LMFDB - Embedded newform 7200.2.h.h.1151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(1151,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("7200.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.h (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:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	7200.1151
Dual form			7200.2.h.h.1151.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421i q^{7} +O(q^{10})$	$q-2.41421i q^{7} +5.41421 q^{11} +6.65685 q^{13} -3.41421i q^{17} -7.24264i q^{19} -2.58579 q^{23} -10.2426i q^{29} +5.24264i q^{31} +6.82843 q^{37} -4.82843i q^{41} +3.58579i q^{43} +7.41421 q^{47} +1.17157 q^{49} +0.828427i q^{53} -5.07107 q^{59} -1.82843 q^{61} +3.24264i q^{67} -11.6569 q^{71} -16.4853 q^{73} -13.0711i q^{77} -12.0000i q^{79} +4.58579 q^{83} +12.0000i q^{89} -16.0711i q^{91} +9.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 16 q^{11} + 4 q^{13} - 16 q^{23} + 16 q^{37} + 24 q^{47} + 16 q^{49} + 8 q^{59} + 4 q^{61} - 24 q^{71} - 32 q^{73} + 24 q^{83} + 36 q^{97}+O(q^{100}) $ 4 * q + 16 * q^11 + 4 * q^13 - 16 * q^23 + 16 * q^37 + 24 * q^47 + 16 * q^49 + 8 * q^59 + 4 * q^61 - 24 * q^71 - 32 * q^73 + 24 * q^83 + 36 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.41421i	− 0.912487i	−0.889855	−	0.456243i	$-0.849195\pi$
$7$	− 2.41421i	− 0.912487i	0.889855	−	0.456243i	$-0.150805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.41421	1.63245	0.816223	−	0.577736i	$-0.196064\pi$
$11$	5.41421	1.63245	0.816223	+	0.577736i	$0.196064\pi$
$12$	0	0
$13$	6.65685	1.84628	0.923140	−	0.384465i	$-0.125614\pi$
$13$	6.65685	1.84628	0.923140	+	0.384465i	$0.125614\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.41421i	− 0.828068i	−0.910261	−	0.414034i	$-0.864119\pi$
$17$	− 3.41421i	− 0.828068i	0.910261	−	0.414034i	$-0.135881\pi$
$18$	0	0
$19$	− 7.24264i	− 1.66158i	−0.556589	−	0.830788i	$-0.687890\pi$
$19$	− 7.24264i	− 1.66158i	0.556589	−	0.830788i	$-0.312110\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.58579	−0.539174	−0.269587	−	0.962976i	$-0.586887\pi$
$23$	−2.58579	−0.539174	−0.269587	+	0.962976i	$0.586887\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 10.2426i	− 1.90201i	−0.309175	−	0.951005i	$-0.600053\pi$
$29$	− 10.2426i	− 1.90201i	0.309175	−	0.951005i	$-0.399947\pi$
$30$	0	0
$31$	5.24264i	0.941606i	0.882238	+	0.470803i	$0.156036\pi$
$31$	5.24264i	0.941606i	−0.882238	+	0.470803i	$0.843964\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.82843	1.12259	0.561293	−	0.827617i	$-0.310304\pi$
$37$	6.82843	1.12259	0.561293	+	0.827617i	$0.310304\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.82843i	− 0.754074i	−0.926198	−	0.377037i	$-0.876943\pi$
$41$	− 4.82843i	− 0.754074i	0.926198	−	0.377037i	$-0.123057\pi$
$42$	0	0
$43$	3.58579i	0.546827i	0.961897	+	0.273414i	$0.0881528\pi$
$43$	3.58579i	0.546827i	−0.961897	+	0.273414i	$0.911847\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.41421	1.08147	0.540737	−	0.841192i	$-0.318145\pi$
$47$	7.41421	1.08147	0.540737	+	0.841192i	$0.318145\pi$
$48$	0	0
$49$	1.17157	0.167368
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.828427i	0.113793i	0.998380	+	0.0568966i	$0.0181205\pi$
$53$	0.828427i	0.113793i	−0.998380	+	0.0568966i	$0.981879\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.07107	−0.660197	−0.330098	−	0.943947i	$-0.607082\pi$
$59$	−5.07107	−0.660197	−0.330098	+	0.943947i	$0.607082\pi$
$60$	0	0
$61$	−1.82843	−0.234106	−0.117053	−	0.993126i	$-0.537345\pi$
$61$	−1.82843	−0.234106	−0.117053	+	0.993126i	$0.537345\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.24264i	0.396152i	0.980187	+	0.198076i	$0.0634692\pi$
$67$	3.24264i	0.396152i	−0.980187	+	0.198076i	$0.936531\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.6569	−1.38341	−0.691707	−	0.722178i	$-0.743141\pi$
$71$	−11.6569	−1.38341	−0.691707	+	0.722178i	$0.743141\pi$
$72$	0	0
$73$	−16.4853	−1.92946	−0.964728	−	0.263248i	$-0.915206\pi$
$73$	−16.4853	−1.92946	−0.964728	+	0.263248i	$0.915206\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 13.0711i	− 1.48959i
$78$	0	0
$79$	− 12.0000i	− 1.35011i	−0.737769	−	0.675053i	$-0.764121\pi$
$79$	− 12.0000i	− 1.35011i	0.737769	−	0.675053i	$-0.235879\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.58579	0.503355	0.251678	−	0.967811i	$-0.419018\pi$
$83$	4.58579	0.503355	0.251678	+	0.967811i	$0.419018\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.0000i	1.27200i	0.771690	+	0.635999i	$0.219412\pi$
$89$	12.0000i	1.27200i	−0.771690	+	0.635999i	$0.780588\pi$
$90$	0	0
$91$	− 16.0711i	− 1.68471i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.00000	0.913812	0.456906	−	0.889515i	$-0.348958\pi$
$97$	9.00000	0.913812	0.456906	+	0.889515i	$0.348958\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 0.828427i	− 0.0824316i	−0.999150	−	0.0412158i	$-0.986877\pi$
$101$	− 0.828427i	− 0.0824316i	0.999150	−	0.0412158i	$-0.0131231\pi$
$102$	0	0
$103$	7.65685i	0.754452i	0.926121	+	0.377226i	$0.123122\pi$
$103$	7.65685i	0.754452i	−0.926121	+	0.377226i	$0.876878\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.3137	−1.67378	−0.836890	−	0.547372i	$-0.815628\pi$
$107$	−17.3137	−1.67378	−0.836890	+	0.547372i	$0.815628\pi$
$108$	0	0
$109$	8.17157	0.782695	0.391347	−	0.920243i	$-0.372009\pi$
$109$	8.17157	0.782695	0.391347	+	0.920243i	$0.372009\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.65685i	0.908440i	0.890889	+	0.454220i	$0.150082\pi$
$113$	9.65685i	0.908440i	−0.890889	+	0.454220i	$0.849918\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.24264	−0.755602
$120$	0	0
$121$	18.3137	1.66488
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.485281i	0.0430618i	0.999768	+	0.0215309i	$0.00685402\pi$
$127$	0.485281i	0.0430618i	−0.999768	+	0.0215309i	$0.993146\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	−17.4853	−1.51617
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.5858i	1.24615i	0.782163	+	0.623074i	$0.214116\pi$
$137$	14.5858i	1.24615i	−0.782163	+	0.623074i	$0.785884\pi$
$138$	0	0
$139$	10.1421i	0.860245i	0.902771	+	0.430122i	$0.141530\pi$
$139$	10.1421i	0.860245i	−0.902771	+	0.430122i	$0.858470\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	36.0416	3.01395
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.10051i	0.172080i	0.996292	+	0.0860400i	$0.0274213\pi$
$149$	2.10051i	0.172080i	−0.996292	+	0.0860400i	$0.972579\pi$
$150$	0	0
$151$	− 8.07107i	− 0.656814i	−0.944536	−	0.328407i	$-0.893488\pi$
$151$	− 8.07107i	− 0.656814i	0.944536	−	0.328407i	$-0.106512\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−7.48528	−0.597390	−0.298695	−	0.954349i	$-0.596551\pi$
$157$	−7.48528	−0.597390	−0.298695	+	0.954349i	$0.596551\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.24264i	0.491989i
$162$	0	0
$163$	21.2426i	1.66385i	0.554887	+	0.831926i	$0.312762\pi$
$163$	21.2426i	1.66385i	−0.554887	+	0.831926i	$0.687238\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.6569	−1.52109	−0.760547	−	0.649283i	$-0.775069\pi$
$167$	−19.6569	−1.52109	−0.760547	+	0.649283i	$0.775069\pi$
$168$	0	0
$169$	31.3137	2.40875
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.2426i	0.778734i	0.921083	+	0.389367i	$0.127306\pi$
$173$	10.2426i	0.778734i	−0.921083	+	0.389367i	$0.872694\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−24.3848	−1.82260	−0.911302	−	0.411739i	$-0.864922\pi$
$179$	−24.3848	−1.82260	−0.911302	+	0.411739i	$0.864922\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 18.4853i	− 1.35178i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.727922	−0.0526706	−0.0263353	−	0.999653i	$-0.508384\pi$
$191$	−0.727922	−0.0526706	−0.0263353	+	0.999653i	$0.508384\pi$
$192$	0	0
$193$	1.48528	0.106913	0.0534564	−	0.998570i	$-0.482976\pi$
$193$	1.48528	0.106913	0.0534564	+	0.998570i	$0.482976\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.58579i	0.469218i	0.972090	+	0.234609i	$0.0753810\pi$
$197$	6.58579i	0.469218i	−0.972090	+	0.234609i	$0.924619\pi$
$198$	0	0
$199$	− 9.92893i	− 0.703843i	−0.936029	−	0.351922i	$-0.885528\pi$
$199$	− 9.92893i	− 0.703843i	0.936029	−	0.351922i	$-0.114472\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−24.7279	−1.73556
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 39.2132i	− 2.71243i
$210$	0	0
$211$	16.2132i	1.11616i	0.829786	+	0.558081i	$0.188462\pi$
$211$	16.2132i	1.11616i	−0.829786	+	0.558081i	$0.811538\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.6569	0.859203
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 22.7279i	− 1.52885i
$222$	0	0
$223$	− 17.7279i	− 1.18715i	−0.804779	−	0.593575i	$-0.797716\pi$
$223$	− 17.7279i	− 1.18715i	0.804779	−	0.593575i	$-0.202284\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.3431	0.819243	0.409622	−	0.912255i	$-0.365661\pi$
$227$	12.3431	0.819243	0.409622	+	0.912255i	$0.365661\pi$
$228$	0	0
$229$	25.4853	1.68411	0.842057	−	0.539388i	$-0.181344\pi$
$229$	25.4853	1.68411	0.842057	+	0.539388i	$0.181344\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 15.1716i	− 0.993923i	−0.867773	−	0.496961i	$-0.834449\pi$
$233$	− 15.1716i	− 0.993923i	0.867773	−	0.496961i	$-0.165551\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−13.9706	−0.899923	−0.449962	−	0.893048i	$-0.648562\pi$
$241$	−13.9706	−0.899923	−0.449962	+	0.893048i	$0.648562\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 48.2132i	− 3.06773i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.31371	0.209159	0.104580	−	0.994517i	$-0.466650\pi$
$251$	3.31371	0.209159	0.104580	+	0.994517i	$0.466650\pi$
$252$	0	0
$253$	−14.0000	−0.880172
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 12.0000i	− 0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$	− 12.0000i	− 0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$	0	0
$259$	− 16.4853i	− 1.02435i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	10.9706	0.676474	0.338237	−	0.941061i	$-0.390169\pi$
$263$	10.9706	0.676474	0.338237	+	0.941061i	$0.390169\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 21.0711i	− 1.28473i	−0.766401	−	0.642363i	$-0.777954\pi$
$269$	− 21.0711i	− 1.28473i	0.766401	−	0.642363i	$-0.222046\pi$
$270$	0	0
$271$	− 12.4853i	− 0.758427i	−0.925309	−	0.379213i	$-0.876195\pi$
$271$	− 12.4853i	− 0.758427i	0.925309	−	0.379213i	$-0.123805\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.0000	0.901263	0.450631	−	0.892710i	$-0.351199\pi$
$277$	15.0000	0.901263	0.450631	+	0.892710i	$0.351199\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.10051i	0.483236i	0.970371	+	0.241618i	$0.0776780\pi$
$281$	8.10051i	0.483236i	−0.970371	+	0.241618i	$0.922322\pi$
$282$	0	0
$283$	15.9289i	0.946877i	0.880827	+	0.473438i	$0.156987\pi$
$283$	15.9289i	0.946877i	−0.880827	+	0.473438i	$0.843013\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.6569	−0.688082
$288$	0	0
$289$	5.34315	0.314303
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.5563i	0.908812i	0.890795	+	0.454406i	$0.150148\pi$
$293$	15.5563i	0.908812i	−0.890795	+	0.454406i	$0.849852\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−17.2132	−0.995465
$300$	0	0
$301$	8.65685	0.498973
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 7.24264i	− 0.413359i	−0.978409	−	0.206680i	$-0.933734\pi$
$307$	− 7.24264i	− 0.413359i	0.978409	−	0.206680i	$-0.0662658\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.3848	−0.702276	−0.351138	−	0.936324i	$-0.614205\pi$
$311$	−12.3848	−0.702276	−0.351138	+	0.936324i	$0.614205\pi$
$312$	0	0
$313$	18.7990	1.06258	0.531291	−	0.847189i	$-0.321707\pi$
$313$	18.7990	1.06258	0.531291	+	0.847189i	$0.321707\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 11.3137i	− 0.635441i	−0.948184	−	0.317721i	$-0.897083\pi$
$317$	− 11.3137i	− 0.635441i	0.948184	−	0.317721i	$-0.102917\pi$
$318$	0	0
$319$	− 55.4558i	− 3.10493i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−24.7279	−1.37590
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 17.8995i	− 0.986831i
$330$	0	0
$331$	− 32.4853i	− 1.78555i	−0.450500	−	0.892776i	$-0.648754\pi$
$331$	− 32.4853i	− 1.78555i	0.450500	−	0.892776i	$-0.351246\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0.514719	0.0280385	0.0140193	−	0.999902i	$-0.495537\pi$
$337$	0.514719	0.0280385	0.0140193	+	0.999902i	$0.495537\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	28.3848i	1.53712i
$342$	0	0
$343$	− 19.7279i	− 1.06521i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.3848	−1.52377	−0.761887	−	0.647710i	$-0.775727\pi$
$347$	−28.3848	−1.52377	−0.761887	+	0.647710i	$0.775727\pi$
$348$	0	0
$349$	−4.97056	−0.266068	−0.133034	−	0.991111i	$-0.542472\pi$
$349$	−4.97056	−0.266068	−0.133034	+	0.991111i	$0.542472\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.8995i	1.05914i	0.848265	+	0.529572i	$0.177647\pi$
$353$	19.8995i	1.05914i	−0.848265	+	0.529572i	$0.822353\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.3137	1.33601	0.668003	−	0.744158i	$-0.267149\pi$
$359$	25.3137	1.33601	0.668003	+	0.744158i	$0.267149\pi$
$360$	0	0
$361$	−33.4558	−1.76083
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.3848i	0.907478i	0.891135	+	0.453739i	$0.149910\pi$
$367$	17.3848i	0.907478i	−0.891135	+	0.453739i	$0.850090\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	−7.00000	−0.362446	−0.181223	−	0.983442i	$-0.558006\pi$
$373$	−7.00000	−0.362446	−0.181223	+	0.983442i	$0.558006\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 68.1838i	− 3.51164i
$378$	0	0
$379$	− 26.5563i	− 1.36411i	−0.731302	−	0.682054i	$-0.761087\pi$
$379$	− 26.5563i	− 1.36411i	0.731302	−	0.682054i	$-0.238913\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.6274	0.747426	0.373713	−	0.927544i	$-0.378084\pi$
$383$	14.6274	0.747426	0.373713	+	0.927544i	$0.378084\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 8.14214i	− 0.412823i	−0.978465	−	0.206411i	$-0.933822\pi$
$389$	− 8.14214i	− 0.412823i	0.978465	−	0.206411i	$-0.0661785\pi$
$390$	0	0
$391$	8.82843i	0.446473i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.00000	0.250943	0.125471	−	0.992097i	$-0.459956\pi$
$397$	5.00000	0.250943	0.125471	+	0.992097i	$0.459956\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 26.4853i	− 1.32261i	−0.750116	−	0.661306i	$-0.770003\pi$
$401$	− 26.4853i	− 1.32261i	0.750116	−	0.661306i	$-0.229997\pi$
$402$	0	0
$403$	34.8995i	1.73847i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	36.9706	1.83256
$408$	0	0
$409$	19.8284	0.980453	0.490226	−	0.871595i	$-0.336914\pi$
$409$	19.8284	0.980453	0.490226	+	0.871595i	$0.336914\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	12.2426i	0.602421i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−30.2843	−1.47948	−0.739742	−	0.672891i	$-0.765052\pi$
$419$	−30.2843	−1.47948	−0.739742	+	0.672891i	$0.765052\pi$
$420$	0	0
$421$	20.4853	0.998392	0.499196	−	0.866489i	$-0.333629\pi$
$421$	20.4853	0.998392	0.499196	+	0.866489i	$0.333629\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.41421i	0.213619i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.41421	−0.0681203	−0.0340601	−	0.999420i	$-0.510844\pi$
$431$	−1.41421	−0.0681203	−0.0340601	+	0.999420i	$0.510844\pi$
$432$	0	0
$433$	−17.4853	−0.840289	−0.420144	−	0.907457i	$-0.638021\pi$
$433$	−17.4853	−0.840289	−0.420144	+	0.907457i	$0.638021\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.7279i	0.895878i
$438$	0	0
$439$	− 7.44365i	− 0.355266i	−0.984097	−	0.177633i	$-0.943156\pi$
$439$	− 7.44365i	− 0.355266i	0.984097	−	0.177633i	$-0.0568440\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.68629	0.412698	0.206349	−	0.978478i	$-0.433842\pi$
$443$	8.68629	0.412698	0.206349	+	0.978478i	$0.433842\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 3.31371i	− 0.156384i	−0.996938	−	0.0781918i	$-0.975085\pi$
$449$	− 3.31371i	− 0.156384i	0.996938	−	0.0781918i	$-0.0249147\pi$
$450$	0	0
$451$	− 26.1421i	− 1.23099i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 21.7990i	− 1.01528i	−0.861569	−	0.507640i	$-0.830518\pi$
$461$	− 21.7990i	− 1.01528i	0.861569	−	0.507640i	$-0.169482\pi$
$462$	0	0
$463$	32.7696i	1.52293i	0.648206	+	0.761465i	$0.275520\pi$
$463$	32.7696i	1.52293i	−0.648206	+	0.761465i	$0.724480\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.3848	−0.850746	−0.425373	−	0.905018i	$-0.639857\pi$
$467$	−18.3848	−0.850746	−0.425373	+	0.905018i	$0.639857\pi$
$468$	0	0
$469$	7.82843	0.361483
$470$	0	0
$471$	0	0
$472$	0	0
$473$	19.4142i	0.892666i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.5858	−0.575059	−0.287530	−	0.957772i	$-0.592834\pi$
$479$	−12.5858	−0.575059	−0.287530	+	0.957772i	$0.592834\pi$
$480$	0	0
$481$	45.4558	2.07261
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.72792i	0.259557i	0.991543	+	0.129778i	$0.0414266\pi$
$487$	5.72792i	0.259557i	−0.991543	+	0.129778i	$0.958573\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	23.3137	1.05213	0.526066	−	0.850443i	$-0.323666\pi$
$491$	23.3137	1.05213	0.526066	+	0.850443i	$0.323666\pi$
$492$	0	0
$493$	−34.9706	−1.57499
$494$	0	0
$495$	0	0
$496$	0	0
$497$	28.1421i	1.26235i
$498$	0	0
$499$	23.0416i	1.03149i	0.856744	+	0.515743i	$0.172484\pi$
$499$	23.0416i	1.03149i	−0.856744	+	0.515743i	$0.827516\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	41.5980	1.85476	0.927381	−	0.374118i	$-0.122054\pi$
$503$	41.5980	1.85476	0.927381	+	0.374118i	$0.122054\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.8995i	1.50257i	0.659979	+	0.751284i	$0.270565\pi$
$509$	33.8995i	1.50257i	−0.659979	+	0.751284i	$0.729435\pi$
$510$	0	0
$511$	39.7990i	1.76060i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	40.1421	1.76545
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 34.2426i	− 1.50020i	−0.661326	−	0.750099i	$-0.730006\pi$
$521$	− 34.2426i	− 1.50020i	0.661326	−	0.750099i	$-0.269994\pi$
$522$	0	0
$523$	21.2426i	0.928876i	0.885606	+	0.464438i	$0.153744\pi$
$523$	21.2426i	0.928876i	−0.885606	+	0.464438i	$0.846256\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17.8995	0.779714
$528$	0	0
$529$	−16.3137	−0.709292
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 32.1421i	− 1.39223i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.34315	0.273219
$540$	0	0
$541$	10.6569	0.458174	0.229087	−	0.973406i	$-0.426426\pi$
$541$	10.6569	0.458174	0.229087	+	0.973406i	$0.426426\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.97056i	0.298040i	0.988834	+	0.149020i	$0.0476118\pi$
$547$	6.97056i	0.298040i	−0.988834	+	0.149020i	$0.952388\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−74.1838	−3.16033
$552$	0	0
$553$	−28.9706	−1.23195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.21320i	− 0.390376i	−0.980766	−	0.195188i	$-0.937468\pi$
$557$	− 9.21320i	− 0.390376i	0.980766	−	0.195188i	$-0.0625317\pi$
$558$	0	0
$559$	23.8701i	1.00960i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.8701	−0.626698	−0.313349	−	0.949638i	$-0.601451\pi$
$563$	−14.8701	−0.626698	−0.313349	+	0.949638i	$0.601451\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.2426i	0.429394i	0.976681	+	0.214697i	$0.0688764\pi$
$569$	10.2426i	0.429394i	−0.976681	+	0.214697i	$0.931124\pi$
$570$	0	0
$571$	− 7.24264i	− 0.303095i	−0.988450	−	0.151548i	$-0.951574\pi$
$571$	− 7.24264i	− 0.303095i	0.988450	−	0.151548i	$-0.0484257\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.0000	0.957503	0.478751	−	0.877951i	$-0.341090\pi$
$577$	23.0000	0.957503	0.478751	+	0.877951i	$0.341090\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 11.0711i	− 0.459305i
$582$	0	0
$583$	4.48528i	0.185761i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.6569	−0.976423	−0.488211	−	0.872725i	$-0.662351\pi$
$587$	−23.6569	−0.976423	−0.488211	+	0.872725i	$0.662351\pi$
$588$	0	0
$589$	37.9706	1.56455
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.79899i	0.402396i	0.979551	+	0.201198i	$0.0644835\pi$
$593$	9.79899i	0.402396i	−0.979551	+	0.201198i	$0.935516\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−25.6569	−1.04831	−0.524155	−	0.851623i	$-0.675619\pi$
$599$	−25.6569	−1.04831	−0.524155	+	0.851623i	$0.675619\pi$
$600$	0	0
$601$	37.7696	1.54065	0.770326	−	0.637650i	$-0.220093\pi$
$601$	37.7696	1.54065	0.770326	+	0.637650i	$0.220093\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 28.9706i	− 1.17588i	−0.808905	−	0.587939i	$-0.799939\pi$
$607$	− 28.9706i	− 1.17588i	0.808905	−	0.587939i	$-0.200061\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	49.3553	1.99670
$612$	0	0
$613$	40.9706	1.65479	0.827393	−	0.561624i	$-0.189823\pi$
$613$	40.9706	1.65479	0.827393	+	0.561624i	$0.189823\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.1716i	1.09389i	0.837170	+	0.546943i	$0.184209\pi$
$617$	27.1716i	1.09389i	−0.837170	+	0.546943i	$0.815791\pi$
$618$	0	0
$619$	2.27208i	0.0913225i	0.998957	+	0.0456613i	$0.0145395\pi$
$619$	2.27208i	0.0913225i	−0.998957	+	0.0456613i	$0.985461\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	28.9706	1.16068
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 23.3137i	− 0.929578i
$630$	0	0
$631$	37.0416i	1.47460i	0.675563	+	0.737302i	$0.263901\pi$
$631$	37.0416i	1.47460i	−0.675563	+	0.737302i	$0.736099\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	7.79899	0.309007
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21.6569i	0.855394i	0.903922	+	0.427697i	$0.140675\pi$
$641$	21.6569i	0.855394i	−0.903922	+	0.427697i	$0.859325\pi$
$642$	0	0
$643$	13.8579i	0.546501i	0.961943	+	0.273250i	$0.0880988\pi$
$643$	13.8579i	0.546501i	−0.961943	+	0.273250i	$0.911901\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.34315	0.249375	0.124687	−	0.992196i	$-0.460207\pi$
$647$	6.34315	0.249375	0.124687	+	0.992196i	$0.460207\pi$
$648$	0	0
$649$	−27.4558	−1.07774
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 25.8995i	− 1.01353i	−0.862086	−	0.506763i	$-0.830842\pi$
$653$	− 25.8995i	− 1.01353i	0.862086	−	0.506763i	$-0.169158\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−38.2843	−1.49134	−0.745672	−	0.666313i	$-0.767871\pi$
$659$	−38.2843	−1.49134	−0.745672	+	0.666313i	$0.767871\pi$
$660$	0	0
$661$	−8.97056	−0.348914	−0.174457	−	0.984665i	$-0.555817\pi$
$661$	−8.97056	−0.348914	−0.174457	+	0.984665i	$0.555817\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	26.4853i	1.02551i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.89949	−0.382166
$672$	0	0
$673$	−0.485281	−0.0187062	−0.00935311	−	0.999956i	$-0.502977\pi$
$673$	−0.485281	−0.0187062	−0.00935311	+	0.999956i	$0.502977\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 11.6152i	− 0.446409i	−0.974772	−	0.223205i	$-0.928348\pi$
$677$	− 11.6152i	− 0.446409i	0.974772	−	0.223205i	$-0.0716518\pi$
$678$	0	0
$679$	− 21.7279i	− 0.833841i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	37.6569	1.44090	0.720450	−	0.693507i	$-0.243935\pi$
$683$	37.6569	1.44090	0.720450	+	0.693507i	$0.243935\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.51472i	0.210094i
$690$	0	0
$691$	− 22.0000i	− 0.836919i	−0.908235	−	0.418460i	$-0.862570\pi$
$691$	− 22.0000i	− 0.836919i	0.908235	−	0.418460i	$-0.137430\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−16.4853	−0.624425
$698$	0	0
$699$	0	0
$700$	0	0
$701$	51.2132i	1.93430i	0.254214	+	0.967148i	$0.418183\pi$
$701$	51.2132i	1.93430i	−0.254214	+	0.967148i	$0.581817\pi$
$702$	0	0
$703$	− 49.4558i	− 1.86526i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.00000	−0.0752177
$708$	0	0
$709$	7.48528	0.281116	0.140558	−	0.990072i	$-0.455110\pi$
$709$	7.48528	0.281116	0.140558	+	0.990072i	$0.455110\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 13.5563i	− 0.507689i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.0416	0.598252	0.299126	−	0.954214i	$-0.403305\pi$
$719$	16.0416	0.598252	0.299126	+	0.954214i	$0.403305\pi$
$720$	0	0
$721$	18.4853	0.688428
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8.07107i	− 0.299339i	−0.988736	−	0.149670i	$-0.952179\pi$
$727$	− 8.07107i	− 0.299339i	0.988736	−	0.149670i	$-0.0478210\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	12.2426	0.452810
$732$	0	0
$733$	7.31371	0.270138	0.135069	−	0.990836i	$-0.456874\pi$
$733$	7.31371	0.270138	0.135069	+	0.990836i	$0.456874\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17.5563i	0.646696i
$738$	0	0
$739$	2.34315i	0.0861940i	0.999071	+	0.0430970i	$0.0137225\pi$
$739$	2.34315i	0.0861940i	−0.999071	+	0.0430970i	$0.986278\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.9706	0.622590	0.311295	−	0.950313i	$-0.399237\pi$
$743$	16.9706	0.622590	0.311295	+	0.950313i	$0.399237\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	41.7990i	1.52730i
$750$	0	0
$751$	− 52.7696i	− 1.92559i	−0.270236	−	0.962794i	$-0.587102\pi$
$751$	− 52.7696i	− 1.92559i	0.270236	−	0.962794i	$-0.412898\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−24.7990	−0.901335	−0.450667	−	0.892692i	$-0.648814\pi$
$757$	−24.7990	−0.901335	−0.450667	+	0.892692i	$0.648814\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20.6863i	0.749877i	0.927050	+	0.374939i	$0.122336\pi$
$761$	20.6863i	0.749877i	−0.927050	+	0.374939i	$0.877664\pi$
$762$	0	0
$763$	− 19.7279i	− 0.714199i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−33.7574	−1.21891
$768$	0	0
$769$	8.79899	0.317300	0.158650	−	0.987335i	$-0.449286\pi$
$769$	8.79899	0.317300	0.158650	+	0.987335i	$0.449286\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	37.1127i	1.33485i	0.744677	+	0.667425i	$0.232604\pi$
$773$	37.1127i	1.33485i	−0.744677	+	0.667425i	$0.767396\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−34.9706	−1.25295
$780$	0	0
$781$	−63.1127	−2.25835
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	38.0711i	1.35709i	0.734560	+	0.678544i	$0.237389\pi$
$787$	38.0711i	1.35709i	−0.734560	+	0.678544i	$0.762611\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	23.3137	0.828940
$792$	0	0
$793$	−12.1716	−0.432225
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 36.2843i	− 1.28525i	−0.766179	−	0.642627i	$-0.777844\pi$
$797$	− 36.2843i	− 1.28525i	0.766179	−	0.642627i	$-0.222156\pi$
$798$	0	0
$799$	− 25.3137i	− 0.895535i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−89.2548	−3.14973
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 10.2010i	− 0.358648i	−0.983790	−	0.179324i	$-0.942609\pi$
$809$	− 10.2010i	− 0.358648i	0.983790	−	0.179324i	$-0.0573911\pi$
$810$	0	0
$811$	31.0416i	1.09002i	0.838430	+	0.545010i	$0.183474\pi$
$811$	31.0416i	1.09002i	−0.838430	+	0.545010i	$0.816526\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	25.9706	0.908595
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 25.4142i	− 0.886962i	−0.896284	−	0.443481i	$-0.853743\pi$
$821$	− 25.4142i	− 0.886962i	0.896284	−	0.443481i	$-0.146257\pi$
$822$	0	0
$823$	− 56.2132i	− 1.95947i	−0.200300	−	0.979735i	$-0.564192\pi$
$823$	− 56.2132i	− 1.95947i	0.200300	−	0.979735i	$-0.435808\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.3431	1.12468	0.562341	−	0.826906i	$-0.309901\pi$
$827$	32.3431	1.12468	0.562341	+	0.826906i	$0.309901\pi$
$828$	0	0
$829$	39.9411	1.38721	0.693606	−	0.720354i	$-0.256021\pi$
$829$	39.9411	1.38721	0.693606	+	0.720354i	$0.256021\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 4.00000i	− 0.138592i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	45.2132	1.56093	0.780467	−	0.625198i	$-0.214982\pi$
$839$	45.2132	1.56093	0.780467	+	0.625198i	$0.214982\pi$
$840$	0	0
$841$	−75.9117	−2.61764
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 44.2132i	− 1.51918i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−17.6569	−0.605269
$852$	0	0
$853$	24.4558	0.837352	0.418676	−	0.908136i	$-0.362494\pi$
$853$	24.4558	0.837352	0.418676	+	0.908136i	$0.362494\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 33.6569i	− 1.14970i	−0.818260	−	0.574848i	$-0.805061\pi$
$857$	− 33.6569i	− 1.14970i	0.818260	−	0.574848i	$-0.194939\pi$
$858$	0	0
$859$	10.0000i	0.341196i	0.985341	+	0.170598i	$0.0545699\pi$
$859$	10.0000i	0.341196i	−0.985341	+	0.170598i	$0.945430\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.3431	−1.30522	−0.652608	−	0.757696i	$-0.726325\pi$
$863$	−38.3431	−1.30522	−0.652608	+	0.757696i	$0.726325\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 64.9706i	− 2.20398i
$870$	0	0
$871$	21.5858i	0.731406i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−16.1716	−0.546075	−0.273038	−	0.962003i	$-0.588028\pi$
$877$	−16.1716	−0.546075	−0.273038	+	0.962003i	$0.588028\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	45.2548i	1.52467i	0.647180	+	0.762337i	$0.275948\pi$
$881$	45.2548i	1.52467i	−0.647180	+	0.762337i	$0.724052\pi$
$882$	0	0
$883$	− 1.24264i	− 0.0418182i	−0.999781	−	0.0209091i	$-0.993344\pi$
$883$	− 1.24264i	− 0.0418182i	0.999781	−	0.0209091i	$-0.00665606\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.0416	0.874392	0.437196	−	0.899366i	$-0.355972\pi$
$887$	26.0416	0.874392	0.437196	+	0.899366i	$0.355972\pi$
$888$	0	0
$889$	1.17157	0.0392933
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 53.6985i	− 1.79695i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	53.6985	1.79094
$900$	0	0
$901$	2.82843	0.0942286
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 19.3137i	− 0.641301i	−0.947198	−	0.320651i	$-0.896098\pi$
$907$	− 19.3137i	− 0.641301i	0.947198	−	0.320651i	$-0.103902\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−26.5269	−0.878876	−0.439438	−	0.898273i	$-0.644822\pi$
$911$	−26.5269	−0.878876	−0.439438	+	0.898273i	$0.644822\pi$
$912$	0	0
$913$	24.8284	0.821701
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.4853i	0.478346i
$918$	0	0
$919$	− 37.5269i	− 1.23790i	−0.785431	−	0.618949i	$-0.787559\pi$
$919$	− 37.5269i	− 1.23790i	0.785431	−	0.618949i	$-0.212441\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−77.5980	−2.55417
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.10051i	0.265769i	0.991132	+	0.132884i	$0.0424239\pi$
$929$	8.10051i	0.265769i	−0.991132	+	0.132884i	$0.957576\pi$
$930$	0	0
$931$	− 8.48528i	− 0.278094i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.5147	−0.735524	−0.367762	−	0.929920i	$-0.619876\pi$
$937$	−22.5147	−0.735524	−0.367762	+	0.929920i	$0.619876\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 14.3848i	− 0.468930i	−0.972125	−	0.234465i	$-0.924666\pi$
$941$	− 14.3848i	− 0.468930i	0.972125	−	0.234465i	$-0.0753339\pi$
$942$	0	0
$943$	12.4853i	0.406577i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−45.8995	−1.49153	−0.745767	−	0.666207i	$-0.767917\pi$
$947$	−45.8995	−1.49153	−0.745767	+	0.666207i	$0.767917\pi$
$948$	0	0
$949$	−109.740	−3.56231
$950$	0	0
$951$	0	0
$952$	0	0
$953$	41.6569i	1.34940i	0.738093	+	0.674699i	$0.235727\pi$
$953$	41.6569i	1.34940i	−0.738093	+	0.674699i	$0.764273\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	35.2132	1.13709
$960$	0	0
$961$	3.51472	0.113378
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	37.9411i	1.22010i	0.792361	+	0.610052i	$0.208852\pi$
$967$	37.9411i	1.22010i	−0.792361	+	0.610052i	$0.791148\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.87006	0.156288	0.0781438	−	0.996942i	$-0.475101\pi$
$971$	4.87006	0.156288	0.0781438	+	0.996942i	$0.475101\pi$
$972$	0	0
$973$	24.4853	0.784962
$974$	0	0
$975$	0	0
$976$	0	0
$977$	55.3553i	1.77097i	0.464664	+	0.885487i	$0.346175\pi$
$977$	55.3553i	1.77097i	−0.464664	+	0.885487i	$0.653825\pi$
$978$	0	0
$979$	64.9706i	2.07647i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.6569	0.563166	0.281583	−	0.959537i	$-0.409140\pi$
$983$	17.6569	0.563166	0.281583	+	0.959537i	$0.409140\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 9.27208i	− 0.294835i
$990$	0	0
$991$	− 0.0710678i	− 0.00225754i	−0.999999	−	0.00112877i	$-0.999641\pi$
$991$	− 0.0710678i	− 0.00225754i	0.999999	−	0.00112877i	$-0.000359299\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−21.9411	−0.694882	−0.347441	−	0.937702i	$-0.612949\pi$
$997$	−21.9411	−0.694882	−0.347441	+	0.937702i	$0.612949\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.h.h.1151.1	yes	4
3.2	odd	2		7200.2.h.b.1151.1	yes	4
4.3	odd	2		7200.2.h.b.1151.4	yes	4
5.2	odd	4		7200.2.o.i.7199.3		4
5.3	odd	4		7200.2.o.g.7199.2		4
5.4	even	2		7200.2.h.g.1151.4	yes	4
12.11	even	2	inner	7200.2.h.h.1151.4	yes	4
15.2	even	4		7200.2.o.h.7199.3		4
15.8	even	4		7200.2.o.f.7199.2		4
15.14	odd	2		7200.2.h.a.1151.4	yes	4
20.3	even	4		7200.2.o.h.7199.4		4
20.7	even	4		7200.2.o.f.7199.1		4
20.19	odd	2		7200.2.h.a.1151.1	✓	4
60.23	odd	4		7200.2.o.i.7199.4		4
60.47	odd	4		7200.2.o.g.7199.1		4
60.59	even	2		7200.2.h.g.1151.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7200.2.h.a.1151.1	✓	4	20.19	odd	2
7200.2.h.a.1151.4	yes	4	15.14	odd	2
7200.2.h.b.1151.1	yes	4	3.2	odd	2
7200.2.h.b.1151.4	yes	4	4.3	odd	2
7200.2.h.g.1151.1	yes	4	60.59	even	2
7200.2.h.g.1151.4	yes	4	5.4	even	2
7200.2.h.h.1151.1	yes	4	1.1	even	1	trivial
7200.2.h.h.1151.4	yes	4	12.11	even	2	inner
7200.2.o.f.7199.1		4	20.7	even	4
7200.2.o.f.7199.2		4	15.8	even	4
7200.2.o.g.7199.1		4	60.47	odd	4
7200.2.o.g.7199.2		4	5.3	odd	4
7200.2.o.h.7199.3		4	15.2	even	4
7200.2.o.h.7199.4		4	20.3	even	4
7200.2.o.i.7199.3		4	5.2	odd	4
7200.2.o.i.7199.4		4	60.23	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:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.41421i q^{7} +O(q^{10})\)	\(q-2.41421i q^{7} +5.41421 q^{11} +6.65685 q^{13} -3.41421i q^{17} -7.24264i q^{19} -2.58579 q^{23} -10.2426i q^{29} +5.24264i q^{31} +6.82843 q^{37} -4.82843i q^{41} +3.58579i q^{43} +7.41421 q^{47} +1.17157 q^{49} +0.828427i q^{53} -5.07107 q^{59} -1.82843 q^{61} +3.24264i q^{67} -11.6569 q^{71} -16.4853 q^{73} -13.0711i q^{77} -12.0000i q^{79} +4.58579 q^{83} +12.0000i q^{89} -16.0711i q^{91} +9.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 16 q^{11} + 4 q^{13} - 16 q^{23} + 16 q^{37} + 24 q^{47} + 16 q^{49} + 8 q^{59} + 4 q^{61} - 24 q^{71} - 32 q^{73} + 24 q^{83} + 36 q^{97}+O(q^{100}) \) 4 * q + 16 * q^11 + 4 * q^13 - 16 * q^23 + 16 * q^37 + 24 * q^47 + 16 * q^49 + 8 * q^59 + 4 * q^61 - 24 * q^71 - 32 * q^73 + 24 * q^83 + 36 * q^97

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