LMFDB - Embedded newform 4730.2.a.be.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4730,2,Mod(1,4730)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4730, 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("4730.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4730 = 2 \cdot 5 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4730.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:	$37.7692401561$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 26 x^{10} + 79 x^{9} + 247 x^{8} - 766 x^{7} - 1023 x^{6} + 3281 x^{5} + 1634 x^{4} + \cdots + 392 $ x^12 - 3x^11 - 26x^10 + 79x^9 + 247x^8 - 766x^7 - 1023x^6 + 3281x^5 + 1634x^4 - 5557x^3 - 483x^2 + 1648*x + 392
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.59957$ of defining polynomial
Character	$\chi$	$=$	4730.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.59957 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.59957 q^{6} +1.31461 q^{7} +1.00000 q^{8} +3.75779 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.59957 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.59957 q^{6} +1.31461 q^{7} +1.00000 q^{8} +3.75779 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.59957 q^{12} -5.57356 q^{13} +1.31461 q^{14} +2.59957 q^{15} +1.00000 q^{16} +6.71697 q^{17} +3.75779 q^{18} -4.17687 q^{19} +1.00000 q^{20} +3.41742 q^{21} -1.00000 q^{22} +7.57356 q^{23} +2.59957 q^{24} +1.00000 q^{25} -5.57356 q^{26} +1.96993 q^{27} +1.31461 q^{28} +3.72343 q^{29} +2.59957 q^{30} +5.06624 q^{31} +1.00000 q^{32} -2.59957 q^{33} +6.71697 q^{34} +1.31461 q^{35} +3.75779 q^{36} -4.21701 q^{37} -4.17687 q^{38} -14.4889 q^{39} +1.00000 q^{40} +8.18305 q^{41} +3.41742 q^{42} +1.00000 q^{43} -1.00000 q^{44} +3.75779 q^{45} +7.57356 q^{46} +9.44879 q^{47} +2.59957 q^{48} -5.27180 q^{49} +1.00000 q^{50} +17.4613 q^{51} -5.57356 q^{52} -7.39275 q^{53} +1.96993 q^{54} -1.00000 q^{55} +1.31461 q^{56} -10.8581 q^{57} +3.72343 q^{58} -9.36203 q^{59} +2.59957 q^{60} -0.668250 q^{61} +5.06624 q^{62} +4.94002 q^{63} +1.00000 q^{64} -5.57356 q^{65} -2.59957 q^{66} -2.23127 q^{67} +6.71697 q^{68} +19.6880 q^{69} +1.31461 q^{70} -12.3895 q^{71} +3.75779 q^{72} +5.06235 q^{73} -4.21701 q^{74} +2.59957 q^{75} -4.17687 q^{76} -1.31461 q^{77} -14.4889 q^{78} +13.7458 q^{79} +1.00000 q^{80} -6.15238 q^{81} +8.18305 q^{82} +17.1925 q^{83} +3.41742 q^{84} +6.71697 q^{85} +1.00000 q^{86} +9.67935 q^{87} -1.00000 q^{88} +13.5657 q^{89} +3.75779 q^{90} -7.32705 q^{91} +7.57356 q^{92} +13.1701 q^{93} +9.44879 q^{94} -4.17687 q^{95} +2.59957 q^{96} -6.57793 q^{97} -5.27180 q^{98} -3.75779 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 12 q^{5} + 3 q^{6} + 8 q^{7} + 12 q^{8} + 25 q^{9}+O(q^{10}) $ 12 * q + 12 * q^2 + 3 * q^3 + 12 * q^4 + 12 * q^5 + 3 * q^6 + 8 * q^7 + 12 * q^8 + 25 * q^9	\( 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 12 q^{5} + 3 q^{6} + 8 q^{7} + 12 q^{8} + 25 q^{9} + 12 q^{10} - 12 q^{11} + 3 q^{12} + 16 q^{13} + 8 q^{14} + 3 q^{15} + 12 q^{16} + 18 q^{17} + 25 q^{18} - 4 q^{19} + 12 q^{20} + 4 q^{21} - 12 q^{22} + 8 q^{23} + 3 q^{24} + 12 q^{25} + 16 q^{26} + 6 q^{27} + 8 q^{28} + 20 q^{29} + 3 q^{30} + 5 q^{31} + 12 q^{32} - 3 q^{33} + 18 q^{34} + 8 q^{35} + 25 q^{36} + 19 q^{37} - 4 q^{38} + 6 q^{39} + 12 q^{40} + 16 q^{41} + 4 q^{42} + 12 q^{43} - 12 q^{44} + 25 q^{45} + 8 q^{46} - q^{47} + 3 q^{48} + 52 q^{49} + 12 q^{50} + q^{51} + 16 q^{52} + 11 q^{53} + 6 q^{54} - 12 q^{55} + 8 q^{56} + 9 q^{57} + 20 q^{58} - 11 q^{59} + 3 q^{60} + 18 q^{61} + 5 q^{62} + 15 q^{63} + 12 q^{64} + 16 q^{65} - 3 q^{66} - 10 q^{67} + 18 q^{68} + 8 q^{70} - 2 q^{71} + 25 q^{72} + 29 q^{73} + 19 q^{74} + 3 q^{75} - 4 q^{76} - 8 q^{77} + 6 q^{78} + 2 q^{79} + 12 q^{80} - 8 q^{81} + 16 q^{82} + 26 q^{83} + 4 q^{84} + 18 q^{85} + 12 q^{86} - 4 q^{87} - 12 q^{88} + 41 q^{89} + 25 q^{90} - 4 q^{91} + 8 q^{92} + 5 q^{93} - q^{94} - 4 q^{95} + 3 q^{96} - 7 q^{97} + 52 q^{98} - 25 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 + 3 * q^3 + 12 * q^4 + 12 * q^5 + 3 * q^6 + 8 * q^7 + 12 * q^8 + 25 * q^9 + 12 * q^10 - 12 * q^11 + 3 * q^12 + 16 * q^13 + 8 * q^14 + 3 * q^15 + 12 * q^16 + 18 * q^17 + 25 * q^18 - 4 * q^19 + 12 * q^20 + 4 * q^21 - 12 * q^22 + 8 * q^23 + 3 * q^24 + 12 * q^25 + 16 * q^26 + 6 * q^27 + 8 * q^28 + 20 * q^29 + 3 * q^30 + 5 * q^31 + 12 * q^32 - 3 * q^33 + 18 * q^34 + 8 * q^35 + 25 * q^36 + 19 * q^37 - 4 * q^38 + 6 * q^39 + 12 * q^40 + 16 * q^41 + 4 * q^42 + 12 * q^43 - 12 * q^44 + 25 * q^45 + 8 * q^46 - q^47 + 3 * q^48 + 52 * q^49 + 12 * q^50 + q^51 + 16 * q^52 + 11 * q^53 + 6 * q^54 - 12 * q^55 + 8 * q^56 + 9 * q^57 + 20 * q^58 - 11 * q^59 + 3 * q^60 + 18 * q^61 + 5 * q^62 + 15 * q^63 + 12 * q^64 + 16 * q^65 - 3 * q^66 - 10 * q^67 + 18 * q^68 + 8 * q^70 - 2 * q^71 + 25 * q^72 + 29 * q^73 + 19 * q^74 + 3 * q^75 - 4 * q^76 - 8 * q^77 + 6 * q^78 + 2 * q^79 + 12 * q^80 - 8 * q^81 + 16 * q^82 + 26 * q^83 + 4 * q^84 + 18 * q^85 + 12 * q^86 - 4 * q^87 - 12 * q^88 + 41 * q^89 + 25 * q^90 - 4 * q^91 + 8 * q^92 + 5 * q^93 - q^94 - 4 * q^95 + 3 * q^96 - 7 * q^97 + 52 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.59957	1.50087	0.750433	−	0.660947i	$-0.229845\pi$
$3$	2.59957	1.50087	0.750433	+	0.660947i	$0.229845\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.59957	1.06127
$7$	1.31461	0.496875	0.248438	−	0.968648i	$-0.420083\pi$
$7$	1.31461	0.496875	0.248438	+	0.968648i	$0.420083\pi$
$8$	1.00000	0.353553
$9$	3.75779	1.25260
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.59957	0.750433
$13$	−5.57356	−1.54583	−0.772914	−	0.634511i	$-0.781202\pi$
$13$	−5.57356	−1.54583	−0.772914	+	0.634511i	$0.781202\pi$
$14$	1.31461	0.351344
$15$	2.59957	0.671207
$16$	1.00000	0.250000
$17$	6.71697	1.62910	0.814552	−	0.580090i	$-0.196983\pi$
$17$	6.71697	1.62910	0.814552	+	0.580090i	$0.196983\pi$
$18$	3.75779	0.885720
$19$	−4.17687	−0.958239	−0.479120	−	0.877750i	$-0.659044\pi$
$19$	−4.17687	−0.958239	−0.479120	+	0.877750i	$0.659044\pi$
$20$	1.00000	0.223607
$21$	3.41742	0.745743
$22$	−1.00000	−0.213201
$23$	7.57356	1.57920	0.789599	−	0.613624i	$-0.210289\pi$
$23$	7.57356	1.57920	0.789599	+	0.613624i	$0.210289\pi$
$24$	2.59957	0.530636
$25$	1.00000	0.200000
$26$	−5.57356	−1.09307
$27$	1.96993	0.379114
$28$	1.31461	0.248438
$29$	3.72343	0.691424	0.345712	−	0.938341i	$-0.387637\pi$
$29$	3.72343	0.691424	0.345712	+	0.938341i	$0.387637\pi$
$30$	2.59957	0.474615
$31$	5.06624	0.909923	0.454962	−	0.890511i	$-0.349653\pi$
$31$	5.06624	0.909923	0.454962	+	0.890511i	$0.349653\pi$
$32$	1.00000	0.176777
$33$	−2.59957	−0.452528
$34$	6.71697	1.15195
$35$	1.31461	0.222209
$36$	3.75779	0.626298
$37$	−4.21701	−0.693273	−0.346636	−	0.938000i	$-0.612676\pi$
$37$	−4.21701	−0.693273	−0.346636	+	0.938000i	$0.612676\pi$
$38$	−4.17687	−0.677578
$39$	−14.4889	−2.32008
$40$	1.00000	0.158114
$41$	8.18305	1.27798	0.638989	−	0.769216i	$-0.279353\pi$
$41$	8.18305	1.27798	0.638989	+	0.769216i	$0.279353\pi$
$42$	3.41742	0.527320
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−1.00000	−0.150756
$45$	3.75779	0.560178
$46$	7.57356	1.11666
$47$	9.44879	1.37825	0.689124	−	0.724644i	$-0.257996\pi$
$47$	9.44879	1.37825	0.689124	+	0.724644i	$0.257996\pi$
$48$	2.59957	0.375216
$49$	−5.27180	−0.753115
$50$	1.00000	0.141421
$51$	17.4613	2.44507
$52$	−5.57356	−0.772914
$53$	−7.39275	−1.01547	−0.507736	−	0.861513i	$-0.669517\pi$
$53$	−7.39275	−1.01547	−0.507736	+	0.861513i	$0.669517\pi$
$54$	1.96993	0.268074
$55$	−1.00000	−0.134840
$56$	1.31461	0.175672
$57$	−10.8581	−1.43819
$58$	3.72343	0.488911
$59$	−9.36203	−1.21883	−0.609416	−	0.792850i	$-0.708596\pi$
$59$	−9.36203	−1.21883	−0.609416	+	0.792850i	$0.708596\pi$
$60$	2.59957	0.335604
$61$	−0.668250	−0.0855607	−0.0427803	−	0.999085i	$-0.513622\pi$
$61$	−0.668250	−0.0855607	−0.0427803	+	0.999085i	$0.513622\pi$
$62$	5.06624	0.643413
$63$	4.94002	0.622384
$64$	1.00000	0.125000
$65$	−5.57356	−0.691315
$66$	−2.59957	−0.319986
$67$	−2.23127	−0.272593	−0.136296	−	0.990668i	$-0.543520\pi$
$67$	−2.23127	−0.272593	−0.136296	+	0.990668i	$0.543520\pi$
$68$	6.71697	0.814552
$69$	19.6880	2.37016
$70$	1.31461	0.157126
$71$	−12.3895	−1.47036	−0.735179	−	0.677873i	$-0.762902\pi$
$71$	−12.3895	−1.47036	−0.735179	+	0.677873i	$0.762902\pi$
$72$	3.75779	0.442860
$73$	5.06235	0.592503	0.296251	−	0.955110i	$-0.404263\pi$
$73$	5.06235	0.592503	0.296251	+	0.955110i	$0.404263\pi$
$74$	−4.21701	−0.490218
$75$	2.59957	0.300173
$76$	−4.17687	−0.479120
$77$	−1.31461	−0.149814
$78$	−14.4889	−1.64054
$79$	13.7458	1.54652	0.773259	−	0.634090i	$-0.218625\pi$
$79$	13.7458	1.54652	0.773259	+	0.634090i	$0.218625\pi$
$80$	1.00000	0.111803
$81$	−6.15238	−0.683598
$82$	8.18305	0.903667
$83$	17.1925	1.88712	0.943560	−	0.331203i	$-0.107454\pi$
$83$	17.1925	1.88712	0.943560	+	0.331203i	$0.107454\pi$
$84$	3.41742	0.372872
$85$	6.71697	0.728558
$86$	1.00000	0.107833
$87$	9.67935	1.03773
$88$	−1.00000	−0.106600
$89$	13.5657	1.43796	0.718982	−	0.695029i	$-0.244608\pi$
$89$	13.5657	1.43796	0.718982	+	0.695029i	$0.244608\pi$
$90$	3.75779	0.396106
$91$	−7.32705	−0.768084
$92$	7.57356	0.789599
$93$	13.1701	1.36567
$94$	9.44879	0.974568
$95$	−4.17687	−0.428538
$96$	2.59957	0.265318
$97$	−6.57793	−0.667887	−0.333944	−	0.942593i	$-0.608380\pi$
$97$	−6.57793	−0.667887	−0.333944	+	0.942593i	$0.608380\pi$
$98$	−5.27180	−0.532533
$99$	−3.75779	−0.377672
$100$	1.00000	0.100000
$101$	17.3922	1.73059	0.865293	−	0.501266i	$-0.167132\pi$
$101$	17.3922	1.73059	0.865293	+	0.501266i	$0.167132\pi$
$102$	17.4613	1.72892
$103$	−11.8833	−1.17090	−0.585450	−	0.810708i	$-0.699082\pi$
$103$	−11.8833	−1.17090	−0.585450	+	0.810708i	$0.699082\pi$
$104$	−5.57356	−0.546533
$105$	3.41742	0.333506
$106$	−7.39275	−0.718047
$107$	−6.38220	−0.616991	−0.308495	−	0.951226i	$-0.599825\pi$
$107$	−6.38220	−0.616991	−0.308495	+	0.951226i	$0.599825\pi$
$108$	1.96993	0.189557
$109$	−13.9123	−1.33256	−0.666279	−	0.745703i	$-0.732114\pi$
$109$	−13.9123	−1.33256	−0.666279	+	0.745703i	$0.732114\pi$
$110$	−1.00000	−0.0953463
$111$	−10.9624	−1.04051
$112$	1.31461	0.124219
$113$	−11.8699	−1.11662	−0.558312	−	0.829631i	$-0.688551\pi$
$113$	−11.8699	−1.11662	−0.558312	+	0.829631i	$0.688551\pi$
$114$	−10.8581	−1.01695
$115$	7.57356	0.706238
$116$	3.72343	0.345712
$117$	−20.9443	−1.93630
$118$	−9.36203	−0.861845
$119$	8.83019	0.809462
$120$	2.59957	0.237308
$121$	1.00000	0.0909091
$122$	−0.668250	−0.0605005
$123$	21.2725	1.91807
$124$	5.06624	0.454962
$125$	1.00000	0.0894427
$126$	4.94002	0.440092
$127$	−14.7236	−1.30651	−0.653254	−	0.757139i	$-0.726597\pi$
$127$	−14.7236	−1.30651	−0.653254	+	0.757139i	$0.726597\pi$
$128$	1.00000	0.0883883
$129$	2.59957	0.228880
$130$	−5.57356	−0.488834
$131$	4.49008	0.392300	0.196150	−	0.980574i	$-0.437156\pi$
$131$	4.49008	0.392300	0.196150	+	0.980574i	$0.437156\pi$
$132$	−2.59957	−0.226264
$133$	−5.49095	−0.476126
$134$	−2.23127	−0.192752
$135$	1.96993	0.169545
$136$	6.71697	0.575976
$137$	−9.15835	−0.782450	−0.391225	−	0.920295i	$-0.627949\pi$
$137$	−9.15835	−0.782450	−0.391225	+	0.920295i	$0.627949\pi$
$138$	19.6880	1.67596
$139$	−1.70694	−0.144781	−0.0723905	−	0.997376i	$-0.523063\pi$
$139$	−1.70694	−0.144781	−0.0723905	+	0.997376i	$0.523063\pi$
$140$	1.31461	0.111105
$141$	24.5628	2.06856
$142$	−12.3895	−1.03970
$143$	5.57356	0.466085
$144$	3.75779	0.313149
$145$	3.72343	0.309214
$146$	5.06235	0.418963
$147$	−13.7044	−1.13032
$148$	−4.21701	−0.346636
$149$	0.849654	0.0696064	0.0348032	−	0.999394i	$-0.488920\pi$
$149$	0.849654	0.0696064	0.0348032	+	0.999394i	$0.488920\pi$
$150$	2.59957	0.212254
$151$	−7.85240	−0.639019	−0.319510	−	0.947583i	$-0.603518\pi$
$151$	−7.85240	−0.639019	−0.319510	+	0.947583i	$0.603518\pi$
$152$	−4.17687	−0.338789
$153$	25.2410	2.04061
$154$	−1.31461	−0.105934
$155$	5.06624	0.406930
$156$	−14.4889	−1.16004
$157$	2.62743	0.209692	0.104846	−	0.994488i	$-0.466565\pi$
$157$	2.62743	0.209692	0.104846	+	0.994488i	$0.466565\pi$
$158$	13.7458	1.09355
$159$	−19.2180	−1.52409
$160$	1.00000	0.0790569
$161$	9.95627	0.784664
$162$	−6.15238	−0.483377
$163$	−17.3419	−1.35832	−0.679159	−	0.733991i	$-0.737655\pi$
$163$	−17.3419	−1.35832	−0.679159	+	0.733991i	$0.737655\pi$
$164$	8.18305	0.638989
$165$	−2.59957	−0.202377
$166$	17.1925	1.33439
$167$	15.9304	1.23273	0.616367	−	0.787459i	$-0.288604\pi$
$167$	15.9304	1.23273	0.616367	+	0.787459i	$0.288604\pi$
$168$	3.41742	0.263660
$169$	18.0646	1.38958
$170$	6.71697	0.515168
$171$	−15.6958	−1.20029
$172$	1.00000	0.0762493
$173$	6.45313	0.490622	0.245311	−	0.969444i	$-0.421110\pi$
$173$	6.45313	0.490622	0.245311	+	0.969444i	$0.421110\pi$
$174$	9.67935	0.733789
$175$	1.31461	0.0993751
$176$	−1.00000	−0.0753778
$177$	−24.3373	−1.82930
$178$	13.5657	1.01679
$179$	−17.1890	−1.28477	−0.642384	−	0.766383i	$-0.722055\pi$
$179$	−17.1890	−1.28477	−0.642384	+	0.766383i	$0.722055\pi$
$180$	3.75779	0.280089
$181$	−26.1524	−1.94389	−0.971947	−	0.235199i	$-0.924426\pi$
$181$	−26.1524	−1.94389	−0.971947	+	0.235199i	$0.924426\pi$
$182$	−7.32705	−0.543117
$183$	−1.73717	−0.128415
$184$	7.57356	0.558330
$185$	−4.21701	−0.310041
$186$	13.1701	0.965676
$187$	−6.71697	−0.491194
$188$	9.44879	0.689124
$189$	2.58969	0.188372
$190$	−4.17687	−0.303022
$191$	1.97353	0.142800	0.0713999	−	0.997448i	$-0.477253\pi$
$191$	1.97353	0.142800	0.0713999	+	0.997448i	$0.477253\pi$
$192$	2.59957	0.187608
$193$	−18.8219	−1.35483	−0.677414	−	0.735602i	$-0.736899\pi$
$193$	−18.8219	−1.35483	−0.677414	+	0.735602i	$0.736899\pi$
$194$	−6.57793	−0.472268
$195$	−14.4889	−1.03757
$196$	−5.27180	−0.376557
$197$	−2.64046	−0.188125	−0.0940625	−	0.995566i	$-0.529985\pi$
$197$	−2.64046	−0.188125	−0.0940625	+	0.995566i	$0.529985\pi$
$198$	−3.75779	−0.267055
$199$	−27.2645	−1.93273	−0.966365	−	0.257173i	$-0.917209\pi$
$199$	−27.2645	−1.93273	−0.966365	+	0.257173i	$0.917209\pi$
$200$	1.00000	0.0707107
$201$	−5.80035	−0.409125
$202$	17.3922	1.22371
$203$	4.89486	0.343552
$204$	17.4613	1.22253
$205$	8.18305	0.571529
$206$	−11.8833	−0.827952
$207$	28.4599	1.97810
$208$	−5.57356	−0.386457
$209$	4.17687	0.288920
$210$	3.41742	0.235825
$211$	−3.07497	−0.211690	−0.105845	−	0.994383i	$-0.533755\pi$
$211$	−3.07497	−0.211690	−0.105845	+	0.994383i	$0.533755\pi$
$212$	−7.39275	−0.507736
$213$	−32.2073	−2.20681
$214$	−6.38220	−0.436278
$215$	1.00000	0.0681994
$216$	1.96993	0.134037
$217$	6.66012	0.452119
$218$	−13.9123	−0.942261
$219$	13.1600	0.889267
$220$	−1.00000	−0.0674200
$221$	−37.4375	−2.51832
$222$	−10.9624	−0.735751
$223$	−17.0264	−1.14017	−0.570086	−	0.821585i	$-0.693090\pi$
$223$	−17.0264	−1.14017	−0.570086	+	0.821585i	$0.693090\pi$
$224$	1.31461	0.0878360
$225$	3.75779	0.250519
$226$	−11.8699	−0.789572
$227$	5.54356	0.367939	0.183969	−	0.982932i	$-0.441105\pi$
$227$	5.54356	0.367939	0.183969	+	0.982932i	$0.441105\pi$
$228$	−10.8581	−0.719094
$229$	−7.85340	−0.518967	−0.259484	−	0.965747i	$-0.583552\pi$
$229$	−7.85340	−0.518967	−0.259484	+	0.965747i	$0.583552\pi$
$230$	7.57356	0.499386
$231$	−3.41742	−0.224850
$232$	3.72343	0.244455
$233$	−19.4216	−1.27235	−0.636174	−	0.771546i	$-0.719484\pi$
$233$	−19.4216	−1.27235	−0.636174	+	0.771546i	$0.719484\pi$
$234$	−20.9443	−1.36917
$235$	9.44879	0.616371
$236$	−9.36203	−0.609416
$237$	35.7331	2.32112
$238$	8.83019	0.572376
$239$	−3.24394	−0.209833	−0.104917	−	0.994481i	$-0.533458\pi$
$239$	−3.24394	−0.209833	−0.104917	+	0.994481i	$0.533458\pi$
$240$	2.59957	0.167802
$241$	−5.23092	−0.336953	−0.168477	−	0.985706i	$-0.553885\pi$
$241$	−5.23092	−0.336953	−0.168477	+	0.985706i	$0.553885\pi$
$242$	1.00000	0.0642824
$243$	−21.9034	−1.40510
$244$	−0.668250	−0.0427803
$245$	−5.27180	−0.336803
$246$	21.2725	1.35628
$247$	23.2800	1.48127
$248$	5.06624	0.321706
$249$	44.6931	2.83231
$250$	1.00000	0.0632456
$251$	10.8220	0.683080	0.341540	−	0.939867i	$-0.389052\pi$
$251$	10.8220	0.683080	0.341540	+	0.939867i	$0.389052\pi$
$252$	4.94002	0.311192
$253$	−7.57356	−0.476146
$254$	−14.7236	−0.923841
$255$	17.4613	1.09347
$256$	1.00000	0.0625000
$257$	17.4637	1.08935	0.544677	−	0.838646i	$-0.316652\pi$
$257$	17.4637	1.08935	0.544677	+	0.838646i	$0.316652\pi$
$258$	2.59957	0.161842
$259$	−5.54372	−0.344470
$260$	−5.57356	−0.345658
$261$	13.9919	0.866076
$262$	4.49008	0.277398
$263$	5.85841	0.361245	0.180622	−	0.983552i	$-0.442189\pi$
$263$	5.85841	0.361245	0.180622	+	0.983552i	$0.442189\pi$
$264$	−2.59957	−0.159993
$265$	−7.39275	−0.454133
$266$	−5.49095	−0.336672
$267$	35.2651	2.15819
$268$	−2.23127	−0.136296
$269$	−22.5619	−1.37563	−0.687813	−	0.725888i	$-0.741429\pi$
$269$	−22.5619	−1.37563	−0.687813	+	0.725888i	$0.741429\pi$
$270$	1.96993	0.119886
$271$	0.0467252	0.00283835	0.00141918	−	0.999999i	$-0.499548\pi$
$271$	0.0467252	0.00283835	0.00141918	+	0.999999i	$0.499548\pi$
$272$	6.71697	0.407276
$273$	−19.0472	−1.15279
$274$	−9.15835	−0.553276
$275$	−1.00000	−0.0603023
$276$	19.6880	1.18508
$277$	−7.87535	−0.473184	−0.236592	−	0.971609i	$-0.576030\pi$
$277$	−7.87535	−0.473184	−0.236592	+	0.971609i	$0.576030\pi$
$278$	−1.70694	−0.102376
$279$	19.0379	1.13977
$280$	1.31461	0.0785629
$281$	−18.3908	−1.09710	−0.548551	−	0.836117i	$-0.684820\pi$
$281$	−18.3908	−1.09710	−0.548551	+	0.836117i	$0.684820\pi$
$282$	24.5628	1.46270
$283$	−26.2707	−1.56163	−0.780815	−	0.624762i	$-0.785196\pi$
$283$	−26.2707	−1.56163	−0.780815	+	0.624762i	$0.785196\pi$
$284$	−12.3895	−0.735179
$285$	−10.8581	−0.643177
$286$	5.57356	0.329572
$287$	10.7575	0.634996
$288$	3.75779	0.221430
$289$	28.1177	1.65398
$290$	3.72343	0.218648
$291$	−17.0998	−1.00241
$292$	5.06235	0.296251
$293$	24.2493	1.41666	0.708330	−	0.705881i	$-0.249449\pi$
$293$	24.2493	1.41666	0.708330	+	0.705881i	$0.249449\pi$
$294$	−13.7044	−0.799260
$295$	−9.36203	−0.545079
$296$	−4.21701	−0.245109
$297$	−1.96993	−0.114307
$298$	0.849654	0.0492191
$299$	−42.2117	−2.44117
$300$	2.59957	0.150087
$301$	1.31461	0.0757728
$302$	−7.85240	−0.451855
$303$	45.2123	2.59738
$304$	−4.17687	−0.239560
$305$	−0.668250	−0.0382639
$306$	25.2410	1.44293
$307$	3.00747	0.171646	0.0858228	−	0.996310i	$-0.472648\pi$
$307$	3.00747	0.171646	0.0858228	+	0.996310i	$0.472648\pi$
$308$	−1.31461	−0.0749068
$309$	−30.8916	−1.75736
$310$	5.06624	0.287743
$311$	15.9087	0.902098	0.451049	−	0.892499i	$-0.351050\pi$
$311$	15.9087	0.902098	0.451049	+	0.892499i	$0.351050\pi$
$312$	−14.4889	−0.820272
$313$	11.4679	0.648204	0.324102	−	0.946022i	$-0.394938\pi$
$313$	11.4679	0.648204	0.324102	+	0.946022i	$0.394938\pi$
$314$	2.62743	0.148275
$315$	4.94002	0.278339
$316$	13.7458	0.773259
$317$	19.5543	1.09828	0.549141	−	0.835730i	$-0.314955\pi$
$317$	19.5543	1.09828	0.549141	+	0.835730i	$0.314955\pi$
$318$	−19.2180	−1.07769
$319$	−3.72343	−0.208472
$320$	1.00000	0.0559017
$321$	−16.5910	−0.926020
$322$	9.95627	0.554841
$323$	−28.0559	−1.56107
$324$	−6.15238	−0.341799
$325$	−5.57356	−0.309166
$326$	−17.3419	−0.960476
$327$	−36.1661	−1.99999
$328$	8.18305	0.451834
$329$	12.4215	0.684817
$330$	−2.59957	−0.143102
$331$	28.3899	1.56045	0.780224	−	0.625500i	$-0.215105\pi$
$331$	28.3899	1.56045	0.780224	+	0.625500i	$0.215105\pi$
$332$	17.1925	0.943560
$333$	−15.8466	−0.868391
$334$	15.9304	0.871674
$335$	−2.23127	−0.121907
$336$	3.41742	0.186436
$337$	−11.4319	−0.622733	−0.311366	−	0.950290i	$-0.600787\pi$
$337$	−11.4319	−0.622733	−0.311366	+	0.950290i	$0.600787\pi$
$338$	18.0646	0.982585
$339$	−30.8566	−1.67590
$340$	6.71697	0.364279
$341$	−5.06624	−0.274352
$342$	−15.6958	−0.848732
$343$	−16.1326	−0.871080
$344$	1.00000	0.0539164
$345$	19.6880	1.05997
$346$	6.45313	0.346922
$347$	−0.216577	−0.0116265	−0.00581323	−	0.999983i	$-0.501850\pi$
$347$	−0.216577	−0.0116265	−0.00581323	+	0.999983i	$0.501850\pi$
$348$	9.67935	0.518867
$349$	28.8333	1.54341	0.771706	−	0.635979i	$-0.219404\pi$
$349$	28.8333	1.54341	0.771706	+	0.635979i	$0.219404\pi$
$350$	1.31461	0.0702688
$351$	−10.9795	−0.586045
$352$	−1.00000	−0.0533002
$353$	8.43021	0.448695	0.224347	−	0.974509i	$-0.427975\pi$
$353$	8.43021	0.448695	0.224347	+	0.974509i	$0.427975\pi$
$354$	−24.3373	−1.29351
$355$	−12.3895	−0.657564
$356$	13.5657	0.718982
$357$	22.9547	1.21489
$358$	−17.1890	−0.908468
$359$	13.8343	0.730149	0.365074	−	0.930978i	$-0.381044\pi$
$359$	13.8343	0.730149	0.365074	+	0.930978i	$0.381044\pi$
$360$	3.75779	0.198053
$361$	−1.55376	−0.0817771
$362$	−26.1524	−1.37454
$363$	2.59957	0.136442
$364$	−7.32705	−0.384042
$365$	5.06235	0.264975
$366$	−1.73717	−0.0908031
$367$	30.3158	1.58247	0.791237	−	0.611510i	$-0.209438\pi$
$367$	30.3158	1.58247	0.791237	+	0.611510i	$0.209438\pi$
$368$	7.57356	0.394799
$369$	30.7502	1.60079
$370$	−4.21701	−0.219232
$371$	−9.71857	−0.504563
$372$	13.1701	0.682836
$373$	−14.0392	−0.726923	−0.363462	−	0.931609i	$-0.618405\pi$
$373$	−14.0392	−0.726923	−0.363462	+	0.931609i	$0.618405\pi$
$374$	−6.71697	−0.347326
$375$	2.59957	0.134241
$376$	9.44879	0.487284
$377$	−20.7528	−1.06882
$378$	2.58969	0.133199
$379$	16.3978	0.842300	0.421150	−	0.906991i	$-0.361627\pi$
$379$	16.3978	0.842300	0.421150	+	0.906991i	$0.361627\pi$
$380$	−4.17687	−0.214269
$381$	−38.2751	−1.96089
$382$	1.97353	0.100975
$383$	−10.9455	−0.559287	−0.279643	−	0.960104i	$-0.590216\pi$
$383$	−10.9455	−0.559287	−0.279643	+	0.960104i	$0.590216\pi$
$384$	2.59957	0.132659
$385$	−1.31461	−0.0669987
$386$	−18.8219	−0.958008
$387$	3.75779	0.191019
$388$	−6.57793	−0.333944
$389$	5.20236	0.263770	0.131885	−	0.991265i	$-0.457897\pi$
$389$	5.20236	0.263770	0.131885	+	0.991265i	$0.457897\pi$
$390$	−14.4889	−0.733674
$391$	50.8714	2.57268
$392$	−5.27180	−0.266266
$393$	11.6723	0.588790
$394$	−2.64046	−0.133024
$395$	13.7458	0.691624
$396$	−3.75779	−0.188836
$397$	−6.79903	−0.341234	−0.170617	−	0.985337i	$-0.554576\pi$
$397$	−6.79903	−0.341234	−0.170617	+	0.985337i	$0.554576\pi$
$398$	−27.2645	−1.36665
$399$	−14.2741	−0.714600
$400$	1.00000	0.0500000
$401$	5.06500	0.252934	0.126467	−	0.991971i	$-0.459636\pi$
$401$	5.06500	0.252934	0.126467	+	0.991971i	$0.459636\pi$
$402$	−5.80035	−0.289295
$403$	−28.2370	−1.40659
$404$	17.3922	0.865293
$405$	−6.15238	−0.305714
$406$	4.89486	0.242928
$407$	4.21701	0.209030
$408$	17.4613	0.864462
$409$	20.5308	1.01518	0.507592	−	0.861598i	$-0.330536\pi$
$409$	20.5308	1.01518	0.507592	+	0.861598i	$0.330536\pi$
$410$	8.18305	0.404132
$411$	−23.8078	−1.17435
$412$	−11.8833	−0.585450
$413$	−12.3074	−0.605608
$414$	28.4599	1.39873
$415$	17.1925	0.843945
$416$	−5.57356	−0.273266
$417$	−4.43733	−0.217297
$418$	4.17687	0.204297
$419$	8.48337	0.414440	0.207220	−	0.978294i	$-0.433558\pi$
$419$	8.48337	0.414440	0.207220	+	0.978294i	$0.433558\pi$
$420$	3.41742	0.166753
$421$	0.573257	0.0279388	0.0139694	−	0.999902i	$-0.495553\pi$
$421$	0.573257	0.0279388	0.0139694	+	0.999902i	$0.495553\pi$
$422$	−3.07497	−0.149687
$423$	35.5066	1.72639
$424$	−7.39275	−0.359023
$425$	6.71697	0.325821
$426$	−32.2073	−1.56045
$427$	−0.878487	−0.0425130
$428$	−6.38220	−0.308495
$429$	14.4889	0.699530
$430$	1.00000	0.0482243
$431$	36.2286	1.74507	0.872536	−	0.488550i	$-0.162474\pi$
$431$	36.2286	1.74507	0.872536	+	0.488550i	$0.162474\pi$
$432$	1.96993	0.0947784
$433$	−18.0485	−0.867356	−0.433678	−	0.901068i	$-0.642784\pi$
$433$	−18.0485	−0.867356	−0.433678	+	0.901068i	$0.642784\pi$
$434$	6.66012	0.319696
$435$	9.67935	0.464089
$436$	−13.9123	−0.666279
$437$	−31.6338	−1.51325
$438$	13.1600	0.628807
$439$	15.7171	0.750137	0.375069	−	0.926997i	$-0.377619\pi$
$439$	15.7171	0.750137	0.375069	+	0.926997i	$0.377619\pi$
$440$	−1.00000	−0.0476731
$441$	−19.8103	−0.943349
$442$	−37.4375	−1.78072
$443$	30.9404	1.47002	0.735012	−	0.678054i	$-0.237176\pi$
$443$	30.9404	1.47002	0.735012	+	0.678054i	$0.237176\pi$
$444$	−10.9624	−0.520254
$445$	13.5657	0.643077
$446$	−17.0264	−0.806223
$447$	2.20874	0.104470
$448$	1.31461	0.0621094
$449$	−10.3368	−0.487823	−0.243911	−	0.969798i	$-0.578431\pi$
$449$	−10.3368	−0.487823	−0.243911	+	0.969798i	$0.578431\pi$
$450$	3.75779	0.177144
$451$	−8.18305	−0.385325
$452$	−11.8699	−0.558312
$453$	−20.4129	−0.959082
$454$	5.54356	0.260172
$455$	−7.32705	−0.343498
$456$	−10.8581	−0.508476
$457$	11.5843	0.541889	0.270945	−	0.962595i	$-0.412664\pi$
$457$	11.5843	0.541889	0.270945	+	0.962595i	$0.412664\pi$
$458$	−7.85340	−0.366965
$459$	13.2320	0.617616
$460$	7.57356	0.353119
$461$	−17.0675	−0.794915	−0.397457	−	0.917621i	$-0.630107\pi$
$461$	−17.0675	−0.794915	−0.397457	+	0.917621i	$0.630107\pi$
$462$	−3.41742	−0.158993
$463$	−36.7991	−1.71020	−0.855099	−	0.518464i	$-0.826504\pi$
$463$	−36.7991	−1.71020	−0.855099	+	0.518464i	$0.826504\pi$
$464$	3.72343	0.172856
$465$	13.1701	0.610747
$466$	−19.4216	−0.899686
$467$	−32.8303	−1.51920	−0.759602	−	0.650388i	$-0.774606\pi$
$467$	−32.8303	−1.51920	−0.759602	+	0.650388i	$0.774606\pi$
$468$	−20.9443	−0.968150
$469$	−2.93325	−0.135445
$470$	9.44879	0.435840
$471$	6.83021	0.314719
$472$	−9.36203	−0.430922
$473$	−1.00000	−0.0459800
$474$	35.7331	1.64128
$475$	−4.17687	−0.191648
$476$	8.83019	0.404731
$477$	−27.7804	−1.27198
$478$	−3.24394	−0.148374
$479$	−17.6284	−0.805464	−0.402732	−	0.915318i	$-0.631939\pi$
$479$	−17.6284	−0.805464	−0.402732	+	0.915318i	$0.631939\pi$
$480$	2.59957	0.118654
$481$	23.5038	1.07168
$482$	−5.23092	−0.238262
$483$	25.8821	1.17768
$484$	1.00000	0.0454545
$485$	−6.57793	−0.298688
$486$	−21.9034	−0.993557
$487$	14.3735	0.651327	0.325663	−	0.945486i	$-0.394412\pi$
$487$	14.3735	0.651327	0.325663	+	0.945486i	$0.394412\pi$
$488$	−0.668250	−0.0302503
$489$	−45.0814	−2.03865
$490$	−5.27180	−0.238156
$491$	0.294792	0.0133038	0.00665188	−	0.999978i	$-0.497883\pi$
$491$	0.294792	0.0133038	0.00665188	+	0.999978i	$0.497883\pi$
$492$	21.2725	0.959037
$493$	25.0102	1.12640
$494$	23.2800	1.04742
$495$	−3.75779	−0.168900
$496$	5.06624	0.227481
$497$	−16.2873	−0.730584
$498$	44.6931	2.00275
$499$	−21.3048	−0.953736	−0.476868	−	0.878975i	$-0.658228\pi$
$499$	−21.3048	−0.953736	−0.476868	+	0.878975i	$0.658228\pi$
$500$	1.00000	0.0447214
$501$	41.4123	1.85017
$502$	10.8220	0.483010
$503$	32.0649	1.42970	0.714852	−	0.699276i	$-0.246494\pi$
$503$	32.0649	1.42970	0.714852	+	0.699276i	$0.246494\pi$
$504$	4.94002	0.220046
$505$	17.3922	0.773942
$506$	−7.57356	−0.336686
$507$	46.9603	2.08558
$508$	−14.7236	−0.653254
$509$	25.5094	1.13068	0.565342	−	0.824857i	$-0.308744\pi$
$509$	25.5094	1.13068	0.565342	+	0.824857i	$0.308744\pi$
$510$	17.4613	0.773198
$511$	6.65501	0.294400
$512$	1.00000	0.0441942
$513$	−8.22815	−0.363282
$514$	17.4637	0.770289
$515$	−11.8833	−0.523643
$516$	2.59957	0.114440
$517$	−9.44879	−0.415557
$518$	−5.54372	−0.243577
$519$	16.7754	0.736358
$520$	−5.57356	−0.244417
$521$	−22.9894	−1.00718	−0.503592	−	0.863942i	$-0.667988\pi$
$521$	−22.9894	−1.00718	−0.503592	+	0.863942i	$0.667988\pi$
$522$	13.9919	0.612408
$523$	0.126856	0.00554702	0.00277351	−	0.999996i	$-0.499117\pi$
$523$	0.126856	0.00554702	0.00277351	+	0.999996i	$0.499117\pi$
$524$	4.49008	0.196150
$525$	3.41742	0.149149
$526$	5.85841	0.255439
$527$	34.0298	1.48236
$528$	−2.59957	−0.113132
$529$	34.3589	1.49386
$530$	−7.39275	−0.321120
$531$	−35.1806	−1.52671
$532$	−5.49095	−0.238063
$533$	−45.6088	−1.97554
$534$	35.2651	1.52607
$535$	−6.38220	−0.275927
$536$	−2.23127	−0.0963762
$537$	−44.6842	−1.92826
$538$	−22.5619	−0.972714
$539$	5.27180	0.227073
$540$	1.96993	0.0847724
$541$	17.5008	0.752418	0.376209	−	0.926535i	$-0.377228\pi$
$541$	17.5008	0.752418	0.376209	+	0.926535i	$0.377228\pi$
$542$	0.0467252	0.00200702
$543$	−67.9852	−2.91752
$544$	6.71697	0.287988
$545$	−13.9123	−0.595938
$546$	−19.0472	−0.815146
$547$	22.3258	0.954584	0.477292	−	0.878745i	$-0.341618\pi$
$547$	22.3258	0.954584	0.477292	+	0.878745i	$0.341618\pi$
$548$	−9.15835	−0.391225
$549$	−2.51114	−0.107173
$550$	−1.00000	−0.0426401
$551$	−15.5523	−0.662550
$552$	19.6880	0.837979
$553$	18.0703	0.768427
$554$	−7.87535	−0.334591
$555$	−10.9624	−0.465330
$556$	−1.70694	−0.0723905
$557$	29.4251	1.24678	0.623391	−	0.781910i	$-0.285754\pi$
$557$	29.4251	1.24678	0.623391	+	0.781910i	$0.285754\pi$
$558$	19.0379	0.805937
$559$	−5.57356	−0.235737
$560$	1.31461	0.0555524
$561$	−17.4613	−0.737215
$562$	−18.3908	−0.775769
$563$	−28.9048	−1.21819	−0.609096	−	0.793097i	$-0.708468\pi$
$563$	−28.9048	−1.21819	−0.609096	+	0.793097i	$0.708468\pi$
$564$	24.5628	1.03428
$565$	−11.8699	−0.499369
$566$	−26.2707	−1.10424
$567$	−8.08798	−0.339663
$568$	−12.3895	−0.519850
$569$	−21.5069	−0.901614	−0.450807	−	0.892621i	$-0.648864\pi$
$569$	−21.5069	−0.901614	−0.450807	+	0.892621i	$0.648864\pi$
$570$	−10.8581	−0.454795
$571$	−35.2843	−1.47660	−0.738301	−	0.674471i	$-0.764372\pi$
$571$	−35.2843	−1.47660	−0.738301	+	0.674471i	$0.764372\pi$
$572$	5.57356	0.233042
$573$	5.13035	0.214323
$574$	10.7575	0.449010
$575$	7.57356	0.315839
$576$	3.75779	0.156575
$577$	15.7311	0.654893	0.327446	−	0.944870i	$-0.393812\pi$
$577$	15.7311	0.654893	0.327446	+	0.944870i	$0.393812\pi$
$578$	28.1177	1.16954
$579$	−48.9289	−2.03341
$580$	3.72343	0.154607
$581$	22.6014	0.937663
$582$	−17.0998	−0.708810
$583$	7.39275	0.306176
$584$	5.06235	0.209481
$585$	−20.9443	−0.865939
$586$	24.2493	1.00173
$587$	−18.1412	−0.748770	−0.374385	−	0.927273i	$-0.622146\pi$
$587$	−18.1412	−0.748770	−0.374385	+	0.927273i	$0.622146\pi$
$588$	−13.7044	−0.565162
$589$	−21.1610	−0.871925
$590$	−9.36203	−0.385429
$591$	−6.86407	−0.282350
$592$	−4.21701	−0.173318
$593$	−35.2994	−1.44957	−0.724785	−	0.688975i	$-0.758061\pi$
$593$	−35.2994	−1.44957	−0.724785	+	0.688975i	$0.758061\pi$
$594$	−1.96993	−0.0808273
$595$	8.83019	0.362002
$596$	0.849654	0.0348032
$597$	−70.8762	−2.90077
$598$	−42.2117	−1.72617
$599$	29.7558	1.21579	0.607894	−	0.794018i	$-0.292014\pi$
$599$	29.7558	1.21579	0.607894	+	0.794018i	$0.292014\pi$
$600$	2.59957	0.106127
$601$	−25.6905	−1.04794	−0.523968	−	0.851738i	$-0.675549\pi$
$601$	−25.6905	−1.04794	−0.523968	+	0.851738i	$0.675549\pi$
$602$	1.31461	0.0535795
$603$	−8.38464	−0.341449
$604$	−7.85240	−0.319510
$605$	1.00000	0.0406558
$606$	45.2123	1.83662
$607$	−24.5178	−0.995146	−0.497573	−	0.867422i	$-0.665775\pi$
$607$	−24.5178	−0.995146	−0.497573	+	0.867422i	$0.665775\pi$
$608$	−4.17687	−0.169394
$609$	12.7246	0.515625
$610$	−0.668250	−0.0270567
$611$	−52.6634	−2.13053
$612$	25.2410	1.02031
$613$	21.9544	0.886732	0.443366	−	0.896341i	$-0.353784\pi$
$613$	21.9544	0.886732	0.443366	+	0.896341i	$0.353784\pi$
$614$	3.00747	0.121372
$615$	21.2725	0.857789
$616$	−1.31461	−0.0529671
$617$	−2.80351	−0.112865	−0.0564325	−	0.998406i	$-0.517973\pi$
$617$	−2.80351	−0.112865	−0.0564325	+	0.998406i	$0.517973\pi$
$618$	−30.8916	−1.24264
$619$	−42.9736	−1.72725	−0.863627	−	0.504132i	$-0.831812\pi$
$619$	−42.9736	−1.72725	−0.863627	+	0.504132i	$0.831812\pi$
$620$	5.06624	0.203465
$621$	14.9194	0.598695
$622$	15.9087	0.637879
$623$	17.8336	0.714489
$624$	−14.4889	−0.580020
$625$	1.00000	0.0400000
$626$	11.4679	0.458349
$627$	10.8581	0.433630
$628$	2.62743	0.104846
$629$	−28.3256	−1.12941
$630$	4.94002	0.196815
$631$	47.3795	1.88615	0.943074	−	0.332584i	$-0.107921\pi$
$631$	47.3795	1.88615	0.943074	+	0.332584i	$0.107921\pi$
$632$	13.7458	0.546777
$633$	−7.99362	−0.317718
$634$	19.5543	0.776602
$635$	−14.7236	−0.584288
$636$	−19.2180	−0.762043
$637$	29.3827	1.16419
$638$	−3.72343	−0.147412
$639$	−46.5570	−1.84177
$640$	1.00000	0.0395285
$641$	32.3462	1.27760	0.638799	−	0.769374i	$-0.279432\pi$
$641$	32.3462	1.27760	0.638799	+	0.769374i	$0.279432\pi$
$642$	−16.5910	−0.654795
$643$	−16.7661	−0.661191	−0.330595	−	0.943773i	$-0.607250\pi$
$643$	−16.7661	−0.661191	−0.330595	+	0.943773i	$0.607250\pi$
$644$	9.95627	0.392332
$645$	2.59957	0.102358
$646$	−28.0559	−1.10385
$647$	−4.18826	−0.164657	−0.0823287	−	0.996605i	$-0.526236\pi$
$647$	−4.18826	−0.164657	−0.0823287	+	0.996605i	$0.526236\pi$
$648$	−6.15238	−0.241688
$649$	9.36203	0.367492
$650$	−5.57356	−0.218613
$651$	17.3135	0.678569
$652$	−17.3419	−0.679159
$653$	−33.8154	−1.32330	−0.661650	−	0.749813i	$-0.730143\pi$
$653$	−33.8154	−1.32330	−0.661650	+	0.749813i	$0.730143\pi$
$654$	−36.1661	−1.41421
$655$	4.49008	0.175442
$656$	8.18305	0.319495
$657$	19.0232	0.742167
$658$	12.4215	0.484239
$659$	13.8008	0.537603	0.268801	−	0.963196i	$-0.413373\pi$
$659$	13.8008	0.537603	0.268801	+	0.963196i	$0.413373\pi$
$660$	−2.59957	−0.101188
$661$	37.5014	1.45864	0.729318	−	0.684175i	$-0.239837\pi$
$661$	37.5014	1.45864	0.729318	+	0.684175i	$0.239837\pi$
$662$	28.3899	1.10340
$663$	−97.3215	−3.77965
$664$	17.1925	0.667197
$665$	−5.49095	−0.212930
$666$	−15.8466	−0.614045
$667$	28.1997	1.09190
$668$	15.9304	0.616367
$669$	−44.2614	−1.71124
$670$	−2.23127	−0.0862015
$671$	0.668250	0.0257975
$672$	3.41742	0.131830
$673$	5.07386	0.195583	0.0977914	−	0.995207i	$-0.468822\pi$
$673$	5.07386	0.195583	0.0977914	+	0.995207i	$0.468822\pi$
$674$	−11.4319	−0.440339
$675$	1.96993	0.0758227
$676$	18.0646	0.694792
$677$	−38.5192	−1.48041	−0.740207	−	0.672379i	$-0.765272\pi$
$677$	−38.5192	−1.48041	−0.740207	+	0.672379i	$0.765272\pi$
$678$	−30.8566	−1.18504
$679$	−8.64740	−0.331857
$680$	6.71697	0.257584
$681$	14.4109	0.552227
$682$	−5.06624	−0.193996
$683$	−28.0937	−1.07498	−0.537488	−	0.843271i	$-0.680627\pi$
$683$	−28.0937	−1.07498	−0.537488	+	0.843271i	$0.680627\pi$
$684$	−15.6958	−0.600144
$685$	−9.15835	−0.349922
$686$	−16.1326	−0.615946
$687$	−20.4155	−0.778900
$688$	1.00000	0.0381246
$689$	41.2039	1.56974
$690$	19.6880	0.749511
$691$	11.9943	0.456284	0.228142	−	0.973628i	$-0.426735\pi$
$691$	11.9943	0.456284	0.228142	+	0.973628i	$0.426735\pi$
$692$	6.45313	0.245311
$693$	−4.94002	−0.187656
$694$	−0.216577	−0.00822115
$695$	−1.70694	−0.0647480
$696$	9.67935	0.366895
$697$	54.9653	2.08196
$698$	28.8333	1.09136
$699$	−50.4878	−1.90962
$700$	1.31461	0.0496875
$701$	3.21670	0.121493	0.0607465	−	0.998153i	$-0.480652\pi$
$701$	3.21670	0.121493	0.0607465	+	0.998153i	$0.480652\pi$
$702$	−10.9795	−0.414396
$703$	17.6139	0.664321
$704$	−1.00000	−0.0376889
$705$	24.5628	0.925090
$706$	8.43021	0.317275
$707$	22.8639	0.859886
$708$	−24.3373	−0.914652
$709$	32.6871	1.22759	0.613794	−	0.789466i	$-0.289642\pi$
$709$	32.6871	1.22759	0.613794	+	0.789466i	$0.289642\pi$
$710$	−12.3895	−0.464968
$711$	51.6536	1.93716
$712$	13.5657	0.508397
$713$	38.3695	1.43695
$714$	22.9547	0.859060
$715$	5.57356	0.208439
$716$	−17.1890	−0.642384
$717$	−8.43287	−0.314931
$718$	13.8343	0.516293
$719$	49.1563	1.83322	0.916611	−	0.399781i	$-0.130914\pi$
$719$	49.1563	1.83322	0.916611	+	0.399781i	$0.130914\pi$
$720$	3.75779	0.140045
$721$	−15.6219	−0.581792
$722$	−1.55376	−0.0578251
$723$	−13.5982	−0.505721
$724$	−26.1524	−0.971947
$725$	3.72343	0.138285
$726$	2.59957	0.0964793
$727$	13.6910	0.507770	0.253885	−	0.967234i	$-0.418291\pi$
$727$	13.6910	0.507770	0.253885	+	0.967234i	$0.418291\pi$
$728$	−7.32705	−0.271559
$729$	−38.4823	−1.42527
$730$	5.06235	0.187366
$731$	6.71697	0.248436
$732$	−1.73717	−0.0642075
$733$	−19.3546	−0.714877	−0.357438	−	0.933937i	$-0.616350\pi$
$733$	−19.3546	−0.714877	−0.357438	+	0.933937i	$0.616350\pi$
$734$	30.3158	1.11898
$735$	−13.7044	−0.505496
$736$	7.57356	0.279165
$737$	2.23127	0.0821899
$738$	30.7502	1.13193
$739$	−11.7315	−0.431550	−0.215775	−	0.976443i	$-0.569228\pi$
$739$	−11.7315	−0.431550	−0.215775	+	0.976443i	$0.569228\pi$
$740$	−4.21701	−0.155020
$741$	60.5182	2.22319
$742$	−9.71857	−0.356780
$743$	−5.47258	−0.200770	−0.100385	−	0.994949i	$-0.532007\pi$
$743$	−5.47258	−0.200770	−0.100385	+	0.994949i	$0.532007\pi$
$744$	13.1701	0.482838
$745$	0.849654	0.0311289
$746$	−14.0392	−0.514012
$747$	64.6057	2.36380
$748$	−6.71697	−0.245597
$749$	−8.39010	−0.306567
$750$	2.59957	0.0949231
$751$	53.8834	1.96623	0.983117	−	0.182980i	$-0.0585743\pi$
$751$	53.8834	1.96623	0.983117	+	0.182980i	$0.0585743\pi$
$752$	9.44879	0.344562
$753$	28.1327	1.02521
$754$	−20.7528	−0.755772
$755$	−7.85240	−0.285778
$756$	2.58969	0.0941861
$757$	46.0984	1.67547	0.837737	−	0.546074i	$-0.183878\pi$
$757$	46.0984	1.67547	0.837737	+	0.546074i	$0.183878\pi$
$758$	16.3978	0.595596
$759$	−19.6880	−0.714631
$760$	−4.17687	−0.151511
$761$	19.9270	0.722353	0.361177	−	0.932497i	$-0.382375\pi$
$761$	19.9270	0.722353	0.361177	+	0.932497i	$0.382375\pi$
$762$	−38.2751	−1.38656
$763$	−18.2892	−0.662115
$764$	1.97353	0.0713999
$765$	25.2410	0.912589
$766$	−10.9455	−0.395475
$767$	52.1799	1.88411
$768$	2.59957	0.0938041
$769$	−24.2110	−0.873070	−0.436535	−	0.899687i	$-0.643795\pi$
$769$	−24.2110	−0.873070	−0.436535	+	0.899687i	$0.643795\pi$
$770$	−1.31461	−0.0473752
$771$	45.3981	1.63497
$772$	−18.8219	−0.677414
$773$	19.7406	0.710019	0.355009	−	0.934863i	$-0.384478\pi$
$773$	19.7406	0.710019	0.355009	+	0.934863i	$0.384478\pi$
$774$	3.75779	0.135071
$775$	5.06624	0.181985
$776$	−6.57793	−0.236134
$777$	−14.4113	−0.517003
$778$	5.20236	0.186514
$779$	−34.1795	−1.22461
$780$	−14.4889	−0.518786
$781$	12.3895	0.443329
$782$	50.8714	1.81916
$783$	7.33491	0.262128
$784$	−5.27180	−0.188279
$785$	2.62743	0.0937771
$786$	11.6723	0.416337
$787$	−33.9623	−1.21063	−0.605313	−	0.795988i	$-0.706952\pi$
$787$	−33.9623	−1.21063	−0.605313	+	0.795988i	$0.706952\pi$
$788$	−2.64046	−0.0940625
$789$	15.2294	0.542180
$790$	13.7458	0.489052
$791$	−15.6042	−0.554823
$792$	−3.75779	−0.133527
$793$	3.72453	0.132262
$794$	−6.79903	−0.241289
$795$	−19.2180	−0.681592
$796$	−27.2645	−0.966365
$797$	−18.5917	−0.658552	−0.329276	−	0.944234i	$-0.606805\pi$
$797$	−18.5917	−0.658552	−0.329276	+	0.944234i	$0.606805\pi$
$798$	−14.2741	−0.505299
$799$	63.4673	2.24531
$800$	1.00000	0.0353553
$801$	50.9771	1.80119
$802$	5.06500	0.178851
$803$	−5.06235	−0.178646
$804$	−5.80035	−0.204563
$805$	9.95627	0.350912
$806$	−28.2370	−0.994606
$807$	−58.6515	−2.06463
$808$	17.3922	0.611855
$809$	17.5314	0.616372	0.308186	−	0.951326i	$-0.400278\pi$
$809$	17.5314	0.616372	0.308186	+	0.951326i	$0.400278\pi$
$810$	−6.15238	−0.216173
$811$	28.8875	1.01438	0.507188	−	0.861836i	$-0.330685\pi$
$811$	28.8875	1.01438	0.507188	+	0.861836i	$0.330685\pi$
$812$	4.89486	0.171776
$813$	0.121466	0.00425999
$814$	4.21701	0.147806
$815$	−17.3419	−0.607459
$816$	17.4613	0.611267
$817$	−4.17687	−0.146130
$818$	20.5308	0.717843
$819$	−27.5335	−0.962099
$820$	8.18305	0.285765
$821$	−2.11062	−0.0736611	−0.0368305	−	0.999322i	$-0.511726\pi$
$821$	−2.11062	−0.0736611	−0.0368305	+	0.999322i	$0.511726\pi$
$822$	−23.8078	−0.830393
$823$	−20.3993	−0.711074	−0.355537	−	0.934662i	$-0.615702\pi$
$823$	−20.3993	−0.711074	−0.355537	+	0.934662i	$0.615702\pi$
$824$	−11.8833	−0.413976
$825$	−2.59957	−0.0905056
$826$	−12.3074	−0.428229
$827$	27.7088	0.963528	0.481764	−	0.876301i	$-0.339996\pi$
$827$	27.7088	0.963528	0.481764	+	0.876301i	$0.339996\pi$
$828$	28.4599	0.989048
$829$	−43.7080	−1.51804	−0.759021	−	0.651066i	$-0.774322\pi$
$829$	−43.7080	−1.51804	−0.759021	+	0.651066i	$0.774322\pi$
$830$	17.1925	0.596760
$831$	−20.4726	−0.710185
$832$	−5.57356	−0.193229
$833$	−35.4106	−1.22690
$834$	−4.43733	−0.153652
$835$	15.9304	0.551295
$836$	4.17687	0.144460
$837$	9.98015	0.344964
$838$	8.48337	0.293053
$839$	−50.1817	−1.73247	−0.866233	−	0.499640i	$-0.833466\pi$
$839$	−50.1817	−1.73247	−0.866233	+	0.499640i	$0.833466\pi$
$840$	3.41742	0.117912
$841$	−15.1360	−0.521932
$842$	0.573257	0.0197557
$843$	−47.8082	−1.64660
$844$	−3.07497	−0.105845
$845$	18.0646	0.621441
$846$	35.5066	1.22074
$847$	1.31461	0.0451705
$848$	−7.39275	−0.253868
$849$	−68.2926	−2.34380
$850$	6.71697	0.230390
$851$	−31.9378	−1.09481
$852$	−32.2073	−1.10340
$853$	13.2315	0.453037	0.226518	−	0.974007i	$-0.427266\pi$
$853$	13.2315	0.453037	0.226518	+	0.974007i	$0.427266\pi$
$854$	−0.878487	−0.0300612
$855$	−15.6958	−0.536785
$856$	−6.38220	−0.218139
$857$	−8.15827	−0.278681	−0.139341	−	0.990245i	$-0.544498\pi$
$857$	−8.15827	−0.278681	−0.139341	+	0.990245i	$0.544498\pi$
$858$	14.4889	0.494643
$859$	41.8387	1.42752	0.713759	−	0.700391i	$-0.246991\pi$
$859$	41.8387	1.42752	0.713759	+	0.700391i	$0.246991\pi$
$860$	1.00000	0.0340997
$861$	27.9650	0.953043
$862$	36.2286	1.23395
$863$	−56.7578	−1.93206	−0.966029	−	0.258432i	$-0.916794\pi$
$863$	−56.7578	−1.93206	−0.966029	+	0.258432i	$0.916794\pi$
$864$	1.96993	0.0670185
$865$	6.45313	0.219413
$866$	−18.0485	−0.613313
$867$	73.0941	2.48241
$868$	6.66012	0.226059
$869$	−13.7458	−0.466293
$870$	9.67935	0.328161
$871$	12.4361	0.421382
$872$	−13.9123	−0.471130
$873$	−24.7185	−0.836594
$874$	−31.6338	−1.07003
$875$	1.31461	0.0444419
$876$	13.1600	0.444634
$877$	47.5158	1.60449	0.802247	−	0.596992i	$-0.203637\pi$
$877$	47.5158	1.60449	0.802247	+	0.596992i	$0.203637\pi$
$878$	15.7171	0.530427
$879$	63.0379	2.12622
$880$	−1.00000	−0.0337100
$881$	45.6699	1.53866	0.769328	−	0.638854i	$-0.220591\pi$
$881$	45.6699	1.53866	0.769328	+	0.638854i	$0.220591\pi$
$882$	−19.8103	−0.667049
$883$	56.9027	1.91493	0.957464	−	0.288553i	$-0.0931743\pi$
$883$	56.9027	1.91493	0.957464	+	0.288553i	$0.0931743\pi$
$884$	−37.4375	−1.25916
$885$	−24.3373	−0.818090
$886$	30.9404	1.03946
$887$	23.3035	0.782456	0.391228	−	0.920294i	$-0.372050\pi$
$887$	23.3035	0.782456	0.391228	+	0.920294i	$0.372050\pi$
$888$	−10.9624	−0.367875
$889$	−19.3558	−0.649172
$890$	13.5657	0.454724
$891$	6.15238	0.206113
$892$	−17.0264	−0.570086
$893$	−39.4664	−1.32069
$894$	2.20874	0.0738713
$895$	−17.1890	−0.574566
$896$	1.31461	0.0439180
$897$	−109.733	−3.66386
$898$	−10.3368	−0.344943
$899$	18.8638	0.629143
$900$	3.75779	0.125260
$901$	−49.6569	−1.65431
$902$	−8.18305	−0.272466
$903$	3.41742	0.113725
$904$	−11.8699	−0.394786
$905$	−26.1524	−0.869336
$906$	−20.4129	−0.678173
$907$	32.4461	1.07736	0.538678	−	0.842512i	$-0.318924\pi$
$907$	32.4461	1.07736	0.538678	+	0.842512i	$0.318924\pi$
$908$	5.54356	0.183969
$909$	65.3562	2.16773
$910$	−7.32705	−0.242889
$911$	−41.2146	−1.36550	−0.682750	−	0.730652i	$-0.739216\pi$
$911$	−41.2146	−1.36550	−0.682750	+	0.730652i	$0.739216\pi$
$912$	−10.8581	−0.359547
$913$	−17.1925	−0.568988
$914$	11.5843	0.383174
$915$	−1.73717	−0.0574289
$916$	−7.85340	−0.259484
$917$	5.90270	0.194924
$918$	13.2320	0.436720
$919$	−36.9652	−1.21937	−0.609684	−	0.792644i	$-0.708704\pi$
$919$	−36.9652	−1.21937	−0.609684	+	0.792644i	$0.708704\pi$
$920$	7.57356	0.249693
$921$	7.81815	0.257617
$922$	−17.0675	−0.562090
$923$	69.0534	2.27292
$924$	−3.41742	−0.112425
$925$	−4.21701	−0.138655
$926$	−36.7991	−1.20929
$927$	−44.6551	−1.46667
$928$	3.72343	0.122228
$929$	−8.40193	−0.275659	−0.137829	−	0.990456i	$-0.544013\pi$
$929$	−8.40193	−0.275659	−0.137829	+	0.990456i	$0.544013\pi$
$930$	13.1701	0.431864
$931$	22.0196	0.721664
$932$	−19.4216	−0.636174
$933$	41.3558	1.35393
$934$	−32.8303	−1.07424
$935$	−6.71697	−0.219668
$936$	−20.9443	−0.684585
$937$	0.466767	0.0152486	0.00762431	−	0.999971i	$-0.497573\pi$
$937$	0.466767	0.0152486	0.00762431	+	0.999971i	$0.497573\pi$
$938$	−2.93325	−0.0957739
$939$	29.8117	0.972866
$940$	9.44879	0.308186
$941$	−16.4977	−0.537809	−0.268905	−	0.963167i	$-0.586662\pi$
$941$	−16.4977	−0.537809	−0.268905	+	0.963167i	$0.586662\pi$
$942$	6.83021	0.222540
$943$	61.9749	2.01818
$944$	−9.36203	−0.304708
$945$	2.58969	0.0842426
$946$	−1.00000	−0.0325128
$947$	−14.0154	−0.455440	−0.227720	−	0.973727i	$-0.573127\pi$
$947$	−14.0154	−0.455440	−0.227720	+	0.973727i	$0.573127\pi$
$948$	35.7331	1.16056
$949$	−28.2153	−0.915908
$950$	−4.17687	−0.135516
$951$	50.8330	1.64837
$952$	8.83019	0.286188
$953$	9.28262	0.300694	0.150347	−	0.988633i	$-0.451961\pi$
$953$	9.28262	0.300694	0.150347	+	0.988633i	$0.451961\pi$
$954$	−27.7804	−0.899423
$955$	1.97353	0.0638620
$956$	−3.24394	−0.104917
$957$	−9.67935	−0.312889
$958$	−17.6284	−0.569549
$959$	−12.0396	−0.388780
$960$	2.59957	0.0839009
$961$	−5.33322	−0.172039
$962$	23.5038	0.757792
$963$	−23.9830	−0.772840
$964$	−5.23092	−0.168477
$965$	−18.8219	−0.605898
$966$	25.8821	0.832742
$967$	19.6189	0.630903	0.315451	−	0.948942i	$-0.397844\pi$
$967$	19.6189	0.630903	0.315451	+	0.948942i	$0.397844\pi$
$968$	1.00000	0.0321412
$969$	−72.9334	−2.34296
$970$	−6.57793	−0.211205
$971$	−32.6615	−1.04816	−0.524079	−	0.851670i	$-0.675590\pi$
$971$	−32.6615	−1.04816	−0.524079	+	0.851670i	$0.675590\pi$
$972$	−21.9034	−0.702551
$973$	−2.24396	−0.0719381
$974$	14.3735	0.460558
$975$	−14.4889	−0.464016
$976$	−0.668250	−0.0213902
$977$	36.8075	1.17758	0.588788	−	0.808287i	$-0.299605\pi$
$977$	36.8075	1.17758	0.588788	+	0.808287i	$0.299605\pi$
$978$	−45.0814	−1.44155
$979$	−13.5657	−0.433562
$980$	−5.27180	−0.168402
$981$	−52.2795	−1.66916
$982$	0.294792	0.00940718
$983$	52.4941	1.67430	0.837151	−	0.546972i	$-0.184220\pi$
$983$	52.4941	1.67430	0.837151	+	0.546972i	$0.184220\pi$
$984$	21.2725	0.678141
$985$	−2.64046	−0.0841320
$986$	25.0102	0.796487
$987$	32.2905	1.02782
$988$	23.2800	0.740637
$989$	7.57356	0.240825
$990$	−3.75779	−0.119430
$991$	−6.94321	−0.220558	−0.110279	−	0.993901i	$-0.535174\pi$
$991$	−6.94321	−0.220558	−0.110279	+	0.993901i	$0.535174\pi$
$992$	5.06624	0.160853
$993$	73.8016	2.34202
$994$	−16.2873	−0.516601
$995$	−27.2645	−0.864343
$996$	44.6931	1.41616
$997$	−45.2883	−1.43429	−0.717147	−	0.696922i	$-0.754553\pi$
$997$	−45.2883	−1.43429	−0.717147	+	0.696922i	$0.754553\pi$
$998$	−21.3048	−0.674393
$999$	−8.30723	−0.262829

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4730.2.a.be.1.10	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4730.2.a.be.1.10	✓	12	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:	\( N \)	\(=\)	\( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4730.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.59957 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.59957 q^{6} +1.31461 q^{7} +1.00000 q^{8} +3.75779 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.59957 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.59957 q^{6} +1.31461 q^{7} +1.00000 q^{8} +3.75779 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.59957 q^{12} -5.57356 q^{13} +1.31461 q^{14} +2.59957 q^{15} +1.00000 q^{16} +6.71697 q^{17} +3.75779 q^{18} -4.17687 q^{19} +1.00000 q^{20} +3.41742 q^{21} -1.00000 q^{22} +7.57356 q^{23} +2.59957 q^{24} +1.00000 q^{25} -5.57356 q^{26} +1.96993 q^{27} +1.31461 q^{28} +3.72343 q^{29} +2.59957 q^{30} +5.06624 q^{31} +1.00000 q^{32} -2.59957 q^{33} +6.71697 q^{34} +1.31461 q^{35} +3.75779 q^{36} -4.21701 q^{37} -4.17687 q^{38} -14.4889 q^{39} +1.00000 q^{40} +8.18305 q^{41} +3.41742 q^{42} +1.00000 q^{43} -1.00000 q^{44} +3.75779 q^{45} +7.57356 q^{46} +9.44879 q^{47} +2.59957 q^{48} -5.27180 q^{49} +1.00000 q^{50} +17.4613 q^{51} -5.57356 q^{52} -7.39275 q^{53} +1.96993 q^{54} -1.00000 q^{55} +1.31461 q^{56} -10.8581 q^{57} +3.72343 q^{58} -9.36203 q^{59} +2.59957 q^{60} -0.668250 q^{61} +5.06624 q^{62} +4.94002 q^{63} +1.00000 q^{64} -5.57356 q^{65} -2.59957 q^{66} -2.23127 q^{67} +6.71697 q^{68} +19.6880 q^{69} +1.31461 q^{70} -12.3895 q^{71} +3.75779 q^{72} +5.06235 q^{73} -4.21701 q^{74} +2.59957 q^{75} -4.17687 q^{76} -1.31461 q^{77} -14.4889 q^{78} +13.7458 q^{79} +1.00000 q^{80} -6.15238 q^{81} +8.18305 q^{82} +17.1925 q^{83} +3.41742 q^{84} +6.71697 q^{85} +1.00000 q^{86} +9.67935 q^{87} -1.00000 q^{88} +13.5657 q^{89} +3.75779 q^{90} -7.32705 q^{91} +7.57356 q^{92} +13.1701 q^{93} +9.44879 q^{94} -4.17687 q^{95} +2.59957 q^{96} -6.57793 q^{97} -5.27180 q^{98} -3.75779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 12 q^{5} + 3 q^{6} + 8 q^{7} + 12 q^{8} + 25 q^{9}+O(q^{10}) \) 12 * q + 12 * q^2 + 3 * q^3 + 12 * q^4 + 12 * q^5 + 3 * q^6 + 8 * q^7 + 12 * q^8 + 25 * q^9	\( 12 q + 12 q^{2} + 3 q^{3} + 12 q^{4} + 12 q^{5} + 3 q^{6} + 8 q^{7} + 12 q^{8} + 25 q^{9} + 12 q^{10} - 12 q^{11} + 3 q^{12} + 16 q^{13} + 8 q^{14} + 3 q^{15} + 12 q^{16} + 18 q^{17} + 25 q^{18} - 4 q^{19} + 12 q^{20} + 4 q^{21} - 12 q^{22} + 8 q^{23} + 3 q^{24} + 12 q^{25} + 16 q^{26} + 6 q^{27} + 8 q^{28} + 20 q^{29} + 3 q^{30} + 5 q^{31} + 12 q^{32} - 3 q^{33} + 18 q^{34} + 8 q^{35} + 25 q^{36} + 19 q^{37} - 4 q^{38} + 6 q^{39} + 12 q^{40} + 16 q^{41} + 4 q^{42} + 12 q^{43} - 12 q^{44} + 25 q^{45} + 8 q^{46} - q^{47} + 3 q^{48} + 52 q^{49} + 12 q^{50} + q^{51} + 16 q^{52} + 11 q^{53} + 6 q^{54} - 12 q^{55} + 8 q^{56} + 9 q^{57} + 20 q^{58} - 11 q^{59} + 3 q^{60} + 18 q^{61} + 5 q^{62} + 15 q^{63} + 12 q^{64} + 16 q^{65} - 3 q^{66} - 10 q^{67} + 18 q^{68} + 8 q^{70} - 2 q^{71} + 25 q^{72} + 29 q^{73} + 19 q^{74} + 3 q^{75} - 4 q^{76} - 8 q^{77} + 6 q^{78} + 2 q^{79} + 12 q^{80} - 8 q^{81} + 16 q^{82} + 26 q^{83} + 4 q^{84} + 18 q^{85} + 12 q^{86} - 4 q^{87} - 12 q^{88} + 41 q^{89} + 25 q^{90} - 4 q^{91} + 8 q^{92} + 5 q^{93} - q^{94} - 4 q^{95} + 3 q^{96} - 7 q^{97} + 52 q^{98} - 25 q^{99}+O(q^{100}) \) 12 * q + 12 * q^2 + 3 * q^3 + 12 * q^4 + 12 * q^5 + 3 * q^6 + 8 * q^7 + 12 * q^8 + 25 * q^9 + 12 * q^10 - 12 * q^11 + 3 * q^12 + 16 * q^13 + 8 * q^14 + 3 * q^15 + 12 * q^16 + 18 * q^17 + 25 * q^18 - 4 * q^19 + 12 * q^20 + 4 * q^21 - 12 * q^22 + 8 * q^23 + 3 * q^24 + 12 * q^25 + 16 * q^26 + 6 * q^27 + 8 * q^28 + 20 * q^29 + 3 * q^30 + 5 * q^31 + 12 * q^32 - 3 * q^33 + 18 * q^34 + 8 * q^35 + 25 * q^36 + 19 * q^37 - 4 * q^38 + 6 * q^39 + 12 * q^40 + 16 * q^41 + 4 * q^42 + 12 * q^43 - 12 * q^44 + 25 * q^45 + 8 * q^46 - q^47 + 3 * q^48 + 52 * q^49 + 12 * q^50 + q^51 + 16 * q^52 + 11 * q^53 + 6 * q^54 - 12 * q^55 + 8 * q^56 + 9 * q^57 + 20 * q^58 - 11 * q^59 + 3 * q^60 + 18 * q^61 + 5 * q^62 + 15 * q^63 + 12 * q^64 + 16 * q^65 - 3 * q^66 - 10 * q^67 + 18 * q^68 + 8 * q^70 - 2 * q^71 + 25 * q^72 + 29 * q^73 + 19 * q^74 + 3 * q^75 - 4 * q^76 - 8 * q^77 + 6 * q^78 + 2 * q^79 + 12 * q^80 - 8 * q^81 + 16 * q^82 + 26 * q^83 + 4 * q^84 + 18 * q^85 + 12 * q^86 - 4 * q^87 - 12 * q^88 + 41 * q^89 + 25 * q^90 - 4 * q^91 + 8 * q^92 + 5 * q^93 - q^94 - 4 * q^95 + 3 * q^96 - 7 * q^97 + 52 * q^98 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more