LMFDB - Embedded newform 6075.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6075,2,Mod(1,6075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6075, 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("6075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6075 = 3^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6075.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:	$48.5091192279$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 243)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	6075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.73205 q^{8} +3.46410 q^{11} -5.00000 q^{13} +1.73205 q^{14} -5.00000 q^{16} -1.00000 q^{19} +6.00000 q^{22} -6.92820 q^{23} -8.66025 q^{26} +1.00000 q^{28} +3.46410 q^{29} +5.00000 q^{31} -5.19615 q^{32} +1.00000 q^{37} -1.73205 q^{38} -3.46410 q^{41} +1.00000 q^{43} +3.46410 q^{44} -12.0000 q^{46} -3.46410 q^{47} -6.00000 q^{49} -5.00000 q^{52} -10.3923 q^{53} -1.73205 q^{56} +6.00000 q^{58} -3.46410 q^{59} +2.00000 q^{61} +8.66025 q^{62} +1.00000 q^{64} -8.00000 q^{67} -10.3923 q^{71} -2.00000 q^{73} +1.73205 q^{74} -1.00000 q^{76} +3.46410 q^{77} -1.00000 q^{79} -6.00000 q^{82} +6.92820 q^{83} +1.73205 q^{86} -6.00000 q^{88} -10.3923 q^{89} -5.00000 q^{91} -6.92820 q^{92} -6.00000 q^{94} -17.0000 q^{97} -10.3923 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{7} - 10 q^{13} - 10 q^{16} - 2 q^{19} + 12 q^{22} + 2 q^{28} + 10 q^{31} + 2 q^{37} + 2 q^{43} - 24 q^{46} - 12 q^{49} - 10 q^{52} + 12 q^{58} + 4 q^{61} + 2 q^{64} - 16 q^{67} - 4 q^{73}+ \cdots - 34 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^7 - 10 * q^13 - 10 * q^16 - 2 * q^19 + 12 * q^22 + 2 * q^28 + 10 * q^31 + 2 * q^37 + 2 * q^43 - 24 * q^46 - 12 * q^49 - 10 * q^52 + 12 * q^58 + 4 * q^61 + 2 * q^64 - 16 * q^67 - 4 * q^73 - 2 * q^76 - 2 * q^79 - 12 * q^82 - 12 * q^88 - 10 * q^91 - 12 * q^94 - 34 * q^97

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.73205	−0.612372
$9$	0	0
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	1.73205	0.462910
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	6.00000	1.27920
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	0	0
$26$	−8.66025	−1.69842
$27$	0	0
$28$	1.00000	0.188982
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	−1.73205	−0.280976
$39$	0	0
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	3.46410	0.522233
$45$	0	0
$46$	−12.0000	−1.76930
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	−5.00000	−0.693375
$53$	−10.3923	−1.42749	−0.713746	−	0.700404i	$-0.753003\pi$
$53$	−10.3923	−1.42749	−0.713746	+	0.700404i	$0.753003\pi$
$54$	0	0
$55$	0	0
$56$	−1.73205	−0.231455
$57$	0	0
$58$	6.00000	0.787839
$59$	−3.46410	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	8.66025	1.09985
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.3923	−1.23334	−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923	−1.23334	−0.616670	+	0.787222i	$0.711519\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	1.73205	0.201347
$75$	0	0
$76$	−1.00000	−0.114708
$77$	3.46410	0.394771
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	0	0
$82$	−6.00000	−0.662589
$83$	6.92820	0.760469	0.380235	−	0.924890i	$-0.375843\pi$
$83$	6.92820	0.760469	0.380235	+	0.924890i	$0.375843\pi$
$84$	0	0
$85$	0	0
$86$	1.73205	0.186772
$87$	0	0
$88$	−6.00000	−0.639602
$89$	−10.3923	−1.10158	−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923	−1.10158	−0.550791	+	0.834643i	$0.685674\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	−6.92820	−0.722315
$93$	0	0
$94$	−6.00000	−0.618853
$95$	0	0
$96$	0	0
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	−10.3923	−1.04978
$99$	0	0
$100$	0	0
$101$	13.8564	1.37876	0.689382	−	0.724398i	$-0.257882\pi$
$101$	13.8564	1.37876	0.689382	+	0.724398i	$0.257882\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	8.66025	0.849208
$105$	0	0
$106$	−18.0000	−1.74831
$107$	10.3923	1.00466	0.502331	−	0.864675i	$-0.332476\pi$
$107$	10.3923	1.00466	0.502331	+	0.864675i	$0.332476\pi$
$108$	0	0
$109$	17.0000	1.62830	0.814152	−	0.580651i	$-0.197202\pi$
$109$	17.0000	1.62830	0.814152	+	0.580651i	$0.197202\pi$
$110$	0	0
$111$	0	0
$112$	−5.00000	−0.472456
$113$	−17.3205	−1.62938	−0.814688	−	0.579899i	$-0.803092\pi$
$113$	−17.3205	−1.62938	−0.814688	+	0.579899i	$0.803092\pi$
$114$	0	0
$115$	0	0
$116$	3.46410	0.321634
$117$	0	0
$118$	−6.00000	−0.552345
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.46410	0.313625
$123$	0	0
$124$	5.00000	0.449013
$125$	0	0
$126$	0	0
$127$	−17.0000	−1.50851	−0.754253	−	0.656584i	$-0.772001\pi$
$127$	−17.0000	−1.50851	−0.754253	+	0.656584i	$0.772001\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	0	0
$131$	−3.46410	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	−13.8564	−1.19701
$135$	0	0
$136$	0	0
$137$	6.92820	0.591916	0.295958	−	0.955201i	$-0.404361\pi$
$137$	6.92820	0.591916	0.295958	+	0.955201i	$0.404361\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	0	0
$141$	0	0
$142$	−18.0000	−1.51053
$143$	−17.3205	−1.44841
$144$	0	0
$145$	0	0
$146$	−3.46410	−0.286691
$147$	0	0
$148$	1.00000	0.0821995
$149$	6.92820	0.567581	0.283790	−	0.958886i	$-0.408408\pi$
$149$	6.92820	0.567581	0.283790	+	0.958886i	$0.408408\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	1.73205	0.140488
$153$	0	0
$154$	6.00000	0.483494
$155$	0	0
$156$	0	0
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	−1.73205	−0.137795
$159$	0	0
$160$	0	0
$161$	−6.92820	−0.546019
$162$	0	0
$163$	1.00000	0.0783260	0.0391630	−	0.999233i	$-0.487531\pi$
$163$	1.00000	0.0783260	0.0391630	+	0.999233i	$0.487531\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	12.0000	0.931381
$167$	24.2487	1.87642	0.938211	−	0.346064i	$-0.112482\pi$
$167$	24.2487	1.87642	0.938211	+	0.346064i	$0.112482\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	1.00000	0.0762493
$173$	−13.8564	−1.05348	−0.526742	−	0.850026i	$-0.676586\pi$
$173$	−13.8564	−1.05348	−0.526742	+	0.850026i	$0.676586\pi$
$174$	0	0
$175$	0	0
$176$	−17.3205	−1.30558
$177$	0	0
$178$	−18.0000	−1.34916
$179$	20.7846	1.55351	0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	−8.66025	−0.641941
$183$	0	0
$184$	12.0000	0.884652
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−3.46410	−0.252646
$189$	0	0
$190$	0	0
$191$	−6.92820	−0.501307	−0.250654	−	0.968077i	$-0.580646\pi$
$191$	−6.92820	−0.501307	−0.250654	+	0.968077i	$0.580646\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−29.4449	−2.11402
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−10.3923	−0.740421	−0.370211	−	0.928948i	$-0.620714\pi$
$197$	−10.3923	−0.740421	−0.370211	+	0.928948i	$0.620714\pi$
$198$	0	0
$199$	−19.0000	−1.34687	−0.673437	−	0.739244i	$-0.735183\pi$
$199$	−19.0000	−1.34687	−0.673437	+	0.739244i	$0.735183\pi$
$200$	0	0
$201$	0	0
$202$	24.0000	1.68863
$203$	3.46410	0.243132
$204$	0	0
$205$	0	0
$206$	−13.8564	−0.965422
$207$	0	0
$208$	25.0000	1.73344
$209$	−3.46410	−0.239617
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	−10.3923	−0.713746
$213$	0	0
$214$	18.0000	1.23045
$215$	0	0
$216$	0	0
$217$	5.00000	0.339422
$218$	29.4449	1.99426
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	−5.19615	−0.347183
$225$	0	0
$226$	−30.0000	−1.99557
$227$	−13.8564	−0.919682	−0.459841	−	0.888001i	$-0.652094\pi$
$227$	−13.8564	−0.919682	−0.459841	+	0.888001i	$0.652094\pi$
$228$	0	0
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	−3.46410	−0.225494
$237$	0	0
$238$	0	0
$239$	6.92820	0.448148	0.224074	−	0.974572i	$-0.428064\pi$
$239$	6.92820	0.448148	0.224074	+	0.974572i	$0.428064\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	1.73205	0.111340
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	5.00000	0.318142
$248$	−8.66025	−0.549927
$249$	0	0
$250$	0	0
$251$	−20.7846	−1.31191	−0.655956	−	0.754799i	$-0.727735\pi$
$251$	−20.7846	−1.31191	−0.655956	+	0.754799i	$0.727735\pi$
$252$	0	0
$253$	−24.0000	−1.50887
$254$	−29.4449	−1.84754
$255$	0	0
$256$	19.0000	1.18750
$257$	3.46410	0.216085	0.108042	−	0.994146i	$-0.465542\pi$
$257$	3.46410	0.216085	0.108042	+	0.994146i	$0.465542\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	0	0
$261$	0	0
$262$	−6.00000	−0.370681
$263$	−13.8564	−0.854423	−0.427211	−	0.904152i	$-0.640504\pi$
$263$	−13.8564	−0.854423	−0.427211	+	0.904152i	$0.640504\pi$
$264$	0	0
$265$	0	0
$266$	−1.73205	−0.106199
$267$	0	0
$268$	−8.00000	−0.488678
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	−22.5167	−1.35046
$279$	0	0
$280$	0	0
$281$	13.8564	0.826604	0.413302	−	0.910594i	$-0.364375\pi$
$281$	13.8564	0.826604	0.413302	+	0.910594i	$0.364375\pi$
$282$	0	0
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	−10.3923	−0.616670
$285$	0	0
$286$	−30.0000	−1.77394
$287$	−3.46410	−0.204479
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	−2.00000	−0.117041
$293$	13.8564	0.809500	0.404750	−	0.914427i	$-0.367359\pi$
$293$	13.8564	0.809500	0.404750	+	0.914427i	$0.367359\pi$
$294$	0	0
$295$	0	0
$296$	−1.73205	−0.100673
$297$	0	0
$298$	12.0000	0.695141
$299$	34.6410	2.00334
$300$	0	0
$301$	1.00000	0.0576390
$302$	−27.7128	−1.59469
$303$	0	0
$304$	5.00000	0.286770
$305$	0	0
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	3.46410	0.197386
$309$	0	0
$310$	0	0
$311$	−13.8564	−0.785725	−0.392862	−	0.919597i	$-0.628515\pi$
$311$	−13.8564	−0.785725	−0.392862	+	0.919597i	$0.628515\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	22.5167	1.27069
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	27.7128	1.55651	0.778253	−	0.627950i	$-0.216106\pi$
$317$	27.7128	1.55651	0.778253	+	0.627950i	$0.216106\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	0	0
$322$	−12.0000	−0.668734
$323$	0	0
$324$	0	0
$325$	0	0
$326$	1.73205	0.0959294
$327$	0	0
$328$	6.00000	0.331295
$329$	−3.46410	−0.190982
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	6.92820	0.380235
$333$	0	0
$334$	42.0000	2.29814
$335$	0	0
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	20.7846	1.13053
$339$	0	0
$340$	0	0
$341$	17.3205	0.937958
$342$	0	0
$343$	−13.0000	−0.701934
$344$	−1.73205	−0.0933859
$345$	0	0
$346$	−24.0000	−1.29025
$347$	24.2487	1.30174	0.650870	−	0.759190i	$-0.274404\pi$
$347$	24.2487	1.30174	0.650870	+	0.759190i	$0.274404\pi$
$348$	0	0
$349$	−1.00000	−0.0535288	−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000	−0.0535288	−0.0267644	+	0.999642i	$0.508520\pi$
$350$	0	0
$351$	0	0
$352$	−18.0000	−0.959403
$353$	17.3205	0.921878	0.460939	−	0.887432i	$-0.347513\pi$
$353$	17.3205	0.921878	0.460939	+	0.887432i	$0.347513\pi$
$354$	0	0
$355$	0	0
$356$	−10.3923	−0.550791
$357$	0	0
$358$	36.0000	1.90266
$359$	−31.1769	−1.64545	−0.822727	−	0.568436i	$-0.807549\pi$
$359$	−31.1769	−1.64545	−0.822727	+	0.568436i	$0.807549\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	29.4449	1.54759
$363$	0	0
$364$	−5.00000	−0.262071
$365$	0	0
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	34.6410	1.80579
$369$	0	0
$370$	0	0
$371$	−10.3923	−0.539542
$372$	0	0
$373$	−23.0000	−1.19089	−0.595447	−	0.803394i	$-0.703025\pi$
$373$	−23.0000	−1.19089	−0.595447	+	0.803394i	$0.703025\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	−17.3205	−0.892052
$378$	0	0
$379$	−19.0000	−0.975964	−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000	−0.975964	−0.487982	+	0.872854i	$0.662267\pi$
$380$	0	0
$381$	0	0
$382$	−12.0000	−0.613973
$383$	−17.3205	−0.885037	−0.442518	−	0.896759i	$-0.645915\pi$
$383$	−17.3205	−0.885037	−0.442518	+	0.896759i	$0.645915\pi$
$384$	0	0
$385$	0	0
$386$	17.3205	0.881591
$387$	0	0
$388$	−17.0000	−0.863044
$389$	−6.92820	−0.351274	−0.175637	−	0.984455i	$-0.556198\pi$
$389$	−6.92820	−0.351274	−0.175637	+	0.984455i	$0.556198\pi$
$390$	0	0
$391$	0	0
$392$	10.3923	0.524891
$393$	0	0
$394$	−18.0000	−0.906827
$395$	0	0
$396$	0	0
$397$	1.00000	0.0501886	0.0250943	−	0.999685i	$-0.492011\pi$
$397$	1.00000	0.0501886	0.0250943	+	0.999685i	$0.492011\pi$
$398$	−32.9090	−1.64958
$399$	0	0
$400$	0	0
$401$	−3.46410	−0.172989	−0.0864945	−	0.996252i	$-0.527566\pi$
$401$	−3.46410	−0.172989	−0.0864945	+	0.996252i	$0.527566\pi$
$402$	0	0
$403$	−25.0000	−1.24534
$404$	13.8564	0.689382
$405$	0	0
$406$	6.00000	0.297775
$407$	3.46410	0.171709
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−3.46410	−0.170457
$414$	0	0
$415$	0	0
$416$	25.9808	1.27381
$417$	0	0
$418$	−6.00000	−0.293470
$419$	38.1051	1.86156	0.930778	−	0.365584i	$-0.119131\pi$
$419$	38.1051	1.86156	0.930778	+	0.365584i	$0.119131\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	8.66025	0.421575
$423$	0	0
$424$	18.0000	0.874157
$425$	0	0
$426$	0	0
$427$	2.00000	0.0967868
$428$	10.3923	0.502331
$429$	0	0
$430$	0	0
$431$	20.7846	1.00116	0.500580	−	0.865690i	$-0.333120\pi$
$431$	20.7846	1.00116	0.500580	+	0.865690i	$0.333120\pi$
$432$	0	0
$433$	1.00000	0.0480569	0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000	0.0480569	0.0240285	+	0.999711i	$0.492351\pi$
$434$	8.66025	0.415705
$435$	0	0
$436$	17.0000	0.814152
$437$	6.92820	0.331421
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.8564	−0.658338	−0.329169	−	0.944271i	$-0.606769\pi$
$443$	−13.8564	−0.658338	−0.329169	+	0.944271i	$0.606769\pi$
$444$	0	0
$445$	0	0
$446$	32.9090	1.55828
$447$	0	0
$448$	1.00000	0.0472456
$449$	10.3923	0.490443	0.245222	−	0.969467i	$-0.421139\pi$
$449$	10.3923	0.490443	0.245222	+	0.969467i	$0.421139\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	−17.3205	−0.814688
$453$	0	0
$454$	−24.0000	−1.12638
$455$	0	0
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	8.66025	0.404667
$459$	0	0
$460$	0	0
$461$	−27.7128	−1.29071	−0.645357	−	0.763881i	$-0.723291\pi$
$461$	−27.7128	−1.29071	−0.645357	+	0.763881i	$0.723291\pi$
$462$	0	0
$463$	31.0000	1.44069	0.720346	−	0.693615i	$-0.243983\pi$
$463$	31.0000	1.44069	0.720346	+	0.693615i	$0.243983\pi$
$464$	−17.3205	−0.804084
$465$	0	0
$466$	0	0
$467$	−10.3923	−0.480899	−0.240449	−	0.970662i	$-0.577295\pi$
$467$	−10.3923	−0.480899	−0.240449	+	0.970662i	$0.577295\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	0	0
$472$	6.00000	0.276172
$473$	3.46410	0.159280
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	12.0000	0.548867
$479$	13.8564	0.633115	0.316558	−	0.948573i	$-0.397473\pi$
$479$	13.8564	0.633115	0.316558	+	0.948573i	$0.397473\pi$
$480$	0	0
$481$	−5.00000	−0.227980
$482$	−32.9090	−1.49896
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	19.0000	0.860972	0.430486	−	0.902597i	$-0.358342\pi$
$487$	19.0000	0.860972	0.430486	+	0.902597i	$0.358342\pi$
$488$	−3.46410	−0.156813
$489$	0	0
$490$	0	0
$491$	38.1051	1.71966	0.859830	−	0.510581i	$-0.170569\pi$
$491$	38.1051	1.71966	0.859830	+	0.510581i	$0.170569\pi$
$492$	0	0
$493$	0	0
$494$	8.66025	0.389643
$495$	0	0
$496$	−25.0000	−1.12253
$497$	−10.3923	−0.466159
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	0	0
$502$	−36.0000	−1.60676
$503$	41.5692	1.85348	0.926740	−	0.375703i	$-0.122599\pi$
$503$	41.5692	1.85348	0.926740	+	0.375703i	$0.122599\pi$
$504$	0	0
$505$	0	0
$506$	−41.5692	−1.84798
$507$	0	0
$508$	−17.0000	−0.754253
$509$	27.7128	1.22835	0.614174	−	0.789170i	$-0.289489\pi$
$509$	27.7128	1.22835	0.614174	+	0.789170i	$0.289489\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	8.66025	0.382733
$513$	0	0
$514$	6.00000	0.264649
$515$	0	0
$516$	0	0
$517$	−12.0000	−0.527759
$518$	1.73205	0.0761019
$519$	0	0
$520$	0	0
$521$	20.7846	0.910590	0.455295	−	0.890341i	$-0.349534\pi$
$521$	20.7846	0.910590	0.455295	+	0.890341i	$0.349534\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	−3.46410	−0.151330
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	17.3205	0.750234
$534$	0	0
$535$	0	0
$536$	13.8564	0.598506
$537$	0	0
$538$	0	0
$539$	−20.7846	−0.895257
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	−27.7128	−1.19037
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	6.92820	0.295958
$549$	0	0
$550$	0	0
$551$	−3.46410	−0.147576
$552$	0	0
$553$	−1.00000	−0.0425243
$554$	−29.4449	−1.25099
$555$	0	0
$556$	−13.0000	−0.551323
$557$	10.3923	0.440336	0.220168	−	0.975462i	$-0.429339\pi$
$557$	10.3923	0.440336	0.220168	+	0.975462i	$0.429339\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	0	0
$562$	24.0000	1.01238
$563$	34.6410	1.45994	0.729972	−	0.683477i	$-0.239533\pi$
$563$	34.6410	1.45994	0.729972	+	0.683477i	$0.239533\pi$
$564$	0	0
$565$	0	0
$566$	22.5167	0.946446
$567$	0	0
$568$	18.0000	0.755263
$569$	24.2487	1.01656	0.508279	−	0.861192i	$-0.330282\pi$
$569$	24.2487	1.01656	0.508279	+	0.861192i	$0.330282\pi$
$570$	0	0
$571$	41.0000	1.71580	0.857898	−	0.513820i	$-0.171770\pi$
$571$	41.0000	1.71580	0.857898	+	0.513820i	$0.171770\pi$
$572$	−17.3205	−0.724207
$573$	0	0
$574$	−6.00000	−0.250435
$575$	0	0
$576$	0	0
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	−29.4449	−1.22474
$579$	0	0
$580$	0	0
$581$	6.92820	0.287430
$582$	0	0
$583$	−36.0000	−1.49097
$584$	3.46410	0.143346
$585$	0	0
$586$	24.0000	0.991431
$587$	6.92820	0.285958	0.142979	−	0.989726i	$-0.454332\pi$
$587$	6.92820	0.285958	0.142979	+	0.989726i	$0.454332\pi$
$588$	0	0
$589$	−5.00000	−0.206021
$590$	0	0
$591$	0	0
$592$	−5.00000	−0.205499
$593$	20.7846	0.853522	0.426761	−	0.904365i	$-0.359655\pi$
$593$	20.7846	0.853522	0.426761	+	0.904365i	$0.359655\pi$
$594$	0	0
$595$	0	0
$596$	6.92820	0.283790
$597$	0	0
$598$	60.0000	2.45358
$599$	−24.2487	−0.990775	−0.495388	−	0.868672i	$-0.664974\pi$
$599$	−24.2487	−0.990775	−0.495388	+	0.868672i	$0.664974\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	1.73205	0.0705931
$603$	0	0
$604$	−16.0000	−0.651031
$605$	0	0
$606$	0	0
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	5.19615	0.210732
$609$	0	0
$610$	0	0
$611$	17.3205	0.700713
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−34.6410	−1.39800
$615$	0	0
$616$	−6.00000	−0.241747
$617$	−6.92820	−0.278919	−0.139459	−	0.990228i	$-0.544536\pi$
$617$	−6.92820	−0.278919	−0.139459	+	0.990228i	$0.544536\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	0	0
$622$	−24.0000	−0.962312
$623$	−10.3923	−0.416359
$624$	0	0
$625$	0	0
$626$	1.73205	0.0692267
$627$	0	0
$628$	13.0000	0.518756
$629$	0	0
$630$	0	0
$631$	17.0000	0.676759	0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000	0.676759	0.338380	+	0.941010i	$0.390121\pi$
$632$	1.73205	0.0688973
$633$	0	0
$634$	48.0000	1.90632
$635$	0	0
$636$	0	0
$637$	30.0000	1.18864
$638$	20.7846	0.822871
$639$	0	0
$640$	0	0
$641$	45.0333	1.77871	0.889355	−	0.457218i	$-0.151154\pi$
$641$	45.0333	1.77871	0.889355	+	0.457218i	$0.151154\pi$
$642$	0	0
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	−6.92820	−0.273009
$645$	0	0
$646$	0	0
$647$	−20.7846	−0.817127	−0.408564	−	0.912730i	$-0.633970\pi$
$647$	−20.7846	−0.817127	−0.408564	+	0.912730i	$0.633970\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	1.00000	0.0391630
$653$	−17.3205	−0.677804	−0.338902	−	0.940822i	$-0.610055\pi$
$653$	−17.3205	−0.677804	−0.338902	+	0.940822i	$0.610055\pi$
$654$	0	0
$655$	0	0
$656$	17.3205	0.676252
$657$	0	0
$658$	−6.00000	−0.233904
$659$	−27.7128	−1.07954	−0.539769	−	0.841813i	$-0.681488\pi$
$659$	−27.7128	−1.07954	−0.539769	+	0.841813i	$0.681488\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−32.9090	−1.27904
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	−24.0000	−0.929284
$668$	24.2487	0.938211
$669$	0	0
$670$	0	0
$671$	6.92820	0.267460
$672$	0	0
$673$	37.0000	1.42625	0.713123	−	0.701039i	$-0.247280\pi$
$673$	37.0000	1.42625	0.713123	+	0.701039i	$0.247280\pi$
$674$	−8.66025	−0.333581
$675$	0	0
$676$	12.0000	0.461538
$677$	17.3205	0.665681	0.332841	−	0.942983i	$-0.391993\pi$
$677$	17.3205	0.665681	0.332841	+	0.942983i	$0.391993\pi$
$678$	0	0
$679$	−17.0000	−0.652400
$680$	0	0
$681$	0	0
$682$	30.0000	1.14876
$683$	41.5692	1.59060	0.795301	−	0.606215i	$-0.207313\pi$
$683$	41.5692	1.59060	0.795301	+	0.606215i	$0.207313\pi$
$684$	0	0
$685$	0	0
$686$	−22.5167	−0.859690
$687$	0	0
$688$	−5.00000	−0.190623
$689$	51.9615	1.97958
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	−13.8564	−0.526742
$693$	0	0
$694$	42.0000	1.59430
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−1.73205	−0.0655591
$699$	0	0
$700$	0	0
$701$	41.5692	1.57005	0.785024	−	0.619466i	$-0.212651\pi$
$701$	41.5692	1.57005	0.785024	+	0.619466i	$0.212651\pi$
$702$	0	0
$703$	−1.00000	−0.0377157
$704$	3.46410	0.130558
$705$	0	0
$706$	30.0000	1.12906
$707$	13.8564	0.521124
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	0	0
$711$	0	0
$712$	18.0000	0.674579
$713$	−34.6410	−1.29732
$714$	0	0
$715$	0	0
$716$	20.7846	0.776757
$717$	0	0
$718$	−54.0000	−2.01526
$719$	−10.3923	−0.387568	−0.193784	−	0.981044i	$-0.562076\pi$
$719$	−10.3923	−0.387568	−0.193784	+	0.981044i	$0.562076\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	−31.1769	−1.16028
$723$	0	0
$724$	17.0000	0.631800
$725$	0	0
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	8.66025	0.320970
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−41.0000	−1.51437	−0.757185	−	0.653201i	$-0.773426\pi$
$733$	−41.0000	−1.51437	−0.757185	+	0.653201i	$0.773426\pi$
$734$	27.7128	1.02290
$735$	0	0
$736$	36.0000	1.32698
$737$	−27.7128	−1.02081
$738$	0	0
$739$	−19.0000	−0.698926	−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000	−0.698926	−0.349463	+	0.936950i	$0.613636\pi$
$740$	0	0
$741$	0	0
$742$	−18.0000	−0.660801
$743$	−6.92820	−0.254171	−0.127086	−	0.991892i	$-0.540562\pi$
$743$	−6.92820	−0.254171	−0.127086	+	0.991892i	$0.540562\pi$
$744$	0	0
$745$	0	0
$746$	−39.8372	−1.45854
$747$	0	0
$748$	0	0
$749$	10.3923	0.379727
$750$	0	0
$751$	5.00000	0.182453	0.0912263	−	0.995830i	$-0.470921\pi$
$751$	5.00000	0.182453	0.0912263	+	0.995830i	$0.470921\pi$
$752$	17.3205	0.631614
$753$	0	0
$754$	−30.0000	−1.09254
$755$	0	0
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	−32.9090	−1.19531
$759$	0	0
$760$	0	0
$761$	27.7128	1.00459	0.502294	−	0.864697i	$-0.332489\pi$
$761$	27.7128	1.00459	0.502294	+	0.864697i	$0.332489\pi$
$762$	0	0
$763$	17.0000	0.615441
$764$	−6.92820	−0.250654
$765$	0	0
$766$	−30.0000	−1.08394
$767$	17.3205	0.625407
$768$	0	0
$769$	−13.0000	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$769$	−13.0000	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$770$	0	0
$771$	0	0
$772$	10.0000	0.359908
$773$	20.7846	0.747570	0.373785	−	0.927515i	$-0.378060\pi$
$773$	20.7846	0.747570	0.373785	+	0.927515i	$0.378060\pi$
$774$	0	0
$775$	0	0
$776$	29.4449	1.05701
$777$	0	0
$778$	−12.0000	−0.430221
$779$	3.46410	0.124114
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	0	0
$784$	30.0000	1.07143
$785$	0	0
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−10.3923	−0.370211
$789$	0	0
$790$	0	0
$791$	−17.3205	−0.615846
$792$	0	0
$793$	−10.0000	−0.355110
$794$	1.73205	0.0614682
$795$	0	0
$796$	−19.0000	−0.673437
$797$	13.8564	0.490819	0.245410	−	0.969419i	$-0.421078\pi$
$797$	13.8564	0.490819	0.245410	+	0.969419i	$0.421078\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	−6.00000	−0.211867
$803$	−6.92820	−0.244491
$804$	0	0
$805$	0	0
$806$	−43.3013	−1.52522
$807$	0	0
$808$	−24.0000	−0.844317
$809$	31.1769	1.09612	0.548061	−	0.836438i	$-0.315366\pi$
$809$	31.1769	1.09612	0.548061	+	0.836438i	$0.315366\pi$
$810$	0	0
$811$	35.0000	1.22902	0.614508	−	0.788911i	$-0.289355\pi$
$811$	35.0000	1.22902	0.614508	+	0.788911i	$0.289355\pi$
$812$	3.46410	0.121566
$813$	0	0
$814$	6.00000	0.210300
$815$	0	0
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	8.66025	0.302799
$819$	0	0
$820$	0	0
$821$	−38.1051	−1.32988	−0.664939	−	0.746898i	$-0.731542\pi$
$821$	−38.1051	−1.32988	−0.664939	+	0.746898i	$0.731542\pi$
$822$	0	0
$823$	−23.0000	−0.801730	−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000	−0.801730	−0.400865	+	0.916137i	$0.631290\pi$
$824$	13.8564	0.482711
$825$	0	0
$826$	−6.00000	−0.208767
$827$	−10.3923	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$828$	0	0
$829$	−19.0000	−0.659897	−0.329949	−	0.943999i	$-0.607031\pi$
$829$	−19.0000	−0.659897	−0.329949	+	0.943999i	$0.607031\pi$
$830$	0	0
$831$	0	0
$832$	−5.00000	−0.173344
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−3.46410	−0.119808
$837$	0	0
$838$	66.0000	2.27993
$839$	3.46410	0.119594	0.0597970	−	0.998211i	$-0.480955\pi$
$839$	3.46410	0.119594	0.0597970	+	0.998211i	$0.480955\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	−32.9090	−1.13412
$843$	0	0
$844$	5.00000	0.172107
$845$	0	0
$846$	0	0
$847$	1.00000	0.0343604
$848$	51.9615	1.78437
$849$	0	0
$850$	0	0
$851$	−6.92820	−0.237496
$852$	0	0
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	3.46410	0.118539
$855$	0	0
$856$	−18.0000	−0.615227
$857$	38.1051	1.30165	0.650823	−	0.759229i	$-0.274424\pi$
$857$	38.1051	1.30165	0.650823	+	0.759229i	$0.274424\pi$
$858$	0	0
$859$	−49.0000	−1.67186	−0.835929	−	0.548837i	$-0.815071\pi$
$859$	−49.0000	−1.67186	−0.835929	+	0.548837i	$0.815071\pi$
$860$	0	0
$861$	0	0
$862$	36.0000	1.22616
$863$	10.3923	0.353758	0.176879	−	0.984233i	$-0.443400\pi$
$863$	10.3923	0.353758	0.176879	+	0.984233i	$0.443400\pi$
$864$	0	0
$865$	0	0
$866$	1.73205	0.0588575
$867$	0	0
$868$	5.00000	0.169711
$869$	−3.46410	−0.117512
$870$	0	0
$871$	40.0000	1.35535
$872$	−29.4449	−0.997129
$873$	0	0
$874$	12.0000	0.405906
$875$	0	0
$876$	0	0
$877$	−23.0000	−0.776655	−0.388327	−	0.921521i	$-0.626947\pi$
$877$	−23.0000	−0.776655	−0.388327	+	0.921521i	$0.626947\pi$
$878$	34.6410	1.16908
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	19.0000	0.639401	0.319700	−	0.947519i	$-0.396418\pi$
$883$	19.0000	0.639401	0.319700	+	0.947519i	$0.396418\pi$
$884$	0	0
$885$	0	0
$886$	−24.0000	−0.806296
$887$	13.8564	0.465253	0.232626	−	0.972566i	$-0.425268\pi$
$887$	13.8564	0.465253	0.232626	+	0.972566i	$0.425268\pi$
$888$	0	0
$889$	−17.0000	−0.570162
$890$	0	0
$891$	0	0
$892$	19.0000	0.636167
$893$	3.46410	0.115922
$894$	0	0
$895$	0	0
$896$	12.1244	0.405046
$897$	0	0
$898$	18.0000	0.600668
$899$	17.3205	0.577671
$900$	0	0
$901$	0	0
$902$	−20.7846	−0.692052
$903$	0	0
$904$	30.0000	0.997785
$905$	0	0
$906$	0	0
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	−13.8564	−0.459841
$909$	0	0
$910$	0	0
$911$	−27.7128	−0.918166	−0.459083	−	0.888393i	$-0.651822\pi$
$911$	−27.7128	−0.918166	−0.459083	+	0.888393i	$0.651822\pi$
$912$	0	0
$913$	24.0000	0.794284
$914$	−29.4449	−0.973950
$915$	0	0
$916$	5.00000	0.165205
$917$	−3.46410	−0.114395
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	0	0
$922$	−48.0000	−1.58080
$923$	51.9615	1.71033
$924$	0	0
$925$	0	0
$926$	53.6936	1.76448
$927$	0	0
$928$	−18.0000	−0.590879
$929$	−6.92820	−0.227307	−0.113653	−	0.993520i	$-0.536255\pi$
$929$	−6.92820	−0.227307	−0.113653	+	0.993520i	$0.536255\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	0	0
$934$	−18.0000	−0.588978
$935$	0	0
$936$	0	0
$937$	37.0000	1.20874	0.604369	−	0.796705i	$-0.293425\pi$
$937$	37.0000	1.20874	0.604369	+	0.796705i	$0.293425\pi$
$938$	−13.8564	−0.452428
$939$	0	0
$940$	0	0
$941$	−3.46410	−0.112926	−0.0564632	−	0.998405i	$-0.517982\pi$
$941$	−3.46410	−0.112926	−0.0564632	+	0.998405i	$0.517982\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	17.3205	0.563735
$945$	0	0
$946$	6.00000	0.195077
$947$	−13.8564	−0.450273	−0.225136	−	0.974327i	$-0.572283\pi$
$947$	−13.8564	−0.450273	−0.225136	+	0.974327i	$0.572283\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	6.92820	0.224074
$957$	0	0
$958$	24.0000	0.775405
$959$	6.92820	0.223723
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−8.66025	−0.279218
$963$	0	0
$964$	−19.0000	−0.611949
$965$	0	0
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	−1.73205	−0.0556702
$969$	0	0
$970$	0	0
$971$	−51.9615	−1.66752	−0.833762	−	0.552124i	$-0.813818\pi$
$971$	−51.9615	−1.66752	−0.833762	+	0.552124i	$0.813818\pi$
$972$	0	0
$973$	−13.0000	−0.416761
$974$	32.9090	1.05447
$975$	0	0
$976$	−10.0000	−0.320092
$977$	3.46410	0.110826	0.0554132	−	0.998464i	$-0.482352\pi$
$977$	3.46410	0.110826	0.0554132	+	0.998464i	$0.482352\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	0	0
$981$	0	0
$982$	66.0000	2.10614
$983$	6.92820	0.220975	0.110488	−	0.993877i	$-0.464759\pi$
$983$	6.92820	0.220975	0.110488	+	0.993877i	$0.464759\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	5.00000	0.159071
$989$	−6.92820	−0.220304
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	−25.9808	−0.824890
$993$	0	0
$994$	−18.0000	−0.570925
$995$	0	0
$996$	0	0
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	−48.4974	−1.53516
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6075.2.a.bm.1.2		2
3.2	odd	2	inner	6075.2.a.bm.1.1		2
5.4	even	2		243.2.a.c.1.1	✓	2
15.14	odd	2		243.2.a.c.1.2	yes	2
20.19	odd	2		3888.2.a.ba.1.1		2
45.4	even	6		243.2.c.d.163.2		4
45.14	odd	6		243.2.c.d.163.1		4
45.29	odd	6		243.2.c.d.82.1		4
45.34	even	6		243.2.c.d.82.2		4
60.59	even	2		3888.2.a.ba.1.2		2
135.4	even	18		729.2.e.n.406.1		12
135.14	odd	18		729.2.e.n.163.1		12
135.29	odd	18		729.2.e.n.568.1		12
135.34	even	18		729.2.e.n.325.1		12
135.49	even	18		729.2.e.n.649.1		12
135.59	odd	18		729.2.e.n.649.2		12
135.74	odd	18		729.2.e.n.325.2		12
135.79	even	18		729.2.e.n.568.2		12
135.94	even	18		729.2.e.n.163.2		12
135.104	odd	18		729.2.e.n.406.2		12
135.119	odd	18		729.2.e.n.82.2		12
135.124	even	18		729.2.e.n.82.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.c.1.1	✓	2	5.4	even	2
243.2.a.c.1.2	yes	2	15.14	odd	2
243.2.c.d.82.1		4	45.29	odd	6
243.2.c.d.82.2		4	45.34	even	6
243.2.c.d.163.1		4	45.14	odd	6
243.2.c.d.163.2		4	45.4	even	6
729.2.e.n.82.1		12	135.124	even	18
729.2.e.n.82.2		12	135.119	odd	18
729.2.e.n.163.1		12	135.14	odd	18
729.2.e.n.163.2		12	135.94	even	18
729.2.e.n.325.1		12	135.34	even	18
729.2.e.n.325.2		12	135.74	odd	18
729.2.e.n.406.1		12	135.4	even	18
729.2.e.n.406.2		12	135.104	odd	18
729.2.e.n.568.1		12	135.29	odd	18
729.2.e.n.568.2		12	135.79	even	18
729.2.e.n.649.1		12	135.49	even	18
729.2.e.n.649.2		12	135.59	odd	18
3888.2.a.ba.1.1		2	20.19	odd	2
3888.2.a.ba.1.2		2	60.59	even	2
6075.2.a.bm.1.1		2	3.2	odd	2	inner
6075.2.a.bm.1.2		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:	\( N \)	\(=\)	\( 6075 = 3^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6075.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.73205 q^{8} +3.46410 q^{11} -5.00000 q^{13} +1.73205 q^{14} -5.00000 q^{16} -1.00000 q^{19} +6.00000 q^{22} -6.92820 q^{23} -8.66025 q^{26} +1.00000 q^{28} +3.46410 q^{29} +5.00000 q^{31} -5.19615 q^{32} +1.00000 q^{37} -1.73205 q^{38} -3.46410 q^{41} +1.00000 q^{43} +3.46410 q^{44} -12.0000 q^{46} -3.46410 q^{47} -6.00000 q^{49} -5.00000 q^{52} -10.3923 q^{53} -1.73205 q^{56} +6.00000 q^{58} -3.46410 q^{59} +2.00000 q^{61} +8.66025 q^{62} +1.00000 q^{64} -8.00000 q^{67} -10.3923 q^{71} -2.00000 q^{73} +1.73205 q^{74} -1.00000 q^{76} +3.46410 q^{77} -1.00000 q^{79} -6.00000 q^{82} +6.92820 q^{83} +1.73205 q^{86} -6.00000 q^{88} -10.3923 q^{89} -5.00000 q^{91} -6.92820 q^{92} -6.00000 q^{94} -17.0000 q^{97} -10.3923 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{7} - 10 q^{13} - 10 q^{16} - 2 q^{19} + 12 q^{22} + 2 q^{28} + 10 q^{31} + 2 q^{37} + 2 q^{43} - 24 q^{46} - 12 q^{49} - 10 q^{52} + 12 q^{58} + 4 q^{61} + 2 q^{64} - 16 q^{67} - 4 q^{73}+ \cdots - 34 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^7 - 10 * q^13 - 10 * q^16 - 2 * q^19 + 12 * q^22 + 2 * q^28 + 10 * q^31 + 2 * q^37 + 2 * q^43 - 24 * q^46 - 12 * q^49 - 10 * q^52 + 12 * q^58 + 4 * q^61 + 2 * q^64 - 16 * q^67 - 4 * q^73 - 2 * q^76 - 2 * q^79 - 12 * q^82 - 12 * q^88 - 10 * q^91 - 12 * q^94 - 34 * q^97

Introduction

L-functions

Modular forms

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Database

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