LMFDB - Embedded newform 3600.3.c.d.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(449,3600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3600.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3600.c (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:	$98.0928951697$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3600.449
Dual form			3600.3.c.d.449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{7} +O(q^{10})$	$q-1.00000i q^{7} -4.24264i q^{11} -7.00000i q^{13} -4.24264 q^{17} -7.00000 q^{19} -29.6985 q^{23} +29.6985i q^{29} -17.0000 q^{31} +16.0000i q^{37} +50.9117i q^{41} +55.0000i q^{43} +46.6690 q^{47} +48.0000 q^{49} +84.8528 q^{53} -55.1543i q^{59} +65.0000 q^{61} -49.0000i q^{67} -50.9117i q^{71} -88.0000i q^{73} -4.24264 q^{77} -40.0000 q^{79} -156.978 q^{83} +101.823i q^{89} -7.00000 q^{91} -41.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 28 q^{19} - 68 q^{31} + 192 q^{49} + 260 q^{61} - 160 q^{79} - 28 q^{91}+O(q^{100}) $ 4 * q - 28 * q^19 - 68 * q^31 + 192 * q^49 + 260 * q^61 - 160 * q^79 - 28 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$7$	− 1.00000i	− 0.142857i	−0.997446	−	0.0714286i	$-0.977244\pi$
$7$	− 1.00000i	− 0.142857i	0.997446	−	0.0714286i	$-0.0227558\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.24264i	− 0.385695i	−0.981229	−	0.192847i	$-0.938228\pi$
$11$	− 4.24264i	− 0.385695i	0.981229	−	0.192847i	$-0.0617722\pi$
$12$	0	0
$13$	− 7.00000i	− 0.538462i	−0.963076	−	0.269231i	$-0.913231\pi$
$13$	− 7.00000i	− 0.538462i	0.963076	−	0.269231i	$-0.0867694\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.24264	−0.249567	−0.124784	−	0.992184i	$-0.539824\pi$
$17$	−4.24264	−0.249567	−0.124784	+	0.992184i	$0.539824\pi$
$18$	0	0
$19$	−7.00000	−0.368421	−0.184211	−	0.982887i	$-0.558973\pi$
$19$	−7.00000	−0.368421	−0.184211	+	0.982887i	$0.558973\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−29.6985	−1.29124	−0.645619	−	0.763659i	$-0.723401\pi$
$23$	−29.6985	−1.29124	−0.645619	+	0.763659i	$0.723401\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	29.6985i	1.02409i	0.858960	+	0.512043i	$0.171111\pi$
$29$	29.6985i	1.02409i	−0.858960	+	0.512043i	$0.828889\pi$
$30$	0	0
$31$	−17.0000	−0.548387	−0.274194	−	0.961675i	$-0.588411\pi$
$31$	−17.0000	−0.548387	−0.274194	+	0.961675i	$0.588411\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	16.0000i	0.432432i	0.976346	+	0.216216i	$0.0693716\pi$
$37$	16.0000i	0.432432i	−0.976346	+	0.216216i	$0.930628\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	50.9117i	1.24175i	0.783910	+	0.620874i	$0.213222\pi$
$41$	50.9117i	1.24175i	−0.783910	+	0.620874i	$0.786778\pi$
$42$	0	0
$43$	55.0000i	1.27907i	0.768762	+	0.639535i	$0.220873\pi$
$43$	55.0000i	1.27907i	−0.768762	+	0.639535i	$0.779127\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	46.6690	0.992958	0.496479	−	0.868049i	$-0.334626\pi$
$47$	46.6690	0.992958	0.496479	+	0.868049i	$0.334626\pi$
$48$	0	0
$49$	48.0000	0.979592
$50$	0	0
$51$	0	0
$52$	0	0
$53$	84.8528	1.60100	0.800498	−	0.599335i	$-0.204568\pi$
$53$	84.8528	1.60100	0.800498	+	0.599335i	$0.204568\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 55.1543i	− 0.934819i	−0.884041	−	0.467410i	$-0.845187\pi$
$59$	− 55.1543i	− 0.934819i	0.884041	−	0.467410i	$-0.154813\pi$
$60$	0	0
$61$	65.0000	1.06557	0.532787	−	0.846249i	$-0.321145\pi$
$61$	65.0000	1.06557	0.532787	+	0.846249i	$0.321145\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 49.0000i	− 0.731343i	−0.930744	−	0.365672i	$-0.880839\pi$
$67$	− 49.0000i	− 0.731343i	0.930744	−	0.365672i	$-0.119161\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 50.9117i	− 0.717066i	−0.933517	−	0.358533i	$-0.883277\pi$
$71$	− 50.9117i	− 0.717066i	0.933517	−	0.358533i	$-0.116723\pi$
$72$	0	0
$73$	− 88.0000i	− 1.20548i	−0.797938	−	0.602740i	$-0.794076\pi$
$73$	− 88.0000i	− 1.20548i	0.797938	−	0.602740i	$-0.205924\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.24264	−0.0550992
$78$	0	0
$79$	−40.0000	−0.506329	−0.253165	−	0.967423i	$-0.581471\pi$
$79$	−40.0000	−0.506329	−0.253165	+	0.967423i	$0.581471\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−156.978	−1.89130	−0.945649	−	0.325190i	$-0.894572\pi$
$83$	−156.978	−1.89130	−0.945649	+	0.325190i	$0.894572\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	101.823i	1.14408i	0.820225	+	0.572041i	$0.193848\pi$
$89$	101.823i	1.14408i	−0.820225	+	0.572041i	$0.806152\pi$
$90$	0	0
$91$	−7.00000	−0.0769231
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 41.0000i	− 0.422680i	−0.977413	−	0.211340i	$-0.932217\pi$
$97$	− 41.0000i	− 0.422680i	0.977413	−	0.211340i	$-0.0677828\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	118.794i	1.17618i	0.808796	+	0.588089i	$0.200119\pi$
$101$	118.794i	1.17618i	−0.808796	+	0.588089i	$0.799881\pi$
$102$	0	0
$103$	82.0000i	0.796117i	0.917360	+	0.398058i	$0.130316\pi$
$103$	82.0000i	0.796117i	−0.917360	+	0.398058i	$0.869684\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	118.794	1.11022	0.555112	−	0.831776i	$-0.312675\pi$
$107$	118.794	1.11022	0.555112	+	0.831776i	$0.312675\pi$
$108$	0	0
$109$	49.0000	0.449541	0.224771	−	0.974412i	$-0.427837\pi$
$109$	49.0000	0.449541	0.224771	+	0.974412i	$0.427837\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	67.8823	0.600728	0.300364	−	0.953825i	$-0.402892\pi$
$113$	67.8823	0.600728	0.300364	+	0.953825i	$0.402892\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.24264i	0.0356524i
$120$	0	0
$121$	103.000	0.851240
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 160.000i	− 1.25984i	−0.776659	−	0.629921i	$-0.783087\pi$
$127$	− 160.000i	− 1.25984i	0.776659	−	0.629921i	$-0.216913\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.9706i	0.129546i	0.997900	+	0.0647731i	$0.0206324\pi$
$131$	16.9706i	0.129546i	−0.997900	+	0.0647731i	$0.979368\pi$
$132$	0	0
$133$	7.00000i	0.0526316i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	89.0955	0.650332	0.325166	−	0.945657i	$-0.394580\pi$
$137$	89.0955	0.650332	0.325166	+	0.945657i	$0.394580\pi$
$138$	0	0
$139$	8.00000	0.0575540	0.0287770	−	0.999586i	$-0.490839\pi$
$139$	8.00000	0.0575540	0.0287770	+	0.999586i	$0.490839\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−29.6985	−0.207682
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	89.0955i	0.597956i	0.954260	+	0.298978i	$0.0966457\pi$
$149$	89.0955i	0.597956i	−0.954260	+	0.298978i	$0.903354\pi$
$150$	0	0
$151$	25.0000	0.165563	0.0827815	−	0.996568i	$-0.473620\pi$
$151$	25.0000	0.165563	0.0827815	+	0.996568i	$0.473620\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 137.000i	− 0.872611i	−0.899798	−	0.436306i	$-0.856287\pi$
$157$	− 137.000i	− 0.872611i	0.899798	−	0.436306i	$-0.143713\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	29.6985i	0.184463i
$162$	0	0
$163$	79.0000i	0.484663i	0.970194	+	0.242331i	$0.0779121\pi$
$163$	79.0000i	0.484663i	−0.970194	+	0.242331i	$0.922088\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	118.794	0.711341	0.355670	−	0.934611i	$-0.384253\pi$
$167$	118.794	0.711341	0.355670	+	0.934611i	$0.384253\pi$
$168$	0	0
$169$	120.000	0.710059
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−131.522	−0.760242	−0.380121	−	0.924937i	$-0.624118\pi$
$173$	−131.522	−0.760242	−0.380121	+	0.924937i	$0.624118\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	148.492i	0.829567i	0.909920	+	0.414783i	$0.136143\pi$
$179$	148.492i	0.829567i	−0.909920	+	0.414783i	$0.863857\pi$
$180$	0	0
$181$	−97.0000	−0.535912	−0.267956	−	0.963431i	$-0.586348\pi$
$181$	−97.0000	−0.535912	−0.267956	+	0.963431i	$0.586348\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	18.0000i	0.0962567i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	207.889i	1.08843i	0.838947	+	0.544213i	$0.183172\pi$
$191$	207.889i	1.08843i	−0.838947	+	0.544213i	$0.816828\pi$
$192$	0	0
$193$	185.000i	0.958549i	0.877665	+	0.479275i	$0.159100\pi$
$193$	185.000i	0.958549i	−0.877665	+	0.479275i	$0.840900\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	258.801	1.31371	0.656856	−	0.754016i	$-0.271886\pi$
$197$	258.801	1.31371	0.656856	+	0.754016i	$0.271886\pi$
$198$	0	0
$199$	311.000	1.56281	0.781407	−	0.624022i	$-0.214502\pi$
$199$	311.000	1.56281	0.781407	+	0.624022i	$0.214502\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	29.6985	0.146298
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	29.6985i	0.142098i
$210$	0	0
$211$	97.0000	0.459716	0.229858	−	0.973224i	$-0.426174\pi$
$211$	97.0000	0.459716	0.229858	+	0.973224i	$0.426174\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	17.0000i	0.0783410i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	29.6985i	0.134382i
$222$	0	0
$223$	− 239.000i	− 1.07175i	−0.844298	−	0.535874i	$-0.819982\pi$
$223$	− 239.000i	− 1.07175i	0.844298	−	0.535874i	$-0.180018\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	152.735	0.672842	0.336421	−	0.941712i	$-0.390784\pi$
$227$	152.735	0.672842	0.336421	+	0.941712i	$0.390784\pi$
$228$	0	0
$229$	−17.0000	−0.0742358	−0.0371179	−	0.999311i	$-0.511818\pi$
$229$	−17.0000	−0.0742358	−0.0371179	+	0.999311i	$0.511818\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−356.382	−1.52954	−0.764768	−	0.644306i	$-0.777146\pi$
$233$	−356.382	−1.52954	−0.764768	+	0.644306i	$0.777146\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	424.264i	1.77516i	0.460650	+	0.887582i	$0.347616\pi$
$239$	424.264i	1.77516i	−0.460650	+	0.887582i	$0.652384\pi$
$240$	0	0
$241$	95.0000	0.394191	0.197095	−	0.980384i	$-0.436849\pi$
$241$	95.0000	0.394191	0.197095	+	0.980384i	$0.436849\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	49.0000i	0.198381i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	237.588i	0.946565i	0.880911	+	0.473283i	$0.156931\pi$
$251$	237.588i	0.946565i	−0.880911	+	0.473283i	$0.843069\pi$
$252$	0	0
$253$	126.000i	0.498024i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	339.411	1.32067	0.660333	−	0.750973i	$-0.270415\pi$
$257$	339.411	1.32067	0.660333	+	0.750973i	$0.270415\pi$
$258$	0	0
$259$	16.0000	0.0617761
$260$	0	0
$261$	0	0
$262$	0	0
$263$	322.441	1.22601	0.613005	−	0.790079i	$-0.289960\pi$
$263$	322.441	1.22601	0.613005	+	0.790079i	$0.289960\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 29.6985i	− 0.110403i	−0.998475	−	0.0552016i	$-0.982420\pi$
$269$	− 29.6985i	− 0.110403i	0.998475	−	0.0552016i	$-0.0175802\pi$
$270$	0	0
$271$	448.000	1.65314	0.826568	−	0.562836i	$-0.190290\pi$
$271$	448.000	1.65314	0.826568	+	0.562836i	$0.190290\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 455.000i	− 1.64260i	−0.570497	−	0.821300i	$-0.693249\pi$
$277$	− 455.000i	− 1.64260i	0.570497	−	0.821300i	$-0.306751\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 97.5807i	− 0.347262i	−0.984811	−	0.173631i	$-0.944450\pi$
$281$	− 97.5807i	− 0.347262i	0.984811	−	0.173631i	$-0.0555501\pi$
$282$	0	0
$283$	− 71.0000i	− 0.250883i	−0.992101	−	0.125442i	$-0.959965\pi$
$283$	− 71.0000i	− 0.250883i	0.992101	−	0.125442i	$-0.0400348\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	50.9117	0.177393
$288$	0	0
$289$	−271.000	−0.937716
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−394.566	−1.34664	−0.673320	−	0.739351i	$-0.735132\pi$
$293$	−394.566	−1.34664	−0.673320	+	0.739351i	$0.735132\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	207.889i	0.695282i
$300$	0	0
$301$	55.0000	0.182724
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 559.000i	− 1.82085i	−0.413678	−	0.910423i	$-0.635756\pi$
$307$	− 559.000i	− 1.82085i	0.413678	−	0.910423i	$-0.364244\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	564.271i	1.81438i	0.420725	+	0.907188i	$0.361776\pi$
$311$	564.271i	1.81438i	−0.420725	+	0.907188i	$0.638224\pi$
$312$	0	0
$313$	383.000i	1.22364i	0.790996	+	0.611821i	$0.209563\pi$
$313$	383.000i	1.22364i	−0.790996	+	0.611821i	$0.790437\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	373.352	1.17777	0.588884	−	0.808218i	$-0.299568\pi$
$317$	373.352	1.17777	0.588884	+	0.808218i	$0.299568\pi$
$318$	0	0
$319$	126.000	0.394984
$320$	0	0
$321$	0	0
$322$	0	0
$323$	29.6985	0.0919458
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 46.6690i	− 0.141851i
$330$	0	0
$331$	40.0000	0.120846	0.0604230	−	0.998173i	$-0.480755\pi$
$331$	40.0000	0.120846	0.0604230	+	0.998173i	$0.480755\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 449.000i	− 1.33234i	−0.745798	−	0.666172i	$-0.767932\pi$
$337$	− 449.000i	− 1.33234i	0.745798	−	0.666172i	$-0.232068\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	72.1249i	0.211510i
$342$	0	0
$343$	− 97.0000i	− 0.282799i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	292.742	0.843637	0.421819	−	0.906680i	$-0.361392\pi$
$347$	292.742	0.843637	0.421819	+	0.906680i	$0.361392\pi$
$348$	0	0
$349$	184.000	0.527221	0.263610	−	0.964629i	$-0.415087\pi$
$349$	184.000	0.527221	0.263610	+	0.964629i	$0.415087\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−267.286	−0.757185	−0.378593	−	0.925563i	$-0.623592\pi$
$353$	−267.286	−0.757185	−0.378593	+	0.925563i	$0.623592\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	220.617i	0.614533i	0.951624	+	0.307266i	$0.0994143\pi$
$359$	220.617i	0.614533i	−0.951624	+	0.307266i	$0.900586\pi$
$360$	0	0
$361$	−312.000	−0.864266
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	263.000i	0.716621i	0.933602	+	0.358311i	$0.116647\pi$
$367$	263.000i	0.716621i	−0.933602	+	0.358311i	$0.883353\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 84.8528i	− 0.228714i
$372$	0	0
$373$	305.000i	0.817694i	0.912603	+	0.408847i	$0.134069\pi$
$373$	305.000i	0.817694i	−0.912603	+	0.408847i	$0.865931\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	207.889	0.551431
$378$	0	0
$379$	497.000	1.31135	0.655673	−	0.755045i	$-0.272385\pi$
$379$	497.000	1.31135	0.655673	+	0.755045i	$0.272385\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	237.588	0.620334	0.310167	−	0.950682i	$-0.399615\pi$
$383$	237.588	0.620334	0.310167	+	0.950682i	$0.399615\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 84.8528i	− 0.218131i	−0.994035	−	0.109065i	$-0.965214\pi$
$389$	− 84.8528i	− 0.218131i	0.994035	−	0.109065i	$-0.0347858\pi$
$390$	0	0
$391$	126.000	0.322251
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	439.000i	1.10579i	0.833250	+	0.552897i	$0.186478\pi$
$397$	439.000i	1.10579i	−0.833250	+	0.552897i	$0.813522\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	254.558i	0.634809i	0.948290	+	0.317405i	$0.102811\pi$
$401$	254.558i	0.634809i	−0.948290	+	0.317405i	$0.897189\pi$
$402$	0	0
$403$	119.000i	0.295285i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	67.8823	0.166787
$408$	0	0
$409$	343.000	0.838631	0.419315	−	0.907841i	$-0.362270\pi$
$409$	343.000	0.838631	0.419315	+	0.907841i	$0.362270\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−55.1543	−0.133546
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 118.794i	− 0.283518i	−0.989901	−	0.141759i	$-0.954724\pi$
$419$	− 118.794i	− 0.283518i	0.989901	−	0.141759i	$-0.0452758\pi$
$420$	0	0
$421$	608.000	1.44418	0.722090	−	0.691799i	$-0.243182\pi$
$421$	608.000	1.44418	0.722090	+	0.691799i	$0.243182\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 65.0000i	− 0.152225i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	445.477i	1.03359i	0.856109	+	0.516795i	$0.172875\pi$
$431$	445.477i	1.03359i	−0.856109	+	0.516795i	$0.827125\pi$
$432$	0	0
$433$	− 337.000i	− 0.778291i	−0.921176	−	0.389145i	$-0.872770\pi$
$433$	− 337.000i	− 0.778291i	0.921176	−	0.389145i	$-0.127230\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	207.889	0.475719
$438$	0	0
$439$	65.0000	0.148064	0.0740319	−	0.997256i	$-0.476413\pi$
$439$	65.0000	0.148064	0.0740319	+	0.997256i	$0.476413\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−576.999	−1.30248	−0.651241	−	0.758871i	$-0.725751\pi$
$443$	−576.999	−1.30248	−0.651241	+	0.758871i	$0.725751\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	237.588i	0.529149i	0.964365	+	0.264574i	$0.0852315\pi$
$449$	237.588i	0.529149i	−0.964365	+	0.264574i	$0.914769\pi$
$450$	0	0
$451$	216.000	0.478936
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 590.000i	− 1.29103i	−0.763748	−	0.645514i	$-0.776643\pi$
$457$	− 590.000i	− 1.29103i	0.763748	−	0.645514i	$-0.223357\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 50.9117i	− 0.110438i	−0.998474	−	0.0552188i	$-0.982414\pi$
$461$	− 50.9117i	− 0.110438i	0.998474	−	0.0552188i	$-0.0175856\pi$
$462$	0	0
$463$	64.0000i	0.138229i	0.997609	+	0.0691145i	$0.0220174\pi$
$463$	64.0000i	0.138229i	−0.997609	+	0.0691145i	$0.977983\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	343.654	0.735876	0.367938	−	0.929850i	$-0.380064\pi$
$467$	343.654	0.735876	0.367938	+	0.929850i	$0.380064\pi$
$468$	0	0
$469$	−49.0000	−0.104478
$470$	0	0
$471$	0	0
$472$	0	0
$473$	233.345	0.493330
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	326.683i	0.682011i	0.940061	+	0.341006i	$0.110768\pi$
$479$	326.683i	0.682011i	−0.940061	+	0.341006i	$0.889232\pi$
$480$	0	0
$481$	112.000	0.232848
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 823.000i	− 1.68994i	−0.534815	−	0.844969i	$-0.679619\pi$
$487$	− 823.000i	− 1.68994i	0.534815	−	0.844969i	$-0.320381\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 475.176i	− 0.967771i	−0.875131	−	0.483886i	$-0.839225\pi$
$491$	− 475.176i	− 0.967771i	0.875131	−	0.483886i	$-0.160775\pi$
$492$	0	0
$493$	− 126.000i	− 0.255578i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−50.9117	−0.102438
$498$	0	0
$499$	−553.000	−1.10822	−0.554108	−	0.832445i	$-0.686941\pi$
$499$	−553.000	−1.10822	−0.554108	+	0.832445i	$0.686941\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−118.794	−0.236171	−0.118085	−	0.993003i	$-0.537676\pi$
$503$	−118.794	−0.236171	−0.118085	+	0.993003i	$0.537676\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	725.492i	1.42533i	0.701506	+	0.712664i	$0.252511\pi$
$509$	725.492i	1.42533i	−0.701506	+	0.712664i	$0.747489\pi$
$510$	0	0
$511$	−88.0000	−0.172211
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 198.000i	− 0.382979i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	649.124i	1.24592i	0.782254	+	0.622960i	$0.214070\pi$
$521$	649.124i	1.24592i	−0.782254	+	0.622960i	$0.785930\pi$
$522$	0	0
$523$	775.000i	1.48184i	0.671596	+	0.740918i	$0.265609\pi$
$523$	775.000i	1.48184i	−0.671596	+	0.740918i	$0.734391\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	72.1249	0.136859
$528$	0	0
$529$	353.000	0.667297
$530$	0	0
$531$	0	0
$532$	0	0
$533$	356.382	0.668634
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 203.647i	− 0.377823i
$540$	0	0
$541$	−175.000	−0.323475	−0.161738	−	0.986834i	$-0.551710\pi$
$541$	−175.000	−0.323475	−0.161738	+	0.986834i	$0.551710\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	266.000i	0.486289i	0.969990	+	0.243144i	$0.0781789\pi$
$547$	266.000i	0.486289i	−0.969990	+	0.243144i	$0.921821\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 207.889i	− 0.377295i
$552$	0	0
$553$	40.0000i	0.0723327i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−292.742	−0.525569	−0.262785	−	0.964854i	$-0.584641\pi$
$557$	−292.742	−0.525569	−0.262785	+	0.964854i	$0.584641\pi$
$558$	0	0
$559$	385.000	0.688730
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1013.99	−1.80105	−0.900525	−	0.434804i	$-0.856818\pi$
$563$	−1013.99	−1.80105	−0.900525	+	0.434804i	$0.856818\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	683.065i	1.20047i	0.799825	+	0.600233i	$0.204925\pi$
$569$	683.065i	1.20047i	−0.799825	+	0.600233i	$0.795075\pi$
$570$	0	0
$571$	−1001.00	−1.75306	−0.876532	−	0.481343i	$-0.840149\pi$
$571$	−1001.00	−1.75306	−0.876532	+	0.481343i	$0.840149\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1081.00i	1.87348i	0.350021	+	0.936742i	$0.386174\pi$
$577$	1081.00i	1.87348i	−0.350021	+	0.936742i	$0.613826\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	156.978i	0.270185i
$582$	0	0
$583$	− 360.000i	− 0.617496i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	661.852	1.12752	0.563758	−	0.825940i	$-0.309355\pi$
$587$	661.852	1.12752	0.563758	+	0.825940i	$0.309355\pi$
$588$	0	0
$589$	119.000	0.202037
$590$	0	0
$591$	0	0
$592$	0	0
$593$	593.970	1.00164	0.500818	−	0.865553i	$-0.333033\pi$
$593$	593.970	1.00164	0.500818	+	0.865553i	$0.333033\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	665.000	1.10649	0.553245	−	0.833019i	$-0.313389\pi$
$601$	665.000	1.10649	0.553245	+	0.833019i	$0.313389\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	296.000i	0.487644i	0.969820	+	0.243822i	$0.0784013\pi$
$607$	296.000i	0.487644i	−0.969820	+	0.243822i	$0.921599\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 326.683i	− 0.534670i
$612$	0	0
$613$	− 952.000i	− 1.55302i	−0.630106	−	0.776509i	$-0.716989\pi$
$613$	− 952.000i	− 1.55302i	0.630106	−	0.776509i	$-0.283011\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	50.9117	0.0825149	0.0412574	−	0.999149i	$-0.486864\pi$
$617$	50.9117	0.0825149	0.0412574	+	0.999149i	$0.486864\pi$
$618$	0	0
$619$	−1033.00	−1.66882	−0.834410	−	0.551144i	$-0.814192\pi$
$619$	−1033.00	−1.66882	−0.834410	+	0.551144i	$0.814192\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	101.823	0.163440
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 67.8823i	− 0.107921i
$630$	0	0
$631$	799.000	1.26624	0.633122	−	0.774052i	$-0.281773\pi$
$631$	799.000	1.26624	0.633122	+	0.774052i	$0.281773\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 336.000i	− 0.527473i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	203.647i	0.317702i	0.987303	+	0.158851i	$0.0507789\pi$
$641$	203.647i	0.317702i	−0.987303	+	0.158851i	$0.949221\pi$
$642$	0	0
$643$	1000.00i	1.55521i	0.628753	+	0.777605i	$0.283566\pi$
$643$	1000.00i	1.55521i	−0.628753	+	0.777605i	$0.716434\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.9411	−0.0524592	−0.0262296	−	0.999656i	$-0.508350\pi$
$647$	−33.9411	−0.0524592	−0.0262296	+	0.999656i	$0.508350\pi$
$648$	0	0
$649$	−234.000	−0.360555
$650$	0	0
$651$	0	0
$652$	0	0
$653$	462.448	0.708190	0.354095	−	0.935210i	$-0.384789\pi$
$653$	462.448	0.708190	0.354095	+	0.935210i	$0.384789\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1238.85i	− 1.87990i	−0.341318	−	0.939948i	$-0.610873\pi$
$659$	− 1238.85i	− 1.87990i	0.341318	−	0.939948i	$-0.389127\pi$
$660$	0	0
$661$	−136.000	−0.205749	−0.102874	−	0.994694i	$-0.532804\pi$
$661$	−136.000	−0.205749	−0.102874	+	0.994694i	$0.532804\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 882.000i	− 1.32234i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 275.772i	− 0.410986i
$672$	0	0
$673$	− 328.000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$673$	− 328.000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29.6985	−0.0438678	−0.0219339	−	0.999759i	$-0.506982\pi$
$677$	−29.6985	−0.0438678	−0.0219339	+	0.999759i	$0.506982\pi$
$678$	0	0
$679$	−41.0000	−0.0603829
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−984.293	−1.44113	−0.720566	−	0.693387i	$-0.756118\pi$
$683$	−984.293	−1.44113	−0.720566	+	0.693387i	$0.756118\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 593.970i	− 0.862075i
$690$	0	0
$691$	−842.000	−1.21852	−0.609262	−	0.792969i	$-0.708534\pi$
$691$	−842.000	−1.21852	−0.609262	+	0.792969i	$0.708534\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 216.000i	− 0.309900i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	335.169i	0.478129i	0.971004	+	0.239065i	$0.0768408\pi$
$701$	335.169i	0.478129i	−0.971004	+	0.239065i	$0.923159\pi$
$702$	0	0
$703$	− 112.000i	− 0.159317i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	118.794	0.168025
$708$	0	0
$709$	985.000	1.38928	0.694640	−	0.719357i	$-0.255564\pi$
$709$	985.000	1.38928	0.694640	+	0.719357i	$0.255564\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	504.874	0.708099
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 267.286i	− 0.371747i	−0.982574	−	0.185874i	$-0.940488\pi$
$719$	− 267.286i	− 0.371747i	0.982574	−	0.185874i	$-0.0595115\pi$
$720$	0	0
$721$	82.0000	0.113731
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	623.000i	0.856946i	0.903555	+	0.428473i	$0.140948\pi$
$727$	623.000i	0.856946i	−0.903555	+	0.428473i	$0.859052\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 233.345i	− 0.319214i
$732$	0	0
$733$	296.000i	0.403820i	0.979404	+	0.201910i	$0.0647148\pi$
$733$	296.000i	0.403820i	−0.979404	+	0.201910i	$0.935285\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−207.889	−0.282075
$738$	0	0
$739$	704.000	0.952639	0.476319	−	0.879272i	$-0.341971\pi$
$739$	704.000	0.952639	0.476319	+	0.879272i	$0.341971\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	712.764	0.959305	0.479653	−	0.877459i	$-0.340763\pi$
$743$	712.764	0.959305	0.479653	+	0.877459i	$0.340763\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 118.794i	− 0.158603i
$750$	0	0
$751$	−656.000	−0.873502	−0.436751	−	0.899582i	$-0.643871\pi$
$751$	−656.000	−0.873502	−0.436751	+	0.899582i	$0.643871\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	937.000i	1.23778i	0.785477	+	0.618890i	$0.212417\pi$
$757$	937.000i	1.23778i	−0.785477	+	0.618890i	$0.787583\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 475.176i	− 0.624410i	−0.950015	−	0.312205i	$-0.898932\pi$
$761$	− 475.176i	− 0.624410i	0.950015	−	0.312205i	$-0.101068\pi$
$762$	0	0
$763$	− 49.0000i	− 0.0642202i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−386.080	−0.503364
$768$	0	0
$769$	−281.000	−0.365410	−0.182705	−	0.983168i	$-0.558485\pi$
$769$	−281.000	−0.365410	−0.182705	+	0.983168i	$0.558485\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	118.794	0.153679	0.0768395	−	0.997043i	$-0.475517\pi$
$773$	118.794	0.153679	0.0768395	+	0.997043i	$0.475517\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 356.382i	− 0.457486i
$780$	0	0
$781$	−216.000	−0.276569
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	95.0000i	0.120712i	0.998177	+	0.0603558i	$0.0192235\pi$
$787$	95.0000i	0.120712i	−0.998177	+	0.0603558i	$0.980776\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 67.8823i	− 0.0858183i
$792$	0	0
$793$	− 455.000i	− 0.573770i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−950.352	−1.19241	−0.596205	−	0.802832i	$-0.703326\pi$
$797$	−950.352	−1.19241	−0.596205	+	0.802832i	$0.703326\pi$
$798$	0	0
$799$	−198.000	−0.247810
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−373.352	−0.464947
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	356.382i	0.440521i	0.975441	+	0.220261i	$0.0706908\pi$
$809$	356.382i	0.440521i	−0.975441	+	0.220261i	$0.929309\pi$
$810$	0	0
$811$	1177.00	1.45129	0.725647	−	0.688067i	$-0.241540\pi$
$811$	1177.00	1.45129	0.725647	+	0.688067i	$0.241540\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 385.000i	− 0.471236i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	564.271i	0.687297i	0.939098	+	0.343649i	$0.111663\pi$
$821$	564.271i	0.687297i	−0.939098	+	0.343649i	$0.888337\pi$
$822$	0	0
$823$	− 641.000i	− 0.778858i	−0.921057	−	0.389429i	$-0.872672\pi$
$823$	− 641.000i	− 0.778858i	0.921057	−	0.389429i	$-0.127328\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−526.087	−0.636140	−0.318070	−	0.948067i	$-0.603035\pi$
$827$	−526.087	−0.636140	−0.318070	+	0.948067i	$0.603035\pi$
$828$	0	0
$829$	−1046.00	−1.26176	−0.630881	−	0.775880i	$-0.717306\pi$
$829$	−1046.00	−1.26176	−0.630881	+	0.775880i	$0.717306\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−203.647	−0.244474
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1387.34i	1.65357i	0.562520	+	0.826784i	$0.309832\pi$
$839$	1387.34i	1.65357i	−0.562520	+	0.826784i	$0.690168\pi$
$840$	0	0
$841$	−41.0000	−0.0487515
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 103.000i	− 0.121606i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 475.176i	− 0.558373i
$852$	0	0
$853$	335.000i	0.392732i	0.980531	+	0.196366i	$0.0629140\pi$
$853$	335.000i	0.392732i	−0.980531	+	0.196366i	$0.937086\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−237.588	−0.277232	−0.138616	−	0.990346i	$-0.544265\pi$
$857$	−237.588	−0.277232	−0.138616	+	0.990346i	$0.544265\pi$
$858$	0	0
$859$	−1414.00	−1.64610	−0.823050	−	0.567969i	$-0.807729\pi$
$859$	−1414.00	−1.64610	−0.823050	+	0.567969i	$0.807729\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1561.29	1.80914	0.904572	−	0.426320i	$-0.140190\pi$
$863$	1561.29	1.80914	0.904572	+	0.426320i	$0.140190\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	169.706i	0.195288i
$870$	0	0
$871$	−343.000	−0.393800
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 425.000i	− 0.484607i	−0.970201	−	0.242303i	$-0.922097\pi$
$877$	− 425.000i	− 0.484607i	0.970201	−	0.242303i	$-0.0779029\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1731.00i	− 1.96481i	−0.186764	−	0.982405i	$-0.559800\pi$
$881$	− 1731.00i	− 1.96481i	0.186764	−	0.982405i	$-0.440200\pi$
$882$	0	0
$883$	385.000i	0.436014i	0.975947	+	0.218007i	$0.0699555\pi$
$883$	385.000i	0.436014i	−0.975947	+	0.218007i	$0.930045\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1141.27	−1.28666	−0.643332	−	0.765588i	$-0.722448\pi$
$887$	−1141.27	−1.28666	−0.643332	+	0.765588i	$0.722448\pi$
$888$	0	0
$889$	−160.000	−0.179978
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−326.683	−0.365827
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 504.874i	− 0.561595i
$900$	0	0
$901$	−360.000	−0.399556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	752.000i	0.829107i	0.910025	+	0.414553i	$0.136062\pi$
$907$	752.000i	0.829107i	−0.910025	+	0.414553i	$0.863938\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1336.43i	1.46699i	0.679693	+	0.733497i	$0.262113\pi$
$911$	1336.43i	1.46699i	−0.679693	+	0.733497i	$0.737887\pi$
$912$	0	0
$913$	666.000i	0.729463i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.9706	0.0185066
$918$	0	0
$919$	719.000	0.782372	0.391186	−	0.920312i	$-0.372065\pi$
$919$	719.000	0.782372	0.391186	+	0.920312i	$0.372065\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−356.382	−0.386112
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1294.01i	− 1.39290i	−0.717605	−	0.696451i	$-0.754761\pi$
$929$	− 1294.01i	− 1.39290i	0.717605	−	0.696451i	$-0.245239\pi$
$930$	0	0
$931$	−336.000	−0.360902
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1471.00i	1.56990i	0.619557	+	0.784952i	$0.287312\pi$
$937$	1471.00i	1.56990i	−0.619557	+	0.784952i	$0.712688\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 895.197i	− 0.951325i	−0.879628	−	0.475663i	$-0.842208\pi$
$941$	− 895.197i	− 0.951325i	0.879628	−	0.475663i	$-0.157792\pi$
$942$	0	0
$943$	− 1512.00i	− 1.60339i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1022.48	−1.07970	−0.539850	−	0.841761i	$-0.681519\pi$
$947$	−1022.48	−1.07970	−0.539850	+	0.841761i	$0.681519\pi$
$948$	0	0
$949$	−616.000	−0.649104
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−950.352	−0.997221	−0.498610	−	0.866826i	$-0.666156\pi$
$953$	−950.352	−0.997221	−0.498610	+	0.866826i	$0.666156\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 89.0955i	− 0.0929045i
$960$	0	0
$961$	−672.000	−0.699272
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1016.00i	1.05067i	0.850895	+	0.525336i	$0.176060\pi$
$967$	1016.00i	1.05067i	−0.850895	+	0.525336i	$0.823940\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1574.02i	− 1.62103i	−0.585718	−	0.810515i	$-0.699187\pi$
$971$	− 1574.02i	− 1.62103i	0.585718	−	0.810515i	$-0.300813\pi$
$972$	0	0
$973$	− 8.00000i	− 0.00822199i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−801.859	−0.820736	−0.410368	−	0.911920i	$-0.634600\pi$
$977$	−801.859	−0.820736	−0.410368	+	0.911920i	$0.634600\pi$
$978$	0	0
$979$	432.000	0.441267
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1866.76	1.89905	0.949523	−	0.313698i	$-0.101568\pi$
$983$	1866.76	1.89905	0.949523	+	0.313698i	$0.101568\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1633.42i	− 1.65158i
$990$	0	0
$991$	1081.00	1.09082	0.545409	−	0.838170i	$-0.316374\pi$
$991$	1081.00	1.09082	0.545409	+	0.838170i	$0.316374\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 248.000i	− 0.248746i	−0.992236	−	0.124373i	$-0.960308\pi$
$997$	− 248.000i	− 0.248746i	0.992236	−	0.124373i	$-0.0396920\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.c.d.449.1		4
3.2	odd	2	inner	3600.3.c.d.449.2		4
4.3	odd	2		900.3.b.a.449.4		4
5.2	odd	4		3600.3.l.g.1601.1		2
5.3	odd	4		3600.3.l.f.1601.1		2
5.4	even	2	inner	3600.3.c.d.449.3		4
12.11	even	2		900.3.b.a.449.3		4
15.2	even	4		3600.3.l.g.1601.2		2
15.8	even	4		3600.3.l.f.1601.2		2
15.14	odd	2	inner	3600.3.c.d.449.4		4
20.3	even	4		900.3.g.b.701.2	yes	2
20.7	even	4		900.3.g.a.701.2	yes	2
20.19	odd	2		900.3.b.a.449.2		4
60.23	odd	4		900.3.g.b.701.1	yes	2
60.47	odd	4		900.3.g.a.701.1	✓	2
60.59	even	2		900.3.b.a.449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.3.b.a.449.1		4	60.59	even	2
900.3.b.a.449.2		4	20.19	odd	2
900.3.b.a.449.3		4	12.11	even	2
900.3.b.a.449.4		4	4.3	odd	2
900.3.g.a.701.1	✓	2	60.47	odd	4
900.3.g.a.701.2	yes	2	20.7	even	4
900.3.g.b.701.1	yes	2	60.23	odd	4
900.3.g.b.701.2	yes	2	20.3	even	4
3600.3.c.d.449.1		4	1.1	even	1	trivial
3600.3.c.d.449.2		4	3.2	odd	2	inner
3600.3.c.d.449.3		4	5.4	even	2	inner
3600.3.c.d.449.4		4	15.14	odd	2	inner
3600.3.l.f.1601.1		2	5.3	odd	4
3600.3.l.f.1601.2		2	15.8	even	4
3600.3.l.g.1601.1		2	5.2	odd	4
3600.3.l.g.1601.2		2	15.2	even	4

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:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{7} +O(q^{10})\)	\(q-1.00000i q^{7} -4.24264i q^{11} -7.00000i q^{13} -4.24264 q^{17} -7.00000 q^{19} -29.6985 q^{23} +29.6985i q^{29} -17.0000 q^{31} +16.0000i q^{37} +50.9117i q^{41} +55.0000i q^{43} +46.6690 q^{47} +48.0000 q^{49} +84.8528 q^{53} -55.1543i q^{59} +65.0000 q^{61} -49.0000i q^{67} -50.9117i q^{71} -88.0000i q^{73} -4.24264 q^{77} -40.0000 q^{79} -156.978 q^{83} +101.823i q^{89} -7.00000 q^{91} -41.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 28 q^{19} - 68 q^{31} + 192 q^{49} + 260 q^{61} - 160 q^{79} - 28 q^{91}+O(q^{100}) \) 4 * q - 28 * q^19 - 68 * q^31 + 192 * q^49 + 260 * q^61 - 160 * q^79 - 28 * q^91

Introduction

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