LMFDB - Embedded newform 1445.2.a.p.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1445,2,Mod(1,1445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1445, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1445 = 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1445.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:	$11.5383830921$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17 $ x^12 - 4x^11 - 10x^10 + 52x^9 + 21x^8 - 232x^7 + 44x^6 + 424x^5 - 137x^4 - 316x^3 + 46x^2 + 92*x + 17
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-2.35190$ of defining polynomial
Character	$\chi$	$=$	1445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.35190 q^{2} -1.56935 q^{3} +3.53144 q^{4} +1.00000 q^{5} -3.69096 q^{6} -3.58212 q^{7} +3.60181 q^{8} -0.537139 q^{9} +2.35190 q^{10} -2.48259 q^{11} -5.54207 q^{12} -1.25948 q^{13} -8.42480 q^{14} -1.56935 q^{15} +1.40821 q^{16} -1.26330 q^{18} -3.63431 q^{19} +3.53144 q^{20} +5.62161 q^{21} -5.83882 q^{22} -8.83293 q^{23} -5.65249 q^{24} +1.00000 q^{25} -2.96218 q^{26} +5.55101 q^{27} -12.6501 q^{28} +8.75919 q^{29} -3.69096 q^{30} -2.44403 q^{31} -3.89165 q^{32} +3.89606 q^{33} -3.58212 q^{35} -1.89688 q^{36} -4.60155 q^{37} -8.54755 q^{38} +1.97657 q^{39} +3.60181 q^{40} +4.32497 q^{41} +13.2215 q^{42} +7.54720 q^{43} -8.76714 q^{44} -0.537139 q^{45} -20.7742 q^{46} -11.3322 q^{47} -2.20997 q^{48} +5.83161 q^{49} +2.35190 q^{50} -4.44779 q^{52} +5.69139 q^{53} +13.0554 q^{54} -2.48259 q^{55} -12.9021 q^{56} +5.70351 q^{57} +20.6008 q^{58} -4.47000 q^{59} -5.54207 q^{60} +0.242871 q^{61} -5.74812 q^{62} +1.92410 q^{63} -11.9692 q^{64} -1.25948 q^{65} +9.16315 q^{66} -7.23278 q^{67} +13.8620 q^{69} -8.42480 q^{70} -1.83778 q^{71} -1.93467 q^{72} +5.47256 q^{73} -10.8224 q^{74} -1.56935 q^{75} -12.8344 q^{76} +8.89296 q^{77} +4.64870 q^{78} +9.03570 q^{79} +1.40821 q^{80} -7.10006 q^{81} +10.1719 q^{82} +7.31575 q^{83} +19.8524 q^{84} +17.7503 q^{86} -13.7462 q^{87} -8.94182 q^{88} -2.19350 q^{89} -1.26330 q^{90} +4.51162 q^{91} -31.1930 q^{92} +3.83554 q^{93} -26.6522 q^{94} -3.63431 q^{95} +6.10736 q^{96} -9.82039 q^{97} +13.7154 q^{98} +1.33350 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} - 8 q^{3} + 12 q^{4} + 12 q^{5} - 8 q^{6} - 16 q^{7} - 12 q^{8} + 12 q^{9} - 4 q^{10} - 16 q^{11} - 16 q^{12} - 8 q^{13} + 16 q^{14} - 8 q^{15} + 12 q^{16} + 4 q^{18} + 12 q^{20} + 16 q^{21}+ \cdots - 56 q^{99}+O(q^{100}) $ 12 * q - 4 * q^2 - 8 * q^3 + 12 * q^4 + 12 * q^5 - 8 * q^6 - 16 * q^7 - 12 * q^8 + 12 * q^9 - 4 * q^10 - 16 * q^11 - 16 * q^12 - 8 * q^13 + 16 * q^14 - 8 * q^15 + 12 * q^16 + 4 * q^18 + 12 * q^20 + 16 * q^21 - 16 * q^22 - 16 * q^23 + 12 * q^25 + 16 * q^26 - 32 * q^27 - 40 * q^28 - 16 * q^29 - 8 * q^30 - 24 * q^31 - 28 * q^32 - 16 * q^35 + 12 * q^36 - 24 * q^37 - 24 * q^38 - 8 * q^39 - 12 * q^40 - 8 * q^41 - 16 * q^43 - 8 * q^44 + 12 * q^45 - 40 * q^46 - 32 * q^47 + 24 * q^48 + 20 * q^49 - 4 * q^50 - 24 * q^52 - 8 * q^54 - 16 * q^55 + 24 * q^56 - 32 * q^57 + 16 * q^58 + 8 * q^59 - 16 * q^60 - 24 * q^61 - 8 * q^62 - 48 * q^63 + 36 * q^64 - 8 * q^65 + 40 * q^66 - 8 * q^67 + 48 * q^69 + 16 * q^70 + 16 * q^71 + 12 * q^72 - 16 * q^73 - 8 * q^75 + 16 * q^76 + 24 * q^77 + 24 * q^78 - 40 * q^79 + 12 * q^80 + 36 * q^81 + 16 * q^82 - 40 * q^83 + 32 * q^84 - 8 * q^86 - 32 * q^87 - 48 * q^88 + 8 * q^89 + 4 * q^90 - 72 * q^91 + 8 * q^92 + 24 * q^93 - 16 * q^94 - 8 * q^96 - 32 * q^97 - 60 * q^98 - 56 * q^99

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 $

$2$	2.35190	1.66305	0.831523	−	0.555490i	$-0.187469\pi$
$2$	2.35190	1.66305	0.831523	+	0.555490i	$0.187469\pi$
$3$	−1.56935	−0.906065	−0.453032	−	0.891494i	$-0.649658\pi$
$3$	−1.56935	−0.906065	−0.453032	+	0.891494i	$0.649658\pi$
$4$	3.53144	1.76572
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−3.69096	−1.50683
$7$	−3.58212	−1.35392	−0.676958	−	0.736022i	$-0.736702\pi$
$7$	−3.58212	−1.35392	−0.676958	+	0.736022i	$0.736702\pi$
$8$	3.60181	1.27343
$9$	−0.537139	−0.179046
$10$	2.35190	0.743737
$11$	−2.48259	−0.748530	−0.374265	−	0.927322i	$-0.622105\pi$
$11$	−2.48259	−0.748530	−0.374265	+	0.927322i	$0.622105\pi$
$12$	−5.54207	−1.59986
$13$	−1.25948	−0.349317	−0.174659	−	0.984629i	$-0.555882\pi$
$13$	−1.25948	−0.349317	−0.174659	+	0.984629i	$0.555882\pi$
$14$	−8.42480	−2.25162
$15$	−1.56935	−0.405205
$16$	1.40821	0.352052
$17$	0	0
$17$	0	0
$18$	−1.26330	−0.297762
$19$	−3.63431	−0.833768	−0.416884	−	0.908960i	$-0.636878\pi$
$19$	−3.63431	−0.833768	−0.416884	+	0.908960i	$0.636878\pi$
$20$	3.53144	0.789655
$21$	5.62161	1.22674
$22$	−5.83882	−1.24484
$23$	−8.83293	−1.84179	−0.920896	−	0.389808i	$-0.872541\pi$
$23$	−8.83293	−1.84179	−0.920896	+	0.389808i	$0.872541\pi$
$24$	−5.65249	−1.15381
$25$	1.00000	0.200000
$26$	−2.96218	−0.580931
$27$	5.55101	1.06829
$28$	−12.6501	−2.39064
$29$	8.75919	1.62654	0.813271	−	0.581885i	$-0.197685\pi$
$29$	8.75919	1.62654	0.813271	+	0.581885i	$0.197685\pi$
$30$	−3.69096	−0.673874
$31$	−2.44403	−0.438961	−0.219480	−	0.975617i	$-0.570436\pi$
$31$	−2.44403	−0.438961	−0.219480	+	0.975617i	$0.570436\pi$
$32$	−3.89165	−0.687953
$33$	3.89606	0.678217
$34$	0	0
$35$	−3.58212	−0.605489
$36$	−1.89688	−0.316146
$37$	−4.60155	−0.756490	−0.378245	−	0.925705i	$-0.623472\pi$
$37$	−4.60155	−0.756490	−0.378245	+	0.925705i	$0.623472\pi$
$38$	−8.54755	−1.38660
$39$	1.97657	0.316504
$40$	3.60181	0.569495
$41$	4.32497	0.675447	0.337724	−	0.941245i	$-0.390343\pi$
$41$	4.32497	0.675447	0.337724	+	0.941245i	$0.390343\pi$
$42$	13.2215	2.04012
$43$	7.54720	1.15094	0.575469	−	0.817824i	$-0.304820\pi$
$43$	7.54720	1.15094	0.575469	+	0.817824i	$0.304820\pi$
$44$	−8.76714	−1.32170
$45$	−0.537139	−0.0800720
$46$	−20.7742	−3.06299
$47$	−11.3322	−1.65297	−0.826484	−	0.562961i	$-0.809662\pi$
$47$	−11.3322	−1.65297	−0.826484	+	0.562961i	$0.809662\pi$
$48$	−2.20997	−0.318982
$49$	5.83161	0.833087
$50$	2.35190	0.332609
$51$	0	0
$52$	−4.44779	−0.616797
$53$	5.69139	0.781772	0.390886	−	0.920439i	$-0.372169\pi$
$53$	5.69139	0.781772	0.390886	+	0.920439i	$0.372169\pi$
$54$	13.0554	1.77662
$55$	−2.48259	−0.334753
$56$	−12.9021	−1.72412
$57$	5.70351	0.755448
$58$	20.6008	2.70501
$59$	−4.47000	−0.581945	−0.290972	−	0.956731i	$-0.593979\pi$
$59$	−4.47000	−0.581945	−0.290972	+	0.956731i	$0.593979\pi$
$60$	−5.54207	−0.715478
$61$	0.242871	0.0310964	0.0155482	−	0.999879i	$-0.495051\pi$
$61$	0.242871	0.0310964	0.0155482	+	0.999879i	$0.495051\pi$
$62$	−5.74812	−0.730012
$63$	1.92410	0.242414
$64$	−11.9692	−1.49615
$65$	−1.25948	−0.156220
$66$	9.16315	1.12791
$67$	−7.23278	−0.883625	−0.441812	−	0.897108i	$-0.645664\pi$
$67$	−7.23278	−0.883625	−0.441812	+	0.897108i	$0.645664\pi$
$68$	0	0
$69$	13.8620	1.66878
$70$	−8.42480	−1.00696
$71$	−1.83778	−0.218105	−0.109052	−	0.994036i	$-0.534782\pi$
$71$	−1.83778	−0.218105	−0.109052	+	0.994036i	$0.534782\pi$
$72$	−1.93467	−0.228003
$73$	5.47256	0.640515	0.320257	−	0.947331i	$-0.396231\pi$
$73$	5.47256	0.640515	0.320257	+	0.947331i	$0.396231\pi$
$74$	−10.8224	−1.25808
$75$	−1.56935	−0.181213
$76$	−12.8344	−1.47220
$77$	8.89296	1.01345
$78$	4.64870	0.526361
$79$	9.03570	1.01660	0.508298	−	0.861181i	$-0.330275\pi$
$79$	9.03570	1.01660	0.508298	+	0.861181i	$0.330275\pi$
$80$	1.40821	0.157442
$81$	−7.10006	−0.788896
$82$	10.1719	1.12330
$83$	7.31575	0.803008	0.401504	−	0.915857i	$-0.368488\pi$
$83$	7.31575	0.803008	0.401504	+	0.915857i	$0.368488\pi$
$84$	19.8524	2.16607
$85$	0	0
$86$	17.7503	1.91406
$87$	−13.7462	−1.47375
$88$	−8.94182	−0.953201
$89$	−2.19350	−0.232510	−0.116255	−	0.993219i	$-0.537089\pi$
$89$	−2.19350	−0.232510	−0.116255	+	0.993219i	$0.537089\pi$
$90$	−1.26330	−0.133163
$91$	4.51162	0.472946
$92$	−31.1930	−3.25209
$93$	3.83554	0.397727
$94$	−26.6522	−2.74896
$95$	−3.63431	−0.372873
$96$	6.10736	0.623330
$97$	−9.82039	−0.997110	−0.498555	−	0.866858i	$-0.666136\pi$
$97$	−9.82039	−0.997110	−0.498555	+	0.866858i	$0.666136\pi$
$98$	13.7154	1.38546
$99$	1.33350	0.134022
$100$	3.53144	0.353144
$101$	−5.46415	−0.543704	−0.271852	−	0.962339i	$-0.587636\pi$
$101$	−5.46415	−0.543704	−0.271852	+	0.962339i	$0.587636\pi$
$102$	0	0
$103$	7.16074	0.705568	0.352784	−	0.935705i	$-0.385235\pi$
$103$	7.16074	0.705568	0.352784	+	0.935705i	$0.385235\pi$
$104$	−4.53641	−0.444832
$105$	5.62161	0.548613
$106$	13.3856	1.30012
$107$	0.623156	0.0602427	0.0301214	−	0.999546i	$-0.490411\pi$
$107$	0.623156	0.0602427	0.0301214	+	0.999546i	$0.490411\pi$
$108$	19.6031	1.88631
$109$	−6.18952	−0.592848	−0.296424	−	0.955056i	$-0.595794\pi$
$109$	−6.18952	−0.592848	−0.296424	+	0.955056i	$0.595794\pi$
$110$	−5.83882	−0.556710
$111$	7.22144	0.685429
$112$	−5.04437	−0.476648
$113$	−16.0596	−1.51076	−0.755380	−	0.655287i	$-0.772548\pi$
$113$	−16.0596	−1.51076	−0.755380	+	0.655287i	$0.772548\pi$
$114$	13.4141	1.25635
$115$	−8.83293	−0.823675
$116$	30.9326	2.87202
$117$	0.676517	0.0625440
$118$	−10.5130	−0.967801
$119$	0	0
$120$	−5.65249	−0.516000
$121$	−4.83672	−0.439702
$122$	0.571208	0.0517147
$123$	−6.78740	−0.611999
$124$	−8.63096	−0.775083
$125$	1.00000	0.0894427
$126$	4.52529	0.403145
$127$	−5.91786	−0.525125	−0.262563	−	0.964915i	$-0.584568\pi$
$127$	−5.91786	−0.525125	−0.262563	+	0.964915i	$0.584568\pi$
$128$	−20.3671	−1.80021
$129$	−11.8442	−1.04282
$130$	−2.96218	−0.259800
$131$	16.1207	1.40848	0.704238	−	0.709964i	$-0.251289\pi$
$131$	16.1207	1.40848	0.704238	+	0.709964i	$0.251289\pi$
$132$	13.7587	1.19754
$133$	13.0186	1.12885
$134$	−17.0108	−1.46951
$135$	5.55101	0.477755
$136$	0	0
$137$	−17.1320	−1.46369	−0.731843	−	0.681473i	$-0.761339\pi$
$137$	−17.1320	−1.46369	−0.731843	+	0.681473i	$0.761339\pi$
$138$	32.6020	2.77526
$139$	2.13307	0.180925	0.0904624	−	0.995900i	$-0.471165\pi$
$139$	2.13307	0.180925	0.0904624	+	0.995900i	$0.471165\pi$
$140$	−12.6501	−1.06913
$141$	17.7842	1.49770
$142$	−4.32229	−0.362718
$143$	3.12678	0.261475
$144$	−0.756403	−0.0630335
$145$	8.75919	0.727412
$146$	12.8709	1.06521
$147$	−9.15184	−0.754831
$148$	−16.2501	−1.33575
$149$	18.5384	1.51872	0.759362	−	0.650669i	$-0.225511\pi$
$149$	18.5384	1.51872	0.759362	+	0.650669i	$0.225511\pi$
$150$	−3.69096	−0.301366
$151$	8.91261	0.725298	0.362649	−	0.931926i	$-0.381872\pi$
$151$	8.91261	0.725298	0.362649	+	0.931926i	$0.381872\pi$
$152$	−13.0901	−1.06175
$153$	0	0
$154$	20.9154	1.68541
$155$	−2.44403	−0.196309
$156$	6.98014	0.558859
$157$	18.2426	1.45592	0.727960	−	0.685620i	$-0.240469\pi$
$157$	18.2426	1.45592	0.727960	+	0.685620i	$0.240469\pi$
$158$	21.2511	1.69064
$159$	−8.93178	−0.708337
$160$	−3.89165	−0.307662
$161$	31.6406	2.49363
$162$	−16.6987	−1.31197
$163$	−9.36510	−0.733531	−0.366766	−	0.930313i	$-0.619535\pi$
$163$	−9.36510	−0.733531	−0.366766	+	0.930313i	$0.619535\pi$
$164$	15.2734	1.19265
$165$	3.89606	0.303308
$166$	17.2059	1.33544
$167$	9.28133	0.718211	0.359106	−	0.933297i	$-0.383082\pi$
$167$	9.28133	0.718211	0.359106	+	0.933297i	$0.383082\pi$
$168$	20.2479	1.56216
$169$	−11.4137	−0.877977
$170$	0	0
$171$	1.95213	0.149283
$172$	26.6525	2.03224
$173$	−4.29978	−0.326906	−0.163453	−	0.986551i	$-0.552263\pi$
$173$	−4.29978	−0.326906	−0.163453	+	0.986551i	$0.552263\pi$
$174$	−32.3298	−2.45092
$175$	−3.58212	−0.270783
$176$	−3.49601	−0.263521
$177$	7.01500	0.527280
$178$	−5.15889	−0.386675
$179$	−0.117678	−0.00879567	−0.00439783	−	0.999990i	$-0.501400\pi$
$179$	−0.117678	−0.00879567	−0.00439783	+	0.999990i	$0.501400\pi$
$180$	−1.89688	−0.141385
$181$	2.32364	0.172715	0.0863574	−	0.996264i	$-0.472477\pi$
$181$	2.32364	0.172715	0.0863574	+	0.996264i	$0.472477\pi$
$182$	10.6109	0.786531
$183$	−0.381149	−0.0281754
$184$	−31.8145	−2.34539
$185$	−4.60155	−0.338313
$186$	9.02082	0.661438
$187$	0	0
$188$	−40.0189	−2.91868
$189$	−19.8844	−1.44638
$190$	−8.54755	−0.620104
$191$	−14.6622	−1.06092	−0.530461	−	0.847710i	$-0.677981\pi$
$191$	−14.6622	−1.06092	−0.530461	+	0.847710i	$0.677981\pi$
$192$	18.7838	1.35561
$193$	8.39606	0.604362	0.302181	−	0.953251i	$-0.402285\pi$
$193$	8.39606	0.604362	0.302181	+	0.953251i	$0.402285\pi$
$194$	−23.0966	−1.65824
$195$	1.97657	0.141545
$196$	20.5940	1.47100
$197$	−6.68150	−0.476037	−0.238019	−	0.971261i	$-0.576498\pi$
$197$	−6.68150	−0.476037	−0.238019	+	0.971261i	$0.576498\pi$
$198$	3.13626	0.222884
$199$	18.8718	1.33779	0.668893	−	0.743359i	$-0.266768\pi$
$199$	18.8718	1.33779	0.668893	+	0.743359i	$0.266768\pi$
$200$	3.60181	0.254686
$201$	11.3508	0.800621
$202$	−12.8512	−0.904204
$203$	−31.3765	−2.20220
$204$	0	0
$205$	4.32497	0.302069
$206$	16.8414	1.17339
$207$	4.74451	0.329766
$208$	−1.77361	−0.122978
$209$	9.02252	0.624101
$210$	13.2215	0.912368
$211$	−6.61973	−0.455721	−0.227860	−	0.973694i	$-0.573173\pi$
$211$	−6.61973	−0.455721	−0.227860	+	0.973694i	$0.573173\pi$
$212$	20.0988	1.38039
$213$	2.88413	0.197617
$214$	1.46560	0.100186
$215$	7.54720	0.514715
$216$	19.9937	1.36040
$217$	8.75482	0.594316
$218$	−14.5571	−0.985934
$219$	−8.58836	−0.580348
$220$	−8.76714	−0.591081
$221$	0	0
$222$	16.9841	1.13990
$223$	2.94266	0.197055	0.0985275	−	0.995134i	$-0.468587\pi$
$223$	2.94266	0.197055	0.0985275	+	0.995134i	$0.468587\pi$
$224$	13.9404	0.931429
$225$	−0.537139	−0.0358093
$226$	−37.7706	−2.51246
$227$	−21.5256	−1.42871	−0.714353	−	0.699786i	$-0.753279\pi$
$227$	−21.5256	−1.42871	−0.714353	+	0.699786i	$0.753279\pi$
$228$	20.1416	1.33391
$229$	−11.9409	−0.789075	−0.394537	−	0.918880i	$-0.629095\pi$
$229$	−11.9409	−0.789075	−0.394537	+	0.918880i	$0.629095\pi$
$230$	−20.7742	−1.36981
$231$	−13.9562	−0.918249
$232$	31.5489	2.07129
$233$	−10.0687	−0.659623	−0.329812	−	0.944047i	$-0.606985\pi$
$233$	−10.0687	−0.659623	−0.329812	+	0.944047i	$0.606985\pi$
$234$	1.59110	0.104014
$235$	−11.3322	−0.739230
$236$	−15.7856	−1.02755
$237$	−14.1802	−0.921101
$238$	0	0
$239$	−16.6253	−1.07540	−0.537701	−	0.843135i	$-0.680707\pi$
$239$	−16.6253	−1.07540	−0.537701	+	0.843135i	$0.680707\pi$
$240$	−2.20997	−0.142653
$241$	5.92200	0.381470	0.190735	−	0.981642i	$-0.438913\pi$
$241$	5.92200	0.381470	0.190735	+	0.981642i	$0.438913\pi$
$242$	−11.3755	−0.731245
$243$	−5.51054	−0.353502
$244$	0.857684	0.0549076
$245$	5.83161	0.372568
$246$	−15.9633	−1.01778
$247$	4.57735	0.291250
$248$	−8.80292	−0.558986
$249$	−11.4810	−0.727577
$250$	2.35190	0.148747
$251$	7.77270	0.490608	0.245304	−	0.969446i	$-0.421112\pi$
$251$	7.77270	0.490608	0.245304	+	0.969446i	$0.421112\pi$
$252$	6.79484	0.428035
$253$	21.9286	1.37864
$254$	−13.9182	−0.873307
$255$	0	0
$256$	−23.9630	−1.49768
$257$	−8.06588	−0.503135	−0.251568	−	0.967840i	$-0.580946\pi$
$257$	−8.06588	−0.503135	−0.251568	+	0.967840i	$0.580946\pi$
$258$	−27.8564	−1.73426
$259$	16.4833	1.02422
$260$	−4.44779	−0.275840
$261$	−4.70491	−0.291226
$262$	37.9144	2.34236
$263$	−21.3252	−1.31497	−0.657483	−	0.753469i	$-0.728379\pi$
$263$	−21.3252	−1.31497	−0.657483	+	0.753469i	$0.728379\pi$
$264$	14.0329	0.863662
$265$	5.69139	0.349619
$266$	30.6184	1.87733
$267$	3.44237	0.210670
$268$	−25.5422	−1.56024
$269$	−4.20185	−0.256191	−0.128096	−	0.991762i	$-0.540886\pi$
$269$	−4.20185	−0.256191	−0.128096	+	0.991762i	$0.540886\pi$
$270$	13.0554	0.794528
$271$	24.7136	1.50124	0.750621	−	0.660733i	$-0.229755\pi$
$271$	24.7136	1.50124	0.750621	+	0.660733i	$0.229755\pi$
$272$	0	0
$273$	−7.08031	−0.428520
$274$	−40.2928	−2.43418
$275$	−2.48259	−0.149706
$276$	48.9527	2.94661
$277$	−20.0627	−1.20545	−0.602725	−	0.797949i	$-0.705918\pi$
$277$	−20.0627	−1.20545	−0.602725	+	0.797949i	$0.705918\pi$
$278$	5.01678	0.300886
$279$	1.31278	0.0785943
$280$	−12.9021	−0.771049
$281$	25.1805	1.50215	0.751073	−	0.660219i	$-0.229537\pi$
$281$	25.1805	1.50215	0.751073	+	0.660219i	$0.229537\pi$
$282$	41.8266	2.49074
$283$	−8.48659	−0.504476	−0.252238	−	0.967665i	$-0.581167\pi$
$283$	−8.48659	−0.504476	−0.252238	+	0.967665i	$0.581167\pi$
$284$	−6.49003	−0.385112
$285$	5.70351	0.337847
$286$	7.35389	0.434845
$287$	−15.4926	−0.914499
$288$	2.09036	0.123175
$289$	0	0
$290$	20.6008	1.20972
$291$	15.4116	0.903446
$292$	19.3260	1.13097
$293$	25.9873	1.51820	0.759098	−	0.650977i	$-0.225640\pi$
$293$	25.9873	1.51820	0.759098	+	0.650977i	$0.225640\pi$
$294$	−21.5242	−1.25532
$295$	−4.47000	−0.260254
$296$	−16.5739	−0.963338
$297$	−13.7809	−0.799649
$298$	43.6005	2.52571
$299$	11.1249	0.643370
$300$	−5.54207	−0.319972
$301$	−27.0350	−1.55827
$302$	20.9616	1.20620
$303$	8.57517	0.492631
$304$	−5.11786	−0.293529
$305$	0.242871	0.0139067
$306$	0	0
$307$	16.9475	0.967245	0.483622	−	0.875277i	$-0.339321\pi$
$307$	16.9475	0.967245	0.483622	+	0.875277i	$0.339321\pi$
$308$	31.4050	1.78947
$309$	−11.2377	−0.639291
$310$	−5.74812	−0.326471
$311$	17.1905	0.974782	0.487391	−	0.873184i	$-0.337949\pi$
$311$	17.1905	0.974782	0.487391	+	0.873184i	$0.337949\pi$
$312$	7.11922	0.403046
$313$	5.91487	0.334328	0.167164	−	0.985929i	$-0.446539\pi$
$313$	5.91487	0.334328	0.167164	+	0.985929i	$0.446539\pi$
$314$	42.9049	2.42126
$315$	1.92410	0.108411
$316$	31.9090	1.79502
$317$	0.640574	0.0359782	0.0179891	−	0.999838i	$-0.494274\pi$
$317$	0.640574	0.0359782	0.0179891	+	0.999838i	$0.494274\pi$
$318$	−21.0067	−1.17800
$319$	−21.7455	−1.21752
$320$	−11.9692	−0.669098
$321$	−0.977950	−0.0545838
$322$	74.4157	4.14702
$323$	0	0
$324$	−25.0735	−1.39297
$325$	−1.25948	−0.0698635
$326$	−22.0258	−1.21990
$327$	9.71352	0.537159
$328$	15.5777	0.860135
$329$	40.5932	2.23798
$330$	9.16315	0.504415
$331$	11.6609	0.640939	0.320469	−	0.947259i	$-0.396159\pi$
$331$	11.6609	0.640939	0.320469	+	0.947259i	$0.396159\pi$
$332$	25.8352	1.41789
$333$	2.47167	0.135447
$334$	21.8288	1.19442
$335$	−7.23278	−0.395169
$336$	7.91638	0.431874
$337$	−30.8806	−1.68217	−0.841087	−	0.540900i	$-0.818084\pi$
$337$	−30.8806	−1.68217	−0.841087	+	0.540900i	$0.818084\pi$
$338$	−26.8439	−1.46012
$339$	25.2031	1.36885
$340$	0	0
$341$	6.06754	0.328576
$342$	4.59122	0.248265
$343$	4.18533	0.225986
$344$	27.1836	1.46564
$345$	13.8620	0.746303
$346$	−10.1127	−0.543660
$347$	0.0149992	0.000805197 0	0.000402598	−	1.00000i	$-0.499872\pi$
$347$	0.0149992	0.000805197 0	0.000402598		1.00000i	$0.499872\pi$
$348$	−48.5441	−2.60224
$349$	−3.65491	−0.195643	−0.0978215	−	0.995204i	$-0.531187\pi$
$349$	−3.65491	−0.195643	−0.0978215	+	0.995204i	$0.531187\pi$
$350$	−8.42480	−0.450325
$351$	−6.99140	−0.373173
$352$	9.66138	0.514953
$353$	3.82333	0.203495	0.101748	−	0.994810i	$-0.467557\pi$
$353$	3.82333	0.203495	0.101748	+	0.994810i	$0.467557\pi$
$354$	16.4986	0.876891
$355$	−1.83778	−0.0975395
$356$	−7.74622	−0.410549
$357$	0	0
$358$	−0.276767	−0.0146276
$359$	−6.69675	−0.353441	−0.176721	−	0.984261i	$-0.556549\pi$
$359$	−6.69675	−0.353441	−0.176721	+	0.984261i	$0.556549\pi$
$360$	−1.93467	−0.101966
$361$	−5.79178	−0.304830
$362$	5.46497	0.287232
$363$	7.59052	0.398399
$364$	15.9325	0.835092
$365$	5.47256	0.286447
$366$	−0.896426	−0.0468569
$367$	−23.0276	−1.20203	−0.601015	−	0.799238i	$-0.705237\pi$
$367$	−23.0276	−1.20203	−0.601015	+	0.799238i	$0.705237\pi$
$368$	−12.4386	−0.648406
$369$	−2.32311	−0.120936
$370$	−10.8224	−0.562630
$371$	−20.3873	−1.05845
$372$	13.5450	0.702275
$373$	23.7303	1.22871	0.614355	−	0.789030i	$-0.289416\pi$
$373$	23.7303	1.22871	0.614355	+	0.789030i	$0.289416\pi$
$374$	0	0
$375$	−1.56935	−0.0810409
$376$	−40.8163	−2.10494
$377$	−11.0320	−0.568179
$378$	−46.7662	−2.40539
$379$	−2.97819	−0.152979	−0.0764897	−	0.997070i	$-0.524371\pi$
$379$	−2.97819	−0.152979	−0.0764897	+	0.997070i	$0.524371\pi$
$380$	−12.8344	−0.658389
$381$	9.28720	0.475798
$382$	−34.4841	−1.76436
$383$	−24.4222	−1.24791	−0.623957	−	0.781459i	$-0.714476\pi$
$383$	−24.4222	−1.24791	−0.623957	+	0.781459i	$0.714476\pi$
$384$	31.9631	1.63111
$385$	8.89296	0.453227
$386$	19.7467	1.00508
$387$	−4.05390	−0.206071
$388$	−34.6802	−1.76062
$389$	1.68867	0.0856191	0.0428095	−	0.999083i	$-0.486369\pi$
$389$	1.68867	0.0856191	0.0428095	+	0.999083i	$0.486369\pi$
$390$	4.64870	0.235396
$391$	0	0
$392$	21.0043	1.06088
$393$	−25.2991	−1.27617
$394$	−15.7142	−0.791671
$395$	9.03570	0.454635
$396$	4.70917	0.236645
$397$	−31.5256	−1.58222	−0.791112	−	0.611671i	$-0.790498\pi$
$397$	−31.5256	−1.58222	−0.791112	+	0.611671i	$0.790498\pi$
$398$	44.3846	2.22480
$399$	−20.4307	−1.02281
$400$	1.40821	0.0704103
$401$	5.06607	0.252987	0.126494	−	0.991967i	$-0.459628\pi$
$401$	5.06607	0.252987	0.126494	+	0.991967i	$0.459628\pi$
$402$	26.6959	1.33147
$403$	3.07821	0.153337
$404$	−19.2964	−0.960029
$405$	−7.10006	−0.352805
$406$	−73.7945	−3.66236
$407$	11.4238	0.566256
$408$	0	0
$409$	−26.1178	−1.29144	−0.645722	−	0.763573i	$-0.723443\pi$
$409$	−26.1178	−1.29144	−0.645722	+	0.763573i	$0.723443\pi$
$410$	10.1719	0.502355
$411$	26.8861	1.32619
$412$	25.2877	1.24584
$413$	16.0121	0.787904
$414$	11.1586	0.548416
$415$	7.31575	0.359116
$416$	4.90146	0.240314
$417$	−3.34754	−0.163930
$418$	21.2201	1.03791
$419$	−7.29855	−0.356557	−0.178279	−	0.983980i	$-0.557053\pi$
$419$	−7.29855	−0.356557	−0.178279	+	0.983980i	$0.557053\pi$
$420$	19.8524	0.968697
$421$	−21.0644	−1.02662	−0.513309	−	0.858204i	$-0.671580\pi$
$421$	−21.0644	−1.02662	−0.513309	+	0.858204i	$0.671580\pi$
$422$	−15.5690	−0.757885
$423$	6.08695	0.295958
$424$	20.4993	0.995533
$425$	0	0
$426$	6.78319	0.328646
$427$	−0.869993	−0.0421019
$428$	2.20064	0.106372
$429$	−4.90702	−0.236913
$430$	17.7503	0.855995
$431$	−21.0432	−1.01362	−0.506809	−	0.862059i	$-0.669175\pi$
$431$	−21.0432	−1.01362	−0.506809	+	0.862059i	$0.669175\pi$
$432$	7.81697	0.376094
$433$	37.1262	1.78417	0.892084	−	0.451869i	$-0.149242\pi$
$433$	37.1262	1.78417	0.892084	+	0.451869i	$0.149242\pi$
$434$	20.5905	0.988375
$435$	−13.7462	−0.659082
$436$	−21.8579	−1.04680
$437$	32.1016	1.53563
$438$	−20.1990	−0.965145
$439$	−0.471326	−0.0224952	−0.0112476	−	0.999937i	$-0.503580\pi$
$439$	−0.471326	−0.0224952	−0.0112476	+	0.999937i	$0.503580\pi$
$440$	−8.94182	−0.426285
$441$	−3.13238	−0.149161
$442$	0	0
$443$	−11.0249	−0.523808	−0.261904	−	0.965094i	$-0.584350\pi$
$443$	−11.0249	−0.523808	−0.261904	+	0.965094i	$0.584350\pi$
$444$	25.5021	1.21028
$445$	−2.19350	−0.103982
$446$	6.92084	0.327711
$447$	−29.0932	−1.37606
$448$	42.8751	2.02566
$449$	−17.2633	−0.814703	−0.407352	−	0.913271i	$-0.633548\pi$
$449$	−17.2633	−0.814703	−0.407352	+	0.913271i	$0.633548\pi$
$450$	−1.26330	−0.0595525
$451$	−10.7372	−0.505593
$452$	−56.7136	−2.66758
$453$	−13.9870	−0.657167
$454$	−50.6262	−2.37600
$455$	4.51162	0.211508
$456$	20.5429	0.962011
$457$	11.2676	0.527078	0.263539	−	0.964649i	$-0.415110\pi$
$457$	11.2676	0.527078	0.263539	+	0.964649i	$0.415110\pi$
$458$	−28.0837	−1.31227
$459$	0	0
$460$	−31.1930	−1.45438
$461$	30.0075	1.39759	0.698794	−	0.715323i	$-0.253720\pi$
$461$	30.0075	1.39759	0.698794	+	0.715323i	$0.253720\pi$
$462$	−32.8235	−1.52709
$463$	−17.1592	−0.797457	−0.398728	−	0.917069i	$-0.630548\pi$
$463$	−17.1592	−0.797457	−0.398728	+	0.917069i	$0.630548\pi$
$464$	12.3348	0.572627
$465$	3.83554	0.177869
$466$	−23.6806	−1.09698
$467$	−26.5914	−1.23051	−0.615253	−	0.788330i	$-0.710946\pi$
$467$	−26.5914	−1.23051	−0.615253	+	0.788330i	$0.710946\pi$
$468$	2.38908	0.110435
$469$	25.9087	1.19635
$470$	−26.6522	−1.22937
$471$	−28.6291	−1.31916
$472$	−16.1001	−0.741066
$473$	−18.7366	−0.861512
$474$	−33.3504	−1.53183
$475$	−3.63431	−0.166754
$476$	0	0
$477$	−3.05707	−0.139974
$478$	−39.1011	−1.78844
$479$	−22.9101	−1.04679	−0.523395	−	0.852090i	$-0.675335\pi$
$479$	−22.9101	−1.04679	−0.523395	+	0.852090i	$0.675335\pi$
$480$	6.10736	0.278761
$481$	5.79557	0.264255
$482$	13.9280	0.634402
$483$	−49.6552	−2.25939
$484$	−17.0806	−0.776392
$485$	−9.82039	−0.445921
$486$	−12.9603	−0.587889
$487$	−15.7135	−0.712048	−0.356024	−	0.934477i	$-0.615868\pi$
$487$	−15.7135	−0.712048	−0.356024	+	0.934477i	$0.615868\pi$
$488$	0.874773	0.0395991
$489$	14.6971	0.664627
$490$	13.7154	0.619597
$491$	−0.859183	−0.0387744	−0.0193872	−	0.999812i	$-0.506172\pi$
$491$	−0.859183	−0.0387744	−0.0193872	+	0.999812i	$0.506172\pi$
$492$	−23.9693	−1.08062
$493$	0	0
$494$	10.7655	0.484362
$495$	1.33350	0.0599363
$496$	−3.44170	−0.154537
$497$	6.58317	0.295296
$498$	−27.0021	−1.20999
$499$	−22.6396	−1.01349	−0.506744	−	0.862097i	$-0.669151\pi$
$499$	−22.6396	−1.01349	−0.506744	+	0.862097i	$0.669151\pi$
$500$	3.53144	0.157931
$501$	−14.5657	−0.650746
$502$	18.2806	0.815904
$503$	17.1150	0.763121	0.381560	−	0.924344i	$-0.375387\pi$
$503$	17.1150	0.763121	0.381560	+	0.924344i	$0.375387\pi$
$504$	6.93023	0.308697
$505$	−5.46415	−0.243152
$506$	51.5739	2.29274
$507$	17.9121	0.795504
$508$	−20.8986	−0.927225
$509$	−29.7081	−1.31679	−0.658393	−	0.752674i	$-0.728764\pi$
$509$	−29.7081	−1.31679	−0.658393	+	0.752674i	$0.728764\pi$
$510$	0	0
$511$	−19.6034	−0.867203
$512$	−15.6244	−0.690508
$513$	−20.1741	−0.890709
$514$	−18.9701	−0.836737
$515$	7.16074	0.315540
$516$	−41.8271	−1.84134
$517$	28.1332	1.23730
$518$	38.7672	1.70333
$519$	6.74786	0.296198
$520$	−4.53641	−0.198935
$521$	38.6564	1.69357	0.846784	−	0.531936i	$-0.178535\pi$
$521$	38.6564	1.69357	0.846784	+	0.531936i	$0.178535\pi$
$522$	−11.0655	−0.484323
$523$	0.599508	0.0262147	0.0131073	−	0.999914i	$-0.495828\pi$
$523$	0.599508	0.0262147	0.0131073	+	0.999914i	$0.495828\pi$
$524$	56.9295	2.48698
$525$	5.62161	0.245347
$526$	−50.1547	−2.18685
$527$	0	0
$528$	5.48646	0.238767
$529$	55.0206	2.39220
$530$	13.3856	0.581433
$531$	2.40101	0.104195
$532$	45.9743	1.99324
$533$	−5.44723	−0.235946
$534$	8.09611	0.350353
$535$	0.623156	0.0269414
$536$	−26.0511	−1.12523
$537$	0.184678	0.00796945
$538$	−9.88234	−0.426058
$539$	−14.4775	−0.623591
$540$	19.6031	0.843582
$541$	−39.7072	−1.70715	−0.853573	−	0.520973i	$-0.825569\pi$
$541$	−39.7072	−1.70715	−0.853573	+	0.520973i	$0.825569\pi$
$542$	58.1239	2.49663
$543$	−3.64660	−0.156491
$544$	0	0
$545$	−6.18952	−0.265130
$546$	−16.6522	−0.712649
$547$	−4.03770	−0.172639	−0.0863197	−	0.996267i	$-0.527511\pi$
$547$	−4.03770	−0.172639	−0.0863197	+	0.996267i	$0.527511\pi$
$548$	−60.5007	−2.58446
$549$	−0.130455	−0.00556770
$550$	−5.83882	−0.248968
$551$	−31.8336	−1.35616
$552$	49.9281	2.12508
$553$	−32.3670	−1.37638
$554$	−47.1854	−2.00472
$555$	7.22144	0.306533
$556$	7.53283	0.319463
$557$	5.42781	0.229984	0.114992	−	0.993366i	$-0.463316\pi$
$557$	5.42781	0.229984	0.114992	+	0.993366i	$0.463316\pi$
$558$	3.08754	0.130706
$559$	−9.50557	−0.402043
$560$	−5.04437	−0.213163
$561$	0	0
$562$	59.2222	2.49814
$563$	−5.18858	−0.218673	−0.109336	−	0.994005i	$-0.534873\pi$
$563$	−5.18858	−0.218673	−0.109336	+	0.994005i	$0.534873\pi$
$564$	62.8037	2.64451
$565$	−16.0596	−0.675632
$566$	−19.9596	−0.838966
$567$	25.4333	1.06810
$568$	−6.61934	−0.277741
$569$	−11.9963	−0.502911	−0.251455	−	0.967869i	$-0.580909\pi$
$569$	−11.9963	−0.502911	−0.251455	+	0.967869i	$0.580909\pi$
$570$	13.4141	0.561855
$571$	−43.5118	−1.82091	−0.910456	−	0.413606i	$-0.864269\pi$
$571$	−43.5118	−1.82091	−0.910456	+	0.413606i	$0.864269\pi$
$572$	11.0421	0.461692
$573$	23.0102	0.961263
$574$	−36.4371	−1.52085
$575$	−8.83293	−0.368358
$576$	6.42912	0.267880
$577$	−27.7097	−1.15357	−0.576784	−	0.816897i	$-0.695693\pi$
$577$	−27.7097	−1.15357	−0.576784	+	0.816897i	$0.695693\pi$
$578$	0	0
$579$	−13.1764	−0.547591
$580$	30.9326	1.28441
$581$	−26.2059	−1.08720
$582$	36.2467	1.50247
$583$	−14.1294	−0.585180
$584$	19.7111	0.815651
$585$	0.676517	0.0279705
$586$	61.1196	2.52483
$587$	4.37916	0.180747	0.0903736	−	0.995908i	$-0.471194\pi$
$587$	4.37916	0.180747	0.0903736	+	0.995908i	$0.471194\pi$
$588$	−32.3192	−1.33282
$589$	8.88237	0.365992
$590$	−10.5130	−0.432814
$591$	10.4856	0.431320
$592$	−6.47993	−0.266324
$593$	−35.8783	−1.47335	−0.736673	−	0.676250i	$-0.763604\pi$
$593$	−35.8783	−1.47335	−0.736673	+	0.676250i	$0.763604\pi$
$594$	−32.4114	−1.32985
$595$	0	0
$596$	65.4673	2.68164
$597$	−29.6165	−1.21212
$598$	26.1647	1.06995
$599$	11.1415	0.455230	0.227615	−	0.973751i	$-0.426907\pi$
$599$	11.1415	0.455230	0.227615	+	0.973751i	$0.426907\pi$
$600$	−5.65249	−0.230762
$601$	20.4505	0.834193	0.417096	−	0.908862i	$-0.363048\pi$
$601$	20.4505	0.834193	0.417096	+	0.908862i	$0.363048\pi$
$602$	−63.5837	−2.59148
$603$	3.88501	0.158210
$604$	31.4744	1.28067
$605$	−4.83672	−0.196641
$606$	20.1680	0.819268
$607$	1.57554	0.0639492	0.0319746	−	0.999489i	$-0.489820\pi$
$607$	1.57554	0.0639492	0.0319746	+	0.999489i	$0.489820\pi$
$608$	14.1435	0.573593
$609$	49.2407	1.99534
$610$	0.571208	0.0231275
$611$	14.2727	0.577410
$612$	0	0
$613$	−22.4800	−0.907959	−0.453980	−	0.891012i	$-0.649996\pi$
$613$	−22.4800	−0.907959	−0.453980	+	0.891012i	$0.649996\pi$
$614$	39.8589	1.60857
$615$	−6.78740	−0.273694
$616$	32.0307	1.29055
$617$	−27.1976	−1.09493	−0.547467	−	0.836827i	$-0.684408\pi$
$617$	−27.1976	−1.09493	−0.547467	+	0.836827i	$0.684408\pi$
$618$	−26.4300	−1.06317
$619$	−14.3702	−0.577588	−0.288794	−	0.957391i	$-0.593254\pi$
$619$	−14.3702	−0.577588	−0.288794	+	0.957391i	$0.593254\pi$
$620$	−8.63096	−0.346628
$621$	−49.0317	−1.96757
$622$	40.4303	1.62111
$623$	7.85738	0.314799
$624$	2.78342	0.111426
$625$	1.00000	0.0400000
$626$	13.9112	0.556003
$627$	−14.1595	−0.565476
$628$	64.4228	2.57075
$629$	0	0
$630$	4.52529	0.180292
$631$	−23.2691	−0.926330	−0.463165	−	0.886272i	$-0.653286\pi$
$631$	−23.2691	−0.926330	−0.463165	+	0.886272i	$0.653286\pi$
$632$	32.5448	1.29456
$633$	10.3887	0.412913
$634$	1.50657	0.0598334
$635$	−5.91786	−0.234843
$636$	−31.5421	−1.25073
$637$	−7.34480	−0.291012
$638$	−51.1434	−2.02478
$639$	0.987146	0.0390509
$640$	−20.3671	−0.805079
$641$	−5.38403	−0.212656	−0.106328	−	0.994331i	$-0.533909\pi$
$641$	−5.38403	−0.212656	−0.106328	+	0.994331i	$0.533909\pi$
$642$	−2.30004	−0.0907754
$643$	−16.1891	−0.638438	−0.319219	−	0.947681i	$-0.603420\pi$
$643$	−16.1891	−0.638438	−0.319219	+	0.947681i	$0.603420\pi$
$644$	111.737	4.40306
$645$	−11.8442	−0.466365
$646$	0	0
$647$	−25.8383	−1.01581	−0.507904	−	0.861413i	$-0.669580\pi$
$647$	−25.8383	−1.01581	−0.507904	+	0.861413i	$0.669580\pi$
$648$	−25.5730	−1.00460
$649$	11.0972	0.435604
$650$	−2.96218	−0.116186
$651$	−13.7394	−0.538489
$652$	−33.0723	−1.29521
$653$	12.4784	0.488319	0.244159	−	0.969735i	$-0.421488\pi$
$653$	12.4784	0.488319	0.244159	+	0.969735i	$0.421488\pi$
$654$	22.8452	0.893320
$655$	16.1207	0.629889
$656$	6.09045	0.237792
$657$	−2.93953	−0.114682
$658$	95.4713	3.72186
$659$	41.7109	1.62483	0.812413	−	0.583082i	$-0.198153\pi$
$659$	41.7109	1.62483	0.812413	+	0.583082i	$0.198153\pi$
$660$	13.7587	0.535557
$661$	18.1720	0.706808	0.353404	−	0.935471i	$-0.385024\pi$
$661$	18.1720	0.706808	0.353404	+	0.935471i	$0.385024\pi$
$662$	27.4252	1.06591
$663$	0	0
$664$	26.3499	1.02257
$665$	13.0186	0.504838
$666$	5.81313	0.225254
$667$	−77.3693	−2.99575
$668$	32.7765	1.26816
$669$	−4.61806	−0.178545
$670$	−17.0108	−0.657184
$671$	−0.602949	−0.0232766
$672$	−21.8773	−0.843936
$673$	6.39677	0.246577	0.123289	−	0.992371i	$-0.460656\pi$
$673$	6.39677	0.246577	0.123289	+	0.992371i	$0.460656\pi$
$674$	−72.6282	−2.79753
$675$	5.55101	0.213659
$676$	−40.3069	−1.55026
$677$	41.8887	1.60991	0.804957	−	0.593334i	$-0.202189\pi$
$677$	41.8887	1.60991	0.804957	+	0.593334i	$0.202189\pi$
$678$	59.2753	2.27645
$679$	35.1779	1.35000
$680$	0	0
$681$	33.7812	1.29450
$682$	14.2703	0.546436
$683$	30.3064	1.15964	0.579820	−	0.814744i	$-0.303123\pi$
$683$	30.3064	1.15964	0.579820	+	0.814744i	$0.303123\pi$
$684$	6.89384	0.263593
$685$	−17.1320	−0.654580
$686$	9.84348	0.375826
$687$	18.7394	0.714953
$688$	10.6280	0.405189
$689$	−7.16820	−0.273087
$690$	32.6020	1.24114
$691$	−5.79281	−0.220369	−0.110184	−	0.993911i	$-0.535144\pi$
$691$	−5.79281	−0.220369	−0.110184	+	0.993911i	$0.535144\pi$
$692$	−15.1844	−0.577225
$693$	−4.77676	−0.181454
$694$	0.0352765	0.00133908
$695$	2.13307	0.0809121
$696$	−49.5113	−1.87672
$697$	0	0
$698$	−8.59600	−0.325363
$699$	15.8013	0.597662
$700$	−12.6501	−0.478128
$701$	5.24783	0.198208	0.0991039	−	0.995077i	$-0.468402\pi$
$701$	5.24783	0.198208	0.0991039	+	0.995077i	$0.468402\pi$
$702$	−16.4431	−0.620604
$703$	16.7235	0.630738
$704$	29.7146	1.11991
$705$	17.7842	0.669790
$706$	8.99209	0.338422
$707$	19.5733	0.736129
$708$	24.7731	0.931030
$709$	41.7572	1.56823	0.784113	−	0.620618i	$-0.213118\pi$
$709$	41.7572	1.56823	0.784113	+	0.620618i	$0.213118\pi$
$710$	−4.32229	−0.162213
$711$	−4.85343	−0.182018
$712$	−7.90055	−0.296086
$713$	21.5879	0.808475
$714$	0	0
$715$	3.12678	0.116935
$716$	−0.415573	−0.0155307
$717$	26.0910	0.974385
$718$	−15.7501	−0.587789
$719$	−10.1819	−0.379719	−0.189860	−	0.981811i	$-0.560803\pi$
$719$	−10.1819	−0.379719	−0.189860	+	0.981811i	$0.560803\pi$
$720$	−0.756403	−0.0281895
$721$	−25.6506	−0.955280
$722$	−13.6217	−0.506947
$723$	−9.29370	−0.345636
$724$	8.20580	0.304966
$725$	8.75919	0.325308
$726$	17.8521	0.662555
$727$	46.0182	1.70672	0.853361	−	0.521321i	$-0.174561\pi$
$727$	46.0182	1.70672	0.853361	+	0.521321i	$0.174561\pi$
$728$	16.2500	0.602264
$729$	29.9482	1.10919
$730$	12.8709	0.476374
$731$	0	0
$732$	−1.34601	−0.0497498
$733$	−2.55919	−0.0945257	−0.0472629	−	0.998882i	$-0.515050\pi$
$733$	−2.55919	−0.0945257	−0.0472629	+	0.998882i	$0.515050\pi$
$734$	−54.1586	−1.99903
$735$	−9.15184	−0.337571
$736$	34.3746	1.26707
$737$	17.9561	0.661420
$738$	−5.46373	−0.201123
$739$	−2.47667	−0.0911057	−0.0455528	−	0.998962i	$-0.514505\pi$
$739$	−2.47667	−0.0911057	−0.0455528	+	0.998962i	$0.514505\pi$
$740$	−16.2501	−0.597366
$741$	−7.18347	−0.263891
$742$	−47.9488	−1.76026
$743$	1.59477	0.0585064	0.0292532	−	0.999572i	$-0.490687\pi$
$743$	1.59477	0.0585064	0.0292532	+	0.999572i	$0.490687\pi$
$744$	13.8149	0.506478
$745$	18.5384	0.679194
$746$	55.8114	2.04340
$747$	−3.92957	−0.143776
$748$	0	0
$749$	−2.23222	−0.0815636
$750$	−3.69096	−0.134775
$751$	12.7441	0.465041	0.232520	−	0.972592i	$-0.425303\pi$
$751$	12.7441	0.465041	0.232520	+	0.972592i	$0.425303\pi$
$752$	−15.9580	−0.581930
$753$	−12.1981	−0.444523
$754$	−25.9463	−0.944908
$755$	8.91261	0.324363
$756$	−70.2207	−2.55390
$757$	−46.0573	−1.67398	−0.836991	−	0.547217i	$-0.815687\pi$
$757$	−46.0573	−1.67398	−0.836991	+	0.547217i	$0.815687\pi$
$758$	−7.00441	−0.254412
$759$	−34.4136	−1.24914
$760$	−13.0901	−0.474827
$761$	−29.7003	−1.07664	−0.538318	−	0.842742i	$-0.680940\pi$
$761$	−29.7003	−1.07664	−0.538318	+	0.842742i	$0.680940\pi$
$762$	21.8426	0.791273
$763$	22.1716	0.802666
$764$	−51.7788	−1.87329
$765$	0	0
$766$	−57.4385	−2.07534
$767$	5.62989	0.203284
$768$	37.6063	1.35700
$769$	43.8299	1.58055	0.790273	−	0.612755i	$-0.209939\pi$
$769$	43.8299	1.58055	0.790273	+	0.612755i	$0.209939\pi$
$770$	20.9154	0.753738
$771$	12.6582	0.455873
$772$	29.6502	1.06713
$773$	−27.9122	−1.00393	−0.501966	−	0.864887i	$-0.667390\pi$
$773$	−27.9122	−1.00393	−0.501966	+	0.864887i	$0.667390\pi$
$774$	−9.53437	−0.342706
$775$	−2.44403	−0.0877922
$776$	−35.3711	−1.26975
$777$	−25.8681	−0.928013
$778$	3.97159	0.142388
$779$	−15.7183	−0.563167
$780$	6.98014	0.249929
$781$	4.56247	0.163258
$782$	0	0
$783$	48.6224	1.73762
$784$	8.21211	0.293289
$785$	18.2426	0.651107
$786$	−59.5010	−2.12233
$787$	−30.0226	−1.07019	−0.535096	−	0.844792i	$-0.679724\pi$
$787$	−30.0226	−1.07019	−0.535096	+	0.844792i	$0.679724\pi$
$788$	−23.5953	−0.840549
$789$	33.4667	1.19144
$790$	21.2511	0.756079
$791$	57.5275	2.04544
$792$	4.80300	0.170667
$793$	−0.305891	−0.0108625
$794$	−74.1451	−2.63131
$795$	−8.93178	−0.316778
$796$	66.6447	2.36216
$797$	−5.16206	−0.182850	−0.0914249	−	0.995812i	$-0.529142\pi$
$797$	−5.16206	−0.182850	−0.0914249	+	0.995812i	$0.529142\pi$
$798$	−48.0509	−1.70099
$799$	0	0
$800$	−3.89165	−0.137591
$801$	1.17821	0.0416301
$802$	11.9149	0.420730
$803$	−13.5861	−0.479445
$804$	40.0846	1.41367
$805$	31.6406	1.11519
$806$	7.23966	0.255006
$807$	6.59417	0.232126
$808$	−19.6808	−0.692369
$809$	22.2343	0.781717	0.390859	−	0.920451i	$-0.372178\pi$
$809$	22.2343	0.781717	0.390859	+	0.920451i	$0.372178\pi$
$810$	−16.6987	−0.586731
$811$	−15.6644	−0.550051	−0.275026	−	0.961437i	$-0.588686\pi$
$811$	−15.6644	−0.550051	−0.275026	+	0.961437i	$0.588686\pi$
$812$	−110.804	−3.88847
$813$	−38.7842	−1.36022
$814$	26.8676	0.941710
$815$	−9.36510	−0.328045
$816$	0	0
$817$	−27.4289	−0.959615
$818$	−61.4266	−2.14773
$819$	−2.42337	−0.0846793
$820$	15.2734	0.533370
$821$	−34.9393	−1.21939	−0.609694	−	0.792637i	$-0.708708\pi$
$821$	−34.9393	−1.21939	−0.609694	+	0.792637i	$0.708708\pi$
$822$	63.2335	2.20552
$823$	−2.44416	−0.0851980	−0.0425990	−	0.999092i	$-0.513564\pi$
$823$	−2.44416	−0.0851980	−0.0425990	+	0.999092i	$0.513564\pi$
$824$	25.7916	0.898492
$825$	3.89606	0.135643
$826$	37.6589	1.31032
$827$	50.8219	1.76725	0.883626	−	0.468194i	$-0.155095\pi$
$827$	50.8219	1.76725	0.883626	+	0.468194i	$0.155095\pi$
$828$	16.7550	0.582275
$829$	41.5846	1.44429	0.722147	−	0.691740i	$-0.243156\pi$
$829$	41.5846	1.44429	0.722147	+	0.691740i	$0.243156\pi$
$830$	17.2059	0.597226
$831$	31.4854	1.09222
$832$	15.0750	0.522631
$833$	0	0
$834$	−7.87308	−0.272623
$835$	9.28133	0.321194
$836$	31.8625	1.10199
$837$	−13.5668	−0.468939
$838$	−17.1655	−0.592971
$839$	4.44768	0.153551	0.0767755	−	0.997048i	$-0.475538\pi$
$839$	4.44768	0.153551	0.0767755	+	0.997048i	$0.475538\pi$
$840$	20.2479	0.698620
$841$	47.7235	1.64564
$842$	−49.5415	−1.70731
$843$	−39.5171	−1.36104
$844$	−23.3772	−0.804676
$845$	−11.4137	−0.392643
$846$	14.3159	0.492191
$847$	17.3257	0.595320
$848$	8.01465	0.275224
$849$	13.3184	0.457088
$850$	0	0
$851$	40.6451	1.39330
$852$	10.1851	0.348937
$853$	17.8866	0.612426	0.306213	−	0.951963i	$-0.400938\pi$
$853$	17.8866	0.612426	0.306213	+	0.951963i	$0.400938\pi$
$854$	−2.04614	−0.0700174
$855$	1.95213	0.0667615
$856$	2.24449	0.0767150
$857$	−38.5585	−1.31713	−0.658566	−	0.752523i	$-0.728837\pi$
$857$	−38.5585	−1.31713	−0.658566	+	0.752523i	$0.728837\pi$
$858$	−11.5408	−0.393997
$859$	3.50985	0.119755	0.0598773	−	0.998206i	$-0.480929\pi$
$859$	3.50985	0.119755	0.0598773	+	0.998206i	$0.480929\pi$
$860$	26.6525	0.908843
$861$	24.3133	0.828595
$862$	−49.4916	−1.68569
$863$	−21.9552	−0.747365	−0.373683	−	0.927557i	$-0.621905\pi$
$863$	−21.9552	−0.747365	−0.373683	+	0.927557i	$0.621905\pi$
$864$	−21.6026	−0.734935
$865$	−4.29978	−0.146197
$866$	87.3171	2.96715
$867$	0	0
$868$	30.9171	1.04940
$869$	−22.4320	−0.760952
$870$	−32.3298	−1.09608
$871$	9.10956	0.308666
$872$	−22.2934	−0.754951
$873$	5.27492	0.178529
$874$	75.4998	2.55382
$875$	−3.58212	−0.121098
$876$	−30.3293	−1.02473
$877$	5.65595	0.190988	0.0954939	−	0.995430i	$-0.469557\pi$
$877$	5.65595	0.190988	0.0954939	+	0.995430i	$0.469557\pi$
$878$	−1.10851	−0.0374105
$879$	−40.7832	−1.37558
$880$	−3.49601	−0.117850
$881$	−34.3915	−1.15868	−0.579340	−	0.815086i	$-0.696689\pi$
$881$	−34.3915	−1.15868	−0.579340	+	0.815086i	$0.696689\pi$
$882$	−7.36706	−0.248062
$883$	58.3115	1.96234	0.981170	−	0.193148i	$-0.0618697\pi$
$883$	58.3115	1.96234	0.981170	+	0.193148i	$0.0618697\pi$
$884$	0	0
$885$	7.01500	0.235807
$886$	−25.9295	−0.871117
$887$	44.5012	1.49420	0.747102	−	0.664710i	$-0.231445\pi$
$887$	44.5012	1.49420	0.747102	+	0.664710i	$0.231445\pi$
$888$	26.0102	0.872846
$889$	21.1985	0.710975
$890$	−5.15889	−0.172927
$891$	17.6266	0.590513
$892$	10.3918	0.347944
$893$	41.1847	1.37819
$894$	−68.4244	−2.28845
$895$	−0.117678	−0.00393354
$896$	72.9573	2.43733
$897$	−17.4589	−0.582935
$898$	−40.6015	−1.35489
$899$	−21.4077	−0.713988
$900$	−1.89688	−0.0632292
$901$	0	0
$902$	−25.2527	−0.840824
$903$	42.4274	1.41190
$904$	−57.8436	−1.92385
$905$	2.32364	0.0772404
$906$	−32.8961	−1.09290
$907$	−35.3726	−1.17453	−0.587264	−	0.809396i	$-0.699795\pi$
$907$	−35.3726	−1.17453	−0.587264	+	0.809396i	$0.699795\pi$
$908$	−76.0165	−2.52270
$909$	2.93501	0.0973482
$910$	10.6109	0.351748
$911$	−49.1810	−1.62944	−0.814719	−	0.579856i	$-0.803109\pi$
$911$	−49.1810	−1.62944	−0.814719	+	0.579856i	$0.803109\pi$
$912$	8.03172	0.265957
$913$	−18.1620	−0.601076
$914$	26.5004	0.876554
$915$	−0.381149	−0.0126004
$916$	−42.1685	−1.39329
$917$	−57.7465	−1.90696
$918$	0	0
$919$	49.6635	1.63825	0.819123	−	0.573618i	$-0.194460\pi$
$919$	49.6635	1.63825	0.819123	+	0.573618i	$0.194460\pi$
$920$	−31.8145	−1.04889
$921$	−26.5966	−0.876387
$922$	70.5747	2.32425
$923$	2.31466	0.0761878
$924$	−49.2854	−1.62137
$925$	−4.60155	−0.151298
$926$	−40.3568	−1.32621
$927$	−3.84631	−0.126329
$928$	−34.0877	−1.11898
$929$	34.1165	1.11933	0.559664	−	0.828720i	$-0.310930\pi$
$929$	34.1165	1.11933	0.559664	+	0.828720i	$0.310930\pi$
$930$	9.02082	0.295804
$931$	−21.1939	−0.694601
$932$	−35.5571	−1.16471
$933$	−26.9779	−0.883216
$934$	−62.5405	−2.04639
$935$	0	0
$936$	2.43668	0.0796455
$937$	−38.5275	−1.25864	−0.629319	−	0.777147i	$-0.716666\pi$
$937$	−38.5275	−1.25864	−0.629319	+	0.777147i	$0.716666\pi$
$938$	60.9347	1.98959
$939$	−9.28250	−0.302923
$940$	−40.0189	−1.30527
$941$	37.1363	1.21061	0.605305	−	0.795994i	$-0.293051\pi$
$941$	37.1363	1.21061	0.605305	+	0.795994i	$0.293051\pi$
$942$	−67.3328	−2.19382
$943$	−38.2022	−1.24403
$944$	−6.29469	−0.204875
$945$	−19.8844	−0.646840
$946$	−44.0667	−1.43273
$947$	−2.23113	−0.0725021	−0.0362510	−	0.999343i	$-0.511542\pi$
$947$	−2.23113	−0.0725021	−0.0362510	+	0.999343i	$0.511542\pi$
$948$	−50.0765	−1.62641
$949$	−6.89259	−0.223743
$950$	−8.54755	−0.277319
$951$	−1.00528	−0.0325986
$952$	0	0
$953$	25.8280	0.836651	0.418326	−	0.908297i	$-0.362617\pi$
$953$	25.8280	0.836651	0.418326	+	0.908297i	$0.362617\pi$
$954$	−7.18992	−0.232782
$955$	−14.6622	−0.474458
$956$	−58.7114	−1.89886
$957$	34.1264	1.10315
$958$	−53.8823	−1.74086
$959$	61.3689	1.98171
$960$	18.7838	0.606246
$961$	−25.0267	−0.807313
$962$	13.6306	0.439469
$963$	−0.334721	−0.0107862
$964$	20.9132	0.673569
$965$	8.39606	0.270279
$966$	−116.784	−3.75747
$967$	−31.4013	−1.00980	−0.504899	−	0.863178i	$-0.668470\pi$
$967$	−31.4013	−1.00980	−0.504899	+	0.863178i	$0.668470\pi$
$968$	−17.4209	−0.559930
$969$	0	0
$970$	−23.0966	−0.741587
$971$	41.1265	1.31981	0.659906	−	0.751348i	$-0.270596\pi$
$971$	41.1265	1.31981	0.659906	+	0.751348i	$0.270596\pi$
$972$	−19.4602	−0.624185
$973$	−7.64093	−0.244957
$974$	−36.9567	−1.18417
$975$	1.97657	0.0633009
$976$	0.342012	0.0109475
$977$	27.8890	0.892250	0.446125	−	0.894971i	$-0.352804\pi$
$977$	27.8890	0.892250	0.446125	+	0.894971i	$0.352804\pi$
$978$	34.5662	1.10530
$979$	5.44557	0.174041
$980$	20.5940	0.657851
$981$	3.32463	0.106147
$982$	−2.02071	−0.0644836
$983$	56.6933	1.80824	0.904118	−	0.427283i	$-0.140529\pi$
$983$	56.6933	1.80824	0.904118	+	0.427283i	$0.140529\pi$
$984$	−24.4469	−0.779338
$985$	−6.68150	−0.212890
$986$	0	0
$987$	−63.7050	−2.02775
$988$	16.1647	0.514266
$989$	−66.6639	−2.11979
$990$	3.13626	0.0996768
$991$	35.8868	1.13998	0.569991	−	0.821651i	$-0.306947\pi$
$991$	35.8868	1.13998	0.569991	+	0.821651i	$0.306947\pi$
$992$	9.51130	0.301984
$993$	−18.3000	−0.580732
$994$	15.4830	0.491090
$995$	18.8718	0.598276
$996$	−40.5444	−1.28470
$997$	−32.9550	−1.04370	−0.521848	−	0.853038i	$-0.674757\pi$
$997$	−32.9550	−1.04370	−0.521848	+	0.853038i	$0.674757\pi$
$998$	−53.2461	−1.68548
$999$	−25.5433	−0.808153

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1445.2.a.p.1.12		12
5.4	even	2		7225.2.a.bs.1.1		12
17.3	odd	16		85.2.l.a.26.6	✓	24
17.4	even	4		1445.2.d.j.866.1		24
17.6	odd	16		85.2.l.a.36.6	yes	24
17.13	even	4		1445.2.d.j.866.2		24
17.16	even	2		1445.2.a.q.1.12		12
51.20	even	16		765.2.be.b.451.1		24
51.23	even	16		765.2.be.b.631.1		24
85.3	even	16		425.2.n.c.349.1		24
85.23	even	16		425.2.n.f.274.6		24
85.37	even	16		425.2.n.f.349.6		24
85.54	odd	16		425.2.m.b.26.1		24
85.57	even	16		425.2.n.c.274.1		24
85.74	odd	16		425.2.m.b.376.1		24
85.84	even	2		7225.2.a.bq.1.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.l.a.26.6	✓	24	17.3	odd	16
85.2.l.a.36.6	yes	24	17.6	odd	16
425.2.m.b.26.1		24	85.54	odd	16
425.2.m.b.376.1		24	85.74	odd	16
425.2.n.c.274.1		24	85.57	even	16
425.2.n.c.349.1		24	85.3	even	16
425.2.n.f.274.6		24	85.23	even	16
425.2.n.f.349.6		24	85.37	even	16
765.2.be.b.451.1		24	51.20	even	16
765.2.be.b.631.1		24	51.23	even	16
1445.2.a.p.1.12		12	1.1	even	1	trivial
1445.2.a.q.1.12		12	17.16	even	2
1445.2.d.j.866.1		24	17.4	even	4
1445.2.d.j.866.2		24	17.13	even	4
7225.2.a.bq.1.1		12	85.84	even	2
7225.2.a.bs.1.1		12	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1445 = 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1445.a (trivial)

\(f(q)\)	\(=\)	\(q+2.35190 q^{2} -1.56935 q^{3} +3.53144 q^{4} +1.00000 q^{5} -3.69096 q^{6} -3.58212 q^{7} +3.60181 q^{8} -0.537139 q^{9} +2.35190 q^{10} -2.48259 q^{11} -5.54207 q^{12} -1.25948 q^{13} -8.42480 q^{14} -1.56935 q^{15} +1.40821 q^{16} -1.26330 q^{18} -3.63431 q^{19} +3.53144 q^{20} +5.62161 q^{21} -5.83882 q^{22} -8.83293 q^{23} -5.65249 q^{24} +1.00000 q^{25} -2.96218 q^{26} +5.55101 q^{27} -12.6501 q^{28} +8.75919 q^{29} -3.69096 q^{30} -2.44403 q^{31} -3.89165 q^{32} +3.89606 q^{33} -3.58212 q^{35} -1.89688 q^{36} -4.60155 q^{37} -8.54755 q^{38} +1.97657 q^{39} +3.60181 q^{40} +4.32497 q^{41} +13.2215 q^{42} +7.54720 q^{43} -8.76714 q^{44} -0.537139 q^{45} -20.7742 q^{46} -11.3322 q^{47} -2.20997 q^{48} +5.83161 q^{49} +2.35190 q^{50} -4.44779 q^{52} +5.69139 q^{53} +13.0554 q^{54} -2.48259 q^{55} -12.9021 q^{56} +5.70351 q^{57} +20.6008 q^{58} -4.47000 q^{59} -5.54207 q^{60} +0.242871 q^{61} -5.74812 q^{62} +1.92410 q^{63} -11.9692 q^{64} -1.25948 q^{65} +9.16315 q^{66} -7.23278 q^{67} +13.8620 q^{69} -8.42480 q^{70} -1.83778 q^{71} -1.93467 q^{72} +5.47256 q^{73} -10.8224 q^{74} -1.56935 q^{75} -12.8344 q^{76} +8.89296 q^{77} +4.64870 q^{78} +9.03570 q^{79} +1.40821 q^{80} -7.10006 q^{81} +10.1719 q^{82} +7.31575 q^{83} +19.8524 q^{84} +17.7503 q^{86} -13.7462 q^{87} -8.94182 q^{88} -2.19350 q^{89} -1.26330 q^{90} +4.51162 q^{91} -31.1930 q^{92} +3.83554 q^{93} -26.6522 q^{94} -3.63431 q^{95} +6.10736 q^{96} -9.82039 q^{97} +13.7154 q^{98} +1.33350 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} - 8 q^{3} + 12 q^{4} + 12 q^{5} - 8 q^{6} - 16 q^{7} - 12 q^{8} + 12 q^{9} - 4 q^{10} - 16 q^{11} - 16 q^{12} - 8 q^{13} + 16 q^{14} - 8 q^{15} + 12 q^{16} + 4 q^{18} + 12 q^{20} + 16 q^{21}+ \cdots - 56 q^{99}+O(q^{100}) \) 12 * q - 4 * q^2 - 8 * q^3 + 12 * q^4 + 12 * q^5 - 8 * q^6 - 16 * q^7 - 12 * q^8 + 12 * q^9 - 4 * q^10 - 16 * q^11 - 16 * q^12 - 8 * q^13 + 16 * q^14 - 8 * q^15 + 12 * q^16 + 4 * q^18 + 12 * q^20 + 16 * q^21 - 16 * q^22 - 16 * q^23 + 12 * q^25 + 16 * q^26 - 32 * q^27 - 40 * q^28 - 16 * q^29 - 8 * q^30 - 24 * q^31 - 28 * q^32 - 16 * q^35 + 12 * q^36 - 24 * q^37 - 24 * q^38 - 8 * q^39 - 12 * q^40 - 8 * q^41 - 16 * q^43 - 8 * q^44 + 12 * q^45 - 40 * q^46 - 32 * q^47 + 24 * q^48 + 20 * q^49 - 4 * q^50 - 24 * q^52 - 8 * q^54 - 16 * q^55 + 24 * q^56 - 32 * q^57 + 16 * q^58 + 8 * q^59 - 16 * q^60 - 24 * q^61 - 8 * q^62 - 48 * q^63 + 36 * q^64 - 8 * q^65 + 40 * q^66 - 8 * q^67 + 48 * q^69 + 16 * q^70 + 16 * q^71 + 12 * q^72 - 16 * q^73 - 8 * q^75 + 16 * q^76 + 24 * q^77 + 24 * q^78 - 40 * q^79 + 12 * q^80 + 36 * q^81 + 16 * q^82 - 40 * q^83 + 32 * q^84 - 8 * q^86 - 32 * q^87 - 48 * q^88 + 8 * q^89 + 4 * q^90 - 72 * q^91 + 8 * q^92 + 24 * q^93 - 16 * q^94 - 8 * q^96 - 32 * q^97 - 60 * q^98 - 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more