LMFDB - Embedded newform 7200.2.o.j.7199.1

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			7199.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	7200.7199
Dual form			7200.2.o.j.7199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.585786 q^{7} +O(q^{10})$	$q+0.585786 q^{7} -5.41421 q^{11} -0.585786i q^{13} -6.82843 q^{17} +0.828427i q^{23} +6.00000i q^{29} -10.4853i q^{31} -5.07107i q^{37} +3.07107i q^{41} +1.17157 q^{43} +5.65685i q^{47} -6.65685 q^{49} +6.82843 q^{53} +9.41421 q^{59} +7.17157 q^{61} +8.00000 q^{67} +5.65685 q^{71} +6.48528i q^{73} -3.17157 q^{77} +2.48528i q^{79} -14.8284i q^{83} +4.24264i q^{89} -0.343146i q^{91} +14.4853i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	0.585786	0.221406	0.110703	−	0.993854i	$-0.464690\pi$
$7$	0.585786	0.221406	0.110703	+	0.993854i	$0.464690\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.41421	−1.63245	−0.816223	−	0.577736i	$-0.803936\pi$
$11$	−5.41421	−1.63245	−0.816223	+	0.577736i	$0.803936\pi$
$12$	0	0
$13$	− 0.585786i	− 0.162468i	−0.996695	−	0.0812340i	$-0.974114\pi$
$13$	− 0.585786i	− 0.162468i	0.996695	−	0.0812340i	$-0.0258861\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.828427i	0.172739i	0.996263	+	0.0863695i	$0.0275266\pi$
$23$	0.828427i	0.172739i	−0.996263	+	0.0863695i	$0.972473\pi$
$24$	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
$31$	− 10.4853i	− 1.88321i	−0.336717	−	0.941606i	$-0.609316\pi$
$31$	− 10.4853i	− 1.88321i	0.336717	−	0.941606i	$-0.390684\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.07107i	− 0.833678i	−0.908980	−	0.416839i	$-0.863138\pi$
$37$	− 5.07107i	− 0.833678i	0.908980	−	0.416839i	$-0.136862\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.07107i	0.479620i	0.970820	+	0.239810i	$0.0770852\pi$
$41$	3.07107i	0.479620i	−0.970820	+	0.239810i	$0.922915\pi$
$42$	0	0
$43$	1.17157	0.178663	0.0893316	−	0.996002i	$-0.471527\pi$
$43$	1.17157	0.178663	0.0893316	+	0.996002i	$0.471527\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.65685i	0.825137i	0.910927	+	0.412568i	$0.135368\pi$
$47$	5.65685i	0.825137i	−0.910927	+	0.412568i	$0.864632\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.82843	0.937957	0.468978	−	0.883210i	$-0.344622\pi$
$53$	6.82843	0.937957	0.468978	+	0.883210i	$0.344622\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.41421	1.22563	0.612813	−	0.790228i	$-0.290038\pi$
$59$	9.41421	1.22563	0.612813	+	0.790228i	$0.290038\pi$
$60$	0	0
$61$	7.17157	0.918226	0.459113	−	0.888378i	$-0.348167\pi$
$61$	7.17157	0.918226	0.459113	+	0.888378i	$0.348167\pi$
$62$	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	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	6.48528i	0.759045i	0.925183	+	0.379522i	$0.123912\pi$
$73$	6.48528i	0.759045i	−0.925183	+	0.379522i	$0.876088\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.17157	−0.361434
$78$	0	0
$79$	2.48528i	0.279616i	0.990179	+	0.139808i	$0.0446485\pi$
$79$	2.48528i	0.279616i	−0.990179	+	0.139808i	$0.955351\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 14.8284i	− 1.62763i	−0.581123	−	0.813816i	$-0.697387\pi$
$83$	− 14.8284i	− 1.62763i	0.581123	−	0.813816i	$-0.302613\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.24264i	0.449719i	0.974391	+	0.224860i	$0.0721923\pi$
$89$	4.24264i	0.449719i	−0.974391	+	0.224860i	$0.927808\pi$
$90$	0	0
$91$	− 0.343146i	− 0.0359714i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.4853i	1.47076i	0.677656	+	0.735379i	$0.262996\pi$
$97$	14.4853i	1.47076i	−0.677656	+	0.735379i	$0.737004\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.828427i	0.0824316i	0.999150	+	0.0412158i	$0.0131231\pi$
$101$	0.828427i	0.0824316i	−0.999150	+	0.0412158i	$0.986877\pi$
$102$	0	0
$103$	13.5563	1.33575	0.667873	−	0.744275i	$-0.267205\pi$
$103$	13.5563	1.33575	0.667873	+	0.744275i	$0.267205\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.17157i	0.886649i	0.896361	+	0.443325i	$0.146201\pi$
$107$	9.17157i	0.886649i	−0.896361	+	0.443325i	$0.853799\pi$
$108$	0	0
$109$	12.8284	1.22874	0.614370	−	0.789018i	$-0.289410\pi$
$109$	12.8284	1.22874	0.614370	+	0.789018i	$0.289410\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.65685	0.908440	0.454220	−	0.890889i	$-0.349918\pi$
$113$	9.65685	0.908440	0.454220	+	0.890889i	$0.349918\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	18.3137	1.66488
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.7279	−1.30689	−0.653446	−	0.756973i	$-0.726677\pi$
$127$	−14.7279	−1.30689	−0.653446	+	0.756973i	$0.726677\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.7279	1.11204	0.556022	−	0.831168i	$-0.312327\pi$
$131$	12.7279	1.11204	0.556022	+	0.831168i	$0.312327\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.31371	−0.283109	−0.141555	−	0.989930i	$-0.545210\pi$
$137$	−3.31371	−0.283109	−0.141555	+	0.989930i	$0.545210\pi$
$138$	0	0
$139$	− 7.65685i	− 0.649446i	−0.945809	−	0.324723i	$-0.894729\pi$
$139$	− 7.65685i	− 0.649446i	0.945809	−	0.324723i	$-0.105271\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.17157i	0.265220i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3.17157i	− 0.259825i	−0.991525	−	0.129913i	$-0.958530\pi$
$149$	− 3.17157i	− 0.259825i	0.991525	−	0.129913i	$-0.0414697\pi$
$150$	0	0
$151$	− 4.34315i	− 0.353440i	−0.984261	−	0.176720i	$-0.943451\pi$
$151$	− 4.34315i	− 0.353440i	0.984261	−	0.176720i	$-0.0565487\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.2426i	1.45592i	0.685619	+	0.727961i	$0.259532\pi$
$157$	18.2426i	1.45592i	−0.685619	+	0.727961i	$0.740468\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.485281i	0.0382455i
$162$	0	0
$163$	−16.4853	−1.29123	−0.645613	−	0.763664i	$-0.723398\pi$
$163$	−16.4853	−1.29123	−0.645613	+	0.763664i	$0.723398\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.7990i	1.37733i	0.725081	+	0.688664i	$0.241802\pi$
$167$	17.7990i	1.37733i	−0.725081	+	0.688664i	$0.758198\pi$
$168$	0	0
$169$	12.6569	0.973604
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.48528	−0.645124	−0.322562	−	0.946548i	$-0.604544\pi$
$173$	−8.48528	−0.645124	−0.322562	+	0.946548i	$0.604544\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.0416	1.79696	0.898478	−	0.439019i	$-0.144674\pi$
$179$	24.0416	1.79696	0.898478	+	0.439019i	$0.144674\pi$
$180$	0	0
$181$	18.4853	1.37400	0.687000	−	0.726657i	$-0.258927\pi$
$181$	18.4853	1.37400	0.687000	+	0.726657i	$0.258927\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	36.9706	2.70356
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.4853	1.48226	0.741131	−	0.671360i	$-0.234290\pi$
$191$	20.4853	1.48226	0.741131	+	0.671360i	$0.234290\pi$
$192$	0	0
$193$	− 14.9706i	− 1.07760i	−0.842432	−	0.538802i	$-0.818877\pi$
$193$	− 14.9706i	− 1.07760i	0.842432	−	0.538802i	$-0.181123\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.31371	−0.0935979	−0.0467989	−	0.998904i	$-0.514902\pi$
$197$	−1.31371	−0.0935979	−0.0467989	+	0.998904i	$0.514902\pi$
$198$	0	0
$199$	− 4.34315i	− 0.307877i	−0.988080	−	0.153939i	$-0.950804\pi$
$199$	− 4.34315i	− 0.307877i	0.988080	−	0.153939i	$-0.0491958\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.51472i	0.246685i
$204$	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
$217$	− 6.14214i	− 0.416955i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000i	0.269069i
$222$	0	0
$223$	−22.7279	−1.52197	−0.760987	−	0.648767i	$-0.775285\pi$
$223$	−22.7279	−1.52197	−0.760987	+	0.648767i	$0.775285\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5.65685i	− 0.375459i	−0.982221	−	0.187729i	$-0.939887\pi$
$227$	− 5.65685i	− 0.375459i	0.982221	−	0.187729i	$-0.0601128\pi$
$228$	0	0
$229$	−18.9706	−1.25361	−0.626805	−	0.779176i	$-0.715638\pi$
$229$	−18.9706	−1.25361	−0.626805	+	0.779176i	$0.715638\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.1421	−1.71263	−0.856314	−	0.516455i	$-0.827251\pi$
$233$	−26.1421	−1.71263	−0.856314	+	0.516455i	$0.827251\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−25.4558	−1.64660	−0.823301	−	0.567605i	$-0.807870\pi$
$239$	−25.4558	−1.64660	−0.823301	+	0.567605i	$0.807870\pi$
$240$	0	0
$241$	−4.97056	−0.320182	−0.160091	−	0.987102i	$-0.551179\pi$
$241$	−4.97056	−0.320182	−0.160091	+	0.987102i	$0.551179\pi$
$242$	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
$251$	5.89949	0.372373	0.186186	−	0.982514i	$-0.440387\pi$
$251$	5.89949	0.372373	0.186186	+	0.982514i	$0.440387\pi$
$252$	0	0
$253$	− 4.48528i	− 0.281987i
$254$	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
$259$	− 2.97056i	− 0.184582i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	26.4853i	1.63315i	0.577238	+	0.816576i	$0.304131\pi$
$263$	26.4853i	1.63315i	−0.577238	+	0.816576i	$0.695869\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.65685i	0.222962i	0.993767	+	0.111481i	$0.0355595\pi$
$269$	3.65685i	0.222962i	−0.993767	+	0.111481i	$0.964441\pi$
$270$	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
$277$	30.7279i	1.84626i	0.384486	+	0.923131i	$0.374379\pi$
$277$	30.7279i	1.84626i	−0.384486	+	0.923131i	$0.625621\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	25.4142i	1.51608i	0.652205	+	0.758042i	$0.273844\pi$
$281$	25.4142i	1.51608i	−0.652205	+	0.758042i	$0.726156\pi$
$282$	0	0
$283$	15.3137	0.910305	0.455153	−	0.890413i	$-0.349585\pi$
$283$	15.3137	0.910305	0.455153	+	0.890413i	$0.349585\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.79899i	0.106191i
$288$	0	0
$289$	29.6274	1.74279
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.68629	−0.390617	−0.195309	−	0.980742i	$-0.562571\pi$
$293$	−6.68629	−0.390617	−0.195309	+	0.980742i	$0.562571\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.485281	0.0280645
$300$	0	0
$301$	0.686292	0.0395572
$302$	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
$311$	2.14214	0.121469	0.0607347	−	0.998154i	$-0.480656\pi$
$311$	2.14214	0.121469	0.0607347	+	0.998154i	$0.480656\pi$
$312$	0	0
$313$	− 3.65685i	− 0.206698i	−0.994645	−	0.103349i	$-0.967044\pi$
$313$	− 3.65685i	− 0.206698i	0.994645	−	0.103349i	$-0.0329558\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−28.6274	−1.60788	−0.803938	−	0.594713i	$-0.797266\pi$
$317$	−28.6274	−1.60788	−0.803938	+	0.594713i	$0.797266\pi$
$318$	0	0
$319$	− 32.4853i	− 1.81883i
$320$	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
$329$	3.31371i	0.182691i
$330$	0	0
$331$	4.97056i	0.273207i	0.990626	+	0.136603i	$0.0436186\pi$
$331$	4.97056i	0.273207i	−0.990626	+	0.136603i	$0.956381\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.51472i	0.518300i	0.965837	+	0.259150i	$0.0834424\pi$
$337$	9.51472i	0.518300i	−0.965837	+	0.259150i	$0.916558\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	56.7696i	3.07424i
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 33.6569i	− 1.80679i	−0.428805	−	0.903397i	$-0.641065\pi$
$347$	− 33.6569i	− 1.80679i	0.428805	−	0.903397i	$-0.358935\pi$
$348$	0	0
$349$	19.4558	1.04145	0.520724	−	0.853725i	$-0.325662\pi$
$349$	19.4558	1.04145	0.520724	+	0.853725i	$0.325662\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.1421	1.60430	0.802152	−	0.597120i	$-0.203688\pi$
$353$	30.1421	1.60430	0.802152	+	0.597120i	$0.203688\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.1716	−0.695169	−0.347585	−	0.937649i	$-0.612998\pi$
$359$	−13.1716	−0.695169	−0.347585	+	0.937649i	$0.612998\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	28.8701	1.50700	0.753502	−	0.657445i	$-0.228363\pi$
$367$	28.8701	1.50700	0.753502	+	0.657445i	$0.228363\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	14.7279i	0.762583i	0.924455	+	0.381291i	$0.124521\pi$
$373$	14.7279i	0.762583i	−0.924455	+	0.381291i	$0.875479\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.51472	0.181017
$378$	0	0
$379$	13.3137i	0.683879i	0.939722	+	0.341940i	$0.111084\pi$
$379$	13.3137i	0.683879i	−0.939722	+	0.341940i	$0.888916\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 21.6569i	− 1.10661i	−0.832978	−	0.553307i	$-0.813366\pi$
$383$	− 21.6569i	− 1.10661i	0.832978	−	0.553307i	$-0.186634\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 23.6569i	− 1.19945i	−0.800206	−	0.599725i	$-0.795277\pi$
$389$	− 23.6569i	− 1.19945i	0.800206	−	0.599725i	$-0.204723\pi$
$390$	0	0
$391$	− 5.65685i	− 0.286079i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.7279i	0.538419i	0.963082	+	0.269209i	$0.0867624\pi$
$397$	10.7279i	0.538419i	−0.963082	+	0.269209i	$0.913238\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.7279i	1.23485i	0.786628	+	0.617427i	$0.211825\pi$
$401$	24.7279i	1.23485i	−0.786628	+	0.617427i	$0.788175\pi$
$402$	0	0
$403$	−6.14214	−0.305962
$404$	0	0
$405$	0	0
$406$	0	0
$407$	27.4558i	1.36094i
$408$	0	0
$409$	−18.2843	−0.904099	−0.452050	−	0.891993i	$-0.649307\pi$
$409$	−18.2843	−0.904099	−0.452050	+	0.891993i	$0.649307\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.51472	0.271362
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.61522	−0.0789088	−0.0394544	−	0.999221i	$-0.512562\pi$
$419$	−1.61522	−0.0789088	−0.0394544	+	0.999221i	$0.512562\pi$
$420$	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
$427$	4.20101	0.203301
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.1421	1.25922	0.629611	−	0.776910i	$-0.283214\pi$
$431$	26.1421	1.25922	0.629611	+	0.776910i	$0.283214\pi$
$432$	0	0
$433$	2.48528i	0.119435i	0.998215	+	0.0597175i	$0.0190200\pi$
$433$	2.48528i	0.119435i	−0.998215	+	0.0597175i	$0.980980\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 38.2843i	− 1.82721i	−0.406604	−	0.913604i	$-0.633287\pi$
$439$	− 38.2843i	− 1.82721i	0.406604	−	0.913604i	$-0.366713\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 18.6274i	− 0.885015i	−0.896765	−	0.442508i	$-0.854089\pi$
$443$	− 18.6274i	− 0.885015i	0.896765	−	0.442508i	$-0.145911\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.4142i	1.01060i	0.862944	+	0.505300i	$0.168618\pi$
$449$	21.4142i	1.01060i	−0.862944	+	0.505300i	$0.831382\pi$
$450$	0	0
$451$	− 16.6274i	− 0.782954i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.4853i	0.677593i	0.940860	+	0.338796i	$0.110020\pi$
$457$	14.4853i	0.677593i	−0.940860	+	0.338796i	$0.889980\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 19.1716i	− 0.892909i	−0.894806	−	0.446455i	$-0.852686\pi$
$461$	− 19.1716i	− 0.892909i	0.894806	−	0.446455i	$-0.147314\pi$
$462$	0	0
$463$	2.92893	0.136119	0.0680595	−	0.997681i	$-0.478319\pi$
$463$	2.92893	0.136119	0.0680595	+	0.997681i	$0.478319\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 31.1127i	− 1.43972i	−0.694117	−	0.719862i	$-0.744205\pi$
$467$	− 31.1127i	− 1.43972i	0.694117	−	0.719862i	$-0.255795\pi$
$468$	0	0
$469$	4.68629	0.216393
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.34315	−0.291658
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.68629	0.214122	0.107061	−	0.994252i	$-0.465856\pi$
$479$	4.68629	0.214122	0.107061	+	0.994252i	$0.465856\pi$
$480$	0	0
$481$	−2.97056	−0.135446
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.7279	0.667386	0.333693	−	0.942682i	$-0.391705\pi$
$487$	14.7279	0.667386	0.333693	+	0.942682i	$0.391705\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.9289	0.763992	0.381996	−	0.924164i	$-0.375237\pi$
$491$	16.9289	0.763992	0.381996	+	0.924164i	$0.375237\pi$
$492$	0	0
$493$	− 40.9706i	− 1.84522i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.31371	0.148640
$498$	0	0
$499$	26.6274i	1.19201i	0.802982	+	0.596003i	$0.203246\pi$
$499$	26.6274i	1.19201i	−0.802982	+	0.596003i	$0.796754\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.2843i	0.904431i	0.891909	+	0.452215i	$0.149366\pi$
$503$	20.2843i	0.904431i	−0.891909	+	0.452215i	$0.850634\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.1421i	0.892784i	0.894837	+	0.446392i	$0.147291\pi$
$509$	20.1421i	0.892784i	−0.894837	+	0.446392i	$0.852709\pi$
$510$	0	0
$511$	3.79899i	0.168057i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 30.6274i	− 1.34699i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 24.7279i	− 1.08335i	−0.840588	−	0.541675i	$-0.817790\pi$
$521$	− 24.7279i	− 1.08335i	0.840588	−	0.541675i	$-0.182210\pi$
$522$	0	0
$523$	24.4853	1.07067	0.535333	−	0.844641i	$-0.320186\pi$
$523$	24.4853	1.07067	0.535333	+	0.844641i	$0.320186\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	71.5980i	3.11886i
$528$	0	0
$529$	22.3137	0.970161
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.79899	0.0779229
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	36.0416	1.55242
$540$	0	0
$541$	23.1716	0.996224	0.498112	−	0.867113i	$-0.334027\pi$
$541$	23.1716	0.996224	0.498112	+	0.867113i	$0.334027\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−0.485281	−0.0207491	−0.0103746	−	0.999946i	$-0.503302\pi$
$547$	−0.485281	−0.0207491	−0.0103746	+	0.999946i	$0.503302\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	1.45584i	0.0619088i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.51472	0.148923	0.0744617	−	0.997224i	$-0.476276\pi$
$557$	3.51472	0.148923	0.0744617	+	0.997224i	$0.476276\pi$
$558$	0	0
$559$	− 0.686292i	− 0.0290270i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17.6569i	0.744148i	0.928203	+	0.372074i	$0.121353\pi$
$563$	17.6569i	0.744148i	−0.928203	+	0.372074i	$0.878647\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 33.2132i	− 1.39237i	−0.717862	−	0.696185i	$-0.754879\pi$
$569$	− 33.2132i	− 1.39237i	0.717862	−	0.696185i	$-0.245121\pi$
$570$	0	0
$571$	− 28.9706i	− 1.21238i	−0.795320	−	0.606190i	$-0.792697\pi$
$571$	− 28.9706i	− 1.21238i	0.795320	−	0.606190i	$-0.207303\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	35.4558i	1.47605i	0.674775	+	0.738023i	$0.264240\pi$
$577$	35.4558i	1.47605i	−0.674775	+	0.738023i	$0.735760\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 8.68629i	− 0.360368i
$582$	0	0
$583$	−36.9706	−1.53116
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.14214i	− 0.0884154i	−0.999022	−	0.0442077i	$-0.985924\pi$
$587$	− 2.14214i	− 0.0884154i	0.999022	−	0.0442077i	$-0.0140763\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21.6569	−0.889340	−0.444670	−	0.895694i	$-0.646679\pi$
$593$	−21.6569	−0.889340	−0.444670	+	0.895694i	$0.646679\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13.8579	0.566217	0.283108	−	0.959088i	$-0.408634\pi$
$599$	13.8579	0.566217	0.283108	+	0.959088i	$0.408634\pi$
$600$	0	0
$601$	39.3137	1.60364	0.801820	−	0.597566i	$-0.203865\pi$
$601$	39.3137	1.60364	0.801820	+	0.597566i	$0.203865\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−39.2132	−1.59161	−0.795807	−	0.605550i	$-0.792953\pi$
$607$	−39.2132	−1.59161	−0.795807	+	0.605550i	$0.792953\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.31371	0.134058
$612$	0	0
$613$	39.2132i	1.58381i	0.610647	+	0.791903i	$0.290910\pi$
$613$	39.2132i	1.58381i	−0.610647	+	0.791903i	$0.709090\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21.1716	−0.852335	−0.426168	−	0.904644i	$-0.640137\pi$
$617$	−21.1716	−0.852335	−0.426168	+	0.904644i	$0.640137\pi$
$618$	0	0
$619$	10.9706i	0.440944i	0.975393	+	0.220472i	$0.0707598\pi$
$619$	10.9706i	0.440944i	−0.975393	+	0.220472i	$0.929240\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.48528i	0.0995707i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	34.6274i	1.38069i
$630$	0	0
$631$	12.8284i	0.510692i	0.966850	+	0.255346i	$0.0821893\pi$
$631$	12.8284i	0.510692i	−0.966850	+	0.255346i	$0.917811\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.89949i	0.154504i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 12.4437i	− 0.491495i	−0.969334	−	0.245747i	$-0.920967\pi$
$641$	− 12.4437i	− 0.491495i	0.969334	−	0.245747i	$-0.0790333\pi$
$642$	0	0
$643$	27.1127	1.06922	0.534610	−	0.845099i	$-0.320458\pi$
$643$	27.1127	1.06922	0.534610	+	0.845099i	$0.320458\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	35.3137i	1.38833i	0.719818	+	0.694163i	$0.244225\pi$
$647$	35.3137i	1.38833i	−0.719818	+	0.694163i	$0.755775\pi$
$648$	0	0
$649$	−50.9706	−2.00077
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.34315	0.326493	0.163246	−	0.986585i	$-0.447803\pi$
$653$	8.34315	0.326493	0.163246	+	0.986585i	$0.447803\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9.41421	0.366726	0.183363	−	0.983045i	$-0.441302\pi$
$659$	9.41421	0.366726	0.183363	+	0.983045i	$0.441302\pi$
$660$	0	0
$661$	44.4264	1.72799	0.863993	−	0.503503i	$-0.167956\pi$
$661$	44.4264	1.72799	0.863993	+	0.503503i	$0.167956\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.97056	−0.192461
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−38.8284	−1.49895
$672$	0	0
$673$	0.544156i	0.0209757i	0.999945	+	0.0104878i	$0.00333844\pi$
$673$	0.544156i	0.0209757i	−0.999945	+	0.0104878i	$0.996662\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.3137	1.12662	0.563309	−	0.826247i	$-0.309528\pi$
$677$	29.3137	1.12662	0.563309	+	0.826247i	$0.309528\pi$
$678$	0	0
$679$	8.48528i	0.325635i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 5.17157i	− 0.197885i	−0.995093	−	0.0989424i	$-0.968454\pi$
$683$	− 5.17157i	− 0.197885i	0.995093	−	0.0989424i	$-0.0315459\pi$
$684$	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
$697$	− 20.9706i	− 0.794317i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 26.4853i	− 1.00034i	−0.865929	−	0.500168i	$-0.833272\pi$
$701$	− 26.4853i	− 1.00034i	0.865929	−	0.500168i	$-0.166728\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.485281i	0.0182509i
$708$	0	0
$709$	26.9706	1.01290	0.506450	−	0.862269i	$-0.330957\pi$
$709$	26.9706	1.01290	0.506450	+	0.862269i	$0.330957\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.68629	0.325304
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−5.17157	−0.192867	−0.0964336	−	0.995339i	$-0.530744\pi$
$719$	−5.17157	−0.192867	−0.0964336	+	0.995339i	$0.530744\pi$
$720$	0	0
$721$	7.94113	0.295743
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−41.0711	−1.52324	−0.761621	−	0.648023i	$-0.775596\pi$
$727$	−41.0711	−1.52324	−0.761621	+	0.648023i	$0.775596\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	− 31.8995i	− 1.17823i	−0.808047	−	0.589117i	$-0.799476\pi$
$733$	− 31.8995i	− 1.17823i	0.808047	−	0.589117i	$-0.200524\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−43.3137	−1.59548
$738$	0	0
$739$	33.6569i	1.23809i	0.785357	+	0.619044i	$0.212480\pi$
$739$	33.6569i	1.23809i	−0.785357	+	0.619044i	$0.787520\pi$
$740$	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
$749$	5.37258i	0.196310i
$750$	0	0
$751$	23.1716i	0.845543i	0.906236	+	0.422771i	$0.138943\pi$
$751$	23.1716i	0.845543i	−0.906236	+	0.422771i	$0.861057\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 0.870058i	− 0.0316228i	−0.999875	−	0.0158114i	$-0.994967\pi$
$757$	− 0.870058i	− 0.0316228i	0.999875	−	0.0158114i	$-0.00503313\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 22.8701i	− 0.829039i	−0.910040	−	0.414519i	$-0.863950\pi$
$761$	− 22.8701i	− 0.829039i	0.910040	−	0.414519i	$-0.136050\pi$
$762$	0	0
$763$	7.51472	0.272051
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.51472i	− 0.199125i
$768$	0	0
$769$	13.6569	0.492479	0.246239	−	0.969209i	$-0.420805\pi$
$769$	13.6569	0.492479	0.246239	+	0.969209i	$0.420805\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.6863	0.672099	0.336050	−	0.941844i	$-0.390909\pi$
$773$	18.6863	0.672099	0.336050	+	0.941844i	$0.390909\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−30.6274	−1.09594
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	13.8579	0.493980	0.246990	−	0.969018i	$-0.420559\pi$
$787$	13.8579	0.493980	0.246990	+	0.969018i	$0.420559\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.65685	0.201135
$792$	0	0
$793$	− 4.20101i	− 0.149182i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13.1716	−0.466561	−0.233281	−	0.972409i	$-0.574946\pi$
$797$	−13.1716	−0.466561	−0.233281	+	0.972409i	$0.574946\pi$
$798$	0	0
$799$	− 38.6274i	− 1.36654i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 35.1127i	− 1.23910i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.0416i	0.423361i	0.977339	+	0.211681i	$0.0678937\pi$
$809$	12.0416i	0.423361i	−0.977339	+	0.211681i	$0.932106\pi$
$810$	0	0
$811$	− 48.6274i	− 1.70754i	−0.520651	−	0.853770i	$-0.674311\pi$
$811$	− 48.6274i	− 1.70754i	0.520651	−	0.853770i	$-0.325689\pi$
$812$	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
$821$	6.68629i	0.233353i	0.993170	+	0.116677i	$0.0372241\pi$
$821$	6.68629i	0.233353i	−0.993170	+	0.116677i	$0.962776\pi$
$822$	0	0
$823$	−2.24264	−0.0781735	−0.0390868	−	0.999236i	$-0.512445\pi$
$823$	−2.24264	−0.0781735	−0.0390868	+	0.999236i	$0.512445\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.62742i	− 0.0913642i	−0.998956	−	0.0456821i	$-0.985454\pi$
$827$	− 2.62742i	− 0.0913642i	0.998956	−	0.0456821i	$-0.0145461\pi$
$828$	0	0
$829$	−31.4558	−1.09251	−0.546253	−	0.837620i	$-0.683946\pi$
$829$	−31.4558	−1.09251	−0.546253	+	0.837620i	$0.683946\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	45.4558	1.57495
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−42.4264	−1.46472	−0.732361	−	0.680916i	$-0.761582\pi$
$839$	−42.4264	−1.46472	−0.732361	+	0.680916i	$0.761582\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.7279	0.368616
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.20101	0.144009
$852$	0	0
$853$	30.2426i	1.03549i	0.855536	+	0.517744i	$0.173228\pi$
$853$	30.2426i	1.03549i	−0.855536	+	0.517744i	$0.826772\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.79899	−0.129771	−0.0648855	−	0.997893i	$-0.520668\pi$
$857$	−3.79899	−0.129771	−0.0648855	+	0.997893i	$0.520668\pi$
$858$	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
$863$	45.7990i	1.55902i	0.626392	+	0.779508i	$0.284531\pi$
$863$	45.7990i	1.55902i	−0.626392	+	0.779508i	$0.715469\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 13.4558i	− 0.456458i
$870$	0	0
$871$	− 4.68629i	− 0.158789i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 29.5563i	− 0.998047i	−0.866588	−	0.499023i	$-0.833692\pi$
$877$	− 29.5563i	− 0.998047i	0.866588	−	0.499023i	$-0.166308\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	35.3553i	1.19115i	0.803299	+	0.595576i	$0.203076\pi$
$881$	35.3553i	1.19115i	−0.803299	+	0.595576i	$0.796924\pi$
$882$	0	0
$883$	42.4264	1.42776	0.713881	−	0.700267i	$-0.246936\pi$
$883$	42.4264	1.42776	0.713881	+	0.700267i	$0.246936\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 7.17157i	− 0.240798i	−0.992726	−	0.120399i	$-0.961583\pi$
$887$	− 7.17157i	− 0.240798i	0.992726	−	0.120399i	$-0.0384174\pi$
$888$	0	0
$889$	−8.62742	−0.289354
$890$	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
$899$	62.9117	2.09822
$900$	0	0
$901$	−46.6274	−1.55338
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	30.1421	1.00085	0.500427	−	0.865779i	$-0.333177\pi$
$907$	30.1421	1.00085	0.500427	+	0.865779i	$0.333177\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−29.6569	−0.982575	−0.491288	−	0.870997i	$-0.663474\pi$
$911$	−29.6569	−0.982575	−0.491288	+	0.870997i	$0.663474\pi$
$912$	0	0
$913$	80.2843i	2.65702i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.45584	0.246214
$918$	0	0
$919$	− 7.17157i	− 0.236568i	−0.992980	−	0.118284i	$-0.962261\pi$
$919$	− 7.17157i	− 0.236568i	0.992980	−	0.118284i	$-0.0377394\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 3.31371i	− 0.109072i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	26.1005i	0.856330i	0.903701	+	0.428165i	$0.140840\pi$
$929$	26.1005i	0.856330i	−0.903701	+	0.428165i	$0.859160\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 18.9706i	− 0.619741i	−0.950779	−	0.309871i	$-0.899714\pi$
$937$	− 18.9706i	− 0.619741i	0.950779	−	0.309871i	$-0.100286\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 14.2843i	− 0.465654i	−0.972518	−	0.232827i	$-0.925202\pi$
$941$	− 14.2843i	− 0.465654i	0.972518	−	0.232827i	$-0.0747976\pi$
$942$	0	0
$943$	−2.54416	−0.0828491
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 27.5980i	− 0.896814i	−0.893830	−	0.448407i	$-0.851992\pi$
$947$	− 27.5980i	− 0.896814i	0.893830	−	0.448407i	$-0.148008\pi$
$948$	0	0
$949$	3.79899	0.123320
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−45.2548	−1.46595	−0.732974	−	0.680257i	$-0.761868\pi$
$953$	−45.2548	−1.46595	−0.732974	+	0.680257i	$0.761868\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.94113	−0.0626822
$960$	0	0
$961$	−78.9411	−2.54649
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.27208	−0.0409073	−0.0204536	−	0.999791i	$-0.506511\pi$
$967$	−1.27208	−0.0409073	−0.0204536	+	0.999791i	$0.506511\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.9289	−0.414909	−0.207455	−	0.978245i	$-0.566518\pi$
$971$	−12.9289	−0.414909	−0.207455	+	0.978245i	$0.566518\pi$
$972$	0	0
$973$	− 4.48528i	− 0.143792i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.8284	1.37020	0.685101	−	0.728448i	$-0.259758\pi$
$977$	42.8284	1.37020	0.685101	+	0.728448i	$0.259758\pi$
$978$	0	0
$979$	− 22.9706i	− 0.734142i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 39.5980i	− 1.26298i	−0.775384	−	0.631490i	$-0.782444\pi$
$983$	− 39.5980i	− 1.26298i	0.775384	−	0.631490i	$-0.217556\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.970563i	0.0308621i
$990$	0	0
$991$	27.6569i	0.878549i	0.898353	+	0.439274i	$0.144764\pi$
$991$	27.6569i	0.878549i	−0.898353	+	0.439274i	$0.855236\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 30.7279i	− 0.973163i	−0.873635	−	0.486582i	$-0.838244\pi$
$997$	− 30.7279i	− 0.973163i	0.873635	−	0.486582i	$-0.161756\pi$
$998$	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.7199.1		4
3.2	odd	2		7200.2.o.m.7199.1		4
4.3	odd	2		7200.2.o.e.7199.3		4
5.2	odd	4		1440.2.h.a.1151.3	yes	4
5.3	odd	4		7200.2.h.c.1151.2		4
5.4	even	2		7200.2.o.b.7199.4		4
12.11	even	2		7200.2.o.b.7199.3		4
15.2	even	4		1440.2.h.d.1151.1	yes	4
15.8	even	4		7200.2.h.i.1151.2		4
15.14	odd	2		7200.2.o.e.7199.4		4
20.3	even	4		7200.2.h.i.1151.3		4
20.7	even	4		1440.2.h.d.1151.4	yes	4
20.19	odd	2		7200.2.o.m.7199.2		4
40.27	even	4		2880.2.h.a.1151.2		4
40.37	odd	4		2880.2.h.d.1151.1		4
60.23	odd	4		7200.2.h.c.1151.3		4
60.47	odd	4		1440.2.h.a.1151.2	✓	4
60.59	even	2	inner	7200.2.o.j.7199.2		4
120.77	even	4		2880.2.h.a.1151.3		4
120.107	odd	4		2880.2.h.d.1151.4		4

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

\(f(q)\)	\(=\)	\(q+0.585786 q^{7} +O(q^{10})\)	\(q+0.585786 q^{7} -5.41421 q^{11} -0.585786i q^{13} -6.82843 q^{17} +0.828427i q^{23} +6.00000i q^{29} -10.4853i q^{31} -5.07107i q^{37} +3.07107i q^{41} +1.17157 q^{43} +5.65685i q^{47} -6.65685 q^{49} +6.82843 q^{53} +9.41421 q^{59} +7.17157 q^{61} +8.00000 q^{67} +5.65685 q^{71} +6.48528i q^{73} -3.17157 q^{77} +2.48528i q^{79} -14.8284i q^{83} +4.24264i q^{89} -0.343146i q^{91} +14.4853i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7}+O(q^{10}) \) 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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