LMFDB - Embedded newform 3450.2.d.g.2899.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(2899,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3450.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.d (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:	$27.5483886973$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3450.2899
Dual form			3450.2.d.g.2899.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} -2.00000 q^{19} +2.00000 q^{21} +1.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +1.00000 q^{36} -10.0000i q^{37} +2.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -2.00000i q^{42} -2.00000i q^{43} +1.00000 q^{46} -1.00000i q^{48} +3.00000 q^{49} +2.00000i q^{52} -12.0000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +2.00000i q^{57} -6.00000i q^{58} -12.0000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +14.0000i q^{67} +1.00000 q^{69} -1.00000i q^{72} -2.00000i q^{73} -10.0000 q^{74} +2.00000 q^{76} +2.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -2.00000 q^{84} -2.00000 q^{86} -6.00000i q^{87} -12.0000 q^{89} +4.00000 q^{91} -1.00000i q^{92} +4.00000i q^{93} -1.00000 q^{96} -10.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{21} + 2 q^{24} - 4 q^{26} + 12 q^{29} - 8 q^{31} + 2 q^{36} - 4 q^{39} - 12 q^{41} + 2 q^{46} + 6 q^{49} + 2 q^{54} - 4 q^{56} - 24 q^{59} - 20 q^{61} - 2 q^{64} + 2 q^{69} - 20 q^{74} + 4 q^{76} + 20 q^{79} + 2 q^{81} - 4 q^{84} - 4 q^{86} - 24 q^{89} + 8 q^{91} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 - 4 * q^19 + 4 * q^21 + 2 * q^24 - 4 * q^26 + 12 * q^29 - 8 * q^31 + 2 * q^36 - 4 * q^39 - 12 * q^41 + 2 * q^46 + 6 * q^49 + 2 * q^54 - 4 * q^56 - 24 * q^59 - 20 * q^61 - 2 * q^64 + 2 * q^69 - 20 * q^74 + 4 * q^76 + 20 * q^79 + 2 * q^81 - 4 * q^84 - 4 * q^86 - 24 * q^89 + 8 * q^91 - 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000i	0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.00000i	0.235702i
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000i	0.192450i
$28$	− 2.00000i	− 0.377964i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	2.00000i	0.324443i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 2.00000i	− 0.308607i
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	0	0
$45$	0	0
$46$	1.00000	0.147442
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	2.00000i	0.277350i
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	2.00000i	0.264906i
$58$	− 6.00000i	− 0.787839i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000i	0.508001i
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	14.0000i	1.71037i	0.518321	+	0.855186i	$0.326557\pi$
$67$	14.0000i	1.71037i	−0.518321	+	0.855186i	$0.673443\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	2.00000i	0.226455i
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−2.00000	−0.215666
$87$	− 6.00000i	− 0.643268i
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	− 1.00000i	− 0.104257i
$93$	4.00000i	0.414781i
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	− 14.0000i	− 1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$	− 14.0000i	− 1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−12.0000	−1.16554
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	2.00000i	0.188982i
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	−6.00000	−0.557086
$117$	2.00000i	0.184900i
$118$	12.0000i	1.10469i
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	10.0000i	0.905357i
$123$	6.00000i	0.541002i
$124$	4.00000	0.359211
$125$	0	0
$126$	−2.00000	−0.178174
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	1.00000i	0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	14.0000	1.20942
$135$	0	0
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	− 1.00000i	− 0.0851257i
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	− 3.00000i	− 0.247436i
$148$	10.0000i	0.821995i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 2.00000i	− 0.162221i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	− 10.0000i	− 0.795557i
$159$	−12.0000	−0.951662
$160$	0	0
$161$	−2.00000	−0.157622
$162$	− 1.00000i	− 0.0785674i
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	0	0
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	2.00000i	0.154303i
$169$	9.00000	0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	2.00000i	0.152499i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	0	0
$177$	12.0000i	0.901975i
$178$	12.0000i	0.899438i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 4.00000i	− 0.296500i
$183$	10.0000i	0.739221i
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	4.00000	0.293294
$187$	0	0
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.00000i	0.0721688i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	18.0000i	1.26648i
$203$	12.0000i	0.842235i
$204$	0	0
$205$	0	0
$206$	−14.0000	−0.975426
$207$	− 1.00000i	− 0.0695048i
$208$	− 2.00000i	− 0.138675i
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	12.0000i	0.824163i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 8.00000i	− 0.543075i
$218$	2.00000i	0.135457i
$219$	−2.00000	−0.135147
$220$	0	0
$221$	0	0
$222$	10.0000i	0.671156i
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−12.0000	−0.798228
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 2.00000i	− 0.132453i
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	12.0000	0.781133
$237$	− 10.0000i	− 0.649570i
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	11.0000i	0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	6.00000	0.382546
$247$	4.00000i	0.254514i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	2.00000i	0.125988i
$253$	0	0
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	2.00000i	0.124515i
$259$	20.0000	1.24274
$260$	0	0
$261$	−6.00000	−0.371391
$262$	12.0000i	0.741362i
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	12.0000i	0.734388i
$268$	− 14.0000i	− 0.855186i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	− 4.00000i	− 0.242091i
$274$	12.0000	0.724947
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	14.0000i	0.841178i	0.907251	+	0.420589i	$0.138177\pi$
$277$	14.0000i	0.841178i	−0.907251	+	0.420589i	$0.861823\pi$
$278$	20.0000i	1.19952i
$279$	4.00000	0.239474
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	10.0000i	0.594438i	0.954809	+	0.297219i	$0.0960592\pi$
$283$	10.0000i	0.594438i	−0.954809	+	0.297219i	$0.903941\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 12.0000i	− 0.708338i
$288$	1.00000i	0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	−10.0000	−0.586210
$292$	2.00000i	0.117041i
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	10.0000	0.581238
$297$	0	0
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	4.00000	0.230556
$302$	− 8.00000i	− 0.460348i
$303$	18.0000i	1.03407i
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	− 2.00000i	− 0.113228i
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−10.0000	−0.562544
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	12.0000i	0.672927i
$319$	0	0
$320$	0	0
$321$	0	0
$322$	2.00000i	0.111456i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−8.00000	−0.443079
$327$	2.00000i	0.110600i
$328$	− 6.00000i	− 0.331295i
$329$	0	0
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	0	0
$333$	10.0000i	0.547997i
$334$	−24.0000	−1.31322
$335$	0	0
$336$	2.00000	0.109109
$337$	− 10.0000i	− 0.544735i	−0.962193	−	0.272367i	$-0.912193\pi$
$337$	− 10.0000i	− 0.544735i	0.962193	−	0.272367i	$-0.0878066\pi$
$338$	− 9.00000i	− 0.489535i
$339$	−12.0000	−0.651751
$340$	0	0
$341$	0	0
$342$	− 2.00000i	− 0.108148i
$343$	20.0000i	1.07990i
$344$	2.00000	0.107833
$345$	0	0
$346$	6.00000	0.322562
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	6.00000i	0.321634i
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	12.0000	0.635999
$357$	0	0
$358$	12.0000i	0.634220i
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 2.00000i	− 0.105118i
$363$	11.0000i	0.577350i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	10.0000	0.522708
$367$	− 10.0000i	− 0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$	− 10.0000i	− 0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$	1.00000i	0.0521286i
$369$	6.00000	0.312348
$370$	0	0
$371$	24.0000	1.24602
$372$	− 4.00000i	− 0.207390i
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	2.00000i	0.102869i
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	12.0000i	0.613973i
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	2.00000i	0.101666i
$388$	10.0000i	0.507673i
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	0	0
$392$	3.00000i	0.151523i
$393$	12.0000i	0.605320i
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	− 10.0000i	− 0.501255i
$399$	−4.00000	−0.200250
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	− 14.0000i	− 0.698257i
$403$	8.00000i	0.398508i
$404$	18.0000	0.895533
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	14.0000i	0.689730i
$413$	− 24.0000i	− 1.18096i
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	20.0000i	0.979404i
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	− 8.00000i	− 0.389434i
$423$	0	0
$424$	12.0000	0.582772
$425$	0	0
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	1.00000i	0.0481125i
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 2.00000i	− 0.0956730i
$438$	2.00000i	0.0955637i
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	− 36.0000i	− 1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$	− 36.0000i	− 1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	10.0000	0.474579
$445$	0	0
$446$	16.0000	0.757622
$447$	0	0
$448$	− 2.00000i	− 0.0944911i
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	0	0
$452$	12.0000i	0.564433i
$453$	− 8.00000i	− 0.375873i
$454$	12.0000	0.563188
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	− 34.0000i	− 1.59045i	−0.606313	−	0.795226i	$-0.707352\pi$
$457$	− 34.0000i	− 1.59045i	0.606313	−	0.795226i	$-0.292648\pi$
$458$	26.0000i	1.21490i
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	−28.0000	−1.29292
$470$	0	0
$471$	14.0000	0.645086
$472$	− 12.0000i	− 0.552345i
$473$	0	0
$474$	−10.0000	−0.459315
$475$	0	0
$476$	0	0
$477$	12.0000i	0.549442i
$478$	− 24.0000i	− 1.09773i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	− 14.0000i	− 0.637683i
$483$	2.00000i	0.0910032i
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	− 10.0000i	− 0.452679i
$489$	−8.00000	−0.361773
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	− 6.00000i	− 0.270501i
$493$	0	0
$494$	4.00000	0.179969
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	12.0000i	0.535586i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	4.00000i	0.177471i
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	− 1.00000i	− 0.0441942i
$513$	− 2.00000i	− 0.0883022i
$514$	−6.00000	−0.264649
$515$	0	0
$516$	2.00000	0.0880451
$517$	0	0
$518$	− 20.0000i	− 0.878750i
$519$	6.00000	0.263371
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	6.00000i	0.262613i
$523$	34.0000i	1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$	34.0000i	1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−12.0000	−0.523225
$527$	0	0
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	12.0000	0.520756
$532$	4.00000i	0.173422i
$533$	12.0000i	0.519778i
$534$	12.0000	0.519291
$535$	0	0
$536$	−14.0000	−0.604708
$537$	12.0000i	0.517838i
$538$	− 6.00000i	− 0.258678i
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	− 20.0000i	− 0.859074i
$543$	− 2.00000i	− 0.0858282i
$544$	0	0
$545$	0	0
$546$	−4.00000	−0.171184
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	− 12.0000i	− 0.512615i
$549$	10.0000	0.426790
$550$	0	0
$551$	−12.0000	−0.511217
$552$	1.00000i	0.0425628i
$553$	20.0000i	0.850487i
$554$	14.0000	0.594803
$555$	0	0
$556$	20.0000	0.848189
$557$	− 36.0000i	− 1.52537i	−0.646771	−	0.762684i	$-0.723881\pi$
$557$	− 36.0000i	− 1.52537i	0.646771	−	0.762684i	$-0.276119\pi$
$558$	− 4.00000i	− 0.169334i
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	12.0000i	0.506189i
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	0	0
$565$	0	0
$566$	10.0000	0.420331
$567$	2.00000i	0.0839921i
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	38.0000	1.59025	0.795125	−	0.606445i	$-0.207405\pi$
$571$	38.0000	1.59025	0.795125	+	0.606445i	$0.207405\pi$
$572$	0	0
$573$	12.0000i	0.501307i
$574$	−12.0000	−0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 34.0000i	− 1.41544i	−0.706494	−	0.707719i	$-0.749724\pi$
$577$	− 34.0000i	− 1.41544i	0.706494	−	0.707719i	$-0.250276\pi$
$578$	− 17.0000i	− 0.707107i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	0	0
$582$	10.0000i	0.414513i
$583$	0	0
$584$	2.00000	0.0827606
$585$	0	0
$586$	−12.0000	−0.495715
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	3.00000i	0.123718i
$589$	8.00000	0.329634
$590$	0	0
$591$	18.0000	0.740421
$592$	− 10.0000i	− 0.410997i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 10.0000i	− 0.409273i
$598$	− 2.00000i	− 0.0817861i
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	− 4.00000i	− 0.163028i
$603$	− 14.0000i	− 0.570124i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	18.0000	0.731200
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	2.00000i	0.0811107i
$609$	12.0000	0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 38.0000i	− 1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$	− 38.0000i	− 1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	− 12.0000i	− 0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$	− 12.0000i	− 0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$	14.0000i	0.563163i
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	24.0000i	0.962312i
$623$	− 24.0000i	− 0.961540i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	22.0000	0.879297
$627$	0	0
$628$	− 14.0000i	− 0.558661i
$629$	0	0
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	10.0000i	0.397779i
$633$	− 8.00000i	− 0.317971i
$634$	−6.00000	−0.238290
$635$	0	0
$636$	12.0000	0.475831
$637$	− 6.00000i	− 0.237729i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−48.0000	−1.89589	−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000	−1.89589	−0.947943	+	0.318440i	$0.896841\pi$
$642$	0	0
$643$	22.0000i	0.867595i	0.901010	+	0.433798i	$0.142827\pi$
$643$	22.0000i	0.867595i	−0.901010	+	0.433798i	$0.857173\pi$
$644$	2.00000	0.0788110
$645$	0	0
$646$	0	0
$647$	48.0000i	1.88707i	0.331266	+	0.943537i	$0.392524\pi$
$647$	48.0000i	1.88707i	−0.331266	+	0.943537i	$0.607476\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	−8.00000	−0.313545
$652$	8.00000i	0.313304i
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	− 32.0000i	− 1.24372i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	10.0000	0.387492
$667$	6.00000i	0.232321i
$668$	24.0000i	0.928588i
$669$	16.0000	0.618596
$670$	0	0
$671$	0	0
$672$	− 2.00000i	− 0.0771517i
$673$	− 38.0000i	− 1.46479i	−0.680879	−	0.732396i	$-0.738402\pi$
$673$	− 38.0000i	− 1.46479i	0.680879	−	0.732396i	$-0.261598\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 48.0000i	− 1.84479i	−0.386248	−	0.922395i	$-0.626229\pi$
$677$	− 48.0000i	− 1.84479i	0.386248	−	0.922395i	$-0.373771\pi$
$678$	12.0000i	0.460857i
$679$	20.0000	0.767530
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	20.0000	0.763604
$687$	26.0000i	0.991962i
$688$	− 2.00000i	− 0.0762493i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	6.00000	0.227429
$697$	0	0
$698$	2.00000i	0.0757011i
$699$	6.00000	0.226941
$700$	0	0
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	20.0000i	0.754314i
$704$	0	0
$705$	0	0
$706$	−30.0000	−1.12906
$707$	− 36.0000i	− 1.35392i
$708$	− 12.0000i	− 0.450988i
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	− 12.0000i	− 0.449719i
$713$	− 4.00000i	− 0.149801i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	− 24.0000i	− 0.896296i
$718$	36.0000i	1.34351i
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	15.0000i	0.558242i
$723$	− 14.0000i	− 0.520666i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	11.0000	0.408248
$727$	26.0000i	0.964287i	0.876092	+	0.482143i	$0.160142\pi$
$727$	26.0000i	0.964287i	−0.876092	+	0.482143i	$0.839858\pi$
$728$	4.00000i	0.148250i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	− 10.0000i	− 0.369611i
$733$	46.0000i	1.69905i	0.527549	+	0.849524i	$0.323111\pi$
$733$	46.0000i	1.69905i	−0.527549	+	0.849524i	$0.676889\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	1.00000	0.0368605
$737$	0	0
$738$	− 6.00000i	− 0.220863i
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	− 24.0000i	− 0.881068i
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	10.0000	0.366126
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	0	0
$753$	12.0000i	0.437304i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	2.00000	0.0727393
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	26.0000i	0.944363i
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	4.00000i	0.144905i
$763$	− 4.00000i	− 0.144810i
$764$	12.0000	0.434145
$765$	0	0
$766$	−12.0000	−0.433578
$767$	24.0000i	0.866590i
$768$	− 1.00000i	− 0.0360844i
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	14.0000i	0.503871i
$773$	− 24.0000i	− 0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$773$	− 24.0000i	− 0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	10.0000	0.358979
$777$	− 20.0000i	− 0.717496i
$778$	12.0000i	0.430221i
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	6.00000i	0.214423i
$784$	3.00000	0.107143
$785$	0	0
$786$	12.0000	0.428026
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	− 18.0000i	− 0.641223i
$789$	−12.0000	−0.427211
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	20.0000i	0.710221i
$794$	14.0000	0.496841
$795$	0	0
$796$	−10.0000	−0.354441
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	4.00000i	0.141598i
$799$	0	0
$800$	0	0
$801$	12.0000	0.423999
$802$	− 24.0000i	− 0.847469i
$803$	0	0
$804$	−14.0000	−0.493742
$805$	0	0
$806$	8.00000	0.281788
$807$	− 6.00000i	− 0.211210i
$808$	− 18.0000i	− 0.633238i
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	− 12.0000i	− 0.421117i
$813$	− 20.0000i	− 0.701431i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.00000i	0.139942i
$818$	− 10.0000i	− 0.349642i
$819$	−4.00000	−0.139771
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	− 12.0000i	− 0.418548i
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	−24.0000	−0.835067
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	1.00000i	0.0347524i
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	2.00000i	0.0693375i
$833$	0	0
$834$	20.0000	0.692543
$835$	0	0
$836$	0	0
$837$	− 4.00000i	− 0.138260i
$838$	0	0
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 2.00000i	− 0.0689246i
$843$	12.0000i	0.413302i
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	− 22.0000i	− 0.755929i
$848$	− 12.0000i	− 0.412082i
$849$	10.0000	0.343199
$850$	0	0
$851$	10.0000	0.342796
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	−20.0000	−0.684386
$855$	0	0
$856$	0	0
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	− 24.0000i	− 0.817443i
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−26.0000	−0.883516
$867$	− 17.0000i	− 0.577350i
$868$	8.00000i	0.271538i
$869$	0	0
$870$	0	0
$871$	28.0000	0.948744
$872$	− 2.00000i	− 0.0677285i
$873$	10.0000i	0.338449i
$874$	−2.00000	−0.0676510
$875$	0	0
$876$	2.00000	0.0675737
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	8.00000i	0.269987i
$879$	−12.0000	−0.404750
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	3.00000i	0.101015i
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	0	0
$885$	0	0
$886$	−36.0000	−1.20944
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	− 10.0000i	− 0.335578i
$889$	8.00000	0.268311
$890$	0	0
$891$	0	0
$892$	− 16.0000i	− 0.535720i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	− 2.00000i	− 0.0667781i
$898$	18.0000i	0.600668i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 4.00000i	− 0.133112i
$904$	12.0000	0.399114
$905$	0	0
$906$	−8.00000	−0.265782
$907$	− 34.0000i	− 1.12895i	−0.825450	−	0.564476i	$-0.809078\pi$
$907$	− 34.0000i	− 1.12895i	0.825450	−	0.564476i	$-0.190922\pi$
$908$	− 12.0000i	− 0.398234i
$909$	18.0000	0.597022
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	2.00000i	0.0662266i
$913$	0	0
$914$	−34.0000	−1.12462
$915$	0	0
$916$	26.0000	0.859064
$917$	− 24.0000i	− 0.792550i
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	30.0000i	0.987997i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	14.0000i	0.459820i
$928$	− 6.00000i	− 0.196960i
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	− 6.00000i	− 0.196537i
$933$	24.0000i	0.785725i
$934$	−12.0000	−0.392652
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	− 46.0000i	− 1.50275i	−0.659873	−	0.751377i	$-0.729390\pi$
$937$	− 46.0000i	− 1.50275i	0.659873	−	0.751377i	$-0.270610\pi$
$938$	28.0000i	0.914232i
$939$	22.0000	0.717943
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	− 14.0000i	− 0.456145i
$943$	− 6.00000i	− 0.195387i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	10.0000i	0.324785i
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	− 24.0000i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$	− 24.0000i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$	12.0000	0.388514
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	0	0
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	20.0000i	0.644826i
$963$	0	0
$964$	−14.0000	−0.450910
$965$	0	0
$966$	2.00000	0.0643489
$967$	− 4.00000i	− 0.128631i	−0.997930	−	0.0643157i	$-0.979514\pi$
$967$	− 4.00000i	− 0.128631i	0.997930	−	0.0643157i	$-0.0204865\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	1.00000i	0.0320750i
$973$	− 40.0000i	− 1.28234i
$974$	8.00000	0.256337
$975$	0	0
$976$	−10.0000	−0.320092
$977$	36.0000i	1.15174i	0.817541	+	0.575871i	$0.195337\pi$
$977$	36.0000i	1.15174i	−0.817541	+	0.575871i	$0.804663\pi$
$978$	8.00000i	0.255812i
$979$	0	0
$980$	0	0
$981$	2.00000	0.0638551
$982$	− 12.0000i	− 0.382935i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 4.00000i	− 0.127257i
$989$	2.00000	0.0635963
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	4.00000i	0.127000i
$993$	− 32.0000i	− 1.01549i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	26.0000i	0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$	26.0000i	0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$	− 40.0000i	− 1.26618i
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.d.g.2899.1		2
5.2	odd	4		3450.2.a.o.1.1		1
5.3	odd	4		138.2.a.b.1.1	✓	1
5.4	even	2	inner	3450.2.d.g.2899.2		2
15.8	even	4		414.2.a.c.1.1		1
20.3	even	4		1104.2.a.b.1.1		1
35.13	even	4		6762.2.a.g.1.1		1
40.3	even	4		4416.2.a.t.1.1		1
40.13	odd	4		4416.2.a.i.1.1		1
60.23	odd	4		3312.2.a.h.1.1		1
115.68	even	4		3174.2.a.d.1.1		1
345.68	odd	4		9522.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.a.b.1.1	✓	1	5.3	odd	4
414.2.a.c.1.1		1	15.8	even	4
1104.2.a.b.1.1		1	20.3	even	4
3174.2.a.d.1.1		1	115.68	even	4
3312.2.a.h.1.1		1	60.23	odd	4
3450.2.a.o.1.1		1	5.2	odd	4
3450.2.d.g.2899.1		2	1.1	even	1	trivial
3450.2.d.g.2899.2		2	5.4	even	2	inner
4416.2.a.i.1.1		1	40.13	odd	4
4416.2.a.t.1.1		1	40.3	even	4
6762.2.a.g.1.1		1	35.13	even	4
9522.2.a.k.1.1		1	345.68	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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} -2.00000 q^{19} +2.00000 q^{21} +1.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +1.00000 q^{36} -10.0000i q^{37} +2.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} -2.00000i q^{42} -2.00000i q^{43} +1.00000 q^{46} -1.00000i q^{48} +3.00000 q^{49} +2.00000i q^{52} -12.0000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +2.00000i q^{57} -6.00000i q^{58} -12.0000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +14.0000i q^{67} +1.00000 q^{69} -1.00000i q^{72} -2.00000i q^{73} -10.0000 q^{74} +2.00000 q^{76} +2.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} -2.00000 q^{84} -2.00000 q^{86} -6.00000i q^{87} -12.0000 q^{89} +4.00000 q^{91} -1.00000i q^{92} +4.00000i q^{93} -1.00000 q^{96} -10.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{21} + 2 q^{24} - 4 q^{26} + 12 q^{29} - 8 q^{31} + 2 q^{36} - 4 q^{39} - 12 q^{41} + 2 q^{46} + 6 q^{49} + 2 q^{54} - 4 q^{56} - 24 q^{59} - 20 q^{61} - 2 q^{64} + 2 q^{69} - 20 q^{74} + 4 q^{76} + 20 q^{79} + 2 q^{81} - 4 q^{84} - 4 q^{86} - 24 q^{89} + 8 q^{91} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 - 4 * q^19 + 4 * q^21 + 2 * q^24 - 4 * q^26 + 12 * q^29 - 8 * q^31 + 2 * q^36 - 4 * q^39 - 12 * q^41 + 2 * q^46 + 6 * q^49 + 2 * q^54 - 4 * q^56 - 24 * q^59 - 20 * q^61 - 2 * q^64 + 2 * q^69 - 20 * q^74 + 4 * q^76 + 20 * q^79 + 2 * q^81 - 4 * q^84 - 4 * q^86 - 24 * q^89 + 8 * q^91 - 2 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more