LMFDB - Embedded newform 1152.3.e.b.1025.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,3,Mod(1025,1152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,0,0,0,-8,0,0,0,0,0,-32,0,0,0,0,0,32,0,0,0,0,0,44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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	$-0.837788\pi$
$5$	− 4.87832i	− 0.975663i	0.872938	−	0.487832i	$-0.162212\pi$
$6$	0	0
$7$	−11.7980	−1.68542	−0.842711	−	0.538366i	$-0.819042\pi$
$7$	−11.7980	−1.68542	−0.842711	+	0.538366i	$0.819042\pi$
$8$	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	$-0.837802\pi$
$13$	−22.6969	−1.74592	−0.872959	+	0.487793i	$0.837802\pi$
$14$	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	$0.0994716\pi$
$23$	14.1421i	0.614875i	−0.951568	+	0.307438i	$0.900528\pi$
$24$	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	$0.112098\pi$
$29$	20.0061i	0.689865i	−0.938628	+	0.344932i	$0.887902\pi$
$30$	0	0
$31$	39.7980	1.28381	0.641903	−	0.766786i	$-0.278145\pi$
$31$	39.7980	1.28381	0.641903	+	0.766786i	$0.278145\pi$
$32$	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	$-0.510343\pi$
$37$	−2.40408	−0.0649752	−0.0324876	+	0.999472i	$0.510343\pi$
$38$	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	$0.146891\pi$
$47$	41.8549i	0.890531i	−0.895399	+	0.445265i	$0.853109\pi$
$48$	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	$-0.824579\pi$
$53$	− 55.5043i	− 1.04725i	0.851949	−	0.523625i	$-0.175421\pi$
$54$	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	$-0.528237\pi$
$61$	−10.8082	−0.177183	−0.0885915	+	0.996068i	$0.528237\pi$
$62$	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	$0.0788072\pi$
$71$	34.7983i	0.490117i	−0.969508	+	0.245059i	$0.921193\pi$
$72$	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	$-0.0923708\pi$
$79$	151.394	1.91638	0.958189	+	0.286136i	$0.0923708\pi$
$80$	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	$-0.997422\pi$
$101$	− 1.63573i	− 0.0161953i	0.999967	−	0.00809766i	$-0.00257759\pi$
$102$	0	0
$103$	−38.9898	−0.378542	−0.189271	−	0.981925i	$-0.560612\pi$
$103$	−38.9898	−0.378542	−0.189271	+	0.981925i	$0.560612\pi$
$104$	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	$-0.548818\pi$
$109$	−33.3031	−0.305533	−0.152766	+	0.988262i	$0.548818\pi$
$110$	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	$-0.442265\pi$
$127$	45.8184	0.360775	0.180387	+	0.983596i	$0.442265\pi$
$128$	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	$0.0856775\pi$
$149$	79.2457i	0.531851i	−0.963994	+	0.265925i	$0.914323\pi$
$150$	0	0
$151$	14.9898	0.0992702	0.0496351	−	0.998767i	$-0.484194\pi$
$151$	14.9898	0.0992702	0.0496351	+	0.998767i	$0.484194\pi$
$152$	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	$-0.824702\pi$
$157$	−267.576	−1.70430	−0.852151	+	0.523295i	$0.824702\pi$
$158$	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	$0.355280\pi$
$167$	300.070i	1.79683i	−0.439150	+	0.898414i	$0.644720\pi$
$168$	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	$0.0288261\pi$
$173$	31.2909i	0.180872i	−0.995902	+	0.0904362i	$0.971174\pi$
$174$	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	$-0.274579\pi$
$181$	235.464	1.30091	0.650454	+	0.759546i	$0.274579\pi$
$182$	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	$0.119921\pi$
$191$	140.535i	0.735787i	−0.929868	+	0.367893i	$0.880079\pi$
$192$	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	$-0.950984\pi$
$197$	− 60.4324i	− 0.306763i	0.988167	−	0.153382i	$-0.0490164\pi$
$198$	0	0
$199$	143.757	0.722398	0.361199	−	0.932489i	$-0.382368\pi$
$199$	143.757	0.722398	0.361199	+	0.932489i	$0.382368\pi$
$200$	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	$-0.311482\pi$
$223$	248.969	1.11645	0.558227	+	0.829688i	$0.311482\pi$
$224$	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	$-0.361029\pi$
$229$	193.666	0.845704	0.422852	+	0.906199i	$0.361029\pi$
$230$	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	$-0.904971\pi$
$239$	− 140.593i	− 0.588255i	0.955766	−	0.294128i	$-0.0950291\pi$
$240$	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	$-0.898714\pi$
$263$	− 164.563i	− 0.625713i	0.949800	−	0.312856i	$-0.101286\pi$
$264$	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	$-0.853157\pi$
$269$	− 239.480i	− 0.890262i	0.895465	−	0.445131i	$-0.146843\pi$
$270$	0	0
$271$	153.778	0.567445	0.283722	−	0.958906i	$-0.408431\pi$
$271$	153.778	0.567445	0.283722	+	0.958906i	$0.408431\pi$
$272$	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	$-0.578499\pi$
$277$	−135.242	−0.488238	−0.244119	+	0.969745i	$0.578499\pi$
$278$	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	$0.157843\pi$
$293$	278.822i	0.951610i	−0.879551	+	0.475805i	$0.842157\pi$
$294$	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	$-0.831601\pi$
$311$	− 313.927i	− 1.00941i	0.863292	−	0.504705i	$-0.168399\pi$
$312$	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	$-0.965345\pi$
$317$	− 68.8888i	− 0.217315i	0.994079	−	0.108657i	$-0.0346551\pi$
$318$	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	$-0.258694\pi$
$349$	479.898	1.37507	0.687533	+	0.726153i	$0.258694\pi$
$350$	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	$0.371527\pi$
$359$	660.308i	1.83930i	−0.392742	+	0.919649i	$0.628473\pi$
$360$	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	$-0.625227\pi$
$367$	−281.373	−0.766685	−0.383343	+	0.923606i	$0.625227\pi$
$368$	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	$-0.189272\pi$
$373$	617.959	1.65673	0.828364	+	0.560191i	$0.189272\pi$
$374$	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	$0.135022\pi$
$383$	315.269i	0.823156i	−0.911375	+	0.411578i	$0.864978\pi$
$384$	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	$0.243831\pi$
$389$	539.364i	1.38654i	−0.720677	+	0.693270i	$0.756169\pi$
$390$	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	$-0.288090\pi$
$397$	490.404	1.23527	0.617637	+	0.786463i	$0.288090\pi$
$398$	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	$-0.473242\pi$
$421$	70.6969	0.167926	0.0839631	+	0.996469i	$0.473242\pi$
$422$	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	$0.126747\pi$
$431$	334.239i	0.775497i	−0.921765	+	0.387749i	$0.873253\pi$
$432$	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	$-0.415123\pi$
$439$	231.353	0.527000	0.263500	+	0.964659i	$0.415123\pi$
$440$	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	$-0.683714\pi$
$461$	− 772.653i	− 1.67604i	0.545641	−	0.838019i	$-0.316286\pi$
$462$	0	0
$463$	−461.818	−0.997448	−0.498724	−	0.866761i	$-0.666198\pi$
$463$	−461.818	−0.997448	−0.498724	+	0.866761i	$0.666198\pi$
$464$	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	$0.0587133\pi$
$479$	175.706i	0.366818i	−0.983037	+	0.183409i	$0.941287\pi$
$480$	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	$-0.752495\pi$
$487$	−694.100	−1.42526	−0.712628	+	0.701542i	$0.752495\pi$
$488$	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	$0.0779777\pi$
$503$	243.986i	0.485063i	−0.970144	+	0.242531i	$0.922022\pi$
$504$	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	$0.127359\pi$
$509$	396.530i	0.779038i	−0.921018	+	0.389519i	$0.872641\pi$
$510$	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	$-0.675251\pi$
$541$	−566.070	−1.04634	−0.523170	+	0.852228i	$0.675251\pi$
$542$	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	$-0.935390\pi$
$557$	− 224.568i	− 0.403174i	0.979471	−	0.201587i	$-0.0646098\pi$
$558$	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	$-0.986157\pi$
$599$	− 52.0835i	− 0.0869507i	0.999054	−	0.0434754i	$-0.0138430\pi$
$600$	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	$-0.555507\pi$
$607$	−210.627	−0.346996	−0.173498	+	0.984834i	$0.555507\pi$
$608$	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	$-0.415754\pi$
$613$	320.706	0.523175	0.261587	+	0.965180i	$0.415754\pi$
$614$	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	$-0.510743\pi$
$631$	−42.5857	−0.0674892	−0.0337446	+	0.999430i	$0.510743\pi$
$632$	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	$-0.859546\pi$
$647$	− 552.628i	− 0.854140i	0.904219	−	0.427070i	$-0.140454\pi$
$648$	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	$0.0925181\pi$
$653$	374.272i	0.573158i	−0.958057	+	0.286579i	$0.907482\pi$
$654$	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	$-0.417321\pi$
$661$	339.535	0.513668	0.256834	+	0.966456i	$0.417321\pi$
$662$	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	$0.0736239\pi$
$677$	310.390i	0.458479i	−0.973370	+	0.229240i	$0.926376\pi$
$678$	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	$0.193633\pi$
$701$	801.223i	1.14297i	−0.820612	+	0.571486i	$0.806367\pi$
$702$	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	$-0.820417\pi$
$709$	−1198.25	−1.69006	−0.845030	+	0.534719i	$0.820417\pi$
$710$	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	$0.316754\pi$
$719$	1206.22i	1.67764i	−0.544409	+	0.838820i	$0.683246\pi$
$720$	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	$-0.387467\pi$
$727$	503.394	0.692426	0.346213	+	0.938156i	$0.387467\pi$
$728$	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	$-0.416373\pi$
$733$	380.736	0.519421	0.259711	+	0.965687i	$0.416373\pi$
$734$	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	$-0.948435\pi$
$743$	− 239.675i	− 0.322577i	0.986907	−	0.161288i	$-0.0515649\pi$
$744$	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	$-0.603766\pi$
$751$	−481.010	−0.640493	−0.320246	+	0.947334i	$0.603766\pi$
$752$	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	$-0.571808\pi$
$757$	−338.656	−0.447366	−0.223683	+	0.974662i	$0.571808\pi$
$758$	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	$0.0838806\pi$
$773$	402.701i	0.520958i	−0.965479	+	0.260479i	$0.916119\pi$
$774$	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	$0.185859\pi$
$797$	878.734i	1.10255i	−0.834323	+	0.551276i	$0.814141\pi$
$798$	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	$0.0280368\pi$
$821$	144.441i	0.175933i	−0.996123	+	0.0879665i	$0.971963\pi$
$822$	0	0
$823$	795.839	0.966997	0.483499	−	0.875345i	$-0.339366\pi$
$823$	795.839	0.966997	0.483499	+	0.875345i	$0.339366\pi$
$824$	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	$-0.754416\pi$
$829$	−1188.54	−1.43370	−0.716849	+	0.697228i	$0.754416\pi$
$830$	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	$0.116040\pi$
$839$	598.256i	0.713058i	−0.934284	+	0.356529i	$0.883960\pi$
$840$	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	$-0.436669\pi$
$853$	337.192	0.395301	0.197651	+	0.980273i	$0.436669\pi$
$854$	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	$-0.914008\pi$
$863$	− 460.632i	− 0.533757i	0.963730	−	0.266879i	$-0.0859923\pi$
$864$	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	$-0.291997\pi$
$877$	1066.32	1.21588	0.607938	+	0.793985i	$0.291997\pi$
$878$	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	$0.00678763\pi$
$887$	37.8259i	0.0426447i	−0.999773	+	0.0213224i	$0.993212\pi$
$888$	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	$-0.933478\pi$
$911$	− 378.008i	− 0.414937i	0.978242	−	0.207469i	$-0.0665225\pi$
$912$	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	$-0.535775\pi$
$919$	−206.141	−0.224310	−0.112155	+	0.993691i	$0.535775\pi$
$920$	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	$-0.859130\pi$
$941$	− 805.965i	− 0.856499i	0.903661	−	0.428249i	$-0.140870\pi$
$942$	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	$-0.312254\pi$
$967$	1075.72	1.11243	0.556213	+	0.831040i	$0.312254\pi$
$968$	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	$0.0340358\pi$
$983$	209.817i	0.213446i	−0.994289	+	0.106723i	$0.965964\pi$
$984$	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	$-0.441894\pi$
$991$	359.798	0.363066	0.181533	+	0.983385i	$0.441894\pi$
$992$	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	$-0.195924\pi$
$997$	1628.06	1.63296	0.816478	+	0.577377i	$0.195924\pi$
$998$	0	0
$999$	0	0

Twists

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

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

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.e (of order \(2\), degree \(1\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more