LMFDB - Embedded newform 28.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [28,4,Mod(1,28)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(28, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("28.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 28 = 2^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	28.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:	$1.65205348016$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	28.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-10.0000 q^{3} -8.00000 q^{5} -7.00000 q^{7} +73.0000 q^{9} +O(q^{10})$

$q-10.0000 q^{3} -8.00000 q^{5} -7.00000 q^{7} +73.0000 q^{9} -40.0000 q^{11} -12.0000 q^{13} +80.0000 q^{15} -58.0000 q^{17} +26.0000 q^{19} +70.0000 q^{21} -64.0000 q^{23} -61.0000 q^{25} -460.000 q^{27} -62.0000 q^{29} +252.000 q^{31} +400.000 q^{33} +56.0000 q^{35} +26.0000 q^{37} +120.000 q^{39} +6.00000 q^{41} +416.000 q^{43} -584.000 q^{45} -396.000 q^{47} +49.0000 q^{49} +580.000 q^{51} -450.000 q^{53} +320.000 q^{55} -260.000 q^{57} +274.000 q^{59} -576.000 q^{61} -511.000 q^{63} +96.0000 q^{65} -476.000 q^{67} +640.000 q^{69} -448.000 q^{71} -158.000 q^{73} +610.000 q^{75} +280.000 q^{77} -936.000 q^{79} +2629.00 q^{81} +530.000 q^{83} +464.000 q^{85} +620.000 q^{87} -390.000 q^{89} +84.0000 q^{91} -2520.00 q^{93} -208.000 q^{95} +214.000 q^{97} -2920.00 q^{99} +O(q^{100})$

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
$3$	−10.0000	−1.92450	−0.962250	−	0.272166i	$-0.912260\pi$
$3$	−10.0000	−1.92450	−0.962250	+	0.272166i	$0.912260\pi$
$4$	0	0
$5$	−8.00000	−0.715542	−0.357771	−	0.933809i	$-0.616463\pi$
$5$	−8.00000	−0.715542	−0.357771	+	0.933809i	$0.616463\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	73.0000	2.70370
$10$	0	0
$11$	−40.0000	−1.09640	−0.548202	−	0.836346i	$-0.684688\pi$
$11$	−40.0000	−1.09640	−0.548202	+	0.836346i	$0.684688\pi$
$12$	0	0
$13$	−12.0000	−0.256015	−0.128008	−	0.991773i	$-0.540858\pi$
$13$	−12.0000	−0.256015	−0.128008	+	0.991773i	$0.540858\pi$
$14$	0	0
$15$	80.0000	1.37706
$16$	0	0
$17$	−58.0000	−0.827474	−0.413737	−	0.910396i	$-0.635777\pi$
$17$	−58.0000	−0.827474	−0.413737	+	0.910396i	$0.635777\pi$
$18$	0	0
$19$	26.0000	0.313937	0.156969	−	0.987604i	$-0.449828\pi$
$19$	26.0000	0.313937	0.156969	+	0.987604i	$0.449828\pi$
$20$	0	0
$21$	70.0000	0.727393
$22$	0	0
$23$	−64.0000	−0.580214	−0.290107	−	0.956994i	$-0.593691\pi$
$23$	−64.0000	−0.580214	−0.290107	+	0.956994i	$0.593691\pi$
$24$	0	0
$25$	−61.0000	−0.488000
$26$	0	0
$27$	−460.000	−3.27878
$28$	0	0
$29$	−62.0000	−0.397004	−0.198502	−	0.980101i	$-0.563608\pi$
$29$	−62.0000	−0.397004	−0.198502	+	0.980101i	$0.563608\pi$
$30$	0	0
$31$	252.000	1.46002	0.730009	−	0.683438i	$-0.239516\pi$
$31$	252.000	1.46002	0.730009	+	0.683438i	$0.239516\pi$
$32$	0	0
$33$	400.000	2.11003
$34$	0	0
$35$	56.0000	0.270449
$36$	0	0
$37$	26.0000	0.115524	0.0577618	−	0.998330i	$-0.481604\pi$
$37$	26.0000	0.115524	0.0577618	+	0.998330i	$0.481604\pi$
$38$	0	0
$39$	120.000	0.492702
$40$	0	0
$41$	6.00000	0.0228547	0.0114273	−	0.999935i	$-0.496362\pi$
$41$	6.00000	0.0228547	0.0114273	+	0.999935i	$0.496362\pi$
$42$	0	0
$43$	416.000	1.47534	0.737668	−	0.675164i	$-0.235927\pi$
$43$	416.000	1.47534	0.737668	+	0.675164i	$0.235927\pi$
$44$	0	0
$45$	−584.000	−1.93461
$46$	0	0
$47$	−396.000	−1.22899	−0.614495	−	0.788921i	$-0.710640\pi$
$47$	−396.000	−1.22899	−0.614495	+	0.788921i	$0.710640\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	580.000	1.59248
$52$	0	0
$53$	−450.000	−1.16627	−0.583134	−	0.812376i	$-0.698174\pi$
$53$	−450.000	−1.16627	−0.583134	+	0.812376i	$0.698174\pi$
$54$	0	0
$55$	320.000	0.784523
$56$	0	0
$57$	−260.000	−0.604173
$58$	0	0
$59$	274.000	0.604606	0.302303	−	0.953212i	$-0.402245\pi$
$59$	274.000	0.604606	0.302303	+	0.953212i	$0.402245\pi$
$60$	0	0
$61$	−576.000	−1.20900	−0.604502	−	0.796604i	$-0.706628\pi$
$61$	−576.000	−1.20900	−0.604502	+	0.796604i	$0.706628\pi$
$62$	0	0
$63$	−511.000	−1.02190
$64$	0	0
$65$	96.0000	0.183190
$66$	0	0
$67$	−476.000	−0.867950	−0.433975	−	0.900925i	$-0.642889\pi$
$67$	−476.000	−0.867950	−0.433975	+	0.900925i	$0.642889\pi$
$68$	0	0
$69$	640.000	1.11662
$70$	0	0
$71$	−448.000	−0.748843	−0.374421	−	0.927259i	$-0.622159\pi$
$71$	−448.000	−0.748843	−0.374421	+	0.927259i	$0.622159\pi$
$72$	0	0
$73$	−158.000	−0.253322	−0.126661	−	0.991946i	$-0.540426\pi$
$73$	−158.000	−0.253322	−0.126661	+	0.991946i	$0.540426\pi$
$74$	0	0
$75$	610.000	0.939156
$76$	0	0
$77$	280.000	0.414402
$78$	0	0
$79$	−936.000	−1.33302	−0.666508	−	0.745498i	$-0.732212\pi$
$79$	−936.000	−1.33302	−0.666508	+	0.745498i	$0.732212\pi$
$80$	0	0
$81$	2629.00	3.60631
$82$	0	0
$83$	530.000	0.700904	0.350452	−	0.936581i	$-0.386028\pi$
$83$	530.000	0.700904	0.350452	+	0.936581i	$0.386028\pi$
$84$	0	0
$85$	464.000	0.592093
$86$	0	0
$87$	620.000	0.764034
$88$	0	0
$89$	−390.000	−0.464493	−0.232247	−	0.972657i	$-0.574608\pi$
$89$	−390.000	−0.464493	−0.232247	+	0.972657i	$0.574608\pi$
$90$	0	0
$91$	84.0000	0.0967648
$92$	0	0
$93$	−2520.00	−2.80980
$94$	0	0
$95$	−208.000	−0.224635
$96$	0	0
$97$	214.000	0.224004	0.112002	−	0.993708i	$-0.464274\pi$
$97$	214.000	0.224004	0.112002	+	0.993708i	$0.464274\pi$
$98$	0	0
$99$	−2920.00	−2.96435
$100$	0	0
$101$	1432.00	1.41079	0.705393	−	0.708817i	$-0.250771\pi$
$101$	1432.00	1.41079	0.705393	+	0.708817i	$0.250771\pi$
$102$	0	0
$103$	764.000	0.730866	0.365433	−	0.930838i	$-0.380921\pi$
$103$	764.000	0.730866	0.365433	+	0.930838i	$0.380921\pi$
$104$	0	0
$105$	−560.000	−0.520480
$106$	0	0
$107$	324.000	0.292731	0.146366	−	0.989231i	$-0.453242\pi$
$107$	324.000	0.292731	0.146366	+	0.989231i	$0.453242\pi$
$108$	0	0
$109$	−1334.00	−1.17224	−0.586119	−	0.810225i	$-0.699345\pi$
$109$	−1334.00	−1.17224	−0.586119	+	0.810225i	$0.699345\pi$
$110$	0	0
$111$	−260.000	−0.222325
$112$	0	0
$113$	1798.00	1.49683	0.748414	−	0.663232i	$-0.230816\pi$
$113$	1798.00	1.49683	0.748414	+	0.663232i	$0.230816\pi$
$114$	0	0
$115$	512.000	0.415167
$116$	0	0
$117$	−876.000	−0.692190
$118$	0	0
$119$	406.000	0.312756
$120$	0	0
$121$	269.000	0.202104
$122$	0	0
$123$	−60.0000	−0.0439839
$124$	0	0
$125$	1488.00	1.06473
$126$	0	0
$127$	−384.000	−0.268303	−0.134152	−	0.990961i	$-0.542831\pi$
$127$	−384.000	−0.268303	−0.134152	+	0.990961i	$0.542831\pi$
$128$	0	0
$129$	−4160.00	−2.83928
$130$	0	0
$131$	−1814.00	−1.20985	−0.604923	−	0.796284i	$-0.706796\pi$
$131$	−1814.00	−1.20985	−0.604923	+	0.796284i	$0.706796\pi$
$132$	0	0
$133$	−182.000	−0.118657
$134$	0	0
$135$	3680.00	2.34610
$136$	0	0
$137$	1666.00	1.03895	0.519474	−	0.854486i	$-0.326128\pi$
$137$	1666.00	1.03895	0.519474	+	0.854486i	$0.326128\pi$
$138$	0	0
$139$	1126.00	0.687094	0.343547	−	0.939135i	$-0.388372\pi$
$139$	1126.00	0.687094	0.343547	+	0.939135i	$0.388372\pi$
$140$	0	0
$141$	3960.00	2.36519
$142$	0	0
$143$	480.000	0.280697
$144$	0	0
$145$	496.000	0.284073
$146$	0	0
$147$	−490.000	−0.274929
$148$	0	0
$149$	2694.00	1.48122	0.740608	−	0.671938i	$-0.234538\pi$
$149$	2694.00	1.48122	0.740608	+	0.671938i	$0.234538\pi$
$150$	0	0
$151$	−2648.00	−1.42709	−0.713547	−	0.700607i	$-0.752912\pi$
$151$	−2648.00	−1.42709	−0.713547	+	0.700607i	$0.752912\pi$
$152$	0	0
$153$	−4234.00	−2.23725
$154$	0	0
$155$	−2016.00	−1.04470
$156$	0	0
$157$	−556.000	−0.282635	−0.141317	−	0.989964i	$-0.545134\pi$
$157$	−556.000	−0.282635	−0.141317	+	0.989964i	$0.545134\pi$
$158$	0	0
$159$	4500.00	2.24449
$160$	0	0
$161$	448.000	0.219300
$162$	0	0
$163$	−328.000	−0.157613	−0.0788066	−	0.996890i	$-0.525111\pi$
$163$	−328.000	−0.157613	−0.0788066	+	0.996890i	$0.525111\pi$
$164$	0	0
$165$	−3200.00	−1.50982
$166$	0	0
$167$	−4268.00	−1.97765	−0.988826	−	0.149077i	$-0.952370\pi$
$167$	−4268.00	−1.97765	−0.988826	+	0.149077i	$0.952370\pi$
$168$	0	0
$169$	−2053.00	−0.934456
$170$	0	0
$171$	1898.00	0.848793
$172$	0	0
$173$	−3476.00	−1.52760	−0.763802	−	0.645451i	$-0.776669\pi$
$173$	−3476.00	−1.52760	−0.763802	+	0.645451i	$0.776669\pi$
$174$	0	0
$175$	427.000	0.184447
$176$	0	0
$177$	−2740.00	−1.16357
$178$	0	0
$179$	2268.00	0.947029	0.473515	−	0.880786i	$-0.342985\pi$
$179$	2268.00	0.947029	0.473515	+	0.880786i	$0.342985\pi$
$180$	0	0
$181$	−276.000	−0.113342	−0.0566710	−	0.998393i	$-0.518049\pi$
$181$	−276.000	−0.113342	−0.0566710	+	0.998393i	$0.518049\pi$
$182$	0	0
$183$	5760.00	2.32673
$184$	0	0
$185$	−208.000	−0.0826620
$186$	0	0
$187$	2320.00	0.907247
$188$	0	0
$189$	3220.00	1.23926
$190$	0	0
$191$	−3000.00	−1.13650	−0.568252	−	0.822854i	$-0.692380\pi$
$191$	−3000.00	−1.13650	−0.568252	+	0.822854i	$0.692380\pi$
$192$	0	0
$193$	3278.00	1.22257	0.611284	−	0.791411i	$-0.290653\pi$
$193$	3278.00	1.22257	0.611284	+	0.791411i	$0.290653\pi$
$194$	0	0
$195$	−960.000	−0.352549
$196$	0	0
$197$	−2362.00	−0.854241	−0.427121	−	0.904195i	$-0.640472\pi$
$197$	−2362.00	−0.854241	−0.427121	+	0.904195i	$0.640472\pi$
$198$	0	0
$199$	−1036.00	−0.369046	−0.184523	−	0.982828i	$-0.559074\pi$
$199$	−1036.00	−0.369046	−0.184523	+	0.982828i	$0.559074\pi$
$200$	0	0
$201$	4760.00	1.67037
$202$	0	0
$203$	434.000	0.150053
$204$	0	0
$205$	−48.0000	−0.0163535
$206$	0	0
$207$	−4672.00	−1.56873
$208$	0	0
$209$	−1040.00	−0.344202
$210$	0	0
$211$	3524.00	1.14977	0.574887	−	0.818233i	$-0.305046\pi$
$211$	3524.00	1.14977	0.574887	+	0.818233i	$0.305046\pi$
$212$	0	0
$213$	4480.00	1.44115
$214$	0	0
$215$	−3328.00	−1.05566
$216$	0	0
$217$	−1764.00	−0.551835
$218$	0	0
$219$	1580.00	0.487518
$220$	0	0
$221$	696.000	0.211846
$222$	0	0
$223$	−1336.00	−0.401189	−0.200595	−	0.979674i	$-0.564287\pi$
$223$	−1336.00	−0.401189	−0.200595	+	0.979674i	$0.564287\pi$
$224$	0	0
$225$	−4453.00	−1.31941
$226$	0	0
$227$	1290.00	0.377182	0.188591	−	0.982056i	$-0.439608\pi$
$227$	1290.00	0.377182	0.188591	+	0.982056i	$0.439608\pi$
$228$	0	0
$229$	5524.00	1.59404	0.797022	−	0.603950i	$-0.206407\pi$
$229$	5524.00	1.59404	0.797022	+	0.603950i	$0.206407\pi$
$230$	0	0
$231$	−2800.00	−0.797517
$232$	0	0
$233$	6314.00	1.77530	0.887648	−	0.460523i	$-0.152338\pi$
$233$	6314.00	1.77530	0.887648	+	0.460523i	$0.152338\pi$
$234$	0	0
$235$	3168.00	0.879394
$236$	0	0
$237$	9360.00	2.56539
$238$	0	0
$239$	−3960.00	−1.07176	−0.535881	−	0.844294i	$-0.680020\pi$
$239$	−3960.00	−1.07176	−0.535881	+	0.844294i	$0.680020\pi$
$240$	0	0
$241$	−7018.00	−1.87581	−0.937903	−	0.346898i	$-0.887235\pi$
$241$	−7018.00	−1.87581	−0.937903	+	0.346898i	$0.887235\pi$
$242$	0	0
$243$	−13870.0	−3.66157
$244$	0	0
$245$	−392.000	−0.102220
$246$	0	0
$247$	−312.000	−0.0803728
$248$	0	0
$249$	−5300.00	−1.34889
$250$	0	0
$251$	−2394.00	−0.602024	−0.301012	−	0.953620i	$-0.597324\pi$
$251$	−2394.00	−0.602024	−0.301012	+	0.953620i	$0.597324\pi$
$252$	0	0
$253$	2560.00	0.636149
$254$	0	0
$255$	−4640.00	−1.13948
$256$	0	0
$257$	−2766.00	−0.671355	−0.335678	−	0.941977i	$-0.608965\pi$
$257$	−2766.00	−0.671355	−0.335678	+	0.941977i	$0.608965\pi$
$258$	0	0
$259$	−182.000	−0.0436638
$260$	0	0
$261$	−4526.00	−1.07338
$262$	0	0
$263$	7968.00	1.86817	0.934084	−	0.357055i	$-0.116219\pi$
$263$	7968.00	1.86817	0.934084	+	0.357055i	$0.116219\pi$
$264$	0	0
$265$	3600.00	0.834514
$266$	0	0
$267$	3900.00	0.893918
$268$	0	0
$269$	−2900.00	−0.657309	−0.328654	−	0.944450i	$-0.606595\pi$
$269$	−2900.00	−0.657309	−0.328654	+	0.944450i	$0.606595\pi$
$270$	0	0
$271$	2640.00	0.591766	0.295883	−	0.955224i	$-0.404386\pi$
$271$	2640.00	0.591766	0.295883	+	0.955224i	$0.404386\pi$
$272$	0	0
$273$	−840.000	−0.186224
$274$	0	0
$275$	2440.00	0.535046
$276$	0	0
$277$	−1522.00	−0.330138	−0.165069	−	0.986282i	$-0.552785\pi$
$277$	−1522.00	−0.330138	−0.165069	+	0.986282i	$0.552785\pi$
$278$	0	0
$279$	18396.0	3.94745
$280$	0	0
$281$	−4534.00	−0.962547	−0.481274	−	0.876570i	$-0.659826\pi$
$281$	−4534.00	−0.962547	−0.481274	+	0.876570i	$0.659826\pi$
$282$	0	0
$283$	4834.00	1.01538	0.507688	−	0.861541i	$-0.330500\pi$
$283$	4834.00	1.01538	0.507688	+	0.861541i	$0.330500\pi$
$284$	0	0
$285$	2080.00	0.432311
$286$	0	0
$287$	−42.0000	−0.00863826
$288$	0	0
$289$	−1549.00	−0.315286
$290$	0	0
$291$	−2140.00	−0.431096
$292$	0	0
$293$	−4656.00	−0.928350	−0.464175	−	0.885744i	$-0.653649\pi$
$293$	−4656.00	−0.928350	−0.464175	+	0.885744i	$0.653649\pi$
$294$	0	0
$295$	−2192.00	−0.432621
$296$	0	0
$297$	18400.0	3.59487
$298$	0	0
$299$	768.000	0.148544
$300$	0	0
$301$	−2912.00	−0.557624
$302$	0	0
$303$	−14320.0	−2.71506
$304$	0	0
$305$	4608.00	0.865093
$306$	0	0
$307$	−7238.00	−1.34558	−0.672792	−	0.739831i	$-0.734905\pi$
$307$	−7238.00	−1.34558	−0.672792	+	0.739831i	$0.734905\pi$
$308$	0	0
$309$	−7640.00	−1.40655
$310$	0	0
$311$	1096.00	0.199834	0.0999171	−	0.994996i	$-0.468142\pi$
$311$	1096.00	0.199834	0.0999171	+	0.994996i	$0.468142\pi$
$312$	0	0
$313$	−3818.00	−0.689476	−0.344738	−	0.938699i	$-0.612032\pi$
$313$	−3818.00	−0.689476	−0.344738	+	0.938699i	$0.612032\pi$
$314$	0	0
$315$	4088.00	0.731215
$316$	0	0
$317$	1998.00	0.354003	0.177001	−	0.984211i	$-0.443360\pi$
$317$	1998.00	0.354003	0.177001	+	0.984211i	$0.443360\pi$
$318$	0	0
$319$	2480.00	0.435277
$320$	0	0
$321$	−3240.00	−0.563362
$322$	0	0
$323$	−1508.00	−0.259775
$324$	0	0
$325$	732.000	0.124936
$326$	0	0
$327$	13340.0	2.25597
$328$	0	0
$329$	2772.00	0.464515
$330$	0	0
$331$	−7936.00	−1.31783	−0.658915	−	0.752217i	$-0.728984\pi$
$331$	−7936.00	−1.31783	−0.658915	+	0.752217i	$0.728984\pi$
$332$	0	0
$333$	1898.00	0.312342
$334$	0	0
$335$	3808.00	0.621055
$336$	0	0
$337$	2766.00	0.447103	0.223551	−	0.974692i	$-0.428235\pi$
$337$	2766.00	0.447103	0.223551	+	0.974692i	$0.428235\pi$
$338$	0	0
$339$	−17980.0	−2.88065
$340$	0	0
$341$	−10080.0	−1.60077
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−5120.00	−0.798990
$346$	0	0
$347$	8352.00	1.29210	0.646050	−	0.763295i	$-0.276420\pi$
$347$	8352.00	1.29210	0.646050	+	0.763295i	$0.276420\pi$
$348$	0	0
$349$	−5924.00	−0.908609	−0.454304	−	0.890847i	$-0.650112\pi$
$349$	−5924.00	−0.908609	−0.454304	+	0.890847i	$0.650112\pi$
$350$	0	0
$351$	5520.00	0.839418
$352$	0	0
$353$	2226.00	0.335632	0.167816	−	0.985818i	$-0.446329\pi$
$353$	2226.00	0.335632	0.167816	+	0.985818i	$0.446329\pi$
$354$	0	0
$355$	3584.00	0.535828
$356$	0	0
$357$	−4060.00	−0.601899
$358$	0	0
$359$	3880.00	0.570414	0.285207	−	0.958466i	$-0.407938\pi$
$359$	3880.00	0.570414	0.285207	+	0.958466i	$0.407938\pi$
$360$	0	0
$361$	−6183.00	−0.901443
$362$	0	0
$363$	−2690.00	−0.388949
$364$	0	0
$365$	1264.00	0.181262
$366$	0	0
$367$	−2584.00	−0.367531	−0.183765	−	0.982970i	$-0.558829\pi$
$367$	−2584.00	−0.367531	−0.183765	+	0.982970i	$0.558829\pi$
$368$	0	0
$369$	438.000	0.0617923
$370$	0	0
$371$	3150.00	0.440808
$372$	0	0
$373$	10534.0	1.46228	0.731139	−	0.682228i	$-0.238989\pi$
$373$	10534.0	1.46228	0.731139	+	0.682228i	$0.238989\pi$
$374$	0	0
$375$	−14880.0	−2.04907
$376$	0	0
$377$	744.000	0.101639
$378$	0	0
$379$	−4472.00	−0.606098	−0.303049	−	0.952975i	$-0.598005\pi$
$379$	−4472.00	−0.606098	−0.303049	+	0.952975i	$0.598005\pi$
$380$	0	0
$381$	3840.00	0.516350
$382$	0	0
$383$	2468.00	0.329266	0.164633	−	0.986355i	$-0.447356\pi$
$383$	2468.00	0.329266	0.164633	+	0.986355i	$0.447356\pi$
$384$	0	0
$385$	−2240.00	−0.296522
$386$	0	0
$387$	30368.0	3.98887
$388$	0	0
$389$	−1046.00	−0.136335	−0.0681675	−	0.997674i	$-0.521715\pi$
$389$	−1046.00	−0.136335	−0.0681675	+	0.997674i	$0.521715\pi$
$390$	0	0
$391$	3712.00	0.480112
$392$	0	0
$393$	18140.0	2.32835
$394$	0	0
$395$	7488.00	0.953828
$396$	0	0
$397$	2124.00	0.268515	0.134258	−	0.990946i	$-0.457135\pi$
$397$	2124.00	0.268515	0.134258	+	0.990946i	$0.457135\pi$
$398$	0	0
$399$	1820.00	0.228356
$400$	0	0
$401$	11598.0	1.44433	0.722165	−	0.691721i	$-0.243147\pi$
$401$	11598.0	1.44433	0.722165	+	0.691721i	$0.243147\pi$
$402$	0	0
$403$	−3024.00	−0.373787
$404$	0	0
$405$	−21032.0	−2.58047
$406$	0	0
$407$	−1040.00	−0.126661
$408$	0	0
$409$	−2770.00	−0.334884	−0.167442	−	0.985882i	$-0.553551\pi$
$409$	−2770.00	−0.334884	−0.167442	+	0.985882i	$0.553551\pi$
$410$	0	0
$411$	−16660.0	−1.99946
$412$	0	0
$413$	−1918.00	−0.228520
$414$	0	0
$415$	−4240.00	−0.501526
$416$	0	0
$417$	−11260.0	−1.32231
$418$	0	0
$419$	9438.00	1.10042	0.550211	−	0.835026i	$-0.314547\pi$
$419$	9438.00	1.10042	0.550211	+	0.835026i	$0.314547\pi$
$420$	0	0
$421$	5550.00	0.642495	0.321248	−	0.946995i	$-0.395898\pi$
$421$	5550.00	0.642495	0.321248	+	0.946995i	$0.395898\pi$
$422$	0	0
$423$	−28908.0	−3.32283
$424$	0	0
$425$	3538.00	0.403808
$426$	0	0
$427$	4032.00	0.456961
$428$	0	0
$429$	−4800.00	−0.540201
$430$	0	0
$431$	−3000.00	−0.335278	−0.167639	−	0.985848i	$-0.553614\pi$
$431$	−3000.00	−0.335278	−0.167639	+	0.985848i	$0.553614\pi$
$432$	0	0
$433$	12926.0	1.43460	0.717302	−	0.696762i	$-0.245377\pi$
$433$	12926.0	1.43460	0.717302	+	0.696762i	$0.245377\pi$
$434$	0	0
$435$	−4960.00	−0.546698
$436$	0	0
$437$	−1664.00	−0.182151
$438$	0	0
$439$	−408.000	−0.0443571	−0.0221786	−	0.999754i	$-0.507060\pi$
$439$	−408.000	−0.0443571	−0.0221786	+	0.999754i	$0.507060\pi$
$440$	0	0
$441$	3577.00	0.386243
$442$	0	0
$443$	−14452.0	−1.54997	−0.774983	−	0.631982i	$-0.782242\pi$
$443$	−14452.0	−1.54997	−0.774983	+	0.631982i	$0.782242\pi$
$444$	0	0
$445$	3120.00	0.332364
$446$	0	0
$447$	−26940.0	−2.85060
$448$	0	0
$449$	10258.0	1.07818	0.539092	−	0.842247i	$-0.318767\pi$
$449$	10258.0	1.07818	0.539092	+	0.842247i	$0.318767\pi$
$450$	0	0
$451$	−240.000	−0.0250580
$452$	0	0
$453$	26480.0	2.74644
$454$	0	0
$455$	−672.000	−0.0692392
$456$	0	0
$457$	−5498.00	−0.562769	−0.281385	−	0.959595i	$-0.590794\pi$
$457$	−5498.00	−0.562769	−0.281385	+	0.959595i	$0.590794\pi$
$458$	0	0
$459$	26680.0	2.71311
$460$	0	0
$461$	−16316.0	−1.64840	−0.824199	−	0.566300i	$-0.808375\pi$
$461$	−16316.0	−1.64840	−0.824199	+	0.566300i	$0.808375\pi$
$462$	0	0
$463$	8944.00	0.897760	0.448880	−	0.893592i	$-0.351823\pi$
$463$	8944.00	0.897760	0.448880	+	0.893592i	$0.351823\pi$
$464$	0	0
$465$	20160.0	2.01053
$466$	0	0
$467$	−9422.00	−0.933615	−0.466807	−	0.884359i	$-0.654596\pi$
$467$	−9422.00	−0.933615	−0.466807	+	0.884359i	$0.654596\pi$
$468$	0	0
$469$	3332.00	0.328054
$470$	0	0
$471$	5560.00	0.543931
$472$	0	0
$473$	−16640.0	−1.61756
$474$	0	0
$475$	−1586.00	−0.153201
$476$	0	0
$477$	−32850.0	−3.15325
$478$	0	0
$479$	13820.0	1.31827	0.659136	−	0.752024i	$-0.270922\pi$
$479$	13820.0	1.31827	0.659136	+	0.752024i	$0.270922\pi$
$480$	0	0
$481$	−312.000	−0.0295758
$482$	0	0
$483$	−4480.00	−0.422044
$484$	0	0
$485$	−1712.00	−0.160284
$486$	0	0
$487$	−13264.0	−1.23419	−0.617094	−	0.786890i	$-0.711690\pi$
$487$	−13264.0	−1.23419	−0.617094	+	0.786890i	$0.711690\pi$
$488$	0	0
$489$	3280.00	0.303327
$490$	0	0
$491$	−5940.00	−0.545964	−0.272982	−	0.962019i	$-0.588010\pi$
$491$	−5940.00	−0.545964	−0.272982	+	0.962019i	$0.588010\pi$
$492$	0	0
$493$	3596.00	0.328511
$494$	0	0
$495$	23360.0	2.12112
$496$	0	0
$497$	3136.00	0.283036
$498$	0	0
$499$	−8252.00	−0.740301	−0.370151	−	0.928972i	$-0.620694\pi$
$499$	−8252.00	−0.740301	−0.370151	+	0.928972i	$0.620694\pi$
$500$	0	0
$501$	42680.0	3.80599
$502$	0	0
$503$	4704.00	0.416980	0.208490	−	0.978024i	$-0.433145\pi$
$503$	4704.00	0.416980	0.208490	+	0.978024i	$0.433145\pi$
$504$	0	0
$505$	−11456.0	−1.00948
$506$	0	0
$507$	20530.0	1.79836
$508$	0	0
$509$	−10788.0	−0.939430	−0.469715	−	0.882818i	$-0.655643\pi$
$509$	−10788.0	−0.939430	−0.469715	+	0.882818i	$0.655643\pi$
$510$	0	0
$511$	1106.00	0.0957467
$512$	0	0
$513$	−11960.0	−1.02933
$514$	0	0
$515$	−6112.00	−0.522965
$516$	0	0
$517$	15840.0	1.34747
$518$	0	0
$519$	34760.0	2.93987
$520$	0	0
$521$	−14586.0	−1.22653	−0.613267	−	0.789876i	$-0.710145\pi$
$521$	−14586.0	−1.22653	−0.613267	+	0.789876i	$0.710145\pi$
$522$	0	0
$523$	26.0000	0.00217381	0.00108690	−	0.999999i	$-0.499654\pi$
$523$	26.0000	0.00217381	0.00108690	+	0.999999i	$0.499654\pi$
$524$	0	0
$525$	−4270.00	−0.354968
$526$	0	0
$527$	−14616.0	−1.20813
$528$	0	0
$529$	−8071.00	−0.663352
$530$	0	0
$531$	20002.0	1.63468
$532$	0	0
$533$	−72.0000	−0.00585116
$534$	0	0
$535$	−2592.00	−0.209462
$536$	0	0
$537$	−22680.0	−1.82256
$538$	0	0
$539$	−1960.00	−0.156629
$540$	0	0
$541$	11214.0	0.891178	0.445589	−	0.895238i	$-0.352994\pi$
$541$	11214.0	0.891178	0.445589	+	0.895238i	$0.352994\pi$
$542$	0	0
$543$	2760.00	0.218127
$544$	0	0
$545$	10672.0	0.838786
$546$	0	0
$547$	−5424.00	−0.423973	−0.211987	−	0.977273i	$-0.567993\pi$
$547$	−5424.00	−0.423973	−0.211987	+	0.977273i	$0.567993\pi$
$548$	0	0
$549$	−42048.0	−3.26879
$550$	0	0
$551$	−1612.00	−0.124634
$552$	0	0
$553$	6552.00	0.503833
$554$	0	0
$555$	2080.00	0.159083
$556$	0	0
$557$	−17618.0	−1.34021	−0.670106	−	0.742265i	$-0.733752\pi$
$557$	−17618.0	−1.34021	−0.670106	+	0.742265i	$0.733752\pi$
$558$	0	0
$559$	−4992.00	−0.377709
$560$	0	0
$561$	−23200.0	−1.74600
$562$	0	0
$563$	−3562.00	−0.266644	−0.133322	−	0.991073i	$-0.542564\pi$
$563$	−3562.00	−0.266644	−0.133322	+	0.991073i	$0.542564\pi$
$564$	0	0
$565$	−14384.0	−1.07104
$566$	0	0
$567$	−18403.0	−1.36306
$568$	0	0
$569$	2838.00	0.209095	0.104548	−	0.994520i	$-0.466661\pi$
$569$	2838.00	0.209095	0.104548	+	0.994520i	$0.466661\pi$
$570$	0	0
$571$	−360.000	−0.0263845	−0.0131922	−	0.999913i	$-0.504199\pi$
$571$	−360.000	−0.0263845	−0.0131922	+	0.999913i	$0.504199\pi$
$572$	0	0
$573$	30000.0	2.18720
$574$	0	0
$575$	3904.00	0.283144
$576$	0	0
$577$	22018.0	1.58860	0.794299	−	0.607527i	$-0.207838\pi$
$577$	22018.0	1.58860	0.794299	+	0.607527i	$0.207838\pi$
$578$	0	0
$579$	−32780.0	−2.35283
$580$	0	0
$581$	−3710.00	−0.264917
$582$	0	0
$583$	18000.0	1.27870
$584$	0	0
$585$	7008.00	0.495291
$586$	0	0
$587$	1454.00	0.102237	0.0511184	−	0.998693i	$-0.483721\pi$
$587$	1454.00	0.102237	0.0511184	+	0.998693i	$0.483721\pi$
$588$	0	0
$589$	6552.00	0.458354
$590$	0	0
$591$	23620.0	1.64399
$592$	0	0
$593$	13818.0	0.956892	0.478446	−	0.878117i	$-0.341200\pi$
$593$	13818.0	0.956892	0.478446	+	0.878117i	$0.341200\pi$
$594$	0	0
$595$	−3248.00	−0.223790
$596$	0	0
$597$	10360.0	0.710229
$598$	0	0
$599$	6696.00	0.456746	0.228373	−	0.973574i	$-0.426659\pi$
$599$	6696.00	0.456746	0.228373	+	0.973574i	$0.426659\pi$
$600$	0	0
$601$	10010.0	0.679395	0.339698	−	0.940535i	$-0.389675\pi$
$601$	10010.0	0.679395	0.339698	+	0.940535i	$0.389675\pi$
$602$	0	0
$603$	−34748.0	−2.34668
$604$	0	0
$605$	−2152.00	−0.144614
$606$	0	0
$607$	2880.00	0.192579	0.0962896	−	0.995353i	$-0.469303\pi$
$607$	2880.00	0.192579	0.0962896	+	0.995353i	$0.469303\pi$
$608$	0	0
$609$	−4340.00	−0.288778
$610$	0	0
$611$	4752.00	0.314640
$612$	0	0
$613$	6522.00	0.429724	0.214862	−	0.976644i	$-0.431070\pi$
$613$	6522.00	0.429724	0.214862	+	0.976644i	$0.431070\pi$
$614$	0	0
$615$	480.000	0.0314723
$616$	0	0
$617$	6614.00	0.431555	0.215778	−	0.976443i	$-0.430771\pi$
$617$	6614.00	0.431555	0.215778	+	0.976443i	$0.430771\pi$
$618$	0	0
$619$	5266.00	0.341936	0.170968	−	0.985277i	$-0.445311\pi$
$619$	5266.00	0.341936	0.170968	+	0.985277i	$0.445311\pi$
$620$	0	0
$621$	29440.0	1.90239
$622$	0	0
$623$	2730.00	0.175562
$624$	0	0
$625$	−4279.00	−0.273856
$626$	0	0
$627$	10400.0	0.662418
$628$	0	0
$629$	−1508.00	−0.0955928
$630$	0	0
$631$	3344.00	0.210971	0.105485	−	0.994421i	$-0.466360\pi$
$631$	3344.00	0.210971	0.105485	+	0.994421i	$0.466360\pi$
$632$	0	0
$633$	−35240.0	−2.21274
$634$	0	0
$635$	3072.00	0.191982
$636$	0	0
$637$	−588.000	−0.0365736
$638$	0	0
$639$	−32704.0	−2.02465
$640$	0	0
$641$	−4882.00	−0.300823	−0.150411	−	0.988623i	$-0.548060\pi$
$641$	−4882.00	−0.300823	−0.150411	+	0.988623i	$0.548060\pi$
$642$	0	0
$643$	−15898.0	−0.975048	−0.487524	−	0.873110i	$-0.662100\pi$
$643$	−15898.0	−0.975048	−0.487524	+	0.873110i	$0.662100\pi$
$644$	0	0
$645$	33280.0	2.03163
$646$	0	0
$647$	6132.00	0.372602	0.186301	−	0.982493i	$-0.440350\pi$
$647$	6132.00	0.372602	0.186301	+	0.982493i	$0.440350\pi$
$648$	0	0
$649$	−10960.0	−0.662893
$650$	0	0
$651$	17640.0	1.06201
$652$	0	0
$653$	−24198.0	−1.45014	−0.725070	−	0.688676i	$-0.758192\pi$
$653$	−24198.0	−1.45014	−0.725070	+	0.688676i	$0.758192\pi$
$654$	0	0
$655$	14512.0	0.865696
$656$	0	0
$657$	−11534.0	−0.684907
$658$	0	0
$659$	17456.0	1.03185	0.515925	−	0.856634i	$-0.327448\pi$
$659$	17456.0	1.03185	0.515925	+	0.856634i	$0.327448\pi$
$660$	0	0
$661$	−656.000	−0.0386013	−0.0193006	−	0.999814i	$-0.506144\pi$
$661$	−656.000	−0.0386013	−0.0193006	+	0.999814i	$0.506144\pi$
$662$	0	0
$663$	−6960.00	−0.407698
$664$	0	0
$665$	1456.00	0.0849041
$666$	0	0
$667$	3968.00	0.230347
$668$	0	0
$669$	13360.0	0.772089
$670$	0	0
$671$	23040.0	1.32556
$672$	0	0
$673$	−18214.0	−1.04324	−0.521618	−	0.853179i	$-0.674671\pi$
$673$	−18214.0	−1.04324	−0.521618	+	0.853179i	$0.674671\pi$
$674$	0	0
$675$	28060.0	1.60004
$676$	0	0
$677$	30252.0	1.71740	0.858699	−	0.512480i	$-0.171273\pi$
$677$	30252.0	1.71740	0.858699	+	0.512480i	$0.171273\pi$
$678$	0	0
$679$	−1498.00	−0.0846656
$680$	0	0
$681$	−12900.0	−0.725887
$682$	0	0
$683$	−10836.0	−0.607069	−0.303534	−	0.952820i	$-0.598167\pi$
$683$	−10836.0	−0.607069	−0.303534	+	0.952820i	$0.598167\pi$
$684$	0	0
$685$	−13328.0	−0.743411
$686$	0	0
$687$	−55240.0	−3.06774
$688$	0	0
$689$	5400.00	0.298583
$690$	0	0
$691$	9578.00	0.527300	0.263650	−	0.964618i	$-0.415074\pi$
$691$	9578.00	0.527300	0.263650	+	0.964618i	$0.415074\pi$
$692$	0	0
$693$	20440.0	1.12042
$694$	0	0
$695$	−9008.00	−0.491644
$696$	0	0
$697$	−348.000	−0.0189117
$698$	0	0
$699$	−63140.0	−3.41656
$700$	0	0
$701$	12442.0	0.670368	0.335184	−	0.942153i	$-0.391202\pi$
$701$	12442.0	0.670368	0.335184	+	0.942153i	$0.391202\pi$
$702$	0	0
$703$	676.000	0.0362672
$704$	0	0
$705$	−31680.0	−1.69239
$706$	0	0
$707$	−10024.0	−0.533227
$708$	0	0
$709$	−25174.0	−1.33347	−0.666734	−	0.745295i	$-0.732308\pi$
$709$	−25174.0	−1.33347	−0.666734	+	0.745295i	$0.732308\pi$
$710$	0	0
$711$	−68328.0	−3.60408
$712$	0	0
$713$	−16128.0	−0.847123
$714$	0	0
$715$	−3840.00	−0.200850
$716$	0	0
$717$	39600.0	2.06261
$718$	0	0
$719$	34188.0	1.77329	0.886646	−	0.462448i	$-0.153029\pi$
$719$	34188.0	1.77329	0.886646	+	0.462448i	$0.153029\pi$
$720$	0	0
$721$	−5348.00	−0.276241
$722$	0	0
$723$	70180.0	3.60999
$724$	0	0
$725$	3782.00	0.193738
$726$	0	0
$727$	−5204.00	−0.265482	−0.132741	−	0.991151i	$-0.542378\pi$
$727$	−5204.00	−0.265482	−0.132741	+	0.991151i	$0.542378\pi$
$728$	0	0
$729$	67717.0	3.44038
$730$	0	0
$731$	−24128.0	−1.22080
$732$	0	0
$733$	−32880.0	−1.65682	−0.828411	−	0.560121i	$-0.810755\pi$
$733$	−32880.0	−1.65682	−0.828411	+	0.560121i	$0.810755\pi$
$734$	0	0
$735$	3920.00	0.196723
$736$	0	0
$737$	19040.0	0.951625
$738$	0	0
$739$	−3912.00	−0.194730	−0.0973648	−	0.995249i	$-0.531041\pi$
$739$	−3912.00	−0.194730	−0.0973648	+	0.995249i	$0.531041\pi$
$740$	0	0
$741$	3120.00	0.154678
$742$	0	0
$743$	−16008.0	−0.790413	−0.395206	−	0.918592i	$-0.629327\pi$
$743$	−16008.0	−0.790413	−0.395206	+	0.918592i	$0.629327\pi$
$744$	0	0
$745$	−21552.0	−1.05987
$746$	0	0
$747$	38690.0	1.89504
$748$	0	0
$749$	−2268.00	−0.110642
$750$	0	0
$751$	9960.00	0.483949	0.241974	−	0.970283i	$-0.422205\pi$
$751$	9960.00	0.483949	0.241974	+	0.970283i	$0.422205\pi$
$752$	0	0
$753$	23940.0	1.15860
$754$	0	0
$755$	21184.0	1.02115
$756$	0	0
$757$	12378.0	0.594301	0.297151	−	0.954831i	$-0.403964\pi$
$757$	12378.0	0.594301	0.297151	+	0.954831i	$0.403964\pi$
$758$	0	0
$759$	−25600.0	−1.22427
$760$	0	0
$761$	34670.0	1.65149	0.825747	−	0.564041i	$-0.190754\pi$
$761$	34670.0	1.65149	0.825747	+	0.564041i	$0.190754\pi$
$762$	0	0
$763$	9338.00	0.443065
$764$	0	0
$765$	33872.0	1.60084
$766$	0	0
$767$	−3288.00	−0.154789
$768$	0	0
$769$	−10898.0	−0.511043	−0.255521	−	0.966803i	$-0.582247\pi$
$769$	−10898.0	−0.511043	−0.255521	+	0.966803i	$0.582247\pi$
$770$	0	0
$771$	27660.0	1.29202
$772$	0	0
$773$	−25808.0	−1.20084	−0.600420	−	0.799685i	$-0.705000\pi$
$773$	−25808.0	−1.20084	−0.600420	+	0.799685i	$0.705000\pi$
$774$	0	0
$775$	−15372.0	−0.712488
$776$	0	0
$777$	1820.00	0.0840311
$778$	0	0
$779$	156.000	0.00717494
$780$	0	0
$781$	17920.0	0.821035
$782$	0	0
$783$	28520.0	1.30169
$784$	0	0
$785$	4448.00	0.202237
$786$	0	0
$787$	−21054.0	−0.953614	−0.476807	−	0.879008i	$-0.658206\pi$
$787$	−21054.0	−0.953614	−0.476807	+	0.879008i	$0.658206\pi$
$788$	0	0
$789$	−79680.0	−3.59529
$790$	0	0
$791$	−12586.0	−0.565748
$792$	0	0
$793$	6912.00	0.309524
$794$	0	0
$795$	−36000.0	−1.60602
$796$	0	0
$797$	−24276.0	−1.07892	−0.539461	−	0.842011i	$-0.681372\pi$
$797$	−24276.0	−1.07892	−0.539461	+	0.842011i	$0.681372\pi$
$798$	0	0
$799$	22968.0	1.01696
$800$	0	0
$801$	−28470.0	−1.25585
$802$	0	0
$803$	6320.00	0.277743
$804$	0	0
$805$	−3584.00	−0.156919
$806$	0	0
$807$	29000.0	1.26499
$808$	0	0
$809$	21526.0	0.935493	0.467747	−	0.883863i	$-0.345066\pi$
$809$	21526.0	0.935493	0.467747	+	0.883863i	$0.345066\pi$
$810$	0	0
$811$	−12806.0	−0.554475	−0.277238	−	0.960801i	$-0.589419\pi$
$811$	−12806.0	−0.554475	−0.277238	+	0.960801i	$0.589419\pi$
$812$	0	0
$813$	−26400.0	−1.13885
$814$	0	0
$815$	2624.00	0.112779
$816$	0	0
$817$	10816.0	0.463163
$818$	0	0
$819$	6132.00	0.261623
$820$	0	0
$821$	13214.0	0.561720	0.280860	−	0.959749i	$-0.409380\pi$
$821$	13214.0	0.561720	0.280860	+	0.959749i	$0.409380\pi$
$822$	0	0
$823$	32248.0	1.36585	0.682925	−	0.730488i	$-0.260708\pi$
$823$	32248.0	1.36585	0.682925	+	0.730488i	$0.260708\pi$
$824$	0	0
$825$	−24400.0	−1.02970
$826$	0	0
$827$	−14316.0	−0.601954	−0.300977	−	0.953631i	$-0.597313\pi$
$827$	−14316.0	−0.601954	−0.300977	+	0.953631i	$0.597313\pi$
$828$	0	0
$829$	25168.0	1.05443	0.527214	−	0.849733i	$-0.323237\pi$
$829$	25168.0	1.05443	0.527214	+	0.849733i	$0.323237\pi$
$830$	0	0
$831$	15220.0	0.635350
$832$	0	0
$833$	−2842.00	−0.118211
$834$	0	0
$835$	34144.0	1.41509
$836$	0	0
$837$	−115920.	−4.78707
$838$	0	0
$839$	9356.00	0.384988	0.192494	−	0.981298i	$-0.438342\pi$
$839$	9356.00	0.384988	0.192494	+	0.981298i	$0.438342\pi$
$840$	0	0
$841$	−20545.0	−0.842388
$842$	0	0
$843$	45340.0	1.85242
$844$	0	0
$845$	16424.0	0.668642
$846$	0	0
$847$	−1883.00	−0.0763880
$848$	0	0
$849$	−48340.0	−1.95409
$850$	0	0
$851$	−1664.00	−0.0670284
$852$	0	0
$853$	2372.00	0.0952119	0.0476059	−	0.998866i	$-0.484841\pi$
$853$	2372.00	0.0952119	0.0476059	+	0.998866i	$0.484841\pi$
$854$	0	0
$855$	−15184.0	−0.607347
$856$	0	0
$857$	11694.0	0.466114	0.233057	−	0.972463i	$-0.425127\pi$
$857$	11694.0	0.466114	0.233057	+	0.972463i	$0.425127\pi$
$858$	0	0
$859$	−20506.0	−0.814500	−0.407250	−	0.913317i	$-0.633512\pi$
$859$	−20506.0	−0.814500	−0.407250	+	0.913317i	$0.633512\pi$
$860$	0	0
$861$	420.000	0.0166243
$862$	0	0
$863$	−28136.0	−1.10980	−0.554902	−	0.831916i	$-0.687244\pi$
$863$	−28136.0	−1.10980	−0.554902	+	0.831916i	$0.687244\pi$
$864$	0	0
$865$	27808.0	1.09306
$866$	0	0
$867$	15490.0	0.606768
$868$	0	0
$869$	37440.0	1.46152
$870$	0	0
$871$	5712.00	0.222209
$872$	0	0
$873$	15622.0	0.605641
$874$	0	0
$875$	−10416.0	−0.402429
$876$	0	0
$877$	−37070.0	−1.42733	−0.713663	−	0.700489i	$-0.752965\pi$
$877$	−37070.0	−1.42733	−0.713663	+	0.700489i	$0.752965\pi$
$878$	0	0
$879$	46560.0	1.78661
$880$	0	0
$881$	−6198.00	−0.237021	−0.118511	−	0.992953i	$-0.537812\pi$
$881$	−6198.00	−0.237021	−0.118511	+	0.992953i	$0.537812\pi$
$882$	0	0
$883$	−31876.0	−1.21485	−0.607425	−	0.794377i	$-0.707798\pi$
$883$	−31876.0	−1.21485	−0.607425	+	0.794377i	$0.707798\pi$
$884$	0	0
$885$	21920.0	0.832579
$886$	0	0
$887$	−132.000	−0.00499676	−0.00249838	−	0.999997i	$-0.500795\pi$
$887$	−132.000	−0.00499676	−0.00249838	+	0.999997i	$0.500795\pi$
$888$	0	0
$889$	2688.00	0.101409
$890$	0	0
$891$	−105160.	−3.95398
$892$	0	0
$893$	−10296.0	−0.385826
$894$	0	0
$895$	−18144.0	−0.677639
$896$	0	0
$897$	−7680.00	−0.285873
$898$	0	0
$899$	−15624.0	−0.579632
$900$	0	0
$901$	26100.0	0.965058
$902$	0	0
$903$	29120.0	1.07315
$904$	0	0
$905$	2208.00	0.0811010
$906$	0	0
$907$	38244.0	1.40008	0.700039	−	0.714104i	$-0.253166\pi$
$907$	38244.0	1.40008	0.700039	+	0.714104i	$0.253166\pi$
$908$	0	0
$909$	104536.	3.81435
$910$	0	0
$911$	−7008.00	−0.254869	−0.127434	−	0.991847i	$-0.540674\pi$
$911$	−7008.00	−0.254869	−0.127434	+	0.991847i	$0.540674\pi$
$912$	0	0
$913$	−21200.0	−0.768475
$914$	0	0
$915$	−46080.0	−1.66487
$916$	0	0
$917$	12698.0	0.457279
$918$	0	0
$919$	36664.0	1.31603	0.658016	−	0.753004i	$-0.271396\pi$
$919$	36664.0	1.31603	0.658016	+	0.753004i	$0.271396\pi$
$920$	0	0
$921$	72380.0	2.58958
$922$	0	0
$923$	5376.00	0.191715
$924$	0	0
$925$	−1586.00	−0.0563755
$926$	0	0
$927$	55772.0	1.97604
$928$	0	0
$929$	45510.0	1.60725	0.803625	−	0.595136i	$-0.202902\pi$
$929$	45510.0	1.60725	0.803625	+	0.595136i	$0.202902\pi$
$930$	0	0
$931$	1274.00	0.0448482
$932$	0	0
$933$	−10960.0	−0.384581
$934$	0	0
$935$	−18560.0	−0.649173
$936$	0	0
$937$	−3838.00	−0.133812	−0.0669061	−	0.997759i	$-0.521313\pi$
$937$	−3838.00	−0.133812	−0.0669061	+	0.997759i	$0.521313\pi$
$938$	0	0
$939$	38180.0	1.32690
$940$	0	0
$941$	−16832.0	−0.583111	−0.291556	−	0.956554i	$-0.594173\pi$
$941$	−16832.0	−0.583111	−0.291556	+	0.956554i	$0.594173\pi$
$942$	0	0
$943$	−384.000	−0.0132606
$944$	0	0
$945$	−25760.0	−0.886744
$946$	0	0
$947$	−40928.0	−1.40442	−0.702208	−	0.711972i	$-0.747802\pi$
$947$	−40928.0	−1.40442	−0.702208	+	0.711972i	$0.747802\pi$
$948$	0	0
$949$	1896.00	0.0648543
$950$	0	0
$951$	−19980.0	−0.681279
$952$	0	0
$953$	−24070.0	−0.818157	−0.409079	−	0.912499i	$-0.634150\pi$
$953$	−24070.0	−0.818157	−0.409079	+	0.912499i	$0.634150\pi$
$954$	0	0
$955$	24000.0	0.813217
$956$	0	0
$957$	−24800.0	−0.837691
$958$	0	0
$959$	−11662.0	−0.392686
$960$	0	0
$961$	33713.0	1.13165
$962$	0	0
$963$	23652.0	0.791459
$964$	0	0
$965$	−26224.0	−0.874798
$966$	0	0
$967$	−17152.0	−0.570394	−0.285197	−	0.958469i	$-0.592059\pi$
$967$	−17152.0	−0.570394	−0.285197	+	0.958469i	$0.592059\pi$
$968$	0	0
$969$	15080.0	0.499937
$970$	0	0
$971$	−32910.0	−1.08767	−0.543837	−	0.839191i	$-0.683029\pi$
$971$	−32910.0	−1.08767	−0.543837	+	0.839191i	$0.683029\pi$
$972$	0	0
$973$	−7882.00	−0.259697
$974$	0	0
$975$	−7320.00	−0.240439
$976$	0	0
$977$	−6822.00	−0.223393	−0.111697	−	0.993742i	$-0.535628\pi$
$977$	−6822.00	−0.223393	−0.111697	+	0.993742i	$0.535628\pi$
$978$	0	0
$979$	15600.0	0.509273
$980$	0	0
$981$	−97382.0	−3.16939
$982$	0	0
$983$	−48420.0	−1.57107	−0.785533	−	0.618820i	$-0.787611\pi$
$983$	−48420.0	−1.57107	−0.785533	+	0.618820i	$0.787611\pi$
$984$	0	0
$985$	18896.0	0.611245
$986$	0	0
$987$	−27720.0	−0.893959
$988$	0	0
$989$	−26624.0	−0.856010
$990$	0	0
$991$	−49216.0	−1.57760	−0.788798	−	0.614652i	$-0.789296\pi$
$991$	−49216.0	−1.57760	−0.788798	+	0.614652i	$0.789296\pi$
$992$	0	0
$993$	79360.0	2.53617
$994$	0	0
$995$	8288.00	0.264068
$996$	0	0
$997$	35264.0	1.12018	0.560091	−	0.828431i	$-0.310766\pi$
$997$	35264.0	1.12018	0.560091	+	0.828431i	$0.310766\pi$
$998$	0	0
$999$	−11960.0	−0.378776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.4.a.a.1.1	✓	1
3.2	odd	2		252.4.a.d.1.1		1
4.3	odd	2		112.4.a.g.1.1		1
5.2	odd	4		700.4.e.a.449.2		2
5.3	odd	4		700.4.e.a.449.1		2
5.4	even	2		700.4.a.n.1.1		1
7.2	even	3		196.4.e.f.165.1		2
7.3	odd	6		196.4.e.a.177.1		2
7.4	even	3		196.4.e.f.177.1		2
7.5	odd	6		196.4.e.a.165.1		2
7.6	odd	2		196.4.a.d.1.1		1
8.3	odd	2		448.4.a.a.1.1		1
8.5	even	2		448.4.a.p.1.1		1
12.11	even	2		1008.4.a.o.1.1		1
21.2	odd	6		1764.4.k.d.361.1		2
21.5	even	6		1764.4.k.m.361.1		2
21.11	odd	6		1764.4.k.d.1549.1		2
21.17	even	6		1764.4.k.m.1549.1		2
21.20	even	2		1764.4.a.c.1.1		1
28.27	even	2		784.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.4.a.a.1.1	✓	1	1.1	even	1	trivial
112.4.a.g.1.1		1	4.3	odd	2
196.4.a.d.1.1		1	7.6	odd	2
196.4.e.a.165.1		2	7.5	odd	6
196.4.e.a.177.1		2	7.3	odd	6
196.4.e.f.165.1		2	7.2	even	3
196.4.e.f.177.1		2	7.4	even	3
252.4.a.d.1.1		1	3.2	odd	2
448.4.a.a.1.1		1	8.3	odd	2
448.4.a.p.1.1		1	8.5	even	2
700.4.a.n.1.1		1	5.4	even	2
700.4.e.a.449.1		2	5.3	odd	4
700.4.e.a.449.2		2	5.2	odd	4
784.4.a.a.1.1		1	28.27	even	2
1008.4.a.o.1.1		1	12.11	even	2
1764.4.a.c.1.1		1	21.20	even	2
1764.4.k.d.361.1		2	21.2	odd	6
1764.4.k.d.1549.1		2	21.11	odd	6
1764.4.k.m.361.1		2	21.5	even	6
1764.4.k.m.1549.1		2	21.17	even	6

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:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	28.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more