LMFDB - Embedded newform 7200.2.o.j.7199.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(7199,7200)]

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, 1]))

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

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

chi := DirichletCharacter("7200.7199");

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.o (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:	$57.4922894553$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			7199.3
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	7200.7199
Dual form			7200.2.o.j.7199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.41421 q^{7} +O(q^{10})$	$q+3.41421 q^{7} -2.58579 q^{11} -3.41421i q^{13} -1.17157 q^{17} -4.82843i q^{23} +6.00000i q^{29} +6.48528i q^{31} +9.07107i q^{37} -11.0711i q^{41} +6.82843 q^{43} -5.65685i q^{47} +4.65685 q^{49} +1.17157 q^{53} +6.58579 q^{59} +12.8284 q^{61} +8.00000 q^{67} -5.65685 q^{71} -10.4853i q^{73} -8.82843 q^{77} -14.4853i q^{79} -9.17157i q^{83} -4.24264i q^{89} -11.6569i q^{91} -2.48528i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7}+O(q^{10}) $ 4 * q + 8 * q^7	$ 4 q + 8 q^{7} - 16 q^{11} - 16 q^{17} + 16 q^{43} - 4 q^{49} + 16 q^{53} + 32 q^{59} + 40 q^{61} + 32 q^{67} - 24 q^{77}+O(q^{100}) $ 4 * q + 8 * q^7 - 16 * q^11 - 16 * q^17 + 16 * q^43 - 4 * q^49 + 16 * q^53 + 32 * q^59 + 40 * q^61 + 32 * q^67 - 24 * q^77

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
$7$	3.41421	1.29045	0.645226	−	0.763992i	$-0.276763\pi$
$7$	3.41421	1.29045	0.645226	+	0.763992i	$0.276763\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.58579	−0.779644	−0.389822	−	0.920890i	$-0.627463\pi$
$11$	−2.58579	−0.779644	−0.389822	+	0.920890i	$0.627463\pi$
$12$	0	0
$13$	− 3.41421i	− 0.946932i	−0.880812	−	0.473466i	$-0.843003\pi$
$13$	− 3.41421i	− 0.946932i	0.880812	−	0.473466i	$-0.156997\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.17157	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$17$	−1.17157	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.82843i	− 1.00680i	−0.864054	−	0.503398i	$-0.832083\pi$
$23$	− 4.82843i	− 1.00680i	0.864054	−	0.503398i	$-0.167917\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	6.48528i	1.16479i	0.812906	+	0.582395i	$0.197884\pi$
$31$	6.48528i	1.16479i	−0.812906	+	0.582395i	$0.802116\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.07107i	1.49127i	0.666352	+	0.745637i	$0.267855\pi$
$37$	9.07107i	1.49127i	−0.666352	+	0.745637i	$0.732145\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 11.0711i	− 1.72901i	−0.502624	−	0.864505i	$-0.667632\pi$
$41$	− 11.0711i	− 1.72901i	0.502624	−	0.864505i	$-0.332368\pi$
$42$	0	0
$43$	6.82843	1.04133	0.520663	−	0.853762i	$-0.325685\pi$
$43$	6.82843	1.04133	0.520663	+	0.853762i	$0.325685\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 5.65685i	− 0.825137i	−0.910927	−	0.412568i	$-0.864632\pi$
$47$	− 5.65685i	− 0.825137i	0.910927	−	0.412568i	$-0.135368\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.17157	0.160928	0.0804640	−	0.996758i	$-0.474360\pi$
$53$	1.17157	0.160928	0.0804640	+	0.996758i	$0.474360\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.58579	0.857396	0.428698	−	0.903448i	$-0.358972\pi$
$59$	6.58579	0.857396	0.428698	+	0.903448i	$0.358972\pi$
$60$	0	0
$61$	12.8284	1.64251	0.821256	−	0.570560i	$-0.193274\pi$
$61$	12.8284	1.64251	0.821256	+	0.570560i	$0.193274\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	− 10.4853i	− 1.22721i	−0.789613	−	0.613605i	$-0.789719\pi$
$73$	− 10.4853i	− 1.22721i	0.789613	−	0.613605i	$-0.210281\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.82843	−1.00609
$78$	0	0
$79$	− 14.4853i	− 1.62972i	−0.579657	−	0.814861i	$-0.696813\pi$
$79$	− 14.4853i	− 1.62972i	0.579657	−	0.814861i	$-0.303187\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 9.17157i	− 1.00671i	−0.864079	−	0.503355i	$-0.832099\pi$
$83$	− 9.17157i	− 1.00671i	0.864079	−	0.503355i	$-0.167901\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 4.24264i	− 0.449719i	−0.974391	−	0.224860i	$-0.927808\pi$
$89$	− 4.24264i	− 0.449719i	0.974391	−	0.224860i	$-0.0721923\pi$
$90$	0	0
$91$	− 11.6569i	− 1.22197i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 2.48528i	− 0.252342i	−0.992009	−	0.126171i	$-0.959731\pi$
$97$	− 2.48528i	− 0.252342i	0.992009	−	0.126171i	$-0.0402688\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 4.82843i	− 0.480446i	−0.970718	−	0.240223i	$-0.922779\pi$
$101$	− 4.82843i	− 0.480446i	0.970718	−	0.240223i	$-0.0772206\pi$
$102$	0	0
$103$	−17.5563	−1.72988	−0.864939	−	0.501877i	$-0.832643\pi$
$103$	−17.5563	−1.72988	−0.864939	+	0.501877i	$0.832643\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.8284i	1.43352i	0.697321	+	0.716759i	$0.254375\pi$
$107$	14.8284i	1.43352i	−0.697321	+	0.716759i	$0.745625\pi$
$108$	0	0
$109$	7.17157	0.686912	0.343456	−	0.939169i	$-0.388402\pi$
$109$	7.17157	0.686912	0.343456	+	0.939169i	$0.388402\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.65685	−0.155864	−0.0779319	−	0.996959i	$-0.524832\pi$
$113$	−1.65685	−0.155864	−0.0779319	+	0.996959i	$0.524832\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−4.31371	−0.392155
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.7279	0.951949	0.475975	−	0.879459i	$-0.342095\pi$
$127$	10.7279	0.951949	0.475975	+	0.879459i	$0.342095\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.7279	−1.11204	−0.556022	−	0.831168i	$-0.687673\pi$
$131$	−12.7279	−1.11204	−0.556022	+	0.831168i	$0.687673\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.3137	1.65008	0.825041	−	0.565073i	$-0.191152\pi$
$137$	19.3137	1.65008	0.825041	+	0.565073i	$0.191152\pi$
$138$	0	0
$139$	3.65685i	0.310170i	0.987901	+	0.155085i	$0.0495652\pi$
$139$	3.65685i	0.310170i	−0.987901	+	0.155085i	$0.950435\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.82843i	0.738270i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 8.82843i	− 0.723253i	−0.932323	−	0.361626i	$-0.882222\pi$
$149$	− 8.82843i	− 0.723253i	0.932323	−	0.361626i	$-0.117778\pi$
$150$	0	0
$151$	− 15.6569i	− 1.27414i	−0.770807	−	0.637068i	$-0.780147\pi$
$151$	− 15.6569i	− 1.27414i	0.770807	−	0.637068i	$-0.219853\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.75736i	0.778722i	0.921085	+	0.389361i	$0.127304\pi$
$157$	9.75736i	0.778722i	−0.921085	+	0.389361i	$0.872696\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 16.4853i	− 1.29922i
$162$	0	0
$163$	0.485281	0.0380102	0.0190051	−	0.999819i	$-0.493950\pi$
$163$	0.485281	0.0380102	0.0190051	+	0.999819i	$0.493950\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 21.7990i	− 1.68686i	−0.537242	−	0.843428i	$-0.680534\pi$
$167$	− 21.7990i	− 1.68686i	0.537242	−	0.843428i	$-0.319466\pi$
$168$	0	0
$169$	1.34315	0.103319
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.48528	0.645124	0.322562	−	0.946548i	$-0.395456\pi$
$173$	8.48528	0.645124	0.322562	+	0.946548i	$0.395456\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−24.0416	−1.79696	−0.898478	−	0.439019i	$-0.855326\pi$
$179$	−24.0416	−1.79696	−0.898478	+	0.439019i	$0.855326\pi$
$180$	0	0
$181$	1.51472	0.112588	0.0562941	−	0.998414i	$-0.482072\pi$
$181$	1.51472	0.112588	0.0562941	+	0.998414i	$0.482072\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.02944	0.221534
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.51472	0.254316	0.127158	−	0.991882i	$-0.459414\pi$
$191$	3.51472	0.254316	0.127158	+	0.991882i	$0.459414\pi$
$192$	0	0
$193$	18.9706i	1.36553i	0.730638	+	0.682765i	$0.239223\pi$
$193$	18.9706i	1.36553i	−0.730638	+	0.682765i	$0.760777\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.3137	1.51854	0.759269	−	0.650776i	$-0.225556\pi$
$197$	21.3137	1.51854	0.759269	+	0.650776i	$0.225556\pi$
$198$	0	0
$199$	− 15.6569i	− 1.10988i	−0.831889	−	0.554942i	$-0.812740\pi$
$199$	− 15.6569i	− 1.10988i	0.831889	−	0.554942i	$-0.187260\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.4853i	1.43778i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 26.0000i	− 1.78991i	−0.446153	−	0.894957i	$-0.647206\pi$
$211$	− 26.0000i	− 1.78991i	0.446153	−	0.894957i	$-0.352794\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	22.1421i	1.50311i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000i	0.269069i
$222$	0	0
$223$	2.72792	0.182675	0.0913376	−	0.995820i	$-0.470886\pi$
$223$	2.72792	0.182675	0.0913376	+	0.995820i	$0.470886\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.65685i	0.375459i	0.982221	+	0.187729i	$0.0601128\pi$
$227$	5.65685i	0.375459i	−0.982221	+	0.187729i	$0.939887\pi$
$228$	0	0
$229$	14.9706	0.989283	0.494641	−	0.869097i	$-0.335299\pi$
$229$	14.9706	0.989283	0.494641	+	0.869097i	$0.335299\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.14214	0.140336	0.0701680	−	0.997535i	$-0.477646\pi$
$233$	2.14214	0.140336	0.0701680	+	0.997535i	$0.477646\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	25.4558	1.64660	0.823301	−	0.567605i	$-0.192130\pi$
$239$	25.4558	1.64660	0.823301	+	0.567605i	$0.192130\pi$
$240$	0	0
$241$	28.9706	1.86616	0.933079	−	0.359671i	$-0.117111\pi$
$241$	28.9706	1.86616	0.933079	+	0.359671i	$0.117111\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−13.8995	−0.877328	−0.438664	−	0.898651i	$-0.644548\pi$
$251$	−13.8995	−0.877328	−0.438664	+	0.898651i	$0.644548\pi$
$252$	0	0
$253$	12.4853i	0.784943i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	30.9706i	1.92442i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.51472i	0.586703i	0.956005	+	0.293351i	$0.0947706\pi$
$263$	9.51472i	0.586703i	−0.956005	+	0.293351i	$0.905229\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 7.65685i	− 0.466847i	−0.972375	−	0.233423i	$-0.925007\pi$
$269$	− 7.65685i	− 0.466847i	0.972375	−	0.233423i	$-0.0749928\pi$
$270$	0	0
$271$	− 22.0000i	− 1.33640i	−0.743980	−	0.668202i	$-0.767064\pi$
$271$	− 22.0000i	− 1.33640i	0.743980	−	0.668202i	$-0.232936\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.27208i	0.316768i	0.987378	+	0.158384i	$0.0506285\pi$
$277$	5.27208i	0.316768i	−0.987378	+	0.158384i	$0.949372\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.5858i	1.34736i	0.739025	+	0.673678i	$0.235286\pi$
$281$	22.5858i	1.34736i	−0.739025	+	0.673678i	$0.764714\pi$
$282$	0	0
$283$	−7.31371	−0.434755	−0.217377	−	0.976088i	$-0.569750\pi$
$283$	−7.31371	−0.434755	−0.217377	+	0.976088i	$0.569750\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 37.7990i	− 2.23120i
$288$	0	0
$289$	−15.6274	−0.919260
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−29.3137	−1.71253	−0.856263	−	0.516541i	$-0.827219\pi$
$293$	−29.3137	−1.71253	−0.856263	+	0.516541i	$0.827219\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−16.4853	−0.953368
$300$	0	0
$301$	23.3137	1.34378
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.0000	1.36975	0.684876	−	0.728659i	$-0.259856\pi$
$307$	24.0000	1.36975	0.684876	+	0.728659i	$0.259856\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.1421	−1.48238	−0.741192	−	0.671293i	$-0.765739\pi$
$311$	−26.1421	−1.48238	−0.741192	+	0.671293i	$0.765739\pi$
$312$	0	0
$313$	7.65685i	0.432791i	0.976306	+	0.216395i	$0.0694301\pi$
$313$	7.65685i	0.432791i	−0.976306	+	0.216395i	$0.930570\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.6274	0.933889	0.466944	−	0.884287i	$-0.345355\pi$
$317$	16.6274	0.933889	0.466944	+	0.884287i	$0.345355\pi$
$318$	0	0
$319$	− 15.5147i	− 0.868657i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 19.3137i	− 1.06480i
$330$	0	0
$331$	− 28.9706i	− 1.59237i	−0.605056	−	0.796183i	$-0.706849\pi$
$331$	− 28.9706i	− 1.59237i	0.605056	−	0.796183i	$-0.293151\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.4853i	1.44275i	0.692547	+	0.721373i	$0.256488\pi$
$337$	26.4853i	1.44275i	−0.692547	+	0.721373i	$0.743512\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 16.7696i	− 0.908122i
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 22.3431i	− 1.19944i	−0.800209	−	0.599721i	$-0.795278\pi$
$347$	− 22.3431i	− 1.19944i	0.800209	−	0.599721i	$-0.204722\pi$
$348$	0	0
$349$	−31.4558	−1.68379	−0.841896	−	0.539639i	$-0.818561\pi$
$349$	−31.4558	−1.68379	−0.841896	+	0.539639i	$0.818561\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.85786	0.0988841	0.0494421	−	0.998777i	$-0.484256\pi$
$353$	1.85786	0.0988841	0.0494421	+	0.998777i	$0.484256\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.8284	−0.993726	−0.496863	−	0.867829i	$-0.665515\pi$
$359$	−18.8284	−0.993726	−0.496863	+	0.867829i	$0.665515\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−24.8701	−1.29821	−0.649103	−	0.760700i	$-0.724856\pi$
$367$	−24.8701	−1.29821	−0.649103	+	0.760700i	$0.724856\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	− 10.7279i	− 0.555471i	−0.960658	−	0.277735i	$-0.910416\pi$
$373$	− 10.7279i	− 0.555471i	0.960658	−	0.277735i	$-0.0895838\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	20.4853	1.05505
$378$	0	0
$379$	− 9.31371i	− 0.478413i	−0.970969	−	0.239207i	$-0.923113\pi$
$379$	− 9.31371i	− 0.478413i	0.970969	−	0.239207i	$-0.0768873\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 10.3431i	− 0.528510i	−0.964453	−	0.264255i	$-0.914874\pi$
$383$	− 10.3431i	− 0.528510i	0.964453	−	0.264255i	$-0.0851261\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 12.3431i	− 0.625822i	−0.949782	−	0.312911i	$-0.898696\pi$
$389$	− 12.3431i	− 0.625822i	0.949782	−	0.312911i	$-0.101304\pi$
$390$	0	0
$391$	5.65685i	0.286079i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 14.7279i	− 0.739173i	−0.929196	−	0.369587i	$-0.879499\pi$
$397$	− 14.7279i	− 0.739173i	0.929196	−	0.369587i	$-0.120501\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 0.727922i	− 0.0363507i	−0.999835	−	0.0181753i	$-0.994214\pi$
$401$	− 0.727922i	− 0.0363507i	0.999835	−	0.0181753i	$-0.00578571\pi$
$402$	0	0
$403$	22.1421	1.10298
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 23.4558i	− 1.16266i
$408$	0	0
$409$	38.2843	1.89304	0.946518	−	0.322652i	$-0.104574\pi$
$409$	38.2843	1.89304	0.946518	+	0.322652i	$0.104574\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	22.4853	1.10643
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−38.3848	−1.87522	−0.937610	−	0.347690i	$-0.886966\pi$
$419$	−38.3848	−1.87522	−0.937610	+	0.347690i	$0.886966\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	43.7990	2.11958
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.14214	−0.103183	−0.0515915	−	0.998668i	$-0.516429\pi$
$431$	−2.14214	−0.103183	−0.0515915	+	0.998668i	$0.516429\pi$
$432$	0	0
$433$	− 14.4853i	− 0.696118i	−0.937473	−	0.348059i	$-0.886841\pi$
$433$	− 14.4853i	− 0.696118i	0.937473	−	0.348059i	$-0.113159\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	18.2843i	0.872661i	0.899787	+	0.436330i	$0.143722\pi$
$439$	18.2843i	0.872661i	−0.899787	+	0.436330i	$0.856278\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.6274i	1.26511i	0.774517	+	0.632553i	$0.217993\pi$
$443$	26.6274i	1.26511i	−0.774517	+	0.632553i	$0.782007\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.5858i	0.877117i	0.898703	+	0.438559i	$0.144511\pi$
$449$	18.5858i	0.877117i	−0.898703	+	0.438559i	$0.855489\pi$
$450$	0	0
$451$	28.6274i	1.34801i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.48528i	− 0.116257i	−0.998309	−	0.0581283i	$-0.981487\pi$
$457$	− 2.48528i	− 0.116257i	0.998309	−	0.0581283i	$-0.0185132\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 24.8284i	− 1.15638i	−0.815904	−	0.578188i	$-0.803760\pi$
$461$	− 24.8284i	− 1.15638i	0.815904	−	0.578188i	$-0.196240\pi$
$462$	0	0
$463$	17.0711	0.793360	0.396680	−	0.917957i	$-0.370162\pi$
$463$	17.0711	0.793360	0.396680	+	0.917957i	$0.370162\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.1127i	1.43972i	0.694117	+	0.719862i	$0.255795\pi$
$467$	31.1127i	1.43972i	−0.694117	+	0.719862i	$0.744205\pi$
$468$	0	0
$469$	27.3137	1.26123
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−17.6569	−0.811863
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.3137	1.24800	0.623998	−	0.781426i	$-0.285507\pi$
$479$	27.3137	1.24800	0.623998	+	0.781426i	$0.285507\pi$
$480$	0	0
$481$	30.9706	1.41214
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−10.7279	−0.486129	−0.243064	−	0.970010i	$-0.578153\pi$
$487$	−10.7279	−0.486129	−0.243064	+	0.970010i	$0.578153\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.0711	1.40222	0.701109	−	0.713054i	$-0.252689\pi$
$491$	31.0711	1.40222	0.701109	+	0.713054i	$0.252689\pi$
$492$	0	0
$493$	− 7.02944i	− 0.316590i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−19.3137	−0.866338
$498$	0	0
$499$	− 18.6274i	− 0.833878i	−0.908935	−	0.416939i	$-0.863103\pi$
$499$	− 18.6274i	− 0.833878i	0.908935	−	0.416939i	$-0.136897\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 36.2843i	− 1.61784i	−0.587922	−	0.808918i	$-0.700054\pi$
$503$	− 36.2843i	− 1.61784i	0.587922	−	0.808918i	$-0.299946\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 8.14214i	− 0.360894i	−0.983585	−	0.180447i	$-0.942246\pi$
$509$	− 8.14214i	− 0.360894i	0.983585	−	0.180447i	$-0.0577544\pi$
$510$	0	0
$511$	− 35.7990i	− 1.58365i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.6274i	0.643313i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.727922i	0.0318908i	0.999873	+	0.0159454i	$0.00507580\pi$
$521$	0.727922i	0.0318908i	−0.999873	+	0.0159454i	$0.994924\pi$
$522$	0	0
$523$	7.51472	0.328596	0.164298	−	0.986411i	$-0.447464\pi$
$523$	7.51472	0.328596	0.164298	+	0.986411i	$0.447464\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 7.59798i	− 0.330973i
$528$	0	0
$529$	−0.313708	−0.0136395
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−37.7990	−1.63726
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12.0416	−0.518670
$540$	0	0
$541$	28.8284	1.23943	0.619715	−	0.784827i	$-0.287248\pi$
$541$	28.8284	1.23943	0.619715	+	0.784827i	$0.287248\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16.4853	0.704860	0.352430	−	0.935838i	$-0.385356\pi$
$547$	16.4853	0.704860	0.352430	+	0.935838i	$0.385356\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 49.4558i	− 2.10308i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.4853	0.867989	0.433995	−	0.900915i	$-0.357104\pi$
$557$	20.4853	0.867989	0.433995	+	0.900915i	$0.357104\pi$
$558$	0	0
$559$	− 23.3137i	− 0.986065i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.34315i	0.267332i	0.991026	+	0.133666i	$0.0426749\pi$
$563$	6.34315i	0.267332i	−0.991026	+	0.133666i	$0.957325\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.21320i	0.386238i	0.981175	+	0.193119i	$0.0618603\pi$
$569$	9.21320i	0.386238i	−0.981175	+	0.193119i	$0.938140\pi$
$570$	0	0
$571$	4.97056i	0.208012i	0.994577	+	0.104006i	$0.0331660\pi$
$571$	4.97056i	0.208012i	−0.994577	+	0.104006i	$0.966834\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 15.4558i	− 0.643435i	−0.946836	−	0.321718i	$-0.895740\pi$
$577$	− 15.4558i	− 0.643435i	0.946836	−	0.321718i	$-0.104260\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 31.3137i	− 1.29911i
$582$	0	0
$583$	−3.02944	−0.125466
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.1421i	1.07900i	0.841985	+	0.539501i	$0.181387\pi$
$587$	26.1421i	1.07900i	−0.841985	+	0.539501i	$0.818613\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−10.3431	−0.424742	−0.212371	−	0.977189i	$-0.568119\pi$
$593$	−10.3431	−0.424742	−0.212371	+	0.977189i	$0.568119\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	42.1421	1.72188	0.860940	−	0.508706i	$-0.169876\pi$
$599$	42.1421	1.72188	0.860940	+	0.508706i	$0.169876\pi$
$600$	0	0
$601$	16.6863	0.680648	0.340324	−	0.940308i	$-0.389463\pi$
$601$	16.6863	0.680648	0.340324	+	0.940308i	$0.389463\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.21320	0.130420	0.0652100	−	0.997872i	$-0.479228\pi$
$607$	3.21320	0.130420	0.0652100	+	0.997872i	$0.479228\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−19.3137	−0.781349
$612$	0	0
$613$	− 3.21320i	− 0.129780i	−0.997892	−	0.0648900i	$-0.979330\pi$
$613$	− 3.21320i	− 0.129780i	0.997892	−	0.0648900i	$-0.0206697\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.8284	−1.08007	−0.540036	−	0.841642i	$-0.681589\pi$
$617$	−26.8284	−1.08007	−0.540036	+	0.841642i	$0.681589\pi$
$618$	0	0
$619$	− 22.9706i	− 0.923265i	−0.887071	−	0.461632i	$-0.847264\pi$
$619$	− 22.9706i	− 0.923265i	0.887071	−	0.461632i	$-0.152736\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 14.4853i	− 0.580341i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 10.6274i	− 0.423743i
$630$	0	0
$631$	7.17157i	0.285496i	0.989759	+	0.142748i	$0.0455938\pi$
$631$	7.17157i	0.285496i	−0.989759	+	0.142748i	$0.954406\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 15.8995i	− 0.629961i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 43.5563i	− 1.72037i	−0.509980	−	0.860186i	$-0.670347\pi$
$641$	− 43.5563i	− 1.72037i	0.509980	−	0.860186i	$-0.329653\pi$
$642$	0	0
$643$	−35.1127	−1.38471	−0.692355	−	0.721557i	$-0.743427\pi$
$643$	−35.1127	−1.38471	−0.692355	+	0.721557i	$0.743427\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.6863i	0.498750i	0.968407	+	0.249375i	$0.0802251\pi$
$647$	12.6863i	0.498750i	−0.968407	+	0.249375i	$0.919775\pi$
$648$	0	0
$649$	−17.0294	−0.668464
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.6569	0.769232	0.384616	−	0.923077i	$-0.374334\pi$
$653$	19.6569	0.769232	0.384616	+	0.923077i	$0.374334\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.58579	0.256546	0.128273	−	0.991739i	$-0.459057\pi$
$659$	6.58579	0.256546	0.128273	+	0.991739i	$0.459057\pi$
$660$	0	0
$661$	−40.4264	−1.57240	−0.786202	−	0.617969i	$-0.787956\pi$
$661$	−40.4264	−1.57240	−0.786202	+	0.617969i	$0.787956\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28.9706	1.12174
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−33.1716	−1.28057
$672$	0	0
$673$	51.4558i	1.98348i	0.128276	+	0.991739i	$0.459056\pi$
$673$	51.4558i	1.98348i	−0.128276	+	0.991739i	$0.540944\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.68629	0.256975	0.128488	−	0.991711i	$-0.458988\pi$
$677$	6.68629	0.256975	0.128488	+	0.991711i	$0.458988\pi$
$678$	0	0
$679$	− 8.48528i	− 0.325635i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 10.8284i	− 0.414338i	−0.978305	−	0.207169i	$-0.933575\pi$
$683$	− 10.8284i	− 0.414338i	0.978305	−	0.207169i	$-0.0664250\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 4.00000i	− 0.152388i
$690$	0	0
$691$	8.00000i	0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$	8.00000i	0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.9706i	0.491295i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 9.51472i	− 0.359366i	−0.983725	−	0.179683i	$-0.942493\pi$
$701$	− 9.51472i	− 0.359366i	0.983725	−	0.179683i	$-0.0575072\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 16.4853i	− 0.619993i
$708$	0	0
$709$	−6.97056	−0.261785	−0.130892	−	0.991397i	$-0.541784\pi$
$709$	−6.97056	−0.261785	−0.130892	+	0.991397i	$0.541784\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	31.3137	1.17271
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.8284	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$719$	−10.8284	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$720$	0	0
$721$	−59.9411	−2.23232
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−26.9289	−0.998739	−0.499369	−	0.866389i	$-0.666435\pi$
$727$	−26.9289	−0.998739	−0.499369	+	0.866389i	$0.666435\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	− 12.1005i	− 0.446942i	−0.974711	−	0.223471i	$-0.928261\pi$
$733$	− 12.1005i	− 0.446942i	0.974711	−	0.223471i	$-0.0717389\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.6863	−0.761989
$738$	0	0
$739$	22.3431i	0.821906i	0.911657	+	0.410953i	$0.134804\pi$
$739$	22.3431i	0.821906i	−0.911657	+	0.410953i	$0.865196\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	50.6274i	1.84989i
$750$	0	0
$751$	28.8284i	1.05196i	0.850496	+	0.525982i	$0.176302\pi$
$751$	28.8284i	1.05196i	−0.850496	+	0.525982i	$0.823698\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	52.8701i	1.92159i	0.277251	+	0.960797i	$0.410577\pi$
$757$	52.8701i	1.92159i	−0.277251	+	0.960797i	$0.589423\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.8701i	1.11904i	0.828817	+	0.559519i	$0.189014\pi$
$761$	30.8701i	1.11904i	−0.828817	+	0.559519i	$0.810986\pi$
$762$	0	0
$763$	24.4853	0.886427
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 22.4853i	− 0.811896i
$768$	0	0
$769$	2.34315	0.0844960	0.0422480	−	0.999107i	$-0.486548\pi$
$769$	2.34315	0.0844960	0.0422480	+	0.999107i	$0.486548\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	41.3137	1.48595	0.742975	−	0.669319i	$-0.233414\pi$
$773$	41.3137	1.48595	0.742975	+	0.669319i	$0.233414\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	14.6274	0.523410
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	42.1421	1.50220	0.751102	−	0.660186i	$-0.229522\pi$
$787$	42.1421	1.50220	0.751102	+	0.660186i	$0.229522\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.65685	−0.201135
$792$	0	0
$793$	− 43.7990i	− 1.55535i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.8284	−0.666937	−0.333469	−	0.942761i	$-0.608219\pi$
$797$	−18.8284	−0.666937	−0.333469	+	0.942761i	$0.608219\pi$
$798$	0	0
$799$	6.62742i	0.234461i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	27.1127i	0.956786i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 36.0416i	− 1.26716i	−0.773679	−	0.633578i	$-0.781586\pi$
$809$	− 36.0416i	− 1.26716i	0.773679	−	0.633578i	$-0.218414\pi$
$810$	0	0
$811$	− 3.37258i	− 0.118427i	−0.998245	−	0.0592137i	$-0.981141\pi$
$811$	− 3.37258i	− 0.118427i	0.998245	−	0.0592137i	$-0.0188593\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29.3137i	1.02306i	0.859267	+	0.511528i	$0.170920\pi$
$821$	29.3137i	1.02306i	−0.859267	+	0.511528i	$0.829080\pi$
$822$	0	0
$823$	6.24264	0.217605	0.108802	−	0.994063i	$-0.465298\pi$
$823$	6.24264	0.217605	0.108802	+	0.994063i	$0.465298\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42.6274i	1.48230i	0.671339	+	0.741150i	$0.265719\pi$
$827$	42.6274i	1.48230i	−0.671339	+	0.741150i	$0.734281\pi$
$828$	0	0
$829$	19.4558	0.675729	0.337865	−	0.941195i	$-0.390295\pi$
$829$	19.4558	0.675729	0.337865	+	0.941195i	$0.390295\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.45584	−0.189034
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	42.4264	1.46472	0.732361	−	0.680916i	$-0.238418\pi$
$839$	42.4264	1.46472	0.732361	+	0.680916i	$0.238418\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−14.7279	−0.506057
$848$	0	0
$849$	0	0
$850$	0	0
$851$	43.7990	1.50141
$852$	0	0
$853$	21.7574i	0.744958i	0.928041	+	0.372479i	$0.121492\pi$
$853$	21.7574i	0.744958i	−0.928041	+	0.372479i	$0.878508\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.7990	1.22287	0.611435	−	0.791295i	$-0.290593\pi$
$857$	35.7990	1.22287	0.611435	+	0.791295i	$0.290593\pi$
$858$	0	0
$859$	26.0000i	0.887109i	0.896248	+	0.443554i	$0.146283\pi$
$859$	26.0000i	0.887109i	−0.896248	+	0.443554i	$0.853717\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.20101i	0.211085i	0.994415	+	0.105542i	$0.0336579\pi$
$863$	6.20101i	0.211085i	−0.994415	+	0.105542i	$0.966342\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	37.4558i	1.27060i
$870$	0	0
$871$	− 27.3137i	− 0.925490i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.55635i	0.0525542i	0.999655	+	0.0262771i	$0.00836522\pi$
$877$	1.55635i	0.0525542i	−0.999655	+	0.0262771i	$0.991635\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 35.3553i	− 1.19115i	−0.803299	−	0.595576i	$-0.796924\pi$
$881$	− 35.3553i	− 1.19115i	0.803299	−	0.595576i	$-0.203076\pi$
$882$	0	0
$883$	−42.4264	−1.42776	−0.713881	−	0.700267i	$-0.753064\pi$
$883$	−42.4264	−1.42776	−0.713881	+	0.700267i	$0.753064\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 12.8284i	− 0.430736i	−0.976533	−	0.215368i	$-0.930905\pi$
$887$	− 12.8284i	− 0.430736i	0.976533	−	0.215368i	$-0.0690952\pi$
$888$	0	0
$889$	36.6274	1.22844
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−38.9117	−1.29778
$900$	0	0
$901$	−1.37258	−0.0457274
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.85786	0.0616894	0.0308447	−	0.999524i	$-0.490180\pi$
$907$	1.85786	0.0616894	0.0308447	+	0.999524i	$0.490180\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−18.3431	−0.607736	−0.303868	−	0.952714i	$-0.598278\pi$
$911$	−18.3431	−0.607736	−0.303868	+	0.952714i	$0.598278\pi$
$912$	0	0
$913$	23.7157i	0.784876i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−43.4558	−1.43504
$918$	0	0
$919$	− 12.8284i	− 0.423171i	−0.977360	−	0.211585i	$-0.932137\pi$
$919$	− 12.8284i	− 0.423171i	0.977360	−	0.211585i	$-0.0678626\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	19.3137i	0.635718i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	45.8995i	1.50591i	0.658070	+	0.752957i	$0.271373\pi$
$929$	45.8995i	1.50591i	−0.658070	+	0.752957i	$0.728627\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.9706i	0.489067i	0.969641	+	0.244533i	$0.0786348\pi$
$937$	14.9706i	0.489067i	−0.969641	+	0.244533i	$0.921365\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	42.2843i	1.37843i	0.724558	+	0.689214i	$0.242044\pi$
$941$	42.2843i	1.37843i	−0.724558	+	0.689214i	$0.757956\pi$
$942$	0	0
$943$	−53.4558	−1.74076
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51.5980i	1.67671i	0.545125	+	0.838355i	$0.316482\pi$
$947$	51.5980i	1.67671i	−0.545125	+	0.838355i	$0.683518\pi$
$948$	0	0
$949$	−35.7990	−1.16208
$950$	0	0
$951$	0	0
$952$	0	0
$953$	45.2548	1.46595	0.732974	−	0.680257i	$-0.238132\pi$
$953$	45.2548	1.46595	0.732974	+	0.680257i	$0.238132\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	65.9411	2.12935
$960$	0	0
$961$	−11.0589	−0.356738
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−26.7279	−0.859512	−0.429756	−	0.902945i	$-0.641400\pi$
$967$	−26.7279	−0.859512	−0.429756	+	0.902945i	$0.641400\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−27.0711	−0.868752	−0.434376	−	0.900732i	$-0.643031\pi$
$971$	−27.0711	−0.868752	−0.434376	+	0.900732i	$0.643031\pi$
$972$	0	0
$973$	12.4853i	0.400260i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.1716	1.18922	0.594612	−	0.804013i	$-0.297306\pi$
$977$	37.1716	1.18922	0.594612	+	0.804013i	$0.297306\pi$
$978$	0	0
$979$	10.9706i	0.350621i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	39.5980i	1.26298i	0.775384	+	0.631490i	$0.217556\pi$
$983$	39.5980i	1.26298i	−0.775384	+	0.631490i	$0.782444\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 32.9706i	− 1.04840i
$990$	0	0
$991$	16.3431i	0.519157i	0.965722	+	0.259579i	$0.0835837\pi$
$991$	16.3431i	0.519157i	−0.965722	+	0.259579i	$0.916416\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 5.27208i	− 0.166968i	−0.996509	−	0.0834842i	$-0.973395\pi$
$997$	− 5.27208i	− 0.166968i	0.996509	−	0.0834842i	$-0.0266048\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.o.j.7199.3		4
3.2	odd	2		7200.2.o.m.7199.3		4
4.3	odd	2		7200.2.o.e.7199.1		4
5.2	odd	4		1440.2.h.a.1151.4	yes	4
5.3	odd	4		7200.2.h.c.1151.1		4
5.4	even	2		7200.2.o.b.7199.2		4
12.11	even	2		7200.2.o.b.7199.1		4
15.2	even	4		1440.2.h.d.1151.2	yes	4
15.8	even	4		7200.2.h.i.1151.1		4
15.14	odd	2		7200.2.o.e.7199.2		4
20.3	even	4		7200.2.h.i.1151.4		4
20.7	even	4		1440.2.h.d.1151.3	yes	4
20.19	odd	2		7200.2.o.m.7199.4		4
40.27	even	4		2880.2.h.a.1151.1		4
40.37	odd	4		2880.2.h.d.1151.2		4
60.23	odd	4		7200.2.h.c.1151.4		4
60.47	odd	4		1440.2.h.a.1151.1	✓	4
60.59	even	2	inner	7200.2.o.j.7199.4		4
120.77	even	4		2880.2.h.a.1151.4		4
120.107	odd	4		2880.2.h.d.1151.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.h.a.1151.1	✓	4	60.47	odd	4
1440.2.h.a.1151.4	yes	4	5.2	odd	4
1440.2.h.d.1151.2	yes	4	15.2	even	4
1440.2.h.d.1151.3	yes	4	20.7	even	4
2880.2.h.a.1151.1		4	40.27	even	4
2880.2.h.a.1151.4		4	120.77	even	4
2880.2.h.d.1151.2		4	40.37	odd	4
2880.2.h.d.1151.3		4	120.107	odd	4
7200.2.h.c.1151.1		4	5.3	odd	4
7200.2.h.c.1151.4		4	60.23	odd	4
7200.2.h.i.1151.1		4	15.8	even	4
7200.2.h.i.1151.4		4	20.3	even	4
7200.2.o.b.7199.1		4	12.11	even	2
7200.2.o.b.7199.2		4	5.4	even	2
7200.2.o.e.7199.1		4	4.3	odd	2
7200.2.o.e.7199.2		4	15.14	odd	2
7200.2.o.j.7199.3		4	1.1	even	1	trivial
7200.2.o.j.7199.4		4	60.59	even	2	inner
7200.2.o.m.7199.3		4	3.2	odd	2
7200.2.o.m.7199.4		4	20.19	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:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.41421 q^{7} +O(q^{10})\)	\(q+3.41421 q^{7} -2.58579 q^{11} -3.41421i q^{13} -1.17157 q^{17} -4.82843i q^{23} +6.00000i q^{29} +6.48528i q^{31} +9.07107i q^{37} -11.0711i q^{41} +6.82843 q^{43} -5.65685i q^{47} +4.65685 q^{49} +1.17157 q^{53} +6.58579 q^{59} +12.8284 q^{61} +8.00000 q^{67} -5.65685 q^{71} -10.4853i q^{73} -8.82843 q^{77} -14.4853i q^{79} -9.17157i q^{83} -4.24264i q^{89} -11.6569i q^{91} -2.48528i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7}+O(q^{10}) \) 4 * q + 8 * q^7	\( 4 q + 8 q^{7} - 16 q^{11} - 16 q^{17} + 16 q^{43} - 4 q^{49} + 16 q^{53} + 32 q^{59} + 40 q^{61} + 32 q^{67} - 24 q^{77}+O(q^{100}) \) 4 * q + 8 * q^7 - 16 * q^11 - 16 * q^17 + 16 * q^43 - 4 * q^49 + 16 * q^53 + 32 * q^59 + 40 * q^61 + 32 * q^67 - 24 * q^77

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