LMFDB - Embedded newform 349.2.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [349,2,Mod(1,349)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(349, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("349.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 349 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	349.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:	$2.78677903054$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 14 x^{15} + 102 x^{14} + 26 x^{13} - 792 x^{12} + 474 x^{11} + 2887 x^{10} + \cdots - 4 $ x^17 - 5x^16 - 14x^15 + 102x^14 + 26x^13 - 792x^12 + 474x^11 + 2887x^10 - 3021x^9 - 4835x^8 + 6673x^7 + 2880x^6 - 5373x^5 - 164x^4 + 1075x^3 + 75x^2 - 41x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.226775$ of defining polynomial
Character	$\chi$	$=$	349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.226775 q^{2} -1.51708 q^{3} -1.94857 q^{4} +2.45530 q^{5} -0.344035 q^{6} +0.487144 q^{7} -0.895438 q^{8} -0.698475 q^{9} +O(q^{10})$	\(q+0.226775 q^{2} -1.51708 q^{3} -1.94857 q^{4} +2.45530 q^{5} -0.344035 q^{6} +0.487144 q^{7} -0.895438 q^{8} -0.698475 q^{9} +0.556801 q^{10} +2.99449 q^{11} +2.95614 q^{12} +1.49100 q^{13} +0.110472 q^{14} -3.72489 q^{15} +3.69408 q^{16} +3.52560 q^{17} -0.158397 q^{18} +6.32323 q^{19} -4.78434 q^{20} -0.739035 q^{21} +0.679075 q^{22} +1.27434 q^{23} +1.35845 q^{24} +1.02851 q^{25} +0.338121 q^{26} +5.61087 q^{27} -0.949235 q^{28} -8.44061 q^{29} -0.844711 q^{30} +10.3672 q^{31} +2.62860 q^{32} -4.54287 q^{33} +0.799517 q^{34} +1.19609 q^{35} +1.36103 q^{36} -10.1239 q^{37} +1.43395 q^{38} -2.26196 q^{39} -2.19857 q^{40} +9.81464 q^{41} -0.167595 q^{42} +11.3047 q^{43} -5.83498 q^{44} -1.71497 q^{45} +0.288989 q^{46} -8.37713 q^{47} -5.60421 q^{48} -6.76269 q^{49} +0.233241 q^{50} -5.34860 q^{51} -2.90532 q^{52} +1.19162 q^{53} +1.27241 q^{54} +7.35238 q^{55} -0.436207 q^{56} -9.59283 q^{57} -1.91412 q^{58} -6.98180 q^{59} +7.25821 q^{60} -12.9039 q^{61} +2.35103 q^{62} -0.340258 q^{63} -6.79207 q^{64} +3.66085 q^{65} -1.03021 q^{66} -0.801136 q^{67} -6.86988 q^{68} -1.93327 q^{69} +0.271242 q^{70} -0.317689 q^{71} +0.625441 q^{72} -4.67967 q^{73} -2.29584 q^{74} -1.56033 q^{75} -12.3213 q^{76} +1.45875 q^{77} -0.512955 q^{78} -2.75962 q^{79} +9.07009 q^{80} -6.41671 q^{81} +2.22572 q^{82} +4.05170 q^{83} +1.44006 q^{84} +8.65641 q^{85} +2.56362 q^{86} +12.8051 q^{87} -2.68138 q^{88} +14.6322 q^{89} -0.388912 q^{90} +0.726329 q^{91} -2.48315 q^{92} -15.7279 q^{93} -1.89972 q^{94} +15.5254 q^{95} -3.98779 q^{96} +16.2908 q^{97} -1.53361 q^{98} -2.09158 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 5 q^{2} + 6 q^{3} + 19 q^{4} + 5 q^{5} - 3 q^{6} + q^{7} + 9 q^{8} + 25 q^{9}+O(q^{10}) $ 17 * q + 5 * q^2 + 6 * q^3 + 19 * q^4 + 5 * q^5 - 3 * q^6 + q^7 + 9 * q^8 + 25 * q^9	\( 17 q + 5 q^{2} + 6 q^{3} + 19 q^{4} + 5 q^{5} - 3 q^{6} + q^{7} + 9 q^{8} + 25 q^{9} - 6 q^{10} + 39 q^{11} + 4 q^{12} - 6 q^{13} + 11 q^{14} + 12 q^{15} + 23 q^{16} + q^{17} + 7 q^{19} + 8 q^{20} - 3 q^{21} - q^{22} + 8 q^{23} - 21 q^{24} + 14 q^{25} + q^{26} + 9 q^{27} - 23 q^{28} + 9 q^{29} - 27 q^{30} - 2 q^{31} + 23 q^{32} - 5 q^{33} - 12 q^{34} + 16 q^{35} + 10 q^{36} - 7 q^{37} - 8 q^{38} - 39 q^{40} + 3 q^{41} - 30 q^{42} + q^{43} + 56 q^{44} - 21 q^{45} - q^{46} + 16 q^{47} - 31 q^{48} + 6 q^{49} - 5 q^{50} + 29 q^{51} - 37 q^{52} + 27 q^{53} - 36 q^{54} - 16 q^{55} + 14 q^{56} - 21 q^{57} - 44 q^{58} + 74 q^{59} - 27 q^{60} - 30 q^{61} - 18 q^{62} - 9 q^{63} - 17 q^{64} - 3 q^{65} - 66 q^{66} + 9 q^{67} - 51 q^{68} - 23 q^{69} - 77 q^{70} + 70 q^{71} - 55 q^{72} - 38 q^{73} + 26 q^{74} + 22 q^{75} - 50 q^{76} - 38 q^{77} - 75 q^{78} + 7 q^{79} - 28 q^{80} + 5 q^{81} - 56 q^{82} + 47 q^{83} - 69 q^{84} - 49 q^{85} + 17 q^{86} - 37 q^{87} - 30 q^{88} + 23 q^{89} - 64 q^{90} + 6 q^{91} + 17 q^{92} - 31 q^{93} - 40 q^{94} - 11 q^{95} - 49 q^{96} - 14 q^{97} + 5 q^{98} + 79 q^{99}+O(q^{100}) \) 17 * q + 5 * q^2 + 6 * q^3 + 19 * q^4 + 5 * q^5 - 3 * q^6 + q^7 + 9 * q^8 + 25 * q^9 - 6 * q^10 + 39 * q^11 + 4 * q^12 - 6 * q^13 + 11 * q^14 + 12 * q^15 + 23 * q^16 + q^17 + 7 * q^19 + 8 * q^20 - 3 * q^21 - q^22 + 8 * q^23 - 21 * q^24 + 14 * q^25 + q^26 + 9 * q^27 - 23 * q^28 + 9 * q^29 - 27 * q^30 - 2 * q^31 + 23 * q^32 - 5 * q^33 - 12 * q^34 + 16 * q^35 + 10 * q^36 - 7 * q^37 - 8 * q^38 - 39 * q^40 + 3 * q^41 - 30 * q^42 + q^43 + 56 * q^44 - 21 * q^45 - q^46 + 16 * q^47 - 31 * q^48 + 6 * q^49 - 5 * q^50 + 29 * q^51 - 37 * q^52 + 27 * q^53 - 36 * q^54 - 16 * q^55 + 14 * q^56 - 21 * q^57 - 44 * q^58 + 74 * q^59 - 27 * q^60 - 30 * q^61 - 18 * q^62 - 9 * q^63 - 17 * q^64 - 3 * q^65 - 66 * q^66 + 9 * q^67 - 51 * q^68 - 23 * q^69 - 77 * q^70 + 70 * q^71 - 55 * q^72 - 38 * q^73 + 26 * q^74 + 22 * q^75 - 50 * q^76 - 38 * q^77 - 75 * q^78 + 7 * q^79 - 28 * q^80 + 5 * q^81 - 56 * q^82 + 47 * q^83 - 69 * q^84 - 49 * q^85 + 17 * q^86 - 37 * q^87 - 30 * q^88 + 23 * q^89 - 64 * q^90 + 6 * q^91 + 17 * q^92 - 31 * q^93 - 40 * q^94 - 11 * q^95 - 49 * q^96 - 14 * q^97 + 5 * q^98 + 79 * 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 $

$2$	0.226775	0.160354	0.0801771	−	0.996781i	$-0.474451\pi$
$2$	0.226775	0.160354	0.0801771	+	0.996781i	$0.474451\pi$
$3$	−1.51708	−0.875885	−0.437943	−	0.899003i	$-0.644293\pi$
$3$	−1.51708	−0.875885	−0.437943	+	0.899003i	$0.644293\pi$
$4$	−1.94857	−0.974287
$5$	2.45530	1.09804	0.549022	−	0.835808i	$-0.315000\pi$
$5$	2.45530	1.09804	0.549022	+	0.835808i	$0.315000\pi$
$6$	−0.344035	−0.140452
$7$	0.487144	0.184123	0.0920615	−	0.995753i	$-0.470654\pi$
$7$	0.487144	0.184123	0.0920615	+	0.995753i	$0.470654\pi$
$8$	−0.895438	−0.316585
$9$	−0.698475	−0.232825
$10$	0.556801	0.176076
$11$	2.99449	0.902872	0.451436	−	0.892303i	$-0.350912\pi$
$11$	2.99449	0.902872	0.451436	+	0.892303i	$0.350912\pi$
$12$	2.95614	0.853363
$13$	1.49100	0.413528	0.206764	−	0.978391i	$-0.433707\pi$
$13$	1.49100	0.413528	0.206764	+	0.978391i	$0.433707\pi$
$14$	0.110472	0.0295249
$15$	−3.72489	−0.961761
$16$	3.69408	0.923521
$17$	3.52560	0.855083	0.427541	−	0.903996i	$-0.359380\pi$
$17$	3.52560	0.855083	0.427541	+	0.903996i	$0.359380\pi$
$18$	−0.158397	−0.0373345
$19$	6.32323	1.45065	0.725324	−	0.688407i	$-0.241690\pi$
$19$	6.32323	1.45065	0.725324	+	0.688407i	$0.241690\pi$
$20$	−4.78434	−1.06981
$21$	−0.739035	−0.161271
$22$	0.679075	0.144779
$23$	1.27434	0.265718	0.132859	−	0.991135i	$-0.457584\pi$
$23$	1.27434	0.265718	0.132859	+	0.991135i	$0.457584\pi$
$24$	1.35845	0.277292
$25$	1.02851	0.205703
$26$	0.338121	0.0663109
$27$	5.61087	1.07981
$28$	−0.949235	−0.179389
$29$	−8.44061	−1.56738	−0.783691	−	0.621151i	$-0.786665\pi$
$29$	−8.44061	−1.56738	−0.783691	+	0.621151i	$0.786665\pi$
$30$	−0.844711	−0.154222
$31$	10.3672	1.86201	0.931006	−	0.365003i	$-0.118932\pi$
$31$	10.3672	1.86201	0.931006	+	0.365003i	$0.118932\pi$
$32$	2.62860	0.464675
$33$	−4.54287	−0.790812
$34$	0.799517	0.137116
$35$	1.19609	0.202175
$36$	1.36103	0.226838
$37$	−10.1239	−1.66436	−0.832178	−	0.554508i	$-0.812907\pi$
$37$	−10.1239	−1.66436	−0.832178	+	0.554508i	$0.812907\pi$
$38$	1.43395	0.232617
$39$	−2.26196	−0.362203
$40$	−2.19857	−0.347625
$41$	9.81464	1.53279	0.766395	−	0.642370i	$-0.222049\pi$
$41$	9.81464	1.53279	0.766395	+	0.642370i	$0.222049\pi$
$42$	−0.167595	−0.0258604
$43$	11.3047	1.72395	0.861973	−	0.506954i	$-0.169229\pi$
$43$	11.3047	1.72395	0.861973	+	0.506954i	$0.169229\pi$
$44$	−5.83498	−0.879656
$45$	−1.71497	−0.255652
$46$	0.288989	0.0426091
$47$	−8.37713	−1.22193	−0.610965	−	0.791658i	$-0.709218\pi$
$47$	−8.37713	−1.22193	−0.610965	+	0.791658i	$0.709218\pi$
$48$	−5.60421	−0.808898
$49$	−6.76269	−0.966099
$50$	0.233241	0.0329853
$51$	−5.34860	−0.748954
$52$	−2.90532	−0.402895
$53$	1.19162	0.163681	0.0818405	−	0.996645i	$-0.473920\pi$
$53$	1.19162	0.163681	0.0818405	+	0.996645i	$0.473920\pi$
$54$	1.27241	0.173153
$55$	7.35238	0.991394
$56$	−0.436207	−0.0582906
$57$	−9.59283	−1.27060
$58$	−1.91412	−0.251336
$59$	−6.98180	−0.908952	−0.454476	−	0.890759i	$-0.650174\pi$
$59$	−6.98180	−0.908952	−0.454476	+	0.890759i	$0.650174\pi$
$60$	7.25821	0.937031
$61$	−12.9039	−1.65218	−0.826088	−	0.563541i	$-0.809439\pi$
$61$	−12.9039	−1.65218	−0.826088	+	0.563541i	$0.809439\pi$
$62$	2.35103	0.298581
$63$	−0.340258	−0.0428684
$64$	−6.79207	−0.849008
$65$	3.66085	0.454072
$66$	−1.03021	−0.126810
$67$	−0.801136	−0.0978743	−0.0489371	−	0.998802i	$-0.515583\pi$
$67$	−0.801136	−0.0978743	−0.0489371	+	0.998802i	$0.515583\pi$
$68$	−6.86988	−0.833096
$69$	−1.93327	−0.232739
$70$	0.271242	0.0324196
$71$	−0.317689	−0.0377027	−0.0188514	−	0.999822i	$-0.506001\pi$
$71$	−0.317689	−0.0377027	−0.0188514	+	0.999822i	$0.506001\pi$
$72$	0.625441	0.0737089
$73$	−4.67967	−0.547714	−0.273857	−	0.961770i	$-0.588300\pi$
$73$	−4.67967	−0.547714	−0.273857	+	0.961770i	$0.588300\pi$
$74$	−2.29584	−0.266887
$75$	−1.56033	−0.180172
$76$	−12.3213	−1.41335
$77$	1.45875	0.166240
$78$	−0.512955	−0.0580808
$79$	−2.75962	−0.310481	−0.155241	−	0.987877i	$-0.549615\pi$
$79$	−2.75962	−0.310481	−0.155241	+	0.987877i	$0.549615\pi$
$80$	9.07009	1.01407
$81$	−6.41671	−0.712968
$82$	2.22572	0.245789
$83$	4.05170	0.444731	0.222366	−	0.974963i	$-0.428622\pi$
$83$	4.05170	0.444731	0.222366	+	0.974963i	$0.428622\pi$
$84$	1.44006	0.157124
$85$	8.65641	0.938919
$86$	2.56362	0.276442
$87$	12.8051	1.37285
$88$	−2.68138	−0.285836
$89$	14.6322	1.55101	0.775506	−	0.631340i	$-0.217495\pi$
$89$	14.6322	1.55101	0.775506	+	0.631340i	$0.217495\pi$
$90$	−0.388912	−0.0409949
$91$	0.726329	0.0761400
$92$	−2.48315	−0.258886
$93$	−15.7279	−1.63091
$94$	−1.89972	−0.195941
$95$	15.5254	1.59288
$96$	−3.98779	−0.407002
$97$	16.2908	1.65408	0.827038	−	0.562146i	$-0.190024\pi$
$97$	16.2908	1.65408	0.827038	+	0.562146i	$0.190024\pi$
$98$	−1.53361	−0.154918
$99$	−2.09158	−0.210211
$100$	−2.00413	−0.200413
$101$	11.1211	1.10659	0.553293	−	0.832987i	$-0.313371\pi$
$101$	11.1211	1.10659	0.553293	+	0.832987i	$0.313371\pi$
$102$	−1.21293	−0.120098
$103$	−9.69564	−0.955340	−0.477670	−	0.878539i	$-0.658518\pi$
$103$	−9.69564	−0.955340	−0.477670	+	0.878539i	$0.658518\pi$
$104$	−1.33509	−0.130917
$105$	−1.81455	−0.177082
$106$	0.270229	0.0262469
$107$	−3.30219	−0.319234	−0.159617	−	0.987179i	$-0.551026\pi$
$107$	−3.30219	−0.319234	−0.159617	+	0.987179i	$0.551026\pi$
$108$	−10.9332	−1.05205
$109$	10.4178	0.997841	0.498921	−	0.866648i	$-0.333730\pi$
$109$	10.4178	0.997841	0.498921	+	0.866648i	$0.333730\pi$
$110$	1.66734	0.158974
$111$	15.3587	1.45779
$112$	1.79955	0.170041
$113$	−4.78449	−0.450087	−0.225043	−	0.974349i	$-0.572252\pi$
$113$	−4.78449	−0.450087	−0.225043	+	0.974349i	$0.572252\pi$
$114$	−2.17541	−0.203746
$115$	3.12889	0.291771
$116$	16.4471	1.52708
$117$	−1.04142	−0.0962796
$118$	−1.58330	−0.145754
$119$	1.71747	0.157440
$120$	3.33540	0.304479
$121$	−2.03304	−0.184822
$122$	−2.92628	−0.264933
$123$	−14.8896	−1.34255
$124$	−20.2013	−1.81413
$125$	−9.75120	−0.872174
$126$	−0.0771619	−0.00687413
$127$	−11.0429	−0.979898	−0.489949	−	0.871751i	$-0.662985\pi$
$127$	−11.0429	−0.979898	−0.489949	+	0.871751i	$0.662985\pi$
$128$	−6.79747	−0.600817
$129$	−17.1501	−1.50998
$130$	0.830189	0.0728124
$131$	−2.66863	−0.233159	−0.116580	−	0.993181i	$-0.537193\pi$
$131$	−2.66863	−0.233159	−0.116580	+	0.993181i	$0.537193\pi$
$132$	8.85212	0.770478
$133$	3.08032	0.267098
$134$	−0.181678	−0.0156945
$135$	13.7764	1.18568
$136$	−3.15695	−0.270706
$137$	3.72454	0.318209	0.159104	−	0.987262i	$-0.449139\pi$
$137$	3.72454	0.318209	0.159104	+	0.987262i	$0.449139\pi$
$138$	−0.438418	−0.0373206
$139$	17.2668	1.46455	0.732277	−	0.681007i	$-0.238458\pi$
$139$	17.2668	1.46455	0.732277	+	0.681007i	$0.238458\pi$
$140$	−2.33066	−0.196977
$141$	12.7088	1.07027
$142$	−0.0720439	−0.00604579
$143$	4.46477	0.373363
$144$	−2.58022	−0.215019
$145$	−20.7242	−1.72106
$146$	−1.06123	−0.0878283
$147$	10.2595	0.846192
$148$	19.7271	1.62156
$149$	12.4704	1.02162	0.510809	−	0.859694i	$-0.329346\pi$
$149$	12.4704	1.02162	0.510809	+	0.859694i	$0.329346\pi$
$150$	−0.353845	−0.0288913
$151$	2.38571	0.194146	0.0970730	−	0.995277i	$-0.469052\pi$
$151$	2.38571	0.194146	0.0970730	+	0.995277i	$0.469052\pi$
$152$	−5.66206	−0.459254
$153$	−2.46254	−0.199085
$154$	0.330807	0.0266572
$155$	25.4547	2.04457
$156$	4.40759	0.352890
$157$	−8.77192	−0.700075	−0.350038	−	0.936736i	$-0.613831\pi$
$157$	−8.77192	−0.700075	−0.350038	+	0.936736i	$0.613831\pi$
$158$	−0.625812	−0.0497870
$159$	−1.80777	−0.143366
$160$	6.45401	0.510234
$161$	0.620787	0.0489249
$162$	−1.45515	−0.114327
$163$	−19.0375	−1.49113	−0.745566	−	0.666432i	$-0.767821\pi$
$163$	−19.0375	−1.49113	−0.745566	+	0.666432i	$0.767821\pi$
$164$	−19.1245	−1.49338
$165$	−11.1541	−0.868348
$166$	0.918823	0.0713145
$167$	3.89997	0.301789	0.150895	−	0.988550i	$-0.451785\pi$
$167$	3.89997	0.301789	0.150895	+	0.988550i	$0.451785\pi$
$168$	0.661760	0.0510559
$169$	−10.7769	−0.828995
$170$	1.96306	0.150560
$171$	−4.41662	−0.337747
$172$	−22.0280	−1.67962
$173$	11.7284	0.891691	0.445846	−	0.895110i	$-0.352903\pi$
$173$	11.7284	0.891691	0.445846	+	0.895110i	$0.352903\pi$
$174$	2.90387	0.220142
$175$	0.501033	0.0378746
$176$	11.0619	0.833821
$177$	10.5919	0.796138
$178$	3.31822	0.248711
$179$	3.24841	0.242797	0.121399	−	0.992604i	$-0.461262\pi$
$179$	3.24841	0.242797	0.121399	+	0.992604i	$0.461262\pi$
$180$	3.34174	0.249079
$181$	−13.6659	−1.01578	−0.507889	−	0.861422i	$-0.669574\pi$
$181$	−13.6659	−1.01578	−0.507889	+	0.861422i	$0.669574\pi$
$182$	0.164713	0.0122094
$183$	19.5762	1.44712
$184$	−1.14109	−0.0841225
$185$	−24.8572	−1.82754
$186$	−3.56670	−0.261523
$187$	10.5574	0.772031
$188$	16.3234	1.19051
$189$	2.73330	0.198818
$190$	3.52078	0.255424
$191$	24.4712	1.77068	0.885338	−	0.464949i	$-0.153927\pi$
$191$	24.4712	1.77068	0.885338	+	0.464949i	$0.153927\pi$
$192$	10.3041	0.743634
$193$	−6.71909	−0.483651	−0.241826	−	0.970320i	$-0.577746\pi$
$193$	−6.71909	−0.483651	−0.241826	+	0.970320i	$0.577746\pi$
$194$	3.69434	0.265238
$195$	−5.55379	−0.397715
$196$	13.1776	0.941257
$197$	−18.0333	−1.28482	−0.642410	−	0.766361i	$-0.722065\pi$
$197$	−18.0333	−1.28482	−0.642410	+	0.766361i	$0.722065\pi$
$198$	−0.474317	−0.0337082
$199$	−15.4466	−1.09498	−0.547489	−	0.836813i	$-0.684416\pi$
$199$	−15.4466	−1.09498	−0.547489	+	0.836813i	$0.684416\pi$
$200$	−0.920969	−0.0651224
$201$	1.21539	0.0857267
$202$	2.52198	0.177446
$203$	−4.11179	−0.288591
$204$	10.4221	0.729696
$205$	24.0979	1.68307
$206$	−2.19873	−0.153193
$207$	−0.890095	−0.0618659
$208$	5.50786	0.381902
$209$	18.9348	1.30975
$210$	−0.411496	−0.0283959
$211$	−6.47035	−0.445437	−0.222718	−	0.974883i	$-0.571493\pi$
$211$	−6.47035	−0.445437	−0.222718	+	0.974883i	$0.571493\pi$
$212$	−2.32195	−0.159472
$213$	0.481959	0.0330233
$214$	−0.748853	−0.0511906
$215$	27.7564	1.89297
$216$	−5.02419	−0.341853
$217$	5.05034	0.342839
$218$	2.36249	0.160008
$219$	7.09943	0.479735
$220$	−14.3266	−0.965902
$221$	5.25665	0.353601
$222$	3.48297	0.233762
$223$	−12.0958	−0.809995	−0.404997	−	0.914318i	$-0.632728\pi$
$223$	−12.0958	−0.809995	−0.404997	+	0.914318i	$0.632728\pi$
$224$	1.28051	0.0855574
$225$	−0.718390	−0.0478927
$226$	−1.08500	−0.0721733
$227$	−1.70774	−0.113347	−0.0566734	−	0.998393i	$-0.518049\pi$
$227$	−1.70774	−0.113347	−0.0566734	+	0.998393i	$0.518049\pi$
$228$	18.6923	1.23793
$229$	−4.07932	−0.269569	−0.134784	−	0.990875i	$-0.543034\pi$
$229$	−4.07932	−0.269569	−0.134784	+	0.990875i	$0.543034\pi$
$230$	0.709555	0.0467867
$231$	−2.21303	−0.145607
$232$	7.55804	0.496209
$233$	−5.69239	−0.372921	−0.186460	−	0.982462i	$-0.559702\pi$
$233$	−5.69239	−0.372921	−0.186460	+	0.982462i	$0.559702\pi$
$234$	−0.236169	−0.0154388
$235$	−20.5684	−1.34173
$236$	13.6045	0.885580
$237$	4.18656	0.271946
$238$	0.389480	0.0252462
$239$	−1.13222	−0.0732369	−0.0366185	−	0.999329i	$-0.511659\pi$
$239$	−1.13222	−0.0732369	−0.0366185	+	0.999329i	$0.511659\pi$
$240$	−13.7600	−0.888207
$241$	6.75030	0.434825	0.217412	−	0.976080i	$-0.430238\pi$
$241$	6.75030	0.434825	0.217412	+	0.976080i	$0.430238\pi$
$242$	−0.461042	−0.0296369
$243$	−7.09798	−0.455335
$244$	25.1442	1.60969
$245$	−16.6045	−1.06082
$246$	−3.37658	−0.215283
$247$	9.42791	0.599884
$248$	−9.28322	−0.589485
$249$	−6.14674	−0.389534
$250$	−2.21133	−0.139857
$251$	12.0446	0.760250	0.380125	−	0.924935i	$-0.375881\pi$
$251$	12.0446	0.760250	0.380125	+	0.924935i	$0.375881\pi$
$252$	0.663017	0.0417661
$253$	3.81600	0.239910
$254$	−2.50425	−0.157131
$255$	−13.1324	−0.822386
$256$	12.0426	0.752665
$257$	−26.0501	−1.62496	−0.812482	−	0.582987i	$-0.801884\pi$
$257$	−26.0501	−1.62496	−0.812482	+	0.582987i	$0.801884\pi$
$258$	−3.88921	−0.242131
$259$	−4.93179	−0.306446
$260$	−7.13343	−0.442397
$261$	5.89555	0.364926
$262$	−0.605179	−0.0373881
$263$	−22.0663	−1.36067	−0.680333	−	0.732903i	$-0.738165\pi$
$263$	−22.0663	−1.36067	−0.680333	+	0.732903i	$0.738165\pi$
$264$	4.06786	0.250359
$265$	2.92578	0.179729
$266$	0.698540	0.0428302
$267$	−22.1982	−1.35851
$268$	1.56107	0.0953576
$269$	22.8677	1.39427	0.697135	−	0.716940i	$-0.254458\pi$
$269$	22.8677	1.39427	0.697135	+	0.716940i	$0.254458\pi$
$270$	3.12414	0.190129
$271$	−14.6057	−0.887236	−0.443618	−	0.896216i	$-0.646305\pi$
$271$	−14.6057	−0.887236	−0.443618	+	0.896216i	$0.646305\pi$
$272$	13.0238	0.789687
$273$	−1.10190	−0.0666899
$274$	0.844632	0.0510261
$275$	3.07987	0.185723
$276$	3.76713	0.226754
$277$	−14.0792	−0.845938	−0.422969	−	0.906144i	$-0.639012\pi$
$277$	−14.0792	−0.845938	−0.422969	+	0.906144i	$0.639012\pi$
$278$	3.91569	0.234847
$279$	−7.24126	−0.433523
$280$	−1.07102	−0.0640057
$281$	−7.01466	−0.418460	−0.209230	−	0.977867i	$-0.567096\pi$
$281$	−7.01466	−0.418460	−0.209230	+	0.977867i	$0.567096\pi$
$282$	2.88203	0.171622
$283$	−5.04053	−0.299629	−0.149814	−	0.988714i	$-0.547868\pi$
$283$	−5.04053	−0.299629	−0.149814	+	0.988714i	$0.547868\pi$
$284$	0.619040	0.0367333
$285$	−23.5533	−1.39518
$286$	1.01250	0.0598703
$287$	4.78114	0.282222
$288$	−1.83601	−0.108188
$289$	−4.57017	−0.268833
$290$	−4.69974	−0.275978
$291$	−24.7144	−1.44878
$292$	9.11869	0.533631
$293$	21.0460	1.22952	0.614760	−	0.788714i	$-0.289253\pi$
$293$	21.0460	1.22952	0.614760	+	0.788714i	$0.289253\pi$
$294$	2.32660	0.135690
$295$	−17.1424	−0.998071
$296$	9.06531	0.526910
$297$	16.8017	0.974933
$298$	2.82798	0.163821
$299$	1.90004	0.109882
$300$	3.04042	0.175539
$301$	5.50700	0.317418
$302$	0.541018	0.0311321
$303$	−16.8715	−0.969243
$304$	23.3585	1.33970
$305$	−31.6830	−1.81416
$306$	−0.558443	−0.0319240
$307$	−10.2103	−0.582732	−0.291366	−	0.956612i	$-0.594110\pi$
$307$	−10.2103	−0.582732	−0.291366	+	0.956612i	$0.594110\pi$
$308$	−2.84247	−0.161965
$309$	14.7090	0.836768
$310$	5.77250	0.327856
$311$	−16.5993	−0.941257	−0.470629	−	0.882331i	$-0.655973\pi$
$311$	−16.5993	−0.941257	−0.470629	+	0.882331i	$0.655973\pi$
$312$	2.02544	0.114668
$313$	−17.3842	−0.982612	−0.491306	−	0.870987i	$-0.663480\pi$
$313$	−17.3842	−0.982612	−0.491306	+	0.870987i	$0.663480\pi$
$314$	−1.98925	−0.112260
$315$	−0.835436	−0.0470715
$316$	5.37732	0.302498
$317$	2.85890	0.160572	0.0802860	−	0.996772i	$-0.474417\pi$
$317$	2.85890	0.160572	0.0802860	+	0.996772i	$0.474417\pi$
$318$	−0.409958	−0.0229893
$319$	−25.2753	−1.41515
$320$	−16.6766	−0.932249
$321$	5.00967	0.279613
$322$	0.140779	0.00784531
$323$	22.2932	1.24042
$324$	12.5034	0.694635
$325$	1.53351	0.0850638
$326$	−4.31723	−0.239109
$327$	−15.8046	−0.873995
$328$	−8.78840	−0.485258
$329$	−4.08086	−0.224985
$330$	−2.52948	−0.139243
$331$	10.1701	0.559001	0.279500	−	0.960146i	$-0.409831\pi$
$331$	10.1701	0.559001	0.279500	+	0.960146i	$0.409831\pi$
$332$	−7.89502	−0.433296
$333$	7.07128	0.387504
$334$	0.884416	0.0483931
$335$	−1.96703	−0.107470
$336$	−2.73006	−0.148937
$337$	10.2191	0.556671	0.278336	−	0.960484i	$-0.410217\pi$
$337$	10.2191	0.556671	0.278336	+	0.960484i	$0.410217\pi$
$338$	−2.44394	−0.132933
$339$	7.25844	0.394224
$340$	−16.8676	−0.914777
$341$	31.0446	1.68116
$342$	−1.00158	−0.0541592
$343$	−6.70441	−0.362004
$344$	−10.1226	−0.545776
$345$	−4.74677	−0.255558
$346$	2.65970	0.142986
$347$	32.1125	1.72389	0.861945	−	0.507002i	$-0.169246\pi$
$347$	32.1125	1.72389	0.861945	+	0.507002i	$0.169246\pi$
$348$	−24.9516	−1.33755
$349$	1.00000	0.0535288
$349$	1.00000	0.0535288
$350$	0.113622	0.00607334
$351$	8.36579	0.446533
$352$	7.87132	0.419543
$353$	−36.0065	−1.91643	−0.958217	−	0.286041i	$-0.907661\pi$
$353$	−36.0065	−1.91643	−0.958217	+	0.286041i	$0.907661\pi$
$354$	2.40198	0.127664
$355$	−0.780022	−0.0413993
$356$	−28.5119	−1.51113
$357$	−2.60554	−0.137900
$358$	0.736658	0.0389336
$359$	0.257743	0.0136031	0.00680157	−	0.999977i	$-0.497835\pi$
$359$	0.257743	0.0136031	0.00680157	+	0.999977i	$0.497835\pi$
$360$	1.53565	0.0809357
$361$	20.9832	1.10438
$362$	−3.09908	−0.162884
$363$	3.08428	0.161883
$364$	−1.41531	−0.0741822
$365$	−11.4900	−0.601415
$366$	4.43940	0.232051
$367$	35.2941	1.84234	0.921168	−	0.389164i	$-0.127236\pi$
$367$	35.2941	1.84234	0.921168	+	0.389164i	$0.127236\pi$
$368$	4.70752	0.245397
$369$	−6.85528	−0.356872
$370$	−5.63699	−0.293053
$371$	0.580488	0.0301374
$372$	30.6470	1.58897
$373$	−2.37074	−0.122752	−0.0613761	−	0.998115i	$-0.519549\pi$
$373$	−2.37074	−0.122752	−0.0613761	+	0.998115i	$0.519549\pi$
$374$	2.39415	0.123798
$375$	14.7933	0.763925
$376$	7.50120	0.386845
$377$	−12.5849	−0.648156
$378$	0.619844	0.0318814
$379$	12.5565	0.644985	0.322492	−	0.946572i	$-0.395479\pi$
$379$	12.5565	0.644985	0.322492	+	0.946572i	$0.395479\pi$
$380$	−30.2525	−1.55192
$381$	16.7529	0.858278
$382$	5.54946	0.283935
$383$	−38.3066	−1.95737	−0.978687	−	0.205357i	$-0.934165\pi$
$383$	−38.3066	−1.95737	−0.978687	+	0.205357i	$0.934165\pi$
$384$	10.3123	0.526247
$385$	3.58166	0.182538
$386$	−1.52372	−0.0775554
$387$	−7.89603	−0.401378
$388$	−31.7437	−1.61154
$389$	−11.3857	−0.577276	−0.288638	−	0.957438i	$-0.593202\pi$
$389$	−11.3857	−0.577276	−0.288638	+	0.957438i	$0.593202\pi$
$390$	−1.25946	−0.0637753
$391$	4.49281	0.227211
$392$	6.05557	0.305852
$393$	4.04852	0.204221
$394$	−4.08951	−0.206026
$395$	−6.77570	−0.340922
$396$	4.07559	0.204806
$397$	−25.4035	−1.27496	−0.637482	−	0.770465i	$-0.720024\pi$
$397$	−25.4035	−1.27496	−0.637482	+	0.770465i	$0.720024\pi$
$398$	−3.50289	−0.175584
$399$	−4.67309	−0.233947
$400$	3.79941	0.189971
$401$	7.87559	0.393288	0.196644	−	0.980475i	$-0.436996\pi$
$401$	7.87559	0.393288	0.196644	+	0.980475i	$0.436996\pi$
$402$	0.275619	0.0137466
$403$	15.4575	0.769994
$404$	−21.6702	−1.07813
$405$	−15.7550	−0.782870
$406$	−0.932451	−0.0462767
$407$	−30.3159	−1.50270
$408$	4.78934	0.237108
$409$	−4.07196	−0.201346	−0.100673	−	0.994920i	$-0.532100\pi$
$409$	−4.07196	−0.201346	−0.100673	+	0.994920i	$0.532100\pi$
$410$	5.46481	0.269887
$411$	−5.65041	−0.278714
$412$	18.8927	0.930774
$413$	−3.40114	−0.167359
$414$	−0.201851	−0.00992045
$415$	9.94814	0.488335
$416$	3.91923	0.192156
$417$	−26.1951	−1.28278
$418$	4.29395	0.210024
$419$	−3.33174	−0.162766	−0.0813831	−	0.996683i	$-0.525934\pi$
$419$	−3.33174	−0.162766	−0.0813831	+	0.996683i	$0.525934\pi$
$420$	3.53579	0.172529
$421$	32.8827	1.60260	0.801302	−	0.598260i	$-0.204141\pi$
$421$	32.8827	1.60260	0.801302	+	0.598260i	$0.204141\pi$
$422$	−1.46731	−0.0714277
$423$	5.85121	0.284496
$424$	−1.06702	−0.0518190
$425$	3.62612	0.175893
$426$	0.109296	0.00529542
$427$	−6.28606	−0.304204
$428$	6.43455	0.311026
$429$	−6.77341	−0.327023
$430$	6.29446	0.303546
$431$	−18.0609	−0.869961	−0.434981	−	0.900440i	$-0.643245\pi$
$431$	−18.0609	−0.869961	−0.434981	+	0.900440i	$0.643245\pi$
$432$	20.7270	0.997230
$433$	−12.2845	−0.590354	−0.295177	−	0.955443i	$-0.595379\pi$
$433$	−12.2845	−0.590354	−0.295177	+	0.955443i	$0.595379\pi$
$434$	1.14529	0.0549757
$435$	31.4403	1.50745
$436$	−20.2998	−0.972183
$437$	8.05795	0.385464
$438$	1.60997	0.0769275
$439$	8.54404	0.407785	0.203892	−	0.978993i	$-0.434641\pi$
$439$	8.54404	0.407785	0.203892	+	0.978993i	$0.434641\pi$
$440$	−6.58359	−0.313861
$441$	4.72357	0.224932
$442$	1.19208	0.0567013
$443$	−10.7618	−0.511308	−0.255654	−	0.966768i	$-0.582291\pi$
$443$	−10.7618	−0.511308	−0.255654	+	0.966768i	$0.582291\pi$
$444$	−29.9276	−1.42030
$445$	35.9265	1.70308
$446$	−2.74303	−0.129886
$447$	−18.9186	−0.894820
$448$	−3.30871	−0.156322
$449$	15.8005	0.745672	0.372836	−	0.927897i	$-0.378385\pi$
$449$	15.8005	0.745672	0.372836	+	0.927897i	$0.378385\pi$
$450$	−0.162913	−0.00767979
$451$	29.3898	1.38391
$452$	9.32292	0.438513
$453$	−3.61930	−0.170050
$454$	−0.387273	−0.0181756
$455$	1.78336	0.0836051
$456$	8.58978	0.402253
$457$	−12.8790	−0.602455	−0.301227	−	0.953552i	$-0.597396\pi$
$457$	−12.8790	−0.602455	−0.301227	+	0.953552i	$0.597396\pi$
$458$	−0.925087	−0.0432265
$459$	19.7817	0.923330
$460$	−6.09688	−0.284268
$461$	−11.7328	−0.546453	−0.273226	−	0.961950i	$-0.588091\pi$
$461$	−11.7328	−0.546453	−0.273226	+	0.961950i	$0.588091\pi$
$462$	−0.501860	−0.0233486
$463$	−5.17464	−0.240486	−0.120243	−	0.992745i	$-0.538367\pi$
$463$	−5.17464	−0.240486	−0.120243	+	0.992745i	$0.538367\pi$
$464$	−31.1803	−1.44751
$465$	−38.6168	−1.79081
$466$	−1.29089	−0.0597994
$467$	12.1127	0.560510	0.280255	−	0.959926i	$-0.409581\pi$
$467$	12.1127	0.560510	0.280255	+	0.959926i	$0.409581\pi$
$468$	2.02929	0.0938040
$469$	−0.390268	−0.0180209
$470$	−4.66440	−0.215153
$471$	13.3077	0.613186
$472$	6.25176	0.287761
$473$	33.8517	1.55650
$474$	0.949406	0.0436077
$475$	6.50352	0.298402
$476$	−3.34662	−0.153392
$477$	−0.832314	−0.0381090
$478$	−0.256758	−0.0117438
$479$	16.6894	0.762556	0.381278	−	0.924460i	$-0.375484\pi$
$479$	16.6894	0.762556	0.381278	+	0.924460i	$0.375484\pi$
$480$	−9.79124	−0.446907
$481$	−15.0947	−0.688258
$482$	1.53080	0.0697260
$483$	−0.941782	−0.0428526
$484$	3.96152	0.180069
$485$	39.9988	1.81625
$486$	−1.60964	−0.0730149
$487$	8.33836	0.377847	0.188924	−	0.981992i	$-0.439500\pi$
$487$	8.33836	0.377847	0.188924	+	0.981992i	$0.439500\pi$
$488$	11.5546	0.523054
$489$	28.8814	1.30606
$490$	−3.76548	−0.170107
$491$	−21.3567	−0.963816	−0.481908	−	0.876222i	$-0.660056\pi$
$491$	−21.3567	−0.963816	−0.481908	+	0.876222i	$0.660056\pi$
$492$	29.0134	1.30803
$493$	−29.7582	−1.34024
$494$	2.13801	0.0961938
$495$	−5.13545	−0.230821
$496$	38.2975	1.71961
$497$	−0.154760	−0.00694194
$498$	−1.39393	−0.0624633
$499$	−1.91439	−0.0857000	−0.0428500	−	0.999082i	$-0.513644\pi$
$499$	−1.91439	−0.0857000	−0.0428500	+	0.999082i	$0.513644\pi$
$500$	19.0009	0.849748
$501$	−5.91656	−0.264333
$502$	2.73142	0.121909
$503$	42.5266	1.89617	0.948083	−	0.318024i	$-0.103019\pi$
$503$	42.5266	1.89617	0.948083	+	0.318024i	$0.103019\pi$
$504$	0.304679	0.0135715
$505$	27.3056	1.21508
$506$	0.865373	0.0384705
$507$	16.3494	0.726104
$508$	21.5179	0.954701
$509$	40.6336	1.80105	0.900527	−	0.434800i	$-0.143181\pi$
$509$	40.6336	1.80105	0.900527	+	0.434800i	$0.143181\pi$
$510$	−2.97811	−0.131873
$511$	−2.27967	−0.100847
$512$	16.3259	0.721510
$513$	35.4788	1.56643
$514$	−5.90752	−0.260570
$515$	−23.8057	−1.04901
$516$	33.4182	1.47115
$517$	−25.0852	−1.10325
$518$	−1.11841	−0.0491399
$519$	−17.7928	−0.781019
$520$	−3.27806	−0.143752
$521$	−16.7753	−0.734937	−0.367469	−	0.930036i	$-0.619775\pi$
$521$	−16.7753	−0.734937	−0.367469	+	0.930036i	$0.619775\pi$
$522$	1.33696	0.0585173
$523$	−23.5004	−1.02760	−0.513801	−	0.857909i	$-0.671763\pi$
$523$	−23.5004	−1.02760	−0.513801	+	0.857909i	$0.671763\pi$
$524$	5.20002	0.227164
$525$	−0.760107	−0.0331738
$526$	−5.00408	−0.218188
$527$	36.5507	1.59217
$528$	−16.7817	−0.730332
$529$	−21.3761	−0.929394
$530$	0.663493	0.0288203
$531$	4.87661	0.211627
$532$	−6.00223	−0.260230
$533$	14.6336	0.633851
$534$	−5.03400	−0.217842
$535$	−8.10787	−0.350534
$536$	0.717367	0.0309855
$537$	−4.92809	−0.212663
$538$	5.18583	0.223577
$539$	−20.2508	−0.872264
$540$	−26.8443	−1.15520
$541$	12.3074	0.529138	0.264569	−	0.964367i	$-0.414770\pi$
$541$	12.3074	0.529138	0.264569	+	0.964367i	$0.414770\pi$
$542$	−3.31222	−0.142272
$543$	20.7322	0.889705
$544$	9.26739	0.397336
$545$	25.5788	1.09567
$546$	−0.249883	−0.0106940
$547$	25.5229	1.09128	0.545639	−	0.838020i	$-0.316287\pi$
$547$	25.5229	1.09128	0.545639	+	0.838020i	$0.316287\pi$
$548$	−7.25753	−0.310026
$549$	9.01306	0.384668
$550$	0.698437	0.0297815
$551$	−53.3719	−2.27372
$552$	1.73113	0.0736816
$553$	−1.34433	−0.0571667
$554$	−3.19282	−0.135650
$555$	37.7103	1.60071
$556$	−33.6457	−1.42689
$557$	−15.2979	−0.648193	−0.324096	−	0.946024i	$-0.605060\pi$
$557$	−15.2979	−0.648193	−0.324096	+	0.946024i	$0.605060\pi$
$558$	−1.64214	−0.0695172
$559$	16.8552	0.712900
$560$	4.41844	0.186713
$561$	−16.0163	−0.676210
$562$	−1.59075	−0.0671017
$563$	12.4755	0.525780	0.262890	−	0.964826i	$-0.415324\pi$
$563$	12.4755	0.525780	0.262890	+	0.964826i	$0.415324\pi$
$564$	−24.7639	−1.04275
$565$	−11.7474	−0.494215
$566$	−1.14307	−0.0480467
$567$	−3.12586	−0.131274
$568$	0.284471	0.0119361
$569$	34.4772	1.44536	0.722680	−	0.691183i	$-0.242910\pi$
$569$	34.4772	1.44536	0.722680	+	0.691183i	$0.242910\pi$
$570$	−5.34130	−0.223722
$571$	43.5325	1.82178	0.910889	−	0.412652i	$-0.135397\pi$
$571$	43.5325	1.82178	0.910889	+	0.412652i	$0.135397\pi$
$572$	−8.69993	−0.363762
$573$	−37.1247	−1.55091
$574$	1.08424	0.0452554
$575$	1.31068	0.0546590
$576$	4.74409	0.197670
$577$	−35.3099	−1.46997	−0.734986	−	0.678082i	$-0.762811\pi$
$577$	−35.3099	−1.46997	−0.734986	+	0.678082i	$0.762811\pi$
$578$	−1.03640	−0.0431085
$579$	10.1934	0.423623
$580$	40.3827	1.67680
$581$	1.97376	0.0818853
$582$	−5.60460	−0.232318
$583$	3.56828	0.147783
$584$	4.19036	0.173398
$585$	−2.55701	−0.105719
$586$	4.77270	0.197159
$587$	32.3564	1.33549	0.667746	−	0.744389i	$-0.267259\pi$
$587$	32.3564	1.33549	0.667746	+	0.744389i	$0.267259\pi$
$588$	−19.9914	−0.824433
$589$	65.5545	2.70113
$590$	−3.88747	−0.160045
$591$	27.3579	1.12536
$592$	−37.3985	−1.53707
$593$	−43.0597	−1.76825	−0.884125	−	0.467250i	$-0.845245\pi$
$593$	−43.0597	−1.76825	−0.884125	+	0.467250i	$0.845245\pi$
$594$	3.81021	0.156335
$595$	4.21691	0.172877
$596$	−24.2995	−0.995348
$597$	23.4336	0.959075
$598$	0.430881	0.0176200
$599$	8.83006	0.360787	0.180393	−	0.983595i	$-0.442263\pi$
$599$	8.83006	0.360787	0.180393	+	0.983595i	$0.442263\pi$
$600$	1.39718	0.0570397
$601$	−13.2588	−0.540838	−0.270419	−	0.962743i	$-0.587162\pi$
$601$	−13.2588	−0.540838	−0.270419	+	0.962743i	$0.587162\pi$
$602$	1.24885	0.0508993
$603$	0.559573	0.0227876
$604$	−4.64872	−0.189154
$605$	−4.99173	−0.202943
$606$	−3.82604	−0.155422
$607$	2.01980	0.0819812	0.0409906	−	0.999160i	$-0.486949\pi$
$607$	2.01980	0.0819812	0.0409906	+	0.999160i	$0.486949\pi$
$608$	16.6212	0.674081
$609$	6.23790	0.252773
$610$	−7.18491	−0.290909
$611$	−12.4903	−0.505302
$612$	4.79844	0.193966
$613$	−27.0390	−1.09209	−0.546047	−	0.837754i	$-0.683868\pi$
$613$	−27.0390	−1.09209	−0.546047	+	0.837754i	$0.683868\pi$
$614$	−2.31544	−0.0934435
$615$	−36.5584	−1.47418
$616$	−1.30622	−0.0526289
$617$	−9.84285	−0.396258	−0.198129	−	0.980176i	$-0.563487\pi$
$617$	−9.84285	−0.396258	−0.198129	+	0.980176i	$0.563487\pi$
$618$	3.33564	0.134179
$619$	33.5512	1.34854	0.674269	−	0.738486i	$-0.264459\pi$
$619$	33.5512	1.34854	0.674269	+	0.738486i	$0.264459\pi$
$620$	−49.6004	−1.99200
$621$	7.15017	0.286926
$622$	−3.76430	−0.150934
$623$	7.12799	0.285577
$624$	−8.35586	−0.334502
$625$	−29.0847	−1.16339
$626$	−3.94230	−0.157566
$627$	−28.7256	−1.14719
$628$	17.0927	0.682074
$629$	−35.6927	−1.42316
$630$	−0.189456	−0.00754810
$631$	−31.0876	−1.23758	−0.618788	−	0.785558i	$-0.712376\pi$
$631$	−31.0876	−1.23758	−0.618788	+	0.785558i	$0.712376\pi$
$632$	2.47107	0.0982937
$633$	9.81602	0.390152
$634$	0.648328	0.0257484
$635$	−27.1136	−1.07597
$636$	3.52258	0.139679
$637$	−10.0831	−0.399509
$638$	−5.73181	−0.226924
$639$	0.221898	0.00877814
$640$	−16.6899	−0.659724
$641$	1.12365	0.0443815	0.0221908	−	0.999754i	$-0.492936\pi$
$641$	1.12365	0.0443815	0.0221908	+	0.999754i	$0.492936\pi$
$642$	1.13607	0.0448371
$643$	−12.8343	−0.506136	−0.253068	−	0.967448i	$-0.581440\pi$
$643$	−12.8343	−0.506136	−0.253068	+	0.967448i	$0.581440\pi$
$644$	−1.20965	−0.0476668
$645$	−42.1086	−1.65803
$646$	5.05553	0.198907
$647$	−22.3934	−0.880377	−0.440188	−	0.897905i	$-0.645088\pi$
$647$	−22.3934	−0.880377	−0.440188	+	0.897905i	$0.645088\pi$
$648$	5.74576	0.225715
$649$	−20.9069	−0.820668
$650$	0.347761	0.0136403
$651$	−7.66176	−0.300288
$652$	37.0959	1.45279
$653$	9.23899	0.361550	0.180775	−	0.983525i	$-0.442139\pi$
$653$	9.23899	0.361550	0.180775	+	0.983525i	$0.442139\pi$
$654$	−3.58408	−0.140149
$655$	−6.55230	−0.256019
$656$	36.2561	1.41556
$657$	3.26864	0.127522
$658$	−0.925438	−0.0360773
$659$	15.9916	0.622945	0.311472	−	0.950255i	$-0.399178\pi$
$659$	15.9916	0.622945	0.311472	+	0.950255i	$0.399178\pi$
$660$	21.7346	0.846019
$661$	−5.59324	−0.217552	−0.108776	−	0.994066i	$-0.534693\pi$
$661$	−5.59324	−0.217552	−0.108776	+	0.994066i	$0.534693\pi$
$662$	2.30633	0.0896381
$663$	−7.97475	−0.309714
$664$	−3.62804	−0.140795
$665$	7.56312	0.293285
$666$	1.60359	0.0621378
$667$	−10.7562	−0.416482
$668$	−7.59938	−0.294029
$669$	18.3503	0.709463
$670$	−0.446073	−0.0172333
$671$	−38.6406	−1.49170
$672$	−1.94263	−0.0749385
$673$	27.9099	1.07585	0.537924	−	0.842993i	$-0.319209\pi$
$673$	27.9099	1.07585	0.537924	+	0.842993i	$0.319209\pi$
$674$	2.31744	0.0892646
$675$	5.77086	0.222120
$676$	20.9996	0.807678
$677$	−24.7563	−0.951460	−0.475730	−	0.879591i	$-0.657816\pi$
$677$	−24.7563	−0.951460	−0.475730	+	0.879591i	$0.657816\pi$
$678$	1.64603	0.0632155
$679$	7.93594	0.304553
$680$	−7.75127	−0.297248
$681$	2.59078	0.0992788
$682$	7.04014	0.269581
$683$	37.7443	1.44424	0.722122	−	0.691766i	$-0.243167\pi$
$683$	37.7443	1.44424	0.722122	+	0.691766i	$0.243167\pi$
$684$	8.60610	0.329063
$685$	9.14487	0.349407
$686$	−1.52039	−0.0580488
$687$	6.18864	0.236111
$688$	41.7604	1.59210
$689$	1.77669	0.0676867
$690$	−1.07645	−0.0409797
$691$	−16.9108	−0.643317	−0.321659	−	0.946856i	$-0.604240\pi$
$691$	−16.9108	−0.643317	−0.321659	+	0.946856i	$0.604240\pi$
$692$	−22.8536	−0.868763
$693$	−1.01890	−0.0387047
$694$	7.28231	0.276433
$695$	42.3953	1.60815
$696$	−11.4661	−0.434623
$697$	34.6025	1.31066
$698$	0.226775	0.00858356
$699$	8.63579	0.326636
$700$	−0.976300	−0.0369007
$701$	−20.7193	−0.782557	−0.391279	−	0.920272i	$-0.627967\pi$
$701$	−20.7193	−0.782557	−0.391279	+	0.920272i	$0.627967\pi$
$702$	1.89715	0.0716034
$703$	−64.0157	−2.41440
$704$	−20.3388	−0.766546
$705$	31.2038	1.17520
$706$	−8.16538	−0.307308
$707$	5.41755	0.203748
$708$	−20.6391	−0.775667
$709$	−31.6515	−1.18870	−0.594348	−	0.804208i	$-0.702590\pi$
$709$	−31.6515	−1.18870	−0.594348	+	0.804208i	$0.702590\pi$
$710$	−0.176890	−0.00663855
$711$	1.92752	0.0722878
$712$	−13.1022	−0.491027
$713$	13.2114	0.494771
$714$	−0.590871	−0.0221128
$715$	10.9624	0.409969
$716$	−6.32976	−0.236554
$717$	1.71766	0.0641471
$718$	0.0584496	0.00218132
$719$	8.81235	0.328645	0.164323	−	0.986407i	$-0.447456\pi$
$719$	8.81235	0.328645	0.164323	+	0.986407i	$0.447456\pi$
$720$	−6.33523	−0.236100
$721$	−4.72317	−0.175900
$722$	4.75847	0.177092
$723$	−10.2407	−0.380857
$724$	26.6290	0.989659
$725$	−8.68127	−0.322414
$726$	0.699437	0.0259585
$727$	20.6470	0.765756	0.382878	−	0.923799i	$-0.374933\pi$
$727$	20.6470	0.765756	0.382878	+	0.923799i	$0.374933\pi$
$728$	−0.650383	−0.0241048
$729$	30.0183	1.11179
$730$	−2.60565	−0.0964394
$731$	39.8557	1.47412
$732$	−38.1457	−1.40991
$733$	−9.53031	−0.352010	−0.176005	−	0.984389i	$-0.556318\pi$
$733$	−9.53031	−0.352010	−0.176005	+	0.984389i	$0.556318\pi$
$734$	8.00382	0.295426
$735$	25.1903	0.929156
$736$	3.34973	0.123473
$737$	−2.39899	−0.0883680
$738$	−1.55461	−0.0572259
$739$	−16.0136	−0.589069	−0.294534	−	0.955641i	$-0.595165\pi$
$739$	−16.0136	−0.589069	−0.294534	+	0.955641i	$0.595165\pi$
$740$	48.4361	1.78055
$741$	−14.3029	−0.525429
$742$	0.131640	0.00483266
$743$	13.8446	0.507908	0.253954	−	0.967216i	$-0.418269\pi$
$743$	13.8446	0.507908	0.253954	+	0.967216i	$0.418269\pi$
$744$	14.0834	0.516321
$745$	30.6187	1.12178
$746$	−0.537624	−0.0196838
$747$	−2.83001	−0.103545
$748$	−20.5718	−0.752179
$749$	−1.60864	−0.0587784
$750$	3.35476	0.122498
$751$	−31.0215	−1.13199	−0.565995	−	0.824409i	$-0.691508\pi$
$751$	−31.0215	−1.13199	−0.565995	+	0.824409i	$0.691508\pi$
$752$	−30.9458	−1.12848
$753$	−18.2726	−0.665892
$754$	−2.85394	−0.103935
$755$	5.85763	0.213181
$756$	−5.32604	−0.193706
$757$	−21.4520	−0.779686	−0.389843	−	0.920881i	$-0.627471\pi$
$757$	−21.4520	−0.779686	−0.389843	+	0.920881i	$0.627471\pi$
$758$	2.84750	0.103426
$759$	−5.78917	−0.210133
$760$	−13.9021	−0.504281
$761$	38.6517	1.40112	0.700562	−	0.713592i	$-0.252933\pi$
$761$	38.6517	1.40112	0.700562	+	0.713592i	$0.252933\pi$
$762$	3.79914	0.137628
$763$	5.07495	0.183726
$764$	−47.6840	−1.72515
$765$	−6.04628	−0.218604
$766$	−8.68697	−0.313873
$767$	−10.4098	−0.375877
$768$	−18.2696	−0.659248
$769$	−1.72383	−0.0621630	−0.0310815	−	0.999517i	$-0.509895\pi$
$769$	−1.72383	−0.0621630	−0.0310815	+	0.999517i	$0.509895\pi$
$770$	0.812232	0.0292708
$771$	39.5201	1.42328
$772$	13.0926	0.471215
$773$	−3.79897	−0.136639	−0.0683197	−	0.997663i	$-0.521764\pi$
$773$	−3.79897	−0.136639	−0.0683197	+	0.997663i	$0.521764\pi$
$774$	−1.79062	−0.0643626
$775$	10.6628	0.383021
$776$	−14.5874	−0.523656
$777$	7.48191	0.268412
$778$	−2.58199	−0.0925687
$779$	62.0602	2.22354
$780$	10.8220	0.387489
$781$	−0.951315	−0.0340407
$782$	1.01886	0.0364343
$783$	−47.3592	−1.69248
$784$	−24.9819	−0.892212
$785$	−21.5377	−0.768714
$786$	0.918103	0.0327477
$787$	−44.3707	−1.58165	−0.790823	−	0.612045i	$-0.790347\pi$
$787$	−44.3707	−1.58165	−0.790823	+	0.612045i	$0.790347\pi$
$788$	35.1392	1.25178
$789$	33.4763	1.19179
$790$	−1.53656	−0.0546683
$791$	−2.33073	−0.0828713
$792$	1.87288	0.0665497
$793$	−19.2397	−0.683221
$794$	−5.76088	−0.204446
$795$	−4.43863	−0.157422
$796$	30.0987	1.06682
$797$	−13.5945	−0.481541	−0.240771	−	0.970582i	$-0.577400\pi$
$797$	−13.5945	−0.481541	−0.240771	+	0.970582i	$0.577400\pi$
$798$	−1.05974	−0.0375144
$799$	−29.5344	−1.04485
$800$	2.70355	0.0955849
$801$	−10.2202	−0.361114
$802$	1.78599	0.0630654
$803$	−14.0132	−0.494516
$804$	−2.36827	−0.0835223
$805$	1.52422	0.0537217
$806$	3.50538	0.123472
$807$	−34.6921	−1.22122
$808$	−9.95821	−0.350329
$809$	−9.90164	−0.348123	−0.174062	−	0.984735i	$-0.555689\pi$
$809$	−9.90164	−0.348123	−0.174062	+	0.984735i	$0.555689\pi$
$810$	−3.57283	−0.125536
$811$	−39.2002	−1.37650	−0.688252	−	0.725472i	$-0.741622\pi$
$811$	−39.2002	−1.37650	−0.688252	+	0.725472i	$0.741622\pi$
$812$	8.01212	0.281170
$813$	22.1581	0.777117
$814$	−6.87488	−0.240964
$815$	−46.7428	−1.63733
$816$	−19.7582	−0.691675
$817$	71.4820	2.50084
$818$	−0.923419	−0.0322866
$819$	−0.507323	−0.0177273
$820$	−46.9566	−1.63979
$821$	0.851334	0.0297118	0.0148559	−	0.999890i	$-0.495271\pi$
$821$	0.851334	0.0297118	0.0148559	+	0.999890i	$0.495271\pi$
$822$	−1.28137	−0.0446930
$823$	44.9153	1.56565	0.782824	−	0.622244i	$-0.213779\pi$
$823$	44.9153	1.56565	0.782824	+	0.622244i	$0.213779\pi$
$824$	8.68184	0.302446
$825$	−4.67240	−0.162672
$826$	−0.771293	−0.0268367
$827$	−11.3616	−0.395081	−0.197540	−	0.980295i	$-0.563295\pi$
$827$	−11.3616	−0.395081	−0.197540	+	0.980295i	$0.563295\pi$
$828$	1.73442	0.0602751
$829$	−4.86817	−0.169079	−0.0845393	−	0.996420i	$-0.526942\pi$
$829$	−4.86817	−0.169079	−0.0845393	+	0.996420i	$0.526942\pi$
$830$	2.25599	0.0783065
$831$	21.3593	0.740945
$832$	−10.1269	−0.351089
$833$	−23.8425	−0.826094
$834$	−5.94040	−0.205699
$835$	9.57562	0.331378
$836$	−36.8959	−1.27607
$837$	58.1693	2.01063
$838$	−0.755555	−0.0261002
$839$	−52.4044	−1.80920	−0.904600	−	0.426262i	$-0.859830\pi$
$839$	−52.4044	−1.80920	−0.904600	+	0.426262i	$0.859830\pi$
$840$	1.62482	0.0560616
$841$	42.2438	1.45668
$842$	7.45697	0.256984
$843$	10.6418	0.366523
$844$	12.6079	0.433983
$845$	−26.4606	−0.910273
$846$	1.32691	0.0456201
$847$	−0.990382	−0.0340299
$848$	4.40193	0.151163
$849$	7.64688	0.262440
$850$	0.822314	0.0282051
$851$	−12.9013	−0.442250
$852$	−0.939132	−0.0321741
$853$	−3.85732	−0.132072	−0.0660361	−	0.997817i	$-0.521035\pi$
$853$	−3.85732	−0.132072	−0.0660361	+	0.997817i	$0.521035\pi$
$854$	−1.42552	−0.0487803
$855$	−10.8441	−0.370862
$856$	2.95690	0.101065
$857$	32.7750	1.11957	0.559786	−	0.828637i	$-0.310883\pi$
$857$	32.7750	1.11957	0.559786	+	0.828637i	$0.310883\pi$
$858$	−1.53604	−0.0524395
$859$	43.6127	1.48805	0.744023	−	0.668154i	$-0.232915\pi$
$859$	43.6127	1.48805	0.744023	+	0.668154i	$0.232915\pi$
$860$	−54.0854	−1.84430
$861$	−7.25336	−0.247194
$862$	−4.09575	−0.139502
$863$	34.6831	1.18063	0.590314	−	0.807174i	$-0.299004\pi$
$863$	34.6831	1.18063	0.590314	+	0.807174i	$0.299004\pi$
$864$	14.7487	0.501763
$865$	28.7967	0.979117
$866$	−2.78581	−0.0946658
$867$	6.93330	0.235467
$868$	−9.84095	−0.334024
$869$	−8.26364	−0.280325
$870$	7.12987	0.241725
$871$	−1.19449	−0.0404738
$872$	−9.32846	−0.315902
$873$	−11.3787	−0.385110
$874$	1.82734	0.0618108
$875$	−4.75024	−0.160587
$876$	−13.8338	−0.467399
$877$	−15.7716	−0.532568	−0.266284	−	0.963895i	$-0.585796\pi$
$877$	−15.7716	−0.532568	−0.266284	+	0.963895i	$0.585796\pi$
$878$	1.93757	0.0653899
$879$	−31.9284	−1.07692
$880$	27.1603	0.915573
$881$	10.7453	0.362019	0.181009	−	0.983481i	$-0.442064\pi$
$881$	10.7453	0.362019	0.181009	+	0.983481i	$0.442064\pi$
$882$	1.07119	0.0360688
$883$	−10.4529	−0.351769	−0.175884	−	0.984411i	$-0.556278\pi$
$883$	−10.4529	−0.351769	−0.175884	+	0.984411i	$0.556278\pi$
$884$	−10.2430	−0.344508
$885$	26.0064	0.874195
$886$	−2.44050	−0.0819903
$887$	−27.4301	−0.921011	−0.460506	−	0.887657i	$-0.652332\pi$
$887$	−27.4301	−0.921011	−0.460506	+	0.887657i	$0.652332\pi$
$888$	−13.7528	−0.461513
$889$	−5.37947	−0.180422
$890$	8.14724	0.273096
$891$	−19.2148	−0.643719
$892$	23.5696	0.789167
$893$	−52.9705	−1.77259
$894$	−4.29027	−0.143488
$895$	7.97583	0.266602
$896$	−3.31135	−0.110624
$897$	−2.88250	−0.0962440
$898$	3.58316	0.119572
$899$	−87.5059	−2.91848
$900$	1.39984	0.0466612
$901$	4.20116	0.139961
$902$	6.66488	0.221916
$903$	−8.35455	−0.278022
$904$	4.28421	0.142491
$905$	−33.5539	−1.11537
$906$	−0.820767	−0.0272682
$907$	6.11450	0.203029	0.101514	−	0.994834i	$-0.467631\pi$
$907$	6.11450	0.203029	0.101514	+	0.994834i	$0.467631\pi$
$908$	3.32766	0.110432
$909$	−7.76778	−0.257641
$910$	0.404421	0.0134064
$911$	−23.1412	−0.766701	−0.383350	−	0.923603i	$-0.625230\pi$
$911$	−23.1412	−0.766701	−0.383350	+	0.923603i	$0.625230\pi$
$912$	−35.4367	−1.17343
$913$	12.1328	0.401536
$914$	−2.92064	−0.0966061
$915$	48.0656	1.58900
$916$	7.94885	0.262637
$917$	−1.30001	−0.0429300
$918$	4.48599	0.148060
$919$	−24.1951	−0.798121	−0.399061	−	0.916925i	$-0.630664\pi$
$919$	−24.1951	−0.798121	−0.399061	+	0.916925i	$0.630664\pi$
$920$	−2.80173	−0.0923703
$921$	15.4898	0.510406
$922$	−2.66071	−0.0876259
$923$	−0.473673	−0.0155911
$924$	4.31225	0.141863
$925$	−10.4125	−0.342362
$926$	−1.17348	−0.0385629
$927$	6.77216	0.222427
$928$	−22.1870	−0.728324
$929$	−25.5140	−0.837086	−0.418543	−	0.908197i	$-0.637459\pi$
$929$	−25.5140	−0.837086	−0.418543	+	0.908197i	$0.637459\pi$
$930$	−8.75733	−0.287164
$931$	−42.7620	−1.40147
$932$	11.0920	0.363332
$933$	25.1824	0.824433
$934$	2.74686	0.0898800
$935$	25.9215	0.847724
$936$	0.932530	0.0304807
$937$	29.3071	0.957421	0.478711	−	0.877973i	$-0.341104\pi$
$937$	29.3071	0.957421	0.478711	+	0.877973i	$0.341104\pi$
$938$	−0.0885031	−0.00288973
$939$	26.3731	0.860655
$940$	40.0790	1.30723
$941$	45.1230	1.47097	0.735485	−	0.677541i	$-0.236955\pi$
$941$	45.1230	1.47097	0.735485	+	0.677541i	$0.236955\pi$
$942$	3.01785	0.0983268
$943$	12.5072	0.407290
$944$	−25.7913	−0.839437
$945$	6.71108	0.218312
$946$	7.67672	0.249592
$947$	22.2824	0.724082	0.362041	−	0.932162i	$-0.382080\pi$
$947$	22.2824	0.724082	0.362041	+	0.932162i	$0.382080\pi$
$948$	−8.15781	−0.264953
$949$	−6.97738	−0.226495
$950$	1.47484	0.0478500
$951$	−4.33718	−0.140643
$952$	−1.53789	−0.0498433
$953$	46.7857	1.51554	0.757769	−	0.652522i	$-0.226289\pi$
$953$	46.7857	1.51554	0.757769	+	0.652522i	$0.226289\pi$
$954$	−0.188748	−0.00611094
$955$	60.0843	1.94428
$956$	2.20620	0.0713537
$957$	38.3446	1.23950
$958$	3.78473	0.122279
$959$	1.81438	0.0585895
$960$	25.2997	0.816543
$961$	76.4798	2.46709
$962$	−3.42310	−0.110365
$963$	2.30649	0.0743257
$964$	−13.1534	−0.423644
$965$	−16.4974	−0.531071
$966$	−0.213573	−0.00687159
$967$	54.4475	1.75092	0.875458	−	0.483295i	$-0.160560\pi$
$967$	54.4475	1.75092	0.875458	+	0.483295i	$0.160560\pi$
$968$	1.82046	0.0585118
$969$	−33.8205	−1.08647
$970$	9.07072	0.291243
$971$	46.1660	1.48154	0.740769	−	0.671760i	$-0.234461\pi$
$971$	46.1660	1.48154	0.740769	+	0.671760i	$0.234461\pi$
$972$	13.8309	0.443627
$973$	8.41143	0.269658
$974$	1.89093	0.0605894
$975$	−2.32645	−0.0745061
$976$	−47.6681	−1.52582
$977$	10.0509	0.321557	0.160779	−	0.986991i	$-0.448600\pi$
$977$	10.0509	0.321557	0.160779	+	0.986991i	$0.448600\pi$
$978$	6.54957	0.209432
$979$	43.8160	1.40037
$980$	32.3550	1.03354
$981$	−7.27655	−0.232322
$982$	−4.84317	−0.154552
$983$	−17.1112	−0.545764	−0.272882	−	0.962048i	$-0.587977\pi$
$983$	−17.1112	−0.545764	−0.272882	+	0.962048i	$0.587977\pi$
$984$	13.3327	0.425030
$985$	−44.2773	−1.41079
$986$	−6.74841	−0.214913
$987$	6.19099	0.197061
$988$	−18.3710	−0.584459
$989$	14.4060	0.458084
$990$	−1.16459	−0.0370132
$991$	−18.2371	−0.579321	−0.289660	−	0.957130i	$-0.593542\pi$
$991$	−18.2371	−0.579321	−0.289660	+	0.957130i	$0.593542\pi$
$992$	27.2514	0.865231
$993$	−15.4289	−0.489620
$994$	−0.0350957	−0.00111317
$995$	−37.9260	−1.20233
$996$	11.9774	0.379517
$997$	55.1592	1.74691	0.873454	−	0.486906i	$-0.161875\pi$
$997$	55.1592	1.74691	0.873454	+	0.486906i	$0.161875\pi$
$998$	−0.434136	−0.0137423
$999$	−56.8039	−1.79719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.2.a.b.1.9	✓	17
3.2	odd	2		3141.2.a.e.1.9		17
4.3	odd	2		5584.2.a.m.1.14		17
5.4	even	2		8725.2.a.m.1.9		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.2.a.b.1.9	✓	17	1.1	even	1	trivial
3141.2.a.e.1.9		17	3.2	odd	2
5584.2.a.m.1.14		17	4.3	odd	2
8725.2.a.m.1.9		17	5.4	even	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:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	349.a (trivial)

\(f(q)\)	\(=\)	\(q+0.226775 q^{2} -1.51708 q^{3} -1.94857 q^{4} +2.45530 q^{5} -0.344035 q^{6} +0.487144 q^{7} -0.895438 q^{8} -0.698475 q^{9} +O(q^{10})\)	\(q+0.226775 q^{2} -1.51708 q^{3} -1.94857 q^{4} +2.45530 q^{5} -0.344035 q^{6} +0.487144 q^{7} -0.895438 q^{8} -0.698475 q^{9} +0.556801 q^{10} +2.99449 q^{11} +2.95614 q^{12} +1.49100 q^{13} +0.110472 q^{14} -3.72489 q^{15} +3.69408 q^{16} +3.52560 q^{17} -0.158397 q^{18} +6.32323 q^{19} -4.78434 q^{20} -0.739035 q^{21} +0.679075 q^{22} +1.27434 q^{23} +1.35845 q^{24} +1.02851 q^{25} +0.338121 q^{26} +5.61087 q^{27} -0.949235 q^{28} -8.44061 q^{29} -0.844711 q^{30} +10.3672 q^{31} +2.62860 q^{32} -4.54287 q^{33} +0.799517 q^{34} +1.19609 q^{35} +1.36103 q^{36} -10.1239 q^{37} +1.43395 q^{38} -2.26196 q^{39} -2.19857 q^{40} +9.81464 q^{41} -0.167595 q^{42} +11.3047 q^{43} -5.83498 q^{44} -1.71497 q^{45} +0.288989 q^{46} -8.37713 q^{47} -5.60421 q^{48} -6.76269 q^{49} +0.233241 q^{50} -5.34860 q^{51} -2.90532 q^{52} +1.19162 q^{53} +1.27241 q^{54} +7.35238 q^{55} -0.436207 q^{56} -9.59283 q^{57} -1.91412 q^{58} -6.98180 q^{59} +7.25821 q^{60} -12.9039 q^{61} +2.35103 q^{62} -0.340258 q^{63} -6.79207 q^{64} +3.66085 q^{65} -1.03021 q^{66} -0.801136 q^{67} -6.86988 q^{68} -1.93327 q^{69} +0.271242 q^{70} -0.317689 q^{71} +0.625441 q^{72} -4.67967 q^{73} -2.29584 q^{74} -1.56033 q^{75} -12.3213 q^{76} +1.45875 q^{77} -0.512955 q^{78} -2.75962 q^{79} +9.07009 q^{80} -6.41671 q^{81} +2.22572 q^{82} +4.05170 q^{83} +1.44006 q^{84} +8.65641 q^{85} +2.56362 q^{86} +12.8051 q^{87} -2.68138 q^{88} +14.6322 q^{89} -0.388912 q^{90} +0.726329 q^{91} -2.48315 q^{92} -15.7279 q^{93} -1.89972 q^{94} +15.5254 q^{95} -3.98779 q^{96} +16.2908 q^{97} -1.53361 q^{98} -2.09158 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 5 q^{2} + 6 q^{3} + 19 q^{4} + 5 q^{5} - 3 q^{6} + q^{7} + 9 q^{8} + 25 q^{9}+O(q^{10}) \) 17 * q + 5 * q^2 + 6 * q^3 + 19 * q^4 + 5 * q^5 - 3 * q^6 + q^7 + 9 * q^8 + 25 * q^9	\( 17 q + 5 q^{2} + 6 q^{3} + 19 q^{4} + 5 q^{5} - 3 q^{6} + q^{7} + 9 q^{8} + 25 q^{9} - 6 q^{10} + 39 q^{11} + 4 q^{12} - 6 q^{13} + 11 q^{14} + 12 q^{15} + 23 q^{16} + q^{17} + 7 q^{19} + 8 q^{20} - 3 q^{21} - q^{22} + 8 q^{23} - 21 q^{24} + 14 q^{25} + q^{26} + 9 q^{27} - 23 q^{28} + 9 q^{29} - 27 q^{30} - 2 q^{31} + 23 q^{32} - 5 q^{33} - 12 q^{34} + 16 q^{35} + 10 q^{36} - 7 q^{37} - 8 q^{38} - 39 q^{40} + 3 q^{41} - 30 q^{42} + q^{43} + 56 q^{44} - 21 q^{45} - q^{46} + 16 q^{47} - 31 q^{48} + 6 q^{49} - 5 q^{50} + 29 q^{51} - 37 q^{52} + 27 q^{53} - 36 q^{54} - 16 q^{55} + 14 q^{56} - 21 q^{57} - 44 q^{58} + 74 q^{59} - 27 q^{60} - 30 q^{61} - 18 q^{62} - 9 q^{63} - 17 q^{64} - 3 q^{65} - 66 q^{66} + 9 q^{67} - 51 q^{68} - 23 q^{69} - 77 q^{70} + 70 q^{71} - 55 q^{72} - 38 q^{73} + 26 q^{74} + 22 q^{75} - 50 q^{76} - 38 q^{77} - 75 q^{78} + 7 q^{79} - 28 q^{80} + 5 q^{81} - 56 q^{82} + 47 q^{83} - 69 q^{84} - 49 q^{85} + 17 q^{86} - 37 q^{87} - 30 q^{88} + 23 q^{89} - 64 q^{90} + 6 q^{91} + 17 q^{92} - 31 q^{93} - 40 q^{94} - 11 q^{95} - 49 q^{96} - 14 q^{97} + 5 q^{98} + 79 q^{99}+O(q^{100}) \) 17 * q + 5 * q^2 + 6 * q^3 + 19 * q^4 + 5 * q^5 - 3 * q^6 + q^7 + 9 * q^8 + 25 * q^9 - 6 * q^10 + 39 * q^11 + 4 * q^12 - 6 * q^13 + 11 * q^14 + 12 * q^15 + 23 * q^16 + q^17 + 7 * q^19 + 8 * q^20 - 3 * q^21 - q^22 + 8 * q^23 - 21 * q^24 + 14 * q^25 + q^26 + 9 * q^27 - 23 * q^28 + 9 * q^29 - 27 * q^30 - 2 * q^31 + 23 * q^32 - 5 * q^33 - 12 * q^34 + 16 * q^35 + 10 * q^36 - 7 * q^37 - 8 * q^38 - 39 * q^40 + 3 * q^41 - 30 * q^42 + q^43 + 56 * q^44 - 21 * q^45 - q^46 + 16 * q^47 - 31 * q^48 + 6 * q^49 - 5 * q^50 + 29 * q^51 - 37 * q^52 + 27 * q^53 - 36 * q^54 - 16 * q^55 + 14 * q^56 - 21 * q^57 - 44 * q^58 + 74 * q^59 - 27 * q^60 - 30 * q^61 - 18 * q^62 - 9 * q^63 - 17 * q^64 - 3 * q^65 - 66 * q^66 + 9 * q^67 - 51 * q^68 - 23 * q^69 - 77 * q^70 + 70 * q^71 - 55 * q^72 - 38 * q^73 + 26 * q^74 + 22 * q^75 - 50 * q^76 - 38 * q^77 - 75 * q^78 + 7 * q^79 - 28 * q^80 + 5 * q^81 - 56 * q^82 + 47 * q^83 - 69 * q^84 - 49 * q^85 + 17 * q^86 - 37 * q^87 - 30 * q^88 + 23 * q^89 - 64 * q^90 + 6 * q^91 + 17 * q^92 - 31 * q^93 - 40 * q^94 - 11 * q^95 - 49 * q^96 - 14 * q^97 + 5 * q^98 + 79 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more