LMFDB - Embedded newform 671.2.a.c.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [671,2,Mod(1,671)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(671, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("671.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 671 = 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	671.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:	$5.35796197563$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} + \cdots - 43 $ x^19 - 5x^18 - 18x^17 + 122x^16 + 78x^15 - 1177x^14 + 387x^13 + 5755x^12 - 4673x^11 - 15053x^10 + 16875x^9 + 20141x^8 - 28019x^7 - 11589x^6 + 21077x^5 + 1674x^4 - 6404x^3 + 241x^2 + 643x - 43
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$0.686089$ of defining polynomial
Character	$\chi$	$=$	671.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.686089 q^{2} -1.33733 q^{3} -1.52928 q^{4} +3.77631 q^{5} -0.917525 q^{6} +4.48234 q^{7} -2.42140 q^{8} -1.21156 q^{9} +O(q^{10})$	\(q+0.686089 q^{2} -1.33733 q^{3} -1.52928 q^{4} +3.77631 q^{5} -0.917525 q^{6} +4.48234 q^{7} -2.42140 q^{8} -1.21156 q^{9} +2.59089 q^{10} +1.00000 q^{11} +2.04515 q^{12} -0.769699 q^{13} +3.07528 q^{14} -5.05016 q^{15} +1.39726 q^{16} -3.80128 q^{17} -0.831238 q^{18} +6.11093 q^{19} -5.77505 q^{20} -5.99435 q^{21} +0.686089 q^{22} -3.48519 q^{23} +3.23821 q^{24} +9.26054 q^{25} -0.528082 q^{26} +5.63223 q^{27} -6.85475 q^{28} +4.50868 q^{29} -3.46486 q^{30} +1.54601 q^{31} +5.80145 q^{32} -1.33733 q^{33} -2.60802 q^{34} +16.9267 q^{35} +1.85281 q^{36} +10.3247 q^{37} +4.19264 q^{38} +1.02934 q^{39} -9.14398 q^{40} -1.01077 q^{41} -4.11266 q^{42} -3.73505 q^{43} -1.52928 q^{44} -4.57523 q^{45} -2.39115 q^{46} -8.78295 q^{47} -1.86860 q^{48} +13.0913 q^{49} +6.35356 q^{50} +5.08355 q^{51} +1.17709 q^{52} +3.35846 q^{53} +3.86421 q^{54} +3.77631 q^{55} -10.8535 q^{56} -8.17230 q^{57} +3.09336 q^{58} +9.89674 q^{59} +7.72312 q^{60} -1.00000 q^{61} +1.06070 q^{62} -5.43061 q^{63} +1.18579 q^{64} -2.90662 q^{65} -0.917525 q^{66} -7.87874 q^{67} +5.81322 q^{68} +4.66084 q^{69} +11.6132 q^{70} -13.3078 q^{71} +2.93367 q^{72} +7.03634 q^{73} +7.08369 q^{74} -12.3844 q^{75} -9.34533 q^{76} +4.48234 q^{77} +0.706218 q^{78} -16.1138 q^{79} +5.27651 q^{80} -3.89745 q^{81} -0.693478 q^{82} +8.42331 q^{83} +9.16704 q^{84} -14.3548 q^{85} -2.56258 q^{86} -6.02958 q^{87} -2.42140 q^{88} +2.88714 q^{89} -3.13901 q^{90} -3.45005 q^{91} +5.32984 q^{92} -2.06751 q^{93} -6.02589 q^{94} +23.0768 q^{95} -7.75844 q^{96} +13.9532 q^{97} +8.98183 q^{98} -1.21156 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 5 q^{2} + 23 q^{4} + q^{6} + 9 q^{7} + 9 q^{8} + 29 q^{9}+O(q^{10}) $ 19 * q + 5 * q^2 + 23 * q^4 + q^6 + 9 * q^7 + 9 * q^8 + 29 * q^9	\( 19 q + 5 q^{2} + 23 q^{4} + q^{6} + 9 q^{7} + 9 q^{8} + 29 q^{9} + 7 q^{10} + 19 q^{11} - 4 q^{12} + 8 q^{13} - 11 q^{14} - 5 q^{15} + 31 q^{16} + 9 q^{17} + 10 q^{18} + 17 q^{19} - 6 q^{20} + 18 q^{21} + 5 q^{22} - 10 q^{23} + 26 q^{24} + 45 q^{25} + 5 q^{26} - 33 q^{27} + 36 q^{28} + 27 q^{29} - 30 q^{30} + 7 q^{31} + 8 q^{32} - 5 q^{34} + 17 q^{35} + 38 q^{36} + 20 q^{37} - 37 q^{38} + 24 q^{39} + 10 q^{40} + 19 q^{41} + 21 q^{42} + 20 q^{43} + 23 q^{44} - 32 q^{45} + 41 q^{46} - 19 q^{47} + 5 q^{48} + 42 q^{49} + 36 q^{50} + 47 q^{51} - 28 q^{52} + 3 q^{53} - 33 q^{54} - 44 q^{56} + 11 q^{57} + 23 q^{58} - 28 q^{59} - 96 q^{60} - 19 q^{61} - 11 q^{62} - 32 q^{63} + 47 q^{64} + 25 q^{65} + q^{66} + 3 q^{67} + 38 q^{68} + 3 q^{70} - 19 q^{71} + 34 q^{72} + 20 q^{73} - 22 q^{74} - 50 q^{75} - 25 q^{76} + 9 q^{77} - 94 q^{78} + 69 q^{79} - 36 q^{80} + 47 q^{81} - 61 q^{82} + q^{83} - 28 q^{84} + 24 q^{85} - 27 q^{86} - 58 q^{87} + 9 q^{88} + 36 q^{90} + 24 q^{91} - 67 q^{92} - 14 q^{93} + 64 q^{94} - 3 q^{95} - 26 q^{96} + 21 q^{97} - 87 q^{98} + 29 q^{99}+O(q^{100}) \) 19 * q + 5 * q^2 + 23 * q^4 + q^6 + 9 * q^7 + 9 * q^8 + 29 * q^9 + 7 * q^10 + 19 * q^11 - 4 * q^12 + 8 * q^13 - 11 * q^14 - 5 * q^15 + 31 * q^16 + 9 * q^17 + 10 * q^18 + 17 * q^19 - 6 * q^20 + 18 * q^21 + 5 * q^22 - 10 * q^23 + 26 * q^24 + 45 * q^25 + 5 * q^26 - 33 * q^27 + 36 * q^28 + 27 * q^29 - 30 * q^30 + 7 * q^31 + 8 * q^32 - 5 * q^34 + 17 * q^35 + 38 * q^36 + 20 * q^37 - 37 * q^38 + 24 * q^39 + 10 * q^40 + 19 * q^41 + 21 * q^42 + 20 * q^43 + 23 * q^44 - 32 * q^45 + 41 * q^46 - 19 * q^47 + 5 * q^48 + 42 * q^49 + 36 * q^50 + 47 * q^51 - 28 * q^52 + 3 * q^53 - 33 * q^54 - 44 * q^56 + 11 * q^57 + 23 * q^58 - 28 * q^59 - 96 * q^60 - 19 * q^61 - 11 * q^62 - 32 * q^63 + 47 * q^64 + 25 * q^65 + q^66 + 3 * q^67 + 38 * q^68 + 3 * q^70 - 19 * q^71 + 34 * q^72 + 20 * q^73 - 22 * q^74 - 50 * q^75 - 25 * q^76 + 9 * q^77 - 94 * q^78 + 69 * q^79 - 36 * q^80 + 47 * q^81 - 61 * q^82 + q^83 - 28 * q^84 + 24 * q^85 - 27 * q^86 - 58 * q^87 + 9 * q^88 + 36 * q^90 + 24 * q^91 - 67 * q^92 - 14 * q^93 + 64 * q^94 - 3 * q^95 - 26 * q^96 + 21 * q^97 - 87 * 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 $

$2$	0.686089	0.485138	0.242569	−	0.970134i	$-0.422010\pi$
$2$	0.686089	0.485138	0.242569	+	0.970134i	$0.422010\pi$
$3$	−1.33733	−0.772106	−0.386053	−	0.922477i	$-0.626162\pi$
$3$	−1.33733	−0.772106	−0.386053	+	0.922477i	$0.626162\pi$
$4$	−1.52928	−0.764641
$5$	3.77631	1.68882	0.844409	−	0.535698i	$-0.179952\pi$
$5$	3.77631	1.68882	0.844409	+	0.535698i	$0.179952\pi$
$6$	−0.917525	−0.374578
$7$	4.48234	1.69416	0.847082	−	0.531462i	$-0.178357\pi$
$7$	4.48234	1.69416	0.847082	+	0.531462i	$0.178357\pi$
$8$	−2.42140	−0.856095
$9$	−1.21156	−0.403853
$10$	2.59089	0.819311
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.04515	0.590383
$13$	−0.769699	−0.213476	−0.106738	−	0.994287i	$-0.534041\pi$
$13$	−0.769699	−0.213476	−0.106738	+	0.994287i	$0.534041\pi$
$14$	3.07528	0.821904
$15$	−5.05016	−1.30395
$16$	1.39726	0.349316
$17$	−3.80128	−0.921945	−0.460972	−	0.887414i	$-0.652499\pi$
$17$	−3.80128	−0.921945	−0.460972	+	0.887414i	$0.652499\pi$
$18$	−0.831238	−0.195925
$19$	6.11093	1.40194	0.700971	−	0.713189i	$-0.252750\pi$
$19$	6.11093	1.40194	0.700971	+	0.713189i	$0.252750\pi$
$20$	−5.77505	−1.29134
$21$	−5.99435	−1.30807
$22$	0.686089	0.146275
$23$	−3.48519	−0.726713	−0.363357	−	0.931650i	$-0.618369\pi$
$23$	−3.48519	−0.726713	−0.363357	+	0.931650i	$0.618369\pi$
$24$	3.23821	0.660996
$25$	9.26054	1.85211
$26$	−0.528082	−0.103565
$27$	5.63223	1.08392
$28$	−6.85475	−1.29543
$29$	4.50868	0.837241	0.418621	−	0.908161i	$-0.362514\pi$
$29$	4.50868	0.837241	0.418621	+	0.908161i	$0.362514\pi$
$30$	−3.46486	−0.632595
$31$	1.54601	0.277671	0.138835	−	0.990315i	$-0.455664\pi$
$31$	1.54601	0.277671	0.138835	+	0.990315i	$0.455664\pi$
$32$	5.80145	1.02556
$33$	−1.33733	−0.232799
$34$	−2.60802	−0.447271
$35$	16.9267	2.86114
$36$	1.85281	0.308802
$37$	10.3247	1.69737	0.848687	−	0.528895i	$-0.177393\pi$
$37$	10.3247	1.69737	0.848687	+	0.528895i	$0.177393\pi$
$38$	4.19264	0.680136
$39$	1.02934	0.164826
$40$	−9.14398	−1.44579
$41$	−1.01077	−0.157856	−0.0789278	−	0.996880i	$-0.525150\pi$
$41$	−1.01077	−0.157856	−0.0789278	+	0.996880i	$0.525150\pi$
$42$	−4.11266	−0.634597
$43$	−3.73505	−0.569590	−0.284795	−	0.958588i	$-0.591925\pi$
$43$	−3.73505	−0.569590	−0.284795	+	0.958588i	$0.591925\pi$
$44$	−1.52928	−0.230548
$45$	−4.57523	−0.682034
$46$	−2.39115	−0.352557
$47$	−8.78295	−1.28112	−0.640562	−	0.767906i	$-0.721299\pi$
$47$	−8.78295	−1.28112	−0.640562	+	0.767906i	$0.721299\pi$
$48$	−1.86860	−0.269709
$49$	13.0913	1.87019
$50$	6.35356	0.898529
$51$	5.08355	0.711839
$52$	1.17709	0.163232
$53$	3.35846	0.461320	0.230660	−	0.973034i	$-0.425912\pi$
$53$	3.35846	0.461320	0.230660	+	0.973034i	$0.425912\pi$
$54$	3.86421	0.525853
$55$	3.77631	0.509198
$56$	−10.8535	−1.45037
$57$	−8.17230	−1.08245
$58$	3.09336	0.406178
$59$	9.89674	1.28845	0.644223	−	0.764838i	$-0.277181\pi$
$59$	9.89674	1.28845	0.644223	+	0.764838i	$0.277181\pi$
$60$	7.72312	0.997051
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	1.06070	0.134709
$63$	−5.43061	−0.684193
$64$	1.18579	0.148223
$65$	−2.90662	−0.360522
$66$	−0.917525	−0.112940
$67$	−7.87874	−0.962542	−0.481271	−	0.876572i	$-0.659825\pi$
$67$	−7.87874	−0.962542	−0.481271	+	0.876572i	$0.659825\pi$
$68$	5.81322	0.704957
$69$	4.66084	0.561099
$70$	11.6132	1.38805
$71$	−13.3078	−1.57934	−0.789671	−	0.613530i	$-0.789749\pi$
$71$	−13.3078	−1.57934	−0.789671	+	0.613530i	$0.789749\pi$
$72$	2.93367	0.345736
$73$	7.03634	0.823541	0.411771	−	0.911288i	$-0.364911\pi$
$73$	7.03634	0.823541	0.411771	+	0.911288i	$0.364911\pi$
$74$	7.08369	0.823462
$75$	−12.3844	−1.43002
$76$	−9.34533	−1.07198
$77$	4.48234	0.510810
$78$	0.706218	0.0799635
$79$	−16.1138	−1.81295	−0.906474	−	0.422261i	$-0.861237\pi$
$79$	−16.1138	−1.81295	−0.906474	+	0.422261i	$0.861237\pi$
$80$	5.27651	0.589931
$81$	−3.89745	−0.433050
$82$	−0.693478	−0.0765819
$83$	8.42331	0.924579	0.462289	−	0.886729i	$-0.347028\pi$
$83$	8.42331	0.924579	0.462289	+	0.886729i	$0.347028\pi$
$84$	9.16704	1.00021
$85$	−14.3548	−1.55700
$86$	−2.56258	−0.276330
$87$	−6.02958	−0.646439
$88$	−2.42140	−0.258122
$89$	2.88714	0.306036	0.153018	−	0.988223i	$-0.451101\pi$
$89$	2.88714	0.306036	0.153018	+	0.988223i	$0.451101\pi$
$90$	−3.13901	−0.330881
$91$	−3.45005	−0.361664
$92$	5.32984	0.555675
$93$	−2.06751	−0.214391
$94$	−6.02589	−0.621523
$95$	23.0768	2.36763
$96$	−7.75844	−0.791842
$97$	13.9532	1.41673	0.708367	−	0.705844i	$-0.249432\pi$
$97$	13.9532	1.41673	0.708367	+	0.705844i	$0.249432\pi$
$98$	8.98183	0.907302
$99$	−1.21156	−0.121766
$100$	−14.1620	−1.41620
$101$	1.01604	0.101100	0.0505498	−	0.998722i	$-0.483903\pi$
$101$	1.01604	0.101100	0.0505498	+	0.998722i	$0.483903\pi$
$102$	3.48777	0.345340
$103$	−12.6258	−1.24406	−0.622028	−	0.782995i	$-0.713691\pi$
$103$	−12.6258	−1.24406	−0.622028	+	0.782995i	$0.713691\pi$
$104$	1.86375	0.182756
$105$	−22.6365	−2.20910
$106$	2.30420	0.223804
$107$	−12.7922	−1.23667	−0.618334	−	0.785916i	$-0.712192\pi$
$107$	−12.7922	−1.23667	−0.618334	+	0.785916i	$0.712192\pi$
$108$	−8.61326	−0.828811
$109$	−18.3052	−1.75332	−0.876661	−	0.481109i	$-0.840234\pi$
$109$	−18.3052	−1.75332	−0.876661	+	0.481109i	$0.840234\pi$
$110$	2.59089	0.247032
$111$	−13.8075	−1.31055
$112$	6.26301	0.591799
$113$	−13.6164	−1.28092	−0.640461	−	0.767990i	$-0.721257\pi$
$113$	−13.6164	−1.28092	−0.640461	+	0.767990i	$0.721257\pi$
$114$	−5.60693	−0.525137
$115$	−13.1612	−1.22729
$116$	−6.89504	−0.640189
$117$	0.932535	0.0862129
$118$	6.79005	0.625075
$119$	−17.0386	−1.56193
$120$	12.2285	1.11630
$121$	1.00000	0.0909091
$122$	−0.686089	−0.0621156
$123$	1.35173	0.121881
$124$	−2.36428	−0.212318
$125$	16.0892	1.43906
$126$	−3.72589	−0.331928
$127$	−8.22802	−0.730119	−0.365059	−	0.930984i	$-0.618951\pi$
$127$	−8.22802	−0.730119	−0.365059	+	0.930984i	$0.618951\pi$
$128$	−10.7894	−0.953653
$129$	4.99498	0.439783
$130$	−1.99420	−0.174903
$131$	−15.6721	−1.36928	−0.684639	−	0.728882i	$-0.740040\pi$
$131$	−15.6721	−1.36928	−0.684639	+	0.728882i	$0.740040\pi$
$132$	2.04515	0.178007
$133$	27.3912	2.37512
$134$	−5.40552	−0.466966
$135$	21.2691	1.83055
$136$	9.20442	0.789272
$137$	3.16502	0.270406	0.135203	−	0.990818i	$-0.456831\pi$
$137$	3.16502	0.270406	0.135203	+	0.990818i	$0.456831\pi$
$138$	3.19775	0.272211
$139$	3.56579	0.302446	0.151223	−	0.988500i	$-0.451679\pi$
$139$	3.56579	0.302446	0.151223	+	0.988500i	$0.451679\pi$
$140$	−25.8857	−2.18774
$141$	11.7457	0.989164
$142$	−9.13032	−0.766200
$143$	−0.769699	−0.0643655
$144$	−1.69287	−0.141072
$145$	17.0262	1.41395
$146$	4.82756	0.399531
$147$	−17.5074	−1.44399
$148$	−15.7894	−1.29788
$149$	13.3424	1.09305	0.546524	−	0.837443i	$-0.315951\pi$
$149$	13.3424	1.09305	0.546524	+	0.837443i	$0.315951\pi$
$150$	−8.49678	−0.693760
$151$	22.1019	1.79863	0.899315	−	0.437302i	$-0.144066\pi$
$151$	22.1019	1.79863	0.899315	+	0.437302i	$0.144066\pi$
$152$	−14.7970	−1.20020
$153$	4.60547	0.372330
$154$	3.07528	0.247813
$155$	5.83820	0.468936
$156$	−1.57415	−0.126033
$157$	−9.85879	−0.786817	−0.393409	−	0.919364i	$-0.628704\pi$
$157$	−9.85879	−0.786817	−0.393409	+	0.919364i	$0.628704\pi$
$158$	−11.0555	−0.879531
$159$	−4.49135	−0.356188
$160$	21.9081	1.73199
$161$	−15.6218	−1.23117
$162$	−2.67400	−0.210089
$163$	−11.6798	−0.914830	−0.457415	−	0.889253i	$-0.651225\pi$
$163$	−11.6798	−0.914830	−0.457415	+	0.889253i	$0.651225\pi$
$164$	1.54575	0.120703
$165$	−5.05016	−0.393155
$166$	5.77914	0.448549
$167$	2.77827	0.214989	0.107494	−	0.994206i	$-0.465717\pi$
$167$	2.77827	0.214989	0.107494	+	0.994206i	$0.465717\pi$
$168$	14.5147	1.11984
$169$	−12.4076	−0.954428
$170$	−9.84868	−0.755360
$171$	−7.40375	−0.566179
$172$	5.71194	0.435531
$173$	−12.5194	−0.951835	−0.475918	−	0.879490i	$-0.657884\pi$
$173$	−12.5194	−0.951835	−0.475918	+	0.879490i	$0.657884\pi$
$174$	−4.13683	−0.313612
$175$	41.5089	3.13778
$176$	1.39726	0.105323
$177$	−13.2352	−0.994817
$178$	1.98084	0.148470
$179$	11.8950	0.889076	0.444538	−	0.895760i	$-0.353368\pi$
$179$	11.8950	0.889076	0.444538	+	0.895760i	$0.353368\pi$
$180$	6.99681	0.521511
$181$	−2.77705	−0.206416	−0.103208	−	0.994660i	$-0.532911\pi$
$181$	−2.77705	−0.206416	−0.103208	+	0.994660i	$0.532911\pi$
$182$	−2.36704	−0.175457
$183$	1.33733	0.0988580
$184$	8.43906	0.622136
$185$	38.9894	2.86656
$186$	−1.41850	−0.104009
$187$	−3.80128	−0.277977
$188$	13.4316	0.979600
$189$	25.2455	1.83634
$190$	15.8327	1.14863
$191$	−10.4918	−0.759157	−0.379578	−	0.925160i	$-0.623931\pi$
$191$	−10.4918	−0.759157	−0.379578	+	0.925160i	$0.623931\pi$
$192$	−1.58578	−0.114444
$193$	16.1834	1.16491	0.582454	−	0.812864i	$-0.302093\pi$
$193$	16.1834	1.16491	0.582454	+	0.812864i	$0.302093\pi$
$194$	9.57315	0.687312
$195$	3.88711	0.278361
$196$	−20.0203	−1.43002
$197$	5.10016	0.363372	0.181686	−	0.983357i	$-0.441845\pi$
$197$	5.10016	0.363372	0.181686	+	0.983357i	$0.441845\pi$
$198$	−0.831238	−0.0590735
$199$	3.56997	0.253069	0.126534	−	0.991962i	$-0.459615\pi$
$199$	3.56997	0.253069	0.126534	+	0.991962i	$0.459615\pi$
$200$	−22.4235	−1.58558
$201$	10.5365	0.743184
$202$	0.697093	0.0490473
$203$	20.2094	1.41842
$204$	−7.77417	−0.544301
$205$	−3.81698	−0.266590
$206$	−8.66242	−0.603540
$207$	4.22252	0.293485
$208$	−1.07547	−0.0745706
$209$	6.11093	0.422702
$210$	−15.5307	−1.07172
$211$	−15.7416	−1.08370	−0.541848	−	0.840476i	$-0.682275\pi$
$211$	−15.7416	−1.08370	−0.541848	+	0.840476i	$0.682275\pi$
$212$	−5.13603	−0.352744
$213$	17.7968	1.21942
$214$	−8.77659	−0.599955
$215$	−14.1047	−0.961934
$216$	−13.6379	−0.927941
$217$	6.92972	0.470420
$218$	−12.5590	−0.850604
$219$	−9.40988	−0.635861
$220$	−5.77505	−0.389354
$221$	2.92584	0.196813
$222$	−9.47320	−0.635799
$223$	15.6928	1.05087	0.525435	−	0.850834i	$-0.323903\pi$
$223$	15.6928	1.05087	0.525435	+	0.850834i	$0.323903\pi$
$224$	26.0041	1.73747
$225$	−11.2197	−0.747980
$226$	−9.34206	−0.621425
$227$	−4.05098	−0.268873	−0.134436	−	0.990922i	$-0.542922\pi$
$227$	−4.05098	−0.268873	−0.134436	+	0.990922i	$0.542922\pi$
$228$	12.4978	0.827684
$229$	7.07401	0.467463	0.233732	−	0.972301i	$-0.424906\pi$
$229$	7.07401	0.467463	0.233732	+	0.972301i	$0.424906\pi$
$230$	−9.02975	−0.595404
$231$	−5.99435	−0.394399
$232$	−10.9173	−0.716758
$233$	−5.99348	−0.392646	−0.196323	−	0.980539i	$-0.562900\pi$
$233$	−5.99348	−0.392646	−0.196323	+	0.980539i	$0.562900\pi$
$234$	0.639803	0.0418252
$235$	−33.1672	−2.16359
$236$	−15.1349	−0.985198
$237$	21.5495	1.39979
$238$	−11.6900	−0.757750
$239$	−6.10444	−0.394864	−0.197432	−	0.980317i	$-0.563260\pi$
$239$	−6.10444	−0.394864	−0.197432	+	0.980317i	$0.563260\pi$
$240$	−7.05641	−0.455489
$241$	3.66176	0.235875	0.117937	−	0.993021i	$-0.462372\pi$
$241$	3.66176	0.235875	0.117937	+	0.993021i	$0.462372\pi$
$242$	0.686089	0.0441035
$243$	−11.6845	−0.749562
$244$	1.52928	0.0979022
$245$	49.4370	3.15842
$246$	0.927407	0.0591293
$247$	−4.70357	−0.299281
$248$	−3.74350	−0.237713
$249$	−11.2647	−0.713872
$250$	11.0386	0.698142
$251$	−28.4222	−1.79399	−0.896996	−	0.442039i	$-0.854255\pi$
$251$	−28.4222	−1.79399	−0.896996	+	0.442039i	$0.854255\pi$
$252$	8.30494	0.523162
$253$	−3.48519	−0.219112
$254$	−5.64516	−0.354209
$255$	19.1971	1.20217
$256$	−9.77403	−0.610877
$257$	14.7040	0.917210	0.458605	−	0.888640i	$-0.348349\pi$
$257$	14.7040	0.917210	0.458605	+	0.888640i	$0.348349\pi$
$258$	3.42700	0.213356
$259$	46.2789	2.87563
$260$	4.44505	0.275670
$261$	−5.46253	−0.338122
$262$	−10.7525	−0.664289
$263$	−13.3143	−0.820996	−0.410498	−	0.911862i	$-0.634645\pi$
$263$	−13.3143	−0.820996	−0.410498	+	0.911862i	$0.634645\pi$
$264$	3.23821	0.199298
$265$	12.6826	0.779085
$266$	18.7928	1.15226
$267$	−3.86105	−0.236292
$268$	12.0488	0.735999
$269$	−23.1975	−1.41438	−0.707190	−	0.707024i	$-0.750037\pi$
$269$	−23.1975	−1.41438	−0.707190	+	0.707024i	$0.750037\pi$
$270$	14.5925	0.888070
$271$	−21.4301	−1.30178	−0.650892	−	0.759171i	$-0.725605\pi$
$271$	−21.4301	−1.30178	−0.650892	+	0.759171i	$0.725605\pi$
$272$	−5.31139	−0.322050
$273$	4.61384	0.279242
$274$	2.17149	0.131184
$275$	9.26054	0.558432
$276$	−7.12774	−0.429039
$277$	11.5072	0.691402	0.345701	−	0.938345i	$-0.387641\pi$
$277$	11.5072	0.691402	0.345701	+	0.938345i	$0.387641\pi$
$278$	2.44645	0.146728
$279$	−1.87308	−0.112138
$280$	−40.9864	−2.44940
$281$	9.24018	0.551223	0.275611	−	0.961269i	$-0.411120\pi$
$281$	9.24018	0.551223	0.275611	+	0.961269i	$0.411120\pi$
$282$	8.05858	0.479881
$283$	25.8120	1.53437	0.767184	−	0.641428i	$-0.221658\pi$
$283$	25.8120	1.53437	0.767184	+	0.641428i	$0.221658\pi$
$284$	20.3513	1.20763
$285$	−30.8612	−1.82806
$286$	−0.528082	−0.0312262
$287$	−4.53061	−0.267433
$288$	−7.02880	−0.414176
$289$	−2.55030	−0.150018
$290$	11.6815	0.685961
$291$	−18.6600	−1.09387
$292$	−10.7605	−0.629713
$293$	12.6913	0.741436	0.370718	−	0.928746i	$-0.379112\pi$
$293$	12.6913	0.741436	0.370718	+	0.928746i	$0.379112\pi$
$294$	−12.0116	−0.700533
$295$	37.3732	2.17595
$296$	−25.0003	−1.45311
$297$	5.63223	0.326815
$298$	9.15405	0.530280
$299$	2.68255	0.155136
$300$	18.9392	1.09345
$301$	−16.7417	−0.964978
$302$	15.1639	0.872585
$303$	−1.35877	−0.0780595
$304$	8.53858	0.489721
$305$	−3.77631	−0.216231
$306$	3.15976	0.180632
$307$	−17.9853	−1.02647	−0.513237	−	0.858247i	$-0.671554\pi$
$307$	−17.9853	−1.02647	−0.513237	+	0.858247i	$0.671554\pi$
$308$	−6.85475	−0.390586
$309$	16.8848	0.960543
$310$	4.00553	0.227499
$311$	20.5838	1.16720	0.583600	−	0.812041i	$-0.301644\pi$
$311$	20.5838	1.16720	0.583600	+	0.812041i	$0.301644\pi$
$312$	−2.49244	−0.141107
$313$	−20.1778	−1.14052	−0.570259	−	0.821465i	$-0.693157\pi$
$313$	−20.1778	−1.14052	−0.570259	+	0.821465i	$0.693157\pi$
$314$	−6.76401	−0.381715
$315$	−20.5077	−1.15548
$316$	24.6426	1.38625
$317$	−15.3821	−0.863948	−0.431974	−	0.901886i	$-0.642183\pi$
$317$	−15.3821	−0.863948	−0.431974	+	0.901886i	$0.642183\pi$
$318$	−3.08147	−0.172800
$319$	4.50868	0.252438
$320$	4.47790	0.250322
$321$	17.1073	0.954838
$322$	−10.7180	−0.597289
$323$	−23.2293	−1.29251
$324$	5.96030	0.331128
$325$	−7.12783	−0.395381
$326$	−8.01336	−0.443819
$327$	24.4800	1.35375
$328$	2.44748	0.135139
$329$	−39.3681	−2.17044
$330$	−3.46486	−0.190734
$331$	17.0595	0.937674	0.468837	−	0.883285i	$-0.344673\pi$
$331$	17.0595	0.937674	0.468837	+	0.883285i	$0.344673\pi$
$332$	−12.8816	−0.706970
$333$	−12.5090	−0.685490
$334$	1.90614	0.104299
$335$	−29.7526	−1.62556
$336$	−8.37569	−0.456931
$337$	−4.88401	−0.266049	−0.133025	−	0.991113i	$-0.542469\pi$
$337$	−4.88401	−0.266049	−0.133025	+	0.991113i	$0.542469\pi$
$338$	−8.51270	−0.463030
$339$	18.2096	0.989008
$340$	21.9525	1.19054
$341$	1.54601	0.0837209
$342$	−5.07963	−0.274675
$343$	27.3035	1.47425
$344$	9.04406	0.487623
$345$	17.6008	0.947595
$346$	−8.58945	−0.461772
$347$	−18.7220	−1.00505	−0.502526	−	0.864562i	$-0.667596\pi$
$347$	−18.7220	−1.00505	−0.502526	+	0.864562i	$0.667596\pi$
$348$	9.22092	0.494293
$349$	−4.28971	−0.229623	−0.114811	−	0.993387i	$-0.536626\pi$
$349$	−4.28971	−0.229623	−0.114811	+	0.993387i	$0.536626\pi$
$350$	28.4788	1.52226
$351$	−4.33512	−0.231392
$352$	5.80145	0.309218
$353$	−30.8366	−1.64127	−0.820634	−	0.571454i	$-0.806380\pi$
$353$	−30.8366	−1.64127	−0.820634	+	0.571454i	$0.806380\pi$
$354$	−9.08051	−0.482624
$355$	−50.2543	−2.66722
$356$	−4.41525	−0.234008
$357$	22.7862	1.20597
$358$	8.16105	0.431325
$359$	17.4702	0.922040	0.461020	−	0.887390i	$-0.347484\pi$
$359$	17.4702	0.922040	0.461020	+	0.887390i	$0.347484\pi$
$360$	11.0785	0.583886
$361$	18.3434	0.965444
$362$	−1.90530	−0.100140
$363$	−1.33733	−0.0701914
$364$	5.27610	0.276543
$365$	26.5714	1.39081
$366$	0.917525	0.0479598
$367$	32.6736	1.70555	0.852774	−	0.522279i	$-0.174918\pi$
$367$	32.6736	1.70555	0.852774	+	0.522279i	$0.174918\pi$
$368$	−4.86974	−0.253853
$369$	1.22461	0.0637505
$370$	26.7502	1.39068
$371$	15.0537	0.781551
$372$	3.16181	0.163932
$373$	15.0278	0.778109	0.389055	−	0.921215i	$-0.372802\pi$
$373$	15.0278	0.778109	0.389055	+	0.921215i	$0.372802\pi$
$374$	−2.60802	−0.134857
$375$	−21.5164	−1.11110
$376$	21.2671	1.09676
$377$	−3.47033	−0.178731
$378$	17.3207	0.890881
$379$	19.3890	0.995947	0.497973	−	0.867192i	$-0.334078\pi$
$379$	19.3890	0.995947	0.497973	+	0.867192i	$0.334078\pi$
$380$	−35.2909	−1.81038
$381$	11.0035	0.563729
$382$	−7.19828	−0.368296
$383$	−29.4718	−1.50594	−0.752970	−	0.658055i	$-0.771379\pi$
$383$	−29.4718	−1.50594	−0.752970	+	0.658055i	$0.771379\pi$
$384$	14.4289	0.736321
$385$	16.9267	0.862665
$386$	11.1033	0.565142
$387$	4.52523	0.230030
$388$	−21.3384	−1.08329
$389$	−12.3906	−0.628229	−0.314114	−	0.949385i	$-0.601708\pi$
$389$	−12.3906	−0.628229	−0.314114	+	0.949385i	$0.601708\pi$
$390$	2.66690	0.135044
$391$	13.2482	0.669990
$392$	−31.6994	−1.60106
$393$	20.9587	1.05723
$394$	3.49917	0.176286
$395$	−60.8509	−3.06174
$396$	1.85281	0.0931074
$397$	−4.89557	−0.245701	−0.122851	−	0.992425i	$-0.539204\pi$
$397$	−4.89557	−0.245701	−0.122851	+	0.992425i	$0.539204\pi$
$398$	2.44932	0.122773
$399$	−36.6310	−1.83384
$400$	12.9394	0.646971
$401$	0.312953	0.0156281	0.00781407	−	0.999969i	$-0.497513\pi$
$401$	0.312953	0.0156281	0.00781407	+	0.999969i	$0.497513\pi$
$402$	7.22895	0.360547
$403$	−1.18996	−0.0592761
$404$	−1.55381	−0.0773048
$405$	−14.7180	−0.731343
$406$	13.8655	0.688132
$407$	10.3247	0.511778
$408$	−12.3093	−0.609402
$409$	2.56948	0.127053	0.0635264	−	0.997980i	$-0.479765\pi$
$409$	2.56948	0.127053	0.0635264	+	0.997980i	$0.479765\pi$
$410$	−2.61879	−0.129333
$411$	−4.23266	−0.208782
$412$	19.3084	0.951256
$413$	44.3605	2.18284
$414$	2.89702	0.142381
$415$	31.8091	1.56145
$416$	−4.46537	−0.218933
$417$	−4.76862	−0.233520
$418$	4.19264	0.205069
$419$	−10.7998	−0.527604	−0.263802	−	0.964577i	$-0.584977\pi$
$419$	−10.7998	−0.527604	−0.263802	+	0.964577i	$0.584977\pi$
$420$	34.6176	1.68917
$421$	25.5815	1.24677	0.623383	−	0.781917i	$-0.285758\pi$
$421$	25.5815	1.24677	0.623383	+	0.781917i	$0.285758\pi$
$422$	−10.8001	−0.525743
$423$	10.6411	0.517386
$424$	−8.13218	−0.394934
$425$	−35.2019	−1.70754
$426$	12.2102	0.591587
$427$	−4.48234	−0.216915
$428$	19.5629	0.945606
$429$	1.02934	0.0496969
$430$	−9.67710	−0.466671
$431$	24.2472	1.16795	0.583973	−	0.811773i	$-0.301497\pi$
$431$	24.2472	1.16795	0.583973	+	0.811773i	$0.301497\pi$
$432$	7.86971	0.378632
$433$	−5.72897	−0.275317	−0.137658	−	0.990480i	$-0.543958\pi$
$433$	−5.72897	−0.275317	−0.137658	+	0.990480i	$0.543958\pi$
$434$	4.75440	0.228219
$435$	−22.7696	−1.09172
$436$	27.9938	1.34066
$437$	−21.2978	−1.01881
$438$	−6.45602	−0.308480
$439$	3.68355	0.175806	0.0879031	−	0.996129i	$-0.471983\pi$
$439$	3.68355	0.175806	0.0879031	+	0.996129i	$0.471983\pi$
$440$	−9.14398	−0.435922
$441$	−15.8609	−0.755282
$442$	2.00739	0.0954816
$443$	−38.8838	−1.84743	−0.923713	−	0.383086i	$-0.874861\pi$
$443$	−38.8838	−1.84743	−0.923713	+	0.383086i	$0.874861\pi$
$444$	21.1156	1.00210
$445$	10.9027	0.516840
$446$	10.7667	0.509817
$447$	−17.8431	−0.843949
$448$	5.31510	0.251115
$449$	10.7057	0.505234	0.252617	−	0.967566i	$-0.418709\pi$
$449$	10.7057	0.505234	0.252617	+	0.967566i	$0.418709\pi$
$450$	−7.69771	−0.362874
$451$	−1.01077	−0.0475953
$452$	20.8233	0.979446
$453$	−29.5575	−1.38873
$454$	−2.77933	−0.130441
$455$	−13.0285	−0.610784
$456$	19.7884	0.926678
$457$	35.1021	1.64201	0.821005	−	0.570922i	$-0.193414\pi$
$457$	35.1021	1.64201	0.821005	+	0.570922i	$0.193414\pi$
$458$	4.85340	0.226785
$459$	−21.4097	−0.999317
$460$	20.1272	0.938434
$461$	7.65121	0.356352	0.178176	−	0.983999i	$-0.442980\pi$
$461$	7.65121	0.356352	0.178176	+	0.983999i	$0.442980\pi$
$462$	−4.11266	−0.191338
$463$	−32.3318	−1.50259	−0.751293	−	0.659968i	$-0.770570\pi$
$463$	−32.3318	−1.50259	−0.751293	+	0.659968i	$0.770570\pi$
$464$	6.29982	0.292462
$465$	−7.80758	−0.362068
$466$	−4.11206	−0.190488
$467$	1.52940	0.0707723	0.0353862	−	0.999374i	$-0.488734\pi$
$467$	1.52940	0.0707723	0.0353862	+	0.999374i	$0.488734\pi$
$468$	−1.42611	−0.0659219
$469$	−35.3152	−1.63070
$470$	−22.7556	−1.04964
$471$	13.1844	0.607506
$472$	−23.9640	−1.10303
$473$	−3.73505	−0.171738
$474$	14.7849	0.679091
$475$	56.5905	2.59655
$476$	26.0568	1.19431
$477$	−4.06897	−0.186305
$478$	−4.18819	−0.191563
$479$	34.2341	1.56419	0.782097	−	0.623157i	$-0.214150\pi$
$479$	34.2341	1.56419	0.782097	+	0.623157i	$0.214150\pi$
$480$	−29.2983	−1.33728
$481$	−7.94693	−0.362349
$482$	2.51230	0.114432
$483$	20.8915	0.950594
$484$	−1.52928	−0.0695128
$485$	52.6917	2.39261
$486$	−8.01663	−0.363642
$487$	35.9633	1.62965	0.814827	−	0.579704i	$-0.196832\pi$
$487$	35.9633	1.62965	0.814827	+	0.579704i	$0.196832\pi$
$488$	2.42140	0.109612
$489$	15.6197	0.706345
$490$	33.9182	1.53227
$491$	23.3680	1.05459	0.527293	−	0.849684i	$-0.323207\pi$
$491$	23.3680	1.05459	0.527293	+	0.849684i	$0.323207\pi$
$492$	−2.06717	−0.0931954
$493$	−17.1387	−0.771890
$494$	−3.22707	−0.145193
$495$	−4.57523	−0.205641
$496$	2.16018	0.0969948
$497$	−59.6499	−2.67567
$498$	−7.72860	−0.346327
$499$	0.906030	0.0405595	0.0202797	−	0.999794i	$-0.493544\pi$
$499$	0.906030	0.0405595	0.0202797	+	0.999794i	$0.493544\pi$
$500$	−24.6048	−1.10036
$501$	−3.71545	−0.165994
$502$	−19.5002	−0.870334
$503$	−15.4553	−0.689116	−0.344558	−	0.938765i	$-0.611971\pi$
$503$	−15.4553	−0.689116	−0.344558	+	0.938765i	$0.611971\pi$
$504$	13.1497	0.585734
$505$	3.83688	0.170739
$506$	−2.39115	−0.106300
$507$	16.5930	0.736919
$508$	12.5830	0.558278
$509$	35.6136	1.57855	0.789273	−	0.614043i	$-0.210458\pi$
$509$	35.6136	1.57855	0.789273	+	0.614043i	$0.210458\pi$
$510$	13.1709	0.583217
$511$	31.5392	1.39521
$512$	14.8728	0.657293
$513$	34.4181	1.51960
$514$	10.0883	0.444974
$515$	−47.6789	−2.10099
$516$	−7.63873	−0.336276
$517$	−8.78295	−0.386274
$518$	31.7515	1.39508
$519$	16.7426	0.734917
$520$	7.03811	0.308641
$521$	−16.6516	−0.729521	−0.364761	−	0.931101i	$-0.618849\pi$
$521$	−16.6516	−0.729521	−0.364761	+	0.931101i	$0.618849\pi$
$522$	−3.74779	−0.164036
$523$	−0.581619	−0.0254324	−0.0127162	−	0.999919i	$-0.504048\pi$
$523$	−0.581619	−0.0254324	−0.0127162	+	0.999919i	$0.504048\pi$
$524$	23.9671	1.04701
$525$	−55.5109	−2.42269
$526$	−9.13481	−0.398297
$527$	−5.87679	−0.255997
$528$	−1.86860	−0.0813203
$529$	−10.8534	−0.471888
$530$	8.70139	0.377964
$531$	−11.9905	−0.520343
$532$	−41.8889	−1.81611
$533$	0.777988	0.0336984
$534$	−2.64902	−0.114634
$535$	−48.3073	−2.08851
$536$	19.0776	0.824027
$537$	−15.9075	−0.686460
$538$	−15.9156	−0.686170
$539$	13.0913	0.563884
$540$	−32.5264	−1.39971
$541$	28.5442	1.22721	0.613607	−	0.789612i	$-0.289718\pi$
$541$	28.5442	1.22721	0.613607	+	0.789612i	$0.289718\pi$
$542$	−14.7029	−0.631545
$543$	3.71382	0.159375
$544$	−22.0529	−0.945511
$545$	−69.1262	−2.96104
$546$	3.16551	0.135471
$547$	−25.6176	−1.09533	−0.547664	−	0.836698i	$-0.684483\pi$
$547$	−25.6176	−1.09533	−0.547664	+	0.836698i	$0.684483\pi$
$548$	−4.84021	−0.206763
$549$	1.21156	0.0517081
$550$	6.35356	0.270917
$551$	27.5522	1.17376
$552$	−11.2858	−0.480354
$553$	−72.2277	−3.07143
$554$	7.89499	0.335426
$555$	−52.1416	−2.21329
$556$	−5.45309	−0.231263
$557$	−10.2215	−0.433100	−0.216550	−	0.976272i	$-0.569480\pi$
$557$	−10.2215	−0.433100	−0.216550	+	0.976272i	$0.569480\pi$
$558$	−1.28510	−0.0544025
$559$	2.87486	0.121594
$560$	23.6511	0.999441
$561$	5.08355	0.214627
$562$	6.33959	0.267419
$563$	1.25185	0.0527590	0.0263795	−	0.999652i	$-0.491602\pi$
$563$	1.25185	0.0527590	0.0263795	+	0.999652i	$0.491602\pi$
$564$	−17.9624	−0.756355
$565$	−51.4198	−2.16325
$566$	17.7094	0.744380
$567$	−17.4697	−0.733658
$568$	32.2235	1.35207
$569$	27.7083	1.16159	0.580796	−	0.814049i	$-0.302742\pi$
$569$	27.7083	1.16159	0.580796	+	0.814049i	$0.302742\pi$
$570$	−21.1735	−0.886861
$571$	38.4361	1.60850	0.804251	−	0.594290i	$-0.202567\pi$
$571$	38.4361	1.60850	0.804251	+	0.594290i	$0.202567\pi$
$572$	1.17709	0.0492164
$573$	14.0309	0.586149
$574$	−3.10840	−0.129742
$575$	−32.2748	−1.34595
$576$	−1.43665	−0.0598604
$577$	−40.3318	−1.67904	−0.839518	−	0.543332i	$-0.817163\pi$
$577$	−40.3318	−1.67904	−0.839518	+	0.543332i	$0.817163\pi$
$578$	−1.74973	−0.0727793
$579$	−21.6425	−0.899432
$580$	−26.0378	−1.08116
$581$	37.7561	1.56639
$582$	−12.8024	−0.530678
$583$	3.35846	0.139093
$584$	−17.0378	−0.705029
$585$	3.52155	0.145598
$586$	8.70739	0.359699
$587$	21.4539	0.885499	0.442750	−	0.896645i	$-0.354003\pi$
$587$	21.4539	0.885499	0.442750	+	0.896645i	$0.354003\pi$
$588$	26.7737	1.10413
$589$	9.44752	0.389278
$590$	25.6414	1.05564
$591$	−6.82058	−0.280561
$592$	14.4264	0.592920
$593$	−4.78901	−0.196661	−0.0983305	−	0.995154i	$-0.531350\pi$
$593$	−4.78901	−0.196661	−0.0983305	+	0.995154i	$0.531350\pi$
$594$	3.86421	0.158551
$595$	−64.3431	−2.63781
$596$	−20.4042	−0.835790
$597$	−4.77422	−0.195396
$598$	1.84047	0.0752624
$599$	−24.9946	−1.02125	−0.510626	−	0.859803i	$-0.670586\pi$
$599$	−24.9946	−1.02125	−0.510626	+	0.859803i	$0.670586\pi$
$600$	29.9875	1.22424
$601$	11.5085	0.469440	0.234720	−	0.972063i	$-0.424583\pi$
$601$	11.5085	0.469440	0.234720	+	0.972063i	$0.424583\pi$
$602$	−11.4863	−0.468148
$603$	9.54556	0.388725
$604$	−33.8001	−1.37531
$605$	3.77631	0.153529
$606$	−0.932241	−0.0378697
$607$	48.6573	1.97494	0.987470	−	0.157807i	$-0.0504424\pi$
$607$	48.6573	1.97494	0.987470	+	0.157807i	$0.0504424\pi$
$608$	35.4523	1.43778
$609$	−27.0266	−1.09517
$610$	−2.59089	−0.104902
$611$	6.76023	0.273490
$612$	−7.04306	−0.284699
$613$	21.1808	0.855484	0.427742	−	0.903901i	$-0.359309\pi$
$613$	21.1808	0.855484	0.427742	+	0.903901i	$0.359309\pi$
$614$	−12.3395	−0.497982
$615$	5.10455	0.205835
$616$	−10.8535	−0.437302
$617$	−10.2709	−0.413489	−0.206745	−	0.978395i	$-0.566287\pi$
$617$	−10.2709	−0.413489	−0.206745	+	0.978395i	$0.566287\pi$
$618$	11.5845	0.465996
$619$	7.70550	0.309710	0.154855	−	0.987937i	$-0.450509\pi$
$619$	7.70550	0.309710	0.154855	+	0.987937i	$0.450509\pi$
$620$	−8.92825	−0.358567
$621$	−19.6294	−0.787701
$622$	14.1223	0.566253
$623$	12.9411	0.518476
$624$	1.43826	0.0575764
$625$	14.4550	0.578199
$626$	−13.8438	−0.553309
$627$	−8.17230	−0.326370
$628$	15.0769	0.601633
$629$	−39.2471	−1.56489
$630$	−14.0701	−0.560567
$631$	−16.0146	−0.637532	−0.318766	−	0.947833i	$-0.603268\pi$
$631$	−16.0146	−0.637532	−0.318766	+	0.947833i	$0.603268\pi$
$632$	39.0181	1.55206
$633$	21.0517	0.836728
$634$	−10.5535	−0.419134
$635$	−31.0716	−1.23304
$636$	6.86854	0.272356
$637$	−10.0764	−0.399241
$638$	3.09336	0.122467
$639$	16.1231	0.637822
$640$	−40.7440	−1.61055
$641$	24.9229	0.984396	0.492198	−	0.870483i	$-0.336194\pi$
$641$	24.9229	0.984396	0.492198	+	0.870483i	$0.336194\pi$
$642$	11.7372	0.463229
$643$	11.8125	0.465838	0.232919	−	0.972496i	$-0.425172\pi$
$643$	11.8125	0.465838	0.232919	+	0.972496i	$0.425172\pi$
$644$	23.8902	0.941404
$645$	18.8626	0.742714
$646$	−15.9374	−0.627048
$647$	−37.8707	−1.48885	−0.744426	−	0.667706i	$-0.767277\pi$
$647$	−37.8707	−1.48885	−0.744426	+	0.667706i	$0.767277\pi$
$648$	9.43729	0.370732
$649$	9.89674	0.388481
$650$	−4.89033	−0.191815
$651$	−9.26729	−0.363214
$652$	17.8616	0.699516
$653$	−33.5190	−1.31170	−0.655850	−	0.754891i	$-0.727690\pi$
$653$	−33.5190	−1.31170	−0.655850	+	0.754891i	$0.727690\pi$
$654$	16.7955	0.656756
$655$	−59.1828	−2.31246
$656$	−1.41231	−0.0551415
$657$	−8.52494	−0.332589
$658$	−27.0101	−1.05296
$659$	27.9658	1.08939	0.544696	−	0.838633i	$-0.316645\pi$
$659$	27.9658	1.08939	0.544696	+	0.838633i	$0.316645\pi$
$660$	7.72312	0.300622
$661$	−18.1170	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$661$	−18.1170	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$662$	11.7043	0.454902
$663$	−3.91280	−0.151961
$664$	−20.3962	−0.791527
$665$	103.438	4.01115
$666$	−8.58230	−0.332557
$667$	−15.7136	−0.608434
$668$	−4.24875	−0.164389
$669$	−20.9864	−0.811382
$670$	−20.4129	−0.788621
$671$	−1.00000	−0.0386046
$672$	−34.7759	−1.34151
$673$	−1.95653	−0.0754186	−0.0377093	−	0.999289i	$-0.512006\pi$
$673$	−1.95653	−0.0754186	−0.0377093	+	0.999289i	$0.512006\pi$
$674$	−3.35087	−0.129071
$675$	52.1575	2.00754
$676$	18.9747	0.729794
$677$	−20.6599	−0.794025	−0.397013	−	0.917813i	$-0.629953\pi$
$677$	−20.6599	−0.794025	−0.397013	+	0.917813i	$0.629953\pi$
$678$	12.4934	0.479806
$679$	62.5430	2.40018
$680$	34.7588	1.33294
$681$	5.41748	0.207598
$682$	1.06070	0.0406162
$683$	−12.0060	−0.459397	−0.229699	−	0.973262i	$-0.573774\pi$
$683$	−12.0060	−0.459397	−0.229699	+	0.973262i	$0.573774\pi$
$684$	11.3224	0.432923
$685$	11.9521	0.456666
$686$	18.7326	0.715214
$687$	−9.46025	−0.360931
$688$	−5.21885	−0.198967
$689$	−2.58500	−0.0984807
$690$	12.0757	0.459715
$691$	−29.6509	−1.12797	−0.563986	−	0.825784i	$-0.690733\pi$
$691$	−29.6509	−1.12797	−0.563986	+	0.825784i	$0.690733\pi$
$692$	19.1457	0.727812
$693$	−5.43061	−0.206292
$694$	−12.8450	−0.487589
$695$	13.4655	0.510777
$696$	14.6000	0.553413
$697$	3.84221	0.145534
$698$	−2.94313	−0.111399
$699$	8.01523	0.303164
$700$	−63.4788	−2.39927
$701$	−2.43910	−0.0921235	−0.0460618	−	0.998939i	$-0.514667\pi$
$701$	−2.43910	−0.0921235	−0.0460618	+	0.998939i	$0.514667\pi$
$702$	−2.97428	−0.112257
$703$	63.0937	2.37962
$704$	1.18579	0.0446910
$705$	44.3553	1.67052
$706$	−21.1567	−0.796242
$707$	4.55422	0.171279
$708$	20.2403	0.760677
$709$	−36.3219	−1.36410	−0.682049	−	0.731307i	$-0.738911\pi$
$709$	−36.3219	−1.36410	−0.682049	+	0.731307i	$0.738911\pi$
$710$	−34.4790	−1.29397
$711$	19.5229	0.732164
$712$	−6.99093	−0.261996
$713$	−5.38813	−0.201787
$714$	15.6333	0.585063
$715$	−2.90662	−0.108702
$716$	−18.1908	−0.679823
$717$	8.16363	0.304876
$718$	11.9861	0.447317
$719$	3.36651	0.125550	0.0627748	−	0.998028i	$-0.480005\pi$
$719$	3.36651	0.125550	0.0627748	+	0.998028i	$0.480005\pi$
$720$	−6.39280	−0.238246
$721$	−56.5931	−2.10764
$722$	12.5852	0.468374
$723$	−4.89697	−0.182120
$724$	4.24688	0.157834
$725$	41.7528	1.55066
$726$	−0.917525	−0.0340526
$727$	8.51027	0.315629	0.157814	−	0.987469i	$-0.449555\pi$
$727$	8.51027	0.315629	0.157814	+	0.987469i	$0.449555\pi$
$728$	8.35396	0.309618
$729$	27.3184	1.01179
$730$	18.2304	0.674736
$731$	14.1980	0.525130
$732$	−2.04515	−0.0755908
$733$	−14.2509	−0.526368	−0.263184	−	0.964746i	$-0.584773\pi$
$733$	−14.2509	−0.526368	−0.263184	+	0.964746i	$0.584773\pi$
$734$	22.4170	0.827427
$735$	−66.1134	−2.43863
$736$	−20.2192	−0.745289
$737$	−7.87874	−0.290217
$738$	0.840190	0.0309278
$739$	4.37346	0.160880	0.0804402	−	0.996759i	$-0.474367\pi$
$739$	4.37346	0.160880	0.0804402	+	0.996759i	$0.474367\pi$
$740$	−59.6258	−2.19189
$741$	6.29021	0.231077
$742$	10.3282	0.379161
$743$	−4.92865	−0.180815	−0.0904074	−	0.995905i	$-0.528817\pi$
$743$	−4.92865	−0.180815	−0.0904074	+	0.995905i	$0.528817\pi$
$744$	5.00628	0.183539
$745$	50.3849	1.84596
$746$	10.3104	0.377491
$747$	−10.2053	−0.373394
$748$	5.81322	0.212552
$749$	−57.3389	−2.09512
$750$	−14.7622	−0.539040
$751$	−35.7468	−1.30442	−0.652210	−	0.758038i	$-0.726158\pi$
$751$	−35.7468	−1.30442	−0.652210	+	0.758038i	$0.726158\pi$
$752$	−12.2721	−0.447517
$753$	38.0097	1.38515
$754$	−2.38095	−0.0867093
$755$	83.4638	3.03756
$756$	−38.6075	−1.40414
$757$	18.0088	0.654540	0.327270	−	0.944931i	$-0.393871\pi$
$757$	18.0088	0.654540	0.327270	+	0.944931i	$0.393871\pi$
$758$	13.3026	0.483172
$759$	4.66084	0.169178
$760$	−55.8782	−2.02691
$761$	49.4533	1.79268	0.896341	−	0.443366i	$-0.146216\pi$
$761$	49.4533	1.79268	0.896341	+	0.443366i	$0.146216\pi$
$762$	7.54942	0.273487
$763$	−82.0501	−2.97041
$764$	16.0448	0.580482
$765$	17.3917	0.628798
$766$	−20.2203	−0.730589
$767$	−7.61751	−0.275052
$768$	13.0711	0.471662
$769$	36.3593	1.31115	0.655575	−	0.755130i	$-0.272426\pi$
$769$	36.3593	1.31115	0.655575	+	0.755130i	$0.272426\pi$
$770$	11.6132	0.418512
$771$	−19.6640	−0.708183
$772$	−24.7490	−0.890736
$773$	16.1097	0.579427	0.289714	−	0.957113i	$-0.406440\pi$
$773$	16.1097	0.579427	0.289714	+	0.957113i	$0.406440\pi$
$774$	3.10471	0.111597
$775$	14.3168	0.514276
$776$	−33.7863	−1.21286
$777$	−61.8900	−2.22029
$778$	−8.50107	−0.304778
$779$	−6.17674	−0.221305
$780$	−5.94448	−0.212846
$781$	−13.3078	−0.476190
$782$	9.08944	0.325038
$783$	25.3939	0.907505
$784$	18.2921	0.653288
$785$	−37.2299	−1.32879
$786$	14.3796	0.512902
$787$	15.6275	0.557061	0.278530	−	0.960427i	$-0.410153\pi$
$787$	15.6275	0.557061	0.278530	+	0.960427i	$0.410153\pi$
$788$	−7.79959	−0.277849
$789$	17.8056	0.633896
$790$	−41.7492	−1.48537
$791$	−61.0333	−2.17009
$792$	2.93367	0.104243
$793$	0.769699	0.0273328
$794$	−3.35880	−0.119199
$795$	−16.9608	−0.601536
$796$	−5.45949	−0.193507
$797$	−37.0889	−1.31376	−0.656879	−	0.753996i	$-0.728124\pi$
$797$	−37.0889	−1.31376	−0.656879	+	0.753996i	$0.728124\pi$
$798$	−25.1321	−0.889668
$799$	33.3864	1.18113
$800$	53.7246	1.89945
$801$	−3.49794	−0.123594
$802$	0.214714	0.00758181
$803$	7.03634	0.248307
$804$	−16.1132	−0.568269
$805$	−58.9929	−2.07923
$806$	−0.816418	−0.0287571
$807$	31.0227	1.09205
$808$	−2.46024	−0.0865508
$809$	24.3691	0.856771	0.428386	−	0.903596i	$-0.359082\pi$
$809$	24.3691	0.856771	0.428386	+	0.903596i	$0.359082\pi$
$810$	−10.0979	−0.354803
$811$	39.6765	1.39323	0.696614	−	0.717446i	$-0.254689\pi$
$811$	39.6765	1.39323	0.696614	+	0.717446i	$0.254689\pi$
$812$	−30.9059	−1.08458
$813$	28.6590	1.00511
$814$	7.08369	0.248283
$815$	−44.1064	−1.54498
$816$	7.10306	0.248657
$817$	−22.8246	−0.798532
$818$	1.76289	0.0616382
$819$	4.17994	0.146059
$820$	5.83724	0.203845
$821$	25.3389	0.884333	0.442167	−	0.896933i	$-0.354210\pi$
$821$	25.3389	0.884333	0.442167	+	0.896933i	$0.354210\pi$
$822$	−2.90399	−0.101288
$823$	−10.4823	−0.365389	−0.182695	−	0.983170i	$-0.558482\pi$
$823$	−10.4823	−0.365389	−0.182695	+	0.983170i	$0.558482\pi$
$824$	30.5721	1.06503
$825$	−12.3844	−0.431168
$826$	30.4353	1.05898
$827$	−29.9510	−1.04150	−0.520749	−	0.853710i	$-0.674347\pi$
$827$	−29.9510	−1.04150	−0.520749	+	0.853710i	$0.674347\pi$
$828$	−6.45742	−0.224411
$829$	−28.6181	−0.993946	−0.496973	−	0.867766i	$-0.665555\pi$
$829$	−28.6181	−0.993946	−0.496973	+	0.867766i	$0.665555\pi$
$830$	21.8239	0.757517
$831$	−15.3889	−0.533836
$832$	−0.912699	−0.0316421
$833$	−49.7638	−1.72421
$834$	−3.27170	−0.113290
$835$	10.4916	0.363077
$836$	−9.34533	−0.323215
$837$	8.70745	0.300974
$838$	−7.40962	−0.255961
$839$	36.0462	1.24445	0.622227	−	0.782837i	$-0.286228\pi$
$839$	36.0462	1.24445	0.622227	+	0.782837i	$0.286228\pi$
$840$	54.8122	1.89120
$841$	−8.67179	−0.299027
$842$	17.5512	0.604854
$843$	−12.3571	−0.425602
$844$	24.0733	0.828638
$845$	−46.8549	−1.61186
$846$	7.30072	0.251004
$847$	4.48234	0.154015
$848$	4.69265	0.161146
$849$	−34.5191	−1.18469
$850$	−24.1516	−0.828394
$851$	−35.9837	−1.23350
$852$	−27.2164	−0.932417
$853$	−42.5109	−1.45554	−0.727772	−	0.685819i	$-0.759444\pi$
$853$	−42.5109	−1.45554	−0.727772	+	0.685819i	$0.759444\pi$
$854$	−3.07528	−0.105234
$855$	−27.9589	−0.956173
$856$	30.9750	1.05871
$857$	8.59466	0.293588	0.146794	−	0.989167i	$-0.453105\pi$
$857$	8.59466	0.293588	0.146794	+	0.989167i	$0.453105\pi$
$858$	0.706218	0.0241099
$859$	−9.56524	−0.326362	−0.163181	−	0.986596i	$-0.552175\pi$
$859$	−9.56524	−0.326362	−0.163181	+	0.986596i	$0.552175\pi$
$860$	21.5701	0.735534
$861$	6.05890	0.206487
$862$	16.6357	0.566616
$863$	40.9448	1.39378	0.696888	−	0.717180i	$-0.254568\pi$
$863$	40.9448	1.39378	0.696888	+	0.717180i	$0.254568\pi$
$864$	32.6751	1.11163
$865$	−47.2773	−1.60748
$866$	−3.93058	−0.133567
$867$	3.41058	0.115829
$868$	−10.5975	−0.359702
$869$	−16.1138	−0.546625
$870$	−15.6220	−0.529634
$871$	6.06426	0.205480
$872$	44.3243	1.50101
$873$	−16.9051	−0.572152
$874$	−14.6122	−0.494264
$875$	72.1170	2.43800
$876$	14.3904	0.486205
$877$	35.9784	1.21490	0.607451	−	0.794357i	$-0.292192\pi$
$877$	35.9784	1.21490	0.607451	+	0.794357i	$0.292192\pi$
$878$	2.52725	0.0852904
$879$	−16.9725	−0.572467
$880$	5.27651	0.177871
$881$	20.3508	0.685636	0.342818	−	0.939402i	$-0.388619\pi$
$881$	20.3508	0.685636	0.342818	+	0.939402i	$0.388619\pi$
$882$	−10.8820	−0.366417
$883$	29.2431	0.984108	0.492054	−	0.870565i	$-0.336246\pi$
$883$	29.2431	0.984108	0.492054	+	0.870565i	$0.336246\pi$
$884$	−4.47443	−0.150491
$885$	−49.9802	−1.68007
$886$	−26.6778	−0.896257
$887$	−7.26531	−0.243945	−0.121973	−	0.992533i	$-0.538922\pi$
$887$	−7.26531	−0.243945	−0.121973	+	0.992533i	$0.538922\pi$
$888$	33.4336	1.12196
$889$	−36.8808	−1.23694
$890$	7.48026	0.250739
$891$	−3.89745	−0.130569
$892$	−23.9987	−0.803538
$893$	−53.6720	−1.79606
$894$	−12.2420	−0.409432
$895$	44.9193	1.50149
$896$	−48.3615	−1.61564
$897$	−3.58744	−0.119781
$898$	7.34509	0.245109
$899$	6.97044	0.232477
$900$	17.1581	0.571936
$901$	−12.7664	−0.425311
$902$	−0.693478	−0.0230903
$903$	22.3892	0.745065
$904$	32.9708	1.09659
$905$	−10.4870	−0.348599
$906$	−20.2791	−0.673727
$907$	24.6557	0.818680	0.409340	−	0.912382i	$-0.365759\pi$
$907$	24.6557	0.818680	0.409340	+	0.912382i	$0.365759\pi$
$908$	6.19509	0.205591
$909$	−1.23099	−0.0408293
$910$	−8.93870	−0.296315
$911$	9.27932	0.307437	0.153719	−	0.988115i	$-0.450875\pi$
$911$	9.27932	0.307437	0.153719	+	0.988115i	$0.450875\pi$
$912$	−11.4189	−0.378116
$913$	8.42331	0.278771
$914$	24.0832	0.796602
$915$	5.05016	0.166953
$916$	−10.8181	−0.357442
$917$	−70.2477	−2.31978
$918$	−14.6889	−0.484807
$919$	−0.138091	−0.00455519	−0.00227759	−	0.999997i	$-0.500725\pi$
$919$	−0.138091	−0.00455519	−0.00227759	+	0.999997i	$0.500725\pi$
$920$	31.8685	1.05067
$921$	24.0522	0.792546
$922$	5.24942	0.172880
$923$	10.2430	0.337152
$924$	9.16704	0.301574
$925$	95.6126	3.14372
$926$	−22.1825	−0.728963
$927$	15.2969	0.502416
$928$	26.1569	0.858642
$929$	−4.82781	−0.158395	−0.0791977	−	0.996859i	$-0.525236\pi$
$929$	−4.82781	−0.158395	−0.0791977	+	0.996859i	$0.525236\pi$
$930$	−5.35670	−0.175653
$931$	80.0002	2.62190
$932$	9.16571	0.300233
$933$	−27.5272	−0.901201
$934$	1.04931	0.0343344
$935$	−14.3548	−0.469453
$936$	−2.25804	−0.0738065
$937$	−16.4702	−0.538058	−0.269029	−	0.963132i	$-0.586703\pi$
$937$	−16.4702	−0.538058	−0.269029	+	0.963132i	$0.586703\pi$
$938$	−24.2294	−0.791117
$939$	26.9843	0.880601
$940$	50.7219	1.65437
$941$	54.5905	1.77960	0.889800	−	0.456350i	$-0.150843\pi$
$941$	54.5905	1.77960	0.889800	+	0.456350i	$0.150843\pi$
$942$	9.04569	0.294725
$943$	3.52273	0.114716
$944$	13.8284	0.450075
$945$	95.3351	3.10125
$946$	−2.56258	−0.0833166
$947$	3.35352	0.108975	0.0544874	−	0.998514i	$-0.482648\pi$
$947$	3.35352	0.108975	0.0544874	+	0.998514i	$0.482648\pi$
$948$	−32.9552	−1.07033
$949$	−5.41586	−0.175806
$950$	38.8262	1.25969
$951$	20.5710	0.667059
$952$	41.2573	1.33716
$953$	19.0563	0.617293	0.308647	−	0.951177i	$-0.400124\pi$
$953$	19.0563	0.617293	0.308647	+	0.951177i	$0.400124\pi$
$954$	−2.79168	−0.0903839
$955$	−39.6202	−1.28208
$956$	9.33541	0.301929
$957$	−6.02958	−0.194909
$958$	23.4876	0.758851
$959$	14.1867	0.458112
$960$	−5.98842	−0.193275
$961$	−28.6099	−0.922899
$962$	−5.45231	−0.175789
$963$	15.4985	0.499432
$964$	−5.59987	−0.180360
$965$	61.1137	1.96732
$966$	14.3334	0.461170
$967$	−25.4500	−0.818415	−0.409208	−	0.912441i	$-0.634195\pi$
$967$	−25.4500	−0.818415	−0.409208	+	0.912441i	$0.634195\pi$
$968$	−2.42140	−0.0778268
$969$	31.0652	0.997957
$970$	36.1512	1.16075
$971$	−29.6230	−0.950648	−0.475324	−	0.879811i	$-0.657669\pi$
$971$	−29.6230	−0.950648	−0.475324	+	0.879811i	$0.657669\pi$
$972$	17.8689	0.573146
$973$	15.9831	0.512393
$974$	24.6741	0.790608
$975$	9.53224	0.305276
$976$	−1.39726	−0.0447253
$977$	7.18850	0.229980	0.114990	−	0.993367i	$-0.463316\pi$
$977$	7.18850	0.229980	0.114990	+	0.993367i	$0.463316\pi$
$978$	10.7165	0.342675
$979$	2.88714	0.0922734
$980$	−75.6031	−2.41505
$981$	22.1778	0.708084
$982$	16.0326	0.511620
$983$	−39.9478	−1.27414	−0.637068	−	0.770807i	$-0.719853\pi$
$983$	−39.9478	−1.27414	−0.637068	+	0.770807i	$0.719853\pi$
$984$	−3.27308	−0.104342
$985$	19.2598	0.613669
$986$	−11.7587	−0.374474
$987$	52.6480	1.67581
$988$	7.19309	0.228843
$989$	13.0174	0.413928
$990$	−3.13901	−0.0997644
$991$	11.8963	0.377897	0.188949	−	0.981987i	$-0.439492\pi$
$991$	11.8963	0.377897	0.188949	+	0.981987i	$0.439492\pi$
$992$	8.96908	0.284768
$993$	−22.8141	−0.723983
$994$	−40.9252	−1.29807
$995$	13.4813	0.427387
$996$	17.2269	0.545856
$997$	−6.22191	−0.197050	−0.0985249	−	0.995135i	$-0.531412\pi$
$997$	−6.22191	−0.197050	−0.0985249	+	0.995135i	$0.531412\pi$
$998$	0.621618	0.0196770
$999$	58.1512	1.83982

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.a.c.1.11	✓	19
3.2	odd	2		6039.2.a.k.1.9		19
11.10	odd	2		7381.2.a.i.1.9		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.c.1.11	✓	19	1.1	even	1	trivial
6039.2.a.k.1.9		19	3.2	odd	2
7381.2.a.i.1.9		19	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.a (trivial)

\(f(q)\)	\(=\)	\(q+0.686089 q^{2} -1.33733 q^{3} -1.52928 q^{4} +3.77631 q^{5} -0.917525 q^{6} +4.48234 q^{7} -2.42140 q^{8} -1.21156 q^{9} +O(q^{10})\)	\(q+0.686089 q^{2} -1.33733 q^{3} -1.52928 q^{4} +3.77631 q^{5} -0.917525 q^{6} +4.48234 q^{7} -2.42140 q^{8} -1.21156 q^{9} +2.59089 q^{10} +1.00000 q^{11} +2.04515 q^{12} -0.769699 q^{13} +3.07528 q^{14} -5.05016 q^{15} +1.39726 q^{16} -3.80128 q^{17} -0.831238 q^{18} +6.11093 q^{19} -5.77505 q^{20} -5.99435 q^{21} +0.686089 q^{22} -3.48519 q^{23} +3.23821 q^{24} +9.26054 q^{25} -0.528082 q^{26} +5.63223 q^{27} -6.85475 q^{28} +4.50868 q^{29} -3.46486 q^{30} +1.54601 q^{31} +5.80145 q^{32} -1.33733 q^{33} -2.60802 q^{34} +16.9267 q^{35} +1.85281 q^{36} +10.3247 q^{37} +4.19264 q^{38} +1.02934 q^{39} -9.14398 q^{40} -1.01077 q^{41} -4.11266 q^{42} -3.73505 q^{43} -1.52928 q^{44} -4.57523 q^{45} -2.39115 q^{46} -8.78295 q^{47} -1.86860 q^{48} +13.0913 q^{49} +6.35356 q^{50} +5.08355 q^{51} +1.17709 q^{52} +3.35846 q^{53} +3.86421 q^{54} +3.77631 q^{55} -10.8535 q^{56} -8.17230 q^{57} +3.09336 q^{58} +9.89674 q^{59} +7.72312 q^{60} -1.00000 q^{61} +1.06070 q^{62} -5.43061 q^{63} +1.18579 q^{64} -2.90662 q^{65} -0.917525 q^{66} -7.87874 q^{67} +5.81322 q^{68} +4.66084 q^{69} +11.6132 q^{70} -13.3078 q^{71} +2.93367 q^{72} +7.03634 q^{73} +7.08369 q^{74} -12.3844 q^{75} -9.34533 q^{76} +4.48234 q^{77} +0.706218 q^{78} -16.1138 q^{79} +5.27651 q^{80} -3.89745 q^{81} -0.693478 q^{82} +8.42331 q^{83} +9.16704 q^{84} -14.3548 q^{85} -2.56258 q^{86} -6.02958 q^{87} -2.42140 q^{88} +2.88714 q^{89} -3.13901 q^{90} -3.45005 q^{91} +5.32984 q^{92} -2.06751 q^{93} -6.02589 q^{94} +23.0768 q^{95} -7.75844 q^{96} +13.9532 q^{97} +8.98183 q^{98} -1.21156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 5 q^{2} + 23 q^{4} + q^{6} + 9 q^{7} + 9 q^{8} + 29 q^{9}+O(q^{10}) \) 19 * q + 5 * q^2 + 23 * q^4 + q^6 + 9 * q^7 + 9 * q^8 + 29 * q^9	\( 19 q + 5 q^{2} + 23 q^{4} + q^{6} + 9 q^{7} + 9 q^{8} + 29 q^{9} + 7 q^{10} + 19 q^{11} - 4 q^{12} + 8 q^{13} - 11 q^{14} - 5 q^{15} + 31 q^{16} + 9 q^{17} + 10 q^{18} + 17 q^{19} - 6 q^{20} + 18 q^{21} + 5 q^{22} - 10 q^{23} + 26 q^{24} + 45 q^{25} + 5 q^{26} - 33 q^{27} + 36 q^{28} + 27 q^{29} - 30 q^{30} + 7 q^{31} + 8 q^{32} - 5 q^{34} + 17 q^{35} + 38 q^{36} + 20 q^{37} - 37 q^{38} + 24 q^{39} + 10 q^{40} + 19 q^{41} + 21 q^{42} + 20 q^{43} + 23 q^{44} - 32 q^{45} + 41 q^{46} - 19 q^{47} + 5 q^{48} + 42 q^{49} + 36 q^{50} + 47 q^{51} - 28 q^{52} + 3 q^{53} - 33 q^{54} - 44 q^{56} + 11 q^{57} + 23 q^{58} - 28 q^{59} - 96 q^{60} - 19 q^{61} - 11 q^{62} - 32 q^{63} + 47 q^{64} + 25 q^{65} + q^{66} + 3 q^{67} + 38 q^{68} + 3 q^{70} - 19 q^{71} + 34 q^{72} + 20 q^{73} - 22 q^{74} - 50 q^{75} - 25 q^{76} + 9 q^{77} - 94 q^{78} + 69 q^{79} - 36 q^{80} + 47 q^{81} - 61 q^{82} + q^{83} - 28 q^{84} + 24 q^{85} - 27 q^{86} - 58 q^{87} + 9 q^{88} + 36 q^{90} + 24 q^{91} - 67 q^{92} - 14 q^{93} + 64 q^{94} - 3 q^{95} - 26 q^{96} + 21 q^{97} - 87 q^{98} + 29 q^{99}+O(q^{100}) \) 19 * q + 5 * q^2 + 23 * q^4 + q^6 + 9 * q^7 + 9 * q^8 + 29 * q^9 + 7 * q^10 + 19 * q^11 - 4 * q^12 + 8 * q^13 - 11 * q^14 - 5 * q^15 + 31 * q^16 + 9 * q^17 + 10 * q^18 + 17 * q^19 - 6 * q^20 + 18 * q^21 + 5 * q^22 - 10 * q^23 + 26 * q^24 + 45 * q^25 + 5 * q^26 - 33 * q^27 + 36 * q^28 + 27 * q^29 - 30 * q^30 + 7 * q^31 + 8 * q^32 - 5 * q^34 + 17 * q^35 + 38 * q^36 + 20 * q^37 - 37 * q^38 + 24 * q^39 + 10 * q^40 + 19 * q^41 + 21 * q^42 + 20 * q^43 + 23 * q^44 - 32 * q^45 + 41 * q^46 - 19 * q^47 + 5 * q^48 + 42 * q^49 + 36 * q^50 + 47 * q^51 - 28 * q^52 + 3 * q^53 - 33 * q^54 - 44 * q^56 + 11 * q^57 + 23 * q^58 - 28 * q^59 - 96 * q^60 - 19 * q^61 - 11 * q^62 - 32 * q^63 + 47 * q^64 + 25 * q^65 + q^66 + 3 * q^67 + 38 * q^68 + 3 * q^70 - 19 * q^71 + 34 * q^72 + 20 * q^73 - 22 * q^74 - 50 * q^75 - 25 * q^76 + 9 * q^77 - 94 * q^78 + 69 * q^79 - 36 * q^80 + 47 * q^81 - 61 * q^82 + q^83 - 28 * q^84 + 24 * q^85 - 27 * q^86 - 58 * q^87 + 9 * q^88 + 36 * q^90 + 24 * q^91 - 67 * q^92 - 14 * q^93 + 64 * q^94 - 3 * q^95 - 26 * q^96 + 21 * q^97 - 87 * q^98 + 29 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more