LMFDB - Embedded newform 1400.2.g.b.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,2,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, 2, names="a")

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

chi := DirichletCharacter("1400.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
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:	$11.1790562830$
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 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1400.449
Dual form			1400.2.g.b.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +2.00000i q^{17} +2.00000 q^{19} +2.00000 q^{21} +8.00000i q^{23} +4.00000i q^{27} -2.00000 q^{29} +4.00000 q^{31} +6.00000i q^{37} -2.00000 q^{41} +8.00000i q^{43} +4.00000i q^{47} -1.00000 q^{49} -4.00000 q^{51} -10.0000i q^{53} +4.00000i q^{57} -6.00000 q^{59} +4.00000 q^{61} +1.00000i q^{63} +12.0000i q^{67} -16.0000 q^{69} -14.0000i q^{73} +8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} -4.00000i q^{87} -10.0000 q^{89} +8.00000i q^{93} +2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 4 q^{19} + 4 q^{21} - 4 q^{29} + 8 q^{31} - 4 q^{41} - 2 q^{49} - 8 q^{51} - 12 q^{59} + 8 q^{61} - 32 q^{69} + 16 q^{79} - 22 q^{81} - 20 q^{89}+O(q^{100}) $ 2 * q - 2 * q^9 + 4 * q^19 + 4 * q^21 - 4 * q^29 + 8 * q^31 - 4 * q^41 - 2 * q^49 - 8 * q^51 - 12 * q^59 + 8 * q^61 - 32 * q^69 + 16 * q^79 - 22 * q^81 - 20 * q^89

Display 100 coefficients 10 coefficients

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
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000i	0.769800i
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.00000i	0.529813i
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	1.00000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	0	0
$69$	−16.0000	−1.92617
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.00000i	− 0.428845i
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	− 12.0000i	− 1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$	− 12.0000i	− 1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	0	0
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	− 4.00000i	− 0.360668i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	0	0
$129$	−16.0000	−1.40872
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	0	0
$133$	− 2.00000i	− 0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	0	0
$139$	−18.0000	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.00000i	− 0.164957i
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	20.0000	1.58610
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	8.00000i	0.608229i	0.952636	+	0.304114i	$0.0983605\pi$
$173$	8.00000i	0.608229i	−0.952636	+	0.304114i	$0.901639\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 12.0000i	− 0.901975i
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	0	0
$183$	8.00000i	0.591377i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	− 18.0000i	− 1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	− 18.0000i	− 1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	0	0
$203$	2.00000i	0.140372i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 8.00000i	− 0.556038i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	0	0
$219$	28.0000	1.89206
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 24.0000i	− 1.60716i	−0.595198	−	0.803579i	$-0.702926\pi$
$223$	− 24.0000i	− 1.60716i	0.595198	−	0.803579i	$-0.297074\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 14.0000i	− 0.929213i	−0.885517	−	0.464606i	$-0.846196\pi$
$227$	− 14.0000i	− 0.929213i	0.885517	−	0.464606i	$-0.153804\pi$
$228$	0	0
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.0000i	1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$	26.0000i	1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	− 10.0000i	− 0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−14.0000	−0.883672	−0.441836	−	0.897096i	$-0.645673\pi$
$251$	−14.0000	−0.883672	−0.441836	+	0.897096i	$0.645673\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 20.0000i	− 1.22398i
$268$	0	0
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 10.0000i	− 0.594438i	−0.954809	−	0.297219i	$-0.903941\pi$
$283$	− 10.0000i	− 0.594438i	0.954809	−	0.297219i	$-0.0960592\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.00000i	0.118056i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	−4.00000	−0.234484
$292$	0	0
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	24.0000i	1.37876i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	0	0
$309$	24.0000	1.36531
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	14.0000i	0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$	14.0000i	0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−24.0000	−1.33955
$322$	0	0
$323$	4.00000i	0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 20.0000i	− 1.10600i
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	− 6.00000i	− 0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	0	0
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.00000i	0.211702i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	− 22.0000i	− 1.15470i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.00000i	− 0.406663i
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	28.0000i	1.41241i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	0	0
$413$	6.00000i	0.295241i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 36.0000i	− 1.76293i
$418$	0	0
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	− 4.00000i	− 0.194487i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.0000i	0.765384i
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.00000i	0.189194i
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	32.0000i	1.50349i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	0	0
$459$	−8.00000	−0.373408
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.0000i	0.457869i
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	16.0000i	0.728025i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	0	0
$489$	−32.0000	−1.44709
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	− 4.00000i	− 0.180151i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	0	0
$503$	− 32.0000i	− 1.42681i	−0.700752	−	0.713405i	$-0.747152\pi$
$503$	− 32.0000i	− 1.42681i	0.700752	−	0.713405i	$-0.252848\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	26.0000i	1.15470i
$508$	0	0
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	0	0
$513$	8.00000i	0.353209i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−16.0000	−0.702322
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	− 34.0000i	− 1.48672i	−0.668894	−	0.743358i	$-0.733232\pi$
$523$	− 34.0000i	− 1.48672i	0.668894	−	0.743358i	$-0.266768\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.00000i	0.348485i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.00000i	0.345225i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	16.0000i	0.686626i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	0	0
$549$	−4.00000	−0.170716
$550$	0	0
$551$	−4.00000	−0.170406
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.0000i	1.10166i	0.834619	+	0.550828i	$0.185688\pi$
$557$	26.0000i	1.10166i	−0.834619	+	0.550828i	$0.814312\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.0000i	1.43293i	0.697623	+	0.716465i	$0.254241\pi$
$563$	34.0000i	1.43293i	−0.697623	+	0.716465i	$0.745759\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	11.0000i	0.461957i
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	− 16.0000i	− 0.668410i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 18.0000i	− 0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$	− 18.0000i	− 0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$	0	0
$579$	36.0000	1.49611
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 10.0000i	− 0.412744i	−0.978474	−	0.206372i	$-0.933834\pi$
$587$	− 10.0000i	− 0.412744i	0.978474	−	0.206372i	$-0.0661657\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	−36.0000	−1.48084
$592$	0	0
$593$	42.0000i	1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$	42.0000i	1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.00000i	0.327418i
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	0	0
$609$	−4.00000	−0.162088
$610$	0	0
$611$	0	0
$612$	0	0
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	−6.00000	−0.241160	−0.120580	−	0.992704i	$-0.538475\pi$
$619$	−6.00000	−0.241160	−0.120580	+	0.992704i	$0.538475\pi$
$620$	0	0
$621$	−32.0000	−1.28412
$622$	0	0
$623$	10.0000i	0.400642i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	0	0
$633$	40.0000i	1.58986i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.0000i	1.41531i	0.706560	+	0.707653i	$0.250246\pi$
$647$	36.0000i	1.41531i	−0.706560	+	0.707653i	$0.749754\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	− 22.0000i	− 0.860927i	−0.902608	−	0.430463i	$-0.858350\pi$
$653$	− 22.0000i	− 0.860927i	0.902608	−	0.430463i	$-0.141650\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.0000i	0.546192i
$658$	0	0
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16.0000i	− 0.619522i
$668$	0	0
$669$	48.0000	1.85579
$670$	0	0
$671$	0	0
$672$	0	0
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.0000i	0.922395i	0.887298	+	0.461197i	$0.152580\pi$
$677$	24.0000i	0.922395i	−0.887298	+	0.461197i	$0.847420\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	28.0000	1.07296
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	32.0000i	1.22088i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.00000i	− 0.151511i
$698$	0	0
$699$	−52.0000	−1.96682
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	12.0000i	0.452589i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 12.0000i	− 0.451306i
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	32.0000i	1.19841i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	32.0000i	1.19506i
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	0	0
$723$	− 4.00000i	− 0.148762i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 20.0000i	− 0.741759i	−0.928681	−	0.370879i	$-0.879056\pi$
$727$	− 20.0000i	− 0.741759i	0.928681	−	0.370879i	$-0.120944\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	52.0000i	1.92066i	0.278859	+	0.960332i	$0.410044\pi$
$733$	52.0000i	1.92066i	−0.278859	+	0.960332i	$0.589956\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 16.0000i	− 0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$	− 16.0000i	− 0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 6.00000i	− 0.219529i
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	− 28.0000i	− 1.02038i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	10.0000i	0.362024i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	36.0000	1.29651
$772$	0	0
$773$	4.00000i	0.143870i	0.997409	+	0.0719350i	$0.0229174\pi$
$773$	4.00000i	0.143870i	−0.997409	+	0.0719350i	$0.977083\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	12.0000i	0.430498i
$778$	0	0
$779$	−4.00000	−0.143315
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 8.00000i	− 0.285897i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.0000i	0.850124i	0.905164	+	0.425062i	$0.139748\pi$
$797$	24.0000i	0.850124i	−0.905164	+	0.425062i	$0.860252\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	48.0000i	1.68968i
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	30.0000	1.05344	0.526721	−	0.850038i	$-0.323421\pi$
$811$	30.0000	1.05344	0.526721	+	0.850038i	$0.323421\pi$
$812$	0	0
$813$	64.0000i	2.24458i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	8.00000i	0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$	8.00000i	0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	44.0000	1.52634
$832$	0	0
$833$	− 2.00000i	− 0.0692959i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.0000i	0.553041i
$838$	0	0
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	− 12.0000i	− 0.413302i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.0000i	0.377964i
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	−48.0000	−1.64542
$852$	0	0
$853$	− 32.0000i	− 1.09566i	−0.836590	−	0.547830i	$-0.815454\pi$
$853$	− 32.0000i	− 1.09566i	0.836590	−	0.547830i	$-0.184546\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 6.00000i	− 0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$	− 6.00000i	− 0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	8.00000i	0.272323i	0.990687	+	0.136162i	$0.0434766\pi$
$863$	8.00000i	0.272323i	−0.990687	+	0.136162i	$0.956523\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	26.0000i	0.883006i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 2.00000i	− 0.0676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	0	0
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	0	0
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 36.0000i	− 1.20876i	−0.796696	−	0.604381i	$-0.793421\pi$
$887$	− 36.0000i	− 1.20876i	0.796696	−	0.604381i	$-0.206579\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.00000i	0.267710i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	16.0000i	0.532447i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 14.0000i	− 0.462321i
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.0000i	0.394132i
$928$	0	0
$929$	2.00000	0.0656179	0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000	0.0656179	0.0328089	+	0.999462i	$0.489555\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	0	0
$933$	− 48.0000i	− 1.57145i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 18.0000i	− 0.588034i	−0.955800	−	0.294017i	$-0.905008\pi$
$937$	− 18.0000i	− 0.588034i	0.955800	−	0.294017i	$-0.0949923\pi$
$938$	0	0
$939$	−28.0000	−0.913745
$940$	0	0
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	− 16.0000i	− 0.521032i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 40.0000i	− 1.29983i	−0.760009	−	0.649913i	$-0.774805\pi$
$947$	− 40.0000i	− 1.29983i	0.760009	−	0.649913i	$-0.225195\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	10.0000i	0.323932i	0.986796	+	0.161966i	$0.0517835\pi$
$953$	10.0000i	0.323932i	−0.986796	+	0.161966i	$0.948217\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24.0000i	0.771788i	0.922543	+	0.385894i	$0.126107\pi$
$967$	24.0000i	0.771788i	−0.922543	+	0.385894i	$0.873893\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	−26.0000	−0.834380	−0.417190	−	0.908819i	$-0.636985\pi$
$971$	−26.0000	−0.834380	−0.417190	+	0.908819i	$0.636985\pi$
$972$	0	0
$973$	18.0000i	0.577054i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	54.0000i	1.72761i	0.503824	+	0.863807i	$0.331926\pi$
$977$	54.0000i	1.72761i	−0.503824	+	0.863807i	$0.668074\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	− 12.0000i	− 0.382741i	−0.981518	−	0.191370i	$-0.938707\pi$
$983$	− 12.0000i	− 0.382741i	0.981518	−	0.191370i	$-0.0612931\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.00000i	0.254643i
$988$	0	0
$989$	−64.0000	−2.03508
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	0	0
$993$	16.0000i	0.507745i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 52.0000i	− 1.64686i	−0.567420	−	0.823428i	$-0.692059\pi$
$997$	− 52.0000i	− 1.64686i	0.567420	−	0.823428i	$-0.307941\pi$
$998$	0	0
$999$	−24.0000	−0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.2.g.b.449.2		2
4.3	odd	2		2800.2.g.g.449.1		2
5.2	odd	4		56.2.a.b.1.1	✓	1
5.3	odd	4		1400.2.a.a.1.1		1
5.4	even	2	inner	1400.2.g.b.449.1		2
15.2	even	4		504.2.a.h.1.1		1
20.3	even	4		2800.2.a.bd.1.1		1
20.7	even	4		112.2.a.a.1.1		1
20.19	odd	2		2800.2.g.g.449.2		2
35.2	odd	12		392.2.i.a.361.1		2
35.12	even	12		392.2.i.e.361.1		2
35.13	even	4		9800.2.a.bj.1.1		1
35.17	even	12		392.2.i.e.177.1		2
35.27	even	4		392.2.a.b.1.1		1
35.32	odd	12		392.2.i.a.177.1		2
40.27	even	4		448.2.a.h.1.1		1
40.37	odd	4		448.2.a.c.1.1		1
55.32	even	4		6776.2.a.h.1.1		1
60.47	odd	4		1008.2.a.m.1.1		1
65.12	odd	4		9464.2.a.h.1.1		1
80.27	even	4		1792.2.b.h.897.2		2
80.37	odd	4		1792.2.b.a.897.1		2
80.67	even	4		1792.2.b.h.897.1		2
80.77	odd	4		1792.2.b.a.897.2		2
105.2	even	12		3528.2.s.a.361.1		2
105.17	odd	12		3528.2.s.ba.3313.1		2
105.32	even	12		3528.2.s.a.3313.1		2
105.47	odd	12		3528.2.s.ba.361.1		2
105.62	odd	4		3528.2.a.b.1.1		1
120.77	even	4		4032.2.a.d.1.1		1
120.107	odd	4		4032.2.a.a.1.1		1
140.27	odd	4		784.2.a.i.1.1		1
140.47	odd	12		784.2.i.b.753.1		2
140.67	even	12		784.2.i.j.177.1		2
140.87	odd	12		784.2.i.b.177.1		2
140.107	even	12		784.2.i.j.753.1		2
280.27	odd	4		3136.2.a.c.1.1		1
280.237	even	4		3136.2.a.w.1.1		1
420.167	even	4		7056.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.a.b.1.1	✓	1	5.2	odd	4
112.2.a.a.1.1		1	20.7	even	4
392.2.a.b.1.1		1	35.27	even	4
392.2.i.a.177.1		2	35.32	odd	12
392.2.i.a.361.1		2	35.2	odd	12
392.2.i.e.177.1		2	35.17	even	12
392.2.i.e.361.1		2	35.12	even	12
448.2.a.c.1.1		1	40.37	odd	4
448.2.a.h.1.1		1	40.27	even	4
504.2.a.h.1.1		1	15.2	even	4
784.2.a.i.1.1		1	140.27	odd	4
784.2.i.b.177.1		2	140.87	odd	12
784.2.i.b.753.1		2	140.47	odd	12
784.2.i.j.177.1		2	140.67	even	12
784.2.i.j.753.1		2	140.107	even	12
1008.2.a.m.1.1		1	60.47	odd	4
1400.2.a.a.1.1		1	5.3	odd	4
1400.2.g.b.449.1		2	5.4	even	2	inner
1400.2.g.b.449.2		2	1.1	even	1	trivial
1792.2.b.a.897.1		2	80.37	odd	4
1792.2.b.a.897.2		2	80.77	odd	4
1792.2.b.h.897.1		2	80.67	even	4
1792.2.b.h.897.2		2	80.27	even	4
2800.2.a.bd.1.1		1	20.3	even	4
2800.2.g.g.449.1		2	4.3	odd	2
2800.2.g.g.449.2		2	20.19	odd	2
3136.2.a.c.1.1		1	280.27	odd	4
3136.2.a.w.1.1		1	280.237	even	4
3528.2.a.b.1.1		1	105.62	odd	4
3528.2.s.a.361.1		2	105.2	even	12
3528.2.s.a.3313.1		2	105.32	even	12
3528.2.s.ba.361.1		2	105.47	odd	12
3528.2.s.ba.3313.1		2	105.17	odd	12
4032.2.a.a.1.1		1	120.107	odd	4
4032.2.a.d.1.1		1	120.77	even	4
6776.2.a.h.1.1		1	55.32	even	4
7056.2.a.c.1.1		1	420.167	even	4
9464.2.a.h.1.1		1	65.12	odd	4
9800.2.a.bj.1.1		1	35.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +2.00000i q^{17} +2.00000 q^{19} +2.00000 q^{21} +8.00000i q^{23} +4.00000i q^{27} -2.00000 q^{29} +4.00000 q^{31} +6.00000i q^{37} -2.00000 q^{41} +8.00000i q^{43} +4.00000i q^{47} -1.00000 q^{49} -4.00000 q^{51} -10.0000i q^{53} +4.00000i q^{57} -6.00000 q^{59} +4.00000 q^{61} +1.00000i q^{63} +12.0000i q^{67} -16.0000 q^{69} -14.0000i q^{73} +8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} -4.00000i q^{87} -10.0000 q^{89} +8.00000i q^{93} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 4 q^{19} + 4 q^{21} - 4 q^{29} + 8 q^{31} - 4 q^{41} - 2 q^{49} - 8 q^{51} - 12 q^{59} + 8 q^{61} - 32 q^{69} + 16 q^{79} - 22 q^{81} - 20 q^{89}+O(q^{100}) \) 2 * q - 2 * q^9 + 4 * q^19 + 4 * q^21 - 4 * q^29 + 8 * q^31 - 4 * q^41 - 2 * q^49 - 8 * q^51 - 12 * q^59 + 8 * q^61 - 32 * q^69 + 16 * q^79 - 22 * q^81 - 20 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more