LMFDB - Embedded newform 3600.3.c.e.449.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3600,3,Mod(449,3600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0, 0,156] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 225)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3600.449
Dual form			3600.3.c.e.449.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{7} -18.3848i q^{11} -1.00000i q^{13} -26.8701 q^{17} +25.0000 q^{19} +4.24264 q^{23} +26.8701i q^{29} +39.0000 q^{31} +32.0000i q^{37} -5.65685i q^{41} -23.0000i q^{43} -32.5269 q^{47} -32.0000 q^{49} -96.1665 q^{53} +9.89949i q^{59} +73.0000 q^{61} -63.0000i q^{67} -62.2254i q^{71} -136.000i q^{73} +165.463 q^{77} -24.0000 q^{79} +46.6690 q^{83} -101.823i q^{89} +9.00000 q^{91} -7.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 100 q^{19} + 156 q^{31} - 128 q^{49} + 292 q^{61} - 96 q^{79} + 36 q^{91}+O(q^{100}) $ 4 * q + 100 * q^19 + 156 * q^31 - 128 * q^49 + 292 * q^61 - 96 * q^79 + 36 * q^91

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
$7$	9.00000i	1.28571i	0.765986	+	0.642857i	$0.222251\pi$
$7$	9.00000i	1.28571i	−0.765986	+	0.642857i	$0.777749\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 18.3848i	− 1.67134i	−0.549229	−	0.835672i	$-0.685079\pi$
$11$	− 18.3848i	− 1.67134i	0.549229	−	0.835672i	$-0.314921\pi$
$12$	0	0
$13$	− 1.00000i	− 0.0769231i	−0.999260	−	0.0384615i	$-0.987754\pi$
$13$	− 1.00000i	− 0.0769231i	0.999260	−	0.0384615i	$-0.0122457\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−26.8701	−1.58059	−0.790296	−	0.612725i	$-0.790073\pi$
$17$	−26.8701	−1.58059	−0.790296	+	0.612725i	$0.790073\pi$
$18$	0	0
$19$	25.0000	1.31579	0.657895	−	0.753110i	$-0.271447\pi$
$19$	25.0000	1.31579	0.657895	+	0.753110i	$0.271447\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.24264	0.184463	0.0922313	−	0.995738i	$-0.470600\pi$
$23$	4.24264	0.184463	0.0922313	+	0.995738i	$0.470600\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	26.8701i	0.926554i	0.886214	+	0.463277i	$0.153326\pi$
$29$	26.8701i	0.926554i	−0.886214	+	0.463277i	$0.846674\pi$
$30$	0	0
$31$	39.0000	1.25806	0.629032	−	0.777379i	$-0.283451\pi$
$31$	39.0000	1.25806	0.629032	+	0.777379i	$0.283451\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	32.0000i	0.864865i	0.901666	+	0.432432i	$0.142345\pi$
$37$	32.0000i	0.864865i	−0.901666	+	0.432432i	$0.857655\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.65685i	− 0.137972i	−0.997618	−	0.0689860i	$-0.978024\pi$
$41$	− 5.65685i	− 0.137972i	0.997618	−	0.0689860i	$-0.0219764\pi$
$42$	0	0
$43$	− 23.0000i	− 0.534884i	−0.963574	−	0.267442i	$-0.913822\pi$
$43$	− 23.0000i	− 0.534884i	0.963574	−	0.267442i	$-0.0861783\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−32.5269	−0.692062	−0.346031	−	0.938223i	$-0.612471\pi$
$47$	−32.5269	−0.692062	−0.346031	+	0.938223i	$0.612471\pi$
$48$	0	0
$49$	−32.0000	−0.653061
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−96.1665	−1.81446	−0.907231	−	0.420632i	$-0.861808\pi$
$53$	−96.1665	−1.81446	−0.907231	+	0.420632i	$0.861808\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.89949i	0.167788i	0.996475	+	0.0838940i	$0.0267357\pi$
$59$	9.89949i	0.167788i	−0.996475	+	0.0838940i	$0.973264\pi$
$60$	0	0
$61$	73.0000	1.19672	0.598361	−	0.801227i	$-0.295819\pi$
$61$	73.0000	1.19672	0.598361	+	0.801227i	$0.295819\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 63.0000i	− 0.940299i	−0.882587	−	0.470149i	$-0.844200\pi$
$67$	− 63.0000i	− 0.940299i	0.882587	−	0.470149i	$-0.155800\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 62.2254i	− 0.876414i	−0.898874	−	0.438207i	$-0.855614\pi$
$71$	− 62.2254i	− 0.876414i	0.898874	−	0.438207i	$-0.144386\pi$
$72$	0	0
$73$	− 136.000i	− 1.86301i	−0.363724	−	0.931507i	$-0.618495\pi$
$73$	− 136.000i	− 1.86301i	0.363724	−	0.931507i	$-0.381505\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	165.463	2.14887
$78$	0	0
$79$	−24.0000	−0.303797	−0.151899	−	0.988396i	$-0.548539\pi$
$79$	−24.0000	−0.303797	−0.151899	+	0.988396i	$0.548539\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	46.6690	0.562278	0.281139	−	0.959667i	$-0.409288\pi$
$83$	46.6690	0.562278	0.281139	+	0.959667i	$0.409288\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 101.823i	− 1.14408i	−0.820225	−	0.572041i	$-0.806152\pi$
$89$	− 101.823i	− 1.14408i	0.820225	−	0.572041i	$-0.193848\pi$
$90$	0	0
$91$	9.00000	0.0989011
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 7.00000i	− 0.0721649i	−0.999349	−	0.0360825i	$-0.988512\pi$
$97$	− 7.00000i	− 0.0721649i	0.999349	−	0.0360825i	$-0.0114879\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 50.9117i	− 0.504076i	−0.967717	−	0.252038i	$-0.918899\pi$
$101$	− 50.9117i	− 0.504076i	0.967717	−	0.252038i	$-0.0811008\pi$
$102$	0	0
$103$	110.000i	1.06796i	0.845497	+	0.533981i	$0.179304\pi$
$103$	110.000i	1.06796i	−0.845497	+	0.533981i	$0.820696\pi$
$104$	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
$109$	25.0000	0.229358	0.114679	−	0.993403i	$-0.463416\pi$
$109$	25.0000	0.229358	0.114679	+	0.993403i	$0.463416\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−45.2548	−0.400485	−0.200243	−	0.979746i	$-0.564173\pi$
$113$	−45.2548	−0.400485	−0.200243	+	0.979746i	$0.564173\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 241.831i	− 2.03219i
$120$	0	0
$121$	−217.000	−1.79339
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	80.0000i	0.629921i	0.949105	+	0.314961i	$0.101991\pi$
$127$	80.0000i	0.629921i	−0.949105	+	0.314961i	$0.898009\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 84.8528i	− 0.647731i	−0.946103	−	0.323866i	$-0.895017\pi$
$131$	− 84.8528i	− 0.647731i	0.946103	−	0.323866i	$-0.104983\pi$
$132$	0	0
$133$	225.000i	1.69173i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.2132	0.154841	0.0774205	−	0.996999i	$-0.475332\pi$
$137$	21.2132	0.154841	0.0774205	+	0.996999i	$0.475332\pi$
$138$	0	0
$139$	104.000	0.748201	0.374101	−	0.927388i	$-0.377951\pi$
$139$	104.000	0.748201	0.374101	+	0.927388i	$0.377951\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−18.3848	−0.128565
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	171.120i	1.14846i	0.818696	+	0.574228i	$0.194698\pi$
$149$	171.120i	1.14846i	−0.818696	+	0.574228i	$0.805302\pi$
$150$	0	0
$151$	17.0000	0.112583	0.0562914	−	0.998414i	$-0.482072\pi$
$151$	17.0000	0.112583	0.0562914	+	0.998414i	$0.482072\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	65.0000i	0.414013i	0.978340	+	0.207006i	$0.0663721\pi$
$157$	65.0000i	0.414013i	−0.978340	+	0.207006i	$0.933628\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	38.1838i	0.237166i
$162$	0	0
$163$	17.0000i	0.104294i	0.998639	+	0.0521472i	$0.0166065\pi$
$163$	17.0000i	0.104294i	−0.998639	+	0.0521472i	$0.983393\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	254.558	1.52430	0.762151	−	0.647399i	$-0.224143\pi$
$167$	254.558	1.52430	0.762151	+	0.647399i	$0.224143\pi$
$168$	0	0
$169$	168.000	0.994083
$170$	0	0
$171$	0	0
$172$	0	0
$173$	173.948	1.00548	0.502741	−	0.864437i	$-0.332325\pi$
$173$	173.948	1.00548	0.502741	+	0.864437i	$0.332325\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 125.865i	− 0.703156i	−0.936159	−	0.351578i	$-0.885645\pi$
$179$	− 125.865i	− 0.703156i	0.936159	−	0.351578i	$-0.114355\pi$
$180$	0	0
$181$	119.000	0.657459	0.328729	−	0.944424i	$-0.393380\pi$
$181$	119.000	0.657459	0.328729	+	0.944424i	$0.393380\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	494.000i	2.64171i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 15.5563i	− 0.0814469i	−0.999170	−	0.0407234i	$-0.987034\pi$
$191$	− 15.5563i	− 0.0814469i	0.999170	−	0.0407234i	$-0.0129663\pi$
$192$	0	0
$193$	− 281.000i	− 1.45596i	−0.685599	−	0.727979i	$-0.740460\pi$
$193$	− 281.000i	− 1.45596i	0.685599	−	0.727979i	$-0.259540\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	224.860	1.14142	0.570711	−	0.821151i	$-0.306668\pi$
$197$	224.860	1.14142	0.570711	+	0.821151i	$0.306668\pi$
$198$	0	0
$199$	175.000	0.879397	0.439698	−	0.898145i	$-0.355085\pi$
$199$	175.000	0.879397	0.439698	+	0.898145i	$0.355085\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−241.831	−1.19128
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 459.619i	− 2.19914i
$210$	0	0
$211$	49.0000	0.232227	0.116114	−	0.993236i	$-0.462956\pi$
$211$	49.0000	0.232227	0.116114	+	0.993236i	$0.462956\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	351.000i	1.61751i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	26.8701i	0.121584i
$222$	0	0
$223$	− 265.000i	− 1.18834i	−0.804339	−	0.594170i	$-0.797480\pi$
$223$	− 265.000i	− 1.18834i	0.804339	−	0.594170i	$-0.202520\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	130.108	0.573161	0.286581	−	0.958056i	$-0.407481\pi$
$227$	130.108	0.573161	0.286581	+	0.958056i	$0.407481\pi$
$228$	0	0
$229$	87.0000	0.379913	0.189956	−	0.981793i	$-0.439165\pi$
$229$	87.0000	0.379913	0.189956	+	0.981793i	$0.439165\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−152.735	−0.655515	−0.327758	−	0.944762i	$-0.606293\pi$
$233$	−152.735	−0.655515	−0.327758	+	0.944762i	$0.606293\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	277.186i	1.15977i	0.814697	+	0.579887i	$0.196903\pi$
$239$	277.186i	1.15977i	−0.814697	+	0.579887i	$0.803097\pi$
$240$	0	0
$241$	415.000	1.72199	0.860996	−	0.508612i	$-0.169841\pi$
$241$	415.000	1.72199	0.860996	+	0.508612i	$0.169841\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 25.0000i	− 0.101215i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	373.352i	1.48746i	0.668480	+	0.743730i	$0.266945\pi$
$251$	373.352i	1.48746i	−0.668480	+	0.743730i	$0.733055\pi$
$252$	0	0
$253$	− 78.0000i	− 0.308300i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	271.529	1.05653	0.528267	−	0.849079i	$-0.322842\pi$
$257$	271.529	1.05653	0.528267	+	0.849079i	$0.322842\pi$
$258$	0	0
$259$	−288.000	−1.11197
$260$	0	0
$261$	0	0
$262$	0	0
$263$	209.304	0.795831	0.397916	−	0.917422i	$-0.369734\pi$
$263$	209.304	0.795831	0.397916	+	0.917422i	$0.369734\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 94.7523i	− 0.352239i	−0.984369	−	0.176120i	$-0.943645\pi$
$269$	− 94.7523i	− 0.352239i	0.984369	−	0.176120i	$-0.0563545\pi$
$270$	0	0
$271$	176.000	0.649446	0.324723	−	0.945809i	$-0.394729\pi$
$271$	176.000	0.649446	0.324723	+	0.945809i	$0.394729\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	303.000i	1.09386i	0.837177	+	0.546931i	$0.184204\pi$
$277$	303.000i	1.09386i	−0.837177	+	0.546931i	$0.815796\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 513.360i	− 1.82690i	−0.406948	−	0.913451i	$-0.633407\pi$
$281$	− 513.360i	− 1.82690i	0.406948	−	0.913451i	$-0.366593\pi$
$282$	0	0
$283$	− 393.000i	− 1.38869i	−0.719641	−	0.694346i	$-0.755694\pi$
$283$	− 393.000i	− 1.38869i	0.719641	−	0.694346i	$-0.244306\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	50.9117	0.177393
$288$	0	0
$289$	433.000	1.49827
$290$	0	0
$291$	0	0
$292$	0	0
$293$	46.6690	0.159280	0.0796400	−	0.996824i	$-0.474623\pi$
$293$	46.6690	0.159280	0.0796400	+	0.996824i	$0.474623\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 4.24264i	− 0.0141894i
$300$	0	0
$301$	207.000	0.687708
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 289.000i	− 0.941368i	−0.882302	−	0.470684i	$-0.844007\pi$
$307$	− 289.000i	− 0.941368i	0.882302	−	0.470684i	$-0.155993\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 462.448i	− 1.48697i	−0.668752	−	0.743485i	$-0.733171\pi$
$311$	− 462.448i	− 1.48697i	0.668752	−	0.743485i	$-0.266829\pi$
$312$	0	0
$313$	481.000i	1.53674i	0.640005	+	0.768371i	$0.278932\pi$
$313$	481.000i	1.53674i	−0.640005	+	0.768371i	$0.721068\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.3137	−0.0356899	−0.0178450	−	0.999841i	$-0.505681\pi$
$317$	−11.3137	−0.0356899	−0.0178450	+	0.999841i	$0.505681\pi$
$318$	0	0
$319$	494.000	1.54859
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−671.751	−2.07973
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 292.742i	− 0.889794i
$330$	0	0
$331$	72.0000	0.217523	0.108761	−	0.994068i	$-0.465312\pi$
$331$	72.0000	0.217523	0.108761	+	0.994068i	$0.465312\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 335.000i	− 0.994065i	−0.867732	−	0.497033i	$-0.834423\pi$
$337$	− 335.000i	− 0.994065i	0.867732	−	0.497033i	$-0.165577\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 717.006i	− 2.10266i
$342$	0	0
$343$	153.000i	0.446064i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	383.252	1.10447	0.552236	−	0.833688i	$-0.313775\pi$
$347$	383.252	1.10447	0.552236	+	0.833688i	$0.313775\pi$
$348$	0	0
$349$	−568.000	−1.62751	−0.813754	−	0.581210i	$-0.802579\pi$
$349$	−568.000	−1.62751	−0.813754	+	0.581210i	$0.802579\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−131.522	−0.372583	−0.186292	−	0.982495i	$-0.559647\pi$
$353$	−131.522	−0.372583	−0.186292	+	0.982495i	$0.559647\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 424.264i	− 1.18179i	−0.806747	−	0.590897i	$-0.798774\pi$
$359$	− 424.264i	− 1.18179i	0.806747	−	0.590897i	$-0.201226\pi$
$360$	0	0
$361$	264.000	0.731302
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	369.000i	1.00545i	0.864447	+	0.502725i	$0.167669\pi$
$367$	369.000i	1.00545i	−0.864447	+	0.502725i	$0.832331\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 865.499i	− 2.33288i
$372$	0	0
$373$	− 313.000i	− 0.839142i	−0.907723	−	0.419571i	$-0.862181\pi$
$373$	− 313.000i	− 0.839142i	0.907723	−	0.419571i	$-0.137819\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	26.8701	0.0712734
$378$	0	0
$379$	−47.0000	−0.124011	−0.0620053	−	0.998076i	$-0.519750\pi$
$379$	−47.0000	−0.124011	−0.0620053	+	0.998076i	$0.519750\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−373.352	−0.974810	−0.487405	−	0.873176i	$-0.662057\pi$
$383$	−373.352	−0.974810	−0.487405	+	0.873176i	$0.662057\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 390.323i	− 1.00340i	−0.865041	−	0.501700i	$-0.832708\pi$
$389$	− 390.323i	− 1.00340i	0.865041	−	0.501700i	$-0.167292\pi$
$390$	0	0
$391$	−114.000	−0.291560
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 47.0000i	− 0.118388i	−0.998247	−	0.0591940i	$-0.981147\pi$
$397$	− 47.0000i	− 0.118388i	0.998247	−	0.0591940i	$-0.0188530\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	152.735i	0.380885i	0.981698	+	0.190443i	$0.0609923\pi$
$401$	152.735i	0.380885i	−0.981698	+	0.190443i	$0.939008\pi$
$402$	0	0
$403$	− 39.0000i	− 0.0967742i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	588.313	1.44549
$408$	0	0
$409$	103.000	0.251834	0.125917	−	0.992041i	$-0.459813\pi$
$409$	103.000	0.251834	0.125917	+	0.992041i	$0.459813\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−89.0955	−0.215727
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 39.5980i	− 0.0945059i	−0.998883	−	0.0472530i	$-0.984953\pi$
$419$	− 39.5980i	− 0.0945059i	0.998883	−	0.0472530i	$-0.0150467\pi$
$420$	0	0
$421$	−560.000	−1.33017	−0.665083	−	0.746769i	$-0.731604\pi$
$421$	−560.000	−1.33017	−0.665083	+	0.746769i	$0.731604\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	657.000i	1.53864i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 479.418i	− 1.11234i	−0.831069	−	0.556170i	$-0.812270\pi$
$431$	− 479.418i	− 1.11234i	0.831069	−	0.556170i	$-0.187730\pi$
$432$	0	0
$433$	257.000i	0.593533i	0.954950	+	0.296767i	$0.0959084\pi$
$433$	257.000i	0.593533i	−0.954950	+	0.296767i	$0.904092\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	106.066	0.242714
$438$	0	0
$439$	−167.000	−0.380410	−0.190205	−	0.981744i	$-0.560915\pi$
$439$	−167.000	−0.380410	−0.190205	+	0.981744i	$0.560915\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−169.706	−0.383083	−0.191541	−	0.981485i	$-0.561349\pi$
$443$	−169.706	−0.383083	−0.191541	+	0.981485i	$0.561349\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	690.136i	1.53705i	0.639819	+	0.768526i	$0.279009\pi$
$449$	690.136i	1.53705i	−0.639819	+	0.768526i	$0.720991\pi$
$450$	0	0
$451$	−104.000	−0.230599
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 50.0000i	− 0.109409i	−0.998503	−	0.0547046i	$-0.982578\pi$
$457$	− 50.0000i	− 0.109409i	0.998503	−	0.0547046i	$-0.0174217\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 152.735i	− 0.331313i	−0.986184	−	0.165656i	$-0.947026\pi$
$461$	− 152.735i	− 0.331313i	0.986184	−	0.165656i	$-0.0529742\pi$
$462$	0	0
$463$	− 240.000i	− 0.518359i	−0.965829	−	0.259179i	$-0.916548\pi$
$463$	− 240.000i	− 0.518359i	0.965829	−	0.259179i	$-0.0834520\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	592.555	1.26886	0.634428	−	0.772982i	$-0.281236\pi$
$467$	592.555	1.26886	0.634428	+	0.772982i	$0.281236\pi$
$468$	0	0
$469$	567.000	1.20896
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−422.850	−0.893974
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 496.389i	− 1.03630i	−0.855289	−	0.518151i	$-0.826620\pi$
$479$	− 496.389i	− 1.03630i	0.855289	−	0.518151i	$-0.173380\pi$
$480$	0	0
$481$	32.0000	0.0665281
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 545.000i	− 1.11910i	−0.828798	−	0.559548i	$-0.810975\pi$
$487$	− 545.000i	− 1.11910i	0.828798	−	0.559548i	$-0.189025\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	905.097i	1.84337i	0.387934	+	0.921687i	$0.373189\pi$
$491$	905.097i	1.84337i	−0.387934	+	0.921687i	$0.626811\pi$
$492$	0	0
$493$	− 722.000i	− 1.46450i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	560.029	1.12682
$498$	0	0
$499$	7.00000	0.0140281	0.00701403	−	0.999975i	$-0.497767\pi$
$499$	7.00000	0.0140281	0.00701403	+	0.999975i	$0.497767\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	786.303	1.56323	0.781613	−	0.623764i	$-0.214397\pi$
$503$	786.303	1.56323	0.781613	+	0.623764i	$0.214397\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 578.413i	− 1.13637i	−0.822900	−	0.568186i	$-0.807645\pi$
$509$	− 578.413i	− 1.13637i	0.822900	−	0.568186i	$-0.192355\pi$
$510$	0	0
$511$	1224.00	2.39530
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	598.000i	1.15667i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	414.365i	0.795325i	0.917532	+	0.397663i	$0.130179\pi$
$521$	414.365i	0.795325i	−0.917532	+	0.397663i	$0.869821\pi$
$522$	0	0
$523$	− 263.000i	− 0.502868i	−0.967874	−	0.251434i	$-0.919098\pi$
$523$	− 263.000i	− 0.502868i	0.967874	−	0.251434i	$-0.0809022\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1047.93	−1.98849
$528$	0	0
$529$	−511.000	−0.965974
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.65685	−0.0106132
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	588.313i	1.09149i
$540$	0	0
$541$	489.000	0.903882	0.451941	−	0.892048i	$-0.350732\pi$
$541$	489.000	0.903882	0.451941	+	0.892048i	$0.350732\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 10.0000i	− 0.0182815i	−0.999958	−	0.00914077i	$-0.997090\pi$
$547$	− 10.0000i	− 0.0182815i	0.999958	−	0.00914077i	$-0.00290964\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	671.751i	1.21915i
$552$	0	0
$553$	− 216.000i	− 0.390597i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	623.668	1.11969	0.559846	−	0.828597i	$-0.310860\pi$
$557$	623.668	1.11969	0.559846	+	0.828597i	$0.310860\pi$
$558$	0	0
$559$	−23.0000	−0.0411449
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−63.6396	−0.113037	−0.0565183	−	0.998402i	$-0.518000\pi$
$563$	−63.6396	−0.113037	−0.0565183	+	0.998402i	$0.518000\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	810.344i	1.42416i	0.702101	+	0.712078i	$0.252246\pi$
$569$	810.344i	1.42416i	−0.702101	+	0.712078i	$0.747754\pi$
$570$	0	0
$571$	−377.000	−0.660245	−0.330123	−	0.943938i	$-0.607090\pi$
$571$	−377.000	−0.660245	−0.330123	+	0.943938i	$0.607090\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	791.000i	1.37088i	0.728127	+	0.685442i	$0.240391\pi$
$577$	791.000i	1.37088i	−0.728127	+	0.685442i	$0.759609\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	420.021i	0.722928i
$582$	0	0
$583$	1768.00i	3.03259i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−356.382	−0.607124	−0.303562	−	0.952812i	$-0.598176\pi$
$587$	−356.382	−0.607124	−0.303562	+	0.952812i	$0.598176\pi$
$588$	0	0
$589$	975.000	1.65535
$590$	0	0
$591$	0	0
$592$	0	0
$593$	435.578	0.734533	0.367266	−	0.930116i	$-0.380294\pi$
$593$	435.578	0.734533	0.367266	+	0.930116i	$0.380294\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1108.74i	− 1.85099i	−0.378759	−	0.925495i	$-0.623649\pi$
$599$	− 1108.74i	− 1.85099i	0.378759	−	0.925495i	$-0.376351\pi$
$600$	0	0
$601$	233.000	0.387687	0.193844	−	0.981032i	$-0.437905\pi$
$601$	233.000	0.387687	0.193844	+	0.981032i	$0.437905\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	776.000i	1.27842i	0.769033	+	0.639209i	$0.220738\pi$
$607$	776.000i	1.27842i	−0.769033	+	0.639209i	$0.779262\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	32.5269i	0.0532355i
$612$	0	0
$613$	− 376.000i	− 0.613377i	−0.951810	−	0.306688i	$-0.900779\pi$
$613$	− 376.000i	− 0.613377i	0.951810	−	0.306688i	$-0.0992210\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	593.970	0.962674	0.481337	−	0.876536i	$-0.340151\pi$
$617$	593.970	0.962674	0.481337	+	0.876536i	$0.340151\pi$
$618$	0	0
$619$	1031.00	1.66559	0.832795	−	0.553582i	$-0.186739\pi$
$619$	1031.00	1.66559	0.832795	+	0.553582i	$0.186739\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	916.410	1.47096
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 859.842i	− 1.36700i
$630$	0	0
$631$	−745.000	−1.18067	−0.590333	−	0.807160i	$-0.701003\pi$
$631$	−745.000	−1.18067	−0.590333	+	0.807160i	$0.701003\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	32.0000i	0.0502355i
$638$	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
$643$	− 1032.00i	− 1.60498i	−0.596668	−	0.802488i	$-0.703509\pi$
$643$	− 1032.00i	− 1.60498i	0.596668	−	0.802488i	$-0.296491\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	56.5685	0.0874321	0.0437160	−	0.999044i	$-0.486080\pi$
$647$	56.5685	0.0874321	0.0437160	+	0.999044i	$0.486080\pi$
$648$	0	0
$649$	182.000	0.280431
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−272.943	−0.417983	−0.208992	−	0.977917i	$-0.567018\pi$
$653$	−272.943	−0.417983	−0.208992	+	0.977917i	$0.567018\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	379.009i	0.575128i	0.957761	+	0.287564i	$0.0928454\pi$
$659$	379.009i	0.575128i	−0.957761	+	0.287564i	$0.907155\pi$
$660$	0	0
$661$	−440.000	−0.665658	−0.332829	−	0.942987i	$-0.608003\pi$
$661$	−440.000	−0.665658	−0.332829	+	0.942987i	$0.608003\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	114.000i	0.170915i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1342.09i	− 2.00013i
$672$	0	0
$673$	264.000i	0.392273i	0.980577	+	0.196137i	$0.0628396\pi$
$673$	264.000i	0.392273i	−0.980577	+	0.196137i	$0.937160\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−538.815	−0.795887	−0.397943	−	0.917410i	$-0.630276\pi$
$677$	−538.815	−0.795887	−0.397943	+	0.917410i	$0.630276\pi$
$678$	0	0
$679$	63.0000	0.0927835
$680$	0	0
$681$	0	0
$682$	0	0
$683$	576.999	0.844801	0.422401	−	0.906409i	$-0.361188\pi$
$683$	576.999	0.844801	0.422401	+	0.906409i	$0.361188\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	96.1665i	0.139574i
$690$	0	0
$691$	182.000	0.263386	0.131693	−	0.991291i	$-0.457959\pi$
$691$	182.000	0.263386	0.131693	+	0.991291i	$0.457959\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	152.000i	0.218077i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	558.614i	0.796882i	0.917194	+	0.398441i	$0.130449\pi$
$701$	558.614i	0.796882i	−0.917194	+	0.398441i	$0.869551\pi$
$702$	0	0
$703$	800.000i	1.13798i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	458.205	0.648098
$708$	0	0
$709$	−607.000	−0.856135	−0.428068	−	0.903747i	$-0.640806\pi$
$709$	−607.000	−0.856135	−0.428068	+	0.903747i	$0.640806\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	165.463	0.232066
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 106.066i	− 0.147519i	−0.997276	−	0.0737594i	$-0.976500\pi$
$719$	− 106.066i	− 0.147519i	0.997276	−	0.0737594i	$-0.0234997\pi$
$720$	0	0
$721$	−990.000	−1.37309
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 71.0000i	− 0.0976616i	−0.998807	−	0.0488308i	$-0.984450\pi$
$727$	− 71.0000i	− 0.0976616i	0.998807	−	0.0488308i	$-0.0155495\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	618.011i	0.845433i
$732$	0	0
$733$	904.000i	1.23329i	0.787242	+	0.616644i	$0.211508\pi$
$733$	904.000i	1.23329i	−0.787242	+	0.616644i	$0.788492\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1158.24	−1.57156
$738$	0	0
$739$	−800.000	−1.08254	−0.541272	−	0.840848i	$-0.682057\pi$
$739$	−800.000	−1.08254	−0.541272	+	0.840848i	$0.682057\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	418.607	0.563401	0.281701	−	0.959502i	$-0.409101\pi$
$743$	418.607	0.563401	0.281701	+	0.959502i	$0.409101\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1069.15i	1.42743i
$750$	0	0
$751$	704.000	0.937417	0.468708	−	0.883353i	$-0.344720\pi$
$751$	704.000	0.937417	0.468708	+	0.883353i	$0.344720\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 977.000i	− 1.29062i	−0.763920	−	0.645310i	$-0.776728\pi$
$757$	− 977.000i	− 1.29062i	0.763920	−	0.645310i	$-0.223272\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 746.705i	− 0.981215i	−0.871381	−	0.490608i	$-0.836775\pi$
$761$	− 746.705i	− 0.981215i	0.871381	−	0.490608i	$-0.163225\pi$
$762$	0	0
$763$	225.000i	0.294889i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.89949	0.0129068
$768$	0	0
$769$	−1337.00	−1.73862	−0.869311	−	0.494266i	$-0.835437\pi$
$769$	−1337.00	−1.73862	−0.869311	+	0.494266i	$0.835437\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1170.97	−1.51484	−0.757418	−	0.652930i	$-0.773540\pi$
$773$	−1170.97	−1.51484	−0.757418	+	0.652930i	$0.773540\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 141.421i	− 0.181542i
$780$	0	0
$781$	−1144.00	−1.46479
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 703.000i	− 0.893266i	−0.894717	−	0.446633i	$-0.852623\pi$
$787$	− 703.000i	− 0.893266i	0.894717	−	0.446633i	$-0.147377\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 407.294i	− 0.514910i
$792$	0	0
$793$	− 73.0000i	− 0.0920555i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1538.66	−1.93057	−0.965285	−	0.261199i	$-0.915882\pi$
$797$	−1538.66	−1.93057	−0.965285	+	0.261199i	$0.915882\pi$
$798$	0	0
$799$	874.000	1.09387
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2500.33	−3.11374
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	73.5391i	0.0909012i	0.998967	+	0.0454506i	$0.0144724\pi$
$809$	73.5391i	0.0909012i	−0.998967	+	0.0454506i	$0.985528\pi$
$810$	0	0
$811$	−1095.00	−1.35018	−0.675092	−	0.737733i	$-0.735896\pi$
$811$	−1095.00	−1.35018	−0.675092	+	0.737733i	$0.735896\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 575.000i	− 0.703794i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1166.73i	1.42110i	0.703645	+	0.710552i	$0.251555\pi$
$821$	1166.73i	1.42110i	−0.703645	+	0.710552i	$0.748445\pi$
$822$	0	0
$823$	− 1015.00i	− 1.23329i	−0.787240	−	0.616646i	$-0.788491\pi$
$823$	− 1015.00i	− 1.23329i	0.787240	−	0.616646i	$-0.211509\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	379.009	0.458294	0.229147	−	0.973392i	$-0.426406\pi$
$827$	379.009	0.458294	0.229147	+	0.973392i	$0.426406\pi$
$828$	0	0
$829$	−150.000	−0.180941	−0.0904704	−	0.995899i	$-0.528837\pi$
$829$	−150.000	−0.180941	−0.0904704	+	0.995899i	$0.528837\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	859.842	1.03222
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	660.438i	0.787173i	0.919288	+	0.393586i	$0.128766\pi$
$839$	660.438i	0.787173i	−0.919288	+	0.393586i	$0.871234\pi$
$840$	0	0
$841$	119.000	0.141498
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1953.00i	− 2.30579i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	135.765i	0.159535i
$852$	0	0
$853$	297.000i	0.348183i	0.984729	+	0.174091i	$0.0556988\pi$
$853$	297.000i	0.348183i	−0.984729	+	0.174091i	$0.944301\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1165.31	1.35976	0.679879	−	0.733325i	$-0.262032\pi$
$857$	1165.31	1.35976	0.679879	+	0.733325i	$0.262032\pi$
$858$	0	0
$859$	−1286.00	−1.49709	−0.748545	−	0.663084i	$-0.769247\pi$
$859$	−1286.00	−1.49709	−0.748545	+	0.663084i	$0.769247\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	633.568	0.734146	0.367073	−	0.930192i	$-0.380360\pi$
$863$	633.568	0.734146	0.367073	+	0.930192i	$0.380360\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	441.235i	0.507750i
$870$	0	0
$871$	−63.0000	−0.0723307
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1599.00i	− 1.82326i	−0.411011	−	0.911631i	$-0.634824\pi$
$877$	− 1599.00i	− 1.82326i	0.411011	−	0.911631i	$-0.365176\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 961.665i	− 1.09156i	−0.837928	−	0.545780i	$-0.816233\pi$
$881$	− 961.665i	− 1.09156i	0.837928	−	0.545780i	$-0.183767\pi$
$882$	0	0
$883$	− 193.000i	− 0.218573i	−0.994010	−	0.109287i	$-0.965143\pi$
$883$	− 193.000i	− 0.218573i	0.994010	−	0.109287i	$-0.0348566\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−700.036	−0.789217	−0.394609	−	0.918849i	$-0.629120\pi$
$887$	−700.036	−0.789217	−0.394609	+	0.918849i	$0.629120\pi$
$888$	0	0
$889$	−720.000	−0.809899
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−813.173	−0.910608
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1047.93i	1.16566i
$900$	0	0
$901$	2584.00	2.86792
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	240.000i	0.264609i	0.991209	+	0.132304i	$0.0422376\pi$
$907$	240.000i	0.264609i	−0.991209	+	0.132304i	$0.957762\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 171.120i	− 0.187837i	−0.995580	−	0.0939187i	$-0.970061\pi$
$911$	− 171.120i	− 0.187837i	0.995580	−	0.0939187i	$-0.0299394\pi$
$912$	0	0
$913$	− 858.000i	− 0.939759i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	763.675	0.832798
$918$	0	0
$919$	343.000	0.373232	0.186616	−	0.982433i	$-0.440248\pi$
$919$	343.000	0.373232	0.186616	+	0.982433i	$0.440248\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−62.2254	−0.0674165
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 719.835i	− 0.774849i	−0.921901	−	0.387424i	$-0.873365\pi$
$929$	− 719.835i	− 0.774849i	0.921901	−	0.387424i	$-0.126635\pi$
$930$	0	0
$931$	−800.000	−0.859291
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	609.000i	0.649947i	0.945723	+	0.324973i	$0.105355\pi$
$937$	609.000i	0.649947i	−0.945723	+	0.324973i	$0.894645\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 111.723i	− 0.118728i	−0.998236	−	0.0593639i	$-0.981093\pi$
$941$	− 111.723i	− 0.118728i	0.998236	−	0.0593639i	$-0.0189072\pi$
$942$	0	0
$943$	− 24.0000i	− 0.0254507i
$944$	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
$949$	−136.000	−0.143309
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1380.27	1.44834	0.724172	−	0.689619i	$-0.242222\pi$
$953$	1380.27	1.44834	0.724172	+	0.689619i	$0.242222\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	190.919i	0.199081i
$960$	0	0
$961$	560.000	0.582726
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 584.000i	− 0.603930i	−0.953319	−	0.301965i	$-0.902357\pi$
$967$	− 584.000i	− 0.603930i	0.953319	−	0.301965i	$-0.0976425\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1030.96i	1.06175i	0.847449	+	0.530876i	$0.178137\pi$
$971$	1030.96i	1.06175i	−0.847449	+	0.530876i	$0.821863\pi$
$972$	0	0
$973$	936.000i	0.961973i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1028.13	−1.05234	−0.526169	−	0.850380i	$-0.676372\pi$
$977$	−1028.13	−1.05234	−0.526169	+	0.850380i	$0.676372\pi$
$978$	0	0
$979$	−1872.00	−1.91216
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1323.70	1.34660	0.673298	−	0.739371i	$-0.264877\pi$
$983$	1323.70	1.34660	0.673298	+	0.739371i	$0.264877\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 97.5807i	− 0.0986661i
$990$	0	0
$991$	961.000	0.969728	0.484864	−	0.874590i	$-0.338869\pi$
$991$	961.000	0.969728	0.484864	+	0.874590i	$0.338869\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 600.000i	− 0.601805i	−0.953655	−	0.300903i	$-0.902712\pi$
$997$	− 600.000i	− 0.601805i	0.953655	−	0.300903i	$-0.0972879\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.c.e.449.3		4
3.2	odd	2	inner	3600.3.c.e.449.4		4
4.3	odd	2		225.3.d.a.224.1		4
5.2	odd	4		3600.3.l.b.1601.1		2
5.3	odd	4		3600.3.l.j.1601.1		2
5.4	even	2	inner	3600.3.c.e.449.1		4
12.11	even	2		225.3.d.a.224.3		4
15.2	even	4		3600.3.l.b.1601.2		2
15.8	even	4		3600.3.l.j.1601.2		2
15.14	odd	2	inner	3600.3.c.e.449.2		4
20.3	even	4		225.3.c.a.26.2	yes	2
20.7	even	4		225.3.c.b.26.1	yes	2
20.19	odd	2		225.3.d.a.224.4		4
60.23	odd	4		225.3.c.a.26.1	✓	2
60.47	odd	4		225.3.c.b.26.2	yes	2
60.59	even	2		225.3.d.a.224.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.3.c.a.26.1	✓	2	60.23	odd	4
225.3.c.a.26.2	yes	2	20.3	even	4
225.3.c.b.26.1	yes	2	20.7	even	4
225.3.c.b.26.2	yes	2	60.47	odd	4
225.3.d.a.224.1		4	4.3	odd	2
225.3.d.a.224.2		4	60.59	even	2
225.3.d.a.224.3		4	12.11	even	2
225.3.d.a.224.4		4	20.19	odd	2
3600.3.c.e.449.1		4	5.4	even	2	inner
3600.3.c.e.449.2		4	15.14	odd	2	inner
3600.3.c.e.449.3		4	1.1	even	1	trivial
3600.3.c.e.449.4		4	3.2	odd	2	inner
3600.3.l.b.1601.1		2	5.2	odd	4
3600.3.l.b.1601.2		2	15.2	even	4
3600.3.l.j.1601.1		2	5.3	odd	4
3600.3.l.j.1601.2		2	15.8	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+9.00000i q^{7} -18.3848i q^{11} -1.00000i q^{13} -26.8701 q^{17} +25.0000 q^{19} +4.24264 q^{23} +26.8701i q^{29} +39.0000 q^{31} +32.0000i q^{37} -5.65685i q^{41} -23.0000i q^{43} -32.5269 q^{47} -32.0000 q^{49} -96.1665 q^{53} +9.89949i q^{59} +73.0000 q^{61} -63.0000i q^{67} -62.2254i q^{71} -136.000i q^{73} +165.463 q^{77} -24.0000 q^{79} +46.6690 q^{83} -101.823i q^{89} +9.00000 q^{91} -7.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 100 q^{19} + 156 q^{31} - 128 q^{49} + 292 q^{61} - 96 q^{79} + 36 q^{91}+O(q^{100}) \) 4 * q + 100 * q^19 + 156 * q^31 - 128 * q^49 + 292 * q^61 - 96 * q^79 + 36 * q^91

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