LMFDB - Embedded newform 3600.3.c.f.449.3

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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 450)
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.f.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.0000i q^{7} +O(q^{10})$	$q+11.0000i q^{7} -4.24264i q^{11} -7.00000i q^{13} +12.7279 q^{17} +29.0000 q^{19} +38.1838 q^{23} +46.6690i q^{29} -29.0000 q^{31} -56.0000i q^{37} -67.8823i q^{41} -5.00000i q^{43} +63.6396 q^{47} -72.0000 q^{49} -67.8823 q^{53} +29.6985i q^{59} -55.0000 q^{61} -37.0000i q^{67} +33.9411i q^{71} -16.0000i q^{73} +46.6690 q^{77} +104.000 q^{79} +29.6985 q^{83} -135.765i q^{89} +77.0000 q^{91} -41.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 116 q^{19} - 116 q^{31} - 288 q^{49} - 220 q^{61} + 416 q^{79} + 308 q^{91}+O(q^{100}) $ 4 * q + 116 * q^19 - 116 * q^31 - 288 * q^49 - 220 * q^61 + 416 * q^79 + 308 * 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
$7$	11.0000i	1.57143i	0.618590	+	0.785714i	$0.287704\pi$
$7$	11.0000i	1.57143i	−0.618590	+	0.785714i	$0.712296\pi$
$8$	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
$17$	12.7279	0.748701	0.374351	−	0.927287i	$-0.377866\pi$
$17$	12.7279	0.748701	0.374351	+	0.927287i	$0.377866\pi$
$18$	0	0
$19$	29.0000	1.52632	0.763158	−	0.646212i	$-0.223648\pi$
$19$	29.0000	1.52632	0.763158	+	0.646212i	$0.223648\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	38.1838	1.66016	0.830082	−	0.557642i	$-0.188294\pi$
$23$	38.1838	1.66016	0.830082	+	0.557642i	$0.188294\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	46.6690i	1.60928i	0.593765	+	0.804639i	$0.297641\pi$
$29$	46.6690i	1.60928i	−0.593765	+	0.804639i	$0.702359\pi$
$30$	0	0
$31$	−29.0000	−0.935484	−0.467742	−	0.883865i	$-0.654932\pi$
$31$	−29.0000	−0.935484	−0.467742	+	0.883865i	$0.654932\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 56.0000i	− 1.51351i	−0.653697	−	0.756757i	$-0.726783\pi$
$37$	− 56.0000i	− 1.51351i	0.653697	−	0.756757i	$-0.273217\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 67.8823i	− 1.65566i	−0.560976	−	0.827832i	$-0.689574\pi$
$41$	− 67.8823i	− 1.65566i	0.560976	−	0.827832i	$-0.310426\pi$
$42$	0	0
$43$	− 5.00000i	− 0.116279i	−0.998308	−	0.0581395i	$-0.981483\pi$
$43$	− 5.00000i	− 0.116279i	0.998308	−	0.0581395i	$-0.0185168\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	63.6396	1.35403	0.677017	−	0.735967i	$-0.263272\pi$
$47$	63.6396	1.35403	0.677017	+	0.735967i	$0.263272\pi$
$48$	0	0
$49$	−72.0000	−1.46939
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−67.8823	−1.28080	−0.640399	−	0.768043i	$-0.721231\pi$
$53$	−67.8823	−1.28080	−0.640399	+	0.768043i	$0.721231\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	29.6985i	0.503364i	0.967810	+	0.251682i	$0.0809837\pi$
$59$	29.6985i	0.503364i	−0.967810	+	0.251682i	$0.919016\pi$
$60$	0	0
$61$	−55.0000	−0.901639	−0.450820	−	0.892615i	$-0.648868\pi$
$61$	−55.0000	−0.901639	−0.450820	+	0.892615i	$0.648868\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 37.0000i	− 0.552239i	−0.961123	−	0.276119i	$-0.910951\pi$
$67$	− 37.0000i	− 0.552239i	0.961123	−	0.276119i	$-0.0890485\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	33.9411i	0.478044i	0.971014	+	0.239022i	$0.0768268\pi$
$71$	33.9411i	0.478044i	−0.971014	+	0.239022i	$0.923173\pi$
$72$	0	0
$73$	− 16.0000i	− 0.219178i	−0.993977	−	0.109589i	$-0.965047\pi$
$73$	− 16.0000i	− 0.219178i	0.993977	−	0.109589i	$-0.0349535\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	46.6690	0.606092
$78$	0	0
$79$	104.000	1.31646	0.658228	−	0.752819i	$-0.271306\pi$
$79$	104.000	1.31646	0.658228	+	0.752819i	$0.271306\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	29.6985	0.357813	0.178907	−	0.983866i	$-0.442744\pi$
$83$	29.6985	0.357813	0.178907	+	0.983866i	$0.442744\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 135.765i	− 1.52544i	−0.646727	−	0.762722i	$-0.723863\pi$
$89$	− 135.765i	− 1.52544i	0.646727	−	0.762722i	$-0.276137\pi$
$90$	0	0
$91$	77.0000	0.846154
$92$	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
$101$	33.9411i	0.336051i	0.985783	+	0.168025i	$0.0537391\pi$
$101$	33.9411i	0.336051i	−0.985783	+	0.168025i	$0.946261\pi$
$102$	0	0
$103$	− 62.0000i	− 0.601942i	−0.953633	−	0.300971i	$-0.902689\pi$
$103$	− 62.0000i	− 0.601942i	0.953633	−	0.300971i	$-0.0973107\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	169.000	1.55046	0.775229	−	0.631680i	$-0.217634\pi$
$109$	169.000	1.55046	0.775229	+	0.631680i	$0.217634\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	33.9411	0.300364	0.150182	−	0.988658i	$-0.452014\pi$
$113$	33.9411	0.300364	0.150182	+	0.988658i	$0.452014\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	140.007i	1.17653i
$120$	0	0
$121$	103.000	0.851240
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 88.0000i	− 0.692913i	−0.938066	−	0.346457i	$-0.887385\pi$
$127$	− 88.0000i	− 0.692913i	0.938066	−	0.346457i	$-0.112615\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	169.706i	1.29546i	0.761869	+	0.647731i	$0.224282\pi$
$131$	169.706i	1.29546i	−0.761869	+	0.647731i	$0.775718\pi$
$132$	0	0
$133$	319.000i	2.39850i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	224.860	1.64131	0.820657	−	0.571421i	$-0.193608\pi$
$137$	224.860	1.64131	0.820657	+	0.571421i	$0.193608\pi$
$138$	0	0
$139$	−208.000	−1.49640	−0.748201	−	0.663472i	$-0.769082\pi$
$139$	−208.000	−1.49640	−0.748201	+	0.663472i	$0.769082\pi$
$140$	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
$149$	190.919i	1.28133i	0.767819	+	0.640667i	$0.221342\pi$
$149$	190.919i	1.28133i	−0.767819	+	0.640667i	$0.778658\pi$
$150$	0	0
$151$	253.000	1.67550	0.837748	−	0.546057i	$-0.183872\pi$
$151$	253.000	1.67550	0.837748	+	0.546057i	$0.183872\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	127.000i	0.808917i	0.914556	+	0.404459i	$0.132540\pi$
$157$	127.000i	0.808917i	−0.914556	+	0.404459i	$0.867460\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	420.021i	2.60883i
$162$	0	0
$163$	19.0000i	0.116564i	0.998300	+	0.0582822i	$0.0185623\pi$
$163$	19.0000i	0.116564i	−0.998300	+	0.0582822i	$0.981438\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−135.765	−0.812961	−0.406481	−	0.913659i	$-0.633244\pi$
$167$	−135.765	−0.812961	−0.406481	+	0.913659i	$0.633244\pi$
$168$	0	0
$169$	120.000	0.710059
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−216.375	−1.25072	−0.625360	−	0.780336i	$-0.715048\pi$
$173$	−216.375	−1.25072	−0.625360	+	0.780336i	$0.715048\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	97.5807i	0.545144i	0.962135	+	0.272572i	$0.0878743\pi$
$179$	97.5807i	0.545144i	−0.962135	+	0.272572i	$0.912126\pi$
$180$	0	0
$181$	143.000	0.790055	0.395028	−	0.918669i	$-0.370735\pi$
$181$	143.000	0.790055	0.395028	+	0.918669i	$0.370735\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 54.0000i	− 0.288770i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	72.1249i	0.377617i	0.982014	+	0.188809i	$0.0604626\pi$
$191$	72.1249i	0.377617i	−0.982014	+	0.188809i	$0.939537\pi$
$192$	0	0
$193$	− 79.0000i	− 0.409326i	−0.978832	−	0.204663i	$-0.934390\pi$
$193$	− 79.0000i	− 0.409326i	0.978832	−	0.204663i	$-0.0656099\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−80.6102	−0.409189	−0.204594	−	0.978847i	$-0.565588\pi$
$197$	−80.6102	−0.409189	−0.204594	+	0.978847i	$0.565588\pi$
$198$	0	0
$199$	−85.0000	−0.427136	−0.213568	−	0.976928i	$-0.568508\pi$
$199$	−85.0000	−0.427136	−0.213568	+	0.976928i	$0.568508\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−513.360	−2.52886
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 123.037i	− 0.588692i
$210$	0	0
$211$	−11.0000	−0.0521327	−0.0260664	−	0.999660i	$-0.508298\pi$
$211$	−11.0000	−0.0521327	−0.0260664	+	0.999660i	$0.508298\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 319.000i	− 1.47005i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 89.0955i	− 0.403147i
$222$	0	0
$223$	181.000i	0.811659i	0.913949	+	0.405830i	$0.133017\pi$
$223$	181.000i	0.811659i	−0.913949	+	0.405830i	$0.866983\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	441.235	1.94376	0.971882	−	0.235467i	$-0.0756618\pi$
$227$	441.235	1.94376	0.971882	+	0.235467i	$0.0756618\pi$
$228$	0	0
$229$	7.00000	0.0305677	0.0152838	−	0.999883i	$-0.495135\pi$
$229$	7.00000	0.0305677	0.0152838	+	0.999883i	$0.495135\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−33.9411	−0.145670	−0.0728350	−	0.997344i	$-0.523205\pi$
$233$	−33.9411	−0.145670	−0.0728350	+	0.997344i	$0.523205\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	33.9411i	0.142013i	0.997476	+	0.0710065i	$0.0226211\pi$
$239$	33.9411i	0.142013i	−0.997476	+	0.0710065i	$0.977379\pi$
$240$	0	0
$241$	47.0000	0.195021	0.0975104	−	0.995235i	$-0.468912\pi$
$241$	47.0000	0.195021	0.0975104	+	0.995235i	$0.468912\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 203.000i	− 0.821862i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 33.9411i	− 0.135224i	−0.997712	−	0.0676118i	$-0.978462\pi$
$251$	− 33.9411i	− 0.135224i	0.997712	−	0.0676118i	$-0.0215379\pi$
$252$	0	0
$253$	− 162.000i	− 0.640316i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	373.352	1.45273	0.726366	−	0.687308i	$-0.241208\pi$
$257$	373.352	1.45273	0.726366	+	0.687308i	$0.241208\pi$
$258$	0	0
$259$	616.000	2.37838
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−33.9411	−0.129054	−0.0645269	−	0.997916i	$-0.520554\pi$
$263$	−33.9411	−0.129054	−0.0645269	+	0.997916i	$0.520554\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	462.448i	1.71914i	0.511021	+	0.859568i	$0.329268\pi$
$269$	462.448i	1.71914i	−0.511021	+	0.859568i	$0.670732\pi$
$270$	0	0
$271$	−200.000	−0.738007	−0.369004	−	0.929428i	$-0.620301\pi$
$271$	−200.000	−0.738007	−0.369004	+	0.929428i	$0.620301\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	73.0000i	0.263538i	0.991280	+	0.131769i	$0.0420657\pi$
$277$	73.0000i	0.263538i	−0.991280	+	0.131769i	$0.957934\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	224.860i	0.800213i	0.916469	+	0.400107i	$0.131027\pi$
$281$	224.860i	0.800213i	−0.916469	+	0.400107i	$0.868973\pi$
$282$	0	0
$283$	493.000i	1.74205i	0.491239	+	0.871025i	$0.336544\pi$
$283$	493.000i	1.74205i	−0.491239	+	0.871025i	$0.663456\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	746.705	2.60176
$288$	0	0
$289$	−127.000	−0.439446
$290$	0	0
$291$	0	0
$292$	0	0
$293$	284.257	0.970160	0.485080	−	0.874470i	$-0.338790\pi$
$293$	284.257	0.970160	0.485080	+	0.874470i	$0.338790\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 267.286i	− 0.893934i
$300$	0	0
$301$	55.0000	0.182724
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 235.000i	− 0.765472i	−0.923858	−	0.382736i	$-0.874982\pi$
$307$	− 235.000i	− 0.765472i	0.923858	−	0.382736i	$-0.125018\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	581.242i	1.86894i	0.356036	+	0.934472i	$0.384128\pi$
$311$	581.242i	1.86894i	−0.356036	+	0.934472i	$0.615872\pi$
$312$	0	0
$313$	− 169.000i	− 0.539936i	−0.962869	−	0.269968i	$-0.912987\pi$
$313$	− 169.000i	− 0.539936i	0.962869	−	0.269968i	$-0.0870131\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−101.823	−0.321209	−0.160605	−	0.987019i	$-0.551344\pi$
$317$	−101.823	−0.321209	−0.160605	+	0.987019i	$0.551344\pi$
$318$	0	0
$319$	198.000	0.620690
$320$	0	0
$321$	0	0
$322$	0	0
$323$	369.110	1.14275
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	700.036i	2.12777i
$330$	0	0
$331$	−176.000	−0.531722	−0.265861	−	0.964011i	$-0.585656\pi$
$331$	−176.000	−0.531722	−0.265861	+	0.964011i	$0.585656\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 617.000i	− 1.83086i	−0.402477	−	0.915430i	$-0.631851\pi$
$337$	− 617.000i	− 1.83086i	0.402477	−	0.915430i	$-0.368149\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	123.037i	0.360811i
$342$	0	0
$343$	− 253.000i	− 0.737609i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	309.713	0.892544	0.446272	−	0.894897i	$-0.352751\pi$
$347$	309.713	0.892544	0.446272	+	0.894897i	$0.352751\pi$
$348$	0	0
$349$	328.000	0.939828	0.469914	−	0.882712i	$-0.344285\pi$
$349$	328.000	0.939828	0.469914	+	0.882712i	$0.344285\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	156.978	0.444696	0.222348	−	0.974967i	$-0.428628\pi$
$353$	156.978	0.444696	0.222348	+	0.974967i	$0.428628\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	305.470i	0.850892i	0.904984	+	0.425446i	$0.139883\pi$
$359$	305.470i	0.850892i	−0.904984	+	0.425446i	$0.860117\pi$
$360$	0	0
$361$	480.000	1.32964
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	59.0000i	0.160763i	0.996764	+	0.0803815i	$0.0256139\pi$
$367$	59.0000i	0.160763i	−0.996764	+	0.0803815i	$0.974386\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 746.705i	− 2.01268i
$372$	0	0
$373$	− 415.000i	− 1.11260i	−0.830981	−	0.556300i	$-0.812220\pi$
$373$	− 415.000i	− 1.11260i	0.830981	−	0.556300i	$-0.187780\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	326.683	0.866534
$378$	0	0
$379$	−235.000	−0.620053	−0.310026	−	0.950728i	$-0.600338\pi$
$379$	−235.000	−0.620053	−0.310026	+	0.950728i	$0.600338\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	101.823	0.265857	0.132929	−	0.991126i	$-0.457562\pi$
$383$	101.823	0.265857	0.132929	+	0.991126i	$0.457562\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	169.706i	0.436261i	0.975920	+	0.218131i	$0.0699959\pi$
$389$	169.706i	0.436261i	−0.975920	+	0.218131i	$0.930004\pi$
$390$	0	0
$391$	486.000	1.24297
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 761.000i	− 1.91688i	−0.285300	−	0.958438i	$-0.592093\pi$
$397$	− 761.000i	− 1.91688i	0.285300	−	0.958438i	$-0.407907\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 475.176i	− 1.18498i	−0.805579	−	0.592488i	$-0.798146\pi$
$401$	− 475.176i	− 1.18498i	0.805579	−	0.592488i	$-0.201854\pi$
$402$	0	0
$403$	203.000i	0.503722i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−237.588	−0.583754
$408$	0	0
$409$	−113.000	−0.276284	−0.138142	−	0.990412i	$-0.544113\pi$
$409$	−113.000	−0.276284	−0.138142	+	0.990412i	$0.544113\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−326.683	−0.791001
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	33.9411i	0.0810051i	0.999179	+	0.0405025i	$0.0128959\pi$
$419$	33.9411i	0.0810051i	−0.999179	+	0.0405025i	$0.987104\pi$
$420$	0	0
$421$	−40.0000	−0.0950119	−0.0475059	−	0.998871i	$-0.515127\pi$
$421$	−40.0000	−0.0950119	−0.0475059	+	0.998871i	$0.515127\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 605.000i	− 1.41686i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 470.933i	− 1.09265i	−0.837573	−	0.546326i	$-0.816026\pi$
$431$	− 470.933i	− 1.09265i	0.837573	−	0.546326i	$-0.183974\pi$
$432$	0	0
$433$	407.000i	0.939954i	0.882679	+	0.469977i	$0.155738\pi$
$433$	407.000i	0.939954i	−0.882679	+	0.469977i	$0.844262\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1107.33	2.53393
$438$	0	0
$439$	461.000	1.05011	0.525057	−	0.851067i	$-0.324044\pi$
$439$	461.000	1.05011	0.525057	+	0.851067i	$0.324044\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−746.705	−1.68556	−0.842782	−	0.538255i	$-0.819084\pi$
$443$	−746.705	−1.68556	−0.842782	+	0.538255i	$0.819084\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 101.823i	− 0.226778i	−0.993551	−	0.113389i	$-0.963829\pi$
$449$	− 101.823i	− 0.226778i	0.993551	−	0.113389i	$-0.0361707\pi$
$450$	0	0
$451$	−288.000	−0.638581
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	850.000i	1.85996i	0.367615	+	0.929978i	$0.380174\pi$
$457$	850.000i	1.85996i	−0.367615	+	0.929978i	$0.619826\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	576.999i	1.25163i	0.779974	+	0.625813i	$0.215232\pi$
$461$	576.999i	1.25163i	−0.779974	+	0.625813i	$0.784768\pi$
$462$	0	0
$463$	− 296.000i	− 0.639309i	−0.947534	−	0.319654i	$-0.896433\pi$
$463$	− 296.000i	− 0.639309i	0.947534	−	0.319654i	$-0.103567\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	377.595	0.808555	0.404277	−	0.914636i	$-0.367523\pi$
$467$	377.595	0.808555	0.404277	+	0.914636i	$0.367523\pi$
$468$	0	0
$469$	407.000	0.867804
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−21.2132	−0.0448482
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	683.065i	1.42602i	0.701152	+	0.713012i	$0.252669\pi$
$479$	683.065i	1.42602i	−0.701152	+	0.713012i	$0.747331\pi$
$480$	0	0
$481$	−392.000	−0.814969
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	221.000i	0.453799i	0.973918	+	0.226899i	$0.0728589\pi$
$487$	221.000i	0.453799i	−0.973918	+	0.226899i	$0.927141\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	916.410i	1.86642i	0.359336	+	0.933208i	$0.383003\pi$
$491$	916.410i	1.86642i	−0.359336	+	0.933208i	$0.616997\pi$
$492$	0	0
$493$	594.000i	1.20487i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−373.352	−0.751212
$498$	0	0
$499$	−661.000	−1.32465	−0.662325	−	0.749217i	$-0.730430\pi$
$499$	−661.000	−1.32465	−0.662325	+	0.749217i	$0.730430\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	67.8823	0.134955	0.0674774	−	0.997721i	$-0.478505\pi$
$503$	67.8823	0.134955	0.0674774	+	0.997721i	$0.478505\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 835.800i	− 1.64204i	−0.570897	−	0.821022i	$-0.693404\pi$
$509$	− 835.800i	− 1.64204i	0.570897	−	0.821022i	$-0.306596\pi$
$510$	0	0
$511$	176.000	0.344423
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 270.000i	− 0.522244i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	224.860i	0.431593i	0.976438	+	0.215797i	$0.0692348\pi$
$521$	224.860i	0.431593i	−0.976438	+	0.215797i	$0.930765\pi$
$522$	0	0
$523$	− 149.000i	− 0.284895i	−0.989802	−	0.142447i	$-0.954503\pi$
$523$	− 149.000i	− 0.284895i	0.989802	−	0.142447i	$-0.0454972\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−369.110	−0.700398
$528$	0	0
$529$	929.000	1.75614
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−475.176	−0.891512
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	305.470i	0.566735i
$540$	0	0
$541$	17.0000	0.0314233	0.0157116	−	0.999877i	$-0.494999\pi$
$541$	17.0000	0.0314233	0.0157116	+	0.999877i	$0.494999\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 454.000i	− 0.829982i	−0.909826	−	0.414991i	$-0.863785\pi$
$547$	− 454.000i	− 0.829982i	0.909826	−	0.414991i	$-0.136215\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1353.40i	2.45627i
$552$	0	0
$553$	1144.00i	2.06872i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	80.6102	0.144722	0.0723610	−	0.997379i	$-0.476947\pi$
$557$	80.6102	0.144722	0.0723610	+	0.997379i	$0.476947\pi$
$558$	0	0
$559$	−35.0000	−0.0626118
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−674.580	−1.19819	−0.599094	−	0.800679i	$-0.704472\pi$
$563$	−674.580	−1.19819	−0.599094	+	0.800679i	$0.704472\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 12.7279i	− 0.0223689i	−0.999937	−	0.0111845i	$-0.996440\pi$
$569$	− 12.7279i	− 0.0223689i	0.999937	−	0.0111845i	$-0.00356020\pi$
$570$	0	0
$571$	787.000	1.37828	0.689142	−	0.724626i	$-0.257988\pi$
$571$	787.000	1.37828	0.689142	+	0.724626i	$0.257988\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	697.000i	1.20797i	0.796995	+	0.603986i	$0.206422\pi$
$577$	697.000i	1.20797i	−0.796995	+	0.603986i	$0.793578\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	326.683i	0.562278i
$582$	0	0
$583$	288.000i	0.493997i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.9411	0.0578213	0.0289107	−	0.999582i	$-0.490796\pi$
$587$	33.9411	0.0578213	0.0289107	+	0.999582i	$0.490796\pi$
$588$	0	0
$589$	−841.000	−1.42784
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−441.235	−0.744072	−0.372036	−	0.928218i	$-0.621340\pi$
$593$	−441.235	−0.744072	−0.372036	+	0.928218i	$0.621340\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 33.9411i	− 0.0566630i	−0.999599	−	0.0283315i	$-0.990981\pi$
$599$	− 33.9411i	− 0.0566630i	0.999599	−	0.0283315i	$-0.00901940\pi$
$600$	0	0
$601$	−607.000	−1.00998	−0.504992	−	0.863124i	$-0.668504\pi$
$601$	−607.000	−1.00998	−0.504992	+	0.863124i	$0.668504\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 568.000i	− 0.935750i	−0.883795	−	0.467875i	$-0.845020\pi$
$607$	− 568.000i	− 0.935750i	0.883795	−	0.467875i	$-0.154980\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 445.477i	− 0.729095i
$612$	0	0
$613$	− 232.000i	− 0.378467i	−0.981932	−	0.189233i	$-0.939400\pi$
$613$	− 232.000i	− 0.378467i	0.981932	−	0.189233i	$-0.0606002\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−814.587	−1.32024	−0.660119	−	0.751161i	$-0.729494\pi$
$617$	−814.587	−1.32024	−0.660119	+	0.751161i	$0.729494\pi$
$618$	0	0
$619$	−109.000	−0.176090	−0.0880452	−	0.996116i	$-0.528062\pi$
$619$	−109.000	−0.176090	−0.0880452	+	0.996116i	$0.528062\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1493.41	2.39713
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 712.764i	− 1.13317i
$630$	0	0
$631$	955.000	1.51347	0.756735	−	0.653721i	$-0.226793\pi$
$631$	955.000	1.51347	0.756735	+	0.653721i	$0.226793\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	504.000i	0.791209i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 67.8823i	− 0.105901i	−0.998597	−	0.0529503i	$-0.983138\pi$
$641$	− 67.8823i	− 0.105901i	0.998597	−	0.0529503i	$-0.0168625\pi$
$642$	0	0
$643$	− 368.000i	− 0.572317i	−0.958182	−	0.286159i	$-0.907622\pi$
$643$	− 368.000i	− 0.572317i	0.958182	−	0.286159i	$-0.0923784\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−543.058	−0.839348	−0.419674	−	0.907675i	$-0.637856\pi$
$647$	−543.058	−0.839348	−0.419674	+	0.907675i	$0.637856\pi$
$648$	0	0
$649$	126.000	0.194145
$650$	0	0
$651$	0	0
$652$	0	0
$653$	598.212	0.916099	0.458049	−	0.888927i	$-0.348548\pi$
$653$	598.212	0.916099	0.458049	+	0.888927i	$0.348548\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	203.647i	0.309024i	0.987991	+	0.154512i	$0.0493805\pi$
$659$	203.647i	0.309024i	−0.987991	+	0.154512i	$0.950619\pi$
$660$	0	0
$661$	872.000	1.31921	0.659607	−	0.751611i	$-0.270723\pi$
$661$	872.000	1.31921	0.659607	+	0.751611i	$0.270723\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1782.00i	2.67166i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	233.345i	0.347757i
$672$	0	0
$673$	− 1120.00i	− 1.66419i	−0.554633	−	0.832095i	$-0.687141\pi$
$673$	− 1120.00i	− 1.66419i	0.554633	−	0.832095i	$-0.312859\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	700.036	1.03403	0.517013	−	0.855977i	$-0.327044\pi$
$677$	700.036	1.03403	0.517013	+	0.855977i	$0.327044\pi$
$678$	0	0
$679$	451.000	0.664212
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−712.764	−1.04358	−0.521789	−	0.853075i	$-0.674735\pi$
$683$	−712.764	−1.04358	−0.521789	+	0.853075i	$0.674735\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	475.176i	0.689660i
$690$	0	0
$691$	−410.000	−0.593343	−0.296671	−	0.954980i	$-0.595877\pi$
$691$	−410.000	−0.593343	−0.296671	+	0.954980i	$0.595877\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 864.000i	− 1.23960i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 190.919i	− 0.272352i	−0.990685	−	0.136176i	$-0.956519\pi$
$701$	− 190.919i	− 0.272352i	0.990685	−	0.136176i	$-0.0434813\pi$
$702$	0	0
$703$	− 1624.00i	− 2.31010i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−373.352	−0.528080
$708$	0	0
$709$	−1103.00	−1.55571	−0.777856	−	0.628442i	$-0.783693\pi$
$709$	−1103.00	−1.55571	−0.777856	+	0.628442i	$0.783693\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1107.33	−1.55306
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	377.595i	0.525167i	0.964909	+	0.262583i	$0.0845745\pi$
$719$	377.595i	0.525167i	−0.964909	+	0.262583i	$0.915425\pi$
$720$	0	0
$721$	682.000	0.945908
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	587.000i	0.807428i	0.914885	+	0.403714i	$0.132281\pi$
$727$	587.000i	0.807428i	−0.914885	+	0.403714i	$0.867719\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 63.6396i	− 0.0870583i
$732$	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
$737$	−156.978	−0.212996
$738$	0	0
$739$	−304.000	−0.411367	−0.205683	−	0.978619i	$-0.565942\pi$
$739$	−304.000	−0.411367	−0.205683	+	0.978619i	$0.565942\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	848.528	1.14203	0.571015	−	0.820940i	$-0.306550\pi$
$743$	848.528	1.14203	0.571015	+	0.820940i	$0.306550\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−728.000	−0.969374	−0.484687	−	0.874688i	$-0.661067\pi$
$751$	−728.000	−0.969374	−0.484687	+	0.874688i	$0.661067\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 815.000i	− 1.07662i	−0.842747	−	0.538309i	$-0.819063\pi$
$757$	− 815.000i	− 1.07662i	0.842747	−	0.538309i	$-0.180937\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 203.647i	− 0.267604i	−0.991008	−	0.133802i	$-0.957281\pi$
$761$	− 203.647i	− 0.267604i	0.991008	−	0.133802i	$-0.0427186\pi$
$762$	0	0
$763$	1859.00i	2.43644i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	207.889	0.271042
$768$	0	0
$769$	−785.000	−1.02081	−0.510403	−	0.859935i	$-0.670504\pi$
$769$	−785.000	−1.02081	−0.510403	+	0.859935i	$0.670504\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	373.352	0.482991	0.241496	−	0.970402i	$-0.422362\pi$
$773$	373.352	0.482991	0.241496	+	0.970402i	$0.422362\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1968.59i	− 2.52707i
$780$	0	0
$781$	144.000	0.184379
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 925.000i	− 1.17535i	−0.809097	−	0.587675i	$-0.800043\pi$
$787$	− 925.000i	− 1.17535i	0.809097	−	0.587675i	$-0.199957\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	373.352i	0.472000i
$792$	0	0
$793$	385.000i	0.485498i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−339.411	−0.425861	−0.212931	−	0.977067i	$-0.568301\pi$
$797$	−339.411	−0.425861	−0.212931	+	0.977067i	$0.568301\pi$
$798$	0	0
$799$	810.000	1.01377
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−67.8823	−0.0845358
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	780.646i	0.964952i	0.875909	+	0.482476i	$0.160262\pi$
$809$	780.646i	0.964952i	−0.875909	+	0.482476i	$0.839738\pi$
$810$	0	0
$811$	−635.000	−0.782984	−0.391492	−	0.920182i	$-0.628041\pi$
$811$	−635.000	−0.782984	−0.391492	+	0.920182i	$0.628041\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 145.000i	− 0.177479i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 861.256i	− 1.04903i	−0.851400	−	0.524516i	$-0.824246\pi$
$821$	− 861.256i	− 1.04903i	0.851400	−	0.524516i	$-0.175754\pi$
$822$	0	0
$823$	763.000i	0.927096i	0.886072	+	0.463548i	$0.153424\pi$
$823$	763.000i	0.927096i	−0.886072	+	0.463548i	$0.846576\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−407.294	−0.492495	−0.246248	−	0.969207i	$-0.579198\pi$
$827$	−407.294	−0.492495	−0.246248	+	0.969207i	$0.579198\pi$
$828$	0	0
$829$	−182.000	−0.219542	−0.109771	−	0.993957i	$-0.535012\pi$
$829$	−182.000	−0.219542	−0.109771	+	0.993957i	$0.535012\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−916.410	−1.10013
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 750.947i	− 0.895051i	−0.894271	−	0.447525i	$-0.852305\pi$
$839$	− 750.947i	− 0.895051i	0.894271	−	0.447525i	$-0.147695\pi$
$840$	0	0
$841$	−1337.00	−1.58977
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1133.00i	1.33766i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 2138.29i	− 2.51268i
$852$	0	0
$853$	− 409.000i	− 0.479484i	−0.970837	−	0.239742i	$-0.922937\pi$
$853$	− 409.000i	− 0.479484i	0.970837	−	0.239742i	$-0.0770629\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	67.8823	0.0792092	0.0396046	−	0.999215i	$-0.487390\pi$
$857$	67.8823	0.0792092	0.0396046	+	0.999215i	$0.487390\pi$
$858$	0	0
$859$	458.000	0.533178	0.266589	−	0.963810i	$-0.414103\pi$
$859$	458.000	0.533178	0.266589	+	0.963810i	$0.414103\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1255.82	1.45518	0.727591	−	0.686011i	$-0.240640\pi$
$863$	1255.82	1.45518	0.727591	+	0.686011i	$0.240640\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 441.235i	− 0.507750i
$870$	0	0
$871$	−259.000	−0.297359
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	895.000i	1.02052i	0.860019	+	0.510262i	$0.170452\pi$
$877$	895.000i	1.02052i	−0.860019	+	0.510262i	$0.829548\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 916.410i	− 1.04019i	−0.854107	−	0.520097i	$-0.825896\pi$
$881$	− 916.410i	− 1.04019i	0.854107	−	0.520097i	$-0.174104\pi$
$882$	0	0
$883$	781.000i	0.884485i	0.896896	+	0.442242i	$0.145817\pi$
$883$	781.000i	0.884485i	−0.896896	+	0.442242i	$0.854183\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	538.815	0.607458	0.303729	−	0.952758i	$-0.401768\pi$
$887$	538.815	0.607458	0.303729	+	0.952758i	$0.401768\pi$
$888$	0	0
$889$	968.000	1.08886
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1845.55	2.06668
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1353.40i	− 1.50545i
$900$	0	0
$901$	−864.000	−0.958935
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 688.000i	− 0.758545i	−0.925285	−	0.379272i	$-0.876174\pi$
$907$	− 688.000i	− 0.758545i	0.925285	−	0.379272i	$-0.123826\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1022.48i	− 1.12237i	−0.827691	−	0.561184i	$-0.810346\pi$
$911$	− 1022.48i	− 1.12237i	0.827691	−	0.561184i	$-0.189654\pi$
$912$	0	0
$913$	− 126.000i	− 0.138007i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1866.76	−2.03573
$918$	0	0
$919$	35.0000	0.0380849	0.0190424	−	0.999819i	$-0.493938\pi$
$919$	35.0000	0.0380849	0.0190424	+	0.999819i	$0.493938\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	237.588	0.257408
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	284.257i	0.305982i	0.988228	+	0.152991i	$0.0488905\pi$
$929$	284.257i	0.305982i	−0.988228	+	0.152991i	$0.951110\pi$
$930$	0	0
$931$	−2088.00	−2.24275
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 377.000i	− 0.402348i	−0.979556	−	0.201174i	$-0.935524\pi$
$937$	− 377.000i	− 0.402348i	0.979556	−	0.201174i	$-0.0644757\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 793.374i	− 0.843118i	−0.906801	−	0.421559i	$-0.861483\pi$
$941$	− 793.374i	− 0.843118i	0.906801	−	0.421559i	$-0.138517\pi$
$942$	0	0
$943$	− 2592.00i	− 2.74867i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1013.99	1.07074	0.535370	−	0.844618i	$-0.320172\pi$
$947$	1013.99	1.07074	0.535370	+	0.844618i	$0.320172\pi$
$948$	0	0
$949$	−112.000	−0.118019
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1120.06	1.17530	0.587648	−	0.809117i	$-0.300054\pi$
$953$	1120.06	1.17530	0.587648	+	0.809117i	$0.300054\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2473.46i	2.57921i
$960$	0	0
$961$	−120.000	−0.124870
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1160.00i	1.19959i	0.800155	+	0.599793i	$0.204750\pi$
$967$	1160.00i	1.19959i	−0.800155	+	0.599793i	$0.795250\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	649.124i	0.668511i	0.942483	+	0.334255i	$0.108485\pi$
$971$	649.124i	0.668511i	−0.942483	+	0.334255i	$0.891515\pi$
$972$	0	0
$973$	− 2288.00i	− 2.35149i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−632.153	−0.647035	−0.323518	−	0.946222i	$-0.604866\pi$
$977$	−632.153	−0.647035	−0.323518	+	0.946222i	$0.604866\pi$
$978$	0	0
$979$	−576.000	−0.588355
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−814.587	−0.828674	−0.414337	−	0.910123i	$-0.635987\pi$
$983$	−814.587	−0.828674	−0.414337	+	0.910123i	$0.635987\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 190.919i	− 0.193042i
$990$	0	0
$991$	157.000	0.158426	0.0792129	−	0.996858i	$-0.474759\pi$
$991$	157.000	0.158426	0.0792129	+	0.996858i	$0.474759\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	616.000i	0.617854i	0.951086	+	0.308927i	$0.0999698\pi$
$997$	616.000i	0.617854i	−0.951086	+	0.308927i	$0.900030\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.c.f.449.3		4
3.2	odd	2	inner	3600.3.c.f.449.4		4
4.3	odd	2		450.3.b.a.449.1		4
5.2	odd	4		3600.3.l.a.1601.1		2
5.3	odd	4		3600.3.l.k.1601.1		2
5.4	even	2	inner	3600.3.c.f.449.1		4
12.11	even	2		450.3.b.a.449.3		4
15.2	even	4		3600.3.l.a.1601.2		2
15.8	even	4		3600.3.l.k.1601.2		2
15.14	odd	2	inner	3600.3.c.f.449.2		4
20.3	even	4		450.3.d.a.251.2	yes	2
20.7	even	4		450.3.d.g.251.1	yes	2
20.19	odd	2		450.3.b.a.449.4		4
60.23	odd	4		450.3.d.a.251.1	✓	2
60.47	odd	4		450.3.d.g.251.2	yes	2
60.59	even	2		450.3.b.a.449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.3.b.a.449.1		4	4.3	odd	2
450.3.b.a.449.2		4	60.59	even	2
450.3.b.a.449.3		4	12.11	even	2
450.3.b.a.449.4		4	20.19	odd	2
450.3.d.a.251.1	✓	2	60.23	odd	4
450.3.d.a.251.2	yes	2	20.3	even	4
450.3.d.g.251.1	yes	2	20.7	even	4
450.3.d.g.251.2	yes	2	60.47	odd	4
3600.3.c.f.449.1		4	5.4	even	2	inner
3600.3.c.f.449.2		4	15.14	odd	2	inner
3600.3.c.f.449.3		4	1.1	even	1	trivial
3600.3.c.f.449.4		4	3.2	odd	2	inner
3600.3.l.a.1601.1		2	5.2	odd	4
3600.3.l.a.1601.2		2	15.2	even	4
3600.3.l.k.1601.1		2	5.3	odd	4
3600.3.l.k.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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+11.0000i q^{7} +O(q^{10})\)	\(q+11.0000i q^{7} -4.24264i q^{11} -7.00000i q^{13} +12.7279 q^{17} +29.0000 q^{19} +38.1838 q^{23} +46.6690i q^{29} -29.0000 q^{31} -56.0000i q^{37} -67.8823i q^{41} -5.00000i q^{43} +63.6396 q^{47} -72.0000 q^{49} -67.8823 q^{53} +29.6985i q^{59} -55.0000 q^{61} -37.0000i q^{67} +33.9411i q^{71} -16.0000i q^{73} +46.6690 q^{77} +104.000 q^{79} +29.6985 q^{83} -135.765i q^{89} +77.0000 q^{91} -41.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 116 q^{19} - 116 q^{31} - 288 q^{49} - 220 q^{61} + 416 q^{79} + 308 q^{91}+O(q^{100}) \) 4 * q + 116 * q^19 - 116 * q^31 - 288 * q^49 - 220 * q^61 + 416 * q^79 + 308 * q^91

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