LMFDB - Embedded newform 7200.2.o.d.7199.2

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:	$\Z[a_1, \ldots, a_{19}]$
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			7199.2
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	7200.7199
Dual form			7200.2.o.d.7199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41421 q^{7} +O(q^{10})$	$q-3.41421 q^{7} -1.41421 q^{11} +6.24264i q^{13} +6.82843 q^{17} +5.65685i q^{19} -8.82843i q^{23} +2.00000i q^{29} -8.82843i q^{31} +7.41421i q^{37} +0.242641i q^{41} +1.17157 q^{43} +8.00000i q^{47} +4.65685 q^{49} -1.17157 q^{53} -2.58579 q^{59} -8.82843 q^{61} +11.3137 q^{67} +2.34315 q^{71} +4.82843i q^{73} +4.82843 q^{77} -10.4853i q^{79} +6.82843i q^{83} +15.0711i q^{89} -21.3137i q^{91} -6.48528i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7}+O(q^{10}) $ 4 * q - 8 * q^7	$ 4 q - 8 q^{7} + 16 q^{17} + 16 q^{43} - 4 q^{49} - 16 q^{53} - 16 q^{59} - 24 q^{61} + 32 q^{71} + 8 q^{77}+O(q^{100}) $ 4 * q - 8 * q^7 + 16 * q^17 + 16 * q^43 - 4 * q^49 - 16 * q^53 - 16 * q^59 - 24 * q^61 + 32 * q^71 + 8 * 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	$-0.723237\pi$
$7$	−3.41421	−1.29045	−0.645226	+	0.763992i	$0.723237\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	6.24264i	1.73140i	0.500566	+	0.865699i	$0.333125\pi$
$13$	6.24264i	1.73140i	−0.500566	+	0.865699i	$0.666875\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.82843	1.65614	0.828068	−	0.560627i	$-0.189440\pi$
$17$	6.82843	1.65614	0.828068	+	0.560627i	$0.189440\pi$
$18$	0	0
$19$	5.65685i	1.29777i	0.760886	+	0.648886i	$0.224765\pi$
$19$	5.65685i	1.29777i	−0.760886	+	0.648886i	$0.775235\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 8.82843i	− 1.84085i	−0.390914	−	0.920427i	$-0.627841\pi$
$23$	− 8.82843i	− 1.84085i	0.390914	−	0.920427i	$-0.372159\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	− 8.82843i	− 1.58563i	−0.609461	−	0.792816i	$-0.708614\pi$
$31$	− 8.82843i	− 1.58563i	0.609461	−	0.792816i	$-0.291386\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.41421i	1.21889i	0.792829	+	0.609445i	$0.208608\pi$
$37$	7.41421i	1.21889i	−0.792829	+	0.609445i	$0.791392\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.242641i	0.0378941i	0.999820	+	0.0189471i	$0.00603140\pi$
$41$	0.242641i	0.0378941i	−0.999820	+	0.0189471i	$0.993969\pi$
$42$	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
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	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	$-0.525640\pi$
$53$	−1.17157	−0.160928	−0.0804640	+	0.996758i	$0.525640\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.58579	−0.336641	−0.168320	−	0.985732i	$-0.553834\pi$
$59$	−2.58579	−0.336641	−0.168320	+	0.985732i	$0.553834\pi$
$60$	0	0
$61$	−8.82843	−1.13036	−0.565182	−	0.824966i	$-0.691194\pi$
$61$	−8.82843	−1.13036	−0.565182	+	0.824966i	$0.691194\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.34315	0.278080	0.139040	−	0.990287i	$-0.455598\pi$
$71$	2.34315	0.278080	0.139040	+	0.990287i	$0.455598\pi$
$72$	0	0
$73$	4.82843i	0.565125i	0.959249	+	0.282562i	$0.0911844\pi$
$73$	4.82843i	0.565125i	−0.959249	+	0.282562i	$0.908816\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.82843	0.550250
$78$	0	0
$79$	− 10.4853i	− 1.17969i	−0.807518	−	0.589843i	$-0.799190\pi$
$79$	− 10.4853i	− 1.17969i	0.807518	−	0.589843i	$-0.200810\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.82843i	0.749517i	0.927122	+	0.374759i	$0.122274\pi$
$83$	6.82843i	0.749517i	−0.927122	+	0.374759i	$0.877726\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.0711i	1.59753i	0.601643	+	0.798765i	$0.294513\pi$
$89$	15.0711i	1.59753i	−0.601643	+	0.798765i	$0.705487\pi$
$90$	0	0
$91$	− 21.3137i	− 2.23428i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.48528i	− 0.658481i	−0.944246	−	0.329240i	$-0.893207\pi$
$97$	− 6.48528i	− 0.658481i	0.944246	−	0.329240i	$-0.106793\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 16.8284i	− 1.67449i	−0.546827	−	0.837246i	$-0.684165\pi$
$101$	− 16.8284i	− 1.67449i	0.546827	−	0.837246i	$-0.315835\pi$
$102$	0	0
$103$	6.24264	0.615106	0.307553	−	0.951531i	$-0.400490\pi$
$103$	6.24264	0.615106	0.307553	+	0.951531i	$0.400490\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.1421i	0.980477i	0.871588	+	0.490239i	$0.163090\pi$
$107$	10.1421i	0.980477i	−0.871588	+	0.490239i	$0.836910\pi$
$108$	0	0
$109$	−3.17157	−0.303782	−0.151891	−	0.988397i	$-0.548536\pi$
$109$	−3.17157	−0.303782	−0.151891	+	0.988397i	$0.548536\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−23.3137	−2.13716
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.4142	−1.36779	−0.683895	−	0.729580i	$-0.739715\pi$
$127$	−15.4142	−1.36779	−0.683895	+	0.729580i	$0.739715\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.8995	−1.21440	−0.607202	−	0.794547i	$-0.707708\pi$
$131$	−13.8995	−1.21440	−0.607202	+	0.794547i	$0.707708\pi$
$132$	0	0
$133$	− 19.3137i	− 1.67471i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.65685	0.483298	0.241649	−	0.970364i	$-0.422312\pi$
$137$	5.65685	0.483298	0.241649	+	0.970364i	$0.422312\pi$
$138$	0	0
$139$	− 8.34315i	− 0.707656i	−0.935310	−	0.353828i	$-0.884880\pi$
$139$	− 8.34315i	− 0.707656i	0.935310	−	0.353828i	$-0.115120\pi$
$140$	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
$149$	− 4.82843i	− 0.395560i	−0.980246	−	0.197780i	$-0.936627\pi$
$149$	− 4.82843i	− 0.395560i	0.980246	−	0.197780i	$-0.0633732\pi$
$150$	0	0
$151$	10.9706i	0.892772i	0.894841	+	0.446386i	$0.147289\pi$
$151$	10.9706i	0.892772i	−0.894841	+	0.446386i	$0.852711\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.100505i	0.00802118i	0.999992	+	0.00401059i	$0.00127661\pi$
$157$	0.100505i	0.00802118i	−0.999992	+	0.00401059i	$0.998723\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	30.1421i	2.37553i
$162$	0	0
$163$	−8.48528	−0.664619	−0.332309	−	0.943170i	$-0.607828\pi$
$163$	−8.48528	−0.664619	−0.332309	+	0.943170i	$0.607828\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 14.4853i	− 1.12090i	−0.828187	−	0.560452i	$-0.810627\pi$
$167$	− 14.4853i	− 1.12090i	0.828187	−	0.560452i	$-0.189373\pi$
$168$	0	0
$169$	−25.9706	−1.99774
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.485281	−0.0368953	−0.0184476	−	0.999830i	$-0.505872\pi$
$173$	−0.485281	−0.0368953	−0.0184476	+	0.999830i	$0.505872\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	13.4142	1.00263	0.501313	−	0.865266i	$-0.332851\pi$
$179$	13.4142	1.00263	0.501313	+	0.865266i	$0.332851\pi$
$180$	0	0
$181$	−24.8284	−1.84548	−0.922741	−	0.385420i	$-0.874057\pi$
$181$	−24.8284	−1.84548	−0.922741	+	0.385420i	$0.874057\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−9.65685	−0.706179
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.17157	0.0847720	0.0423860	−	0.999101i	$-0.486504\pi$
$191$	1.17157	0.0847720	0.0423860	+	0.999101i	$0.486504\pi$
$192$	0	0
$193$	− 4.34315i	− 0.312626i	−0.987708	−	0.156313i	$-0.950039\pi$
$193$	− 4.34315i	− 0.312626i	0.987708	−	0.156313i	$-0.0499609\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	15.6569i	1.10988i	0.831889	+	0.554942i	$0.187260\pi$
$199$	15.6569i	1.10988i	−0.831889	+	0.554942i	$0.812740\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 6.82843i	− 0.479262i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 8.00000i	− 0.553372i
$210$	0	0
$211$	− 6.00000i	− 0.413057i	−0.978441	−	0.206529i	$-0.933783\pi$
$211$	− 6.00000i	− 0.413057i	0.978441	−	0.206529i	$-0.0662166\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	30.1421i	2.04618i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	42.6274i	2.86743i
$222$	0	0
$223$	−4.10051	−0.274590	−0.137295	−	0.990530i	$-0.543841\pi$
$223$	−4.10051	−0.274590	−0.137295	+	0.990530i	$0.543841\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	28.2843i	1.87729i	0.344881	+	0.938647i	$0.387919\pi$
$227$	28.2843i	1.87729i	−0.344881	+	0.938647i	$0.612081\pi$
$228$	0	0
$229$	−7.65685	−0.505979	−0.252990	−	0.967469i	$-0.581414\pi$
$229$	−7.65685	−0.505979	−0.252990	+	0.967469i	$0.581414\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.4853	−0.817938	−0.408969	−	0.912548i	$-0.634112\pi$
$233$	−12.4853	−0.817938	−0.408969	+	0.912548i	$0.634112\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.14214	−0.397302	−0.198651	−	0.980070i	$-0.563656\pi$
$239$	−6.14214	−0.397302	−0.198651	+	0.980070i	$0.563656\pi$
$240$	0	0
$241$	−14.3431	−0.923923	−0.461962	−	0.886900i	$-0.652854\pi$
$241$	−14.3431	−0.923923	−0.461962	+	0.886900i	$0.652854\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−35.3137	−2.24696
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.7279	−1.30833	−0.654167	−	0.756350i	$-0.726981\pi$
$251$	−20.7279	−1.30833	−0.654167	+	0.756350i	$0.726981\pi$
$252$	0	0
$253$	12.4853i	0.784943i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.97056	0.559568	0.279784	−	0.960063i	$-0.409737\pi$
$257$	8.97056	0.559568	0.279784	+	0.960063i	$0.409737\pi$
$258$	0	0
$259$	− 25.3137i	− 1.57292i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.8284i	1.53099i	0.643444	+	0.765493i	$0.277505\pi$
$263$	24.8284i	1.53099i	−0.643444	+	0.765493i	$0.722495\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.9706i	0.668887i	0.942416	+	0.334444i	$0.108548\pi$
$269$	10.9706i	0.668887i	−0.942416	+	0.334444i	$0.891452\pi$
$270$	0	0
$271$	1.31371i	0.0798021i	0.999204	+	0.0399011i	$0.0127043\pi$
$271$	1.31371i	0.0798021i	−0.999204	+	0.0399011i	$0.987296\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	32.8701i	1.97497i	0.157712	+	0.987485i	$0.449588\pi$
$277$	32.8701i	1.97497i	−0.157712	+	0.987485i	$0.550412\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.89949i	0.113314i	0.998394	+	0.0566572i	$0.0180442\pi$
$281$	1.89949i	0.113314i	−0.998394	+	0.0566572i	$0.981956\pi$
$282$	0	0
$283$	−15.3137	−0.910305	−0.455153	−	0.890413i	$-0.650415\pi$
$283$	−15.3137	−0.910305	−0.455153	+	0.890413i	$0.650415\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 0.828427i	− 0.0489005i
$288$	0	0
$289$	29.6274	1.74279
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	55.1127	3.18725
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−19.3137	−1.10229	−0.551146	−	0.834409i	$-0.685809\pi$
$307$	−19.3137	−1.10229	−0.551146	+	0.834409i	$0.685809\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−28.4853	−1.61525	−0.807626	−	0.589695i	$-0.799248\pi$
$311$	−28.4853	−1.61525	−0.807626	+	0.589695i	$0.799248\pi$
$312$	0	0
$313$	− 22.2843i	− 1.25958i	−0.776765	−	0.629791i	$-0.783141\pi$
$313$	− 22.2843i	− 1.25958i	0.776765	−	0.629791i	$-0.216859\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.31371	0.298448	0.149224	−	0.988803i	$-0.452323\pi$
$317$	5.31371	0.298448	0.149224	+	0.988803i	$0.452323\pi$
$318$	0	0
$319$	− 2.82843i	− 0.158362i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	38.6274i	2.14929i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 27.3137i	− 1.50585i
$330$	0	0
$331$	− 8.68629i	− 0.477442i	−0.971088	−	0.238721i	$-0.923272\pi$
$331$	− 8.68629i	− 0.477442i	0.971088	−	0.238721i	$-0.0767281\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.1716i	0.608554i	0.952584	+	0.304277i	$0.0984149\pi$
$337$	11.1716i	0.608554i	−0.952584	+	0.304277i	$0.901585\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12.4853i	0.676116i
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.9706i	0.696296i	0.937440	+	0.348148i	$0.113189\pi$
$347$	12.9706i	0.696296i	−0.937440	+	0.348148i	$0.886811\pi$
$348$	0	0
$349$	−3.17157	−0.169770	−0.0848852	−	0.996391i	$-0.527052\pi$
$349$	−3.17157	−0.169770	−0.0848852	+	0.996391i	$0.527052\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.4853	1.72902	0.864509	−	0.502618i	$-0.167630\pi$
$353$	32.4853	1.72902	0.864509	+	0.502618i	$0.167630\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.1127	−1.64207	−0.821033	−	0.570881i	$-0.806602\pi$
$359$	−31.1127	−1.64207	−0.821033	+	0.570881i	$0.806602\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−33.0711	−1.72630	−0.863148	−	0.504951i	$-0.831510\pi$
$367$	−33.0711	−1.72630	−0.863148	+	0.504951i	$0.831510\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	− 17.0711i	− 0.883906i	−0.897038	−	0.441953i	$-0.854286\pi$
$373$	− 17.0711i	− 0.883906i	0.897038	−	0.441953i	$-0.145714\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.4853	−0.643025
$378$	0	0
$379$	− 29.3137i	− 1.50574i	−0.658167	−	0.752872i	$-0.728668\pi$
$379$	− 29.3137i	− 1.50574i	0.658167	−	0.752872i	$-0.271332\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 14.6274i	− 0.747426i	−0.927544	−	0.373713i	$-0.878084\pi$
$383$	− 14.6274i	− 0.747426i	0.927544	−	0.373713i	$-0.121916\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.65685i	0.388218i	0.980980	+	0.194109i	$0.0621815\pi$
$389$	7.65685i	0.388218i	−0.980980	+	0.194109i	$0.937818\pi$
$390$	0	0
$391$	− 60.2843i	− 3.04871i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 16.3848i	− 0.822328i	−0.911561	−	0.411164i	$-0.865122\pi$
$397$	− 16.3848i	− 0.822328i	0.911561	−	0.411164i	$-0.134878\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 16.7279i	− 0.835353i	−0.908596	−	0.417676i	$-0.862845\pi$
$401$	− 16.7279i	− 0.835353i	0.908596	−	0.417676i	$-0.137155\pi$
$402$	0	0
$403$	55.1127	2.74536
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 10.4853i	− 0.519736i
$408$	0	0
$409$	−11.6569	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$409$	−11.6569	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.82843	0.434418
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.2426	−1.77057	−0.885284	−	0.465050i	$-0.846036\pi$
$419$	−36.2426	−1.77057	−0.885284	+	0.465050i	$0.846036\pi$
$420$	0	0
$421$	27.9411	1.36177	0.680884	−	0.732392i	$-0.261596\pi$
$421$	27.9411	1.36177	0.680884	+	0.732392i	$0.261596\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	30.1421	1.45868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.4853	−0.601395	−0.300697	−	0.953720i	$-0.597219\pi$
$431$	−12.4853	−0.601395	−0.300697	+	0.953720i	$0.597219\pi$
$432$	0	0
$433$	− 7.17157i	− 0.344644i	−0.985041	−	0.172322i	$-0.944873\pi$
$433$	− 7.17157i	− 0.344644i	0.985041	−	0.172322i	$-0.0551269\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	49.9411	2.38901
$438$	0	0
$439$	4.34315i	0.207287i	0.994615	+	0.103644i	$0.0330501\pi$
$439$	4.34315i	0.207287i	−0.994615	+	0.103644i	$0.966950\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 15.3137i	− 0.727576i	−0.931482	−	0.363788i	$-0.881483\pi$
$443$	− 15.3137i	− 0.727576i	0.931482	−	0.363788i	$-0.118517\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17.2132i	0.812341i	0.913797	+	0.406171i	$0.133136\pi$
$449$	17.2132i	0.812341i	−0.913797	+	0.406171i	$0.866864\pi$
$450$	0	0
$451$	− 0.343146i	− 0.0161581i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.1421i	1.50355i	0.659422	+	0.751773i	$0.270801\pi$
$457$	32.1421i	1.50355i	−0.659422	+	0.751773i	$0.729199\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	33.7990i	1.57418i	0.616841	+	0.787088i	$0.288412\pi$
$461$	33.7990i	1.57418i	−0.616841	+	0.787088i	$0.711588\pi$
$462$	0	0
$463$	−20.3848	−0.947361	−0.473680	−	0.880697i	$-0.657075\pi$
$463$	−20.3848	−0.947361	−0.473680	+	0.880697i	$0.657075\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.7990i	0.545992i	0.962015	+	0.272996i	$0.0880146\pi$
$467$	11.7990i	0.545992i	−0.962015	+	0.272996i	$0.911985\pi$
$468$	0	0
$469$	−38.6274	−1.78365
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.65685	−0.0761822
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−46.2843	−2.11038
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.04163	0.273772	0.136886	−	0.990587i	$-0.456291\pi$
$487$	6.04163	0.273772	0.136886	+	0.990587i	$0.456291\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.9289	0.583475	0.291737	−	0.956498i	$-0.405767\pi$
$491$	12.9289	0.583475	0.291737	+	0.956498i	$0.405767\pi$
$492$	0	0
$493$	13.6569i	0.615074i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	12.9706i	0.580642i	0.956929	+	0.290321i	$0.0937621\pi$
$499$	12.9706i	0.580642i	−0.956929	+	0.290321i	$0.906238\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 20.1421i	− 0.892784i	−0.894837	−	0.446392i	$-0.852709\pi$
$509$	− 20.1421i	− 0.892784i	0.894837	−	0.446392i	$-0.147291\pi$
$510$	0	0
$511$	− 16.4853i	− 0.729266i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 11.3137i	− 0.497576i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 7.27208i	− 0.318596i	−0.987231	−	0.159298i	$-0.949077\pi$
$521$	− 7.27208i	− 0.318596i	0.987231	−	0.159298i	$-0.0509230\pi$
$522$	0	0
$523$	27.7990	1.21556	0.607782	−	0.794104i	$-0.292059\pi$
$523$	27.7990	1.21556	0.607782	+	0.794104i	$0.292059\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 60.2843i	− 2.62602i
$528$	0	0
$529$	−54.9411	−2.38874
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.51472	−0.0656097
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.58579	−0.283670
$540$	0	0
$541$	−4.14214	−0.178084	−0.0890422	−	0.996028i	$-0.528381\pi$
$541$	−4.14214	−0.178084	−0.0890422	+	0.996028i	$0.528381\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.20101	0.179622	0.0898111	−	0.995959i	$-0.471374\pi$
$547$	4.20101	0.179622	0.0898111	+	0.995959i	$0.471374\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	35.7990i	1.52233i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−20.4853	−0.867989	−0.433995	−	0.900915i	$-0.642896\pi$
$557$	−20.4853	−0.867989	−0.433995	+	0.900915i	$0.642896\pi$
$558$	0	0
$559$	7.31371i	0.309337i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 22.3431i	− 0.941651i	−0.882226	−	0.470826i	$-0.843956\pi$
$563$	− 22.3431i	− 0.941651i	0.882226	−	0.470826i	$-0.156044\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.5858i	1.44991i	0.688795	+	0.724956i	$0.258140\pi$
$569$	34.5858i	1.44991i	−0.688795	+	0.724956i	$0.741860\pi$
$570$	0	0
$571$	34.6274i	1.44911i	0.689216	+	0.724556i	$0.257955\pi$
$571$	34.6274i	1.44911i	−0.689216	+	0.724556i	$0.742045\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 27.4558i	− 1.14300i	−0.820601	−	0.571501i	$-0.806361\pi$
$577$	− 27.4558i	− 1.14300i	0.820601	−	0.571501i	$-0.193639\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 23.3137i	− 0.967216i
$582$	0	0
$583$	1.65685	0.0686199
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.4558i	1.54597i	0.634425	+	0.772984i	$0.281237\pi$
$587$	37.4558i	1.54597i	−0.634425	+	0.772984i	$0.718763\pi$
$588$	0	0
$589$	49.9411	2.05779
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14.6274	0.600676	0.300338	−	0.953833i	$-0.402901\pi$
$593$	14.6274	0.600676	0.300338	+	0.953833i	$0.402901\pi$
$594$	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
$601$	45.9411	1.87398	0.936989	−	0.349359i	$-0.113601\pi$
$601$	45.9411	1.87398	0.936989	+	0.349359i	$0.113601\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10.0416	0.407577	0.203789	−	0.979015i	$-0.434674\pi$
$607$	10.0416	0.407577	0.203789	+	0.979015i	$0.434674\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−49.9411	−2.02040
$612$	0	0
$613$	21.0711i	0.851052i	0.904946	+	0.425526i	$0.139911\pi$
$613$	21.0711i	0.851052i	−0.904946	+	0.425526i	$0.860089\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.1716	0.530268	0.265134	−	0.964212i	$-0.414584\pi$
$617$	13.1716	0.530268	0.265134	+	0.964212i	$0.414584\pi$
$618$	0	0
$619$	− 26.9706i	− 1.08404i	−0.840366	−	0.542019i	$-0.817660\pi$
$619$	− 26.9706i	− 1.08404i	0.840366	−	0.542019i	$-0.182340\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 51.4558i	− 2.06153i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	50.6274i	2.01865i
$630$	0	0
$631$	− 16.1421i	− 0.642608i	−0.946976	−	0.321304i	$-0.895879\pi$
$631$	− 16.1421i	− 0.642608i	0.946976	−	0.321304i	$-0.104121\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	29.0711i	1.15184i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 14.8701i	− 0.587332i	−0.955908	−	0.293666i	$-0.905125\pi$
$641$	− 14.8701i	− 0.587332i	0.955908	−	0.293666i	$-0.0948753\pi$
$642$	0	0
$643$	1.17157	0.0462023	0.0231012	−	0.999733i	$-0.492646\pi$
$643$	1.17157	0.0462023	0.0231012	+	0.999733i	$0.492646\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.0294i	0.590868i	0.955363	+	0.295434i	$0.0954643\pi$
$647$	15.0294i	0.590868i	−0.955363	+	0.295434i	$0.904536\pi$
$648$	0	0
$649$	3.65685	0.143544
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.2843	−1.18512	−0.592558	−	0.805528i	$-0.701882\pi$
$653$	−30.2843	−1.18512	−0.592558	+	0.805528i	$0.701882\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4.04163	0.157440	0.0787198	−	0.996897i	$-0.474917\pi$
$659$	4.04163	0.157440	0.0787198	+	0.996897i	$0.474917\pi$
$660$	0	0
$661$	−28.1421	−1.09460	−0.547301	−	0.836936i	$-0.684345\pi$
$661$	−28.1421	−1.09460	−0.547301	+	0.836936i	$0.684345\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	17.6569	0.683676
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.4853	0.481989
$672$	0	0
$673$	42.7696i	1.64865i	0.566120	+	0.824323i	$0.308444\pi$
$673$	42.7696i	1.64865i	−0.566120	+	0.824323i	$0.691556\pi$
$674$	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
$679$	22.1421i	0.849737i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19.7990i	0.757587i	0.925481	+	0.378794i	$0.123661\pi$
$683$	19.7990i	0.757587i	−0.925481	+	0.378794i	$0.876339\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 7.31371i	− 0.278630i
$690$	0	0
$691$	− 29.6569i	− 1.12820i	−0.825707	−	0.564100i	$-0.809223\pi$
$691$	− 29.6569i	− 1.12820i	0.825707	−	0.564100i	$-0.190777\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.65685i	0.0627578i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 21.5147i	− 0.812600i	−0.913740	−	0.406300i	$-0.866819\pi$
$701$	− 21.5147i	− 0.812600i	0.913740	−	0.406300i	$-0.133181\pi$
$702$	0	0
$703$	−41.9411	−1.58184
$704$	0	0
$705$	0	0
$706$	0	0
$707$	57.4558i	2.16085i
$708$	0	0
$709$	31.6569	1.18890	0.594449	−	0.804133i	$-0.297370\pi$
$709$	31.6569	1.18890	0.594449	+	0.804133i	$0.297370\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−77.9411	−2.91892
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27.7990	−1.03673	−0.518364	−	0.855160i	$-0.673459\pi$
$719$	−27.7990	−1.03673	−0.518364	+	0.855160i	$0.673459\pi$
$720$	0	0
$721$	−21.3137	−0.793764
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−9.75736	−0.361880	−0.180940	−	0.983494i	$-0.557914\pi$
$727$	−9.75736	−0.361880	−0.180940	+	0.983494i	$0.557914\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	2.24264i	0.0828338i	0.999142	+	0.0414169i	$0.0131872\pi$
$733$	2.24264i	0.0828338i	−0.999142	+	0.0414169i	$0.986813\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	− 12.0000i	− 0.441427i	−0.975339	−	0.220714i	$-0.929161\pi$
$739$	− 12.0000i	− 0.441427i	0.975339	−	0.220714i	$-0.0708386\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.02944i	0.257885i	0.991652	+	0.128943i	$0.0411583\pi$
$743$	7.02944i	0.257885i	−0.991652	+	0.128943i	$0.958842\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 34.6274i	− 1.26526i
$750$	0	0
$751$	− 21.7990i	− 0.795456i	−0.917503	−	0.397728i	$-0.869799\pi$
$751$	− 21.7990i	− 0.795456i	0.917503	−	0.397728i	$-0.130201\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3.41421i	− 0.124092i	−0.998073	−	0.0620459i	$-0.980237\pi$
$757$	− 3.41421i	− 0.124092i	0.998073	−	0.0620459i	$-0.0197625\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.8701i	1.40904i	0.709685	+	0.704519i	$0.248837\pi$
$761$	38.8701i	1.40904i	−0.709685	+	0.704519i	$0.751163\pi$
$762$	0	0
$763$	10.8284	0.392015
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 16.1421i	− 0.582859i
$768$	0	0
$769$	31.5980	1.13945	0.569726	−	0.821835i	$-0.307049\pi$
$769$	31.5980	1.13945	0.569726	+	0.821835i	$0.307049\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	43.2548	1.55577	0.777884	−	0.628408i	$-0.216293\pi$
$773$	43.2548	1.55577	0.777884	+	0.628408i	$0.216293\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.37258	−0.0491779
$780$	0	0
$781$	−3.31371	−0.118574
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−26.1421	−0.931866	−0.465933	−	0.884820i	$-0.654281\pi$
$787$	−26.1421	−0.931866	−0.465933	+	0.884820i	$0.654281\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13.6569	0.485582
$792$	0	0
$793$	− 55.1127i	− 1.95711i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.485281	−0.0171895	−0.00859477	−	0.999963i	$-0.502736\pi$
$797$	−0.485281	−0.0171895	−0.00859477	+	0.999963i	$0.502736\pi$
$798$	0	0
$799$	54.6274i	1.93258i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 6.82843i	− 0.240970i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 10.1005i	− 0.355115i	−0.984110	−	0.177557i	$-0.943180\pi$
$809$	− 10.1005i	− 0.355115i	0.984110	−	0.177557i	$-0.0568195\pi$
$810$	0	0
$811$	− 12.6274i	− 0.443409i	−0.975114	−	0.221704i	$-0.928838\pi$
$811$	− 12.6274i	− 0.443409i	0.975114	−	0.221704i	$-0.0711620\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.62742i	0.231864i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 5.31371i	− 0.185450i	−0.995692	−	0.0927249i	$-0.970442\pi$
$821$	− 5.31371i	− 0.185450i	0.995692	−	0.0927249i	$-0.0295577\pi$
$822$	0	0
$823$	−36.8701	−1.28521	−0.642605	−	0.766198i	$-0.722146\pi$
$823$	−36.8701	−1.28521	−0.642605	+	0.766198i	$0.722146\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	50.6274i	1.76049i	0.474522	+	0.880244i	$0.342621\pi$
$827$	50.6274i	1.76049i	−0.474522	+	0.880244i	$0.657379\pi$
$828$	0	0
$829$	−13.5147	−0.469386	−0.234693	−	0.972070i	$-0.575408\pi$
$829$	−13.5147	−0.469386	−0.234693	+	0.972070i	$0.575408\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	31.7990	1.10177
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.4853	−0.845326	−0.422663	−	0.906287i	$-0.638905\pi$
$839$	−24.4853	−0.845326	−0.422663	+	0.906287i	$0.638905\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	30.7279	1.05582
$848$	0	0
$849$	0	0
$850$	0	0
$851$	65.4558	2.24380
$852$	0	0
$853$	− 47.2132i	− 1.61655i	−0.588806	−	0.808275i	$-0.700402\pi$
$853$	− 47.2132i	− 1.61655i	0.588806	−	0.808275i	$-0.299598\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.4853	1.65623	0.828113	−	0.560561i	$-0.189415\pi$
$857$	48.4853	1.65623	0.828113	+	0.560561i	$0.189415\pi$
$858$	0	0
$859$	− 10.0000i	− 0.341196i	−0.985341	−	0.170598i	$-0.945430\pi$
$859$	− 10.0000i	− 0.341196i	0.985341	−	0.170598i	$-0.0545699\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	50.7696i	1.72822i	0.503307	+	0.864108i	$0.332117\pi$
$863$	50.7696i	1.72822i	−0.503307	+	0.864108i	$0.667883\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14.8284i	0.503020i
$870$	0	0
$871$	70.6274i	2.39312i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.58579i	0.154851i	0.996998	+	0.0774255i	$0.0246700\pi$
$877$	4.58579i	0.154851i	−0.996998	+	0.0774255i	$0.975330\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 25.4142i	− 0.856227i	−0.903725	−	0.428113i	$-0.859178\pi$
$881$	− 25.4142i	− 0.856227i	0.903725	−	0.428113i	$-0.140822\pi$
$882$	0	0
$883$	−33.4558	−1.12588	−0.562939	−	0.826498i	$-0.690330\pi$
$883$	−33.4558	−1.12588	−0.562939	+	0.826498i	$0.690330\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 28.1421i	− 0.944920i	−0.881352	−	0.472460i	$-0.843366\pi$
$887$	− 28.1421i	− 0.944920i	0.881352	−	0.472460i	$-0.156634\pi$
$888$	0	0
$889$	52.6274	1.76507
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−45.2548	−1.51440
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.6569	0.588889
$900$	0	0
$901$	−8.00000	−0.266519
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−31.1127	−1.03308	−0.516540	−	0.856263i	$-0.672780\pi$
$907$	−31.1127	−1.03308	−0.516540	+	0.856263i	$0.672780\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	36.2843	1.20215	0.601076	−	0.799192i	$-0.294739\pi$
$911$	36.2843	1.20215	0.601076	+	0.799192i	$0.294739\pi$
$912$	0	0
$913$	− 9.65685i	− 0.319595i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	47.4558	1.56713
$918$	0	0
$919$	26.4853i	0.873669i	0.899542	+	0.436834i	$0.143900\pi$
$919$	26.4853i	0.873669i	−0.899542	+	0.436834i	$0.856100\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14.6274i	0.481467i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 32.7279i	− 1.07377i	−0.843656	−	0.536884i	$-0.819601\pi$
$929$	− 32.7279i	− 1.07377i	0.843656	−	0.536884i	$-0.180399\pi$
$930$	0	0
$931$	26.3431i	0.863362i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 42.2843i	− 1.38137i	−0.723157	−	0.690683i	$-0.757310\pi$
$937$	− 42.2843i	− 1.38137i	0.723157	−	0.690683i	$-0.242690\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 46.9706i	− 1.53120i	−0.643319	−	0.765598i	$-0.722443\pi$
$941$	− 46.9706i	− 1.53120i	0.643319	−	0.765598i	$-0.277557\pi$
$942$	0	0
$943$	2.14214	0.0697575
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 44.9706i	− 1.46135i	−0.682727	−	0.730673i	$-0.739206\pi$
$947$	− 44.9706i	− 1.46135i	0.682727	−	0.730673i	$-0.260794\pi$
$948$	0	0
$949$	−30.1421	−0.978455
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.34315	0.0759019	0.0379510	−	0.999280i	$-0.487917\pi$
$953$	2.34315	0.0759019	0.0379510	+	0.999280i	$0.487917\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−19.3137	−0.623672
$960$	0	0
$961$	−46.9411	−1.51423
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−19.8995	−0.639925	−0.319962	−	0.947430i	$-0.603670\pi$
$967$	−19.8995	−0.639925	−0.319962	+	0.947430i	$0.603670\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	29.6985	0.953070	0.476535	−	0.879156i	$-0.341893\pi$
$971$	29.6985	0.953070	0.476535	+	0.879156i	$0.341893\pi$
$972$	0	0
$973$	28.4853i	0.913196i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.4853	0.783354	0.391677	−	0.920103i	$-0.371895\pi$
$977$	24.4853	0.783354	0.391677	+	0.920103i	$0.371895\pi$
$978$	0	0
$979$	− 21.3137i	− 0.681189i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 19.3137i	− 0.616012i	−0.951384	−	0.308006i	$-0.900338\pi$
$983$	− 19.3137i	− 0.616012i	0.951384	−	0.308006i	$-0.0996616\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 10.3431i	− 0.328893i
$990$	0	0
$991$	− 0.343146i	− 0.0109004i	−0.999985	−	0.00545019i	$-0.998265\pi$
$991$	− 0.343146i	− 0.0109004i	0.999985	−	0.00545019i	$-0.00173486\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 34.2426i	− 1.08448i	−0.840225	−	0.542238i	$-0.817577\pi$
$997$	− 34.2426i	− 1.08448i	0.840225	−	0.542238i	$-0.182423\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.o.d.7199.2		4
3.2	odd	2		7200.2.o.c.7199.2		4
4.3	odd	2		7200.2.o.l.7199.4		4
5.2	odd	4		7200.2.h.e.1151.1		4
5.3	odd	4		1440.2.h.c.1151.2	yes	4
5.4	even	2		7200.2.o.k.7199.3		4
12.11	even	2		7200.2.o.k.7199.4		4
15.2	even	4		7200.2.h.f.1151.1		4
15.8	even	4		1440.2.h.b.1151.4	yes	4
15.14	odd	2		7200.2.o.l.7199.3		4
20.3	even	4		1440.2.h.b.1151.1	✓	4
20.7	even	4		7200.2.h.f.1151.4		4
20.19	odd	2		7200.2.o.c.7199.1		4
40.3	even	4		2880.2.h.b.1151.3		4
40.13	odd	4		2880.2.h.c.1151.4		4
60.23	odd	4		1440.2.h.c.1151.3	yes	4
60.47	odd	4		7200.2.h.e.1151.4		4
60.59	even	2	inner	7200.2.o.d.7199.1		4
120.53	even	4		2880.2.h.b.1151.2		4
120.83	odd	4		2880.2.h.c.1151.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.h.b.1151.1	✓	4	20.3	even	4
1440.2.h.b.1151.4	yes	4	15.8	even	4
1440.2.h.c.1151.2	yes	4	5.3	odd	4
1440.2.h.c.1151.3	yes	4	60.23	odd	4
2880.2.h.b.1151.2		4	120.53	even	4
2880.2.h.b.1151.3		4	40.3	even	4
2880.2.h.c.1151.1		4	120.83	odd	4
2880.2.h.c.1151.4		4	40.13	odd	4
7200.2.h.e.1151.1		4	5.2	odd	4
7200.2.h.e.1151.4		4	60.47	odd	4
7200.2.h.f.1151.1		4	15.2	even	4
7200.2.h.f.1151.4		4	20.7	even	4
7200.2.o.c.7199.1		4	20.19	odd	2
7200.2.o.c.7199.2		4	3.2	odd	2
7200.2.o.d.7199.1		4	60.59	even	2	inner
7200.2.o.d.7199.2		4	1.1	even	1	trivial
7200.2.o.k.7199.3		4	5.4	even	2
7200.2.o.k.7199.4		4	12.11	even	2
7200.2.o.l.7199.3		4	15.14	odd	2
7200.2.o.l.7199.4		4	4.3	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)

\(f(q)\)	\(=\)	\(q-3.41421 q^{7} +O(q^{10})\)	\(q-3.41421 q^{7} -1.41421 q^{11} +6.24264i q^{13} +6.82843 q^{17} +5.65685i q^{19} -8.82843i q^{23} +2.00000i q^{29} -8.82843i q^{31} +7.41421i q^{37} +0.242641i q^{41} +1.17157 q^{43} +8.00000i q^{47} +4.65685 q^{49} -1.17157 q^{53} -2.58579 q^{59} -8.82843 q^{61} +11.3137 q^{67} +2.34315 q^{71} +4.82843i q^{73} +4.82843 q^{77} -10.4853i q^{79} +6.82843i q^{83} +15.0711i q^{89} -21.3137i q^{91} -6.48528i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7}+O(q^{10}) \) 4 * q - 8 * q^7	\( 4 q - 8 q^{7} + 16 q^{17} + 16 q^{43} - 4 q^{49} - 16 q^{53} - 16 q^{59} - 24 q^{61} + 32 q^{71} + 8 q^{77}+O(q^{100}) \) 4 * q - 8 * q^7 + 16 * q^17 + 16 * q^43 - 4 * q^49 - 16 * q^53 - 16 * q^59 - 24 * q^61 + 32 * q^71 + 8 * q^77

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