LMFDB - Embedded newform 3381.2.a.bh.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 100x^{3} - 17x^{2} - 28x - 2 $ x^10 - 4x^9 - 6x^8 + 36x^7 + x^6 - 100x^5 + 26x^4 + 100x^3 - 17x^2 - 28x - 2
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			1.7
Root			$-0.488009$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.488009 q^{2} +1.00000 q^{3} -1.76185 q^{4} +0.529828 q^{5} +0.488009 q^{6} -1.83582 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.488009 q^{2} +1.00000 q^{3} -1.76185 q^{4} +0.529828 q^{5} +0.488009 q^{6} -1.83582 q^{8} +1.00000 q^{9} +0.258561 q^{10} -1.22947 q^{11} -1.76185 q^{12} +1.15081 q^{13} +0.529828 q^{15} +2.62780 q^{16} -6.79418 q^{17} +0.488009 q^{18} +2.34174 q^{19} -0.933476 q^{20} -0.599993 q^{22} +1.00000 q^{23} -1.83582 q^{24} -4.71928 q^{25} +0.561605 q^{26} +1.00000 q^{27} -5.30228 q^{29} +0.258561 q^{30} +10.3026 q^{31} +4.95402 q^{32} -1.22947 q^{33} -3.31562 q^{34} -1.76185 q^{36} +1.58592 q^{37} +1.14279 q^{38} +1.15081 q^{39} -0.972666 q^{40} +6.36361 q^{41} -10.2438 q^{43} +2.16614 q^{44} +0.529828 q^{45} +0.488009 q^{46} -11.8065 q^{47} +2.62780 q^{48} -2.30305 q^{50} -6.79418 q^{51} -2.02755 q^{52} -12.2750 q^{53} +0.488009 q^{54} -0.651408 q^{55} +2.34174 q^{57} -2.58756 q^{58} -10.4312 q^{59} -0.933476 q^{60} -9.66077 q^{61} +5.02778 q^{62} -2.83799 q^{64} +0.609730 q^{65} -0.599993 q^{66} +15.1775 q^{67} +11.9703 q^{68} +1.00000 q^{69} -7.49613 q^{71} -1.83582 q^{72} -11.1935 q^{73} +0.773944 q^{74} -4.71928 q^{75} -4.12579 q^{76} +0.561605 q^{78} +11.6816 q^{79} +1.39228 q^{80} +1.00000 q^{81} +3.10550 q^{82} -10.9393 q^{83} -3.59975 q^{85} -4.99905 q^{86} -5.30228 q^{87} +2.25708 q^{88} +7.21683 q^{89} +0.258561 q^{90} -1.76185 q^{92} +10.3026 q^{93} -5.76167 q^{94} +1.24072 q^{95} +4.95402 q^{96} -8.00904 q^{97} -1.22947 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{2} + 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} - 12 q^{8} + 10 q^{9}+O(q^{10}) $ 10 * q - 4 * q^2 + 10 * q^3 + 8 * q^4 - 4 * q^5 - 4 * q^6 - 12 * q^8 + 10 * q^9	$ 10 q - 4 q^{2} + 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} - 12 q^{8} + 10 q^{9} - 8 q^{10} - 2 q^{11} + 8 q^{12} - 16 q^{13} - 4 q^{15} + 4 q^{16} - 12 q^{17} - 4 q^{18} - 26 q^{19} - 8 q^{22} + 10 q^{23} - 12 q^{24} + 14 q^{25} + 12 q^{26} + 10 q^{27} - 16 q^{29} - 8 q^{30} - 20 q^{31} - 8 q^{32} - 2 q^{33} + 4 q^{34} + 8 q^{36} + 8 q^{37} + 8 q^{38} - 16 q^{39} + 12 q^{40} - 22 q^{41} - 4 q^{43} - 24 q^{44} - 4 q^{45} - 4 q^{46} - 6 q^{47} + 4 q^{48} - 48 q^{50} - 12 q^{51} - 24 q^{52} - 30 q^{53} - 4 q^{54} - 48 q^{55} - 26 q^{57} + 24 q^{58} - 42 q^{59} - 14 q^{61} - 40 q^{62} + 8 q^{64} - 44 q^{65} - 8 q^{66} - 8 q^{68} + 10 q^{69} + 8 q^{71} - 12 q^{72} - 24 q^{73} + 8 q^{74} + 14 q^{75} - 32 q^{76} + 12 q^{78} + 32 q^{79} - 28 q^{80} + 10 q^{81} + 64 q^{82} - 28 q^{83} - 4 q^{85} - 4 q^{86} - 16 q^{87} + 20 q^{88} - 8 q^{90} + 8 q^{92} - 20 q^{93} - 8 q^{94} - 16 q^{95} - 8 q^{96} + 12 q^{97} - 2 q^{99}+O(q^{100}) $ 10 * q - 4 * q^2 + 10 * q^3 + 8 * q^4 - 4 * q^5 - 4 * q^6 - 12 * q^8 + 10 * q^9 - 8 * q^10 - 2 * q^11 + 8 * q^12 - 16 * q^13 - 4 * q^15 + 4 * q^16 - 12 * q^17 - 4 * q^18 - 26 * q^19 - 8 * q^22 + 10 * q^23 - 12 * q^24 + 14 * q^25 + 12 * q^26 + 10 * q^27 - 16 * q^29 - 8 * q^30 - 20 * q^31 - 8 * q^32 - 2 * q^33 + 4 * q^34 + 8 * q^36 + 8 * q^37 + 8 * q^38 - 16 * q^39 + 12 * q^40 - 22 * q^41 - 4 * q^43 - 24 * q^44 - 4 * q^45 - 4 * q^46 - 6 * q^47 + 4 * q^48 - 48 * q^50 - 12 * q^51 - 24 * q^52 - 30 * q^53 - 4 * q^54 - 48 * q^55 - 26 * q^57 + 24 * q^58 - 42 * q^59 - 14 * q^61 - 40 * q^62 + 8 * q^64 - 44 * q^65 - 8 * q^66 - 8 * q^68 + 10 * q^69 + 8 * q^71 - 12 * q^72 - 24 * q^73 + 8 * q^74 + 14 * q^75 - 32 * q^76 + 12 * q^78 + 32 * q^79 - 28 * q^80 + 10 * q^81 + 64 * q^82 - 28 * q^83 - 4 * q^85 - 4 * q^86 - 16 * q^87 + 20 * q^88 - 8 * q^90 + 8 * q^92 - 20 * q^93 - 8 * q^94 - 16 * q^95 - 8 * q^96 + 12 * q^97 - 2 * 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 $

$2$	0.488009	0.345074	0.172537	−	0.985003i	$-0.444803\pi$
$2$	0.488009	0.345074	0.172537	+	0.985003i	$0.444803\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.76185	−0.880924
$5$	0.529828	0.236946	0.118473	−	0.992957i	$-0.462200\pi$
$5$	0.529828	0.236946	0.118473	+	0.992957i	$0.462200\pi$
$6$	0.488009	0.199229
$7$	0	0
$7$	0	0
$8$	−1.83582	−0.649059
$9$	1.00000	0.333333
$10$	0.258561	0.0817641
$11$	−1.22947	−0.370700	−0.185350	−	0.982673i	$-0.559342\pi$
$11$	−1.22947	−0.370700	−0.185350	+	0.982673i	$0.559342\pi$
$12$	−1.76185	−0.508601
$13$	1.15081	0.319177	0.159588	−	0.987184i	$-0.448983\pi$
$13$	1.15081	0.319177	0.159588	+	0.987184i	$0.448983\pi$
$14$	0	0
$15$	0.529828	0.136801
$16$	2.62780	0.656950
$17$	−6.79418	−1.64783	−0.823916	−	0.566712i	$-0.808215\pi$
$17$	−6.79418	−1.64783	−0.823916	+	0.566712i	$0.808215\pi$
$18$	0.488009	0.115025
$19$	2.34174	0.537232	0.268616	−	0.963247i	$-0.413434\pi$
$19$	2.34174	0.537232	0.268616	+	0.963247i	$0.413434\pi$
$20$	−0.933476	−0.208731
$21$	0	0
$22$	−0.599993	−0.127919
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.83582	−0.374734
$25$	−4.71928	−0.943857
$26$	0.561605	0.110140
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.30228	−0.984610	−0.492305	−	0.870423i	$-0.663845\pi$
$29$	−5.30228	−0.984610	−0.492305	+	0.870423i	$0.663845\pi$
$30$	0.258561	0.0472065
$31$	10.3026	1.85041	0.925204	−	0.379470i	$-0.123894\pi$
$31$	10.3026	1.85041	0.925204	+	0.379470i	$0.123894\pi$
$32$	4.95402	0.875755
$33$	−1.22947	−0.214023
$34$	−3.31562	−0.568625
$35$	0	0
$36$	−1.76185	−0.293641
$37$	1.58592	0.260724	0.130362	−	0.991466i	$-0.458386\pi$
$37$	1.58592	0.260724	0.130362	+	0.991466i	$0.458386\pi$
$38$	1.14279	0.185385
$39$	1.15081	0.184277
$40$	−0.972666	−0.153792
$41$	6.36361	0.993828	0.496914	−	0.867800i	$-0.334466\pi$
$41$	6.36361	0.993828	0.496914	+	0.867800i	$0.334466\pi$
$42$	0	0
$43$	−10.2438	−1.56216	−0.781079	−	0.624432i	$-0.785330\pi$
$43$	−10.2438	−1.56216	−0.781079	+	0.624432i	$0.785330\pi$
$44$	2.16614	0.326558
$45$	0.529828	0.0789821
$46$	0.488009	0.0719530
$47$	−11.8065	−1.72215	−0.861077	−	0.508475i	$-0.830209\pi$
$47$	−11.8065	−1.72215	−0.861077	+	0.508475i	$0.830209\pi$
$48$	2.62780	0.379290
$49$	0	0
$50$	−2.30305	−0.325701
$51$	−6.79418	−0.951376
$52$	−2.02755	−0.281170
$53$	−12.2750	−1.68610	−0.843050	−	0.537835i	$-0.819242\pi$
$53$	−12.2750	−1.68610	−0.843050	+	0.537835i	$0.819242\pi$
$54$	0.488009	0.0664096
$55$	−0.651408	−0.0878358
$56$	0	0
$57$	2.34174	0.310171
$58$	−2.58756	−0.339764
$59$	−10.4312	−1.35802	−0.679012	−	0.734127i	$-0.737592\pi$
$59$	−10.4312	−1.35802	−0.679012	+	0.734127i	$0.737592\pi$
$60$	−0.933476	−0.120511
$61$	−9.66077	−1.23693	−0.618467	−	0.785810i	$-0.712246\pi$
$61$	−9.66077	−1.23693	−0.618467	+	0.785810i	$0.712246\pi$
$62$	5.02778	0.638529
$63$	0	0
$64$	−2.83799	−0.354749
$65$	0.609730	0.0756277
$66$	−0.599993	−0.0738540
$67$	15.1775	1.85423	0.927115	−	0.374777i	$-0.122281\pi$
$67$	15.1775	1.85423	0.927115	+	0.374777i	$0.122281\pi$
$68$	11.9703	1.45161
$69$	1.00000	0.120386
$70$	0	0
$71$	−7.49613	−0.889627	−0.444813	−	0.895623i	$-0.646730\pi$
$71$	−7.49613	−0.889627	−0.444813	+	0.895623i	$0.646730\pi$
$72$	−1.83582	−0.216353
$73$	−11.1935	−1.31010	−0.655049	−	0.755586i	$-0.727352\pi$
$73$	−11.1935	−1.31010	−0.655049	+	0.755586i	$0.727352\pi$
$74$	0.773944	0.0899692
$75$	−4.71928	−0.544936
$76$	−4.12579	−0.473260
$77$	0	0
$78$	0.561605	0.0635892
$79$	11.6816	1.31428	0.657142	−	0.753767i	$-0.271765\pi$
$79$	11.6816	1.31428	0.657142	+	0.753767i	$0.271765\pi$
$80$	1.39228	0.155662
$81$	1.00000	0.111111
$82$	3.10550	0.342945
$83$	−10.9393	−1.20075	−0.600375	−	0.799719i	$-0.704982\pi$
$83$	−10.9393	−1.20075	−0.600375	+	0.799719i	$0.704982\pi$
$84$	0	0
$85$	−3.59975	−0.390447
$86$	−4.99905	−0.539061
$87$	−5.30228	−0.568465
$88$	2.25708	0.240606
$89$	7.21683	0.764983	0.382491	−	0.923959i	$-0.375066\pi$
$89$	7.21683	0.764983	0.382491	+	0.923959i	$0.375066\pi$
$90$	0.258561	0.0272547
$91$	0	0
$92$	−1.76185	−0.183685
$93$	10.3026	1.06833
$94$	−5.76167	−0.594271
$95$	1.24072	0.127295
$96$	4.95402	0.505618
$97$	−8.00904	−0.813195	−0.406597	−	0.913607i	$-0.633285\pi$
$97$	−8.00904	−0.813195	−0.406597	+	0.913607i	$0.633285\pi$
$98$	0	0
$99$	−1.22947	−0.123567
$100$	8.31465	0.831465
$101$	−5.74251	−0.571401	−0.285701	−	0.958319i	$-0.592226\pi$
$101$	−5.74251	−0.571401	−0.285701	+	0.958319i	$0.592226\pi$
$102$	−3.31562	−0.328296
$103$	−1.58251	−0.155929	−0.0779645	−	0.996956i	$-0.524842\pi$
$103$	−1.58251	−0.155929	−0.0779645	+	0.996956i	$0.524842\pi$
$104$	−2.11267	−0.207164
$105$	0	0
$106$	−5.99031	−0.581830
$107$	17.0391	1.64723	0.823617	−	0.567147i	$-0.191953\pi$
$107$	17.0391	1.64723	0.823617	+	0.567147i	$0.191953\pi$
$108$	−1.76185	−0.169534
$109$	−3.40999	−0.326618	−0.163309	−	0.986575i	$-0.552217\pi$
$109$	−3.40999	−0.326618	−0.163309	+	0.986575i	$0.552217\pi$
$110$	−0.317893	−0.0303099
$111$	1.58592	0.150529
$112$	0	0
$113$	7.79701	0.733481	0.366740	−	0.930323i	$-0.380474\pi$
$113$	7.79701	0.733481	0.366740	+	0.930323i	$0.380474\pi$
$114$	1.14279	0.107032
$115$	0.529828	0.0494067
$116$	9.34182	0.867366
$117$	1.15081	0.106392
$118$	−5.09051	−0.468619
$119$	0	0
$120$	−0.972666	−0.0887918
$121$	−9.48840	−0.862582
$122$	−4.71454	−0.426835
$123$	6.36361	0.573787
$124$	−18.1517	−1.63007
$125$	−5.14955	−0.460589
$126$	0	0
$127$	−2.12218	−0.188313	−0.0941565	−	0.995557i	$-0.530015\pi$
$127$	−2.12218	−0.188313	−0.0941565	+	0.995557i	$0.530015\pi$
$128$	−11.2930	−0.998170
$129$	−10.2438	−0.901912
$130$	0.297554	0.0260972
$131$	−5.47146	−0.478044	−0.239022	−	0.971014i	$-0.576827\pi$
$131$	−5.47146	−0.478044	−0.239022	+	0.971014i	$0.576827\pi$
$132$	2.16614	0.188538
$133$	0	0
$134$	7.40677	0.639848
$135$	0.529828	0.0456003
$136$	12.4729	1.06954
$137$	10.3012	0.880090	0.440045	−	0.897976i	$-0.354962\pi$
$137$	10.3012	0.880090	0.440045	+	0.897976i	$0.354962\pi$
$138$	0.488009	0.0415421
$139$	−20.2827	−1.72035	−0.860176	−	0.509997i	$-0.829646\pi$
$139$	−20.2827	−1.72035	−0.860176	+	0.509997i	$0.829646\pi$
$140$	0	0
$141$	−11.8065	−0.994286
$142$	−3.65818	−0.306987
$143$	−1.41489	−0.118319
$144$	2.62780	0.218983
$145$	−2.80930	−0.233299
$146$	−5.46252	−0.452082
$147$	0	0
$148$	−2.79415	−0.229678
$149$	−6.08821	−0.498765	−0.249383	−	0.968405i	$-0.580228\pi$
$149$	−6.08821	−0.498765	−0.249383	+	0.968405i	$0.580228\pi$
$150$	−2.30305	−0.188043
$151$	−17.0653	−1.38875	−0.694376	−	0.719612i	$-0.744320\pi$
$151$	−17.0653	−1.38875	−0.694376	+	0.719612i	$0.744320\pi$
$152$	−4.29900	−0.348695
$153$	−6.79418	−0.549277
$154$	0	0
$155$	5.45862	0.438447
$156$	−2.02755	−0.162334
$157$	−0.204125	−0.0162909	−0.00814545	−	0.999967i	$-0.502593\pi$
$157$	−0.204125	−0.0162909	−0.00814545	+	0.999967i	$0.502593\pi$
$158$	5.70073	0.453526
$159$	−12.2750	−0.973470
$160$	2.62478	0.207507
$161$	0	0
$162$	0.488009	0.0383416
$163$	20.6054	1.61394	0.806970	−	0.590593i	$-0.201106\pi$
$163$	20.6054	1.61394	0.806970	+	0.590593i	$0.201106\pi$
$164$	−11.2117	−0.875487
$165$	−0.651408	−0.0507120
$166$	−5.33850	−0.414348
$167$	13.5408	1.04782	0.523908	−	0.851775i	$-0.324474\pi$
$167$	13.5408	1.04782	0.523908	+	0.851775i	$0.324474\pi$
$168$	0	0
$169$	−11.6756	−0.898126
$170$	−1.75671	−0.134733
$171$	2.34174	0.179077
$172$	18.0479	1.37614
$173$	4.97475	0.378224	0.189112	−	0.981956i	$-0.439439\pi$
$173$	4.97475	0.378224	0.189112	+	0.981956i	$0.439439\pi$
$174$	−2.58756	−0.196163
$175$	0	0
$176$	−3.23080	−0.243531
$177$	−10.4312	−0.784056
$178$	3.52188	0.263976
$179$	−6.35192	−0.474765	−0.237382	−	0.971416i	$-0.576289\pi$
$179$	−6.35192	−0.474765	−0.237382	+	0.971416i	$0.576289\pi$
$180$	−0.933476	−0.0695772
$181$	−9.74302	−0.724193	−0.362096	−	0.932141i	$-0.617939\pi$
$181$	−9.74302	−0.724193	−0.362096	+	0.932141i	$0.617939\pi$
$182$	0	0
$183$	−9.66077	−0.714145
$184$	−1.83582	−0.135338
$185$	0.840265	0.0617775
$186$	5.02778	0.368655
$187$	8.35326	0.610851
$188$	20.8012	1.51709
$189$	0	0
$190$	0.605482	0.0439263
$191$	5.93676	0.429569	0.214785	−	0.976661i	$-0.431095\pi$
$191$	5.93676	0.429569	0.214785	+	0.976661i	$0.431095\pi$
$192$	−2.83799	−0.204815
$193$	−0.689373	−0.0496221	−0.0248111	−	0.999692i	$-0.507898\pi$
$193$	−0.689373	−0.0496221	−0.0248111	+	0.999692i	$0.507898\pi$
$194$	−3.90848	−0.280613
$195$	0.609730	0.0436637
$196$	0	0
$197$	−12.1451	−0.865302	−0.432651	−	0.901561i	$-0.642422\pi$
$197$	−12.1451	−0.865302	−0.432651	+	0.901561i	$0.642422\pi$
$198$	−0.599993	−0.0426397
$199$	20.2213	1.43345	0.716725	−	0.697356i	$-0.245640\pi$
$199$	20.2213	1.43345	0.716725	+	0.697356i	$0.245640\pi$
$200$	8.66373	0.612618
$201$	15.1775	1.07054
$202$	−2.80240	−0.197176
$203$	0	0
$204$	11.9703	0.838090
$205$	3.37162	0.235484
$206$	−0.772278	−0.0538071
$207$	1.00000	0.0695048
$208$	3.02409	0.209683
$209$	−2.87910	−0.199152
$210$	0	0
$211$	6.39015	0.439916	0.219958	−	0.975509i	$-0.429408\pi$
$211$	6.39015	0.439916	0.219958	+	0.975509i	$0.429408\pi$
$212$	21.6267	1.48533
$213$	−7.49613	−0.513626
$214$	8.31524	0.568418
$215$	−5.42743	−0.370147
$216$	−1.83582	−0.124911
$217$	0	0
$218$	−1.66410	−0.112707
$219$	−11.1935	−0.756386
$220$	1.14768	0.0773767
$221$	−7.81880	−0.525949
$222$	0.773944	0.0519437
$223$	−25.8012	−1.72778	−0.863889	−	0.503682i	$-0.831978\pi$
$223$	−25.8012	−1.72778	−0.863889	+	0.503682i	$0.831978\pi$
$224$	0	0
$225$	−4.71928	−0.314619
$226$	3.80501	0.253105
$227$	−14.3656	−0.953477	−0.476739	−	0.879045i	$-0.658181\pi$
$227$	−14.3656	−0.953477	−0.476739	+	0.879045i	$0.658181\pi$
$228$	−4.12579	−0.273237
$229$	−14.1552	−0.935403	−0.467702	−	0.883886i	$-0.654918\pi$
$229$	−14.1552	−0.935403	−0.467702	+	0.883886i	$0.654918\pi$
$230$	0.258561	0.0170490
$231$	0	0
$232$	9.73402	0.639069
$233$	−17.7970	−1.16592	−0.582961	−	0.812500i	$-0.698106\pi$
$233$	−17.7970	−1.16592	−0.582961	+	0.812500i	$0.698106\pi$
$234$	0.561605	0.0367132
$235$	−6.25541	−0.408058
$236$	18.3781	1.19632
$237$	11.6816	0.758802
$238$	0	0
$239$	15.9717	1.03312	0.516561	−	0.856251i	$-0.327212\pi$
$239$	15.9717	1.03312	0.516561	+	0.856251i	$0.327212\pi$
$240$	1.39228	0.0898714
$241$	4.75480	0.306284	0.153142	−	0.988204i	$-0.451061\pi$
$241$	4.75480	0.306284	0.153142	+	0.988204i	$0.451061\pi$
$242$	−4.63042	−0.297655
$243$	1.00000	0.0641500
$244$	17.0208	1.08965
$245$	0	0
$246$	3.10550	0.197999
$247$	2.69489	0.171472
$248$	−18.9137	−1.20102
$249$	−10.9393	−0.693253
$250$	−2.51302	−0.158938
$251$	−27.3606	−1.72699	−0.863494	−	0.504359i	$-0.831729\pi$
$251$	−27.3606	−1.72699	−0.863494	+	0.504359i	$0.831729\pi$
$252$	0	0
$253$	−1.22947	−0.0772962
$254$	−1.03564	−0.0649820
$255$	−3.59975	−0.225425
$256$	0.164896	0.0103060
$257$	23.1469	1.44387	0.721933	−	0.691963i	$-0.243254\pi$
$257$	23.1469	1.44387	0.721933	+	0.691963i	$0.243254\pi$
$258$	−4.99905	−0.311227
$259$	0	0
$260$	−1.07425	−0.0666222
$261$	−5.30228	−0.328203
$262$	−2.67012	−0.164961
$263$	9.91237	0.611223	0.305611	−	0.952156i	$-0.401139\pi$
$263$	9.91237	0.611223	0.305611	+	0.952156i	$0.401139\pi$
$264$	2.25708	0.138914
$265$	−6.50363	−0.399515
$266$	0	0
$267$	7.21683	0.441663
$268$	−26.7405	−1.63344
$269$	−6.72908	−0.410279	−0.205140	−	0.978733i	$-0.565765\pi$
$269$	−6.72908	−0.410279	−0.205140	+	0.978733i	$0.565765\pi$
$270$	0.258561	0.0157355
$271$	−1.75395	−0.106545	−0.0532723	−	0.998580i	$-0.516965\pi$
$271$	−1.75395	−0.106545	−0.0532723	+	0.998580i	$0.516965\pi$
$272$	−17.8538	−1.08254
$273$	0	0
$274$	5.02708	0.303697
$275$	5.80222	0.349887
$276$	−1.76185	−0.106051
$277$	30.5460	1.83533	0.917665	−	0.397354i	$-0.130072\pi$
$277$	30.5460	1.83533	0.917665	+	0.397354i	$0.130072\pi$
$278$	−9.89812	−0.593650
$279$	10.3026	0.616803
$280$	0	0
$281$	−0.225769	−0.0134683	−0.00673414	−	0.999977i	$-0.502144\pi$
$281$	−0.225769	−0.0134683	−0.00673414	+	0.999977i	$0.502144\pi$
$282$	−5.76167	−0.343103
$283$	−18.6558	−1.10897	−0.554486	−	0.832193i	$-0.687085\pi$
$283$	−18.6558	−1.10897	−0.554486	+	0.832193i	$0.687085\pi$
$284$	13.2070	0.783693
$285$	1.24072	0.0734938
$286$	−0.690477	−0.0408287
$287$	0	0
$288$	4.95402	0.291918
$289$	29.1609	1.71535
$290$	−1.37096	−0.0805057
$291$	−8.00904	−0.469498
$292$	19.7212	1.15410
$293$	−18.1136	−1.05821	−0.529103	−	0.848558i	$-0.677472\pi$
$293$	−18.1136	−1.05821	−0.529103	+	0.848558i	$0.677472\pi$
$294$	0	0
$295$	−5.52673	−0.321779
$296$	−2.91146	−0.169225
$297$	−1.22947	−0.0713412
$298$	−2.97110	−0.172111
$299$	1.15081	0.0665529
$300$	8.31465	0.480047
$301$	0	0
$302$	−8.32801	−0.479223
$303$	−5.74251	−0.329899
$304$	6.15362	0.352934
$305$	−5.11854	−0.293087
$306$	−3.31562	−0.189542
$307$	16.7437	0.955615	0.477808	−	0.878464i	$-0.341432\pi$
$307$	16.7437	0.955615	0.477808	+	0.878464i	$0.341432\pi$
$308$	0	0
$309$	−1.58251	−0.0900257
$310$	2.66386	0.151297
$311$	−8.28378	−0.469730	−0.234865	−	0.972028i	$-0.575465\pi$
$311$	−8.28378	−0.469730	−0.234865	+	0.972028i	$0.575465\pi$
$312$	−2.11267	−0.119606
$313$	3.45260	0.195152	0.0975762	−	0.995228i	$-0.468891\pi$
$313$	3.45260	0.195152	0.0975762	+	0.995228i	$0.468891\pi$
$314$	−0.0996146	−0.00562158
$315$	0	0
$316$	−20.5812	−1.15778
$317$	4.96629	0.278935	0.139467	−	0.990227i	$-0.455461\pi$
$317$	4.96629	0.278935	0.139467	+	0.990227i	$0.455461\pi$
$318$	−5.99031	−0.335920
$319$	6.51901	0.364994
$320$	−1.50365	−0.0840565
$321$	17.0391	0.951030
$322$	0	0
$323$	−15.9102	−0.885268
$324$	−1.76185	−0.0978804
$325$	−5.43099	−0.301257
$326$	10.0556	0.556929
$327$	−3.40999	−0.188573
$328$	−11.6824	−0.645053
$329$	0	0
$330$	−0.317893	−0.0174994
$331$	16.4700	0.905273	0.452637	−	0.891695i	$-0.350483\pi$
$331$	16.4700	0.905273	0.452637	+	0.891695i	$0.350483\pi$
$332$	19.2735	1.05777
$333$	1.58592	0.0869080
$334$	6.60802	0.361575
$335$	8.04148	0.439353
$336$	0	0
$337$	19.3145	1.05213	0.526065	−	0.850444i	$-0.323667\pi$
$337$	19.3145	1.05213	0.526065	+	0.850444i	$0.323667\pi$
$338$	−5.69782	−0.309920
$339$	7.79701	0.423475
$340$	6.34221	0.343954
$341$	−12.6668	−0.685945
$342$	1.14279	0.0617950
$343$	0	0
$344$	18.8056	1.01393
$345$	0.529828	0.0285250
$346$	2.42773	0.130515
$347$	−11.0847	−0.595056	−0.297528	−	0.954713i	$-0.596162\pi$
$347$	−11.0847	−0.595056	−0.297528	+	0.954713i	$0.596162\pi$
$348$	9.34182	0.500774
$349$	15.6263	0.836456	0.418228	−	0.908342i	$-0.362651\pi$
$349$	15.6263	0.836456	0.418228	+	0.908342i	$0.362651\pi$
$350$	0	0
$351$	1.15081	0.0614256
$352$	−6.09083	−0.324642
$353$	5.74170	0.305600	0.152800	−	0.988257i	$-0.451171\pi$
$353$	5.74170	0.305600	0.152800	+	0.988257i	$0.451171\pi$
$354$	−5.09051	−0.270558
$355$	−3.97166	−0.210794
$356$	−12.7150	−0.673891
$357$	0	0
$358$	−3.09979	−0.163829
$359$	−22.4827	−1.18659	−0.593296	−	0.804985i	$-0.702173\pi$
$359$	−22.4827	−1.18659	−0.593296	+	0.804985i	$0.702173\pi$
$360$	−0.972666	−0.0512640
$361$	−13.5163	−0.711382
$362$	−4.75468	−0.249900
$363$	−9.48840	−0.498012
$364$	0	0
$365$	−5.93062	−0.310423
$366$	−4.71454	−0.246433
$367$	13.8006	0.720383	0.360192	−	0.932878i	$-0.382711\pi$
$367$	13.8006	0.720383	0.360192	+	0.932878i	$0.382711\pi$
$368$	2.62780	0.136984
$369$	6.36361	0.331276
$370$	0.410057	0.0213178
$371$	0	0
$372$	−18.1517	−0.941120
$373$	38.4020	1.98838	0.994189	−	0.107647i	$-0.0343317\pi$
$373$	38.4020	1.98838	0.994189	+	0.107647i	$0.0343317\pi$
$374$	4.07646	0.210789
$375$	−5.14955	−0.265921
$376$	21.6745	1.11778
$377$	−6.10191	−0.314264
$378$	0	0
$379$	−1.85059	−0.0950582	−0.0475291	−	0.998870i	$-0.515135\pi$
$379$	−1.85059	−0.0950582	−0.0475291	+	0.998870i	$0.515135\pi$
$380$	−2.18596	−0.112137
$381$	−2.12218	−0.108723
$382$	2.89719	0.148233
$383$	8.46055	0.432314	0.216157	−	0.976359i	$-0.430648\pi$
$383$	8.46055	0.432314	0.216157	+	0.976359i	$0.430648\pi$
$384$	−11.2930	−0.576294
$385$	0	0
$386$	−0.336420	−0.0171233
$387$	−10.2438	−0.520719
$388$	14.1107	0.716362
$389$	−14.8630	−0.753584	−0.376792	−	0.926298i	$-0.622973\pi$
$389$	−14.8630	−0.753584	−0.376792	+	0.926298i	$0.622973\pi$
$390$	0.297554	0.0150672
$391$	−6.79418	−0.343597
$392$	0	0
$393$	−5.47146	−0.275999
$394$	−5.92691	−0.298594
$395$	6.18924	0.311415
$396$	2.16614	0.108853
$397$	−20.4780	−1.02776	−0.513879	−	0.857862i	$-0.671792\pi$
$397$	−20.4780	−1.02776	−0.513879	+	0.857862i	$0.671792\pi$
$398$	9.86817	0.494647
$399$	0	0
$400$	−12.4013	−0.620067
$401$	5.68921	0.284106	0.142053	−	0.989859i	$-0.454630\pi$
$401$	5.68921	0.284106	0.142053	+	0.989859i	$0.454630\pi$
$402$	7.40677	0.369416
$403$	11.8564	0.590607
$404$	10.1174	0.503361
$405$	0.529828	0.0263274
$406$	0	0
$407$	−1.94984	−0.0966502
$408$	12.4729	0.617499
$409$	9.45837	0.467686	0.233843	−	0.972274i	$-0.424870\pi$
$409$	9.45837	0.467686	0.233843	+	0.972274i	$0.424870\pi$
$410$	1.64538	0.0812595
$411$	10.3012	0.508120
$412$	2.78814	0.137362
$413$	0	0
$414$	0.488009	0.0239843
$415$	−5.79597	−0.284513
$416$	5.70113	0.279521
$417$	−20.2827	−0.993246
$418$	−1.40503	−0.0687221
$419$	33.9358	1.65787	0.828936	−	0.559343i	$-0.188946\pi$
$419$	33.9358	1.65787	0.828936	+	0.559343i	$0.188946\pi$
$420$	0	0
$421$	20.8461	1.01598	0.507990	−	0.861363i	$-0.330389\pi$
$421$	20.8461	1.01598	0.507990	+	0.861363i	$0.330389\pi$
$422$	3.11845	0.151804
$423$	−11.8065	−0.574051
$424$	22.5346	1.09438
$425$	32.0637	1.55532
$426$	−3.65818	−0.177239
$427$	0	0
$428$	−30.0203	−1.45109
$429$	−1.41489	−0.0683113
$430$	−2.64863	−0.127728
$431$	−7.76349	−0.373954	−0.186977	−	0.982364i	$-0.559869\pi$
$431$	−7.76349	−0.373954	−0.186977	+	0.982364i	$0.559869\pi$
$432$	2.62780	0.126430
$433$	34.9372	1.67897	0.839486	−	0.543381i	$-0.182856\pi$
$433$	34.9372	1.67897	0.839486	+	0.543381i	$0.182856\pi$
$434$	0	0
$435$	−2.80930	−0.134696
$436$	6.00788	0.287725
$437$	2.34174	0.112021
$438$	−5.46252	−0.261010
$439$	−24.0130	−1.14608	−0.573040	−	0.819528i	$-0.694236\pi$
$439$	−24.0130	−1.14608	−0.573040	+	0.819528i	$0.694236\pi$
$440$	1.19586	0.0570106
$441$	0	0
$442$	−3.81565	−0.181492
$443$	38.1749	1.81375	0.906873	−	0.421404i	$-0.138462\pi$
$443$	38.1749	1.81375	0.906873	+	0.421404i	$0.138462\pi$
$444$	−2.79415	−0.132605
$445$	3.82368	0.181260
$446$	−12.5912	−0.596212
$447$	−6.08821	−0.287962
$448$	0	0
$449$	−12.8912	−0.608371	−0.304186	−	0.952613i	$-0.598384\pi$
$449$	−12.8912	−0.608371	−0.304186	+	0.952613i	$0.598384\pi$
$450$	−2.30305	−0.108567
$451$	−7.82387	−0.368412
$452$	−13.7371	−0.646140
$453$	−17.0653	−0.801797
$454$	−7.01054	−0.329021
$455$	0	0
$456$	−4.29900	−0.201319
$457$	19.5526	0.914631	0.457315	−	0.889305i	$-0.348811\pi$
$457$	19.5526	0.914631	0.457315	+	0.889305i	$0.348811\pi$
$458$	−6.90787	−0.322784
$459$	−6.79418	−0.317125
$460$	−0.933476	−0.0435235
$461$	21.9571	1.02264	0.511322	−	0.859389i	$-0.329156\pi$
$461$	21.9571	1.02264	0.511322	+	0.859389i	$0.329156\pi$
$462$	0	0
$463$	11.3775	0.528757	0.264378	−	0.964419i	$-0.414833\pi$
$463$	11.3775	0.528757	0.264378	+	0.964419i	$0.414833\pi$
$464$	−13.9333	−0.646839
$465$	5.45862	0.253138
$466$	−8.68510	−0.402330
$467$	18.3868	0.850837	0.425419	−	0.904997i	$-0.360127\pi$
$467$	18.3868	0.850837	0.425419	+	0.904997i	$0.360127\pi$
$468$	−2.02755	−0.0937234
$469$	0	0
$470$	−3.05269	−0.140810
$471$	−0.204125	−0.00940556
$472$	19.1497	0.881437
$473$	12.5944	0.579091
$474$	5.70073	0.261843
$475$	−11.0513	−0.507070
$476$	0	0
$477$	−12.2750	−0.562033
$478$	7.79432	0.356504
$479$	−16.0629	−0.733934	−0.366967	−	0.930234i	$-0.619604\pi$
$479$	−16.0629	−0.733934	−0.366967	+	0.930234i	$0.619604\pi$
$480$	2.62478	0.119804
$481$	1.82509	0.0832170
$482$	2.32039	0.105691
$483$	0	0
$484$	16.7171	0.759869
$485$	−4.24341	−0.192683
$486$	0.488009	0.0221365
$487$	25.0151	1.13354	0.566770	−	0.823876i	$-0.308193\pi$
$487$	25.0151	1.13354	0.566770	+	0.823876i	$0.308193\pi$
$488$	17.7354	0.802843
$489$	20.6054	0.931808
$490$	0	0
$491$	−3.88844	−0.175483	−0.0877413	−	0.996143i	$-0.527965\pi$
$491$	−3.88844	−0.175483	−0.0877413	+	0.996143i	$0.527965\pi$
$492$	−11.2117	−0.505463
$493$	36.0247	1.62247
$494$	1.31513	0.0591706
$495$	−0.651408	−0.0292786
$496$	27.0733	1.21563
$497$	0	0
$498$	−5.33850	−0.239224
$499$	−18.6852	−0.836464	−0.418232	−	0.908340i	$-0.637350\pi$
$499$	−18.6852	−0.836464	−0.418232	+	0.908340i	$0.637350\pi$
$500$	9.07271	0.405744
$501$	13.5408	0.604957
$502$	−13.3522	−0.595939
$503$	11.5905	0.516793	0.258396	−	0.966039i	$-0.416806\pi$
$503$	11.5905	0.516793	0.258396	+	0.966039i	$0.416806\pi$
$504$	0	0
$505$	−3.04254	−0.135391
$506$	−0.599993	−0.0266729
$507$	−11.6756	−0.518533
$508$	3.73896	0.165889
$509$	−8.14711	−0.361114	−0.180557	−	0.983564i	$-0.557790\pi$
$509$	−8.14711	−0.361114	−0.180557	+	0.983564i	$0.557790\pi$
$510$	−1.75671	−0.0777884
$511$	0	0
$512$	22.6665	1.00173
$513$	2.34174	0.103390
$514$	11.2959	0.498241
$515$	−0.838456	−0.0369468
$516$	18.0479	0.794516
$517$	14.5157	0.638402
$518$	0	0
$519$	4.97475	0.218367
$520$	−1.11935	−0.0490868
$521$	−15.7494	−0.689993	−0.344997	−	0.938604i	$-0.612120\pi$
$521$	−15.7494	−0.689993	−0.344997	+	0.938604i	$0.612120\pi$
$522$	−2.58756	−0.113255
$523$	−30.8710	−1.34989	−0.674947	−	0.737866i	$-0.735834\pi$
$523$	−30.8710	−1.34989	−0.674947	+	0.737866i	$0.735834\pi$
$524$	9.63987	0.421120
$525$	0	0
$526$	4.83732	0.210917
$527$	−69.9980	−3.04916
$528$	−3.23080	−0.140603
$529$	1.00000	0.0434783
$530$	−3.17383	−0.137862
$531$	−10.4312	−0.452675
$532$	0	0
$533$	7.32329	0.317207
$534$	3.52188	0.152407
$535$	9.02779	0.390306
$536$	−27.8631	−1.20350
$537$	−6.35192	−0.274106
$538$	−3.28385	−0.141577
$539$	0	0
$540$	−0.933476	−0.0401704
$541$	−20.9433	−0.900421	−0.450211	−	0.892922i	$-0.648651\pi$
$541$	−20.9433	−0.900421	−0.450211	+	0.892922i	$0.648651\pi$
$542$	−0.855941	−0.0367658
$543$	−9.74302	−0.418113
$544$	−33.6585	−1.44310
$545$	−1.80671	−0.0773908
$546$	0	0
$547$	9.21005	0.393793	0.196897	−	0.980424i	$-0.436914\pi$
$547$	9.21005	0.393793	0.196897	+	0.980424i	$0.436914\pi$
$548$	−18.1491	−0.775292
$549$	−9.66077	−0.412312
$550$	2.83154	0.120737
$551$	−12.4166	−0.528964
$552$	−1.83582	−0.0781375
$553$	0	0
$554$	14.9067	0.633326
$555$	0.840265	0.0356673
$556$	35.7349	1.51550
$557$	−33.6854	−1.42730	−0.713648	−	0.700504i	$-0.752958\pi$
$557$	−33.6854	−1.42730	−0.713648	+	0.700504i	$0.752958\pi$
$558$	5.02778	0.212843
$559$	−11.7886	−0.498604
$560$	0	0
$561$	8.35326	0.352675
$562$	−0.110178	−0.00464756
$563$	15.6505	0.659589	0.329795	−	0.944053i	$-0.393021\pi$
$563$	15.6505	0.659589	0.329795	+	0.944053i	$0.393021\pi$
$564$	20.8012	0.875890
$565$	4.13107	0.173795
$566$	−9.10420	−0.382678
$567$	0	0
$568$	13.7615	0.577420
$569$	23.9863	1.00556	0.502779	−	0.864415i	$-0.332311\pi$
$569$	23.9863	1.00556	0.502779	+	0.864415i	$0.332311\pi$
$570$	0.605482	0.0253608
$571$	−28.9012	−1.20948	−0.604738	−	0.796425i	$-0.706722\pi$
$571$	−28.9012	−1.20948	−0.604738	+	0.796425i	$0.706722\pi$
$572$	2.49281	0.104230
$573$	5.93676	0.248012
$574$	0	0
$575$	−4.71928	−0.196808
$576$	−2.83799	−0.118250
$577$	21.3891	0.890441	0.445220	−	0.895421i	$-0.353125\pi$
$577$	21.3891	0.890441	0.445220	+	0.895421i	$0.353125\pi$
$578$	14.2308	0.591923
$579$	−0.689373	−0.0286494
$580$	4.94955	0.205519
$581$	0	0
$582$	−3.90848	−0.162012
$583$	15.0918	0.625037
$584$	20.5492	0.850331
$585$	0.609730	0.0252092
$586$	−8.83959	−0.365160
$587$	1.91410	0.0790035	0.0395017	−	0.999220i	$-0.487423\pi$
$587$	1.91410	0.0790035	0.0395017	+	0.999220i	$0.487423\pi$
$588$	0	0
$589$	24.1261	0.994098
$590$	−2.69709	−0.111038
$591$	−12.1451	−0.499582
$592$	4.16748	0.171283
$593$	−13.6823	−0.561865	−0.280932	−	0.959728i	$-0.590644\pi$
$593$	−13.6823	−0.561865	−0.280932	+	0.959728i	$0.590644\pi$
$594$	−0.599993	−0.0246180
$595$	0	0
$596$	10.7265	0.439374
$597$	20.2213	0.827602
$598$	0.561605	0.0229657
$599$	−48.6547	−1.98798	−0.993988	−	0.109490i	$-0.965078\pi$
$599$	−48.6547	−1.98798	−0.993988	+	0.109490i	$0.965078\pi$
$600$	8.66373	0.353695
$601$	16.6250	0.678147	0.339074	−	0.940760i	$-0.389886\pi$
$601$	16.6250	0.678147	0.339074	+	0.940760i	$0.389886\pi$
$602$	0	0
$603$	15.1775	0.618077
$604$	30.0664	1.22338
$605$	−5.02722	−0.204385
$606$	−2.80240	−0.113840
$607$	39.8095	1.61582	0.807908	−	0.589308i	$-0.200600\pi$
$607$	39.8095	1.61582	0.807908	+	0.589308i	$0.200600\pi$
$608$	11.6010	0.470484
$609$	0	0
$610$	−2.49790	−0.101137
$611$	−13.5870	−0.549671
$612$	11.9703	0.483871
$613$	15.9213	0.643055	0.321528	−	0.946900i	$-0.395804\pi$
$613$	15.9213	0.643055	0.321528	+	0.946900i	$0.395804\pi$
$614$	8.17109	0.329758
$615$	3.37162	0.135957
$616$	0	0
$617$	34.0485	1.37074	0.685370	−	0.728195i	$-0.259641\pi$
$617$	34.0485	1.37074	0.685370	+	0.728195i	$0.259641\pi$
$618$	−0.772278	−0.0310656
$619$	−30.5180	−1.22662	−0.613311	−	0.789841i	$-0.710163\pi$
$619$	−30.5180	−1.22662	−0.613311	+	0.789841i	$0.710163\pi$
$620$	−9.61726	−0.386238
$621$	1.00000	0.0401286
$622$	−4.04256	−0.162092
$623$	0	0
$624$	3.02409	0.121061
$625$	20.8680	0.834722
$626$	1.68490	0.0673421
$627$	−2.87910	−0.114980
$628$	0.359636	0.0143510
$629$	−10.7750	−0.429629
$630$	0	0
$631$	12.8855	0.512964	0.256482	−	0.966549i	$-0.417437\pi$
$631$	12.8855	0.512964	0.256482	+	0.966549i	$0.417437\pi$
$632$	−21.4453	−0.853047
$633$	6.39015	0.253986
$634$	2.42359	0.0962532
$635$	−1.12439	−0.0446200
$636$	21.6267	0.857553
$637$	0	0
$638$	3.18133	0.125950
$639$	−7.49613	−0.296542
$640$	−5.98335	−0.236513
$641$	−19.1370	−0.755864	−0.377932	−	0.925833i	$-0.623365\pi$
$641$	−19.1370	−0.755864	−0.377932	+	0.925833i	$0.623365\pi$
$642$	8.31524	0.328176
$643$	−18.1262	−0.714826	−0.357413	−	0.933946i	$-0.616341\pi$
$643$	−18.1262	−0.714826	−0.357413	+	0.933946i	$0.616341\pi$
$644$	0	0
$645$	−5.42743	−0.213705
$646$	−7.76433	−0.305483
$647$	29.7058	1.16786	0.583928	−	0.811806i	$-0.301515\pi$
$647$	29.7058	1.16786	0.583928	+	0.811806i	$0.301515\pi$
$648$	−1.83582	−0.0721176
$649$	12.8248	0.503419
$650$	−2.65037	−0.103956
$651$	0	0
$652$	−36.3036	−1.42176
$653$	−25.3995	−0.993959	−0.496980	−	0.867762i	$-0.665558\pi$
$653$	−25.3995	−0.993959	−0.496980	+	0.867762i	$0.665558\pi$
$654$	−1.66410	−0.0650716
$655$	−2.89893	−0.113271
$656$	16.7223	0.652896
$657$	−11.1935	−0.436700
$658$	0	0
$659$	38.6690	1.50633	0.753164	−	0.657832i	$-0.228527\pi$
$659$	38.6690	1.50633	0.753164	+	0.657832i	$0.228527\pi$
$660$	1.14768	0.0446734
$661$	6.26718	0.243765	0.121883	−	0.992545i	$-0.461107\pi$
$661$	6.26718	0.243765	0.121883	+	0.992545i	$0.461107\pi$
$662$	8.03751	0.312387
$663$	−7.81880	−0.303657
$664$	20.0826	0.779357
$665$	0	0
$666$	0.773944	0.0299897
$667$	−5.30228	−0.205305
$668$	−23.8568	−0.923046
$669$	−25.8012	−0.997533
$670$	3.92431	0.151609
$671$	11.8776	0.458531
$672$	0	0
$673$	−4.39319	−0.169345	−0.0846725	−	0.996409i	$-0.526984\pi$
$673$	−4.39319	−0.169345	−0.0846725	+	0.996409i	$0.526984\pi$
$674$	9.42567	0.363063
$675$	−4.71928	−0.181645
$676$	20.5707	0.791181
$677$	18.7904	0.722174	0.361087	−	0.932532i	$-0.382406\pi$
$677$	18.7904	0.722174	0.361087	+	0.932532i	$0.382406\pi$
$678$	3.80501	0.146130
$679$	0	0
$680$	6.60847	0.253423
$681$	−14.3656	−0.550490
$682$	−6.18151	−0.236702
$683$	−9.79287	−0.374714	−0.187357	−	0.982292i	$-0.559992\pi$
$683$	−9.79287	−0.374714	−0.187357	+	0.982292i	$0.559992\pi$
$684$	−4.12579	−0.157753
$685$	5.45786	0.208534
$686$	0	0
$687$	−14.1552	−0.540055
$688$	−26.9185	−1.02626
$689$	−14.1262	−0.538164
$690$	0.258561	0.00984324
$691$	−34.3477	−1.30665	−0.653323	−	0.757079i	$-0.726626\pi$
$691$	−34.3477	−1.30665	−0.653323	+	0.757079i	$0.726626\pi$
$692$	−8.76476	−0.333186
$693$	0	0
$694$	−5.40942	−0.205339
$695$	−10.7463	−0.407631
$696$	9.73402	0.368967
$697$	−43.2355	−1.63766
$698$	7.62577	0.288640
$699$	−17.7970	−0.673145
$700$	0	0
$701$	36.0260	1.36068	0.680341	−	0.732895i	$-0.261832\pi$
$701$	36.0260	1.36068	0.680341	+	0.732895i	$0.261832\pi$
$702$	0.561605	0.0211964
$703$	3.71381	0.140069
$704$	3.48923	0.131505
$705$	−6.25541	−0.235592
$706$	2.80200	0.105455
$707$	0	0
$708$	18.3781	0.690693
$709$	42.8099	1.60776	0.803880	−	0.594792i	$-0.202766\pi$
$709$	42.8099	1.60776	0.803880	+	0.594792i	$0.202766\pi$
$710$	−1.93820	−0.0727395
$711$	11.6816	0.438095
$712$	−13.2488	−0.496519
$713$	10.3026	0.385837
$714$	0	0
$715$	−0.749645	−0.0280352
$716$	11.1911	0.418232
$717$	15.9717	0.596473
$718$	−10.9718	−0.409462
$719$	−22.4457	−0.837084	−0.418542	−	0.908197i	$-0.637459\pi$
$719$	−22.4457	−0.837084	−0.418542	+	0.908197i	$0.637459\pi$
$720$	1.39228	0.0518873
$721$	0	0
$722$	−6.59606	−0.245480
$723$	4.75480	0.176833
$724$	17.1657	0.637958
$725$	25.0230	0.929330
$726$	−4.63042	−0.171851
$727$	−20.8139	−0.771946	−0.385973	−	0.922510i	$-0.626134\pi$
$727$	−20.8139	−0.771946	−0.385973	+	0.922510i	$0.626134\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−2.89420	−0.107119
$731$	69.5980	2.57417
$732$	17.0208	0.629107
$733$	11.0833	0.409369	0.204685	−	0.978828i	$-0.434383\pi$
$733$	11.0833	0.409369	0.204685	+	0.978828i	$0.434383\pi$
$734$	6.73480	0.248586
$735$	0	0
$736$	4.95402	0.182608
$737$	−18.6603	−0.687362
$738$	3.10550	0.114315
$739$	39.3320	1.44685	0.723425	−	0.690403i	$-0.242567\pi$
$739$	39.3320	1.44685	0.723425	+	0.690403i	$0.242567\pi$
$740$	−1.48042	−0.0544213
$741$	2.69489	0.0989993
$742$	0	0
$743$	−32.0266	−1.17494	−0.587471	−	0.809245i	$-0.699876\pi$
$743$	−32.0266	−1.17494	−0.587471	+	0.809245i	$0.699876\pi$
$744$	−18.9137	−0.693411
$745$	−3.22570	−0.118181
$746$	18.7405	0.686139
$747$	−10.9393	−0.400250
$748$	−14.7172	−0.538113
$749$	0	0
$750$	−2.51302	−0.0917627
$751$	−28.9345	−1.05584	−0.527918	−	0.849295i	$-0.677027\pi$
$751$	−28.9345	−1.05584	−0.527918	+	0.849295i	$0.677027\pi$
$752$	−31.0251	−1.13137
$753$	−27.3606	−0.997077
$754$	−2.97779	−0.108445
$755$	−9.04166	−0.329060
$756$	0	0
$757$	29.2265	1.06225	0.531127	−	0.847292i	$-0.321769\pi$
$757$	29.2265	1.06225	0.531127	+	0.847292i	$0.321769\pi$
$758$	−0.903103	−0.0328022
$759$	−1.22947	−0.0446270
$760$	−2.27773	−0.0826220
$761$	−42.0019	−1.52257	−0.761284	−	0.648418i	$-0.775431\pi$
$761$	−42.0019	−1.52257	−0.761284	+	0.648418i	$0.775431\pi$
$762$	−1.03564	−0.0375174
$763$	0	0
$764$	−10.4597	−0.378418
$765$	−3.59975	−0.130149
$766$	4.12882	0.149181
$767$	−12.0043	−0.433450
$768$	0.164896	0.00595019
$769$	28.2042	1.01707	0.508535	−	0.861041i	$-0.330187\pi$
$769$	28.2042	1.01707	0.508535	+	0.861041i	$0.330187\pi$
$770$	0	0
$771$	23.1469	0.833616
$772$	1.21457	0.0437133
$773$	47.9059	1.72305	0.861527	−	0.507711i	$-0.169508\pi$
$773$	47.9059	1.72305	0.861527	+	0.507711i	$0.169508\pi$
$774$	−4.99905	−0.179687
$775$	−48.6210	−1.74652
$776$	14.7031	0.527811
$777$	0	0
$778$	−7.25328	−0.260043
$779$	14.9019	0.533916
$780$	−1.07425	−0.0384644
$781$	9.21627	0.329784
$782$	−3.31562	−0.118566
$783$	−5.30228	−0.189488
$784$	0	0
$785$	−0.108151	−0.00386007
$786$	−2.67012	−0.0952401
$787$	20.4022	0.727259	0.363629	−	0.931544i	$-0.381538\pi$
$787$	20.4022	0.727259	0.363629	+	0.931544i	$0.381538\pi$
$788$	21.3978	0.762265
$789$	9.91237	0.352890
$790$	3.02041	0.107461
$791$	0	0
$792$	2.25708	0.0802019
$793$	−11.1177	−0.394801
$794$	−9.99343	−0.354653
$795$	−6.50363	−0.230660
$796$	−35.6268	−1.26276
$797$	−7.31640	−0.259160	−0.129580	−	0.991569i	$-0.541363\pi$
$797$	−7.31640	−0.259160	−0.129580	+	0.991569i	$0.541363\pi$
$798$	0	0
$799$	80.2155	2.83782
$800$	−23.3794	−0.826587
$801$	7.21683	0.254994
$802$	2.77639	0.0980376
$803$	13.7621	0.485653
$804$	−26.7405	−0.943064
$805$	0	0
$806$	5.78601	0.203803
$807$	−6.72908	−0.236875
$808$	10.5422	0.370873
$809$	−22.8009	−0.801635	−0.400818	−	0.916158i	$-0.631274\pi$
$809$	−22.8009	−0.801635	−0.400818	+	0.916158i	$0.631274\pi$
$810$	0.258561	0.00908490
$811$	−26.0156	−0.913531	−0.456766	−	0.889587i	$-0.650992\pi$
$811$	−26.0156	−0.913531	−0.456766	+	0.889587i	$0.650992\pi$
$812$	0	0
$813$	−1.75395	−0.0615136
$814$	−0.951542	−0.0333515
$815$	10.9173	0.382417
$816$	−17.8538	−0.625007
$817$	−23.9882	−0.839241
$818$	4.61577	0.161387
$819$	0	0
$820$	−5.94027	−0.207443
$821$	−7.49849	−0.261699	−0.130850	−	0.991402i	$-0.541770\pi$
$821$	−7.49849	−0.261699	−0.130850	+	0.991402i	$0.541770\pi$
$822$	5.02708	0.175339
$823$	−22.8235	−0.795579	−0.397789	−	0.917477i	$-0.630223\pi$
$823$	−22.8235	−0.795579	−0.397789	+	0.917477i	$0.630223\pi$
$824$	2.90519	0.101207
$825$	5.80222	0.202007
$826$	0	0
$827$	−38.4557	−1.33724	−0.668618	−	0.743606i	$-0.733114\pi$
$827$	−38.4557	−1.33724	−0.668618	+	0.743606i	$0.733114\pi$
$828$	−1.76185	−0.0612284
$829$	−38.4417	−1.33513	−0.667567	−	0.744550i	$-0.732664\pi$
$829$	−38.4417	−1.33513	−0.667567	+	0.744550i	$0.732664\pi$
$830$	−2.82849	−0.0981782
$831$	30.5460	1.05963
$832$	−3.26598	−0.113228
$833$	0	0
$834$	−9.89812	−0.342744
$835$	7.17428	0.248276
$836$	5.07254	0.175437
$837$	10.3026	0.356111
$838$	16.5610	0.572090
$839$	−17.7730	−0.613594	−0.306797	−	0.951775i	$-0.599257\pi$
$839$	−17.7730	−0.613594	−0.306797	+	0.951775i	$0.599257\pi$
$840$	0	0
$841$	−0.885774	−0.0305439
$842$	10.1731	0.350588
$843$	−0.225769	−0.00777591
$844$	−11.2585	−0.387533
$845$	−6.18608	−0.212808
$846$	−5.76167	−0.198090
$847$	0	0
$848$	−32.2562	−1.10768
$849$	−18.6558	−0.640265
$850$	15.6474	0.536700
$851$	1.58592	0.0543647
$852$	13.2070	0.452465
$853$	14.8174	0.507337	0.253668	−	0.967291i	$-0.418363\pi$
$853$	14.8174	0.507337	0.253668	+	0.967291i	$0.418363\pi$
$854$	0	0
$855$	1.24072	0.0424317
$856$	−31.2807	−1.06915
$857$	−11.4619	−0.391532	−0.195766	−	0.980651i	$-0.562719\pi$
$857$	−11.4619	−0.391532	−0.195766	+	0.980651i	$0.562719\pi$
$858$	−0.690477	−0.0235725
$859$	37.3084	1.27295	0.636473	−	0.771299i	$-0.280393\pi$
$859$	37.3084	1.27295	0.636473	+	0.771299i	$0.280393\pi$
$860$	9.56230	0.326072
$861$	0	0
$862$	−3.78866	−0.129042
$863$	−20.6216	−0.701966	−0.350983	−	0.936382i	$-0.614153\pi$
$863$	−20.6216	−0.701966	−0.350983	+	0.936382i	$0.614153\pi$
$864$	4.95402	0.168539
$865$	2.63576	0.0896186
$866$	17.0496	0.579371
$867$	29.1609	0.990358
$868$	0	0
$869$	−14.3622	−0.487204
$870$	−1.37096	−0.0464800
$871$	17.4664	0.591827
$872$	6.26011	0.211994
$873$	−8.00904	−0.271065
$874$	1.14279	0.0386554
$875$	0	0
$876$	19.7212	0.666318
$877$	−35.5379	−1.20003	−0.600015	−	0.799989i	$-0.704839\pi$
$877$	−35.5379	−1.20003	−0.600015	+	0.799989i	$0.704839\pi$
$878$	−11.7186	−0.395483
$879$	−18.1136	−0.610956
$880$	−1.71177	−0.0577038
$881$	38.2118	1.28739	0.643694	−	0.765283i	$-0.277401\pi$
$881$	38.2118	1.28739	0.643694	+	0.765283i	$0.277401\pi$
$882$	0	0
$883$	57.3513	1.93002	0.965012	−	0.262206i	$-0.0844498\pi$
$883$	57.3513	1.93002	0.965012	+	0.262206i	$0.0844498\pi$
$884$	13.7755	0.463321
$885$	−5.52673	−0.185779
$886$	18.6297	0.625878
$887$	−23.2067	−0.779203	−0.389602	−	0.920983i	$-0.627387\pi$
$887$	−23.2067	−0.779203	−0.389602	+	0.920983i	$0.627387\pi$
$888$	−2.91146	−0.0977022
$889$	0	0
$890$	1.86599	0.0625481
$891$	−1.22947	−0.0411888
$892$	45.4578	1.52204
$893$	−27.6477	−0.925196
$894$	−2.97110	−0.0993685
$895$	−3.36542	−0.112494
$896$	0	0
$897$	1.15081	0.0384244
$898$	−6.29100	−0.209933
$899$	−54.6275	−1.82193
$900$	8.31465	0.277155
$901$	83.3986	2.77841
$902$	−3.81812	−0.127130
$903$	0	0
$904$	−14.3139	−0.476072
$905$	−5.16212	−0.171595
$906$	−8.32801	−0.276680
$907$	−8.43338	−0.280026	−0.140013	−	0.990150i	$-0.544714\pi$
$907$	−8.43338	−0.280026	−0.140013	+	0.990150i	$0.544714\pi$
$908$	25.3100	0.839941
$909$	−5.74251	−0.190467
$910$	0	0
$911$	38.4319	1.27330	0.636652	−	0.771151i	$-0.280319\pi$
$911$	38.4319	1.27330	0.636652	+	0.771151i	$0.280319\pi$
$912$	6.15362	0.203767
$913$	13.4496	0.445117
$914$	9.54183	0.315616
$915$	−5.11854	−0.169214
$916$	24.9393	0.824019
$917$	0	0
$918$	−3.31562	−0.109432
$919$	11.7107	0.386299	0.193150	−	0.981169i	$-0.438130\pi$
$919$	11.7107	0.386299	0.193150	+	0.981169i	$0.438130\pi$
$920$	−0.972666	−0.0320678
$921$	16.7437	0.551725
$922$	10.7153	0.352888
$923$	−8.62660	−0.283948
$924$	0	0
$925$	−7.48441	−0.246086
$926$	5.55232	0.182460
$927$	−1.58251	−0.0519764
$928$	−26.2676	−0.862277
$929$	−56.3116	−1.84753	−0.923763	−	0.382966i	$-0.874903\pi$
$929$	−56.3116	−1.84753	−0.923763	+	0.382966i	$0.874903\pi$
$930$	2.66386	0.0873513
$931$	0	0
$932$	31.3556	1.02709
$933$	−8.28378	−0.271199
$934$	8.97290	0.293602
$935$	4.42579	0.144739
$936$	−2.11267	−0.0690548
$937$	−56.5397	−1.84707	−0.923535	−	0.383514i	$-0.874714\pi$
$937$	−56.5397	−1.84707	−0.923535	+	0.383514i	$0.874714\pi$
$938$	0	0
$939$	3.45260	0.112671
$940$	11.0211	0.359468
$941$	−29.9234	−0.975474	−0.487737	−	0.872991i	$-0.662177\pi$
$941$	−29.9234	−0.975474	−0.487737	+	0.872991i	$0.662177\pi$
$942$	−0.0996146	−0.00324562
$943$	6.36361	0.207228
$944$	−27.4111	−0.892154
$945$	0	0
$946$	6.14618	0.199830
$947$	16.6859	0.542218	0.271109	−	0.962549i	$-0.412610\pi$
$947$	16.6859	0.542218	0.271109	+	0.962549i	$0.412610\pi$
$948$	−20.5812	−0.668447
$949$	−12.8816	−0.418153
$950$	−5.39315	−0.174977
$951$	4.96629	0.161043
$952$	0	0
$953$	−53.5813	−1.73567	−0.867834	−	0.496854i	$-0.834489\pi$
$953$	−53.5813	−1.73567	−0.867834	+	0.496854i	$0.834489\pi$
$954$	−5.99031	−0.193943
$955$	3.14546	0.101785
$956$	−28.1396	−0.910101
$957$	6.51901	0.210730
$958$	−7.83886	−0.253262
$959$	0	0
$960$	−1.50365	−0.0485300
$961$	75.1443	2.42401
$962$	0.890661	0.0287161
$963$	17.0391	0.549078
$964$	−8.37723	−0.269812
$965$	−0.365249	−0.0117578
$966$	0	0
$967$	20.4391	0.657276	0.328638	−	0.944456i	$-0.393410\pi$
$967$	20.4391	0.657276	0.328638	+	0.944456i	$0.393410\pi$
$968$	17.4190	0.559866
$969$	−15.9102	−0.511110
$970$	−2.07082	−0.0664901
$971$	−18.7747	−0.602509	−0.301255	−	0.953544i	$-0.597405\pi$
$971$	−18.7747	−0.602509	−0.301255	+	0.953544i	$0.597405\pi$
$972$	−1.76185	−0.0565113
$973$	0	0
$974$	12.2076	0.391156
$975$	−5.43099	−0.173931
$976$	−25.3866	−0.812604
$977$	55.6654	1.78089	0.890446	−	0.455088i	$-0.150392\pi$
$977$	55.6654	1.78089	0.890446	+	0.455088i	$0.150392\pi$
$978$	10.0556	0.321543
$979$	−8.87289	−0.283579
$980$	0	0
$981$	−3.40999	−0.108873
$982$	−1.89759	−0.0605546
$983$	−27.5075	−0.877353	−0.438677	−	0.898645i	$-0.644553\pi$
$983$	−27.5075	−0.877353	−0.438677	+	0.898645i	$0.644553\pi$
$984$	−11.6824	−0.372422
$985$	−6.43481	−0.205030
$986$	17.5804	0.559873
$987$	0	0
$988$	−4.74799	−0.151054
$989$	−10.2438	−0.325733
$990$	−0.317893	−0.0101033
$991$	−1.05755	−0.0335943	−0.0167971	−	0.999859i	$-0.505347\pi$
$991$	−1.05755	−0.0335943	−0.0167971	+	0.999859i	$0.505347\pi$
$992$	51.0395	1.62050
$993$	16.4700	0.522660
$994$	0	0
$995$	10.7138	0.339650
$996$	19.2735	0.610703
$997$	−22.4724	−0.711707	−0.355854	−	0.934542i	$-0.615810\pi$
$997$	−22.4724	−0.711707	−0.355854	+	0.934542i	$0.615810\pi$
$998$	−9.11854	−0.288642
$999$	1.58592	0.0501763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bh.1.7	yes	10
7.6	odd	2		3381.2.a.bg.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3381.2.a.bg.1.7	✓	10	7.6	odd	2
3381.2.a.bh.1.7	yes	10	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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q+0.488009 q^{2} +1.00000 q^{3} -1.76185 q^{4} +0.529828 q^{5} +0.488009 q^{6} -1.83582 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.488009 q^{2} +1.00000 q^{3} -1.76185 q^{4} +0.529828 q^{5} +0.488009 q^{6} -1.83582 q^{8} +1.00000 q^{9} +0.258561 q^{10} -1.22947 q^{11} -1.76185 q^{12} +1.15081 q^{13} +0.529828 q^{15} +2.62780 q^{16} -6.79418 q^{17} +0.488009 q^{18} +2.34174 q^{19} -0.933476 q^{20} -0.599993 q^{22} +1.00000 q^{23} -1.83582 q^{24} -4.71928 q^{25} +0.561605 q^{26} +1.00000 q^{27} -5.30228 q^{29} +0.258561 q^{30} +10.3026 q^{31} +4.95402 q^{32} -1.22947 q^{33} -3.31562 q^{34} -1.76185 q^{36} +1.58592 q^{37} +1.14279 q^{38} +1.15081 q^{39} -0.972666 q^{40} +6.36361 q^{41} -10.2438 q^{43} +2.16614 q^{44} +0.529828 q^{45} +0.488009 q^{46} -11.8065 q^{47} +2.62780 q^{48} -2.30305 q^{50} -6.79418 q^{51} -2.02755 q^{52} -12.2750 q^{53} +0.488009 q^{54} -0.651408 q^{55} +2.34174 q^{57} -2.58756 q^{58} -10.4312 q^{59} -0.933476 q^{60} -9.66077 q^{61} +5.02778 q^{62} -2.83799 q^{64} +0.609730 q^{65} -0.599993 q^{66} +15.1775 q^{67} +11.9703 q^{68} +1.00000 q^{69} -7.49613 q^{71} -1.83582 q^{72} -11.1935 q^{73} +0.773944 q^{74} -4.71928 q^{75} -4.12579 q^{76} +0.561605 q^{78} +11.6816 q^{79} +1.39228 q^{80} +1.00000 q^{81} +3.10550 q^{82} -10.9393 q^{83} -3.59975 q^{85} -4.99905 q^{86} -5.30228 q^{87} +2.25708 q^{88} +7.21683 q^{89} +0.258561 q^{90} -1.76185 q^{92} +10.3026 q^{93} -5.76167 q^{94} +1.24072 q^{95} +4.95402 q^{96} -8.00904 q^{97} -1.22947 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} + 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} - 12 q^{8} + 10 q^{9}+O(q^{10}) \) 10 * q - 4 * q^2 + 10 * q^3 + 8 * q^4 - 4 * q^5 - 4 * q^6 - 12 * q^8 + 10 * q^9	\( 10 q - 4 q^{2} + 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} - 12 q^{8} + 10 q^{9} - 8 q^{10} - 2 q^{11} + 8 q^{12} - 16 q^{13} - 4 q^{15} + 4 q^{16} - 12 q^{17} - 4 q^{18} - 26 q^{19} - 8 q^{22} + 10 q^{23} - 12 q^{24} + 14 q^{25} + 12 q^{26} + 10 q^{27} - 16 q^{29} - 8 q^{30} - 20 q^{31} - 8 q^{32} - 2 q^{33} + 4 q^{34} + 8 q^{36} + 8 q^{37} + 8 q^{38} - 16 q^{39} + 12 q^{40} - 22 q^{41} - 4 q^{43} - 24 q^{44} - 4 q^{45} - 4 q^{46} - 6 q^{47} + 4 q^{48} - 48 q^{50} - 12 q^{51} - 24 q^{52} - 30 q^{53} - 4 q^{54} - 48 q^{55} - 26 q^{57} + 24 q^{58} - 42 q^{59} - 14 q^{61} - 40 q^{62} + 8 q^{64} - 44 q^{65} - 8 q^{66} - 8 q^{68} + 10 q^{69} + 8 q^{71} - 12 q^{72} - 24 q^{73} + 8 q^{74} + 14 q^{75} - 32 q^{76} + 12 q^{78} + 32 q^{79} - 28 q^{80} + 10 q^{81} + 64 q^{82} - 28 q^{83} - 4 q^{85} - 4 q^{86} - 16 q^{87} + 20 q^{88} - 8 q^{90} + 8 q^{92} - 20 q^{93} - 8 q^{94} - 16 q^{95} - 8 q^{96} + 12 q^{97} - 2 q^{99}+O(q^{100}) \) 10 * q - 4 * q^2 + 10 * q^3 + 8 * q^4 - 4 * q^5 - 4 * q^6 - 12 * q^8 + 10 * q^9 - 8 * q^10 - 2 * q^11 + 8 * q^12 - 16 * q^13 - 4 * q^15 + 4 * q^16 - 12 * q^17 - 4 * q^18 - 26 * q^19 - 8 * q^22 + 10 * q^23 - 12 * q^24 + 14 * q^25 + 12 * q^26 + 10 * q^27 - 16 * q^29 - 8 * q^30 - 20 * q^31 - 8 * q^32 - 2 * q^33 + 4 * q^34 + 8 * q^36 + 8 * q^37 + 8 * q^38 - 16 * q^39 + 12 * q^40 - 22 * q^41 - 4 * q^43 - 24 * q^44 - 4 * q^45 - 4 * q^46 - 6 * q^47 + 4 * q^48 - 48 * q^50 - 12 * q^51 - 24 * q^52 - 30 * q^53 - 4 * q^54 - 48 * q^55 - 26 * q^57 + 24 * q^58 - 42 * q^59 - 14 * q^61 - 40 * q^62 + 8 * q^64 - 44 * q^65 - 8 * q^66 - 8 * q^68 + 10 * q^69 + 8 * q^71 - 12 * q^72 - 24 * q^73 + 8 * q^74 + 14 * q^75 - 32 * q^76 + 12 * q^78 + 32 * q^79 - 28 * q^80 + 10 * q^81 + 64 * q^82 - 28 * q^83 - 4 * q^85 - 4 * q^86 - 16 * q^87 + 20 * q^88 - 8 * q^90 + 8 * q^92 - 20 * q^93 - 8 * q^94 - 16 * q^95 - 8 * q^96 + 12 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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