LMFDB - Embedded newform 1344.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.a (trivial)

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:	yes
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +2.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +2.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} -12.0000 q^{11} +66.0000 q^{13} +6.00000 q^{15} -70.0000 q^{17} +92.0000 q^{19} +21.0000 q^{21} +16.0000 q^{23} -121.000 q^{25} +27.0000 q^{27} +122.000 q^{29} +64.0000 q^{31} -36.0000 q^{33} +14.0000 q^{35} +306.000 q^{37} +198.000 q^{39} +50.0000 q^{41} -20.0000 q^{43} +18.0000 q^{45} -176.000 q^{47} +49.0000 q^{49} -210.000 q^{51} -526.000 q^{53} -24.0000 q^{55} +276.000 q^{57} -540.000 q^{59} +818.000 q^{61} +63.0000 q^{63} +132.000 q^{65} +228.000 q^{67} +48.0000 q^{69} +864.000 q^{71} +106.000 q^{73} -363.000 q^{75} -84.0000 q^{77} +736.000 q^{79} +81.0000 q^{81} +588.000 q^{83} -140.000 q^{85} +366.000 q^{87} +146.000 q^{89} +462.000 q^{91} +192.000 q^{93} +184.000 q^{95} -1214.00 q^{97} -108.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	2.00000	0.178885	0.0894427	−	0.995992i	$-0.471491\pi$
$5$	2.00000	0.178885	0.0894427	+	0.995992i	$0.471491\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	66.0000	1.40809	0.704043	−	0.710158i	$-0.251376\pi$
$13$	66.0000	1.40809	0.704043	+	0.710158i	$0.251376\pi$
$14$	0	0
$15$	6.00000	0.103280
$16$	0	0
$17$	−70.0000	−0.998676	−0.499338	−	0.866407i	$-0.666423\pi$
$17$	−70.0000	−0.998676	−0.499338	+	0.866407i	$0.666423\pi$
$18$	0	0
$19$	92.0000	1.11086	0.555428	−	0.831565i	$-0.312555\pi$
$19$	92.0000	1.11086	0.555428	+	0.831565i	$0.312555\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	16.0000	0.145054	0.0725268	−	0.997366i	$-0.476894\pi$
$23$	16.0000	0.145054	0.0725268	+	0.997366i	$0.476894\pi$
$24$	0	0
$25$	−121.000	−0.968000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	122.000	0.781201	0.390601	−	0.920560i	$-0.372267\pi$
$29$	122.000	0.781201	0.390601	+	0.920560i	$0.372267\pi$
$30$	0	0
$31$	64.0000	0.370798	0.185399	−	0.982663i	$-0.440642\pi$
$31$	64.0000	0.370798	0.185399	+	0.982663i	$0.440642\pi$
$32$	0	0
$33$	−36.0000	−0.189903
$34$	0	0
$35$	14.0000	0.0676123
$36$	0	0
$37$	306.000	1.35962	0.679812	−	0.733386i	$-0.262061\pi$
$37$	306.000	1.35962	0.679812	+	0.733386i	$0.262061\pi$
$38$	0	0
$39$	198.000	0.812958
$40$	0	0
$41$	50.0000	0.190456	0.0952279	−	0.995455i	$-0.469642\pi$
$41$	50.0000	0.190456	0.0952279	+	0.995455i	$0.469642\pi$
$42$	0	0
$43$	−20.0000	−0.0709296	−0.0354648	−	0.999371i	$-0.511291\pi$
$43$	−20.0000	−0.0709296	−0.0354648	+	0.999371i	$0.511291\pi$
$44$	0	0
$45$	18.0000	0.0596285
$46$	0	0
$47$	−176.000	−0.546218	−0.273109	−	0.961983i	$-0.588052\pi$
$47$	−176.000	−0.546218	−0.273109	+	0.961983i	$0.588052\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−210.000	−0.576586
$52$	0	0
$53$	−526.000	−1.36324	−0.681619	−	0.731707i	$-0.738724\pi$
$53$	−526.000	−1.36324	−0.681619	+	0.731707i	$0.738724\pi$
$54$	0	0
$55$	−24.0000	−0.0588393
$56$	0	0
$57$	276.000	0.641353
$58$	0	0
$59$	−540.000	−1.19156	−0.595780	−	0.803148i	$-0.703157\pi$
$59$	−540.000	−1.19156	−0.595780	+	0.803148i	$0.703157\pi$
$60$	0	0
$61$	818.000	1.71695	0.858477	−	0.512852i	$-0.171411\pi$
$61$	818.000	1.71695	0.858477	+	0.512852i	$0.171411\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	132.000	0.251886
$66$	0	0
$67$	228.000	0.415741	0.207870	−	0.978156i	$-0.433347\pi$
$67$	228.000	0.415741	0.207870	+	0.978156i	$0.433347\pi$
$68$	0	0
$69$	48.0000	0.0837467
$70$	0	0
$71$	864.000	1.44420	0.722098	−	0.691791i	$-0.243178\pi$
$71$	864.000	1.44420	0.722098	+	0.691791i	$0.243178\pi$
$72$	0	0
$73$	106.000	0.169950	0.0849751	−	0.996383i	$-0.472919\pi$
$73$	106.000	0.169950	0.0849751	+	0.996383i	$0.472919\pi$
$74$	0	0
$75$	−363.000	−0.558875
$76$	0	0
$77$	−84.0000	−0.124321
$78$	0	0
$79$	736.000	1.04818	0.524092	−	0.851662i	$-0.324405\pi$
$79$	736.000	1.04818	0.524092	+	0.851662i	$0.324405\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	588.000	0.777607	0.388804	−	0.921321i	$-0.372888\pi$
$83$	588.000	0.777607	0.388804	+	0.921321i	$0.372888\pi$
$84$	0	0
$85$	−140.000	−0.178649
$86$	0	0
$87$	366.000	0.451027
$88$	0	0
$89$	146.000	0.173887	0.0869436	−	0.996213i	$-0.472290\pi$
$89$	146.000	0.173887	0.0869436	+	0.996213i	$0.472290\pi$
$90$	0	0
$91$	462.000	0.532206
$92$	0	0
$93$	192.000	0.214080
$94$	0	0
$95$	184.000	0.198716
$96$	0	0
$97$	−1214.00	−1.27075	−0.635376	−	0.772203i	$-0.719155\pi$
$97$	−1214.00	−1.27075	−0.635376	+	0.772203i	$0.719155\pi$
$98$	0	0
$99$	−108.000	−0.109640
$100$	0	0
$101$	−846.000	−0.833467	−0.416733	−	0.909029i	$-0.636825\pi$
$101$	−846.000	−0.833467	−0.416733	+	0.909029i	$0.636825\pi$
$102$	0	0
$103$	−168.000	−0.160714	−0.0803570	−	0.996766i	$-0.525606\pi$
$103$	−168.000	−0.160714	−0.0803570	+	0.996766i	$0.525606\pi$
$104$	0	0
$105$	42.0000	0.0390360
$106$	0	0
$107$	708.000	0.639672	0.319836	−	0.947473i	$-0.396372\pi$
$107$	708.000	0.639672	0.319836	+	0.947473i	$0.396372\pi$
$108$	0	0
$109$	−646.000	−0.567666	−0.283833	−	0.958874i	$-0.591606\pi$
$109$	−646.000	−0.567666	−0.283833	+	0.958874i	$0.591606\pi$
$110$	0	0
$111$	918.000	0.784979
$112$	0	0
$113$	1938.00	1.61338	0.806689	−	0.590976i	$-0.201257\pi$
$113$	1938.00	1.61338	0.806689	+	0.590976i	$0.201257\pi$
$114$	0	0
$115$	32.0000	0.0259480
$116$	0	0
$117$	594.000	0.469362
$118$	0	0
$119$	−490.000	−0.377464
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	150.000	0.109960
$124$	0	0
$125$	−492.000	−0.352047
$126$	0	0
$127$	224.000	0.156510	0.0782551	−	0.996933i	$-0.475065\pi$
$127$	224.000	0.156510	0.0782551	+	0.996933i	$0.475065\pi$
$128$	0	0
$129$	−60.0000	−0.0409512
$130$	0	0
$131$	2588.00	1.72607	0.863033	−	0.505148i	$-0.168562\pi$
$131$	2588.00	1.72607	0.863033	+	0.505148i	$0.168562\pi$
$132$	0	0
$133$	644.000	0.419864
$134$	0	0
$135$	54.0000	0.0344265
$136$	0	0
$137$	490.000	0.305573	0.152787	−	0.988259i	$-0.451175\pi$
$137$	490.000	0.305573	0.152787	+	0.988259i	$0.451175\pi$
$138$	0	0
$139$	1716.00	1.04712	0.523558	−	0.851990i	$-0.324604\pi$
$139$	1716.00	1.04712	0.523558	+	0.851990i	$0.324604\pi$
$140$	0	0
$141$	−528.000	−0.315359
$142$	0	0
$143$	−792.000	−0.463149
$144$	0	0
$145$	244.000	0.139745
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	2386.00	1.31187	0.655935	−	0.754817i	$-0.272274\pi$
$149$	2386.00	1.31187	0.655935	+	0.754817i	$0.272274\pi$
$150$	0	0
$151$	−104.000	−0.0560490	−0.0280245	−	0.999607i	$-0.508922\pi$
$151$	−104.000	−0.0560490	−0.0280245	+	0.999607i	$0.508922\pi$
$152$	0	0
$153$	−630.000	−0.332892
$154$	0	0
$155$	128.000	0.0663304
$156$	0	0
$157$	−1566.00	−0.796054	−0.398027	−	0.917374i	$-0.630305\pi$
$157$	−1566.00	−0.796054	−0.398027	+	0.917374i	$0.630305\pi$
$158$	0	0
$159$	−1578.00	−0.787066
$160$	0	0
$161$	112.000	0.0548251
$162$	0	0
$163$	1076.00	0.517048	0.258524	−	0.966005i	$-0.416764\pi$
$163$	1076.00	0.517048	0.258524	+	0.966005i	$0.416764\pi$
$164$	0	0
$165$	−72.0000	−0.0339709
$166$	0	0
$167$	−2760.00	−1.27889	−0.639447	−	0.768835i	$-0.720837\pi$
$167$	−2760.00	−1.27889	−0.639447	+	0.768835i	$0.720837\pi$
$168$	0	0
$169$	2159.00	0.982704
$170$	0	0
$171$	828.000	0.370285
$172$	0	0
$173$	−1558.00	−0.684697	−0.342348	−	0.939573i	$-0.611222\pi$
$173$	−1558.00	−0.684697	−0.342348	+	0.939573i	$0.611222\pi$
$174$	0	0
$175$	−847.000	−0.365870
$176$	0	0
$177$	−1620.00	−0.687947
$178$	0	0
$179$	3452.00	1.44142	0.720711	−	0.693235i	$-0.243815\pi$
$179$	3452.00	1.44142	0.720711	+	0.693235i	$0.243815\pi$
$180$	0	0
$181$	1162.00	0.477187	0.238593	−	0.971120i	$-0.423314\pi$
$181$	1162.00	0.477187	0.238593	+	0.971120i	$0.423314\pi$
$182$	0	0
$183$	2454.00	0.991284
$184$	0	0
$185$	612.000	0.243217
$186$	0	0
$187$	840.000	0.328486
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	1928.00	0.730394	0.365197	−	0.930930i	$-0.381002\pi$
$191$	1928.00	0.730394	0.365197	+	0.930930i	$0.381002\pi$
$192$	0	0
$193$	−318.000	−0.118602	−0.0593009	−	0.998240i	$-0.518887\pi$
$193$	−318.000	−0.118602	−0.0593009	+	0.998240i	$0.518887\pi$
$194$	0	0
$195$	396.000	0.145426
$196$	0	0
$197$	−4062.00	−1.46906	−0.734532	−	0.678574i	$-0.762598\pi$
$197$	−4062.00	−1.46906	−0.734532	+	0.678574i	$0.762598\pi$
$198$	0	0
$199$	5480.00	1.95209	0.976047	−	0.217558i	$-0.0698090\pi$
$199$	5480.00	1.95209	0.976047	+	0.217558i	$0.0698090\pi$
$200$	0	0
$201$	684.000	0.240028
$202$	0	0
$203$	854.000	0.295266
$204$	0	0
$205$	100.000	0.0340698
$206$	0	0
$207$	144.000	0.0483512
$208$	0	0
$209$	−1104.00	−0.365384
$210$	0	0
$211$	−4892.00	−1.59611	−0.798055	−	0.602585i	$-0.794138\pi$
$211$	−4892.00	−1.59611	−0.798055	+	0.602585i	$0.794138\pi$
$212$	0	0
$213$	2592.00	0.833807
$214$	0	0
$215$	−40.0000	−0.0126883
$216$	0	0
$217$	448.000	0.140148
$218$	0	0
$219$	318.000	0.0981208
$220$	0	0
$221$	−4620.00	−1.40622
$222$	0	0
$223$	4320.00	1.29726	0.648629	−	0.761105i	$-0.275343\pi$
$223$	4320.00	1.29726	0.648629	+	0.761105i	$0.275343\pi$
$224$	0	0
$225$	−1089.00	−0.322667
$226$	0	0
$227$	5516.00	1.61282	0.806409	−	0.591358i	$-0.201408\pi$
$227$	5516.00	1.61282	0.806409	+	0.591358i	$0.201408\pi$
$228$	0	0
$229$	−1670.00	−0.481907	−0.240954	−	0.970537i	$-0.577460\pi$
$229$	−1670.00	−0.481907	−0.240954	+	0.970537i	$0.577460\pi$
$230$	0	0
$231$	−252.000	−0.0717765
$232$	0	0
$233$	−3926.00	−1.10387	−0.551933	−	0.833888i	$-0.686110\pi$
$233$	−3926.00	−1.10387	−0.551933	+	0.833888i	$0.686110\pi$
$234$	0	0
$235$	−352.000	−0.0977104
$236$	0	0
$237$	2208.00	0.605169
$238$	0	0
$239$	3640.00	0.985155	0.492577	−	0.870269i	$-0.336055\pi$
$239$	3640.00	0.985155	0.492577	+	0.870269i	$0.336055\pi$
$240$	0	0
$241$	5650.00	1.51016	0.755080	−	0.655633i	$-0.227598\pi$
$241$	5650.00	1.51016	0.755080	+	0.655633i	$0.227598\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	98.0000	0.0255551
$246$	0	0
$247$	6072.00	1.56418
$248$	0	0
$249$	1764.00	0.448952
$250$	0	0
$251$	−4652.00	−1.16985	−0.584924	−	0.811088i	$-0.698876\pi$
$251$	−4652.00	−1.16985	−0.584924	+	0.811088i	$0.698876\pi$
$252$	0	0
$253$	−192.000	−0.0477112
$254$	0	0
$255$	−420.000	−0.103143
$256$	0	0
$257$	−6006.00	−1.45776	−0.728879	−	0.684642i	$-0.759958\pi$
$257$	−6006.00	−1.45776	−0.728879	+	0.684642i	$0.759958\pi$
$258$	0	0
$259$	2142.00	0.513890
$260$	0	0
$261$	1098.00	0.260400
$262$	0	0
$263$	5040.00	1.18167	0.590836	−	0.806792i	$-0.298798\pi$
$263$	5040.00	1.18167	0.590836	+	0.806792i	$0.298798\pi$
$264$	0	0
$265$	−1052.00	−0.243864
$266$	0	0
$267$	438.000	0.100394
$268$	0	0
$269$	−5478.00	−1.24163	−0.620817	−	0.783956i	$-0.713199\pi$
$269$	−5478.00	−1.24163	−0.620817	+	0.783956i	$0.713199\pi$
$270$	0	0
$271$	−2176.00	−0.487759	−0.243879	−	0.969806i	$-0.578420\pi$
$271$	−2176.00	−0.487759	−0.243879	+	0.969806i	$0.578420\pi$
$272$	0	0
$273$	1386.00	0.307269
$274$	0	0
$275$	1452.00	0.318396
$276$	0	0
$277$	4658.00	1.01037	0.505184	−	0.863011i	$-0.331425\pi$
$277$	4658.00	1.01037	0.505184	+	0.863011i	$0.331425\pi$
$278$	0	0
$279$	576.000	0.123599
$280$	0	0
$281$	−4614.00	−0.979531	−0.489765	−	0.871854i	$-0.662918\pi$
$281$	−4614.00	−0.979531	−0.489765	+	0.871854i	$0.662918\pi$
$282$	0	0
$283$	−1244.00	−0.261301	−0.130650	−	0.991429i	$-0.541707\pi$
$283$	−1244.00	−0.261301	−0.130650	+	0.991429i	$0.541707\pi$
$284$	0	0
$285$	552.000	0.114729
$286$	0	0
$287$	350.000	0.0719855
$288$	0	0
$289$	−13.0000	−0.00264604
$290$	0	0
$291$	−3642.00	−0.733669
$292$	0	0
$293$	6354.00	1.26691	0.633455	−	0.773780i	$-0.281636\pi$
$293$	6354.00	1.26691	0.633455	+	0.773780i	$0.281636\pi$
$294$	0	0
$295$	−1080.00	−0.213153
$296$	0	0
$297$	−324.000	−0.0633010
$298$	0	0
$299$	1056.00	0.204248
$300$	0	0
$301$	−140.000	−0.0268089
$302$	0	0
$303$	−2538.00	−0.481202
$304$	0	0
$305$	1636.00	0.307138
$306$	0	0
$307$	3740.00	0.695287	0.347643	−	0.937627i	$-0.386982\pi$
$307$	3740.00	0.695287	0.347643	+	0.937627i	$0.386982\pi$
$308$	0	0
$309$	−504.000	−0.0927882
$310$	0	0
$311$	−2184.00	−0.398210	−0.199105	−	0.979978i	$-0.563803\pi$
$311$	−2184.00	−0.398210	−0.199105	+	0.979978i	$0.563803\pi$
$312$	0	0
$313$	4442.00	0.802162	0.401081	−	0.916043i	$-0.368635\pi$
$313$	4442.00	0.802162	0.401081	+	0.916043i	$0.368635\pi$
$314$	0	0
$315$	126.000	0.0225374
$316$	0	0
$317$	4314.00	0.764348	0.382174	−	0.924090i	$-0.375175\pi$
$317$	4314.00	0.764348	0.382174	+	0.924090i	$0.375175\pi$
$318$	0	0
$319$	−1464.00	−0.256954
$320$	0	0
$321$	2124.00	0.369315
$322$	0	0
$323$	−6440.00	−1.10938
$324$	0	0
$325$	−7986.00	−1.36303
$326$	0	0
$327$	−1938.00	−0.327742
$328$	0	0
$329$	−1232.00	−0.206451
$330$	0	0
$331$	−7988.00	−1.32647	−0.663233	−	0.748413i	$-0.730816\pi$
$331$	−7988.00	−1.32647	−0.663233	+	0.748413i	$0.730816\pi$
$332$	0	0
$333$	2754.00	0.453208
$334$	0	0
$335$	456.000	0.0743700
$336$	0	0
$337$	−10606.0	−1.71438	−0.857189	−	0.515001i	$-0.827791\pi$
$337$	−10606.0	−1.71438	−0.857189	+	0.515001i	$0.827791\pi$
$338$	0	0
$339$	5814.00	0.931484
$340$	0	0
$341$	−768.000	−0.121963
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	96.0000	0.0149811
$346$	0	0
$347$	8340.00	1.29024	0.645122	−	0.764080i	$-0.276807\pi$
$347$	8340.00	1.29024	0.645122	+	0.764080i	$0.276807\pi$
$348$	0	0
$349$	10498.0	1.61016	0.805079	−	0.593168i	$-0.202123\pi$
$349$	10498.0	1.61016	0.805079	+	0.593168i	$0.202123\pi$
$350$	0	0
$351$	1782.00	0.270986
$352$	0	0
$353$	−7878.00	−1.18783	−0.593914	−	0.804528i	$-0.702418\pi$
$353$	−7878.00	−1.18783	−0.593914	+	0.804528i	$0.702418\pi$
$354$	0	0
$355$	1728.00	0.258346
$356$	0	0
$357$	−1470.00	−0.217929
$358$	0	0
$359$	11856.0	1.74300	0.871498	−	0.490399i	$-0.163149\pi$
$359$	11856.0	1.74300	0.871498	+	0.490399i	$0.163149\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	0	0
$363$	−3561.00	−0.514887
$364$	0	0
$365$	212.000	0.0304016
$366$	0	0
$367$	−5488.00	−0.780576	−0.390288	−	0.920693i	$-0.627625\pi$
$367$	−5488.00	−0.780576	−0.390288	+	0.920693i	$0.627625\pi$
$368$	0	0
$369$	450.000	0.0634853
$370$	0	0
$371$	−3682.00	−0.515256
$372$	0	0
$373$	7570.00	1.05083	0.525415	−	0.850846i	$-0.323910\pi$
$373$	7570.00	1.05083	0.525415	+	0.850846i	$0.323910\pi$
$374$	0	0
$375$	−1476.00	−0.203254
$376$	0	0
$377$	8052.00	1.10000
$378$	0	0
$379$	13596.0	1.84269	0.921345	−	0.388746i	$-0.127092\pi$
$379$	13596.0	1.84269	0.921345	+	0.388746i	$0.127092\pi$
$380$	0	0
$381$	672.000	0.0903612
$382$	0	0
$383$	−8592.00	−1.14629	−0.573147	−	0.819452i	$-0.694278\pi$
$383$	−8592.00	−1.14629	−0.573147	+	0.819452i	$0.694278\pi$
$384$	0	0
$385$	−168.000	−0.0222392
$386$	0	0
$387$	−180.000	−0.0236432
$388$	0	0
$389$	610.000	0.0795070	0.0397535	−	0.999210i	$-0.487343\pi$
$389$	610.000	0.0795070	0.0397535	+	0.999210i	$0.487343\pi$
$390$	0	0
$391$	−1120.00	−0.144861
$392$	0	0
$393$	7764.00	0.996545
$394$	0	0
$395$	1472.00	0.187505
$396$	0	0
$397$	−910.000	−0.115042	−0.0575209	−	0.998344i	$-0.518320\pi$
$397$	−910.000	−0.115042	−0.0575209	+	0.998344i	$0.518320\pi$
$398$	0	0
$399$	1932.00	0.242408
$400$	0	0
$401$	−8094.00	−1.00797	−0.503984	−	0.863713i	$-0.668133\pi$
$401$	−8094.00	−1.00797	−0.503984	+	0.863713i	$0.668133\pi$
$402$	0	0
$403$	4224.00	0.522115
$404$	0	0
$405$	162.000	0.0198762
$406$	0	0
$407$	−3672.00	−0.447210
$408$	0	0
$409$	10122.0	1.22372	0.611859	−	0.790967i	$-0.290422\pi$
$409$	10122.0	1.22372	0.611859	+	0.790967i	$0.290422\pi$
$410$	0	0
$411$	1470.00	0.176423
$412$	0	0
$413$	−3780.00	−0.450367
$414$	0	0
$415$	1176.00	0.139103
$416$	0	0
$417$	5148.00	0.604553
$418$	0	0
$419$	−4228.00	−0.492963	−0.246481	−	0.969148i	$-0.579274\pi$
$419$	−4228.00	−0.492963	−0.246481	+	0.969148i	$0.579274\pi$
$420$	0	0
$421$	9218.00	1.06712	0.533560	−	0.845762i	$-0.320854\pi$
$421$	9218.00	1.06712	0.533560	+	0.845762i	$0.320854\pi$
$422$	0	0
$423$	−1584.00	−0.182073
$424$	0	0
$425$	8470.00	0.966718
$426$	0	0
$427$	5726.00	0.648947
$428$	0	0
$429$	−2376.00	−0.267399
$430$	0	0
$431$	9192.00	1.02729	0.513646	−	0.858002i	$-0.328294\pi$
$431$	9192.00	1.02729	0.513646	+	0.858002i	$0.328294\pi$
$432$	0	0
$433$	−3614.00	−0.401103	−0.200552	−	0.979683i	$-0.564273\pi$
$433$	−3614.00	−0.401103	−0.200552	+	0.979683i	$0.564273\pi$
$434$	0	0
$435$	732.000	0.0806821
$436$	0	0
$437$	1472.00	0.161133
$438$	0	0
$439$	−4344.00	−0.472273	−0.236136	−	0.971720i	$-0.575881\pi$
$439$	−4344.00	−0.472273	−0.236136	+	0.971720i	$0.575881\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−10876.0	−1.16644	−0.583222	−	0.812313i	$-0.698208\pi$
$443$	−10876.0	−1.16644	−0.583222	+	0.812313i	$0.698208\pi$
$444$	0	0
$445$	292.000	0.0311059
$446$	0	0
$447$	7158.00	0.757409
$448$	0	0
$449$	−15550.0	−1.63441	−0.817205	−	0.576347i	$-0.804478\pi$
$449$	−15550.0	−1.63441	−0.817205	+	0.576347i	$0.804478\pi$
$450$	0	0
$451$	−600.000	−0.0626450
$452$	0	0
$453$	−312.000	−0.0323599
$454$	0	0
$455$	924.000	0.0952039
$456$	0	0
$457$	7834.00	0.801880	0.400940	−	0.916104i	$-0.368684\pi$
$457$	7834.00	0.801880	0.400940	+	0.916104i	$0.368684\pi$
$458$	0	0
$459$	−1890.00	−0.192195
$460$	0	0
$461$	−15990.0	−1.61546	−0.807732	−	0.589550i	$-0.799305\pi$
$461$	−15990.0	−1.61546	−0.807732	+	0.589550i	$0.799305\pi$
$462$	0	0
$463$	−12448.0	−1.24948	−0.624738	−	0.780834i	$-0.714794\pi$
$463$	−12448.0	−1.24948	−0.624738	+	0.780834i	$0.714794\pi$
$464$	0	0
$465$	384.000	0.0382959
$466$	0	0
$467$	−13860.0	−1.37337	−0.686686	−	0.726955i	$-0.740935\pi$
$467$	−13860.0	−1.37337	−0.686686	+	0.726955i	$0.740935\pi$
$468$	0	0
$469$	1596.00	0.157135
$470$	0	0
$471$	−4698.00	−0.459602
$472$	0	0
$473$	240.000	0.0233303
$474$	0	0
$475$	−11132.0	−1.07531
$476$	0	0
$477$	−4734.00	−0.454413
$478$	0	0
$479$	−14368.0	−1.37054	−0.685272	−	0.728287i	$-0.740317\pi$
$479$	−14368.0	−1.37054	−0.685272	+	0.728287i	$0.740317\pi$
$480$	0	0
$481$	20196.0	1.91447
$482$	0	0
$483$	336.000	0.0316533
$484$	0	0
$485$	−2428.00	−0.227319
$486$	0	0
$487$	−17240.0	−1.60415	−0.802073	−	0.597226i	$-0.796269\pi$
$487$	−17240.0	−1.60415	−0.802073	+	0.597226i	$0.796269\pi$
$488$	0	0
$489$	3228.00	0.298518
$490$	0	0
$491$	−18156.0	−1.66878	−0.834388	−	0.551178i	$-0.814179\pi$
$491$	−18156.0	−1.66878	−0.834388	+	0.551178i	$0.814179\pi$
$492$	0	0
$493$	−8540.00	−0.780167
$494$	0	0
$495$	−216.000	−0.0196131
$496$	0	0
$497$	6048.00	0.545855
$498$	0	0
$499$	−1084.00	−0.0972475	−0.0486238	−	0.998817i	$-0.515484\pi$
$499$	−1084.00	−0.0972475	−0.0486238	+	0.998817i	$0.515484\pi$
$500$	0	0
$501$	−8280.00	−0.738369
$502$	0	0
$503$	−1048.00	−0.0928986	−0.0464493	−	0.998921i	$-0.514791\pi$
$503$	−1048.00	−0.0928986	−0.0464493	+	0.998921i	$0.514791\pi$
$504$	0	0
$505$	−1692.00	−0.149095
$506$	0	0
$507$	6477.00	0.567364
$508$	0	0
$509$	−5046.00	−0.439411	−0.219705	−	0.975566i	$-0.570510\pi$
$509$	−5046.00	−0.439411	−0.219705	+	0.975566i	$0.570510\pi$
$510$	0	0
$511$	742.000	0.0642351
$512$	0	0
$513$	2484.00	0.213784
$514$	0	0
$515$	−336.000	−0.0287494
$516$	0	0
$517$	2112.00	0.179663
$518$	0	0
$519$	−4674.00	−0.395310
$520$	0	0
$521$	−3390.00	−0.285064	−0.142532	−	0.989790i	$-0.545524\pi$
$521$	−3390.00	−0.285064	−0.142532	+	0.989790i	$0.545524\pi$
$522$	0	0
$523$	−13052.0	−1.09125	−0.545625	−	0.838029i	$-0.683708\pi$
$523$	−13052.0	−1.09125	−0.545625	+	0.838029i	$0.683708\pi$
$524$	0	0
$525$	−2541.00	−0.211235
$526$	0	0
$527$	−4480.00	−0.370307
$528$	0	0
$529$	−11911.0	−0.978959
$530$	0	0
$531$	−4860.00	−0.397187
$532$	0	0
$533$	3300.00	0.268178
$534$	0	0
$535$	1416.00	0.114428
$536$	0	0
$537$	10356.0	0.832206
$538$	0	0
$539$	−588.000	−0.0469888
$540$	0	0
$541$	−16966.0	−1.34829	−0.674145	−	0.738599i	$-0.735488\pi$
$541$	−16966.0	−1.34829	−0.674145	+	0.738599i	$0.735488\pi$
$542$	0	0
$543$	3486.00	0.275504
$544$	0	0
$545$	−1292.00	−0.101547
$546$	0	0
$547$	11348.0	0.887030	0.443515	−	0.896267i	$-0.353731\pi$
$547$	11348.0	0.887030	0.443515	+	0.896267i	$0.353731\pi$
$548$	0	0
$549$	7362.00	0.572318
$550$	0	0
$551$	11224.0	0.867801
$552$	0	0
$553$	5152.00	0.396176
$554$	0	0
$555$	1836.00	0.140421
$556$	0	0
$557$	−16118.0	−1.22611	−0.613053	−	0.790041i	$-0.710059\pi$
$557$	−16118.0	−1.22611	−0.613053	+	0.790041i	$0.710059\pi$
$558$	0	0
$559$	−1320.00	−0.0998749
$560$	0	0
$561$	2520.00	0.189651
$562$	0	0
$563$	15244.0	1.14113	0.570567	−	0.821251i	$-0.306724\pi$
$563$	15244.0	1.14113	0.570567	+	0.821251i	$0.306724\pi$
$564$	0	0
$565$	3876.00	0.288610
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	−18918.0	−1.39382	−0.696910	−	0.717158i	$-0.745442\pi$
$569$	−18918.0	−1.39382	−0.696910	+	0.717158i	$0.745442\pi$
$570$	0	0
$571$	19372.0	1.41978	0.709889	−	0.704314i	$-0.248745\pi$
$571$	19372.0	1.41978	0.709889	+	0.704314i	$0.248745\pi$
$572$	0	0
$573$	5784.00	0.421693
$574$	0	0
$575$	−1936.00	−0.140412
$576$	0	0
$577$	−7230.00	−0.521644	−0.260822	−	0.965387i	$-0.583994\pi$
$577$	−7230.00	−0.521644	−0.260822	+	0.965387i	$0.583994\pi$
$578$	0	0
$579$	−954.000	−0.0684748
$580$	0	0
$581$	4116.00	0.293908
$582$	0	0
$583$	6312.00	0.448399
$584$	0	0
$585$	1188.00	0.0839620
$586$	0	0
$587$	7396.00	0.520044	0.260022	−	0.965603i	$-0.416270\pi$
$587$	7396.00	0.520044	0.260022	+	0.965603i	$0.416270\pi$
$588$	0	0
$589$	5888.00	0.411903
$590$	0	0
$591$	−12186.0	−0.848164
$592$	0	0
$593$	714.000	0.0494443	0.0247221	−	0.999694i	$-0.492130\pi$
$593$	714.000	0.0494443	0.0247221	+	0.999694i	$0.492130\pi$
$594$	0	0
$595$	−980.000	−0.0675228
$596$	0	0
$597$	16440.0	1.12704
$598$	0	0
$599$	−11536.0	−0.786892	−0.393446	−	0.919348i	$-0.628717\pi$
$599$	−11536.0	−0.786892	−0.393446	+	0.919348i	$0.628717\pi$
$600$	0	0
$601$	4138.00	0.280853	0.140426	−	0.990091i	$-0.455153\pi$
$601$	4138.00	0.280853	0.140426	+	0.990091i	$0.455153\pi$
$602$	0	0
$603$	2052.00	0.138580
$604$	0	0
$605$	−2374.00	−0.159532
$606$	0	0
$607$	−6848.00	−0.457911	−0.228955	−	0.973437i	$-0.573531\pi$
$607$	−6848.00	−0.457911	−0.228955	+	0.973437i	$0.573531\pi$
$608$	0	0
$609$	2562.00	0.170472
$610$	0	0
$611$	−11616.0	−0.769121
$612$	0	0
$613$	12722.0	0.838233	0.419116	−	0.907932i	$-0.362340\pi$
$613$	12722.0	0.838233	0.419116	+	0.907932i	$0.362340\pi$
$614$	0	0
$615$	300.000	0.0196702
$616$	0	0
$617$	−24726.0	−1.61334	−0.806670	−	0.591002i	$-0.798733\pi$
$617$	−24726.0	−1.61334	−0.806670	+	0.591002i	$0.798733\pi$
$618$	0	0
$619$	−23964.0	−1.55605	−0.778025	−	0.628234i	$-0.783778\pi$
$619$	−23964.0	−1.55605	−0.778025	+	0.628234i	$0.783778\pi$
$620$	0	0
$621$	432.000	0.0279156
$622$	0	0
$623$	1022.00	0.0657232
$624$	0	0
$625$	14141.0	0.905024
$626$	0	0
$627$	−3312.00	−0.210955
$628$	0	0
$629$	−21420.0	−1.35782
$630$	0	0
$631$	−17224.0	−1.08665	−0.543325	−	0.839522i	$-0.682835\pi$
$631$	−17224.0	−1.08665	−0.543325	+	0.839522i	$0.682835\pi$
$632$	0	0
$633$	−14676.0	−0.921514
$634$	0	0
$635$	448.000	0.0279974
$636$	0	0
$637$	3234.00	0.201155
$638$	0	0
$639$	7776.00	0.481399
$640$	0	0
$641$	−18190.0	−1.12085	−0.560423	−	0.828207i	$-0.689361\pi$
$641$	−18190.0	−1.12085	−0.560423	+	0.828207i	$0.689361\pi$
$642$	0	0
$643$	−14116.0	−0.865755	−0.432878	−	0.901453i	$-0.642502\pi$
$643$	−14116.0	−0.865755	−0.432878	+	0.901453i	$0.642502\pi$
$644$	0	0
$645$	−120.000	−0.00732557
$646$	0	0
$647$	4056.00	0.246457	0.123229	−	0.992378i	$-0.460675\pi$
$647$	4056.00	0.246457	0.123229	+	0.992378i	$0.460675\pi$
$648$	0	0
$649$	6480.00	0.391930
$650$	0	0
$651$	1344.00	0.0809148
$652$	0	0
$653$	16490.0	0.988214	0.494107	−	0.869401i	$-0.335495\pi$
$653$	16490.0	0.988214	0.494107	+	0.869401i	$0.335495\pi$
$654$	0	0
$655$	5176.00	0.308768
$656$	0	0
$657$	954.000	0.0566501
$658$	0	0
$659$	4988.00	0.294848	0.147424	−	0.989073i	$-0.452902\pi$
$659$	4988.00	0.294848	0.147424	+	0.989073i	$0.452902\pi$
$660$	0	0
$661$	−4982.00	−0.293158	−0.146579	−	0.989199i	$-0.546826\pi$
$661$	−4982.00	−0.293158	−0.146579	+	0.989199i	$0.546826\pi$
$662$	0	0
$663$	−13860.0	−0.811882
$664$	0	0
$665$	1288.00	0.0751075
$666$	0	0
$667$	1952.00	0.113316
$668$	0	0
$669$	12960.0	0.748972
$670$	0	0
$671$	−9816.00	−0.564743
$672$	0	0
$673$	−16190.0	−0.927309	−0.463654	−	0.886016i	$-0.653462\pi$
$673$	−16190.0	−0.927309	−0.463654	+	0.886016i	$0.653462\pi$
$674$	0	0
$675$	−3267.00	−0.186292
$676$	0	0
$677$	23202.0	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$677$	23202.0	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$678$	0	0
$679$	−8498.00	−0.480299
$680$	0	0
$681$	16548.0	0.931161
$682$	0	0
$683$	−13452.0	−0.753626	−0.376813	−	0.926289i	$-0.622980\pi$
$683$	−13452.0	−0.753626	−0.376813	+	0.926289i	$0.622980\pi$
$684$	0	0
$685$	980.000	0.0546626
$686$	0	0
$687$	−5010.00	−0.278229
$688$	0	0
$689$	−34716.0	−1.91956
$690$	0	0
$691$	−31220.0	−1.71876	−0.859381	−	0.511336i	$-0.829151\pi$
$691$	−31220.0	−1.71876	−0.859381	+	0.511336i	$0.829151\pi$
$692$	0	0
$693$	−756.000	−0.0414402
$694$	0	0
$695$	3432.00	0.187314
$696$	0	0
$697$	−3500.00	−0.190204
$698$	0	0
$699$	−11778.0	−0.637317
$700$	0	0
$701$	−33542.0	−1.80722	−0.903612	−	0.428352i	$-0.859094\pi$
$701$	−33542.0	−1.80722	−0.903612	+	0.428352i	$0.859094\pi$
$702$	0	0
$703$	28152.0	1.51035
$704$	0	0
$705$	−1056.00	−0.0564131
$706$	0	0
$707$	−5922.00	−0.315021
$708$	0	0
$709$	28562.0	1.51293	0.756466	−	0.654033i	$-0.226924\pi$
$709$	28562.0	1.51293	0.756466	+	0.654033i	$0.226924\pi$
$710$	0	0
$711$	6624.00	0.349394
$712$	0	0
$713$	1024.00	0.0537856
$714$	0	0
$715$	−1584.00	−0.0828507
$716$	0	0
$717$	10920.0	0.568779
$718$	0	0
$719$	1248.00	0.0647323	0.0323662	−	0.999476i	$-0.489696\pi$
$719$	1248.00	0.0647323	0.0323662	+	0.999476i	$0.489696\pi$
$720$	0	0
$721$	−1176.00	−0.0607441
$722$	0	0
$723$	16950.0	0.871891
$724$	0	0
$725$	−14762.0	−0.756203
$726$	0	0
$727$	−4216.00	−0.215079	−0.107540	−	0.994201i	$-0.534297\pi$
$727$	−4216.00	−0.215079	−0.107540	+	0.994201i	$0.534297\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	1400.00	0.0708357
$732$	0	0
$733$	−12670.0	−0.638441	−0.319220	−	0.947680i	$-0.603421\pi$
$733$	−12670.0	−0.638441	−0.319220	+	0.947680i	$0.603421\pi$
$734$	0	0
$735$	294.000	0.0147542
$736$	0	0
$737$	−2736.00	−0.136746
$738$	0	0
$739$	−25996.0	−1.29402	−0.647008	−	0.762483i	$-0.723980\pi$
$739$	−25996.0	−1.29402	−0.647008	+	0.762483i	$0.723980\pi$
$740$	0	0
$741$	18216.0	0.903079
$742$	0	0
$743$	−24368.0	−1.20320	−0.601598	−	0.798799i	$-0.705469\pi$
$743$	−24368.0	−1.20320	−0.601598	+	0.798799i	$0.705469\pi$
$744$	0	0
$745$	4772.00	0.234675
$746$	0	0
$747$	5292.00	0.259202
$748$	0	0
$749$	4956.00	0.241773
$750$	0	0
$751$	17600.0	0.855171	0.427585	−	0.903975i	$-0.359364\pi$
$751$	17600.0	0.855171	0.427585	+	0.903975i	$0.359364\pi$
$752$	0	0
$753$	−13956.0	−0.675412
$754$	0	0
$755$	−208.000	−0.0100264
$756$	0	0
$757$	−17182.0	−0.824954	−0.412477	−	0.910968i	$-0.635336\pi$
$757$	−17182.0	−0.824954	−0.412477	+	0.910968i	$0.635336\pi$
$758$	0	0
$759$	−576.000	−0.0275461
$760$	0	0
$761$	−7038.00	−0.335253	−0.167626	−	0.985851i	$-0.553610\pi$
$761$	−7038.00	−0.335253	−0.167626	+	0.985851i	$0.553610\pi$
$762$	0	0
$763$	−4522.00	−0.214558
$764$	0	0
$765$	−1260.00	−0.0595495
$766$	0	0
$767$	−35640.0	−1.67782
$768$	0	0
$769$	−18254.0	−0.855990	−0.427995	−	0.903781i	$-0.640780\pi$
$769$	−18254.0	−0.855990	−0.427995	+	0.903781i	$0.640780\pi$
$770$	0	0
$771$	−18018.0	−0.841637
$772$	0	0
$773$	−2990.00	−0.139124	−0.0695620	−	0.997578i	$-0.522160\pi$
$773$	−2990.00	−0.139124	−0.0695620	+	0.997578i	$0.522160\pi$
$774$	0	0
$775$	−7744.00	−0.358933
$776$	0	0
$777$	6426.00	0.296694
$778$	0	0
$779$	4600.00	0.211569
$780$	0	0
$781$	−10368.0	−0.475027
$782$	0	0
$783$	3294.00	0.150342
$784$	0	0
$785$	−3132.00	−0.142402
$786$	0	0
$787$	−10804.0	−0.489353	−0.244677	−	0.969605i	$-0.578682\pi$
$787$	−10804.0	−0.489353	−0.244677	+	0.969605i	$0.578682\pi$
$788$	0	0
$789$	15120.0	0.682239
$790$	0	0
$791$	13566.0	0.609800
$792$	0	0
$793$	53988.0	2.41762
$794$	0	0
$795$	−3156.00	−0.140795
$796$	0	0
$797$	−11238.0	−0.499461	−0.249730	−	0.968315i	$-0.580342\pi$
$797$	−11238.0	−0.499461	−0.249730	+	0.968315i	$0.580342\pi$
$798$	0	0
$799$	12320.0	0.545495
$800$	0	0
$801$	1314.00	0.0579624
$802$	0	0
$803$	−1272.00	−0.0559003
$804$	0	0
$805$	224.000	0.00980741
$806$	0	0
$807$	−16434.0	−0.716858
$808$	0	0
$809$	35034.0	1.52253	0.761267	−	0.648439i	$-0.224578\pi$
$809$	35034.0	1.52253	0.761267	+	0.648439i	$0.224578\pi$
$810$	0	0
$811$	9252.00	0.400594	0.200297	−	0.979735i	$-0.435809\pi$
$811$	9252.00	0.400594	0.200297	+	0.979735i	$0.435809\pi$
$812$	0	0
$813$	−6528.00	−0.281608
$814$	0	0
$815$	2152.00	0.0924924
$816$	0	0
$817$	−1840.00	−0.0787925
$818$	0	0
$819$	4158.00	0.177402
$820$	0	0
$821$	−18318.0	−0.778688	−0.389344	−	0.921092i	$-0.627298\pi$
$821$	−18318.0	−0.778688	−0.389344	+	0.921092i	$0.627298\pi$
$822$	0	0
$823$	30200.0	1.27911	0.639554	−	0.768746i	$-0.279119\pi$
$823$	30200.0	1.27911	0.639554	+	0.768746i	$0.279119\pi$
$824$	0	0
$825$	4356.00	0.183826
$826$	0	0
$827$	−9612.00	−0.404162	−0.202081	−	0.979369i	$-0.564770\pi$
$827$	−9612.00	−0.404162	−0.202081	+	0.979369i	$0.564770\pi$
$828$	0	0
$829$	−3806.00	−0.159455	−0.0797273	−	0.996817i	$-0.525405\pi$
$829$	−3806.00	−0.159455	−0.0797273	+	0.996817i	$0.525405\pi$
$830$	0	0
$831$	13974.0	0.583337
$832$	0	0
$833$	−3430.00	−0.142668
$834$	0	0
$835$	−5520.00	−0.228775
$836$	0	0
$837$	1728.00	0.0713601
$838$	0	0
$839$	−19176.0	−0.789069	−0.394535	−	0.918881i	$-0.629094\pi$
$839$	−19176.0	−0.789069	−0.394535	+	0.918881i	$0.629094\pi$
$840$	0	0
$841$	−9505.00	−0.389725
$842$	0	0
$843$	−13842.0	−0.565532
$844$	0	0
$845$	4318.00	0.175791
$846$	0	0
$847$	−8309.00	−0.337073
$848$	0	0
$849$	−3732.00	−0.150862
$850$	0	0
$851$	4896.00	0.197218
$852$	0	0
$853$	38234.0	1.53471	0.767355	−	0.641223i	$-0.221573\pi$
$853$	38234.0	1.53471	0.767355	+	0.641223i	$0.221573\pi$
$854$	0	0
$855$	1656.00	0.0662386
$856$	0	0
$857$	34818.0	1.38782	0.693909	−	0.720063i	$-0.255887\pi$
$857$	34818.0	1.38782	0.693909	+	0.720063i	$0.255887\pi$
$858$	0	0
$859$	15764.0	0.626148	0.313074	−	0.949729i	$-0.398641\pi$
$859$	15764.0	0.626148	0.313074	+	0.949729i	$0.398641\pi$
$860$	0	0
$861$	1050.00	0.0415609
$862$	0	0
$863$	−1832.00	−0.0722619	−0.0361309	−	0.999347i	$-0.511503\pi$
$863$	−1832.00	−0.0722619	−0.0361309	+	0.999347i	$0.511503\pi$
$864$	0	0
$865$	−3116.00	−0.122482
$866$	0	0
$867$	−39.0000	−0.00152769
$868$	0	0
$869$	−8832.00	−0.344770
$870$	0	0
$871$	15048.0	0.585398
$872$	0	0
$873$	−10926.0	−0.423584
$874$	0	0
$875$	−3444.00	−0.133061
$876$	0	0
$877$	−18374.0	−0.707464	−0.353732	−	0.935347i	$-0.615088\pi$
$877$	−18374.0	−0.707464	−0.353732	+	0.935347i	$0.615088\pi$
$878$	0	0
$879$	19062.0	0.731451
$880$	0	0
$881$	40490.0	1.54840	0.774201	−	0.632939i	$-0.218152\pi$
$881$	40490.0	1.54840	0.774201	+	0.632939i	$0.218152\pi$
$882$	0	0
$883$	548.000	0.0208852	0.0104426	−	0.999945i	$-0.496676\pi$
$883$	548.000	0.0208852	0.0104426	+	0.999945i	$0.496676\pi$
$884$	0	0
$885$	−3240.00	−0.123064
$886$	0	0
$887$	−1272.00	−0.0481506	−0.0240753	−	0.999710i	$-0.507664\pi$
$887$	−1272.00	−0.0481506	−0.0240753	+	0.999710i	$0.507664\pi$
$888$	0	0
$889$	1568.00	0.0591553
$890$	0	0
$891$	−972.000	−0.0365468
$892$	0	0
$893$	−16192.0	−0.606769
$894$	0	0
$895$	6904.00	0.257849
$896$	0	0
$897$	3168.00	0.117922
$898$	0	0
$899$	7808.00	0.289668
$900$	0	0
$901$	36820.0	1.36143
$902$	0	0
$903$	−420.000	−0.0154781
$904$	0	0
$905$	2324.00	0.0853617
$906$	0	0
$907$	15100.0	0.552797	0.276399	−	0.961043i	$-0.410859\pi$
$907$	15100.0	0.552797	0.276399	+	0.961043i	$0.410859\pi$
$908$	0	0
$909$	−7614.00	−0.277822
$910$	0	0
$911$	53192.0	1.93450	0.967250	−	0.253825i	$-0.0816889\pi$
$911$	53192.0	1.93450	0.967250	+	0.253825i	$0.0816889\pi$
$912$	0	0
$913$	−7056.00	−0.255772
$914$	0	0
$915$	4908.00	0.177326
$916$	0	0
$917$	18116.0	0.652392
$918$	0	0
$919$	−53624.0	−1.92480	−0.962401	−	0.271634i	$-0.912436\pi$
$919$	−53624.0	−1.92480	−0.962401	+	0.271634i	$0.912436\pi$
$920$	0	0
$921$	11220.0	0.401424
$922$	0	0
$923$	57024.0	2.03355
$924$	0	0
$925$	−37026.0	−1.31612
$926$	0	0
$927$	−1512.00	−0.0535713
$928$	0	0
$929$	−28966.0	−1.02297	−0.511487	−	0.859291i	$-0.670905\pi$
$929$	−28966.0	−1.02297	−0.511487	+	0.859291i	$0.670905\pi$
$930$	0	0
$931$	4508.00	0.158694
$932$	0	0
$933$	−6552.00	−0.229907
$934$	0	0
$935$	1680.00	0.0587614
$936$	0	0
$937$	50874.0	1.77373	0.886863	−	0.462033i	$-0.152880\pi$
$937$	50874.0	1.77373	0.886863	+	0.462033i	$0.152880\pi$
$938$	0	0
$939$	13326.0	0.463128
$940$	0	0
$941$	−16566.0	−0.573896	−0.286948	−	0.957946i	$-0.592641\pi$
$941$	−16566.0	−0.573896	−0.286948	+	0.957946i	$0.592641\pi$
$942$	0	0
$943$	800.000	0.0276263
$944$	0	0
$945$	378.000	0.0130120
$946$	0	0
$947$	38428.0	1.31863	0.659315	−	0.751867i	$-0.270846\pi$
$947$	38428.0	1.31863	0.659315	+	0.751867i	$0.270846\pi$
$948$	0	0
$949$	6996.00	0.239304
$950$	0	0
$951$	12942.0	0.441297
$952$	0	0
$953$	746.000	0.0253571	0.0126785	−	0.999920i	$-0.495964\pi$
$953$	746.000	0.0253571	0.0126785	+	0.999920i	$0.495964\pi$
$954$	0	0
$955$	3856.00	0.130657
$956$	0	0
$957$	−4392.00	−0.148352
$958$	0	0
$959$	3430.00	0.115496
$960$	0	0
$961$	−25695.0	−0.862509
$962$	0	0
$963$	6372.00	0.213224
$964$	0	0
$965$	−636.000	−0.0212161
$966$	0	0
$967$	9432.00	0.313664	0.156832	−	0.987625i	$-0.449872\pi$
$967$	9432.00	0.313664	0.156832	+	0.987625i	$0.449872\pi$
$968$	0	0
$969$	−19320.0	−0.640503
$970$	0	0
$971$	−3452.00	−0.114089	−0.0570443	−	0.998372i	$-0.518168\pi$
$971$	−3452.00	−0.114089	−0.0570443	+	0.998372i	$0.518168\pi$
$972$	0	0
$973$	12012.0	0.395773
$974$	0	0
$975$	−23958.0	−0.786944
$976$	0	0
$977$	−18078.0	−0.591982	−0.295991	−	0.955191i	$-0.595650\pi$
$977$	−18078.0	−0.591982	−0.295991	+	0.955191i	$0.595650\pi$
$978$	0	0
$979$	−1752.00	−0.0571953
$980$	0	0
$981$	−5814.00	−0.189222
$982$	0	0
$983$	792.000	0.0256977	0.0128489	−	0.999917i	$-0.495910\pi$
$983$	792.000	0.0256977	0.0128489	+	0.999917i	$0.495910\pi$
$984$	0	0
$985$	−8124.00	−0.262794
$986$	0	0
$987$	−3696.00	−0.119195
$988$	0	0
$989$	−320.000	−0.0102886
$990$	0	0
$991$	−14976.0	−0.480049	−0.240024	−	0.970767i	$-0.577155\pi$
$991$	−14976.0	−0.480049	−0.240024	+	0.970767i	$0.577155\pi$
$992$	0	0
$993$	−23964.0	−0.765835
$994$	0	0
$995$	10960.0	0.349201
$996$	0	0
$997$	17114.0	0.543637	0.271818	−	0.962349i	$-0.412375\pi$
$997$	17114.0	0.543637	0.271818	+	0.962349i	$0.412375\pi$
$998$	0	0
$999$	8262.00	0.261660

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.u.1.1		1
4.3	odd	2		1344.4.a.h.1.1		1
8.3	odd	2		336.4.a.i.1.1		1
8.5	even	2		168.4.a.b.1.1	✓	1
24.5	odd	2		504.4.a.d.1.1		1
24.11	even	2		1008.4.a.k.1.1		1
56.13	odd	2		1176.4.a.l.1.1		1
56.27	even	2		2352.4.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.a.b.1.1	✓	1	8.5	even	2
336.4.a.i.1.1		1	8.3	odd	2
504.4.a.d.1.1		1	24.5	odd	2
1008.4.a.k.1.1		1	24.11	even	2
1176.4.a.l.1.1		1	56.13	odd	2
1344.4.a.h.1.1		1	4.3	odd	2
1344.4.a.u.1.1		1	1.1	even	1	trivial
2352.4.a.k.1.1		1	56.27	even	2

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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