LMFDB - Embedded newform 1152.3.b.g.703.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(703,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1152.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.b (of order $2$, degree $1$, 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:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.4
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.703
Dual form			1152.3.b.g.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{5} +11.3137i q^{7} +O(q^{10})$	$q+4.00000i q^{5} +11.3137i q^{7} +14.1421 q^{11} +20.0000i q^{13} +10.0000 q^{17} -14.1421 q^{19} +11.3137i q^{23} +9.00000 q^{25} -20.0000i q^{29} -45.2548 q^{35} -20.0000i q^{37} +30.0000 q^{41} -2.82843 q^{43} +67.8823i q^{47} -79.0000 q^{49} -60.0000i q^{53} +56.5685i q^{55} -42.4264 q^{59} -28.0000i q^{61} -80.0000 q^{65} -82.0244 q^{67} -56.5685i q^{71} -10.0000 q^{73} +160.000i q^{77} +113.137i q^{79} +25.4558 q^{83} +40.0000i q^{85} -22.0000 q^{89} -226.274 q^{91} -56.5685i q^{95} +150.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 40 q^{17} + 36 q^{25} + 120 q^{41} - 316 q^{49} - 320 q^{65} - 40 q^{73} - 88 q^{89} + 600 q^{97}+O(q^{100}) $ 4 * q + 40 * q^17 + 36 * q^25 + 120 * q^41 - 316 * q^49 - 320 * q^65 - 40 * q^73 - 88 * q^89 + 600 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	4.00000i	0.800000i	0.916515	+	0.400000i	$0.130990\pi$
$5$	4.00000i	0.800000i	−0.916515	+	0.400000i	$0.869010\pi$
$6$	0	0
$7$	11.3137i	1.61624i	0.589015	+	0.808122i	$0.299516\pi$
$7$	11.3137i	1.61624i	−0.589015	+	0.808122i	$0.700484\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	14.1421	1.28565	0.642824	−	0.766014i	$-0.277763\pi$
$11$	14.1421	1.28565	0.642824	+	0.766014i	$0.277763\pi$
$12$	0	0
$13$	20.0000i	1.53846i	0.638971	+	0.769231i	$0.279360\pi$
$13$	20.0000i	1.53846i	−0.638971	+	0.769231i	$0.720640\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	10.0000	0.588235	0.294118	−	0.955769i	$-0.404974\pi$
$17$	10.0000	0.588235	0.294118	+	0.955769i	$0.404974\pi$
$18$	0	0
$19$	−14.1421	−0.744323	−0.372161	−	0.928168i	$-0.621383\pi$
$19$	−14.1421	−0.744323	−0.372161	+	0.928168i	$0.621383\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	11.3137i	0.491900i	0.969282	+	0.245950i	$0.0791000\pi$
$23$	11.3137i	0.491900i	−0.969282	+	0.245950i	$0.920900\pi$
$24$	0	0
$25$	9.00000	0.360000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 20.0000i	− 0.689655i	−0.938666	−	0.344828i	$-0.887937\pi$
$29$	− 20.0000i	− 0.689655i	0.938666	−	0.344828i	$-0.112063\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−45.2548	−1.29300
$36$	0	0
$37$	− 20.0000i	− 0.540541i	−0.962784	−	0.270270i	$-0.912887\pi$
$37$	− 20.0000i	− 0.540541i	0.962784	−	0.270270i	$-0.0871131\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	30.0000	0.731707	0.365854	−	0.930672i	$-0.380777\pi$
$41$	30.0000	0.731707	0.365854	+	0.930672i	$0.380777\pi$
$42$	0	0
$43$	−2.82843	−0.0657774	−0.0328887	−	0.999459i	$-0.510471\pi$
$43$	−2.82843	−0.0657774	−0.0328887	+	0.999459i	$0.510471\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	67.8823i	1.44430i	0.691735	+	0.722152i	$0.256847\pi$
$47$	67.8823i	1.44430i	−0.691735	+	0.722152i	$0.743153\pi$
$48$	0	0
$49$	−79.0000	−1.61224
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 60.0000i	− 1.13208i	−0.824379	−	0.566038i	$-0.808476\pi$
$53$	− 60.0000i	− 1.13208i	0.824379	−	0.566038i	$-0.191524\pi$
$54$	0	0
$55$	56.5685i	1.02852i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−42.4264	−0.719092	−0.359546	−	0.933127i	$-0.617068\pi$
$59$	−42.4264	−0.719092	−0.359546	+	0.933127i	$0.617068\pi$
$60$	0	0
$61$	− 28.0000i	− 0.459016i	−0.973307	−	0.229508i	$-0.926288\pi$
$61$	− 28.0000i	− 0.459016i	0.973307	−	0.229508i	$-0.0737118\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−80.0000	−1.23077
$66$	0	0
$67$	−82.0244	−1.22424	−0.612122	−	0.790763i	$-0.709684\pi$
$67$	−82.0244	−1.22424	−0.612122	+	0.790763i	$0.709684\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 56.5685i	− 0.796740i	−0.917225	−	0.398370i	$-0.869576\pi$
$71$	− 56.5685i	− 0.796740i	0.917225	−	0.398370i	$-0.130424\pi$
$72$	0	0
$73$	−10.0000	−0.136986	−0.0684932	−	0.997652i	$-0.521819\pi$
$73$	−10.0000	−0.136986	−0.0684932	+	0.997652i	$0.521819\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	160.000i	2.07792i
$78$	0	0
$79$	113.137i	1.43211i	0.698041	+	0.716057i	$0.254055\pi$
$79$	113.137i	1.43211i	−0.698041	+	0.716057i	$0.745945\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	25.4558	0.306697	0.153348	−	0.988172i	$-0.450994\pi$
$83$	25.4558	0.306697	0.153348	+	0.988172i	$0.450994\pi$
$84$	0	0
$85$	40.0000i	0.470588i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−22.0000	−0.247191	−0.123596	−	0.992333i	$-0.539443\pi$
$89$	−22.0000	−0.247191	−0.123596	+	0.992333i	$0.539443\pi$
$90$	0	0
$91$	−226.274	−2.48653
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 56.5685i	− 0.595458i
$96$	0	0
$97$	150.000	1.54639	0.773196	−	0.634167i	$-0.218657\pi$
$97$	150.000	1.54639	0.773196	+	0.634167i	$0.218657\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 140.000i	− 1.38614i	−0.720871	−	0.693069i	$-0.756258\pi$
$101$	− 140.000i	− 1.38614i	0.720871	−	0.693069i	$-0.243742\pi$
$102$	0	0
$103$	101.823i	0.988576i	0.869298	+	0.494288i	$0.164571\pi$
$103$	101.823i	0.988576i	−0.869298	+	0.494288i	$0.835429\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−144.250	−1.34813	−0.674064	−	0.738673i	$-0.735453\pi$
$107$	−144.250	−1.34813	−0.674064	+	0.738673i	$0.735453\pi$
$108$	0	0
$109$	68.0000i	0.623853i	0.950106	+	0.311927i	$0.100974\pi$
$109$	68.0000i	0.623853i	−0.950106	+	0.311927i	$0.899026\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	190.000	1.68142	0.840708	−	0.541489i	$-0.182139\pi$
$113$	190.000	1.68142	0.840708	+	0.541489i	$0.182139\pi$
$114$	0	0
$115$	−45.2548	−0.393520
$116$	0	0
$117$	0	0
$118$	0	0
$119$	113.137i	0.950732i
$120$	0	0
$121$	79.0000	0.652893
$122$	0	0
$123$	0	0
$124$	0	0
$125$	136.000i	1.08800i
$126$	0	0
$127$	− 45.2548i	− 0.356337i	−0.984000	−	0.178169i	$-0.942983\pi$
$127$	− 45.2548i	− 0.356337i	0.984000	−	0.178169i	$-0.0570173\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.1421	0.107955	0.0539776	−	0.998542i	$-0.482810\pi$
$131$	14.1421	0.107955	0.0539776	+	0.998542i	$0.482810\pi$
$132$	0	0
$133$	− 160.000i	− 1.20301i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−130.000	−0.948905	−0.474453	−	0.880281i	$-0.657354\pi$
$137$	−130.000	−0.948905	−0.474453	+	0.880281i	$0.657354\pi$
$138$	0	0
$139$	42.4264	0.305226	0.152613	−	0.988286i	$-0.451231\pi$
$139$	42.4264	0.305226	0.152613	+	0.988286i	$0.451231\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	282.843i	1.97792i
$144$	0	0
$145$	80.0000	0.551724
$146$	0	0
$147$	0	0
$148$	0	0
$149$	52.0000i	0.348993i	0.984658	+	0.174497i	$0.0558298\pi$
$149$	52.0000i	0.348993i	−0.984658	+	0.174497i	$0.944170\pi$
$150$	0	0
$151$	169.706i	1.12388i	0.827179	+	0.561939i	$0.189945\pi$
$151$	169.706i	1.12388i	−0.827179	+	0.561939i	$0.810055\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	100.000i	0.636943i	0.947932	+	0.318471i	$0.103169\pi$
$157$	100.000i	0.636943i	−0.947932	+	0.318471i	$0.896831\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−128.000	−0.795031
$162$	0	0
$163$	110.309	0.676740	0.338370	−	0.941013i	$-0.390124\pi$
$163$	110.309	0.676740	0.338370	+	0.941013i	$0.390124\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	124.451i	0.745214i	0.927989	+	0.372607i	$0.121536\pi$
$167$	124.451i	0.745214i	−0.927989	+	0.372607i	$0.878464\pi$
$168$	0	0
$169$	−231.000	−1.36686
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 20.0000i	− 0.115607i	−0.998328	−	0.0578035i	$-0.981590\pi$
$173$	− 20.0000i	− 0.115607i	0.998328	−	0.0578035i	$-0.0184097\pi$
$174$	0	0
$175$	101.823i	0.581848i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−42.4264	−0.237019	−0.118510	−	0.992953i	$-0.537812\pi$
$179$	−42.4264	−0.237019	−0.118510	+	0.992953i	$0.537812\pi$
$180$	0	0
$181$	− 180.000i	− 0.994475i	−0.867614	−	0.497238i	$-0.834348\pi$
$181$	− 180.000i	− 0.994475i	0.867614	−	0.497238i	$-0.165652\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	80.0000	0.432432
$186$	0	0
$187$	141.421	0.756264
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 226.274i	− 1.18468i	−0.805688	−	0.592341i	$-0.798204\pi$
$191$	− 226.274i	− 1.18468i	0.805688	−	0.592341i	$-0.201796\pi$
$192$	0	0
$193$	−170.000	−0.880829	−0.440415	−	0.897795i	$-0.645169\pi$
$193$	−170.000	−0.880829	−0.440415	+	0.897795i	$0.645169\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 220.000i	− 1.11675i	−0.829588	−	0.558376i	$-0.811425\pi$
$197$	− 220.000i	− 1.11675i	0.829588	−	0.558376i	$-0.188575\pi$
$198$	0	0
$199$	− 169.706i	− 0.852792i	−0.904537	−	0.426396i	$-0.859783\pi$
$199$	− 169.706i	− 0.852792i	0.904537	−	0.426396i	$-0.140217\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	226.274	1.11465
$204$	0	0
$205$	120.000i	0.585366i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−200.000	−0.956938
$210$	0	0
$211$	−410.122	−1.94371	−0.971853	−	0.235588i	$-0.924298\pi$
$211$	−410.122	−1.94371	−0.971853	+	0.235588i	$0.924298\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 11.3137i	− 0.0526219i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	200.000i	0.904977i
$222$	0	0
$223$	181.019i	0.811746i	0.913930	+	0.405873i	$0.133032\pi$
$223$	181.019i	0.811746i	−0.913930	+	0.405873i	$0.866968\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−223.446	−0.984342	−0.492171	−	0.870498i	$-0.663797\pi$
$227$	−223.446	−0.984342	−0.492171	+	0.870498i	$0.663797\pi$
$228$	0	0
$229$	− 260.000i	− 1.13537i	−0.823245	−	0.567686i	$-0.807839\pi$
$229$	− 260.000i	− 1.13537i	0.823245	−	0.567686i	$-0.192161\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.0000	0.0429185	0.0214592	−	0.999770i	$-0.493169\pi$
$233$	10.0000	0.0429185	0.0214592	+	0.999770i	$0.493169\pi$
$234$	0	0
$235$	−271.529	−1.15544
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 339.411i	− 1.42013i	−0.704136	−	0.710065i	$-0.748665\pi$
$239$	− 339.411i	− 1.42013i	0.704136	−	0.710065i	$-0.251335\pi$
$240$	0	0
$241$	−10.0000	−0.0414938	−0.0207469	−	0.999785i	$-0.506604\pi$
$241$	−10.0000	−0.0414938	−0.0207469	+	0.999785i	$0.506604\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 316.000i	− 1.28980i
$246$	0	0
$247$	− 282.843i	− 1.14511i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	183.848	0.732461	0.366231	−	0.930524i	$-0.380648\pi$
$251$	183.848	0.732461	0.366231	+	0.930524i	$0.380648\pi$
$252$	0	0
$253$	160.000i	0.632411i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	350.000	1.36187	0.680934	−	0.732345i	$-0.261574\pi$
$257$	350.000	1.36187	0.680934	+	0.732345i	$0.261574\pi$
$258$	0	0
$259$	226.274	0.873645
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 237.588i	− 0.903376i	−0.892176	−	0.451688i	$-0.850822\pi$
$263$	− 237.588i	− 0.903376i	0.892176	−	0.451688i	$-0.149178\pi$
$264$	0	0
$265$	240.000	0.905660
$266$	0	0
$267$	0	0
$268$	0	0
$269$	188.000i	0.698885i	0.936958	+	0.349442i	$0.113629\pi$
$269$	188.000i	0.698885i	−0.936958	+	0.349442i	$0.886371\pi$
$270$	0	0
$271$	− 113.137i	− 0.417480i	−0.977971	−	0.208740i	$-0.933064\pi$
$271$	− 113.137i	− 0.417480i	0.977971	−	0.208740i	$-0.0669362\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	127.279	0.462834
$276$	0	0
$277$	540.000i	1.94946i	0.223390	+	0.974729i	$0.428288\pi$
$277$	540.000i	1.94946i	−0.223390	+	0.974729i	$0.571712\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	490.000	1.74377	0.871886	−	0.489709i	$-0.162897\pi$
$281$	490.000	1.74377	0.871886	+	0.489709i	$0.162897\pi$
$282$	0	0
$283$	483.661	1.70905	0.854525	−	0.519411i	$-0.173848\pi$
$283$	483.661	1.70905	0.854525	+	0.519411i	$0.173848\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	339.411i	1.18262i
$288$	0	0
$289$	−189.000	−0.653979
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 380.000i	− 1.29693i	−0.761245	−	0.648464i	$-0.775412\pi$
$293$	− 380.000i	− 1.29693i	0.761245	−	0.648464i	$-0.224588\pi$
$294$	0	0
$295$	− 169.706i	− 0.575273i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−226.274	−0.756770
$300$	0	0
$301$	− 32.0000i	− 0.106312i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	112.000	0.367213
$306$	0	0
$307$	483.661	1.57544	0.787722	−	0.616031i	$-0.211261\pi$
$307$	483.661	1.57544	0.787722	+	0.616031i	$0.211261\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 169.706i	− 0.545677i	−0.962060	−	0.272839i	$-0.912037\pi$
$311$	− 169.706i	− 0.545677i	0.962060	−	0.272839i	$-0.0879625\pi$
$312$	0	0
$313$	130.000	0.415335	0.207668	−	0.978199i	$-0.433413\pi$
$313$	130.000	0.415335	0.207668	+	0.978199i	$0.433413\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	460.000i	1.45110i	0.688167	+	0.725552i	$0.258416\pi$
$317$	460.000i	1.45110i	−0.688167	+	0.725552i	$0.741584\pi$
$318$	0	0
$319$	− 282.843i	− 0.886654i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−141.421	−0.437837
$324$	0	0
$325$	180.000i	0.553846i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−768.000	−2.33435
$330$	0	0
$331$	325.269	0.982686	0.491343	−	0.870966i	$-0.336506\pi$
$331$	325.269	0.982686	0.491343	+	0.870966i	$0.336506\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 328.098i	− 0.979396i
$336$	0	0
$337$	450.000	1.33531	0.667656	−	0.744470i	$-0.267298\pi$
$337$	450.000	1.33531	0.667656	+	0.744470i	$0.267298\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 339.411i	− 0.989537i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	115.966	0.334195	0.167097	−	0.985940i	$-0.446561\pi$
$347$	115.966	0.334195	0.167097	+	0.985940i	$0.446561\pi$
$348$	0	0
$349$	− 140.000i	− 0.401146i	−0.979679	−	0.200573i	$-0.935720\pi$
$349$	− 140.000i	− 0.401146i	0.979679	−	0.200573i	$-0.0642804\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−130.000	−0.368272	−0.184136	−	0.982901i	$-0.558949\pi$
$353$	−130.000	−0.368272	−0.184136	+	0.982901i	$0.558949\pi$
$354$	0	0
$355$	226.274	0.637392
$356$	0	0
$357$	0	0
$358$	0	0
$359$	622.254i	1.73330i	0.498918	+	0.866649i	$0.333731\pi$
$359$	622.254i	1.73330i	−0.498918	+	0.866649i	$0.666269\pi$
$360$	0	0
$361$	−161.000	−0.445983
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 40.0000i	− 0.109589i
$366$	0	0
$367$	− 610.940i	− 1.66469i	−0.554260	−	0.832344i	$-0.686999\pi$
$367$	− 610.940i	− 1.66469i	0.554260	−	0.832344i	$-0.313001\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	678.823	1.82971
$372$	0	0
$373$	− 420.000i	− 1.12601i	−0.826455	−	0.563003i	$-0.809646\pi$
$373$	− 420.000i	− 1.12601i	0.826455	−	0.563003i	$-0.190354\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	400.000	1.06101
$378$	0	0
$379$	−296.985	−0.783601	−0.391801	−	0.920050i	$-0.628148\pi$
$379$	−296.985	−0.783601	−0.391801	+	0.920050i	$0.628148\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 181.019i	− 0.472635i	−0.971676	−	0.236318i	$-0.924059\pi$
$383$	− 181.019i	− 0.472635i	0.971676	−	0.236318i	$-0.0759406\pi$
$384$	0	0
$385$	−640.000	−1.66234
$386$	0	0
$387$	0	0
$388$	0	0
$389$	372.000i	0.956298i	0.878279	+	0.478149i	$0.158692\pi$
$389$	372.000i	0.956298i	−0.878279	+	0.478149i	$0.841308\pi$
$390$	0	0
$391$	113.137i	0.289353i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−452.548	−1.14569
$396$	0	0
$397$	100.000i	0.251889i	0.992037	+	0.125945i	$0.0401962\pi$
$397$	100.000i	0.251889i	−0.992037	+	0.125945i	$0.959804\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−502.000	−1.25187	−0.625935	−	0.779875i	$-0.715283\pi$
$401$	−502.000	−1.25187	−0.625935	+	0.779875i	$0.715283\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 282.843i	− 0.694945i
$408$	0	0
$409$	−190.000	−0.464548	−0.232274	−	0.972650i	$-0.574617\pi$
$409$	−190.000	−0.464548	−0.232274	+	0.972650i	$0.574617\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 480.000i	− 1.16223i
$414$	0	0
$415$	101.823i	0.245358i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	806.102	1.92387	0.961935	−	0.273278i	$-0.0881077\pi$
$419$	806.102	1.92387	0.961935	+	0.273278i	$0.0881077\pi$
$420$	0	0
$421$	− 292.000i	− 0.693587i	−0.937942	−	0.346793i	$-0.887271\pi$
$421$	− 292.000i	− 0.693587i	0.937942	−	0.346793i	$-0.112729\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	90.0000	0.211765
$426$	0	0
$427$	316.784	0.741883
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 339.411i	− 0.787497i	−0.919218	−	0.393749i	$-0.871178\pi$
$431$	− 339.411i	− 0.787497i	0.919218	−	0.393749i	$-0.128822\pi$
$432$	0	0
$433$	−10.0000	−0.0230947	−0.0115473	−	0.999933i	$-0.503676\pi$
$433$	−10.0000	−0.0230947	−0.0115473	+	0.999933i	$0.503676\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 160.000i	− 0.366133i
$438$	0	0
$439$	169.706i	0.386573i	0.981142	+	0.193287i	$0.0619147\pi$
$439$	169.706i	0.386573i	−0.981142	+	0.193287i	$0.938085\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	534.573	1.20671	0.603355	−	0.797473i	$-0.293830\pi$
$443$	534.573	1.20671	0.603355	+	0.797473i	$0.293830\pi$
$444$	0	0
$445$	− 88.0000i	− 0.197753i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−150.000	−0.334076	−0.167038	−	0.985950i	$-0.553420\pi$
$449$	−150.000	−0.334076	−0.167038	+	0.985950i	$0.553420\pi$
$450$	0	0
$451$	424.264	0.940719
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 905.097i	− 1.98922i
$456$	0	0
$457$	290.000	0.634573	0.317287	−	0.948330i	$-0.397228\pi$
$457$	290.000	0.634573	0.317287	+	0.948330i	$0.397228\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	60.0000i	0.130152i	0.997880	+	0.0650759i	$0.0207290\pi$
$461$	60.0000i	0.130152i	−0.997880	+	0.0650759i	$0.979271\pi$
$462$	0	0
$463$	− 67.8823i	− 0.146614i	−0.997309	−	0.0733070i	$-0.976645\pi$
$463$	− 67.8823i	− 0.146614i	0.997309	−	0.0733070i	$-0.0233553\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−280.014	−0.599602	−0.299801	−	0.954002i	$-0.596920\pi$
$467$	−280.014	−0.599602	−0.299801	+	0.954002i	$0.596920\pi$
$468$	0	0
$469$	− 928.000i	− 1.97868i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−40.0000	−0.0845666
$474$	0	0
$475$	−127.279	−0.267956
$476$	0	0
$477$	0	0
$478$	0	0
$479$	905.097i	1.88955i	0.327713	+	0.944777i	$0.393722\pi$
$479$	905.097i	1.88955i	−0.327713	+	0.944777i	$0.606278\pi$
$480$	0	0
$481$	400.000	0.831601
$482$	0	0
$483$	0	0
$484$	0	0
$485$	600.000i	1.23711i
$486$	0	0
$487$	237.588i	0.487860i	0.969793	+	0.243930i	$0.0784367\pi$
$487$	237.588i	0.487860i	−0.969793	+	0.243930i	$0.921563\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	749.533	1.52654	0.763272	−	0.646077i	$-0.223592\pi$
$491$	749.533	1.52654	0.763272	+	0.646077i	$0.223592\pi$
$492$	0	0
$493$	− 200.000i	− 0.405680i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	640.000	1.28773
$498$	0	0
$499$	155.563	0.311750	0.155875	−	0.987777i	$-0.450180\pi$
$499$	155.563	0.311750	0.155875	+	0.987777i	$0.450180\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	237.588i	0.472342i	0.971712	+	0.236171i	$0.0758925\pi$
$503$	237.588i	0.472342i	−0.971712	+	0.236171i	$0.924107\pi$
$504$	0	0
$505$	560.000	1.10891
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 900.000i	− 1.76817i	−0.467323	−	0.884086i	$-0.654782\pi$
$509$	− 900.000i	− 1.76817i	0.467323	−	0.884086i	$-0.345218\pi$
$510$	0	0
$511$	− 113.137i	− 0.221403i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−407.294	−0.790861
$516$	0	0
$517$	960.000i	1.85687i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−642.000	−1.23225	−0.616123	−	0.787650i	$-0.711298\pi$
$521$	−642.000	−1.23225	−0.616123	+	0.787650i	$0.711298\pi$
$522$	0	0
$523$	596.798	1.14111	0.570553	−	0.821261i	$-0.306729\pi$
$523$	596.798	1.14111	0.570553	+	0.821261i	$0.306729\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	401.000	0.758034
$530$	0	0
$531$	0	0
$532$	0	0
$533$	600.000i	1.12570i
$534$	0	0
$535$	− 576.999i	− 1.07850i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1117.23	−2.07278
$540$	0	0
$541$	580.000i	1.07209i	0.844190	+	0.536044i	$0.180082\pi$
$541$	580.000i	1.07209i	−0.844190	+	0.536044i	$0.819918\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−272.000	−0.499083
$546$	0	0
$547$	−398.808	−0.729083	−0.364541	−	0.931187i	$-0.618774\pi$
$547$	−398.808	−0.729083	−0.364541	+	0.931187i	$0.618774\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	282.843i	0.513326i
$552$	0	0
$553$	−1280.00	−2.31465
$554$	0	0
$555$	0	0
$556$	0	0
$557$	460.000i	0.825853i	0.910764	+	0.412926i	$0.135493\pi$
$557$	460.000i	0.825853i	−0.910764	+	0.412926i	$0.864507\pi$
$558$	0	0
$559$	− 56.5685i	− 0.101196i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−619.426	−1.10022	−0.550111	−	0.835091i	$-0.685415\pi$
$563$	−619.426	−1.10022	−0.550111	+	0.835091i	$0.685415\pi$
$564$	0	0
$565$	760.000i	1.34513i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	670.000	1.17750	0.588752	−	0.808314i	$-0.299619\pi$
$569$	670.000	1.17750	0.588752	+	0.808314i	$0.299619\pi$
$570$	0	0
$571$	−127.279	−0.222906	−0.111453	−	0.993770i	$-0.535550\pi$
$571$	−127.279	−0.222906	−0.111453	+	0.993770i	$0.535550\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	101.823i	0.177084i
$576$	0	0
$577$	610.000	1.05719	0.528596	−	0.848873i	$-0.322719\pi$
$577$	610.000	1.05719	0.528596	+	0.848873i	$0.322719\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	288.000i	0.495697i
$582$	0	0
$583$	− 848.528i	− 1.45545i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	511.945	0.872139	0.436069	−	0.899913i	$-0.356370\pi$
$587$	511.945	0.872139	0.436069	+	0.899913i	$0.356370\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	670.000	1.12985	0.564924	−	0.825143i	$-0.308905\pi$
$593$	670.000	1.12985	0.564924	+	0.825143i	$0.308905\pi$
$594$	0	0
$595$	−452.548	−0.760585
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 848.528i	− 1.41657i	−0.705924	−	0.708287i	$-0.749468\pi$
$599$	− 848.528i	− 1.41657i	0.705924	−	0.708287i	$-0.250532\pi$
$600$	0	0
$601$	470.000	0.782030	0.391015	−	0.920384i	$-0.372124\pi$
$601$	470.000	0.782030	0.391015	+	0.920384i	$0.372124\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	316.000i	0.522314i
$606$	0	0
$607$	45.2548i	0.0745549i	0.999305	+	0.0372775i	$0.0118685\pi$
$607$	45.2548i	0.0745549i	−0.999305	+	0.0372775i	$0.988131\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1357.65	−2.22200
$612$	0	0
$613$	540.000i	0.880914i	0.897774	+	0.440457i	$0.145183\pi$
$613$	540.000i	0.880914i	−0.897774	+	0.440457i	$0.854817\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−950.000	−1.53971	−0.769854	−	0.638220i	$-0.779671\pi$
$617$	−950.000	−1.53971	−0.769854	+	0.638220i	$0.779671\pi$
$618$	0	0
$619$	−466.690	−0.753943	−0.376971	−	0.926225i	$-0.623034\pi$
$619$	−466.690	−0.753943	−0.376971	+	0.926225i	$0.623034\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 248.902i	− 0.399521i
$624$	0	0
$625$	−319.000	−0.510400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 200.000i	− 0.317965i
$630$	0	0
$631$	848.528i	1.34474i	0.740217	+	0.672368i	$0.234723\pi$
$631$	848.528i	1.34474i	−0.740217	+	0.672368i	$0.765277\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	181.019	0.285070
$636$	0	0
$637$	− 1580.00i	− 2.48038i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−470.000	−0.733229	−0.366615	−	0.930373i	$-0.619483\pi$
$641$	−470.000	−0.733229	−0.366615	+	0.930373i	$0.619483\pi$
$642$	0	0
$643$	−534.573	−0.831373	−0.415686	−	0.909508i	$-0.636459\pi$
$643$	−534.573	−0.831373	−0.415686	+	0.909508i	$0.636459\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	667.509i	1.03170i	0.856679	+	0.515849i	$0.172524\pi$
$647$	667.509i	1.03170i	−0.856679	+	0.515849i	$0.827476\pi$
$648$	0	0
$649$	−600.000	−0.924499
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 900.000i	− 1.37825i	−0.724640	−	0.689127i	$-0.757994\pi$
$653$	− 900.000i	− 1.37825i	0.724640	−	0.689127i	$-0.242006\pi$
$654$	0	0
$655$	56.5685i	0.0863642i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	523.259	0.794020	0.397010	−	0.917814i	$-0.370048\pi$
$659$	523.259	0.794020	0.397010	+	0.917814i	$0.370048\pi$
$660$	0	0
$661$	172.000i	0.260212i	0.991500	+	0.130106i	$0.0415317\pi$
$661$	172.000i	0.260212i	−0.991500	+	0.130106i	$0.958468\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	640.000	0.962406
$666$	0	0
$667$	226.274	0.339242
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 395.980i	− 0.590134i
$672$	0	0
$673$	−170.000	−0.252600	−0.126300	−	0.991992i	$-0.540310\pi$
$673$	−170.000	−0.252600	−0.126300	+	0.991992i	$0.540310\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 300.000i	− 0.443131i	−0.975145	−	0.221566i	$-0.928883\pi$
$677$	− 300.000i	− 0.443131i	0.975145	−	0.221566i	$-0.0711167\pi$
$678$	0	0
$679$	1697.06i	2.49935i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−110.309	−0.161506	−0.0807530	−	0.996734i	$-0.525733\pi$
$683$	−110.309	−0.161506	−0.0807530	+	0.996734i	$0.525733\pi$
$684$	0	0
$685$	− 520.000i	− 0.759124i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1200.00	1.74165
$690$	0	0
$691$	−183.848	−0.266060	−0.133030	−	0.991112i	$-0.542471\pi$
$691$	−183.848	−0.266060	−0.133030	+	0.991112i	$0.542471\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	169.706i	0.244181i
$696$	0	0
$697$	300.000	0.430416
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 548.000i	− 0.781740i	−0.920446	−	0.390870i	$-0.872174\pi$
$701$	− 548.000i	− 0.781740i	0.920446	−	0.390870i	$-0.127826\pi$
$702$	0	0
$703$	282.843i	0.402337i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1583.92	2.24034
$708$	0	0
$709$	460.000i	0.648801i	0.945920	+	0.324401i	$0.105163\pi$
$709$	460.000i	0.648801i	−0.945920	+	0.324401i	$0.894837\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−1131.37	−1.58234
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1244.51i	1.73089i	0.501006	+	0.865444i	$0.332963\pi$
$719$	1244.51i	1.73089i	−0.501006	+	0.865444i	$0.667037\pi$
$720$	0	0
$721$	−1152.00	−1.59778
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 180.000i	− 0.248276i
$726$	0	0
$727$	1029.55i	1.41616i	0.706133	+	0.708079i	$0.250438\pi$
$727$	1029.55i	1.41616i	−0.706133	+	0.708079i	$0.749562\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−28.2843	−0.0386926
$732$	0	0
$733$	660.000i	0.900409i	0.892925	+	0.450205i	$0.148649\pi$
$733$	660.000i	0.900409i	−0.892925	+	0.450205i	$0.851351\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1160.00	−1.57395
$738$	0	0
$739$	−636.396	−0.861158	−0.430579	−	0.902553i	$-0.641691\pi$
$739$	−636.396	−0.861158	−0.430579	+	0.902553i	$0.641691\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 237.588i	− 0.319768i	−0.987136	−	0.159884i	$-0.948888\pi$
$743$	− 237.588i	− 0.319768i	0.987136	−	0.159884i	$-0.0511121\pi$
$744$	0	0
$745$	−208.000	−0.279195
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1632.00i	− 2.17891i
$750$	0	0
$751$	791.960i	1.05454i	0.849698	+	0.527270i	$0.176784\pi$
$751$	791.960i	1.05454i	−0.849698	+	0.527270i	$0.823216\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−678.823	−0.899103
$756$	0	0
$757$	− 660.000i	− 0.871863i	−0.899980	−	0.435931i	$-0.856419\pi$
$757$	− 660.000i	− 0.871863i	0.899980	−	0.435931i	$-0.143581\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1022.00	1.34297	0.671485	−	0.741018i	$-0.265657\pi$
$761$	1022.00	1.34297	0.671485	+	0.741018i	$0.265657\pi$
$762$	0	0
$763$	−769.332	−1.00830
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 848.528i	− 1.10629i
$768$	0	0
$769$	−362.000	−0.470741	−0.235371	−	0.971906i	$-0.575630\pi$
$769$	−362.000	−0.470741	−0.235371	+	0.971906i	$0.575630\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	820.000i	1.06080i	0.847747	+	0.530401i	$0.177959\pi$
$773$	820.000i	1.06080i	−0.847747	+	0.530401i	$0.822041\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−424.264	−0.544627
$780$	0	0
$781$	− 800.000i	− 1.02433i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−400.000	−0.509554
$786$	0	0
$787$	766.504	0.973956	0.486978	−	0.873414i	$-0.338099\pi$
$787$	766.504	0.973956	0.486978	+	0.873414i	$0.338099\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2149.60i	2.71758i
$792$	0	0
$793$	560.000	0.706179
$794$	0	0
$795$	0	0
$796$	0	0
$797$	860.000i	1.07905i	0.841971	+	0.539523i	$0.181395\pi$
$797$	860.000i	1.07905i	−0.841971	+	0.539523i	$0.818605\pi$
$798$	0	0
$799$	678.823i	0.849590i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−141.421	−0.176116
$804$	0	0
$805$	− 512.000i	− 0.636025i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−482.000	−0.595797	−0.297899	−	0.954598i	$-0.596286\pi$
$809$	−482.000	−0.595797	−0.297899	+	0.954598i	$0.596286\pi$
$810$	0	0
$811$	890.955	1.09859	0.549294	−	0.835629i	$-0.314897\pi$
$811$	890.955	1.09859	0.549294	+	0.835629i	$0.314897\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	441.235i	0.541392i
$816$	0	0
$817$	40.0000	0.0489596
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1508.00i	1.83678i	0.395671	+	0.918392i	$0.370512\pi$
$821$	1508.00i	1.83678i	−0.395671	+	0.918392i	$0.629488\pi$
$822$	0	0
$823$	− 916.410i	− 1.11350i	−0.830680	−	0.556750i	$-0.812048\pi$
$823$	− 916.410i	− 1.11350i	0.830680	−	0.556750i	$-0.187952\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−845.700	−1.02261	−0.511306	−	0.859399i	$-0.670838\pi$
$827$	−845.700	−1.02261	−0.511306	+	0.859399i	$0.670838\pi$
$828$	0	0
$829$	1092.00i	1.31725i	0.752471	+	0.658625i	$0.228862\pi$
$829$	1092.00i	1.31725i	−0.752471	+	0.658625i	$0.771138\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−790.000	−0.948379
$834$	0	0
$835$	−497.803	−0.596171
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 282.843i	− 0.337119i	−0.985691	−	0.168559i	$-0.946088\pi$
$839$	− 282.843i	− 0.337119i	0.985691	−	0.168559i	$-0.0539115\pi$
$840$	0	0
$841$	441.000	0.524376
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 924.000i	− 1.09349i
$846$	0	0
$847$	893.783i	1.05523i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	226.274	0.265892
$852$	0	0
$853$	1500.00i	1.75850i	0.476361	+	0.879250i	$0.341955\pi$
$853$	1500.00i	1.75850i	−0.476361	+	0.879250i	$0.658045\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−290.000	−0.338390	−0.169195	−	0.985583i	$-0.554117\pi$
$857$	−290.000	−0.338390	−0.169195	+	0.985583i	$0.554117\pi$
$858$	0	0
$859$	−975.807	−1.13598	−0.567990	−	0.823035i	$-0.692279\pi$
$859$	−975.807	−1.13598	−0.567990	+	0.823035i	$0.692279\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	633.568i	0.734146i	0.930192	+	0.367073i	$0.119640\pi$
$863$	633.568i	0.734146i	−0.930192	+	0.367073i	$0.880360\pi$
$864$	0	0
$865$	80.0000	0.0924855
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1600.00i	1.84120i
$870$	0	0
$871$	− 1640.49i	− 1.88345i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1538.66	−1.75847
$876$	0	0
$877$	100.000i	0.114025i	0.998373	+	0.0570125i	$0.0181575\pi$
$877$	100.000i	0.114025i	−0.998373	+	0.0570125i	$0.981842\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	510.000	0.578888	0.289444	−	0.957195i	$-0.406530\pi$
$881$	510.000	0.578888	0.289444	+	0.957195i	$0.406530\pi$
$882$	0	0
$883$	−398.808	−0.451651	−0.225826	−	0.974168i	$-0.572508\pi$
$883$	−398.808	−0.451651	−0.225826	+	0.974168i	$0.572508\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 350.725i	− 0.395406i	−0.980262	−	0.197703i	$-0.936652\pi$
$887$	− 350.725i	− 0.395406i	0.980262	−	0.197703i	$-0.0633481\pi$
$888$	0	0
$889$	512.000	0.575928
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 960.000i	− 1.07503i
$894$	0	0
$895$	− 169.706i	− 0.189615i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 600.000i	− 0.665927i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	720.000	0.795580
$906$	0	0
$907$	−1383.10	−1.52492	−0.762459	−	0.647036i	$-0.776008\pi$
$907$	−1383.10	−1.52492	−0.762459	+	0.647036i	$0.776008\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	565.685i	0.620950i	0.950582	+	0.310475i	$0.100488\pi$
$911$	565.685i	0.620950i	−0.950582	+	0.310475i	$0.899512\pi$
$912$	0	0
$913$	360.000	0.394304
$914$	0	0
$915$	0	0
$916$	0	0
$917$	160.000i	0.174482i
$918$	0	0
$919$	622.254i	0.677099i	0.940949	+	0.338549i	$0.109936\pi$
$919$	622.254i	0.677099i	−0.940949	+	0.338549i	$0.890064\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1131.37	1.22575
$924$	0	0
$925$	− 180.000i	− 0.194595i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	490.000	0.527449	0.263724	−	0.964598i	$-0.415049\pi$
$929$	490.000	0.527449	0.263724	+	0.964598i	$0.415049\pi$
$930$	0	0
$931$	1117.23	1.20003
$932$	0	0
$933$	0	0
$934$	0	0
$935$	565.685i	0.605011i
$936$	0	0
$937$	−10.0000	−0.0106724	−0.00533618	−	0.999986i	$-0.501699\pi$
$937$	−10.0000	−0.0106724	−0.00533618	+	0.999986i	$0.501699\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	700.000i	0.743889i	0.928255	+	0.371945i	$0.121309\pi$
$941$	700.000i	0.743889i	−0.928255	+	0.371945i	$0.878691\pi$
$942$	0	0
$943$	339.411i	0.359927i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−709.935	−0.749668	−0.374834	−	0.927092i	$-0.622300\pi$
$947$	−709.935	−0.749668	−0.374834	+	0.927092i	$0.622300\pi$
$948$	0	0
$949$	− 200.000i	− 0.210748i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	670.000	0.703043	0.351522	−	0.936180i	$-0.385664\pi$
$953$	670.000	0.703043	0.351522	+	0.936180i	$0.385664\pi$
$954$	0	0
$955$	905.097	0.947745
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1470.78i	− 1.53366i
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 680.000i	− 0.704663i
$966$	0	0
$967$	1006.92i	1.04128i	0.853776	+	0.520641i	$0.174307\pi$
$967$	1006.92i	1.04128i	−0.853776	+	0.520641i	$0.825693\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−608.112	−0.626274	−0.313137	−	0.949708i	$-0.601380\pi$
$971$	−608.112	−0.626274	−0.313137	+	0.949708i	$0.601380\pi$
$972$	0	0
$973$	480.000i	0.493320i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	330.000	0.337769	0.168884	−	0.985636i	$-0.445984\pi$
$977$	330.000	0.337769	0.168884	+	0.985636i	$0.445984\pi$
$978$	0	0
$979$	−311.127	−0.317801
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 576.999i	− 0.586978i	−0.955962	−	0.293489i	$-0.905184\pi$
$983$	− 576.999i	− 0.586978i	0.955962	−	0.293489i	$-0.0948164\pi$
$984$	0	0
$985$	880.000	0.893401
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 32.0000i	− 0.0323559i
$990$	0	0
$991$	− 226.274i	− 0.228329i	−0.993462	−	0.114165i	$-0.963581\pi$
$991$	− 226.274i	− 0.228329i	0.993462	−	0.114165i	$-0.0364191\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	678.823	0.682234
$996$	0	0
$997$	620.000i	0.621866i	0.950432	+	0.310933i	$0.100641\pi$
$997$	620.000i	0.621866i	−0.950432	+	0.310933i	$0.899359\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.b.g.703.4		4
3.2	odd	2		128.3.d.c.63.1	✓	4
4.3	odd	2	inner	1152.3.b.g.703.3		4
8.3	odd	2	inner	1152.3.b.g.703.1		4
8.5	even	2	inner	1152.3.b.g.703.2		4
12.11	even	2		128.3.d.c.63.3	yes	4
16.3	odd	4		2304.3.g.m.1279.2		2
16.5	even	4		2304.3.g.h.1279.1		2
16.11	odd	4		2304.3.g.h.1279.2		2
16.13	even	4		2304.3.g.m.1279.1		2
24.5	odd	2		128.3.d.c.63.4	yes	4
24.11	even	2		128.3.d.c.63.2	yes	4
48.5	odd	4		256.3.c.f.255.2		2
48.11	even	4		256.3.c.f.255.1		2
48.29	odd	4		256.3.c.c.255.1		2
48.35	even	4		256.3.c.c.255.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.d.c.63.1	✓	4	3.2	odd	2
128.3.d.c.63.2	yes	4	24.11	even	2
128.3.d.c.63.3	yes	4	12.11	even	2
128.3.d.c.63.4	yes	4	24.5	odd	2
256.3.c.c.255.1		2	48.29	odd	4
256.3.c.c.255.2		2	48.35	even	4
256.3.c.f.255.1		2	48.11	even	4
256.3.c.f.255.2		2	48.5	odd	4
1152.3.b.g.703.1		4	8.3	odd	2	inner
1152.3.b.g.703.2		4	8.5	even	2	inner
1152.3.b.g.703.3		4	4.3	odd	2	inner
1152.3.b.g.703.4		4	1.1	even	1	trivial
2304.3.g.h.1279.1		2	16.5	even	4
2304.3.g.h.1279.2		2	16.11	odd	4
2304.3.g.m.1279.1		2	16.13	even	4
2304.3.g.m.1279.2		2	16.3	odd	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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{5} +11.3137i q^{7} +O(q^{10})\)	\(q+4.00000i q^{5} +11.3137i q^{7} +14.1421 q^{11} +20.0000i q^{13} +10.0000 q^{17} -14.1421 q^{19} +11.3137i q^{23} +9.00000 q^{25} -20.0000i q^{29} -45.2548 q^{35} -20.0000i q^{37} +30.0000 q^{41} -2.82843 q^{43} +67.8823i q^{47} -79.0000 q^{49} -60.0000i q^{53} +56.5685i q^{55} -42.4264 q^{59} -28.0000i q^{61} -80.0000 q^{65} -82.0244 q^{67} -56.5685i q^{71} -10.0000 q^{73} +160.000i q^{77} +113.137i q^{79} +25.4558 q^{83} +40.0000i q^{85} -22.0000 q^{89} -226.274 q^{91} -56.5685i q^{95} +150.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 40 q^{17} + 36 q^{25} + 120 q^{41} - 316 q^{49} - 320 q^{65} - 40 q^{73} - 88 q^{89} + 600 q^{97}+O(q^{100}) \) 4 * q + 40 * q^17 + 36 * q^25 + 120 * q^41 - 316 * q^49 - 320 * q^65 - 40 * q^73 - 88 * q^89 + 600 * q^97

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