LMFDB - Embedded newform 1800.4.f.a.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,4,Mod(649,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1800.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1800.f (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:	$106.203438010$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1800.649
Dual form			1800.4.f.a.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{7} +O(q^{10})$	$q-4.00000i q^{7} -72.0000 q^{11} -6.00000i q^{13} +38.0000i q^{17} -52.0000 q^{19} -152.000i q^{23} -78.0000 q^{29} +120.000 q^{31} +150.000i q^{37} -362.000 q^{41} -484.000i q^{43} +280.000i q^{47} +327.000 q^{49} +670.000i q^{53} +696.000 q^{59} +222.000 q^{61} +4.00000i q^{67} -96.0000 q^{71} +178.000i q^{73} +288.000i q^{77} +632.000 q^{79} +612.000i q^{83} +994.000 q^{89} -24.0000 q^{91} -1634.00i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 144 q^{11} - 104 q^{19} - 156 q^{29} + 240 q^{31} - 724 q^{41} + 654 q^{49} + 1392 q^{59} + 444 q^{61} - 192 q^{71} + 1264 q^{79} + 1988 q^{89} - 48 q^{91}+O(q^{100}) $ 2 * q - 144 * q^11 - 104 * q^19 - 156 * q^29 + 240 * q^31 - 724 * q^41 + 654 * q^49 + 1392 * q^59 + 444 * q^61 - 192 * q^71 + 1264 * q^79 + 1988 * q^89 - 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$7$	− 4.00000i	− 0.215980i	−0.994152	−	0.107990i	$-0.965559\pi$
$7$	− 4.00000i	− 0.215980i	0.994152	−	0.107990i	$-0.0344414\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−72.0000	−1.97353	−0.986764	−	0.162160i	$-0.948154\pi$
$11$	−72.0000	−1.97353	−0.986764	+	0.162160i	$0.948154\pi$
$12$	0	0
$13$	− 6.00000i	− 0.128008i	−0.997950	−	0.0640039i	$-0.979613\pi$
$13$	− 6.00000i	− 0.128008i	0.997950	−	0.0640039i	$-0.0203870\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	38.0000i	0.542138i	0.962560	+	0.271069i	$0.0873772\pi$
$17$	38.0000i	0.542138i	−0.962560	+	0.271069i	$0.912623\pi$
$18$	0	0
$19$	−52.0000	−0.627875	−0.313937	−	0.949444i	$-0.601648\pi$
$19$	−52.0000	−0.627875	−0.313937	+	0.949444i	$0.601648\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 152.000i	− 1.37801i	−0.724757	−	0.689004i	$-0.758048\pi$
$23$	− 152.000i	− 1.37801i	0.724757	−	0.689004i	$-0.241952\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−78.0000	−0.499456	−0.249728	−	0.968316i	$-0.580341\pi$
$29$	−78.0000	−0.499456	−0.249728	+	0.968316i	$0.580341\pi$
$30$	0	0
$31$	120.000	0.695246	0.347623	−	0.937634i	$-0.386989\pi$
$31$	120.000	0.695246	0.347623	+	0.937634i	$0.386989\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	150.000i	0.666482i	0.942842	+	0.333241i	$0.108142\pi$
$37$	150.000i	0.666482i	−0.942842	+	0.333241i	$0.891858\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−362.000	−1.37890	−0.689450	−	0.724333i	$-0.742148\pi$
$41$	−362.000	−1.37890	−0.689450	+	0.724333i	$0.742148\pi$
$42$	0	0
$43$	− 484.000i	− 1.71650i	−0.513236	−	0.858248i	$-0.671553\pi$
$43$	− 484.000i	− 1.71650i	0.513236	−	0.858248i	$-0.328447\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	280.000i	0.868983i	0.900676	+	0.434491i	$0.143072\pi$
$47$	280.000i	0.868983i	−0.900676	+	0.434491i	$0.856928\pi$
$48$	0	0
$49$	327.000	0.953353
$50$	0	0
$51$	0	0
$52$	0	0
$53$	670.000i	1.73644i	0.496175	+	0.868222i	$0.334737\pi$
$53$	670.000i	1.73644i	−0.496175	+	0.868222i	$0.665263\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	696.000	1.53579	0.767894	−	0.640577i	$-0.221305\pi$
$59$	696.000	1.53579	0.767894	+	0.640577i	$0.221305\pi$
$60$	0	0
$61$	222.000	0.465970	0.232985	−	0.972480i	$-0.425151\pi$
$61$	222.000	0.465970	0.232985	+	0.972480i	$0.425151\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000i	0.00729370i	0.999993	+	0.00364685i	$0.00116083\pi$
$67$	4.00000i	0.00729370i	−0.999993	+	0.00364685i	$0.998839\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−96.0000	−0.160466	−0.0802331	−	0.996776i	$-0.525566\pi$
$71$	−96.0000	−0.160466	−0.0802331	+	0.996776i	$0.525566\pi$
$72$	0	0
$73$	178.000i	0.285388i	0.989767	+	0.142694i	$0.0455765\pi$
$73$	178.000i	0.285388i	−0.989767	+	0.142694i	$0.954424\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	288.000i	0.426242i
$78$	0	0
$79$	632.000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	632.000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	612.000i	0.809346i	0.914461	+	0.404673i	$0.132615\pi$
$83$	612.000i	0.809346i	−0.914461	+	0.404673i	$0.867385\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	994.000	1.18386	0.591931	−	0.805988i	$-0.298366\pi$
$89$	994.000	1.18386	0.591931	+	0.805988i	$0.298366\pi$
$90$	0	0
$91$	−24.0000	−0.0276471
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1634.00i	− 1.71039i	−0.518309	−	0.855194i	$-0.673438\pi$
$97$	− 1634.00i	− 1.71039i	0.518309	−	0.855194i	$-0.326562\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−890.000	−0.876815	−0.438407	−	0.898776i	$-0.644457\pi$
$101$	−890.000	−0.876815	−0.438407	+	0.898776i	$0.644457\pi$
$102$	0	0
$103$	− 524.000i	− 0.501274i	−0.968081	−	0.250637i	$-0.919360\pi$
$103$	− 524.000i	− 0.501274i	0.968081	−	0.250637i	$-0.0806401\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	932.000i	0.842055i	0.907048	+	0.421027i	$0.138330\pi$
$107$	932.000i	0.842055i	−0.907048	+	0.421027i	$0.861670\pi$
$108$	0	0
$109$	−446.000	−0.391918	−0.195959	−	0.980612i	$-0.562782\pi$
$109$	−446.000	−0.391918	−0.195959	+	0.980612i	$0.562782\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	786.000i	0.654342i	0.944965	+	0.327171i	$0.106095\pi$
$113$	786.000i	0.654342i	−0.944965	+	0.327171i	$0.893905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	152.000	0.117091
$120$	0	0
$121$	3853.00	2.89482
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 716.000i	− 0.500273i	−0.968211	−	0.250137i	$-0.919524\pi$
$127$	− 716.000i	− 0.500273i	0.968211	−	0.250137i	$-0.0804756\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	808.000	0.538895	0.269448	−	0.963015i	$-0.413159\pi$
$131$	808.000	0.538895	0.269448	+	0.963015i	$0.413159\pi$
$132$	0	0
$133$	208.000i	0.135608i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1770.00i	− 1.10381i	−0.833909	−	0.551903i	$-0.813902\pi$
$137$	− 1770.00i	− 1.10381i	0.833909	−	0.551903i	$-0.186098\pi$
$138$	0	0
$139$	924.000	0.563832	0.281916	−	0.959439i	$-0.409030\pi$
$139$	924.000	0.563832	0.281916	+	0.959439i	$0.409030\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	432.000i	0.252627i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3198.00	−1.75832	−0.879162	−	0.476522i	$-0.841897\pi$
$149$	−3198.00	−1.75832	−0.879162	+	0.476522i	$0.841897\pi$
$150$	0	0
$151$	−3384.00	−1.82375	−0.911874	−	0.410470i	$-0.865365\pi$
$151$	−3384.00	−1.82375	−0.911874	+	0.410470i	$0.865365\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3302.00i	1.67852i	0.543727	+	0.839262i	$0.317013\pi$
$157$	3302.00i	1.67852i	−0.543727	+	0.839262i	$0.682987\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−608.000	−0.297622
$162$	0	0
$163$	2252.00i	1.08215i	0.840975	+	0.541074i	$0.181982\pi$
$163$	2252.00i	1.08215i	−0.840975	+	0.541074i	$0.818018\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 184.000i	− 0.0852596i	−0.999091	−	0.0426298i	$-0.986426\pi$
$167$	− 184.000i	− 0.0852596i	0.999091	−	0.0426298i	$-0.0135736\pi$
$168$	0	0
$169$	2161.00	0.983614
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2646.00i	1.16284i	0.813603	+	0.581421i	$0.197503\pi$
$173$	2646.00i	1.16284i	−0.813603	+	0.581421i	$0.802497\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−608.000	−0.253877	−0.126939	−	0.991911i	$-0.540515\pi$
$179$	−608.000	−0.253877	−0.126939	+	0.991911i	$0.540515\pi$
$180$	0	0
$181$	2246.00	0.922342	0.461171	−	0.887311i	$-0.347430\pi$
$181$	2246.00	0.922342	0.461171	+	0.887311i	$0.347430\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2736.00i	− 1.06993i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3848.00	1.45776	0.728878	−	0.684643i	$-0.240042\pi$
$191$	3848.00	1.45776	0.728878	+	0.684643i	$0.240042\pi$
$192$	0	0
$193$	2058.00i	0.767555i	0.923426	+	0.383777i	$0.125377\pi$
$193$	2058.00i	0.767555i	−0.923426	+	0.383777i	$0.874623\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3838.00i	− 1.38805i	−0.719950	−	0.694026i	$-0.755835\pi$
$197$	− 3838.00i	− 1.38805i	0.719950	−	0.694026i	$-0.244165\pi$
$198$	0	0
$199$	1992.00	0.709594	0.354797	−	0.934943i	$-0.384550\pi$
$199$	1992.00	0.709594	0.354797	+	0.934943i	$0.384550\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	312.000i	0.107872i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3744.00	1.23913
$210$	0	0
$211$	4764.00	1.55435	0.777174	−	0.629286i	$-0.216653\pi$
$211$	4764.00	1.55435	0.777174	+	0.629286i	$0.216653\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 480.000i	− 0.150159i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	228.000	0.0693979
$222$	0	0
$223$	4092.00i	1.22879i	0.788998	+	0.614396i	$0.210600\pi$
$223$	4092.00i	1.22879i	−0.788998	+	0.614396i	$0.789400\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	468.000i	0.136838i	0.997657	+	0.0684191i	$0.0217955\pi$
$227$	468.000i	0.136838i	−0.997657	+	0.0684191i	$0.978205\pi$
$228$	0	0
$229$	5586.00	1.61194	0.805968	−	0.591959i	$-0.201645\pi$
$229$	5586.00	1.61194	0.805968	+	0.591959i	$0.201645\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1058.00i	0.297476i	0.988877	+	0.148738i	$0.0475211\pi$
$233$	1058.00i	0.297476i	−0.988877	+	0.148738i	$0.952479\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6840.00	1.85123	0.925613	−	0.378472i	$-0.123550\pi$
$239$	6840.00	1.85123	0.925613	+	0.378472i	$0.123550\pi$
$240$	0	0
$241$	−6430.00	−1.71864	−0.859321	−	0.511437i	$-0.829113\pi$
$241$	−6430.00	−1.71864	−0.859321	+	0.511437i	$0.829113\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	312.000i	0.0803728i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6352.00	1.59735	0.798675	−	0.601763i	$-0.205535\pi$
$251$	6352.00	1.59735	0.798675	+	0.601763i	$0.205535\pi$
$252$	0	0
$253$	10944.0i	2.71954i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1422.00i	0.345144i	0.984997	+	0.172572i	$0.0552077\pi$
$257$	1422.00i	0.345144i	−0.984997	+	0.172572i	$0.944792\pi$
$258$	0	0
$259$	600.000	0.143947
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 7224.00i	− 1.69373i	−0.531808	−	0.846865i	$-0.678487\pi$
$263$	− 7224.00i	− 1.69373i	0.531808	−	0.846865i	$-0.321513\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3186.00	0.722133	0.361067	−	0.932540i	$-0.382413\pi$
$269$	3186.00	0.722133	0.361067	+	0.932540i	$0.382413\pi$
$270$	0	0
$271$	−256.000	−0.0573834	−0.0286917	−	0.999588i	$-0.509134\pi$
$271$	−256.000	−0.0573834	−0.0286917	+	0.999588i	$0.509134\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5942.00i	1.28888i	0.764654	+	0.644441i	$0.222910\pi$
$277$	5942.00i	1.28888i	−0.764654	+	0.644441i	$0.777090\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3202.00	−0.679770	−0.339885	−	0.940467i	$-0.610388\pi$
$281$	−3202.00	−0.679770	−0.339885	+	0.940467i	$0.610388\pi$
$282$	0	0
$283$	3940.00i	0.827593i	0.910370	+	0.413796i	$0.135797\pi$
$283$	3940.00i	0.827593i	−0.910370	+	0.413796i	$0.864203\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1448.00i	0.297814i
$288$	0	0
$289$	3469.00	0.706086
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1826.00i	− 0.364082i	−0.983291	−	0.182041i	$-0.941730\pi$
$293$	− 1826.00i	− 0.364082i	0.983291	−	0.182041i	$-0.0582704\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−912.000	−0.176396
$300$	0	0
$301$	−1936.00	−0.370728
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6580.00i	− 1.22326i	−0.791144	−	0.611629i	$-0.790514\pi$
$307$	− 6580.00i	− 1.22326i	0.791144	−	0.611629i	$-0.209486\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5728.00	1.04439	0.522195	−	0.852826i	$-0.325113\pi$
$311$	5728.00	1.04439	0.522195	+	0.852826i	$0.325113\pi$
$312$	0	0
$313$	− 1742.00i	− 0.314580i	−0.987552	−	0.157290i	$-0.949724\pi$
$313$	− 1742.00i	− 0.314580i	0.987552	−	0.157290i	$-0.0502758\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8746.00i	1.54960i	0.632204	+	0.774802i	$0.282150\pi$
$317$	8746.00i	1.54960i	−0.632204	+	0.774802i	$0.717850\pi$
$318$	0	0
$319$	5616.00	0.985692
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1976.00i	− 0.340395i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1120.00	0.187683
$330$	0	0
$331$	−2564.00	−0.425771	−0.212885	−	0.977077i	$-0.568286\pi$
$331$	−2564.00	−0.425771	−0.212885	+	0.977077i	$0.568286\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4166.00i	0.673402i	0.941612	+	0.336701i	$0.109311\pi$
$337$	4166.00i	0.673402i	−0.941612	+	0.336701i	$0.890689\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8640.00	−1.37209
$342$	0	0
$343$	− 2680.00i	− 0.421885i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9444.00i	− 1.46104i	−0.682892	−	0.730519i	$-0.739278\pi$
$347$	− 9444.00i	− 1.46104i	0.682892	−	0.730519i	$-0.260722\pi$
$348$	0	0
$349$	9218.00	1.41383	0.706917	−	0.707296i	$-0.250085\pi$
$349$	9218.00	1.41383	0.706917	+	0.707296i	$0.250085\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4698.00i	0.708355i	0.935178	+	0.354177i	$0.115239\pi$
$353$	4698.00i	0.708355i	−0.935178	+	0.354177i	$0.884761\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6056.00	−0.890316	−0.445158	−	0.895452i	$-0.646852\pi$
$359$	−6056.00	−0.890316	−0.445158	+	0.895452i	$0.646852\pi$
$360$	0	0
$361$	−4155.00	−0.605773
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8228.00i	1.17029i	0.810927	+	0.585147i	$0.198963\pi$
$367$	8228.00i	1.17029i	−0.810927	+	0.585147i	$0.801037\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2680.00	0.375037
$372$	0	0
$373$	5954.00i	0.826505i	0.910616	+	0.413253i	$0.135607\pi$
$373$	5954.00i	0.826505i	−0.910616	+	0.413253i	$0.864393\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	468.000i	0.0639343i
$378$	0	0
$379$	−5284.00	−0.716150	−0.358075	−	0.933693i	$-0.616567\pi$
$379$	−5284.00	−0.716150	−0.358075	+	0.933693i	$0.616567\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 9832.00i	− 1.31173i	−0.754879	−	0.655864i	$-0.772305\pi$
$383$	− 9832.00i	− 1.31173i	0.754879	−	0.655864i	$-0.227695\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−222.000	−0.0289353	−0.0144677	−	0.999895i	$-0.504605\pi$
$389$	−222.000	−0.0289353	−0.0144677	+	0.999895i	$0.504605\pi$
$390$	0	0
$391$	5776.00	0.747071
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 12098.0i	− 1.52942i	−0.644372	−	0.764712i	$-0.722881\pi$
$397$	− 12098.0i	− 1.52942i	0.644372	−	0.764712i	$-0.277119\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5958.00	0.741966	0.370983	−	0.928640i	$-0.379021\pi$
$401$	5958.00	0.741966	0.370983	+	0.928640i	$0.379021\pi$
$402$	0	0
$403$	− 720.000i	− 0.0889969i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 10800.0i	− 1.31532i
$408$	0	0
$409$	−1930.00	−0.233331	−0.116665	−	0.993171i	$-0.537221\pi$
$409$	−1930.00	−0.233331	−0.116665	+	0.993171i	$0.537221\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2784.00i	− 0.331699i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4744.00	0.553125	0.276563	−	0.960996i	$-0.410805\pi$
$419$	4744.00	0.553125	0.276563	+	0.960996i	$0.410805\pi$
$420$	0	0
$421$	1614.00	0.186845	0.0934223	−	0.995627i	$-0.470219\pi$
$421$	1614.00	0.186845	0.0934223	+	0.995627i	$0.470219\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 888.000i	− 0.100640i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9296.00	−1.03892	−0.519458	−	0.854496i	$-0.673866\pi$
$431$	−9296.00	−1.03892	−0.519458	+	0.854496i	$0.673866\pi$
$432$	0	0
$433$	− 3494.00i	− 0.387785i	−0.981023	−	0.193893i	$-0.937889\pi$
$433$	− 3494.00i	− 0.387785i	0.981023	−	0.193893i	$-0.0621113\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7904.00i	0.865216i
$438$	0	0
$439$	12584.0	1.36811	0.684056	−	0.729429i	$-0.260214\pi$
$439$	12584.0	1.36811	0.684056	+	0.729429i	$0.260214\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12852.0i	1.37837i	0.724586	+	0.689184i	$0.242031\pi$
$443$	12852.0i	1.37837i	−0.724586	+	0.689184i	$0.757969\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14458.0	1.51963	0.759816	−	0.650138i	$-0.225289\pi$
$449$	14458.0	1.51963	0.759816	+	0.650138i	$0.225289\pi$
$450$	0	0
$451$	26064.0	2.72130
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4310.00i	0.441167i	0.975368	+	0.220583i	$0.0707961\pi$
$457$	4310.00i	0.441167i	−0.975368	+	0.220583i	$0.929204\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5338.00	−0.539296	−0.269648	−	0.962959i	$-0.586907\pi$
$461$	−5338.00	−0.539296	−0.269648	+	0.962959i	$0.586907\pi$
$462$	0	0
$463$	1156.00i	0.116034i	0.998316	+	0.0580171i	$0.0184778\pi$
$463$	1156.00i	0.116034i	−0.998316	+	0.0580171i	$0.981522\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5948.00i	0.589380i	0.955593	+	0.294690i	$0.0952164\pi$
$467$	5948.00i	0.589380i	−0.955593	+	0.294690i	$0.904784\pi$
$468$	0	0
$469$	16.0000	0.00157529
$470$	0	0
$471$	0	0
$472$	0	0
$473$	34848.0i	3.38755i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6888.00	0.657037	0.328519	−	0.944498i	$-0.393451\pi$
$479$	6888.00	0.657037	0.328519	+	0.944498i	$0.393451\pi$
$480$	0	0
$481$	900.000	0.0853149
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 2892.00i	− 0.269095i	−0.990907	−	0.134547i	$-0.957042\pi$
$487$	− 2892.00i	− 0.269095i	0.990907	−	0.134547i	$-0.0429580\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4096.00	−0.376476	−0.188238	−	0.982123i	$-0.560278\pi$
$491$	−4096.00	−0.376476	−0.188238	+	0.982123i	$0.560278\pi$
$492$	0	0
$493$	− 2964.00i	− 0.270775i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	384.000i	0.0346575i
$498$	0	0
$499$	−11060.0	−0.992212	−0.496106	−	0.868262i	$-0.665237\pi$
$499$	−11060.0	−0.992212	−0.496106	+	0.868262i	$0.665237\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9648.00i	0.855235i	0.903960	+	0.427617i	$0.140647\pi$
$503$	9648.00i	0.855235i	−0.903960	+	0.427617i	$0.859353\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−10062.0	−0.876209	−0.438104	−	0.898924i	$-0.644350\pi$
$509$	−10062.0	−0.876209	−0.438104	+	0.898924i	$0.644350\pi$
$510$	0	0
$511$	712.000	0.0616380
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 20160.0i	− 1.71496i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7966.00	0.669859	0.334930	−	0.942243i	$-0.391287\pi$
$521$	7966.00	0.669859	0.334930	+	0.942243i	$0.391287\pi$
$522$	0	0
$523$	7668.00i	0.641106i	0.947231	+	0.320553i	$0.103869\pi$
$523$	7668.00i	0.641106i	−0.947231	+	0.320553i	$0.896131\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4560.00i	0.376920i
$528$	0	0
$529$	−10937.0	−0.898907
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2172.00i	0.176510i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−23544.0	−1.88147
$540$	0	0
$541$	6590.00	0.523708	0.261854	−	0.965107i	$-0.415666\pi$
$541$	6590.00	0.523708	0.261854	+	0.965107i	$0.415666\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4700.00i	0.367381i	0.982984	+	0.183691i	$0.0588044\pi$
$547$	4700.00i	0.367381i	−0.982984	+	0.183691i	$0.941196\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4056.00	0.313596
$552$	0	0
$553$	− 2528.00i	− 0.194397i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 15766.0i	− 1.19933i	−0.800251	−	0.599665i	$-0.795300\pi$
$557$	− 15766.0i	− 1.19933i	0.800251	−	0.599665i	$-0.204700\pi$
$558$	0	0
$559$	−2904.00	−0.219725
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 22788.0i	− 1.70586i	−0.522025	−	0.852930i	$-0.674823\pi$
$563$	− 22788.0i	− 1.70586i	0.522025	−	0.852930i	$-0.325177\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3358.00	−0.247407	−0.123704	−	0.992319i	$-0.539477\pi$
$569$	−3358.00	−0.247407	−0.123704	+	0.992319i	$0.539477\pi$
$570$	0	0
$571$	−11444.0	−0.838733	−0.419366	−	0.907817i	$-0.637748\pi$
$571$	−11444.0	−0.838733	−0.419366	+	0.907817i	$0.637748\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	10622.0i	0.766377i	0.923670	+	0.383189i	$0.125174\pi$
$577$	10622.0i	0.766377i	−0.923670	+	0.383189i	$0.874826\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2448.00	0.174802
$582$	0	0
$583$	− 48240.0i	− 3.42692i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6588.00i	− 0.463230i	−0.972808	−	0.231615i	$-0.925599\pi$
$587$	− 6588.00i	− 0.463230i	0.972808	−	0.231615i	$-0.0744009\pi$
$588$	0	0
$589$	−6240.00	−0.436528
$590$	0	0
$591$	0	0
$592$	0	0
$593$	11362.0i	0.786815i	0.919364	+	0.393408i	$0.128704\pi$
$593$	11362.0i	0.786815i	−0.919364	+	0.393408i	$0.871296\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1624.00	0.110776	0.0553880	−	0.998465i	$-0.482360\pi$
$599$	1624.00	0.110776	0.0553880	+	0.998465i	$0.482360\pi$
$600$	0	0
$601$	−14950.0	−1.01468	−0.507340	−	0.861746i	$-0.669371\pi$
$601$	−14950.0	−1.01468	−0.507340	+	0.861746i	$0.669371\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8244.00i	− 0.551258i	−0.961264	−	0.275629i	$-0.911114\pi$
$607$	− 8244.00i	− 0.551258i	0.961264	−	0.275629i	$-0.0888861\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1680.00	0.111237
$612$	0	0
$613$	6698.00i	0.441321i	0.975351	+	0.220660i	$0.0708213\pi$
$613$	6698.00i	0.441321i	−0.975351	+	0.220660i	$0.929179\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22670.0i	1.47919i	0.673053	+	0.739595i	$0.264983\pi$
$617$	22670.0i	1.47919i	−0.673053	+	0.739595i	$0.735017\pi$
$618$	0	0
$619$	10060.0	0.653224	0.326612	−	0.945159i	$-0.394093\pi$
$619$	10060.0	0.653224	0.326612	+	0.945159i	$0.394093\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 3976.00i	− 0.255690i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5700.00	−0.361326
$630$	0	0
$631$	10240.0	0.646035	0.323017	−	0.946393i	$-0.395303\pi$
$631$	10240.0	0.646035	0.323017	+	0.946393i	$0.395303\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1962.00i	− 0.122037i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13218.0	−0.814477	−0.407238	−	0.913322i	$-0.633508\pi$
$641$	−13218.0	−0.814477	−0.407238	+	0.913322i	$0.633508\pi$
$642$	0	0
$643$	− 23412.0i	− 1.43589i	−0.696098	−	0.717946i	$-0.745082\pi$
$643$	− 23412.0i	− 1.43589i	0.696098	−	0.717946i	$-0.254918\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 15264.0i	− 0.927496i	−0.885967	−	0.463748i	$-0.846504\pi$
$647$	− 15264.0i	− 0.927496i	0.885967	−	0.463748i	$-0.153496\pi$
$648$	0	0
$649$	−50112.0	−3.03092
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1482.00i	− 0.0888134i	−0.999014	−	0.0444067i	$-0.985860\pi$
$653$	− 1482.00i	− 0.0888134i	0.999014	−	0.0444067i	$-0.0141397\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18920.0	−1.11839	−0.559195	−	0.829036i	$-0.688890\pi$
$659$	−18920.0	−1.11839	−0.559195	+	0.829036i	$0.688890\pi$
$660$	0	0
$661$	−24218.0	−1.42507	−0.712535	−	0.701637i	$-0.752453\pi$
$661$	−24218.0	−1.42507	−0.712535	+	0.701637i	$0.752453\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11856.0i	0.688255i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−15984.0	−0.919606
$672$	0	0
$673$	890.000i	0.0509762i	0.999675	+	0.0254881i	$0.00811399\pi$
$673$	890.000i	0.0509762i	−0.999675	+	0.0254881i	$0.991886\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29250.0i	1.66052i	0.557380	+	0.830258i	$0.311807\pi$
$677$	29250.0i	1.66052i	−0.557380	+	0.830258i	$0.688193\pi$
$678$	0	0
$679$	−6536.00	−0.369409
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 14580.0i	− 0.816820i	−0.912799	−	0.408410i	$-0.866083\pi$
$683$	− 14580.0i	− 0.816820i	0.912799	−	0.408410i	$-0.133917\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4020.00	0.222278
$690$	0	0
$691$	23668.0	1.30300	0.651500	−	0.758649i	$-0.274140\pi$
$691$	23668.0	1.30300	0.651500	+	0.758649i	$0.274140\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 13756.0i	− 0.747555i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−32402.0	−1.74580	−0.872901	−	0.487898i	$-0.837764\pi$
$701$	−32402.0	−1.74580	−0.872901	+	0.487898i	$0.837764\pi$
$702$	0	0
$703$	− 7800.00i	− 0.418467i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3560.00i	0.189374i
$708$	0	0
$709$	30626.0	1.62226	0.811131	−	0.584865i	$-0.198852\pi$
$709$	30626.0	1.62226	0.811131	+	0.584865i	$0.198852\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 18240.0i	− 0.958055i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	13440.0	0.697117	0.348559	−	0.937287i	$-0.386671\pi$
$719$	13440.0	0.697117	0.348559	+	0.937287i	$0.386671\pi$
$720$	0	0
$721$	−2096.00	−0.108265
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	24820.0i	1.26619i	0.774073	+	0.633097i	$0.218217\pi$
$727$	24820.0i	1.26619i	−0.774073	+	0.633097i	$0.781783\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	18392.0	0.930578
$732$	0	0
$733$	21986.0i	1.10787i	0.832559	+	0.553937i	$0.186875\pi$
$733$	21986.0i	1.10787i	−0.832559	+	0.553937i	$0.813125\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 288.000i	− 0.0143943i
$738$	0	0
$739$	−4420.00	−0.220017	−0.110008	−	0.993931i	$-0.535088\pi$
$739$	−4420.00	−0.220017	−0.110008	+	0.993931i	$0.535088\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 34560.0i	− 1.70644i	−0.521553	−	0.853219i	$-0.674647\pi$
$743$	− 34560.0i	− 1.70644i	0.521553	−	0.853219i	$-0.325353\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3728.00	0.181867
$750$	0	0
$751$	−24792.0	−1.20462	−0.602312	−	0.798261i	$-0.705754\pi$
$751$	−24792.0	−1.20462	−0.602312	+	0.798261i	$0.705754\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2166.00i	0.103996i	0.998647	+	0.0519978i	$0.0165589\pi$
$757$	2166.00i	0.103996i	−0.998647	+	0.0519978i	$0.983441\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10622.0	0.505975	0.252988	−	0.967470i	$-0.418587\pi$
$761$	10622.0	0.505975	0.252988	+	0.967470i	$0.418587\pi$
$762$	0	0
$763$	1784.00i	0.0846463i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 4176.00i	− 0.196593i
$768$	0	0
$769$	−29826.0	−1.39864	−0.699319	−	0.714809i	$-0.746513\pi$
$769$	−29826.0	−1.39864	−0.699319	+	0.714809i	$0.746513\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 6386.00i	− 0.297139i	−0.988902	−	0.148570i	$-0.952533\pi$
$773$	− 6386.00i	− 0.297139i	0.988902	−	0.148570i	$-0.0474669\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18824.0	0.865776
$780$	0	0
$781$	6912.00	0.316685
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 3516.00i	− 0.159253i	−0.996825	−	0.0796263i	$-0.974627\pi$
$787$	− 3516.00i	− 0.159253i	0.996825	−	0.0796263i	$-0.0253727\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3144.00	0.141325
$792$	0	0
$793$	− 1332.00i	− 0.0596478i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 25030.0i	− 1.11243i	−0.831038	−	0.556216i	$-0.812253\pi$
$797$	− 25030.0i	− 1.11243i	0.831038	−	0.556216i	$-0.187747\pi$
$798$	0	0
$799$	−10640.0	−0.471109
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 12816.0i	− 0.563221i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7962.00	0.346019	0.173009	−	0.984920i	$-0.444651\pi$
$809$	7962.00	0.346019	0.173009	+	0.984920i	$0.444651\pi$
$810$	0	0
$811$	−34668.0	−1.50106	−0.750529	−	0.660837i	$-0.770201\pi$
$811$	−34668.0	−1.50106	−0.750529	+	0.660837i	$0.770201\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	25168.0i	1.07774i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−250.000	−0.0106274	−0.00531368	−	0.999986i	$-0.501691\pi$
$821$	−250.000	−0.0106274	−0.00531368	+	0.999986i	$0.501691\pi$
$822$	0	0
$823$	− 6388.00i	− 0.270561i	−0.990807	−	0.135280i	$-0.956806\pi$
$823$	− 6388.00i	− 0.270561i	0.990807	−	0.135280i	$-0.0431936\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3932.00i	0.165331i	0.996577	+	0.0826657i	$0.0263434\pi$
$827$	3932.00i	0.165331i	−0.996577	+	0.0826657i	$0.973657\pi$
$828$	0	0
$829$	25906.0	1.08535	0.542673	−	0.839944i	$-0.317412\pi$
$829$	25906.0	1.08535	0.542673	+	0.839944i	$0.317412\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	12426.0i	0.516849i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9944.00	−0.409184	−0.204592	−	0.978847i	$-0.565587\pi$
$839$	−9944.00	−0.409184	−0.204592	+	0.978847i	$0.565587\pi$
$840$	0	0
$841$	−18305.0	−0.750543
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 15412.0i	− 0.625221i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	22800.0	0.918418
$852$	0	0
$853$	− 14630.0i	− 0.587247i	−0.955921	−	0.293623i	$-0.905139\pi$
$853$	− 14630.0i	− 0.587247i	0.955921	−	0.293623i	$-0.0948612\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	478.000i	0.0190527i	0.999955	+	0.00952635i	$0.00303238\pi$
$857$	478.000i	0.0190527i	−0.999955	+	0.00952635i	$0.996968\pi$
$858$	0	0
$859$	−24132.0	−0.958525	−0.479263	−	0.877672i	$-0.659096\pi$
$859$	−24132.0	−0.958525	−0.479263	+	0.877672i	$0.659096\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 15776.0i	− 0.622273i	−0.950365	−	0.311136i	$-0.899290\pi$
$863$	− 15776.0i	− 0.622273i	0.950365	−	0.311136i	$-0.100710\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−45504.0	−1.77631
$870$	0	0
$871$	24.0000	0.000933650 0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33542.0i	1.29149i	0.763555	+	0.645743i	$0.223452\pi$
$877$	33542.0i	1.29149i	−0.763555	+	0.645743i	$0.776548\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−22858.0	−0.874127	−0.437063	−	0.899431i	$-0.643981\pi$
$881$	−22858.0	−0.874127	−0.437063	+	0.899431i	$0.643981\pi$
$882$	0	0
$883$	2764.00i	0.105341i	0.998612	+	0.0526704i	$0.0167733\pi$
$883$	2764.00i	0.105341i	−0.998612	+	0.0526704i	$0.983227\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6216.00i	0.235302i	0.993055	+	0.117651i	$0.0375364\pi$
$887$	6216.00i	0.235302i	−0.993055	+	0.117651i	$0.962464\pi$
$888$	0	0
$889$	−2864.00	−0.108049
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 14560.0i	− 0.545612i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9360.00	−0.347245
$900$	0	0
$901$	−25460.0	−0.941394
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	18884.0i	0.691326i	0.938359	+	0.345663i	$0.112346\pi$
$907$	18884.0i	0.691326i	−0.938359	+	0.345663i	$0.887654\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	15232.0	0.553961	0.276981	−	0.960876i	$-0.410666\pi$
$911$	15232.0	0.553961	0.276981	+	0.960876i	$0.410666\pi$
$912$	0	0
$913$	− 44064.0i	− 1.59727i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3232.00i	− 0.116390i
$918$	0	0
$919$	−7744.00	−0.277966	−0.138983	−	0.990295i	$-0.544383\pi$
$919$	−7744.00	−0.277966	−0.138983	+	0.990295i	$0.544383\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	576.000i	0.0205409i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	22266.0	0.786355	0.393177	−	0.919463i	$-0.371376\pi$
$929$	22266.0	0.786355	0.393177	+	0.919463i	$0.371376\pi$
$930$	0	0
$931$	−17004.0	−0.598586
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 16202.0i	− 0.564884i	−0.959284	−	0.282442i	$-0.908856\pi$
$937$	− 16202.0i	− 0.564884i	0.959284	−	0.282442i	$-0.0911445\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	53494.0	1.85319	0.926596	−	0.376057i	$-0.122720\pi$
$941$	53494.0	1.85319	0.926596	+	0.376057i	$0.122720\pi$
$942$	0	0
$943$	55024.0i	1.90014i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2332.00i	− 0.0800209i	−0.999199	−	0.0400105i	$-0.987261\pi$
$947$	− 2332.00i	− 0.0800209i	0.999199	−	0.0400105i	$-0.0127391\pi$
$948$	0	0
$949$	1068.00	0.0365319
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 15414.0i	− 0.523933i	−0.965077	−	0.261967i	$-0.915629\pi$
$953$	− 15414.0i	− 0.523933i	0.965077	−	0.261967i	$-0.0843710\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−7080.00	−0.238400
$960$	0	0
$961$	−15391.0	−0.516633
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 35012.0i	− 1.16433i	−0.813070	−	0.582167i	$-0.802205\pi$
$967$	− 35012.0i	− 1.16433i	0.813070	−	0.582167i	$-0.197795\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11360.0	−0.375448	−0.187724	−	0.982222i	$-0.560111\pi$
$971$	−11360.0	−0.375448	−0.187724	+	0.982222i	$0.560111\pi$
$972$	0	0
$973$	− 3696.00i	− 0.121776i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24586.0i	− 0.805093i	−0.915400	−	0.402546i	$-0.868125\pi$
$977$	− 24586.0i	− 0.805093i	0.915400	−	0.402546i	$-0.131875\pi$
$978$	0	0
$979$	−71568.0	−2.33639
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 8832.00i	− 0.286569i	−0.989682	−	0.143284i	$-0.954234\pi$
$983$	− 8832.00i	− 0.286569i	0.989682	−	0.143284i	$-0.0457663\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−73568.0	−2.36535
$990$	0	0
$991$	−22912.0	−0.734434	−0.367217	−	0.930135i	$-0.619689\pi$
$991$	−22912.0	−0.734434	−0.367217	+	0.930135i	$0.619689\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10974.0i	0.348596i	0.984693	+	0.174298i	$0.0557656\pi$
$997$	10974.0i	0.348596i	−0.984693	+	0.174298i	$0.944234\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.f.a.649.1		2
3.2	odd	2		600.4.f.i.49.1		2
5.2	odd	4		360.4.a.l.1.1		1
5.3	odd	4		1800.4.a.n.1.1		1
5.4	even	2	inner	1800.4.f.a.649.2		2
12.11	even	2		1200.4.f.a.49.2		2
15.2	even	4		120.4.a.a.1.1	✓	1
15.8	even	4		600.4.a.l.1.1		1
15.14	odd	2		600.4.f.i.49.2		2
20.7	even	4		720.4.a.v.1.1		1
60.23	odd	4		1200.4.a.k.1.1		1
60.47	odd	4		240.4.a.h.1.1		1
60.59	even	2		1200.4.f.a.49.1		2
120.77	even	4		960.4.a.bf.1.1		1
120.107	odd	4		960.4.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.a.1.1	✓	1	15.2	even	4
240.4.a.h.1.1		1	60.47	odd	4
360.4.a.l.1.1		1	5.2	odd	4
600.4.a.l.1.1		1	15.8	even	4
600.4.f.i.49.1		2	3.2	odd	2
600.4.f.i.49.2		2	15.14	odd	2
720.4.a.v.1.1		1	20.7	even	4
960.4.a.o.1.1		1	120.107	odd	4
960.4.a.bf.1.1		1	120.77	even	4
1200.4.a.k.1.1		1	60.23	odd	4
1200.4.f.a.49.1		2	60.59	even	2
1200.4.f.a.49.2		2	12.11	even	2
1800.4.a.n.1.1		1	5.3	odd	4
1800.4.f.a.649.1		2	1.1	even	1	trivial
1800.4.f.a.649.2		2	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{7} +O(q^{10})\)	\(q-4.00000i q^{7} -72.0000 q^{11} -6.00000i q^{13} +38.0000i q^{17} -52.0000 q^{19} -152.000i q^{23} -78.0000 q^{29} +120.000 q^{31} +150.000i q^{37} -362.000 q^{41} -484.000i q^{43} +280.000i q^{47} +327.000 q^{49} +670.000i q^{53} +696.000 q^{59} +222.000 q^{61} +4.00000i q^{67} -96.0000 q^{71} +178.000i q^{73} +288.000i q^{77} +632.000 q^{79} +612.000i q^{83} +994.000 q^{89} -24.0000 q^{91} -1634.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 144 q^{11} - 104 q^{19} - 156 q^{29} + 240 q^{31} - 724 q^{41} + 654 q^{49} + 1392 q^{59} + 444 q^{61} - 192 q^{71} + 1264 q^{79} + 1988 q^{89} - 48 q^{91}+O(q^{100}) \) 2 * q - 144 * q^11 - 104 * q^19 - 156 * q^29 + 240 * q^31 - 724 * q^41 + 654 * q^49 + 1392 * q^59 + 444 * q^61 - 192 * q^71 + 1264 * q^79 + 1988 * q^89 - 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more