LMFDB - Embedded newform 1152.3.e.h.1025.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(1025,1152)]

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([0, 0, 1]))

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

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

chi := DirichletCharacter("1152.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.e (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:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.4
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.1025
Dual form			1152.3.e.h.1025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.87832i q^{5} +11.7980 q^{7} +O(q^{10})$	$q+4.87832i q^{5} +11.7980 q^{7} +9.75663i q^{11} +22.6969 q^{13} -22.1988i q^{17} +17.7980 q^{19} -14.1421i q^{23} +1.20204 q^{25} -20.0061i q^{29} -39.7980 q^{31} +57.5542i q^{35} +2.40408 q^{37} +64.3395i q^{41} -3.19184 q^{43} -41.8549i q^{47} +90.1918 q^{49} +55.5043i q^{53} -47.5959 q^{55} -111.423i q^{59} +10.8082 q^{61} +110.723i q^{65} +18.2020 q^{67} -34.7983i q^{71} +87.5959 q^{73} +115.108i q^{77} -151.394 q^{79} +61.8112i q^{83} +108.293 q^{85} -72.9532i q^{89} +267.778 q^{91} +86.8241i q^{95} -87.3735 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7}+O(q^{10}) $ 4 * q + 8 * q^7	$ 4 q + 8 q^{7} + 32 q^{13} + 32 q^{19} + 44 q^{25} - 120 q^{31} + 88 q^{37} + 144 q^{43} + 204 q^{49} - 112 q^{55} + 200 q^{61} + 112 q^{67} + 272 q^{73} - 488 q^{79} + 296 q^{85} + 640 q^{91} + 160 q^{97}+O(q^{100}) $ 4 * q + 8 * q^7 + 32 * q^13 + 32 * q^19 + 44 * q^25 - 120 * q^31 + 88 * q^37 + 144 * q^43 + 204 * q^49 - 112 * q^55 + 200 * q^61 + 112 * q^67 + 272 * q^73 - 488 * q^79 + 296 * q^85 + 640 * q^91 + 160 * 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
$5$	4.87832i	0.975663i	0.872938	+	0.487832i	$0.162212\pi$
$5$	4.87832i	0.975663i	−0.872938	+	0.487832i	$0.837788\pi$
$6$	0	0
$7$	11.7980	1.68542	0.842711	−	0.538366i	$-0.180958\pi$
$7$	11.7980	1.68542	0.842711	+	0.538366i	$0.180958\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	9.75663i	0.886966i	0.896283	+	0.443483i	$0.146257\pi$
$11$	9.75663i	0.886966i	−0.896283	+	0.443483i	$0.853743\pi$
$12$	0	0
$13$	22.6969	1.74592	0.872959	−	0.487793i	$-0.162198\pi$
$13$	22.6969	1.74592	0.872959	+	0.487793i	$0.162198\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 22.1988i	− 1.30581i	−0.757438	−	0.652907i	$-0.773549\pi$
$17$	− 22.1988i	− 1.30581i	0.757438	−	0.652907i	$-0.226451\pi$
$18$	0	0
$19$	17.7980	0.936735	0.468367	−	0.883534i	$-0.344842\pi$
$19$	17.7980	0.936735	0.468367	+	0.883534i	$0.344842\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 14.1421i	− 0.614875i	−0.951568	−	0.307438i	$-0.900528\pi$
$23$	− 14.1421i	− 0.614875i	0.951568	−	0.307438i	$-0.0994716\pi$
$24$	0	0
$25$	1.20204	0.0480816
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 20.0061i	− 0.689865i	−0.938628	−	0.344932i	$-0.887902\pi$
$29$	− 20.0061i	− 0.689865i	0.938628	−	0.344932i	$-0.112098\pi$
$30$	0	0
$31$	−39.7980	−1.28381	−0.641903	−	0.766786i	$-0.721855\pi$
$31$	−39.7980	−1.28381	−0.641903	+	0.766786i	$0.721855\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	57.5542i	1.64440i
$36$	0	0
$37$	2.40408	0.0649752	0.0324876	−	0.999472i	$-0.489657\pi$
$37$	2.40408	0.0649752	0.0324876	+	0.999472i	$0.489657\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	64.3395i	1.56926i	0.619967	+	0.784628i	$0.287146\pi$
$41$	64.3395i	1.56926i	−0.619967	+	0.784628i	$0.712854\pi$
$42$	0	0
$43$	−3.19184	−0.0742287	−0.0371144	−	0.999311i	$-0.511817\pi$
$43$	−3.19184	−0.0742287	−0.0371144	+	0.999311i	$0.511817\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 41.8549i	− 0.890531i	−0.895399	−	0.445265i	$-0.853109\pi$
$47$	− 41.8549i	− 0.890531i	0.895399	−	0.445265i	$-0.146891\pi$
$48$	0	0
$49$	90.1918	1.84065
$50$	0	0
$51$	0	0
$52$	0	0
$53$	55.5043i	1.04725i	0.851949	+	0.523625i	$0.175421\pi$
$53$	55.5043i	1.04725i	−0.851949	+	0.523625i	$0.824579\pi$
$54$	0	0
$55$	−47.5959	−0.865380
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 111.423i	− 1.88852i	−0.329200	−	0.944260i	$-0.606779\pi$
$59$	− 111.423i	− 1.88852i	0.329200	−	0.944260i	$-0.393221\pi$
$60$	0	0
$61$	10.8082	0.177183	0.0885915	−	0.996068i	$-0.471763\pi$
$61$	10.8082	0.177183	0.0885915	+	0.996068i	$0.471763\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	110.723i	1.70343i
$66$	0	0
$67$	18.2020	0.271672	0.135836	−	0.990731i	$-0.456628\pi$
$67$	18.2020	0.271672	0.135836	+	0.990731i	$0.456628\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 34.7983i	− 0.490117i	−0.969508	−	0.245059i	$-0.921193\pi$
$71$	− 34.7983i	− 0.490117i	0.969508	−	0.245059i	$-0.0788072\pi$
$72$	0	0
$73$	87.5959	1.19994	0.599972	−	0.800021i	$-0.295178\pi$
$73$	87.5959	1.19994	0.599972	+	0.800021i	$0.295178\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	115.108i	1.49491i
$78$	0	0
$79$	−151.394	−1.91638	−0.958189	−	0.286136i	$-0.907629\pi$
$79$	−151.394	−1.91638	−0.958189	+	0.286136i	$0.907629\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	61.8112i	0.744714i	0.928090	+	0.372357i	$0.121450\pi$
$83$	61.8112i	0.744714i	−0.928090	+	0.372357i	$0.878550\pi$
$84$	0	0
$85$	108.293	1.27403
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 72.9532i	− 0.819699i	−0.912153	−	0.409850i	$-0.865581\pi$
$89$	− 72.9532i	− 0.819699i	0.912153	−	0.409850i	$-0.134419\pi$
$90$	0	0
$91$	267.778	2.94261
$92$	0	0
$93$	0	0
$94$	0	0
$95$	86.8241i	0.913937i
$96$	0	0
$97$	−87.3735	−0.900757	−0.450379	−	0.892838i	$-0.648711\pi$
$97$	−87.3735	−0.900757	−0.450379	+	0.892838i	$0.648711\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.63573i	0.0161953i	0.999967	+	0.00809766i	$0.00257759\pi$
$101$	1.63573i	0.0161953i	−0.999967	+	0.00809766i	$0.997422\pi$
$102$	0	0
$103$	38.9898	0.378542	0.189271	−	0.981925i	$-0.439388\pi$
$103$	38.9898	0.378542	0.189271	+	0.981925i	$0.439388\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	205.303i	1.91872i	0.282179	+	0.959362i	$0.408943\pi$
$107$	205.303i	1.91872i	−0.282179	+	0.959362i	$0.591057\pi$
$108$	0	0
$109$	33.3031	0.305533	0.152766	−	0.988262i	$-0.451182\pi$
$109$	33.3031	0.305533	0.152766	+	0.988262i	$0.451182\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	93.8951i	0.830930i	0.909609	+	0.415465i	$0.136381\pi$
$113$	93.8951i	0.830930i	−0.909609	+	0.415465i	$0.863619\pi$
$114$	0	0
$115$	68.9898	0.599911
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 261.901i	− 2.20085i
$120$	0	0
$121$	25.8082	0.213291
$122$	0	0
$123$	0	0
$124$	0	0
$125$	127.822i	1.02257i
$126$	0	0
$127$	−45.8184	−0.360775	−0.180387	−	0.983596i	$-0.557735\pi$
$127$	−45.8184	−0.360775	−0.180387	+	0.983596i	$0.557735\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	194.304i	1.48324i	0.670821	+	0.741619i	$0.265942\pi$
$131$	194.304i	1.48324i	−0.670821	+	0.741619i	$0.734058\pi$
$132$	0	0
$133$	209.980	1.57879
$134$	0	0
$135$	0	0
$136$	0	0
$137$	90.3668i	0.659612i	0.944049	+	0.329806i	$0.106983\pi$
$137$	90.3668i	0.659612i	−0.944049	+	0.329806i	$0.893017\pi$
$138$	0	0
$139$	8.22245	0.0591543	0.0295772	−	0.999563i	$-0.490584\pi$
$139$	8.22245	0.0591543	0.0295772	+	0.999563i	$0.490584\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	221.446i	1.54857i
$144$	0	0
$145$	97.5959	0.673075
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 79.2457i	− 0.531851i	−0.963994	−	0.265925i	$-0.914323\pi$
$149$	− 79.2457i	− 0.531851i	0.963994	−	0.265925i	$-0.0856775\pi$
$150$	0	0
$151$	−14.9898	−0.0992702	−0.0496351	−	0.998767i	$-0.515806\pi$
$151$	−14.9898	−0.0992702	−0.0496351	+	0.998767i	$0.515806\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 194.147i	− 1.25256i
$156$	0	0
$157$	267.576	1.70430	0.852151	−	0.523295i	$-0.175298\pi$
$157$	267.576	1.70430	0.852151	+	0.523295i	$0.175298\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 166.848i	− 1.03633i
$162$	0	0
$163$	−261.171	−1.60228	−0.801139	−	0.598478i	$-0.795772\pi$
$163$	−261.171	−1.60228	−0.801139	+	0.598478i	$0.795772\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 300.070i	− 1.79683i	−0.439150	−	0.898414i	$-0.644720\pi$
$167$	− 300.070i	− 1.79683i	0.439150	−	0.898414i	$-0.355280\pi$
$168$	0	0
$169$	346.151	2.04823
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 31.2909i	− 0.180872i	−0.995902	−	0.0904362i	$-0.971174\pi$
$173$	− 31.2909i	− 0.180872i	0.995902	−	0.0904362i	$-0.0288261\pi$
$174$	0	0
$175$	14.1816	0.0810379
$176$	0	0
$177$	0	0
$178$	0	0
$179$	66.4825i	0.371410i	0.982606	+	0.185705i	$0.0594570\pi$
$179$	66.4825i	0.371410i	−0.982606	+	0.185705i	$0.940543\pi$
$180$	0	0
$181$	−235.464	−1.30091	−0.650454	−	0.759546i	$-0.725421\pi$
$181$	−235.464	−1.30091	−0.650454	+	0.759546i	$0.725421\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.7279i	0.0633939i
$186$	0	0
$187$	216.586	1.15821
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 140.535i	− 0.735787i	−0.929868	−	0.367893i	$-0.880079\pi$
$191$	− 140.535i	− 0.735787i	0.929868	−	0.367893i	$-0.119921\pi$
$192$	0	0
$193$	293.151	1.51892	0.759459	−	0.650556i	$-0.225464\pi$
$193$	293.151	1.51892	0.759459	+	0.650556i	$0.225464\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	60.4324i	0.306763i	0.988167	+	0.153382i	$0.0490164\pi$
$197$	60.4324i	0.306763i	−0.988167	+	0.153382i	$0.950984\pi$
$198$	0	0
$199$	−143.757	−0.722398	−0.361199	−	0.932489i	$-0.617632\pi$
$199$	−143.757	−0.722398	−0.361199	+	0.932489i	$0.617632\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 236.031i	− 1.16271i
$204$	0	0
$205$	−313.868	−1.53107
$206$	0	0
$207$	0	0
$208$	0	0
$209$	173.648i	0.830852i
$210$	0	0
$211$	−255.778	−1.21222	−0.606108	−	0.795382i	$-0.707270\pi$
$211$	−255.778	−1.21222	−0.606108	+	0.795382i	$0.707270\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 15.5708i	− 0.0724222i
$216$	0	0
$217$	−469.535	−2.16375
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 503.845i	− 2.27984i
$222$	0	0
$223$	−248.969	−1.11645	−0.558227	−	0.829688i	$-0.688518\pi$
$223$	−248.969	−1.11645	−0.558227	+	0.829688i	$0.688518\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	277.600i	1.22291i	0.791280	+	0.611454i	$0.209415\pi$
$227$	277.600i	1.22291i	−0.791280	+	0.611454i	$0.790585\pi$
$228$	0	0
$229$	−193.666	−0.845704	−0.422852	−	0.906199i	$-0.638971\pi$
$229$	−193.666	−0.845704	−0.422852	+	0.906199i	$0.638971\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	150.435i	0.645643i	0.946460	+	0.322821i	$0.104631\pi$
$233$	150.435i	0.645643i	−0.946460	+	0.322821i	$0.895369\pi$
$234$	0	0
$235$	204.182	0.868858
$236$	0	0
$237$	0	0
$238$	0	0
$239$	140.593i	0.588255i	0.955766	+	0.294128i	$0.0950291\pi$
$239$	140.593i	0.588255i	−0.955766	+	0.294128i	$0.904971\pi$
$240$	0	0
$241$	−228.182	−0.946812	−0.473406	−	0.880844i	$-0.656976\pi$
$241$	−228.182	−0.946812	−0.473406	+	0.880844i	$0.656976\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	439.984i	1.79585i
$246$	0	0
$247$	403.959	1.63546
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 214.546i	− 0.854766i	−0.904071	−	0.427383i	$-0.859436\pi$
$251$	− 214.546i	− 0.854766i	0.904071	−	0.427383i	$-0.140564\pi$
$252$	0	0
$253$	137.980	0.545374
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 382.652i	− 1.48892i	−0.667669	−	0.744458i	$-0.732708\pi$
$257$	− 382.652i	− 1.48892i	0.667669	−	0.744458i	$-0.267292\pi$
$258$	0	0
$259$	28.3633	0.109511
$260$	0	0
$261$	0	0
$262$	0	0
$263$	164.563i	0.625713i	0.949800	+	0.312856i	$0.101286\pi$
$263$	164.563i	0.625713i	−0.949800	+	0.312856i	$0.898714\pi$
$264$	0	0
$265$	−270.767	−1.02176
$266$	0	0
$267$	0	0
$268$	0	0
$269$	239.480i	0.890262i	0.895465	+	0.445131i	$0.146843\pi$
$269$	239.480i	0.890262i	−0.895465	+	0.445131i	$0.853157\pi$
$270$	0	0
$271$	−153.778	−0.567445	−0.283722	−	0.958906i	$-0.591569\pi$
$271$	−153.778	−0.567445	−0.283722	+	0.958906i	$0.591569\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	11.7279i	0.0426468i
$276$	0	0
$277$	135.242	0.488238	0.244119	−	0.969745i	$-0.421501\pi$
$277$	135.242	0.488238	0.244119	+	0.969745i	$0.421501\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	63.6396i	0.226475i	0.993568	+	0.113238i	$0.0361222\pi$
$281$	63.6396i	0.226475i	−0.993568	+	0.113238i	$0.963878\pi$
$282$	0	0
$283$	−333.576	−1.17871	−0.589356	−	0.807873i	$-0.700618\pi$
$283$	−333.576	−1.17871	−0.589356	+	0.807873i	$0.700618\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	759.075i	2.64486i
$288$	0	0
$289$	−203.788	−0.705148
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 278.822i	− 0.951610i	−0.879551	−	0.475805i	$-0.842157\pi$
$293$	− 278.822i	− 0.951610i	0.879551	−	0.475805i	$-0.157843\pi$
$294$	0	0
$295$	543.555	1.84256
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 320.983i	− 1.07352i
$300$	0	0
$301$	−37.6571	−0.125107
$302$	0	0
$303$	0	0
$304$	0	0
$305$	52.7256i	0.172871i
$306$	0	0
$307$	99.7367	0.324875	0.162438	−	0.986719i	$-0.448064\pi$
$307$	99.7367	0.324875	0.162438	+	0.986719i	$0.448064\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	313.927i	1.00941i	0.863292	+	0.504705i	$0.168399\pi$
$311$	313.927i	1.00941i	−0.863292	+	0.504705i	$0.831601\pi$
$312$	0	0
$313$	−52.4245	−0.167490	−0.0837452	−	0.996487i	$-0.526688\pi$
$313$	−52.4245	−0.167490	−0.0837452	+	0.996487i	$0.526688\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	68.8888i	0.217315i	0.994079	+	0.108657i	$0.0346551\pi$
$317$	68.8888i	0.217315i	−0.994079	+	0.108657i	$0.965345\pi$
$318$	0	0
$319$	195.192	0.611887
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 395.094i	− 1.22320i
$324$	0	0
$325$	27.2827	0.0839466
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 493.803i	− 1.50092i
$330$	0	0
$331$	−114.202	−0.345021	−0.172511	−	0.985008i	$-0.555188\pi$
$331$	−114.202	−0.345021	−0.172511	+	0.985008i	$0.555188\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	88.7953i	0.265061i
$336$	0	0
$337$	−203.192	−0.602943	−0.301472	−	0.953475i	$-0.597478\pi$
$337$	−203.192	−0.602943	−0.301472	+	0.953475i	$0.597478\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 388.294i	− 1.13869i
$342$	0	0
$343$	485.980	1.41685
$344$	0	0
$345$	0	0
$346$	0	0
$347$	79.5524i	0.229258i	0.993408	+	0.114629i	$0.0365679\pi$
$347$	79.5524i	0.229258i	−0.993408	+	0.114629i	$0.963432\pi$
$348$	0	0
$349$	−479.898	−1.37507	−0.687533	−	0.726153i	$-0.741306\pi$
$349$	−479.898	−1.37507	−0.687533	+	0.726153i	$0.741306\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	132.922i	0.376549i	0.982116	+	0.188274i	$0.0602894\pi$
$353$	132.922i	0.376549i	−0.982116	+	0.188274i	$0.939711\pi$
$354$	0	0
$355$	169.757	0.478189
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 660.308i	− 1.83930i	−0.392742	−	0.919649i	$-0.628473\pi$
$359$	− 660.308i	− 1.83930i	0.392742	−	0.919649i	$-0.371527\pi$
$360$	0	0
$361$	−44.2327	−0.122528
$362$	0	0
$363$	0	0
$364$	0	0
$365$	427.320i	1.17074i
$366$	0	0
$367$	281.373	0.766685	0.383343	−	0.923606i	$-0.374773\pi$
$367$	281.373	0.766685	0.383343	+	0.923606i	$0.374773\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	654.837i	1.76506i
$372$	0	0
$373$	−617.959	−1.65673	−0.828364	−	0.560191i	$-0.810728\pi$
$373$	−617.959	−1.65673	−0.828364	+	0.560191i	$0.810728\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 454.077i	− 1.20445i
$378$	0	0
$379$	−695.292	−1.83454	−0.917272	−	0.398262i	$-0.869613\pi$
$379$	−695.292	−1.83454	−0.917272	+	0.398262i	$0.869613\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 315.269i	− 0.823156i	−0.911375	−	0.411578i	$-0.864978\pi$
$383$	− 315.269i	− 0.823156i	0.911375	−	0.411578i	$-0.135022\pi$
$384$	0	0
$385$	−561.535	−1.45853
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 539.364i	− 1.38654i	−0.720677	−	0.693270i	$-0.756169\pi$
$389$	− 539.364i	− 1.38654i	0.720677	−	0.693270i	$-0.243831\pi$
$390$	0	0
$391$	−313.939	−0.802912
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 738.547i	− 1.86974i
$396$	0	0
$397$	−490.404	−1.23527	−0.617637	−	0.786463i	$-0.711910\pi$
$397$	−490.404	−1.23527	−0.617637	+	0.786463i	$0.711910\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 460.891i	− 1.14935i	−0.818380	−	0.574677i	$-0.805128\pi$
$401$	− 460.891i	− 1.14935i	0.818380	−	0.574677i	$-0.194872\pi$
$402$	0	0
$403$	−903.292	−2.24142
$404$	0	0
$405$	0	0
$406$	0	0
$407$	23.4557i	0.0576308i
$408$	0	0
$409$	535.292	1.30878	0.654391	−	0.756156i	$-0.272925\pi$
$409$	535.292	1.30878	0.654391	+	0.756156i	$0.272925\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1314.56i	− 3.18296i
$414$	0	0
$415$	−301.535	−0.726590
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 405.993i	− 0.968958i	−0.874803	−	0.484479i	$-0.839009\pi$
$419$	− 405.993i	− 0.968958i	0.874803	−	0.484479i	$-0.160991\pi$
$420$	0	0
$421$	−70.6969	−0.167926	−0.0839631	−	0.996469i	$-0.526758\pi$
$421$	−70.6969	−0.167926	−0.0839631	+	0.996469i	$0.526758\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 26.6839i	− 0.0627856i
$426$	0	0
$427$	127.514	0.298628
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 334.239i	− 0.775497i	−0.921765	−	0.387749i	$-0.873253\pi$
$431$	− 334.239i	− 0.775497i	0.921765	−	0.387749i	$-0.126747\pi$
$432$	0	0
$433$	88.3429	0.204025	0.102013	−	0.994783i	$-0.467472\pi$
$433$	88.3429	0.204025	0.102013	+	0.994783i	$0.467472\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 251.701i	− 0.575975i
$438$	0	0
$439$	−231.353	−0.527000	−0.263500	−	0.964659i	$-0.584877\pi$
$439$	−231.353	−0.527000	−0.263500	+	0.964659i	$0.584877\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	715.091i	1.61420i	0.590414	+	0.807101i	$0.298965\pi$
$443$	715.091i	1.61420i	−0.590414	+	0.807101i	$0.701035\pi$
$444$	0	0
$445$	355.889	0.799750
$446$	0	0
$447$	0	0
$448$	0	0
$449$	576.455i	1.28386i	0.766761	+	0.641932i	$0.221867\pi$
$449$	576.455i	1.28386i	−0.766761	+	0.641932i	$0.778133\pi$
$450$	0	0
$451$	−627.737	−1.39188
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1306.30i	2.87100i
$456$	0	0
$457$	−168.990	−0.369781	−0.184890	−	0.982759i	$-0.559193\pi$
$457$	−168.990	−0.369781	−0.184890	+	0.982759i	$0.559193\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	772.653i	1.67604i	0.545641	+	0.838019i	$0.316286\pi$
$461$	772.653i	1.67604i	−0.545641	+	0.838019i	$0.683714\pi$
$462$	0	0
$463$	461.818	0.997448	0.498724	−	0.866761i	$-0.333802\pi$
$463$	461.818	0.997448	0.498724	+	0.866761i	$0.333802\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 516.016i	− 1.10496i	−0.833526	−	0.552480i	$-0.813682\pi$
$467$	− 516.016i	− 1.10496i	0.833526	−	0.552480i	$-0.186318\pi$
$468$	0	0
$469$	214.747	0.457883
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 31.1416i	− 0.0658384i
$474$	0	0
$475$	21.3939	0.0450397
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 175.706i	− 0.366818i	−0.983037	−	0.183409i	$-0.941287\pi$
$479$	− 175.706i	− 0.366818i	0.983037	−	0.183409i	$-0.0587133\pi$
$480$	0	0
$481$	54.5653	0.113441
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 426.235i	− 0.878836i
$486$	0	0
$487$	694.100	1.42526	0.712628	−	0.701542i	$-0.247505\pi$
$487$	694.100	1.42526	0.712628	+	0.701542i	$0.247505\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 512.173i	− 1.04312i	−0.853214	−	0.521561i	$-0.825350\pi$
$491$	− 512.173i	− 1.04312i	0.853214	−	0.521561i	$-0.174650\pi$
$492$	0	0
$493$	−444.111	−0.900834
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 410.549i	− 0.826054i
$498$	0	0
$499$	132.849	0.266230	0.133115	−	0.991101i	$-0.457502\pi$
$499$	132.849	0.266230	0.133115	+	0.991101i	$0.457502\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 243.986i	− 0.485063i	−0.970144	−	0.242531i	$-0.922022\pi$
$503$	− 243.986i	− 0.485063i	0.970144	−	0.242531i	$-0.0779777\pi$
$504$	0	0
$505$	−7.97959	−0.0158012
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 396.530i	− 0.779038i	−0.921018	−	0.389519i	$-0.872641\pi$
$509$	− 396.530i	− 0.779038i	0.921018	−	0.389519i	$-0.127359\pi$
$510$	0	0
$511$	1033.45	2.02241
$512$	0	0
$513$	0	0
$514$	0	0
$515$	190.205i	0.369329i
$516$	0	0
$517$	408.363	0.789871
$518$	0	0
$519$	0	0
$520$	0	0
$521$	76.4527i	0.146742i	0.997305	+	0.0733711i	$0.0233757\pi$
$521$	76.4527i	0.146742i	−0.997305	+	0.0733711i	$0.976624\pi$
$522$	0	0
$523$	298.524	0.570793	0.285396	−	0.958410i	$-0.407875\pi$
$523$	298.524	0.570793	0.285396	+	0.958410i	$0.407875\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	883.468i	1.67641i
$528$	0	0
$529$	329.000	0.621928
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1460.31i	2.73979i
$534$	0	0
$535$	−1001.53	−1.87203
$536$	0	0
$537$	0	0
$538$	0	0
$539$	879.968i	1.63259i
$540$	0	0
$541$	566.070	1.04634	0.523170	−	0.852228i	$-0.324749\pi$
$541$	566.070	1.04634	0.523170	+	0.852228i	$0.324749\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	162.463i	0.298097i
$546$	0	0
$547$	500.727	0.915405	0.457702	−	0.889105i	$-0.348672\pi$
$547$	500.727	0.915405	0.457702	+	0.889105i	$0.348672\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 356.067i	− 0.646220i
$552$	0	0
$553$	−1786.14	−3.22991
$554$	0	0
$555$	0	0
$556$	0	0
$557$	224.568i	0.403174i	0.979471	+	0.201587i	$0.0646098\pi$
$557$	224.568i	0.403174i	−0.979471	+	0.201587i	$0.935390\pi$
$558$	0	0
$559$	−72.4449	−0.129597
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 708.092i	− 1.25771i	−0.777521	−	0.628856i	$-0.783523\pi$
$563$	− 708.092i	− 1.25771i	0.777521	−	0.628856i	$-0.216477\pi$
$564$	0	0
$565$	−458.050	−0.810708
$566$	0	0
$567$	0	0
$568$	0	0
$569$	627.453i	1.10273i	0.834264	+	0.551365i	$0.185893\pi$
$569$	627.453i	1.10273i	−0.834264	+	0.551365i	$0.814107\pi$
$570$	0	0
$571$	−25.2531	−0.0442260	−0.0221130	−	0.999755i	$-0.507039\pi$
$571$	−25.2531	−0.0442260	−0.0221130	+	0.999755i	$0.507039\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 16.9994i	− 0.0295642i
$576$	0	0
$577$	158.000	0.273830	0.136915	−	0.990583i	$-0.456281\pi$
$577$	158.000	0.273830	0.136915	+	0.990583i	$0.456281\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	729.246i	1.25516i
$582$	0	0
$583$	−541.535	−0.928876
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1060.63i	1.80687i	0.428728	+	0.903434i	$0.358962\pi$
$587$	1060.63i	1.80687i	−0.428728	+	0.903434i	$0.641038\pi$
$588$	0	0
$589$	−708.322	−1.20258
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1051.50i	− 1.77319i	−0.462545	−	0.886596i	$-0.653064\pi$
$593$	− 1051.50i	− 1.77319i	0.462545	−	0.886596i	$-0.346936\pi$
$594$	0	0
$595$	1277.63	2.14729
$596$	0	0
$597$	0	0
$598$	0	0
$599$	52.0835i	0.0869507i	0.999054	+	0.0434754i	$0.0138430\pi$
$599$	52.0835i	0.0869507i	−0.999054	+	0.0434754i	$0.986157\pi$
$600$	0	0
$601$	66.0000	0.109817	0.0549085	−	0.998491i	$-0.482513\pi$
$601$	66.0000	0.109817	0.0549085	+	0.998491i	$0.482513\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	125.900i	0.208100i
$606$	0	0
$607$	210.627	0.346996	0.173498	−	0.984834i	$-0.444493\pi$
$607$	210.627	0.346996	0.173498	+	0.984834i	$0.444493\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 949.979i	− 1.55479i
$612$	0	0
$613$	−320.706	−0.523175	−0.261587	−	0.965180i	$-0.584246\pi$
$613$	−320.706	−0.523175	−0.261587	+	0.965180i	$0.584246\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 233.316i	− 0.378146i	−0.981963	−	0.189073i	$-0.939452\pi$
$617$	− 233.316i	− 0.378146i	0.981963	−	0.189073i	$-0.0605484\pi$
$618$	0	0
$619$	206.020	0.332828	0.166414	−	0.986056i	$-0.446781\pi$
$619$	206.020	0.332828	0.166414	+	0.986056i	$0.446781\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 860.699i	− 1.38154i
$624$	0	0
$625$	−593.504	−0.949607
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 53.3678i	− 0.0848455i
$630$	0	0
$631$	42.5857	0.0674892	0.0337446	−	0.999430i	$-0.489257\pi$
$631$	42.5857	0.0674892	0.0337446	+	0.999430i	$0.489257\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 223.516i	− 0.351994i
$636$	0	0
$637$	2047.08	3.21362
$638$	0	0
$639$	0	0
$640$	0	0
$641$	160.021i	0.249643i	0.992179	+	0.124821i	$0.0398358\pi$
$641$	160.021i	0.249643i	−0.992179	+	0.124821i	$0.960164\pi$
$642$	0	0
$643$	997.980	1.55207	0.776034	−	0.630691i	$-0.217229\pi$
$643$	997.980	1.55207	0.776034	+	0.630691i	$0.217229\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	552.628i	0.854140i	0.904219	+	0.427070i	$0.140454\pi$
$647$	552.628i	0.854140i	−0.904219	+	0.427070i	$0.859546\pi$
$648$	0	0
$649$	1087.11	1.67505
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 374.272i	− 0.573158i	−0.958057	−	0.286579i	$-0.907482\pi$
$653$	− 374.272i	− 0.573158i	0.958057	−	0.286579i	$-0.0925181\pi$
$654$	0	0
$655$	−947.878	−1.44714
$656$	0	0
$657$	0	0
$658$	0	0
$659$	251.759i	0.382032i	0.981587	+	0.191016i	$0.0611782\pi$
$659$	251.759i	0.382032i	−0.981587	+	0.191016i	$0.938822\pi$
$660$	0	0
$661$	−339.535	−0.513668	−0.256834	−	0.966456i	$-0.582679\pi$
$661$	−339.535	−0.513668	−0.256834	+	0.966456i	$0.582679\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1024.35i	1.54037i
$666$	0	0
$667$	−282.929	−0.424181
$668$	0	0
$669$	0	0
$670$	0	0
$671$	105.451i	0.157155i
$672$	0	0
$673$	1049.03	1.55873	0.779367	−	0.626567i	$-0.215541\pi$
$673$	1049.03	1.55873	0.779367	+	0.626567i	$0.215541\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 310.390i	− 0.458479i	−0.973370	−	0.229240i	$-0.926376\pi$
$677$	− 310.390i	− 0.458479i	0.973370	−	0.229240i	$-0.0736239\pi$
$678$	0	0
$679$	−1030.83	−1.51816
$680$	0	0
$681$	0	0
$682$	0	0
$683$	746.233i	1.09258i	0.837596	+	0.546291i	$0.183961\pi$
$683$	746.233i	1.09258i	−0.837596	+	0.546291i	$0.816039\pi$
$684$	0	0
$685$	−440.838	−0.643559
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1259.78i	1.82841i
$690$	0	0
$691$	−776.241	−1.12336	−0.561679	−	0.827355i	$-0.689845\pi$
$691$	−776.241	−1.12336	−0.561679	+	0.827355i	$0.689845\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	40.1117i	0.0577147i
$696$	0	0
$697$	1428.26	2.04916
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 801.223i	− 1.14297i	−0.820612	−	0.571486i	$-0.806367\pi$
$701$	− 801.223i	− 1.14297i	0.820612	−	0.571486i	$-0.193633\pi$
$702$	0	0
$703$	42.7878	0.0608645
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.2982i	0.0272959i
$708$	0	0
$709$	1198.25	1.69006	0.845030	−	0.534719i	$-0.179583\pi$
$709$	1198.25	1.69006	0.845030	+	0.534719i	$0.179583\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	562.828i	0.789380i
$714$	0	0
$715$	−1080.28	−1.51088
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1206.22i	− 1.67764i	−0.544409	−	0.838820i	$-0.683246\pi$
$719$	− 1206.22i	− 1.67764i	0.544409	−	0.838820i	$-0.316754\pi$
$720$	0	0
$721$	460.000	0.638003
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 24.0481i	− 0.0331698i
$726$	0	0
$727$	−503.394	−0.692426	−0.346213	−	0.938156i	$-0.612533\pi$
$727$	−503.394	−0.692426	−0.346213	+	0.938156i	$0.612533\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	70.8550i	0.0969289i
$732$	0	0
$733$	−380.736	−0.519421	−0.259711	−	0.965687i	$-0.583627\pi$
$733$	−380.736	−0.519421	−0.259711	+	0.965687i	$0.583627\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	177.591i	0.240964i
$738$	0	0
$739$	−717.775	−0.971279	−0.485640	−	0.874159i	$-0.661413\pi$
$739$	−717.775	−0.971279	−0.485640	+	0.874159i	$0.661413\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	239.675i	0.322577i	0.986907	+	0.161288i	$0.0515649\pi$
$743$	239.675i	0.322577i	−0.986907	+	0.161288i	$0.948435\pi$
$744$	0	0
$745$	386.586	0.518907
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2422.16i	3.23386i
$750$	0	0
$751$	481.010	0.640493	0.320246	−	0.947334i	$-0.396234\pi$
$751$	481.010	0.640493	0.320246	+	0.947334i	$0.396234\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 73.1249i	− 0.0968542i
$756$	0	0
$757$	338.656	0.447366	0.223683	−	0.974662i	$-0.428192\pi$
$757$	338.656	0.447366	0.223683	+	0.974662i	$0.428192\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 525.945i	− 0.691123i	−0.938396	−	0.345561i	$-0.887688\pi$
$761$	− 525.945i	− 0.691123i	0.938396	−	0.345561i	$-0.112312\pi$
$762$	0	0
$763$	392.908	0.514952
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2528.95i	− 3.29720i
$768$	0	0
$769$	248.506	0.323155	0.161577	−	0.986860i	$-0.448342\pi$
$769$	248.506	0.323155	0.161577	+	0.986860i	$0.448342\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 402.701i	− 0.520958i	−0.965479	−	0.260479i	$-0.916119\pi$
$773$	− 402.701i	− 0.520958i	0.965479	−	0.260479i	$-0.0838806\pi$
$774$	0	0
$775$	−47.8388	−0.0617275
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1145.11i	1.46998i
$780$	0	0
$781$	339.514	0.434717
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1305.32i	1.66283i
$786$	0	0
$787$	1258.97	1.59971	0.799853	−	0.600195i	$-0.204910\pi$
$787$	1258.97	1.59971	0.799853	+	0.600195i	$0.204910\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1107.77i	1.40047i
$792$	0	0
$793$	245.312	0.309347
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 878.734i	− 1.10255i	−0.834323	−	0.551276i	$-0.814141\pi$
$797$	− 878.734i	− 1.10255i	0.834323	−	0.551276i	$-0.185859\pi$
$798$	0	0
$799$	−929.131	−1.16287
$800$	0	0
$801$	0	0
$802$	0	0
$803$	854.641i	1.06431i
$804$	0	0
$805$	813.939	1.01110
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1199.39i	− 1.48256i	−0.671195	−	0.741281i	$-0.734218\pi$
$809$	− 1199.39i	− 1.48256i	0.671195	−	0.741281i	$-0.265782\pi$
$810$	0	0
$811$	30.2837	0.0373412	0.0186706	−	0.999826i	$-0.494057\pi$
$811$	30.2837	0.0373412	0.0186706	+	0.999826i	$0.494057\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1274.08i	− 1.56328i
$816$	0	0
$817$	−56.8082	−0.0695326
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 144.441i	− 0.175933i	−0.996123	−	0.0879665i	$-0.971963\pi$
$821$	− 144.441i	− 0.175933i	0.996123	−	0.0879665i	$-0.0280368\pi$
$822$	0	0
$823$	−795.839	−0.966997	−0.483499	−	0.875345i	$-0.660634\pi$
$823$	−795.839	−0.966997	−0.483499	+	0.875345i	$0.660634\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 567.541i	− 0.686265i	−0.939287	−	0.343133i	$-0.888512\pi$
$827$	− 567.541i	− 0.686265i	0.939287	−	0.343133i	$-0.111488\pi$
$828$	0	0
$829$	1188.54	1.43370	0.716849	−	0.697228i	$-0.245584\pi$
$829$	1188.54	1.43370	0.716849	+	0.697228i	$0.245584\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2002.15i	− 2.40354i
$834$	0	0
$835$	1463.84	1.75310
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 598.256i	− 0.713058i	−0.934284	−	0.356529i	$-0.883960\pi$
$839$	− 598.256i	− 0.713058i	0.934284	−	0.356529i	$-0.116040\pi$
$840$	0	0
$841$	440.757	0.524087
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1688.63i	1.99838i
$846$	0	0
$847$	304.484	0.359485
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 33.9989i	− 0.0399517i
$852$	0	0
$853$	−337.192	−0.395301	−0.197651	−	0.980273i	$-0.563331\pi$
$853$	−337.192	−0.395301	−0.197651	+	0.980273i	$0.563331\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	913.497i	1.06592i	0.846139	+	0.532962i	$0.178921\pi$
$857$	913.497i	1.06592i	−0.846139	+	0.532962i	$0.821079\pi$
$858$	0	0
$859$	748.645	0.871531	0.435765	−	0.900060i	$-0.356478\pi$
$859$	748.645	0.871531	0.435765	+	0.900060i	$0.356478\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	460.632i	0.533757i	0.963730	+	0.266879i	$0.0859923\pi$
$863$	460.632i	0.533757i	−0.963730	+	0.266879i	$0.914008\pi$
$864$	0	0
$865$	152.647	0.176470
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1477.09i	− 1.69976i
$870$	0	0
$871$	413.131	0.474318
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1508.04i	1.72347i
$876$	0	0
$877$	−1066.32	−1.21588	−0.607938	−	0.793985i	$-0.708003\pi$
$877$	−1066.32	−1.21588	−0.607938	+	0.793985i	$0.708003\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	586.899i	0.666173i	0.942896	+	0.333087i	$0.108090\pi$
$881$	586.899i	0.666173i	−0.942896	+	0.333087i	$0.891910\pi$
$882$	0	0
$883$	1242.95	1.40764	0.703820	−	0.710378i	$-0.251476\pi$
$883$	1242.95	1.40764	0.703820	+	0.710378i	$0.251476\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 37.8259i	− 0.0426447i	−0.999773	−	0.0213224i	$-0.993212\pi$
$887$	− 37.8259i	− 0.0426447i	0.999773	−	0.0213224i	$-0.00678763\pi$
$888$	0	0
$889$	−540.563	−0.608058
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 744.933i	− 0.834191i
$894$	0	0
$895$	−324.322	−0.362371
$896$	0	0
$897$	0	0
$898$	0	0
$899$	796.201i	0.885652i
$900$	0	0
$901$	1232.13	1.36751
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1148.67i	− 1.26925i
$906$	0	0
$907$	−763.837	−0.842157	−0.421079	−	0.907024i	$-0.638348\pi$
$907$	−763.837	−0.842157	−0.421079	+	0.907024i	$0.638348\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	378.008i	0.414937i	0.978242	+	0.207469i	$0.0665225\pi$
$911$	378.008i	0.414937i	−0.978242	+	0.207469i	$0.933478\pi$
$912$	0	0
$913$	−603.069	−0.660536
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2292.39i	2.49988i
$918$	0	0
$919$	206.141	0.224310	0.112155	−	0.993691i	$-0.464225\pi$
$919$	206.141	0.224310	0.112155	+	0.993691i	$0.464225\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 789.815i	− 0.855704i
$924$	0	0
$925$	2.88981	0.00312411
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1793.22i	1.93027i	0.261752	+	0.965135i	$0.415700\pi$
$929$	1793.22i	1.93027i	−0.261752	+	0.965135i	$0.584300\pi$
$930$	0	0
$931$	1605.23	1.72420
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1056.57i	1.13002i
$936$	0	0
$937$	322.767	0.344469	0.172234	−	0.985056i	$-0.444901\pi$
$937$	322.767	0.344469	0.172234	+	0.985056i	$0.444901\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	805.965i	0.856499i	0.903661	+	0.428249i	$0.140870\pi$
$941$	805.965i	0.856499i	−0.903661	+	0.428249i	$0.859130\pi$
$942$	0	0
$943$	909.898	0.964897
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1197.31i	− 1.26432i	−0.774839	−	0.632158i	$-0.782169\pi$
$947$	− 1197.31i	− 1.26432i	0.774839	−	0.632158i	$-0.217831\pi$
$948$	0	0
$949$	1988.16	2.09500
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1109.71i	− 1.16444i	−0.813030	−	0.582222i	$-0.802184\pi$
$953$	− 1109.71i	− 1.16444i	0.813030	−	0.582222i	$-0.197816\pi$
$954$	0	0
$955$	685.576	0.717880
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1066.14i	1.11172i
$960$	0	0
$961$	622.878	0.648156
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1430.08i	1.48195i
$966$	0	0
$967$	−1075.72	−1.11243	−0.556213	−	0.831040i	$-0.687746\pi$
$967$	−1075.72	−1.11243	−0.556213	+	0.831040i	$0.687746\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	958.163i	0.986779i	0.869808	+	0.493390i	$0.164242\pi$
$971$	958.163i	0.986779i	−0.869808	+	0.493390i	$0.835758\pi$
$972$	0	0
$973$	97.0081	0.0997000
$974$	0	0
$975$	0	0
$976$	0	0
$977$	116.792i	0.119542i	0.998212	+	0.0597709i	$0.0190370\pi$
$977$	116.792i	0.119542i	−0.998212	+	0.0597709i	$0.980963\pi$
$978$	0	0
$979$	711.778	0.727046
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 209.817i	− 0.213446i	−0.994289	−	0.106723i	$-0.965964\pi$
$983$	− 209.817i	− 0.213446i	0.994289	−	0.106723i	$-0.0340358\pi$
$984$	0	0
$985$	−294.808	−0.299298
$986$	0	0
$987$	0	0
$988$	0	0
$989$	45.1394i	0.0456414i
$990$	0	0
$991$	−359.798	−0.363066	−0.181533	−	0.983385i	$-0.558106\pi$
$991$	−359.798	−0.363066	−0.181533	+	0.983385i	$0.558106\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 701.293i	− 0.704817i
$996$	0	0
$997$	−1628.06	−1.63296	−0.816478	−	0.577377i	$-0.804076\pi$
$997$	−1628.06	−1.63296	−0.816478	+	0.577377i	$0.804076\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.e.h.1025.4	yes	4
3.2	odd	2	inner	1152.3.e.h.1025.1	yes	4
4.3	odd	2		1152.3.e.d.1025.4	yes	4
8.3	odd	2		1152.3.e.b.1025.1	✓	4
8.5	even	2		1152.3.e.f.1025.1	yes	4
12.11	even	2		1152.3.e.d.1025.1	yes	4
16.3	odd	4		2304.3.h.k.2177.8		8
16.5	even	4		2304.3.h.i.2177.1		8
16.11	odd	4		2304.3.h.k.2177.1		8
16.13	even	4		2304.3.h.i.2177.8		8
24.5	odd	2		1152.3.e.f.1025.4	yes	4
24.11	even	2		1152.3.e.b.1025.4	yes	4
48.5	odd	4		2304.3.h.i.2177.7		8
48.11	even	4		2304.3.h.k.2177.7		8
48.29	odd	4		2304.3.h.i.2177.2		8
48.35	even	4		2304.3.h.k.2177.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.e.b.1025.1	✓	4	8.3	odd	2
1152.3.e.b.1025.4	yes	4	24.11	even	2
1152.3.e.d.1025.1	yes	4	12.11	even	2
1152.3.e.d.1025.4	yes	4	4.3	odd	2
1152.3.e.f.1025.1	yes	4	8.5	even	2
1152.3.e.f.1025.4	yes	4	24.5	odd	2
1152.3.e.h.1025.1	yes	4	3.2	odd	2	inner
1152.3.e.h.1025.4	yes	4	1.1	even	1	trivial
2304.3.h.i.2177.1		8	16.5	even	4
2304.3.h.i.2177.2		8	48.29	odd	4
2304.3.h.i.2177.7		8	48.5	odd	4
2304.3.h.i.2177.8		8	16.13	even	4
2304.3.h.k.2177.1		8	16.11	odd	4
2304.3.h.k.2177.2		8	48.35	even	4
2304.3.h.k.2177.7		8	48.11	even	4
2304.3.h.k.2177.8		8	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.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.87832i q^{5} +11.7980 q^{7} +O(q^{10})\)	\(q+4.87832i q^{5} +11.7980 q^{7} +9.75663i q^{11} +22.6969 q^{13} -22.1988i q^{17} +17.7980 q^{19} -14.1421i q^{23} +1.20204 q^{25} -20.0061i q^{29} -39.7980 q^{31} +57.5542i q^{35} +2.40408 q^{37} +64.3395i q^{41} -3.19184 q^{43} -41.8549i q^{47} +90.1918 q^{49} +55.5043i q^{53} -47.5959 q^{55} -111.423i q^{59} +10.8082 q^{61} +110.723i q^{65} +18.2020 q^{67} -34.7983i q^{71} +87.5959 q^{73} +115.108i q^{77} -151.394 q^{79} +61.8112i q^{83} +108.293 q^{85} -72.9532i q^{89} +267.778 q^{91} +86.8241i q^{95} -87.3735 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7}+O(q^{10}) \) 4 * q + 8 * q^7	\( 4 q + 8 q^{7} + 32 q^{13} + 32 q^{19} + 44 q^{25} - 120 q^{31} + 88 q^{37} + 144 q^{43} + 204 q^{49} - 112 q^{55} + 200 q^{61} + 112 q^{67} + 272 q^{73} - 488 q^{79} + 296 q^{85} + 640 q^{91} + 160 q^{97}+O(q^{100}) \) 4 * q + 8 * q^7 + 32 * q^13 + 32 * q^19 + 44 * q^25 - 120 * q^31 + 88 * q^37 + 144 * q^43 + 204 * q^49 - 112 * q^55 + 200 * q^61 + 112 * q^67 + 272 * q^73 - 488 * q^79 + 296 * q^85 + 640 * q^91 + 160 * q^97

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