LMFDB - Embedded newform 1400.4.g.e.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,4,Mod(449,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("1400.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1400.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$82.6026740080$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1400.449
Dual form			1400.4.g.e.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{3} -7.00000i q^{7} +11.0000 q^{9} +20.0000 q^{11} -10.0000i q^{13} +14.0000i q^{17} -12.0000 q^{19} -28.0000 q^{21} +104.000i q^{23} -152.000i q^{27} +122.000 q^{29} +224.000 q^{31} -80.0000i q^{33} -158.000i q^{37} -40.0000 q^{39} +378.000 q^{41} +404.000i q^{43} -112.000i q^{47} -49.0000 q^{49} +56.0000 q^{51} +270.000i q^{53} +48.0000i q^{57} -324.000 q^{59} -186.000 q^{61} -77.0000i q^{63} -156.000i q^{67} +416.000 q^{69} -360.000 q^{71} -102.000i q^{73} -140.000i q^{77} +912.000 q^{79} -311.000 q^{81} +1068.00i q^{83} -488.000i q^{87} +1590.00 q^{89} -70.0000 q^{91} -896.000i q^{93} -866.000i q^{97} +220.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 22 q^{9} + 40 q^{11} - 24 q^{19} - 56 q^{21} + 244 q^{29} + 448 q^{31} - 80 q^{39} + 756 q^{41} - 98 q^{49} + 112 q^{51} - 648 q^{59} - 372 q^{61} + 832 q^{69} - 720 q^{71} + 1824 q^{79} - 622 q^{81}+ \cdots + 440 q^{99}+O(q^{100}) $ 2 * q + 22 * q^9 + 40 * q^11 - 24 * q^19 - 56 * q^21 + 244 * q^29 + 448 * q^31 - 80 * q^39 + 756 * q^41 - 98 * q^49 + 112 * q^51 - 648 * q^59 - 372 * q^61 + 832 * q^69 - 720 * q^71 + 1824 * q^79 - 622 * q^81 + 3180 * q^89 - 140 * q^91 + 440 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 4.00000i	− 0.769800i	−0.922958	−	0.384900i	$-0.874236\pi$
$3$	− 4.00000i	− 0.769800i	0.922958	−	0.384900i	$-0.125764\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	0	0
$9$	11.0000	0.407407
$10$	0	0
$11$	20.0000	0.548202	0.274101	−	0.961701i	$-0.411620\pi$
$11$	20.0000	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$12$	0	0
$13$	− 10.0000i	− 0.213346i	−0.994294	−	0.106673i	$-0.965980\pi$
$13$	− 10.0000i	− 0.213346i	0.994294	−	0.106673i	$-0.0340198\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	14.0000i	0.199735i	0.995001	+	0.0998676i	$0.0318419\pi$
$17$	14.0000i	0.199735i	−0.995001	+	0.0998676i	$0.968158\pi$
$18$	0	0
$19$	−12.0000	−0.144894	−0.0724471	−	0.997372i	$-0.523081\pi$
$19$	−12.0000	−0.144894	−0.0724471	+	0.997372i	$0.523081\pi$
$20$	0	0
$21$	−28.0000	−0.290957
$22$	0	0
$23$	104.000i	0.942848i	0.881907	+	0.471424i	$0.156260\pi$
$23$	104.000i	0.942848i	−0.881907	+	0.471424i	$0.843740\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 152.000i	− 1.08342i
$28$	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
$31$	224.000	1.29779	0.648897	−	0.760877i	$-0.275231\pi$
$31$	224.000	1.29779	0.648897	+	0.760877i	$0.275231\pi$
$32$	0	0
$33$	− 80.0000i	− 0.422006i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 158.000i	− 0.702028i	−0.936370	−	0.351014i	$-0.885837\pi$
$37$	− 158.000i	− 0.702028i	0.936370	−	0.351014i	$-0.114163\pi$
$38$	0	0
$39$	−40.0000	−0.164234
$40$	0	0
$41$	378.000	1.43985	0.719923	−	0.694054i	$-0.244177\pi$
$41$	378.000	1.43985	0.719923	+	0.694054i	$0.244177\pi$
$42$	0	0
$43$	404.000i	1.43278i	0.697701	+	0.716389i	$0.254206\pi$
$43$	404.000i	1.43278i	−0.697701	+	0.716389i	$0.745794\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 112.000i	− 0.347593i	−0.984782	−	0.173797i	$-0.944396\pi$
$47$	− 112.000i	− 0.347593i	0.984782	−	0.173797i	$-0.0556035\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	56.0000	0.153756
$52$	0	0
$53$	270.000i	0.699761i	0.936794	+	0.349881i	$0.113778\pi$
$53$	270.000i	0.699761i	−0.936794	+	0.349881i	$0.886222\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	48.0000i	0.111540i
$58$	0	0
$59$	−324.000	−0.714936	−0.357468	−	0.933925i	$-0.616360\pi$
$59$	−324.000	−0.714936	−0.357468	+	0.933925i	$0.616360\pi$
$60$	0	0
$61$	−186.000	−0.390408	−0.195204	−	0.980763i	$-0.562537\pi$
$61$	−186.000	−0.390408	−0.195204	+	0.980763i	$0.562537\pi$
$62$	0	0
$63$	− 77.0000i	− 0.153986i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 156.000i	− 0.284454i	−0.989834	−	0.142227i	$-0.954574\pi$
$67$	− 156.000i	− 0.284454i	0.989834	−	0.142227i	$-0.0454263\pi$
$68$	0	0
$69$	416.000	0.725805
$70$	0	0
$71$	−360.000	−0.601748	−0.300874	−	0.953664i	$-0.597278\pi$
$71$	−360.000	−0.601748	−0.300874	+	0.953664i	$0.597278\pi$
$72$	0	0
$73$	− 102.000i	− 0.163537i	−0.996651	−	0.0817685i	$-0.973943\pi$
$73$	− 102.000i	− 0.163537i	0.996651	−	0.0817685i	$-0.0260568\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 140.000i	− 0.207201i
$78$	0	0
$79$	912.000	1.29884	0.649418	−	0.760432i	$-0.275013\pi$
$79$	912.000	1.29884	0.649418	+	0.760432i	$0.275013\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	1068.00i	1.41239i	0.708018	+	0.706194i	$0.249589\pi$
$83$	1068.00i	1.41239i	−0.708018	+	0.706194i	$0.750411\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 488.000i	− 0.601369i
$88$	0	0
$89$	1590.00	1.89370	0.946852	−	0.321669i	$-0.104244\pi$
$89$	1590.00	1.89370	0.946852	+	0.321669i	$0.104244\pi$
$90$	0	0
$91$	−70.0000	−0.0806373
$92$	0	0
$93$	− 896.000i	− 0.999042i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 866.000i	− 0.906484i	−0.891387	−	0.453242i	$-0.850267\pi$
$97$	− 866.000i	− 0.906484i	0.891387	−	0.453242i	$-0.149733\pi$
$98$	0	0
$99$	220.000	0.223342
$100$	0	0
$101$	702.000	0.691600	0.345800	−	0.938308i	$-0.387608\pi$
$101$	702.000	0.691600	0.345800	+	0.938308i	$0.387608\pi$
$102$	0	0
$103$	− 296.000i	− 0.283163i	−0.989927	−	0.141581i	$-0.954781\pi$
$103$	− 296.000i	− 0.283163i	0.989927	−	0.141581i	$-0.0452187\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	44.0000i	0.0397537i	0.999802	+	0.0198768i	$0.00632741\pi$
$107$	44.0000i	0.0397537i	−0.999802	+	0.0198768i	$0.993673\pi$
$108$	0	0
$109$	650.000	0.571181	0.285590	−	0.958352i	$-0.407810\pi$
$109$	650.000	0.571181	0.285590	+	0.958352i	$0.407810\pi$
$110$	0	0
$111$	−632.000	−0.540421
$112$	0	0
$113$	− 942.000i	− 0.784212i	−0.919920	−	0.392106i	$-0.871747\pi$
$113$	− 942.000i	− 0.784212i	0.919920	−	0.392106i	$-0.128253\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 110.000i	− 0.0869188i
$118$	0	0
$119$	98.0000	0.0754928
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	− 1512.00i	− 1.10839i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 736.000i	− 0.514248i	−0.966378	−	0.257124i	$-0.917225\pi$
$127$	− 736.000i	− 0.514248i	0.966378	−	0.257124i	$-0.0827748\pi$
$128$	0	0
$129$	1616.00	1.10295
$130$	0	0
$131$	924.000	0.616261	0.308131	−	0.951344i	$-0.400297\pi$
$131$	924.000	0.616261	0.308131	+	0.951344i	$0.400297\pi$
$132$	0	0
$133$	84.0000i	0.0547648i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 474.000i	− 0.295595i	−0.989018	−	0.147798i	$-0.952782\pi$
$137$	− 474.000i	− 0.295595i	0.989018	−	0.147798i	$-0.0472184\pi$
$138$	0	0
$139$	−1012.00	−0.617530	−0.308765	−	0.951138i	$-0.599916\pi$
$139$	−1012.00	−0.617530	−0.308765	+	0.951138i	$0.599916\pi$
$140$	0	0
$141$	−448.000	−0.267577
$142$	0	0
$143$	− 200.000i	− 0.116957i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	196.000i	0.109971i
$148$	0	0
$149$	−302.000	−0.166046	−0.0830228	−	0.996548i	$-0.526457\pi$
$149$	−302.000	−0.166046	−0.0830228	+	0.996548i	$0.526457\pi$
$150$	0	0
$151$	−2104.00	−1.13391	−0.566957	−	0.823747i	$-0.691880\pi$
$151$	−2104.00	−1.13391	−0.566957	+	0.823747i	$0.691880\pi$
$152$	0	0
$153$	154.000i	0.0813736i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1190.00i	− 0.604919i	−0.953162	−	0.302460i	$-0.902192\pi$
$157$	− 1190.00i	− 0.604919i	0.953162	−	0.302460i	$-0.0978078\pi$
$158$	0	0
$159$	1080.00	0.538677
$160$	0	0
$161$	728.000	0.356363
$162$	0	0
$163$	− 1732.00i	− 0.832274i	−0.909302	−	0.416137i	$-0.863384\pi$
$163$	− 1732.00i	− 0.832274i	0.909302	−	0.416137i	$-0.136616\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2264.00i	− 1.04906i	−0.851391	−	0.524532i	$-0.824240\pi$
$167$	− 2264.00i	− 1.04906i	0.851391	−	0.524532i	$-0.175760\pi$
$168$	0	0
$169$	2097.00	0.954483
$170$	0	0
$171$	−132.000	−0.0590309
$172$	0	0
$173$	− 1066.00i	− 0.468477i	−0.972179	−	0.234238i	$-0.924740\pi$
$173$	− 1066.00i	− 0.468477i	0.972179	−	0.234238i	$-0.0752596\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1296.00i	0.550358i
$178$	0	0
$179$	2356.00	0.983775	0.491887	−	0.870659i	$-0.336307\pi$
$179$	2356.00	0.983775	0.491887	+	0.870659i	$0.336307\pi$
$180$	0	0
$181$	2158.00	0.886204	0.443102	−	0.896471i	$-0.353878\pi$
$181$	2158.00	0.886204	0.443102	+	0.896471i	$0.353878\pi$
$182$	0	0
$183$	744.000i	0.300536i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	280.000i	0.109495i
$188$	0	0
$189$	−1064.00	−0.409495
$190$	0	0
$191$	−2688.00	−1.01831	−0.509154	−	0.860675i	$-0.670042\pi$
$191$	−2688.00	−1.01831	−0.509154	+	0.860675i	$0.670042\pi$
$192$	0	0
$193$	− 4414.00i	− 1.64625i	−0.567859	−	0.823126i	$-0.692228\pi$
$193$	− 4414.00i	− 1.64625i	0.567859	−	0.823126i	$-0.307772\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3774.00i	− 1.36491i	−0.730930	−	0.682453i	$-0.760913\pi$
$197$	− 3774.00i	− 1.36491i	0.730930	−	0.682453i	$-0.239087\pi$
$198$	0	0
$199$	−664.000	−0.236531	−0.118266	−	0.992982i	$-0.537733\pi$
$199$	−664.000	−0.236531	−0.118266	+	0.992982i	$0.537733\pi$
$200$	0	0
$201$	−624.000	−0.218973
$202$	0	0
$203$	− 854.000i	− 0.295266i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1144.00i	0.384123i
$208$	0	0
$209$	−240.000	−0.0794313
$210$	0	0
$211$	1260.00	0.411099	0.205550	−	0.978647i	$-0.434102\pi$
$211$	1260.00	0.411099	0.205550	+	0.978647i	$0.434102\pi$
$212$	0	0
$213$	1440.00i	0.463226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1568.00i	− 0.490520i
$218$	0	0
$219$	−408.000	−0.125891
$220$	0	0
$221$	140.000	0.0426128
$222$	0	0
$223$	− 1152.00i	− 0.345936i	−0.984927	−	0.172968i	$-0.944664\pi$
$223$	− 1152.00i	− 0.345936i	0.984927	−	0.172968i	$-0.0553356\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2588.00i	− 0.756703i	−0.925662	−	0.378352i	$-0.876491\pi$
$227$	− 2588.00i	− 0.756703i	0.925662	−	0.378352i	$-0.123509\pi$
$228$	0	0
$229$	−766.000	−0.221042	−0.110521	−	0.993874i	$-0.535252\pi$
$229$	−766.000	−0.221042	−0.110521	+	0.993874i	$0.535252\pi$
$230$	0	0
$231$	−560.000	−0.159503
$232$	0	0
$233$	6138.00i	1.72581i	0.505366	+	0.862905i	$0.331357\pi$
$233$	6138.00i	1.72581i	−0.505366	+	0.862905i	$0.668643\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3648.00i	− 0.999844i
$238$	0	0
$239$	−3856.00	−1.04361	−0.521807	−	0.853063i	$-0.674742\pi$
$239$	−3856.00	−1.04361	−0.521807	+	0.853063i	$0.674742\pi$
$240$	0	0
$241$	2578.00	0.689060	0.344530	−	0.938775i	$-0.388038\pi$
$241$	2578.00	0.689060	0.344530	+	0.938775i	$0.388038\pi$
$242$	0	0
$243$	− 2860.00i	− 0.755017i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	120.000i	0.0309126i
$248$	0	0
$249$	4272.00	1.08726
$250$	0	0
$251$	−1404.00	−0.353067	−0.176533	−	0.984295i	$-0.556488\pi$
$251$	−1404.00	−0.353067	−0.176533	+	0.984295i	$0.556488\pi$
$252$	0	0
$253$	2080.00i	0.516871i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2946.00i	− 0.715044i	−0.933905	−	0.357522i	$-0.883622\pi$
$257$	− 2946.00i	− 0.715044i	0.933905	−	0.357522i	$-0.116378\pi$
$258$	0	0
$259$	−1106.00	−0.265342
$260$	0	0
$261$	1342.00	0.318267
$262$	0	0
$263$	− 3080.00i	− 0.722133i	−0.932540	−	0.361066i	$-0.882413\pi$
$263$	− 3080.00i	− 0.722133i	0.932540	−	0.361066i	$-0.117587\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6360.00i	− 1.45777i
$268$	0	0
$269$	−2294.00	−0.519954	−0.259977	−	0.965615i	$-0.583715\pi$
$269$	−2294.00	−0.519954	−0.259977	+	0.965615i	$0.583715\pi$
$270$	0	0
$271$	−3728.00	−0.835645	−0.417823	−	0.908529i	$-0.637207\pi$
$271$	−3728.00	−0.835645	−0.417823	+	0.908529i	$0.637207\pi$
$272$	0	0
$273$	280.000i	0.0620746i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2734.00i	− 0.593033i	−0.955028	−	0.296516i	$-0.904175\pi$
$277$	− 2734.00i	− 0.593033i	0.955028	−	0.296516i	$-0.0958250\pi$
$278$	0	0
$279$	2464.00	0.528731
$280$	0	0
$281$	1034.00	0.219513	0.109757	−	0.993958i	$-0.464993\pi$
$281$	1034.00	0.219513	0.109757	+	0.993958i	$0.464993\pi$
$282$	0	0
$283$	− 5180.00i	− 1.08805i	−0.839068	−	0.544027i	$-0.816899\pi$
$283$	− 5180.00i	− 1.08805i	0.839068	−	0.544027i	$-0.183101\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2646.00i	− 0.544211i
$288$	0	0
$289$	4717.00	0.960106
$290$	0	0
$291$	−3464.00	−0.697812
$292$	0	0
$293$	7934.00i	1.58194i	0.611853	+	0.790971i	$0.290424\pi$
$293$	7934.00i	1.58194i	−0.611853	+	0.790971i	$0.709576\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 3040.00i	− 0.593935i
$298$	0	0
$299$	1040.00	0.201153
$300$	0	0
$301$	2828.00	0.541539
$302$	0	0
$303$	− 2808.00i	− 0.532394i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3444.00i	0.640259i	0.947374	+	0.320129i	$0.103726\pi$
$307$	3444.00i	0.640259i	−0.947374	+	0.320129i	$0.896274\pi$
$308$	0	0
$309$	−1184.00	−0.217979
$310$	0	0
$311$	7656.00	1.39592	0.697961	−	0.716135i	$-0.254091\pi$
$311$	7656.00	1.39592	0.697961	+	0.716135i	$0.254091\pi$
$312$	0	0
$313$	− 3798.00i	− 0.685865i	−0.939360	−	0.342932i	$-0.888580\pi$
$313$	− 3798.00i	− 0.685865i	0.939360	−	0.342932i	$-0.111420\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6822.00i	− 1.20871i	−0.796714	−	0.604356i	$-0.793430\pi$
$317$	− 6822.00i	− 1.20871i	0.796714	−	0.604356i	$-0.206570\pi$
$318$	0	0
$319$	2440.00	0.428256
$320$	0	0
$321$	176.000	0.0306024
$322$	0	0
$323$	− 168.000i	− 0.0289405i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2600.00i	− 0.439695i
$328$	0	0
$329$	−784.000	−0.131378
$330$	0	0
$331$	−6572.00	−1.09133	−0.545664	−	0.838004i	$-0.683723\pi$
$331$	−6572.00	−1.09133	−0.545664	+	0.838004i	$0.683723\pi$
$332$	0	0
$333$	− 1738.00i	− 0.286011i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11022.0i	1.78162i	0.454374	+	0.890811i	$0.349863\pi$
$337$	11022.0i	1.78162i	−0.454374	+	0.890811i	$0.650137\pi$
$338$	0	0
$339$	−3768.00	−0.603686
$340$	0	0
$341$	4480.00	0.711453
$342$	0	0
$343$	343.000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10908.0i	1.68753i	0.536715	+	0.843764i	$0.319665\pi$
$347$	10908.0i	1.68753i	−0.536715	+	0.843764i	$0.680335\pi$
$348$	0	0
$349$	12058.0	1.84943	0.924713	−	0.380664i	$-0.124305\pi$
$349$	12058.0	1.84943	0.924713	+	0.380664i	$0.124305\pi$
$350$	0	0
$351$	−1520.00	−0.231144
$352$	0	0
$353$	1570.00i	0.236721i	0.992971	+	0.118361i	$0.0377639\pi$
$353$	1570.00i	0.236721i	−0.992971	+	0.118361i	$0.962236\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 392.000i	− 0.0581144i
$358$	0	0
$359$	−120.000	−0.0176417	−0.00882083	−	0.999961i	$-0.502808\pi$
$359$	−120.000	−0.0176417	−0.00882083	+	0.999961i	$0.502808\pi$
$360$	0	0
$361$	−6715.00	−0.979006
$362$	0	0
$363$	3724.00i	0.538455i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4336.00i	− 0.616723i	−0.951269	−	0.308362i	$-0.900219\pi$
$367$	− 4336.00i	− 0.616723i	0.951269	−	0.308362i	$-0.0997806\pi$
$368$	0	0
$369$	4158.00	0.586604
$370$	0	0
$371$	1890.00	0.264485
$372$	0	0
$373$	7758.00i	1.07693i	0.842649	+	0.538464i	$0.180995\pi$
$373$	7758.00i	1.07693i	−0.842649	+	0.538464i	$0.819005\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1220.00i	− 0.166666i
$378$	0	0
$379$	−10660.0	−1.44477	−0.722384	−	0.691492i	$-0.756954\pi$
$379$	−10660.0	−1.44477	−0.722384	+	0.691492i	$0.756954\pi$
$380$	0	0
$381$	−2944.00	−0.395868
$382$	0	0
$383$	− 14304.0i	− 1.90836i	−0.299240	−	0.954178i	$-0.596733\pi$
$383$	− 14304.0i	− 1.90836i	0.299240	−	0.954178i	$-0.403267\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4444.00i	0.583724i
$388$	0	0
$389$	11970.0	1.56016	0.780081	−	0.625678i	$-0.215178\pi$
$389$	11970.0	1.56016	0.780081	+	0.625678i	$0.215178\pi$
$390$	0	0
$391$	−1456.00	−0.188320
$392$	0	0
$393$	− 3696.00i	− 0.474398i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4106.00i	0.519079i	0.965733	+	0.259539i	$0.0835707\pi$
$397$	4106.00i	0.519079i	−0.965733	+	0.259539i	$0.916429\pi$
$398$	0	0
$399$	336.000	0.0421580
$400$	0	0
$401$	−3790.00	−0.471979	−0.235989	−	0.971756i	$-0.575833\pi$
$401$	−3790.00	−0.471979	−0.235989	+	0.971756i	$0.575833\pi$
$402$	0	0
$403$	− 2240.00i	− 0.276879i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3160.00i	− 0.384854i
$408$	0	0
$409$	13366.0	1.61591	0.807954	−	0.589246i	$-0.200575\pi$
$409$	13366.0	1.61591	0.807954	+	0.589246i	$0.200575\pi$
$410$	0	0
$411$	−1896.00	−0.227549
$412$	0	0
$413$	2268.00i	0.270220i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4048.00i	0.475375i
$418$	0	0
$419$	−6524.00	−0.760664	−0.380332	−	0.924850i	$-0.624190\pi$
$419$	−6524.00	−0.760664	−0.380332	+	0.924850i	$0.624190\pi$
$420$	0	0
$421$	−2978.00	−0.344748	−0.172374	−	0.985032i	$-0.555144\pi$
$421$	−2978.00	−0.344748	−0.172374	+	0.985032i	$0.555144\pi$
$422$	0	0
$423$	− 1232.00i	− 0.141612i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1302.00i	0.147560i
$428$	0	0
$429$	−800.000	−0.0900335
$430$	0	0
$431$	15760.0	1.76133	0.880664	−	0.473741i	$-0.157097\pi$
$431$	15760.0	1.76133	0.880664	+	0.473741i	$0.157097\pi$
$432$	0	0
$433$	10322.0i	1.14560i	0.819696	+	0.572799i	$0.194142\pi$
$433$	10322.0i	1.14560i	−0.819696	+	0.572799i	$0.805858\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1248.00i	− 0.136613i
$438$	0	0
$439$	344.000	0.0373991	0.0186996	−	0.999825i	$-0.494047\pi$
$439$	344.000	0.0373991	0.0186996	+	0.999825i	$0.494047\pi$
$440$	0	0
$441$	−539.000	−0.0582011
$442$	0	0
$443$	− 252.000i	− 0.0270268i	−0.999909	−	0.0135134i	$-0.995698\pi$
$443$	− 252.000i	− 0.0270268i	0.999909	−	0.0135134i	$-0.00430158\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1208.00i	0.127822i
$448$	0	0
$449$	11966.0	1.25771	0.628854	−	0.777524i	$-0.283525\pi$
$449$	11966.0	1.25771	0.628854	+	0.777524i	$0.283525\pi$
$450$	0	0
$451$	7560.00	0.789327
$452$	0	0
$453$	8416.00i	0.872888i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15846.0i	1.62198i	0.585060	+	0.810990i	$0.301071\pi$
$457$	15846.0i	1.62198i	−0.585060	+	0.810990i	$0.698929\pi$
$458$	0	0
$459$	2128.00	0.216398
$460$	0	0
$461$	5430.00	0.548591	0.274295	−	0.961645i	$-0.411555\pi$
$461$	5430.00	0.548591	0.274295	+	0.961645i	$0.411555\pi$
$462$	0	0
$463$	8912.00i	0.894548i	0.894397	+	0.447274i	$0.147605\pi$
$463$	8912.00i	0.894548i	−0.894397	+	0.447274i	$0.852395\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11028.0i	1.09275i	0.837540	+	0.546376i	$0.183993\pi$
$467$	11028.0i	1.09275i	−0.837540	+	0.546376i	$0.816007\pi$
$468$	0	0
$469$	−1092.00	−0.107514
$470$	0	0
$471$	−4760.00	−0.465667
$472$	0	0
$473$	8080.00i	0.785452i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2970.00i	0.285088i
$478$	0	0
$479$	−5856.00	−0.558596	−0.279298	−	0.960204i	$-0.590102\pi$
$479$	−5856.00	−0.558596	−0.279298	+	0.960204i	$0.590102\pi$
$480$	0	0
$481$	−1580.00	−0.149775
$482$	0	0
$483$	− 2912.00i	− 0.274328i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 10264.0i	− 0.955044i	−0.878620	−	0.477522i	$-0.841535\pi$
$487$	− 10264.0i	− 0.955044i	0.878620	−	0.477522i	$-0.158465\pi$
$488$	0	0
$489$	−6928.00	−0.640685
$490$	0	0
$491$	2804.00	0.257725	0.128862	−	0.991663i	$-0.458867\pi$
$491$	2804.00	0.257725	0.128862	+	0.991663i	$0.458867\pi$
$492$	0	0
$493$	1708.00i	0.156033i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2520.00i	0.227440i
$498$	0	0
$499$	−8716.00	−0.781927	−0.390964	−	0.920406i	$-0.627858\pi$
$499$	−8716.00	−0.781927	−0.390964	+	0.920406i	$0.627858\pi$
$500$	0	0
$501$	−9056.00	−0.807569
$502$	0	0
$503$	2504.00i	0.221964i	0.993822	+	0.110982i	$0.0353996\pi$
$503$	2504.00i	0.221964i	−0.993822	+	0.110982i	$0.964600\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 8388.00i	− 0.734762i
$508$	0	0
$509$	−17094.0	−1.48856	−0.744281	−	0.667866i	$-0.767208\pi$
$509$	−17094.0	−1.48856	−0.744281	+	0.667866i	$0.767208\pi$
$510$	0	0
$511$	−714.000	−0.0618112
$512$	0	0
$513$	1824.00i	0.156982i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2240.00i	− 0.190551i
$518$	0	0
$519$	−4264.00	−0.360634
$520$	0	0
$521$	9882.00	0.830976	0.415488	−	0.909599i	$-0.363611\pi$
$521$	9882.00	0.830976	0.415488	+	0.909599i	$0.363611\pi$
$522$	0	0
$523$	− 7532.00i	− 0.629735i	−0.949136	−	0.314867i	$-0.898040\pi$
$523$	− 7532.00i	− 0.629735i	0.949136	−	0.314867i	$-0.101960\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3136.00i	0.259215i
$528$	0	0
$529$	1351.00	0.111038
$530$	0	0
$531$	−3564.00	−0.291270
$532$	0	0
$533$	− 3780.00i	− 0.307186i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9424.00i	− 0.757310i
$538$	0	0
$539$	−980.000	−0.0783146
$540$	0	0
$541$	−17338.0	−1.37785	−0.688927	−	0.724831i	$-0.741918\pi$
$541$	−17338.0	−1.37785	−0.688927	+	0.724831i	$0.741918\pi$
$542$	0	0
$543$	− 8632.00i	− 0.682200i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 764.000i	− 0.0597190i	−0.999554	−	0.0298595i	$-0.990494\pi$
$547$	− 764.000i	− 0.0597190i	0.999554	−	0.0298595i	$-0.00950598\pi$
$548$	0	0
$549$	−2046.00	−0.159055
$550$	0	0
$551$	−1464.00	−0.113191
$552$	0	0
$553$	− 6384.00i	− 0.490914i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2442.00i	0.185765i	0.995677	+	0.0928823i	$0.0296080\pi$
$557$	2442.00i	0.185765i	−0.995677	+	0.0928823i	$0.970392\pi$
$558$	0	0
$559$	4040.00	0.305678
$560$	0	0
$561$	1120.00	0.0842895
$562$	0	0
$563$	8972.00i	0.671625i	0.941929	+	0.335812i	$0.109011\pi$
$563$	8972.00i	0.671625i	−0.941929	+	0.335812i	$0.890989\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2177.00i	0.161244i
$568$	0	0
$569$	−8682.00	−0.639663	−0.319832	−	0.947474i	$-0.603626\pi$
$569$	−8682.00	−0.639663	−0.319832	+	0.947474i	$0.603626\pi$
$570$	0	0
$571$	15524.0	1.13776	0.568878	−	0.822422i	$-0.307377\pi$
$571$	15524.0	1.13776	0.568878	+	0.822422i	$0.307377\pi$
$572$	0	0
$573$	10752.0i	0.783894i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	19774.0i	1.42669i	0.700811	+	0.713347i	$0.252822\pi$
$577$	19774.0i	1.42669i	−0.700811	+	0.713347i	$0.747178\pi$
$578$	0	0
$579$	−17656.0	−1.26729
$580$	0	0
$581$	7476.00	0.533833
$582$	0	0
$583$	5400.00i	0.383611i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6700.00i	0.471105i	0.971862	+	0.235552i	$0.0756899\pi$
$587$	6700.00i	0.471105i	−0.971862	+	0.235552i	$0.924310\pi$
$588$	0	0
$589$	−2688.00	−0.188043
$590$	0	0
$591$	−15096.0	−1.05070
$592$	0	0
$593$	− 1294.00i	− 0.0896091i	−0.998996	−	0.0448046i	$-0.985733\pi$
$593$	− 1294.00i	− 0.0896091i	0.998996	−	0.0448046i	$-0.0142665\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2656.00i	0.182082i
$598$	0	0
$599$	−19720.0	−1.34514	−0.672569	−	0.740035i	$-0.734809\pi$
$599$	−19720.0	−1.34514	−0.672569	+	0.740035i	$0.734809\pi$
$600$	0	0
$601$	21450.0	1.45585	0.727923	−	0.685659i	$-0.240486\pi$
$601$	21450.0	1.45585	0.727923	+	0.685659i	$0.240486\pi$
$602$	0	0
$603$	− 1716.00i	− 0.115889i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	19264.0i	1.28814i	0.764966	+	0.644071i	$0.222756\pi$
$607$	19264.0i	1.28814i	−0.764966	+	0.644071i	$0.777244\pi$
$608$	0	0
$609$	−3416.00	−0.227296
$610$	0	0
$611$	−1120.00	−0.0741577
$612$	0	0
$613$	13150.0i	0.866433i	0.901290	+	0.433217i	$0.142621\pi$
$613$	13150.0i	0.866433i	−0.901290	+	0.433217i	$0.857379\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15238.0i	0.994261i	0.867676	+	0.497130i	$0.165613\pi$
$617$	15238.0i	0.994261i	−0.867676	+	0.497130i	$0.834387\pi$
$618$	0	0
$619$	−23764.0	−1.54306	−0.771531	−	0.636191i	$-0.780509\pi$
$619$	−23764.0	−1.54306	−0.771531	+	0.636191i	$0.780509\pi$
$620$	0	0
$621$	15808.0	1.02150
$622$	0	0
$623$	− 11130.0i	− 0.715753i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	960.000i	0.0611463i
$628$	0	0
$629$	2212.00	0.140220
$630$	0	0
$631$	−2008.00	−0.126683	−0.0633417	−	0.997992i	$-0.520176\pi$
$631$	−2008.00	−0.126683	−0.0633417	+	0.997992i	$0.520176\pi$
$632$	0	0
$633$	− 5040.00i	− 0.316464i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	490.000i	0.0304780i
$638$	0	0
$639$	−3960.00	−0.245157
$640$	0	0
$641$	13058.0	0.804618	0.402309	−	0.915504i	$-0.368208\pi$
$641$	13058.0	0.804618	0.402309	+	0.915504i	$0.368208\pi$
$642$	0	0
$643$	24188.0i	1.48349i	0.670684	+	0.741743i	$0.266001\pi$
$643$	24188.0i	1.48349i	−0.670684	+	0.741743i	$0.733999\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 6648.00i	− 0.403956i	−0.979390	−	0.201978i	$-0.935263\pi$
$647$	− 6648.00i	− 0.403956i	0.979390	−	0.201978i	$-0.0647370\pi$
$648$	0	0
$649$	−6480.00	−0.391930
$650$	0	0
$651$	−6272.00	−0.377602
$652$	0	0
$653$	9814.00i	0.588134i	0.955785	+	0.294067i	$0.0950089\pi$
$653$	9814.00i	0.588134i	−0.955785	+	0.294067i	$0.904991\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1122.00i	− 0.0666262i
$658$	0	0
$659$	−26540.0	−1.56882	−0.784409	−	0.620243i	$-0.787034\pi$
$659$	−26540.0	−1.56882	−0.784409	+	0.620243i	$0.787034\pi$
$660$	0	0
$661$	−1330.00	−0.0782617	−0.0391309	−	0.999234i	$-0.512459\pi$
$661$	−1330.00	−0.0782617	−0.0391309	+	0.999234i	$0.512459\pi$
$662$	0	0
$663$	− 560.000i	− 0.0328033i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12688.0i	0.736554i
$668$	0	0
$669$	−4608.00	−0.266301
$670$	0	0
$671$	−3720.00	−0.214022
$672$	0	0
$673$	− 17950.0i	− 1.02812i	−0.857756	−	0.514058i	$-0.828142\pi$
$673$	− 17950.0i	− 1.02812i	0.857756	−	0.514058i	$-0.171858\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 23742.0i	− 1.34783i	−0.738810	−	0.673914i	$-0.764612\pi$
$677$	− 23742.0i	− 1.34783i	0.738810	−	0.673914i	$-0.235388\pi$
$678$	0	0
$679$	−6062.00	−0.342619
$680$	0	0
$681$	−10352.0	−0.582510
$682$	0	0
$683$	6356.00i	0.356084i	0.984023	+	0.178042i	$0.0569763\pi$
$683$	6356.00i	0.356084i	−0.984023	+	0.178042i	$0.943024\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3064.00i	0.170159i
$688$	0	0
$689$	2700.00	0.149291
$690$	0	0
$691$	−3156.00	−0.173748	−0.0868740	−	0.996219i	$-0.527688\pi$
$691$	−3156.00	−0.173748	−0.0868740	+	0.996219i	$0.527688\pi$
$692$	0	0
$693$	− 1540.00i	− 0.0844152i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5292.00i	0.287588i
$698$	0	0
$699$	24552.0	1.32853
$700$	0	0
$701$	−18330.0	−0.987610	−0.493805	−	0.869573i	$-0.664394\pi$
$701$	−18330.0	−0.987610	−0.493805	+	0.869573i	$0.664394\pi$
$702$	0	0
$703$	1896.00i	0.101720i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4914.00i	− 0.261400i
$708$	0	0
$709$	−19070.0	−1.01014	−0.505070	−	0.863079i	$-0.668533\pi$
$709$	−19070.0	−1.01014	−0.505070	+	0.863079i	$0.668533\pi$
$710$	0	0
$711$	10032.0	0.529155
$712$	0	0
$713$	23296.0i	1.22362i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15424.0i	0.803375i
$718$	0	0
$719$	4176.00	0.216604	0.108302	−	0.994118i	$-0.465459\pi$
$719$	4176.00	0.216604	0.108302	+	0.994118i	$0.465459\pi$
$720$	0	0
$721$	−2072.00	−0.107025
$722$	0	0
$723$	− 10312.0i	− 0.530439i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10520.0i	0.536678i	0.963324	+	0.268339i	$0.0864748\pi$
$727$	10520.0i	0.536678i	−0.963324	+	0.268339i	$0.913525\pi$
$728$	0	0
$729$	−19837.0	−1.00782
$730$	0	0
$731$	−5656.00	−0.286176
$732$	0	0
$733$	− 6874.00i	− 0.346381i	−0.984888	−	0.173190i	$-0.944592\pi$
$733$	− 6874.00i	− 0.346381i	0.984888	−	0.173190i	$-0.0554076\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3120.00i	− 0.155939i
$738$	0	0
$739$	−11804.0	−0.587574	−0.293787	−	0.955871i	$-0.594916\pi$
$739$	−11804.0	−0.587574	−0.293787	+	0.955871i	$0.594916\pi$
$740$	0	0
$741$	480.000	0.0237965
$742$	0	0
$743$	− 20648.0i	− 1.01952i	−0.860317	−	0.509759i	$-0.829735\pi$
$743$	− 20648.0i	− 1.01952i	0.860317	−	0.509759i	$-0.170265\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11748.0i	0.575417i
$748$	0	0
$749$	308.000	0.0150255
$750$	0	0
$751$	1232.00	0.0598619	0.0299310	−	0.999552i	$-0.490471\pi$
$751$	1232.00	0.0598619	0.0299310	+	0.999552i	$0.490471\pi$
$752$	0	0
$753$	5616.00i	0.271791i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	11250.0i	0.540143i	0.962840	+	0.270071i	$0.0870473\pi$
$757$	11250.0i	0.540143i	−0.962840	+	0.270071i	$0.912953\pi$
$758$	0	0
$759$	8320.00	0.397888
$760$	0	0
$761$	14698.0	0.700134	0.350067	−	0.936725i	$-0.386159\pi$
$761$	14698.0	0.700134	0.350067	+	0.936725i	$0.386159\pi$
$762$	0	0
$763$	− 4550.00i	− 0.215886i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3240.00i	0.152529i
$768$	0	0
$769$	30014.0	1.40745	0.703727	−	0.710470i	$-0.251518\pi$
$769$	30014.0	1.40745	0.703727	+	0.710470i	$0.251518\pi$
$770$	0	0
$771$	−11784.0	−0.550441
$772$	0	0
$773$	− 34018.0i	− 1.58285i	−0.611267	−	0.791425i	$-0.709340\pi$
$773$	− 34018.0i	− 1.58285i	0.611267	−	0.791425i	$-0.290660\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4424.00i	0.204260i
$778$	0	0
$779$	−4536.00	−0.208625
$780$	0	0
$781$	−7200.00	−0.329880
$782$	0	0
$783$	− 18544.0i	− 0.846371i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18516.0i	0.838658i	0.907834	+	0.419329i	$0.137735\pi$
$787$	18516.0i	0.838658i	−0.907834	+	0.419329i	$0.862265\pi$
$788$	0	0
$789$	−12320.0	−0.555898
$790$	0	0
$791$	−6594.00	−0.296404
$792$	0	0
$793$	1860.00i	0.0832920i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38618.0i	1.71634i	0.513369	+	0.858168i	$0.328397\pi$
$797$	38618.0i	1.71634i	−0.513369	+	0.858168i	$0.671603\pi$
$798$	0	0
$799$	1568.00	0.0694266
$800$	0	0
$801$	17490.0	0.771509
$802$	0	0
$803$	− 2040.00i	− 0.0896514i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	9176.00i	0.400261i
$808$	0	0
$809$	−19514.0	−0.848054	−0.424027	−	0.905650i	$-0.639384\pi$
$809$	−19514.0	−0.848054	−0.424027	+	0.905650i	$0.639384\pi$
$810$	0	0
$811$	−31020.0	−1.34311	−0.671553	−	0.740956i	$-0.734373\pi$
$811$	−31020.0	−1.34311	−0.671553	+	0.740956i	$0.734373\pi$
$812$	0	0
$813$	14912.0i	0.643280i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4848.00i	− 0.207601i
$818$	0	0
$819$	−770.000	−0.0328522
$820$	0	0
$821$	3726.00	0.158390	0.0791951	−	0.996859i	$-0.474765\pi$
$821$	3726.00	0.158390	0.0791951	+	0.996859i	$0.474765\pi$
$822$	0	0
$823$	32904.0i	1.39363i	0.717249	+	0.696817i	$0.245401\pi$
$823$	32904.0i	1.39363i	−0.717249	+	0.696817i	$0.754599\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39868.0i	1.67636i	0.545397	+	0.838178i	$0.316379\pi$
$827$	39868.0i	1.67636i	−0.545397	+	0.838178i	$0.683621\pi$
$828$	0	0
$829$	26810.0	1.12322	0.561610	−	0.827402i	$-0.310182\pi$
$829$	26810.0	1.12322	0.561610	+	0.827402i	$0.310182\pi$
$830$	0	0
$831$	−10936.0	−0.456517
$832$	0	0
$833$	− 686.000i	− 0.0285336i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 34048.0i	− 1.40606i
$838$	0	0
$839$	37096.0	1.52646	0.763228	−	0.646130i	$-0.223613\pi$
$839$	37096.0	1.52646	0.763228	+	0.646130i	$0.223613\pi$
$840$	0	0
$841$	−9505.00	−0.389725
$842$	0	0
$843$	− 4136.00i	− 0.168982i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6517.00i	0.264376i
$848$	0	0
$849$	−20720.0	−0.837584
$850$	0	0
$851$	16432.0	0.661906
$852$	0	0
$853$	− 14386.0i	− 0.577453i	−0.957412	−	0.288726i	$-0.906768\pi$
$853$	− 14386.0i	− 0.577453i	0.957412	−	0.288726i	$-0.0932318\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	118.000i	0.00470339i	0.999997	+	0.00235169i	$0.000748568\pi$
$857$	118.000i	0.00470339i	−0.999997	+	0.00235169i	$0.999251\pi$
$858$	0	0
$859$	25116.0	0.997610	0.498805	−	0.866714i	$-0.333772\pi$
$859$	25116.0	0.997610	0.498805	+	0.866714i	$0.333772\pi$
$860$	0	0
$861$	−10584.0	−0.418934
$862$	0	0
$863$	− 11776.0i	− 0.464496i	−0.972657	−	0.232248i	$-0.925392\pi$
$863$	− 11776.0i	− 0.464496i	0.972657	−	0.232248i	$-0.0746080\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 18868.0i	− 0.739090i
$868$	0	0
$869$	18240.0	0.712025
$870$	0	0
$871$	−1560.00	−0.0606872
$872$	0	0
$873$	− 9526.00i	− 0.369308i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 24694.0i	− 0.950806i	−0.879768	−	0.475403i	$-0.842302\pi$
$877$	− 24694.0i	− 0.950806i	0.879768	−	0.475403i	$-0.157698\pi$
$878$	0	0
$879$	31736.0	1.21778
$880$	0	0
$881$	32850.0	1.25624	0.628118	−	0.778118i	$-0.283825\pi$
$881$	32850.0	1.25624	0.628118	+	0.778118i	$0.283825\pi$
$882$	0	0
$883$	47340.0i	1.80421i	0.431516	+	0.902105i	$0.357979\pi$
$883$	47340.0i	1.80421i	−0.431516	+	0.902105i	$0.642021\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1992.00i	− 0.0754057i	−0.999289	−	0.0377028i	$-0.987996\pi$
$887$	− 1992.00i	− 0.0754057i	0.999289	−	0.0377028i	$-0.0120040\pi$
$888$	0	0
$889$	−5152.00	−0.194367
$890$	0	0
$891$	−6220.00	−0.233870
$892$	0	0
$893$	1344.00i	0.0503642i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4160.00i	− 0.154848i
$898$	0	0
$899$	27328.0	1.01384
$900$	0	0
$901$	−3780.00	−0.139767
$902$	0	0
$903$	− 11312.0i	− 0.416877i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	13516.0i	0.494809i	0.968912	+	0.247404i	$0.0795776\pi$
$907$	13516.0i	0.494809i	−0.968912	+	0.247404i	$0.920422\pi$
$908$	0	0
$909$	7722.00	0.281763
$910$	0	0
$911$	−3408.00	−0.123943	−0.0619715	−	0.998078i	$-0.519739\pi$
$911$	−3408.00	−0.123943	−0.0619715	+	0.998078i	$0.519739\pi$
$912$	0	0
$913$	21360.0i	0.774275i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6468.00i	− 0.232925i
$918$	0	0
$919$	−39240.0	−1.40850	−0.704248	−	0.709954i	$-0.748716\pi$
$919$	−39240.0	−1.40850	−0.704248	+	0.709954i	$0.748716\pi$
$920$	0	0
$921$	13776.0	0.492871
$922$	0	0
$923$	3600.00i	0.128381i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 3256.00i	− 0.115363i
$928$	0	0
$929$	−21922.0	−0.774206	−0.387103	−	0.922036i	$-0.626524\pi$
$929$	−21922.0	−0.774206	−0.387103	+	0.922036i	$0.626524\pi$
$930$	0	0
$931$	588.000	0.0206992
$932$	0	0
$933$	− 30624.0i	− 1.07458i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 186.000i	− 0.00648490i	−0.999995	−	0.00324245i	$-0.998968\pi$
$937$	− 186.000i	− 0.00648490i	0.999995	−	0.00324245i	$-0.00103211\pi$
$938$	0	0
$939$	−15192.0	−0.527979
$940$	0	0
$941$	−18282.0	−0.633343	−0.316672	−	0.948535i	$-0.602565\pi$
$941$	−18282.0	−0.633343	−0.316672	+	0.948535i	$0.602565\pi$
$942$	0	0
$943$	39312.0i	1.35756i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16980.0i	0.582657i	0.956623	+	0.291328i	$0.0940972\pi$
$947$	16980.0i	0.582657i	−0.956623	+	0.291328i	$0.905903\pi$
$948$	0	0
$949$	−1020.00	−0.0348900
$950$	0	0
$951$	−27288.0	−0.930467
$952$	0	0
$953$	− 44310.0i	− 1.50613i	−0.657946	−	0.753065i	$-0.728575\pi$
$953$	− 44310.0i	− 1.50613i	0.657946	−	0.753065i	$-0.271425\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 9760.00i	− 0.329672i
$958$	0	0
$959$	−3318.00	−0.111725
$960$	0	0
$961$	20385.0	0.684267
$962$	0	0
$963$	484.000i	0.0161959i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 51512.0i	− 1.71304i	−0.516110	−	0.856522i	$-0.672620\pi$
$967$	− 51512.0i	− 1.71304i	0.516110	−	0.856522i	$-0.327380\pi$
$968$	0	0
$969$	−672.000	−0.0222784
$970$	0	0
$971$	1844.00	0.0609442	0.0304721	−	0.999536i	$-0.490299\pi$
$971$	1844.00	0.0609442	0.0304721	+	0.999536i	$0.490299\pi$
$972$	0	0
$973$	7084.00i	0.233405i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 34162.0i	− 1.11867i	−0.828942	−	0.559334i	$-0.811057\pi$
$977$	− 34162.0i	− 1.11867i	0.828942	−	0.559334i	$-0.188943\pi$
$978$	0	0
$979$	31800.0	1.03813
$980$	0	0
$981$	7150.00	0.232703
$982$	0	0
$983$	− 216.000i	− 0.00700847i	−0.999994	−	0.00350424i	$-0.998885\pi$
$983$	− 216.000i	− 0.00700847i	0.999994	−	0.00350424i	$-0.00111544\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3136.00i	0.101135i
$988$	0	0
$989$	−42016.0	−1.35089
$990$	0	0
$991$	52832.0	1.69351	0.846753	−	0.531987i	$-0.178554\pi$
$991$	52832.0	1.69351	0.846753	+	0.531987i	$0.178554\pi$
$992$	0	0
$993$	26288.0i	0.840105i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 5054.00i	− 0.160543i	−0.996773	−	0.0802717i	$-0.974421\pi$
$997$	− 5054.00i	− 0.160543i	0.996773	−	0.0802717i	$-0.0255788\pi$
$998$	0	0
$999$	−24016.0	−0.760593

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.4.g.e.449.1		2
5.2	odd	4		280.4.a.a.1.1	✓	1
5.3	odd	4		1400.4.a.g.1.1		1
5.4	even	2	inner	1400.4.g.e.449.2		2
20.7	even	4		560.4.a.l.1.1		1
35.27	even	4		1960.4.a.g.1.1		1
40.27	even	4		2240.4.a.l.1.1		1
40.37	odd	4		2240.4.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.4.a.a.1.1	✓	1	5.2	odd	4
560.4.a.l.1.1		1	20.7	even	4
1400.4.a.g.1.1		1	5.3	odd	4
1400.4.g.e.449.1		2	1.1	even	1	trivial
1400.4.g.e.449.2		2	5.4	even	2	inner
1960.4.a.g.1.1		1	35.27	even	4
2240.4.a.l.1.1		1	40.27	even	4
2240.4.a.z.1.1		1	40.37	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1400.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{3} -7.00000i q^{7} +11.0000 q^{9} +20.0000 q^{11} -10.0000i q^{13} +14.0000i q^{17} -12.0000 q^{19} -28.0000 q^{21} +104.000i q^{23} -152.000i q^{27} +122.000 q^{29} +224.000 q^{31} -80.0000i q^{33} -158.000i q^{37} -40.0000 q^{39} +378.000 q^{41} +404.000i q^{43} -112.000i q^{47} -49.0000 q^{49} +56.0000 q^{51} +270.000i q^{53} +48.0000i q^{57} -324.000 q^{59} -186.000 q^{61} -77.0000i q^{63} -156.000i q^{67} +416.000 q^{69} -360.000 q^{71} -102.000i q^{73} -140.000i q^{77} +912.000 q^{79} -311.000 q^{81} +1068.00i q^{83} -488.000i q^{87} +1590.00 q^{89} -70.0000 q^{91} -896.000i q^{93} -866.000i q^{97} +220.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 22 q^{9} + 40 q^{11} - 24 q^{19} - 56 q^{21} + 244 q^{29} + 448 q^{31} - 80 q^{39} + 756 q^{41} - 98 q^{49} + 112 q^{51} - 648 q^{59} - 372 q^{61} + 832 q^{69} - 720 q^{71} + 1824 q^{79} - 622 q^{81}+ \cdots + 440 q^{99}+O(q^{100}) \) 2 * q + 22 * q^9 + 40 * q^11 - 24 * q^19 - 56 * q^21 + 244 * q^29 + 448 * q^31 - 80 * q^39 + 756 * q^41 - 98 * q^49 + 112 * q^51 - 648 * q^59 - 372 * q^61 + 832 * q^69 - 720 * q^71 + 1824 * q^79 - 622 * q^81 + 3180 * q^89 - 140 * q^91 + 440 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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