9 * q^14 - 27 * q^15 + 23 * q^16 - 46 * q^17 - 9 * q^18 + 17 * q^19 - 53 * q^20 + 40 * q^21 - 15 * q^22 - 62 * q^23 - 18 * q^24 + 35 * q^25 - 29 * q^26 + 40 * q^27 + 33 * q^28 - 47 * q^29 - 14 * q^30 + q^31 - 45 * q^32 - 29 * q^33 + 9 * q^34 - 27 * q^35 + 33 * q^36 - 49 * q^37 - 52 * q^38 - 40 * q^39 - 12 * q^40 - 49 * q^41 - 9 * q^42 - 45 * q^43 - 68 * q^44 - 27 * q^45 - 36 * q^46 - 88 * q^47 + 23 * q^48 + 40 * q^49 - 32 * q^50 - 46 * q^51 - 34 * q^52 - 96 * q^53 - 9 * q^54 + 26 * q^55 - 18 * q^56 + 17 * q^57 + 12 * q^58 - 45 * q^59 - 53 * q^60 - 23 * q^61 - 15 * q^62 + 40 * q^63 + 50 * q^64 - 32 * q^65 - 15 * q^66 - 31 * q^67 - 76 * q^68 - 62 * q^69 - 14 * q^70 - 89 * q^71 - 18 * q^72 + 4 * q^73 + 10 * q^74 + 35 * q^75 + 51 * q^76 - 29 * q^77 - 29 * q^78 - 44 * q^79 - 53 * q^80 + 40 * q^81 - 15 * q^82 - 60 * q^83 + 33 * q^84 - 24 * q^85 - 9 * q^86 - 47 * q^87 - 64 * q^88 - 39 * q^89 - 14 * q^90 - 40 * q^91 - 80 * q^92 + q^93 + 51 * q^94 - 71 * q^95 - 45 * q^96 - 21 * q^97 - 9 * q^98 - 29 * 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 $

$2$	−1.83203	−1.29544	−0.647722	−	0.761877i	$-0.724278\pi$
$2$	−1.83203	−1.29544	−0.647722	+	0.761877i	$0.724278\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.35635	0.678175
$5$	1.36968	0.612538	0.306269	−	0.951945i	$-0.400919\pi$
$5$	1.36968	0.612538	0.306269	+	0.951945i	$0.400919\pi$
$6$	−1.83203	−0.747925
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.17919	0.416906
$9$	1.00000	0.333333
$10$	−2.50930	−0.793509
$11$	−5.37273	−1.61994	−0.809970	−	0.586472i	$-0.800517\pi$
$11$	−5.37273	−1.61994	−0.809970	+	0.586472i	$0.800517\pi$
$12$	1.35635	0.391545
$13$	−1.02371	−0.283927	−0.141964	−	0.989872i	$-0.545342\pi$
$13$	−1.02371	−0.283927	−0.141964	+	0.989872i	$0.545342\pi$
$14$	−1.83203	−0.489632
$15$	1.36968	0.353649
$16$	−4.87301	−1.21825
$17$	5.32646	1.29186	0.645928	−	0.763398i	$-0.276471\pi$
$17$	5.32646	1.29186	0.645928	+	0.763398i	$0.276471\pi$
$18$	−1.83203	−0.431815
$19$	−1.77346	−0.406860	−0.203430	−	0.979090i	$-0.565209\pi$
$19$	−1.77346	−0.406860	−0.203430	+	0.979090i	$0.565209\pi$
$20$	1.85776	0.415408
$21$	1.00000	0.218218
$22$	9.84303	2.09854
$23$	−1.40182	−0.292299	−0.146150	−	0.989263i	$-0.546688\pi$
$23$	−1.40182	−0.292299	−0.146150	+	0.989263i	$0.546688\pi$
$24$	1.17919	0.240701
$25$	−3.12398	−0.624797
$26$	1.87548	0.367812
$27$	1.00000	0.192450
$28$	1.35635	0.256326
$29$	4.08916	0.759338	0.379669	−	0.925122i	$-0.376038\pi$
$29$	4.08916	0.759338	0.379669	+	0.925122i	$0.376038\pi$
$30$	−2.50930	−0.458133
$31$	4.59327	0.824975	0.412488	−	0.910963i	$-0.364660\pi$
$31$	4.59327	0.824975	0.412488	+	0.910963i	$0.364660\pi$
$32$	6.56915	1.16127
$33$	−5.37273	−0.935273
$34$	−9.75826	−1.67353
$35$	1.36968	0.231518
$36$	1.35635	0.226058
$37$	−8.65651	−1.42312	−0.711561	−	0.702625i	$-0.752011\pi$
$37$	−8.65651	−1.42312	−0.711561	+	0.702625i	$0.752011\pi$
$38$	3.24904	0.527064
$39$	−1.02371	−0.163925
$40$	1.61511	0.255371
$41$	−10.0319	−1.56672	−0.783362	−	0.621566i	$-0.786497\pi$
$41$	−10.0319	−1.56672	−0.783362	+	0.621566i	$0.786497\pi$
$42$	−1.83203	−0.282689
$43$	−3.93042	−0.599384	−0.299692	−	0.954036i	$-0.596884\pi$
$43$	−3.93042	−0.599384	−0.299692	+	0.954036i	$0.596884\pi$
$44$	−7.28731	−1.09860
$45$	1.36968	0.204179
$46$	2.56818	0.378657
$47$	12.4813	1.82058	0.910290	−	0.413970i	$-0.135858\pi$
$47$	12.4813	1.82058	0.910290	+	0.413970i	$0.135858\pi$
$48$	−4.87301	−0.703359
$49$	1.00000	0.142857
$50$	5.72325	0.809389
$51$	5.32646	0.745853
$52$	−1.38851	−0.192552
$53$	−3.44770	−0.473578	−0.236789	−	0.971561i	$-0.576095\pi$
$53$	−3.44770	−0.473578	−0.236789	+	0.971561i	$0.576095\pi$
$54$	−1.83203	−0.249308
$55$	−7.35891	−0.992275
$56$	1.17919	0.157576
$57$	−1.77346	−0.234901
$58$	−7.49149	−0.983680
$59$	−12.1040	−1.57581	−0.787904	−	0.615799i	$-0.788833\pi$
$59$	−12.1040	−1.57581	−0.787904	+	0.615799i	$0.788833\pi$
$60$	1.85776	0.239836
$61$	7.33415	0.939041	0.469521	−	0.882921i	$-0.344427\pi$
$61$	7.33415	0.939041	0.469521	+	0.882921i	$0.344427\pi$
$62$	−8.41503	−1.06871
$63$	1.00000	0.125988
$64$	−2.28889	−0.286111
$65$	−1.40216	−0.173916
$66$	9.84303	1.21159
$67$	10.8300	1.32309	0.661547	−	0.749904i	$-0.269900\pi$
$67$	10.8300	1.32309	0.661547	+	0.749904i	$0.269900\pi$
$68$	7.22455	0.876105
$69$	−1.40182	−0.168759
$70$	−2.50930	−0.299918
$71$	−9.83352	−1.16702	−0.583512	−	0.812104i	$-0.698322\pi$
$71$	−9.83352	−1.16702	−0.583512	+	0.812104i	$0.698322\pi$
$72$	1.17919	0.138969
$73$	10.0412	1.17523	0.587616	−	0.809140i	$-0.300067\pi$
$73$	10.0412	1.17523	0.587616	+	0.809140i	$0.300067\pi$
$74$	15.8590	1.84357
$75$	−3.12398	−0.360727
$76$	−2.40543	−0.275922
$77$	−5.37273	−0.612280
$78$	1.87548	0.212356
$79$	2.35585	0.265053	0.132527	−	0.991179i	$-0.457691\pi$
$79$	2.35585	0.265053	0.132527	+	0.991179i	$0.457691\pi$
$80$	−6.67446	−0.746227
$81$	1.00000	0.111111
$82$	18.3788	2.02960
$83$	4.42317	0.485506	0.242753	−	0.970088i	$-0.421950\pi$
$83$	4.42317	0.485506	0.242753	+	0.970088i	$0.421950\pi$
$84$	1.35635	0.147990
$85$	7.29553	0.791312
$86$	7.20067	0.776469
$87$	4.08916	0.438404
$88$	−6.33546	−0.675362
$89$	7.35599	0.779734	0.389867	−	0.920871i	$-0.372521\pi$
$89$	7.35599	0.779734	0.389867	+	0.920871i	$0.372521\pi$
$90$	−2.50930	−0.264503
$91$	−1.02371	−0.107314
$92$	−1.90136	−0.198230
$93$	4.59327	0.476300
$94$	−22.8661	−2.35846
$95$	−2.42907	−0.249217
$96$	6.56915	0.670462
$97$	−14.9568	−1.51864	−0.759318	−	0.650720i	$-0.774467\pi$
$97$	−14.9568	−1.51864	−0.759318	+	0.650720i	$0.774467\pi$
$98$	−1.83203	−0.185063
$99$	−5.37273	−0.539980
$100$	−4.23722	−0.423722
$101$	−14.4377	−1.43660	−0.718300	−	0.695733i	$-0.755080\pi$
$101$	−14.4377	−1.43660	−0.718300	+	0.695733i	$0.755080\pi$
$102$	−9.75826	−0.966211
$103$	10.1657	1.00166	0.500829	−	0.865546i	$-0.333028\pi$
$103$	10.1657	1.00166	0.500829	+	0.865546i	$0.333028\pi$
$104$	−1.20715	−0.118371
$105$	1.36968	0.133667
$106$	6.31630	0.613493
$107$	−12.0088	−1.16093	−0.580466	−	0.814285i	$-0.697129\pi$
$107$	−12.0088	−1.16093	−0.580466	+	0.814285i	$0.697129\pi$
$108$	1.35635	0.130515
$109$	−6.75836	−0.647333	−0.323667	−	0.946171i	$-0.604916\pi$
$109$	−6.75836	−0.647333	−0.323667	+	0.946171i	$0.604916\pi$
$110$	13.4818	1.28544
$111$	−8.65651	−0.821639
$112$	−4.87301	−0.460457
$113$	16.3543	1.53849	0.769243	−	0.638956i	$-0.220634\pi$
$113$	16.3543	1.53849	0.769243	+	0.638956i	$0.220634\pi$
$114$	3.24904	0.304301
$115$	−1.92004	−0.179044
$116$	5.54634	0.514965
$117$	−1.02371	−0.0946423
$118$	22.1750	2.04137
$119$	5.32646	0.488276
$120$	1.61511	0.147438
$121$	17.8662	1.62420
$122$	−13.4364	−1.21648
$123$	−10.0319	−0.904549
$124$	6.23008	0.559478
$125$	−11.1272	−0.995250
$126$	−1.83203	−0.163211
$127$	−0.455225	−0.0403947	−0.0201973	−	0.999796i	$-0.506429\pi$
$127$	−0.455225	−0.0403947	−0.0201973	+	0.999796i	$0.506429\pi$
$128$	−8.94498	−0.790632
$129$	−3.93042	−0.346055
$130$	2.56880	0.225299
$131$	−17.3279	−1.51394	−0.756972	−	0.653447i	$-0.773322\pi$
$131$	−17.3279	−1.51394	−0.756972	+	0.653447i	$0.773322\pi$
$132$	−7.28731	−0.634279
$133$	−1.77346	−0.153779
$134$	−19.8409	−1.71399
$135$	1.36968	0.117883
$136$	6.28090	0.538582
$137$	9.35838	0.799540	0.399770	−	0.916615i	$-0.369090\pi$
$137$	9.35838	0.799540	0.399770	+	0.916615i	$0.369090\pi$
$138$	2.56818	0.218618
$139$	5.95363	0.504980	0.252490	−	0.967600i	$-0.418751\pi$
$139$	5.95363	0.504980	0.252490	+	0.967600i	$0.418751\pi$
$140$	1.85776	0.157010
$141$	12.4813	1.05111
$142$	18.0153	1.51181
$143$	5.50014	0.459945
$144$	−4.87301	−0.406085
$145$	5.60083	0.465124
$146$	−18.3958	−1.52245
$147$	1.00000	0.0824786
$148$	−11.7413	−0.965126
$149$	−22.9114	−1.87697	−0.938486	−	0.345318i	$-0.887771\pi$
$149$	−22.9114	−1.87697	−0.938486	+	0.345318i	$0.887771\pi$
$150$	5.72325	0.467301
$151$	−14.1324	−1.15008	−0.575038	−	0.818127i	$-0.695013\pi$
$151$	−14.1324	−1.15008	−0.575038	+	0.818127i	$0.695013\pi$
$152$	−2.09124	−0.169622
$153$	5.32646	0.430619
$154$	9.84303	0.793174
$155$	6.29130	0.505329
$156$	−1.38851	−0.111170
$157$	−22.8193	−1.82118	−0.910589	−	0.413312i	$-0.864372\pi$
$157$	−22.8193	−1.82118	−0.910589	+	0.413312i	$0.864372\pi$
$158$	−4.31599	−0.343362
$159$	−3.44770	−0.273420
$160$	8.99762	0.711325
$161$	−1.40182	−0.110479
$162$	−1.83203	−0.143938
$163$	8.03727	0.629528	0.314764	−	0.949170i	$-0.398075\pi$
$163$	8.03727	0.629528	0.314764	+	0.949170i	$0.398075\pi$
$164$	−13.6068	−1.06251
$165$	−7.35891	−0.572890
$166$	−8.10339	−0.628945
$167$	−0.331679	−0.0256661	−0.0128331	−	0.999918i	$-0.504085\pi$
$167$	−0.331679	−0.0256661	−0.0128331	+	0.999918i	$0.504085\pi$
$168$	1.17919	0.0909763
$169$	−11.9520	−0.919385
$170$	−13.3657	−1.02510
$171$	−1.77346	−0.135620
$172$	−5.33103	−0.406488
$173$	−17.9465	−1.36445	−0.682223	−	0.731144i	$-0.738987\pi$
$173$	−17.9465	−1.36445	−0.682223	+	0.731144i	$0.738987\pi$
$174$	−7.49149	−0.567928
$175$	−3.12398	−0.236151
$176$	26.1814	1.97350
$177$	−12.1040	−0.909793
$178$	−13.4764	−1.01010
$179$	−18.7597	−1.40217	−0.701084	−	0.713078i	$-0.747300\pi$
$179$	−18.7597	−1.40217	−0.701084	+	0.713078i	$0.747300\pi$
$180$	1.85776	0.138469
$181$	−6.79349	−0.504956	−0.252478	−	0.967603i	$-0.581246\pi$
$181$	−6.79349	−0.504956	−0.252478	+	0.967603i	$0.581246\pi$
$182$	1.87548	0.139020
$183$	7.33415	0.542156
$184$	−1.65301	−0.121861
$185$	−11.8566	−0.871716
$186$	−8.41503	−0.617020
$187$	−28.6176	−2.09273
$188$	16.9290	1.23467
$189$	1.00000	0.0727393
$190$	4.45014	0.322847
$191$	5.22838	0.378312	0.189156	−	0.981947i	$-0.439425\pi$
$191$	5.22838	0.378312	0.189156	+	0.981947i	$0.439425\pi$
$192$	−2.28889	−0.165186
$193$	5.49311	0.395403	0.197701	−	0.980262i	$-0.436652\pi$
$193$	5.49311	0.395403	0.197701	+	0.980262i	$0.436652\pi$
$194$	27.4014	1.96731
$195$	−1.40216	−0.100411
$196$	1.35635	0.0968822
$197$	−16.6356	−1.18524	−0.592620	−	0.805483i	$-0.701906\pi$
$197$	−16.6356	−1.18524	−0.592620	+	0.805483i	$0.701906\pi$
$198$	9.84303	0.699514
$199$	−19.4693	−1.38014	−0.690070	−	0.723743i	$-0.742420\pi$
$199$	−19.4693	−1.38014	−0.690070	+	0.723743i	$0.742420\pi$
$200$	−3.68376	−0.260481
$201$	10.8300	0.763888
$202$	26.4503	1.86104
$203$	4.08916	0.287003
$204$	7.22455	0.505819
$205$	−13.7405	−0.959679
$206$	−18.6240	−1.29759
$207$	−1.40182	−0.0974330
$208$	4.98857	0.345895
$209$	9.52833	0.659088
$210$	−2.50930	−0.173158
$211$	−12.8682	−0.885883	−0.442941	−	0.896551i	$-0.646065\pi$
$211$	−12.8682	−0.885883	−0.442941	+	0.896551i	$0.646065\pi$
$212$	−4.67629	−0.321169
$213$	−9.83352	−0.673782
$214$	22.0005	1.50392
$215$	−5.38341	−0.367146
$216$	1.17919	0.0802336
$217$	4.59327	0.311811
$218$	12.3815	0.838584
$219$	10.0412	0.678520
$220$	−9.98126	−0.672937
$221$	−5.45277	−0.366793
$222$	15.8590	1.06439
$223$	21.1070	1.41343	0.706716	−	0.707497i	$-0.250176\pi$
$223$	21.1070	1.41343	0.706716	+	0.707497i	$0.250176\pi$
$224$	6.56915	0.438920
$225$	−3.12398	−0.208266
$226$	−29.9617	−1.99302
$227$	−9.03329	−0.599561	−0.299780	−	0.954008i	$-0.596913\pi$
$227$	−9.03329	−0.599561	−0.299780	+	0.954008i	$0.596913\pi$
$228$	−2.40543	−0.159304
$229$	26.5921	1.75726	0.878628	−	0.477508i	$-0.158460\pi$
$229$	26.5921	1.75726	0.878628	+	0.477508i	$0.158460\pi$
$230$	3.51758	0.231942
$231$	−5.37273	−0.353500
$232$	4.82189	0.316573
$233$	20.9747	1.37410	0.687049	−	0.726611i	$-0.258906\pi$
$233$	20.9747	1.37410	0.687049	+	0.726611i	$0.258906\pi$
$234$	1.87548	0.122604
$235$	17.0953	1.11518
$236$	−16.4173	−1.06867
$237$	2.35585	0.153029
$238$	−9.75826	−0.632534
$239$	7.55623	0.488772	0.244386	−	0.969678i	$-0.421414\pi$
$239$	7.55623	0.488772	0.244386	+	0.969678i	$0.421414\pi$
$240$	−6.67446	−0.430834
$241$	−1.72950	−0.111407	−0.0557033	−	0.998447i	$-0.517740\pi$
$241$	−1.72950	−0.111407	−0.0557033	+	0.998447i	$0.517740\pi$
$242$	−32.7316	−2.10407
$243$	1.00000	0.0641500
$244$	9.94768	0.636835
$245$	1.36968	0.0875055
$246$	18.3788	1.17179
$247$	1.81552	0.115518
$248$	5.41633	0.343937
$249$	4.42317	0.280307
$250$	20.3855	1.28929
$251$	−19.7499	−1.24660	−0.623301	−	0.781982i	$-0.714209\pi$
$251$	−19.7499	−1.24660	−0.623301	+	0.781982i	$0.714209\pi$
$252$	1.35635	0.0854421
$253$	7.53159	0.473507
$254$	0.833988	0.0523290
$255$	7.29553	0.456864
$256$	20.9653	1.31033
$257$	−22.6588	−1.41342	−0.706710	−	0.707504i	$-0.749821\pi$
$257$	−22.6588	−1.41342	−0.706710	+	0.707504i	$0.749821\pi$
$258$	7.20067	0.448294
$259$	−8.65651	−0.537889
$260$	−1.90182	−0.117946
$261$	4.08916	0.253113
$262$	31.7453	1.96123
$263$	20.9413	1.29130	0.645648	−	0.763635i	$-0.276587\pi$
$263$	20.9413	1.29130	0.645648	+	0.763635i	$0.276587\pi$
$264$	−6.33546	−0.389921
$265$	−4.72223	−0.290085
$266$	3.24904	0.199211
$267$	7.35599	0.450180
$268$	14.6893	0.897289
$269$	−10.8246	−0.659988	−0.329994	−	0.943983i	$-0.607047\pi$
$269$	−10.8246	−0.659988	−0.329994	+	0.943983i	$0.607047\pi$
$270$	−2.50930	−0.152711
$271$	18.5637	1.12767	0.563834	−	0.825888i	$-0.309326\pi$
$271$	18.5637	1.12767	0.563834	+	0.825888i	$0.309326\pi$
$272$	−25.9559	−1.57381
$273$	−1.02371	−0.0619580
$274$	−17.1449	−1.03576
$275$	16.7843	1.01213
$276$	−1.90136	−0.114448
$277$	22.1234	1.32927	0.664634	−	0.747169i	$-0.268587\pi$
$277$	22.1234	1.32927	0.664634	+	0.747169i	$0.268587\pi$
$278$	−10.9072	−0.654173
$279$	4.59327	0.274992
$280$	1.61511	0.0965211
$281$	0.846393	0.0504916	0.0252458	−	0.999681i	$-0.491963\pi$
$281$	0.846393	0.0504916	0.0252458	+	0.999681i	$0.491963\pi$
$282$	−22.8661	−1.36166
$283$	24.8454	1.47691	0.738453	−	0.674305i	$-0.235557\pi$
$283$	24.8454	1.47691	0.738453	+	0.674305i	$0.235557\pi$
$284$	−13.3377	−0.791447
$285$	−2.42907	−0.143886
$286$	−10.0764	−0.595833
$287$	−10.0319	−0.592166
$288$	6.56915	0.387091
$289$	11.3712	0.668892
$290$	−10.2609	−0.602542
$291$	−14.9568	−0.876785
$292$	13.6194	0.797013
$293$	−4.76042	−0.278107	−0.139053	−	0.990285i	$-0.544406\pi$
$293$	−4.76042	−0.278107	−0.139053	+	0.990285i	$0.544406\pi$
$294$	−1.83203	−0.106846
$295$	−16.5786	−0.965242
$296$	−10.2076	−0.593308
$297$	−5.37273	−0.311758
$298$	41.9744	2.43151
$299$	1.43506	0.0829916
$300$	−4.23722	−0.244636
$301$	−3.93042	−0.226546
$302$	25.8910	1.48986
$303$	−14.4377	−0.829422
$304$	8.64210	0.495658
$305$	10.0454	0.575199
$306$	−9.75826	−0.557842
$307$	−30.5240	−1.74209	−0.871047	−	0.491199i	$-0.836559\pi$
$307$	−30.5240	−1.74209	−0.871047	+	0.491199i	$0.836559\pi$
$308$	−7.28731	−0.415233
$309$	10.1657	0.578308
$310$	−11.5259	−0.654626
$311$	12.0719	0.684537	0.342268	−	0.939602i	$-0.388805\pi$
$311$	12.0719	0.684537	0.342268	+	0.939602i	$0.388805\pi$
$312$	−1.20715	−0.0683414
$313$	−15.0766	−0.852182	−0.426091	−	0.904680i	$-0.640110\pi$
$313$	−15.0766	−0.852182	−0.426091	+	0.904680i	$0.640110\pi$
$314$	41.8058	2.35924
$315$	1.36968	0.0771726
$316$	3.19535	0.179753
$317$	4.05893	0.227972	0.113986	−	0.993482i	$-0.463638\pi$
$317$	4.05893	0.227972	0.113986	+	0.993482i	$0.463638\pi$
$318$	6.31630	0.354201
$319$	−21.9700	−1.23008
$320$	−3.13504	−0.175254
$321$	−12.0088	−0.670264
$322$	2.56818	0.143119
$323$	−9.44626	−0.525604
$324$	1.35635	0.0753528
$325$	3.19806	0.177397
$326$	−14.7246	−0.815518
$327$	−6.75836	−0.373738
$328$	−11.8295	−0.653176
$329$	12.4813	0.688115
$330$	13.4818	0.742147
$331$	−12.5632	−0.690535	−0.345268	−	0.938504i	$-0.612212\pi$
$331$	−12.5632	−0.690535	−0.345268	+	0.938504i	$0.612212\pi$
$332$	5.99936	0.329258
$333$	−8.65651	−0.474374
$334$	0.607648	0.0332490
$335$	14.8336	0.810446
$336$	−4.87301	−0.265845
$337$	12.3953	0.675213	0.337607	−	0.941287i	$-0.390383\pi$
$337$	12.3953	0.675213	0.337607	+	0.941287i	$0.390383\pi$
$338$	21.8965	1.19101
$339$	16.3543	0.888245
$340$	9.89530	0.536648
$341$	−24.6784	−1.33641
$342$	3.24904	0.175688
$343$	1.00000	0.0539949
$344$	−4.63471	−0.249887
$345$	−1.92004	−0.103371
$346$	32.8786	1.76756
$347$	−5.82527	−0.312717	−0.156358	−	0.987700i	$-0.549975\pi$
$347$	−5.82527	−0.312717	−0.156358	+	0.987700i	$0.549975\pi$
$348$	5.54634	0.297315
$349$	−5.73187	−0.306820	−0.153410	−	0.988163i	$-0.549025\pi$
$349$	−5.73187	−0.306820	−0.153410	+	0.988163i	$0.549025\pi$
$350$	5.72325	0.305920
$351$	−1.02371	−0.0546418
$352$	−35.2943	−1.88119
$353$	3.44188	0.183193	0.0915965	−	0.995796i	$-0.470803\pi$
$353$	3.44188	0.183193	0.0915965	+	0.995796i	$0.470803\pi$
$354$	22.1750	1.17859
$355$	−13.4688	−0.714847
$356$	9.97731	0.528796
$357$	5.32646	0.281906
$358$	34.3685	1.81643
$359$	24.3919	1.28736	0.643678	−	0.765297i	$-0.277408\pi$
$359$	24.3919	1.28736	0.643678	+	0.765297i	$0.277408\pi$
$360$	1.61511	0.0851236
$361$	−15.8548	−0.834465
$362$	12.4459	0.654143
$363$	17.8662	0.937735
$364$	−1.38851	−0.0727779
$365$	13.7532	0.719875
$366$	−13.4364	−0.702332
$367$	8.25850	0.431090	0.215545	−	0.976494i	$-0.430847\pi$
$367$	8.25850	0.431090	0.215545	+	0.976494i	$0.430847\pi$
$368$	6.83108	0.356094
$369$	−10.0319	−0.522241
$370$	21.7217	1.12926
$371$	−3.44770	−0.178996
$372$	6.23008	0.323015
$373$	−2.38131	−0.123300	−0.0616498	−	0.998098i	$-0.519636\pi$
$373$	−2.38131	−0.123300	−0.0616498	+	0.998098i	$0.519636\pi$
$374$	52.4285	2.71101
$375$	−11.1272	−0.574608
$376$	14.7178	0.759011
$377$	−4.18613	−0.215597
$378$	−1.83203	−0.0942297
$379$	−36.0234	−1.85040	−0.925198	−	0.379485i	$-0.876101\pi$
$379$	−36.0234	−1.85040	−0.925198	+	0.379485i	$0.876101\pi$
$380$	−3.29467	−0.169013
$381$	−0.455225	−0.0233219
$382$	−9.57857	−0.490082
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−8.94498	−0.456472
$385$	−7.35891	−0.375045
$386$	−10.0636	−0.512222
$387$	−3.93042	−0.199795
$388$	−20.2867	−1.02990
$389$	−10.7325	−0.544158	−0.272079	−	0.962275i	$-0.587711\pi$
$389$	−10.7325	−0.544158	−0.272079	+	0.962275i	$0.587711\pi$
$390$	2.56880	0.130076
$391$	−7.46672	−0.377608
$392$	1.17919	0.0595580
$393$	−17.3279	−0.874077
$394$	30.4770	1.53541
$395$	3.22675	0.162355
$396$	−7.28731	−0.366201
$397$	29.5005	1.48059	0.740293	−	0.672284i	$-0.234687\pi$
$397$	29.5005	1.48059	0.740293	+	0.672284i	$0.234687\pi$
$398$	35.6684	1.78789
$399$	−1.77346	−0.0887841
$400$	15.2232	0.761161
$401$	4.90214	0.244801	0.122401	−	0.992481i	$-0.460941\pi$
$401$	4.90214	0.244801	0.122401	+	0.992481i	$0.460941\pi$
$402$	−19.8409	−0.989575
$403$	−4.70219	−0.234233
$404$	−19.5825	−0.974267
$405$	1.36968	0.0680598
$406$	−7.49149	−0.371796
$407$	46.5091	2.30537
$408$	6.28090	0.310951
$409$	−15.7275	−0.777677	−0.388838	−	0.921306i	$-0.627124\pi$
$409$	−15.7275	−0.777677	−0.388838	+	0.921306i	$0.627124\pi$
$410$	25.1731	1.24321
$411$	9.35838	0.461615
$412$	13.7883	0.679300
$413$	−12.1040	−0.595599
$414$	2.56818	0.126219
$415$	6.05831	0.297391
$416$	−6.72493	−0.329717
$417$	5.95363	0.291550
$418$	−17.4562	−0.853812
$419$	−32.4883	−1.58716	−0.793579	−	0.608468i	$-0.791784\pi$
$419$	−32.4883	−1.58716	−0.793579	+	0.608468i	$0.791784\pi$
$420$	1.85776	0.0906496
$421$	24.3348	1.18600	0.593002	−	0.805201i	$-0.297943\pi$
$421$	24.3348	1.18600	0.593002	+	0.805201i	$0.297943\pi$
$422$	23.5750	1.14761
$423$	12.4813	0.606860
$424$	−4.06548	−0.197437
$425$	−16.6398	−0.807147
$426$	18.0153	0.872847
$427$	7.33415	0.354924
$428$	−16.2881	−0.787315
$429$	5.50014	0.265549
$430$	9.86260	0.475617
$431$	−26.2637	−1.26508	−0.632540	−	0.774528i	$-0.717988\pi$
$431$	−26.2637	−1.26508	−0.632540	+	0.774528i	$0.717988\pi$
$432$	−4.87301	−0.234453
$433$	−30.5903	−1.47008	−0.735038	−	0.678026i	$-0.762836\pi$
$433$	−30.5903	−1.47008	−0.735038	+	0.678026i	$0.762836\pi$
$434$	−8.41503	−0.403934
$435$	5.60083	0.268539
$436$	−9.16670	−0.439005
$437$	2.48607	0.118925
$438$	−18.3958	−0.878985
$439$	−0.0261766	−0.00124934	−0.000624670	−	1.00000i	$-0.500199\pi$
$439$	−0.0261766	−0.00124934	−0.000624670		1.00000i	$0.500199\pi$
$440$	−8.67754	−0.413685
$441$	1.00000	0.0476190
$442$	9.98966	0.475160
$443$	10.1546	0.482461	0.241231	−	0.970468i	$-0.422449\pi$
$443$	10.1546	0.482461	0.241231	+	0.970468i	$0.422449\pi$
$444$	−11.7413	−0.557216
$445$	10.0753	0.477617
$446$	−38.6688	−1.83102
$447$	−22.9114	−1.08367
$448$	−2.28889	−0.108140
$449$	−5.07530	−0.239518	−0.119759	−	0.992803i	$-0.538212\pi$
$449$	−5.07530	−0.239518	−0.119759	+	0.992803i	$0.538212\pi$
$450$	5.72325	0.269796
$451$	53.8989	2.53800
$452$	22.1822	1.04336
$453$	−14.1324	−0.663996
$454$	16.5493	0.776697
$455$	−1.40216	−0.0657342
$456$	−2.09124	−0.0979314
$457$	16.5149	0.772535	0.386267	−	0.922387i	$-0.373764\pi$
$457$	16.5149	0.772535	0.386267	+	0.922387i	$0.373764\pi$
$458$	−48.7176	−2.27643
$459$	5.32646	0.248618
$460$	−2.60424	−0.121424
$461$	0.813773	0.0379012	0.0189506	−	0.999820i	$-0.493967\pi$
$461$	0.813773	0.0379012	0.0189506	+	0.999820i	$0.493967\pi$
$462$	9.84303	0.457939
$463$	17.1019	0.794791	0.397395	−	0.917647i	$-0.369914\pi$
$463$	17.1019	0.794791	0.397395	+	0.917647i	$0.369914\pi$
$464$	−19.9265	−0.925067
$465$	6.29130	0.291752
$466$	−38.4264	−1.78007
$467$	−1.52775	−0.0706960	−0.0353480	−	0.999375i	$-0.511254\pi$
$467$	−1.52775	−0.0706960	−0.0353480	+	0.999375i	$0.511254\pi$
$468$	−1.38851	−0.0641841
$469$	10.8300	0.500082
$470$	−31.3192	−1.44465
$471$	−22.8193	−1.05146
$472$	−14.2729	−0.656963
$473$	21.1171	0.970966
$474$	−4.31599	−0.198240
$475$	5.54026	0.254205
$476$	7.22455	0.331137
$477$	−3.44770	−0.157859
$478$	−13.8433	−0.633177
$479$	10.0629	0.459786	0.229893	−	0.973216i	$-0.426162\pi$
$479$	10.0629	0.459786	0.229893	+	0.973216i	$0.426162\pi$
$480$	8.99762	0.410683
$481$	8.86178	0.404063
$482$	3.16849	0.144321
$483$	−1.40182	−0.0637849
$484$	24.2329	1.10150
$485$	−20.4860	−0.930223
$486$	−1.83203	−0.0831028
$487$	−13.8571	−0.627925	−0.313963	−	0.949435i	$-0.601657\pi$
$487$	−13.8571	−0.627925	−0.313963	+	0.949435i	$0.601657\pi$
$488$	8.64834	0.391492
$489$	8.03727	0.363458
$490$	−2.50930	−0.113358
$491$	−11.2779	−0.508965	−0.254482	−	0.967077i	$-0.581905\pi$
$491$	−11.2779	−0.508965	−0.254482	+	0.967077i	$0.581905\pi$
$492$	−13.6068	−0.613442
$493$	21.7808	0.980956
$494$	−3.32609	−0.149648
$495$	−7.35891	−0.330758
$496$	−22.3831	−1.00503
$497$	−9.83352	−0.441094
$498$	−8.10339	−0.363122
$499$	40.9168	1.83169	0.915845	−	0.401533i	$-0.131523\pi$
$499$	40.9168	1.83169	0.915845	+	0.401533i	$0.131523\pi$
$500$	−15.0924	−0.674954
$501$	−0.331679	−0.0148183
$502$	36.1825	1.61490
$503$	−34.5288	−1.53956	−0.769782	−	0.638307i	$-0.779635\pi$
$503$	−34.5288	−1.53956	−0.769782	+	0.638307i	$0.779635\pi$
$504$	1.17919	0.0525252
$505$	−19.7749	−0.879973
$506$	−13.7981	−0.613402
$507$	−11.9520	−0.530807
$508$	−0.617444	−0.0273947
$509$	−24.2363	−1.07426	−0.537128	−	0.843501i	$-0.680491\pi$
$509$	−24.2363	−1.07426	−0.537128	+	0.843501i	$0.680491\pi$
$510$	−13.3657	−0.591842
$511$	10.0412	0.444196
$512$	−20.5192	−0.906828
$513$	−1.77346	−0.0783002
$514$	41.5118	1.83101
$515$	13.9238	0.613555
$516$	−5.33103	−0.234686
$517$	−67.0585	−2.94923
$518$	15.8590	0.696805
$519$	−17.9465	−0.787763
$520$	−1.65341	−0.0725067
$521$	−32.1802	−1.40984	−0.704920	−	0.709286i	$-0.749017\pi$
$521$	−32.1802	−1.40984	−0.704920	+	0.709286i	$0.749017\pi$
$522$	−7.49149	−0.327893
$523$	4.33651	0.189622	0.0948112	−	0.995495i	$-0.469775\pi$
$523$	4.33651	0.189622	0.0948112	+	0.995495i	$0.469775\pi$
$524$	−23.5027	−1.02672
$525$	−3.12398	−0.136342
$526$	−38.3652	−1.67280
$527$	24.4659	1.06575
$528$	26.1814	1.13940
$529$	−21.0349	−0.914561
$530$	8.65130	0.375788
$531$	−12.1040	−0.525269
$532$	−2.40543	−0.104289
$533$	10.2698	0.444835
$534$	−13.4764	−0.583182
$535$	−16.4481	−0.711115
$536$	12.7706	0.551606
$537$	−18.7597	−0.809542
$538$	19.8311	0.854978
$539$	−5.37273	−0.231420
$540$	1.85776	0.0799454
$541$	−13.7373	−0.590612	−0.295306	−	0.955403i	$-0.595422\pi$
$541$	−13.7373	−0.590612	−0.295306	+	0.955403i	$0.595422\pi$
$542$	−34.0094	−1.46083
$543$	−6.79349	−0.291537
$544$	34.9903	1.50020
$545$	−9.25677	−0.396516
$546$	1.87548	0.0802631
$547$	12.2197	0.522475	0.261238	−	0.965275i	$-0.415869\pi$
$547$	12.2197	0.522475	0.261238	+	0.965275i	$0.415869\pi$
$548$	12.6932	0.542228
$549$	7.33415	0.313014
$550$	−30.7495	−1.31116
$551$	−7.25197	−0.308944
$552$	−1.65301	−0.0703566
$553$	2.35585	0.100181
$554$	−40.5309	−1.72199
$555$	−11.8566	−0.503286
$556$	8.07520	0.342465
$557$	−12.5575	−0.532077	−0.266038	−	0.963962i	$-0.585715\pi$
$557$	−12.5575	−0.532077	−0.266038	+	0.963962i	$0.585715\pi$
$558$	−8.41503	−0.356237
$559$	4.02363	0.170181
$560$	−6.67446	−0.282047
$561$	−28.6176	−1.20824
$562$	−1.55062	−0.0654090
$563$	15.2809	0.644014	0.322007	−	0.946737i	$-0.395643\pi$
$563$	15.2809	0.644014	0.322007	+	0.946737i	$0.395643\pi$
$564$	16.9290	0.712839
$565$	22.4002	0.942382
$566$	−45.5176	−1.91325
$567$	1.00000	0.0419961
$568$	−11.5956	−0.486539
$569$	−34.8671	−1.46170	−0.730852	−	0.682536i	$-0.760877\pi$
$569$	−34.8671	−1.46170	−0.730852	+	0.682536i	$0.760877\pi$
$570$	4.45014	0.186396
$571$	24.8621	1.04045	0.520223	−	0.854030i	$-0.325849\pi$
$571$	24.8621	1.04045	0.520223	+	0.854030i	$0.325849\pi$
$572$	7.46012	0.311923
$573$	5.22838	0.218419
$574$	18.3788	0.767118
$575$	4.37925	0.182628
$576$	−2.28889	−0.0953704
$577$	14.9193	0.621098	0.310549	−	0.950557i	$-0.399487\pi$
$577$	14.9193	0.621098	0.310549	+	0.950557i	$0.399487\pi$
$578$	−20.8324	−0.866512
$579$	5.49311	0.228286
$580$	7.59670	0.315436
$581$	4.42317	0.183504
$582$	27.4014	1.13583
$583$	18.5236	0.767167
$584$	11.8404	0.489961
$585$	−1.40216	−0.0579721
$586$	8.72125	0.360272
$587$	2.40366	0.0992098	0.0496049	−	0.998769i	$-0.484204\pi$
$587$	2.40366	0.0992098	0.0496049	+	0.998769i	$0.484204\pi$
$588$	1.35635	0.0559350
$589$	−8.14598	−0.335649
$590$	30.3725	1.25042
$591$	−16.6356	−0.684298
$592$	42.1833	1.73372
$593$	−9.74521	−0.400188	−0.200094	−	0.979777i	$-0.564125\pi$
$593$	−9.74521	−0.400188	−0.200094	+	0.979777i	$0.564125\pi$
$594$	9.84303	0.403864
$595$	7.29553	0.299088
$596$	−31.0758	−1.27292
$597$	−19.4693	−0.796824
$598$	−2.62908	−0.107511
$599$	16.9735	0.693518	0.346759	−	0.937954i	$-0.387282\pi$
$599$	16.9735	0.693518	0.346759	+	0.937954i	$0.387282\pi$
$600$	−3.68376	−0.150389
$601$	37.9082	1.54631	0.773153	−	0.634219i	$-0.218678\pi$
$601$	37.9082	1.54631	0.773153	+	0.634219i	$0.218678\pi$
$602$	7.20067	0.293478
$603$	10.8300	0.441031
$604$	−19.1684	−0.779953
$605$	24.4710	0.994888
$606$	26.4503	1.07447
$607$	4.09793	0.166330	0.0831650	−	0.996536i	$-0.473497\pi$
$607$	4.09793	0.166330	0.0831650	+	0.996536i	$0.473497\pi$
$608$	−11.6501	−0.472475
$609$	4.08916	0.165701
$610$	−18.4036	−0.745138
$611$	−12.7772	−0.516912
$612$	7.22455	0.292035
$613$	−6.44422	−0.260279	−0.130140	−	0.991496i	$-0.541543\pi$
$613$	−6.44422	−0.260279	−0.130140	+	0.991496i	$0.541543\pi$
$614$	55.9210	2.25679
$615$	−13.7405	−0.554071
$616$	−6.33546	−0.255263
$617$	−42.7782	−1.72219	−0.861093	−	0.508448i	$-0.830219\pi$
$617$	−42.7782	−1.72219	−0.861093	+	0.508448i	$0.830219\pi$
$618$	−18.6240	−0.749166
$619$	−9.14776	−0.367680	−0.183840	−	0.982956i	$-0.558853\pi$
$619$	−9.14776	−0.367680	−0.183840	+	0.982956i	$0.558853\pi$
$620$	8.53321	0.342702
$621$	−1.40182	−0.0562530
$622$	−22.1162	−0.886779
$623$	7.35599	0.294712
$624$	4.98857	0.199703
$625$	0.379189	0.0151676
$626$	27.6209	1.10395
$627$	9.52833	0.380525
$628$	−30.9510	−1.23508
$629$	−46.1085	−1.83847
$630$	−2.50930	−0.0999728
$631$	25.3094	1.00755	0.503776	−	0.863835i	$-0.331944\pi$
$631$	25.3094	1.00755	0.503776	+	0.863835i	$0.331944\pi$
$632$	2.77799	0.110502
$633$	−12.8682	−0.511465
$634$	−7.43609	−0.295325
$635$	−0.623511	−0.0247433
$636$	−4.67629	−0.185427
$637$	−1.02371	−0.0405610
$638$	40.2498	1.59350
$639$	−9.83352	−0.389008
$640$	−12.2517	−0.484293
$641$	−21.6537	−0.855269	−0.427634	−	0.903952i	$-0.640653\pi$
$641$	−21.6537	−0.855269	−0.427634	+	0.903952i	$0.640653\pi$
$642$	22.0005	0.868290
$643$	38.7969	1.53000	0.765001	−	0.644029i	$-0.222738\pi$
$643$	38.7969	1.53000	0.765001	+	0.644029i	$0.222738\pi$
$644$	−1.90136	−0.0749239
$645$	−5.38341	−0.211972
$646$	17.3059	0.680891
$647$	40.2175	1.58111	0.790557	−	0.612388i	$-0.209791\pi$
$647$	40.2175	1.58111	0.790557	+	0.612388i	$0.209791\pi$
$648$	1.17919	0.0463229
$649$	65.0316	2.55271
$650$	−5.85896	−0.229807
$651$	4.59327	0.180024
$652$	10.9014	0.426930
$653$	26.2613	1.02769	0.513843	−	0.857884i	$-0.328221\pi$
$653$	26.2613	1.02769	0.513843	+	0.857884i	$0.328221\pi$
$654$	12.3815	0.484157
$655$	−23.7336	−0.927349
$656$	48.8857	1.90867
$657$	10.0412	0.391744
$658$	−22.8661	−0.891414
$659$	−14.3000	−0.557049	−0.278524	−	0.960429i	$-0.589845\pi$
$659$	−14.3000	−0.557049	−0.278524	+	0.960429i	$0.589845\pi$
$660$	−9.98126	−0.388520
$661$	−22.9798	−0.893812	−0.446906	−	0.894581i	$-0.647474\pi$
$661$	−22.9798	−0.893812	−0.446906	+	0.894581i	$0.647474\pi$
$662$	23.0162	0.894550
$663$	−5.45277	−0.211768
$664$	5.21574	0.202410
$665$	−2.42907	−0.0941953
$666$	15.8590	0.614525
$667$	−5.73226	−0.221954
$668$	−0.449873	−0.0174061
$669$	21.1070	0.816045
$670$	−27.1757	−1.04989
$671$	−39.4044	−1.52119
$672$	6.56915	0.253411
$673$	−22.3785	−0.862629	−0.431314	−	0.902202i	$-0.641950\pi$
$673$	−22.3785	−0.862629	−0.431314	+	0.902202i	$0.641950\pi$
$674$	−22.7086	−0.874701
$675$	−3.12398	−0.120242
$676$	−16.2111	−0.623505
$677$	−28.3482	−1.08951	−0.544754	−	0.838596i	$-0.683377\pi$
$677$	−28.3482	−1.08951	−0.544754	+	0.838596i	$0.683377\pi$
$678$	−29.9617	−1.15067
$679$	−14.9568	−0.573990
$680$	8.60280	0.329902
$681$	−9.03329	−0.346157
$682$	45.2117	1.73125
$683$	7.66385	0.293249	0.146625	−	0.989192i	$-0.453159\pi$
$683$	7.66385	0.293249	0.146625	+	0.989192i	$0.453159\pi$
$684$	−2.40543	−0.0919741
$685$	12.8180	0.489749
$686$	−1.83203	−0.0699474
$687$	26.5921	1.01455
$688$	19.1530	0.730202
$689$	3.52945	0.134461
$690$	3.51758	0.133912
$691$	−8.66947	−0.329802	−0.164901	−	0.986310i	$-0.552730\pi$
$691$	−8.66947	−0.329802	−0.164901	+	0.986310i	$0.552730\pi$
$692$	−24.3417	−0.925333
$693$	−5.37273	−0.204093
$694$	10.6721	0.405107
$695$	8.15455	0.309320
$696$	4.82189	0.182773
$697$	−53.4347	−2.02398
$698$	10.5010	0.397468
$699$	20.9747	0.793336
$700$	−4.23722	−0.160152
$701$	−16.7240	−0.631658	−0.315829	−	0.948816i	$-0.602283\pi$
$701$	−16.7240	−0.631658	−0.315829	+	0.948816i	$0.602283\pi$
$702$	1.87548	0.0707854
$703$	15.3520	0.579011
$704$	12.2976	0.463483
$705$	17.0953	0.643847
$706$	−6.30565	−0.237316
$707$	−14.4377	−0.542984
$708$	−16.4173	−0.616999
$709$	6.88351	0.258516	0.129258	−	0.991611i	$-0.458741\pi$
$709$	6.88351	0.258516	0.129258	+	0.991611i	$0.458741\pi$
$710$	24.6752	0.926044
$711$	2.35585	0.0883512
$712$	8.67410	0.325076
$713$	−6.43892	−0.241140
$714$	−9.75826	−0.365194
$715$	7.53342	0.281734
$716$	−25.4448	−0.950916
$717$	7.55623	0.282193
$718$	−44.6868	−1.66770
$719$	−47.5863	−1.77467	−0.887335	−	0.461126i	$-0.847446\pi$
$719$	−47.5863	−1.77467	−0.887335	+	0.461126i	$0.847446\pi$
$720$	−6.67446	−0.248742
$721$	10.1657	0.378591
$722$	29.0466	1.08100
$723$	−1.72950	−0.0643206
$724$	−9.21436	−0.342449
$725$	−12.7745	−0.474432
$726$	−32.7316	−1.21478
$727$	24.1128	0.894294	0.447147	−	0.894460i	$-0.352440\pi$
$727$	24.1128	0.894294	0.447147	+	0.894460i	$0.352440\pi$
$728$	−1.20715	−0.0447400
$729$	1.00000	0.0370370
$730$	−25.1963	−0.932557
$731$	−20.9352	−0.774318
$732$	9.94768	0.367677
$733$	2.33283	0.0861652	0.0430826	−	0.999072i	$-0.486282\pi$
$733$	2.33283	0.0861652	0.0430826	+	0.999072i	$0.486282\pi$
$734$	−15.1299	−0.558453
$735$	1.36968	0.0505213
$736$	−9.20875	−0.339439
$737$	−58.1866	−2.14333
$738$	18.3788	0.676534
$739$	−51.7442	−1.90344	−0.951720	−	0.306966i	$-0.900686\pi$
$739$	−51.7442	−1.90344	−0.951720	+	0.306966i	$0.900686\pi$
$740$	−16.0817	−0.591177
$741$	1.81552	0.0666946
$742$	6.31630	0.231879
$743$	12.6645	0.464614	0.232307	−	0.972643i	$-0.425373\pi$
$743$	12.6645	0.464614	0.232307	+	0.972643i	$0.425373\pi$
$744$	5.41633	0.198572
$745$	−31.3812	−1.14972
$746$	4.36264	0.159728
$747$	4.42317	0.161835
$748$	−38.8156	−1.41924
$749$	−12.0088	−0.438791
$750$	20.3855	0.744373
$751$	3.43076	0.125190	0.0625952	−	0.998039i	$-0.480062\pi$
$751$	3.43076	0.125190	0.0625952	+	0.998039i	$0.480062\pi$
$752$	−60.8214	−2.21793
$753$	−19.7499	−0.719726
$754$	7.66914	0.279293
$755$	−19.3568	−0.704465
$756$	1.35635	0.0493300
$757$	−27.1521	−0.986860	−0.493430	−	0.869785i	$-0.664257\pi$
$757$	−27.1521	−0.986860	−0.493430	+	0.869785i	$0.664257\pi$
$758$	65.9960	2.39708
$759$	7.53159	0.273379
$760$	−2.86433	−0.103900
$761$	−36.9466	−1.33931	−0.669657	−	0.742671i	$-0.733559\pi$
$761$	−36.9466	−1.33931	−0.669657	+	0.742671i	$0.733559\pi$
$762$	0.833988	0.0302122
$763$	−6.75836	−0.244669
$764$	7.09151	0.256562
$765$	7.29553	0.263771
$766$	1.83203	0.0661941
$767$	12.3910	0.447414
$768$	20.9653	0.756520
$769$	−33.8668	−1.22127	−0.610634	−	0.791913i	$-0.709085\pi$
$769$	−33.8668	−1.22127	−0.610634	+	0.791913i	$0.709085\pi$
$770$	13.4818	0.485850
$771$	−22.6588	−0.816038
$772$	7.45058	0.268152
$773$	9.49981	0.341684	0.170842	−	0.985298i	$-0.445351\pi$
$773$	9.49981	0.341684	0.170842	+	0.985298i	$0.445351\pi$
$774$	7.20067	0.258823
$775$	−14.3493	−0.515442
$776$	−17.6369	−0.633128
$777$	−8.65651	−0.310550
$778$	19.6623	0.704926
$779$	17.7912	0.637437
$780$	−1.90182	−0.0680960
$781$	52.8329	1.89051
$782$	13.6793	0.489171
$783$	4.08916	0.146135
$784$	−4.87301	−0.174036
$785$	−31.2551	−1.11554
$786$	31.7453	1.13232
$787$	6.78751	0.241949	0.120974	−	0.992656i	$-0.461398\pi$
$787$	6.78751	0.241949	0.120974	+	0.992656i	$0.461398\pi$
$788$	−22.5637	−0.803800
$789$	20.9413	0.745531
$790$	−5.91152	−0.210322
$791$	16.3543	0.581493
$792$	−6.33546	−0.225121
$793$	−7.50807	−0.266619
$794$	−54.0459	−1.91802
$795$	−4.72223	−0.167480
$796$	−26.4072	−0.935977
$797$	48.7620	1.72724	0.863620	−	0.504143i	$-0.168192\pi$
$797$	48.7620	1.72724	0.863620	+	0.504143i	$0.168192\pi$
$798$	3.24904	0.115015
$799$	66.4810	2.35193
$800$	−20.5219	−0.725560
$801$	7.35599	0.259911
$802$	−8.98089	−0.317126
$803$	−53.9486	−1.90380
$804$	14.6893	0.518050
$805$	−1.92004	−0.0676724
$806$	8.61458	0.303436
$807$	−10.8246	−0.381044
$808$	−17.0247	−0.598927
$809$	−16.8722	−0.593193	−0.296597	−	0.955003i	$-0.595852\pi$
$809$	−16.8722	−0.593193	−0.296597	+	0.955003i	$0.595852\pi$
$810$	−2.50930	−0.0881677
$811$	−24.8293	−0.871875	−0.435938	−	0.899977i	$-0.643583\pi$
$811$	−24.8293	−0.871875	−0.435938	+	0.899977i	$0.643583\pi$
$812$	5.54634	0.194638
$813$	18.5637	0.651059
$814$	−85.2063	−2.98648
$815$	11.0085	0.385610
$816$	−25.9559	−0.908639
$817$	6.97045	0.243865
$818$	28.8134	1.00744
$819$	−1.02371	−0.0357714
$820$	−18.6369	−0.650830
$821$	54.6197	1.90624	0.953121	−	0.302591i	$-0.0978515\pi$
$821$	54.6197	1.90624	0.953121	+	0.302591i	$0.0978515\pi$
$822$	−17.1449	−0.597996
$823$	−7.67920	−0.267680	−0.133840	−	0.991003i	$-0.542731\pi$
$823$	−7.67920	−0.267680	−0.133840	+	0.991003i	$0.542731\pi$
$824$	11.9873	0.417597
$825$	16.7843	0.584355
$826$	22.1750	0.771565
$827$	25.8186	0.897802	0.448901	−	0.893581i	$-0.351816\pi$
$827$	25.8186	0.897802	0.448901	+	0.893581i	$0.351816\pi$
$828$	−1.90136	−0.0660767
$829$	−11.1010	−0.385554	−0.192777	−	0.981243i	$-0.561749\pi$
$829$	−11.1010	−0.385554	−0.192777	+	0.981243i	$0.561749\pi$
$830$	−11.0990	−0.385253
$831$	22.1234	0.767453
$832$	2.34317	0.0812347
$833$	5.32646	0.184551
$834$	−10.9072	−0.377687
$835$	−0.454294	−0.0157215
$836$	12.9238	0.446977
$837$	4.59327	0.158767
$838$	59.5197	2.05607
$839$	−6.01410	−0.207630	−0.103815	−	0.994597i	$-0.533105\pi$
$839$	−6.01410	−0.207630	−0.103815	+	0.994597i	$0.533105\pi$
$840$	1.61511	0.0557265
$841$	−12.2787	−0.423405
$842$	−44.5821	−1.53640
$843$	0.846393	0.0291513
$844$	−17.4538	−0.600784
$845$	−16.3704	−0.563159
$846$	−22.8661	−0.786154
$847$	17.8662	0.613892
$848$	16.8007	0.576938
$849$	24.8454	0.852692
$850$	30.4846	1.04561
$851$	12.1348	0.415977
$852$	−13.3377	−0.456942
$853$	−25.1527	−0.861213	−0.430606	−	0.902540i	$-0.641700\pi$
$853$	−25.1527	−0.861213	−0.430606	+	0.902540i	$0.641700\pi$
$854$	−13.4364	−0.459785
$855$	−2.42907	−0.0830724
$856$	−14.1606	−0.483999
$857$	−26.7629	−0.914205	−0.457102	−	0.889414i	$-0.651113\pi$
$857$	−26.7629	−0.914205	−0.457102	+	0.889414i	$0.651113\pi$
$858$	−10.0764	−0.344004
$859$	25.4908	0.869735	0.434867	−	0.900494i	$-0.356795\pi$
$859$	25.4908	0.869735	0.434867	+	0.900494i	$0.356795\pi$
$860$	−7.30180	−0.248989
$861$	−10.0319	−0.341887
$862$	48.1161	1.63884
$863$	43.5009	1.48079	0.740394	−	0.672174i	$-0.234639\pi$
$863$	43.5009	1.48079	0.740394	+	0.672174i	$0.234639\pi$
$864$	6.56915	0.223487
$865$	−24.5809	−0.835775
$866$	56.0425	1.90440
$867$	11.3712	0.386185
$868$	6.23008	0.211463
$869$	−12.6573	−0.429371
$870$	−10.2609	−0.347878
$871$	−11.0868	−0.375662
$872$	−7.96937	−0.269877
$873$	−14.9568	−0.506212
$874$	−4.55456	−0.154060
$875$	−11.1272	−0.376169
$876$	13.6194	0.460156
$877$	−0.402151	−0.0135797	−0.00678984	−	0.999977i	$-0.502161\pi$
$877$	−0.402151	−0.0135797	−0.00678984	+	0.999977i	$0.502161\pi$
$878$	0.0479564	0.00161845
$879$	−4.76042	−0.160565
$880$	35.8601	1.20884
$881$	−14.6455	−0.493419	−0.246709	−	0.969089i	$-0.579349\pi$
$881$	−14.6455	−0.493419	−0.246709	+	0.969089i	$0.579349\pi$
$882$	−1.83203	−0.0616878
$883$	−42.8969	−1.44360	−0.721798	−	0.692104i	$-0.756684\pi$
$883$	−42.8969	−1.44360	−0.721798	+	0.692104i	$0.756684\pi$
$884$	−7.39587	−0.248750
$885$	−16.5786	−0.557283
$886$	−18.6036	−0.625001
$887$	12.4150	0.416853	0.208427	−	0.978038i	$-0.433166\pi$
$887$	12.4150	0.416853	0.208427	+	0.978038i	$0.433166\pi$
$888$	−10.2076	−0.342546
$889$	−0.455225	−0.0152678
$890$	−18.4584	−0.618726
$891$	−5.37273	−0.179993
$892$	28.6285	0.958555
$893$	−22.1350	−0.740721
$894$	41.9744	1.40383
$895$	−25.6948	−0.858882
$896$	−8.94498	−0.298831
$897$	1.43506	0.0479152
$898$	9.29813	0.310283
$899$	18.7826	0.626436
$900$	−4.23722	−0.141241
$901$	−18.3640	−0.611794
$902$	−98.7446	−3.28783
$903$	−3.93042	−0.130796
$904$	19.2848	0.641404
$905$	−9.30489	−0.309305
$906$	25.8910	0.860170
$907$	35.4901	1.17843	0.589214	−	0.807977i	$-0.299438\pi$
$907$	35.4901	1.17843	0.589214	+	0.807977i	$0.299438\pi$
$908$	−12.2523	−0.406607
$909$	−14.4377	−0.478867
$910$	2.56880	0.0851549
$911$	14.7706	0.489371	0.244686	−	0.969602i	$-0.421315\pi$
$911$	14.7706	0.489371	0.244686	+	0.969602i	$0.421315\pi$
$912$	8.64210	0.286168
$913$	−23.7645	−0.786490
$914$	−30.2559	−1.00078
$915$	10.0454	0.332091
$916$	36.0682	1.19173
$917$	−17.3279	−0.572217
$918$	−9.75826	−0.322070
$919$	5.47277	0.180530	0.0902650	−	0.995918i	$-0.471229\pi$
$919$	5.47277	0.180530	0.0902650	+	0.995918i	$0.471229\pi$
$920$	−2.26409	−0.0746447
$921$	−30.5240	−1.00580
$922$	−1.49086	−0.0490989
$923$	10.0667	0.331350
$924$	−7.28731	−0.239735
$925$	27.0428	0.889161
$926$	−31.3312	−1.02961
$927$	10.1657	0.333886
$928$	26.8623	0.881800
$929$	48.9708	1.60668	0.803340	−	0.595520i	$-0.203054\pi$
$929$	48.9708	1.60668	0.803340	+	0.595520i	$0.203054\pi$
$930$	−11.5259	−0.377948
$931$	−1.77346	−0.0581228
$932$	28.4491	0.931880
$933$	12.0719	0.395218
$934$	2.79890	0.0915827
$935$	−39.1969	−1.28188
$936$	−1.20715	−0.0394570
$937$	41.9730	1.37120	0.685598	−	0.727980i	$-0.259541\pi$
$937$	41.9730	1.37120	0.685598	+	0.727980i	$0.259541\pi$
$938$	−19.8409	−0.647829
$939$	−15.0766	−0.492007
$940$	23.1872	0.756285
$941$	−21.0564	−0.686418	−0.343209	−	0.939259i	$-0.611514\pi$
$941$	−21.0564	−0.686418	−0.343209	+	0.939259i	$0.611514\pi$
$942$	41.8058	1.36211
$943$	14.0629	0.457952
$944$	58.9830	1.91973
$945$	1.36968	0.0445556
$946$	−38.6873	−1.25783
$947$	−1.41194	−0.0458817	−0.0229409	−	0.999737i	$-0.507303\pi$
$947$	−1.41194	−0.0458817	−0.0229409	+	0.999737i	$0.507303\pi$
$948$	3.19535	0.103780
$949$	−10.2793	−0.333680
$950$	−10.1499	−0.329308
$951$	4.05893	0.131620
$952$	6.28090	0.203565
$953$	−40.5174	−1.31249	−0.656244	−	0.754549i	$-0.727856\pi$
$953$	−40.5174	−1.31249	−0.656244	+	0.754549i	$0.727856\pi$
$954$	6.31630	0.204498
$955$	7.16119	0.231731
$956$	10.2489	0.331473
$957$	−21.9700	−0.710188
$958$	−18.4356	−0.595627
$959$	9.35838	0.302198
$960$	−3.13504	−0.101183
$961$	−9.90188	−0.319415
$962$	−16.2351	−0.523440
$963$	−12.0088	−0.386977
$964$	−2.34580	−0.0755532
$965$	7.52379	0.242199
$966$	2.56818	0.0826298
$967$	18.0750	0.581254	0.290627	−	0.956836i	$-0.406136\pi$
$967$	18.0750	0.581254	0.290627	+	0.956836i	$0.406136\pi$
$968$	21.0677	0.677140
$969$	−9.44626	−0.303458
$970$	37.5311	1.20505
$971$	−32.1364	−1.03131	−0.515653	−	0.856797i	$-0.672451\pi$
$971$	−32.1364	−1.03131	−0.515653	+	0.856797i	$0.672451\pi$
$972$	1.35635	0.0435050
$973$	5.95363	0.190864
$974$	25.3867	0.813442
$975$	3.19806	0.102420
$976$	−35.7394	−1.14399
$977$	−17.3808	−0.556061	−0.278030	−	0.960572i	$-0.589682\pi$
$977$	−17.3808	−0.556061	−0.278030	+	0.960572i	$0.589682\pi$
$978$	−14.7246	−0.470840
$979$	−39.5218	−1.26312
$980$	1.85776	0.0593441
$981$	−6.75836	−0.215778
$982$	20.6615	0.659335
$983$	−0.338783	−0.0108055	−0.00540275	−	0.999985i	$-0.501720\pi$
$983$	−0.338783	−0.0108055	−0.00540275	+	0.999985i	$0.501720\pi$
$984$	−11.8295	−0.377112
$985$	−22.7854	−0.726004
$986$	−39.9031	−1.27077
$987$	12.4813	0.397283
$988$	2.46248	0.0783418
$989$	5.50974	0.175199
$990$	13.4818	0.428479
$991$	5.98599	0.190151	0.0950756	−	0.995470i	$-0.469691\pi$
$991$	5.98599	0.190151	0.0950756	+	0.995470i	$0.469691\pi$
$992$	30.1739	0.958022
$993$	−12.5632	−0.398681
$994$	18.0153	0.571412
$995$	−26.6666	−0.845389
$996$	5.99936	0.190097
$997$	−42.9503	−1.36025	−0.680125	−	0.733096i	$-0.738074\pi$
$997$	−42.9503	−1.36025	−0.680125	+	0.733096i	$0.738074\pi$
$998$	−74.9611	−2.37285
$999$	−8.65651	−0.273880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.n.1.10	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.n.1.10	✓	40	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-1.83203 q^{2} +1.00000 q^{3} +1.35635 q^{4} +1.36968 q^{5} -1.83203 q^{6} +1.00000 q^{7} +1.17919 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.83203 q^{2} +1.00000 q^{3} +1.35635 q^{4} +1.36968 q^{5} -1.83203 q^{6} +1.00000 q^{7} +1.17919 q^{8} +1.00000 q^{9} -2.50930 q^{10} -5.37273 q^{11} +1.35635 q^{12} -1.02371 q^{13} -1.83203 q^{14} +1.36968 q^{15} -4.87301 q^{16} +5.32646 q^{17} -1.83203 q^{18} -1.77346 q^{19} +1.85776 q^{20} +1.00000 q^{21} +9.84303 q^{22} -1.40182 q^{23} +1.17919 q^{24} -3.12398 q^{25} +1.87548 q^{26} +1.00000 q^{27} +1.35635 q^{28} +4.08916 q^{29} -2.50930 q^{30} +4.59327 q^{31} +6.56915 q^{32} -5.37273 q^{33} -9.75826 q^{34} +1.36968 q^{35} +1.35635 q^{36} -8.65651 q^{37} +3.24904 q^{38} -1.02371 q^{39} +1.61511 q^{40} -10.0319 q^{41} -1.83203 q^{42} -3.93042 q^{43} -7.28731 q^{44} +1.36968 q^{45} +2.56818 q^{46} +12.4813 q^{47} -4.87301 q^{48} +1.00000 q^{49} +5.72325 q^{50} +5.32646 q^{51} -1.38851 q^{52} -3.44770 q^{53} -1.83203 q^{54} -7.35891 q^{55} +1.17919 q^{56} -1.77346 q^{57} -7.49149 q^{58} -12.1040 q^{59} +1.85776 q^{60} +7.33415 q^{61} -8.41503 q^{62} +1.00000 q^{63} -2.28889 q^{64} -1.40216 q^{65} +9.84303 q^{66} +10.8300 q^{67} +7.22455 q^{68} -1.40182 q^{69} -2.50930 q^{70} -9.83352 q^{71} +1.17919 q^{72} +10.0412 q^{73} +15.8590 q^{74} -3.12398 q^{75} -2.40543 q^{76} -5.37273 q^{77} +1.87548 q^{78} +2.35585 q^{79} -6.67446 q^{80} +1.00000 q^{81} +18.3788 q^{82} +4.42317 q^{83} +1.35635 q^{84} +7.29553 q^{85} +7.20067 q^{86} +4.08916 q^{87} -6.33546 q^{88} +7.35599 q^{89} -2.50930 q^{90} -1.02371 q^{91} -1.90136 q^{92} +4.59327 q^{93} -22.8661 q^{94} -2.42907 q^{95} +6.56915 q^{96} -14.9568 q^{97} -1.83203 q^{98} -5.37273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9 - 14 * q^10 - 29 * q^11 + 33 * q^12 - 40 * q^13 - 9 * q^14 - 27 * q^15 + 23 * q^16 - 46 * q^17 - 9 * q^18 + 17 * q^19 - 53 * q^20 + 40 * q^21 - 15 * q^22 - 62 * q^23 - 18 * q^24 + 35 * q^25 - 29 * q^26 + 40 * q^27 + 33 * q^28 - 47 * q^29 - 14 * q^30 + q^31 - 45 * q^32 - 29 * q^33 + 9 * q^34 - 27 * q^35 + 33 * q^36 - 49 * q^37 - 52 * q^38 - 40 * q^39 - 12 * q^40 - 49 * q^41 - 9 * q^42 - 45 * q^43 - 68 * q^44 - 27 * q^45 - 36 * q^46 - 88 * q^47 + 23 * q^48 + 40 * q^49 - 32 * q^50 - 46 * q^51 - 34 * q^52 - 96 * q^53 - 9 * q^54 + 26 * q^55 - 18 * q^56 + 17 * q^57 + 12 * q^58 - 45 * q^59 - 53 * q^60 - 23 * q^61 - 15 * q^62 + 40 * q^63 + 50 * q^64 - 32 * q^65 - 15 * q^66 - 31 * q^67 - 76 * q^68 - 62 * q^69 - 14 * q^70 - 89 * q^71 - 18 * q^72 + 4 * q^73 + 10 * q^74 + 35 * q^75 + 51 * q^76 - 29 * q^77 - 29 * q^78 - 44 * q^79 - 53 * q^80 + 40 * q^81 - 15 * q^82 - 60 * q^83 + 33 * q^84 - 24 * q^85 - 9 * q^86 - 47 * q^87 - 64 * q^88 - 39 * q^89 - 14 * q^90 - 40 * q^91 - 80 * q^92 + q^93 + 51 * q^94 - 71 * q^95 - 45 * q^96 - 21 * q^97 - 9 * q^98 - 29 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more