LMFDB - Embedded newform 1350.4.c.v.649.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(649,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1350.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.c (of order $2$, degree $1$, not 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:	$79.6525785077$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 25 $ x^4 + 11*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.3
Root			$-2.79129i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.v.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} -18.7477i q^{7} -8.00000i q^{8} +O(q^{10})$	$q+2.00000i q^{2} -4.00000 q^{4} -18.7477i q^{7} -8.00000i q^{8} +39.9909 q^{11} +33.7477i q^{13} +37.4955 q^{14} +16.0000 q^{16} -53.7386i q^{17} -91.7386 q^{19} +79.9818i q^{22} +80.7386i q^{23} -67.4955 q^{26} +74.9909i q^{28} +141.234 q^{29} +264.955 q^{31} +32.0000i q^{32} +107.477 q^{34} -61.2432i q^{37} -183.477i q^{38} -314.973 q^{41} +236.261i q^{43} -159.964 q^{44} -161.477 q^{46} -243.261i q^{47} -8.47727 q^{49} -134.991i q^{52} -191.739i q^{53} -149.982 q^{56} +282.468i q^{58} -312.441 q^{59} -550.648 q^{61} +529.909i q^{62} -64.0000 q^{64} -571.270i q^{67} +214.955i q^{68} -183.784 q^{71} +125.027i q^{73} +122.486 q^{74} +366.955 q^{76} -749.739i q^{77} +429.216 q^{79} -629.945i q^{82} -1091.48i q^{83} -472.523 q^{86} -319.927i q^{88} +1233.59 q^{89} +632.693 q^{91} -322.955i q^{92} +486.523 q^{94} -1532.44i q^{97} -16.9545i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4}+O(q^{10}) $ 4 * q - 16 * q^4	$ 4 q - 16 q^{4} - 60 q^{11} + 40 q^{14} + 64 q^{16} - 92 q^{19} - 160 q^{26} + 180 q^{29} - 40 q^{31} - 120 q^{34} - 600 q^{41} + 240 q^{44} - 96 q^{46} + 516 q^{49} - 160 q^{56} + 180 q^{59} + 272 q^{61} - 256 q^{64} - 1560 q^{71} + 160 q^{74} + 368 q^{76} + 892 q^{79} - 2440 q^{86} + 1140 q^{89} + 1156 q^{91} + 2496 q^{94}+O(q^{100}) $ 4 * q - 16 * q^4 - 60 * q^11 + 40 * q^14 + 64 * q^16 - 92 * q^19 - 160 * q^26 + 180 * q^29 - 40 * q^31 - 120 * q^34 - 600 * q^41 + 240 * q^44 - 96 * q^46 + 516 * q^49 - 160 * q^56 + 180 * q^59 + 272 * q^61 - 256 * q^64 - 1560 * q^71 + 160 * q^74 + 368 * q^76 + 892 * q^79 - 2440 * q^86 + 1140 * q^89 + 1156 * q^91 + 2496 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 18.7477i	− 1.01228i	−0.862451	−	0.506141i	$-0.831072\pi$
$7$	− 18.7477i	− 1.01228i	0.862451	−	0.506141i	$-0.168928\pi$
$8$	− 8.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	39.9909	1.09616	0.548078	−	0.836427i	$-0.315360\pi$
$11$	39.9909	1.09616	0.548078	+	0.836427i	$0.315360\pi$
$12$	0	0
$13$	33.7477i	0.719995i	0.932953	+	0.359998i	$0.117222\pi$
$13$	33.7477i	0.719995i	−0.932953	+	0.359998i	$0.882778\pi$
$14$	37.4955	0.715792
$15$	0	0
$16$	16.0000	0.250000
$17$	− 53.7386i	− 0.766678i	−0.923608	−	0.383339i	$-0.874774\pi$
$17$	− 53.7386i	− 0.766678i	0.923608	−	0.383339i	$-0.125226\pi$
$18$	0	0
$19$	−91.7386	−1.10770	−0.553850	−	0.832617i	$-0.686842\pi$
$19$	−91.7386	−1.10770	−0.553850	+	0.832617i	$0.686842\pi$
$20$	0	0
$21$	0	0
$22$	79.9818i	0.775099i
$23$	80.7386i	0.731964i	0.930622	+	0.365982i	$0.119267\pi$
$23$	80.7386i	0.731964i	−0.930622	+	0.365982i	$0.880733\pi$
$24$	0	0
$25$	0	0
$26$	−67.4955	−0.509113
$27$	0	0
$28$	74.9909i	0.506141i
$29$	141.234	0.904362	0.452181	−	0.891926i	$-0.350646\pi$
$29$	141.234	0.904362	0.452181	+	0.891926i	$0.350646\pi$
$30$	0	0
$31$	264.955	1.53507	0.767536	−	0.641006i	$-0.221483\pi$
$31$	264.955	1.53507	0.767536	+	0.641006i	$0.221483\pi$
$32$	32.0000i	0.176777i
$33$	0	0
$34$	107.477	0.542124
$35$	0	0
$36$	0	0
$37$	− 61.2432i	− 0.272117i	−0.990701	−	0.136058i	$-0.956557\pi$
$37$	− 61.2432i	− 0.272117i	0.990701	−	0.136058i	$-0.0434435\pi$
$38$	− 183.477i	− 0.783262i
$39$	0	0
$40$	0	0
$41$	−314.973	−1.19977	−0.599884	−	0.800087i	$-0.704787\pi$
$41$	−314.973	−1.19977	−0.599884	+	0.800087i	$0.704787\pi$
$42$	0	0
$43$	236.261i	0.837896i	0.908010	+	0.418948i	$0.137601\pi$
$43$	236.261i	0.837896i	−0.908010	+	0.418948i	$0.862399\pi$
$44$	−159.964	−0.548078
$45$	0	0
$46$	−161.477	−0.517577
$47$	− 243.261i	− 0.754964i	−0.926017	−	0.377482i	$-0.876790\pi$
$47$	− 243.261i	− 0.754964i	0.926017	−	0.377482i	$-0.123210\pi$
$48$	0	0
$49$	−8.47727	−0.0247151
$50$	0	0
$51$	0	0
$52$	− 134.991i	− 0.359998i
$53$	− 191.739i	− 0.496931i	−0.968641	−	0.248465i	$-0.920074\pi$
$53$	− 191.739i	− 0.496931i	0.968641	−	0.248465i	$-0.0799262\pi$
$54$	0	0
$55$	0	0
$56$	−149.982	−0.357896
$57$	0	0
$58$	282.468i	0.639481i
$59$	−312.441	−0.689430	−0.344715	−	0.938707i	$-0.612024\pi$
$59$	−312.441	−0.689430	−0.344715	+	0.938707i	$0.612024\pi$
$60$	0	0
$61$	−550.648	−1.15579	−0.577895	−	0.816111i	$-0.696126\pi$
$61$	−550.648	−1.15579	−0.577895	+	0.816111i	$0.696126\pi$
$62$	529.909i	1.08546i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 571.270i	− 1.04167i	−0.853658	−	0.520834i	$-0.825621\pi$
$67$	− 571.270i	− 1.04167i	0.853658	−	0.520834i	$-0.174379\pi$
$68$	214.955i	0.383339i
$69$	0	0
$70$	0	0
$71$	−183.784	−0.307199	−0.153600	−	0.988133i	$-0.549087\pi$
$71$	−183.784	−0.307199	−0.153600	+	0.988133i	$0.549087\pi$
$72$	0	0
$73$	125.027i	0.200457i	0.994964	+	0.100228i	$0.0319573\pi$
$73$	125.027i	0.200457i	−0.994964	+	0.100228i	$0.968043\pi$
$74$	122.486	0.192416
$75$	0	0
$76$	366.955	0.553850
$77$	− 749.739i	− 1.10962i
$78$	0	0
$79$	429.216	0.611273	0.305636	−	0.952148i	$-0.401131\pi$
$79$	429.216	0.611273	0.305636	+	0.952148i	$0.401131\pi$
$80$	0	0
$81$	0	0
$82$	− 629.945i	− 0.848364i
$83$	− 1091.48i	− 1.44344i	−0.692187	−	0.721718i	$-0.743353\pi$
$83$	− 1091.48i	− 1.44344i	0.692187	−	0.721718i	$-0.256647\pi$
$84$	0	0
$85$	0	0
$86$	−472.523	−0.592482
$87$	0	0
$88$	− 319.927i	− 0.387550i
$89$	1233.59	1.46922	0.734610	−	0.678489i	$-0.237365\pi$
$89$	1233.59	1.46922	0.734610	+	0.678489i	$0.237365\pi$
$90$	0	0
$91$	632.693	0.728838
$92$	− 322.955i	− 0.365982i
$93$	0	0
$94$	486.523	0.533840
$95$	0	0
$96$	0	0
$97$	− 1532.44i	− 1.60408i	−0.597270	−	0.802040i	$-0.703748\pi$
$97$	− 1532.44i	− 1.60408i	0.597270	−	0.802040i	$-0.296252\pi$
$98$	− 16.9545i	− 0.0174762i
$99$	0	0
$100$	0	0
$101$	358.639	0.353326	0.176663	−	0.984271i	$-0.443470\pi$
$101$	358.639	0.353326	0.176663	+	0.984271i	$0.443470\pi$
$102$	0	0
$103$	− 437.514i	− 0.418539i	−0.977858	−	0.209269i	$-0.932891\pi$
$103$	− 437.514i	− 0.418539i	0.977858	−	0.209269i	$-0.0671086\pi$
$104$	269.982	0.254557
$105$	0	0
$106$	383.477	0.351383
$107$	− 922.818i	− 0.833759i	−0.908962	−	0.416879i	$-0.863124\pi$
$107$	− 922.818i	− 0.833759i	0.908962	−	0.416879i	$-0.136876\pi$
$108$	0	0
$109$	1219.39	1.07152	0.535762	−	0.844369i	$-0.320025\pi$
$109$	1219.39	1.07152	0.535762	+	0.844369i	$0.320025\pi$
$110$	0	0
$111$	0	0
$112$	− 299.964i	− 0.253071i
$113$	− 463.045i	− 0.385484i	−0.981250	−	0.192742i	$-0.938262\pi$
$113$	− 463.045i	− 0.385484i	0.981250	−	0.192742i	$-0.0617380\pi$
$114$	0	0
$115$	0	0
$116$	−564.936	−0.452181
$117$	0	0
$118$	− 624.882i	− 0.487500i
$119$	−1007.48	−0.776095
$120$	0	0
$121$	268.273	0.201557
$122$	− 1101.30i	− 0.817267i
$123$	0	0
$124$	−1059.82	−0.767536
$125$	0	0
$126$	0	0
$127$	− 207.677i	− 0.145105i	−0.997365	−	0.0725527i	$-0.976885\pi$
$127$	− 207.677i	− 0.145105i	0.997365	−	0.0725527i	$-0.0231145\pi$
$128$	− 128.000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−2567.34	−1.71229	−0.856144	−	0.516738i	$-0.827146\pi$
$131$	−2567.34	−1.71229	−0.856144	+	0.516738i	$0.827146\pi$
$132$	0	0
$133$	1719.89i	1.12130i
$134$	1142.54	0.736571
$135$	0	0
$136$	−429.909	−0.271062
$137$	− 15.9205i	− 0.00992830i	−0.999988	−	0.00496415i	$-0.998420\pi$
$137$	− 15.9205i	− 0.00992830i	0.999988	−	0.00496415i	$-0.00158014\pi$
$138$	0	0
$139$	2839.65	1.73278	0.866388	−	0.499372i	$-0.166436\pi$
$139$	2839.65	1.73278	0.866388	+	0.499372i	$0.166436\pi$
$140$	0	0
$141$	0	0
$142$	− 367.568i	− 0.217223i
$143$	1349.60i	0.789227i
$144$	0	0
$145$	0	0
$146$	−250.055	−0.141744
$147$	0	0
$148$	244.973i	0.136058i
$149$	−202.377	−0.111271	−0.0556355	−	0.998451i	$-0.517718\pi$
$149$	−202.377	−0.111271	−0.0556355	+	0.998451i	$0.517718\pi$
$150$	0	0
$151$	3657.81	1.97131	0.985656	−	0.168767i	$-0.0539786\pi$
$151$	3657.81	1.97131	0.985656	+	0.168767i	$0.0539786\pi$
$152$	733.909i	0.391631i
$153$	0	0
$154$	1499.48	0.784619
$155$	0	0
$156$	0	0
$157$	− 2272.29i	− 1.15508i	−0.816361	−	0.577542i	$-0.804012\pi$
$157$	− 2272.29i	− 1.15508i	0.816361	−	0.577542i	$-0.195988\pi$
$158$	858.432i	0.432235i
$159$	0	0
$160$	0	0
$161$	1513.67	0.740954
$162$	0	0
$163$	2754.68i	1.32370i	0.749636	+	0.661851i	$0.230229\pi$
$163$	2754.68i	1.32370i	−0.749636	+	0.661851i	$0.769771\pi$
$164$	1259.89	0.599884
$165$	0	0
$166$	2182.95	1.02066
$167$	1067.85i	0.494808i	0.968912	+	0.247404i	$0.0795774\pi$
$167$	1067.85i	0.494808i	−0.968912	+	0.247404i	$0.920423\pi$
$168$	0	0
$169$	1058.09	0.481607
$170$	0	0
$171$	0	0
$172$	− 945.045i	− 0.418948i
$173$	− 2963.47i	− 1.30236i	−0.758924	−	0.651180i	$-0.774274\pi$
$173$	− 2963.47i	− 1.30236i	0.758924	−	0.651180i	$-0.225726\pi$
$174$	0	0
$175$	0	0
$176$	639.855	0.274039
$177$	0	0
$178$	2467.19i	1.03890i
$179$	−3087.14	−1.28907	−0.644536	−	0.764574i	$-0.722949\pi$
$179$	−3087.14	−1.28907	−0.644536	+	0.764574i	$0.722949\pi$
$180$	0	0
$181$	4080.03	1.67551	0.837753	−	0.546050i	$-0.183869\pi$
$181$	4080.03	1.67551	0.837753	+	0.546050i	$0.183869\pi$
$182$	1265.39i	0.515366i
$183$	0	0
$184$	645.909	0.258788
$185$	0	0
$186$	0	0
$187$	− 2149.06i	− 0.840399i
$188$	973.045i	0.377482i
$189$	0	0
$190$	0	0
$191$	−2672.01	−1.01225	−0.506126	−	0.862460i	$-0.668923\pi$
$191$	−2672.01	−1.01225	−0.506126	+	0.862460i	$0.668923\pi$
$192$	0	0
$193$	− 4509.45i	− 1.68185i	−0.541149	−	0.840927i	$-0.682011\pi$
$193$	− 4509.45i	− 1.68185i	0.541149	−	0.840927i	$-0.317989\pi$
$194$	3064.88	1.13426
$195$	0	0
$196$	33.9091	0.0123575
$197$	2251.82i	0.814393i	0.913341	+	0.407196i	$0.133494\pi$
$197$	2251.82i	0.814393i	−0.913341	+	0.407196i	$0.866506\pi$
$198$	0	0
$199$	−1548.93	−0.551763	−0.275882	−	0.961192i	$-0.588970\pi$
$199$	−1548.93	−0.551763	−0.275882	+	0.961192i	$0.588970\pi$
$200$	0	0
$201$	0	0
$202$	717.277i	0.249839i
$203$	− 2647.82i	− 0.915470i
$204$	0	0
$205$	0	0
$206$	875.027	0.295952
$207$	0	0
$208$	539.964i	0.179999i
$209$	−3668.71	−1.21421
$210$	0	0
$211$	2230.61	0.727781	0.363890	−	0.931442i	$-0.381448\pi$
$211$	2230.61	0.727781	0.363890	+	0.931442i	$0.381448\pi$
$212$	766.955i	0.248465i
$213$	0	0
$214$	1845.64	0.589557
$215$	0	0
$216$	0	0
$217$	− 4967.30i	− 1.55393i
$218$	2438.77i	0.757681i
$219$	0	0
$220$	0	0
$221$	1813.56	0.552005
$222$	0	0
$223$	2069.04i	0.621314i	0.950522	+	0.310657i	$0.100549\pi$
$223$	2069.04i	0.621314i	−0.950522	+	0.310657i	$0.899451\pi$
$224$	599.927	0.178948
$225$	0	0
$226$	926.091	0.272578
$227$	− 5240.45i	− 1.53225i	−0.642690	−	0.766126i	$-0.722182\pi$
$227$	− 5240.45i	− 1.53225i	0.642690	−	0.766126i	$-0.277818\pi$
$228$	0	0
$229$	3557.69	1.02663	0.513317	−	0.858199i	$-0.328417\pi$
$229$	3557.69	1.02663	0.513317	+	0.858199i	$0.328417\pi$
$230$	0	0
$231$	0	0
$232$	− 1129.87i	− 0.319740i
$233$	35.4887i	0.00997828i	0.999988	+	0.00498914i	$0.00158810\pi$
$233$	35.4887i	0.00997828i	−0.999988	+	0.00498914i	$0.998412\pi$
$234$	0	0
$235$	0	0
$236$	1249.76	0.344715
$237$	0	0
$238$	− 2014.95i	− 0.548782i
$239$	1170.62	0.316826	0.158413	−	0.987373i	$-0.449362\pi$
$239$	1170.62	0.316826	0.158413	+	0.987373i	$0.449362\pi$
$240$	0	0
$241$	−3319.50	−0.887252	−0.443626	−	0.896212i	$-0.646308\pi$
$241$	−3319.50	−0.887252	−0.443626	+	0.896212i	$0.646308\pi$
$242$	536.545i	0.142523i
$243$	0	0
$244$	2202.59	0.577895
$245$	0	0
$246$	0	0
$247$	− 3095.97i	− 0.797538i
$248$	− 2119.64i	− 0.542730i
$249$	0	0
$250$	0	0
$251$	5852.48	1.47173	0.735867	−	0.677126i	$-0.236775\pi$
$251$	5852.48	1.47173	0.735867	+	0.677126i	$0.236775\pi$
$252$	0	0
$253$	3228.81i	0.802346i
$254$	415.355	0.102605
$255$	0	0
$256$	256.000	0.0625000
$257$	− 4239.62i	− 1.02903i	−0.857482	−	0.514515i	$-0.827972\pi$
$257$	− 4239.62i	− 1.02903i	0.857482	−	0.514515i	$-0.172028\pi$
$258$	0	0
$259$	−1148.17	−0.275459
$260$	0	0
$261$	0	0
$262$	− 5134.68i	− 1.21077i
$263$	6702.60i	1.57148i	0.618555	+	0.785742i	$0.287719\pi$
$263$	6702.60i	1.57148i	−0.618555	+	0.785742i	$0.712281\pi$
$264$	0	0
$265$	0	0
$266$	−3439.78	−0.792882
$267$	0	0
$268$	2285.08i	0.520834i
$269$	−3182.94	−0.721440	−0.360720	−	0.932674i	$-0.617469\pi$
$269$	−3182.94	−0.721440	−0.360720	+	0.932674i	$0.617469\pi$
$270$	0	0
$271$	−2349.73	−0.526700	−0.263350	−	0.964700i	$-0.584827\pi$
$271$	−2349.73	−0.526700	−0.263350	+	0.964700i	$0.584827\pi$
$272$	− 859.818i	− 0.191670i
$273$	0	0
$274$	31.8409	0.00702037
$275$	0	0
$276$	0	0
$277$	− 6654.79i	− 1.44349i	−0.692157	−	0.721747i	$-0.743340\pi$
$277$	− 6654.79i	− 1.44349i	0.692157	−	0.721747i	$-0.256660\pi$
$278$	5679.30i	1.22526i
$279$	0	0
$280$	0	0
$281$	6245.88	1.32597	0.662986	−	0.748632i	$-0.269289\pi$
$281$	6245.88	1.32597	0.662986	+	0.748632i	$0.269289\pi$
$282$	0	0
$283$	2360.43i	0.495806i	0.968785	+	0.247903i	$0.0797415\pi$
$283$	2360.43i	0.495806i	−0.968785	+	0.247903i	$0.920258\pi$
$284$	735.136	0.153600
$285$	0	0
$286$	−2699.20	−0.558068
$287$	5905.02i	1.21450i
$288$	0	0
$289$	2025.16	0.412204
$290$	0	0
$291$	0	0
$292$	− 500.109i	− 0.100228i
$293$	− 5834.08i	− 1.16324i	−0.813459	−	0.581622i	$-0.802418\pi$
$293$	− 5834.08i	− 1.16324i	0.813459	−	0.581622i	$-0.197582\pi$
$294$	0	0
$295$	0	0
$296$	−489.945	−0.0962078
$297$	0	0
$298$	− 404.755i	− 0.0786805i
$299$	−2724.75	−0.527010
$300$	0	0
$301$	4429.36	0.848187
$302$	7315.61i	1.39393i
$303$	0	0
$304$	−1467.82	−0.276925
$305$	0	0
$306$	0	0
$307$	− 1460.05i	− 0.271432i	−0.990748	−	0.135716i	$-0.956666\pi$
$307$	− 1460.05i	− 0.271432i	0.990748	−	0.135716i	$-0.0433335\pi$
$308$	2998.95i	0.554809i
$309$	0	0
$310$	0	0
$311$	4200.24	0.765833	0.382916	−	0.923783i	$-0.374920\pi$
$311$	4200.24	0.765833	0.382916	+	0.923783i	$0.374920\pi$
$312$	0	0
$313$	5890.15i	1.06368i	0.846846	+	0.531838i	$0.178499\pi$
$313$	5890.15i	1.06368i	−0.846846	+	0.531838i	$0.821501\pi$
$314$	4544.57	0.816768
$315$	0	0
$316$	−1716.86	−0.305636
$317$	1507.03i	0.267014i	0.991048	+	0.133507i	$0.0426239\pi$
$317$	1507.03i	0.267014i	−0.991048	+	0.133507i	$0.957376\pi$
$318$	0	0
$319$	5648.08	0.991322
$320$	0	0
$321$	0	0
$322$	3027.33i	0.523934i
$323$	4929.91i	0.849249i
$324$	0	0
$325$	0	0
$326$	−5509.36	−0.935998
$327$	0	0
$328$	2519.78i	0.424182i
$329$	−4560.60	−0.764237
$330$	0	0
$331$	−3256.75	−0.540807	−0.270404	−	0.962747i	$-0.587157\pi$
$331$	−3256.75	−0.540807	−0.270404	+	0.962747i	$0.587157\pi$
$332$	4365.91i	0.721718i
$333$	0	0
$334$	−2135.70	−0.349882
$335$	0	0
$336$	0	0
$337$	7698.71i	1.24444i	0.782843	+	0.622219i	$0.213769\pi$
$337$	7698.71i	1.24444i	−0.782843	+	0.622219i	$0.786231\pi$
$338$	2116.18i	0.340548i
$339$	0	0
$340$	0	0
$341$	10595.8	1.68268
$342$	0	0
$343$	− 6271.54i	− 0.987263i
$344$	1890.09	0.296241
$345$	0	0
$346$	5926.93	0.920907
$347$	− 9545.73i	− 1.47678i	−0.674376	−	0.738388i	$-0.735587\pi$
$347$	− 9545.73i	− 1.47678i	0.674376	−	0.738388i	$-0.264413\pi$
$348$	0	0
$349$	2926.48	0.448856	0.224428	−	0.974491i	$-0.427949\pi$
$349$	2926.48	0.448856	0.224428	+	0.974491i	$0.427949\pi$
$350$	0	0
$351$	0	0
$352$	1279.71i	0.193775i
$353$	− 4679.69i	− 0.705595i	−0.935700	−	0.352797i	$-0.885231\pi$
$353$	− 4679.69i	− 0.705595i	0.935700	−	0.352797i	$-0.114769\pi$
$354$	0	0
$355$	0	0
$356$	−4934.37	−0.734610
$357$	0	0
$358$	− 6174.28i	− 0.911511i
$359$	−12252.8	−1.80133	−0.900667	−	0.434509i	$-0.856922\pi$
$359$	−12252.8	−1.80133	−0.900667	+	0.434509i	$0.856922\pi$
$360$	0	0
$361$	1556.98	0.226998
$362$	8160.07i	1.18476i
$363$	0	0
$364$	−2530.77	−0.364419
$365$	0	0
$366$	0	0
$367$	− 11666.1i	− 1.65931i	−0.558280	−	0.829653i	$-0.688538\pi$
$367$	− 11666.1i	− 1.65931i	0.558280	−	0.829653i	$-0.311462\pi$
$368$	1291.82i	0.182991i
$369$	0	0
$370$	0	0
$371$	−3594.66	−0.503034
$372$	0	0
$373$	− 1543.32i	− 0.214237i	−0.994246	−	0.107118i	$-0.965838\pi$
$373$	− 1543.32i	− 0.214237i	0.994246	−	0.107118i	$-0.0341623\pi$
$374$	4298.11	0.594252
$375$	0	0
$376$	−1946.09	−0.266920
$377$	4766.33i	0.651136i
$378$	0	0
$379$	1421.25	0.192625	0.0963123	−	0.995351i	$-0.469295\pi$
$379$	1421.25	0.192625	0.0963123	+	0.995351i	$0.469295\pi$
$380$	0	0
$381$	0	0
$382$	− 5344.03i	− 0.715770i
$383$	− 8785.42i	− 1.17210i	−0.810275	−	0.586050i	$-0.800682\pi$
$383$	− 8785.42i	− 1.17210i	0.810275	−	0.586050i	$-0.199318\pi$
$384$	0	0
$385$	0	0
$386$	9018.91	1.18925
$387$	0	0
$388$	6129.76i	0.802040i
$389$	5555.06	0.724043	0.362022	−	0.932170i	$-0.382087\pi$
$389$	5555.06	0.724043	0.362022	+	0.932170i	$0.382087\pi$
$390$	0	0
$391$	4338.78	0.561181
$392$	67.8182i	0.00873810i
$393$	0	0
$394$	−4503.64	−0.575863
$395$	0	0
$396$	0	0
$397$	11483.1i	1.45169i	0.687856	+	0.725847i	$0.258552\pi$
$397$	11483.1i	1.45169i	−0.687856	+	0.725847i	$0.741448\pi$
$398$	− 3097.86i	− 0.390155i
$399$	0	0
$400$	0	0
$401$	−12002.9	−1.49475	−0.747376	−	0.664401i	$-0.768687\pi$
$401$	−12002.9	−1.49475	−0.747376	+	0.664401i	$0.768687\pi$
$402$	0	0
$403$	8941.61i	1.10524i
$404$	−1434.55	−0.176663
$405$	0	0
$406$	5295.64	0.647335
$407$	− 2449.17i	− 0.298282i
$408$	0	0
$409$	−9770.32	−1.18120	−0.590600	−	0.806964i	$-0.701109\pi$
$409$	−9770.32	−1.18120	−0.590600	+	0.806964i	$0.701109\pi$
$410$	0	0
$411$	0	0
$412$	1750.05i	0.209269i
$413$	5857.56i	0.697897i
$414$	0	0
$415$	0	0
$416$	−1079.93	−0.127278
$417$	0	0
$418$	− 7337.42i	− 0.858577i
$419$	8055.79	0.939262	0.469631	−	0.882863i	$-0.344387\pi$
$419$	8055.79	0.939262	0.469631	+	0.882863i	$0.344387\pi$
$420$	0	0
$421$	4967.99	0.575119	0.287559	−	0.957763i	$-0.407156\pi$
$421$	4967.99	0.575119	0.287559	+	0.957763i	$0.407156\pi$
$422$	4461.23i	0.514619i
$423$	0	0
$424$	−1533.91	−0.175692
$425$	0	0
$426$	0	0
$427$	10323.4i	1.16999i
$428$	3691.27i	0.416879i
$429$	0	0
$430$	0	0
$431$	10826.5	1.20997	0.604984	−	0.796238i	$-0.293179\pi$
$431$	10826.5	1.20997	0.604984	+	0.796238i	$0.293179\pi$
$432$	0	0
$433$	2038.63i	0.226259i	0.993580	+	0.113130i	$0.0360875\pi$
$433$	2038.63i	0.226259i	−0.993580	+	0.113130i	$0.963912\pi$
$434$	9934.59	1.09879
$435$	0	0
$436$	−4877.55	−0.535762
$437$	− 7406.85i	− 0.810796i
$438$	0	0
$439$	−11648.4	−1.26640	−0.633200	−	0.773988i	$-0.718259\pi$
$439$	−11648.4	−1.26640	−0.633200	+	0.773988i	$0.718259\pi$
$440$	0	0
$441$	0	0
$442$	3627.11i	0.390326i
$443$	− 4156.59i	− 0.445791i	−0.974842	−	0.222896i	$-0.928449\pi$
$443$	− 4156.59i	− 0.445791i	0.974842	−	0.222896i	$-0.0715509\pi$
$444$	0	0
$445$	0	0
$446$	−4138.08	−0.439336
$447$	0	0
$448$	1199.85i	0.126535i
$449$	−14987.5	−1.57529	−0.787645	−	0.616129i	$-0.788700\pi$
$449$	−14987.5	−1.57529	−0.787645	+	0.616129i	$0.788700\pi$
$450$	0	0
$451$	−12596.0	−1.31513
$452$	1852.18i	0.192742i
$453$	0	0
$454$	10480.9	1.08347
$455$	0	0
$456$	0	0
$457$	13215.5i	1.35272i	0.736571	+	0.676361i	$0.236444\pi$
$457$	13215.5i	1.35272i	−0.736571	+	0.676361i	$0.763556\pi$
$458$	7115.39i	0.725939i
$459$	0	0
$460$	0	0
$461$	6365.55	0.643109	0.321554	−	0.946891i	$-0.395795\pi$
$461$	6365.55	0.643109	0.321554	+	0.946891i	$0.395795\pi$
$462$	0	0
$463$	− 13954.3i	− 1.40068i	−0.713811	−	0.700338i	$-0.753033\pi$
$463$	− 13954.3i	− 1.40068i	0.713811	−	0.700338i	$-0.246967\pi$
$464$	2259.75	0.226091
$465$	0	0
$466$	−70.9773	−0.00705571
$467$	− 15289.1i	− 1.51498i	−0.652848	−	0.757489i	$-0.726426\pi$
$467$	− 15289.1i	− 1.51498i	0.652848	−	0.757489i	$-0.273574\pi$
$468$	0	0
$469$	−10710.0	−1.05446
$470$	0	0
$471$	0	0
$472$	2499.53i	0.243750i
$473$	9448.31i	0.918464i
$474$	0	0
$475$	0	0
$476$	4029.91	0.388047
$477$	0	0
$478$	2341.25i	0.224030i
$479$	−8268.35	−0.788706	−0.394353	−	0.918959i	$-0.629031\pi$
$479$	−8268.35	−0.788706	−0.394353	+	0.918959i	$0.629031\pi$
$480$	0	0
$481$	2066.82	0.195923
$482$	− 6639.00i	− 0.627382i
$483$	0	0
$484$	−1073.09	−0.100779
$485$	0	0
$486$	0	0
$487$	7144.78i	0.664806i	0.943137	+	0.332403i	$0.107859\pi$
$487$	7144.78i	0.664806i	−0.943137	+	0.332403i	$0.892141\pi$
$488$	4405.18i	0.408634i
$489$	0	0
$490$	0	0
$491$	−19527.2	−1.79481	−0.897405	−	0.441209i	$-0.854550\pi$
$491$	−19527.2	−1.79481	−0.897405	+	0.441209i	$0.854550\pi$
$492$	0	0
$493$	− 7589.73i	− 0.693355i
$494$	6191.94	0.563945
$495$	0	0
$496$	4239.27	0.383768
$497$	3445.53i	0.310972i
$498$	0	0
$499$	12923.7	1.15941	0.579704	−	0.814827i	$-0.303168\pi$
$499$	12923.7	1.15941	0.579704	+	0.814827i	$0.303168\pi$
$500$	0	0
$501$	0	0
$502$	11705.0i	1.04067i
$503$	− 14125.0i	− 1.25209i	−0.779787	−	0.626045i	$-0.784672\pi$
$503$	− 14125.0i	− 1.25209i	0.779787	−	0.626045i	$-0.215328\pi$
$504$	0	0
$505$	0	0
$506$	−6457.62	−0.567345
$507$	0	0
$508$	830.709i	0.0725527i
$509$	−13955.7	−1.21527	−0.607636	−	0.794215i	$-0.707882\pi$
$509$	−13955.7	−1.21527	−0.607636	+	0.794215i	$0.707882\pi$
$510$	0	0
$511$	2343.98	0.202919
$512$	512.000i	0.0441942i
$513$	0	0
$514$	8479.25	0.727634
$515$	0	0
$516$	0	0
$517$	− 9728.24i	− 0.827558i
$518$	− 2296.34i	− 0.194779i
$519$	0	0
$520$	0	0
$521$	19825.1	1.66709	0.833544	−	0.552452i	$-0.186308\pi$
$521$	19825.1	1.66709	0.833544	+	0.552452i	$0.186308\pi$
$522$	0	0
$523$	− 14258.7i	− 1.19214i	−0.802933	−	0.596070i	$-0.796728\pi$
$523$	− 14258.7i	− 1.19214i	0.802933	−	0.596070i	$-0.203272\pi$
$524$	10269.4	0.856144
$525$	0	0
$526$	−13405.2	−1.11121
$527$	− 14238.3i	− 1.17691i
$528$	0	0
$529$	5648.27	0.464229
$530$	0	0
$531$	0	0
$532$	− 6879.56i	− 0.560652i
$533$	− 10629.6i	− 0.863827i
$534$	0	0
$535$	0	0
$536$	−4570.16	−0.368285
$537$	0	0
$538$	− 6365.88i	− 0.510135i
$539$	−339.014	−0.0270916
$540$	0	0
$541$	−19037.6	−1.51292	−0.756462	−	0.654037i	$-0.773074\pi$
$541$	−19037.6	−1.51292	−0.756462	+	0.654037i	$0.773074\pi$
$542$	− 4699.45i	− 0.372433i
$543$	0	0
$544$	1719.64	0.135531
$545$	0	0
$546$	0	0
$547$	11176.9i	0.873657i	0.899545	+	0.436828i	$0.143898\pi$
$547$	11176.9i	0.873657i	−0.899545	+	0.436828i	$0.856102\pi$
$548$	63.6819i	0.00496415i
$549$	0	0
$550$	0	0
$551$	−12956.6	−1.00176
$552$	0	0
$553$	− 8046.82i	− 0.618781i
$554$	13309.6	1.02070
$555$	0	0
$556$	−11358.6	−0.866388
$557$	19583.4i	1.48972i	0.667219	+	0.744862i	$0.267485\pi$
$557$	19583.4i	1.48972i	−0.667219	+	0.744862i	$0.732515\pi$
$558$	0	0
$559$	−7973.28	−0.603281
$560$	0	0
$561$	0	0
$562$	12491.8i	0.937603i
$563$	− 26341.1i	− 1.97184i	−0.167230	−	0.985918i	$-0.553482\pi$
$563$	− 26341.1i	− 1.97184i	0.167230	−	0.985918i	$-0.446518\pi$
$564$	0	0
$565$	0	0
$566$	−4720.87	−0.350588
$567$	0	0
$568$	1470.27i	0.108611i
$569$	5794.46	0.426918	0.213459	−	0.976952i	$-0.431527\pi$
$569$	5794.46	0.426918	0.213459	+	0.976952i	$0.431527\pi$
$570$	0	0
$571$	−17512.7	−1.28351	−0.641756	−	0.766909i	$-0.721794\pi$
$571$	−17512.7	−1.28351	−0.641756	+	0.766909i	$0.721794\pi$
$572$	− 5398.41i	− 0.394613i
$573$	0	0
$574$	−11810.0	−0.858784
$575$	0	0
$576$	0	0
$577$	7908.96i	0.570631i	0.958434	+	0.285316i	$0.0920984\pi$
$577$	7908.96i	0.570631i	−0.958434	+	0.285316i	$0.907902\pi$
$578$	4050.32i	0.291472i
$579$	0	0
$580$	0	0
$581$	−20462.7	−1.46116
$582$	0	0
$583$	− 7667.80i	− 0.544713i
$584$	1000.22	0.0708721
$585$	0	0
$586$	11668.2	0.822538
$587$	769.500i	0.0541067i	0.999634	+	0.0270534i	$0.00861241\pi$
$587$	769.500i	0.0541067i	−0.999634	+	0.0270534i	$0.991388\pi$
$588$	0	0
$589$	−24306.6	−1.70040
$590$	0	0
$591$	0	0
$592$	− 979.891i	− 0.0680292i
$593$	13608.5i	0.942385i	0.882030	+	0.471192i	$0.156176\pi$
$593$	13608.5i	0.942385i	−0.882030	+	0.471192i	$0.843824\pi$
$594$	0	0
$595$	0	0
$596$	809.509	0.0556355
$597$	0	0
$598$	− 5449.49i	− 0.372653i
$599$	−1242.90	−0.0847805	−0.0423902	−	0.999101i	$-0.513497\pi$
$599$	−1242.90	−0.0847805	−0.0423902	+	0.999101i	$0.513497\pi$
$600$	0	0
$601$	−8879.95	−0.602697	−0.301349	−	0.953514i	$-0.597437\pi$
$601$	−8879.95	−0.602697	−0.301349	+	0.953514i	$0.597437\pi$
$602$	8858.73i	0.599759i
$603$	0	0
$604$	−14631.2	−0.985656
$605$	0	0
$606$	0	0
$607$	17859.3i	1.19421i	0.802163	+	0.597106i	$0.203683\pi$
$607$	17859.3i	1.19421i	−0.802163	+	0.597106i	$0.796317\pi$
$608$	− 2935.64i	− 0.195815i
$609$	0	0
$610$	0	0
$611$	8209.52	0.543570
$612$	0	0
$613$	− 8241.30i	− 0.543006i	−0.962438	−	0.271503i	$-0.912479\pi$
$613$	− 8241.30i	− 0.543006i	0.962438	−	0.271503i	$-0.0875207\pi$
$614$	2920.11	0.191932
$615$	0	0
$616$	−5997.91	−0.392309
$617$	− 6854.93i	− 0.447276i	−0.974672	−	0.223638i	$-0.928207\pi$
$617$	− 6854.93i	− 0.447276i	0.974672	−	0.223638i	$-0.0717933\pi$
$618$	0	0
$619$	1857.93	0.120641	0.0603203	−	0.998179i	$-0.480788\pi$
$619$	1857.93	0.120641	0.0603203	+	0.998179i	$0.480788\pi$
$620$	0	0
$621$	0	0
$622$	8400.49i	0.541525i
$623$	− 23127.1i	− 1.48727i
$624$	0	0
$625$	0	0
$626$	−11780.3	−0.752133
$627$	0	0
$628$	9089.15i	0.577542i
$629$	−3291.12	−0.208626
$630$	0	0
$631$	−8079.78	−0.509748	−0.254874	−	0.966974i	$-0.582034\pi$
$631$	−8079.78	−0.509748	−0.254874	+	0.966974i	$0.582034\pi$
$632$	− 3433.73i	− 0.216118i
$633$	0	0
$634$	−3014.07	−0.188807
$635$	0	0
$636$	0	0
$637$	− 286.089i	− 0.0177947i
$638$	11296.2i	0.700971i
$639$	0	0
$640$	0	0
$641$	−21810.4	−1.34393	−0.671966	−	0.740582i	$-0.734550\pi$
$641$	−21810.4	−1.34393	−0.671966	+	0.740582i	$0.734550\pi$
$642$	0	0
$643$	10487.6i	0.643220i	0.946872	+	0.321610i	$0.104224\pi$
$643$	10487.6i	0.643220i	−0.946872	+	0.321610i	$0.895776\pi$
$644$	−6054.66	−0.370477
$645$	0	0
$646$	−9859.82	−0.600510
$647$	23644.5i	1.43672i	0.695669	+	0.718362i	$0.255108\pi$
$647$	23644.5i	1.43672i	−0.695669	+	0.718362i	$0.744892\pi$
$648$	0	0
$649$	−12494.8	−0.755722
$650$	0	0
$651$	0	0
$652$	− 11018.7i	− 0.661851i
$653$	− 9685.08i	− 0.580408i	−0.956965	−	0.290204i	$-0.906277\pi$
$653$	− 9685.08i	− 0.580408i	0.956965	−	0.290204i	$-0.0937232\pi$
$654$	0	0
$655$	0	0
$656$	−5039.56	−0.299942
$657$	0	0
$658$	− 9121.20i	− 0.540397i
$659$	15248.7	0.901374	0.450687	−	0.892682i	$-0.351179\pi$
$659$	15248.7	0.901374	0.450687	+	0.892682i	$0.351179\pi$
$660$	0	0
$661$	−1423.58	−0.0837683	−0.0418841	−	0.999122i	$-0.513336\pi$
$661$	−1423.58	−0.0837683	−0.0418841	+	0.999122i	$0.513336\pi$
$662$	− 6513.50i	− 0.382408i
$663$	0	0
$664$	−8731.82	−0.510332
$665$	0	0
$666$	0	0
$667$	11403.0i	0.661961i
$668$	− 4271.41i	− 0.247404i
$669$	0	0
$670$	0	0
$671$	−22020.9	−1.26693
$672$	0	0
$673$	16652.6i	0.953805i	0.878956	+	0.476902i	$0.158240\pi$
$673$	16652.6i	0.953805i	−0.878956	+	0.476902i	$0.841760\pi$
$674$	−15397.4	−0.879950
$675$	0	0
$676$	−4232.36	−0.240804
$677$	− 30296.0i	− 1.71989i	−0.510384	−	0.859947i	$-0.670497\pi$
$677$	− 30296.0i	− 1.71989i	0.510384	−	0.859947i	$-0.329503\pi$
$678$	0	0
$679$	−28729.8	−1.62378
$680$	0	0
$681$	0	0
$682$	21191.5i	1.18983i
$683$	21943.5i	1.22935i	0.788780	+	0.614675i	$0.210713\pi$
$683$	21943.5i	1.22935i	−0.788780	+	0.614675i	$0.789287\pi$
$684$	0	0
$685$	0	0
$686$	12543.1	0.698101
$687$	0	0
$688$	3780.18i	0.209474i
$689$	6470.74	0.357788
$690$	0	0
$691$	16153.5	0.889300	0.444650	−	0.895704i	$-0.353328\pi$
$691$	16153.5	0.889300	0.444650	+	0.895704i	$0.353328\pi$
$692$	11853.9i	0.651180i
$693$	0	0
$694$	19091.5	1.04424
$695$	0	0
$696$	0	0
$697$	16926.2i	0.919836i
$698$	5852.95i	0.317389i
$699$	0	0
$700$	0	0
$701$	31969.9	1.72252	0.861261	−	0.508163i	$-0.169675\pi$
$701$	31969.9	1.72252	0.861261	+	0.508163i	$0.169675\pi$
$702$	0	0
$703$	5618.37i	0.301423i
$704$	−2559.42	−0.137019
$705$	0	0
$706$	9359.39	0.498931
$707$	− 6723.66i	− 0.357665i
$708$	0	0
$709$	27367.1	1.44964	0.724819	−	0.688939i	$-0.241923\pi$
$709$	27367.1	1.44964	0.724819	+	0.688939i	$0.241923\pi$
$710$	0	0
$711$	0	0
$712$	− 9868.75i	− 0.519448i
$713$	21392.1i	1.12362i
$714$	0	0
$715$	0	0
$716$	12348.6	0.644536
$717$	0	0
$718$	− 24505.6i	− 1.27374i
$719$	−3379.36	−0.175283	−0.0876417	−	0.996152i	$-0.527933\pi$
$719$	−3379.36	−0.175283	−0.0876417	+	0.996152i	$0.527933\pi$
$720$	0	0
$721$	−8202.39	−0.423679
$722$	3113.95i	0.160512i
$723$	0	0
$724$	−16320.1	−0.837753
$725$	0	0
$726$	0	0
$727$	− 2831.94i	− 0.144472i	−0.997388	−	0.0722358i	$-0.976987\pi$
$727$	− 2831.94i	− 0.144472i	0.997388	−	0.0722358i	$-0.0230134\pi$
$728$	− 5061.55i	− 0.257683i
$729$	0	0
$730$	0	0
$731$	12696.4	0.642397
$732$	0	0
$733$	25298.0i	1.27476i	0.770548	+	0.637382i	$0.219983\pi$
$733$	25298.0i	1.27476i	−0.770548	+	0.637382i	$0.780017\pi$
$734$	23332.2	1.17331
$735$	0	0
$736$	−2583.64	−0.129394
$737$	− 22845.6i	− 1.14183i
$738$	0	0
$739$	−7545.60	−0.375601	−0.187801	−	0.982207i	$-0.560136\pi$
$739$	−7545.60	−0.375601	−0.187801	+	0.982207i	$0.560136\pi$
$740$	0	0
$741$	0	0
$742$	− 7189.33i	− 0.355699i
$743$	− 26471.0i	− 1.30704i	−0.756911	−	0.653518i	$-0.773293\pi$
$743$	− 26471.0i	− 1.30704i	0.756911	−	0.653518i	$-0.226707\pi$
$744$	0	0
$745$	0	0
$746$	3086.65	0.151488
$747$	0	0
$748$	8596.23i	0.420199i
$749$	−17300.7	−0.843999
$750$	0	0
$751$	−20997.8	−1.02027	−0.510133	−	0.860095i	$-0.670404\pi$
$751$	−20997.8	−1.02027	−0.510133	+	0.860095i	$0.670404\pi$
$752$	− 3892.18i	− 0.188741i
$753$	0	0
$754$	−9532.66	−0.460423
$755$	0	0
$756$	0	0
$757$	37735.2i	1.81177i	0.423524	+	0.905885i	$0.360793\pi$
$757$	37735.2i	1.81177i	−0.423524	+	0.905885i	$0.639207\pi$
$758$	2842.50i	0.136206i
$759$	0	0
$760$	0	0
$761$	−4449.85	−0.211967	−0.105984	−	0.994368i	$-0.533799\pi$
$761$	−4449.85	−0.211967	−0.105984	+	0.994368i	$0.533799\pi$
$762$	0	0
$763$	− 22860.7i	− 1.08468i
$764$	10688.1	0.506126
$765$	0	0
$766$	17570.8	0.828799
$767$	− 10544.2i	− 0.496386i
$768$	0	0
$769$	−6269.70	−0.294007	−0.147003	−	0.989136i	$-0.546963\pi$
$769$	−6269.70	−0.294007	−0.147003	+	0.989136i	$0.546963\pi$
$770$	0	0
$771$	0	0
$772$	18037.8i	0.840927i
$773$	23242.7i	1.08148i	0.841190	+	0.540739i	$0.181855\pi$
$773$	23242.7i	1.08148i	−0.841190	+	0.540739i	$0.818145\pi$
$774$	0	0
$775$	0	0
$776$	−12259.5	−0.567128
$777$	0	0
$778$	11110.1i	0.511976i
$779$	28895.2	1.32898
$780$	0	0
$781$	−7349.69	−0.336738
$782$	8677.57i	0.396815i
$783$	0	0
$784$	−135.636	−0.00617877
$785$	0	0
$786$	0	0
$787$	32584.4i	1.47587i	0.674873	+	0.737934i	$0.264198\pi$
$787$	32584.4i	1.47587i	−0.674873	+	0.737934i	$0.735802\pi$
$788$	− 9007.27i	− 0.407196i
$789$	0	0
$790$	0	0
$791$	−8681.05	−0.390218
$792$	0	0
$793$	− 18583.1i	− 0.832163i
$794$	−22966.3	−1.02650
$795$	0	0
$796$	6195.73	0.275882
$797$	− 6338.31i	− 0.281699i	−0.990031	−	0.140850i	$-0.955017\pi$
$797$	− 6338.31i	− 0.281699i	0.990031	−	0.140850i	$-0.0449834\pi$
$798$	0	0
$799$	−13072.5	−0.578815
$800$	0	0
$801$	0	0
$802$	− 24005.8i	− 1.05695i
$803$	4999.95i	0.219732i
$804$	0	0
$805$	0	0
$806$	−17883.2	−0.781526
$807$	0	0
$808$	− 2869.11i	− 0.124919i
$809$	−17364.4	−0.754635	−0.377317	−	0.926084i	$-0.623153\pi$
$809$	−17364.4	−0.754635	−0.377317	+	0.926084i	$0.623153\pi$
$810$	0	0
$811$	8146.53	0.352729	0.176365	−	0.984325i	$-0.443566\pi$
$811$	8146.53	0.352729	0.176365	+	0.984325i	$0.443566\pi$
$812$	10591.3i	0.457735i
$813$	0	0
$814$	4898.34	0.210917
$815$	0	0
$816$	0	0
$817$	− 21674.3i	− 0.928137i
$818$	− 19540.6i	− 0.835235i
$819$	0	0
$820$	0	0
$821$	17433.3	0.741078	0.370539	−	0.928817i	$-0.379173\pi$
$821$	17433.3	0.741078	0.370539	+	0.928817i	$0.379173\pi$
$822$	0	0
$823$	29692.7i	1.25762i	0.777559	+	0.628810i	$0.216458\pi$
$823$	29692.7i	1.25762i	−0.777559	+	0.628810i	$0.783542\pi$
$824$	−3500.11	−0.147976
$825$	0	0
$826$	−11715.1	−0.493488
$827$	9694.27i	0.407621i	0.979010	+	0.203811i	$0.0653327\pi$
$827$	9694.27i	0.407621i	−0.979010	+	0.203811i	$0.934667\pi$
$828$	0	0
$829$	10975.2	0.459814	0.229907	−	0.973213i	$-0.426158\pi$
$829$	10975.2	0.459814	0.229907	+	0.973213i	$0.426158\pi$
$830$	0	0
$831$	0	0
$832$	− 2159.85i	− 0.0899994i
$833$	455.557i	0.0189485i
$834$	0	0
$835$	0	0
$836$	14674.8	0.607105
$837$	0	0
$838$	16111.6i	0.664159i
$839$	−5314.17	−0.218671	−0.109336	−	0.994005i	$-0.534872\pi$
$839$	−5314.17	−0.218671	−0.109336	+	0.994005i	$0.534872\pi$
$840$	0	0
$841$	−4441.93	−0.182128
$842$	9935.98i	0.406670i
$843$	0	0
$844$	−8922.45	−0.363890
$845$	0	0
$846$	0	0
$847$	− 5029.50i	− 0.204033i
$848$	− 3067.82i	− 0.124233i
$849$	0	0
$850$	0	0
$851$	4944.69	0.199180
$852$	0	0
$853$	− 12904.3i	− 0.517978i	−0.965880	−	0.258989i	$-0.916611\pi$
$853$	− 12904.3i	− 0.517978i	0.965880	−	0.258989i	$-0.0833893\pi$
$854$	−20646.8	−0.827305
$855$	0	0
$856$	−7382.55	−0.294778
$857$	33810.1i	1.34764i	0.738893	+	0.673822i	$0.235349\pi$
$857$	33810.1i	1.34764i	−0.738893	+	0.673822i	$0.764651\pi$
$858$	0	0
$859$	13512.0	0.536698	0.268349	−	0.963322i	$-0.413522\pi$
$859$	13512.0	0.536698	0.268349	+	0.963322i	$0.413522\pi$
$860$	0	0
$861$	0	0
$862$	21653.1i	0.855577i
$863$	8599.41i	0.339197i	0.985513	+	0.169599i	$0.0542471\pi$
$863$	8599.41i	0.339197i	−0.985513	+	0.169599i	$0.945753\pi$
$864$	0	0
$865$	0	0
$866$	−4077.25	−0.159989
$867$	0	0
$868$	19869.2i	0.776963i
$869$	17164.7	0.670050
$870$	0	0
$871$	19279.1	0.749996
$872$	− 9755.09i	− 0.378841i
$873$	0	0
$874$	14813.7	0.573319
$875$	0	0
$876$	0	0
$877$	− 35722.7i	− 1.37545i	−0.725971	−	0.687725i	$-0.758610\pi$
$877$	− 35722.7i	− 1.37545i	0.725971	−	0.687725i	$-0.241390\pi$
$878$	− 23296.9i	− 0.895480i
$879$	0	0
$880$	0	0
$881$	44841.2	1.71480	0.857400	−	0.514650i	$-0.172078\pi$
$881$	44841.2	1.71480	0.857400	+	0.514650i	$0.172078\pi$
$882$	0	0
$883$	22688.4i	0.864694i	0.901707	+	0.432347i	$0.142315\pi$
$883$	22688.4i	0.864694i	−0.901707	+	0.432347i	$0.857685\pi$
$884$	−7254.23	−0.276002
$885$	0	0
$886$	8313.18	0.315222
$887$	14717.2i	0.557110i	0.960420	+	0.278555i	$0.0898554\pi$
$887$	14717.2i	0.557110i	−0.960420	+	0.278555i	$0.910145\pi$
$888$	0	0
$889$	−3893.48	−0.146888
$890$	0	0
$891$	0	0
$892$	− 8276.15i	− 0.310657i
$893$	22316.5i	0.836273i
$894$	0	0
$895$	0	0
$896$	−2399.71	−0.0894739
$897$	0	0
$898$	− 29975.1i	− 1.11390i
$899$	37420.6	1.38826
$900$	0	0
$901$	−10303.8	−0.380986
$902$	− 25192.1i	− 0.929939i
$903$	0	0
$904$	−3704.36	−0.136289
$905$	0	0
$906$	0	0
$907$	− 43973.3i	− 1.60982i	−0.593394	−	0.804912i	$-0.702213\pi$
$907$	− 43973.3i	− 1.60982i	0.593394	−	0.804912i	$-0.297787\pi$
$908$	20961.8i	0.766126i
$909$	0	0
$910$	0	0
$911$	−46506.3	−1.69135	−0.845677	−	0.533695i	$-0.820803\pi$
$911$	−46506.3	−1.69135	−0.845677	+	0.533695i	$0.820803\pi$
$912$	0	0
$913$	− 43649.2i	− 1.58223i
$914$	−26430.9	−0.956518
$915$	0	0
$916$	−14230.8	−0.513317
$917$	48131.8i	1.73332i
$918$	0	0
$919$	−22676.1	−0.813945	−0.406973	−	0.913440i	$-0.633416\pi$
$919$	−22676.1	−0.813945	−0.406973	+	0.913440i	$0.633416\pi$
$920$	0	0
$921$	0	0
$922$	12731.1i	0.454747i
$923$	− 6202.30i	− 0.221182i
$924$	0	0
$925$	0	0
$926$	27908.7	0.990427
$927$	0	0
$928$	4519.49i	0.159870i
$929$	−31583.2	−1.11541	−0.557703	−	0.830040i	$-0.688317\pi$
$929$	−31583.2	−1.11541	−0.557703	+	0.830040i	$0.688317\pi$
$930$	0	0
$931$	777.693	0.0273769
$932$	− 141.955i	− 0.00498914i
$933$	0	0
$934$	30578.2	1.07125
$935$	0	0
$936$	0	0
$937$	6038.94i	0.210548i	0.994443	+	0.105274i	$0.0335720\pi$
$937$	6038.94i	0.210548i	−0.994443	+	0.105274i	$0.966428\pi$
$938$	− 21420.0i	− 0.745618i
$939$	0	0
$940$	0	0
$941$	40751.5	1.41175	0.705877	−	0.708335i	$-0.250553\pi$
$941$	40751.5	1.41175	0.705877	+	0.708335i	$0.250553\pi$
$942$	0	0
$943$	− 25430.5i	− 0.878187i
$944$	−4999.05	−0.172357
$945$	0	0
$946$	−18896.6	−0.649452
$947$	− 18283.2i	− 0.627377i	−0.949526	−	0.313688i	$-0.898435\pi$
$947$	− 18283.2i	− 0.627377i	0.949526	−	0.313688i	$-0.101565\pi$
$948$	0	0
$949$	−4219.39	−0.144328
$950$	0	0
$951$	0	0
$952$	8059.82i	0.274391i
$953$	31486.4i	1.07025i	0.844774	+	0.535123i	$0.179735\pi$
$953$	31486.4i	1.07025i	−0.844774	+	0.535123i	$0.820265\pi$
$954$	0	0
$955$	0	0
$956$	−4682.50	−0.158413
$957$	0	0
$958$	− 16536.7i	− 0.557700i
$959$	−298.473	−0.0100502
$960$	0	0
$961$	40409.9	1.35645
$962$	4133.64i	0.138538i
$963$	0	0
$964$	13278.0	0.443626
$965$	0	0
$966$	0	0
$967$	− 30578.2i	− 1.01689i	−0.861096	−	0.508443i	$-0.830221\pi$
$967$	− 30578.2i	− 1.01689i	0.861096	−	0.508443i	$-0.169779\pi$
$968$	− 2146.18i	− 0.0712613i
$969$	0	0
$970$	0	0
$971$	8854.87	0.292653	0.146327	−	0.989236i	$-0.453255\pi$
$971$	8854.87	0.292653	0.146327	+	0.989236i	$0.453255\pi$
$972$	0	0
$973$	− 53236.9i	− 1.75406i
$974$	−14289.6	−0.470089
$975$	0	0
$976$	−8810.36	−0.288948
$977$	24452.9i	0.800736i	0.916355	+	0.400368i	$0.131118\pi$
$977$	24452.9i	0.800736i	−0.916355	+	0.400368i	$0.868882\pi$
$978$	0	0
$979$	49332.5	1.61049
$980$	0	0
$981$	0	0
$982$	− 39054.4i	− 1.26912i
$983$	− 5512.42i	− 0.178860i	−0.995993	−	0.0894298i	$-0.971496\pi$
$983$	− 5512.42i	− 0.178860i	0.995993	−	0.0894298i	$-0.0285045\pi$
$984$	0	0
$985$	0	0
$986$	15179.5	0.490276
$987$	0	0
$988$	12383.9i	0.398769i
$989$	−19075.4	−0.613309
$990$	0	0
$991$	22724.4	0.728420	0.364210	−	0.931317i	$-0.381339\pi$
$991$	22724.4	0.728420	0.364210	+	0.931317i	$0.381339\pi$
$992$	8478.55i	0.271365i
$993$	0	0
$994$	−6891.07	−0.219891
$995$	0	0
$996$	0	0
$997$	− 10218.0i	− 0.324582i	−0.986743	−	0.162291i	$-0.948112\pi$
$997$	− 10218.0i	− 0.324582i	0.986743	−	0.162291i	$-0.0518883\pi$
$998$	25847.4i	0.819825i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.v.649.3		4
3.2	odd	2		1350.4.c.ba.649.1		4
5.2	odd	4		1350.4.a.bg.1.2	yes	2
5.3	odd	4		1350.4.a.bl.1.1	yes	2
5.4	even	2	inner	1350.4.c.v.649.2		4
15.2	even	4		1350.4.a.bn.1.2	yes	2
15.8	even	4		1350.4.a.be.1.1	✓	2
15.14	odd	2		1350.4.c.ba.649.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.4.a.be.1.1	✓	2	15.8	even	4
1350.4.a.bg.1.2	yes	2	5.2	odd	4
1350.4.a.bl.1.1	yes	2	5.3	odd	4
1350.4.a.bn.1.2	yes	2	15.2	even	4
1350.4.c.v.649.2		4	5.4	even	2	inner
1350.4.c.v.649.3		4	1.1	even	1	trivial
1350.4.c.ba.649.1		4	3.2	odd	2
1350.4.c.ba.649.4		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} -18.7477i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} -18.7477i q^{7} -8.00000i q^{8} +39.9909 q^{11} +33.7477i q^{13} +37.4955 q^{14} +16.0000 q^{16} -53.7386i q^{17} -91.7386 q^{19} +79.9818i q^{22} +80.7386i q^{23} -67.4955 q^{26} +74.9909i q^{28} +141.234 q^{29} +264.955 q^{31} +32.0000i q^{32} +107.477 q^{34} -61.2432i q^{37} -183.477i q^{38} -314.973 q^{41} +236.261i q^{43} -159.964 q^{44} -161.477 q^{46} -243.261i q^{47} -8.47727 q^{49} -134.991i q^{52} -191.739i q^{53} -149.982 q^{56} +282.468i q^{58} -312.441 q^{59} -550.648 q^{61} +529.909i q^{62} -64.0000 q^{64} -571.270i q^{67} +214.955i q^{68} -183.784 q^{71} +125.027i q^{73} +122.486 q^{74} +366.955 q^{76} -749.739i q^{77} +429.216 q^{79} -629.945i q^{82} -1091.48i q^{83} -472.523 q^{86} -319.927i q^{88} +1233.59 q^{89} +632.693 q^{91} -322.955i q^{92} +486.523 q^{94} -1532.44i q^{97} -16.9545i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4}+O(q^{10}) \) 4 * q - 16 * q^4	\( 4 q - 16 q^{4} - 60 q^{11} + 40 q^{14} + 64 q^{16} - 92 q^{19} - 160 q^{26} + 180 q^{29} - 40 q^{31} - 120 q^{34} - 600 q^{41} + 240 q^{44} - 96 q^{46} + 516 q^{49} - 160 q^{56} + 180 q^{59} + 272 q^{61} - 256 q^{64} - 1560 q^{71} + 160 q^{74} + 368 q^{76} + 892 q^{79} - 2440 q^{86} + 1140 q^{89} + 1156 q^{91} + 2496 q^{94}+O(q^{100}) \) 4 * q - 16 * q^4 - 60 * q^11 + 40 * q^14 + 64 * q^16 - 92 * q^19 - 160 * q^26 + 180 * q^29 - 40 * q^31 - 120 * q^34 - 600 * q^41 + 240 * q^44 - 96 * q^46 + 516 * q^49 - 160 * q^56 + 180 * q^59 + 272 * q^61 - 256 * q^64 - 1560 * q^71 + 160 * q^74 + 368 * q^76 + 892 * q^79 - 2440 * q^86 + 1140 * q^89 + 1156 * q^91 + 2496 * q^94

Introduction

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