LMFDB - Embedded newform 1575.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.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:	$12.5764383184$
Analytic rank:	$1$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{7} -6.56155 q^{8} +O(q^{10})$	$q-2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{7} -6.56155 q^{8} +1.56155 q^{11} -0.438447 q^{13} -2.56155 q^{14} +7.68466 q^{16} -0.438447 q^{17} -7.12311 q^{19} -4.00000 q^{22} +3.12311 q^{23} +1.12311 q^{26} +4.56155 q^{28} -6.68466 q^{29} -6.56155 q^{32} +1.12311 q^{34} -6.00000 q^{37} +18.2462 q^{38} -5.12311 q^{41} -0.876894 q^{43} +7.12311 q^{44} -8.00000 q^{46} -8.68466 q^{47} +1.00000 q^{49} -2.00000 q^{52} -5.12311 q^{53} -6.56155 q^{56} +17.1231 q^{58} +4.00000 q^{59} +15.3693 q^{61} +1.43845 q^{64} -10.2462 q^{67} -2.00000 q^{68} -8.00000 q^{71} +12.2462 q^{73} +15.3693 q^{74} -32.4924 q^{76} +1.56155 q^{77} -2.43845 q^{79} +13.1231 q^{82} +4.00000 q^{83} +2.24621 q^{86} -10.2462 q^{88} +1.12311 q^{89} -0.438447 q^{91} +14.2462 q^{92} +22.2462 q^{94} -5.80776 q^{97} -2.56155 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 5 q^{4} + 2 q^{7} - 9 q^{8}+O(q^{10}) $ 2 * q - q^2 + 5 * q^4 + 2 * q^7 - 9 * q^8	$ 2 q - q^{2} + 5 q^{4} + 2 q^{7} - 9 q^{8} - q^{11} - 5 q^{13} - q^{14} + 3 q^{16} - 5 q^{17} - 6 q^{19} - 8 q^{22} - 2 q^{23} - 6 q^{26} + 5 q^{28} - q^{29} - 9 q^{32} - 6 q^{34} - 12 q^{37} + 20 q^{38} - 2 q^{41} - 10 q^{43} + 6 q^{44} - 16 q^{46} - 5 q^{47} + 2 q^{49} - 4 q^{52} - 2 q^{53} - 9 q^{56} + 26 q^{58} + 8 q^{59} + 6 q^{61} + 7 q^{64} - 4 q^{67} - 4 q^{68} - 16 q^{71} + 8 q^{73} + 6 q^{74} - 32 q^{76} - q^{77} - 9 q^{79} + 18 q^{82} + 8 q^{83} - 12 q^{86} - 4 q^{88} - 6 q^{89} - 5 q^{91} + 12 q^{92} + 28 q^{94} + 9 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 + 5 * q^4 + 2 * q^7 - 9 * q^8 - q^11 - 5 * q^13 - q^14 + 3 * q^16 - 5 * q^17 - 6 * q^19 - 8 * q^22 - 2 * q^23 - 6 * q^26 + 5 * q^28 - q^29 - 9 * q^32 - 6 * q^34 - 12 * q^37 + 20 * q^38 - 2 * q^41 - 10 * q^43 + 6 * q^44 - 16 * q^46 - 5 * q^47 + 2 * q^49 - 4 * q^52 - 2 * q^53 - 9 * q^56 + 26 * q^58 + 8 * q^59 + 6 * q^61 + 7 * q^64 - 4 * q^67 - 4 * q^68 - 16 * q^71 + 8 * q^73 + 6 * q^74 - 32 * q^76 - q^77 - 9 * q^79 + 18 * q^82 + 8 * q^83 - 12 * q^86 - 4 * q^88 - 6 * q^89 - 5 * q^91 + 12 * q^92 + 28 * q^94 + 9 * q^97 - q^98

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 $

$2$	−2.56155	−1.81129	−0.905646	−	0.424035i	$-0.860613\pi$
$2$	−2.56155	−1.81129	−0.905646	+	0.424035i	$0.860613\pi$
$3$	0	0
$3$	0	0
$4$	4.56155	2.28078
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−6.56155	−2.31986
$9$	0	0
$10$	0	0
$11$	1.56155	0.470826	0.235413	−	0.971895i	$-0.424356\pi$
$11$	1.56155	0.470826	0.235413	+	0.971895i	$0.424356\pi$
$12$	0	0
$13$	−0.438447	−0.121603	−0.0608017	−	0.998150i	$-0.519366\pi$
$13$	−0.438447	−0.121603	−0.0608017	+	0.998150i	$0.519366\pi$
$14$	−2.56155	−0.684604
$15$	0	0
$16$	7.68466	1.92116
$17$	−0.438447	−0.106339	−0.0531695	−	0.998586i	$-0.516932\pi$
$17$	−0.438447	−0.106339	−0.0531695	+	0.998586i	$0.516932\pi$
$18$	0	0
$19$	−7.12311	−1.63415	−0.817076	−	0.576530i	$-0.804407\pi$
$19$	−7.12311	−1.63415	−0.817076	+	0.576530i	$0.804407\pi$
$20$	0	0
$21$	0	0
$22$	−4.00000	−0.852803
$23$	3.12311	0.651213	0.325606	−	0.945505i	$-0.394432\pi$
$23$	3.12311	0.651213	0.325606	+	0.945505i	$0.394432\pi$
$24$	0	0
$25$	0	0
$26$	1.12311	0.220259
$27$	0	0
$28$	4.56155	0.862052
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−6.56155	−1.15993
$33$	0	0
$34$	1.12311	0.192611
$35$	0	0
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	18.2462	2.95993
$39$	0	0
$40$	0	0
$41$	−5.12311	−0.800095	−0.400047	−	0.916494i	$-0.631006\pi$
$41$	−5.12311	−0.800095	−0.400047	+	0.916494i	$0.631006\pi$
$42$	0	0
$43$	−0.876894	−0.133725	−0.0668626	−	0.997762i	$-0.521299\pi$
$43$	−0.876894	−0.133725	−0.0668626	+	0.997762i	$0.521299\pi$
$44$	7.12311	1.07385
$45$	0	0
$46$	−8.00000	−1.17954
$47$	−8.68466	−1.26679	−0.633394	−	0.773830i	$-0.718339\pi$
$47$	−8.68466	−1.26679	−0.633394	+	0.773830i	$0.718339\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−5.12311	−0.703713	−0.351856	−	0.936054i	$-0.614449\pi$
$53$	−5.12311	−0.703713	−0.351856	+	0.936054i	$0.614449\pi$
$54$	0	0
$55$	0	0
$56$	−6.56155	−0.876824
$57$	0	0
$58$	17.1231	2.24837
$59$	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
$61$	15.3693	1.96784	0.983920	−	0.178611i	$-0.0571605\pi$
$61$	15.3693	1.96784	0.983920	+	0.178611i	$0.0571605\pi$
$62$	0	0
$63$	0	0
$64$	1.43845	0.179806
$65$	0	0
$66$	0	0
$67$	−10.2462	−1.25177	−0.625887	−	0.779914i	$-0.715263\pi$
$67$	−10.2462	−1.25177	−0.625887	+	0.779914i	$0.715263\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	12.2462	1.43331	0.716655	−	0.697428i	$-0.245672\pi$
$73$	12.2462	1.43331	0.716655	+	0.697428i	$0.245672\pi$
$74$	15.3693	1.78665
$75$	0	0
$76$	−32.4924	−3.72714
$77$	1.56155	0.177955
$78$	0	0
$79$	−2.43845	−0.274347	−0.137173	−	0.990547i	$-0.543802\pi$
$79$	−2.43845	−0.274347	−0.137173	+	0.990547i	$0.543802\pi$
$80$	0	0
$81$	0	0
$82$	13.1231	1.44920
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	2.24621	0.242215
$87$	0	0
$88$	−10.2462	−1.09225
$89$	1.12311	0.119049	0.0595245	−	0.998227i	$-0.481042\pi$
$89$	1.12311	0.119049	0.0595245	+	0.998227i	$0.481042\pi$
$90$	0	0
$91$	−0.438447	−0.0459618
$92$	14.2462	1.48527
$93$	0	0
$94$	22.2462	2.29452
$95$	0	0
$96$	0	0
$97$	−5.80776	−0.589689	−0.294845	−	0.955545i	$-0.595268\pi$
$97$	−5.80776	−0.589689	−0.294845	+	0.955545i	$0.595268\pi$
$98$	−2.56155	−0.258756
$99$	0	0
$100$	0	0
$101$	16.2462	1.61656	0.808279	−	0.588799i	$-0.200399\pi$
$101$	16.2462	1.61656	0.808279	+	0.588799i	$0.200399\pi$
$102$	0	0
$103$	−5.56155	−0.547996	−0.273998	−	0.961730i	$-0.588346\pi$
$103$	−5.56155	−0.547996	−0.273998	+	0.961730i	$0.588346\pi$
$104$	2.87689	0.282103
$105$	0	0
$106$	13.1231	1.27463
$107$	13.3693	1.29246	0.646230	−	0.763142i	$-0.276345\pi$
$107$	13.3693	1.29246	0.646230	+	0.763142i	$0.276345\pi$
$108$	0	0
$109$	5.31534	0.509117	0.254559	−	0.967057i	$-0.418070\pi$
$109$	5.31534	0.509117	0.254559	+	0.967057i	$0.418070\pi$
$110$	0	0
$111$	0	0
$112$	7.68466	0.726132
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	−30.4924	−2.83115
$117$	0	0
$118$	−10.2462	−0.943240
$119$	−0.438447	−0.0401924
$120$	0	0
$121$	−8.56155	−0.778323
$122$	−39.3693	−3.56433
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.24621	0.554262	0.277131	−	0.960832i	$-0.410616\pi$
$127$	6.24621	0.554262	0.277131	+	0.960832i	$0.410616\pi$
$128$	9.43845	0.834249
$129$	0	0
$130$	0	0
$131$	0.876894	0.0766146	0.0383073	−	0.999266i	$-0.487803\pi$
$131$	0.876894	0.0766146	0.0383073	+	0.999266i	$0.487803\pi$
$132$	0	0
$133$	−7.12311	−0.617652
$134$	26.2462	2.26733
$135$	0	0
$136$	2.87689	0.246692
$137$	−17.1231	−1.46293	−0.731463	−	0.681881i	$-0.761162\pi$
$137$	−17.1231	−1.46293	−0.731463	+	0.681881i	$0.761162\pi$
$138$	0	0
$139$	−15.1231	−1.28273	−0.641363	−	0.767238i	$-0.721631\pi$
$139$	−15.1231	−1.28273	−0.641363	+	0.767238i	$0.721631\pi$
$140$	0	0
$141$	0	0
$142$	20.4924	1.71969
$143$	−0.684658	−0.0572540
$144$	0	0
$145$	0	0
$146$	−31.3693	−2.59614
$147$	0	0
$148$	−27.3693	−2.24974
$149$	−12.2462	−1.00325	−0.501624	−	0.865086i	$-0.667264\pi$
$149$	−12.2462	−1.00325	−0.501624	+	0.865086i	$0.667264\pi$
$150$	0	0
$151$	−6.93087	−0.564026	−0.282013	−	0.959411i	$-0.591002\pi$
$151$	−6.93087	−0.564026	−0.282013	+	0.959411i	$0.591002\pi$
$152$	46.7386	3.79100
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	−20.2462	−1.61582	−0.807912	−	0.589303i	$-0.799402\pi$
$157$	−20.2462	−1.61582	−0.807912	+	0.589303i	$0.799402\pi$
$158$	6.24621	0.496922
$159$	0	0
$160$	0	0
$161$	3.12311	0.246135
$162$	0	0
$163$	7.12311	0.557925	0.278962	−	0.960302i	$-0.410010\pi$
$163$	7.12311	0.557925	0.278962	+	0.960302i	$0.410010\pi$
$164$	−23.3693	−1.82484
$165$	0	0
$166$	−10.2462	−0.795260
$167$	−6.93087	−0.536327	−0.268163	−	0.963373i	$-0.586417\pi$
$167$	−6.93087	−0.536327	−0.268163	+	0.963373i	$0.586417\pi$
$168$	0	0
$169$	−12.8078	−0.985213
$170$	0	0
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−4.43845	−0.337449	−0.168724	−	0.985663i	$-0.553965\pi$
$173$	−4.43845	−0.337449	−0.168724	+	0.985663i	$0.553965\pi$
$174$	0	0
$175$	0	0
$176$	12.0000	0.904534
$177$	0	0
$178$	−2.87689	−0.215632
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−17.6155	−1.30935	−0.654676	−	0.755910i	$-0.727195\pi$
$181$	−17.6155	−1.30935	−0.654676	+	0.755910i	$0.727195\pi$
$182$	1.12311	0.0832501
$183$	0	0
$184$	−20.4924	−1.51072
$185$	0	0
$186$	0	0
$187$	−0.684658	−0.0500672
$188$	−39.6155	−2.88926
$189$	0	0
$190$	0	0
$191$	13.5616	0.981280	0.490640	−	0.871363i	$-0.336763\pi$
$191$	13.5616	0.981280	0.490640	+	0.871363i	$0.336763\pi$
$192$	0	0
$193$	−19.3693	−1.39423	−0.697117	−	0.716957i	$-0.745534\pi$
$193$	−19.3693	−1.39423	−0.697117	+	0.716957i	$0.745534\pi$
$194$	14.8769	1.06810
$195$	0	0
$196$	4.56155	0.325825
$197$	1.12311	0.0800180	0.0400090	−	0.999199i	$-0.487261\pi$
$197$	1.12311	0.0800180	0.0400090	+	0.999199i	$0.487261\pi$
$198$	0	0
$199$	−1.75379	−0.124323	−0.0621614	−	0.998066i	$-0.519799\pi$
$199$	−1.75379	−0.124323	−0.0621614	+	0.998066i	$0.519799\pi$
$200$	0	0
$201$	0	0
$202$	−41.6155	−2.92806
$203$	−6.68466	−0.469171
$204$	0	0
$205$	0	0
$206$	14.2462	0.992581
$207$	0	0
$208$	−3.36932	−0.233620
$209$	−11.1231	−0.769401
$210$	0	0
$211$	14.0540	0.967516	0.483758	−	0.875202i	$-0.339272\pi$
$211$	14.0540	0.967516	0.483758	+	0.875202i	$0.339272\pi$
$212$	−23.3693	−1.60501
$213$	0	0
$214$	−34.2462	−2.34102
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−13.6155	−0.922160
$219$	0	0
$220$	0	0
$221$	0.192236	0.0129312
$222$	0	0
$223$	2.43845	0.163291	0.0816453	−	0.996661i	$-0.473983\pi$
$223$	2.43845	0.163291	0.0816453	+	0.996661i	$0.473983\pi$
$224$	−6.56155	−0.438412
$225$	0	0
$226$	35.8617	2.38549
$227$	11.3153	0.751026	0.375513	−	0.926817i	$-0.377467\pi$
$227$	11.3153	0.751026	0.375513	+	0.926817i	$0.377467\pi$
$228$	0	0
$229$	10.8769	0.718765	0.359383	−	0.933190i	$-0.382987\pi$
$229$	10.8769	0.718765	0.359383	+	0.933190i	$0.382987\pi$
$230$	0	0
$231$	0	0
$232$	43.8617	2.87966
$233$	5.12311	0.335626	0.167813	−	0.985819i	$-0.446330\pi$
$233$	5.12311	0.335626	0.167813	+	0.985819i	$0.446330\pi$
$234$	0	0
$235$	0	0
$236$	18.2462	1.18773
$237$	0	0
$238$	1.12311	0.0728001
$239$	−19.8078	−1.28126	−0.640629	−	0.767851i	$-0.721326\pi$
$239$	−19.8078	−1.28126	−0.640629	+	0.767851i	$0.721326\pi$
$240$	0	0
$241$	−4.24621	−0.273523	−0.136761	−	0.990604i	$-0.543669\pi$
$241$	−4.24621	−0.273523	−0.136761	+	0.990604i	$0.543669\pi$
$242$	21.9309	1.40977
$243$	0	0
$244$	70.1080	4.48820
$245$	0	0
$246$	0	0
$247$	3.12311	0.198718
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.87689	0.560305	0.280152	−	0.959956i	$-0.409615\pi$
$251$	8.87689	0.560305	0.280152	+	0.959956i	$0.409615\pi$
$252$	0	0
$253$	4.87689	0.306608
$254$	−16.0000	−1.00393
$255$	0	0
$256$	−27.0540	−1.69087
$257$	−10.4924	−0.654499	−0.327250	−	0.944938i	$-0.606122\pi$
$257$	−10.4924	−0.654499	−0.327250	+	0.944938i	$0.606122\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	0	0
$262$	−2.24621	−0.138771
$263$	−12.8769	−0.794023	−0.397012	−	0.917814i	$-0.629953\pi$
$263$	−12.8769	−0.794023	−0.397012	+	0.917814i	$0.629953\pi$
$264$	0	0
$265$	0	0
$266$	18.2462	1.11875
$267$	0	0
$268$	−46.7386	−2.85502
$269$	20.7386	1.26446	0.632228	−	0.774782i	$-0.282140\pi$
$269$	20.7386	1.26446	0.632228	+	0.774782i	$0.282140\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−3.36932	−0.204295
$273$	0	0
$274$	43.8617	2.64978
$275$	0	0
$276$	0	0
$277$	0.246211	0.0147934	0.00739670	−	0.999973i	$-0.497646\pi$
$277$	0.246211	0.0147934	0.00739670	+	0.999973i	$0.497646\pi$
$278$	38.7386	2.32339
$279$	0	0
$280$	0	0
$281$	−12.4384	−0.742016	−0.371008	−	0.928630i	$-0.620988\pi$
$281$	−12.4384	−0.742016	−0.371008	+	0.928630i	$0.620988\pi$
$282$	0	0
$283$	11.3153	0.672627	0.336314	−	0.941750i	$-0.390820\pi$
$283$	11.3153	0.672627	0.336314	+	0.941750i	$0.390820\pi$
$284$	−36.4924	−2.16543
$285$	0	0
$286$	1.75379	0.103704
$287$	−5.12311	−0.302407
$288$	0	0
$289$	−16.8078	−0.988692
$290$	0	0
$291$	0	0
$292$	55.8617	3.26906
$293$	−2.68466	−0.156839	−0.0784197	−	0.996920i	$-0.524987\pi$
$293$	−2.68466	−0.156839	−0.0784197	+	0.996920i	$0.524987\pi$
$294$	0	0
$295$	0	0
$296$	39.3693	2.28830
$297$	0	0
$298$	31.3693	1.81718
$299$	−1.36932	−0.0791896
$300$	0	0
$301$	−0.876894	−0.0505434
$302$	17.7538	1.02162
$303$	0	0
$304$	−54.7386	−3.13948
$305$	0	0
$306$	0	0
$307$	19.3153	1.10238	0.551192	−	0.834378i	$-0.314173\pi$
$307$	19.3153	1.10238	0.551192	+	0.834378i	$0.314173\pi$
$308$	7.12311	0.405877
$309$	0	0
$310$	0	0
$311$	−31.6155	−1.79275	−0.896376	−	0.443294i	$-0.853810\pi$
$311$	−31.6155	−1.79275	−0.896376	+	0.443294i	$0.853810\pi$
$312$	0	0
$313$	22.3002	1.26048	0.630241	−	0.776400i	$-0.282956\pi$
$313$	22.3002	1.26048	0.630241	+	0.776400i	$0.282956\pi$
$314$	51.8617	2.92673
$315$	0	0
$316$	−11.1231	−0.625724
$317$	10.4924	0.589313	0.294657	−	0.955603i	$-0.404795\pi$
$317$	10.4924	0.589313	0.294657	+	0.955603i	$0.404795\pi$
$318$	0	0
$319$	−10.4384	−0.584441
$320$	0	0
$321$	0	0
$322$	−8.00000	−0.445823
$323$	3.12311	0.173774
$324$	0	0
$325$	0	0
$326$	−18.2462	−1.01056
$327$	0	0
$328$	33.6155	1.85611
$329$	−8.68466	−0.478801
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	18.2462	1.00139
$333$	0	0
$334$	17.7538	0.971444
$335$	0	0
$336$	0	0
$337$	1.50758	0.0821230	0.0410615	−	0.999157i	$-0.486926\pi$
$337$	1.50758	0.0821230	0.0410615	+	0.999157i	$0.486926\pi$
$338$	32.8078	1.78451
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.75379	0.310223
$345$	0	0
$346$	11.3693	0.611218
$347$	7.12311	0.382388	0.191194	−	0.981552i	$-0.438764\pi$
$347$	7.12311	0.382388	0.191194	+	0.981552i	$0.438764\pi$
$348$	0	0
$349$	10.4924	0.561646	0.280823	−	0.959760i	$-0.409393\pi$
$349$	10.4924	0.561646	0.280823	+	0.959760i	$0.409393\pi$
$350$	0	0
$351$	0	0
$352$	−10.2462	−0.546125
$353$	5.80776	0.309116	0.154558	−	0.987984i	$-0.450605\pi$
$353$	5.80776	0.309116	0.154558	+	0.987984i	$0.450605\pi$
$354$	0	0
$355$	0	0
$356$	5.12311	0.271524
$357$	0	0
$358$	51.2311	2.70765
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	31.7386	1.67045
$362$	45.1231	2.37162
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	8.68466	0.453335	0.226668	−	0.973972i	$-0.427217\pi$
$367$	8.68466	0.453335	0.226668	+	0.973972i	$0.427217\pi$
$368$	24.0000	1.25109
$369$	0	0
$370$	0	0
$371$	−5.12311	−0.265978
$372$	0	0
$373$	−4.63068	−0.239768	−0.119884	−	0.992788i	$-0.538252\pi$
$373$	−4.63068	−0.239768	−0.119884	+	0.992788i	$0.538252\pi$
$374$	1.75379	0.0906863
$375$	0	0
$376$	56.9848	2.93877
$377$	2.93087	0.150947
$378$	0	0
$379$	−16.4924	−0.847159	−0.423579	−	0.905859i	$-0.639227\pi$
$379$	−16.4924	−0.847159	−0.423579	+	0.905859i	$0.639227\pi$
$380$	0	0
$381$	0	0
$382$	−34.7386	−1.77738
$383$	6.24621	0.319166	0.159583	−	0.987184i	$-0.448985\pi$
$383$	6.24621	0.319166	0.159583	+	0.987184i	$0.448985\pi$
$384$	0	0
$385$	0	0
$386$	49.6155	2.52536
$387$	0	0
$388$	−26.4924	−1.34495
$389$	24.9309	1.26405	0.632023	−	0.774950i	$-0.282225\pi$
$389$	24.9309	1.26405	0.632023	+	0.774950i	$0.282225\pi$
$390$	0	0
$391$	−1.36932	−0.0692493
$392$	−6.56155	−0.331408
$393$	0	0
$394$	−2.87689	−0.144936
$395$	0	0
$396$	0	0
$397$	−27.5616	−1.38327	−0.691637	−	0.722245i	$-0.743110\pi$
$397$	−27.5616	−1.38327	−0.691637	+	0.722245i	$0.743110\pi$
$398$	4.49242	0.225185
$399$	0	0
$400$	0	0
$401$	−31.5616	−1.57611	−0.788054	−	0.615606i	$-0.788911\pi$
$401$	−31.5616	−1.57611	−0.788054	+	0.615606i	$0.788911\pi$
$402$	0	0
$403$	0	0
$404$	74.1080	3.68701
$405$	0	0
$406$	17.1231	0.849805
$407$	−9.36932	−0.464420
$408$	0	0
$409$	6.49242	0.321030	0.160515	−	0.987033i	$-0.448685\pi$
$409$	6.49242	0.321030	0.160515	+	0.987033i	$0.448685\pi$
$410$	0	0
$411$	0	0
$412$	−25.3693	−1.24986
$413$	4.00000	0.196827
$414$	0	0
$415$	0	0
$416$	2.87689	0.141051
$417$	0	0
$418$	28.4924	1.39361
$419$	−26.2462	−1.28221	−0.641106	−	0.767453i	$-0.721524\pi$
$419$	−26.2462	−1.28221	−0.641106	+	0.767453i	$0.721524\pi$
$420$	0	0
$421$	−2.68466	−0.130842	−0.0654211	−	0.997858i	$-0.520839\pi$
$421$	−2.68466	−0.130842	−0.0654211	+	0.997858i	$0.520839\pi$
$422$	−36.0000	−1.75245
$423$	0	0
$424$	33.6155	1.63251
$425$	0	0
$426$	0	0
$427$	15.3693	0.743773
$428$	60.9848	2.94781
$429$	0	0
$430$	0	0
$431$	19.8078	0.954106	0.477053	−	0.878874i	$-0.341705\pi$
$431$	19.8078	0.954106	0.477053	+	0.878874i	$0.341705\pi$
$432$	0	0
$433$	−8.24621	−0.396288	−0.198144	−	0.980173i	$-0.563491\pi$
$433$	−8.24621	−0.396288	−0.198144	+	0.980173i	$0.563491\pi$
$434$	0	0
$435$	0	0
$436$	24.2462	1.16118
$437$	−22.2462	−1.06418
$438$	0	0
$439$	9.36932	0.447173	0.223587	−	0.974684i	$-0.428223\pi$
$439$	9.36932	0.447173	0.223587	+	0.974684i	$0.428223\pi$
$440$	0	0
$441$	0	0
$442$	−0.492423	−0.0234221
$443$	−2.63068	−0.124988	−0.0624938	−	0.998045i	$-0.519905\pi$
$443$	−2.63068	−0.124988	−0.0624938	+	0.998045i	$0.519905\pi$
$444$	0	0
$445$	0	0
$446$	−6.24621	−0.295767
$447$	0	0
$448$	1.43845	0.0679602
$449$	1.80776	0.0853137	0.0426568	−	0.999090i	$-0.486418\pi$
$449$	1.80776	0.0853137	0.0426568	+	0.999090i	$0.486418\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	−63.8617	−3.00380
$453$	0	0
$454$	−28.9848	−1.36033
$455$	0	0
$456$	0	0
$457$	17.1231	0.800985	0.400493	−	0.916300i	$-0.368839\pi$
$457$	17.1231	0.800985	0.400493	+	0.916300i	$0.368839\pi$
$458$	−27.8617	−1.30189
$459$	0	0
$460$	0	0
$461$	13.1231	0.611204	0.305602	−	0.952159i	$-0.401142\pi$
$461$	13.1231	0.611204	0.305602	+	0.952159i	$0.401142\pi$
$462$	0	0
$463$	−12.4924	−0.580572	−0.290286	−	0.956940i	$-0.593750\pi$
$463$	−12.4924	−0.580572	−0.290286	+	0.956940i	$0.593750\pi$
$464$	−51.3693	−2.38476
$465$	0	0
$466$	−13.1231	−0.607916
$467$	22.4384	1.03833	0.519164	−	0.854675i	$-0.326243\pi$
$467$	22.4384	1.03833	0.519164	+	0.854675i	$0.326243\pi$
$468$	0	0
$469$	−10.2462	−0.473126
$470$	0	0
$471$	0	0
$472$	−26.2462	−1.20808
$473$	−1.36932	−0.0629613
$474$	0	0
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	0	0
$478$	50.7386	2.32073
$479$	−4.87689	−0.222831	−0.111415	−	0.993774i	$-0.535538\pi$
$479$	−4.87689	−0.222831	−0.111415	+	0.993774i	$0.535538\pi$
$480$	0	0
$481$	2.63068	0.119949
$482$	10.8769	0.495429
$483$	0	0
$484$	−39.0540	−1.77518
$485$	0	0
$486$	0	0
$487$	3.12311	0.141521	0.0707607	−	0.997493i	$-0.477457\pi$
$487$	3.12311	0.141521	0.0707607	+	0.997493i	$0.477457\pi$
$488$	−100.847	−4.56511
$489$	0	0
$490$	0	0
$491$	41.1771	1.85830	0.929148	−	0.369708i	$-0.120542\pi$
$491$	41.1771	1.85830	0.929148	+	0.369708i	$0.120542\pi$
$492$	0	0
$493$	2.93087	0.132000
$494$	−8.00000	−0.359937
$495$	0	0
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	41.1771	1.84334	0.921670	−	0.387976i	$-0.126826\pi$
$499$	41.1771	1.84334	0.921670	+	0.387976i	$0.126826\pi$
$500$	0	0
$501$	0	0
$502$	−22.7386	−1.01487
$503$	38.9309	1.73584	0.867921	−	0.496703i	$-0.165456\pi$
$503$	38.9309	1.73584	0.867921	+	0.496703i	$0.165456\pi$
$504$	0	0
$505$	0	0
$506$	−12.4924	−0.555356
$507$	0	0
$508$	28.4924	1.26415
$509$	11.7538	0.520978	0.260489	−	0.965477i	$-0.416116\pi$
$509$	11.7538	0.520978	0.260489	+	0.965477i	$0.416116\pi$
$510$	0	0
$511$	12.2462	0.541740
$512$	50.4233	2.22842
$513$	0	0
$514$	26.8769	1.18549
$515$	0	0
$516$	0	0
$517$	−13.5616	−0.596436
$518$	15.3693	0.675289
$519$	0	0
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	−40.4924	−1.77061	−0.885305	−	0.465011i	$-0.846050\pi$
$523$	−40.4924	−1.77061	−0.885305	+	0.465011i	$0.846050\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	32.9848	1.43821
$527$	0	0
$528$	0	0
$529$	−13.2462	−0.575922
$530$	0	0
$531$	0	0
$532$	−32.4924	−1.40873
$533$	2.24621	0.0972942
$534$	0	0
$535$	0	0
$536$	67.2311	2.90394
$537$	0	0
$538$	−53.1231	−2.29030
$539$	1.56155	0.0672608
$540$	0	0
$541$	−37.8078	−1.62548	−0.812741	−	0.582625i	$-0.802026\pi$
$541$	−37.8078	−1.62548	−0.812741	+	0.582625i	$0.802026\pi$
$542$	40.9848	1.76045
$543$	0	0
$544$	2.87689	0.123346
$545$	0	0
$546$	0	0
$547$	2.24621	0.0960411	0.0480205	−	0.998846i	$-0.484709\pi$
$547$	2.24621	0.0960411	0.0480205	+	0.998846i	$0.484709\pi$
$548$	−78.1080	−3.33661
$549$	0	0
$550$	0	0
$551$	47.6155	2.02849
$552$	0	0
$553$	−2.43845	−0.103693
$554$	−0.630683	−0.0267952
$555$	0	0
$556$	−68.9848	−2.92561
$557$	−13.1231	−0.556044	−0.278022	−	0.960575i	$-0.589679\pi$
$557$	−13.1231	−0.556044	−0.278022	+	0.960575i	$0.589679\pi$
$558$	0	0
$559$	0.384472	0.0162614
$560$	0	0
$561$	0	0
$562$	31.8617	1.34401
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	0	0
$565$	0	0
$566$	−28.9848	−1.21832
$567$	0	0
$568$	52.4924	2.20253
$569$	30.9848	1.29895	0.649476	−	0.760382i	$-0.274988\pi$
$569$	30.9848	1.29895	0.649476	+	0.760382i	$0.274988\pi$
$570$	0	0
$571$	40.4924	1.69456	0.847278	−	0.531150i	$-0.178240\pi$
$571$	40.4924	1.69456	0.847278	+	0.531150i	$0.178240\pi$
$572$	−3.12311	−0.130584
$573$	0	0
$574$	13.1231	0.547748
$575$	0	0
$576$	0	0
$577$	24.0540	1.00138	0.500690	−	0.865627i	$-0.333080\pi$
$577$	24.0540	1.00138	0.500690	+	0.865627i	$0.333080\pi$
$578$	43.0540	1.79081
$579$	0	0
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	−8.00000	−0.331326
$584$	−80.3542	−3.32508
$585$	0	0
$586$	6.87689	0.284082
$587$	−26.2462	−1.08330	−0.541649	−	0.840605i	$-0.682200\pi$
$587$	−26.2462	−1.08330	−0.541649	+	0.840605i	$0.682200\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−46.1080	−1.89503
$593$	−27.5616	−1.13182	−0.565909	−	0.824468i	$-0.691475\pi$
$593$	−27.5616	−1.13182	−0.565909	+	0.824468i	$0.691475\pi$
$594$	0	0
$595$	0	0
$596$	−55.8617	−2.28819
$597$	0	0
$598$	3.50758	0.143436
$599$	11.8078	0.482452	0.241226	−	0.970469i	$-0.422450\pi$
$599$	11.8078	0.482452	0.241226	+	0.970469i	$0.422450\pi$
$600$	0	0
$601$	6.49242	0.264831	0.132416	−	0.991194i	$-0.457727\pi$
$601$	6.49242	0.264831	0.132416	+	0.991194i	$0.457727\pi$
$602$	2.24621	0.0915487
$603$	0	0
$604$	−31.6155	−1.28642
$605$	0	0
$606$	0	0
$607$	42.0540	1.70692	0.853459	−	0.521160i	$-0.174500\pi$
$607$	42.0540	1.70692	0.853459	+	0.521160i	$0.174500\pi$
$608$	46.7386	1.89550
$609$	0	0
$610$	0	0
$611$	3.80776	0.154046
$612$	0	0
$613$	−40.7386	−1.64542	−0.822709	−	0.568463i	$-0.807538\pi$
$613$	−40.7386	−1.64542	−0.822709	+	0.568463i	$0.807538\pi$
$614$	−49.4773	−1.99674
$615$	0	0
$616$	−10.2462	−0.412832
$617$	32.2462	1.29818	0.649092	−	0.760710i	$-0.275149\pi$
$617$	32.2462	1.29818	0.649092	+	0.760710i	$0.275149\pi$
$618$	0	0
$619$	32.1080	1.29053	0.645264	−	0.763960i	$-0.276747\pi$
$619$	32.1080	1.29053	0.645264	+	0.763960i	$0.276747\pi$
$620$	0	0
$621$	0	0
$622$	80.9848	3.24720
$623$	1.12311	0.0449963
$624$	0	0
$625$	0	0
$626$	−57.1231	−2.28310
$627$	0	0
$628$	−92.3542	−3.68533
$629$	2.63068	0.104892
$630$	0	0
$631$	−11.8078	−0.470060	−0.235030	−	0.971988i	$-0.575519\pi$
$631$	−11.8078	−0.470060	−0.235030	+	0.971988i	$0.575519\pi$
$632$	16.0000	0.636446
$633$	0	0
$634$	−26.8769	−1.06742
$635$	0	0
$636$	0	0
$637$	−0.438447	−0.0173719
$638$	26.7386	1.05859
$639$	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
$643$	−1.56155	−0.0615816	−0.0307908	−	0.999526i	$-0.509803\pi$
$643$	−1.56155	−0.0615816	−0.0307908	+	0.999526i	$0.509803\pi$
$644$	14.2462	0.561379
$645$	0	0
$646$	−8.00000	−0.314756
$647$	36.4924	1.43467	0.717333	−	0.696731i	$-0.245363\pi$
$647$	36.4924	1.43467	0.717333	+	0.696731i	$0.245363\pi$
$648$	0	0
$649$	6.24621	0.245185
$650$	0	0
$651$	0	0
$652$	32.4924	1.27250
$653$	−33.2311	−1.30043	−0.650216	−	0.759750i	$-0.725322\pi$
$653$	−33.2311	−1.30043	−0.650216	+	0.759750i	$0.725322\pi$
$654$	0	0
$655$	0	0
$656$	−39.3693	−1.53711
$657$	0	0
$658$	22.2462	0.867248
$659$	−9.17708	−0.357488	−0.178744	−	0.983896i	$-0.557203\pi$
$659$	−9.17708	−0.357488	−0.178744	+	0.983896i	$0.557203\pi$
$660$	0	0
$661$	−5.12311	−0.199266	−0.0996329	−	0.995024i	$-0.531767\pi$
$661$	−5.12311	−0.199266	−0.0996329	+	0.995024i	$0.531767\pi$
$662$	−30.7386	−1.19469
$663$	0	0
$664$	−26.2462	−1.01855
$665$	0	0
$666$	0	0
$667$	−20.8769	−0.808357
$668$	−31.6155	−1.22324
$669$	0	0
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	−31.8617	−1.22818	−0.614090	−	0.789236i	$-0.710477\pi$
$673$	−31.8617	−1.22818	−0.614090	+	0.789236i	$0.710477\pi$
$674$	−3.86174	−0.148749
$675$	0	0
$676$	−58.4233	−2.24705
$677$	4.93087	0.189509	0.0947544	−	0.995501i	$-0.469793\pi$
$677$	4.93087	0.189509	0.0947544	+	0.995501i	$0.469793\pi$
$678$	0	0
$679$	−5.80776	−0.222882
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.73863	−0.257847	−0.128923	−	0.991655i	$-0.541152\pi$
$683$	−6.73863	−0.257847	−0.128923	+	0.991655i	$0.541152\pi$
$684$	0	0
$685$	0	0
$686$	−2.56155	−0.0978005
$687$	0	0
$688$	−6.73863	−0.256908
$689$	2.24621	0.0855738
$690$	0	0
$691$	−24.4924	−0.931736	−0.465868	−	0.884854i	$-0.654258\pi$
$691$	−24.4924	−0.931736	−0.465868	+	0.884854i	$0.654258\pi$
$692$	−20.2462	−0.769645
$693$	0	0
$694$	−18.2462	−0.692617
$695$	0	0
$696$	0	0
$697$	2.24621	0.0850813
$698$	−26.8769	−1.01731
$699$	0	0
$700$	0	0
$701$	−28.9309	−1.09270	−0.546352	−	0.837556i	$-0.683984\pi$
$701$	−28.9309	−1.09270	−0.546352	+	0.837556i	$0.683984\pi$
$702$	0	0
$703$	42.7386	1.61192
$704$	2.24621	0.0846573
$705$	0	0
$706$	−14.8769	−0.559899
$707$	16.2462	0.611002
$708$	0	0
$709$	27.1771	1.02066	0.510328	−	0.859980i	$-0.329524\pi$
$709$	27.1771	1.02066	0.510328	+	0.859980i	$0.329524\pi$
$710$	0	0
$711$	0	0
$712$	−7.36932	−0.276177
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−91.2311	−3.40946
$717$	0	0
$718$	20.4924	0.764770
$719$	−8.38447	−0.312688	−0.156344	−	0.987703i	$-0.549971\pi$
$719$	−8.38447	−0.312688	−0.156344	+	0.987703i	$0.549971\pi$
$720$	0	0
$721$	−5.56155	−0.207123
$722$	−81.3002	−3.02568
$723$	0	0
$724$	−80.3542	−2.98634
$725$	0	0
$726$	0	0
$727$	−52.4924	−1.94684	−0.973418	−	0.229035i	$-0.926443\pi$
$727$	−52.4924	−1.94684	−0.973418	+	0.229035i	$0.926443\pi$
$728$	2.87689	0.106625
$729$	0	0
$730$	0	0
$731$	0.384472	0.0142202
$732$	0	0
$733$	−6.68466	−0.246903	−0.123452	−	0.992351i	$-0.539396\pi$
$733$	−6.68466	−0.246903	−0.123452	+	0.992351i	$0.539396\pi$
$734$	−22.2462	−0.821123
$735$	0	0
$736$	−20.4924	−0.755361
$737$	−16.0000	−0.589368
$738$	0	0
$739$	34.9309	1.28495	0.642476	−	0.766305i	$-0.277907\pi$
$739$	34.9309	1.28495	0.642476	+	0.766305i	$0.277907\pi$
$740$	0	0
$741$	0	0
$742$	13.1231	0.481764
$743$	−32.9848	−1.21010	−0.605048	−	0.796189i	$-0.706846\pi$
$743$	−32.9848	−1.21010	−0.605048	+	0.796189i	$0.706846\pi$
$744$	0	0
$745$	0	0
$746$	11.8617	0.434289
$747$	0	0
$748$	−3.12311	−0.114192
$749$	13.3693	0.488504
$750$	0	0
$751$	17.0691	0.622861	0.311431	−	0.950269i	$-0.399192\pi$
$751$	17.0691	0.622861	0.311431	+	0.950269i	$0.399192\pi$
$752$	−66.7386	−2.43371
$753$	0	0
$754$	−7.50758	−0.273410
$755$	0	0
$756$	0	0
$757$	−39.3693	−1.43090	−0.715451	−	0.698663i	$-0.753779\pi$
$757$	−39.3693	−1.43090	−0.715451	+	0.698663i	$0.753779\pi$
$758$	42.2462	1.53445
$759$	0	0
$760$	0	0
$761$	−48.2462	−1.74892	−0.874462	−	0.485094i	$-0.838785\pi$
$761$	−48.2462	−1.74892	−0.874462	+	0.485094i	$0.838785\pi$
$762$	0	0
$763$	5.31534	0.192428
$764$	61.8617	2.23808
$765$	0	0
$766$	−16.0000	−0.578103
$767$	−1.75379	−0.0633256
$768$	0	0
$769$	−42.4924	−1.53232	−0.766158	−	0.642652i	$-0.777834\pi$
$769$	−42.4924	−1.53232	−0.766158	+	0.642652i	$0.777834\pi$
$770$	0	0
$771$	0	0
$772$	−88.3542	−3.17994
$773$	36.9309	1.32831	0.664156	−	0.747594i	$-0.268791\pi$
$773$	36.9309	1.32831	0.664156	+	0.747594i	$0.268791\pi$
$774$	0	0
$775$	0	0
$776$	38.1080	1.36800
$777$	0	0
$778$	−63.8617	−2.28955
$779$	36.4924	1.30748
$780$	0	0
$781$	−12.4924	−0.447014
$782$	3.50758	0.125431
$783$	0	0
$784$	7.68466	0.274452
$785$	0	0
$786$	0	0
$787$	49.1771	1.75297	0.876487	−	0.481426i	$-0.159881\pi$
$787$	49.1771	1.75297	0.876487	+	0.481426i	$0.159881\pi$
$788$	5.12311	0.182503
$789$	0	0
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−6.73863	−0.239296
$794$	70.6004	2.50551
$795$	0	0
$796$	−8.00000	−0.283552
$797$	24.0540	0.852036	0.426018	−	0.904715i	$-0.359916\pi$
$797$	24.0540	0.852036	0.426018	+	0.904715i	$0.359916\pi$
$798$	0	0
$799$	3.80776	0.134709
$800$	0	0
$801$	0	0
$802$	80.8466	2.85479
$803$	19.1231	0.674840
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−106.600	−3.75019
$809$	−16.5464	−0.581740	−0.290870	−	0.956763i	$-0.593945\pi$
$809$	−16.5464	−0.581740	−0.290870	+	0.956763i	$0.593945\pi$
$810$	0	0
$811$	19.6155	0.688794	0.344397	−	0.938824i	$-0.388083\pi$
$811$	19.6155	0.688794	0.344397	+	0.938824i	$0.388083\pi$
$812$	−30.4924	−1.07007
$813$	0	0
$814$	24.0000	0.841200
$815$	0	0
$816$	0	0
$817$	6.24621	0.218527
$818$	−16.6307	−0.581478
$819$	0	0
$820$	0	0
$821$	21.4233	0.747678	0.373839	−	0.927494i	$-0.378041\pi$
$821$	21.4233	0.747678	0.373839	+	0.927494i	$0.378041\pi$
$822$	0	0
$823$	36.4924	1.27205	0.636023	−	0.771670i	$-0.280578\pi$
$823$	36.4924	1.27205	0.636023	+	0.771670i	$0.280578\pi$
$824$	36.4924	1.27127
$825$	0	0
$826$	−10.2462	−0.356511
$827$	−5.36932	−0.186709	−0.0933547	−	0.995633i	$-0.529759\pi$
$827$	−5.36932	−0.186709	−0.0933547	+	0.995633i	$0.529759\pi$
$828$	0	0
$829$	34.8769	1.21132	0.605662	−	0.795722i	$-0.292908\pi$
$829$	34.8769	1.21132	0.605662	+	0.795722i	$0.292908\pi$
$830$	0	0
$831$	0	0
$832$	−0.630683	−0.0218650
$833$	−0.438447	−0.0151913
$834$	0	0
$835$	0	0
$836$	−50.7386	−1.75483
$837$	0	0
$838$	67.2311	2.32246
$839$	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
$842$	6.87689	0.236993
$843$	0	0
$844$	64.1080	2.20669
$845$	0	0
$846$	0	0
$847$	−8.56155	−0.294178
$848$	−39.3693	−1.35195
$849$	0	0
$850$	0	0
$851$	−18.7386	−0.642352
$852$	0	0
$853$	7.26137	0.248624	0.124312	−	0.992243i	$-0.460328\pi$
$853$	7.26137	0.248624	0.124312	+	0.992243i	$0.460328\pi$
$854$	−39.3693	−1.34719
$855$	0	0
$856$	−87.7235	−2.99833
$857$	−15.7538	−0.538139	−0.269070	−	0.963121i	$-0.586716\pi$
$857$	−15.7538	−0.538139	−0.269070	+	0.963121i	$0.586716\pi$
$858$	0	0
$859$	−16.4924	−0.562714	−0.281357	−	0.959603i	$-0.590785\pi$
$859$	−16.4924	−0.562714	−0.281357	+	0.959603i	$0.590785\pi$
$860$	0	0
$861$	0	0
$862$	−50.7386	−1.72816
$863$	−25.7538	−0.876669	−0.438335	−	0.898812i	$-0.644431\pi$
$863$	−25.7538	−0.876669	−0.438335	+	0.898812i	$0.644431\pi$
$864$	0	0
$865$	0	0
$866$	21.1231	0.717792
$867$	0	0
$868$	0	0
$869$	−3.80776	−0.129170
$870$	0	0
$871$	4.49242	0.152220
$872$	−34.8769	−1.18108
$873$	0	0
$874$	56.9848	1.92754
$875$	0	0
$876$	0	0
$877$	40.2462	1.35902	0.679509	−	0.733667i	$-0.262193\pi$
$877$	40.2462	1.35902	0.679509	+	0.733667i	$0.262193\pi$
$878$	−24.0000	−0.809961
$879$	0	0
$880$	0	0
$881$	11.8617	0.399632	0.199816	−	0.979833i	$-0.435966\pi$
$881$	11.8617	0.399632	0.199816	+	0.979833i	$0.435966\pi$
$882$	0	0
$883$	8.49242	0.285793	0.142896	−	0.989738i	$-0.454358\pi$
$883$	8.49242	0.285793	0.142896	+	0.989738i	$0.454358\pi$
$884$	0.876894	0.0294931
$885$	0	0
$886$	6.73863	0.226389
$887$	20.4924	0.688068	0.344034	−	0.938957i	$-0.388206\pi$
$887$	20.4924	0.688068	0.344034	+	0.938957i	$0.388206\pi$
$888$	0	0
$889$	6.24621	0.209491
$890$	0	0
$891$	0	0
$892$	11.1231	0.372429
$893$	61.8617	2.07012
$894$	0	0
$895$	0	0
$896$	9.43845	0.315316
$897$	0	0
$898$	−4.63068	−0.154528
$899$	0	0
$900$	0	0
$901$	2.24621	0.0748321
$902$	20.4924	0.682323
$903$	0	0
$904$	91.8617	3.05528
$905$	0	0
$906$	0	0
$907$	24.1080	0.800491	0.400246	−	0.916408i	$-0.368925\pi$
$907$	24.1080	0.800491	0.400246	+	0.916408i	$0.368925\pi$
$908$	51.6155	1.71292
$909$	0	0
$910$	0	0
$911$	28.4924	0.943996	0.471998	−	0.881600i	$-0.343533\pi$
$911$	28.4924	0.943996	0.471998	+	0.881600i	$0.343533\pi$
$912$	0	0
$913$	6.24621	0.206719
$914$	−43.8617	−1.45082
$915$	0	0
$916$	49.6155	1.63934
$917$	0.876894	0.0289576
$918$	0	0
$919$	40.3002	1.32938	0.664690	−	0.747119i	$-0.268564\pi$
$919$	40.3002	1.32938	0.664690	+	0.747119i	$0.268564\pi$
$920$	0	0
$921$	0	0
$922$	−33.6155	−1.10707
$923$	3.50758	0.115453
$924$	0	0
$925$	0	0
$926$	32.0000	1.05159
$927$	0	0
$928$	43.8617	1.43983
$929$	−22.1080	−0.725338	−0.362669	−	0.931918i	$-0.618134\pi$
$929$	−22.1080	−0.725338	−0.362669	+	0.931918i	$0.618134\pi$
$930$	0	0
$931$	−7.12311	−0.233450
$932$	23.3693	0.765487
$933$	0	0
$934$	−57.4773	−1.88071
$935$	0	0
$936$	0	0
$937$	55.6695	1.81864	0.909322	−	0.416094i	$-0.136601\pi$
$937$	55.6695	1.81864	0.909322	+	0.416094i	$0.136601\pi$
$938$	26.2462	0.856969
$939$	0	0
$940$	0	0
$941$	−43.8617	−1.42985	−0.714926	−	0.699200i	$-0.753540\pi$
$941$	−43.8617	−1.42985	−0.714926	+	0.699200i	$0.753540\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	30.7386	1.00046
$945$	0	0
$946$	3.50758	0.114041
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	−5.36932	−0.174295
$950$	0	0
$951$	0	0
$952$	2.87689	0.0932407
$953$	−33.1231	−1.07296	−0.536481	−	0.843912i	$-0.680247\pi$
$953$	−33.1231	−1.07296	−0.536481	+	0.843912i	$0.680247\pi$
$954$	0	0
$955$	0	0
$956$	−90.3542	−2.92226
$957$	0	0
$958$	12.4924	0.403612
$959$	−17.1231	−0.552934
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−6.73863	−0.217262
$963$	0	0
$964$	−19.3693	−0.623844
$965$	0	0
$966$	0	0
$967$	35.1231	1.12948	0.564741	−	0.825268i	$-0.308976\pi$
$967$	35.1231	1.12948	0.564741	+	0.825268i	$0.308976\pi$
$968$	56.1771	1.80560
$969$	0	0
$970$	0	0
$971$	−49.4773	−1.58780	−0.793901	−	0.608048i	$-0.791953\pi$
$971$	−49.4773	−1.58780	−0.793901	+	0.608048i	$0.791953\pi$
$972$	0	0
$973$	−15.1231	−0.484825
$974$	−8.00000	−0.256337
$975$	0	0
$976$	118.108	3.78054
$977$	33.2311	1.06316	0.531578	−	0.847009i	$-0.321599\pi$
$977$	33.2311	1.06316	0.531578	+	0.847009i	$0.321599\pi$
$978$	0	0
$979$	1.75379	0.0560513
$980$	0	0
$981$	0	0
$982$	−105.477	−3.36591
$983$	51.4233	1.64015	0.820074	−	0.572257i	$-0.193932\pi$
$983$	51.4233	1.64015	0.820074	+	0.572257i	$0.193932\pi$
$984$	0	0
$985$	0	0
$986$	−7.50758	−0.239090
$987$	0	0
$988$	14.2462	0.453232
$989$	−2.73863	−0.0870835
$990$	0	0
$991$	12.4924	0.396835	0.198417	−	0.980118i	$-0.436420\pi$
$991$	12.4924	0.396835	0.198417	+	0.980118i	$0.436420\pi$
$992$	0	0
$993$	0	0
$994$	20.4924	0.649980
$995$	0	0
$996$	0	0
$997$	2.68466	0.0850240	0.0425120	−	0.999096i	$-0.486464\pi$
$997$	2.68466	0.0850240	0.0425120	+	0.999096i	$0.486464\pi$
$998$	−105.477	−3.33882
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.a.p.1.1		2
3.2	odd	2		175.2.a.f.1.2		2
5.2	odd	4		1575.2.d.e.1324.1		4
5.3	odd	4		1575.2.d.e.1324.4		4
5.4	even	2		315.2.a.e.1.2		2
12.11	even	2		2800.2.a.bi.1.2		2
15.2	even	4		175.2.b.b.99.4		4
15.8	even	4		175.2.b.b.99.1		4
15.14	odd	2		35.2.a.b.1.1	✓	2
20.19	odd	2		5040.2.a.bt.1.1		2
21.20	even	2		1225.2.a.s.1.2		2
35.34	odd	2		2205.2.a.x.1.2		2
60.23	odd	4		2800.2.g.t.449.3		4
60.47	odd	4		2800.2.g.t.449.2		4
60.59	even	2		560.2.a.i.1.1		2
105.44	odd	6		245.2.e.i.116.2		4
105.59	even	6		245.2.e.h.226.2		4
105.62	odd	4		1225.2.b.f.99.4		4
105.74	odd	6		245.2.e.i.226.2		4
105.83	odd	4		1225.2.b.f.99.1		4
105.89	even	6		245.2.e.h.116.2		4
105.104	even	2		245.2.a.d.1.1		2
120.29	odd	2		2240.2.a.bh.1.1		2
120.59	even	2		2240.2.a.bd.1.2		2
165.164	even	2		4235.2.a.m.1.2		2
195.194	odd	2		5915.2.a.l.1.2		2
420.419	odd	2		3920.2.a.bs.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.a.b.1.1	✓	2	15.14	odd	2
175.2.a.f.1.2		2	3.2	odd	2
175.2.b.b.99.1		4	15.8	even	4
175.2.b.b.99.4		4	15.2	even	4
245.2.a.d.1.1		2	105.104	even	2
245.2.e.h.116.2		4	105.89	even	6
245.2.e.h.226.2		4	105.59	even	6
245.2.e.i.116.2		4	105.44	odd	6
245.2.e.i.226.2		4	105.74	odd	6
315.2.a.e.1.2		2	5.4	even	2
560.2.a.i.1.1		2	60.59	even	2
1225.2.a.s.1.2		2	21.20	even	2
1225.2.b.f.99.1		4	105.83	odd	4
1225.2.b.f.99.4		4	105.62	odd	4
1575.2.a.p.1.1		2	1.1	even	1	trivial
1575.2.d.e.1324.1		4	5.2	odd	4
1575.2.d.e.1324.4		4	5.3	odd	4
2205.2.a.x.1.2		2	35.34	odd	2
2240.2.a.bd.1.2		2	120.59	even	2
2240.2.a.bh.1.1		2	120.29	odd	2
2800.2.a.bi.1.2		2	12.11	even	2
2800.2.g.t.449.2		4	60.47	odd	4
2800.2.g.t.449.3		4	60.23	odd	4
3920.2.a.bs.1.2		2	420.419	odd	2
4235.2.a.m.1.2		2	165.164	even	2
5040.2.a.bt.1.1		2	20.19	odd	2
5915.2.a.l.1.2		2	195.194	odd	2

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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{7} -6.56155 q^{8} +O(q^{10})\)	\(q-2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{7} -6.56155 q^{8} +1.56155 q^{11} -0.438447 q^{13} -2.56155 q^{14} +7.68466 q^{16} -0.438447 q^{17} -7.12311 q^{19} -4.00000 q^{22} +3.12311 q^{23} +1.12311 q^{26} +4.56155 q^{28} -6.68466 q^{29} -6.56155 q^{32} +1.12311 q^{34} -6.00000 q^{37} +18.2462 q^{38} -5.12311 q^{41} -0.876894 q^{43} +7.12311 q^{44} -8.00000 q^{46} -8.68466 q^{47} +1.00000 q^{49} -2.00000 q^{52} -5.12311 q^{53} -6.56155 q^{56} +17.1231 q^{58} +4.00000 q^{59} +15.3693 q^{61} +1.43845 q^{64} -10.2462 q^{67} -2.00000 q^{68} -8.00000 q^{71} +12.2462 q^{73} +15.3693 q^{74} -32.4924 q^{76} +1.56155 q^{77} -2.43845 q^{79} +13.1231 q^{82} +4.00000 q^{83} +2.24621 q^{86} -10.2462 q^{88} +1.12311 q^{89} -0.438447 q^{91} +14.2462 q^{92} +22.2462 q^{94} -5.80776 q^{97} -2.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 5 q^{4} + 2 q^{7} - 9 q^{8}+O(q^{10}) \) 2 * q - q^2 + 5 * q^4 + 2 * q^7 - 9 * q^8	\( 2 q - q^{2} + 5 q^{4} + 2 q^{7} - 9 q^{8} - q^{11} - 5 q^{13} - q^{14} + 3 q^{16} - 5 q^{17} - 6 q^{19} - 8 q^{22} - 2 q^{23} - 6 q^{26} + 5 q^{28} - q^{29} - 9 q^{32} - 6 q^{34} - 12 q^{37} + 20 q^{38} - 2 q^{41} - 10 q^{43} + 6 q^{44} - 16 q^{46} - 5 q^{47} + 2 q^{49} - 4 q^{52} - 2 q^{53} - 9 q^{56} + 26 q^{58} + 8 q^{59} + 6 q^{61} + 7 q^{64} - 4 q^{67} - 4 q^{68} - 16 q^{71} + 8 q^{73} + 6 q^{74} - 32 q^{76} - q^{77} - 9 q^{79} + 18 q^{82} + 8 q^{83} - 12 q^{86} - 4 q^{88} - 6 q^{89} - 5 q^{91} + 12 q^{92} + 28 q^{94} + 9 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 + 5 * q^4 + 2 * q^7 - 9 * q^8 - q^11 - 5 * q^13 - q^14 + 3 * q^16 - 5 * q^17 - 6 * q^19 - 8 * q^22 - 2 * q^23 - 6 * q^26 + 5 * q^28 - q^29 - 9 * q^32 - 6 * q^34 - 12 * q^37 + 20 * q^38 - 2 * q^41 - 10 * q^43 + 6 * q^44 - 16 * q^46 - 5 * q^47 + 2 * q^49 - 4 * q^52 - 2 * q^53 - 9 * q^56 + 26 * q^58 + 8 * q^59 + 6 * q^61 + 7 * q^64 - 4 * q^67 - 4 * q^68 - 16 * q^71 + 8 * q^73 + 6 * q^74 - 32 * q^76 - q^77 - 9 * q^79 + 18 * q^82 + 8 * q^83 - 12 * q^86 - 4 * q^88 - 6 * q^89 - 5 * q^91 + 12 * q^92 + 28 * q^94 + 9 * q^97 - q^98

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