LMFDB - Embedded newform 448.4.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.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:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	448.1

$q$-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	$-0.0877398\pi$
$3$	10.0000	1.92450	0.962250	+	0.272166i	$0.0877398\pi$
$4$	0	0
$5$	8.00000	0.715542	0.357771	−	0.933809i	$-0.383537\pi$
$5$	8.00000	0.715542	0.357771	+	0.933809i	$0.383537\pi$
$6$	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	$-0.315312\pi$
$11$	40.0000	1.09640	0.548202	+	0.836346i	$0.315312\pi$
$12$	0	0
$13$	12.0000	0.256015	0.128008	−	0.991773i	$-0.459142\pi$
$13$	12.0000	0.256015	0.128008	+	0.991773i	$0.459142\pi$
$14$	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	$-0.550172\pi$
$19$	−26.0000	−0.313937	−0.156969	+	0.987604i	$0.550172\pi$
$20$	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	$-0.436392\pi$
$29$	62.0000	0.397004	0.198502	+	0.980101i	$0.436392\pi$
$30$	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	$-0.518396\pi$
$37$	−26.0000	−0.115524	−0.0577618	+	0.998330i	$0.518396\pi$
$38$	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	$-0.764073\pi$
$43$	−416.000	−1.47534	−0.737668	+	0.675164i	$0.764073\pi$
$44$	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	$-0.301826\pi$
$53$	450.000	1.16627	0.583134	+	0.812376i	$0.301826\pi$
$54$	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	$-0.597755\pi$
$59$	−274.000	−0.604606	−0.302303	+	0.953212i	$0.597755\pi$
$60$	0	0
$61$	576.000	1.20900	0.604502	−	0.796604i	$-0.293372\pi$
$61$	576.000	1.20900	0.604502	+	0.796604i	$0.293372\pi$
$62$	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	$-0.357111\pi$
$67$	476.000	0.867950	0.433975	+	0.900925i	$0.357111\pi$
$68$	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	$-0.613972\pi$
$83$	−530.000	−0.700904	−0.350452	+	0.936581i	$0.613972\pi$
$84$	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	$-0.749229\pi$
$101$	−1432.00	−1.41079	−0.705393	+	0.708817i	$0.749229\pi$
$102$	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	$-0.546758\pi$
$107$	−324.000	−0.292731	−0.146366	+	0.989231i	$0.546758\pi$
$108$	0	0
$109$	1334.00	1.17224	0.586119	−	0.810225i	$-0.300655\pi$
$109$	1334.00	1.17224	0.586119	+	0.810225i	$0.300655\pi$
$110$	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	$-0.293204\pi$
$131$	1814.00	1.20985	0.604923	+	0.796284i	$0.293204\pi$
$132$	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	$-0.611628\pi$
$139$	−1126.00	−0.687094	−0.343547	+	0.939135i	$0.611628\pi$
$140$	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	$-0.765462\pi$
$149$	−2694.00	−1.48122	−0.740608	+	0.671938i	$0.765462\pi$
$150$	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	$-0.454866\pi$
$157$	556.000	0.282635	0.141317	+	0.989964i	$0.454866\pi$
$158$	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	$-0.474889\pi$
$163$	328.000	0.157613	0.0788066	+	0.996890i	$0.474889\pi$
$164$	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	$-0.223331\pi$
$173$	3476.00	1.52760	0.763802	+	0.645451i	$0.223331\pi$
$174$	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	$-0.657015\pi$
$179$	−2268.00	−0.947029	−0.473515	+	0.880786i	$0.657015\pi$
$180$	0	0
$181$	276.000	0.113342	0.0566710	−	0.998393i	$-0.481951\pi$
$181$	276.000	0.113342	0.0566710	+	0.998393i	$0.481951\pi$
$182$	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	$-0.359528\pi$
$197$	2362.00	0.854241	0.427121	+	0.904195i	$0.359528\pi$
$198$	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	$-0.694954\pi$
$211$	−3524.00	−1.14977	−0.574887	+	0.818233i	$0.694954\pi$
$212$	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	$-0.560392\pi$
$227$	−1290.00	−0.377182	−0.188591	+	0.982056i	$0.560392\pi$
$228$	0	0
$229$	−5524.00	−1.59404	−0.797022	−	0.603950i	$-0.793593\pi$
$229$	−5524.00	−1.59404	−0.797022	+	0.603950i	$0.793593\pi$
$230$	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	$-0.402676\pi$
$251$	2394.00	0.602024	0.301012	+	0.953620i	$0.402676\pi$
$252$	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	$-0.393405\pi$
$269$	2900.00	0.657309	0.328654	+	0.944450i	$0.393405\pi$
$270$	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	$-0.447215\pi$
$277$	1522.00	0.330138	0.165069	+	0.986282i	$0.447215\pi$
$278$	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	$-0.669500\pi$
$283$	−4834.00	−1.01538	−0.507688	+	0.861541i	$0.669500\pi$
$284$	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	$-0.346351\pi$
$293$	4656.00	0.928350	0.464175	+	0.885744i	$0.346351\pi$
$294$	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	$-0.265095\pi$
$307$	7238.00	1.34558	0.672792	+	0.739831i	$0.265095\pi$
$308$	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	$-0.556640\pi$
$317$	−1998.00	−0.354003	−0.177001	+	0.984211i	$0.556640\pi$
$318$	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	$-0.271016\pi$
$331$	7936.00	1.31783	0.658915	+	0.752217i	$0.271016\pi$
$332$	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	$-0.723580\pi$
$347$	−8352.00	−1.29210	−0.646050	+	0.763295i	$0.723580\pi$
$348$	0	0
$349$	5924.00	0.908609	0.454304	−	0.890847i	$-0.349888\pi$
$349$	5924.00	0.908609	0.454304	+	0.890847i	$0.349888\pi$
$350$	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	$-0.761011\pi$
$373$	−10534.0	−1.46228	−0.731139	+	0.682228i	$0.761011\pi$
$374$	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	$-0.401995\pi$
$379$	4472.00	0.606098	0.303049	+	0.952975i	$0.401995\pi$
$380$	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	$-0.478285\pi$
$389$	1046.00	0.136335	0.0681675	+	0.997674i	$0.478285\pi$
$390$	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	$-0.542865\pi$
$397$	−2124.00	−0.268515	−0.134258	+	0.990946i	$0.542865\pi$
$398$	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	$-0.685453\pi$
$419$	−9438.00	−1.10042	−0.550211	+	0.835026i	$0.685453\pi$
$420$	0	0
$421$	−5550.00	−0.642495	−0.321248	−	0.946995i	$-0.604102\pi$
$421$	−5550.00	−0.642495	−0.321248	+	0.946995i	$0.604102\pi$
$422$	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	$-0.217758\pi$
$443$	14452.0	1.54997	0.774983	+	0.631982i	$0.217758\pi$
$444$	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	$-0.191625\pi$
$461$	16316.0	1.64840	0.824199	+	0.566300i	$0.191625\pi$
$462$	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	$-0.345404\pi$
$467$	9422.00	0.933615	0.466807	+	0.884359i	$0.345404\pi$
$468$	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	$-0.411990\pi$
$491$	5940.00	0.545964	0.272982	+	0.962019i	$0.411990\pi$
$492$	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	$-0.379306\pi$
$499$	8252.00	0.740301	0.370151	+	0.928972i	$0.379306\pi$
$500$	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	$-0.344357\pi$
$509$	10788.0	0.939430	0.469715	+	0.882818i	$0.344357\pi$
$510$	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	$-0.500346\pi$
$523$	−26.0000	−0.00217381	−0.00108690	+	0.999999i	$0.500346\pi$
$524$	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	$-0.647006\pi$
$541$	−11214.0	−0.891178	−0.445589	+	0.895238i	$0.647006\pi$
$542$	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	$-0.432007\pi$
$547$	5424.00	0.423973	0.211987	+	0.977273i	$0.432007\pi$
$548$	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	$-0.266248\pi$
$557$	17618.0	1.34021	0.670106	+	0.742265i	$0.266248\pi$
$558$	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	$-0.457436\pi$
$563$	3562.00	0.266644	0.133322	+	0.991073i	$0.457436\pi$
$564$	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	$-0.495801\pi$
$571$	360.000	0.0263845	0.0131922	+	0.999913i	$0.495801\pi$
$572$	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	$-0.516279\pi$
$587$	−1454.00	−0.102237	−0.0511184	+	0.998693i	$0.516279\pi$
$588$	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	$-0.568930\pi$
$613$	−6522.00	−0.429724	−0.214862	+	0.976644i	$0.568930\pi$
$614$	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	$-0.554689\pi$
$619$	−5266.00	−0.341936	−0.170968	+	0.985277i	$0.554689\pi$
$620$	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	$-0.337900\pi$
$643$	15898.0	0.975048	0.487524	+	0.873110i	$0.337900\pi$
$644$	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	$-0.241808\pi$
$653$	24198.0	1.45014	0.725070	+	0.688676i	$0.241808\pi$
$654$	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	$-0.672552\pi$
$659$	−17456.0	−1.03185	−0.515925	+	0.856634i	$0.672552\pi$
$660$	0	0
$661$	656.000	0.0386013	0.0193006	−	0.999814i	$-0.493856\pi$
$661$	656.000	0.0386013	0.0193006	+	0.999814i	$0.493856\pi$
$662$	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	$-0.828727\pi$
$677$	−30252.0	−1.71740	−0.858699	+	0.512480i	$0.828727\pi$
$678$	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	$-0.401833\pi$
$683$	10836.0	0.607069	0.303534	+	0.952820i	$0.401833\pi$
$684$	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	$-0.584926\pi$
$691$	−9578.00	−0.527300	−0.263650	+	0.964618i	$0.584926\pi$
$692$	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	$-0.608798\pi$
$701$	−12442.0	−0.670368	−0.335184	+	0.942153i	$0.608798\pi$
$702$	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	$-0.267692\pi$
$709$	25174.0	1.33347	0.666734	+	0.745295i	$0.267692\pi$
$710$	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	$-0.189245\pi$
$733$	32880.0	1.65682	0.828411	+	0.560121i	$0.189245\pi$
$734$	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	$-0.468959\pi$
$739$	3912.00	0.194730	0.0973648	+	0.995249i	$0.468959\pi$
$740$	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	$-0.596036\pi$
$757$	−12378.0	−0.594301	−0.297151	+	0.954831i	$0.596036\pi$
$758$	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	$-0.295000\pi$
$773$	25808.0	1.20084	0.600420	+	0.799685i	$0.295000\pi$
$774$	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	$-0.341794\pi$
$787$	21054.0	0.953614	0.476807	+	0.879008i	$0.341794\pi$
$788$	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	$-0.318628\pi$
$797$	24276.0	1.07892	0.539461	+	0.842011i	$0.318628\pi$
$798$	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	$-0.410581\pi$
$811$	12806.0	0.554475	0.277238	+	0.960801i	$0.410581\pi$
$812$	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	$-0.590620\pi$
$821$	−13214.0	−0.561720	−0.280860	+	0.959749i	$0.590620\pi$
$822$	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	$-0.402687\pi$
$827$	14316.0	0.601954	0.300977	+	0.953631i	$0.402687\pi$
$828$	0	0
$829$	−25168.0	−1.05443	−0.527214	−	0.849733i	$-0.676763\pi$
$829$	−25168.0	−1.05443	−0.527214	+	0.849733i	$0.676763\pi$
$830$	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	$-0.515159\pi$
$853$	−2372.00	−0.0952119	−0.0476059	+	0.998866i	$0.515159\pi$
$854$	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	$-0.366488\pi$
$859$	20506.0	0.814500	0.407250	+	0.913317i	$0.366488\pi$
$860$	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	$-0.247035\pi$
$877$	37070.0	1.42733	0.713663	+	0.700489i	$0.247035\pi$
$878$	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	$-0.292202\pi$
$883$	31876.0	1.21485	0.607425	+	0.794377i	$0.292202\pi$
$884$	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	$-0.746834\pi$
$907$	−38244.0	−1.40008	−0.700039	+	0.714104i	$0.746834\pi$
$908$	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	$-0.405827\pi$
$941$	16832.0	0.583111	0.291556	+	0.956554i	$0.405827\pi$
$942$	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	$-0.252198\pi$
$947$	40928.0	1.40442	0.702208	+	0.711972i	$0.252198\pi$
$948$	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	$-0.316971\pi$
$971$	32910.0	1.08767	0.543837	+	0.839191i	$0.316971\pi$
$972$	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	$-0.689234\pi$
$997$	−35264.0	−1.12018	−0.560091	+	0.828431i	$0.689234\pi$
$998$	0	0
$999$	−11960.0	−0.378776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.a.p.1.1		1
4.3	odd	2		448.4.a.a.1.1		1
8.3	odd	2		112.4.a.g.1.1		1
8.5	even	2		28.4.a.a.1.1	✓	1
24.5	odd	2		252.4.a.d.1.1		1
24.11	even	2		1008.4.a.o.1.1		1
40.13	odd	4		700.4.e.a.449.1		2
40.29	even	2		700.4.a.n.1.1		1
40.37	odd	4		700.4.e.a.449.2		2
56.5	odd	6		196.4.e.a.165.1		2
56.13	odd	2		196.4.a.d.1.1		1
56.27	even	2		784.4.a.a.1.1		1
56.37	even	6		196.4.e.f.165.1		2
56.45	odd	6		196.4.e.a.177.1		2
56.53	even	6		196.4.e.f.177.1		2
168.5	even	6		1764.4.k.m.361.1		2
168.53	odd	6		1764.4.k.d.1549.1		2
168.101	even	6		1764.4.k.m.1549.1		2
168.125	even	2		1764.4.a.c.1.1		1
168.149	odd	6		1764.4.k.d.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.4.a.a.1.1	✓	1	8.5	even	2
112.4.a.g.1.1		1	8.3	odd	2
196.4.a.d.1.1		1	56.13	odd	2
196.4.e.a.165.1		2	56.5	odd	6
196.4.e.a.177.1		2	56.45	odd	6
196.4.e.f.165.1		2	56.37	even	6
196.4.e.f.177.1		2	56.53	even	6
252.4.a.d.1.1		1	24.5	odd	2
448.4.a.a.1.1		1	4.3	odd	2
448.4.a.p.1.1		1	1.1	even	1	trivial
700.4.a.n.1.1		1	40.29	even	2
700.4.e.a.449.1		2	40.13	odd	4
700.4.e.a.449.2		2	40.37	odd	4
784.4.a.a.1.1		1	56.27	even	2
1008.4.a.o.1.1		1	24.11	even	2
1764.4.a.c.1.1		1	168.125	even	2
1764.4.k.d.361.1		2	168.149	odd	6
1764.4.k.d.1549.1		2	168.53	odd	6
1764.4.k.m.361.1		2	168.5	even	6
1764.4.k.m.1549.1		2	168.101	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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