LMFDB - Embedded newform 1400.2.g.a.449.1

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 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			1400.2.g.a.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -1.00000i q^{7} -6.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -1.00000i q^{7} -6.00000 q^{9} -5.00000 q^{11} -5.00000i q^{13} +7.00000i q^{17} +2.00000 q^{19} -3.00000 q^{21} -2.00000i q^{23} +9.00000i q^{27} -7.00000 q^{29} +4.00000 q^{31} +15.0000i q^{33} +6.00000i q^{37} -15.0000 q^{39} -12.0000 q^{41} -2.00000i q^{43} -1.00000i q^{47} -1.00000 q^{49} +21.0000 q^{51} -6.00000i q^{57} +4.00000 q^{59} +4.00000 q^{61} +6.00000i q^{63} -8.00000i q^{67} -6.00000 q^{69} +6.00000i q^{73} +5.00000i q^{77} +3.00000 q^{79} +9.00000 q^{81} -4.00000i q^{83} +21.0000i q^{87} -5.00000 q^{91} -12.0000i q^{93} -13.0000i q^{97} +30.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{9}+O(q^{10}) $ 2 * q - 12 * q^9	$ 2 q - 12 q^{9} - 10 q^{11} + 4 q^{19} - 6 q^{21} - 14 q^{29} + 8 q^{31} - 30 q^{39} - 24 q^{41} - 2 q^{49} + 42 q^{51} + 8 q^{59} + 8 q^{61} - 12 q^{69} + 6 q^{79} + 18 q^{81} - 10 q^{91} + 60 q^{99}+O(q^{100}) $ 2 * q - 12 * q^9 - 10 * q^11 + 4 * q^19 - 6 * q^21 - 14 * q^29 + 8 * q^31 - 30 * q^39 - 24 * q^41 - 2 * q^49 + 42 * q^51 + 8 * q^59 + 8 * q^61 - 12 * q^69 + 6 * q^79 + 18 * q^81 - 10 * q^91 + 60 * q^99

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
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	− 5.00000i	− 1.38675i	−0.720577	−	0.693375i	$-0.756123\pi$
$13$	− 5.00000i	− 1.38675i	0.720577	−	0.693375i	$-0.243877\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.00000i	1.69775i	0.528594	+	0.848875i	$0.322719\pi$
$17$	7.00000i	1.69775i	−0.528594	+	0.848875i	$0.677281\pi$
$18$	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
$21$	−3.00000	−0.654654
$22$	0	0
$23$	− 2.00000i	− 0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$	− 2.00000i	− 0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	9.00000i	1.73205i
$28$	0	0
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	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
$33$	15.0000i	2.61116i
$34$	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
$39$	−15.0000	−2.40192
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.00000i	− 0.145865i	−0.997337	−	0.0729325i	$-0.976764\pi$
$47$	− 1.00000i	− 0.145865i	0.997337	−	0.0729325i	$-0.0232358\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	21.0000	2.94059
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6.00000i	− 0.794719i
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	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
$63$	6.00000i	0.755929i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.00000i	0.569803i
$78$	0	0
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	21.0000i	2.25144i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	0	0
$93$	− 12.0000i	− 1.24434i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 13.0000i	− 1.31995i	−0.751288	−	0.659975i	$-0.770567\pi$
$97$	− 13.0000i	− 1.31995i	0.751288	−	0.659975i	$-0.229433\pi$
$98$	0	0
$99$	30.0000	3.01511
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	13.0000i	1.28093i	0.767988	+	0.640464i	$0.221258\pi$
$103$	13.0000i	1.28093i	−0.767988	+	0.640464i	$0.778742\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 18.0000i	− 1.74013i	−0.492941	−	0.870063i	$-0.664078\pi$
$107$	− 18.0000i	− 1.74013i	0.492941	−	0.870063i	$-0.335922\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	18.0000	1.70848
$112$	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
$117$	30.0000i	2.77350i
$118$	0	0
$119$	7.00000	0.641689
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	36.0000i	3.24601i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	− 2.00000i	− 0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.00000i	0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$	8.00000i	0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$	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
$141$	−3.00000	−0.252646
$142$	0	0
$143$	25.0000i	2.09061i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.00000i	0.247436i
$148$	0	0
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	−19.0000	−1.54620	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000	−1.54620	−0.773099	+	0.634285i	$0.781294\pi$
$152$	0	0
$153$	− 42.0000i	− 3.39550i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	− 14.0000i	− 1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$	− 14.0000i	− 1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.00000i	0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$	3.00000i	0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−12.0000	−0.917663
$172$	0	0
$173$	− 7.00000i	− 0.532200i	−0.963945	−	0.266100i	$-0.914265\pi$
$173$	− 7.00000i	− 0.532200i	0.963945	−	0.266100i	$-0.0857352\pi$
$174$	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
$183$	− 12.0000i	− 0.887066i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 35.0000i	− 2.55945i
$188$	0	0
$189$	9.00000	0.654654
$190$	0	0
$191$	−13.0000	−0.940647	−0.470323	−	0.882494i	$-0.655863\pi$
$191$	−13.0000	−0.940647	−0.470323	+	0.882494i	$0.655863\pi$
$192$	0	0
$193$	− 8.00000i	− 0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$	− 8.00000i	− 0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.00000i	0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$	8.00000i	0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$	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
$203$	7.00000i	0.491304i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.0000i	0.834058i
$208$	0	0
$209$	−10.0000	−0.691714
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	0	0
$219$	18.0000	1.21633
$220$	0	0
$221$	35.0000	2.35435
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 9.00000i	− 0.597351i	−0.954355	−	0.298675i	$-0.903455\pi$
$227$	− 9.00000i	− 0.597351i	0.954355	−	0.298675i	$-0.0965448\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	15.0000	0.986928
$232$	0	0
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 9.00000i	− 0.584613i
$238$	0	0
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 10.0000i	− 0.636285i
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	10.0000i	0.628695i
$254$	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
$261$	42.0000	2.59973
$262$	0	0
$263$	− 30.0000i	− 1.84988i	−0.380114	−	0.924940i	$-0.624115\pi$
$263$	− 30.0000i	− 1.84988i	0.380114	−	0.924940i	$-0.375885\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	15.0000i	0.907841i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	−24.0000	−1.43684
$280$	0	0
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	0	0
$283$	− 25.0000i	− 1.48610i	−0.669238	−	0.743048i	$-0.733379\pi$
$283$	− 25.0000i	− 1.48610i	0.669238	−	0.743048i	$-0.266621\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000i	0.708338i
$288$	0	0
$289$	−32.0000	−1.88235
$290$	0	0
$291$	−39.0000	−2.28622
$292$	0	0
$293$	13.0000i	0.759468i	0.925096	+	0.379734i	$0.123985\pi$
$293$	13.0000i	0.759468i	−0.925096	+	0.379734i	$0.876015\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 45.0000i	− 2.61116i
$298$	0	0
$299$	−10.0000	−0.578315
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	54.0000i	3.10222i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.0000i	0.970241i	0.874447	+	0.485121i	$0.161224\pi$
$307$	17.0000i	0.970241i	−0.874447	+	0.485121i	$0.838776\pi$
$308$	0	0
$309$	39.0000	2.21863
$310$	0	0
$311$	−34.0000	−1.92796	−0.963982	−	0.265969i	$-0.914308\pi$
$311$	−34.0000	−1.92796	−0.963982	+	0.265969i	$0.914308\pi$
$312$	0	0
$313$	− 1.00000i	− 0.0565233i	−0.999601	−	0.0282617i	$-0.991003\pi$
$313$	− 1.00000i	− 0.0565233i	0.999601	−	0.0282617i	$-0.00899717\pi$
$314$	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
$319$	35.0000	1.95962
$320$	0	0
$321$	−54.0000	−3.01399
$322$	0	0
$323$	14.0000i	0.778981i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.0000i	0.829502i
$328$	0	0
$329$	−1.00000	−0.0551318
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	− 36.0000i	− 1.97279i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 34.0000i	− 1.85210i	−0.377403	−	0.926049i	$-0.623183\pi$
$337$	− 34.0000i	− 1.85210i	0.377403	−	0.926049i	$-0.376817\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.0000i	− 1.39575i	−0.716218	−	0.697877i	$-0.754128\pi$
$347$	− 26.0000i	− 1.39575i	0.716218	−	0.697877i	$-0.245872\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	45.0000	2.40192
$352$	0	0
$353$	5.00000i	0.266123i	0.991108	+	0.133062i	$0.0424808\pi$
$353$	5.00000i	0.266123i	−0.991108	+	0.133062i	$0.957519\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 21.0000i	− 1.11144i
$358$	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
$363$	− 42.0000i	− 2.20443i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.00000i	0.365397i	0.983169	+	0.182699i	$0.0584832\pi$
$367$	7.00000i	0.365397i	−0.983169	+	0.182699i	$0.941517\pi$
$368$	0	0
$369$	72.0000	3.74817
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.0000i	0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$	16.0000i	0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	35.0000i	1.80259i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	36.0000	1.84434
$382$	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
$387$	12.0000i	0.609994i
$388$	0	0
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	14.0000	0.708010
$392$	0	0
$393$	18.0000i	0.907980i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.0000i	1.15434i	0.816625	+	0.577168i	$0.195842\pi$
$397$	23.0000i	1.15434i	−0.816625	+	0.577168i	$0.804158\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	0	0
$403$	− 20.0000i	− 0.996271i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 30.0000i	− 1.48704i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	24.0000	1.18383
$412$	0	0
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	54.0000i	2.64439i
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−3.00000	−0.146211	−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000	−0.146211	−0.0731055	+	0.997324i	$0.523291\pi$
$422$	0	0
$423$	6.00000i	0.291730i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	0	0
$429$	75.0000	3.62103
$430$	0	0
$431$	−25.0000	−1.20421	−0.602104	−	0.798418i	$-0.705671\pi$
$431$	−25.0000	−1.20421	−0.602104	+	0.798418i	$0.705671\pi$
$432$	0	0
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.00000i	− 0.191346i
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	6.00000i	0.285069i	0.989790	+	0.142534i	$0.0455251\pi$
$443$	6.00000i	0.285069i	−0.989790	+	0.142534i	$0.954475\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 66.0000i	− 3.12169i
$448$	0	0
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	60.0000	2.82529
$452$	0	0
$453$	57.0000i	2.67809i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000i	0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$	8.00000i	0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	0	0
$459$	−63.0000	−2.94059
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	− 36.0000i	− 1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$	− 36.0000i	− 1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 11.0000i	− 0.509019i	−0.967070	−	0.254510i	$-0.918086\pi$
$467$	− 11.0000i	− 0.509019i	0.967070	−	0.254510i	$-0.0819141\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−30.0000	−1.38233
$472$	0	0
$473$	10.0000i	0.459800i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−34.0000	−1.55350	−0.776750	−	0.629809i	$-0.783133\pi$
$479$	−34.0000	−1.55350	−0.776750	+	0.629809i	$0.783133\pi$
$480$	0	0
$481$	30.0000	1.36788
$482$	0	0
$483$	6.00000i	0.273009i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 22.0000i	− 0.996915i	−0.866914	−	0.498458i	$-0.833900\pi$
$487$	− 22.0000i	− 0.996915i	0.866914	−	0.498458i	$-0.166100\pi$
$488$	0	0
$489$	−42.0000	−1.89931
$490$	0	0
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	0	0
$493$	− 49.0000i	− 2.20685i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.00000	0.0447661	0.0223831	−	0.999749i	$-0.492875\pi$
$499$	1.00000	0.0447661	0.0223831	+	0.999749i	$0.492875\pi$
$500$	0	0
$501$	9.00000	0.402090
$502$	0	0
$503$	3.00000i	0.133763i	0.997761	+	0.0668817i	$0.0213050\pi$
$503$	3.00000i	0.133763i	−0.997761	+	0.0668817i	$0.978695\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	36.0000i	1.59882i
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	18.0000i	0.794719i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.00000i	0.219900i
$518$	0	0
$519$	−21.0000	−0.921798
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	− 44.0000i	− 1.92399i	−0.273075	−	0.961993i	$-0.588041\pi$
$523$	− 44.0000i	− 1.92399i	0.273075	−	0.961993i	$-0.411959\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.0000i	1.21970i
$528$	0	0
$529$	19.0000	0.826087
$530$	0	0
$531$	−24.0000	−1.04151
$532$	0	0
$533$	60.0000i	2.59889i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 12.0000i	− 0.517838i
$538$	0	0
$539$	5.00000	0.215365
$540$	0	0
$541$	7.00000	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$542$	0	0
$543$	− 24.0000i	− 1.02994i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	0	0
$549$	−24.0000	−1.02430
$550$	0	0
$551$	−14.0000	−0.596420
$552$	0	0
$553$	− 3.00000i	− 0.127573i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.0000i	0.677942i	0.940797	+	0.338971i	$0.110079\pi$
$557$	16.0000i	0.677942i	−0.940797	+	0.338971i	$0.889921\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	0	0
$561$	−105.000	−4.43310
$562$	0	0
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 9.00000i	− 0.377964i
$568$	0	0
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	39.0000i	1.62925i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	37.0000i	1.54033i	0.637845	+	0.770165i	$0.279826\pi$
$577$	37.0000i	1.54033i	−0.637845	+	0.770165i	$0.720174\pi$
$578$	0	0
$579$	−24.0000	−0.997406
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	24.0000	0.987228
$592$	0	0
$593$	27.0000i	1.10876i	0.832265	+	0.554379i	$0.187044\pi$
$593$	27.0000i	1.10876i	−0.832265	+	0.554379i	$0.812956\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 12.0000i	− 0.491127i
$598$	0	0
$599$	−15.0000	−0.612883	−0.306442	−	0.951889i	$-0.599138\pi$
$599$	−15.0000	−0.612883	−0.306442	+	0.951889i	$0.599138\pi$
$600$	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
$603$	48.0000i	1.95471i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 3.00000i	− 0.121766i	−0.998145	−	0.0608831i	$-0.980608\pi$
$607$	− 3.00000i	− 0.121766i	0.998145	−	0.0608831i	$-0.0193917\pi$
$608$	0	0
$609$	21.0000	0.850963
$610$	0	0
$611$	−5.00000	−0.202278
$612$	0	0
$613$	− 30.0000i	− 1.21169i	−0.795583	−	0.605844i	$-0.792835\pi$
$613$	− 30.0000i	− 1.21169i	0.795583	−	0.605844i	$-0.207165\pi$
$614$	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
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	18.0000	0.722315
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	30.0000i	1.19808i
$628$	0	0
$629$	−42.0000	−1.67465
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	− 15.0000i	− 0.596196i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.00000i	0.198107i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	21.0000i	0.828159i	0.910241	+	0.414080i	$0.135896\pi$
$643$	21.0000i	0.828159i	−0.910241	+	0.414080i	$0.864104\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	−12.0000	−0.470317
$652$	0	0
$653$	38.0000i	1.48705i	0.668705	+	0.743527i	$0.266849\pi$
$653$	38.0000i	1.48705i	−0.668705	+	0.743527i	$0.733151\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 36.0000i	− 1.40449i
$658$	0	0
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	− 105.000i	− 4.07786i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.0000i	0.542082i
$668$	0	0
$669$	−57.0000	−2.20375
$670$	0	0
$671$	−20.0000	−0.772091
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.0000i	1.11456i	0.830324	+	0.557280i	$0.188155\pi$
$677$	29.0000i	1.11456i	−0.830324	+	0.557280i	$0.811845\pi$
$678$	0	0
$679$	−13.0000	−0.498894
$680$	0	0
$681$	−27.0000	−1.03464
$682$	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
$687$	12.0000i	0.457829i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	0	0
$693$	− 30.0000i	− 1.13961i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 84.0000i	− 3.18173i
$698$	0	0
$699$	−72.0000	−2.72329
$700$	0	0
$701$	35.0000	1.32193	0.660966	−	0.750416i	$-0.270147\pi$
$701$	35.0000	1.32193	0.660966	+	0.750416i	$0.270147\pi$
$702$	0	0
$703$	12.0000i	0.452589i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.0000i	0.676960i
$708$	0	0
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	−18.0000	−0.675053
$712$	0	0
$713$	− 8.00000i	− 0.299602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	27.0000i	1.00833i
$718$	0	0
$719$	14.0000	0.522112	0.261056	−	0.965324i	$-0.415929\pi$
$719$	14.0000	0.522112	0.261056	+	0.965324i	$0.415929\pi$
$720$	0	0
$721$	13.0000	0.484145
$722$	0	0
$723$	66.0000i	2.45457i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	40.0000i	1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$	40.0000i	1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	14.0000	0.517809
$732$	0	0
$733$	− 33.0000i	− 1.21888i	−0.792831	−	0.609441i	$-0.791394\pi$
$733$	− 33.0000i	− 1.21888i	0.792831	−	0.609441i	$-0.208606\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	40.0000i	1.47342i
$738$	0	0
$739$	−51.0000	−1.87607	−0.938033	−	0.346547i	$-0.887354\pi$
$739$	−51.0000	−1.87607	−0.938033	+	0.346547i	$0.887354\pi$
$740$	0	0
$741$	−30.0000	−1.10208
$742$	0	0
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	24.0000i	0.878114i
$748$	0	0
$749$	−18.0000	−0.657706
$750$	0	0
$751$	−3.00000	−0.109472	−0.0547358	−	0.998501i	$-0.517432\pi$
$751$	−3.00000	−0.109472	−0.0547358	+	0.998501i	$0.517432\pi$
$752$	0	0
$753$	− 18.0000i	− 0.655956i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	40.0000i	1.45382i	0.686730	+	0.726912i	$0.259045\pi$
$757$	40.0000i	1.45382i	−0.686730	+	0.726912i	$0.740955\pi$
$758$	0	0
$759$	30.0000	1.08893
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	5.00000i	0.181012i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 20.0000i	− 0.722158i
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	−54.0000	−1.94476
$772$	0	0
$773$	− 21.0000i	− 0.755318i	−0.925945	−	0.377659i	$-0.876729\pi$
$773$	− 21.0000i	− 0.755318i	0.925945	−	0.377659i	$-0.123271\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 18.0000i	− 0.645746i
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 63.0000i	− 2.25144i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 17.0000i	− 0.605985i	−0.952993	−	0.302992i	$-0.902014\pi$
$787$	− 17.0000i	− 0.605985i	0.952993	−	0.302992i	$-0.0979856\pi$
$788$	0	0
$789$	−90.0000	−3.20408
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	− 20.0000i	− 0.710221i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39.0000i	1.38145i	0.723117	+	0.690725i	$0.242709\pi$
$797$	39.0000i	1.38145i	−0.723117	+	0.690725i	$0.757291\pi$
$798$	0	0
$799$	7.00000	0.247642
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 30.0000i	− 1.05868i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	78.0000i	2.74573i
$808$	0	0
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	−10.0000	−0.351147	−0.175574	−	0.984466i	$-0.556178\pi$
$811$	−10.0000	−0.351147	−0.175574	+	0.984466i	$0.556178\pi$
$812$	0	0
$813$	− 36.0000i	− 1.26258i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4.00000i	− 0.139942i
$818$	0	0
$819$	30.0000	1.04828
$820$	0	0
$821$	1.00000	0.0349002	0.0174501	−	0.999848i	$-0.494445\pi$
$821$	1.00000	0.0349002	0.0174501	+	0.999848i	$0.494445\pi$
$822$	0	0
$823$	28.0000i	0.976019i	0.872838	+	0.488009i	$0.162277\pi$
$823$	28.0000i	0.976019i	−0.872838	+	0.488009i	$0.837723\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 6.00000i	− 0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$	− 6.00000i	− 0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	0	0
$833$	− 7.00000i	− 0.242536i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	36.0000i	1.24434i
$838$	0	0
$839$	34.0000	1.17381	0.586905	−	0.809656i	$-0.300346\pi$
$839$	34.0000	1.17381	0.586905	+	0.809656i	$0.300346\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	0	0
$843$	− 57.0000i	− 1.96318i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 14.0000i	− 0.481046i
$848$	0	0
$849$	−75.0000	−2.57399
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	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
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	36.0000	1.22688
$862$	0	0
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	96.0000i	3.26033i
$868$	0	0
$869$	−15.0000	−0.508840
$870$	0	0
$871$	−40.0000	−1.35535
$872$	0	0
$873$	78.0000i	2.63990i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	0	0
$879$	39.0000	1.31544
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	− 40.0000i	− 1.34611i	−0.739594	−	0.673054i	$-0.764982\pi$
$883$	− 40.0000i	− 1.34611i	0.739594	−	0.673054i	$-0.235018\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 16.0000i	− 0.537227i	−0.963248	−	0.268614i	$-0.913434\pi$
$887$	− 16.0000i	− 0.537227i	0.963248	−	0.268614i	$-0.0865655\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	−45.0000	−1.50756
$892$	0	0
$893$	− 2.00000i	− 0.0669274i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	30.0000i	1.00167i
$898$	0	0
$899$	−28.0000	−0.933852
$900$	0	0
$901$	0	0
$902$	0	0
$903$	6.00000i	0.199667i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	18.0000i	0.597680i	0.954303	+	0.298840i	$0.0965997\pi$
$907$	18.0000i	0.597680i	−0.954303	+	0.298840i	$0.903400\pi$
$908$	0	0
$909$	108.000	3.58213
$910$	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
$913$	20.0000i	0.661903i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.00000i	0.198137i
$918$	0	0
$919$	55.0000	1.81428	0.907141	−	0.420826i	$-0.138260\pi$
$919$	55.0000	1.81428	0.907141	+	0.420826i	$0.138260\pi$
$920$	0	0
$921$	51.0000	1.68051
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 78.0000i	− 2.56186i
$928$	0	0
$929$	32.0000	1.04989	0.524943	−	0.851137i	$-0.324087\pi$
$929$	32.0000	1.04989	0.524943	+	0.851137i	$0.324087\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	0	0
$933$	102.000i	3.33933i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 13.0000i	− 0.424691i	−0.977195	−	0.212346i	$-0.931890\pi$
$937$	− 13.0000i	− 0.424691i	0.977195	−	0.212346i	$-0.0681103\pi$
$938$	0	0
$939$	−3.00000	−0.0979013
$940$	0	0
$941$	28.0000	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$942$	0	0
$943$	24.0000i	0.781548i
$944$	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
$949$	30.0000	0.973841
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 105.000i	− 3.39417i
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	108.000i	3.48025i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 6.00000i	− 0.192947i	−0.995336	−	0.0964735i	$-0.969244\pi$
$967$	− 6.00000i	− 0.192947i	0.995336	−	0.0964735i	$-0.0307563\pi$
$968$	0	0
$969$	42.0000	1.34923
$970$	0	0
$971$	−16.0000	−0.513464	−0.256732	−	0.966483i	$-0.582646\pi$
$971$	−16.0000	−0.513464	−0.256732	+	0.966483i	$0.582646\pi$
$972$	0	0
$973$	18.0000i	0.577054i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 26.0000i	− 0.831814i	−0.909407	−	0.415907i	$-0.863464\pi$
$977$	− 26.0000i	− 0.831814i	0.909407	−	0.415907i	$-0.136536\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	30.0000	0.957826
$982$	0	0
$983$	3.00000i	0.0956851i	0.998855	+	0.0478426i	$0.0152346\pi$
$983$	3.00000i	0.0956851i	−0.998855	+	0.0478426i	$0.984765\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.00000i	0.0954911i
$988$	0	0
$989$	−4.00000	−0.127193
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	36.0000i	1.14243i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.00000i	0.0950110i	0.998871	+	0.0475055i	$0.0151272\pi$
$997$	3.00000i	0.0950110i	−0.998871	+	0.0475055i	$0.984873\pi$
$998$	0	0
$999$	−54.0000	−1.70848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.2.g.a.449.1		2
4.3	odd	2		2800.2.g.b.449.2		2
5.2	odd	4		280.2.a.a.1.1	✓	1
5.3	odd	4		1400.2.a.n.1.1		1
5.4	even	2	inner	1400.2.g.a.449.2		2
15.2	even	4		2520.2.a.i.1.1		1
20.3	even	4		2800.2.a.c.1.1		1
20.7	even	4		560.2.a.f.1.1		1
20.19	odd	2		2800.2.g.b.449.1		2
35.2	odd	12		1960.2.q.o.361.1		2
35.12	even	12		1960.2.q.a.361.1		2
35.13	even	4		9800.2.a.a.1.1		1
35.17	even	12		1960.2.q.a.961.1		2
35.27	even	4		1960.2.a.o.1.1		1
35.32	odd	12		1960.2.q.o.961.1		2
40.27	even	4		2240.2.a.a.1.1		1
40.37	odd	4		2240.2.a.z.1.1		1
60.47	odd	4		5040.2.a.a.1.1		1
140.27	odd	4		3920.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.a.a.1.1	✓	1	5.2	odd	4
560.2.a.f.1.1		1	20.7	even	4
1400.2.a.n.1.1		1	5.3	odd	4
1400.2.g.a.449.1		2	1.1	even	1	trivial
1400.2.g.a.449.2		2	5.4	even	2	inner
1960.2.a.o.1.1		1	35.27	even	4
1960.2.q.a.361.1		2	35.12	even	12
1960.2.q.a.961.1		2	35.17	even	12
1960.2.q.o.361.1		2	35.2	odd	12
1960.2.q.o.961.1		2	35.32	odd	12
2240.2.a.a.1.1		1	40.27	even	4
2240.2.a.z.1.1		1	40.37	odd	4
2520.2.a.i.1.1		1	15.2	even	4
2800.2.a.c.1.1		1	20.3	even	4
2800.2.g.b.449.1		2	20.19	odd	2
2800.2.g.b.449.2		2	4.3	odd	2
3920.2.a.c.1.1		1	140.27	odd	4
5040.2.a.a.1.1		1	60.47	odd	4
9800.2.a.a.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)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -1.00000i q^{7} -6.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -1.00000i q^{7} -6.00000 q^{9} -5.00000 q^{11} -5.00000i q^{13} +7.00000i q^{17} +2.00000 q^{19} -3.00000 q^{21} -2.00000i q^{23} +9.00000i q^{27} -7.00000 q^{29} +4.00000 q^{31} +15.0000i q^{33} +6.00000i q^{37} -15.0000 q^{39} -12.0000 q^{41} -2.00000i q^{43} -1.00000i q^{47} -1.00000 q^{49} +21.0000 q^{51} -6.00000i q^{57} +4.00000 q^{59} +4.00000 q^{61} +6.00000i q^{63} -8.00000i q^{67} -6.00000 q^{69} +6.00000i q^{73} +5.00000i q^{77} +3.00000 q^{79} +9.00000 q^{81} -4.00000i q^{83} +21.0000i q^{87} -5.00000 q^{91} -12.0000i q^{93} -13.0000i q^{97} +30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{9}+O(q^{10}) \) 2 * q - 12 * q^9	\( 2 q - 12 q^{9} - 10 q^{11} + 4 q^{19} - 6 q^{21} - 14 q^{29} + 8 q^{31} - 30 q^{39} - 24 q^{41} - 2 q^{49} + 42 q^{51} + 8 q^{59} + 8 q^{61} - 12 q^{69} + 6 q^{79} + 18 q^{81} - 10 q^{91} + 60 q^{99}+O(q^{100}) \) 2 * q - 12 * q^9 - 10 * q^11 + 4 * q^19 - 6 * q^21 - 14 * q^29 + 8 * q^31 - 30 * q^39 - 24 * q^41 - 2 * q^49 + 42 * q^51 + 8 * q^59 + 8 * q^61 - 12 * q^69 + 6 * q^79 + 18 * q^81 - 10 * q^91 + 60 * q^99

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