LMFDB - Embedded newform 1152.4.c.b.1151.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,4,Mod(1151,1152)]

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, 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("1152.1151");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1152.c (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:	$67.9702003266$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 4 x^{10} + 12 x^{9} + 88 x^{8} + 356 x^{7} + 1278 x^{6} + 3320 x^{5} + 8177 x^{4} + \cdots + 98 $ x^12 - 4x^11 - 4x^10 + 12x^9 + 88x^8 + 356x^7 + 1278x^6 + 3320x^5 + 8177x^4 + 7752x^3 + 8062x^2 - 1708*x + 98
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{33}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.12
Root			$-0.843160 - 2.03557i$ of defining polynomial
Character	$\chi$	$=$	1152.1151
Dual form			1152.4.c.b.1151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+19.4649i q^{5} -22.2014i q^{7} +31.9589 q^{11} -12.7921 q^{13} +1.79939i q^{17} -55.3042i q^{19} -152.218 q^{23} -253.882 q^{25} +292.254i q^{29} -271.560i q^{31} +432.148 q^{35} -114.334 q^{37} +300.192i q^{41} +371.823i q^{43} +35.2663 q^{47} -149.902 q^{49} -391.098i q^{53} +622.077i q^{55} -22.8408 q^{59} -662.958 q^{61} -248.997i q^{65} -926.894i q^{67} -1071.12 q^{71} +9.27078 q^{73} -709.534i q^{77} +565.805i q^{79} -566.393 q^{83} -35.0248 q^{85} -911.883i q^{89} +284.003i q^{91} +1076.49 q^{95} +466.815 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 240 q^{23} - 300 q^{25} + 864 q^{35} - 264 q^{37} - 624 q^{47} + 132 q^{49} + 1632 q^{59} - 312 q^{61} - 4080 q^{71} + 432 q^{73} + 3744 q^{83} + 1704 q^{85} - 5856 q^{95} + 96 q^{97}+O(q^{100}) $ 12 * q - 240 * q^23 - 300 * q^25 + 864 * q^35 - 264 * q^37 - 624 * q^47 + 132 * q^49 + 1632 * q^59 - 312 * q^61 - 4080 * q^71 + 432 * q^73 + 3744 * q^83 + 1704 * q^85 - 5856 * q^95 + 96 * q^97

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
$5$	19.4649i	1.74099i	0.492176	+	0.870496i	$0.336202\pi$
$5$	19.4649i	1.74099i	−0.492176	+	0.870496i	$0.663798\pi$
$6$	0	0
$7$	− 22.2014i	− 1.19876i	−0.800463	−	0.599382i	$-0.795413\pi$
$7$	− 22.2014i	− 1.19876i	0.800463	−	0.599382i	$-0.204587\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	31.9589	0.875999	0.437999	−	0.898975i	$-0.355687\pi$
$11$	31.9589	0.875999	0.437999	+	0.898975i	$0.355687\pi$
$12$	0	0
$13$	−12.7921	−0.272915	−0.136457	−	0.990646i	$-0.543572\pi$
$13$	−12.7921	−0.272915	−0.136457	+	0.990646i	$0.543572\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.79939i	0.0256715i	0.999918	+	0.0128357i	$0.00408585\pi$
$17$	1.79939i	0.0256715i	−0.999918	+	0.0128357i	$0.995914\pi$
$18$	0	0
$19$	− 55.3042i	− 0.667772i	−0.942614	−	0.333886i	$-0.891640\pi$
$19$	− 55.3042i	− 0.667772i	0.942614	−	0.333886i	$-0.108360\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−152.218	−1.37999	−0.689994	−	0.723815i	$-0.742387\pi$
$23$	−152.218	−1.37999	−0.689994	+	0.723815i	$0.742387\pi$
$24$	0	0
$25$	−253.882	−2.03105
$26$	0	0
$27$	0	0
$28$	0	0
$29$	292.254i	1.87139i	0.352811	+	0.935695i	$0.385226\pi$
$29$	292.254i	1.87139i	−0.352811	+	0.935695i	$0.614774\pi$
$30$	0	0
$31$	− 271.560i	− 1.57334i	−0.617372	−	0.786672i	$-0.711803\pi$
$31$	− 271.560i	− 1.57334i	0.617372	−	0.786672i	$-0.288197\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	432.148	2.08704
$36$	0	0
$37$	−114.334	−0.508013	−0.254006	−	0.967203i	$-0.581748\pi$
$37$	−114.334	−0.508013	−0.254006	+	0.967203i	$0.581748\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	300.192i	1.14347i	0.820439	+	0.571734i	$0.193729\pi$
$41$	300.192i	1.14347i	−0.820439	+	0.571734i	$0.806271\pi$
$42$	0	0
$43$	371.823i	1.31866i	0.751853	+	0.659331i	$0.229160\pi$
$43$	371.823i	1.31866i	−0.751853	+	0.659331i	$0.770840\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	35.2663	0.109449	0.0547246	−	0.998501i	$-0.482572\pi$
$47$	35.2663	0.109449	0.0547246	+	0.998501i	$0.482572\pi$
$48$	0	0
$49$	−149.902	−0.437033
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 391.098i	− 1.01361i	−0.862060	−	0.506806i	$-0.830826\pi$
$53$	− 391.098i	− 1.01361i	0.862060	−	0.506806i	$-0.169174\pi$
$54$	0	0
$55$	622.077i	1.52511i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−22.8408	−0.0504003	−0.0252001	−	0.999682i	$-0.508022\pi$
$59$	−22.8408	−0.0504003	−0.0252001	+	0.999682i	$0.508022\pi$
$60$	0	0
$61$	−662.958	−1.39153	−0.695763	−	0.718272i	$-0.744933\pi$
$61$	−662.958	−1.39153	−0.695763	+	0.718272i	$0.744933\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 248.997i	− 0.475142i
$66$	0	0
$67$	− 926.894i	− 1.69012i	−0.534671	−	0.845060i	$-0.679564\pi$
$67$	− 926.894i	− 1.69012i	0.534671	−	0.845060i	$-0.320436\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1071.12	−1.79040	−0.895198	−	0.445669i	$-0.852966\pi$
$71$	−1071.12	−1.79040	−0.895198	+	0.445669i	$0.852966\pi$
$72$	0	0
$73$	9.27078	0.0148639	0.00743193	−	0.999972i	$-0.497634\pi$
$73$	9.27078	0.0148639	0.00743193	+	0.999972i	$0.497634\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 709.534i	− 1.05011i
$78$	0	0
$79$	565.805i	0.805798i	0.915245	+	0.402899i	$0.131997\pi$
$79$	565.805i	0.805798i	−0.915245	+	0.402899i	$0.868003\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−566.393	−0.749032	−0.374516	−	0.927220i	$-0.622191\pi$
$83$	−566.393	−0.749032	−0.374516	+	0.927220i	$0.622191\pi$
$84$	0	0
$85$	−35.0248	−0.0446938
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 911.883i	− 1.08606i	−0.839713	−	0.543031i	$-0.817277\pi$
$89$	− 911.883i	− 1.08606i	0.839713	−	0.543031i	$-0.182723\pi$
$90$	0	0
$91$	284.003i	0.327160i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1076.49	1.16259
$96$	0	0
$97$	466.815	0.488638	0.244319	−	0.969695i	$-0.421436\pi$
$97$	466.815	0.488638	0.244319	+	0.969695i	$0.421436\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 615.098i	− 0.605986i	−0.952993	−	0.302993i	$-0.902014\pi$
$101$	− 615.098i	− 0.605986i	0.952993	−	0.302993i	$-0.0979858\pi$
$102$	0	0
$103$	1834.29i	1.75474i	0.479818	+	0.877368i	$0.340703\pi$
$103$	1834.29i	1.75474i	−0.479818	+	0.877368i	$0.659297\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−565.082	−0.510547	−0.255273	−	0.966869i	$-0.582166\pi$
$107$	−565.082	−0.510547	−0.255273	+	0.966869i	$0.582166\pi$
$108$	0	0
$109$	−314.783	−0.276612	−0.138306	−	0.990390i	$-0.544166\pi$
$109$	−314.783	−0.276612	−0.138306	+	0.990390i	$0.544166\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2245.29i	− 1.86920i	−0.355703	−	0.934599i	$-0.615759\pi$
$113$	− 2245.29i	− 1.86920i	0.355703	−	0.934599i	$-0.384241\pi$
$114$	0	0
$115$	− 2962.91i	− 2.40255i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	39.9489	0.0307740
$120$	0	0
$121$	−309.626	−0.232626
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 2508.66i	− 1.79505i
$126$	0	0
$127$	1020.85i	0.713273i	0.934243	+	0.356637i	$0.116077\pi$
$127$	1020.85i	0.713273i	−0.934243	+	0.356637i	$0.883923\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1164.80	0.776864	0.388432	−	0.921477i	$-0.373017\pi$
$131$	1164.80	0.776864	0.388432	+	0.921477i	$0.373017\pi$
$132$	0	0
$133$	−1227.83	−0.800500
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1915.28i	− 1.19440i	−0.802091	−	0.597202i	$-0.796279\pi$
$137$	− 1915.28i	− 1.19440i	0.802091	−	0.597202i	$-0.203721\pi$
$138$	0	0
$139$	− 3016.58i	− 1.84074i	−0.391051	−	0.920369i	$-0.627888\pi$
$139$	− 3016.58i	− 1.84074i	0.391051	−	0.920369i	$-0.372112\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−408.822	−0.239073
$144$	0	0
$145$	−5688.70	−3.25807
$146$	0	0
$147$	0	0
$148$	0	0
$149$	39.8288i	0.0218987i	0.999940	+	0.0109493i	$0.00348535\pi$
$149$	39.8288i	0.0218987i	−0.999940	+	0.0109493i	$0.996515\pi$
$150$	0	0
$151$	− 2021.62i	− 1.08952i	−0.838592	−	0.544760i	$-0.816621\pi$
$151$	− 2021.62i	− 1.08952i	0.838592	−	0.544760i	$-0.183379\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5285.89	2.73918
$156$	0	0
$157$	1260.53	0.640772	0.320386	−	0.947287i	$-0.396187\pi$
$157$	1260.53	0.640772	0.320386	+	0.947287i	$0.396187\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3379.46i	1.65428i
$162$	0	0
$163$	1803.46i	0.866611i	0.901247	+	0.433305i	$0.142653\pi$
$163$	1803.46i	0.866611i	−0.901247	+	0.433305i	$0.857347\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2030.46	−0.940847	−0.470424	−	0.882441i	$-0.655899\pi$
$167$	−2030.46	−0.940847	−0.470424	+	0.882441i	$0.655899\pi$
$168$	0	0
$169$	−2033.36	−0.925518
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1541.47i	0.677432i	0.940889	+	0.338716i	$0.109993\pi$
$173$	1541.47i	0.677432i	−0.940889	+	0.338716i	$0.890007\pi$
$174$	0	0
$175$	5636.53i	2.43475i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2660.38	−1.11087	−0.555436	−	0.831560i	$-0.687448\pi$
$179$	−2660.38	−1.11087	−0.555436	+	0.831560i	$0.687448\pi$
$180$	0	0
$181$	−3890.04	−1.59748	−0.798741	−	0.601675i	$-0.794500\pi$
$181$	−3890.04	−1.59748	−0.798741	+	0.601675i	$0.794500\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2225.51i	− 0.884446i
$186$	0	0
$187$	57.5065i	0.0224882i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1980.04	−0.750107	−0.375053	−	0.927003i	$-0.622376\pi$
$191$	−1980.04	−0.750107	−0.375053	+	0.927003i	$0.622376\pi$
$192$	0	0
$193$	3510.71	1.30936	0.654679	−	0.755907i	$-0.272804\pi$
$193$	3510.71	1.30936	0.654679	+	0.755907i	$0.272804\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	582.273i	0.210585i	0.994441	+	0.105292i	$0.0335779\pi$
$197$	582.273i	0.210585i	−0.994441	+	0.105292i	$0.966422\pi$
$198$	0	0
$199$	− 3859.19i	− 1.37473i	−0.726313	−	0.687364i	$-0.758768\pi$
$199$	− 3859.19i	− 1.37473i	0.726313	−	0.687364i	$-0.241232\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6488.46	2.24335
$204$	0	0
$205$	−5843.21	−1.99077
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1767.47i	− 0.584967i
$210$	0	0
$211$	− 3152.87i	− 1.02868i	−0.857585	−	0.514342i	$-0.828036\pi$
$211$	− 3152.87i	− 1.02868i	0.857585	−	0.514342i	$-0.171964\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7237.49	−2.29578
$216$	0	0
$217$	−6029.02	−1.88607
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 23.0179i	− 0.00700612i
$222$	0	0
$223$	3930.70i	1.18035i	0.807274	+	0.590177i	$0.200942\pi$
$223$	3930.70i	1.18035i	−0.807274	+	0.590177i	$0.799058\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−604.951	−0.176881	−0.0884406	−	0.996081i	$-0.528188\pi$
$227$	−604.951	−0.176881	−0.0884406	+	0.996081i	$0.528188\pi$
$228$	0	0
$229$	−3047.32	−0.879355	−0.439678	−	0.898156i	$-0.644907\pi$
$229$	−3047.32	−0.879355	−0.439678	+	0.898156i	$0.644907\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5242.57i	1.47404i	0.675869	+	0.737022i	$0.263768\pi$
$233$	5242.57i	1.47404i	−0.675869	+	0.737022i	$0.736232\pi$
$234$	0	0
$235$	686.454i	0.190550i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3218.81	−0.871162	−0.435581	−	0.900150i	$-0.643457\pi$
$239$	−3218.81	−0.871162	−0.435581	+	0.900150i	$0.643457\pi$
$240$	0	0
$241$	−4991.45	−1.33414	−0.667069	−	0.744996i	$-0.732452\pi$
$241$	−4991.45	−1.33414	−0.667069	+	0.744996i	$0.732452\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2917.83i	− 0.760872i
$246$	0	0
$247$	707.458i	0.182245i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1930.14	0.485376	0.242688	−	0.970104i	$-0.421971\pi$
$251$	1930.14	0.485376	0.242688	+	0.970104i	$0.421971\pi$
$252$	0	0
$253$	−4864.74	−1.20887
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 136.590i	− 0.0331526i	−0.999863	−	0.0165763i	$-0.994723\pi$
$257$	− 136.590i	− 0.0331526i	0.999863	−	0.0165763i	$-0.00527665\pi$
$258$	0	0
$259$	2538.39i	0.608987i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1726.50	0.404793	0.202397	−	0.979304i	$-0.435127\pi$
$263$	1726.50	0.404793	0.202397	+	0.979304i	$0.435127\pi$
$264$	0	0
$265$	7612.67	1.76469
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 3174.73i	− 0.719579i	−0.933033	−	0.359790i	$-0.882848\pi$
$269$	− 3174.73i	− 0.719579i	0.933033	−	0.359790i	$-0.117152\pi$
$270$	0	0
$271$	2453.18i	0.549889i	0.961460	+	0.274945i	$0.0886596\pi$
$271$	2453.18i	0.549889i	−0.961460	+	0.274945i	$0.911340\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8113.79	−1.77920
$276$	0	0
$277$	4665.27	1.01195	0.505973	−	0.862549i	$-0.331134\pi$
$277$	4665.27	1.01195	0.505973	+	0.862549i	$0.331134\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 1304.96i	− 0.277038i	−0.990360	−	0.138519i	$-0.955766\pi$
$281$	− 1304.96i	− 0.277038i	0.990360	−	0.138519i	$-0.0442342\pi$
$282$	0	0
$283$	− 1487.84i	− 0.312518i	−0.987716	−	0.156259i	$-0.950057\pi$
$283$	− 1487.84i	− 0.312518i	0.987716	−	0.156259i	$-0.0499435\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6664.69	1.37075
$288$	0	0
$289$	4909.76	0.999341
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1936.50i	− 0.386114i	−0.981188	−	0.193057i	$-0.938160\pi$
$293$	− 1936.50i	− 0.386114i	0.981188	−	0.193057i	$-0.0618402\pi$
$294$	0	0
$295$	− 444.593i	− 0.0877465i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1947.19	0.376619
$300$	0	0
$301$	8254.99	1.58076
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 12904.4i	− 2.42263i
$306$	0	0
$307$	− 5884.32i	− 1.09393i	−0.837156	−	0.546964i	$-0.815783\pi$
$307$	− 5884.32i	− 1.09393i	0.837156	−	0.546964i	$-0.184217\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1160.58	0.211610	0.105805	−	0.994387i	$-0.466258\pi$
$311$	1160.58	0.211610	0.105805	+	0.994387i	$0.466258\pi$
$312$	0	0
$313$	−6605.01	−1.19277	−0.596385	−	0.802698i	$-0.703397\pi$
$313$	−6605.01	−1.19277	−0.596385	+	0.802698i	$0.703397\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2661.31i	0.471526i	0.971811	+	0.235763i	$0.0757590\pi$
$317$	2661.31i	0.471526i	−0.971811	+	0.235763i	$0.924241\pi$
$318$	0	0
$319$	9340.14i	1.63933i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	99.5137	0.0171427
$324$	0	0
$325$	3247.68	0.554304
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 782.961i	− 0.131204i
$330$	0	0
$331$	4233.72i	0.703040i	0.936180	+	0.351520i	$0.114335\pi$
$331$	4233.72i	0.703040i	−0.936180	+	0.351520i	$0.885665\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	18041.9	2.94249
$336$	0	0
$337$	−4641.65	−0.750286	−0.375143	−	0.926967i	$-0.622406\pi$
$337$	−4641.65	−0.750286	−0.375143	+	0.926967i	$0.622406\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 8678.77i	− 1.37825i
$342$	0	0
$343$	− 4287.04i	− 0.674864i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4554.37	0.704587	0.352293	−	0.935890i	$-0.385402\pi$
$347$	4554.37	0.704587	0.352293	+	0.935890i	$0.385402\pi$
$348$	0	0
$349$	−747.324	−0.114623	−0.0573113	−	0.998356i	$-0.518253\pi$
$349$	−747.324	−0.114623	−0.0573113	+	0.998356i	$0.518253\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6594.80i	0.994351i	0.867650	+	0.497175i	$0.165629\pi$
$353$	6594.80i	0.994351i	−0.867650	+	0.497175i	$0.834371\pi$
$354$	0	0
$355$	− 20849.2i	− 3.11706i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13093.3	−1.92489	−0.962447	−	0.271471i	$-0.912490\pi$
$359$	−13093.3	−1.92489	−0.962447	+	0.271471i	$0.912490\pi$
$360$	0	0
$361$	3800.44	0.554081
$362$	0	0
$363$	0	0
$364$	0	0
$365$	180.455i	0.0258779i
$366$	0	0
$367$	− 12114.5i	− 1.72309i	−0.507682	−	0.861544i	$-0.669498\pi$
$367$	− 12114.5i	− 1.72309i	0.507682	−	0.861544i	$-0.330502\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8682.92	−1.21508
$372$	0	0
$373$	12969.7	1.80039	0.900196	−	0.435486i	$-0.143423\pi$
$373$	12969.7	1.80039	0.900196	+	0.435486i	$0.143423\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3738.55i	− 0.510730i
$378$	0	0
$379$	12527.1i	1.69783i	0.528532	+	0.848913i	$0.322743\pi$
$379$	12527.1i	1.69783i	−0.528532	+	0.848913i	$0.677257\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6966.21	0.929391	0.464695	−	0.885471i	$-0.346164\pi$
$383$	6966.21	0.929391	0.464695	+	0.885471i	$0.346164\pi$
$384$	0	0
$385$	13811.0	1.82824
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13156.5i	1.71481i	0.514645	+	0.857403i	$0.327924\pi$
$389$	13156.5i	1.71481i	−0.514645	+	0.857403i	$0.672076\pi$
$390$	0	0
$391$	− 273.899i	− 0.0354263i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11013.3	−1.40289
$396$	0	0
$397$	−1737.94	−0.219709	−0.109855	−	0.993948i	$-0.535039\pi$
$397$	−1737.94	−0.219709	−0.109855	+	0.993948i	$0.535039\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	506.378i	0.0630606i	0.999503	+	0.0315303i	$0.0100381\pi$
$401$	506.378i	0.0630606i	−0.999503	+	0.0315303i	$0.989962\pi$
$402$	0	0
$403$	3473.83i	0.429389i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3654.01	−0.445019
$408$	0	0
$409$	9910.80	1.19818	0.599092	−	0.800680i	$-0.295528\pi$
$409$	9910.80	1.19818	0.599092	+	0.800680i	$0.295528\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	507.098i	0.0604180i
$414$	0	0
$415$	− 11024.8i	− 1.30406i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4058.63	−0.473215	−0.236607	−	0.971605i	$-0.576036\pi$
$419$	−4058.63	−0.473215	−0.236607	+	0.971605i	$0.576036\pi$
$420$	0	0
$421$	−7852.53	−0.909048	−0.454524	−	0.890735i	$-0.650191\pi$
$421$	−7852.53	−0.909048	−0.454524	+	0.890735i	$0.650191\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 456.831i	− 0.0521401i
$426$	0	0
$427$	14718.6i	1.66811i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10717.2	1.19775	0.598873	−	0.800844i	$-0.295615\pi$
$431$	10717.2	1.19775	0.598873	+	0.800844i	$0.295615\pi$
$432$	0	0
$433$	−5190.51	−0.576074	−0.288037	−	0.957619i	$-0.593003\pi$
$433$	−5190.51	−0.576074	−0.288037	+	0.957619i	$0.593003\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8418.32i	0.921516i
$438$	0	0
$439$	2101.72i	0.228495i	0.993452	+	0.114248i	$0.0364457\pi$
$439$	2101.72i	0.228495i	−0.993452	+	0.114248i	$0.963554\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−586.526	−0.0629045	−0.0314523	−	0.999505i	$-0.510013\pi$
$443$	−586.526	−0.0629045	−0.0314523	+	0.999505i	$0.510013\pi$
$444$	0	0
$445$	17749.7	1.89082
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16423.7i	1.72624i	0.504995	+	0.863122i	$0.331494\pi$
$449$	16423.7i	1.72624i	−0.504995	+	0.863122i	$0.668506\pi$
$450$	0	0
$451$	9593.83i	1.00168i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5528.08	−0.569583
$456$	0	0
$457$	−5904.11	−0.604338	−0.302169	−	0.953254i	$-0.597711\pi$
$457$	−5904.11	−0.604338	−0.302169	+	0.953254i	$0.597711\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9501.12i	0.959894i	0.877298	+	0.479947i	$0.159344\pi$
$461$	9501.12i	0.959894i	−0.877298	+	0.479947i	$0.840656\pi$
$462$	0	0
$463$	1341.81i	0.134685i	0.997730	+	0.0673424i	$0.0214520\pi$
$463$	1341.81i	0.134685i	−0.997730	+	0.0673424i	$0.978548\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19268.8	1.90933	0.954663	−	0.297689i	$-0.0962159\pi$
$467$	19268.8	1.90933	0.954663	+	0.297689i	$0.0962159\pi$
$468$	0	0
$469$	−20578.3	−2.02605
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11883.1i	1.15515i
$474$	0	0
$475$	14040.7i	1.35628i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12832.2	−1.22404	−0.612021	−	0.790841i	$-0.709643\pi$
$479$	−12832.2	−1.22404	−0.612021	+	0.790841i	$0.709643\pi$
$480$	0	0
$481$	1462.58	0.138644
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9086.49i	0.850714i
$486$	0	0
$487$	− 15784.6i	− 1.46872i	−0.678758	−	0.734362i	$-0.737481\pi$
$487$	− 15784.6i	− 1.46872i	0.678758	−	0.734362i	$-0.262519\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15306.9	1.40691	0.703453	−	0.710742i	$-0.251640\pi$
$491$	15306.9	1.40691	0.703453	+	0.710742i	$0.251640\pi$
$492$	0	0
$493$	−525.878	−0.0480413
$494$	0	0
$495$	0	0
$496$	0	0
$497$	23780.3i	2.14626i
$498$	0	0
$499$	10479.3i	0.940117i	0.882635	+	0.470058i	$0.155767\pi$
$499$	10479.3i	0.940117i	−0.882635	+	0.470058i	$0.844233\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9919.58	−0.879308	−0.439654	−	0.898167i	$-0.644899\pi$
$503$	−9919.58	−0.879308	−0.439654	+	0.898167i	$0.644899\pi$
$504$	0	0
$505$	11972.8	1.05502
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 10028.1i	− 0.873254i	−0.899643	−	0.436627i	$-0.856173\pi$
$509$	− 10028.1i	− 0.873254i	0.899643	−	0.436627i	$-0.143827\pi$
$510$	0	0
$511$	− 205.824i	− 0.0178183i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−35704.2	−3.05498
$516$	0	0
$517$	1127.07	0.0958774
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10216.7i	0.859122i	0.903038	+	0.429561i	$0.141332\pi$
$521$	10216.7i	0.859122i	−0.903038	+	0.429561i	$0.858668\pi$
$522$	0	0
$523$	12295.9i	1.02803i	0.857780	+	0.514017i	$0.171843\pi$
$523$	12295.9i	1.02803i	−0.857780	+	0.514017i	$0.828157\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	488.641	0.0403900
$528$	0	0
$529$	11003.4	0.904365
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3840.09i	− 0.312069i
$534$	0	0
$535$	− 10999.3i	− 0.888858i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4790.72	−0.382841
$540$	0	0
$541$	9786.98	0.777773	0.388887	−	0.921286i	$-0.372860\pi$
$541$	9786.98	0.777773	0.388887	+	0.921286i	$0.372860\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 6127.22i	− 0.481580i
$546$	0	0
$547$	351.890i	0.0275059i	0.999905	+	0.0137530i	$0.00437784\pi$
$547$	351.890i	0.0275059i	−0.999905	+	0.0137530i	$0.995622\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	16162.9	1.24966
$552$	0	0
$553$	12561.7	0.965960
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22996.2i	1.74933i	0.484724	+	0.874667i	$0.338920\pi$
$557$	22996.2i	1.74933i	−0.484724	+	0.874667i	$0.661080\pi$
$558$	0	0
$559$	− 4756.40i	− 0.359882i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	590.483	0.0442023	0.0221011	−	0.999756i	$-0.492964\pi$
$563$	590.483	0.0442023	0.0221011	+	0.999756i	$0.492964\pi$
$564$	0	0
$565$	43704.4	3.25426
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8863.10i	0.653006i	0.945196	+	0.326503i	$0.105870\pi$
$569$	8863.10i	0.653006i	−0.945196	+	0.326503i	$0.894130\pi$
$570$	0	0
$571$	− 86.0835i	− 0.00630907i	−0.999995	−	0.00315454i	$-0.998996\pi$
$571$	− 86.0835i	− 0.00630907i	0.999995	−	0.00315454i	$-0.00100412\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	38645.4	2.80283
$576$	0	0
$577$	−10676.7	−0.770320	−0.385160	−	0.922850i	$-0.625854\pi$
$577$	−10676.7	−0.770320	−0.385160	+	0.922850i	$0.625854\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12574.7i	0.897912i
$582$	0	0
$583$	− 12499.1i	− 0.887922i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14828.7	1.04267	0.521335	−	0.853352i	$-0.325434\pi$
$587$	14828.7	1.04267	0.521335	+	0.853352i	$0.325434\pi$
$588$	0	0
$589$	−15018.4	−1.05063
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 7625.52i	− 0.528065i	−0.964514	−	0.264033i	$-0.914947\pi$
$593$	− 7625.52i	− 0.528065i	0.964514	−	0.264033i	$-0.0850526\pi$
$594$	0	0
$595$	777.600i	0.0535773i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2515.53	0.171589	0.0857945	−	0.996313i	$-0.472657\pi$
$599$	2515.53	0.171589	0.0857945	+	0.996313i	$0.472657\pi$
$600$	0	0
$601$	−884.652	−0.0600428	−0.0300214	−	0.999549i	$-0.509558\pi$
$601$	−884.652	−0.0600428	−0.0300214	+	0.999549i	$0.509558\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 6026.83i	− 0.405001i
$606$	0	0
$607$	− 3262.44i	− 0.218152i	−0.994033	−	0.109076i	$-0.965211\pi$
$607$	− 3262.44i	− 0.218152i	0.994033	−	0.109076i	$-0.0347892\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−451.130	−0.0298703
$612$	0	0
$613$	7103.23	0.468021	0.234010	−	0.972234i	$-0.424815\pi$
$613$	7103.23	0.468021	0.234010	+	0.972234i	$0.424815\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 17015.2i	− 1.11022i	−0.831776	−	0.555111i	$-0.812676\pi$
$617$	− 17015.2i	− 1.11022i	0.831776	−	0.555111i	$-0.187324\pi$
$618$	0	0
$619$	− 14796.3i	− 0.960763i	−0.877060	−	0.480381i	$-0.840498\pi$
$619$	− 14796.3i	− 0.960763i	0.877060	−	0.480381i	$-0.159502\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−20245.1	−1.30193
$624$	0	0
$625$	17095.7	1.09412
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 205.732i	− 0.0130414i
$630$	0	0
$631$	− 6518.78i	− 0.411265i	−0.978629	−	0.205633i	$-0.934075\pi$
$631$	− 6518.78i	− 0.411265i	0.978629	−	0.205633i	$-0.0659252\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19870.7	−1.24180
$636$	0	0
$637$	1917.57	0.119273
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 23065.6i	− 1.42127i	−0.703559	−	0.710637i	$-0.748407\pi$
$641$	− 23065.6i	− 1.42127i	0.703559	−	0.710637i	$-0.251593\pi$
$642$	0	0
$643$	11825.3i	0.725264i	0.931932	+	0.362632i	$0.118122\pi$
$643$	11825.3i	0.725264i	−0.931932	+	0.362632i	$0.881878\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14019.2	0.851857	0.425929	−	0.904757i	$-0.359947\pi$
$647$	14019.2	0.851857	0.425929	+	0.904757i	$0.359947\pi$
$648$	0	0
$649$	−729.968	−0.0441506
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 7701.14i	− 0.461514i	−0.973011	−	0.230757i	$-0.925880\pi$
$653$	− 7701.14i	− 0.461514i	0.973011	−	0.230757i	$-0.0741203\pi$
$654$	0	0
$655$	22672.7i	1.35251i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24026.7	−1.42026	−0.710128	−	0.704073i	$-0.751363\pi$
$659$	−24026.7	−1.42026	−0.710128	+	0.704073i	$0.751363\pi$
$660$	0	0
$661$	1278.87	0.0752530	0.0376265	−	0.999292i	$-0.488020\pi$
$661$	1278.87	0.0752530	0.0376265	+	0.999292i	$0.488020\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 23899.6i	− 1.39366i
$666$	0	0
$667$	− 44486.5i	− 2.58249i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−21187.4	−1.21897
$672$	0	0
$673$	−2951.17	−0.169033	−0.0845164	−	0.996422i	$-0.526935\pi$
$673$	−2951.17	−0.169033	−0.0845164	+	0.996422i	$0.526935\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5263.47i	0.298806i	0.988776	+	0.149403i	$0.0477351\pi$
$677$	5263.47i	0.298806i	−0.988776	+	0.149403i	$0.952265\pi$
$678$	0	0
$679$	− 10363.9i	− 0.585761i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−14057.0	−0.787517	−0.393759	−	0.919214i	$-0.628825\pi$
$683$	−14057.0	−0.787517	−0.393759	+	0.919214i	$0.628825\pi$
$684$	0	0
$685$	37280.7	2.07945
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5002.96i	0.276629i
$690$	0	0
$691$	16916.0i	0.931282i	0.884974	+	0.465641i	$0.154176\pi$
$691$	16916.0i	0.931282i	−0.884974	+	0.465641i	$0.845824\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	58717.3	3.20471
$696$	0	0
$697$	−540.162	−0.0293545
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3743.98i	0.201723i	0.994900	+	0.100862i	$0.0321600\pi$
$701$	3743.98i	0.201723i	−0.994900	+	0.100862i	$0.967840\pi$
$702$	0	0
$703$	6323.18i	0.339237i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13656.0	−0.726433
$708$	0	0
$709$	−5993.31	−0.317466	−0.158733	−	0.987322i	$-0.550741\pi$
$709$	−5993.31	−0.317466	−0.158733	+	0.987322i	$0.550741\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	41336.4i	2.17119i
$714$	0	0
$715$	− 7957.68i	− 0.416224i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9158.35	0.475033	0.237517	−	0.971383i	$-0.423667\pi$
$719$	9158.35	0.475033	0.237517	+	0.971383i	$0.423667\pi$
$720$	0	0
$721$	40723.8	2.10351
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 74198.0i	− 3.80089i
$726$	0	0
$727$	− 20102.5i	− 1.02553i	−0.858529	−	0.512765i	$-0.828621\pi$
$727$	− 20102.5i	− 1.02553i	0.858529	−	0.512765i	$-0.171379\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−669.053	−0.0338520
$732$	0	0
$733$	−8493.27	−0.427975	−0.213988	−	0.976836i	$-0.568645\pi$
$733$	−8493.27	−0.427975	−0.213988	+	0.976836i	$0.568645\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 29622.5i	− 1.48054i
$738$	0	0
$739$	3413.95i	0.169938i	0.996384	+	0.0849689i	$0.0270791\pi$
$739$	3413.95i	0.169938i	−0.996384	+	0.0849689i	$0.972921\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−199.861	−0.00986836	−0.00493418	−	0.999988i	$-0.501571\pi$
$743$	−199.861	−0.00986836	−0.00493418	+	0.999988i	$0.501571\pi$
$744$	0	0
$745$	−775.263	−0.0381254
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12545.6i	0.612025i
$750$	0	0
$751$	− 1351.41i	− 0.0656641i	−0.999461	−	0.0328321i	$-0.989547\pi$
$751$	− 1351.41i	− 0.0656641i	0.999461	−	0.0328321i	$-0.0104526\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	39350.7	1.89684
$756$	0	0
$757$	33084.3	1.58847	0.794233	−	0.607613i	$-0.207873\pi$
$757$	33084.3	1.58847	0.794233	+	0.607613i	$0.207873\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	15677.9i	0.746811i	0.927668	+	0.373406i	$0.121810\pi$
$761$	15677.9i	0.746811i	−0.927668	+	0.373406i	$0.878190\pi$
$762$	0	0
$763$	6988.63i	0.331593i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	292.182	0.0137550
$768$	0	0
$769$	−39525.9	−1.85350	−0.926750	−	0.375679i	$-0.877410\pi$
$769$	−39525.9	−1.85350	−0.926750	+	0.375679i	$0.877410\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	29192.3i	1.35831i	0.733994	+	0.679156i	$0.237654\pi$
$773$	29192.3i	1.35831i	−0.733994	+	0.679156i	$0.762346\pi$
$774$	0	0
$775$	68944.1i	3.19554i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16601.9	0.763575
$780$	0	0
$781$	−34231.7	−1.56838
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24536.1i	1.11558i
$786$	0	0
$787$	24107.0i	1.09190i	0.837819	+	0.545949i	$0.183831\pi$
$787$	24107.0i	1.09190i	−0.837819	+	0.545949i	$0.816169\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−49848.7	−2.24073
$792$	0	0
$793$	8480.62	0.379768
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8464.67i	0.376203i	0.982150	+	0.188102i	$0.0602334\pi$
$797$	8464.67i	0.376203i	−0.982150	+	0.188102i	$0.939767\pi$
$798$	0	0
$799$	63.4576i	0.00280972i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	296.284	0.0130207
$804$	0	0
$805$	−65780.8	−2.88008
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 29597.4i	− 1.28626i	−0.765755	−	0.643132i	$-0.777635\pi$
$809$	− 29597.4i	− 1.28626i	0.765755	−	0.643132i	$-0.222365\pi$
$810$	0	0
$811$	14572.2i	0.630949i	0.948934	+	0.315474i	$0.102164\pi$
$811$	14572.2i	0.630949i	−0.948934	+	0.315474i	$0.897836\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−35104.0	−1.50876
$816$	0	0
$817$	20563.4	0.880565
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7894.06i	0.335572i	0.985823	+	0.167786i	$0.0536618\pi$
$821$	7894.06i	0.335572i	−0.985823	+	0.167786i	$0.946338\pi$
$822$	0	0
$823$	2643.21i	0.111952i	0.998432	+	0.0559760i	$0.0178270\pi$
$823$	2643.21i	0.111952i	−0.998432	+	0.0559760i	$0.982173\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11809.6	0.496567	0.248284	−	0.968687i	$-0.420133\pi$
$827$	11809.6	0.496567	0.248284	+	0.968687i	$0.420133\pi$
$828$	0	0
$829$	−13461.7	−0.563987	−0.281993	−	0.959416i	$-0.590996\pi$
$829$	−13461.7	−0.563987	−0.281993	+	0.959416i	$0.590996\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 269.732i	− 0.0112193i
$834$	0	0
$835$	− 39522.6i	− 1.63801i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12513.7	−0.514922	−0.257461	−	0.966289i	$-0.582886\pi$
$839$	−12513.7	−0.514922	−0.257461	+	0.966289i	$0.582886\pi$
$840$	0	0
$841$	−61023.7	−2.50210
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 39579.1i	− 1.61132i
$846$	0	0
$847$	6874.13i	0.278864i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	17403.8	0.701051
$852$	0	0
$853$	−25549.9	−1.02557	−0.512786	−	0.858517i	$-0.671386\pi$
$853$	−25549.9	−1.02557	−0.512786	+	0.858517i	$0.671386\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	32706.4i	1.30365i	0.758370	+	0.651825i	$0.225996\pi$
$857$	32706.4i	1.30365i	−0.758370	+	0.651825i	$0.774004\pi$
$858$	0	0
$859$	27072.8i	1.07534i	0.843157	+	0.537668i	$0.180695\pi$
$859$	27072.8i	1.07534i	−0.843157	+	0.537668i	$0.819305\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−28641.5	−1.12974	−0.564870	−	0.825180i	$-0.691074\pi$
$863$	−28641.5	−1.12974	−0.564870	+	0.825180i	$0.691074\pi$
$864$	0	0
$865$	−30004.5	−1.17940
$866$	0	0
$867$	0	0
$868$	0	0
$869$	18082.5i	0.705877i
$870$	0	0
$871$	11856.9i	0.461259i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−55695.9	−2.15185
$876$	0	0
$877$	−7094.49	−0.273163	−0.136581	−	0.990629i	$-0.543612\pi$
$877$	−7094.49	−0.273163	−0.136581	+	0.990629i	$0.543612\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	43494.2i	1.66329i	0.555310	+	0.831643i	$0.312600\pi$
$881$	43494.2i	1.66329i	−0.555310	+	0.831643i	$0.687400\pi$
$882$	0	0
$883$	1247.80i	0.0475560i	0.999717	+	0.0237780i	$0.00756948\pi$
$883$	1247.80i	0.0475560i	−0.999717	+	0.0237780i	$0.992431\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7320.80	0.277123	0.138562	−	0.990354i	$-0.455752\pi$
$887$	7320.80	0.277123	0.138562	+	0.990354i	$0.455752\pi$
$888$	0	0
$889$	22664.3	0.855046
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1950.38i	− 0.0730871i
$894$	0	0
$895$	− 51783.9i	− 1.93402i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	79364.7	2.94434
$900$	0	0
$901$	703.736	0.0260209
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 75719.1i	− 2.78120i
$906$	0	0
$907$	− 40054.6i	− 1.46636i	−0.680034	−	0.733181i	$-0.738035\pi$
$907$	− 40054.6i	− 1.46636i	0.680034	−	0.733181i	$-0.261965\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−11863.6	−0.431458	−0.215729	−	0.976453i	$-0.569213\pi$
$911$	−11863.6	−0.431458	−0.215729	+	0.976453i	$0.569213\pi$
$912$	0	0
$913$	−18101.3	−0.656151
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 25860.2i	− 0.931277i
$918$	0	0
$919$	41184.9i	1.47831i	0.673537	+	0.739153i	$0.264774\pi$
$919$	41184.9i	1.47831i	−0.673537	+	0.739153i	$0.735226\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13701.8	0.488625
$924$	0	0
$925$	29027.4	1.03180
$926$	0	0
$927$	0	0
$928$	0	0
$929$	15430.9i	0.544962i	0.962161	+	0.272481i	$0.0878442\pi$
$929$	15430.9i	0.544962i	−0.962161	+	0.272481i	$0.912156\pi$
$930$	0	0
$931$	8290.24i	0.291839i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1119.36	−0.0391517
$936$	0	0
$937$	−19757.6	−0.688852	−0.344426	−	0.938813i	$-0.611926\pi$
$937$	−19757.6	−0.688852	−0.344426	+	0.938813i	$0.611926\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	47029.5i	1.62924i	0.579993	+	0.814622i	$0.303055\pi$
$941$	47029.5i	1.62924i	−0.579993	+	0.814622i	$0.696945\pi$
$942$	0	0
$943$	− 45694.7i	− 1.57797i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13487.8	0.462825	0.231412	−	0.972856i	$-0.425665\pi$
$947$	13487.8	0.462825	0.231412	+	0.972856i	$0.425665\pi$
$948$	0	0
$949$	−118.593	−0.00405657
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 8080.06i	− 0.274647i	−0.990526	−	0.137324i	$-0.956150\pi$
$953$	− 8080.06i	− 0.274647i	0.990526	−	0.137324i	$-0.0438500\pi$
$954$	0	0
$955$	− 38541.2i	− 1.30593i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−42521.9	−1.43181
$960$	0	0
$961$	−43953.9	−1.47541
$962$	0	0
$963$	0	0
$964$	0	0
$965$	68335.5i	2.27958i
$966$	0	0
$967$	− 28145.7i	− 0.935994i	−0.883730	−	0.467997i	$-0.844976\pi$
$967$	− 28145.7i	− 0.935994i	0.883730	−	0.467997i	$-0.155024\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17202.4	−0.568539	−0.284269	−	0.958744i	$-0.591751\pi$
$971$	−17202.4	−0.568539	−0.284269	+	0.958744i	$0.591751\pi$
$972$	0	0
$973$	−66972.2	−2.20661
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24457.7i	− 0.800893i	−0.916320	−	0.400446i	$-0.868855\pi$
$977$	− 24457.7i	− 0.800893i	0.916320	−	0.400446i	$-0.131145\pi$
$978$	0	0
$979$	− 29142.8i	− 0.951388i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15491.0	0.502632	0.251316	−	0.967905i	$-0.419137\pi$
$983$	15491.0	0.502632	0.251316	+	0.967905i	$0.419137\pi$
$984$	0	0
$985$	−11333.9	−0.366627
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 56598.2i	− 1.81974i
$990$	0	0
$991$	− 7771.50i	− 0.249112i	−0.992213	−	0.124556i	$-0.960249\pi$
$991$	− 7771.50i	− 0.249112i	0.992213	−	0.124556i	$-0.0397506\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	75118.7	2.39339
$996$	0	0
$997$	−53099.2	−1.68673	−0.843364	−	0.537343i	$-0.819428\pi$
$997$	−53099.2	−1.68673	−0.843364	+	0.537343i	$0.819428\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.4.c.b.1151.12	yes	12
3.2	odd	2		1152.4.c.c.1151.1	yes	12
4.3	odd	2		1152.4.c.c.1151.12	yes	12
8.3	odd	2		1152.4.c.d.1151.1	yes	12
8.5	even	2		1152.4.c.a.1151.1	✓	12
12.11	even	2	inner	1152.4.c.b.1151.1	yes	12
16.3	odd	4		2304.4.f.k.1151.11		12
16.5	even	4		2304.4.f.j.1151.2		12
16.11	odd	4		2304.4.f.i.1151.1		12
16.13	even	4		2304.4.f.l.1151.12		12
24.5	odd	2		1152.4.c.d.1151.12	yes	12
24.11	even	2		1152.4.c.a.1151.12	yes	12
48.5	odd	4		2304.4.f.k.1151.12		12
48.11	even	4		2304.4.f.l.1151.11		12
48.29	odd	4		2304.4.f.i.1151.2		12
48.35	even	4		2304.4.f.j.1151.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.c.a.1151.1	✓	12	8.5	even	2
1152.4.c.a.1151.12	yes	12	24.11	even	2
1152.4.c.b.1151.1	yes	12	12.11	even	2	inner
1152.4.c.b.1151.12	yes	12	1.1	even	1	trivial
1152.4.c.c.1151.1	yes	12	3.2	odd	2
1152.4.c.c.1151.12	yes	12	4.3	odd	2
1152.4.c.d.1151.1	yes	12	8.3	odd	2
1152.4.c.d.1151.12	yes	12	24.5	odd	2
2304.4.f.i.1151.1		12	16.11	odd	4
2304.4.f.i.1151.2		12	48.29	odd	4
2304.4.f.j.1151.1		12	48.35	even	4
2304.4.f.j.1151.2		12	16.5	even	4
2304.4.f.k.1151.11		12	16.3	odd	4
2304.4.f.k.1151.12		12	48.5	odd	4
2304.4.f.l.1151.11		12	48.11	even	4
2304.4.f.l.1151.12		12	16.13	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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1152.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+19.4649i q^{5} -22.2014i q^{7} +31.9589 q^{11} -12.7921 q^{13} +1.79939i q^{17} -55.3042i q^{19} -152.218 q^{23} -253.882 q^{25} +292.254i q^{29} -271.560i q^{31} +432.148 q^{35} -114.334 q^{37} +300.192i q^{41} +371.823i q^{43} +35.2663 q^{47} -149.902 q^{49} -391.098i q^{53} +622.077i q^{55} -22.8408 q^{59} -662.958 q^{61} -248.997i q^{65} -926.894i q^{67} -1071.12 q^{71} +9.27078 q^{73} -709.534i q^{77} +565.805i q^{79} -566.393 q^{83} -35.0248 q^{85} -911.883i q^{89} +284.003i q^{91} +1076.49 q^{95} +466.815 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 240 q^{23} - 300 q^{25} + 864 q^{35} - 264 q^{37} - 624 q^{47} + 132 q^{49} + 1632 q^{59} - 312 q^{61} - 4080 q^{71} + 432 q^{73} + 3744 q^{83} + 1704 q^{85} - 5856 q^{95} + 96 q^{97}+O(q^{100}) \) 12 * q - 240 * q^23 - 300 * q^25 + 864 * q^35 - 264 * q^37 - 624 * q^47 + 132 * q^49 + 1632 * q^59 - 312 * q^61 - 4080 * q^71 + 432 * q^73 + 3744 * q^83 + 1704 * q^85 - 5856 * q^95 + 96 * q^97

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