LMFDB - Embedded newform 1380.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(1,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} -3.12311 q^{11} +2.00000 q^{13} -1.00000 q^{15} +3.56155 q^{17} -2.00000 q^{19} +1.56155 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.68466 q^{29} +4.68466 q^{31} +3.12311 q^{33} -1.56155 q^{35} +2.43845 q^{37} -2.00000 q^{39} -2.68466 q^{41} +1.00000 q^{45} +4.00000 q^{47} -4.56155 q^{49} -3.56155 q^{51} +7.56155 q^{53} -3.12311 q^{55} +2.00000 q^{57} +3.56155 q^{59} +9.12311 q^{61} -1.56155 q^{63} +2.00000 q^{65} +8.68466 q^{67} -1.00000 q^{69} +0.438447 q^{71} -4.24621 q^{73} -1.00000 q^{75} +4.87689 q^{77} -2.87689 q^{79} +1.00000 q^{81} +12.6847 q^{83} +3.56155 q^{85} -6.68466 q^{87} -5.12311 q^{89} -3.12311 q^{91} -4.68466 q^{93} -2.00000 q^{95} +11.1231 q^{97} -3.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} - 2 q^{15} + 3 q^{17} - 4 q^{19} - q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} + q^{29} - 3 q^{31} - 2 q^{33} + q^{35} + 9 q^{37} - 4 q^{39} + 7 q^{41} + 2 q^{45} + 8 q^{47} - 5 q^{49} - 3 q^{51} + 11 q^{53} + 2 q^{55} + 4 q^{57} + 3 q^{59} + 10 q^{61} + q^{63} + 4 q^{65} + 5 q^{67} - 2 q^{69} + 5 q^{71} + 8 q^{73} - 2 q^{75} + 18 q^{77} - 14 q^{79} + 2 q^{81} + 13 q^{83} + 3 q^{85} - q^{87} - 2 q^{89} + 2 q^{91} + 3 q^{93} - 4 q^{95} + 14 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 - 2 * q^15 + 3 * q^17 - 4 * q^19 - q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 + q^29 - 3 * q^31 - 2 * q^33 + q^35 + 9 * q^37 - 4 * q^39 + 7 * q^41 + 2 * q^45 + 8 * q^47 - 5 * q^49 - 3 * q^51 + 11 * q^53 + 2 * q^55 + 4 * q^57 + 3 * q^59 + 10 * q^61 + q^63 + 4 * q^65 + 5 * q^67 - 2 * q^69 + 5 * q^71 + 8 * q^73 - 2 * q^75 + 18 * q^77 - 14 * q^79 + 2 * q^81 + 13 * q^83 + 3 * q^85 - q^87 - 2 * q^89 + 2 * q^91 + 3 * q^93 - 4 * q^95 + 14 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.12311	−0.941652	−0.470826	−	0.882226i	$-0.656044\pi$
$11$	−3.12311	−0.941652	−0.470826	+	0.882226i	$0.656044\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	3.56155	0.863803	0.431902	−	0.901921i	$-0.357843\pi$
$17$	3.56155	0.863803	0.431902	+	0.901921i	$0.357843\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	1.56155	0.340759
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.68466	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$29$	6.68466	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$30$	0	0
$31$	4.68466	0.841389	0.420695	−	0.907202i	$-0.361786\pi$
$31$	4.68466	0.841389	0.420695	+	0.907202i	$0.361786\pi$
$32$	0	0
$33$	3.12311	0.543663
$34$	0	0
$35$	−1.56155	−0.263951
$36$	0	0
$37$	2.43845	0.400878	0.200439	−	0.979706i	$-0.435763\pi$
$37$	2.43845	0.400878	0.200439	+	0.979706i	$0.435763\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−2.68466	−0.419273	−0.209637	−	0.977779i	$-0.567228\pi$
$41$	−2.68466	−0.419273	−0.209637	+	0.977779i	$0.567228\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−3.56155	−0.498717
$52$	0	0
$53$	7.56155	1.03866	0.519330	−	0.854574i	$-0.326182\pi$
$53$	7.56155	1.03866	0.519330	+	0.854574i	$0.326182\pi$
$54$	0	0
$55$	−3.12311	−0.421119
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	3.56155	0.463675	0.231837	−	0.972755i	$-0.425526\pi$
$59$	3.56155	0.463675	0.231837	+	0.972755i	$0.425526\pi$
$60$	0	0
$61$	9.12311	1.16809	0.584047	−	0.811720i	$-0.301468\pi$
$61$	9.12311	1.16809	0.584047	+	0.811720i	$0.301468\pi$
$62$	0	0
$63$	−1.56155	−0.196737
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	8.68466	1.06100	0.530500	−	0.847685i	$-0.322004\pi$
$67$	8.68466	1.06100	0.530500	+	0.847685i	$0.322004\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0.438447	0.0520341	0.0260171	−	0.999661i	$-0.491718\pi$
$71$	0.438447	0.0520341	0.0260171	+	0.999661i	$0.491718\pi$
$72$	0	0
$73$	−4.24621	−0.496981	−0.248491	−	0.968634i	$-0.579935\pi$
$73$	−4.24621	−0.496981	−0.248491	+	0.968634i	$0.579935\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	4.87689	0.555774
$78$	0	0
$79$	−2.87689	−0.323676	−0.161838	−	0.986817i	$-0.551742\pi$
$79$	−2.87689	−0.323676	−0.161838	+	0.986817i	$0.551742\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.6847	1.39232	0.696161	−	0.717886i	$-0.254890\pi$
$83$	12.6847	1.39232	0.696161	+	0.717886i	$0.254890\pi$
$84$	0	0
$85$	3.56155	0.386305
$86$	0	0
$87$	−6.68466	−0.716671
$88$	0	0
$89$	−5.12311	−0.543048	−0.271524	−	0.962432i	$-0.587528\pi$
$89$	−5.12311	−0.543048	−0.271524	+	0.962432i	$0.587528\pi$
$90$	0	0
$91$	−3.12311	−0.327390
$92$	0	0
$93$	−4.68466	−0.485776
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	11.1231	1.12938	0.564690	−	0.825303i	$-0.308996\pi$
$97$	11.1231	1.12938	0.564690	+	0.825303i	$0.308996\pi$
$98$	0	0
$99$	−3.12311	−0.313884
$100$	0	0
$101$	−1.80776	−0.179879	−0.0899396	−	0.995947i	$-0.528667\pi$
$101$	−1.80776	−0.179879	−0.0899396	+	0.995947i	$0.528667\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	1.56155	0.152392
$106$	0	0
$107$	12.6847	1.22627	0.613136	−	0.789977i	$-0.289908\pi$
$107$	12.6847	1.22627	0.613136	+	0.789977i	$0.289908\pi$
$108$	0	0
$109$	0.246211	0.0235828	0.0117914	−	0.999930i	$-0.496247\pi$
$109$	0.246211	0.0235828	0.0117914	+	0.999930i	$0.496247\pi$
$110$	0	0
$111$	−2.43845	−0.231447
$112$	0	0
$113$	7.56155	0.711331	0.355666	−	0.934613i	$-0.384254\pi$
$113$	7.56155	0.711331	0.355666	+	0.934613i	$0.384254\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−5.56155	−0.509827
$120$	0	0
$121$	−1.24621	−0.113292
$122$	0	0
$123$	2.68466	0.242067
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.24621	0.554262	0.277131	−	0.960832i	$-0.410616\pi$
$127$	6.24621	0.554262	0.277131	+	0.960832i	$0.410616\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.3693	0.993342	0.496671	−	0.867939i	$-0.334556\pi$
$131$	11.3693	0.993342	0.496671	+	0.867939i	$0.334556\pi$
$132$	0	0
$133$	3.12311	0.270808
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	22.4924	1.92166	0.960829	−	0.277143i	$-0.0893876\pi$
$137$	22.4924	1.92166	0.960829	+	0.277143i	$0.0893876\pi$
$138$	0	0
$139$	−18.9309	−1.60570	−0.802848	−	0.596184i	$-0.796683\pi$
$139$	−18.9309	−1.60570	−0.802848	+	0.596184i	$0.796683\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	−6.24621	−0.522334
$144$	0	0
$145$	6.68466	0.555131
$146$	0	0
$147$	4.56155	0.376231
$148$	0	0
$149$	−7.36932	−0.603718	−0.301859	−	0.953353i	$-0.597607\pi$
$149$	−7.36932	−0.603718	−0.301859	+	0.953353i	$0.597607\pi$
$150$	0	0
$151$	6.24621	0.508309	0.254155	−	0.967164i	$-0.418203\pi$
$151$	6.24621	0.508309	0.254155	+	0.967164i	$0.418203\pi$
$152$	0	0
$153$	3.56155	0.287934
$154$	0	0
$155$	4.68466	0.376281
$156$	0	0
$157$	−10.9309	−0.872378	−0.436189	−	0.899855i	$-0.643672\pi$
$157$	−10.9309	−0.872378	−0.436189	+	0.899855i	$0.643672\pi$
$158$	0	0
$159$	−7.56155	−0.599670
$160$	0	0
$161$	−1.56155	−0.123068
$162$	0	0
$163$	−21.3693	−1.67377	−0.836887	−	0.547376i	$-0.815627\pi$
$163$	−21.3693	−1.67377	−0.836887	+	0.547376i	$0.815627\pi$
$164$	0	0
$165$	3.12311	0.243133
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	14.2462	1.08312	0.541560	−	0.840662i	$-0.317834\pi$
$173$	14.2462	1.08312	0.541560	+	0.840662i	$0.317834\pi$
$174$	0	0
$175$	−1.56155	−0.118042
$176$	0	0
$177$	−3.56155	−0.267703
$178$	0	0
$179$	−5.12311	−0.382919	−0.191459	−	0.981501i	$-0.561322\pi$
$179$	−5.12311	−0.382919	−0.191459	+	0.981501i	$0.561322\pi$
$180$	0	0
$181$	−1.12311	−0.0834798	−0.0417399	−	0.999129i	$-0.513290\pi$
$181$	−1.12311	−0.0834798	−0.0417399	+	0.999129i	$0.513290\pi$
$182$	0	0
$183$	−9.12311	−0.674399
$184$	0	0
$185$	2.43845	0.179278
$186$	0	0
$187$	−11.1231	−0.813402
$188$	0	0
$189$	1.56155	0.113586
$190$	0	0
$191$	12.4924	0.903920	0.451960	−	0.892038i	$-0.350725\pi$
$191$	12.4924	0.903920	0.451960	+	0.892038i	$0.350725\pi$
$192$	0	0
$193$	−13.1231	−0.944622	−0.472311	−	0.881432i	$-0.656580\pi$
$193$	−13.1231	−0.944622	−0.472311	+	0.881432i	$0.656580\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−1.36932	−0.0975598	−0.0487799	−	0.998810i	$-0.515533\pi$
$197$	−1.36932	−0.0975598	−0.0487799	+	0.998810i	$0.515533\pi$
$198$	0	0
$199$	−20.2462	−1.43522	−0.717608	−	0.696447i	$-0.754763\pi$
$199$	−20.2462	−1.43522	−0.717608	+	0.696447i	$0.754763\pi$
$200$	0	0
$201$	−8.68466	−0.612569
$202$	0	0
$203$	−10.4384	−0.732635
$204$	0	0
$205$	−2.68466	−0.187505
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	6.24621	0.432059
$210$	0	0
$211$	6.93087	0.477141	0.238570	−	0.971125i	$-0.423321\pi$
$211$	6.93087	0.477141	0.238570	+	0.971125i	$0.423321\pi$
$212$	0	0
$213$	−0.438447	−0.0300419
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7.31534	−0.496598
$218$	0	0
$219$	4.24621	0.286932
$220$	0	0
$221$	7.12311	0.479152
$222$	0	0
$223$	−12.8769	−0.862301	−0.431150	−	0.902280i	$-0.641892\pi$
$223$	−12.8769	−0.862301	−0.431150	+	0.902280i	$0.641892\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	−0.630683	−0.0416767	−0.0208384	−	0.999783i	$-0.506634\pi$
$229$	−0.630683	−0.0416767	−0.0208384	+	0.999783i	$0.506634\pi$
$230$	0	0
$231$	−4.87689	−0.320876
$232$	0	0
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	2.87689	0.186874
$238$	0	0
$239$	1.80776	0.116935	0.0584673	−	0.998289i	$-0.481379\pi$
$239$	1.80776	0.116935	0.0584673	+	0.998289i	$0.481379\pi$
$240$	0	0
$241$	18.4924	1.19120	0.595601	−	0.803281i	$-0.296914\pi$
$241$	18.4924	1.19120	0.595601	+	0.803281i	$0.296914\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.56155	−0.291427
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−12.6847	−0.803858
$250$	0	0
$251$	−13.3693	−0.843864	−0.421932	−	0.906628i	$-0.638648\pi$
$251$	−13.3693	−0.843864	−0.421932	+	0.906628i	$0.638648\pi$
$252$	0	0
$253$	−3.12311	−0.196348
$254$	0	0
$255$	−3.56155	−0.223033
$256$	0	0
$257$	−16.4924	−1.02877	−0.514385	−	0.857560i	$-0.671980\pi$
$257$	−16.4924	−1.02877	−0.514385	+	0.857560i	$0.671980\pi$
$258$	0	0
$259$	−3.80776	−0.236603
$260$	0	0
$261$	6.68466	0.413770
$262$	0	0
$263$	−22.9309	−1.41398	−0.706989	−	0.707225i	$-0.749947\pi$
$263$	−22.9309	−1.41398	−0.706989	+	0.707225i	$0.749947\pi$
$264$	0	0
$265$	7.56155	0.464502
$266$	0	0
$267$	5.12311	0.313529
$268$	0	0
$269$	−16.4384	−1.00227	−0.501135	−	0.865369i	$-0.667084\pi$
$269$	−16.4384	−1.00227	−0.501135	+	0.865369i	$0.667084\pi$
$270$	0	0
$271$	−22.9309	−1.39295	−0.696476	−	0.717581i	$-0.745250\pi$
$271$	−22.9309	−1.39295	−0.696476	+	0.717581i	$0.745250\pi$
$272$	0	0
$273$	3.12311	0.189019
$274$	0	0
$275$	−3.12311	−0.188330
$276$	0	0
$277$	−8.24621	−0.495467	−0.247733	−	0.968828i	$-0.579686\pi$
$277$	−8.24621	−0.495467	−0.247733	+	0.968828i	$0.579686\pi$
$278$	0	0
$279$	4.68466	0.280463
$280$	0	0
$281$	0.246211	0.0146877	0.00734387	−	0.999973i	$-0.497662\pi$
$281$	0.246211	0.0146877	0.00734387	+	0.999973i	$0.497662\pi$
$282$	0	0
$283$	−14.4384	−0.858277	−0.429138	−	0.903239i	$-0.641183\pi$
$283$	−14.4384	−0.858277	−0.429138	+	0.903239i	$0.641183\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	4.19224	0.247460
$288$	0	0
$289$	−4.31534	−0.253844
$290$	0	0
$291$	−11.1231	−0.652048
$292$	0	0
$293$	8.93087	0.521747	0.260873	−	0.965373i	$-0.415989\pi$
$293$	8.93087	0.521747	0.260873	+	0.965373i	$0.415989\pi$
$294$	0	0
$295$	3.56155	0.207362
$296$	0	0
$297$	3.12311	0.181221
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.80776	0.103853
$304$	0	0
$305$	9.12311	0.522388
$306$	0	0
$307$	0.876894	0.0500470	0.0250235	−	0.999687i	$-0.492034\pi$
$307$	0.876894	0.0500470	0.0250235	+	0.999687i	$0.492034\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.36932	−0.417876	−0.208938	−	0.977929i	$-0.567001\pi$
$311$	−7.36932	−0.417876	−0.208938	+	0.977929i	$0.567001\pi$
$312$	0	0
$313$	9.56155	0.540451	0.270225	−	0.962797i	$-0.412902\pi$
$313$	9.56155	0.540451	0.270225	+	0.962797i	$0.412902\pi$
$314$	0	0
$315$	−1.56155	−0.0879835
$316$	0	0
$317$	−4.00000	−0.224662	−0.112331	−	0.993671i	$-0.535832\pi$
$317$	−4.00000	−0.224662	−0.112331	+	0.993671i	$0.535832\pi$
$318$	0	0
$319$	−20.8769	−1.16888
$320$	0	0
$321$	−12.6847	−0.707989
$322$	0	0
$323$	−7.12311	−0.396340
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−0.246211	−0.0136155
$328$	0	0
$329$	−6.24621	−0.344365
$330$	0	0
$331$	−1.56155	−0.0858307	−0.0429154	−	0.999079i	$-0.513665\pi$
$331$	−1.56155	−0.0858307	−0.0429154	+	0.999079i	$0.513665\pi$
$332$	0	0
$333$	2.43845	0.133626
$334$	0	0
$335$	8.68466	0.474494
$336$	0	0
$337$	−12.8769	−0.701449	−0.350725	−	0.936479i	$-0.614065\pi$
$337$	−12.8769	−0.701449	−0.350725	+	0.936479i	$0.614065\pi$
$338$	0	0
$339$	−7.56155	−0.410687
$340$	0	0
$341$	−14.6307	−0.792296
$342$	0	0
$343$	18.0540	0.974823
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	10.6847	0.571937	0.285968	−	0.958239i	$-0.407685\pi$
$349$	10.6847	0.571937	0.285968	+	0.958239i	$0.407685\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	−14.7386	−0.784458	−0.392229	−	0.919868i	$-0.628296\pi$
$353$	−14.7386	−0.784458	−0.392229	+	0.919868i	$0.628296\pi$
$354$	0	0
$355$	0.438447	0.0232704
$356$	0	0
$357$	5.56155	0.294349
$358$	0	0
$359$	−10.2462	−0.540774	−0.270387	−	0.962752i	$-0.587152\pi$
$359$	−10.2462	−0.540774	−0.270387	+	0.962752i	$0.587152\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	1.24621	0.0654091
$364$	0	0
$365$	−4.24621	−0.222257
$366$	0	0
$367$	−2.43845	−0.127286	−0.0636430	−	0.997973i	$-0.520272\pi$
$367$	−2.43845	−0.127286	−0.0636430	+	0.997973i	$0.520272\pi$
$368$	0	0
$369$	−2.68466	−0.139758
$370$	0	0
$371$	−11.8078	−0.613029
$372$	0	0
$373$	−5.36932	−0.278013	−0.139006	−	0.990291i	$-0.544391\pi$
$373$	−5.36932	−0.278013	−0.139006	+	0.990291i	$0.544391\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	13.3693	0.688555
$378$	0	0
$379$	31.3693	1.61133	0.805667	−	0.592369i	$-0.201807\pi$
$379$	31.3693	1.61133	0.805667	+	0.592369i	$0.201807\pi$
$380$	0	0
$381$	−6.24621	−0.320003
$382$	0	0
$383$	−10.4384	−0.533380	−0.266690	−	0.963782i	$-0.585930\pi$
$383$	−10.4384	−0.533380	−0.266690	+	0.963782i	$0.585930\pi$
$384$	0	0
$385$	4.87689	0.248550
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	3.56155	0.180115
$392$	0	0
$393$	−11.3693	−0.573506
$394$	0	0
$395$	−2.87689	−0.144752
$396$	0	0
$397$	25.1231	1.26089	0.630446	−	0.776233i	$-0.282872\pi$
$397$	25.1231	1.26089	0.630446	+	0.776233i	$0.282872\pi$
$398$	0	0
$399$	−3.12311	−0.156351
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	9.36932	0.466719
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−7.61553	−0.377488
$408$	0	0
$409$	−4.43845	−0.219467	−0.109734	−	0.993961i	$-0.535000\pi$
$409$	−4.43845	−0.219467	−0.109734	+	0.993961i	$0.535000\pi$
$410$	0	0
$411$	−22.4924	−1.10947
$412$	0	0
$413$	−5.56155	−0.273666
$414$	0	0
$415$	12.6847	0.622665
$416$	0	0
$417$	18.9309	0.927049
$418$	0	0
$419$	27.6155	1.34911	0.674553	−	0.738226i	$-0.264336\pi$
$419$	27.6155	1.34911	0.674553	+	0.738226i	$0.264336\pi$
$420$	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
$423$	4.00000	0.194487
$424$	0	0
$425$	3.56155	0.172761
$426$	0	0
$427$	−14.2462	−0.689422
$428$	0	0
$429$	6.24621	0.301570
$430$	0	0
$431$	−14.2462	−0.686216	−0.343108	−	0.939296i	$-0.611480\pi$
$431$	−14.2462	−0.686216	−0.343108	+	0.939296i	$0.611480\pi$
$432$	0	0
$433$	6.93087	0.333076	0.166538	−	0.986035i	$-0.446741\pi$
$433$	6.93087	0.333076	0.166538	+	0.986035i	$0.446741\pi$
$434$	0	0
$435$	−6.68466	−0.320505
$436$	0	0
$437$	−2.00000	−0.0956730
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−4.56155	−0.217217
$442$	0	0
$443$	20.4924	0.973624	0.486812	−	0.873507i	$-0.338160\pi$
$443$	20.4924	0.973624	0.486812	+	0.873507i	$0.338160\pi$
$444$	0	0
$445$	−5.12311	−0.242858
$446$	0	0
$447$	7.36932	0.348557
$448$	0	0
$449$	20.4384	0.964550	0.482275	−	0.876020i	$-0.339811\pi$
$449$	20.4384	0.964550	0.482275	+	0.876020i	$0.339811\pi$
$450$	0	0
$451$	8.38447	0.394809
$452$	0	0
$453$	−6.24621	−0.293473
$454$	0	0
$455$	−3.12311	−0.146413
$456$	0	0
$457$	18.0540	0.844529	0.422265	−	0.906473i	$-0.361235\pi$
$457$	18.0540	0.844529	0.422265	+	0.906473i	$0.361235\pi$
$458$	0	0
$459$	−3.56155	−0.166239
$460$	0	0
$461$	20.7386	0.965894	0.482947	−	0.875649i	$-0.339566\pi$
$461$	20.7386	0.965894	0.482947	+	0.875649i	$0.339566\pi$
$462$	0	0
$463$	−14.6307	−0.679946	−0.339973	−	0.940435i	$-0.610418\pi$
$463$	−14.6307	−0.679946	−0.339973	+	0.940435i	$0.610418\pi$
$464$	0	0
$465$	−4.68466	−0.217246
$466$	0	0
$467$	28.6847	1.32737	0.663684	−	0.748013i	$-0.268992\pi$
$467$	28.6847	1.32737	0.663684	+	0.748013i	$0.268992\pi$
$468$	0	0
$469$	−13.5616	−0.626214
$470$	0	0
$471$	10.9309	0.503668
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	7.56155	0.346220
$478$	0	0
$479$	0.492423	0.0224994	0.0112497	−	0.999937i	$-0.496419\pi$
$479$	0.492423	0.0224994	0.0112497	+	0.999937i	$0.496419\pi$
$480$	0	0
$481$	4.87689	0.222367
$482$	0	0
$483$	1.56155	0.0710531
$484$	0	0
$485$	11.1231	0.505074
$486$	0	0
$487$	−9.75379	−0.441986	−0.220993	−	0.975275i	$-0.570930\pi$
$487$	−9.75379	−0.441986	−0.220993	+	0.975275i	$0.570930\pi$
$488$	0	0
$489$	21.3693	0.966354
$490$	0	0
$491$	41.4233	1.86941	0.934704	−	0.355428i	$-0.115665\pi$
$491$	41.4233	1.86941	0.934704	+	0.355428i	$0.115665\pi$
$492$	0	0
$493$	23.8078	1.07225
$494$	0	0
$495$	−3.12311	−0.140373
$496$	0	0
$497$	−0.684658	−0.0307111
$498$	0	0
$499$	−41.1771	−1.84334	−0.921670	−	0.387976i	$-0.873174\pi$
$499$	−41.1771	−1.84334	−0.921670	+	0.387976i	$0.873174\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	−16.6847	−0.743932	−0.371966	−	0.928246i	$-0.621316\pi$
$503$	−16.6847	−0.743932	−0.371966	+	0.928246i	$0.621316\pi$
$504$	0	0
$505$	−1.80776	−0.0804444
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−20.7386	−0.919224	−0.459612	−	0.888120i	$-0.652011\pi$
$509$	−20.7386	−0.919224	−0.459612	+	0.888120i	$0.652011\pi$
$510$	0	0
$511$	6.63068	0.293324
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−12.4924	−0.549416
$518$	0	0
$519$	−14.2462	−0.625339
$520$	0	0
$521$	−9.50758	−0.416535	−0.208267	−	0.978072i	$-0.566782\pi$
$521$	−9.50758	−0.416535	−0.208267	+	0.978072i	$0.566782\pi$
$522$	0	0
$523$	−20.9848	−0.917603	−0.458802	−	0.888539i	$-0.651721\pi$
$523$	−20.9848	−0.917603	−0.458802	+	0.888539i	$0.651721\pi$
$524$	0	0
$525$	1.56155	0.0681518
$526$	0	0
$527$	16.6847	0.726795
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.56155	0.154558
$532$	0	0
$533$	−5.36932	−0.232571
$534$	0	0
$535$	12.6847	0.548406
$536$	0	0
$537$	5.12311	0.221078
$538$	0	0
$539$	14.2462	0.613628
$540$	0	0
$541$	27.7538	1.19323	0.596614	−	0.802528i	$-0.296512\pi$
$541$	27.7538	1.19323	0.596614	+	0.802528i	$0.296512\pi$
$542$	0	0
$543$	1.12311	0.0481971
$544$	0	0
$545$	0.246211	0.0105465
$546$	0	0
$547$	10.2462	0.438096	0.219048	−	0.975714i	$-0.429705\pi$
$547$	10.2462	0.438096	0.219048	+	0.975714i	$0.429705\pi$
$548$	0	0
$549$	9.12311	0.389365
$550$	0	0
$551$	−13.3693	−0.569552
$552$	0	0
$553$	4.49242	0.191037
$554$	0	0
$555$	−2.43845	−0.103506
$556$	0	0
$557$	12.0540	0.510743	0.255372	−	0.966843i	$-0.417802\pi$
$557$	12.0540	0.510743	0.255372	+	0.966843i	$0.417802\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	11.1231	0.469618
$562$	0	0
$563$	3.31534	0.139725	0.0698625	−	0.997557i	$-0.477744\pi$
$563$	3.31534	0.139725	0.0698625	+	0.997557i	$0.477744\pi$
$564$	0	0
$565$	7.56155	0.318117
$566$	0	0
$567$	−1.56155	−0.0655791
$568$	0	0
$569$	−38.4924	−1.61369	−0.806843	−	0.590766i	$-0.798826\pi$
$569$	−38.4924	−1.61369	−0.806843	+	0.590766i	$0.798826\pi$
$570$	0	0
$571$	−11.3693	−0.475791	−0.237896	−	0.971291i	$-0.576458\pi$
$571$	−11.3693	−0.475791	−0.237896	+	0.971291i	$0.576458\pi$
$572$	0	0
$573$	−12.4924	−0.521878
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	2.87689	0.119767	0.0598833	−	0.998205i	$-0.480927\pi$
$577$	2.87689	0.119767	0.0598833	+	0.998205i	$0.480927\pi$
$578$	0	0
$579$	13.1231	0.545378
$580$	0	0
$581$	−19.8078	−0.821765
$582$	0	0
$583$	−23.6155	−0.978055
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	−11.6155	−0.479424	−0.239712	−	0.970844i	$-0.577053\pi$
$587$	−11.6155	−0.479424	−0.239712	+	0.970844i	$0.577053\pi$
$588$	0	0
$589$	−9.36932	−0.386056
$590$	0	0
$591$	1.36932	0.0563262
$592$	0	0
$593$	−5.75379	−0.236280	−0.118140	−	0.992997i	$-0.537693\pi$
$593$	−5.75379	−0.236280	−0.118140	+	0.992997i	$0.537693\pi$
$594$	0	0
$595$	−5.56155	−0.228001
$596$	0	0
$597$	20.2462	0.828622
$598$	0	0
$599$	−5.12311	−0.209324	−0.104662	−	0.994508i	$-0.533376\pi$
$599$	−5.12311	−0.209324	−0.104662	+	0.994508i	$0.533376\pi$
$600$	0	0
$601$	−38.7926	−1.58238	−0.791192	−	0.611568i	$-0.790539\pi$
$601$	−38.7926	−1.58238	−0.791192	+	0.611568i	$0.790539\pi$
$602$	0	0
$603$	8.68466	0.353667
$604$	0	0
$605$	−1.24621	−0.0506657
$606$	0	0
$607$	2.73863	0.111158	0.0555789	−	0.998454i	$-0.482300\pi$
$607$	2.73863	0.111158	0.0555789	+	0.998454i	$0.482300\pi$
$608$	0	0
$609$	10.4384	0.422987
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−25.8617	−1.04455	−0.522273	−	0.852778i	$-0.674916\pi$
$613$	−25.8617	−1.04455	−0.522273	+	0.852778i	$0.674916\pi$
$614$	0	0
$615$	2.68466	0.108256
$616$	0	0
$617$	29.4233	1.18454	0.592269	−	0.805741i	$-0.298232\pi$
$617$	29.4233	1.18454	0.592269	+	0.805741i	$0.298232\pi$
$618$	0	0
$619$	39.3693	1.58239	0.791193	−	0.611566i	$-0.209460\pi$
$619$	39.3693	1.58239	0.791193	+	0.611566i	$0.209460\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	8.00000	0.320513
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.24621	−0.249450
$628$	0	0
$629$	8.68466	0.346280
$630$	0	0
$631$	8.63068	0.343582	0.171791	−	0.985133i	$-0.445045\pi$
$631$	8.63068	0.343582	0.171791	+	0.985133i	$0.445045\pi$
$632$	0	0
$633$	−6.93087	−0.275477
$634$	0	0
$635$	6.24621	0.247873
$636$	0	0
$637$	−9.12311	−0.361471
$638$	0	0
$639$	0.438447	0.0173447
$640$	0	0
$641$	−36.7386	−1.45109	−0.725544	−	0.688175i	$-0.758412\pi$
$641$	−36.7386	−1.45109	−0.725544	+	0.688175i	$0.758412\pi$
$642$	0	0
$643$	−9.06913	−0.357652	−0.178826	−	0.983881i	$-0.557230\pi$
$643$	−9.06913	−0.357652	−0.178826	+	0.983881i	$0.557230\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.8617	0.702217	0.351109	−	0.936335i	$-0.385805\pi$
$647$	17.8617	0.702217	0.351109	+	0.936335i	$0.385805\pi$
$648$	0	0
$649$	−11.1231	−0.436620
$650$	0	0
$651$	7.31534	0.286711
$652$	0	0
$653$	−0.384472	−0.0150455	−0.00752277	−	0.999972i	$-0.502395\pi$
$653$	−0.384472	−0.0150455	−0.00752277	+	0.999972i	$0.502395\pi$
$654$	0	0
$655$	11.3693	0.444236
$656$	0	0
$657$	−4.24621	−0.165660
$658$	0	0
$659$	−1.36932	−0.0533410	−0.0266705	−	0.999644i	$-0.508490\pi$
$659$	−1.36932	−0.0533410	−0.0266705	+	0.999644i	$0.508490\pi$
$660$	0	0
$661$	46.9848	1.82750	0.913749	−	0.406278i	$-0.133174\pi$
$661$	46.9848	1.82750	0.913749	+	0.406278i	$0.133174\pi$
$662$	0	0
$663$	−7.12311	−0.276638
$664$	0	0
$665$	3.12311	0.121109
$666$	0	0
$667$	6.68466	0.258831
$668$	0	0
$669$	12.8769	0.497849
$670$	0	0
$671$	−28.4924	−1.09994
$672$	0	0
$673$	38.8769	1.49859	0.749297	−	0.662234i	$-0.230392\pi$
$673$	38.8769	1.49859	0.749297	+	0.662234i	$0.230392\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−39.5616	−1.52047	−0.760237	−	0.649646i	$-0.774917\pi$
$677$	−39.5616	−1.52047	−0.760237	+	0.649646i	$0.774917\pi$
$678$	0	0
$679$	−17.3693	−0.666573
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	−13.7538	−0.526274	−0.263137	−	0.964758i	$-0.584757\pi$
$683$	−13.7538	−0.526274	−0.263137	+	0.964758i	$0.584757\pi$
$684$	0	0
$685$	22.4924	0.859391
$686$	0	0
$687$	0.630683	0.0240621
$688$	0	0
$689$	15.1231	0.576144
$690$	0	0
$691$	50.7386	1.93019	0.965094	−	0.261903i	$-0.0843499\pi$
$691$	50.7386	1.93019	0.965094	+	0.261903i	$0.0843499\pi$
$692$	0	0
$693$	4.87689	0.185258
$694$	0	0
$695$	−18.9309	−0.718089
$696$	0	0
$697$	−9.56155	−0.362170
$698$	0	0
$699$	8.00000	0.302588
$700$	0	0
$701$	1.61553	0.0610177	0.0305088	−	0.999534i	$-0.490287\pi$
$701$	1.61553	0.0610177	0.0305088	+	0.999534i	$0.490287\pi$
$702$	0	0
$703$	−4.87689	−0.183936
$704$	0	0
$705$	−4.00000	−0.150649
$706$	0	0
$707$	2.82292	0.106167
$708$	0	0
$709$	−11.3693	−0.426984	−0.213492	−	0.976945i	$-0.568484\pi$
$709$	−11.3693	−0.426984	−0.213492	+	0.976945i	$0.568484\pi$
$710$	0	0
$711$	−2.87689	−0.107892
$712$	0	0
$713$	4.68466	0.175442
$714$	0	0
$715$	−6.24621	−0.233595
$716$	0	0
$717$	−1.80776	−0.0675122
$718$	0	0
$719$	−12.9309	−0.482240	−0.241120	−	0.970495i	$-0.577515\pi$
$719$	−12.9309	−0.482240	−0.241120	+	0.970495i	$0.577515\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−18.4924	−0.687741
$724$	0	0
$725$	6.68466	0.248262
$726$	0	0
$727$	−1.17708	−0.0436555	−0.0218278	−	0.999762i	$-0.506949\pi$
$727$	−1.17708	−0.0436555	−0.0218278	+	0.999762i	$0.506949\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	33.1771	1.22542	0.612712	−	0.790306i	$-0.290079\pi$
$733$	33.1771	1.22542	0.612712	+	0.790306i	$0.290079\pi$
$734$	0	0
$735$	4.56155	0.168255
$736$	0	0
$737$	−27.1231	−0.999092
$738$	0	0
$739$	37.6695	1.38570	0.692848	−	0.721084i	$-0.256356\pi$
$739$	37.6695	1.38570	0.692848	+	0.721084i	$0.256356\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	−7.36932	−0.269991
$746$	0	0
$747$	12.6847	0.464107
$748$	0	0
$749$	−19.8078	−0.723760
$750$	0	0
$751$	−15.7538	−0.574864	−0.287432	−	0.957801i	$-0.592802\pi$
$751$	−15.7538	−0.574864	−0.287432	+	0.957801i	$0.592802\pi$
$752$	0	0
$753$	13.3693	0.487205
$754$	0	0
$755$	6.24621	0.227323
$756$	0	0
$757$	5.56155	0.202138	0.101069	−	0.994879i	$-0.467774\pi$
$757$	5.56155	0.202138	0.101069	+	0.994879i	$0.467774\pi$
$758$	0	0
$759$	3.12311	0.113362
$760$	0	0
$761$	42.3002	1.53338	0.766690	−	0.642017i	$-0.221902\pi$
$761$	42.3002	1.53338	0.766690	+	0.642017i	$0.221902\pi$
$762$	0	0
$763$	−0.384472	−0.0139188
$764$	0	0
$765$	3.56155	0.128768
$766$	0	0
$767$	7.12311	0.257200
$768$	0	0
$769$	−11.3693	−0.409988	−0.204994	−	0.978763i	$-0.565717\pi$
$769$	−11.3693	−0.409988	−0.204994	+	0.978763i	$0.565717\pi$
$770$	0	0
$771$	16.4924	0.593960
$772$	0	0
$773$	28.2462	1.01595	0.507973	−	0.861373i	$-0.330395\pi$
$773$	28.2462	1.01595	0.507973	+	0.861373i	$0.330395\pi$
$774$	0	0
$775$	4.68466	0.168278
$776$	0	0
$777$	3.80776	0.136603
$778$	0	0
$779$	5.36932	0.192376
$780$	0	0
$781$	−1.36932	−0.0489980
$782$	0	0
$783$	−6.68466	−0.238890
$784$	0	0
$785$	−10.9309	−0.390139
$786$	0	0
$787$	35.4233	1.26270	0.631352	−	0.775496i	$-0.282500\pi$
$787$	35.4233	1.26270	0.631352	+	0.775496i	$0.282500\pi$
$788$	0	0
$789$	22.9309	0.816361
$790$	0	0
$791$	−11.8078	−0.419836
$792$	0	0
$793$	18.2462	0.647942
$794$	0	0
$795$	−7.56155	−0.268181
$796$	0	0
$797$	−22.3002	−0.789913	−0.394957	−	0.918700i	$-0.629240\pi$
$797$	−22.3002	−0.789913	−0.394957	+	0.918700i	$0.629240\pi$
$798$	0	0
$799$	14.2462	0.503995
$800$	0	0
$801$	−5.12311	−0.181016
$802$	0	0
$803$	13.2614	0.467983
$804$	0	0
$805$	−1.56155	−0.0550375
$806$	0	0
$807$	16.4384	0.578661
$808$	0	0
$809$	16.4384	0.577945	0.288973	−	0.957337i	$-0.406686\pi$
$809$	16.4384	0.577945	0.288973	+	0.957337i	$0.406686\pi$
$810$	0	0
$811$	28.6847	1.00725	0.503627	−	0.863921i	$-0.331998\pi$
$811$	28.6847	1.00725	0.503627	+	0.863921i	$0.331998\pi$
$812$	0	0
$813$	22.9309	0.804221
$814$	0	0
$815$	−21.3693	−0.748535
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−3.12311	−0.109130
$820$	0	0
$821$	−48.2462	−1.68380	−0.841902	−	0.539630i	$-0.818564\pi$
$821$	−48.2462	−1.68380	−0.841902	+	0.539630i	$0.818564\pi$
$822$	0	0
$823$	2.73863	0.0954628	0.0477314	−	0.998860i	$-0.484801\pi$
$823$	2.73863	0.0954628	0.0477314	+	0.998860i	$0.484801\pi$
$824$	0	0
$825$	3.12311	0.108733
$826$	0	0
$827$	25.5616	0.888862	0.444431	−	0.895813i	$-0.353406\pi$
$827$	25.5616	0.888862	0.444431	+	0.895813i	$0.353406\pi$
$828$	0	0
$829$	24.5464	0.852532	0.426266	−	0.904598i	$-0.359829\pi$
$829$	24.5464	0.852532	0.426266	+	0.904598i	$0.359829\pi$
$830$	0	0
$831$	8.24621	0.286058
$832$	0	0
$833$	−16.2462	−0.562898
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	−4.68466	−0.161925
$838$	0	0
$839$	28.8769	0.996941	0.498471	−	0.866907i	$-0.333895\pi$
$839$	28.8769	0.996941	0.498471	+	0.866907i	$0.333895\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	0	0
$843$	−0.246211	−0.00847997
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	1.94602	0.0668662
$848$	0	0
$849$	14.4384	0.495526
$850$	0	0
$851$	2.43845	0.0835889
$852$	0	0
$853$	48.7386	1.66878	0.834390	−	0.551175i	$-0.185820\pi$
$853$	48.7386	1.66878	0.834390	+	0.551175i	$0.185820\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	0	0
$857$	14.6307	0.499775	0.249887	−	0.968275i	$-0.419606\pi$
$857$	14.6307	0.499775	0.249887	+	0.968275i	$0.419606\pi$
$858$	0	0
$859$	20.1922	0.688950	0.344475	−	0.938795i	$-0.388057\pi$
$859$	20.1922	0.688950	0.344475	+	0.938795i	$0.388057\pi$
$860$	0	0
$861$	−4.19224	−0.142871
$862$	0	0
$863$	−12.8769	−0.438335	−0.219167	−	0.975687i	$-0.570334\pi$
$863$	−12.8769	−0.438335	−0.219167	+	0.975687i	$0.570334\pi$
$864$	0	0
$865$	14.2462	0.484386
$866$	0	0
$867$	4.31534	0.146557
$868$	0	0
$869$	8.98485	0.304790
$870$	0	0
$871$	17.3693	0.588537
$872$	0	0
$873$	11.1231	0.376460
$874$	0	0
$875$	−1.56155	−0.0527901
$876$	0	0
$877$	−56.7386	−1.91593	−0.957964	−	0.286889i	$-0.907379\pi$
$877$	−56.7386	−1.91593	−0.957964	+	0.286889i	$0.907379\pi$
$878$	0	0
$879$	−8.93087	−0.301231
$880$	0	0
$881$	−21.6155	−0.728246	−0.364123	−	0.931351i	$-0.618631\pi$
$881$	−21.6155	−0.728246	−0.364123	+	0.931351i	$0.618631\pi$
$882$	0	0
$883$	−35.2311	−1.18562	−0.592810	−	0.805343i	$-0.701981\pi$
$883$	−35.2311	−1.18562	−0.592810	+	0.805343i	$0.701981\pi$
$884$	0	0
$885$	−3.56155	−0.119720
$886$	0	0
$887$	−0.384472	−0.0129093	−0.00645465	−	0.999979i	$-0.502055\pi$
$887$	−0.384472	−0.0129093	−0.00645465	+	0.999979i	$0.502055\pi$
$888$	0	0
$889$	−9.75379	−0.327132
$890$	0	0
$891$	−3.12311	−0.104628
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	−5.12311	−0.171247
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	31.3153	1.04443
$900$	0	0
$901$	26.9309	0.897197
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.12311	−0.0373333
$906$	0	0
$907$	−50.5464	−1.67837	−0.839183	−	0.543849i	$-0.816966\pi$
$907$	−50.5464	−1.67837	−0.839183	+	0.543849i	$0.816966\pi$
$908$	0	0
$909$	−1.80776	−0.0599597
$910$	0	0
$911$	29.7538	0.985787	0.492894	−	0.870090i	$-0.335939\pi$
$911$	29.7538	0.985787	0.492894	+	0.870090i	$0.335939\pi$
$912$	0	0
$913$	−39.6155	−1.31108
$914$	0	0
$915$	−9.12311	−0.301601
$916$	0	0
$917$	−17.7538	−0.586282
$918$	0	0
$919$	−22.8769	−0.754639	−0.377320	−	0.926083i	$-0.623154\pi$
$919$	−22.8769	−0.754639	−0.377320	+	0.926083i	$0.623154\pi$
$920$	0	0
$921$	−0.876894	−0.0288947
$922$	0	0
$923$	0.876894	0.0288633
$924$	0	0
$925$	2.43845	0.0801756
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.93087	0.293012	0.146506	−	0.989210i	$-0.453197\pi$
$929$	8.93087	0.293012	0.146506	+	0.989210i	$0.453197\pi$
$930$	0	0
$931$	9.12311	0.298998
$932$	0	0
$933$	7.36932	0.241261
$934$	0	0
$935$	−11.1231	−0.363764
$936$	0	0
$937$	−32.1080	−1.04892	−0.524461	−	0.851435i	$-0.675733\pi$
$937$	−32.1080	−1.04892	−0.524461	+	0.851435i	$0.675733\pi$
$938$	0	0
$939$	−9.56155	−0.312029
$940$	0	0
$941$	−37.6155	−1.22623	−0.613116	−	0.789993i	$-0.710084\pi$
$941$	−37.6155	−1.22623	−0.613116	+	0.789993i	$0.710084\pi$
$942$	0	0
$943$	−2.68466	−0.0874245
$944$	0	0
$945$	1.56155	0.0507973
$946$	0	0
$947$	−11.5076	−0.373946	−0.186973	−	0.982365i	$-0.559868\pi$
$947$	−11.5076	−0.373946	−0.186973	+	0.982365i	$0.559868\pi$
$948$	0	0
$949$	−8.49242	−0.275676
$950$	0	0
$951$	4.00000	0.129709
$952$	0	0
$953$	−34.9848	−1.13327	−0.566635	−	0.823969i	$-0.691755\pi$
$953$	−34.9848	−1.13327	−0.566635	+	0.823969i	$0.691755\pi$
$954$	0	0
$955$	12.4924	0.404245
$956$	0	0
$957$	20.8769	0.674854
$958$	0	0
$959$	−35.1231	−1.13418
$960$	0	0
$961$	−9.05398	−0.292064
$962$	0	0
$963$	12.6847	0.408757
$964$	0	0
$965$	−13.1231	−0.422448
$966$	0	0
$967$	−12.4924	−0.401729	−0.200865	−	0.979619i	$-0.564375\pi$
$967$	−12.4924	−0.401729	−0.200865	+	0.979619i	$0.564375\pi$
$968$	0	0
$969$	7.12311	0.228827
$970$	0	0
$971$	−2.24621	−0.0720843	−0.0360422	−	0.999350i	$-0.511475\pi$
$971$	−2.24621	−0.0720843	−0.0360422	+	0.999350i	$0.511475\pi$
$972$	0	0
$973$	29.5616	0.947700
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	−9.80776	−0.313778	−0.156889	−	0.987616i	$-0.550147\pi$
$977$	−9.80776	−0.313778	−0.156889	+	0.987616i	$0.550147\pi$
$978$	0	0
$979$	16.0000	0.511362
$980$	0	0
$981$	0.246211	0.00786092
$982$	0	0
$983$	25.6695	0.818730	0.409365	−	0.912371i	$-0.365750\pi$
$983$	25.6695	0.818730	0.409365	+	0.912371i	$0.365750\pi$
$984$	0	0
$985$	−1.36932	−0.0436301
$986$	0	0
$987$	6.24621	0.198819
$988$	0	0
$989$	0	0
$990$	0	0
$991$	23.4233	0.744065	0.372033	−	0.928220i	$-0.378661\pi$
$991$	23.4233	0.744065	0.372033	+	0.928220i	$0.378661\pi$
$992$	0	0
$993$	1.56155	0.0495544
$994$	0	0
$995$	−20.2462	−0.641848
$996$	0	0
$997$	−34.8769	−1.10456	−0.552281	−	0.833658i	$-0.686243\pi$
$997$	−34.8769	−1.10456	−0.552281	+	0.833658i	$0.686243\pi$
$998$	0	0
$999$	−2.43845	−0.0771491

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.a.g.1.1	✓	2
3.2	odd	2		4140.2.a.n.1.1		2
4.3	odd	2		5520.2.a.br.1.2		2
5.2	odd	4		6900.2.f.p.6349.3		4
5.3	odd	4		6900.2.f.p.6349.2		4
5.4	even	2		6900.2.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.g.1.1	✓	2	1.1	even	1	trivial
4140.2.a.n.1.1		2	3.2	odd	2
5520.2.a.br.1.2		2	4.3	odd	2
6900.2.a.u.1.2		2	5.4	even	2
6900.2.f.p.6349.2		4	5.3	odd	4
6900.2.f.p.6349.3		4	5.2	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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} -3.12311 q^{11} +2.00000 q^{13} -1.00000 q^{15} +3.56155 q^{17} -2.00000 q^{19} +1.56155 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.68466 q^{29} +4.68466 q^{31} +3.12311 q^{33} -1.56155 q^{35} +2.43845 q^{37} -2.00000 q^{39} -2.68466 q^{41} +1.00000 q^{45} +4.00000 q^{47} -4.56155 q^{49} -3.56155 q^{51} +7.56155 q^{53} -3.12311 q^{55} +2.00000 q^{57} +3.56155 q^{59} +9.12311 q^{61} -1.56155 q^{63} +2.00000 q^{65} +8.68466 q^{67} -1.00000 q^{69} +0.438447 q^{71} -4.24621 q^{73} -1.00000 q^{75} +4.87689 q^{77} -2.87689 q^{79} +1.00000 q^{81} +12.6847 q^{83} +3.56155 q^{85} -6.68466 q^{87} -5.12311 q^{89} -3.12311 q^{91} -4.68466 q^{93} -2.00000 q^{95} +11.1231 q^{97} -3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} - 2 q^{15} + 3 q^{17} - 4 q^{19} - q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} + q^{29} - 3 q^{31} - 2 q^{33} + q^{35} + 9 q^{37} - 4 q^{39} + 7 q^{41} + 2 q^{45} + 8 q^{47} - 5 q^{49} - 3 q^{51} + 11 q^{53} + 2 q^{55} + 4 q^{57} + 3 q^{59} + 10 q^{61} + q^{63} + 4 q^{65} + 5 q^{67} - 2 q^{69} + 5 q^{71} + 8 q^{73} - 2 q^{75} + 18 q^{77} - 14 q^{79} + 2 q^{81} + 13 q^{83} + 3 q^{85} - q^{87} - 2 q^{89} + 2 q^{91} + 3 q^{93} - 4 q^{95} + 14 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 - 2 * q^15 + 3 * q^17 - 4 * q^19 - q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 + q^29 - 3 * q^31 - 2 * q^33 + q^35 + 9 * q^37 - 4 * q^39 + 7 * q^41 + 2 * q^45 + 8 * q^47 - 5 * q^49 - 3 * q^51 + 11 * q^53 + 2 * q^55 + 4 * q^57 + 3 * q^59 + 10 * q^61 + q^63 + 4 * q^65 + 5 * q^67 - 2 * q^69 + 5 * q^71 + 8 * q^73 - 2 * q^75 + 18 * q^77 - 14 * q^79 + 2 * q^81 + 13 * q^83 + 3 * q^85 - q^87 - 2 * q^89 + 2 * q^91 + 3 * q^93 - 4 * q^95 + 14 * q^97 + 2 * q^99

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