LMFDB - Embedded newform 3600.2.o.b.3599.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,2,Mod(3599,3600)]

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

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

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

chi := DirichletCharacter("3600.3599");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.7461447277$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3599.6
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	3600.3599
Dual form			3600.2.o.b.3599.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{7} +O(q^{10})$	$q+1.73205 q^{7} -2.44949 q^{11} +1.00000i q^{13} -4.24264 q^{17} +1.73205i q^{19} -2.44949i q^{23} -4.24264i q^{29} +5.19615i q^{31} +4.00000i q^{37} -5.19615 q^{43} +12.2474i q^{47} -4.00000 q^{49} -12.2474 q^{59} -7.00000 q^{61} +8.66025 q^{67} -9.79796 q^{71} -8.00000i q^{73} -4.24264 q^{77} +6.92820i q^{79} -7.34847i q^{83} +1.73205i q^{91} +11.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 32 q^{49} - 56 q^{61}+O(q^{100}) $ 8 * q - 32 * q^49 - 56 * q^61

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
$7$	1.73205	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$7$	1.73205	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.24264	−1.02899	−0.514496	−	0.857493i	$-0.672021\pi$
$17$	−4.24264	−1.02899	−0.514496	+	0.857493i	$0.672021\pi$
$18$	0	0
$19$	1.73205i	0.397360i	0.980064	+	0.198680i	$0.0636654\pi$
$19$	1.73205i	0.397360i	−0.980064	+	0.198680i	$0.936335\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 2.44949i	− 0.510754i	−0.966842	−	0.255377i	$-0.917800\pi$
$23$	− 2.44949i	− 0.510754i	0.966842	−	0.255377i	$-0.0821996\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 4.24264i	− 0.787839i	−0.919145	−	0.393919i	$-0.871119\pi$
$29$	− 4.24264i	− 0.787839i	0.919145	−	0.393919i	$-0.128881\pi$
$30$	0	0
$31$	5.19615i	0.933257i	0.884454	+	0.466628i	$0.154531\pi$
$31$	5.19615i	0.933257i	−0.884454	+	0.466628i	$0.845469\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−5.19615	−0.792406	−0.396203	−	0.918163i	$-0.629672\pi$
$43$	−5.19615	−0.792406	−0.396203	+	0.918163i	$0.629672\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.2474i	1.78647i	0.449586	+	0.893237i	$0.351571\pi$
$47$	12.2474i	1.78647i	−0.449586	+	0.893237i	$0.648429\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.2474	−1.59448	−0.797241	−	0.603661i	$-0.793708\pi$
$59$	−12.2474	−1.59448	−0.797241	+	0.603661i	$0.793708\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.66025	1.05802	0.529009	−	0.848616i	$-0.322564\pi$
$67$	8.66025	1.05802	0.529009	+	0.848616i	$0.322564\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.79796	−1.16280	−0.581402	−	0.813617i	$-0.697496\pi$
$71$	−9.79796	−1.16280	−0.581402	+	0.813617i	$0.697496\pi$
$72$	0	0
$73$	− 8.00000i	− 0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$	− 8.00000i	− 0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.24264	−0.483494
$78$	0	0
$79$	6.92820i	0.779484i	0.920924	+	0.389742i	$0.127436\pi$
$79$	6.92820i	0.779484i	−0.920924	+	0.389742i	$0.872564\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 7.34847i	− 0.806599i	−0.915068	−	0.403300i	$-0.867863\pi$
$83$	− 7.34847i	− 0.806599i	0.915068	−	0.403300i	$-0.132137\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	1.73205i	0.181568i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.0000i	1.11688i	0.829545	+	0.558440i	$0.188600\pi$
$97$	11.0000i	1.11688i	−0.829545	+	0.558440i	$0.811400\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 16.9706i	− 1.68863i	−0.535844	−	0.844317i	$-0.680006\pi$
$101$	− 16.9706i	− 1.68863i	0.535844	−	0.844317i	$-0.319994\pi$
$102$	0	0
$103$	3.46410	0.341328	0.170664	−	0.985329i	$-0.445409\pi$
$103$	3.46410	0.341328	0.170664	+	0.985329i	$0.445409\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.5959i	1.89441i	0.320630	+	0.947204i	$0.396105\pi$
$107$	19.5959i	1.89441i	−0.320630	+	0.947204i	$0.603895\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.9706	1.59646	0.798228	−	0.602355i	$-0.205771\pi$
$113$	16.9706	1.59646	0.798228	+	0.602355i	$0.205771\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.34847	−0.673633
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.7846	1.84434	0.922168	−	0.386790i	$-0.126416\pi$
$127$	20.7846	1.84434	0.922168	+	0.386790i	$0.126416\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.79796	−0.856052	−0.428026	−	0.903767i	$-0.640791\pi$
$131$	−9.79796	−0.856052	−0.428026	+	0.903767i	$0.640791\pi$
$132$	0	0
$133$	3.00000i	0.260133i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.7279	−1.08742	−0.543710	−	0.839273i	$-0.682981\pi$
$137$	−12.7279	−1.08742	−0.543710	+	0.839273i	$0.682981\pi$
$138$	0	0
$139$	13.8564i	1.17529i	0.809121	+	0.587643i	$0.199944\pi$
$139$	13.8564i	1.17529i	−0.809121	+	0.587643i	$0.800056\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 2.44949i	− 0.204837i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 12.7279i	− 1.04271i	−0.853339	−	0.521356i	$-0.825426\pi$
$149$	− 12.7279i	− 1.04271i	0.853339	−	0.521356i	$-0.174574\pi$
$150$	0	0
$151$	8.66025i	0.704761i	0.935857	+	0.352381i	$0.114628\pi$
$151$	8.66025i	0.704761i	−0.935857	+	0.352381i	$0.885372\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 5.00000i	− 0.399043i	−0.979893	−	0.199522i	$-0.936061\pi$
$157$	− 5.00000i	− 0.399043i	0.979893	−	0.199522i	$-0.0639388\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4.24264i	− 0.334367i
$162$	0	0
$163$	−12.1244	−0.949653	−0.474826	−	0.880079i	$-0.657489\pi$
$163$	−12.1244	−0.949653	−0.474826	+	0.880079i	$0.657489\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.5959i	1.51638i	0.652035	+	0.758189i	$0.273915\pi$
$167$	19.5959i	1.51638i	−0.652035	+	0.758189i	$0.726085\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.7279	−0.967686	−0.483843	−	0.875155i	$-0.660759\pi$
$173$	−12.7279	−0.967686	−0.483843	+	0.875155i	$0.660759\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.2474	−0.915417	−0.457709	−	0.889102i	$-0.651330\pi$
$179$	−12.2474	−0.915417	−0.457709	+	0.889102i	$0.651330\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	10.3923	0.759961
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.34847	−0.531717	−0.265858	−	0.964012i	$-0.585655\pi$
$191$	−7.34847	−0.531717	−0.265858	+	0.964012i	$0.585655\pi$
$192$	0	0
$193$	− 7.00000i	− 0.503871i	−0.967744	−	0.251936i	$-0.918933\pi$
$193$	− 7.00000i	− 0.503871i	0.967744	−	0.251936i	$-0.0810671\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.7279	−0.906827	−0.453413	−	0.891300i	$-0.649794\pi$
$197$	−12.7279	−0.906827	−0.453413	+	0.891300i	$0.649794\pi$
$198$	0	0
$199$	− 1.73205i	− 0.122782i	−0.998114	−	0.0613909i	$-0.980446\pi$
$199$	− 1.73205i	− 0.122782i	0.998114	−	0.0613909i	$-0.0195536\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 7.34847i	− 0.515761i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4.24264i	− 0.293470i
$210$	0	0
$211$	22.5167i	1.55011i	0.631893	+	0.775055i	$0.282278\pi$
$211$	22.5167i	1.55011i	−0.631893	+	0.775055i	$0.717722\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.00000i	0.610960i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 4.24264i	− 0.285391i
$222$	0	0
$223$	−15.5885	−1.04388	−0.521940	−	0.852982i	$-0.674792\pi$
$223$	−15.5885	−1.04388	−0.521940	+	0.852982i	$0.674792\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.79796i	0.650313i	0.945660	+	0.325157i	$0.105417\pi$
$227$	9.79796i	0.650313i	−0.945660	+	0.325157i	$0.894583\pi$
$228$	0	0
$229$	−25.0000	−1.65205	−0.826023	−	0.563636i	$-0.809402\pi$
$229$	−25.0000	−1.65205	−0.826023	+	0.563636i	$0.809402\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.9706	1.11178	0.555889	−	0.831256i	$-0.312378\pi$
$233$	16.9706	1.11178	0.555889	+	0.831256i	$0.312378\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.79796	−0.633777	−0.316889	−	0.948463i	$-0.602638\pi$
$239$	−9.79796	−0.633777	−0.316889	+	0.948463i	$0.602638\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.73205	−0.110208
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.79796	−0.618442	−0.309221	−	0.950990i	$-0.600068\pi$
$251$	−9.79796	−0.618442	−0.309221	+	0.950990i	$0.600068\pi$
$252$	0	0
$253$	6.00000i	0.377217i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.9706	−1.05859	−0.529297	−	0.848436i	$-0.677544\pi$
$257$	−16.9706	−1.05859	−0.529297	+	0.848436i	$0.677544\pi$
$258$	0	0
$259$	6.92820i	0.430498i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 9.79796i	− 0.604168i	−0.953281	−	0.302084i	$-0.902318\pi$
$263$	− 9.79796i	− 0.604168i	0.953281	−	0.302084i	$-0.0976823\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 12.7279i	− 0.776035i	−0.921652	−	0.388018i	$-0.873160\pi$
$269$	− 12.7279i	− 0.776035i	0.921652	−	0.388018i	$-0.126840\pi$
$270$	0	0
$271$	− 6.92820i	− 0.420858i	−0.977609	−	0.210429i	$-0.932514\pi$
$271$	− 6.92820i	− 0.420858i	0.977609	−	0.210429i	$-0.0674861\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.0000i	1.02143i	0.859750	+	0.510716i	$0.170619\pi$
$277$	17.0000i	1.02143i	−0.859750	+	0.510716i	$0.829381\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.2132i	1.26547i	0.774367	+	0.632737i	$0.218068\pi$
$281$	21.2132i	1.26547i	−0.774367	+	0.632737i	$0.781932\pi$
$282$	0	0
$283$	25.9808	1.54440	0.772198	−	0.635382i	$-0.219157\pi$
$283$	25.9808	1.54440	0.772198	+	0.635382i	$0.219157\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−21.2132	−1.23929	−0.619644	−	0.784883i	$-0.712723\pi$
$293$	−21.2132	−1.23929	−0.619644	+	0.784883i	$0.712723\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.44949	0.141658
$300$	0	0
$301$	−9.00000	−0.518751
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.19615	0.296560	0.148280	−	0.988945i	$-0.452626\pi$
$307$	5.19615	0.296560	0.148280	+	0.988945i	$0.452626\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.1464	−0.972285	−0.486142	−	0.873880i	$-0.661596\pi$
$311$	−17.1464	−0.972285	−0.486142	+	0.873880i	$0.661596\pi$
$312$	0	0
$313$	− 13.0000i	− 0.734803i	−0.930062	−	0.367402i	$-0.880247\pi$
$313$	− 13.0000i	− 0.734803i	0.930062	−	0.367402i	$-0.119753\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.9706	0.953162	0.476581	−	0.879131i	$-0.341876\pi$
$317$	16.9706	0.953162	0.476581	+	0.879131i	$0.341876\pi$
$318$	0	0
$319$	10.3923i	0.581857i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 7.34847i	− 0.408880i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	21.2132i	1.16952i
$330$	0	0
$331$	− 13.8564i	− 0.761617i	−0.924654	−	0.380808i	$-0.875646\pi$
$331$	− 13.8564i	− 0.761617i	0.924654	−	0.380808i	$-0.124354\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.0000i	0.599208i	0.954064	+	0.299604i	$0.0968546\pi$
$337$	11.0000i	0.599208i	−0.954064	+	0.299604i	$0.903145\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 12.7279i	− 0.689256i
$342$	0	0
$343$	−19.0526	−1.02874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.34847i	0.394486i	0.980355	+	0.197243i	$0.0631989\pi$
$347$	7.34847i	0.394486i	−0.980355	+	0.197243i	$0.936801\pi$
$348$	0	0
$349$	4.00000	0.214115	0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.2132	1.12906	0.564532	−	0.825411i	$-0.309057\pi$
$353$	21.2132	1.12906	0.564532	+	0.825411i	$0.309057\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.79796	−0.517116	−0.258558	−	0.965996i	$-0.583247\pi$
$359$	−9.79796	−0.517116	−0.258558	+	0.965996i	$0.583247\pi$
$360$	0	0
$361$	16.0000	0.842105
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.73205	0.0904123	0.0452062	−	0.998978i	$-0.485606\pi$
$367$	1.73205	0.0904123	0.0452062	+	0.998978i	$0.485606\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	29.0000i	1.50156i	0.660551	+	0.750782i	$0.270323\pi$
$373$	29.0000i	1.50156i	−0.660551	+	0.750782i	$0.729677\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.24264	0.218507
$378$	0	0
$379$	− 19.0526i	− 0.978664i	−0.872098	−	0.489332i	$-0.837241\pi$
$379$	− 19.0526i	− 0.978664i	0.872098	−	0.489332i	$-0.162759\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.79796i	0.500652i	0.968162	+	0.250326i	$0.0805379\pi$
$383$	9.79796i	0.500652i	−0.968162	+	0.250326i	$0.919462\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 16.9706i	− 0.860442i	−0.902724	−	0.430221i	$-0.858436\pi$
$389$	− 16.9706i	− 0.860442i	0.902724	−	0.430221i	$-0.141564\pi$
$390$	0	0
$391$	10.3923i	0.525561i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 17.0000i	− 0.853206i	−0.904439	−	0.426603i	$-0.859710\pi$
$397$	− 17.0000i	− 0.853206i	0.904439	−	0.426603i	$-0.140290\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−5.19615	−0.258839
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 9.79796i	− 0.485667i
$408$	0	0
$409$	−1.00000	−0.0494468	−0.0247234	−	0.999694i	$-0.507871\pi$
$409$	−1.00000	−0.0494468	−0.0247234	+	0.999694i	$0.507871\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−21.2132	−1.04383
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	29.3939	1.43598	0.717992	−	0.696051i	$-0.245061\pi$
$419$	29.3939	1.43598	0.717992	+	0.696051i	$0.245061\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−12.1244	−0.586739
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.8434	1.53384	0.766921	−	0.641742i	$-0.221788\pi$
$431$	31.8434	1.53384	0.766921	+	0.641742i	$0.221788\pi$
$432$	0	0
$433$	− 37.0000i	− 1.77811i	−0.457804	−	0.889053i	$-0.651364\pi$
$433$	− 37.0000i	− 1.77811i	0.457804	−	0.889053i	$-0.348636\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.24264	0.202953
$438$	0	0
$439$	− 5.19615i	− 0.247999i	−0.992282	−	0.123999i	$-0.960428\pi$
$439$	− 5.19615i	− 0.247999i	0.992282	−	0.123999i	$-0.0395721\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.5959i	0.931030i	0.885040	+	0.465515i	$0.154131\pi$
$443$	19.5959i	0.931030i	−0.885040	+	0.465515i	$0.845869\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16.9706i	0.800890i	0.916321	+	0.400445i	$0.131145\pi$
$449$	16.9706i	0.800890i	−0.916321	+	0.400445i	$0.868855\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 16.9706i	− 0.790398i	−0.918596	−	0.395199i	$-0.870676\pi$
$461$	− 16.9706i	− 0.790398i	0.918596	−	0.395199i	$-0.129324\pi$
$462$	0	0
$463$	34.6410	1.60990	0.804952	−	0.593340i	$-0.202191\pi$
$463$	34.6410	1.60990	0.804952	+	0.593340i	$0.202191\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12.2474i	− 0.566744i	−0.959010	−	0.283372i	$-0.908547\pi$
$467$	− 12.2474i	− 0.566744i	0.959010	−	0.283372i	$-0.0914532\pi$
$468$	0	0
$469$	15.0000	0.692636
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.7279	0.585230
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7.34847	−0.335760	−0.167880	−	0.985807i	$-0.553692\pi$
$479$	−7.34847	−0.335760	−0.167880	+	0.985807i	$0.553692\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−8.66025	−0.392434	−0.196217	−	0.980561i	$-0.562866\pi$
$487$	−8.66025	−0.392434	−0.196217	+	0.980561i	$0.562866\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	29.3939	1.32653	0.663264	−	0.748386i	$-0.269171\pi$
$491$	29.3939	1.32653	0.663264	+	0.748386i	$0.269171\pi$
$492$	0	0
$493$	18.0000i	0.810679i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.9706	−0.761234
$498$	0	0
$499$	5.19615i	0.232612i	0.993213	+	0.116306i	$0.0371053\pi$
$499$	5.19615i	0.232612i	−0.993213	+	0.116306i	$0.962895\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 4.24264i	− 0.188052i	−0.995570	−	0.0940259i	$-0.970026\pi$
$509$	− 4.24264i	− 0.188052i	0.995570	−	0.0940259i	$-0.0299736\pi$
$510$	0	0
$511$	− 13.8564i	− 0.612971i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 30.0000i	− 1.31940i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21.2132i	0.929367i	0.885477	+	0.464684i	$0.153832\pi$
$521$	21.2132i	0.929367i	−0.885477	+	0.464684i	$0.846168\pi$
$522$	0	0
$523$	−19.0526	−0.833110	−0.416555	−	0.909110i	$-0.636763\pi$
$523$	−19.0526	−0.833110	−0.416555	+	0.909110i	$0.636763\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 22.0454i	− 0.960313i
$528$	0	0
$529$	17.0000	0.739130
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.79796	0.422028
$540$	0	0
$541$	13.0000	0.558914	0.279457	−	0.960158i	$-0.409846\pi$
$541$	13.0000	0.558914	0.279457	+	0.960158i	$0.409846\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−3.46410	−0.148114	−0.0740571	−	0.997254i	$-0.523595\pi$
$547$	−3.46410	−0.148114	−0.0740571	+	0.997254i	$0.523595\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.34847	0.313055
$552$	0	0
$553$	12.0000i	0.510292i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	46.6690	1.97743	0.988716	−	0.149805i	$-0.0478647\pi$
$557$	46.6690	1.97743	0.988716	+	0.149805i	$0.0478647\pi$
$558$	0	0
$559$	− 5.19615i	− 0.219774i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17.1464i	0.722636i	0.932443	+	0.361318i	$0.117673\pi$
$563$	17.1464i	0.722636i	−0.932443	+	0.361318i	$0.882327\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	38.1838i	1.60075i	0.599502	+	0.800373i	$0.295365\pi$
$569$	38.1838i	1.60075i	−0.599502	+	0.800373i	$0.704635\pi$
$570$	0	0
$571$	− 1.73205i	− 0.0724841i	−0.999343	−	0.0362420i	$-0.988461\pi$
$571$	− 1.73205i	− 0.0724841i	0.999343	−	0.0362420i	$-0.0115387\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.0000i	0.541197i	0.962692	+	0.270599i	$0.0872216\pi$
$577$	13.0000i	0.541197i	−0.962692	+	0.270599i	$0.912778\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 12.7279i	− 0.528043i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 29.3939i	− 1.21322i	−0.795001	−	0.606608i	$-0.792530\pi$
$587$	− 29.3939i	− 1.21322i	0.795001	−	0.606608i	$-0.207470\pi$
$588$	0	0
$589$	−9.00000	−0.370839
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16.9706	0.696897	0.348449	−	0.937328i	$-0.386709\pi$
$593$	16.9706	0.696897	0.348449	+	0.937328i	$0.386709\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	29.3939	1.20100	0.600501	−	0.799624i	$-0.294968\pi$
$599$	29.3939	1.20100	0.600501	+	0.799624i	$0.294968\pi$
$600$	0	0
$601$	41.0000	1.67242	0.836212	−	0.548406i	$-0.184765\pi$
$601$	41.0000	1.67242	0.836212	+	0.548406i	$0.184765\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−34.6410	−1.40604	−0.703018	−	0.711172i	$-0.748165\pi$
$607$	−34.6410	−1.40604	−0.703018	+	0.711172i	$0.748165\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.2474	−0.495479
$612$	0	0
$613$	− 4.00000i	− 0.161558i	−0.996732	−	0.0807792i	$-0.974259\pi$
$613$	− 4.00000i	− 0.161558i	0.996732	−	0.0807792i	$-0.0257409\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.9411	1.36642	0.683209	−	0.730223i	$-0.260584\pi$
$617$	33.9411	1.36642	0.683209	+	0.730223i	$0.260584\pi$
$618$	0	0
$619$	− 29.4449i	− 1.18349i	−0.806126	−	0.591744i	$-0.798439\pi$
$619$	− 29.4449i	− 1.18349i	0.806126	−	0.591744i	$-0.201561\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 16.9706i	− 0.676661i
$630$	0	0
$631$	− 43.3013i	− 1.72380i	−0.507081	−	0.861898i	$-0.669276\pi$
$631$	− 43.3013i	− 1.72380i	0.507081	−	0.861898i	$-0.330724\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4.00000i	− 0.158486i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.9411i	1.34059i	0.742093	+	0.670297i	$0.233833\pi$
$641$	33.9411i	1.34059i	−0.742093	+	0.670297i	$0.766167\pi$
$642$	0	0
$643$	41.5692	1.63933	0.819665	−	0.572843i	$-0.194160\pi$
$643$	41.5692	1.63933	0.819665	+	0.572843i	$0.194160\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 39.1918i	− 1.54079i	−0.637567	−	0.770395i	$-0.720059\pi$
$647$	− 39.1918i	− 1.54079i	0.637567	−	0.770395i	$-0.279941\pi$
$648$	0	0
$649$	30.0000	1.17760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−46.6690	−1.82630	−0.913150	−	0.407623i	$-0.866358\pi$
$653$	−46.6690	−1.82630	−0.913150	+	0.407623i	$0.866358\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−10.3923	−0.402392
$668$	0	0
$669$	0	0
$670$	0	0
$671$	17.1464	0.661931
$672$	0	0
$673$	− 16.0000i	− 0.616755i	−0.951264	−	0.308377i	$-0.900214\pi$
$673$	− 16.0000i	− 0.616755i	0.951264	−	0.308377i	$-0.0997859\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.24264	0.163058	0.0815290	−	0.996671i	$-0.474020\pi$
$677$	4.24264	0.163058	0.0815290	+	0.996671i	$0.474020\pi$
$678$	0	0
$679$	19.0526i	0.731170i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.79796i	0.374908i	0.982273	+	0.187454i	$0.0600236\pi$
$683$	9.79796i	0.374908i	−0.982273	+	0.187454i	$0.939976\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 31.1769i	− 1.18603i	−0.805193	−	0.593013i	$-0.797938\pi$
$691$	− 31.1769i	− 1.18603i	0.805193	−	0.593013i	$-0.202062\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12.7279i	0.480727i	0.970683	+	0.240363i	$0.0772666\pi$
$701$	12.7279i	0.480727i	−0.970683	+	0.240363i	$0.922733\pi$
$702$	0	0
$703$	−6.92820	−0.261302
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 29.3939i	− 1.10547i
$708$	0	0
$709$	−43.0000	−1.61490	−0.807449	−	0.589937i	$-0.799153\pi$
$709$	−43.0000	−1.61490	−0.807449	+	0.589937i	$0.799153\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	12.7279	0.476664
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	41.6413	1.55296	0.776480	−	0.630142i	$-0.217003\pi$
$719$	41.6413	1.55296	0.776480	+	0.630142i	$0.217003\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−25.9808	−0.963573	−0.481787	−	0.876289i	$-0.660012\pi$
$727$	−25.9808	−0.963573	−0.481787	+	0.876289i	$0.660012\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	22.0454	0.815379
$732$	0	0
$733$	52.0000i	1.92066i	0.278859	+	0.960332i	$0.410044\pi$
$733$	52.0000i	1.92066i	−0.278859	+	0.960332i	$0.589956\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21.2132	−0.781398
$738$	0	0
$739$	− 41.5692i	− 1.52915i	−0.644536	−	0.764574i	$-0.722949\pi$
$739$	− 41.5692i	− 1.52915i	0.644536	−	0.764574i	$-0.277051\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 9.79796i	− 0.359452i	−0.983717	−	0.179726i	$-0.942479\pi$
$743$	− 9.79796i	− 0.359452i	0.983717	−	0.179726i	$-0.0575212\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	33.9411i	1.24018i
$750$	0	0
$751$	34.6410i	1.26407i	0.774940	+	0.632034i	$0.217780\pi$
$751$	34.6410i	1.26407i	−0.774940	+	0.632034i	$0.782220\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.00000i	0.181728i	0.995863	+	0.0908640i	$0.0289629\pi$
$757$	5.00000i	0.181728i	−0.995863	+	0.0908640i	$0.971037\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.9411i	1.23036i	0.788385	+	0.615182i	$0.210918\pi$
$761$	33.9411i	1.23036i	−0.788385	+	0.615182i	$0.789082\pi$
$762$	0	0
$763$	−12.1244	−0.438931
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 12.2474i	− 0.442230i
$768$	0	0
$769$	7.00000	0.252426	0.126213	−	0.992003i	$-0.459718\pi$
$769$	7.00000	0.252426	0.126213	+	0.992003i	$0.459718\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−50.9117	−1.83117	−0.915583	−	0.402129i	$-0.868270\pi$
$773$	−50.9117	−1.83117	−0.915583	+	0.402129i	$0.868270\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−53.6936	−1.91397	−0.956985	−	0.290139i	$-0.906299\pi$
$787$	−53.6936	−1.91397	−0.956985	+	0.290139i	$0.906299\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	29.3939	1.04513
$792$	0	0
$793$	− 7.00000i	− 0.248577i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.9411	1.20226	0.601128	−	0.799153i	$-0.294718\pi$
$797$	33.9411	1.20226	0.601128	+	0.799153i	$0.294718\pi$
$798$	0	0
$799$	− 51.9615i	− 1.83827i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	19.5959i	0.691525i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16.9706i	0.596653i	0.954464	+	0.298327i	$0.0964285\pi$
$809$	16.9706i	0.596653i	−0.954464	+	0.298327i	$0.903572\pi$
$810$	0	0
$811$	− 5.19615i	− 0.182462i	−0.995830	−	0.0912308i	$-0.970920\pi$
$811$	− 5.19615i	− 0.182462i	0.995830	−	0.0912308i	$-0.0290801\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.00000i	− 0.314870i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.24264i	0.148069i	0.997256	+	0.0740346i	$0.0235875\pi$
$821$	4.24264i	0.148069i	−0.997256	+	0.0740346i	$0.976412\pi$
$822$	0	0
$823$	43.3013	1.50939	0.754694	−	0.656077i	$-0.227785\pi$
$823$	43.3013	1.50939	0.754694	+	0.656077i	$0.227785\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 19.5959i	− 0.681417i	−0.940169	−	0.340708i	$-0.889333\pi$
$827$	− 19.5959i	− 0.681417i	0.940169	−	0.340708i	$-0.110667\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	16.9706	0.587995
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.9444	0.930224	0.465112	−	0.885252i	$-0.346014\pi$
$839$	26.9444	0.930224	0.465112	+	0.885252i	$0.346014\pi$
$840$	0	0
$841$	11.0000	0.379310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8.66025	−0.297570
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.79796	0.335870
$852$	0	0
$853$	− 25.0000i	− 0.855984i	−0.903783	−	0.427992i	$-0.859221\pi$
$853$	− 25.0000i	− 0.855984i	0.903783	−	0.427992i	$-0.140779\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.9411	1.15941	0.579703	−	0.814828i	$-0.303168\pi$
$857$	33.9411	1.15941	0.579703	+	0.814828i	$0.303168\pi$
$858$	0	0
$859$	45.0333i	1.53652i	0.640140	+	0.768259i	$0.278876\pi$
$859$	45.0333i	1.53652i	−0.640140	+	0.768259i	$0.721124\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29.3939i	1.00058i	0.865858	+	0.500290i	$0.166773\pi$
$863$	29.3939i	1.00058i	−0.865858	+	0.500290i	$0.833227\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 16.9706i	− 0.575687i
$870$	0	0
$871$	8.66025i	0.293442i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 53.0000i	− 1.78968i	−0.446384	−	0.894841i	$-0.647289\pi$
$877$	− 53.0000i	− 1.78968i	0.446384	−	0.894841i	$-0.352711\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.9706i	0.571753i	0.958267	+	0.285876i	$0.0922847\pi$
$881$	16.9706i	0.571753i	−0.958267	+	0.285876i	$0.907715\pi$
$882$	0	0
$883$	−43.3013	−1.45720	−0.728602	−	0.684937i	$-0.759830\pi$
$883$	−43.3013	−1.45720	−0.728602	+	0.684937i	$0.759830\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 26.9444i	− 0.904704i	−0.891839	−	0.452352i	$-0.850585\pi$
$887$	− 26.9444i	− 0.904704i	0.891839	−	0.452352i	$-0.149415\pi$
$888$	0	0
$889$	36.0000	1.20740
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−21.2132	−0.709873
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	22.0454	0.735256
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	13.8564	0.460094	0.230047	−	0.973179i	$-0.426112\pi$
$907$	13.8564	0.460094	0.230047	+	0.973179i	$0.426112\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12.2474	−0.405776	−0.202888	−	0.979202i	$-0.565033\pi$
$911$	−12.2474	−0.405776	−0.202888	+	0.979202i	$0.565033\pi$
$912$	0	0
$913$	18.0000i	0.595713i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−16.9706	−0.560417
$918$	0	0
$919$	− 36.3731i	− 1.19984i	−0.800061	−	0.599918i	$-0.795200\pi$
$919$	− 36.3731i	− 1.19984i	0.800061	−	0.599918i	$-0.204800\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 9.79796i	− 0.322504i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 4.24264i	− 0.139197i	−0.997575	−	0.0695983i	$-0.977828\pi$
$929$	− 4.24264i	− 0.139197i	0.997575	−	0.0695983i	$-0.0221717\pi$
$930$	0	0
$931$	− 6.92820i	− 0.227063i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 5.00000i	− 0.163343i	−0.996659	−	0.0816714i	$-0.973974\pi$
$937$	− 5.00000i	− 0.163343i	0.996659	−	0.0816714i	$-0.0260258\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.24264i	0.138306i	0.997606	+	0.0691531i	$0.0220297\pi$
$941$	4.24264i	0.138306i	−0.997606	+	0.0691531i	$0.977970\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 36.7423i	− 1.19397i	−0.802254	−	0.596983i	$-0.796366\pi$
$947$	− 36.7423i	− 1.19397i	0.802254	−	0.596983i	$-0.203634\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.9706	0.549730	0.274865	−	0.961483i	$-0.411367\pi$
$953$	16.9706	0.549730	0.274865	+	0.961483i	$0.411367\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−22.0454	−0.711883
$960$	0	0
$961$	4.00000	0.129032
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−6.92820	−0.222796	−0.111398	−	0.993776i	$-0.535533\pi$
$967$	−6.92820	−0.222796	−0.111398	+	0.993776i	$0.535533\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.2474	0.393039	0.196520	−	0.980500i	$-0.437036\pi$
$971$	12.2474	0.393039	0.196520	+	0.980500i	$0.437036\pi$
$972$	0	0
$973$	24.0000i	0.769405i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21.2132	−0.678671	−0.339335	−	0.940666i	$-0.610202\pi$
$977$	−21.2132	−0.678671	−0.339335	+	0.940666i	$0.610202\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	39.1918i	1.25003i	0.780615	+	0.625013i	$0.214906\pi$
$983$	39.1918i	1.25003i	−0.780615	+	0.625013i	$0.785094\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.7279i	0.404724i
$990$	0	0
$991$	− 60.6218i	− 1.92571i	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	− 60.6218i	− 1.92571i	0.270011	−	0.962857i	$-0.412973\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.00000i	0.126681i	0.997992	+	0.0633406i	$0.0201755\pi$
$997$	4.00000i	0.126681i	−0.997992	+	0.0633406i	$0.979825\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.2.o.b.3599.6		8
3.2	odd	2	inner	3600.2.o.b.3599.8		8
4.3	odd	2	inner	3600.2.o.b.3599.4		8
5.2	odd	4		3600.2.h.h.1151.3	yes	4
5.3	odd	4		3600.2.h.e.1151.1	✓	4
5.4	even	2	inner	3600.2.o.b.3599.1		8
12.11	even	2	inner	3600.2.o.b.3599.2		8
15.2	even	4		3600.2.h.h.1151.4	yes	4
15.8	even	4		3600.2.h.e.1151.2	yes	4
15.14	odd	2	inner	3600.2.o.b.3599.3		8
20.3	even	4		3600.2.h.e.1151.4	yes	4
20.7	even	4		3600.2.h.h.1151.2	yes	4
20.19	odd	2	inner	3600.2.o.b.3599.7		8
60.23	odd	4		3600.2.h.e.1151.3	yes	4
60.47	odd	4		3600.2.h.h.1151.1	yes	4
60.59	even	2	inner	3600.2.o.b.3599.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3600.2.h.e.1151.1	✓	4	5.3	odd	4
3600.2.h.e.1151.2	yes	4	15.8	even	4
3600.2.h.e.1151.3	yes	4	60.23	odd	4
3600.2.h.e.1151.4	yes	4	20.3	even	4
3600.2.h.h.1151.1	yes	4	60.47	odd	4
3600.2.h.h.1151.2	yes	4	20.7	even	4
3600.2.h.h.1151.3	yes	4	5.2	odd	4
3600.2.h.h.1151.4	yes	4	15.2	even	4
3600.2.o.b.3599.1		8	5.4	even	2	inner
3600.2.o.b.3599.2		8	12.11	even	2	inner
3600.2.o.b.3599.3		8	15.14	odd	2	inner
3600.2.o.b.3599.4		8	4.3	odd	2	inner
3600.2.o.b.3599.5		8	60.59	even	2	inner
3600.2.o.b.3599.6		8	1.1	even	1	trivial
3600.2.o.b.3599.7		8	20.19	odd	2	inner
3600.2.o.b.3599.8		8	3.2	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:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3600.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{7} +O(q^{10})\)	\(q+1.73205 q^{7} -2.44949 q^{11} +1.00000i q^{13} -4.24264 q^{17} +1.73205i q^{19} -2.44949i q^{23} -4.24264i q^{29} +5.19615i q^{31} +4.00000i q^{37} -5.19615 q^{43} +12.2474i q^{47} -4.00000 q^{49} -12.2474 q^{59} -7.00000 q^{61} +8.66025 q^{67} -9.79796 q^{71} -8.00000i q^{73} -4.24264 q^{77} +6.92820i q^{79} -7.34847i q^{83} +1.73205i q^{91} +11.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 32 q^{49} - 56 q^{61}+O(q^{100}) \) 8 * q - 32 * q^49 - 56 * q^61

Introduction

L-functions

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