LMFDB - Embedded newform 2240.2.b.h.1121.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1121,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("2240.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
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} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 362 x^{8} - 794 x^{7} + 1463 x^{6} - 2048 x^{5} + 2258 x^{4} + \cdots + 61 $ x^12 - 6x^11 + 35x^10 - 120x^9 + 362x^8 - 794x^7 + 1463x^6 - 2048x^5 + 2258x^4 - 1822x^3 + 1035x^2 - 364*x + 61
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.2
Root			$0.500000 - 2.14588i$ of defining polynomial
Character	$\chi$	$=$	2240.1121
Dual form			2240.2.b.h.1121.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.13466i q^{3} +1.00000i q^{5} +1.00000 q^{7} -6.82607 q^{9} +O(q^{10})$	$q-3.13466i q^{3} +1.00000i q^{5} +1.00000 q^{7} -6.82607 q^{9} +4.07158i q^{11} -2.86968i q^{13} +3.13466 q^{15} +7.69437 q^{17} +2.79216i q^{19} -3.13466i q^{21} +7.79136 q^{23} -1.00000 q^{25} +11.9934i q^{27} -9.95657i q^{29} -0.232455 q^{31} +12.7630 q^{33} +1.00000i q^{35} -4.98614i q^{37} -8.99545 q^{39} -3.91752 q^{41} -9.54585i q^{43} -6.82607i q^{45} -3.90855 q^{47} +1.00000 q^{49} -24.1192i q^{51} -3.85958i q^{53} -4.07158 q^{55} +8.75248 q^{57} +7.86790i q^{59} -10.9651i q^{61} -6.82607 q^{63} +2.86968 q^{65} +11.5177i q^{67} -24.4232i q^{69} +13.0745 q^{71} +5.99406 q^{73} +3.13466i q^{75} +4.07158i q^{77} +0.872456 q^{79} +17.1171 q^{81} -8.33875i q^{83} +7.69437i q^{85} -31.2104 q^{87} -5.79136 q^{89} -2.86968i q^{91} +0.728666i q^{93} -2.79216 q^{95} -10.6226 q^{97} -27.7929i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7} - 20 q^{9}+O(q^{10}) $ 12 * q + 12 * q^7 - 20 * q^9	$ 12 q + 12 q^{7} - 20 q^{9} + 16 q^{17} + 8 q^{23} - 12 q^{25} - 8 q^{31} + 72 q^{33} - 32 q^{39} - 32 q^{47} + 12 q^{49} + 8 q^{55} + 8 q^{57} - 20 q^{63} + 8 q^{65} + 48 q^{71} + 8 q^{73} - 16 q^{79} + 92 q^{81} - 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) $ 12 * q + 12 * q^7 - 20 * q^9 + 16 * q^17 + 8 * q^23 - 12 * q^25 - 8 * q^31 + 72 * q^33 - 32 * q^39 - 32 * q^47 + 12 * q^49 + 8 * q^55 + 8 * q^57 - 20 * q^63 + 8 * q^65 + 48 * q^71 + 8 * q^73 - 16 * q^79 + 92 * q^81 - 32 * q^87 + 16 * q^89 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$1$	$-1$	$1$

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	0
$2$	0	0
$3$	− 3.13466i	− 1.80979i	−0.425630	−	0.904897i	$-0.639947\pi$
$3$	− 3.13466i	− 1.80979i	0.425630	−	0.904897i	$-0.360053\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−6.82607	−2.27536
$10$	0	0
$11$	4.07158i	1.22763i	0.789451	+	0.613813i	$0.210365\pi$
$11$	4.07158i	1.22763i	−0.789451	+	0.613813i	$0.789635\pi$
$12$	0	0
$13$	− 2.86968i	− 0.795905i	−0.917406	−	0.397952i	$-0.869721\pi$
$13$	− 2.86968i	− 0.795905i	0.917406	−	0.397952i	$-0.130279\pi$
$14$	0	0
$15$	3.13466	0.809365
$16$	0	0
$17$	7.69437	1.86616	0.933079	−	0.359672i	$-0.117111\pi$
$17$	7.69437	1.86616	0.933079	+	0.359672i	$0.117111\pi$
$18$	0	0
$19$	2.79216i	0.640566i	0.947322	+	0.320283i	$0.103778\pi$
$19$	2.79216i	0.640566i	−0.947322	+	0.320283i	$0.896222\pi$
$20$	0	0
$21$	− 3.13466i	− 0.684038i
$22$	0	0
$23$	7.79136	1.62461	0.812305	−	0.583233i	$-0.198212\pi$
$23$	7.79136	1.62461	0.812305	+	0.583233i	$0.198212\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	11.9934i	2.30814i
$28$	0	0
$29$	− 9.95657i	− 1.84889i	−0.381318	−	0.924444i	$-0.624530\pi$
$29$	− 9.95657i	− 1.84889i	0.381318	−	0.924444i	$-0.375470\pi$
$30$	0	0
$31$	−0.232455	−0.0417501	−0.0208751	−	0.999782i	$-0.506645\pi$
$31$	−0.232455	−0.0417501	−0.0208751	+	0.999782i	$0.506645\pi$
$32$	0	0
$33$	12.7630	2.22175
$34$	0	0
$35$	1.00000i	0.169031i
$36$	0	0
$37$	− 4.98614i	− 0.819717i	−0.912149	−	0.409859i	$-0.865578\pi$
$37$	− 4.98614i	− 0.819717i	0.912149	−	0.409859i	$-0.134422\pi$
$38$	0	0
$39$	−8.99545	−1.44042
$40$	0	0
$41$	−3.91752	−0.611814	−0.305907	−	0.952061i	$-0.598960\pi$
$41$	−3.91752	−0.611814	−0.305907	+	0.952061i	$0.598960\pi$
$42$	0	0
$43$	− 9.54585i	− 1.45573i	−0.685721	−	0.727865i	$-0.740513\pi$
$43$	− 9.54585i	− 1.45573i	0.685721	−	0.727865i	$-0.259487\pi$
$44$	0	0
$45$	− 6.82607i	− 1.01757i
$46$	0	0
$47$	−3.90855	−0.570121	−0.285061	−	0.958509i	$-0.592014\pi$
$47$	−3.90855	−0.570121	−0.285061	+	0.958509i	$0.592014\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 24.1192i	− 3.37736i
$52$	0	0
$53$	− 3.85958i	− 0.530154i	−0.964227	−	0.265077i	$-0.914603\pi$
$53$	− 3.85958i	− 0.530154i	0.964227	−	0.265077i	$-0.0853973\pi$
$54$	0	0
$55$	−4.07158	−0.549011
$56$	0	0
$57$	8.75248	1.15929
$58$	0	0
$59$	7.86790i	1.02431i	0.858892	+	0.512157i	$0.171153\pi$
$59$	7.86790i	1.02431i	−0.858892	+	0.512157i	$0.828847\pi$
$60$	0	0
$61$	− 10.9651i	− 1.40393i	−0.712210	−	0.701966i	$-0.752306\pi$
$61$	− 10.9651i	− 1.40393i	0.712210	−	0.701966i	$-0.247694\pi$
$62$	0	0
$63$	−6.82607	−0.860004
$64$	0	0
$65$	2.86968	0.355939
$66$	0	0
$67$	11.5177i	1.40711i	0.710643	+	0.703553i	$0.248404\pi$
$67$	11.5177i	1.40711i	−0.710643	+	0.703553i	$0.751596\pi$
$68$	0	0
$69$	− 24.4232i	− 2.94021i
$70$	0	0
$71$	13.0745	1.55166	0.775830	−	0.630942i	$-0.217331\pi$
$71$	13.0745	1.55166	0.775830	+	0.630942i	$0.217331\pi$
$72$	0	0
$73$	5.99406	0.701552	0.350776	−	0.936459i	$-0.385918\pi$
$73$	5.99406	0.701552	0.350776	+	0.936459i	$0.385918\pi$
$74$	0	0
$75$	3.13466i	0.361959i
$76$	0	0
$77$	4.07158i	0.463999i
$78$	0	0
$79$	0.872456	0.0981589	0.0490795	−	0.998795i	$-0.484371\pi$
$79$	0.872456	0.0981589	0.0490795	+	0.998795i	$0.484371\pi$
$80$	0	0
$81$	17.1171	1.90190
$82$	0	0
$83$	− 8.33875i	− 0.915296i	−0.889133	−	0.457648i	$-0.848692\pi$
$83$	− 8.33875i	− 0.915296i	0.889133	−	0.457648i	$-0.151308\pi$
$84$	0	0
$85$	7.69437i	0.834571i
$86$	0	0
$87$	−31.2104	−3.34611
$88$	0	0
$89$	−5.79136	−0.613883	−0.306941	−	0.951728i	$-0.599306\pi$
$89$	−5.79136	−0.613883	−0.306941	+	0.951728i	$0.599306\pi$
$90$	0	0
$91$	− 2.86968i	− 0.300824i
$92$	0	0
$93$	0.728666i	0.0755592i
$94$	0	0
$95$	−2.79216	−0.286470
$96$	0	0
$97$	−10.6226	−1.07856	−0.539279	−	0.842127i	$-0.681303\pi$
$97$	−10.6226	−1.07856	−0.539279	+	0.842127i	$0.681303\pi$
$98$	0	0
$99$	− 27.7929i	− 2.79329i
$100$	0	0
$101$	17.2257i	1.71402i	0.515298	+	0.857011i	$0.327681\pi$
$101$	17.2257i	1.71402i	−0.515298	+	0.857011i	$0.672319\pi$
$102$	0	0
$103$	3.90855	0.385121	0.192561	−	0.981285i	$-0.438321\pi$
$103$	3.90855	0.385121	0.192561	+	0.981285i	$0.438321\pi$
$104$	0	0
$105$	3.13466	0.305911
$106$	0	0
$107$	− 3.60452i	− 0.348462i	−0.984705	−	0.174231i	$-0.944256\pi$
$107$	− 3.60452i	− 0.348462i	0.984705	−	0.174231i	$-0.0557440\pi$
$108$	0	0
$109$	8.80667i	0.843526i	0.906706	+	0.421763i	$0.138589\pi$
$109$	8.80667i	0.843526i	−0.906706	+	0.421763i	$0.861411\pi$
$110$	0	0
$111$	−15.6299	−1.48352
$112$	0	0
$113$	5.43908	0.511665	0.255833	−	0.966721i	$-0.417650\pi$
$113$	5.43908	0.511665	0.255833	+	0.966721i	$0.417650\pi$
$114$	0	0
$115$	7.79136i	0.726548i
$116$	0	0
$117$	19.5886i	1.81097i
$118$	0	0
$119$	7.69437	0.705341
$120$	0	0
$121$	−5.57773	−0.507066
$122$	0	0
$123$	12.2801i	1.10726i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	5.38283	0.477649	0.238825	−	0.971063i	$-0.423238\pi$
$127$	5.38283	0.477649	0.238825	+	0.971063i	$0.423238\pi$
$128$	0	0
$129$	−29.9230	−2.63457
$130$	0	0
$131$	− 10.8008i	− 0.943673i	−0.881686	−	0.471836i	$-0.843591\pi$
$131$	− 10.8008i	− 0.943673i	0.881686	−	0.471836i	$-0.156409\pi$
$132$	0	0
$133$	2.79216i	0.242111i
$134$	0	0
$135$	−11.9934	−1.03223
$136$	0	0
$137$	10.9651	0.936808	0.468404	−	0.883514i	$-0.344829\pi$
$137$	10.9651	0.936808	0.468404	+	0.883514i	$0.344829\pi$
$138$	0	0
$139$	2.40141i	0.203685i	0.994801	+	0.101843i	$0.0324738\pi$
$139$	2.40141i	0.203685i	−0.994801	+	0.101843i	$0.967526\pi$
$140$	0	0
$141$	12.2520i	1.03180i
$142$	0	0
$143$	11.6841	0.977073
$144$	0	0
$145$	9.95657	0.826848
$146$	0	0
$147$	− 3.13466i	− 0.258542i
$148$	0	0
$149$	− 6.46329i	− 0.529494i	−0.964318	−	0.264747i	$-0.914712\pi$
$149$	− 6.46329i	− 0.529494i	0.964318	−	0.264747i	$-0.0852884\pi$
$150$	0	0
$151$	6.45678	0.525446	0.262723	−	0.964871i	$-0.415380\pi$
$151$	6.45678	0.525446	0.262723	+	0.964871i	$0.415380\pi$
$152$	0	0
$153$	−52.5223	−4.24618
$154$	0	0
$155$	− 0.232455i	− 0.0186712i
$156$	0	0
$157$	5.25667i	0.419528i	0.977752	+	0.209764i	$0.0672696\pi$
$157$	5.25667i	0.419528i	−0.977752	+	0.209764i	$0.932730\pi$
$158$	0	0
$159$	−12.0985	−0.959470
$160$	0	0
$161$	7.79136	0.614045
$162$	0	0
$163$	1.94327i	0.152209i	0.997100	+	0.0761044i	$0.0242482\pi$
$163$	1.94327i	0.152209i	−0.997100	+	0.0761044i	$0.975752\pi$
$164$	0	0
$165$	12.7630i	0.993598i
$166$	0	0
$167$	0.419913	0.0324938	0.0162469	−	0.999868i	$-0.494828\pi$
$167$	0.419913	0.0324938	0.0162469	+	0.999868i	$0.494828\pi$
$168$	0	0
$169$	4.76496	0.366536
$170$	0	0
$171$	− 19.0595i	− 1.45752i
$172$	0	0
$173$	− 15.3128i	− 1.16421i	−0.813114	−	0.582104i	$-0.802230\pi$
$173$	− 15.3128i	− 1.16421i	0.813114	−	0.582104i	$-0.197770\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	24.6632	1.85380
$178$	0	0
$179$	− 6.19122i	− 0.462753i	−0.972864	−	0.231377i	$-0.925677\pi$
$179$	− 6.19122i	− 0.462753i	0.972864	−	0.231377i	$-0.0743230\pi$
$180$	0	0
$181$	− 10.3446i	− 0.768912i	−0.923143	−	0.384456i	$-0.874389\pi$
$181$	− 10.3446i	− 0.768912i	0.923143	−	0.384456i	$-0.125611\pi$
$182$	0	0
$183$	−34.3717	−2.54083
$184$	0	0
$185$	4.98614	0.366589
$186$	0	0
$187$	31.3282i	2.29094i
$188$	0	0
$189$	11.9934i	0.872393i
$190$	0	0
$191$	−17.3266	−1.25371	−0.626856	−	0.779135i	$-0.715659\pi$
$191$	−17.3266	−1.25371	−0.626856	+	0.779135i	$0.715659\pi$
$192$	0	0
$193$	−25.2039	−1.81422	−0.907108	−	0.420897i	$-0.861715\pi$
$193$	−25.2039	−1.81422	−0.907108	+	0.420897i	$0.861715\pi$
$194$	0	0
$195$	− 8.99545i	− 0.644177i
$196$	0	0
$197$	− 23.4680i	− 1.67203i	−0.548707	−	0.836014i	$-0.684880\pi$
$197$	− 23.4680i	− 1.67203i	0.548707	−	0.836014i	$-0.315120\pi$
$198$	0	0
$199$	8.26099	0.585606	0.292803	−	0.956173i	$-0.405412\pi$
$199$	8.26099	0.585606	0.292803	+	0.956173i	$0.405412\pi$
$200$	0	0
$201$	36.1039	2.54657
$202$	0	0
$203$	− 9.95657i	− 0.698814i
$204$	0	0
$205$	− 3.91752i	− 0.273611i
$206$	0	0
$207$	−53.1844	−3.69657
$208$	0	0
$209$	−11.3685	−0.786376
$210$	0	0
$211$	− 4.20050i	− 0.289174i	−0.989492	−	0.144587i	$-0.953815\pi$
$211$	− 4.20050i	− 0.289174i	0.989492	−	0.144587i	$-0.0461854\pi$
$212$	0	0
$213$	− 40.9842i	− 2.80819i
$214$	0	0
$215$	9.54585	0.651022
$216$	0	0
$217$	−0.232455	−0.0157801
$218$	0	0
$219$	− 18.7893i	− 1.26967i
$220$	0	0
$221$	− 22.0803i	− 1.48528i
$222$	0	0
$223$	−2.89315	−0.193739	−0.0968697	−	0.995297i	$-0.530883\pi$
$223$	−2.89315	−0.193739	−0.0968697	+	0.995297i	$0.530883\pi$
$224$	0	0
$225$	6.82607	0.455072
$226$	0	0
$227$	− 3.26914i	− 0.216981i	−0.994098	−	0.108490i	$-0.965398\pi$
$227$	− 3.26914i	− 0.216981i	0.994098	−	0.108490i	$-0.0346017\pi$
$228$	0	0
$229$	15.7195i	1.03877i	0.854540	+	0.519386i	$0.173839\pi$
$229$	15.7195i	1.03877i	−0.854540	+	0.519386i	$0.826161\pi$
$230$	0	0
$231$	12.7630	0.839743
$232$	0	0
$233$	16.1154	1.05576	0.527879	−	0.849320i	$-0.322988\pi$
$233$	16.1154	1.05576	0.527879	+	0.849320i	$0.322988\pi$
$234$	0	0
$235$	− 3.90855i	− 0.254966i
$236$	0	0
$237$	− 2.73485i	− 0.177648i
$238$	0	0
$239$	17.1445	1.10899	0.554494	−	0.832188i	$-0.312912\pi$
$239$	17.1445	1.10899	0.554494	+	0.832188i	$0.312912\pi$
$240$	0	0
$241$	−4.90488	−0.315951	−0.157975	−	0.987443i	$-0.550497\pi$
$241$	−4.90488	−0.315951	−0.157975	+	0.987443i	$0.550497\pi$
$242$	0	0
$243$	− 17.6758i	− 1.13390i
$244$	0	0
$245$	1.00000i	0.0638877i
$246$	0	0
$247$	8.01261	0.509830
$248$	0	0
$249$	−26.1391	−1.65650
$250$	0	0
$251$	17.4597i	1.10205i	0.834489	+	0.551024i	$0.185763\pi$
$251$	17.4597i	1.10205i	−0.834489	+	0.551024i	$0.814237\pi$
$252$	0	0
$253$	31.7231i	1.99441i
$254$	0	0
$255$	24.1192	1.51040
$256$	0	0
$257$	−0.622321	−0.0388193	−0.0194096	−	0.999812i	$-0.506179\pi$
$257$	−0.622321	−0.0388193	−0.0194096	+	0.999812i	$0.506179\pi$
$258$	0	0
$259$	− 4.98614i	− 0.309824i
$260$	0	0
$261$	67.9643i	4.20688i
$262$	0	0
$263$	−32.3923	−1.99739	−0.998696	−	0.0510429i	$-0.983745\pi$
$263$	−32.3923	−1.99739	−0.998696	+	0.0510429i	$0.983745\pi$
$264$	0	0
$265$	3.85958	0.237092
$266$	0	0
$267$	18.1539i	1.11100i
$268$	0	0
$269$	− 12.8298i	− 0.782244i	−0.920339	−	0.391122i	$-0.872087\pi$
$269$	− 12.8298i	− 0.782244i	0.920339	−	0.391122i	$-0.127913\pi$
$270$	0	0
$271$	−8.29751	−0.504038	−0.252019	−	0.967722i	$-0.581095\pi$
$271$	−8.29751	−0.504038	−0.252019	+	0.967722i	$0.581095\pi$
$272$	0	0
$273$	−8.99545	−0.544429
$274$	0	0
$275$	− 4.07158i	− 0.245525i
$276$	0	0
$277$	− 19.3642i	− 1.16348i	−0.813374	−	0.581741i	$-0.802372\pi$
$277$	− 19.3642i	− 1.16348i	0.813374	−	0.581741i	$-0.197628\pi$
$278$	0	0
$279$	1.58675	0.0949965
$280$	0	0
$281$	−2.50829	−0.149632	−0.0748161	−	0.997197i	$-0.523837\pi$
$281$	−2.50829	−0.149632	−0.0748161	+	0.997197i	$0.523837\pi$
$282$	0	0
$283$	9.05932i	0.538521i	0.963067	+	0.269260i	$0.0867792\pi$
$283$	9.05932i	0.538521i	−0.963067	+	0.269260i	$0.913221\pi$
$284$	0	0
$285$	8.75248i	0.518452i
$286$	0	0
$287$	−3.91752	−0.231244
$288$	0	0
$289$	42.2033	2.48255
$290$	0	0
$291$	33.2981i	1.95197i
$292$	0	0
$293$	− 19.0128i	− 1.11074i	−0.831603	−	0.555371i	$-0.812576\pi$
$293$	− 19.0128i	− 1.11074i	0.831603	−	0.555371i	$-0.187424\pi$
$294$	0	0
$295$	−7.86790	−0.458087
$296$	0	0
$297$	−48.8321	−2.83353
$298$	0	0
$299$	− 22.3587i	− 1.29303i
$300$	0	0
$301$	− 9.54585i	− 0.550214i
$302$	0	0
$303$	53.9967	3.10203
$304$	0	0
$305$	10.9651	0.627858
$306$	0	0
$307$	27.9149i	1.59319i	0.604516	+	0.796593i	$0.293367\pi$
$307$	27.9149i	1.59319i	−0.604516	+	0.796593i	$0.706633\pi$
$308$	0	0
$309$	− 12.2520i	− 0.696991i
$310$	0	0
$311$	−0.995357	−0.0564415	−0.0282207	−	0.999602i	$-0.508984\pi$
$311$	−0.995357	−0.0564415	−0.0282207	+	0.999602i	$0.508984\pi$
$312$	0	0
$313$	−4.69245	−0.265233	−0.132616	−	0.991167i	$-0.542338\pi$
$313$	−4.69245	−0.265233	−0.132616	+	0.991167i	$0.542338\pi$
$314$	0	0
$315$	− 6.82607i	− 0.384606i
$316$	0	0
$317$	1.32928i	0.0746595i	0.999303	+	0.0373298i	$0.0118852\pi$
$317$	1.32928i	0.0746595i	−0.999303	+	0.0373298i	$0.988115\pi$
$318$	0	0
$319$	40.5389	2.26974
$320$	0	0
$321$	−11.2989	−0.630646
$322$	0	0
$323$	21.4839i	1.19540i
$324$	0	0
$325$	2.86968i	0.159181i
$326$	0	0
$327$	27.6059	1.52661
$328$	0	0
$329$	−3.90855	−0.215486
$330$	0	0
$331$	− 0.850445i	− 0.0467447i	−0.999727	−	0.0233723i	$-0.992560\pi$
$331$	− 0.850445i	− 0.0467447i	0.999727	−	0.0233723i	$-0.00744033\pi$
$332$	0	0
$333$	34.0358i	1.86515i
$334$	0	0
$335$	−11.5177	−0.629277
$336$	0	0
$337$	−1.98257	−0.107997	−0.0539987	−	0.998541i	$-0.517197\pi$
$337$	−1.98257	−0.107997	−0.0539987	+	0.998541i	$0.517197\pi$
$338$	0	0
$339$	− 17.0496i	− 0.926009i
$340$	0	0
$341$	− 0.946457i	− 0.0512535i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	24.4232	1.31490
$346$	0	0
$347$	15.8282i	0.849703i	0.905263	+	0.424851i	$0.139674\pi$
$347$	15.8282i	0.849703i	−0.905263	+	0.424851i	$0.860326\pi$
$348$	0	0
$349$	− 5.81595i	− 0.311321i	−0.987811	−	0.155660i	$-0.950249\pi$
$349$	− 5.81595i	− 0.311321i	0.987811	−	0.155660i	$-0.0497505\pi$
$350$	0	0
$351$	34.4172	1.83706
$352$	0	0
$353$	28.4270	1.51302	0.756508	−	0.653985i	$-0.226904\pi$
$353$	28.4270	1.51302	0.756508	+	0.653985i	$0.226904\pi$
$354$	0	0
$355$	13.0745i	0.693924i
$356$	0	0
$357$	− 24.1192i	− 1.27652i
$358$	0	0
$359$	−32.3907	−1.70952	−0.854758	−	0.519027i	$-0.826294\pi$
$359$	−32.3907	−1.70952	−0.854758	+	0.519027i	$0.826294\pi$
$360$	0	0
$361$	11.2038	0.589675
$362$	0	0
$363$	17.4843i	0.917686i
$364$	0	0
$365$	5.99406i	0.313744i
$366$	0	0
$367$	7.14858	0.373153	0.186576	−	0.982440i	$-0.440261\pi$
$367$	7.14858	0.373153	0.186576	+	0.982440i	$0.440261\pi$
$368$	0	0
$369$	26.7413	1.39209
$370$	0	0
$371$	− 3.85958i	− 0.200379i
$372$	0	0
$373$	34.7310i	1.79830i	0.437640	+	0.899150i	$0.355814\pi$
$373$	34.7310i	1.79830i	−0.437640	+	0.899150i	$0.644186\pi$
$374$	0	0
$375$	−3.13466	−0.161873
$376$	0	0
$377$	−28.5721	−1.47154
$378$	0	0
$379$	− 9.22799i	− 0.474010i	−0.971508	−	0.237005i	$-0.923834\pi$
$379$	− 9.22799i	− 0.474010i	0.971508	−	0.237005i	$-0.0761658\pi$
$380$	0	0
$381$	− 16.8733i	− 0.864447i
$382$	0	0
$383$	−6.71562	−0.343152	−0.171576	−	0.985171i	$-0.554886\pi$
$383$	−6.71562	−0.343152	−0.171576	+	0.985171i	$0.554886\pi$
$384$	0	0
$385$	−4.07158	−0.207507
$386$	0	0
$387$	65.1607i	3.31230i
$388$	0	0
$389$	10.5635i	0.535588i	0.963476	+	0.267794i	$0.0862947\pi$
$389$	10.5635i	0.535588i	−0.963476	+	0.267794i	$0.913705\pi$
$390$	0	0
$391$	59.9496	3.03178
$392$	0	0
$393$	−33.8569	−1.70785
$394$	0	0
$395$	0.872456i	0.0438980i
$396$	0	0
$397$	16.3585i	0.821008i	0.911859	+	0.410504i	$0.134647\pi$
$397$	16.3585i	0.821008i	−0.911859	+	0.410504i	$0.865353\pi$
$398$	0	0
$399$	8.75248	0.438172
$400$	0	0
$401$	18.2429	0.911008	0.455504	−	0.890234i	$-0.349459\pi$
$401$	18.2429	0.911008	0.455504	+	0.890234i	$0.349459\pi$
$402$	0	0
$403$	0.667070i	0.0332291i
$404$	0	0
$405$	17.1171i	0.850553i
$406$	0	0
$407$	20.3015	1.00631
$408$	0	0
$409$	−33.7302	−1.66785	−0.833927	−	0.551875i	$-0.813912\pi$
$409$	−33.7302	−1.66785	−0.833927	+	0.551875i	$0.813912\pi$
$410$	0	0
$411$	− 34.3717i	− 1.69543i
$412$	0	0
$413$	7.86790i	0.387154i
$414$	0	0
$415$	8.33875	0.409333
$416$	0	0
$417$	7.52760	0.368628
$418$	0	0
$419$	27.7029i	1.35338i	0.736269	+	0.676689i	$0.236586\pi$
$419$	27.7029i	1.35338i	−0.736269	+	0.676689i	$0.763414\pi$
$420$	0	0
$421$	− 26.6253i	− 1.29764i	−0.760943	−	0.648819i	$-0.775263\pi$
$421$	− 26.6253i	− 1.29764i	0.760943	−	0.648819i	$-0.224737\pi$
$422$	0	0
$423$	26.6801	1.29723
$424$	0	0
$425$	−7.69437	−0.373232
$426$	0	0
$427$	− 10.9651i	− 0.530637i
$428$	0	0
$429$	− 36.6256i	− 1.76830i
$430$	0	0
$431$	6.24973	0.301039	0.150519	−	0.988607i	$-0.451905\pi$
$431$	6.24973	0.301039	0.150519	+	0.988607i	$0.451905\pi$
$432$	0	0
$433$	29.8155	1.43284	0.716420	−	0.697670i	$-0.245780\pi$
$433$	29.8155	1.43284	0.716420	+	0.697670i	$0.245780\pi$
$434$	0	0
$435$	− 31.2104i	− 1.49643i
$436$	0	0
$437$	21.7547i	1.04067i
$438$	0	0
$439$	21.0945	1.00678	0.503392	−	0.864058i	$-0.332085\pi$
$439$	21.0945	1.00678	0.503392	+	0.864058i	$0.332085\pi$
$440$	0	0
$441$	−6.82607	−0.325051
$442$	0	0
$443$	36.5359i	1.73587i	0.496675	+	0.867937i	$0.334554\pi$
$443$	36.5359i	1.73587i	−0.496675	+	0.867937i	$0.665446\pi$
$444$	0	0
$445$	− 5.79136i	− 0.274537i
$446$	0	0
$447$	−20.2602	−0.958275
$448$	0	0
$449$	−22.9644	−1.08376	−0.541879	−	0.840456i	$-0.682287\pi$
$449$	−22.9644	−1.08376	−0.541879	+	0.840456i	$0.682287\pi$
$450$	0	0
$451$	− 15.9505i	− 0.751079i
$452$	0	0
$453$	− 20.2398i	− 0.950949i
$454$	0	0
$455$	2.86968	0.134532
$456$	0	0
$457$	−1.59652	−0.0746822	−0.0373411	−	0.999303i	$-0.511889\pi$
$457$	−1.59652	−0.0746822	−0.0373411	+	0.999303i	$0.511889\pi$
$458$	0	0
$459$	92.2818i	4.30735i
$460$	0	0
$461$	9.34977i	0.435462i	0.976009	+	0.217731i	$0.0698656\pi$
$461$	9.34977i	0.435462i	−0.976009	+	0.217731i	$0.930134\pi$
$462$	0	0
$463$	42.1170	1.95734	0.978670	−	0.205436i	$-0.0658613\pi$
$463$	42.1170	1.95734	0.978670	+	0.205436i	$0.0658613\pi$
$464$	0	0
$465$	−0.728666	−0.0337911
$466$	0	0
$467$	− 34.5548i	− 1.59901i	−0.600661	−	0.799503i	$-0.705096\pi$
$467$	− 34.5548i	− 1.59901i	0.600661	−	0.799503i	$-0.294904\pi$
$468$	0	0
$469$	11.5177i	0.531836i
$470$	0	0
$471$	16.4779	0.759260
$472$	0	0
$473$	38.8667	1.78709
$474$	0	0
$475$	− 2.79216i	− 0.128113i
$476$	0	0
$477$	26.3458i	1.20629i
$478$	0	0
$479$	15.5529	0.710630	0.355315	−	0.934747i	$-0.384374\pi$
$479$	15.5529	0.710630	0.355315	+	0.934747i	$0.384374\pi$
$480$	0	0
$481$	−14.3086	−0.652417
$482$	0	0
$483$	− 24.4232i	− 1.11130i
$484$	0	0
$485$	− 10.6226i	− 0.482346i
$486$	0	0
$487$	16.0868	0.728961	0.364480	−	0.931211i	$-0.381247\pi$
$487$	16.0868	0.728961	0.364480	+	0.931211i	$0.381247\pi$
$488$	0	0
$489$	6.09149	0.275467
$490$	0	0
$491$	− 19.4429i	− 0.877445i	−0.898623	−	0.438722i	$-0.855431\pi$
$491$	− 19.4429i	− 0.877445i	0.898623	−	0.438722i	$-0.144569\pi$
$492$	0	0
$493$	− 76.6095i	− 3.45032i
$494$	0	0
$495$	27.7929	1.24920
$496$	0	0
$497$	13.0745	0.586473
$498$	0	0
$499$	6.85663i	0.306945i	0.988153	+	0.153472i	$0.0490456\pi$
$499$	6.85663i	0.306945i	−0.988153	+	0.153472i	$0.950954\pi$
$500$	0	0
$501$	− 1.31628i	− 0.0588072i
$502$	0	0
$503$	−20.7137	−0.923580	−0.461790	−	0.886989i	$-0.652793\pi$
$503$	−20.7137	−0.923580	−0.461790	+	0.886989i	$0.652793\pi$
$504$	0	0
$505$	−17.2257	−0.766534
$506$	0	0
$507$	− 14.9365i	− 0.663354i
$508$	0	0
$509$	− 17.5146i	− 0.776319i	−0.921592	−	0.388160i	$-0.873111\pi$
$509$	− 17.5146i	− 0.776319i	0.921592	−	0.388160i	$-0.126889\pi$
$510$	0	0
$511$	5.99406	0.265162
$512$	0	0
$513$	−33.4876	−1.47851
$514$	0	0
$515$	3.90855i	0.172231i
$516$	0	0
$517$	− 15.9140i	− 0.699896i
$518$	0	0
$519$	−48.0003	−2.10698
$520$	0	0
$521$	−18.9413	−0.829831	−0.414916	−	0.909860i	$-0.636189\pi$
$521$	−18.9413	−0.829831	−0.414916	+	0.909860i	$0.636189\pi$
$522$	0	0
$523$	8.81407i	0.385412i	0.981257	+	0.192706i	$0.0617264\pi$
$523$	8.81407i	0.385412i	−0.981257	+	0.192706i	$0.938274\pi$
$524$	0	0
$525$	3.13466i	0.136808i
$526$	0	0
$527$	−1.78859	−0.0779123
$528$	0	0
$529$	37.7052	1.63936
$530$	0	0
$531$	− 53.7069i	− 2.33068i
$532$	0	0
$533$	11.2420i	0.486945i
$534$	0	0
$535$	3.60452	0.155837
$536$	0	0
$537$	−19.4073	−0.837488
$538$	0	0
$539$	4.07158i	0.175375i
$540$	0	0
$541$	25.1371i	1.08073i	0.841431	+	0.540364i	$0.181713\pi$
$541$	25.1371i	1.08073i	−0.841431	+	0.540364i	$0.818287\pi$
$542$	0	0
$543$	−32.4269	−1.39157
$544$	0	0
$545$	−8.80667	−0.377236
$546$	0	0
$547$	18.5973i	0.795163i	0.917567	+	0.397581i	$0.130150\pi$
$547$	18.5973i	0.795163i	−0.917567	+	0.397581i	$0.869850\pi$
$548$	0	0
$549$	74.8483i	3.19445i
$550$	0	0
$551$	27.8004	1.18434
$552$	0	0
$553$	0.872456	0.0371006
$554$	0	0
$555$	− 15.6299i	− 0.663450i
$556$	0	0
$557$	24.1967i	1.02525i	0.858613	+	0.512624i	$0.171326\pi$
$557$	24.1967i	1.02525i	−0.858613	+	0.512624i	$0.828674\pi$
$558$	0	0
$559$	−27.3935	−1.15862
$560$	0	0
$561$	98.2031	4.14614
$562$	0	0
$563$	10.1127i	0.426198i	0.977031	+	0.213099i	$0.0683558\pi$
$563$	10.1127i	0.426198i	−0.977031	+	0.213099i	$0.931644\pi$
$564$	0	0
$565$	5.43908i	0.228824i
$566$	0	0
$567$	17.1171	0.718849
$568$	0	0
$569$	−4.63003	−0.194101	−0.0970505	−	0.995279i	$-0.530941\pi$
$569$	−4.63003	−0.194101	−0.0970505	+	0.995279i	$0.530941\pi$
$570$	0	0
$571$	− 25.7318i	− 1.07684i	−0.842676	−	0.538420i	$-0.819021\pi$
$571$	− 25.7318i	− 1.07684i	0.842676	−	0.538420i	$-0.180979\pi$
$572$	0	0
$573$	54.3131i	2.26896i
$574$	0	0
$575$	−7.79136	−0.324922
$576$	0	0
$577$	−34.8525	−1.45093	−0.725464	−	0.688260i	$-0.758375\pi$
$577$	−34.8525	−1.45093	−0.725464	+	0.688260i	$0.758375\pi$
$578$	0	0
$579$	79.0056i	3.28336i
$580$	0	0
$581$	− 8.33875i	− 0.345950i
$582$	0	0
$583$	15.7146	0.650831
$584$	0	0
$585$	−19.5886	−0.809889
$586$	0	0
$587$	− 26.5473i	− 1.09572i	−0.836569	−	0.547862i	$-0.815442\pi$
$587$	− 26.5473i	− 1.09572i	0.836569	−	0.547862i	$-0.184558\pi$
$588$	0	0
$589$	− 0.649052i	− 0.0267437i
$590$	0	0
$591$	−73.5643	−3.02603
$592$	0	0
$593$	4.15928	0.170801	0.0854005	−	0.996347i	$-0.472783\pi$
$593$	4.15928	0.170801	0.0854005	+	0.996347i	$0.472783\pi$
$594$	0	0
$595$	7.69437i	0.315438i
$596$	0	0
$597$	− 25.8954i	− 1.05983i
$598$	0	0
$599$	30.0612	1.22827	0.614134	−	0.789202i	$-0.289506\pi$
$599$	30.0612	1.22827	0.614134	+	0.789202i	$0.289506\pi$
$600$	0	0
$601$	7.60964	0.310404	0.155202	−	0.987883i	$-0.450397\pi$
$601$	7.60964	0.310404	0.155202	+	0.987883i	$0.450397\pi$
$602$	0	0
$603$	− 78.6204i	− 3.20167i
$604$	0	0
$605$	− 5.57773i	− 0.226767i
$606$	0	0
$607$	−32.7256	−1.32829	−0.664146	−	0.747603i	$-0.731205\pi$
$607$	−32.7256	−1.32829	−0.664146	+	0.747603i	$0.731205\pi$
$608$	0	0
$609$	−31.2104	−1.26471
$610$	0	0
$611$	11.2163i	0.453762i
$612$	0	0
$613$	30.2155i	1.22039i	0.792251	+	0.610196i	$0.208909\pi$
$613$	30.2155i	1.22039i	−0.792251	+	0.610196i	$0.791091\pi$
$614$	0	0
$615$	−12.2801	−0.495181
$616$	0	0
$617$	−16.5315	−0.665531	−0.332766	−	0.943010i	$-0.607982\pi$
$617$	−16.5315	−0.665531	−0.332766	+	0.943010i	$0.607982\pi$
$618$	0	0
$619$	30.9167i	1.24265i	0.783554	+	0.621324i	$0.213405\pi$
$619$	30.9167i	1.24265i	−0.783554	+	0.621324i	$0.786595\pi$
$620$	0	0
$621$	93.4451i	3.74982i
$622$	0	0
$623$	−5.79136	−0.232026
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	35.6364i	1.42318i
$628$	0	0
$629$	− 38.3652i	− 1.52972i
$630$	0	0
$631$	−4.70749	−0.187402	−0.0937012	−	0.995600i	$-0.529870\pi$
$631$	−4.70749	−0.187402	−0.0937012	+	0.995600i	$0.529870\pi$
$632$	0	0
$633$	−13.1671	−0.523347
$634$	0	0
$635$	5.38283i	0.213611i
$636$	0	0
$637$	− 2.86968i	− 0.113701i
$638$	0	0
$639$	−89.2477	−3.53058
$640$	0	0
$641$	30.3531	1.19888	0.599438	−	0.800421i	$-0.295391\pi$
$641$	30.3531	1.19888	0.599438	+	0.800421i	$0.295391\pi$
$642$	0	0
$643$	26.5539i	1.04718i	0.851970	+	0.523591i	$0.175408\pi$
$643$	26.5539i	1.04718i	−0.851970	+	0.523591i	$0.824592\pi$
$644$	0	0
$645$	− 29.9230i	− 1.17822i
$646$	0	0
$647$	−12.0929	−0.475421	−0.237710	−	0.971336i	$-0.576397\pi$
$647$	−12.0929	−0.475421	−0.237710	+	0.971336i	$0.576397\pi$
$648$	0	0
$649$	−32.0348	−1.25747
$650$	0	0
$651$	0.728666i	0.0285587i
$652$	0	0
$653$	31.5455i	1.23447i	0.786779	+	0.617235i	$0.211747\pi$
$653$	31.5455i	1.23447i	−0.786779	+	0.617235i	$0.788253\pi$
$654$	0	0
$655$	10.8008	0.422023
$656$	0	0
$657$	−40.9159	−1.59628
$658$	0	0
$659$	23.3195i	0.908398i	0.890900	+	0.454199i	$0.150074\pi$
$659$	23.3195i	0.908398i	−0.890900	+	0.454199i	$0.849926\pi$
$660$	0	0
$661$	48.3614i	1.88104i	0.339740	+	0.940519i	$0.389661\pi$
$661$	48.3614i	1.88104i	−0.339740	+	0.940519i	$0.610339\pi$
$662$	0	0
$663$	−69.2143	−2.68806
$664$	0	0
$665$	−2.79216	−0.108275
$666$	0	0
$667$	− 77.5752i	− 3.00372i
$668$	0	0
$669$	9.06902i	0.350629i
$670$	0	0
$671$	44.6451	1.72350
$672$	0	0
$673$	−25.4774	−0.982083	−0.491041	−	0.871136i	$-0.663384\pi$
$673$	−25.4774	−0.982083	−0.491041	+	0.871136i	$0.663384\pi$
$674$	0	0
$675$	− 11.9934i	− 0.461627i
$676$	0	0
$677$	− 19.1635i	− 0.736513i	−0.929724	−	0.368256i	$-0.879955\pi$
$677$	− 19.1635i	− 0.736513i	0.929724	−	0.368256i	$-0.120045\pi$
$678$	0	0
$679$	−10.6226	−0.407657
$680$	0	0
$681$	−10.2476	−0.392691
$682$	0	0
$683$	− 13.8961i	− 0.531718i	−0.964012	−	0.265859i	$-0.914344\pi$
$683$	− 13.8961i	− 0.531718i	0.964012	−	0.265859i	$-0.0856556\pi$
$684$	0	0
$685$	10.9651i	0.418953i
$686$	0	0
$687$	49.2752	1.87997
$688$	0	0
$689$	−11.0757	−0.421952
$690$	0	0
$691$	39.9837i	1.52105i	0.649308	+	0.760525i	$0.275059\pi$
$691$	39.9837i	1.52105i	−0.649308	+	0.760525i	$0.724941\pi$
$692$	0	0
$693$	− 27.7929i	− 1.05576i
$694$	0	0
$695$	−2.40141	−0.0910907
$696$	0	0
$697$	−30.1428	−1.14174
$698$	0	0
$699$	− 50.5164i	− 1.91070i
$700$	0	0
$701$	10.7313i	0.405317i	0.979249	+	0.202659i	$0.0649582\pi$
$701$	10.7313i	0.405317i	−0.979249	+	0.202659i	$0.935042\pi$
$702$	0	0
$703$	13.9221	0.525083
$704$	0	0
$705$	−12.2520	−0.461436
$706$	0	0
$707$	17.2257i	0.647840i
$708$	0	0
$709$	23.7245i	0.890991i	0.895284	+	0.445495i	$0.146972\pi$
$709$	23.7245i	0.890991i	−0.895284	+	0.445495i	$0.853028\pi$
$710$	0	0
$711$	−5.95545	−0.223347
$712$	0	0
$713$	−1.81114	−0.0678277
$714$	0	0
$715$	11.6841i	0.436961i
$716$	0	0
$717$	− 53.7422i	− 2.00704i
$718$	0	0
$719$	−43.5164	−1.62289	−0.811444	−	0.584430i	$-0.801318\pi$
$719$	−43.5164	−1.62289	−0.811444	+	0.584430i	$0.801318\pi$
$720$	0	0
$721$	3.90855	0.145562
$722$	0	0
$723$	15.3751i	0.571806i
$724$	0	0
$725$	9.95657i	0.369778i
$726$	0	0
$727$	0.207057	0.00767931	0.00383966	−	0.999993i	$-0.498778\pi$
$727$	0.207057	0.00767931	0.00383966	+	0.999993i	$0.498778\pi$
$728$	0	0
$729$	−4.05645	−0.150239
$730$	0	0
$731$	− 73.4493i	− 2.71662i
$732$	0	0
$733$	− 0.739133i	− 0.0273005i	−0.999907	−	0.0136502i	$-0.995655\pi$
$733$	− 0.739133i	− 0.0273005i	0.999907	−	0.0136502i	$-0.00434514\pi$
$734$	0	0
$735$	3.13466	0.115624
$736$	0	0
$737$	−46.8950	−1.72740
$738$	0	0
$739$	43.7486i	1.60932i	0.593738	+	0.804659i	$0.297652\pi$
$739$	43.7486i	1.60932i	−0.593738	+	0.804659i	$0.702348\pi$
$740$	0	0
$741$	− 25.1168i	− 0.922687i
$742$	0	0
$743$	−22.0721	−0.809748	−0.404874	−	0.914373i	$-0.632685\pi$
$743$	−22.0721	−0.809748	−0.404874	+	0.914373i	$0.632685\pi$
$744$	0	0
$745$	6.46329	0.236797
$746$	0	0
$747$	56.9209i	2.08263i
$748$	0	0
$749$	− 3.60452i	− 0.131706i
$750$	0	0
$751$	47.2686	1.72486	0.862429	−	0.506179i	$-0.168942\pi$
$751$	47.2686	1.72486	0.862429	+	0.506179i	$0.168942\pi$
$752$	0	0
$753$	54.7302	1.99448
$754$	0	0
$755$	6.45678i	0.234986i
$756$	0	0
$757$	0.312591i	0.0113613i	0.999984	+	0.00568066i	$0.00180822\pi$
$757$	0.312591i	0.0113613i	−0.999984	+	0.00568066i	$0.998192\pi$
$758$	0	0
$759$	99.4410	3.60948
$760$	0	0
$761$	5.74491	0.208253	0.104126	−	0.994564i	$-0.466795\pi$
$761$	5.74491	0.208253	0.104126	+	0.994564i	$0.466795\pi$
$762$	0	0
$763$	8.80667i	0.318823i
$764$	0	0
$765$	− 52.5223i	− 1.89895i
$766$	0	0
$767$	22.5783	0.815256
$768$	0	0
$769$	3.97307	0.143273	0.0716363	−	0.997431i	$-0.477178\pi$
$769$	3.97307	0.143273	0.0716363	+	0.997431i	$0.477178\pi$
$770$	0	0
$771$	1.95076i	0.0702550i
$772$	0	0
$773$	24.1261i	0.867756i	0.900972	+	0.433878i	$0.142855\pi$
$773$	24.1261i	0.867756i	−0.900972	+	0.433878i	$0.857145\pi$
$774$	0	0
$775$	0.232455	0.00835002
$776$	0	0
$777$	−15.6299	−0.560718
$778$	0	0
$779$	− 10.9384i	− 0.391907i
$780$	0	0
$781$	53.2339i	1.90486i
$782$	0	0
$783$	119.413	4.26749
$784$	0	0
$785$	−5.25667	−0.187619
$786$	0	0
$787$	− 14.9252i	− 0.532025i	−0.963970	−	0.266013i	$-0.914294\pi$
$787$	− 14.9252i	− 0.532025i	0.963970	−	0.266013i	$-0.0857063\pi$
$788$	0	0
$789$	101.539i	3.61487i
$790$	0	0
$791$	5.43908	0.193391
$792$	0	0
$793$	−31.4662	−1.11740
$794$	0	0
$795$	− 12.0985i	− 0.429088i
$796$	0	0
$797$	7.25881i	0.257120i	0.991702	+	0.128560i	$0.0410355\pi$
$797$	7.25881i	0.257120i	−0.991702	+	0.128560i	$0.958964\pi$
$798$	0	0
$799$	−30.0738	−1.06394
$800$	0	0
$801$	39.5322	1.39680
$802$	0	0
$803$	24.4053i	0.861244i
$804$	0	0
$805$	7.79136i	0.274609i
$806$	0	0
$807$	−40.2169	−1.41570
$808$	0	0
$809$	17.4223	0.612534	0.306267	−	0.951946i	$-0.400920\pi$
$809$	17.4223	0.612534	0.306267	+	0.951946i	$0.400920\pi$
$810$	0	0
$811$	− 8.54412i	− 0.300025i	−0.988684	−	0.150012i	$-0.952069\pi$
$811$	− 8.54412i	− 0.300025i	0.988684	−	0.150012i	$-0.0479313\pi$
$812$	0	0
$813$	26.0098i	0.912205i
$814$	0	0
$815$	−1.94327	−0.0680698
$816$	0	0
$817$	26.6536	0.932491
$818$	0	0
$819$	19.5886i	0.684482i
$820$	0	0
$821$	− 24.2641i	− 0.846821i	−0.905938	−	0.423411i	$-0.860833\pi$
$821$	− 24.2641i	− 0.846821i	0.905938	−	0.423411i	$-0.139167\pi$
$822$	0	0
$823$	−39.8067	−1.38758	−0.693788	−	0.720180i	$-0.744059\pi$
$823$	−39.8067	−1.38758	−0.693788	+	0.720180i	$0.744059\pi$
$824$	0	0
$825$	−12.7630	−0.444350
$826$	0	0
$827$	6.33123i	0.220158i	0.993923	+	0.110079i	$0.0351105\pi$
$827$	6.33123i	0.220158i	−0.993923	+	0.110079i	$0.964890\pi$
$828$	0	0
$829$	28.3161i	0.983460i	0.870748	+	0.491730i	$0.163635\pi$
$829$	28.3161i	0.983460i	−0.870748	+	0.491730i	$0.836365\pi$
$830$	0	0
$831$	−60.7001	−2.10566
$832$	0	0
$833$	7.69437	0.266594
$834$	0	0
$835$	0.419913i	0.0145317i
$836$	0	0
$837$	− 2.78793i	− 0.0963649i
$838$	0	0
$839$	−36.0742	−1.24542	−0.622710	−	0.782453i	$-0.713968\pi$
$839$	−36.0742	−1.24542	−0.622710	+	0.782453i	$0.713968\pi$
$840$	0	0
$841$	−70.1332	−2.41839
$842$	0	0
$843$	7.86264i	0.270804i
$844$	0	0
$845$	4.76496i	0.163920i
$846$	0	0
$847$	−5.57773	−0.191653
$848$	0	0
$849$	28.3979	0.974612
$850$	0	0
$851$	− 38.8488i	− 1.33172i
$852$	0	0
$853$	16.0460i	0.549405i	0.961529	+	0.274702i	$0.0885793\pi$
$853$	16.0460i	0.549405i	−0.961529	+	0.274702i	$0.911421\pi$
$854$	0	0
$855$	19.0595	0.651822
$856$	0	0
$857$	46.0559	1.57324	0.786620	−	0.617437i	$-0.211829\pi$
$857$	46.0559	1.57324	0.786620	+	0.617437i	$0.211829\pi$
$858$	0	0
$859$	− 9.62040i	− 0.328244i	−0.986440	−	0.164122i	$-0.947521\pi$
$859$	− 9.62040i	− 0.328244i	0.986440	−	0.164122i	$-0.0524790\pi$
$860$	0	0
$861$	12.2801i	0.418504i
$862$	0	0
$863$	41.1023	1.39914	0.699570	−	0.714564i	$-0.253375\pi$
$863$	41.1023	1.39914	0.699570	+	0.714564i	$0.253375\pi$
$864$	0	0
$865$	15.3128	0.520650
$866$	0	0
$867$	− 132.293i	− 4.49290i
$868$	0	0
$869$	3.55227i	0.120502i
$870$	0	0
$871$	33.0519	1.11992
$872$	0	0
$873$	72.5104	2.45411
$874$	0	0
$875$	− 1.00000i	− 0.0338062i
$876$	0	0
$877$	− 5.23672i	− 0.176831i	−0.996084	−	0.0884157i	$-0.971820\pi$
$877$	− 5.23672i	− 0.176831i	0.996084	−	0.0884157i	$-0.0281804\pi$
$878$	0	0
$879$	−59.5987	−2.01021
$880$	0	0
$881$	−43.0181	−1.44932	−0.724659	−	0.689108i	$-0.758003\pi$
$881$	−43.0181	−1.44932	−0.724659	+	0.689108i	$0.758003\pi$
$882$	0	0
$883$	32.8329i	1.10492i	0.833541	+	0.552458i	$0.186310\pi$
$883$	32.8329i	1.10492i	−0.833541	+	0.552458i	$0.813690\pi$
$884$	0	0
$885$	24.6632i	0.829044i
$886$	0	0
$887$	−20.8267	−0.699292	−0.349646	−	0.936882i	$-0.613698\pi$
$887$	−20.8267	−0.699292	−0.349646	+	0.936882i	$0.613698\pi$
$888$	0	0
$889$	5.38283	0.180534
$890$	0	0
$891$	69.6934i	2.33482i
$892$	0	0
$893$	− 10.9133i	− 0.365201i
$894$	0	0
$895$	6.19122	0.206950
$896$	0	0
$897$	−70.0867	−2.34013
$898$	0	0
$899$	2.31445i	0.0771913i
$900$	0	0
$901$	− 29.6970i	− 0.989351i
$902$	0	0
$903$	−29.9230	−0.995774
$904$	0	0
$905$	10.3446	0.343868
$906$	0	0
$907$	− 49.2109i	− 1.63402i	−0.576622	−	0.817011i	$-0.695629\pi$
$907$	− 49.2109i	− 1.63402i	0.576622	−	0.817011i	$-0.304371\pi$
$908$	0	0
$909$	− 117.584i	− 3.90001i
$910$	0	0
$911$	−42.0717	−1.39390	−0.696949	−	0.717121i	$-0.745460\pi$
$911$	−42.0717	−1.39390	−0.696949	+	0.717121i	$0.745460\pi$
$912$	0	0
$913$	33.9518	1.12364
$914$	0	0
$915$	− 34.3717i	− 1.13629i
$916$	0	0
$917$	− 10.8008i	− 0.356675i
$918$	0	0
$919$	−36.2172	−1.19469	−0.597347	−	0.801983i	$-0.703778\pi$
$919$	−36.2172	−1.19469	−0.597347	+	0.801983i	$0.703778\pi$
$920$	0	0
$921$	87.5036	2.88334
$922$	0	0
$923$	− 37.5196i	− 1.23497i
$924$	0	0
$925$	4.98614i	0.163943i
$926$	0	0
$927$	−26.6801	−0.876289
$928$	0	0
$929$	−12.7342	−0.417796	−0.208898	−	0.977937i	$-0.566988\pi$
$929$	−12.7342	−0.417796	−0.208898	+	0.977937i	$0.566988\pi$
$930$	0	0
$931$	2.79216i	0.0915095i
$932$	0	0
$933$	3.12010i	0.102148i
$934$	0	0
$935$	−31.3282	−1.02454
$936$	0	0
$937$	−35.2176	−1.15051	−0.575254	−	0.817975i	$-0.695097\pi$
$937$	−35.2176	−1.15051	−0.575254	+	0.817975i	$0.695097\pi$
$938$	0	0
$939$	14.7092i	0.480017i
$940$	0	0
$941$	− 30.2193i	− 0.985121i	−0.870278	−	0.492561i	$-0.836061\pi$
$941$	− 30.2193i	− 0.985121i	0.870278	−	0.492561i	$-0.163939\pi$
$942$	0	0
$943$	−30.5228	−0.993959
$944$	0	0
$945$	−11.9934	−0.390146
$946$	0	0
$947$	7.23680i	0.235164i	0.993063	+	0.117582i	$0.0375144\pi$
$947$	7.23680i	0.235164i	−0.993063	+	0.117582i	$0.962486\pi$
$948$	0	0
$949$	− 17.2010i	− 0.558369i
$950$	0	0
$951$	4.16682	0.135118
$952$	0	0
$953$	44.1717	1.43086	0.715431	−	0.698683i	$-0.246230\pi$
$953$	44.1717	1.43086	0.715431	+	0.698683i	$0.246230\pi$
$954$	0	0
$955$	− 17.3266i	− 0.560677i
$956$	0	0
$957$	− 127.076i	− 4.10777i
$958$	0	0
$959$	10.9651	0.354080
$960$	0	0
$961$	−30.9460	−0.998257
$962$	0	0
$963$	24.6047i	0.792877i
$964$	0	0
$965$	− 25.2039i	− 0.811342i
$966$	0	0
$967$	36.4144	1.17101	0.585505	−	0.810669i	$-0.300896\pi$
$967$	36.4144	1.17101	0.585505	+	0.810669i	$0.300896\pi$
$968$	0	0
$969$	67.3448	2.16343
$970$	0	0
$971$	13.8144i	0.443326i	0.975123	+	0.221663i	$0.0711486\pi$
$971$	13.8144i	0.443326i	−0.975123	+	0.221663i	$0.928851\pi$
$972$	0	0
$973$	2.40141i	0.0769857i
$974$	0	0
$975$	8.99545	0.288085
$976$	0	0
$977$	0.883020	0.0282503	0.0141252	−	0.999900i	$-0.495504\pi$
$977$	0.883020	0.0282503	0.0141252	+	0.999900i	$0.495504\pi$
$978$	0	0
$979$	− 23.5799i	− 0.753618i
$980$	0	0
$981$	− 60.1150i	− 1.91932i
$982$	0	0
$983$	−51.6614	−1.64774	−0.823871	−	0.566777i	$-0.808190\pi$
$983$	−51.6614	−1.64774	−0.823871	+	0.566777i	$0.808190\pi$
$984$	0	0
$985$	23.4680	0.747754
$986$	0	0
$987$	12.2520i	0.389985i
$988$	0	0
$989$	− 74.3752i	− 2.36499i
$990$	0	0
$991$	−16.1941	−0.514423	−0.257211	−	0.966355i	$-0.582804\pi$
$991$	−16.1941	−0.514423	−0.257211	+	0.966355i	$0.582804\pi$
$992$	0	0
$993$	−2.66585	−0.0845983
$994$	0	0
$995$	8.26099i	0.261891i
$996$	0	0
$997$	− 19.0128i	− 0.602142i	−0.953602	−	0.301071i	$-0.902656\pi$
$997$	− 19.0128i	− 0.602142i	0.953602	−	0.301071i	$-0.0973442\pi$
$998$	0	0
$999$	59.8010	1.89202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.b.h.1121.2	yes	12
4.3	odd	2		2240.2.b.g.1121.11	yes	12
8.3	odd	2		2240.2.b.g.1121.2	✓	12
8.5	even	2	inner	2240.2.b.h.1121.11	yes	12
16.3	odd	4		8960.2.a.ch.1.1		6
16.5	even	4		8960.2.a.cb.1.1		6
16.11	odd	4		8960.2.a.cc.1.6		6
16.13	even	4		8960.2.a.ce.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.b.g.1121.2	✓	12	8.3	odd	2
2240.2.b.g.1121.11	yes	12	4.3	odd	2
2240.2.b.h.1121.2	yes	12	1.1	even	1	trivial
2240.2.b.h.1121.11	yes	12	8.5	even	2	inner
8960.2.a.cb.1.1		6	16.5	even	4
8960.2.a.cc.1.6		6	16.11	odd	4
8960.2.a.ce.1.6		6	16.13	even	4
8960.2.a.ch.1.1		6	16.3	odd	4

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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.13466i q^{3} +1.00000i q^{5} +1.00000 q^{7} -6.82607 q^{9} +O(q^{10})\)	\(q-3.13466i q^{3} +1.00000i q^{5} +1.00000 q^{7} -6.82607 q^{9} +4.07158i q^{11} -2.86968i q^{13} +3.13466 q^{15} +7.69437 q^{17} +2.79216i q^{19} -3.13466i q^{21} +7.79136 q^{23} -1.00000 q^{25} +11.9934i q^{27} -9.95657i q^{29} -0.232455 q^{31} +12.7630 q^{33} +1.00000i q^{35} -4.98614i q^{37} -8.99545 q^{39} -3.91752 q^{41} -9.54585i q^{43} -6.82607i q^{45} -3.90855 q^{47} +1.00000 q^{49} -24.1192i q^{51} -3.85958i q^{53} -4.07158 q^{55} +8.75248 q^{57} +7.86790i q^{59} -10.9651i q^{61} -6.82607 q^{63} +2.86968 q^{65} +11.5177i q^{67} -24.4232i q^{69} +13.0745 q^{71} +5.99406 q^{73} +3.13466i q^{75} +4.07158i q^{77} +0.872456 q^{79} +17.1171 q^{81} -8.33875i q^{83} +7.69437i q^{85} -31.2104 q^{87} -5.79136 q^{89} -2.86968i q^{91} +0.728666i q^{93} -2.79216 q^{95} -10.6226 q^{97} -27.7929i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} - 20 q^{9}+O(q^{10}) \) 12 * q + 12 * q^7 - 20 * q^9	\( 12 q + 12 q^{7} - 20 q^{9} + 16 q^{17} + 8 q^{23} - 12 q^{25} - 8 q^{31} + 72 q^{33} - 32 q^{39} - 32 q^{47} + 12 q^{49} + 8 q^{55} + 8 q^{57} - 20 q^{63} + 8 q^{65} + 48 q^{71} + 8 q^{73} - 16 q^{79} + 92 q^{81} - 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) \) 12 * q + 12 * q^7 - 20 * q^9 + 16 * q^17 + 8 * q^23 - 12 * q^25 - 8 * q^31 + 72 * q^33 - 32 * q^39 - 32 * q^47 + 12 * q^49 + 8 * q^55 + 8 * q^57 - 20 * q^63 + 8 * q^65 + 48 * q^71 + 8 * q^73 - 16 * q^79 + 92 * q^81 - 32 * q^87 + 16 * q^89 + 32 * q^97

Introduction

L-functions

Modular forms

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Database

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