LMFDB - Embedded newform 5625.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, 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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{4} -4.47214 q^{7} -3.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} -1.00000 q^{4} -4.47214 q^{7} -3.00000 q^{8} -3.23607 q^{11} -3.38197 q^{13} -4.47214 q^{14} -1.00000 q^{16} -2.85410 q^{17} -3.23607 q^{19} -3.23607 q^{22} -4.47214 q^{23} -3.38197 q^{26} +4.47214 q^{28} -4.38197 q^{29} +7.23607 q^{31} +5.00000 q^{32} -2.85410 q^{34} -8.09017 q^{37} -3.23607 q^{38} +1.38197 q^{41} +5.70820 q^{43} +3.23607 q^{44} -4.47214 q^{46} -5.23607 q^{47} +13.0000 q^{49} +3.38197 q^{52} +1.38197 q^{53} +13.4164 q^{56} -4.38197 q^{58} +4.00000 q^{59} -0.618034 q^{61} +7.23607 q^{62} +7.00000 q^{64} -5.23607 q^{67} +2.85410 q^{68} +0.763932 q^{71} -3.09017 q^{73} -8.09017 q^{74} +3.23607 q^{76} +14.4721 q^{77} +1.38197 q^{82} +3.52786 q^{83} +5.70820 q^{86} +9.70820 q^{88} -7.61803 q^{89} +15.1246 q^{91} +4.47214 q^{92} -5.23607 q^{94} -8.85410 q^{97} +13.0000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8	$ 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{11} - 9 q^{13} - 2 q^{16} + q^{17} - 2 q^{19} - 2 q^{22} - 9 q^{26} - 11 q^{29} + 10 q^{31} + 10 q^{32} + q^{34} - 5 q^{37} - 2 q^{38} + 5 q^{41} - 2 q^{43} + 2 q^{44} - 6 q^{47} + 26 q^{49} + 9 q^{52} + 5 q^{53} - 11 q^{58} + 8 q^{59} + q^{61} + 10 q^{62} + 14 q^{64} - 6 q^{67} - q^{68} + 6 q^{71} + 5 q^{73} - 5 q^{74} + 2 q^{76} + 20 q^{77} + 5 q^{82} + 16 q^{83} - 2 q^{86} + 6 q^{88} - 13 q^{89} - 10 q^{91} - 6 q^{94} - 11 q^{97} + 26 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^11 - 9 * q^13 - 2 * q^16 + q^17 - 2 * q^19 - 2 * q^22 - 9 * q^26 - 11 * q^29 + 10 * q^31 + 10 * q^32 + q^34 - 5 * q^37 - 2 * q^38 + 5 * q^41 - 2 * q^43 + 2 * q^44 - 6 * q^47 + 26 * q^49 + 9 * q^52 + 5 * q^53 - 11 * q^58 + 8 * q^59 + q^61 + 10 * q^62 + 14 * q^64 - 6 * q^67 - q^68 + 6 * q^71 + 5 * q^73 - 5 * q^74 + 2 * q^76 + 20 * q^77 + 5 * q^82 + 16 * q^83 - 2 * q^86 + 6 * q^88 - 13 * q^89 - 10 * q^91 - 6 * q^94 - 11 * q^97 + 26 * 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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.47214	−1.69031	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	−4.47214	−1.69031	−0.845154	+	0.534522i	$0.820491\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	0	0
$11$	−3.23607	−0.975711	−0.487856	−	0.872924i	$-0.662221\pi$
$11$	−3.23607	−0.975711	−0.487856	+	0.872924i	$0.662221\pi$
$12$	0	0
$13$	−3.38197	−0.937989	−0.468994	−	0.883201i	$-0.655384\pi$
$13$	−3.38197	−0.937989	−0.468994	+	0.883201i	$0.655384\pi$
$14$	−4.47214	−1.19523
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.85410	−0.692221	−0.346111	−	0.938194i	$-0.612498\pi$
$17$	−2.85410	−0.692221	−0.346111	+	0.938194i	$0.612498\pi$
$18$	0	0
$19$	−3.23607	−0.742405	−0.371202	−	0.928552i	$-0.621054\pi$
$19$	−3.23607	−0.742405	−0.371202	+	0.928552i	$0.621054\pi$
$20$	0	0
$21$	0	0
$22$	−3.23607	−0.689932
$23$	−4.47214	−0.932505	−0.466252	−	0.884652i	$-0.654396\pi$
$23$	−4.47214	−0.932505	−0.466252	+	0.884652i	$0.654396\pi$
$24$	0	0
$25$	0	0
$26$	−3.38197	−0.663258
$27$	0	0
$28$	4.47214	0.845154
$29$	−4.38197	−0.813711	−0.406855	−	0.913493i	$-0.633375\pi$
$29$	−4.38197	−0.813711	−0.406855	+	0.913493i	$0.633375\pi$
$30$	0	0
$31$	7.23607	1.29964	0.649818	−	0.760090i	$-0.274845\pi$
$31$	7.23607	1.29964	0.649818	+	0.760090i	$0.274845\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−2.85410	−0.489474
$35$	0	0
$36$	0	0
$37$	−8.09017	−1.33002	−0.665008	−	0.746836i	$-0.731572\pi$
$37$	−8.09017	−1.33002	−0.665008	+	0.746836i	$0.731572\pi$
$38$	−3.23607	−0.524960
$39$	0	0
$40$	0	0
$41$	1.38197	0.215827	0.107913	−	0.994160i	$-0.465583\pi$
$41$	1.38197	0.215827	0.107913	+	0.994160i	$0.465583\pi$
$42$	0	0
$43$	5.70820	0.870493	0.435246	−	0.900311i	$-0.356661\pi$
$43$	5.70820	0.870493	0.435246	+	0.900311i	$0.356661\pi$
$44$	3.23607	0.487856
$45$	0	0
$46$	−4.47214	−0.659380
$47$	−5.23607	−0.763759	−0.381880	−	0.924212i	$-0.624723\pi$
$47$	−5.23607	−0.763759	−0.381880	+	0.924212i	$0.624723\pi$
$48$	0	0
$49$	13.0000	1.85714
$50$	0	0
$51$	0	0
$52$	3.38197	0.468994
$53$	1.38197	0.189828	0.0949138	−	0.995485i	$-0.469742\pi$
$53$	1.38197	0.189828	0.0949138	+	0.995485i	$0.469742\pi$
$54$	0	0
$55$	0	0
$56$	13.4164	1.79284
$57$	0	0
$58$	−4.38197	−0.575380
$59$	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
$61$	−0.618034	−0.0791311	−0.0395656	−	0.999217i	$-0.512597\pi$
$61$	−0.618034	−0.0791311	−0.0395656	+	0.999217i	$0.512597\pi$
$62$	7.23607	0.918982
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	−5.23607	−0.639688	−0.319844	−	0.947470i	$-0.603630\pi$
$67$	−5.23607	−0.639688	−0.319844	+	0.947470i	$0.603630\pi$
$68$	2.85410	0.346111
$69$	0	0
$70$	0	0
$71$	0.763932	0.0906621	0.0453310	−	0.998972i	$-0.485566\pi$
$71$	0.763932	0.0906621	0.0453310	+	0.998972i	$0.485566\pi$
$72$	0	0
$73$	−3.09017	−0.361677	−0.180839	−	0.983513i	$-0.557881\pi$
$73$	−3.09017	−0.361677	−0.180839	+	0.983513i	$0.557881\pi$
$74$	−8.09017	−0.940463
$75$	0	0
$76$	3.23607	0.371202
$77$	14.4721	1.64925
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	1.38197	0.152613
$83$	3.52786	0.387233	0.193617	−	0.981077i	$-0.437978\pi$
$83$	3.52786	0.387233	0.193617	+	0.981077i	$0.437978\pi$
$84$	0	0
$85$	0	0
$86$	5.70820	0.615531
$87$	0	0
$88$	9.70820	1.03490
$89$	−7.61803	−0.807510	−0.403755	−	0.914867i	$-0.632295\pi$
$89$	−7.61803	−0.807510	−0.403755	+	0.914867i	$0.632295\pi$
$90$	0	0
$91$	15.1246	1.58549
$92$	4.47214	0.466252
$93$	0	0
$94$	−5.23607	−0.540059
$95$	0	0
$96$	0	0
$97$	−8.85410	−0.898998	−0.449499	−	0.893281i	$-0.648397\pi$
$97$	−8.85410	−0.898998	−0.449499	+	0.893281i	$0.648397\pi$
$98$	13.0000	1.31320
$99$	0	0
$100$	0	0
$101$	16.5623	1.64801	0.824006	−	0.566582i	$-0.191734\pi$
$101$	16.5623	1.64801	0.824006	+	0.566582i	$0.191734\pi$
$102$	0	0
$103$	1.23607	0.121793	0.0608967	−	0.998144i	$-0.480604\pi$
$103$	1.23607	0.121793	0.0608967	+	0.998144i	$0.480604\pi$
$104$	10.1459	0.994887
$105$	0	0
$106$	1.38197	0.134228
$107$	16.4721	1.59242	0.796211	−	0.605019i	$-0.206835\pi$
$107$	16.4721	1.59242	0.796211	+	0.605019i	$0.206835\pi$
$108$	0	0
$109$	−19.0902	−1.82851	−0.914253	−	0.405143i	$-0.867222\pi$
$109$	−19.0902	−1.82851	−0.914253	+	0.405143i	$0.867222\pi$
$110$	0	0
$111$	0	0
$112$	4.47214	0.422577
$113$	2.67376	0.251526	0.125763	−	0.992060i	$-0.459862\pi$
$113$	2.67376	0.251526	0.125763	+	0.992060i	$0.459862\pi$
$114$	0	0
$115$	0	0
$116$	4.38197	0.406855
$117$	0	0
$118$	4.00000	0.368230
$119$	12.7639	1.17007
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	−0.618034	−0.0559542
$123$	0	0
$124$	−7.23607	−0.649818
$125$	0	0
$126$	0	0
$127$	−9.70820	−0.861464	−0.430732	−	0.902480i	$-0.641745\pi$
$127$	−9.70820	−0.861464	−0.430732	+	0.902480i	$0.641745\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	0	0
$131$	−14.1803	−1.23894	−0.619471	−	0.785020i	$-0.712653\pi$
$131$	−14.1803	−1.23894	−0.619471	+	0.785020i	$0.712653\pi$
$132$	0	0
$133$	14.4721	1.25489
$134$	−5.23607	−0.452327
$135$	0	0
$136$	8.56231	0.734212
$137$	−5.38197	−0.459812	−0.229906	−	0.973213i	$-0.573842\pi$
$137$	−5.38197	−0.459812	−0.229906	+	0.973213i	$0.573842\pi$
$138$	0	0
$139$	−5.05573	−0.428821	−0.214411	−	0.976744i	$-0.568783\pi$
$139$	−5.05573	−0.428821	−0.214411	+	0.976744i	$0.568783\pi$
$140$	0	0
$141$	0	0
$142$	0.763932	0.0641078
$143$	10.9443	0.915206
$144$	0	0
$145$	0	0
$146$	−3.09017	−0.255744
$147$	0	0
$148$	8.09017	0.665008
$149$	12.1459	0.995031	0.497515	−	0.867455i	$-0.334246\pi$
$149$	12.1459	0.995031	0.497515	+	0.867455i	$0.334246\pi$
$150$	0	0
$151$	16.4721	1.34048	0.670242	−	0.742143i	$-0.266190\pi$
$151$	16.4721	1.34048	0.670242	+	0.742143i	$0.266190\pi$
$152$	9.70820	0.787439
$153$	0	0
$154$	14.4721	1.16620
$155$	0	0
$156$	0	0
$157$	13.7984	1.10123	0.550615	−	0.834759i	$-0.314393\pi$
$157$	13.7984	1.10123	0.550615	+	0.834759i	$0.314393\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	20.0000	1.57622
$162$	0	0
$163$	−2.94427	−0.230613	−0.115307	−	0.993330i	$-0.536785\pi$
$163$	−2.94427	−0.230613	−0.115307	+	0.993330i	$0.536785\pi$
$164$	−1.38197	−0.107913
$165$	0	0
$166$	3.52786	0.273815
$167$	−23.4164	−1.81202	−0.906008	−	0.423261i	$-0.860886\pi$
$167$	−23.4164	−1.81202	−0.906008	+	0.423261i	$0.860886\pi$
$168$	0	0
$169$	−1.56231	−0.120177
$170$	0	0
$171$	0	0
$172$	−5.70820	−0.435246
$173$	9.90983	0.753430	0.376715	−	0.926329i	$-0.377054\pi$
$173$	9.90983	0.753430	0.376715	+	0.926329i	$0.377054\pi$
$174$	0	0
$175$	0	0
$176$	3.23607	0.243928
$177$	0	0
$178$	−7.61803	−0.570996
$179$	−6.18034	−0.461940	−0.230970	−	0.972961i	$-0.574190\pi$
$179$	−6.18034	−0.461940	−0.230970	+	0.972961i	$0.574190\pi$
$180$	0	0
$181$	−9.79837	−0.728307	−0.364154	−	0.931339i	$-0.618642\pi$
$181$	−9.79837	−0.728307	−0.364154	+	0.931339i	$0.618642\pi$
$182$	15.1246	1.12111
$183$	0	0
$184$	13.4164	0.989071
$185$	0	0
$186$	0	0
$187$	9.23607	0.675408
$188$	5.23607	0.381880
$189$	0	0
$190$	0	0
$191$	24.6525	1.78379	0.891895	−	0.452242i	$-0.149376\pi$
$191$	24.6525	1.78379	0.891895	+	0.452242i	$0.149376\pi$
$192$	0	0
$193$	−7.67376	−0.552369	−0.276185	−	0.961105i	$-0.589070\pi$
$193$	−7.67376	−0.552369	−0.276185	+	0.961105i	$0.589070\pi$
$194$	−8.85410	−0.635687
$195$	0	0
$196$	−13.0000	−0.928571
$197$	5.38197	0.383449	0.191725	−	0.981449i	$-0.438592\pi$
$197$	5.38197	0.383449	0.191725	+	0.981449i	$0.438592\pi$
$198$	0	0
$199$	−18.6525	−1.32224	−0.661119	−	0.750281i	$-0.729918\pi$
$199$	−18.6525	−1.32224	−0.661119	+	0.750281i	$0.729918\pi$
$200$	0	0
$201$	0	0
$202$	16.5623	1.16532
$203$	19.5967	1.37542
$204$	0	0
$205$	0	0
$206$	1.23607	0.0861209
$207$	0	0
$208$	3.38197	0.234497
$209$	10.4721	0.724373
$210$	0	0
$211$	−17.8885	−1.23150	−0.615749	−	0.787942i	$-0.711146\pi$
$211$	−17.8885	−1.23150	−0.615749	+	0.787942i	$0.711146\pi$
$212$	−1.38197	−0.0949138
$213$	0	0
$214$	16.4721	1.12601
$215$	0	0
$216$	0	0
$217$	−32.3607	−2.19679
$218$	−19.0902	−1.29295
$219$	0	0
$220$	0	0
$221$	9.65248	0.649296
$222$	0	0
$223$	14.1803	0.949586	0.474793	−	0.880098i	$-0.342523\pi$
$223$	14.1803	0.949586	0.474793	+	0.880098i	$0.342523\pi$
$224$	−22.3607	−1.49404
$225$	0	0
$226$	2.67376	0.177856
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	−24.9787	−1.65064	−0.825320	−	0.564665i	$-0.809005\pi$
$229$	−24.9787	−1.65064	−0.825320	+	0.564665i	$0.809005\pi$
$230$	0	0
$231$	0	0
$232$	13.1459	0.863070
$233$	14.6180	0.957659	0.478830	−	0.877908i	$-0.341061\pi$
$233$	14.6180	0.957659	0.478830	+	0.877908i	$0.341061\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	12.7639	0.827363
$239$	7.05573	0.456397	0.228199	−	0.973615i	$-0.426716\pi$
$239$	7.05573	0.456397	0.228199	+	0.973615i	$0.426716\pi$
$240$	0	0
$241$	−1.03444	−0.0666343	−0.0333171	−	0.999445i	$-0.510607\pi$
$241$	−1.03444	−0.0666343	−0.0333171	+	0.999445i	$0.510607\pi$
$242$	−0.527864	−0.0339324
$243$	0	0
$244$	0.618034	0.0395656
$245$	0	0
$246$	0	0
$247$	10.9443	0.696367
$248$	−21.7082	−1.37847
$249$	0	0
$250$	0	0
$251$	26.9443	1.70071	0.850354	−	0.526212i	$-0.176388\pi$
$251$	26.9443	1.70071	0.850354	+	0.526212i	$0.176388\pi$
$252$	0	0
$253$	14.4721	0.909855
$254$	−9.70820	−0.609147
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−12.7984	−0.798341	−0.399170	−	0.916877i	$-0.630702\pi$
$257$	−12.7984	−0.798341	−0.399170	+	0.916877i	$0.630702\pi$
$258$	0	0
$259$	36.1803	2.24814
$260$	0	0
$261$	0	0
$262$	−14.1803	−0.876064
$263$	11.8885	0.733079	0.366540	−	0.930402i	$-0.380542\pi$
$263$	11.8885	0.733079	0.366540	+	0.930402i	$0.380542\pi$
$264$	0	0
$265$	0	0
$266$	14.4721	0.887344
$267$	0	0
$268$	5.23607	0.319844
$269$	0.145898	0.00889556	0.00444778	−	0.999990i	$-0.498584\pi$
$269$	0.145898	0.00889556	0.00444778	+	0.999990i	$0.498584\pi$
$270$	0	0
$271$	22.1803	1.34736	0.673680	−	0.739023i	$-0.264713\pi$
$271$	22.1803	1.34736	0.673680	+	0.739023i	$0.264713\pi$
$272$	2.85410	0.173055
$273$	0	0
$274$	−5.38197	−0.325136
$275$	0	0
$276$	0	0
$277$	5.79837	0.348391	0.174195	−	0.984711i	$-0.444268\pi$
$277$	5.79837	0.348391	0.174195	+	0.984711i	$0.444268\pi$
$278$	−5.05573	−0.303222
$279$	0	0
$280$	0	0
$281$	17.3820	1.03692	0.518461	−	0.855102i	$-0.326505\pi$
$281$	17.3820	1.03692	0.518461	+	0.855102i	$0.326505\pi$
$282$	0	0
$283$	−0.291796	−0.0173455	−0.00867274	−	0.999962i	$-0.502761\pi$
$283$	−0.291796	−0.0173455	−0.00867274	+	0.999962i	$0.502761\pi$
$284$	−0.763932	−0.0453310
$285$	0	0
$286$	10.9443	0.647148
$287$	−6.18034	−0.364814
$288$	0	0
$289$	−8.85410	−0.520830
$290$	0	0
$291$	0	0
$292$	3.09017	0.180839
$293$	3.79837	0.221903	0.110952	−	0.993826i	$-0.464610\pi$
$293$	3.79837	0.221903	0.110952	+	0.993826i	$0.464610\pi$
$294$	0	0
$295$	0	0
$296$	24.2705	1.41069
$297$	0	0
$298$	12.1459	0.703593
$299$	15.1246	0.874679
$300$	0	0
$301$	−25.5279	−1.47140
$302$	16.4721	0.947865
$303$	0	0
$304$	3.23607	0.185601
$305$	0	0
$306$	0	0
$307$	−1.34752	−0.0769073	−0.0384536	−	0.999260i	$-0.512243\pi$
$307$	−1.34752	−0.0769073	−0.0384536	+	0.999260i	$0.512243\pi$
$308$	−14.4721	−0.824626
$309$	0	0
$310$	0	0
$311$	−4.29180	−0.243365	−0.121683	−	0.992569i	$-0.538829\pi$
$311$	−4.29180	−0.243365	−0.121683	+	0.992569i	$0.538829\pi$
$312$	0	0
$313$	−8.47214	−0.478873	−0.239437	−	0.970912i	$-0.576963\pi$
$313$	−8.47214	−0.478873	−0.239437	+	0.970912i	$0.576963\pi$
$314$	13.7984	0.778687
$315$	0	0
$316$	0	0
$317$	−24.8328	−1.39475	−0.697375	−	0.716706i	$-0.745649\pi$
$317$	−24.8328	−1.39475	−0.697375	+	0.716706i	$0.745649\pi$
$318$	0	0
$319$	14.1803	0.793947
$320$	0	0
$321$	0	0
$322$	20.0000	1.11456
$323$	9.23607	0.513909
$324$	0	0
$325$	0	0
$326$	−2.94427	−0.163068
$327$	0	0
$328$	−4.14590	−0.228919
$329$	23.4164	1.29099
$330$	0	0
$331$	10.9443	0.601552	0.300776	−	0.953695i	$-0.402754\pi$
$331$	10.9443	0.601552	0.300776	+	0.953695i	$0.402754\pi$
$332$	−3.52786	−0.193617
$333$	0	0
$334$	−23.4164	−1.28129
$335$	0	0
$336$	0	0
$337$	14.9443	0.814066	0.407033	−	0.913413i	$-0.366563\pi$
$337$	14.9443	0.814066	0.407033	+	0.913413i	$0.366563\pi$
$338$	−1.56231	−0.0849782
$339$	0	0
$340$	0	0
$341$	−23.4164	−1.26807
$342$	0	0
$343$	−26.8328	−1.44884
$344$	−17.1246	−0.923297
$345$	0	0
$346$	9.90983	0.532756
$347$	−33.7082	−1.80955	−0.904776	−	0.425889i	$-0.859962\pi$
$347$	−33.7082	−1.80955	−0.904776	+	0.425889i	$0.859962\pi$
$348$	0	0
$349$	25.0344	1.34006	0.670031	−	0.742333i	$-0.266281\pi$
$349$	25.0344	1.34006	0.670031	+	0.742333i	$0.266281\pi$
$350$	0	0
$351$	0	0
$352$	−16.1803	−0.862415
$353$	−30.3607	−1.61594	−0.807968	−	0.589226i	$-0.799433\pi$
$353$	−30.3607	−1.61594	−0.807968	+	0.589226i	$0.799433\pi$
$354$	0	0
$355$	0	0
$356$	7.61803	0.403755
$357$	0	0
$358$	−6.18034	−0.326641
$359$	10.5836	0.558581	0.279290	−	0.960207i	$-0.409901\pi$
$359$	10.5836	0.558581	0.279290	+	0.960207i	$0.409901\pi$
$360$	0	0
$361$	−8.52786	−0.448835
$362$	−9.79837	−0.514991
$363$	0	0
$364$	−15.1246	−0.792745
$365$	0	0
$366$	0	0
$367$	6.00000	0.313197	0.156599	−	0.987662i	$-0.449947\pi$
$367$	6.00000	0.313197	0.156599	+	0.987662i	$0.449947\pi$
$368$	4.47214	0.233126
$369$	0	0
$370$	0	0
$371$	−6.18034	−0.320867
$372$	0	0
$373$	−29.4164	−1.52312	−0.761562	−	0.648092i	$-0.775567\pi$
$373$	−29.4164	−1.52312	−0.761562	+	0.648092i	$0.775567\pi$
$374$	9.23607	0.477586
$375$	0	0
$376$	15.7082	0.810089
$377$	14.8197	0.763251
$378$	0	0
$379$	23.4164	1.20282	0.601410	−	0.798941i	$-0.294606\pi$
$379$	23.4164	1.20282	0.601410	+	0.798941i	$0.294606\pi$
$380$	0	0
$381$	0	0
$382$	24.6525	1.26133
$383$	20.8328	1.06451	0.532254	−	0.846585i	$-0.321345\pi$
$383$	20.8328	1.06451	0.532254	+	0.846585i	$0.321345\pi$
$384$	0	0
$385$	0	0
$386$	−7.67376	−0.390584
$387$	0	0
$388$	8.85410	0.449499
$389$	37.0902	1.88055	0.940273	−	0.340421i	$-0.110570\pi$
$389$	37.0902	1.88055	0.940273	+	0.340421i	$0.110570\pi$
$390$	0	0
$391$	12.7639	0.645500
$392$	−39.0000	−1.96980
$393$	0	0
$394$	5.38197	0.271140
$395$	0	0
$396$	0	0
$397$	−10.9443	−0.549277	−0.274639	−	0.961548i	$-0.588558\pi$
$397$	−10.9443	−0.549277	−0.274639	+	0.961548i	$0.588558\pi$
$398$	−18.6525	−0.934964
$399$	0	0
$400$	0	0
$401$	−33.4508	−1.67046	−0.835228	−	0.549904i	$-0.814664\pi$
$401$	−33.4508	−1.67046	−0.835228	+	0.549904i	$0.814664\pi$
$402$	0	0
$403$	−24.4721	−1.21904
$404$	−16.5623	−0.824006
$405$	0	0
$406$	19.5967	0.972570
$407$	26.1803	1.29771
$408$	0	0
$409$	25.7984	1.27565	0.637824	−	0.770182i	$-0.279835\pi$
$409$	25.7984	1.27565	0.637824	+	0.770182i	$0.279835\pi$
$410$	0	0
$411$	0	0
$412$	−1.23607	−0.0608967
$413$	−17.8885	−0.880238
$414$	0	0
$415$	0	0
$416$	−16.9098	−0.829073
$417$	0	0
$418$	10.4721	0.512209
$419$	−32.9443	−1.60943	−0.804717	−	0.593659i	$-0.797683\pi$
$419$	−32.9443	−1.60943	−0.804717	+	0.593659i	$0.797683\pi$
$420$	0	0
$421$	23.1459	1.12806	0.564031	−	0.825754i	$-0.309250\pi$
$421$	23.1459	1.12806	0.564031	+	0.825754i	$0.309250\pi$
$422$	−17.8885	−0.870801
$423$	0	0
$424$	−4.14590	−0.201343
$425$	0	0
$426$	0	0
$427$	2.76393	0.133756
$428$	−16.4721	−0.796211
$429$	0	0
$430$	0	0
$431$	10.6525	0.513112	0.256556	−	0.966529i	$-0.417412\pi$
$431$	10.6525	0.513112	0.256556	+	0.966529i	$0.417412\pi$
$432$	0	0
$433$	−24.9787	−1.20040	−0.600200	−	0.799850i	$-0.704912\pi$
$433$	−24.9787	−1.20040	−0.600200	+	0.799850i	$0.704912\pi$
$434$	−32.3607	−1.55336
$435$	0	0
$436$	19.0902	0.914253
$437$	14.4721	0.692296
$438$	0	0
$439$	−6.18034	−0.294972	−0.147486	−	0.989064i	$-0.547118\pi$
$439$	−6.18034	−0.294972	−0.147486	+	0.989064i	$0.547118\pi$
$440$	0	0
$441$	0	0
$442$	9.65248	0.459121
$443$	−30.7639	−1.46164	−0.730819	−	0.682571i	$-0.760862\pi$
$443$	−30.7639	−1.46164	−0.730819	+	0.682571i	$0.760862\pi$
$444$	0	0
$445$	0	0
$446$	14.1803	0.671459
$447$	0	0
$448$	−31.3050	−1.47902
$449$	7.79837	0.368028	0.184014	−	0.982924i	$-0.441091\pi$
$449$	7.79837	0.368028	0.184014	+	0.982924i	$0.441091\pi$
$450$	0	0
$451$	−4.47214	−0.210585
$452$	−2.67376	−0.125763
$453$	0	0
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	−22.3607	−1.04599	−0.522994	−	0.852336i	$-0.675185\pi$
$457$	−22.3607	−1.04599	−0.522994	+	0.852336i	$0.675185\pi$
$458$	−24.9787	−1.16718
$459$	0	0
$460$	0	0
$461$	8.20163	0.381988	0.190994	−	0.981591i	$-0.438829\pi$
$461$	8.20163	0.381988	0.190994	+	0.981591i	$0.438829\pi$
$462$	0	0
$463$	−23.2361	−1.07987	−0.539936	−	0.841706i	$-0.681552\pi$
$463$	−23.2361	−1.07987	−0.539936	+	0.841706i	$0.681552\pi$
$464$	4.38197	0.203428
$465$	0	0
$466$	14.6180	0.677167
$467$	5.81966	0.269302	0.134651	−	0.990893i	$-0.457009\pi$
$467$	5.81966	0.269302	0.134651	+	0.990893i	$0.457009\pi$
$468$	0	0
$469$	23.4164	1.08127
$470$	0	0
$471$	0	0
$472$	−12.0000	−0.552345
$473$	−18.4721	−0.849350
$474$	0	0
$475$	0	0
$476$	−12.7639	−0.585034
$477$	0	0
$478$	7.05573	0.322721
$479$	−1.05573	−0.0482374	−0.0241187	−	0.999709i	$-0.507678\pi$
$479$	−1.05573	−0.0482374	−0.0241187	+	0.999709i	$0.507678\pi$
$480$	0	0
$481$	27.3607	1.24754
$482$	−1.03444	−0.0471175
$483$	0	0
$484$	0.527864	0.0239938
$485$	0	0
$486$	0	0
$487$	3.70820	0.168035	0.0840174	−	0.996464i	$-0.473225\pi$
$487$	3.70820	0.168035	0.0840174	+	0.996464i	$0.473225\pi$
$488$	1.85410	0.0839313
$489$	0	0
$490$	0	0
$491$	29.8885	1.34885	0.674426	−	0.738343i	$-0.264391\pi$
$491$	29.8885	1.34885	0.674426	+	0.738343i	$0.264391\pi$
$492$	0	0
$493$	12.5066	0.563268
$494$	10.9443	0.492406
$495$	0	0
$496$	−7.23607	−0.324909
$497$	−3.41641	−0.153247
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	0	0
$501$	0	0
$502$	26.9443	1.20258
$503$	35.4164	1.57914	0.789570	−	0.613661i	$-0.210304\pi$
$503$	35.4164	1.57914	0.789570	+	0.613661i	$0.210304\pi$
$504$	0	0
$505$	0	0
$506$	14.4721	0.643365
$507$	0	0
$508$	9.70820	0.430732
$509$	−26.8541	−1.19029	−0.595144	−	0.803619i	$-0.702905\pi$
$509$	−26.8541	−1.19029	−0.595144	+	0.803619i	$0.702905\pi$
$510$	0	0
$511$	13.8197	0.611346
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−12.7984	−0.564512
$515$	0	0
$516$	0	0
$517$	16.9443	0.745208
$518$	36.1803	1.58967
$519$	0	0
$520$	0	0
$521$	−2.03444	−0.0891305	−0.0445653	−	0.999006i	$-0.514190\pi$
$521$	−2.03444	−0.0891305	−0.0445653	+	0.999006i	$0.514190\pi$
$522$	0	0
$523$	12.2918	0.537483	0.268741	−	0.963212i	$-0.413392\pi$
$523$	12.2918	0.537483	0.268741	+	0.963212i	$0.413392\pi$
$524$	14.1803	0.619471
$525$	0	0
$526$	11.8885	0.518365
$527$	−20.6525	−0.899636
$528$	0	0
$529$	−3.00000	−0.130435
$530$	0	0
$531$	0	0
$532$	−14.4721	−0.627447
$533$	−4.67376	−0.202443
$534$	0	0
$535$	0	0
$536$	15.7082	0.678491
$537$	0	0
$538$	0.145898	0.00629011
$539$	−42.0689	−1.81204
$540$	0	0
$541$	−35.3262	−1.51879	−0.759397	−	0.650628i	$-0.774506\pi$
$541$	−35.3262	−1.51879	−0.759397	+	0.650628i	$0.774506\pi$
$542$	22.1803	0.952727
$543$	0	0
$544$	−14.2705	−0.611843
$545$	0	0
$546$	0	0
$547$	2.36068	0.100935	0.0504677	−	0.998726i	$-0.483929\pi$
$547$	2.36068	0.100935	0.0504677	+	0.998726i	$0.483929\pi$
$548$	5.38197	0.229906
$549$	0	0
$550$	0	0
$551$	14.1803	0.604103
$552$	0	0
$553$	0	0
$554$	5.79837	0.246349
$555$	0	0
$556$	5.05573	0.214411
$557$	22.2705	0.943632	0.471816	−	0.881697i	$-0.343599\pi$
$557$	22.2705	0.943632	0.471816	+	0.881697i	$0.343599\pi$
$558$	0	0
$559$	−19.3050	−0.816512
$560$	0	0
$561$	0	0
$562$	17.3820	0.733214
$563$	−17.5279	−0.738711	−0.369356	−	0.929288i	$-0.620422\pi$
$563$	−17.5279	−0.738711	−0.369356	+	0.929288i	$0.620422\pi$
$564$	0	0
$565$	0	0
$566$	−0.291796	−0.0122651
$567$	0	0
$568$	−2.29180	−0.0961616
$569$	30.2705	1.26901	0.634503	−	0.772920i	$-0.281205\pi$
$569$	30.2705	1.26901	0.634503	+	0.772920i	$0.281205\pi$
$570$	0	0
$571$	−21.2361	−0.888702	−0.444351	−	0.895853i	$-0.646566\pi$
$571$	−21.2361	−0.888702	−0.444351	+	0.895853i	$0.646566\pi$
$572$	−10.9443	−0.457603
$573$	0	0
$574$	−6.18034	−0.257962
$575$	0	0
$576$	0	0
$577$	30.9443	1.28823	0.644113	−	0.764930i	$-0.277226\pi$
$577$	30.9443	1.28823	0.644113	+	0.764930i	$0.277226\pi$
$578$	−8.85410	−0.368282
$579$	0	0
$580$	0	0
$581$	−15.7771	−0.654544
$582$	0	0
$583$	−4.47214	−0.185217
$584$	9.27051	0.383616
$585$	0	0
$586$	3.79837	0.156909
$587$	−14.5836	−0.601929	−0.300965	−	0.953635i	$-0.597309\pi$
$587$	−14.5836	−0.601929	−0.300965	+	0.953635i	$0.597309\pi$
$588$	0	0
$589$	−23.4164	−0.964856
$590$	0	0
$591$	0	0
$592$	8.09017	0.332504
$593$	−32.7426	−1.34458	−0.672290	−	0.740288i	$-0.734689\pi$
$593$	−32.7426	−1.34458	−0.672290	+	0.740288i	$0.734689\pi$
$594$	0	0
$595$	0	0
$596$	−12.1459	−0.497515
$597$	0	0
$598$	15.1246	0.618491
$599$	12.4721	0.509598	0.254799	−	0.966994i	$-0.417991\pi$
$599$	12.4721	0.509598	0.254799	+	0.966994i	$0.417991\pi$
$600$	0	0
$601$	−24.3262	−0.992288	−0.496144	−	0.868240i	$-0.665251\pi$
$601$	−24.3262	−0.992288	−0.496144	+	0.868240i	$0.665251\pi$
$602$	−25.5279	−1.04044
$603$	0	0
$604$	−16.4721	−0.670242
$605$	0	0
$606$	0	0
$607$	39.2361	1.59254	0.796271	−	0.604940i	$-0.206803\pi$
$607$	39.2361	1.59254	0.796271	+	0.604940i	$0.206803\pi$
$608$	−16.1803	−0.656199
$609$	0	0
$610$	0	0
$611$	17.7082	0.716397
$612$	0	0
$613$	−39.5066	−1.59566	−0.797828	−	0.602885i	$-0.794018\pi$
$613$	−39.5066	−1.59566	−0.797828	+	0.602885i	$0.794018\pi$
$614$	−1.34752	−0.0543816
$615$	0	0
$616$	−43.4164	−1.74930
$617$	26.9787	1.08612	0.543061	−	0.839693i	$-0.317265\pi$
$617$	26.9787	1.08612	0.543061	+	0.839693i	$0.317265\pi$
$618$	0	0
$619$	28.3607	1.13991	0.569956	−	0.821675i	$-0.306960\pi$
$619$	28.3607	1.13991	0.569956	+	0.821675i	$0.306960\pi$
$620$	0	0
$621$	0	0
$622$	−4.29180	−0.172085
$623$	34.0689	1.36494
$624$	0	0
$625$	0	0
$626$	−8.47214	−0.338615
$627$	0	0
$628$	−13.7984	−0.550615
$629$	23.0902	0.920665
$630$	0	0
$631$	10.2918	0.409710	0.204855	−	0.978792i	$-0.434328\pi$
$631$	10.2918	0.409710	0.204855	+	0.978792i	$0.434328\pi$
$632$	0	0
$633$	0	0
$634$	−24.8328	−0.986237
$635$	0	0
$636$	0	0
$637$	−43.9656	−1.74198
$638$	14.1803	0.561405
$639$	0	0
$640$	0	0
$641$	−38.9443	−1.53821	−0.769103	−	0.639125i	$-0.779297\pi$
$641$	−38.9443	−1.53821	−0.769103	+	0.639125i	$0.779297\pi$
$642$	0	0
$643$	−13.8885	−0.547711	−0.273855	−	0.961771i	$-0.588299\pi$
$643$	−13.8885	−0.547711	−0.273855	+	0.961771i	$0.588299\pi$
$644$	−20.0000	−0.788110
$645$	0	0
$646$	9.23607	0.363388
$647$	10.3607	0.407320	0.203660	−	0.979042i	$-0.434716\pi$
$647$	10.3607	0.407320	0.203660	+	0.979042i	$0.434716\pi$
$648$	0	0
$649$	−12.9443	−0.508107
$650$	0	0
$651$	0	0
$652$	2.94427	0.115307
$653$	45.1033	1.76503	0.882515	−	0.470285i	$-0.155849\pi$
$653$	45.1033	1.76503	0.882515	+	0.470285i	$0.155849\pi$
$654$	0	0
$655$	0	0
$656$	−1.38197	−0.0539567
$657$	0	0
$658$	23.4164	0.912867
$659$	6.58359	0.256460	0.128230	−	0.991744i	$-0.459070\pi$
$659$	6.58359	0.256460	0.128230	+	0.991744i	$0.459070\pi$
$660$	0	0
$661$	25.4164	0.988584	0.494292	−	0.869296i	$-0.335427\pi$
$661$	25.4164	0.988584	0.494292	+	0.869296i	$0.335427\pi$
$662$	10.9443	0.425361
$663$	0	0
$664$	−10.5836	−0.410723
$665$	0	0
$666$	0	0
$667$	19.5967	0.758789
$668$	23.4164	0.906008
$669$	0	0
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	−29.7984	−1.14864	−0.574321	−	0.818630i	$-0.694734\pi$
$673$	−29.7984	−1.14864	−0.574321	+	0.818630i	$0.694734\pi$
$674$	14.9443	0.575632
$675$	0	0
$676$	1.56231	0.0600887
$677$	37.4164	1.43803	0.719015	−	0.694995i	$-0.244593\pi$
$677$	37.4164	1.43803	0.719015	+	0.694995i	$0.244593\pi$
$678$	0	0
$679$	39.5967	1.51958
$680$	0	0
$681$	0	0
$682$	−23.4164	−0.896661
$683$	7.41641	0.283781	0.141890	−	0.989882i	$-0.454682\pi$
$683$	7.41641	0.283781	0.141890	+	0.989882i	$0.454682\pi$
$684$	0	0
$685$	0	0
$686$	−26.8328	−1.02448
$687$	0	0
$688$	−5.70820	−0.217623
$689$	−4.67376	−0.178056
$690$	0	0
$691$	−21.2361	−0.807858	−0.403929	−	0.914790i	$-0.632356\pi$
$691$	−21.2361	−0.807858	−0.403929	+	0.914790i	$0.632356\pi$
$692$	−9.90983	−0.376715
$693$	0	0
$694$	−33.7082	−1.27955
$695$	0	0
$696$	0	0
$697$	−3.94427	−0.149400
$698$	25.0344	0.947568
$699$	0	0
$700$	0	0
$701$	−0.437694	−0.0165315	−0.00826574	−	0.999966i	$-0.502631\pi$
$701$	−0.437694	−0.0165315	−0.00826574	+	0.999966i	$0.502631\pi$
$702$	0	0
$703$	26.1803	0.987410
$704$	−22.6525	−0.853747
$705$	0	0
$706$	−30.3607	−1.14264
$707$	−74.0689	−2.78565
$708$	0	0
$709$	−6.72949	−0.252731	−0.126366	−	0.991984i	$-0.540331\pi$
$709$	−6.72949	−0.252731	−0.126366	+	0.991984i	$0.540331\pi$
$710$	0	0
$711$	0	0
$712$	22.8541	0.856494
$713$	−32.3607	−1.21192
$714$	0	0
$715$	0	0
$716$	6.18034	0.230970
$717$	0	0
$718$	10.5836	0.394976
$719$	35.8885	1.33842	0.669208	−	0.743075i	$-0.266633\pi$
$719$	35.8885	1.33842	0.669208	+	0.743075i	$0.266633\pi$
$720$	0	0
$721$	−5.52786	−0.205868
$722$	−8.52786	−0.317374
$723$	0	0
$724$	9.79837	0.364154
$725$	0	0
$726$	0	0
$727$	−15.3475	−0.569208	−0.284604	−	0.958645i	$-0.591862\pi$
$727$	−15.3475	−0.569208	−0.284604	+	0.958645i	$0.591862\pi$
$728$	−45.3738	−1.68167
$729$	0	0
$730$	0	0
$731$	−16.2918	−0.602574
$732$	0	0
$733$	−2.58359	−0.0954272	−0.0477136	−	0.998861i	$-0.515193\pi$
$733$	−2.58359	−0.0954272	−0.0477136	+	0.998861i	$0.515193\pi$
$734$	6.00000	0.221464
$735$	0	0
$736$	−22.3607	−0.824226
$737$	16.9443	0.624150
$738$	0	0
$739$	17.7082	0.651407	0.325703	−	0.945472i	$-0.394399\pi$
$739$	17.7082	0.651407	0.325703	+	0.945472i	$0.394399\pi$
$740$	0	0
$741$	0	0
$742$	−6.18034	−0.226887
$743$	0.875388	0.0321149	0.0160574	−	0.999871i	$-0.494889\pi$
$743$	0.875388	0.0321149	0.0160574	+	0.999871i	$0.494889\pi$
$744$	0	0
$745$	0	0
$746$	−29.4164	−1.07701
$747$	0	0
$748$	−9.23607	−0.337704
$749$	−73.6656	−2.69168
$750$	0	0
$751$	5.34752	0.195134	0.0975670	−	0.995229i	$-0.468894\pi$
$751$	5.34752	0.195134	0.0975670	+	0.995229i	$0.468894\pi$
$752$	5.23607	0.190940
$753$	0	0
$754$	14.8197	0.539700
$755$	0	0
$756$	0	0
$757$	−27.3820	−0.995214	−0.497607	−	0.867402i	$-0.665788\pi$
$757$	−27.3820	−0.995214	−0.497607	+	0.867402i	$0.665788\pi$
$758$	23.4164	0.850522
$759$	0	0
$760$	0	0
$761$	9.74265	0.353171	0.176585	−	0.984285i	$-0.443495\pi$
$761$	9.74265	0.353171	0.176585	+	0.984285i	$0.443495\pi$
$762$	0	0
$763$	85.3738	3.09074
$764$	−24.6525	−0.891895
$765$	0	0
$766$	20.8328	0.752720
$767$	−13.5279	−0.488463
$768$	0	0
$769$	−5.63932	−0.203359	−0.101680	−	0.994817i	$-0.532422\pi$
$769$	−5.63932	−0.203359	−0.101680	+	0.994817i	$0.532422\pi$
$770$	0	0
$771$	0	0
$772$	7.67376	0.276185
$773$	−35.9230	−1.29206	−0.646030	−	0.763312i	$-0.723572\pi$
$773$	−35.9230	−1.29206	−0.646030	+	0.763312i	$0.723572\pi$
$774$	0	0
$775$	0	0
$776$	26.5623	0.953531
$777$	0	0
$778$	37.0902	1.32975
$779$	−4.47214	−0.160231
$780$	0	0
$781$	−2.47214	−0.0884600
$782$	12.7639	0.456437
$783$	0	0
$784$	−13.0000	−0.464286
$785$	0	0
$786$	0	0
$787$	−8.18034	−0.291598	−0.145799	−	0.989314i	$-0.546575\pi$
$787$	−8.18034	−0.291598	−0.145799	+	0.989314i	$0.546575\pi$
$788$	−5.38197	−0.191725
$789$	0	0
$790$	0	0
$791$	−11.9574	−0.425157
$792$	0	0
$793$	2.09017	0.0742241
$794$	−10.9443	−0.388398
$795$	0	0
$796$	18.6525	0.661119
$797$	38.1033	1.34969	0.674845	−	0.737960i	$-0.264211\pi$
$797$	38.1033	1.34969	0.674845	+	0.737960i	$0.264211\pi$
$798$	0	0
$799$	14.9443	0.528690
$800$	0	0
$801$	0	0
$802$	−33.4508	−1.18119
$803$	10.0000	0.352892
$804$	0	0
$805$	0	0
$806$	−24.4721	−0.861994
$807$	0	0
$808$	−49.6869	−1.74798
$809$	23.2148	0.816188	0.408094	−	0.912940i	$-0.366194\pi$
$809$	23.2148	0.816188	0.408094	+	0.912940i	$0.366194\pi$
$810$	0	0
$811$	9.23607	0.324322	0.162161	−	0.986764i	$-0.448154\pi$
$811$	9.23607	0.324322	0.162161	+	0.986764i	$0.448154\pi$
$812$	−19.5967	−0.687711
$813$	0	0
$814$	26.1803	0.917620
$815$	0	0
$816$	0	0
$817$	−18.4721	−0.646258
$818$	25.7984	0.902019
$819$	0	0
$820$	0	0
$821$	−5.05573	−0.176446	−0.0882231	−	0.996101i	$-0.528119\pi$
$821$	−5.05573	−0.176446	−0.0882231	+	0.996101i	$0.528119\pi$
$822$	0	0
$823$	14.7639	0.514638	0.257319	−	0.966326i	$-0.417161\pi$
$823$	14.7639	0.514638	0.257319	+	0.966326i	$0.417161\pi$
$824$	−3.70820	−0.129181
$825$	0	0
$826$	−17.8885	−0.622422
$827$	22.0689	0.767410	0.383705	−	0.923456i	$-0.374648\pi$
$827$	22.0689	0.767410	0.383705	+	0.923456i	$0.374648\pi$
$828$	0	0
$829$	−26.5066	−0.920611	−0.460306	−	0.887760i	$-0.652260\pi$
$829$	−26.5066	−0.920611	−0.460306	+	0.887760i	$0.652260\pi$
$830$	0	0
$831$	0	0
$832$	−23.6738	−0.820740
$833$	−37.1033	−1.28555
$834$	0	0
$835$	0	0
$836$	−10.4721	−0.362186
$837$	0	0
$838$	−32.9443	−1.13804
$839$	44.0689	1.52143	0.760713	−	0.649088i	$-0.224849\pi$
$839$	44.0689	1.52143	0.760713	+	0.649088i	$0.224849\pi$
$840$	0	0
$841$	−9.79837	−0.337875
$842$	23.1459	0.797660
$843$	0	0
$844$	17.8885	0.615749
$845$	0	0
$846$	0	0
$847$	2.36068	0.0811139
$848$	−1.38197	−0.0474569
$849$	0	0
$850$	0	0
$851$	36.1803	1.24025
$852$	0	0
$853$	−29.8541	−1.02218	−0.511092	−	0.859526i	$-0.670759\pi$
$853$	−29.8541	−1.02218	−0.511092	+	0.859526i	$0.670759\pi$
$854$	2.76393	0.0945798
$855$	0	0
$856$	−49.4164	−1.68902
$857$	−3.52786	−0.120510	−0.0602548	−	0.998183i	$-0.519191\pi$
$857$	−3.52786	−0.120510	−0.0602548	+	0.998183i	$0.519191\pi$
$858$	0	0
$859$	3.41641	0.116566	0.0582832	−	0.998300i	$-0.481437\pi$
$859$	3.41641	0.116566	0.0582832	+	0.998300i	$0.481437\pi$
$860$	0	0
$861$	0	0
$862$	10.6525	0.362825
$863$	51.1246	1.74030	0.870151	−	0.492785i	$-0.164021\pi$
$863$	51.1246	1.74030	0.870151	+	0.492785i	$0.164021\pi$
$864$	0	0
$865$	0	0
$866$	−24.9787	−0.848811
$867$	0	0
$868$	32.3607	1.09839
$869$	0	0
$870$	0	0
$871$	17.7082	0.600020
$872$	57.2705	1.93942
$873$	0	0
$874$	14.4721	0.489527
$875$	0	0
$876$	0	0
$877$	−18.2148	−0.615069	−0.307535	−	0.951537i	$-0.599504\pi$
$877$	−18.2148	−0.615069	−0.307535	+	0.951537i	$0.599504\pi$
$878$	−6.18034	−0.208576
$879$	0	0
$880$	0	0
$881$	−25.0557	−0.844149	−0.422074	−	0.906561i	$-0.638698\pi$
$881$	−25.0557	−0.844149	−0.422074	+	0.906561i	$0.638698\pi$
$882$	0	0
$883$	0.180340	0.00606892	0.00303446	−	0.999995i	$-0.499034\pi$
$883$	0.180340	0.00606892	0.00303446	+	0.999995i	$0.499034\pi$
$884$	−9.65248	−0.324648
$885$	0	0
$886$	−30.7639	−1.03353
$887$	17.1246	0.574988	0.287494	−	0.957782i	$-0.407178\pi$
$887$	17.1246	0.574988	0.287494	+	0.957782i	$0.407178\pi$
$888$	0	0
$889$	43.4164	1.45614
$890$	0	0
$891$	0	0
$892$	−14.1803	−0.474793
$893$	16.9443	0.567018
$894$	0	0
$895$	0	0
$896$	13.4164	0.448211
$897$	0	0
$898$	7.79837	0.260235
$899$	−31.7082	−1.05753
$900$	0	0
$901$	−3.94427	−0.131403
$902$	−4.47214	−0.148906
$903$	0	0
$904$	−8.02129	−0.266784
$905$	0	0
$906$	0	0
$907$	33.1246	1.09988	0.549942	−	0.835203i	$-0.314650\pi$
$907$	33.1246	1.09988	0.549942	+	0.835203i	$0.314650\pi$
$908$	20.0000	0.663723
$909$	0	0
$910$	0	0
$911$	4.18034	0.138501	0.0692504	−	0.997599i	$-0.477939\pi$
$911$	4.18034	0.138501	0.0692504	+	0.997599i	$0.477939\pi$
$912$	0	0
$913$	−11.4164	−0.377828
$914$	−22.3607	−0.739626
$915$	0	0
$916$	24.9787	0.825320
$917$	63.4164	2.09419
$918$	0	0
$919$	−49.1935	−1.62274	−0.811372	−	0.584530i	$-0.801279\pi$
$919$	−49.1935	−1.62274	−0.811372	+	0.584530i	$0.801279\pi$
$920$	0	0
$921$	0	0
$922$	8.20163	0.270106
$923$	−2.58359	−0.0850400
$924$	0	0
$925$	0	0
$926$	−23.2361	−0.763585
$927$	0	0
$928$	−21.9098	−0.719225
$929$	−2.09017	−0.0685763	−0.0342881	−	0.999412i	$-0.510916\pi$
$929$	−2.09017	−0.0685763	−0.0342881	+	0.999412i	$0.510916\pi$
$930$	0	0
$931$	−42.0689	−1.37875
$932$	−14.6180	−0.478830
$933$	0	0
$934$	5.81966	0.190425
$935$	0	0
$936$	0	0
$937$	−11.6869	−0.381795	−0.190897	−	0.981610i	$-0.561140\pi$
$937$	−11.6869	−0.381795	−0.190897	+	0.981610i	$0.561140\pi$
$938$	23.4164	0.764573
$939$	0	0
$940$	0	0
$941$	−5.32624	−0.173630	−0.0868152	−	0.996224i	$-0.527669\pi$
$941$	−5.32624	−0.173630	−0.0868152	+	0.996224i	$0.527669\pi$
$942$	0	0
$943$	−6.18034	−0.201260
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−18.4721	−0.600581
$947$	29.4164	0.955905	0.477952	−	0.878386i	$-0.341379\pi$
$947$	29.4164	0.955905	0.477952	+	0.878386i	$0.341379\pi$
$948$	0	0
$949$	10.4508	0.339249
$950$	0	0
$951$	0	0
$952$	−38.2918	−1.24104
$953$	10.0902	0.326853	0.163426	−	0.986556i	$-0.447745\pi$
$953$	10.0902	0.326853	0.163426	+	0.986556i	$0.447745\pi$
$954$	0	0
$955$	0	0
$956$	−7.05573	−0.228199
$957$	0	0
$958$	−1.05573	−0.0341090
$959$	24.0689	0.777225
$960$	0	0
$961$	21.3607	0.689054
$962$	27.3607	0.882144
$963$	0	0
$964$	1.03444	0.0333171
$965$	0	0
$966$	0	0
$967$	−34.8328	−1.12015	−0.560074	−	0.828443i	$-0.689227\pi$
$967$	−34.8328	−1.12015	−0.560074	+	0.828443i	$0.689227\pi$
$968$	1.58359	0.0508986
$969$	0	0
$970$	0	0
$971$	6.58359	0.211278	0.105639	−	0.994405i	$-0.466311\pi$
$971$	6.58359	0.211278	0.105639	+	0.994405i	$0.466311\pi$
$972$	0	0
$973$	22.6099	0.724840
$974$	3.70820	0.118819
$975$	0	0
$976$	0.618034	0.0197828
$977$	−45.7426	−1.46344	−0.731718	−	0.681607i	$-0.761281\pi$
$977$	−45.7426	−1.46344	−0.731718	+	0.681607i	$0.761281\pi$
$978$	0	0
$979$	24.6525	0.787897
$980$	0	0
$981$	0	0
$982$	29.8885	0.953782
$983$	21.7082	0.692384	0.346192	−	0.938164i	$-0.387475\pi$
$983$	21.7082	0.692384	0.346192	+	0.938164i	$0.387475\pi$
$984$	0	0
$985$	0	0
$986$	12.5066	0.398291
$987$	0	0
$988$	−10.9443	−0.348184
$989$	−25.5279	−0.811739
$990$	0	0
$991$	−3.12461	−0.0992566	−0.0496283	−	0.998768i	$-0.515804\pi$
$991$	−3.12461	−0.0992566	−0.0496283	+	0.998768i	$0.515804\pi$
$992$	36.1803	1.14873
$993$	0	0
$994$	−3.41641	−0.108362
$995$	0	0
$996$	0	0
$997$	−9.41641	−0.298221	−0.149110	−	0.988821i	$-0.547641\pi$
$997$	−9.41641	−0.298221	−0.149110	+	0.988821i	$0.547641\pi$
$998$	−6.00000	−0.189927
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.h.1.1		2
3.2	odd	2		1875.2.a.a.1.1		2
5.4	even	2		5625.2.a.a.1.2		2
15.2	even	4		1875.2.b.b.1249.1		4
15.8	even	4		1875.2.b.b.1249.4		4
15.14	odd	2		1875.2.a.d.1.2		2
25.4	even	10		225.2.h.a.91.1		4
25.19	even	10		225.2.h.a.136.1		4
75.8	even	20		375.2.i.a.199.1		8
75.17	even	20		375.2.i.a.199.2		8
75.29	odd	10		75.2.g.a.16.1	✓	4
75.44	odd	10		75.2.g.a.61.1	yes	4
75.47	even	20		375.2.i.a.49.1		8
75.53	even	20		375.2.i.a.49.2		8
75.56	odd	10		375.2.g.a.301.1		4
75.71	odd	10		375.2.g.a.76.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.a.16.1	✓	4	75.29	odd	10
75.2.g.a.61.1	yes	4	75.44	odd	10
225.2.h.a.91.1		4	25.4	even	10
225.2.h.a.136.1		4	25.19	even	10
375.2.g.a.76.1		4	75.71	odd	10
375.2.g.a.301.1		4	75.56	odd	10
375.2.i.a.49.1		8	75.47	even	20
375.2.i.a.49.2		8	75.53	even	20
375.2.i.a.199.1		8	75.8	even	20
375.2.i.a.199.2		8	75.17	even	20
1875.2.a.a.1.1		2	3.2	odd	2
1875.2.a.d.1.2		2	15.14	odd	2
1875.2.b.b.1249.1		4	15.2	even	4
1875.2.b.b.1249.4		4	15.8	even	4
5625.2.a.a.1.2		2	5.4	even	2
5625.2.a.h.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{4} -4.47214 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{4} -4.47214 q^{7} -3.00000 q^{8} -3.23607 q^{11} -3.38197 q^{13} -4.47214 q^{14} -1.00000 q^{16} -2.85410 q^{17} -3.23607 q^{19} -3.23607 q^{22} -4.47214 q^{23} -3.38197 q^{26} +4.47214 q^{28} -4.38197 q^{29} +7.23607 q^{31} +5.00000 q^{32} -2.85410 q^{34} -8.09017 q^{37} -3.23607 q^{38} +1.38197 q^{41} +5.70820 q^{43} +3.23607 q^{44} -4.47214 q^{46} -5.23607 q^{47} +13.0000 q^{49} +3.38197 q^{52} +1.38197 q^{53} +13.4164 q^{56} -4.38197 q^{58} +4.00000 q^{59} -0.618034 q^{61} +7.23607 q^{62} +7.00000 q^{64} -5.23607 q^{67} +2.85410 q^{68} +0.763932 q^{71} -3.09017 q^{73} -8.09017 q^{74} +3.23607 q^{76} +14.4721 q^{77} +1.38197 q^{82} +3.52786 q^{83} +5.70820 q^{86} +9.70820 q^{88} -7.61803 q^{89} +15.1246 q^{91} +4.47214 q^{92} -5.23607 q^{94} -8.85410 q^{97} +13.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{11} - 9 q^{13} - 2 q^{16} + q^{17} - 2 q^{19} - 2 q^{22} - 9 q^{26} - 11 q^{29} + 10 q^{31} + 10 q^{32} + q^{34} - 5 q^{37} - 2 q^{38} + 5 q^{41} - 2 q^{43} + 2 q^{44} - 6 q^{47} + 26 q^{49} + 9 q^{52} + 5 q^{53} - 11 q^{58} + 8 q^{59} + q^{61} + 10 q^{62} + 14 q^{64} - 6 q^{67} - q^{68} + 6 q^{71} + 5 q^{73} - 5 q^{74} + 2 q^{76} + 20 q^{77} + 5 q^{82} + 16 q^{83} - 2 q^{86} + 6 q^{88} - 13 q^{89} - 10 q^{91} - 6 q^{94} - 11 q^{97} + 26 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^11 - 9 * q^13 - 2 * q^16 + q^17 - 2 * q^19 - 2 * q^22 - 9 * q^26 - 11 * q^29 + 10 * q^31 + 10 * q^32 + q^34 - 5 * q^37 - 2 * q^38 + 5 * q^41 - 2 * q^43 + 2 * q^44 - 6 * q^47 + 26 * q^49 + 9 * q^52 + 5 * q^53 - 11 * q^58 + 8 * q^59 + q^61 + 10 * q^62 + 14 * q^64 - 6 * q^67 - q^68 + 6 * q^71 + 5 * q^73 - 5 * q^74 + 2 * q^76 + 20 * q^77 + 5 * q^82 + 16 * q^83 - 2 * q^86 + 6 * q^88 - 13 * q^89 - 10 * q^91 - 6 * q^94 - 11 * q^97 + 26 * q^98

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