LMFDB - Embedded newform 7200.2.k.s.3601.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(3601,7200)]

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([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("7200.3601");

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.k (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:	$8$
Coefficient field:	8.0.214798336.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 $ x^8 - 2x^7 - 2x^5 + 9x^4 - 4x^3 - 16*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3601.3
Root			$1.23291 + 0.692769i$ of defining polynomial
Character	$\chi$	$=$	7200.3601
Dual form			7200.2.k.s.3601.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0802864 q^{7} +O(q^{10})$	$q-0.0802864 q^{7} -2.41649i q^{11} +5.26785i q^{13} +0.255918 q^{17} -6.95864i q^{19} -1.64542 q^{23} -4.51516i q^{29} -8.29484 q^{31} +2.67241i q^{37} +8.11921 q^{41} +4.08890i q^{43} +5.70272 q^{47} -6.99355 q^{49} -11.5627i q^{53} +12.6963i q^{59} +11.9403i q^{61} +7.27979i q^{67} -11.3481 q^{71} -12.0779 q^{73} +0.194011i q^{77} -5.50539 q^{79} -9.20811i q^{83} -11.9173 q^{89} -0.422937i q^{91} +8.50539 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7}+O(q^{10}) $ 8 * q + 8 * q^7	$ 8 q + 8 q^{7} - 8 q^{23} - 8 q^{31} - 40 q^{71} + 16 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 - 8 * q^23 - 8 * q^31 - 40 * q^71 + 16 * q^73 + 16 * q^79 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.0802864	−0.0303454	−0.0151727	−	0.999885i	$-0.504830\pi$
$7$	−0.0802864	−0.0303454	−0.0151727	+	0.999885i	$0.504830\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.41649i	− 0.728599i	−0.931282	−	0.364300i	$-0.881308\pi$
$11$	− 2.41649i	− 0.728599i	0.931282	−	0.364300i	$-0.118692\pi$
$12$	0	0
$13$	5.26785i	1.46104i	0.682892	+	0.730520i	$0.260722\pi$
$13$	5.26785i	1.46104i	−0.682892	+	0.730520i	$0.739278\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.255918	0.0620692	0.0310346	−	0.999518i	$-0.490120\pi$
$17$	0.255918	0.0620692	0.0310346	+	0.999518i	$0.490120\pi$
$18$	0	0
$19$	− 6.95864i	− 1.59642i	−0.602378	−	0.798211i	$-0.705780\pi$
$19$	− 6.95864i	− 1.59642i	0.602378	−	0.798211i	$-0.294220\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.64542	−0.343093	−0.171546	−	0.985176i	$-0.554876\pi$
$23$	−1.64542	−0.343093	−0.171546	+	0.985176i	$0.554876\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 4.51516i	− 0.838444i	−0.907884	−	0.419222i	$-0.862303\pi$
$29$	− 4.51516i	− 0.838444i	0.907884	−	0.419222i	$-0.137697\pi$
$30$	0	0
$31$	−8.29484	−1.48980	−0.744899	−	0.667177i	$-0.767502\pi$
$31$	−8.29484	−1.48980	−0.744899	+	0.667177i	$0.767502\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.67241i	0.439341i	0.975574	+	0.219671i	$0.0704983\pi$
$37$	2.67241i	0.439341i	−0.975574	+	0.219671i	$0.929502\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.11921	1.26801	0.634004	−	0.773330i	$-0.281410\pi$
$41$	8.11921	1.26801	0.634004	+	0.773330i	$0.281410\pi$
$42$	0	0
$43$	4.08890i	0.623551i	0.950156	+	0.311776i	$0.100924\pi$
$43$	4.08890i	0.623551i	−0.950156	+	0.311776i	$0.899076\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.70272	0.831827	0.415914	−	0.909404i	$-0.363462\pi$
$47$	5.70272	0.831827	0.415914	+	0.909404i	$0.363462\pi$
$48$	0	0
$49$	−6.99355	−0.999079
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.5627i	− 1.58826i	−0.607749	−	0.794129i	$-0.707927\pi$
$53$	− 11.5627i	− 1.58826i	0.607749	−	0.794129i	$-0.292073\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.6963i	1.65291i	0.563000	+	0.826457i	$0.309647\pi$
$59$	12.6963i	1.65291i	−0.563000	+	0.826457i	$0.690353\pi$
$60$	0	0
$61$	11.9403i	1.52879i	0.644746	+	0.764397i	$0.276963\pi$
$61$	11.9403i	1.52879i	−0.644746	+	0.764397i	$0.723037\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.27979i	0.889367i	0.895688	+	0.444684i	$0.146684\pi$
$67$	7.27979i	0.889367i	−0.895688	+	0.444684i	$0.853316\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.3481	−1.34678	−0.673388	−	0.739289i	$-0.735162\pi$
$71$	−11.3481	−1.34678	−0.673388	+	0.739289i	$0.735162\pi$
$72$	0	0
$73$	−12.0779	−1.41361	−0.706803	−	0.707411i	$-0.749863\pi$
$73$	−12.0779	−1.41361	−0.706803	+	0.707411i	$0.749863\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.194011i	0.0221096i
$78$	0	0
$79$	−5.50539	−0.619405	−0.309702	−	0.950834i	$-0.600229\pi$
$79$	−5.50539	−0.619405	−0.309702	+	0.950834i	$0.600229\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 9.20811i	− 1.01072i	−0.862908	−	0.505361i	$-0.831359\pi$
$83$	− 9.20811i	− 1.01072i	0.862908	−	0.505361i	$-0.168641\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.9173	−1.26323	−0.631615	−	0.775283i	$-0.717607\pi$
$89$	−11.9173	−1.26323	−0.631615	+	0.775283i	$0.717607\pi$
$90$	0	0
$91$	− 0.422937i	− 0.0443358i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.50539	0.863592	0.431796	−	0.901971i	$-0.357880\pi$
$97$	8.50539	0.863592	0.431796	+	0.901971i	$0.357880\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 7.56270i	− 0.752516i	−0.926515	−	0.376258i	$-0.877211\pi$
$101$	− 7.56270i	− 0.752516i	0.926515	−	0.376258i	$-0.122789\pi$
$102$	0	0
$103$	1.78544	0.175925	0.0879624	−	0.996124i	$-0.471964\pi$
$103$	1.78544	0.175925	0.0879624	+	0.996124i	$0.471964\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 10.4705i	− 1.01222i	−0.862469	−	0.506110i	$-0.831083\pi$
$107$	− 10.4705i	− 1.01222i	0.862469	−	0.506110i	$-0.168917\pi$
$108$	0	0
$109$	− 3.64298i	− 0.348934i	−0.984663	−	0.174467i	$-0.944180\pi$
$109$	− 3.64298i	− 0.348934i	0.984663	−	0.174467i	$-0.0558203\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.83298	−0.830937	−0.415468	−	0.909608i	$-0.636382\pi$
$113$	−8.83298	−0.830937	−0.415468	+	0.909608i	$0.636382\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.0205467	−0.00188351
$120$	0	0
$121$	5.16057	0.469143
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.69628	0.771670	0.385835	−	0.922568i	$-0.373913\pi$
$127$	8.69628	0.771670	0.385835	+	0.922568i	$0.373913\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 10.7916i	− 0.942868i	−0.881901	−	0.471434i	$-0.843736\pi$
$131$	− 10.7916i	− 0.942868i	0.881901	−	0.471434i	$-0.156264\pi$
$132$	0	0
$133$	0.558684i	0.0484440i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.5421	−0.986112	−0.493056	−	0.869997i	$-0.664120\pi$
$137$	−11.5421	−0.986112	−0.493056	+	0.869997i	$0.664120\pi$
$138$	0	0
$139$	− 0.214558i	− 0.0181986i	−0.999959	−	0.00909928i	$-0.997104\pi$
$139$	− 0.214558i	− 0.0181986i	0.999959	−	0.00909928i	$-0.00289643\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.7297	1.06451
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.0475i	1.88813i	0.329762	+	0.944064i	$0.393031\pi$
$149$	23.0475i	1.88813i	−0.329762	+	0.944064i	$0.606969\pi$
$150$	0	0
$151$	−9.48573	−0.771938	−0.385969	−	0.922512i	$-0.626133\pi$
$151$	−9.48573	−0.771938	−0.385969	+	0.922512i	$0.626133\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.34413i	− 0.506316i	−0.967425	−	0.253158i	$-0.918531\pi$
$157$	− 6.34413i	− 0.506316i	0.967425	−	0.253158i	$-0.0814693\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.132104	0.0104113
$162$	0	0
$163$	12.4100i	0.972030i	0.873951	+	0.486015i	$0.161550\pi$
$163$	12.4100i	0.972030i	−0.873951	+	0.486015i	$0.838450\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.2654	1.80033	0.900166	−	0.435547i	$-0.143445\pi$
$167$	23.2654	1.80033	0.900166	+	0.435547i	$0.143445\pi$
$168$	0	0
$169$	−14.7503	−1.13464
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 8.63897i	− 0.656809i	−0.944537	−	0.328404i	$-0.893489\pi$
$173$	− 8.63897i	− 0.656809i	0.944537	−	0.328404i	$-0.106511\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 9.40544i	− 0.702996i	−0.936189	−	0.351498i	$-0.885672\pi$
$179$	− 9.40544i	− 0.702996i	0.936189	−	0.351498i	$-0.114328\pi$
$180$	0	0
$181$	− 6.43487i	− 0.478300i	−0.970983	−	0.239150i	$-0.923131\pi$
$181$	− 6.43487i	− 0.478300i	0.970983	−	0.239150i	$-0.0768688\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.618423i	− 0.0452236i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.56270	−0.402503	−0.201251	−	0.979540i	$-0.564501\pi$
$191$	−5.56270	−0.402503	−0.201251	+	0.979540i	$0.564501\pi$
$192$	0	0
$193$	−18.4227	−1.32609	−0.663046	−	0.748578i	$-0.730737\pi$
$193$	−18.4227	−1.32609	−0.663046	+	0.748578i	$0.730737\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.0239i	− 1.28415i	−0.766643	−	0.642074i	$-0.778074\pi$
$197$	− 18.0239i	− 1.28415i	0.766643	−	0.642074i	$-0.221926\pi$
$198$	0	0
$199$	−20.1214	−1.42637	−0.713183	−	0.700977i	$-0.752747\pi$
$199$	−20.1214	−1.42637	−0.713183	+	0.700977i	$0.752747\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.362505i	0.0254429i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−16.8155	−1.16315
$210$	0	0
$211$	3.25592i	0.224147i	0.993700	+	0.112073i	$0.0357492\pi$
$211$	3.25592i	0.224147i	−0.993700	+	0.112073i	$0.964251\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.665963	0.0452085
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.34814i	0.0906856i
$222$	0	0
$223$	−26.9911	−1.80746	−0.903730	−	0.428104i	$-0.859182\pi$
$223$	−26.9911	−1.80746	−0.903730	+	0.428104i	$0.859182\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.8219i	1.31563i	0.753180	+	0.657814i	$0.228519\pi$
$227$	19.8219i	1.31563i	−0.753180	+	0.657814i	$0.771481\pi$
$228$	0	0
$229$	− 21.6797i	− 1.43264i	−0.697773	−	0.716319i	$-0.745826\pi$
$229$	− 21.6797i	− 1.43264i	0.697773	−	0.716319i	$-0.254174\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.2733	−1.13161	−0.565807	−	0.824538i	$-0.691435\pi$
$233$	−17.2733	−1.13161	−0.565807	+	0.824538i	$0.691435\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.3718	−1.05900	−0.529502	−	0.848309i	$-0.677621\pi$
$239$	−16.3718	−1.05900	−0.529502	+	0.848309i	$0.677621\pi$
$240$	0	0
$241$	−6.82654	−0.439736	−0.219868	−	0.975530i	$-0.570563\pi$
$241$	−6.82654	−0.439736	−0.219868	+	0.975530i	$0.570563\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	36.6571	2.33243
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.96969i	0.187445i	0.995598	+	0.0937225i	$0.0298766\pi$
$251$	2.96969i	0.187445i	−0.995598	+	0.0937225i	$0.970123\pi$
$252$	0	0
$253$	3.97613i	0.249977i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.03031	−0.313782	−0.156891	−	0.987616i	$-0.550147\pi$
$257$	−5.03031	−0.313782	−0.156891	+	0.987616i	$0.550147\pi$
$258$	0	0
$259$	− 0.214558i	− 0.0133320i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.70585	−0.166850	−0.0834248	−	0.996514i	$-0.526586\pi$
$263$	−2.70585	−0.166850	−0.0834248	+	0.996514i	$0.526586\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 22.3718i	− 1.36403i	−0.731337	−	0.682017i	$-0.761103\pi$
$269$	− 22.3718i	− 1.36403i	0.731337	−	0.682017i	$-0.238897\pi$
$270$	0	0
$271$	0.869741	0.0528330	0.0264165	−	0.999651i	$-0.491590\pi$
$271$	0.869741	0.0528330	0.0264165	+	0.999651i	$0.491590\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.6733i	1.72281i	0.507918	+	0.861406i	$0.330415\pi$
$277$	28.6733i	1.72281i	−0.507918	+	0.861406i	$0.669585\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.1429	−0.903349	−0.451674	−	0.892183i	$-0.649173\pi$
$281$	−15.1429	−0.903349	−0.451674	+	0.892183i	$0.649173\pi$
$282$	0	0
$283$	6.23225i	0.370469i	0.982694	+	0.185234i	$0.0593044\pi$
$283$	6.23225i	0.370469i	−0.982694	+	0.185234i	$0.940696\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.651862	−0.0384782
$288$	0	0
$289$	−16.9345	−0.996147
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.5054i	1.25636i	0.778069	+	0.628179i	$0.216200\pi$
$293$	21.5054i	1.25636i	−0.778069	+	0.628179i	$0.783800\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 8.66781i	− 0.501272i
$300$	0	0
$301$	− 0.328283i	− 0.0189219i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 3.57706i	− 0.204154i	−0.994777	−	0.102077i	$-0.967451\pi$
$307$	− 3.57706i	− 0.204154i	0.994777	−	0.102077i	$-0.0325488\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.49461	−0.141456	−0.0707282	−	0.997496i	$-0.522532\pi$
$311$	−2.49461	−0.141456	−0.0707282	+	0.997496i	$0.522532\pi$
$312$	0	0
$313$	9.57246	0.541068	0.270534	−	0.962710i	$-0.412800\pi$
$313$	9.57246	0.541068	0.270534	+	0.962710i	$0.412800\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.16702i	0.177877i	0.996037	+	0.0889387i	$0.0283475\pi$
$317$	3.16702i	0.177877i	−0.996037	+	0.0889387i	$0.971652\pi$
$318$	0	0
$319$	−10.9108	−0.610889
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1.78084i	− 0.0990887i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.457851	−0.0252421
$330$	0	0
$331$	16.5118i	0.907573i	0.891111	+	0.453786i	$0.149927\pi$
$331$	16.5118i	0.907573i	−0.891111	+	0.453786i	$0.850073\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.8330	−0.644584	−0.322292	−	0.946640i	$-0.604453\pi$
$337$	−11.8330	−0.644584	−0.322292	+	0.946640i	$0.604453\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	20.0444i	1.08547i
$342$	0	0
$343$	1.12349	0.0606628
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.9713i	1.28684i	0.765511	+	0.643422i	$0.222486\pi$
$347$	23.9713i	1.28684i	−0.765511	+	0.643422i	$0.777514\pi$
$348$	0	0
$349$	8.91570i	0.477247i	0.971112	+	0.238623i	$0.0766961\pi$
$349$	8.91570i	0.477247i	−0.971112	+	0.238623i	$0.923304\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.35606	0.391524	0.195762	−	0.980651i	$-0.437282\pi$
$353$	7.35606	0.391524	0.195762	+	0.980651i	$0.437282\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.2114	1.33061	0.665304	−	0.746572i	$-0.268302\pi$
$359$	25.2114	1.33061	0.665304	+	0.746572i	$0.268302\pi$
$360$	0	0
$361$	−29.4227	−1.54856
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.86573	−0.306189	−0.153094	−	0.988212i	$-0.548924\pi$
$367$	−5.86573	−0.306189	−0.153094	+	0.988212i	$0.548924\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.928327i	0.0481963i
$372$	0	0
$373$	− 27.5063i	− 1.42422i	−0.702067	−	0.712111i	$-0.747739\pi$
$373$	− 27.5063i	− 1.42422i	0.702067	−	0.712111i	$-0.252261\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	23.7852	1.22500
$378$	0	0
$379$	11.7549i	0.603807i	0.953339	+	0.301903i	$0.0976220\pi$
$379$	11.7549i	0.603807i	−0.953339	+	0.301903i	$0.902378\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−34.3335	−1.75436	−0.877180	−	0.480162i	$-0.840578\pi$
$383$	−34.3335	−1.75436	−0.877180	+	0.480162i	$0.840578\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.89515i	0.146790i	0.997303	+	0.0733951i	$0.0233834\pi$
$389$	2.89515i	0.146790i	−0.997303	+	0.0733951i	$0.976617\pi$
$390$	0	0
$391$	−0.421092	−0.0212955
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 22.9099i	− 1.14982i	−0.818218	−	0.574909i	$-0.805038\pi$
$397$	− 22.9099i	− 1.14982i	0.818218	−	0.574909i	$-0.194962\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.4337	0.620910	0.310455	−	0.950588i	$-0.399519\pi$
$401$	12.4337	0.620910	0.310455	+	0.950588i	$0.399519\pi$
$402$	0	0
$403$	− 43.6960i	− 2.17665i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.45785	0.320104
$408$	0	0
$409$	−32.0886	−1.58668	−0.793340	−	0.608778i	$-0.791660\pi$
$409$	−32.0886	−1.58668	−0.793340	+	0.608778i	$0.791660\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.01934i	− 0.0501583i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 16.1364i	− 0.788317i	−0.919043	−	0.394158i	$-0.871036\pi$
$419$	− 16.1364i	− 0.788317i	0.919043	−	0.394158i	$-0.128964\pi$
$420$	0	0
$421$	− 28.7675i	− 1.40204i	−0.713141	−	0.701021i	$-0.752728\pi$
$421$	− 28.7675i	− 1.40204i	0.713141	−	0.701021i	$-0.247272\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 0.958640i	− 0.0463918i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.7297	1.19119	0.595594	−	0.803285i	$-0.296917\pi$
$431$	24.7297	1.19119	0.595594	+	0.803285i	$0.296917\pi$
$432$	0	0
$433$	−4.48816	−0.215687	−0.107844	−	0.994168i	$-0.534395\pi$
$433$	−4.48816	−0.215687	−0.107844	+	0.994168i	$0.534395\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11.4499i	0.547721i
$438$	0	0
$439$	−5.96081	−0.284494	−0.142247	−	0.989831i	$-0.545433\pi$
$439$	−5.96081	−0.284494	−0.142247	+	0.989831i	$0.545433\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.2924i	0.679053i	0.940597	+	0.339526i	$0.110267\pi$
$443$	14.2924i	0.679053i	−0.940597	+	0.339526i	$0.889733\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−24.5529	−1.15872	−0.579362	−	0.815070i	$-0.696698\pi$
$449$	−24.5529	−1.15872	−0.579362	+	0.815070i	$0.696698\pi$
$450$	0	0
$451$	− 19.6200i	− 0.923870i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.9108	1.35239	0.676196	−	0.736722i	$-0.263627\pi$
$457$	28.9108	1.35239	0.676196	+	0.736722i	$0.263627\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 4.35458i	− 0.202813i	−0.994845	−	0.101407i	$-0.967666\pi$
$461$	− 4.35458i	− 0.202813i	0.994845	−	0.101407i	$-0.0323343\pi$
$462$	0	0
$463$	−11.1303	−0.517267	−0.258634	−	0.965976i	$-0.583272\pi$
$463$	−11.1303	−0.517267	−0.258634	+	0.965976i	$0.583272\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 19.1257i	− 0.885030i	−0.896761	−	0.442515i	$-0.854086\pi$
$467$	− 19.1257i	− 0.885030i	0.896761	−	0.442515i	$-0.145914\pi$
$468$	0	0
$469$	− 0.584467i	− 0.0269882i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.88079	0.454319
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−25.6358	−1.17133	−0.585666	−	0.810553i	$-0.699167\pi$
$479$	−25.6358	−1.17133	−0.585666	+	0.810553i	$0.699167\pi$
$480$	0	0
$481$	−14.0779	−0.641895
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.8434	0.581992	0.290996	−	0.956724i	$-0.406013\pi$
$487$	12.8434	0.581992	0.290996	+	0.956724i	$0.406013\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.9887i	0.766689i	0.923605	+	0.383344i	$0.125228\pi$
$491$	16.9887i	0.766689i	−0.923605	+	0.383344i	$0.874772\pi$
$492$	0	0
$493$	− 1.15551i	− 0.0520415i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.911101	0.0408684
$498$	0	0
$499$	14.0521i	0.629060i	0.949248	+	0.314530i	$0.101847\pi$
$499$	14.0521i	0.629060i	−0.949248	+	0.314530i	$0.898153\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.53258	−0.425037	−0.212518	−	0.977157i	$-0.568167\pi$
$503$	−9.53258	−0.425037	−0.212518	+	0.977157i	$0.568167\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 30.3450i	− 1.34502i	−0.740088	−	0.672510i	$-0.765216\pi$
$509$	− 30.3450i	− 1.34502i	0.740088	−	0.672510i	$-0.234784\pi$
$510$	0	0
$511$	0.969687	0.0428964
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 13.7806i	− 0.606069i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.4245	−0.631949	−0.315975	−	0.948768i	$-0.602331\pi$
$521$	−14.4245	−0.631949	−0.315975	+	0.948768i	$0.602331\pi$
$522$	0	0
$523$	− 28.2207i	− 1.23401i	−0.786961	−	0.617003i	$-0.788346\pi$
$523$	− 28.2207i	− 1.23401i	0.786961	−	0.617003i	$-0.211654\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.12280	−0.0924706
$528$	0	0
$529$	−20.2926	−0.882287
$530$	0	0
$531$	0	0
$532$	0	0
$533$	42.7708i	1.85261i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	16.8999i	0.727928i
$540$	0	0
$541$	− 13.4695i	− 0.579100i	−0.957163	−	0.289550i	$-0.906494\pi$
$541$	− 13.4695i	− 0.579100i	0.957163	−	0.289550i	$-0.0935056\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4.42773i	− 0.189316i	−0.995510	−	0.0946581i	$-0.969824\pi$
$547$	− 4.42773i	− 0.189316i	0.995510	−	0.0946581i	$-0.0301758\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−31.4193	−1.33851
$552$	0	0
$553$	0.442008	0.0187961
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.2017i	1.70340i	0.524030	+	0.851700i	$0.324428\pi$
$557$	40.2017i	1.70340i	−0.524030	+	0.851700i	$0.675572\pi$
$558$	0	0
$559$	−21.5397	−0.911033
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13.1128i	0.552637i	0.961066	+	0.276319i	$0.0891145\pi$
$563$	13.1128i	0.552637i	−0.961066	+	0.276319i	$0.910885\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.0257	0.462222	0.231111	−	0.972927i	$-0.425764\pi$
$569$	11.0257	0.462222	0.231111	+	0.972927i	$0.425764\pi$
$570$	0	0
$571$	− 45.6960i	− 1.91232i	−0.292847	−	0.956159i	$-0.594603\pi$
$571$	− 45.6960i	− 1.91232i	0.292847	−	0.956159i	$-0.405397\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.2685	0.718899	0.359449	−	0.933165i	$-0.382965\pi$
$577$	17.2685	0.718899	0.359449	+	0.933165i	$0.382965\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.739286i	0.0306707i
$582$	0	0
$583$	−27.9411	−1.15720
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 34.7155i	− 1.43286i	−0.697657	−	0.716432i	$-0.745774\pi$
$587$	− 34.7155i	− 1.43286i	0.697657	−	0.716432i	$-0.254226\pi$
$588$	0	0
$589$	57.7208i	2.37835i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.34022	−0.383557	−0.191778	−	0.981438i	$-0.561425\pi$
$593$	−9.34022	−0.383557	−0.191778	+	0.981438i	$0.561425\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13.9110	−0.568389	−0.284195	−	0.958767i	$-0.591726\pi$
$599$	−13.9110	−0.568389	−0.284195	+	0.958767i	$0.591726\pi$
$600$	0	0
$601$	11.7330	0.478600	0.239300	−	0.970946i	$-0.423082\pi$
$601$	11.7330	0.478600	0.239300	+	0.970946i	$0.423082\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−10.8158	−0.438998	−0.219499	−	0.975613i	$-0.570442\pi$
$607$	−10.8158	−0.438998	−0.219499	+	0.975613i	$0.570442\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.0411i	1.21533i
$612$	0	0
$613$	− 17.9632i	− 0.725528i	−0.931881	−	0.362764i	$-0.881833\pi$
$613$	− 17.9632i	− 0.725528i	0.931881	−	0.362764i	$-0.118167\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.3576	0.497500	0.248750	−	0.968568i	$-0.419980\pi$
$617$	12.3576	0.497500	0.248750	+	0.968568i	$0.419980\pi$
$618$	0	0
$619$	− 4.99540i	− 0.200782i	−0.994948	−	0.100391i	$-0.967991\pi$
$619$	− 4.99540i	− 0.200782i	0.994948	−	0.100391i	$-0.0320094\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.956795	0.0383332
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.683917i	0.0272696i
$630$	0	0
$631$	17.9674	0.715273	0.357636	−	0.933861i	$-0.383583\pi$
$631$	17.9674	0.715273	0.357636	+	0.933861i	$0.383583\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 36.8410i	− 1.45969i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27.3638	−1.08081	−0.540403	−	0.841406i	$-0.681728\pi$
$641$	−27.3638	−1.08081	−0.540403	+	0.841406i	$0.681728\pi$
$642$	0	0
$643$	− 2.27518i	− 0.0897245i	−0.998993	−	0.0448623i	$-0.985715\pi$
$643$	− 2.27518i	− 0.0897245i	0.998993	−	0.0448623i	$-0.0142849\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.4769	−0.490516	−0.245258	−	0.969458i	$-0.578873\pi$
$647$	−12.4769	−0.490516	−0.245258	+	0.969458i	$0.578873\pi$
$648$	0	0
$649$	30.6804	1.20431
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 29.3055i	− 1.14681i	−0.819271	−	0.573406i	$-0.805622\pi$
$653$	− 29.3055i	− 1.14681i	0.819271	−	0.573406i	$-0.194378\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18.6009i	0.724589i	0.932064	+	0.362295i	$0.118007\pi$
$659$	18.6009i	0.724589i	−0.932064	+	0.362295i	$0.881993\pi$
$660$	0	0
$661$	16.1318i	0.627456i	0.949513	+	0.313728i	$0.101578\pi$
$661$	16.1318i	0.627456i	−0.949513	+	0.313728i	$0.898422\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.42931i	0.287664i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	28.8535	1.11388
$672$	0	0
$673$	34.1385	1.31594	0.657971	−	0.753043i	$-0.271415\pi$
$673$	34.1385	1.31594	0.657971	+	0.753043i	$0.271415\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 12.1940i	− 0.468654i	−0.972158	−	0.234327i	$-0.924711\pi$
$677$	− 12.1940i	− 0.468654i	0.972158	−	0.234327i	$-0.0752886\pi$
$678$	0	0
$679$	−0.682867	−0.0262060
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21.8567i	0.836322i	0.908373	+	0.418161i	$0.137325\pi$
$683$	21.8567i	0.836322i	−0.908373	+	0.418161i	$0.862675\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	60.9106	2.32051
$690$	0	0
$691$	− 6.17780i	− 0.235015i	−0.993072	−	0.117507i	$-0.962510\pi$
$691$	− 6.17780i	− 0.235015i	0.993072	−	0.117507i	$-0.0374903\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.07785	0.0787043
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 33.9746i	− 1.28320i	−0.767038	−	0.641601i	$-0.778270\pi$
$701$	− 33.9746i	− 1.28320i	0.767038	−	0.641601i	$-0.221730\pi$
$702$	0	0
$703$	18.5963	0.701374
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.607181i	0.0228354i
$708$	0	0
$709$	− 22.3441i	− 0.839151i	−0.907720	−	0.419576i	$-0.862179\pi$
$709$	− 22.3441i	− 0.839151i	0.907720	−	0.419576i	$-0.137821\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	13.6485	0.511139
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−17.8427	−0.665422	−0.332711	−	0.943029i	$-0.607963\pi$
$719$	−17.8427	−0.665422	−0.332711	+	0.943029i	$0.607963\pi$
$720$	0	0
$721$	−0.143347	−0.00533851
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−23.9148	−0.886953	−0.443476	−	0.896286i	$-0.646255\pi$
$727$	−23.9148	−0.886953	−0.443476	+	0.896286i	$0.646255\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.04642i	0.0387034i
$732$	0	0
$733$	15.6789i	0.579112i	0.957161	+	0.289556i	$0.0935076\pi$
$733$	15.6789i	0.579112i	−0.957161	+	0.289556i	$0.906492\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17.5915	0.647992
$738$	0	0
$739$	− 22.3083i	− 0.820622i	−0.911946	−	0.410311i	$-0.865420\pi$
$739$	− 22.3083i	− 0.820622i	0.911946	−	0.410311i	$-0.134580\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9.78057	−0.358814	−0.179407	−	0.983775i	$-0.557418\pi$
$743$	−9.78057	−0.358814	−0.179407	+	0.983775i	$0.557418\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.840636i	0.0307162i
$750$	0	0
$751$	8.05399	0.293894	0.146947	−	0.989144i	$-0.453055\pi$
$751$	8.05399	0.293894	0.146947	+	0.989144i	$0.453055\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	28.4889i	1.03545i	0.855549	+	0.517723i	$0.173220\pi$
$757$	28.4889i	1.03545i	−0.855549	+	0.517723i	$0.826780\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21.5005	0.779393	0.389697	−	0.920943i	$-0.372580\pi$
$761$	21.5005	0.779393	0.389697	+	0.920943i	$0.372580\pi$
$762$	0	0
$763$	0.292482i	0.0105886i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−66.8821	−2.41497
$768$	0	0
$769$	23.5596	0.849580	0.424790	−	0.905292i	$-0.360348\pi$
$769$	23.5596	0.849580	0.424790	+	0.905292i	$0.360348\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.1655i	1.12094i	0.828173	+	0.560472i	$0.189380\pi$
$773$	31.1655i	1.12094i	−0.828173	+	0.560472i	$0.810620\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 56.4987i	− 2.02428i
$780$	0	0
$781$	27.4227i	0.981260i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.07812i	0.181015i	0.995896	+	0.0905077i	$0.0288490\pi$
$787$	5.07812i	0.181015i	−0.995896	+	0.0905077i	$0.971151\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.709168	0.0252151
$792$	0	0
$793$	−62.8995	−2.23363
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.43418i	0.0508012i	0.999677	+	0.0254006i	$0.00808613\pi$
$797$	1.43418i	0.0508012i	−0.999677	+	0.0254006i	$0.991914\pi$
$798$	0	0
$799$	1.45943	0.0516309
$800$	0	0
$801$	0	0
$802$	0	0
$803$	29.1860i	1.02995i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23.7153	−0.833787	−0.416894	−	0.908955i	$-0.636881\pi$
$809$	−23.7153	−0.833787	−0.416894	+	0.908955i	$0.636881\pi$
$810$	0	0
$811$	47.8394i	1.67987i	0.542689	+	0.839933i	$0.317406\pi$
$811$	47.8394i	1.67987i	−0.542689	+	0.839933i	$0.682594\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	28.4532	0.995451
$818$	0	0
$819$	0	0
$820$	0	0
$821$	51.9216i	1.81208i	0.423195	+	0.906038i	$0.360908\pi$
$821$	51.9216i	1.81208i	−0.423195	+	0.906038i	$0.639092\pi$
$822$	0	0
$823$	27.8542	0.970937	0.485469	−	0.874254i	$-0.338649\pi$
$823$	27.8542	0.970937	0.485469	+	0.874254i	$0.338649\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.9365i	1.52782i	0.645321	+	0.763912i	$0.276724\pi$
$827$	43.9365i	1.52782i	−0.645321	+	0.763912i	$0.723276\pi$
$828$	0	0
$829$	41.6898i	1.44795i	0.689827	+	0.723974i	$0.257686\pi$
$829$	41.6898i	1.44795i	−0.689827	+	0.723974i	$0.742314\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.78978	−0.0620121
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25.4733	−0.879437	−0.439719	−	0.898136i	$-0.644922\pi$
$839$	−25.4733	−0.879437	−0.439719	+	0.898136i	$0.644922\pi$
$840$	0	0
$841$	8.61336	0.297012
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.414324	−0.0142363
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 4.39722i	− 0.150735i
$852$	0	0
$853$	12.5366i	0.429245i	0.976697	+	0.214622i	$0.0688521\pi$
$853$	12.5366i	0.429245i	−0.976697	+	0.214622i	$0.931148\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.61409	0.0892956	0.0446478	−	0.999003i	$-0.485783\pi$
$857$	2.61409	0.0892956	0.0446478	+	0.999003i	$0.485783\pi$
$858$	0	0
$859$	− 31.8438i	− 1.08650i	−0.839573	−	0.543248i	$-0.817195\pi$
$859$	− 31.8438i	− 1.08650i	0.839573	−	0.543248i	$-0.182805\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22.3335	0.760241	0.380121	−	0.924937i	$-0.375882\pi$
$863$	22.3335	0.760241	0.380121	+	0.924937i	$0.375882\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	13.3037i	0.451298i
$870$	0	0
$871$	−38.3488	−1.29940
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	37.4408i	1.26429i	0.774852	+	0.632143i	$0.217824\pi$
$877$	37.4408i	1.26429i	−0.774852	+	0.632143i	$0.782176\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−53.1952	−1.79219	−0.896096	−	0.443860i	$-0.853609\pi$
$881$	−53.1952	−1.79219	−0.896096	+	0.443860i	$0.853609\pi$
$882$	0	0
$883$	36.0907i	1.21455i	0.794493	+	0.607274i	$0.207737\pi$
$883$	36.0907i	1.21455i	−0.794493	+	0.607274i	$0.792263\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	43.3018	1.45393	0.726966	−	0.686673i	$-0.240930\pi$
$887$	43.3018	1.45393	0.726966	+	0.686673i	$0.240930\pi$
$888$	0	0
$889$	−0.698192	−0.0234166
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 39.6832i	− 1.32795i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	37.4525i	1.24911i
$900$	0	0
$901$	− 2.95910i	− 0.0985820i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 8.75026i	− 0.290548i	−0.989391	−	0.145274i	$-0.953594\pi$
$907$	− 8.75026i	− 0.290548i	0.989391	−	0.145274i	$-0.0464063\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−13.7438	−0.455353	−0.227676	−	0.973737i	$-0.573113\pi$
$911$	−13.7438	−0.455353	−0.227676	+	0.973737i	$0.573113\pi$
$912$	0	0
$913$	−22.2513	−0.736411
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.866420i	0.0286117i
$918$	0	0
$919$	0.989347	0.0326355	0.0163178	−	0.999867i	$-0.494806\pi$
$919$	0.989347	0.0326355	0.0163178	+	0.999867i	$0.494806\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 59.7803i	− 1.96769i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.49434	0.278690	0.139345	−	0.990244i	$-0.455500\pi$
$929$	8.49434	0.278690	0.139345	+	0.990244i	$0.455500\pi$
$930$	0	0
$931$	48.6656i	1.59495i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.6912	−0.610615	−0.305308	−	0.952254i	$-0.598759\pi$
$937$	−18.6912	−0.610615	−0.305308	+	0.952254i	$0.598759\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.03170i	0.0988305i	0.998778	+	0.0494152i	$0.0157358\pi$
$941$	3.03170i	0.0988305i	−0.998778	+	0.0494152i	$0.984264\pi$
$942$	0	0
$943$	−13.3595	−0.435045
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 16.8327i	− 0.546990i	−0.961873	−	0.273495i	$-0.911820\pi$
$947$	− 16.8327i	− 0.546990i	0.961873	−	0.273495i	$-0.0881797\pi$
$948$	0	0
$949$	− 63.6243i	− 2.06533i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.73948	0.0563473	0.0281737	−	0.999603i	$-0.491031\pi$
$953$	1.73948	0.0563473	0.0281737	+	0.999603i	$0.491031\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.926677	0.0299240
$960$	0	0
$961$	37.8044	1.21950
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−17.3399	−0.557615	−0.278808	−	0.960347i	$-0.589939\pi$
$967$	−17.3399	−0.557615	−0.278808	+	0.960347i	$0.589939\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 45.4054i	− 1.45713i	−0.684977	−	0.728565i	$-0.740188\pi$
$971$	− 45.4054i	− 1.45713i	0.684977	−	0.728565i	$-0.259812\pi$
$972$	0	0
$973$	0.0172261i	0 0.000552243i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43.4336	1.38957	0.694783	−	0.719220i	$-0.255501\pi$
$977$	43.4336	1.38957	0.694783	+	0.719220i	$0.255501\pi$
$978$	0	0
$979$	28.7980i	0.920388i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	47.4465	1.51331	0.756655	−	0.653815i	$-0.226832\pi$
$983$	47.4465	1.51331	0.756655	+	0.653815i	$0.226832\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 6.72794i	− 0.213936i
$990$	0	0
$991$	28.8434	0.916242	0.458121	−	0.888890i	$-0.348523\pi$
$991$	28.8434	0.916242	0.458121	+	0.888890i	$0.348523\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33.1449i	1.04971i	0.851192	+	0.524855i	$0.175881\pi$
$997$	33.1449i	1.04971i	−0.851192	+	0.524855i	$0.824119\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.k.s.3601.3		8
3.2	odd	2		2400.2.k.e.1201.2		8
4.3	odd	2		1800.2.k.q.901.7		8
5.2	odd	4		7200.2.d.s.2449.4		8
5.3	odd	4		7200.2.d.t.2449.5		8
5.4	even	2		7200.2.k.r.3601.5		8
8.3	odd	2		1800.2.k.q.901.8		8
8.5	even	2	inner	7200.2.k.s.3601.4		8
12.11	even	2		600.2.k.e.301.2	yes	8
15.2	even	4		2400.2.d.g.49.4		8
15.8	even	4		2400.2.d.h.49.5		8
15.14	odd	2		2400.2.k.d.1201.7		8
20.3	even	4		1800.2.d.t.1549.3		8
20.7	even	4		1800.2.d.s.1549.6		8
20.19	odd	2		1800.2.k.t.901.2		8
24.5	odd	2		2400.2.k.e.1201.6		8
24.11	even	2		600.2.k.e.301.1	yes	8
40.3	even	4		1800.2.d.s.1549.5		8
40.13	odd	4		7200.2.d.s.2449.5		8
40.19	odd	2		1800.2.k.t.901.1		8
40.27	even	4		1800.2.d.t.1549.4		8
40.29	even	2		7200.2.k.r.3601.6		8
40.37	odd	4		7200.2.d.t.2449.4		8
60.23	odd	4		600.2.d.g.349.6		8
60.47	odd	4		600.2.d.h.349.3		8
60.59	even	2		600.2.k.d.301.7	✓	8
120.29	odd	2		2400.2.k.d.1201.3		8
120.53	even	4		2400.2.d.g.49.5		8
120.59	even	2		600.2.k.d.301.8	yes	8
120.77	even	4		2400.2.d.h.49.4		8
120.83	odd	4		600.2.d.h.349.4		8
120.107	odd	4		600.2.d.g.349.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.d.g.349.5		8	120.107	odd	4
600.2.d.g.349.6		8	60.23	odd	4
600.2.d.h.349.3		8	60.47	odd	4
600.2.d.h.349.4		8	120.83	odd	4
600.2.k.d.301.7	✓	8	60.59	even	2
600.2.k.d.301.8	yes	8	120.59	even	2
600.2.k.e.301.1	yes	8	24.11	even	2
600.2.k.e.301.2	yes	8	12.11	even	2
1800.2.d.s.1549.5		8	40.3	even	4
1800.2.d.s.1549.6		8	20.7	even	4
1800.2.d.t.1549.3		8	20.3	even	4
1800.2.d.t.1549.4		8	40.27	even	4
1800.2.k.q.901.7		8	4.3	odd	2
1800.2.k.q.901.8		8	8.3	odd	2
1800.2.k.t.901.1		8	40.19	odd	2
1800.2.k.t.901.2		8	20.19	odd	2
2400.2.d.g.49.4		8	15.2	even	4
2400.2.d.g.49.5		8	120.53	even	4
2400.2.d.h.49.4		8	120.77	even	4
2400.2.d.h.49.5		8	15.8	even	4
2400.2.k.d.1201.3		8	120.29	odd	2
2400.2.k.d.1201.7		8	15.14	odd	2
2400.2.k.e.1201.2		8	3.2	odd	2
2400.2.k.e.1201.6		8	24.5	odd	2
7200.2.d.s.2449.4		8	5.2	odd	4
7200.2.d.s.2449.5		8	40.13	odd	4
7200.2.d.t.2449.4		8	40.37	odd	4
7200.2.d.t.2449.5		8	5.3	odd	4
7200.2.k.r.3601.5		8	5.4	even	2
7200.2.k.r.3601.6		8	40.29	even	2
7200.2.k.s.3601.3		8	1.1	even	1	trivial
7200.2.k.s.3601.4		8	8.5	even	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.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.0802864 q^{7} +O(q^{10})\)	\(q-0.0802864 q^{7} -2.41649i q^{11} +5.26785i q^{13} +0.255918 q^{17} -6.95864i q^{19} -1.64542 q^{23} -4.51516i q^{29} -8.29484 q^{31} +2.67241i q^{37} +8.11921 q^{41} +4.08890i q^{43} +5.70272 q^{47} -6.99355 q^{49} -11.5627i q^{53} +12.6963i q^{59} +11.9403i q^{61} +7.27979i q^{67} -11.3481 q^{71} -12.0779 q^{73} +0.194011i q^{77} -5.50539 q^{79} -9.20811i q^{83} -11.9173 q^{89} -0.422937i q^{91} +8.50539 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7}+O(q^{10}) \) 8 * q + 8 * q^7	\( 8 q + 8 q^{7} - 8 q^{23} - 8 q^{31} - 40 q^{71} + 16 q^{73} + 16 q^{79} + 8 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 - 8 * q^23 - 8 * q^31 - 40 * q^71 + 16 * q^73 + 16 * q^79 + 8 * q^97

Introduction

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