LMFDB - Embedded newform 675.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(1,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.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:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +3.00000 q^{7} -2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +3.00000 q^{7} -2.82843 q^{8} +4.24264 q^{11} +3.00000 q^{13} +4.24264 q^{14} -4.00000 q^{16} -2.82843 q^{17} +1.00000 q^{19} +6.00000 q^{22} +7.07107 q^{23} +4.24264 q^{26} -4.24264 q^{29} +2.00000 q^{31} -4.00000 q^{34} +9.00000 q^{37} +1.41421 q^{38} -4.24264 q^{41} +6.00000 q^{43} +10.0000 q^{46} -2.82843 q^{47} +2.00000 q^{49} -9.89949 q^{53} -8.48528 q^{56} -6.00000 q^{58} -8.48528 q^{59} -13.0000 q^{61} +2.82843 q^{62} +8.00000 q^{64} -3.00000 q^{67} -12.7279 q^{71} +9.00000 q^{73} +12.7279 q^{74} +12.7279 q^{77} -5.00000 q^{79} -6.00000 q^{82} -1.41421 q^{83} +8.48528 q^{86} -12.0000 q^{88} +9.00000 q^{91} -4.00000 q^{94} +3.00000 q^{97} +2.82843 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{7}+O(q^{10}) $ 2 * q + 6 * q^7	$ 2 q + 6 q^{7} + 6 q^{13} - 8 q^{16} + 2 q^{19} + 12 q^{22} + 4 q^{31} - 8 q^{34} + 18 q^{37} + 12 q^{43} + 20 q^{46} + 4 q^{49} - 12 q^{58} - 26 q^{61} + 16 q^{64} - 6 q^{67} + 18 q^{73} - 10 q^{79} - 12 q^{82} - 24 q^{88} + 18 q^{91} - 8 q^{94} + 6 q^{97}+O(q^{100}) $ 2 * q + 6 * q^7 + 6 * q^13 - 8 * q^16 + 2 * q^19 + 12 * q^22 + 4 * q^31 - 8 * q^34 + 18 * q^37 + 12 * q^43 + 20 * q^46 + 4 * q^49 - 12 * q^58 - 26 * q^61 + 16 * q^64 - 6 * q^67 + 18 * q^73 - 10 * q^79 - 12 * q^82 - 24 * q^88 + 18 * q^91 - 8 * q^94 + 6 * q^97

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 $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−2.82843	−1.00000
$9$	0	0
$10$	0	0
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	4.24264	1.13389
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	6.00000	1.27920
$23$	7.07107	1.47442	0.737210	−	0.675664i	$-0.236143\pi$
$23$	7.07107	1.47442	0.737210	+	0.675664i	$0.236143\pi$
$24$	0	0
$25$	0	0
$26$	4.24264	0.832050
$27$	0	0
$28$	0	0
$29$	−4.24264	−0.787839	−0.393919	−	0.919145i	$-0.628881\pi$
$29$	−4.24264	−0.787839	−0.393919	+	0.919145i	$0.628881\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	0	0
$37$	9.00000	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$37$	9.00000	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$38$	1.41421	0.229416
$39$	0	0
$40$	0	0
$41$	−4.24264	−0.662589	−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264	−0.662589	−0.331295	+	0.943527i	$0.607485\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	0	0
$46$	10.0000	1.47442
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.89949	−1.35980	−0.679900	−	0.733305i	$-0.737977\pi$
$53$	−9.89949	−1.35980	−0.679900	+	0.733305i	$0.737977\pi$
$54$	0	0
$55$	0	0
$56$	−8.48528	−1.13389
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−8.48528	−1.10469	−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528	−1.10469	−0.552345	+	0.833616i	$0.686267\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	2.82843	0.359211
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.7279	−1.51053	−0.755263	−	0.655422i	$-0.772491\pi$
$71$	−12.7279	−1.51053	−0.755263	+	0.655422i	$0.772491\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	12.7279	1.47959
$75$	0	0
$76$	0	0
$77$	12.7279	1.45048
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	0	0
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−1.41421	−0.155230	−0.0776151	−	0.996983i	$-0.524731\pi$
$83$	−1.41421	−0.155230	−0.0776151	+	0.996983i	$0.524731\pi$
$84$	0	0
$85$	0	0
$86$	8.48528	0.914991
$87$	0	0
$88$	−12.0000	−1.27920
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	9.00000	0.943456
$92$	0	0
$93$	0	0
$94$	−4.00000	−0.412568
$95$	0	0
$96$	0	0
$97$	3.00000	0.304604	0.152302	−	0.988334i	$-0.451331\pi$
$97$	3.00000	0.304604	0.152302	+	0.988334i	$0.451331\pi$
$98$	2.82843	0.285714
$99$	0	0
$100$	0	0
$101$	−8.48528	−0.844317	−0.422159	−	0.906522i	$-0.638727\pi$
$101$	−8.48528	−0.844317	−0.422159	+	0.906522i	$0.638727\pi$
$102$	0	0
$103$	3.00000	0.295599	0.147799	−	0.989017i	$-0.452781\pi$
$103$	3.00000	0.295599	0.147799	+	0.989017i	$0.452781\pi$
$104$	−8.48528	−0.832050
$105$	0	0
$106$	−14.0000	−1.35980
$107$	−7.07107	−0.683586	−0.341793	−	0.939775i	$-0.611034\pi$
$107$	−7.07107	−0.683586	−0.341793	+	0.939775i	$0.611034\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	0	0
$112$	−12.0000	−1.13389
$113$	−1.41421	−0.133038	−0.0665190	−	0.997785i	$-0.521189\pi$
$113$	−1.41421	−0.133038	−0.0665190	+	0.997785i	$0.521189\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−12.0000	−1.10469
$119$	−8.48528	−0.777844
$120$	0	0
$121$	7.00000	0.636364
$122$	−18.3848	−1.66448
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	11.3137	1.00000
$129$	0	0
$130$	0	0
$131$	8.48528	0.741362	0.370681	−	0.928760i	$-0.379124\pi$
$131$	8.48528	0.741362	0.370681	+	0.928760i	$0.379124\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	−4.24264	−0.366508
$135$	0	0
$136$	8.00000	0.685994
$137$	1.41421	0.120824	0.0604122	−	0.998174i	$-0.480758\pi$
$137$	1.41421	0.120824	0.0604122	+	0.998174i	$0.480758\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	0	0
$142$	−18.0000	−1.51053
$143$	12.7279	1.06436
$144$	0	0
$145$	0	0
$146$	12.7279	1.05337
$147$	0	0
$148$	0	0
$149$	16.9706	1.39028	0.695141	−	0.718873i	$-0.255342\pi$
$149$	16.9706	1.39028	0.695141	+	0.718873i	$0.255342\pi$
$150$	0	0
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	−2.82843	−0.229416
$153$	0	0
$154$	18.0000	1.45048
$155$	0	0
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	−7.07107	−0.562544
$159$	0	0
$160$	0	0
$161$	21.2132	1.67183
$162$	0	0
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	0	0
$165$	0	0
$166$	−2.00000	−0.155230
$167$	5.65685	0.437741	0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685	0.437741	0.218870	+	0.975754i	$0.429763\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.1421	−1.07521	−0.537603	−	0.843198i	$-0.680670\pi$
$173$	−14.1421	−1.07521	−0.537603	+	0.843198i	$0.680670\pi$
$174$	0	0
$175$	0	0
$176$	−16.9706	−1.27920
$177$	0	0
$178$	0	0
$179$	12.7279	0.951330	0.475665	−	0.879627i	$-0.342208\pi$
$179$	12.7279	0.951330	0.475665	+	0.879627i	$0.342208\pi$
$180$	0	0
$181$	−1.00000	−0.0743294	−0.0371647	−	0.999309i	$-0.511833\pi$
$181$	−1.00000	−0.0743294	−0.0371647	+	0.999309i	$0.511833\pi$
$182$	12.7279	0.943456
$183$	0	0
$184$	−20.0000	−1.47442
$185$	0	0
$186$	0	0
$187$	−12.0000	−0.877527
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.9706	1.22795	0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706	1.22795	0.613973	+	0.789327i	$0.289570\pi$
$192$	0	0
$193$	3.00000	0.215945	0.107972	−	0.994154i	$-0.465564\pi$
$193$	3.00000	0.215945	0.107972	+	0.994154i	$0.465564\pi$
$194$	4.24264	0.304604
$195$	0	0
$196$	0	0
$197$	22.6274	1.61214	0.806068	−	0.591822i	$-0.201591\pi$
$197$	22.6274	1.61214	0.806068	+	0.591822i	$0.201591\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	0	0
$202$	−12.0000	−0.844317
$203$	−12.7279	−0.893325
$204$	0	0
$205$	0	0
$206$	4.24264	0.295599
$207$	0	0
$208$	−12.0000	−0.832050
$209$	4.24264	0.293470
$210$	0	0
$211$	−7.00000	−0.481900	−0.240950	−	0.970538i	$-0.577459\pi$
$211$	−7.00000	−0.481900	−0.240950	+	0.970538i	$0.577459\pi$
$212$	0	0
$213$	0	0
$214$	−10.0000	−0.683586
$215$	0	0
$216$	0	0
$217$	6.00000	0.407307
$218$	−11.3137	−0.766261
$219$	0	0
$220$	0	0
$221$	−8.48528	−0.570782
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	0	0
$225$	0	0
$226$	−2.00000	−0.133038
$227$	−24.0416	−1.59570	−0.797850	−	0.602857i	$-0.794029\pi$
$227$	−24.0416	−1.59570	−0.797850	+	0.602857i	$0.794029\pi$
$228$	0	0
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	0	0
$232$	12.0000	0.787839
$233$	19.7990	1.29707	0.648537	−	0.761183i	$-0.275381\pi$
$233$	19.7990	1.29707	0.648537	+	0.761183i	$0.275381\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	−12.0000	−0.777844
$239$	4.24264	0.274434	0.137217	−	0.990541i	$-0.456184\pi$
$239$	4.24264	0.274434	0.137217	+	0.990541i	$0.456184\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	9.89949	0.636364
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.00000	0.190885
$248$	−5.65685	−0.359211
$249$	0	0
$250$	0	0
$251$	−12.7279	−0.803379	−0.401690	−	0.915776i	$-0.631577\pi$
$251$	−12.7279	−0.803379	−0.401690	+	0.915776i	$0.631577\pi$
$252$	0	0
$253$	30.0000	1.88608
$254$	−25.4558	−1.59724
$255$	0	0
$256$	0	0
$257$	−19.7990	−1.23503	−0.617514	−	0.786560i	$-0.711860\pi$
$257$	−19.7990	−1.23503	−0.617514	+	0.786560i	$0.711860\pi$
$258$	0	0
$259$	27.0000	1.67770
$260$	0	0
$261$	0	0
$262$	12.0000	0.741362
$263$	2.82843	0.174408	0.0872041	−	0.996190i	$-0.472207\pi$
$263$	2.82843	0.174408	0.0872041	+	0.996190i	$0.472207\pi$
$264$	0	0
$265$	0	0
$266$	4.24264	0.260133
$267$	0	0
$268$	0	0
$269$	−25.4558	−1.55207	−0.776035	−	0.630690i	$-0.782772\pi$
$269$	−25.4558	−1.55207	−0.776035	+	0.630690i	$0.782772\pi$
$270$	0	0
$271$	−19.0000	−1.15417	−0.577084	−	0.816685i	$-0.695809\pi$
$271$	−19.0000	−1.15417	−0.577084	+	0.816685i	$0.695809\pi$
$272$	11.3137	0.685994
$273$	0	0
$274$	2.00000	0.120824
$275$	0	0
$276$	0	0
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	18.3848	1.10265
$279$	0	0
$280$	0	0
$281$	−8.48528	−0.506189	−0.253095	−	0.967442i	$-0.581448\pi$
$281$	−8.48528	−0.506189	−0.253095	+	0.967442i	$0.581448\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	0	0
$285$	0	0
$286$	18.0000	1.06436
$287$	−12.7279	−0.751305
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.07107	0.413096	0.206548	−	0.978436i	$-0.433777\pi$
$293$	7.07107	0.413096	0.206548	+	0.978436i	$0.433777\pi$
$294$	0	0
$295$	0	0
$296$	−25.4558	−1.47959
$297$	0	0
$298$	24.0000	1.39028
$299$	21.2132	1.22679
$300$	0	0
$301$	18.0000	1.03750
$302$	−1.41421	−0.0813788
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	−21.0000	−1.18699	−0.593495	−	0.804838i	$-0.702252\pi$
$313$	−21.0000	−1.18699	−0.593495	+	0.804838i	$0.702252\pi$
$314$	8.48528	0.478852
$315$	0	0
$316$	0	0
$317$	−11.3137	−0.635441	−0.317721	−	0.948184i	$-0.602917\pi$
$317$	−11.3137	−0.635441	−0.317721	+	0.948184i	$0.602917\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	0	0
$322$	30.0000	1.67183
$323$	−2.82843	−0.157378
$324$	0	0
$325$	0	0
$326$	−12.7279	−0.704934
$327$	0	0
$328$	12.0000	0.662589
$329$	−8.48528	−0.467809
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	0	0
$334$	8.00000	0.437741
$335$	0	0
$336$	0	0
$337$	33.0000	1.79762	0.898812	−	0.438334i	$-0.144431\pi$
$337$	33.0000	1.79762	0.898812	+	0.438334i	$0.144431\pi$
$338$	−5.65685	−0.307692
$339$	0	0
$340$	0	0
$341$	8.48528	0.459504
$342$	0	0
$343$	−15.0000	−0.809924
$344$	−16.9706	−0.914991
$345$	0	0
$346$	−20.0000	−1.07521
$347$	26.8701	1.44246	0.721230	−	0.692696i	$-0.243577\pi$
$347$	26.8701	1.44246	0.721230	+	0.692696i	$0.243577\pi$
$348$	0	0
$349$	−5.00000	−0.267644	−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000	−0.267644	−0.133822	+	0.991005i	$0.542725\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.2843	1.50542	0.752710	−	0.658352i	$-0.228746\pi$
$353$	28.2843	1.50542	0.752710	+	0.658352i	$0.228746\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	18.0000	0.951330
$359$	12.7279	0.671754	0.335877	−	0.941906i	$-0.390967\pi$
$359$	12.7279	0.671754	0.335877	+	0.941906i	$0.390967\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−1.41421	−0.0743294
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−15.0000	−0.782994	−0.391497	−	0.920179i	$-0.628043\pi$
$367$	−15.0000	−0.782994	−0.391497	+	0.920179i	$0.628043\pi$
$368$	−28.2843	−1.47442
$369$	0	0
$370$	0	0
$371$	−29.6985	−1.54187
$372$	0	0
$373$	21.0000	1.08734	0.543669	−	0.839299i	$-0.317035\pi$
$373$	21.0000	1.08734	0.543669	+	0.839299i	$0.317035\pi$
$374$	−16.9706	−0.877527
$375$	0	0
$376$	8.00000	0.412568
$377$	−12.7279	−0.655521
$378$	0	0
$379$	−35.0000	−1.79783	−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000	−1.79783	−0.898915	+	0.438124i	$0.855643\pi$
$380$	0	0
$381$	0	0
$382$	24.0000	1.22795
$383$	−5.65685	−0.289052	−0.144526	−	0.989501i	$-0.546166\pi$
$383$	−5.65685	−0.289052	−0.144526	+	0.989501i	$0.546166\pi$
$384$	0	0
$385$	0	0
$386$	4.24264	0.215945
$387$	0	0
$388$	0	0
$389$	33.9411	1.72088	0.860442	−	0.509549i	$-0.170188\pi$
$389$	33.9411	1.72088	0.860442	+	0.509549i	$0.170188\pi$
$390$	0	0
$391$	−20.0000	−1.01144
$392$	−5.65685	−0.285714
$393$	0	0
$394$	32.0000	1.61214
$395$	0	0
$396$	0	0
$397$	36.0000	1.80679	0.903394	−	0.428811i	$-0.141067\pi$
$397$	36.0000	1.80679	0.903394	+	0.428811i	$0.141067\pi$
$398$	−15.5563	−0.779769
$399$	0	0
$400$	0	0
$401$	−4.24264	−0.211867	−0.105934	−	0.994373i	$-0.533783\pi$
$401$	−4.24264	−0.211867	−0.105934	+	0.994373i	$0.533783\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	0	0
$405$	0	0
$406$	−18.0000	−0.893325
$407$	38.1838	1.89270
$408$	0	0
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−25.4558	−1.25260
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	6.00000	0.293470
$419$	−33.9411	−1.65813	−0.829066	−	0.559150i	$-0.811127\pi$
$419$	−33.9411	−1.65813	−0.829066	+	0.559150i	$0.811127\pi$
$420$	0	0
$421$	23.0000	1.12095	0.560476	−	0.828171i	$-0.310618\pi$
$421$	23.0000	1.12095	0.560476	+	0.828171i	$0.310618\pi$
$422$	−9.89949	−0.481900
$423$	0	0
$424$	28.0000	1.35980
$425$	0	0
$426$	0	0
$427$	−39.0000	−1.88734
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.7279	0.613082	0.306541	−	0.951857i	$-0.400828\pi$
$431$	12.7279	0.613082	0.306541	+	0.951857i	$0.400828\pi$
$432$	0	0
$433$	−36.0000	−1.73005	−0.865025	−	0.501729i	$-0.832697\pi$
$433$	−36.0000	−1.73005	−0.865025	+	0.501729i	$0.832697\pi$
$434$	8.48528	0.407307
$435$	0	0
$436$	0	0
$437$	7.07107	0.338255
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	0	0
$442$	−12.0000	−0.570782
$443$	−14.1421	−0.671913	−0.335957	−	0.941877i	$-0.609060\pi$
$443$	−14.1421	−0.671913	−0.335957	+	0.941877i	$0.609060\pi$
$444$	0	0
$445$	0	0
$446$	8.48528	0.401790
$447$	0	0
$448$	24.0000	1.13389
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	0	0
$453$	0	0
$454$	−34.0000	−1.59570
$455$	0	0
$456$	0	0
$457$	12.0000	0.561336	0.280668	−	0.959805i	$-0.409444\pi$
$457$	12.0000	0.561336	0.280668	+	0.959805i	$0.409444\pi$
$458$	22.6274	1.05731
$459$	0	0
$460$	0	0
$461$	16.9706	0.790398	0.395199	−	0.918596i	$-0.370676\pi$
$461$	16.9706	0.790398	0.395199	+	0.918596i	$0.370676\pi$
$462$	0	0
$463$	−15.0000	−0.697109	−0.348555	−	0.937288i	$-0.613327\pi$
$463$	−15.0000	−0.697109	−0.348555	+	0.937288i	$0.613327\pi$
$464$	16.9706	0.787839
$465$	0	0
$466$	28.0000	1.29707
$467$	22.6274	1.04707	0.523536	−	0.852004i	$-0.324613\pi$
$467$	22.6274	1.04707	0.523536	+	0.852004i	$0.324613\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	0	0
$472$	24.0000	1.10469
$473$	25.4558	1.17046
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	6.00000	0.274434
$479$	−4.24264	−0.193851	−0.0969256	−	0.995292i	$-0.530901\pi$
$479$	−4.24264	−0.193851	−0.0969256	+	0.995292i	$0.530901\pi$
$480$	0	0
$481$	27.0000	1.23109
$482$	7.07107	0.322078
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9.00000	−0.407829	−0.203914	−	0.978989i	$-0.565366\pi$
$487$	−9.00000	−0.407829	−0.203914	+	0.978989i	$0.565366\pi$
$488$	36.7696	1.66448
$489$	0	0
$490$	0	0
$491$	21.2132	0.957338	0.478669	−	0.877995i	$-0.341119\pi$
$491$	21.2132	0.957338	0.478669	+	0.877995i	$0.341119\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	4.24264	0.190885
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−38.1838	−1.71278
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	0	0
$502$	−18.0000	−0.803379
$503$	−35.3553	−1.57642	−0.788208	−	0.615409i	$-0.788991\pi$
$503$	−35.3553	−1.57642	−0.788208	+	0.615409i	$0.788991\pi$
$504$	0	0
$505$	0	0
$506$	42.4264	1.88608
$507$	0	0
$508$	0	0
$509$	29.6985	1.31636	0.658181	−	0.752860i	$-0.271326\pi$
$509$	29.6985	1.31636	0.658181	+	0.752860i	$0.271326\pi$
$510$	0	0
$511$	27.0000	1.19441
$512$	−22.6274	−1.00000
$513$	0	0
$514$	−28.0000	−1.23503
$515$	0	0
$516$	0	0
$517$	−12.0000	−0.527759
$518$	38.1838	1.67770
$519$	0	0
$520$	0	0
$521$	12.7279	0.557620	0.278810	−	0.960346i	$-0.410060\pi$
$521$	12.7279	0.557620	0.278810	+	0.960346i	$0.410060\pi$
$522$	0	0
$523$	−9.00000	−0.393543	−0.196771	−	0.980449i	$-0.563046\pi$
$523$	−9.00000	−0.393543	−0.196771	+	0.980449i	$0.563046\pi$
$524$	0	0
$525$	0	0
$526$	4.00000	0.174408
$527$	−5.65685	−0.246416
$528$	0	0
$529$	27.0000	1.17391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.7279	−0.551308
$534$	0	0
$535$	0	0
$536$	8.48528	0.366508
$537$	0	0
$538$	−36.0000	−1.55207
$539$	8.48528	0.365487
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	−26.8701	−1.15417
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.00000	0.128271	0.0641354	−	0.997941i	$-0.479571\pi$
$547$	3.00000	0.128271	0.0641354	+	0.997941i	$0.479571\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.24264	−0.180743
$552$	0	0
$553$	−15.0000	−0.637865
$554$	16.9706	0.721010
$555$	0	0
$556$	0	0
$557$	−19.7990	−0.838910	−0.419455	−	0.907776i	$-0.637779\pi$
$557$	−19.7990	−0.838910	−0.419455	+	0.907776i	$0.637779\pi$
$558$	0	0
$559$	18.0000	0.761319
$560$	0	0
$561$	0	0
$562$	−12.0000	−0.506189
$563$	11.3137	0.476816	0.238408	−	0.971165i	$-0.423374\pi$
$563$	11.3137	0.476816	0.238408	+	0.971165i	$0.423374\pi$
$564$	0	0
$565$	0	0
$566$	−8.48528	−0.356663
$567$	0	0
$568$	36.0000	1.51053
$569$	−42.4264	−1.77861	−0.889304	−	0.457317i	$-0.848810\pi$
$569$	−42.4264	−1.77861	−0.889304	+	0.457317i	$0.848810\pi$
$570$	0	0
$571$	−25.0000	−1.04622	−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000	−1.04622	−0.523109	+	0.852266i	$0.675228\pi$
$572$	0	0
$573$	0	0
$574$	−18.0000	−0.751305
$575$	0	0
$576$	0	0
$577$	−45.0000	−1.87337	−0.936687	−	0.350167i	$-0.886125\pi$
$577$	−45.0000	−1.87337	−0.936687	+	0.350167i	$0.886125\pi$
$578$	−12.7279	−0.529412
$579$	0	0
$580$	0	0
$581$	−4.24264	−0.176014
$582$	0	0
$583$	−42.0000	−1.73946
$584$	−25.4558	−1.05337
$585$	0	0
$586$	10.0000	0.413096
$587$	26.8701	1.10905	0.554523	−	0.832168i	$-0.312901\pi$
$587$	26.8701	1.10905	0.554523	+	0.832168i	$0.312901\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	0	0
$591$	0	0
$592$	−36.0000	−1.47959
$593$	−9.89949	−0.406524	−0.203262	−	0.979124i	$-0.565154\pi$
$593$	−9.89949	−0.406524	−0.203262	+	0.979124i	$0.565154\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	30.0000	1.22679
$599$	29.6985	1.21345	0.606724	−	0.794913i	$-0.292483\pi$
$599$	29.6985	1.21345	0.606724	+	0.794913i	$0.292483\pi$
$600$	0	0
$601$	−40.0000	−1.63163	−0.815817	−	0.578310i	$-0.803712\pi$
$601$	−40.0000	−1.63163	−0.815817	+	0.578310i	$0.803712\pi$
$602$	25.4558	1.03750
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−39.0000	−1.58296	−0.791481	−	0.611194i	$-0.790689\pi$
$607$	−39.0000	−1.58296	−0.791481	+	0.611194i	$0.790689\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.48528	−0.343278
$612$	0	0
$613$	9.00000	0.363507	0.181753	−	0.983344i	$-0.441823\pi$
$613$	9.00000	0.363507	0.181753	+	0.983344i	$0.441823\pi$
$614$	0	0
$615$	0	0
$616$	−36.0000	−1.45048
$617$	−7.07107	−0.284670	−0.142335	−	0.989819i	$-0.545461\pi$
$617$	−7.07107	−0.284670	−0.142335	+	0.989819i	$0.545461\pi$
$618$	0	0
$619$	19.0000	0.763674	0.381837	−	0.924230i	$-0.375291\pi$
$619$	19.0000	0.763674	0.381837	+	0.924230i	$0.375291\pi$
$620$	0	0
$621$	0	0
$622$	12.0000	0.481156
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−29.6985	−1.18699
$627$	0	0
$628$	0	0
$629$	−25.4558	−1.01499
$630$	0	0
$631$	17.0000	0.676759	0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000	0.676759	0.338380	+	0.941010i	$0.390121\pi$
$632$	14.1421	0.562544
$633$	0	0
$634$	−16.0000	−0.635441
$635$	0	0
$636$	0	0
$637$	6.00000	0.237729
$638$	−25.4558	−1.00781
$639$	0	0
$640$	0	0
$641$	16.9706	0.670297	0.335148	−	0.942165i	$-0.391214\pi$
$641$	16.9706	0.670297	0.335148	+	0.942165i	$0.391214\pi$
$642$	0	0
$643$	−42.0000	−1.65632	−0.828159	−	0.560493i	$-0.810612\pi$
$643$	−42.0000	−1.65632	−0.828159	+	0.560493i	$0.810612\pi$
$644$	0	0
$645$	0	0
$646$	−4.00000	−0.157378
$647$	9.89949	0.389189	0.194595	−	0.980884i	$-0.437661\pi$
$647$	9.89949	0.389189	0.194595	+	0.980884i	$0.437661\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.65685	−0.221370	−0.110685	−	0.993856i	$-0.535304\pi$
$653$	−5.65685	−0.221370	−0.110685	+	0.993856i	$0.535304\pi$
$654$	0	0
$655$	0	0
$656$	16.9706	0.662589
$657$	0	0
$658$	−12.0000	−0.467809
$659$	−16.9706	−0.661079	−0.330540	−	0.943792i	$-0.607231\pi$
$659$	−16.9706	−0.661079	−0.330540	+	0.943792i	$0.607231\pi$
$660$	0	0
$661$	23.0000	0.894596	0.447298	−	0.894385i	$-0.352386\pi$
$661$	23.0000	0.894596	0.447298	+	0.894385i	$0.352386\pi$
$662$	24.0416	0.934405
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	0	0
$667$	−30.0000	−1.16160
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−55.1543	−2.12921
$672$	0	0
$673$	15.0000	0.578208	0.289104	−	0.957298i	$-0.406643\pi$
$673$	15.0000	0.578208	0.289104	+	0.957298i	$0.406643\pi$
$674$	46.6690	1.79762
$675$	0	0
$676$	0	0
$677$	48.0833	1.84799	0.923995	−	0.382405i	$-0.124904\pi$
$677$	48.0833	1.84799	0.923995	+	0.382405i	$0.124904\pi$
$678$	0	0
$679$	9.00000	0.345388
$680$	0	0
$681$	0	0
$682$	12.0000	0.459504
$683$	15.5563	0.595247	0.297624	−	0.954683i	$-0.403806\pi$
$683$	15.5563	0.595247	0.297624	+	0.954683i	$0.403806\pi$
$684$	0	0
$685$	0	0
$686$	−21.2132	−0.809924
$687$	0	0
$688$	−24.0000	−0.914991
$689$	−29.6985	−1.13142
$690$	0	0
$691$	50.0000	1.90209	0.951045	−	0.309053i	$-0.100012\pi$
$691$	50.0000	1.90209	0.951045	+	0.309053i	$0.100012\pi$
$692$	0	0
$693$	0	0
$694$	38.0000	1.44246
$695$	0	0
$696$	0	0
$697$	12.0000	0.454532
$698$	−7.07107	−0.267644
$699$	0	0
$700$	0	0
$701$	−38.1838	−1.44218	−0.721090	−	0.692841i	$-0.756359\pi$
$701$	−38.1838	−1.44218	−0.721090	+	0.692841i	$0.756359\pi$
$702$	0	0
$703$	9.00000	0.339441
$704$	33.9411	1.27920
$705$	0	0
$706$	40.0000	1.50542
$707$	−25.4558	−0.957366
$708$	0	0
$709$	−17.0000	−0.638448	−0.319224	−	0.947679i	$-0.603422\pi$
$709$	−17.0000	−0.638448	−0.319224	+	0.947679i	$0.603422\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	14.1421	0.529627
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	18.0000	0.671754
$719$	−12.7279	−0.474671	−0.237336	−	0.971428i	$-0.576274\pi$
$719$	−12.7279	−0.474671	−0.237336	+	0.971428i	$0.576274\pi$
$720$	0	0
$721$	9.00000	0.335178
$722$	−25.4558	−0.947368
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	−25.4558	−0.943456
$729$	0	0
$730$	0	0
$731$	−16.9706	−0.627679
$732$	0	0
$733$	−24.0000	−0.886460	−0.443230	−	0.896408i	$-0.646168\pi$
$733$	−24.0000	−0.886460	−0.443230	+	0.896408i	$0.646168\pi$
$734$	−21.2132	−0.782994
$735$	0	0
$736$	0	0
$737$	−12.7279	−0.468839
$738$	0	0
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	0	0
$742$	−42.0000	−1.54187
$743$	−14.1421	−0.518825	−0.259412	−	0.965767i	$-0.583529\pi$
$743$	−14.1421	−0.518825	−0.259412	+	0.965767i	$0.583529\pi$
$744$	0	0
$745$	0	0
$746$	29.6985	1.08734
$747$	0	0
$748$	0	0
$749$	−21.2132	−0.775114
$750$	0	0
$751$	−7.00000	−0.255434	−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000	−0.255434	−0.127717	+	0.991811i	$0.540765\pi$
$752$	11.3137	0.412568
$753$	0	0
$754$	−18.0000	−0.655521
$755$	0	0
$756$	0	0
$757$	27.0000	0.981332	0.490666	−	0.871348i	$-0.336754\pi$
$757$	27.0000	0.981332	0.490666	+	0.871348i	$0.336754\pi$
$758$	−49.4975	−1.79783
$759$	0	0
$760$	0	0
$761$	−4.24264	−0.153796	−0.0768978	−	0.997039i	$-0.524502\pi$
$761$	−4.24264	−0.153796	−0.0768978	+	0.997039i	$0.524502\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	0	0
$765$	0	0
$766$	−8.00000	−0.289052
$767$	−25.4558	−0.919157
$768$	0	0
$769$	−11.0000	−0.396670	−0.198335	−	0.980134i	$-0.563553\pi$
$769$	−11.0000	−0.396670	−0.198335	+	0.980134i	$0.563553\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	41.0122	1.47511	0.737553	−	0.675289i	$-0.235981\pi$
$773$	41.0122	1.47511	0.737553	+	0.675289i	$0.235981\pi$
$774$	0	0
$775$	0	0
$776$	−8.48528	−0.304604
$777$	0	0
$778$	48.0000	1.72088
$779$	−4.24264	−0.152008
$780$	0	0
$781$	−54.0000	−1.93227
$782$	−28.2843	−1.01144
$783$	0	0
$784$	−8.00000	−0.285714
$785$	0	0
$786$	0	0
$787$	15.0000	0.534692	0.267346	−	0.963601i	$-0.413853\pi$
$787$	15.0000	0.534692	0.267346	+	0.963601i	$0.413853\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.24264	−0.150851
$792$	0	0
$793$	−39.0000	−1.38493
$794$	50.9117	1.80679
$795$	0	0
$796$	0	0
$797$	−19.7990	−0.701316	−0.350658	−	0.936504i	$-0.614042\pi$
$797$	−19.7990	−0.701316	−0.350658	+	0.936504i	$0.614042\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	0	0
$802$	−6.00000	−0.211867
$803$	38.1838	1.34748
$804$	0	0
$805$	0	0
$806$	8.48528	0.298881
$807$	0	0
$808$	24.0000	0.844317
$809$	25.4558	0.894980	0.447490	−	0.894289i	$-0.352318\pi$
$809$	25.4558	0.894980	0.447490	+	0.894289i	$0.352318\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	0	0
$814$	54.0000	1.89270
$815$	0	0
$816$	0	0
$817$	6.00000	0.209913
$818$	35.3553	1.23617
$819$	0	0
$820$	0	0
$821$	−8.48528	−0.296138	−0.148069	−	0.988977i	$-0.547306\pi$
$821$	−8.48528	−0.296138	−0.148069	+	0.988977i	$0.547306\pi$
$822$	0	0
$823$	39.0000	1.35945	0.679727	−	0.733465i	$-0.262098\pi$
$823$	39.0000	1.35945	0.679727	+	0.733465i	$0.262098\pi$
$824$	−8.48528	−0.295599
$825$	0	0
$826$	−36.0000	−1.25260
$827$	−2.82843	−0.0983540	−0.0491770	−	0.998790i	$-0.515660\pi$
$827$	−2.82843	−0.0983540	−0.0491770	+	0.998790i	$0.515660\pi$
$828$	0	0
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	0	0
$831$	0	0
$832$	24.0000	0.832050
$833$	−5.65685	−0.195998
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−48.0000	−1.65813
$839$	−42.4264	−1.46472	−0.732361	−	0.680916i	$-0.761582\pi$
$839$	−42.4264	−1.46472	−0.732361	+	0.680916i	$0.761582\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	32.5269	1.12095
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	21.0000	0.721569
$848$	39.5980	1.35980
$849$	0	0
$850$	0	0
$851$	63.6396	2.18154
$852$	0	0
$853$	33.0000	1.12990	0.564949	−	0.825126i	$-0.308896\pi$
$853$	33.0000	1.12990	0.564949	+	0.825126i	$0.308896\pi$
$854$	−55.1543	−1.88734
$855$	0	0
$856$	20.0000	0.683586
$857$	26.8701	0.917864	0.458932	−	0.888471i	$-0.348232\pi$
$857$	26.8701	0.917864	0.458932	+	0.888471i	$0.348232\pi$
$858$	0	0
$859$	−47.0000	−1.60362	−0.801810	−	0.597580i	$-0.796129\pi$
$859$	−47.0000	−1.60362	−0.801810	+	0.597580i	$0.796129\pi$
$860$	0	0
$861$	0	0
$862$	18.0000	0.613082
$863$	28.2843	0.962808	0.481404	−	0.876499i	$-0.340127\pi$
$863$	28.2843	0.962808	0.481404	+	0.876499i	$0.340127\pi$
$864$	0	0
$865$	0	0
$866$	−50.9117	−1.73005
$867$	0	0
$868$	0	0
$869$	−21.2132	−0.719609
$870$	0	0
$871$	−9.00000	−0.304953
$872$	22.6274	0.766261
$873$	0	0
$874$	10.0000	0.338255
$875$	0	0
$876$	0	0
$877$	15.0000	0.506514	0.253257	−	0.967399i	$-0.418498\pi$
$877$	15.0000	0.506514	0.253257	+	0.967399i	$0.418498\pi$
$878$	−19.7990	−0.668184
$879$	0	0
$880$	0	0
$881$	−38.1838	−1.28644	−0.643222	−	0.765680i	$-0.722403\pi$
$881$	−38.1838	−1.28644	−0.643222	+	0.765680i	$0.722403\pi$
$882$	0	0
$883$	27.0000	0.908622	0.454311	−	0.890843i	$-0.349885\pi$
$883$	27.0000	0.908622	0.454311	+	0.890843i	$0.349885\pi$
$884$	0	0
$885$	0	0
$886$	−20.0000	−0.671913
$887$	−45.2548	−1.51951	−0.759754	−	0.650210i	$-0.774681\pi$
$887$	−45.2548	−1.51951	−0.759754	+	0.650210i	$0.774681\pi$
$888$	0	0
$889$	−54.0000	−1.81110
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.82843	−0.0946497
$894$	0	0
$895$	0	0
$896$	33.9411	1.13389
$897$	0	0
$898$	0	0
$899$	−8.48528	−0.283000
$900$	0	0
$901$	28.0000	0.932815
$902$	−25.4558	−0.847587
$903$	0	0
$904$	4.00000	0.133038
$905$	0	0
$906$	0	0
$907$	3.00000	0.0996134	0.0498067	−	0.998759i	$-0.484139\pi$
$907$	3.00000	0.0996134	0.0498067	+	0.998759i	$0.484139\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	42.4264	1.40565	0.702825	−	0.711363i	$-0.251922\pi$
$911$	42.4264	1.40565	0.702825	+	0.711363i	$0.251922\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	16.9706	0.561336
$915$	0	0
$916$	0	0
$917$	25.4558	0.840626
$918$	0	0
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	0	0
$921$	0	0
$922$	24.0000	0.790398
$923$	−38.1838	−1.25683
$924$	0	0
$925$	0	0
$926$	−21.2132	−0.697109
$927$	0	0
$928$	0	0
$929$	46.6690	1.53116	0.765581	−	0.643340i	$-0.222452\pi$
$929$	46.6690	1.53116	0.765581	+	0.643340i	$0.222452\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	0	0
$934$	32.0000	1.04707
$935$	0	0
$936$	0	0
$937$	−9.00000	−0.294017	−0.147009	−	0.989135i	$-0.546964\pi$
$937$	−9.00000	−0.294017	−0.147009	+	0.989135i	$0.546964\pi$
$938$	−12.7279	−0.415581
$939$	0	0
$940$	0	0
$941$	21.2132	0.691531	0.345765	−	0.938321i	$-0.387619\pi$
$941$	21.2132	0.691531	0.345765	+	0.938321i	$0.387619\pi$
$942$	0	0
$943$	−30.0000	−0.976934
$944$	33.9411	1.10469
$945$	0	0
$946$	36.0000	1.17046
$947$	−36.7696	−1.19485	−0.597425	−	0.801925i	$-0.703809\pi$
$947$	−36.7696	−1.19485	−0.597425	+	0.801925i	$0.703809\pi$
$948$	0	0
$949$	27.0000	0.876457
$950$	0	0
$951$	0	0
$952$	24.0000	0.777844
$953$	−5.65685	−0.183243	−0.0916217	−	0.995794i	$-0.529205\pi$
$953$	−5.65685	−0.183243	−0.0916217	+	0.995794i	$0.529205\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−6.00000	−0.193851
$959$	4.24264	0.137002
$960$	0	0
$961$	−27.0000	−0.870968
$962$	38.1838	1.23109
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	33.0000	1.06121	0.530604	−	0.847620i	$-0.321965\pi$
$967$	33.0000	1.06121	0.530604	+	0.847620i	$0.321965\pi$
$968$	−19.7990	−0.636364
$969$	0	0
$970$	0	0
$971$	25.4558	0.816917	0.408458	−	0.912777i	$-0.366066\pi$
$971$	25.4558	0.816917	0.408458	+	0.912777i	$0.366066\pi$
$972$	0	0
$973$	39.0000	1.25028
$974$	−12.7279	−0.407829
$975$	0	0
$976$	52.0000	1.66448
$977$	1.41421	0.0452447	0.0226224	−	0.999744i	$-0.492798\pi$
$977$	1.41421	0.0452447	0.0226224	+	0.999744i	$0.492798\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	30.0000	0.957338
$983$	−52.3259	−1.66894	−0.834469	−	0.551056i	$-0.814225\pi$
$983$	−52.3259	−1.66894	−0.834469	+	0.551056i	$0.814225\pi$
$984$	0	0
$985$	0	0
$986$	16.9706	0.540453
$987$	0	0
$988$	0	0
$989$	42.4264	1.34908
$990$	0	0
$991$	53.0000	1.68360	0.841800	−	0.539789i	$-0.181496\pi$
$991$	53.0000	1.68360	0.841800	+	0.539789i	$0.181496\pi$
$992$	0	0
$993$	0	0
$994$	−54.0000	−1.71278
$995$	0	0
$996$	0	0
$997$	48.0000	1.52018	0.760088	−	0.649821i	$-0.225156\pi$
$997$	48.0000	1.52018	0.760088	+	0.649821i	$0.225156\pi$
$998$	22.6274	0.716258
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.a.m.1.2		2
3.2	odd	2	inner	675.2.a.m.1.1		2
5.2	odd	4		135.2.b.b.109.3	yes	4
5.3	odd	4		135.2.b.b.109.1	✓	4
5.4	even	2		675.2.a.l.1.1		2
15.2	even	4		135.2.b.b.109.2	yes	4
15.8	even	4		135.2.b.b.109.4	yes	4
15.14	odd	2		675.2.a.l.1.2		2
20.3	even	4		2160.2.f.k.1729.1		4
20.7	even	4		2160.2.f.k.1729.2		4
45.2	even	12		405.2.j.f.109.1		8
45.7	odd	12		405.2.j.f.109.4		8
45.13	odd	12		405.2.j.f.379.4		8
45.22	odd	12		405.2.j.f.379.2		8
45.23	even	12		405.2.j.f.379.1		8
45.32	even	12		405.2.j.f.379.3		8
45.38	even	12		405.2.j.f.109.3		8
45.43	odd	12		405.2.j.f.109.2		8
60.23	odd	4		2160.2.f.k.1729.4		4
60.47	odd	4		2160.2.f.k.1729.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.b.b.109.1	✓	4	5.3	odd	4
135.2.b.b.109.2	yes	4	15.2	even	4
135.2.b.b.109.3	yes	4	5.2	odd	4
135.2.b.b.109.4	yes	4	15.8	even	4
405.2.j.f.109.1		8	45.2	even	12
405.2.j.f.109.2		8	45.43	odd	12
405.2.j.f.109.3		8	45.38	even	12
405.2.j.f.109.4		8	45.7	odd	12
405.2.j.f.379.1		8	45.23	even	12
405.2.j.f.379.2		8	45.22	odd	12
405.2.j.f.379.3		8	45.32	even	12
405.2.j.f.379.4		8	45.13	odd	12
675.2.a.l.1.1		2	5.4	even	2
675.2.a.l.1.2		2	15.14	odd	2
675.2.a.m.1.1		2	3.2	odd	2	inner
675.2.a.m.1.2		2	1.1	even	1	trivial
2160.2.f.k.1729.1		4	20.3	even	4
2160.2.f.k.1729.2		4	20.7	even	4
2160.2.f.k.1729.3		4	60.47	odd	4
2160.2.f.k.1729.4		4	60.23	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +3.00000 q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +3.00000 q^{7} -2.82843 q^{8} +4.24264 q^{11} +3.00000 q^{13} +4.24264 q^{14} -4.00000 q^{16} -2.82843 q^{17} +1.00000 q^{19} +6.00000 q^{22} +7.07107 q^{23} +4.24264 q^{26} -4.24264 q^{29} +2.00000 q^{31} -4.00000 q^{34} +9.00000 q^{37} +1.41421 q^{38} -4.24264 q^{41} +6.00000 q^{43} +10.0000 q^{46} -2.82843 q^{47} +2.00000 q^{49} -9.89949 q^{53} -8.48528 q^{56} -6.00000 q^{58} -8.48528 q^{59} -13.0000 q^{61} +2.82843 q^{62} +8.00000 q^{64} -3.00000 q^{67} -12.7279 q^{71} +9.00000 q^{73} +12.7279 q^{74} +12.7279 q^{77} -5.00000 q^{79} -6.00000 q^{82} -1.41421 q^{83} +8.48528 q^{86} -12.0000 q^{88} +9.00000 q^{91} -4.00000 q^{94} +3.00000 q^{97} +2.82843 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{7}+O(q^{10}) \) 2 * q + 6 * q^7	\( 2 q + 6 q^{7} + 6 q^{13} - 8 q^{16} + 2 q^{19} + 12 q^{22} + 4 q^{31} - 8 q^{34} + 18 q^{37} + 12 q^{43} + 20 q^{46} + 4 q^{49} - 12 q^{58} - 26 q^{61} + 16 q^{64} - 6 q^{67} + 18 q^{73} - 10 q^{79} - 12 q^{82} - 24 q^{88} + 18 q^{91} - 8 q^{94} + 6 q^{97}+O(q^{100}) \) 2 * q + 6 * q^7 + 6 * q^13 - 8 * q^16 + 2 * q^19 + 12 * q^22 + 4 * q^31 - 8 * q^34 + 18 * q^37 + 12 * q^43 + 20 * q^46 + 4 * q^49 - 12 * q^58 - 26 * q^61 + 16 * q^64 - 6 * q^67 + 18 * q^73 - 10 * q^79 - 12 * q^82 - 24 * q^88 + 18 * q^91 - 8 * q^94 + 6 * q^97

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