LMFDB - Embedded newform 4641.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4641,2,Mod(1,4641)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4641, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4641.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4641 = 3 \cdot 7 \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4641.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$37.0585715781$
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} - x^{16} - 29 x^{15} + 26 x^{14} + 339 x^{13} - 266 x^{12} - 2047 x^{11} + 1356 x^{10} + \cdots + 18 $ x^17 - x^16 - 29x^15 + 26x^14 + 339x^13 - 266x^12 - 2047x^11 + 1356x^10 + 6792x^9 - 3573x^8 - 12125x^7 + 4456x^6 + 10521x^5 - 1858x^4 - 3551x^3 + 34x^2 + 323*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.74789$ of defining polynomial
Character	$\chi$	$=$	4641.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.74789 q^{2} -1.00000 q^{3} +5.55089 q^{4} +3.19340 q^{5} +2.74789 q^{6} -1.00000 q^{7} -9.75745 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.74789 q^{2} -1.00000 q^{3} +5.55089 q^{4} +3.19340 q^{5} +2.74789 q^{6} -1.00000 q^{7} -9.75745 q^{8} +1.00000 q^{9} -8.77510 q^{10} +2.18144 q^{11} -5.55089 q^{12} +1.00000 q^{13} +2.74789 q^{14} -3.19340 q^{15} +15.7106 q^{16} -1.00000 q^{17} -2.74789 q^{18} +6.42896 q^{19} +17.7262 q^{20} +1.00000 q^{21} -5.99434 q^{22} +6.85786 q^{23} +9.75745 q^{24} +5.19778 q^{25} -2.74789 q^{26} -1.00000 q^{27} -5.55089 q^{28} +1.54181 q^{29} +8.77510 q^{30} +6.88659 q^{31} -23.6561 q^{32} -2.18144 q^{33} +2.74789 q^{34} -3.19340 q^{35} +5.55089 q^{36} +9.14042 q^{37} -17.6661 q^{38} -1.00000 q^{39} -31.1594 q^{40} -0.820796 q^{41} -2.74789 q^{42} -3.65489 q^{43} +12.1089 q^{44} +3.19340 q^{45} -18.8446 q^{46} -3.68699 q^{47} -15.7106 q^{48} +1.00000 q^{49} -14.2829 q^{50} +1.00000 q^{51} +5.55089 q^{52} +5.48782 q^{53} +2.74789 q^{54} +6.96619 q^{55} +9.75745 q^{56} -6.42896 q^{57} -4.23672 q^{58} -6.30807 q^{59} -17.7262 q^{60} -5.55600 q^{61} -18.9236 q^{62} -1.00000 q^{63} +33.5831 q^{64} +3.19340 q^{65} +5.99434 q^{66} -11.3869 q^{67} -5.55089 q^{68} -6.85786 q^{69} +8.77510 q^{70} +11.0891 q^{71} -9.75745 q^{72} +2.81537 q^{73} -25.1169 q^{74} -5.19778 q^{75} +35.6865 q^{76} -2.18144 q^{77} +2.74789 q^{78} +13.9348 q^{79} +50.1702 q^{80} +1.00000 q^{81} +2.25546 q^{82} -15.7758 q^{83} +5.55089 q^{84} -3.19340 q^{85} +10.0432 q^{86} -1.54181 q^{87} -21.2853 q^{88} -1.83695 q^{89} -8.77510 q^{90} -1.00000 q^{91} +38.0672 q^{92} -6.88659 q^{93} +10.1314 q^{94} +20.5302 q^{95} +23.6561 q^{96} +14.5576 q^{97} -2.74789 q^{98} +2.18144 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + q^{2} - 17 q^{3} + 25 q^{4} - 4 q^{5} - q^{6} - 17 q^{7} + 6 q^{8} + 17 q^{9}+O(q^{10}) $ 17 * q + q^2 - 17 * q^3 + 25 * q^4 - 4 * q^5 - q^6 - 17 * q^7 + 6 * q^8 + 17 * q^9	\( 17 q + q^{2} - 17 q^{3} + 25 q^{4} - 4 q^{5} - q^{6} - 17 q^{7} + 6 q^{8} + 17 q^{9} - q^{10} + 4 q^{11} - 25 q^{12} + 17 q^{13} - q^{14} + 4 q^{15} + 53 q^{16} - 17 q^{17} + q^{18} + 13 q^{19} - 5 q^{20} + 17 q^{21} + 4 q^{22} + 8 q^{23} - 6 q^{24} + 37 q^{25} + q^{26} - 17 q^{27} - 25 q^{28} - 3 q^{29} + q^{30} + 10 q^{31} + 18 q^{32} - 4 q^{33} - q^{34} + 4 q^{35} + 25 q^{36} - 25 q^{38} - 17 q^{39} - 12 q^{41} + q^{42} + 29 q^{43} + 12 q^{44} - 4 q^{45} + 19 q^{46} - 4 q^{47} - 53 q^{48} + 17 q^{49} + 9 q^{50} + 17 q^{51} + 25 q^{52} - 8 q^{53} - q^{54} + 27 q^{55} - 6 q^{56} - 13 q^{57} + 2 q^{58} + 25 q^{59} + 5 q^{60} - 5 q^{61} - 37 q^{62} - 17 q^{63} + 94 q^{64} - 4 q^{65} - 4 q^{66} + 15 q^{67} - 25 q^{68} - 8 q^{69} + q^{70} + 32 q^{71} + 6 q^{72} - 15 q^{73} + 16 q^{74} - 37 q^{75} + 3 q^{76} - 4 q^{77} - q^{78} + 27 q^{79} + 41 q^{80} + 17 q^{81} - 8 q^{82} - 24 q^{83} + 25 q^{84} + 4 q^{85} + 53 q^{86} + 3 q^{87} - 9 q^{88} - 8 q^{89} - q^{90} - 17 q^{91} + 14 q^{92} - 10 q^{93} + 51 q^{94} - 9 q^{95} - 18 q^{96} - 22 q^{97} + q^{98} + 4 q^{99}+O(q^{100}) \) 17 * q + q^2 - 17 * q^3 + 25 * q^4 - 4 * q^5 - q^6 - 17 * q^7 + 6 * q^8 + 17 * q^9 - q^10 + 4 * q^11 - 25 * q^12 + 17 * q^13 - q^14 + 4 * q^15 + 53 * q^16 - 17 * q^17 + q^18 + 13 * q^19 - 5 * q^20 + 17 * q^21 + 4 * q^22 + 8 * q^23 - 6 * q^24 + 37 * q^25 + q^26 - 17 * q^27 - 25 * q^28 - 3 * q^29 + q^30 + 10 * q^31 + 18 * q^32 - 4 * q^33 - q^34 + 4 * q^35 + 25 * q^36 - 25 * q^38 - 17 * q^39 - 12 * q^41 + q^42 + 29 * q^43 + 12 * q^44 - 4 * q^45 + 19 * q^46 - 4 * q^47 - 53 * q^48 + 17 * q^49 + 9 * q^50 + 17 * q^51 + 25 * q^52 - 8 * q^53 - q^54 + 27 * q^55 - 6 * q^56 - 13 * q^57 + 2 * q^58 + 25 * q^59 + 5 * q^60 - 5 * q^61 - 37 * q^62 - 17 * q^63 + 94 * q^64 - 4 * q^65 - 4 * q^66 + 15 * q^67 - 25 * q^68 - 8 * q^69 + q^70 + 32 * q^71 + 6 * q^72 - 15 * q^73 + 16 * q^74 - 37 * q^75 + 3 * q^76 - 4 * q^77 - q^78 + 27 * q^79 + 41 * q^80 + 17 * q^81 - 8 * q^82 - 24 * q^83 + 25 * q^84 + 4 * q^85 + 53 * q^86 + 3 * q^87 - 9 * q^88 - 8 * q^89 - q^90 - 17 * q^91 + 14 * q^92 - 10 * q^93 + 51 * q^94 - 9 * q^95 - 18 * q^96 - 22 * q^97 + q^98 + 4 * 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 $

$2$	−2.74789	−1.94305	−0.971525	−	0.236936i	$-0.923857\pi$
$2$	−2.74789	−1.94305	−0.971525	+	0.236936i	$0.923857\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.55089	2.77545
$5$	3.19340	1.42813	0.714065	−	0.700079i	$-0.246852\pi$
$5$	3.19340	1.42813	0.714065	+	0.700079i	$0.246852\pi$
$6$	2.74789	1.12182
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−9.75745	−3.44978
$9$	1.00000	0.333333
$10$	−8.77510	−2.77493
$11$	2.18144	0.657728	0.328864	−	0.944377i	$-0.393334\pi$
$11$	2.18144	0.657728	0.328864	+	0.944377i	$0.393334\pi$
$12$	−5.55089	−1.60240
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.74789	0.734404
$15$	−3.19340	−0.824531
$16$	15.7106	3.92765
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.74789	−0.647684
$19$	6.42896	1.47490	0.737452	−	0.675399i	$-0.236029\pi$
$19$	6.42896	1.47490	0.737452	+	0.675399i	$0.236029\pi$
$20$	17.7262	3.96370
$21$	1.00000	0.218218
$22$	−5.99434	−1.27800
$23$	6.85786	1.42996	0.714982	−	0.699143i	$-0.246435\pi$
$23$	6.85786	1.42996	0.714982	+	0.699143i	$0.246435\pi$
$24$	9.75745	1.99173
$25$	5.19778	1.03956
$26$	−2.74789	−0.538905
$27$	−1.00000	−0.192450
$28$	−5.55089	−1.04902
$29$	1.54181	0.286307	0.143153	−	0.989700i	$-0.454276\pi$
$29$	1.54181	0.286307	0.143153	+	0.989700i	$0.454276\pi$
$30$	8.77510	1.60211
$31$	6.88659	1.23687	0.618434	−	0.785837i	$-0.287767\pi$
$31$	6.88659	1.23687	0.618434	+	0.785837i	$0.287767\pi$
$32$	−23.6561	−4.18185
$33$	−2.18144	−0.379739
$34$	2.74789	0.471259
$35$	−3.19340	−0.539782
$36$	5.55089	0.925149
$37$	9.14042	1.50268	0.751338	−	0.659917i	$-0.229409\pi$
$37$	9.14042	1.50268	0.751338	+	0.659917i	$0.229409\pi$
$38$	−17.6661	−2.86581
$39$	−1.00000	−0.160128
$40$	−31.1594	−4.92674
$41$	−0.820796	−0.128187	−0.0640934	−	0.997944i	$-0.520416\pi$
$41$	−0.820796	−0.128187	−0.0640934	+	0.997944i	$0.520416\pi$
$42$	−2.74789	−0.424008
$43$	−3.65489	−0.557366	−0.278683	−	0.960383i	$-0.589898\pi$
$43$	−3.65489	−0.557366	−0.278683	+	0.960383i	$0.589898\pi$
$44$	12.1089	1.82549
$45$	3.19340	0.476043
$46$	−18.8446	−2.77849
$47$	−3.68699	−0.537803	−0.268902	−	0.963168i	$-0.586661\pi$
$47$	−3.68699	−0.537803	−0.268902	+	0.963168i	$0.586661\pi$
$48$	−15.7106	−2.26763
$49$	1.00000	0.142857
$50$	−14.2829	−2.01991
$51$	1.00000	0.140028
$52$	5.55089	0.769770
$53$	5.48782	0.753810	0.376905	−	0.926252i	$-0.376988\pi$
$53$	5.48782	0.753810	0.376905	+	0.926252i	$0.376988\pi$
$54$	2.74789	0.373940
$55$	6.96619	0.939321
$56$	9.75745	1.30389
$57$	−6.42896	−0.851537
$58$	−4.23672	−0.556309
$59$	−6.30807	−0.821240	−0.410620	−	0.911806i	$-0.634688\pi$
$59$	−6.30807	−0.821240	−0.410620	+	0.911806i	$0.634688\pi$
$60$	−17.7262	−2.28844
$61$	−5.55600	−0.711373	−0.355687	−	0.934605i	$-0.615753\pi$
$61$	−5.55600	−0.711373	−0.355687	+	0.934605i	$0.615753\pi$
$62$	−18.9236	−2.40330
$63$	−1.00000	−0.125988
$64$	33.5831	4.19789
$65$	3.19340	0.396092
$66$	5.99434	0.737853
$67$	−11.3869	−1.39113	−0.695566	−	0.718462i	$-0.744847\pi$
$67$	−11.3869	−1.39113	−0.695566	+	0.718462i	$0.744847\pi$
$68$	−5.55089	−0.673144
$69$	−6.85786	−0.825590
$70$	8.77510	1.04882
$71$	11.0891	1.31603	0.658017	−	0.753003i	$-0.271395\pi$
$71$	11.0891	1.31603	0.658017	+	0.753003i	$0.271395\pi$
$72$	−9.75745	−1.14993
$73$	2.81537	0.329514	0.164757	−	0.986334i	$-0.447316\pi$
$73$	2.81537	0.329514	0.164757	+	0.986334i	$0.447316\pi$
$74$	−25.1169	−2.91978
$75$	−5.19778	−0.600188
$76$	35.6865	4.09352
$77$	−2.18144	−0.248598
$78$	2.74789	0.311137
$79$	13.9348	1.56779	0.783896	−	0.620892i	$-0.213229\pi$
$79$	13.9348	1.56779	0.783896	+	0.620892i	$0.213229\pi$
$80$	50.1702	5.60920
$81$	1.00000	0.111111
$82$	2.25546	0.249074
$83$	−15.7758	−1.73161	−0.865807	−	0.500378i	$-0.833194\pi$
$83$	−15.7758	−1.73161	−0.865807	+	0.500378i	$0.833194\pi$
$84$	5.55089	0.605652
$85$	−3.19340	−0.346372
$86$	10.0432	1.08299
$87$	−1.54181	−0.165299
$88$	−21.2853	−2.26902
$89$	−1.83695	−0.194716	−0.0973581	−	0.995249i	$-0.531039\pi$
$89$	−1.83695	−0.194716	−0.0973581	+	0.995249i	$0.531039\pi$
$90$	−8.77510	−0.924976
$91$	−1.00000	−0.104828
$92$	38.0672	3.96878
$93$	−6.88659	−0.714106
$94$	10.1314	1.04498
$95$	20.5302	2.10636
$96$	23.6561	2.41439
$97$	14.5576	1.47811	0.739053	−	0.673648i	$-0.235274\pi$
$97$	14.5576	1.47811	0.739053	+	0.673648i	$0.235274\pi$
$98$	−2.74789	−0.277579
$99$	2.18144	0.219243
$100$	28.8523	2.88523
$101$	−8.13819	−0.809780	−0.404890	−	0.914365i	$-0.632690\pi$
$101$	−8.13819	−0.809780	−0.404890	+	0.914365i	$0.632690\pi$
$102$	−2.74789	−0.272082
$103$	9.23549	0.910000	0.455000	−	0.890492i	$-0.349639\pi$
$103$	9.23549	0.910000	0.455000	+	0.890492i	$0.349639\pi$
$104$	−9.75745	−0.956797
$105$	3.19340	0.311644
$106$	−15.0799	−1.46469
$107$	13.2537	1.28128	0.640640	−	0.767841i	$-0.278669\pi$
$107$	13.2537	1.28128	0.640640	+	0.767841i	$0.278669\pi$
$108$	−5.55089	−0.534135
$109$	4.48151	0.429251	0.214625	−	0.976696i	$-0.431147\pi$
$109$	4.48151	0.429251	0.214625	+	0.976696i	$0.431147\pi$
$110$	−19.1423	−1.82515
$111$	−9.14042	−0.867571
$112$	−15.7106	−1.48451
$113$	−4.76701	−0.448443	−0.224221	−	0.974538i	$-0.571984\pi$
$113$	−4.76701	−0.448443	−0.224221	+	0.974538i	$0.571984\pi$
$114$	17.6661	1.65458
$115$	21.8999	2.04217
$116$	8.55842	0.794629
$117$	1.00000	0.0924500
$118$	17.3339	1.59571
$119$	1.00000	0.0916698
$120$	31.1594	2.84445
$121$	−6.24134	−0.567394
$122$	15.2673	1.38223
$123$	0.820796	0.0740087
$124$	38.2267	3.43286
$125$	0.631580	0.0564903
$126$	2.74789	0.244801
$127$	−8.74443	−0.775942	−0.387971	−	0.921671i	$-0.626824\pi$
$127$	−8.74443	−0.775942	−0.387971	+	0.921671i	$0.626824\pi$
$128$	−44.9704	−3.97486
$129$	3.65489	0.321795
$130$	−8.77510	−0.769627
$131$	2.43749	0.212965	0.106482	−	0.994315i	$-0.466041\pi$
$131$	2.43749	0.212965	0.106482	+	0.994315i	$0.466041\pi$
$132$	−12.1089	−1.05395
$133$	−6.42896	−0.557462
$134$	31.2900	2.70304
$135$	−3.19340	−0.274844
$136$	9.75745	0.836695
$137$	−8.37945	−0.715905	−0.357952	−	0.933740i	$-0.616525\pi$
$137$	−8.37945	−0.715905	−0.357952	+	0.933740i	$0.616525\pi$
$138$	18.8446	1.60416
$139$	2.19001	0.185754	0.0928770	−	0.995678i	$-0.470394\pi$
$139$	2.19001	0.185754	0.0928770	+	0.995678i	$0.470394\pi$
$140$	−17.7262	−1.49814
$141$	3.68699	0.310501
$142$	−30.4716	−2.55712
$143$	2.18144	0.182421
$144$	15.7106	1.30922
$145$	4.92361	0.408884
$146$	−7.73633	−0.640263
$147$	−1.00000	−0.0824786
$148$	50.7375	4.17060
$149$	−22.5524	−1.84757	−0.923784	−	0.382915i	$-0.874920\pi$
$149$	−22.5524	−1.84757	−0.923784	+	0.382915i	$0.874920\pi$
$150$	14.2829	1.16619
$151$	20.5120	1.66924	0.834621	−	0.550824i	$-0.185686\pi$
$151$	20.5120	1.66924	0.834621	+	0.550824i	$0.185686\pi$
$152$	−62.7303	−5.08810
$153$	−1.00000	−0.0808452
$154$	5.99434	0.483038
$155$	21.9916	1.76641
$156$	−5.55089	−0.444427
$157$	4.09131	0.326522	0.163261	−	0.986583i	$-0.447799\pi$
$157$	4.09131	0.326522	0.163261	+	0.986583i	$0.447799\pi$
$158$	−38.2914	−3.04630
$159$	−5.48782	−0.435213
$160$	−75.5433	−5.97222
$161$	−6.85786	−0.540475
$162$	−2.74789	−0.215895
$163$	14.8454	1.16278	0.581392	−	0.813623i	$-0.302508\pi$
$163$	14.8454	1.16278	0.581392	+	0.813623i	$0.302508\pi$
$164$	−4.55615	−0.355776
$165$	−6.96619	−0.542317
$166$	43.3500	3.36461
$167$	4.43703	0.343348	0.171674	−	0.985154i	$-0.445082\pi$
$167$	4.43703	0.343348	0.171674	+	0.985154i	$0.445082\pi$
$168$	−9.75745	−0.752804
$169$	1.00000	0.0769231
$170$	8.77510	0.673019
$171$	6.42896	0.491635
$172$	−20.2879	−1.54694
$173$	−21.5834	−1.64096	−0.820479	−	0.571677i	$-0.806293\pi$
$173$	−21.5834	−1.64096	−0.820479	+	0.571677i	$0.806293\pi$
$174$	4.23672	0.321185
$175$	−5.19778	−0.392915
$176$	34.2717	2.58333
$177$	6.30807	0.474143
$178$	5.04773	0.378343
$179$	4.51339	0.337346	0.168673	−	0.985672i	$-0.446052\pi$
$179$	4.51339	0.337346	0.168673	+	0.985672i	$0.446052\pi$
$180$	17.7262	1.32123
$181$	4.94830	0.367805	0.183902	−	0.982945i	$-0.441127\pi$
$181$	4.94830	0.367805	0.183902	+	0.982945i	$0.441127\pi$
$182$	2.74789	0.203687
$183$	5.55600	0.410712
$184$	−66.9153	−4.93306
$185$	29.1890	2.14602
$186$	18.9236	1.38754
$187$	−2.18144	−0.159522
$188$	−20.4661	−1.49264
$189$	1.00000	0.0727393
$190$	−56.4147	−4.09276
$191$	−24.2673	−1.75592	−0.877960	−	0.478734i	$-0.841096\pi$
$191$	−24.2673	−1.75592	−0.877960	+	0.478734i	$0.841096\pi$
$192$	−33.5831	−2.42365
$193$	−27.1575	−1.95484	−0.977421	−	0.211302i	$-0.932229\pi$
$193$	−27.1575	−1.95484	−0.977421	+	0.211302i	$0.932229\pi$
$194$	−40.0028	−2.87203
$195$	−3.19340	−0.228684
$196$	5.55089	0.396492
$197$	−18.3405	−1.30671	−0.653354	−	0.757052i	$-0.726639\pi$
$197$	−18.3405	−1.30671	−0.653354	+	0.757052i	$0.726639\pi$
$198$	−5.99434	−0.425999
$199$	10.4607	0.741536	0.370768	−	0.928726i	$-0.379095\pi$
$199$	10.4607	0.741536	0.370768	+	0.928726i	$0.379095\pi$
$200$	−50.7171	−3.58624
$201$	11.3869	0.803171
$202$	22.3628	1.57344
$203$	−1.54181	−0.108214
$204$	5.55089	0.388640
$205$	−2.62113	−0.183067
$206$	−25.3781	−1.76818
$207$	6.85786	0.476654
$208$	15.7106	1.08933
$209$	14.0244	0.970086
$210$	−8.77510	−0.605539
$211$	3.17499	0.218575	0.109288	−	0.994010i	$-0.465143\pi$
$211$	3.17499	0.218575	0.109288	+	0.994010i	$0.465143\pi$
$212$	30.4623	2.09216
$213$	−11.0891	−0.759813
$214$	−36.4196	−2.48959
$215$	−11.6715	−0.795991
$216$	9.75745	0.663911
$217$	−6.88659	−0.467492
$218$	−12.3147	−0.834056
$219$	−2.81537	−0.190245
$220$	38.6686	2.60703
$221$	−1.00000	−0.0672673
$222$	25.1169	1.68573
$223$	−19.0465	−1.27545	−0.637724	−	0.770265i	$-0.720124\pi$
$223$	−19.0465	−1.27545	−0.637724	+	0.770265i	$0.720124\pi$
$224$	23.6561	1.58059
$225$	5.19778	0.346518
$226$	13.0992	0.871347
$227$	−19.9507	−1.32417	−0.662086	−	0.749428i	$-0.730329\pi$
$227$	−19.9507	−1.32417	−0.662086	+	0.749428i	$0.730329\pi$
$228$	−35.6865	−2.36339
$229$	−14.3408	−0.947670	−0.473835	−	0.880614i	$-0.657131\pi$
$229$	−14.3408	−0.947670	−0.473835	+	0.880614i	$0.657131\pi$
$230$	−60.1784	−3.96805
$231$	2.18144	0.143528
$232$	−15.0441	−0.987696
$233$	15.4483	1.01205	0.506025	−	0.862519i	$-0.331114\pi$
$233$	15.4483	1.01205	0.506025	+	0.862519i	$0.331114\pi$
$234$	−2.74789	−0.179635
$235$	−11.7740	−0.768053
$236$	−35.0154	−2.27931
$237$	−13.9348	−0.905165
$238$	−2.74789	−0.178119
$239$	−0.817777	−0.0528976	−0.0264488	−	0.999650i	$-0.508420\pi$
$239$	−0.817777	−0.0528976	−0.0264488	+	0.999650i	$0.508420\pi$
$240$	−50.1702	−3.23847
$241$	22.3686	1.44089	0.720443	−	0.693514i	$-0.243938\pi$
$241$	22.3686	1.44089	0.720443	+	0.693514i	$0.243938\pi$
$242$	17.1505	1.10248
$243$	−1.00000	−0.0641500
$244$	−30.8408	−1.97438
$245$	3.19340	0.204019
$246$	−2.25546	−0.143803
$247$	6.42896	0.409065
$248$	−67.1955	−4.26692
$249$	15.7758	0.999748
$250$	−1.73551	−0.109763
$251$	7.49611	0.473150	0.236575	−	0.971613i	$-0.423975\pi$
$251$	7.49611	0.473150	0.236575	+	0.971613i	$0.423975\pi$
$252$	−5.55089	−0.349673
$253$	14.9600	0.940526
$254$	24.0287	1.50770
$255$	3.19340	0.199978
$256$	56.4075	3.52547
$257$	7.57843	0.472729	0.236365	−	0.971664i	$-0.424044\pi$
$257$	7.57843	0.472729	0.236365	+	0.971664i	$0.424044\pi$
$258$	−10.0432	−0.625265
$259$	−9.14042	−0.567958
$260$	17.7262	1.09933
$261$	1.54181	0.0954357
$262$	−6.69796	−0.413801
$263$	−20.8668	−1.28670	−0.643350	−	0.765572i	$-0.722456\pi$
$263$	−20.8668	−1.28670	−0.643350	+	0.765572i	$0.722456\pi$
$264$	21.2853	1.31002
$265$	17.5248	1.07654
$266$	17.6661	1.08318
$267$	1.83695	0.112419
$268$	−63.2075	−3.86101
$269$	10.8238	0.659941	0.329971	−	0.943991i	$-0.392961\pi$
$269$	10.8238	0.659941	0.329971	+	0.943991i	$0.392961\pi$
$270$	8.77510	0.534035
$271$	−10.2763	−0.624243	−0.312122	−	0.950042i	$-0.601040\pi$
$271$	−10.2763	−0.624243	−0.312122	+	0.950042i	$0.601040\pi$
$272$	−15.7106	−0.952596
$273$	1.00000	0.0605228
$274$	23.0258	1.39104
$275$	11.3386	0.683744
$276$	−38.0672	−2.29138
$277$	1.53882	0.0924586	0.0462293	−	0.998931i	$-0.485280\pi$
$277$	1.53882	0.0924586	0.0462293	+	0.998931i	$0.485280\pi$
$278$	−6.01790	−0.360929
$279$	6.88659	0.412289
$280$	31.1594	1.86213
$281$	2.16425	0.129108	0.0645541	−	0.997914i	$-0.479437\pi$
$281$	2.16425	0.129108	0.0645541	+	0.997914i	$0.479437\pi$
$282$	−10.1314	−0.603319
$283$	21.0074	1.24876	0.624380	−	0.781121i	$-0.285352\pi$
$283$	21.0074	1.24876	0.624380	+	0.781121i	$0.285352\pi$
$284$	61.5544	3.65258
$285$	−20.5302	−1.21610
$286$	−5.99434	−0.354453
$287$	0.820796	0.0484501
$288$	−23.6561	−1.39395
$289$	1.00000	0.0588235
$290$	−13.5295	−0.794482
$291$	−14.5576	−0.853384
$292$	15.6278	0.914549
$293$	14.2242	0.830986	0.415493	−	0.909596i	$-0.363609\pi$
$293$	14.2242	0.830986	0.415493	+	0.909596i	$0.363609\pi$
$294$	2.74789	0.160260
$295$	−20.1442	−1.17284
$296$	−89.1873	−5.18390
$297$	−2.18144	−0.126580
$298$	61.9716	3.58992
$299$	6.85786	0.396600
$300$	−28.8523	−1.66579
$301$	3.65489	0.210664
$302$	−56.3647	−3.24342
$303$	8.13819	0.467527
$304$	101.003	5.79291
$305$	−17.7425	−1.01593
$306$	2.74789	0.157086
$307$	4.61456	0.263367	0.131683	−	0.991292i	$-0.457962\pi$
$307$	4.61456	0.263367	0.131683	+	0.991292i	$0.457962\pi$
$308$	−12.1089	−0.689969
$309$	−9.23549	−0.525388
$310$	−60.4305	−3.43222
$311$	11.1673	0.633240	0.316620	−	0.948552i	$-0.397452\pi$
$311$	11.1673	0.633240	0.316620	+	0.948552i	$0.397452\pi$
$312$	9.75745	0.552407
$313$	−18.2034	−1.02892	−0.514458	−	0.857516i	$-0.672007\pi$
$313$	−18.2034	−1.02892	−0.514458	+	0.857516i	$0.672007\pi$
$314$	−11.2425	−0.634449
$315$	−3.19340	−0.179927
$316$	77.3508	4.35132
$317$	−18.2821	−1.02682	−0.513411	−	0.858143i	$-0.671618\pi$
$317$	−18.2821	−1.02682	−0.513411	+	0.858143i	$0.671618\pi$
$318$	15.0799	0.845640
$319$	3.36336	0.188312
$320$	107.244	5.99513
$321$	−13.2537	−0.739747
$322$	18.8446	1.05017
$323$	−6.42896	−0.357717
$324$	5.55089	0.308383
$325$	5.19778	0.288321
$326$	−40.7936	−2.25935
$327$	−4.48151	−0.247828
$328$	8.00888	0.442216
$329$	3.68699	0.203271
$330$	19.1423	1.05375
$331$	−14.3075	−0.786412	−0.393206	−	0.919450i	$-0.628634\pi$
$331$	−14.3075	−0.786412	−0.393206	+	0.919450i	$0.628634\pi$
$332$	−87.5695	−4.80600
$333$	9.14042	0.500892
$334$	−12.1925	−0.667143
$335$	−36.3629	−1.98672
$336$	15.7106	0.857084
$337$	35.0615	1.90992	0.954962	−	0.296729i	$-0.0958959\pi$
$337$	35.0615	1.90992	0.954962	+	0.296729i	$0.0958959\pi$
$338$	−2.74789	−0.149465
$339$	4.76701	0.258909
$340$	−17.7262	−0.961338
$341$	15.0226	0.813522
$342$	−17.6661	−0.955271
$343$	−1.00000	−0.0539949
$344$	35.6624	1.92279
$345$	−21.8999	−1.17905
$346$	59.3089	3.18846
$347$	22.2075	1.19216	0.596081	−	0.802924i	$-0.296724\pi$
$347$	22.2075	1.19216	0.596081	+	0.802924i	$0.296724\pi$
$348$	−8.55842	−0.458780
$349$	−25.9733	−1.39032	−0.695159	−	0.718856i	$-0.744666\pi$
$349$	−25.9733	−1.39032	−0.695159	+	0.718856i	$0.744666\pi$
$350$	14.2829	0.763454
$351$	−1.00000	−0.0533761
$352$	−51.6043	−2.75052
$353$	0.855007	0.0455074	0.0227537	−	0.999741i	$-0.492757\pi$
$353$	0.855007	0.0455074	0.0227537	+	0.999741i	$0.492757\pi$
$354$	−17.3339	−0.921285
$355$	35.4119	1.87947
$356$	−10.1967	−0.540424
$357$	−1.00000	−0.0529256
$358$	−12.4023	−0.655481
$359$	20.0297	1.05713	0.528563	−	0.848894i	$-0.322731\pi$
$359$	20.0297	1.05713	0.528563	+	0.848894i	$0.322731\pi$
$360$	−31.1594	−1.64225
$361$	22.3315	1.17534
$362$	−13.5974	−0.714663
$363$	6.24134	0.327585
$364$	−5.55089	−0.290946
$365$	8.99059	0.470589
$366$	−15.2673	−0.798033
$367$	−25.3401	−1.32274	−0.661370	−	0.750060i	$-0.730025\pi$
$367$	−25.3401	−1.32274	−0.661370	+	0.750060i	$0.730025\pi$
$368$	107.741	5.61640
$369$	−0.820796	−0.0427289
$370$	−80.2081	−4.16982
$371$	−5.48782	−0.284914
$372$	−38.2267	−1.98196
$373$	16.9337	0.876792	0.438396	−	0.898782i	$-0.355547\pi$
$373$	16.9337	0.876792	0.438396	+	0.898782i	$0.355547\pi$
$374$	5.99434	0.309960
$375$	−0.631580	−0.0326147
$376$	35.9757	1.85530
$377$	1.54181	0.0794073
$378$	−2.74789	−0.141336
$379$	−7.09967	−0.364686	−0.182343	−	0.983235i	$-0.558368\pi$
$379$	−7.09967	−0.364686	−0.182343	+	0.983235i	$0.558368\pi$
$380$	113.961	5.84607
$381$	8.74443	0.447991
$382$	66.6838	3.41184
$383$	−6.11383	−0.312402	−0.156201	−	0.987725i	$-0.549925\pi$
$383$	−6.11383	−0.312402	−0.156201	+	0.987725i	$0.549925\pi$
$384$	44.9704	2.29489
$385$	−6.96619	−0.355030
$386$	74.6258	3.79836
$387$	−3.65489	−0.185789
$388$	80.8079	4.10240
$389$	0.533783	0.0270639	0.0135319	−	0.999908i	$-0.495693\pi$
$389$	0.533783	0.0270639	0.0135319	+	0.999908i	$0.495693\pi$
$390$	8.77510	0.444344
$391$	−6.85786	−0.346817
$392$	−9.75745	−0.492826
$393$	−2.43749	−0.122955
$394$	50.3977	2.53900
$395$	44.4995	2.23901
$396$	12.1089	0.608496
$397$	−34.9582	−1.75450	−0.877251	−	0.480032i	$-0.840625\pi$
$397$	−34.9582	−1.75450	−0.877251	+	0.480032i	$0.840625\pi$
$398$	−28.7447	−1.44084
$399$	6.42896	0.321851
$400$	81.6602	4.08301
$401$	17.2449	0.861171	0.430585	−	0.902550i	$-0.358307\pi$
$401$	17.2449	0.861171	0.430585	+	0.902550i	$0.358307\pi$
$402$	−31.2900	−1.56060
$403$	6.88659	0.343045
$404$	−45.1742	−2.24750
$405$	3.19340	0.158681
$406$	4.23672	0.210265
$407$	19.9393	0.988352
$408$	−9.75745	−0.483066
$409$	−24.2529	−1.19923	−0.599614	−	0.800290i	$-0.704679\pi$
$409$	−24.2529	−1.19923	−0.599614	+	0.800290i	$0.704679\pi$
$410$	7.20257	0.355709
$411$	8.37945	0.413328
$412$	51.2652	2.52565
$413$	6.30807	0.310400
$414$	−18.8446	−0.926164
$415$	−50.3782	−2.47297
$416$	−23.6561	−1.15984
$417$	−2.19001	−0.107245
$418$	−38.5374	−1.88493
$419$	25.0053	1.22159	0.610794	−	0.791790i	$-0.290850\pi$
$419$	25.0053	1.22159	0.610794	+	0.791790i	$0.290850\pi$
$420$	17.7262	0.864950
$421$	−37.7228	−1.83850	−0.919249	−	0.393677i	$-0.871203\pi$
$421$	−37.7228	−1.83850	−0.919249	+	0.393677i	$0.871203\pi$
$422$	−8.72453	−0.424703
$423$	−3.68699	−0.179268
$424$	−53.5472	−2.60048
$425$	−5.19778	−0.252129
$426$	30.4716	1.47636
$427$	5.55600	0.268874
$428$	73.5696	3.55612
$429$	−2.18144	−0.105321
$430$	32.0720	1.54665
$431$	17.6620	0.850749	0.425374	−	0.905018i	$-0.360142\pi$
$431$	17.6620	0.850749	0.425374	+	0.905018i	$0.360142\pi$
$432$	−15.7106	−0.755877
$433$	−4.24606	−0.204053	−0.102026	−	0.994782i	$-0.532533\pi$
$433$	−4.24606	−0.204053	−0.102026	+	0.994782i	$0.532533\pi$
$434$	18.9236	0.908361
$435$	−4.92361	−0.236069
$436$	24.8764	1.19136
$437$	44.0889	2.10906
$438$	7.73633	0.369656
$439$	−34.3595	−1.63989	−0.819945	−	0.572442i	$-0.805996\pi$
$439$	−34.3595	−1.63989	−0.819945	+	0.572442i	$0.805996\pi$
$440$	−67.9723	−3.24045
$441$	1.00000	0.0476190
$442$	2.74789	0.130704
$443$	17.8072	0.846047	0.423024	−	0.906119i	$-0.360969\pi$
$443$	17.8072	0.846047	0.423024	+	0.906119i	$0.360969\pi$
$444$	−50.7375	−2.40790
$445$	−5.86611	−0.278080
$446$	52.3376	2.47826
$447$	22.5524	1.06669
$448$	−33.5831	−1.58665
$449$	−29.0314	−1.37008	−0.685038	−	0.728507i	$-0.740215\pi$
$449$	−29.0314	−1.37008	−0.685038	+	0.728507i	$0.740215\pi$
$450$	−14.2829	−0.673303
$451$	−1.79051	−0.0843120
$452$	−26.4612	−1.24463
$453$	−20.5120	−0.963738
$454$	54.8222	2.57293
$455$	−3.19340	−0.149709
$456$	62.7303	2.93761
$457$	6.38991	0.298907	0.149454	−	0.988769i	$-0.452249\pi$
$457$	6.38991	0.298907	0.149454	+	0.988769i	$0.452249\pi$
$458$	39.4071	1.84137
$459$	1.00000	0.0466760
$460$	121.564	5.66794
$461$	9.66218	0.450013	0.225006	−	0.974357i	$-0.427760\pi$
$461$	9.66218	0.450013	0.225006	+	0.974357i	$0.427760\pi$
$462$	−5.99434	−0.278882
$463$	36.7020	1.70569	0.852843	−	0.522167i	$-0.174876\pi$
$463$	36.7020	1.70569	0.852843	+	0.522167i	$0.174876\pi$
$464$	24.2228	1.12451
$465$	−21.9916	−1.01984
$466$	−42.4501	−1.96646
$467$	22.5575	1.04383	0.521917	−	0.852996i	$-0.325217\pi$
$467$	22.5575	1.04383	0.521917	+	0.852996i	$0.325217\pi$
$468$	5.55089	0.256590
$469$	11.3869	0.525799
$470$	32.3537	1.49237
$471$	−4.09131	−0.188518
$472$	61.5507	2.83310
$473$	−7.97291	−0.366595
$474$	38.2914	1.75878
$475$	33.4163	1.53325
$476$	5.55089	0.254425
$477$	5.48782	0.251270
$478$	2.24716	0.102783
$479$	4.38299	0.200264	0.100132	−	0.994974i	$-0.468074\pi$
$479$	4.38299	0.200264	0.100132	+	0.994974i	$0.468074\pi$
$480$	75.5433	3.44806
$481$	9.14042	0.416768
$482$	−61.4664	−2.79972
$483$	6.85786	0.312044
$484$	−34.6450	−1.57477
$485$	46.4883	2.11093
$486$	2.74789	0.124647
$487$	22.3118	1.01104	0.505521	−	0.862814i	$-0.331300\pi$
$487$	22.3118	1.01104	0.505521	+	0.862814i	$0.331300\pi$
$488$	54.2124	2.45408
$489$	−14.8454	−0.671334
$490$	−8.77510	−0.396418
$491$	−8.81559	−0.397842	−0.198921	−	0.980016i	$-0.563744\pi$
$491$	−8.81559	−0.397842	−0.198921	+	0.980016i	$0.563744\pi$
$492$	4.55615	0.205407
$493$	−1.54181	−0.0694396
$494$	−17.6661	−0.794834
$495$	6.96619	0.313107
$496$	108.192	4.85799
$497$	−11.0891	−0.497414
$498$	−43.3500	−1.94256
$499$	6.44110	0.288343	0.144172	−	0.989553i	$-0.453948\pi$
$499$	6.44110	0.288343	0.144172	+	0.989553i	$0.453948\pi$
$500$	3.50583	0.156786
$501$	−4.43703	−0.198232
$502$	−20.5985	−0.919355
$503$	−20.1934	−0.900379	−0.450189	−	0.892933i	$-0.648644\pi$
$503$	−20.1934	−0.900379	−0.450189	+	0.892933i	$0.648644\pi$
$504$	9.75745	0.434631
$505$	−25.9885	−1.15647
$506$	−41.1084	−1.82749
$507$	−1.00000	−0.0444116
$508$	−48.5394	−2.15359
$509$	9.50281	0.421205	0.210602	−	0.977572i	$-0.432457\pi$
$509$	9.50281	0.421205	0.210602	+	0.977572i	$0.432457\pi$
$510$	−8.77510	−0.388568
$511$	−2.81537	−0.124545
$512$	−65.0606	−2.87530
$513$	−6.42896	−0.283846
$514$	−20.8247	−0.918537
$515$	29.4926	1.29960
$516$	20.2879	0.893125
$517$	−8.04294	−0.353728
$518$	25.1169	1.10357
$519$	21.5834	0.947407
$520$	−31.1594	−1.36643
$521$	25.7202	1.12682	0.563412	−	0.826176i	$-0.309488\pi$
$521$	25.7202	1.12682	0.563412	+	0.826176i	$0.309488\pi$
$522$	−4.23672	−0.185436
$523$	19.4915	0.852303	0.426151	−	0.904652i	$-0.359869\pi$
$523$	19.4915	0.852303	0.426151	+	0.904652i	$0.359869\pi$
$524$	13.5303	0.591072
$525$	5.19778	0.226850
$526$	57.3395	2.50012
$527$	−6.88659	−0.299984
$528$	−34.2717	−1.49148
$529$	24.0303	1.04479
$530$	−48.1562	−2.09177
$531$	−6.30807	−0.273747
$532$	−35.6865	−1.54720
$533$	−0.820796	−0.0355526
$534$	−5.04773	−0.218437
$535$	42.3242	1.82983
$536$	111.107	4.79910
$537$	−4.51339	−0.194767
$538$	−29.7427	−1.28230
$539$	2.18144	0.0939611
$540$	−17.7262	−0.762814
$541$	−26.7163	−1.14862	−0.574312	−	0.818637i	$-0.694730\pi$
$541$	−26.7163	−1.14862	−0.574312	+	0.818637i	$0.694730\pi$
$542$	28.2382	1.21294
$543$	−4.94830	−0.212352
$544$	23.6561	1.01425
$545$	14.3112	0.613026
$546$	−2.74789	−0.117599
$547$	−15.9199	−0.680688	−0.340344	−	0.940301i	$-0.610543\pi$
$547$	−15.9199	−0.680688	−0.340344	+	0.940301i	$0.610543\pi$
$548$	−46.5134	−1.98696
$549$	−5.55600	−0.237124
$550$	−31.1573	−1.32855
$551$	9.91224	0.422275
$552$	66.9153	2.84810
$553$	−13.9348	−0.592570
$554$	−4.22850	−0.179652
$555$	−29.1890	−1.23900
$556$	12.1565	0.515550
$557$	23.5685	0.998632	0.499316	−	0.866420i	$-0.333585\pi$
$557$	23.5685	0.998632	0.499316	+	0.866420i	$0.333585\pi$
$558$	−18.9236	−0.801099
$559$	−3.65489	−0.154585
$560$	−50.1702	−2.12008
$561$	2.18144	0.0921003
$562$	−5.94711	−0.250864
$563$	−9.92737	−0.418389	−0.209194	−	0.977874i	$-0.567084\pi$
$563$	−9.92737	−0.418389	−0.209194	+	0.977874i	$0.567084\pi$
$564$	20.4661	0.861778
$565$	−15.2230	−0.640435
$566$	−57.7260	−2.42640
$567$	−1.00000	−0.0419961
$568$	−108.201	−4.54003
$569$	0.671970	0.0281704	0.0140852	−	0.999901i	$-0.495516\pi$
$569$	0.671970	0.0281704	0.0140852	+	0.999901i	$0.495516\pi$
$570$	56.4147	2.36295
$571$	9.40298	0.393502	0.196751	−	0.980453i	$-0.436961\pi$
$571$	9.40298	0.393502	0.196751	+	0.980453i	$0.436961\pi$
$572$	12.1089	0.506299
$573$	24.2673	1.01378
$574$	−2.25546	−0.0941409
$575$	35.6456	1.48653
$576$	33.5831	1.39930
$577$	11.6650	0.485618	0.242809	−	0.970074i	$-0.421931\pi$
$577$	11.6650	0.485618	0.242809	+	0.970074i	$0.421931\pi$
$578$	−2.74789	−0.114297
$579$	27.1575	1.12863
$580$	27.3304	1.13483
$581$	15.7758	0.654489
$582$	40.0028	1.65817
$583$	11.9713	0.495802
$584$	−27.4708	−1.13675
$585$	3.19340	0.132031
$586$	−39.0865	−1.61465
$587$	44.7851	1.84848	0.924238	−	0.381816i	$-0.124701\pi$
$587$	44.7851	1.84848	0.924238	+	0.381816i	$0.124701\pi$
$588$	−5.55089	−0.228915
$589$	44.2736	1.82426
$590$	55.3539	2.27888
$591$	18.3405	0.754428
$592$	143.602	5.90199
$593$	5.62920	0.231164	0.115582	−	0.993298i	$-0.463127\pi$
$593$	5.62920	0.231164	0.115582	+	0.993298i	$0.463127\pi$
$594$	5.99434	0.245951
$595$	3.19340	0.130916
$596$	−125.186	−5.12782
$597$	−10.4607	−0.428126
$598$	−18.8446	−0.770615
$599$	−3.11663	−0.127342	−0.0636710	−	0.997971i	$-0.520281\pi$
$599$	−3.11663	−0.127342	−0.0636710	+	0.997971i	$0.520281\pi$
$600$	50.7171	2.07052
$601$	21.1064	0.860947	0.430473	−	0.902603i	$-0.358347\pi$
$601$	21.1064	0.860947	0.430473	+	0.902603i	$0.358347\pi$
$602$	−10.0432	−0.409332
$603$	−11.3869	−0.463711
$604$	113.860	4.63289
$605$	−19.9311	−0.810313
$606$	−22.3628	−0.908429
$607$	−0.691413	−0.0280636	−0.0140318	−	0.999902i	$-0.504467\pi$
$607$	−0.691413	−0.0280636	−0.0140318	+	0.999902i	$0.504467\pi$
$608$	−152.084	−6.16782
$609$	1.54181	0.0624773
$610$	48.7545	1.97401
$611$	−3.68699	−0.149160
$612$	−5.55089	−0.224381
$613$	−25.8809	−1.04532	−0.522660	−	0.852541i	$-0.675060\pi$
$613$	−25.8809	−1.04532	−0.522660	+	0.852541i	$0.675060\pi$
$614$	−12.6803	−0.511735
$615$	2.62113	0.105694
$616$	21.2853	0.857607
$617$	−33.2798	−1.33980	−0.669898	−	0.742453i	$-0.733662\pi$
$617$	−33.2798	−1.33980	−0.669898	+	0.742453i	$0.733662\pi$
$618$	25.3781	1.02086
$619$	23.0616	0.926923	0.463461	−	0.886117i	$-0.346607\pi$
$619$	23.0616	0.926923	0.463461	+	0.886117i	$0.346607\pi$
$620$	122.073	4.90257
$621$	−6.85786	−0.275197
$622$	−30.6865	−1.23042
$623$	1.83695	0.0735958
$624$	−15.7106	−0.628928
$625$	−23.9720	−0.958880
$626$	50.0208	1.99923
$627$	−14.0244	−0.560079
$628$	22.7104	0.906245
$629$	−9.14042	−0.364453
$630$	8.77510	0.349608
$631$	−35.0653	−1.39593	−0.697964	−	0.716133i	$-0.745910\pi$
$631$	−35.0653	−1.39593	−0.697964	+	0.716133i	$0.745910\pi$
$632$	−135.969	−5.40854
$633$	−3.17499	−0.126195
$634$	50.2371	1.99517
$635$	−27.9244	−1.10815
$636$	−30.4623	−1.20791
$637$	1.00000	0.0396214
$638$	−9.24214	−0.365900
$639$	11.0891	0.438678
$640$	−143.608	−5.67662
$641$	7.93422	0.313383	0.156691	−	0.987648i	$-0.449917\pi$
$641$	7.93422	0.313383	0.156691	+	0.987648i	$0.449917\pi$
$642$	36.4196	1.43737
$643$	−0.242294	−0.00955515	−0.00477757	−	0.999989i	$-0.501521\pi$
$643$	−0.242294	−0.00955515	−0.00477757	+	0.999989i	$0.501521\pi$
$644$	−38.0672	−1.50006
$645$	11.6715	0.459566
$646$	17.6661	0.695062
$647$	48.5764	1.90973	0.954867	−	0.297033i	$-0.0959972\pi$
$647$	48.5764	1.90973	0.954867	+	0.297033i	$0.0959972\pi$
$648$	−9.75745	−0.383309
$649$	−13.7606	−0.540153
$650$	−14.2829	−0.560222
$651$	6.88659	0.269907
$652$	82.4054	3.22724
$653$	−44.1229	−1.72666	−0.863331	−	0.504638i	$-0.831626\pi$
$653$	−44.1229	−1.72666	−0.863331	+	0.504638i	$0.831626\pi$
$654$	12.3147	0.481542
$655$	7.78388	0.304141
$656$	−12.8952	−0.503473
$657$	2.81537	0.109838
$658$	−10.1314	−0.394965
$659$	−41.2866	−1.60830	−0.804150	−	0.594427i	$-0.797379\pi$
$659$	−41.2866	−1.60830	−0.804150	+	0.594427i	$0.797379\pi$
$660$	−38.6686	−1.50517
$661$	−7.97913	−0.310352	−0.155176	−	0.987887i	$-0.549595\pi$
$661$	−7.97913	−0.310352	−0.155176	+	0.987887i	$0.549595\pi$
$662$	39.3155	1.52804
$663$	1.00000	0.0388368
$664$	153.931	5.97369
$665$	−20.5302	−0.796128
$666$	−25.1169	−0.973259
$667$	10.5735	0.409408
$668$	24.6295	0.952944
$669$	19.0465	0.736380
$670$	99.9213	3.86029
$671$	−12.1201	−0.467890
$672$	−23.6561	−0.912554
$673$	33.7716	1.30180	0.650900	−	0.759163i	$-0.274392\pi$
$673$	33.7716	1.30180	0.650900	+	0.759163i	$0.274392\pi$
$674$	−96.3452	−3.71108
$675$	−5.19778	−0.200063
$676$	5.55089	0.213496
$677$	−5.24737	−0.201673	−0.100836	−	0.994903i	$-0.532152\pi$
$677$	−5.24737	−0.201673	−0.100836	+	0.994903i	$0.532152\pi$
$678$	−13.0992	−0.503072
$679$	−14.5576	−0.558671
$680$	31.1594	1.19491
$681$	19.9507	0.764511
$682$	−41.2806	−1.58071
$683$	32.8558	1.25719	0.628597	−	0.777731i	$-0.283630\pi$
$683$	32.8558	1.25719	0.628597	+	0.777731i	$0.283630\pi$
$684$	35.6865	1.36451
$685$	−26.7589	−1.02241
$686$	2.74789	0.104915
$687$	14.3408	0.547138
$688$	−57.4206	−2.18914
$689$	5.48782	0.209069
$690$	60.1784	2.29095
$691$	−35.7860	−1.36136	−0.680682	−	0.732579i	$-0.738316\pi$
$691$	−35.7860	−1.36136	−0.680682	+	0.732579i	$0.738316\pi$
$692$	−119.807	−4.55439
$693$	−2.18144	−0.0828659
$694$	−61.0238	−2.31643
$695$	6.99356	0.265281
$696$	15.0441	0.570247
$697$	0.820796	0.0310899
$698$	71.3717	2.70146
$699$	−15.4483	−0.584307
$700$	−28.8523	−1.09051
$701$	−12.5152	−0.472691	−0.236345	−	0.971669i	$-0.575950\pi$
$701$	−12.5152	−0.472691	−0.236345	+	0.971669i	$0.575950\pi$
$702$	2.74789	0.103712
$703$	58.7634	2.21630
$704$	73.2594	2.76107
$705$	11.7740	0.443436
$706$	−2.34946	−0.0884233
$707$	8.13819	0.306068
$708$	35.0154	1.31596
$709$	4.13905	0.155445	0.0777227	−	0.996975i	$-0.475235\pi$
$709$	4.13905	0.155445	0.0777227	+	0.996975i	$0.475235\pi$
$710$	−97.3080	−3.65190
$711$	13.9348	0.522598
$712$	17.9239	0.671728
$713$	47.2273	1.76868
$714$	2.74789	0.102837
$715$	6.96619	0.260521
$716$	25.0533	0.936287
$717$	0.817777	0.0305404
$718$	−55.0393	−2.05405
$719$	10.4046	0.388027	0.194013	−	0.980999i	$-0.437849\pi$
$719$	10.4046	0.388027	0.194013	+	0.980999i	$0.437849\pi$
$720$	50.1702	1.86973
$721$	−9.23549	−0.343947
$722$	−61.3645	−2.28375
$723$	−22.3686	−0.831896
$724$	27.4675	1.02082
$725$	8.01399	0.297632
$726$	−17.1505	−0.636515
$727$	38.4654	1.42660	0.713302	−	0.700857i	$-0.247199\pi$
$727$	38.4654	1.42660	0.713302	+	0.700857i	$0.247199\pi$
$728$	9.75745	0.361635
$729$	1.00000	0.0370370
$730$	−24.7051	−0.914379
$731$	3.65489	0.135181
$732$	30.8408	1.13991
$733$	16.0162	0.591572	0.295786	−	0.955254i	$-0.404418\pi$
$733$	16.0162	0.591572	0.295786	+	0.955254i	$0.404418\pi$
$734$	69.6316	2.57015
$735$	−3.19340	−0.117790
$736$	−162.230	−5.97989
$737$	−24.8398	−0.914987
$738$	2.25546	0.0830245
$739$	22.0960	0.812815	0.406407	−	0.913692i	$-0.366781\pi$
$739$	22.0960	0.812815	0.406407	+	0.913692i	$0.366781\pi$
$740$	162.025	5.95615
$741$	−6.42896	−0.236174
$742$	15.0799	0.553601
$743$	−40.9387	−1.50190	−0.750948	−	0.660361i	$-0.770403\pi$
$743$	−40.9387	−1.50190	−0.750948	+	0.660361i	$0.770403\pi$
$744$	67.1955	2.46351
$745$	−72.0188	−2.63857
$746$	−46.5319	−1.70365
$747$	−15.7758	−0.577205
$748$	−12.1089	−0.442746
$749$	−13.2537	−0.484278
$750$	1.73551	0.0633719
$751$	−48.8056	−1.78094	−0.890471	−	0.455040i	$-0.849625\pi$
$751$	−48.8056	−1.78094	−0.890471	+	0.455040i	$0.849625\pi$
$752$	−57.9249	−2.11230
$753$	−7.49611	−0.273173
$754$	−4.23672	−0.154292
$755$	65.5029	2.38390
$756$	5.55089	0.201884
$757$	27.8282	1.01143	0.505717	−	0.862700i	$-0.331228\pi$
$757$	27.8282	1.01143	0.505717	+	0.862700i	$0.331228\pi$
$758$	19.5091	0.708603
$759$	−14.9600	−0.543013
$760$	−200.323	−7.26646
$761$	−39.1240	−1.41825	−0.709123	−	0.705085i	$-0.750909\pi$
$761$	−39.1240	−1.41825	−0.709123	+	0.705085i	$0.750909\pi$
$762$	−24.0287	−0.870468
$763$	−4.48151	−0.162242
$764$	−134.705	−4.87346
$765$	−3.19340	−0.115457
$766$	16.8001	0.607013
$767$	−6.30807	−0.227771
$768$	−56.4075	−2.03543
$769$	−21.0953	−0.760716	−0.380358	−	0.924839i	$-0.624199\pi$
$769$	−21.0953	−0.760716	−0.380358	+	0.924839i	$0.624199\pi$
$770$	19.1423	0.689841
$771$	−7.57843	−0.272930
$772$	−150.748	−5.42556
$773$	−22.2483	−0.800214	−0.400107	−	0.916468i	$-0.631027\pi$
$773$	−22.2483	−0.800214	−0.400107	+	0.916468i	$0.631027\pi$
$774$	10.0432	0.360997
$775$	35.7949	1.28579
$776$	−142.046	−5.09914
$777$	9.14042	0.327911
$778$	−1.46678	−0.0525864
$779$	−5.27687	−0.189063
$780$	−17.7262	−0.634699
$781$	24.1902	0.865592
$782$	18.8446	0.673883
$783$	−1.54181	−0.0550998
$784$	15.7106	0.561093
$785$	13.0652	0.466316
$786$	6.69796	0.238908
$787$	−7.39372	−0.263558	−0.131779	−	0.991279i	$-0.542069\pi$
$787$	−7.39372	−0.263558	−0.131779	+	0.991279i	$0.542069\pi$
$788$	−101.806	−3.62670
$789$	20.8668	0.742876
$790$	−122.280	−4.35051
$791$	4.76701	0.169495
$792$	−21.2853	−0.756339
$793$	−5.55600	−0.197299
$794$	96.0612	3.40909
$795$	−17.5248	−0.621540
$796$	58.0659	2.05809
$797$	−33.7670	−1.19609	−0.598044	−	0.801463i	$-0.704055\pi$
$797$	−33.7670	−1.19609	−0.598044	+	0.801463i	$0.704055\pi$
$798$	−17.6661	−0.625372
$799$	3.68699	0.130436
$800$	−122.959	−4.34726
$801$	−1.83695	−0.0649054
$802$	−47.3872	−1.67330
$803$	6.14155	0.216731
$804$	63.2075	2.22916
$805$	−21.8999	−0.771869
$806$	−18.9236	−0.666554
$807$	−10.8238	−0.381017
$808$	79.4080	2.79356
$809$	44.6391	1.56943	0.784713	−	0.619859i	$-0.212810\pi$
$809$	44.6391	1.56943	0.784713	+	0.619859i	$0.212810\pi$
$810$	−8.77510	−0.308325
$811$	−17.2316	−0.605082	−0.302541	−	0.953136i	$-0.597835\pi$
$811$	−17.2316	−0.605082	−0.302541	+	0.953136i	$0.597835\pi$
$812$	−8.55842	−0.300342
$813$	10.2763	0.360407
$814$	−54.7908	−1.92042
$815$	47.4074	1.66061
$816$	15.7106	0.549981
$817$	−23.4972	−0.822061
$818$	66.6442	2.33016
$819$	−1.00000	−0.0349428
$820$	−14.5496	−0.508094
$821$	39.3704	1.37404	0.687019	−	0.726640i	$-0.258919\pi$
$821$	39.3704	1.37404	0.687019	+	0.726640i	$0.258919\pi$
$822$	−23.0258	−0.803117
$823$	−53.5702	−1.86734	−0.933669	−	0.358136i	$-0.883412\pi$
$823$	−53.5702	−1.86734	−0.933669	+	0.358136i	$0.883412\pi$
$824$	−90.1148	−3.13930
$825$	−11.3386	−0.394760
$826$	−17.3339	−0.603122
$827$	32.8895	1.14368	0.571840	−	0.820365i	$-0.306230\pi$
$827$	32.8895	1.14368	0.571840	+	0.820365i	$0.306230\pi$
$828$	38.0672	1.32293
$829$	15.9390	0.553586	0.276793	−	0.960930i	$-0.410728\pi$
$829$	15.9390	0.553586	0.276793	+	0.960930i	$0.410728\pi$
$830$	138.434	4.80511
$831$	−1.53882	−0.0533810
$832$	33.5831	1.16428
$833$	−1.00000	−0.0346479
$834$	6.01790	0.208383
$835$	14.1692	0.490346
$836$	77.8477	2.69242
$837$	−6.88659	−0.238035
$838$	−68.7117	−2.37361
$839$	8.91069	0.307631	0.153816	−	0.988100i	$-0.450844\pi$
$839$	8.91069	0.307631	0.153816	+	0.988100i	$0.450844\pi$
$840$	−31.1594	−1.07510
$841$	−26.6228	−0.918028
$842$	103.658	3.57229
$843$	−2.16425	−0.0745407
$844$	17.6240	0.606644
$845$	3.19340	0.109856
$846$	10.1314	0.348326
$847$	6.24134	0.214455
$848$	86.2170	2.96070
$849$	−21.0074	−0.720972
$850$	14.2829	0.489900
$851$	62.6838	2.14877
$852$	−61.5544	−2.10882
$853$	47.9422	1.64151	0.820754	−	0.571281i	$-0.193553\pi$
$853$	47.9422	1.64151	0.820754	+	0.571281i	$0.193553\pi$
$854$	−15.2673	−0.522435
$855$	20.5302	0.702119
$856$	−129.322	−4.42014
$857$	19.9512	0.681519	0.340759	−	0.940151i	$-0.389316\pi$
$857$	19.9512	0.681519	0.340759	+	0.940151i	$0.389316\pi$
$858$	5.99434	0.204643
$859$	36.9983	1.26236	0.631182	−	0.775635i	$-0.282570\pi$
$859$	36.9983	1.26236	0.631182	+	0.775635i	$0.282570\pi$
$860$	−64.7873	−2.20923
$861$	−0.820796	−0.0279727
$862$	−48.5332	−1.65305
$863$	14.0144	0.477057	0.238528	−	0.971136i	$-0.423335\pi$
$863$	14.0144	0.477057	0.238528	+	0.971136i	$0.423335\pi$
$864$	23.6561	0.804797
$865$	−68.9244	−2.34350
$866$	11.6677	0.396484
$867$	−1.00000	−0.0339618
$868$	−38.2267	−1.29750
$869$	30.3980	1.03118
$870$	13.5295	0.458694
$871$	−11.3869	−0.385831
$872$	−43.7281	−1.48082
$873$	14.5576	0.492702
$874$	−121.151	−4.09801
$875$	−0.631580	−0.0213513
$876$	−15.6278	−0.528015
$877$	38.8007	1.31021	0.655103	−	0.755540i	$-0.272625\pi$
$877$	38.8007	1.31021	0.655103	+	0.755540i	$0.272625\pi$
$878$	94.4161	3.18639
$879$	−14.2242	−0.479770
$880$	109.443	3.68932
$881$	−39.0840	−1.31677	−0.658386	−	0.752680i	$-0.728761\pi$
$881$	−39.0840	−1.31677	−0.658386	+	0.752680i	$0.728761\pi$
$882$	−2.74789	−0.0925262
$883$	−4.73005	−0.159179	−0.0795894	−	0.996828i	$-0.525361\pi$
$883$	−4.73005	−0.159179	−0.0795894	+	0.996828i	$0.525361\pi$
$884$	−5.55089	−0.186697
$885$	20.1442	0.677138
$886$	−48.9323	−1.64391
$887$	−21.5770	−0.724485	−0.362243	−	0.932084i	$-0.617989\pi$
$887$	−21.5770	−0.724485	−0.362243	+	0.932084i	$0.617989\pi$
$888$	89.1873	2.99293
$889$	8.74443	0.293279
$890$	16.1194	0.540324
$891$	2.18144	0.0730809
$892$	−105.725	−3.53993
$893$	−23.7035	−0.793209
$894$	−61.9716	−2.07264
$895$	14.4130	0.481775
$896$	44.9704	1.50236
$897$	−6.85786	−0.228977
$898$	79.7751	2.66213
$899$	10.6178	0.354124
$900$	28.8523	0.961743
$901$	−5.48782	−0.182826
$902$	4.92013	0.163823
$903$	−3.65489	−0.121627
$904$	46.5139	1.54703
$905$	15.8019	0.525273
$906$	56.3647	1.87259
$907$	26.5405	0.881263	0.440632	−	0.897688i	$-0.354755\pi$
$907$	26.5405	0.881263	0.440632	+	0.897688i	$0.354755\pi$
$908$	−110.744	−3.67517
$909$	−8.13819	−0.269927
$910$	8.77510	0.290892
$911$	32.6789	1.08270	0.541350	−	0.840797i	$-0.317913\pi$
$911$	32.6789	1.08270	0.541350	+	0.840797i	$0.317913\pi$
$912$	−101.003	−3.34454
$913$	−34.4138	−1.13893
$914$	−17.5587	−0.580792
$915$	17.7425	0.586549
$916$	−79.6045	−2.63021
$917$	−2.43749	−0.0804931
$918$	−2.74789	−0.0906938
$919$	35.1283	1.15878	0.579388	−	0.815052i	$-0.303291\pi$
$919$	35.1283	1.15878	0.579388	+	0.815052i	$0.303291\pi$
$920$	−213.687	−7.04505
$921$	−4.61456	−0.152055
$922$	−26.5506	−0.874398
$923$	11.0891	0.365002
$924$	12.1089	0.398354
$925$	47.5099	1.56212
$926$	−100.853	−3.31423
$927$	9.23549	0.303333
$928$	−36.4732	−1.19729
$929$	−33.8329	−1.11002	−0.555011	−	0.831843i	$-0.687286\pi$
$929$	−33.8329	−1.11002	−0.555011	+	0.831843i	$0.687286\pi$
$930$	60.4305	1.98159
$931$	6.42896	0.210701
$932$	85.7516	2.80889
$933$	−11.1673	−0.365602
$934$	−61.9854	−2.02822
$935$	−6.96619	−0.227819
$936$	−9.75745	−0.318932
$937$	−57.7836	−1.88771	−0.943854	−	0.330362i	$-0.892829\pi$
$937$	−57.7836	−1.88771	−0.943854	+	0.330362i	$0.892829\pi$
$938$	−31.2900	−1.02165
$939$	18.2034	0.594045
$940$	−65.3564	−2.13169
$941$	−16.4996	−0.537872	−0.268936	−	0.963158i	$-0.586672\pi$
$941$	−16.4996	−0.537872	−0.268936	+	0.963158i	$0.586672\pi$
$942$	11.2425	0.366299
$943$	−5.62891	−0.183302
$944$	−99.1036	−3.22555
$945$	3.19340	0.103881
$946$	21.9087	0.712312
$947$	43.7792	1.42263	0.711316	−	0.702872i	$-0.248099\pi$
$947$	43.7792	1.42263	0.711316	+	0.702872i	$0.248099\pi$
$948$	−77.3508	−2.51224
$949$	2.81537	0.0913908
$950$	−91.8243	−2.97917
$951$	18.2821	0.592836
$952$	−9.75745	−0.316241
$953$	35.5631	1.15200	0.576001	−	0.817449i	$-0.304613\pi$
$953$	35.5631	1.15200	0.576001	+	0.817449i	$0.304613\pi$
$954$	−15.0799	−0.488231
$955$	−77.4951	−2.50768
$956$	−4.53939	−0.146814
$957$	−3.36336	−0.108722
$958$	−12.0440	−0.389123
$959$	8.37945	0.270587
$960$	−107.244	−3.46129
$961$	16.4251	0.529841
$962$	−25.1169	−0.809800
$963$	13.2537	0.427093
$964$	124.166	3.99910
$965$	−86.7247	−2.79177
$966$	−18.8446	−0.606316
$967$	−5.70278	−0.183389	−0.0916945	−	0.995787i	$-0.529228\pi$
$967$	−5.70278	−0.183389	−0.0916945	+	0.995787i	$0.529228\pi$
$968$	60.8996	1.95739
$969$	6.42896	0.206528
$970$	−127.745	−4.10164
$971$	44.1221	1.41595	0.707973	−	0.706239i	$-0.249610\pi$
$971$	44.1221	1.41595	0.707973	+	0.706239i	$0.249610\pi$
$972$	−5.55089	−0.178045
$973$	−2.19001	−0.0702084
$974$	−61.3102	−1.96451
$975$	−5.19778	−0.166462
$976$	−87.2882	−2.79403
$977$	−5.81018	−0.185884	−0.0929421	−	0.995672i	$-0.529627\pi$
$977$	−5.81018	−0.185884	−0.0929421	+	0.995672i	$0.529627\pi$
$978$	40.7936	1.30444
$979$	−4.00719	−0.128070
$980$	17.7262	0.566242
$981$	4.48151	0.143084
$982$	24.2242	0.773027
$983$	−1.15649	−0.0368864	−0.0184432	−	0.999830i	$-0.505871\pi$
$983$	−1.15649	−0.0368864	−0.0184432	+	0.999830i	$0.505871\pi$
$984$	−8.00888	−0.255314
$985$	−58.5685	−1.86615
$986$	4.23672	0.134925
$987$	−3.68699	−0.117358
$988$	35.6865	1.13534
$989$	−25.0647	−0.797013
$990$	−19.1423	−0.608383
$991$	9.97625	0.316906	0.158453	−	0.987367i	$-0.449349\pi$
$991$	9.97625	0.316906	0.158453	+	0.987367i	$0.449349\pi$
$992$	−162.910	−5.17239
$993$	14.3075	0.454035
$994$	30.4716	0.966501
$995$	33.4050	1.05901
$996$	87.5695	2.77475
$997$	−21.7430	−0.688607	−0.344304	−	0.938858i	$-0.611885\pi$
$997$	−21.7430	−0.688607	−0.344304	+	0.938858i	$0.611885\pi$
$998$	−17.6994	−0.560265
$999$	−9.14042	−0.289190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4641.2.a.ba.1.1	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4641.2.a.ba.1.1	✓	17	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4641 = 3 \cdot 7 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4641.a (trivial)

\(f(q)\)	\(=\)	\(q-2.74789 q^{2} -1.00000 q^{3} +5.55089 q^{4} +3.19340 q^{5} +2.74789 q^{6} -1.00000 q^{7} -9.75745 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.74789 q^{2} -1.00000 q^{3} +5.55089 q^{4} +3.19340 q^{5} +2.74789 q^{6} -1.00000 q^{7} -9.75745 q^{8} +1.00000 q^{9} -8.77510 q^{10} +2.18144 q^{11} -5.55089 q^{12} +1.00000 q^{13} +2.74789 q^{14} -3.19340 q^{15} +15.7106 q^{16} -1.00000 q^{17} -2.74789 q^{18} +6.42896 q^{19} +17.7262 q^{20} +1.00000 q^{21} -5.99434 q^{22} +6.85786 q^{23} +9.75745 q^{24} +5.19778 q^{25} -2.74789 q^{26} -1.00000 q^{27} -5.55089 q^{28} +1.54181 q^{29} +8.77510 q^{30} +6.88659 q^{31} -23.6561 q^{32} -2.18144 q^{33} +2.74789 q^{34} -3.19340 q^{35} +5.55089 q^{36} +9.14042 q^{37} -17.6661 q^{38} -1.00000 q^{39} -31.1594 q^{40} -0.820796 q^{41} -2.74789 q^{42} -3.65489 q^{43} +12.1089 q^{44} +3.19340 q^{45} -18.8446 q^{46} -3.68699 q^{47} -15.7106 q^{48} +1.00000 q^{49} -14.2829 q^{50} +1.00000 q^{51} +5.55089 q^{52} +5.48782 q^{53} +2.74789 q^{54} +6.96619 q^{55} +9.75745 q^{56} -6.42896 q^{57} -4.23672 q^{58} -6.30807 q^{59} -17.7262 q^{60} -5.55600 q^{61} -18.9236 q^{62} -1.00000 q^{63} +33.5831 q^{64} +3.19340 q^{65} +5.99434 q^{66} -11.3869 q^{67} -5.55089 q^{68} -6.85786 q^{69} +8.77510 q^{70} +11.0891 q^{71} -9.75745 q^{72} +2.81537 q^{73} -25.1169 q^{74} -5.19778 q^{75} +35.6865 q^{76} -2.18144 q^{77} +2.74789 q^{78} +13.9348 q^{79} +50.1702 q^{80} +1.00000 q^{81} +2.25546 q^{82} -15.7758 q^{83} +5.55089 q^{84} -3.19340 q^{85} +10.0432 q^{86} -1.54181 q^{87} -21.2853 q^{88} -1.83695 q^{89} -8.77510 q^{90} -1.00000 q^{91} +38.0672 q^{92} -6.88659 q^{93} +10.1314 q^{94} +20.5302 q^{95} +23.6561 q^{96} +14.5576 q^{97} -2.74789 q^{98} +2.18144 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + q^{2} - 17 q^{3} + 25 q^{4} - 4 q^{5} - q^{6} - 17 q^{7} + 6 q^{8} + 17 q^{9}+O(q^{10}) \) 17 * q + q^2 - 17 * q^3 + 25 * q^4 - 4 * q^5 - q^6 - 17 * q^7 + 6 * q^8 + 17 * q^9	\( 17 q + q^{2} - 17 q^{3} + 25 q^{4} - 4 q^{5} - q^{6} - 17 q^{7} + 6 q^{8} + 17 q^{9} - q^{10} + 4 q^{11} - 25 q^{12} + 17 q^{13} - q^{14} + 4 q^{15} + 53 q^{16} - 17 q^{17} + q^{18} + 13 q^{19} - 5 q^{20} + 17 q^{21} + 4 q^{22} + 8 q^{23} - 6 q^{24} + 37 q^{25} + q^{26} - 17 q^{27} - 25 q^{28} - 3 q^{29} + q^{30} + 10 q^{31} + 18 q^{32} - 4 q^{33} - q^{34} + 4 q^{35} + 25 q^{36} - 25 q^{38} - 17 q^{39} - 12 q^{41} + q^{42} + 29 q^{43} + 12 q^{44} - 4 q^{45} + 19 q^{46} - 4 q^{47} - 53 q^{48} + 17 q^{49} + 9 q^{50} + 17 q^{51} + 25 q^{52} - 8 q^{53} - q^{54} + 27 q^{55} - 6 q^{56} - 13 q^{57} + 2 q^{58} + 25 q^{59} + 5 q^{60} - 5 q^{61} - 37 q^{62} - 17 q^{63} + 94 q^{64} - 4 q^{65} - 4 q^{66} + 15 q^{67} - 25 q^{68} - 8 q^{69} + q^{70} + 32 q^{71} + 6 q^{72} - 15 q^{73} + 16 q^{74} - 37 q^{75} + 3 q^{76} - 4 q^{77} - q^{78} + 27 q^{79} + 41 q^{80} + 17 q^{81} - 8 q^{82} - 24 q^{83} + 25 q^{84} + 4 q^{85} + 53 q^{86} + 3 q^{87} - 9 q^{88} - 8 q^{89} - q^{90} - 17 q^{91} + 14 q^{92} - 10 q^{93} + 51 q^{94} - 9 q^{95} - 18 q^{96} - 22 q^{97} + q^{98} + 4 q^{99}+O(q^{100}) \) 17 * q + q^2 - 17 * q^3 + 25 * q^4 - 4 * q^5 - q^6 - 17 * q^7 + 6 * q^8 + 17 * q^9 - q^10 + 4 * q^11 - 25 * q^12 + 17 * q^13 - q^14 + 4 * q^15 + 53 * q^16 - 17 * q^17 + q^18 + 13 * q^19 - 5 * q^20 + 17 * q^21 + 4 * q^22 + 8 * q^23 - 6 * q^24 + 37 * q^25 + q^26 - 17 * q^27 - 25 * q^28 - 3 * q^29 + q^30 + 10 * q^31 + 18 * q^32 - 4 * q^33 - q^34 + 4 * q^35 + 25 * q^36 - 25 * q^38 - 17 * q^39 - 12 * q^41 + q^42 + 29 * q^43 + 12 * q^44 - 4 * q^45 + 19 * q^46 - 4 * q^47 - 53 * q^48 + 17 * q^49 + 9 * q^50 + 17 * q^51 + 25 * q^52 - 8 * q^53 - q^54 + 27 * q^55 - 6 * q^56 - 13 * q^57 + 2 * q^58 + 25 * q^59 + 5 * q^60 - 5 * q^61 - 37 * q^62 - 17 * q^63 + 94 * q^64 - 4 * q^65 - 4 * q^66 + 15 * q^67 - 25 * q^68 - 8 * q^69 + q^70 + 32 * q^71 + 6 * q^72 - 15 * q^73 + 16 * q^74 - 37 * q^75 + 3 * q^76 - 4 * q^77 - q^78 + 27 * q^79 + 41 * q^80 + 17 * q^81 - 8 * q^82 - 24 * q^83 + 25 * q^84 + 4 * q^85 + 53 * q^86 + 3 * q^87 - 9 * q^88 - 8 * q^89 - q^90 - 17 * q^91 + 14 * q^92 - 10 * q^93 + 51 * q^94 - 9 * q^95 - 18 * q^96 - 22 * q^97 + q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more