LMFDB - Embedded newform 7200.2.b.i.4751.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(4751,7200)]

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, 1, 1, 0]))

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

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

chi := DirichletCharacter("7200.4751");

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.b (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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6x^{14} + 28x^{12} + 16x^{10} - 40x^{8} + 610x^{6} + 1625x^{4} - 524x^{2} + 1444 $ x^16 - 6x^14 + 28x^12 + 16x^10 - 40x^8 + 610x^6 + 1625x^4 - 524*x^2 + 1444
Coefficient ring:	$\Z[a_1, \ldots, a_{53}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4751.4
Root			$2.05580 - 0.953651i$ of defining polynomial
Character	$\chi$	$=$	7200.4751
Dual form			7200.2.b.i.4751.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.02045i q^{7} +O(q^{10})$	$q-3.02045i q^{7} +3.62258i q^{11} +1.69614i q^{13} +6.60421i q^{17} +5.12311 q^{19} -6.67026 q^{23} -6.82867 q^{29} -1.73642i q^{31} -0.371834i q^{37} -5.83095i q^{41} -5.24477 q^{43} +0.525853 q^{47} -2.12311 q^{49} -10.0054 q^{53} -4.86270i q^{59} +13.4347 q^{67} +2.45567 q^{71} -14.5845 q^{73} +10.9418 q^{77} -14.1051i q^{79} -5.79119i q^{83} -10.2477i q^{89} +5.12311 q^{91} -9.33976 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{19} + 32 q^{49} + 16 q^{91}+O(q^{100}) $ 16 * q + 16 * q^19 + 32 * q^49 + 16 * q^91

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
$7$	− 3.02045i	− 1.14162i	−0.821081	−	0.570811i	$-0.806629\pi$
$7$	− 3.02045i	− 1.14162i	0.821081	−	0.570811i	$-0.193371\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.62258i	1.09225i	0.837704	+	0.546125i	$0.183898\pi$
$11$	3.62258i	1.09225i	−0.837704	+	0.546125i	$0.816102\pi$
$12$	0	0
$13$	1.69614i	0.470425i	0.971944	+	0.235212i	$0.0755786\pi$
$13$	1.69614i	0.470425i	−0.971944	+	0.235212i	$0.924421\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.60421i	1.60176i	0.598828	+	0.800878i	$0.295633\pi$
$17$	6.60421i	1.60176i	−0.598828	+	0.800878i	$0.704367\pi$
$18$	0	0
$19$	5.12311	1.17532	0.587661	−	0.809108i	$-0.300049\pi$
$19$	5.12311	1.17532	0.587661	+	0.809108i	$0.300049\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.67026	−1.39085	−0.695423	−	0.718601i	$-0.744783\pi$
$23$	−6.67026	−1.39085	−0.695423	+	0.718601i	$0.744783\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.82867	−1.26805	−0.634026	−	0.773312i	$-0.718599\pi$
$29$	−6.82867	−1.26805	−0.634026	+	0.773312i	$0.718599\pi$
$30$	0	0
$31$	− 1.73642i	− 0.311870i	−0.987767	−	0.155935i	$-0.950161\pi$
$31$	− 1.73642i	− 0.311870i	0.987767	−	0.155935i	$-0.0498391\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.371834i	− 0.0611292i	−0.999533	−	0.0305646i	$-0.990269\pi$
$37$	− 0.371834i	− 0.0611292i	0.999533	−	0.0305646i	$-0.00973052\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.83095i	− 0.910642i	−0.890327	−	0.455321i	$-0.849525\pi$
$41$	− 5.83095i	− 0.910642i	0.890327	−	0.455321i	$-0.150475\pi$
$42$	0	0
$43$	−5.24477	−0.799819	−0.399910	−	0.916555i	$-0.630959\pi$
$43$	−5.24477	−0.799819	−0.399910	+	0.916555i	$0.630959\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.525853	0.0767035	0.0383518	−	0.999264i	$-0.487789\pi$
$47$	0.525853	0.0767035	0.0383518	+	0.999264i	$0.487789\pi$
$48$	0	0
$49$	−2.12311	−0.303301
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.0054	−1.37435	−0.687173	−	0.726493i	$-0.741149\pi$
$53$	−10.0054	−1.37435	−0.687173	+	0.726493i	$0.741149\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4.86270i	− 0.633069i	−0.948581	−	0.316535i	$-0.897481\pi$
$59$	− 4.86270i	− 0.633069i	0.948581	−	0.316535i	$-0.102519\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.4347	1.64132	0.820658	−	0.571420i	$-0.193607\pi$
$67$	13.4347	1.64132	0.820658	+	0.571420i	$0.193607\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.45567	0.291434	0.145717	−	0.989326i	$-0.453451\pi$
$71$	2.45567	0.291434	0.145717	+	0.989326i	$0.453451\pi$
$72$	0	0
$73$	−14.5845	−1.70699	−0.853495	−	0.521102i	$-0.825521\pi$
$73$	−14.5845	−1.70699	−0.853495	+	0.521102i	$0.825521\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.9418	1.24694
$78$	0	0
$79$	− 14.1051i	− 1.58695i	−0.608603	−	0.793475i	$-0.708270\pi$
$79$	− 14.1051i	− 1.58695i	0.608603	−	0.793475i	$-0.291730\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 5.79119i	− 0.635666i	−0.948147	−	0.317833i	$-0.897045\pi$
$83$	− 5.79119i	− 0.635666i	0.948147	−	0.317833i	$-0.102955\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 10.2477i	− 1.08625i	−0.839651	−	0.543126i	$-0.817240\pi$
$89$	− 10.2477i	− 1.08625i	0.839651	−	0.543126i	$-0.182760\pi$
$90$	0	0
$91$	5.12311	0.537047
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.33976	−0.948309	−0.474154	−	0.880442i	$-0.657246\pi$
$97$	−9.33976	−0.948309	−0.474154	+	0.880442i	$0.657246\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.37300	−0.435129	−0.217565	−	0.976046i	$-0.569811\pi$
$101$	−4.37300	−0.435129	−0.217565	+	0.976046i	$0.569811\pi$
$102$	0	0
$103$	− 8.31768i	− 0.819565i	−0.912183	−	0.409782i	$-0.865605\pi$
$103$	− 8.31768i	− 0.819565i	0.912183	−	0.409782i	$-0.134395\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.41722i	0.717050i	0.933520	+	0.358525i	$0.116720\pi$
$107$	7.41722i	0.717050i	−0.933520	+	0.358525i	$0.883280\pi$
$108$	0	0
$109$	9.65719i	0.924991i	0.886622	+	0.462496i	$0.153046\pi$
$109$	9.65719i	0.924991i	−0.886622	+	0.462496i	$0.846954\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.813015i	0.0764820i	0.999269	+	0.0382410i	$0.0121755\pi$
$113$	0.813015i	0.0764820i	−0.999269	+	0.0382410i	$0.987825\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	19.9477	1.82860
$120$	0	0
$121$	−2.12311	−0.193010
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 15.1022i	− 1.34011i	−0.742313	−	0.670054i	$-0.766271\pi$
$127$	− 15.1022i	− 1.34011i	0.742313	−	0.670054i	$-0.233729\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 6.45101i	− 0.563627i	−0.959469	−	0.281814i	$-0.909064\pi$
$131$	− 6.45101i	− 0.563627i	0.959469	−	0.281814i	$-0.0909360\pi$
$132$	0	0
$133$	− 15.4741i	− 1.34177i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.60421i	0.564235i	0.959380	+	0.282118i	$0.0910369\pi$
$137$	6.60421i	0.564235i	−0.959380	+	0.282118i	$0.908963\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.14441	−0.513821
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.5747	−1.27593	−0.637963	−	0.770067i	$-0.720223\pi$
$149$	−15.5747	−1.27593	−0.637963	+	0.770067i	$0.720223\pi$
$150$	0	0
$151$	− 10.6323i	− 0.865243i	−0.901576	−	0.432622i	$-0.857589\pi$
$151$	− 10.6323i	− 0.865243i	0.901576	−	0.432622i	$-0.142411\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 9.06134i	− 0.723174i	−0.932338	−	0.361587i	$-0.882235\pi$
$157$	− 9.06134i	− 0.723174i	0.932338	−	0.361587i	$-0.117765\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	20.1472i	1.58782i
$162$	0	0
$163$	−10.4895	−0.821604	−0.410802	−	0.911725i	$-0.634751\pi$
$163$	−10.4895	−0.821604	−0.410802	+	0.911725i	$0.634751\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.79741	−0.371235	−0.185617	−	0.982622i	$-0.559429\pi$
$167$	−4.79741	−0.371235	−0.185617	+	0.982622i	$0.559429\pi$
$168$	0	0
$169$	10.1231	0.778700
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.33513	0.253565	0.126783	−	0.991931i	$-0.459535\pi$
$173$	3.33513	0.253565	0.126783	+	0.991931i	$0.459535\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.9413i	1.56523i	0.622506	+	0.782615i	$0.286114\pi$
$179$	20.9413i	1.56523i	−0.622506	+	0.782615i	$0.713886\pi$
$180$	0	0
$181$	6.18435i	0.459679i	0.973229	+	0.229840i	$0.0738202\pi$
$181$	6.18435i	0.459679i	−0.973229	+	0.229840i	$0.926180\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−23.9243	−1.74952
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.4920	1.26568	0.632838	−	0.774284i	$-0.281890\pi$
$191$	17.4920	1.26568	0.632838	+	0.774284i	$0.281890\pi$
$192$	0	0
$193$	−15.7343	−1.13258	−0.566290	−	0.824206i	$-0.691622\pi$
$193$	−15.7343	−1.13258	−0.566290	+	0.824206i	$0.691622\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.6929	−1.75930	−0.879649	−	0.475623i	$-0.842223\pi$
$197$	−24.6929	−1.75930	−0.879649	+	0.475623i	$0.842223\pi$
$198$	0	0
$199$	10.6323i	0.753703i	0.926274	+	0.376851i	$0.122993\pi$
$199$	10.6323i	0.753703i	−0.926274	+	0.376851i	$0.877007\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.6256i	1.44764i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	18.5589i	1.28374i
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.24477	−0.356038
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−11.2017	−0.753505
$222$	0	0
$223$	− 28.6714i	− 1.91998i	−0.280040	−	0.959988i	$-0.590348\pi$
$223$	− 28.6714i	− 1.91998i	0.280040	−	0.959988i	$-0.409652\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 13.2084i	− 0.876673i	−0.898811	−	0.438337i	$-0.855568\pi$
$227$	− 13.2084i	− 0.876673i	0.898811	−	0.438337i	$-0.144432\pi$
$228$	0	0
$229$	22.0259i	1.45551i	0.685836	+	0.727756i	$0.259437\pi$
$229$	22.0259i	1.45551i	−0.685836	+	0.727756i	$0.740563\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.4386i	1.40449i	0.711934	+	0.702246i	$0.247819\pi$
$233$	21.4386i	1.40449i	−0.711934	+	0.702246i	$0.752181\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.91134	0.317688	0.158844	−	0.987304i	$-0.449223\pi$
$239$	4.91134	0.317688	0.158844	+	0.987304i	$0.449223\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.68951i	0.552900i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 9.27944i	− 0.585713i	−0.956156	−	0.292856i	$-0.905394\pi$
$251$	− 9.27944i	− 0.585713i	0.956156	−	0.292856i	$-0.0946058\pi$
$252$	0	0
$253$	− 24.1636i	− 1.51915i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 12.3954i	− 0.773204i	−0.922247	−	0.386602i	$-0.873649\pi$
$257$	− 12.3954i	− 0.773204i	0.922247	−	0.386602i	$-0.126351\pi$
$258$	0	0
$259$	−1.12311	−0.0697864
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.2888	−0.757761	−0.378881	−	0.925446i	$-0.623691\pi$
$263$	−12.2888	−0.757761	−0.378881	+	0.925446i	$0.623691\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.91733	−0.116902	−0.0584508	−	0.998290i	$-0.518616\pi$
$269$	−1.91733	−0.116902	−0.0584508	+	0.998290i	$0.518616\pi$
$270$	0	0
$271$	26.4738i	1.60817i	0.594514	+	0.804085i	$0.297344\pi$
$271$	26.4738i	1.60817i	−0.594514	+	0.804085i	$0.702656\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.8730i	0.713379i	0.934223	+	0.356689i	$0.116095\pi$
$277$	11.8730i	0.713379i	−0.934223	+	0.356689i	$0.883905\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 8.65938i	− 0.516575i	−0.966068	−	0.258288i	$-0.916842\pi$
$281$	− 8.65938i	− 0.516575i	0.966068	−	0.258288i	$-0.0831582\pi$
$282$	0	0
$283$	−15.7343	−0.935307	−0.467654	−	0.883912i	$-0.654900\pi$
$283$	−15.7343	−0.935307	−0.467654	+	0.883912i	$0.654900\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−17.6121	−1.03961
$288$	0	0
$289$	−26.6155	−1.56562
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−16.9710	−0.991454	−0.495727	−	0.868478i	$-0.665098\pi$
$293$	−16.9710	−0.991454	−0.495727	+	0.868478i	$0.665098\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 11.3137i	− 0.654289i
$300$	0	0
$301$	15.8415i	0.913091i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−18.6795	−1.06610	−0.533048	−	0.846085i	$-0.678954\pi$
$307$	−18.6795	−1.06610	−0.533048	+	0.846085i	$0.678954\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−22.4033	−1.27038	−0.635188	−	0.772358i	$-0.719077\pi$
$311$	−22.4033	−1.27038	−0.635188	+	0.772358i	$0.719077\pi$
$312$	0	0
$313$	21.6247	1.22230	0.611151	−	0.791514i	$-0.290707\pi$
$313$	21.6247	1.22230	0.611151	+	0.791514i	$0.290707\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.115279	−0.00647473	−0.00323737	−	0.999995i	$-0.501030\pi$
$317$	−0.115279	−0.00647473	−0.00323737	+	0.999995i	$0.501030\pi$
$318$	0	0
$319$	− 24.7374i	− 1.38503i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	33.8340i	1.88258i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.58831i	− 0.0875664i
$330$	0	0
$331$	−23.3693	−1.28449	−0.642247	−	0.766498i	$-0.721998\pi$
$331$	−23.3693	−1.28449	−0.642247	+	0.766498i	$0.721998\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.2850	0.669205	0.334602	−	0.942359i	$-0.391398\pi$
$337$	12.2850	0.669205	0.334602	+	0.942359i	$0.391398\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.29033	0.340640
$342$	0	0
$343$	− 14.7304i	− 0.795367i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.79119i	0.310887i	0.987845	+	0.155444i	$0.0496807\pi$
$347$	5.79119i	0.310887i	−0.987845	+	0.155444i	$0.950319\pi$
$348$	0	0
$349$	− 6.94568i	− 0.371794i	−0.982569	−	0.185897i	$-0.940481\pi$
$349$	− 6.94568i	− 0.371794i	0.982569	−	0.185897i	$-0.0595190\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	33.0210i	1.75753i	0.477253	+	0.878766i	$0.341633\pi$
$353$	33.0210i	1.75753i	−0.477253	+	0.878766i	$0.658367\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.9477	1.05280	0.526399	−	0.850238i	$-0.323542\pi$
$359$	19.9477	1.05280	0.526399	+	0.850238i	$0.323542\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	19.9819i	1.04304i	0.853238	+	0.521522i	$0.174636\pi$
$367$	19.9819i	1.04304i	−0.853238	+	0.521522i	$0.825364\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	30.2208i	1.56898i
$372$	0	0
$373$	− 5.66906i	− 0.293533i	−0.989171	−	0.146766i	$-0.953113\pi$
$373$	− 5.66906i	− 0.293533i	0.989171	−	0.146766i	$-0.0468866\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 11.5824i	− 0.596523i
$378$	0	0
$379$	16.4924	0.847159	0.423579	−	0.905859i	$-0.360773\pi$
$379$	16.4924	0.847159	0.423579	+	0.905859i	$0.360773\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.6875	0.750498	0.375249	−	0.926924i	$-0.377557\pi$
$383$	14.6875	0.750498	0.375249	+	0.926924i	$0.377557\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13.1190	0.665159	0.332580	−	0.943075i	$-0.392081\pi$
$389$	13.1190	0.665159	0.332580	+	0.943075i	$0.392081\pi$
$390$	0	0
$391$	− 44.0518i	− 2.22779i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 33.2249i	− 1.66751i	−0.552134	−	0.833756i	$-0.686186\pi$
$397$	− 33.2249i	− 1.66751i	0.552134	−	0.833756i	$-0.313814\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 34.4634i	− 1.72102i	−0.509433	−	0.860510i	$-0.670145\pi$
$401$	− 34.4634i	− 1.72102i	0.509433	−	0.860510i	$-0.329855\pi$
$402$	0	0
$403$	2.94521	0.146712
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.34700	0.0667683
$408$	0	0
$409$	5.12311	0.253321	0.126661	−	0.991946i	$-0.459574\pi$
$409$	5.12311	0.253321	0.126661	+	0.991946i	$0.459574\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.6875	−0.722726
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 8.38752i	− 0.409757i	−0.978787	−	0.204878i	$-0.934320\pi$
$419$	− 8.38752i	− 0.409757i	0.978787	−	0.204878i	$-0.0656799\pi$
$420$	0	0
$421$	− 15.0802i	− 0.734965i	−0.930030	−	0.367482i	$-0.880220\pi$
$421$	− 15.0802i	− 0.734965i	0.930030	−	0.367482i	$-0.119780\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.9477	0.960845	0.480422	−	0.877037i	$-0.340483\pi$
$431$	19.9477	0.960845	0.480422	+	0.877037i	$0.340483\pi$
$432$	0	0
$433$	−10.4895	−0.504095	−0.252047	−	0.967715i	$-0.581104\pi$
$433$	−10.4895	−0.504095	−0.252047	+	0.967715i	$0.581104\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−34.1725	−1.63469
$438$	0	0
$439$	23.0010i	1.09778i	0.835895	+	0.548889i	$0.184949\pi$
$439$	23.0010i	1.09778i	−0.835895	+	0.548889i	$0.815051\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.62603i	0.0772550i	0.999254	+	0.0386275i	$0.0122986\pi$
$443$	1.62603i	0.0772550i	−0.999254	+	0.0386275i	$0.987701\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 9.89949i	− 0.467186i	−0.972334	−	0.233593i	$-0.924952\pi$
$449$	− 9.89949i	− 0.467186i	0.972334	−	0.233593i	$-0.0750483\pi$
$450$	0	0
$451$	21.1231	0.994648
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.8293	−0.927575	−0.463788	−	0.885946i	$-0.653510\pi$
$457$	−19.8293	−0.927575	−0.463788	+	0.885946i	$0.653510\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.5747	0.725384	0.362692	−	0.931909i	$-0.381858\pi$
$461$	15.5747	0.725384	0.362692	+	0.931909i	$0.381858\pi$
$462$	0	0
$463$	17.7509i	0.824952i	0.910968	+	0.412476i	$0.135336\pi$
$463$	17.7509i	0.824952i	−0.910968	+	0.412476i	$0.864664\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.0429i	1.29767i	0.760930	+	0.648834i	$0.224743\pi$
$467$	28.0429i	1.29767i	−0.760930	+	0.648834i	$0.775257\pi$
$468$	0	0
$469$	− 40.5790i	− 1.87376i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 18.9996i	− 0.873603i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−37.4396	−1.71066	−0.855331	−	0.518083i	$-0.826646\pi$
$479$	−37.4396	−1.71066	−0.855331	+	0.518083i	$0.826646\pi$
$480$	0	0
$481$	0.630683	0.0287567
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.50778i	− 0.204267i	−0.994771	−	0.102134i	$-0.967433\pi$
$487$	− 4.50778i	− 0.204267i	0.994771	−	0.102134i	$-0.0325669\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 10.5196i	− 0.474741i	−0.971419	−	0.237370i	$-0.923714\pi$
$491$	− 10.5196i	− 0.474741i	0.971419	−	0.237370i	$-0.0762855\pi$
$492$	0	0
$493$	− 45.0979i	− 2.03111i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.41722i	− 0.332708i
$498$	0	0
$499$	25.1231	1.12466	0.562332	−	0.826911i	$-0.309904\pi$
$499$	25.1231	1.12466	0.562332	+	0.826911i	$0.309904\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16.5604	−0.738391	−0.369195	−	0.929352i	$-0.620367\pi$
$503$	−16.5604	−0.738391	−0.369195	+	0.929352i	$0.620367\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.37300	0.193830	0.0969148	−	0.995293i	$-0.469103\pi$
$509$	4.37300	0.193830	0.0969148	+	0.995293i	$0.469103\pi$
$510$	0	0
$511$	44.0518i	1.94874i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.90495i	0.0837794i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.3162i	0.627206i	0.949554	+	0.313603i	$0.101536\pi$
$521$	14.3162i	0.627206i	−0.949554	+	0.313603i	$0.898464\pi$
$522$	0	0
$523$	−18.6795	−0.816798	−0.408399	−	0.912803i	$-0.633913\pi$
$523$	−18.6795	−0.816798	−0.408399	+	0.912803i	$0.633913\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.4677	0.499540
$528$	0	0
$529$	21.4924	0.934453
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9.89012	0.428389
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 7.69113i	− 0.331280i
$540$	0	0
$541$	− 37.8674i	− 1.62805i	−0.580831	−	0.814024i	$-0.697272\pi$
$541$	− 37.8674i	− 1.62805i	0.580831	−	0.814024i	$-0.302728\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−32.1143	−1.37311	−0.686553	−	0.727079i	$-0.740877\pi$
$547$	−32.1143	−1.37311	−0.686553	+	0.727079i	$0.740877\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−34.9840	−1.49037
$552$	0	0
$553$	−42.6038	−1.81170
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.18425	0.389149	0.194575	−	0.980888i	$-0.437667\pi$
$557$	9.18425	0.389149	0.194575	+	0.980888i	$0.437667\pi$
$558$	0	0
$559$	− 8.89586i	− 0.376255i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.79119i	0.244070i	0.992526	+	0.122035i	$0.0389419\pi$
$563$	5.79119i	0.244070i	−0.992526	+	0.122035i	$0.961058\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 6.72287i	− 0.281837i	−0.990021	−	0.140919i	$-0.954994\pi$
$569$	− 6.72287i	− 0.281837i	0.990021	−	0.140919i	$-0.0450056\pi$
$570$	0	0
$571$	23.8617	0.998583	0.499291	−	0.866434i	$-0.333594\pi$
$571$	23.8617	0.998583	0.499291	+	0.866434i	$0.333594\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.24477	0.218342	0.109171	−	0.994023i	$-0.465180\pi$
$577$	5.24477	0.218342	0.109171	+	0.994023i	$0.465180\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17.4920	−0.725690
$582$	0	0
$583$	− 36.2454i	− 1.50113i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7.41722i	− 0.306141i	−0.988215	−	0.153071i	$-0.951084\pi$
$587$	− 7.41722i	− 0.306141i	0.988215	−	0.153071i	$-0.0489162\pi$
$588$	0	0
$589$	− 8.89586i	− 0.366548i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 23.9778i	− 0.984649i	−0.870412	−	0.492325i	$-0.836147\pi$
$593$	− 23.9778i	− 0.984649i	0.870412	−	0.492325i	$-0.163853\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−29.7703	−1.21638	−0.608191	−	0.793790i	$-0.708105\pi$
$599$	−29.7703	−1.21638	−0.608191	+	0.793790i	$0.708105\pi$
$600$	0	0
$601$	1.36932	0.0558556	0.0279278	−	0.999610i	$-0.491109\pi$
$601$	1.36932	0.0558556	0.0279278	+	0.999610i	$0.491109\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 43.4018i	− 1.76162i	−0.473467	−	0.880812i	$-0.656998\pi$
$607$	− 43.4018i	− 1.76162i	0.473467	−	0.880812i	$-0.343002\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.891921i	0.0360832i
$612$	0	0
$613$	22.6305i	0.914036i	0.889457	+	0.457018i	$0.151083\pi$
$613$	22.6305i	0.914036i	−0.889457	+	0.457018i	$0.848917\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.3954i	0.499020i	0.968372	+	0.249510i	$0.0802695\pi$
$617$	12.3954i	0.499020i	−0.968372	+	0.249510i	$0.919730\pi$
$618$	0	0
$619$	−30.7386	−1.23549	−0.617745	−	0.786378i	$-0.711954\pi$
$619$	−30.7386	−1.23549	−0.617745	+	0.786378i	$0.711954\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−30.9526	−1.24009
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.45567	0.0979139
$630$	0	0
$631$	− 10.6323i	− 0.423265i	−0.977349	−	0.211632i	$-0.932122\pi$
$631$	− 10.6323i	− 0.423265i	0.977349	−	0.211632i	$-0.0678779\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.60109i	− 0.142680i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.07107i	0.279290i	0.990202	+	0.139645i	$0.0445962\pi$
$641$	7.07107i	0.279290i	−0.990202	+	0.139645i	$0.955404\pi$
$642$	0	0
$643$	−37.3590	−1.47330	−0.736648	−	0.676276i	$-0.763593\pi$
$643$	−37.3590	−1.47330	−0.736648	+	0.676276i	$0.763593\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.27156	0.167932	0.0839661	−	0.996469i	$-0.473241\pi$
$647$	4.27156	0.167932	0.0839661	+	0.996469i	$0.473241\pi$
$648$	0	0
$649$	17.6155	0.691470
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.5312	−0.412120	−0.206060	−	0.978539i	$-0.566064\pi$
$653$	−10.5312	−0.412120	−0.206060	+	0.978539i	$0.566064\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27.4901i	1.07086i	0.844579	+	0.535431i	$0.179851\pi$
$659$	27.4901i	1.07086i	−0.844579	+	0.535431i	$0.820149\pi$
$660$	0	0
$661$	33.6333i	1.30818i	0.756416	+	0.654091i	$0.226949\pi$
$661$	33.6333i	1.30818i	−0.756416	+	0.654091i	$0.773051\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	45.5490	1.76366
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	44.7037	1.71810	0.859052	−	0.511889i	$-0.171054\pi$
$677$	44.7037	1.71810	0.859052	+	0.511889i	$0.171054\pi$
$678$	0	0
$679$	28.2102i	1.08261i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 32.2080i	− 1.23241i	−0.787588	−	0.616203i	$-0.788670\pi$
$683$	− 32.2080i	− 1.23241i	0.787588	−	0.616203i	$-0.211330\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 16.9706i	− 0.646527i
$690$	0	0
$691$	14.8769	0.565944	0.282972	−	0.959128i	$-0.408680\pi$
$691$	14.8769	0.565944	0.282972	+	0.959128i	$0.408680\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	38.5088	1.45862
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.28434	0.350665	0.175332	−	0.984509i	$-0.443900\pi$
$701$	9.28434	0.350665	0.175332	+	0.984509i	$0.443900\pi$
$702$	0	0
$703$	− 1.90495i	− 0.0718464i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.2084i	0.496753i
$708$	0	0
$709$	15.0802i	0.566349i	0.959068	+	0.283175i	$0.0913876\pi$
$709$	15.0802i	0.566349i	−0.959068	+	0.283175i	$0.908612\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11.5824i	0.433764i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−47.2623	−1.76259	−0.881294	−	0.472569i	$-0.843327\pi$
$719$	−47.2623	−1.76259	−0.881294	+	0.472569i	$0.843327\pi$
$720$	0	0
$721$	−25.1231	−0.935633
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	5.66906i	0.210254i	0.994459	+	0.105127i	$0.0335249\pi$
$727$	5.66906i	0.210254i	−0.994459	+	0.105127i	$0.966475\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 34.6375i	− 1.28111i
$732$	0	0
$733$	23.9548i	0.884790i	0.896820	+	0.442395i	$0.145871\pi$
$733$	23.9548i	0.884790i	−0.896820	+	0.442395i	$0.854129\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	48.6685i	1.79273i
$738$	0	0
$739$	−5.61553	−0.206571	−0.103285	−	0.994652i	$-0.532935\pi$
$739$	−5.61553	−0.206571	−0.103285	+	0.994652i	$0.532935\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.6875	0.538833	0.269417	−	0.963024i	$-0.413169\pi$
$743$	14.6875	0.538833	0.269417	+	0.963024i	$0.413169\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	22.4033	0.818600
$750$	0	0
$751$	− 8.68210i	− 0.316814i	−0.987374	−	0.158407i	$-0.949364\pi$
$751$	− 8.68210i	− 0.316814i	0.987374	−	0.158407i	$-0.0506359\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.69614i	− 0.0616473i	−0.999525	−	0.0308236i	$-0.990187\pi$
$757$	− 1.69614i	− 0.0616473i	0.999525	−	0.0308236i	$-0.00981303\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32.3314i	1.17201i	0.810307	+	0.586006i	$0.199300\pi$
$761$	32.3314i	1.17201i	−0.810307	+	0.586006i	$0.800700\pi$
$762$	0	0
$763$	29.1690	1.05599
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.24782	0.297812
$768$	0	0
$769$	15.6155	0.563110	0.281555	−	0.959545i	$-0.409150\pi$
$769$	15.6155	0.563110	0.281555	+	0.959545i	$0.409150\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−16.1498	−0.580868	−0.290434	−	0.956895i	$-0.593800\pi$
$773$	−16.1498	−0.580868	−0.290434	+	0.956895i	$0.593800\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 29.8726i	− 1.07030i
$780$	0	0
$781$	8.89586i	0.318319i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.18998	0.291941	0.145971	−	0.989289i	$-0.453369\pi$
$787$	8.18998	0.291941	0.145971	+	0.989289i	$0.453369\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.45567	0.0873135
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.1671	0.856042	0.428021	−	0.903769i	$-0.359211\pi$
$797$	24.1671	0.856042	0.428021	+	0.903769i	$0.359211\pi$
$798$	0	0
$799$	3.47284i	0.122860i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 52.8336i	− 1.86446i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	37.6400i	1.32335i	0.749789	+	0.661677i	$0.230155\pi$
$809$	37.6400i	1.32335i	−0.749789	+	0.661677i	$0.769845\pi$
$810$	0	0
$811$	−49.4773	−1.73738	−0.868691	−	0.495354i	$-0.835038\pi$
$811$	−49.4773	−1.73738	−0.868691	+	0.495354i	$0.835038\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−26.8695	−0.940045
$818$	0	0
$819$	0	0
$820$	0	0
$821$	37.9780	1.32544	0.662720	−	0.748867i	$-0.269402\pi$
$821$	37.9780	1.32544	0.662720	+	0.748867i	$0.269402\pi$
$822$	0	0
$823$	32.0636i	1.11767i	0.829279	+	0.558834i	$0.188751\pi$
$823$	32.0636i	1.11767i	−0.829279	+	0.558834i	$0.811249\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 4.16516i	− 0.144837i	−0.997374	−	0.0724184i	$-0.976928\pi$
$827$	− 4.16516i	− 0.144837i	0.997374	−	0.0724184i	$-0.0230717\pi$
$828$	0	0
$829$	− 6.18435i	− 0.214791i	−0.994216	−	0.107396i	$-0.965749\pi$
$829$	− 6.18435i	− 0.214791i	0.994216	−	0.107396i	$-0.0342512\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 14.0214i	− 0.485814i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15.0363	0.519111	0.259556	−	0.965728i	$-0.416424\pi$
$839$	15.0363	0.519111	0.259556	+	0.965728i	$0.416424\pi$
$840$	0	0
$841$	17.6307	0.607955
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.41273i	0.220344i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.48023i	0.0850213i
$852$	0	0
$853$	22.3044i	0.763689i	0.924227	+	0.381844i	$0.124711\pi$
$853$	22.3044i	0.763689i	−0.924227	+	0.381844i	$0.875289\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.7694i	0.367875i	0.982938	+	0.183937i	$0.0588843\pi$
$857$	10.7694i	0.367875i	−0.982938	+	0.183937i	$0.941116\pi$
$858$	0	0
$859$	−50.2462	−1.71438	−0.857189	−	0.515001i	$-0.827791\pi$
$859$	−50.2462	−1.71438	−0.857189	+	0.515001i	$0.827791\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.43971	0.219210	0.109605	−	0.993975i	$-0.465041\pi$
$863$	6.43971	0.219210	0.109605	+	0.993975i	$0.465041\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	51.0970	1.73335
$870$	0	0
$871$	22.7872i	0.772116i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.789443i	0.0266576i	0.999911	+	0.0133288i	$0.00424281\pi$
$877$	0.789443i	0.0266576i	−0.999911	+	0.0133288i	$0.995757\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.83095i	0.196450i	0.995164	+	0.0982249i	$0.0313164\pi$
$881$	5.83095i	0.196450i	−0.995164	+	0.0982249i	$0.968684\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.1379	0.609012	0.304506	−	0.952510i	$-0.401509\pi$
$887$	18.1379	0.609012	0.304506	+	0.952510i	$0.401509\pi$
$888$	0	0
$889$	−45.6155	−1.52990
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.69400	0.0901513
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11.8574i	0.395468i
$900$	0	0
$901$	− 66.0777i	− 2.20137i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.1143	1.06634	0.533168	−	0.846009i	$-0.321001\pi$
$907$	32.1143	1.06634	0.533168	+	0.846009i	$0.321001\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.5807	0.416816	0.208408	−	0.978042i	$-0.433172\pi$
$911$	12.5807	0.416816	0.208408	+	0.978042i	$0.433172\pi$
$912$	0	0
$913$	20.9791	0.694306
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19.4849	−0.643449
$918$	0	0
$919$	49.2611i	1.62497i	0.582981	+	0.812486i	$0.301886\pi$
$919$	49.2611i	1.62497i	−0.582981	+	0.812486i	$0.698114\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.16516i	0.137098i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 8.65938i	− 0.284105i	−0.989859	−	0.142053i	$-0.954630\pi$
$929$	− 8.65938i	− 0.284105i	0.989859	−	0.142053i	$-0.0453702\pi$
$930$	0	0
$931$	−10.8769	−0.356476
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.37300	0.142556	0.0712778	−	0.997457i	$-0.477292\pi$
$941$	4.37300	0.142556	0.0712778	+	0.997457i	$0.477292\pi$
$942$	0	0
$943$	38.8940i	1.26656i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 22.2517i	− 0.723082i	−0.932356	−	0.361541i	$-0.882251\pi$
$947$	− 22.2517i	− 0.723082i	0.932356	−	0.361541i	$-0.117749\pi$
$948$	0	0
$949$	− 24.7374i	− 0.803010i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 0.813015i	− 0.0263361i	−0.999913	−	0.0131681i	$-0.995808\pi$
$953$	− 0.813015i	− 0.0263361i	0.999913	−	0.0131681i	$-0.00419165\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	19.9477	0.644143
$960$	0	0
$961$	27.9848	0.902737
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 23.7917i	− 0.765091i	−0.923937	−	0.382546i	$-0.875048\pi$
$967$	− 23.7917i	− 0.765091i	0.923937	−	0.382546i	$-0.124952\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.6032i	1.04629i	0.852244	+	0.523144i	$0.175241\pi$
$971$	32.6032i	1.04629i	−0.852244	+	0.523144i	$0.824759\pi$
$972$	0	0
$973$	36.2454i	1.16197i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.43904i	− 0.0780319i	−0.999239	−	0.0390160i	$-0.987578\pi$
$977$	− 2.43904i	− 0.0780319i	0.999239	−	0.0390160i	$-0.0124223\pi$
$978$	0	0
$979$	37.1231	1.18646
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−12.5841	−0.401371	−0.200685	−	0.979656i	$-0.564317\pi$
$983$	−12.5841	−0.401371	−0.200685	+	0.979656i	$0.564317\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	34.9840	1.11243
$990$	0	0
$991$	10.6323i	0.337746i	0.985638	+	0.168873i	$0.0540127\pi$
$991$	10.6323i	0.337746i	−0.985638	+	0.168873i	$0.945987\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	32.4813i	1.02869i	0.857583	+	0.514346i	$0.171965\pi$
$997$	32.4813i	1.02869i	−0.857583	+	0.514346i	$0.828035\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.b.i.4751.4		16
3.2	odd	2	inner	7200.2.b.i.4751.1		16
4.3	odd	2		1800.2.b.g.251.2		16
5.2	odd	4		1440.2.m.c.719.2		16
5.3	odd	4		1440.2.m.c.719.3		16
5.4	even	2	inner	7200.2.b.i.4751.15		16
8.3	odd	2	inner	7200.2.b.i.4751.16		16
8.5	even	2		1800.2.b.g.251.13		16
12.11	even	2		1800.2.b.g.251.16		16
15.2	even	4		1440.2.m.c.719.16		16
15.8	even	4		1440.2.m.c.719.13		16
15.14	odd	2	inner	7200.2.b.i.4751.14		16
20.3	even	4		360.2.m.c.179.7	yes	16
20.7	even	4		360.2.m.c.179.9	yes	16
20.19	odd	2		1800.2.b.g.251.15		16
24.5	odd	2		1800.2.b.g.251.3		16
24.11	even	2	inner	7200.2.b.i.4751.13		16
40.3	even	4		1440.2.m.c.719.14		16
40.13	odd	4		360.2.m.c.179.6	yes	16
40.19	odd	2	inner	7200.2.b.i.4751.3		16
40.27	even	4		1440.2.m.c.719.15		16
40.29	even	2		1800.2.b.g.251.4		16
40.37	odd	4		360.2.m.c.179.12	yes	16
60.23	odd	4		360.2.m.c.179.10	yes	16
60.47	odd	4		360.2.m.c.179.8	yes	16
60.59	even	2		1800.2.b.g.251.1		16
120.29	odd	2		1800.2.b.g.251.14		16
120.53	even	4		360.2.m.c.179.11	yes	16
120.59	even	2	inner	7200.2.b.i.4751.2		16
120.77	even	4		360.2.m.c.179.5	✓	16
120.83	odd	4		1440.2.m.c.719.4		16
120.107	odd	4		1440.2.m.c.719.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.m.c.179.5	✓	16	120.77	even	4
360.2.m.c.179.6	yes	16	40.13	odd	4
360.2.m.c.179.7	yes	16	20.3	even	4
360.2.m.c.179.8	yes	16	60.47	odd	4
360.2.m.c.179.9	yes	16	20.7	even	4
360.2.m.c.179.10	yes	16	60.23	odd	4
360.2.m.c.179.11	yes	16	120.53	even	4
360.2.m.c.179.12	yes	16	40.37	odd	4
1440.2.m.c.719.1		16	120.107	odd	4
1440.2.m.c.719.2		16	5.2	odd	4
1440.2.m.c.719.3		16	5.3	odd	4
1440.2.m.c.719.4		16	120.83	odd	4
1440.2.m.c.719.13		16	15.8	even	4
1440.2.m.c.719.14		16	40.3	even	4
1440.2.m.c.719.15		16	40.27	even	4
1440.2.m.c.719.16		16	15.2	even	4
1800.2.b.g.251.1		16	60.59	even	2
1800.2.b.g.251.2		16	4.3	odd	2
1800.2.b.g.251.3		16	24.5	odd	2
1800.2.b.g.251.4		16	40.29	even	2
1800.2.b.g.251.13		16	8.5	even	2
1800.2.b.g.251.14		16	120.29	odd	2
1800.2.b.g.251.15		16	20.19	odd	2
1800.2.b.g.251.16		16	12.11	even	2
7200.2.b.i.4751.1		16	3.2	odd	2	inner
7200.2.b.i.4751.2		16	120.59	even	2	inner
7200.2.b.i.4751.3		16	40.19	odd	2	inner
7200.2.b.i.4751.4		16	1.1	even	1	trivial
7200.2.b.i.4751.13		16	24.11	even	2	inner
7200.2.b.i.4751.14		16	15.14	odd	2	inner
7200.2.b.i.4751.15		16	5.4	even	2	inner
7200.2.b.i.4751.16		16	8.3	odd	2	inner

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.02045i q^{7} +O(q^{10})\)	\(q-3.02045i q^{7} +3.62258i q^{11} +1.69614i q^{13} +6.60421i q^{17} +5.12311 q^{19} -6.67026 q^{23} -6.82867 q^{29} -1.73642i q^{31} -0.371834i q^{37} -5.83095i q^{41} -5.24477 q^{43} +0.525853 q^{47} -2.12311 q^{49} -10.0054 q^{53} -4.86270i q^{59} +13.4347 q^{67} +2.45567 q^{71} -14.5845 q^{73} +10.9418 q^{77} -14.1051i q^{79} -5.79119i q^{83} -10.2477i q^{89} +5.12311 q^{91} -9.33976 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{19} + 32 q^{49} + 16 q^{91}+O(q^{100}) \) 16 * q + 16 * q^19 + 32 * q^49 + 16 * q^91

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