LMFDB - Embedded newform 7500.2.a.n.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7500.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.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:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} + \cdots - 4 $ x^12 - 6x^11 - 11x^10 + 94x^9 + 27x^8 - 460x^7 + 55x^6 + 812x^5 - 127x^4 - 512x^3 + 40x^2 + 72*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{3} $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.434428$ of defining polynomial
Character	$\chi$	$=$	7500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.04684 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.04684 q^{7} +1.00000 q^{9} +6.28891 q^{11} -1.00629 q^{13} -4.69165 q^{17} -5.97554 q^{19} +1.04684 q^{21} +8.05101 q^{23} +1.00000 q^{27} +6.91519 q^{29} -9.52414 q^{31} +6.28891 q^{33} +7.69791 q^{37} -1.00629 q^{39} +1.56932 q^{41} +9.94897 q^{43} -4.84659 q^{47} -5.90413 q^{49} -4.69165 q^{51} +2.82635 q^{53} -5.97554 q^{57} +4.08209 q^{59} +3.16053 q^{61} +1.04684 q^{63} +3.17036 q^{67} +8.05101 q^{69} +6.50896 q^{71} -0.367806 q^{73} +6.58345 q^{77} -3.32009 q^{79} +1.00000 q^{81} +13.8579 q^{83} +6.91519 q^{87} -2.50645 q^{89} -1.05342 q^{91} -9.52414 q^{93} +0.182689 q^{97} +6.28891 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9	$ 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{17} + 10 q^{19} + 8 q^{21} + 18 q^{23} + 12 q^{27} + 8 q^{29} - 2 q^{31} + 2 q^{33} + 4 q^{37} + 10 q^{41} + 28 q^{43} + 22 q^{47} + 28 q^{49} + 8 q^{51} + 16 q^{53} + 10 q^{57} - 2 q^{59} + 34 q^{61} + 8 q^{63} + 32 q^{67} + 18 q^{69} + 24 q^{73} + 18 q^{77} + 6 q^{79} + 12 q^{81} + 28 q^{83} + 8 q^{87} + 10 q^{89} + 20 q^{91} - 2 q^{93} + 16 q^{97} + 2 q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9 + 2 * q^11 + 8 * q^17 + 10 * q^19 + 8 * q^21 + 18 * q^23 + 12 * q^27 + 8 * q^29 - 2 * q^31 + 2 * q^33 + 4 * q^37 + 10 * q^41 + 28 * q^43 + 22 * q^47 + 28 * q^49 + 8 * q^51 + 16 * q^53 + 10 * q^57 - 2 * q^59 + 34 * q^61 + 8 * q^63 + 32 * q^67 + 18 * q^69 + 24 * q^73 + 18 * q^77 + 6 * q^79 + 12 * q^81 + 28 * q^83 + 8 * q^87 + 10 * q^89 + 20 * q^91 - 2 * q^93 + 16 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.04684	0.395667	0.197833	−	0.980236i	$-0.436610\pi$
$7$	1.04684	0.395667	0.197833	+	0.980236i	$0.436610\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.28891	1.89618	0.948088	−	0.318008i	$-0.103014\pi$
$11$	6.28891	1.89618	0.948088	+	0.318008i	$0.103014\pi$
$12$	0	0
$13$	−1.00629	−0.279096	−0.139548	−	0.990215i	$-0.544565\pi$
$13$	−1.00629	−0.279096	−0.139548	+	0.990215i	$0.544565\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.69165	−1.13789	−0.568946	−	0.822375i	$-0.692649\pi$
$17$	−4.69165	−1.13789	−0.568946	+	0.822375i	$0.692649\pi$
$18$	0	0
$19$	−5.97554	−1.37088	−0.685441	−	0.728128i	$-0.740391\pi$
$19$	−5.97554	−1.37088	−0.685441	+	0.728128i	$0.740391\pi$
$20$	0	0
$21$	1.04684	0.228438
$22$	0	0
$23$	8.05101	1.67875	0.839376	−	0.543551i	$-0.182920\pi$
$23$	8.05101	1.67875	0.839376	+	0.543551i	$0.182920\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.91519	1.28412	0.642059	−	0.766655i	$-0.278080\pi$
$29$	6.91519	1.28412	0.642059	+	0.766655i	$0.278080\pi$
$30$	0	0
$31$	−9.52414	−1.71059	−0.855293	−	0.518145i	$-0.826623\pi$
$31$	−9.52414	−1.71059	−0.855293	+	0.518145i	$0.826623\pi$
$32$	0	0
$33$	6.28891	1.09476
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.69791	1.26553	0.632765	−	0.774344i	$-0.281920\pi$
$37$	7.69791	1.26553	0.632765	+	0.774344i	$0.281920\pi$
$38$	0	0
$39$	−1.00629	−0.161136
$40$	0	0
$41$	1.56932	0.245086	0.122543	−	0.992463i	$-0.460895\pi$
$41$	1.56932	0.245086	0.122543	+	0.992463i	$0.460895\pi$
$42$	0	0
$43$	9.94897	1.51720	0.758602	−	0.651555i	$-0.225883\pi$
$43$	9.94897	1.51720	0.758602	+	0.651555i	$0.225883\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.84659	−0.706948	−0.353474	−	0.935444i	$-0.615000\pi$
$47$	−4.84659	−0.706948	−0.353474	+	0.935444i	$0.615000\pi$
$48$	0	0
$49$	−5.90413	−0.843448
$50$	0	0
$51$	−4.69165	−0.656962
$52$	0	0
$53$	2.82635	0.388229	0.194114	−	0.980979i	$-0.437817\pi$
$53$	2.82635	0.388229	0.194114	+	0.980979i	$0.437817\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.97554	−0.791479
$58$	0	0
$59$	4.08209	0.531443	0.265721	−	0.964050i	$-0.414390\pi$
$59$	4.08209	0.531443	0.265721	+	0.964050i	$0.414390\pi$
$60$	0	0
$61$	3.16053	0.404665	0.202332	−	0.979317i	$-0.435148\pi$
$61$	3.16053	0.404665	0.202332	+	0.979317i	$0.435148\pi$
$62$	0	0
$63$	1.04684	0.131889
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.17036	0.387321	0.193660	−	0.981069i	$-0.437964\pi$
$67$	3.17036	0.387321	0.193660	+	0.981069i	$0.437964\pi$
$68$	0	0
$69$	8.05101	0.969228
$70$	0	0
$71$	6.50896	0.772472	0.386236	−	0.922400i	$-0.373775\pi$
$71$	6.50896	0.772472	0.386236	+	0.922400i	$0.373775\pi$
$72$	0	0
$73$	−0.367806	−0.0430485	−0.0215242	−	0.999768i	$-0.506852\pi$
$73$	−0.367806	−0.0430485	−0.0215242	+	0.999768i	$0.506852\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.58345	0.750254
$78$	0	0
$79$	−3.32009	−0.373539	−0.186770	−	0.982404i	$-0.559802\pi$
$79$	−3.32009	−0.373539	−0.186770	+	0.982404i	$0.559802\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.8579	1.52110	0.760549	−	0.649281i	$-0.224930\pi$
$83$	13.8579	1.52110	0.760549	+	0.649281i	$0.224930\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.91519	0.741386
$88$	0	0
$89$	−2.50645	−0.265683	−0.132842	−	0.991137i	$-0.542410\pi$
$89$	−2.50645	−0.265683	−0.132842	+	0.991137i	$0.542410\pi$
$90$	0	0
$91$	−1.05342	−0.110429
$92$	0	0
$93$	−9.52414	−0.987607
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.182689	0.0185493	0.00927465	−	0.999957i	$-0.497048\pi$
$97$	0.182689	0.0185493	0.00927465	+	0.999957i	$0.497048\pi$
$98$	0	0
$99$	6.28891	0.632059
$100$	0	0
$101$	−5.96970	−0.594008	−0.297004	−	0.954876i	$-0.595987\pi$
$101$	−5.96970	−0.594008	−0.297004	+	0.954876i	$0.595987\pi$
$102$	0	0
$103$	−8.95864	−0.882721	−0.441360	−	0.897330i	$-0.645504\pi$
$103$	−8.95864	−0.882721	−0.441360	+	0.897330i	$0.645504\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.69495	−0.260530	−0.130265	−	0.991479i	$-0.541583\pi$
$107$	−2.69495	−0.260530	−0.130265	+	0.991479i	$0.541583\pi$
$108$	0	0
$109$	10.2999	0.986549	0.493275	−	0.869874i	$-0.335800\pi$
$109$	10.2999	0.986549	0.493275	+	0.869874i	$0.335800\pi$
$110$	0	0
$111$	7.69791	0.730654
$112$	0	0
$113$	−3.38070	−0.318029	−0.159015	−	0.987276i	$-0.550832\pi$
$113$	−3.38070	−0.318029	−0.159015	+	0.987276i	$0.550832\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00629	−0.0930319
$118$	0	0
$119$	−4.91139	−0.450226
$120$	0	0
$121$	28.5503	2.59548
$122$	0	0
$123$	1.56932	0.141501
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−5.97421	−0.530126	−0.265063	−	0.964231i	$-0.585393\pi$
$127$	−5.97421	−0.530126	−0.265063	+	0.964231i	$0.585393\pi$
$128$	0	0
$129$	9.94897	0.875958
$130$	0	0
$131$	6.20733	0.542337	0.271168	−	0.962532i	$-0.412590\pi$
$131$	6.20733	0.542337	0.271168	+	0.962532i	$0.412590\pi$
$132$	0	0
$133$	−6.25541	−0.542413
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.9777	1.10876	0.554381	−	0.832263i	$-0.312955\pi$
$137$	12.9777	1.10876	0.554381	+	0.832263i	$0.312955\pi$
$138$	0	0
$139$	3.99948	0.339231	0.169616	−	0.985510i	$-0.445747\pi$
$139$	3.99948	0.339231	0.169616	+	0.985510i	$0.445747\pi$
$140$	0	0
$141$	−4.84659	−0.408156
$142$	0	0
$143$	−6.32849	−0.529215
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5.90413	−0.486965
$148$	0	0
$149$	5.68762	0.465948	0.232974	−	0.972483i	$-0.425154\pi$
$149$	5.68762	0.465948	0.232974	+	0.972483i	$0.425154\pi$
$150$	0	0
$151$	−5.51150	−0.448519	−0.224260	−	0.974529i	$-0.571996\pi$
$151$	−5.51150	−0.448519	−0.224260	+	0.974529i	$0.571996\pi$
$152$	0	0
$153$	−4.69165	−0.379297
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.0217	1.35848	0.679242	−	0.733915i	$-0.262309\pi$
$157$	17.0217	1.35848	0.679242	+	0.733915i	$0.262309\pi$
$158$	0	0
$159$	2.82635	0.224144
$160$	0	0
$161$	8.42809	0.664227
$162$	0	0
$163$	12.2958	0.963078	0.481539	−	0.876425i	$-0.340078\pi$
$163$	12.2958	0.963078	0.481539	+	0.876425i	$0.340078\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.42858	−0.420076	−0.210038	−	0.977693i	$-0.567359\pi$
$167$	−5.42858	−0.420076	−0.210038	+	0.977693i	$0.567359\pi$
$168$	0	0
$169$	−11.9874	−0.922106
$170$	0	0
$171$	−5.97554	−0.456961
$172$	0	0
$173$	10.7456	0.816973	0.408486	−	0.912764i	$-0.366057\pi$
$173$	10.7456	0.816973	0.408486	+	0.912764i	$0.366057\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.08209	0.306829
$178$	0	0
$179$	−19.3062	−1.44302	−0.721508	−	0.692406i	$-0.756551\pi$
$179$	−19.3062	−1.44302	−0.721508	+	0.692406i	$0.756551\pi$
$180$	0	0
$181$	16.7512	1.24511	0.622555	−	0.782576i	$-0.286095\pi$
$181$	16.7512	1.24511	0.622555	+	0.782576i	$0.286095\pi$
$182$	0	0
$183$	3.16053	0.233633
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−29.5053	−2.15764
$188$	0	0
$189$	1.04684	0.0761461
$190$	0	0
$191$	−11.9454	−0.864336	−0.432168	−	0.901793i	$-0.642251\pi$
$191$	−11.9454	−0.864336	−0.432168	+	0.901793i	$0.642251\pi$
$192$	0	0
$193$	26.4298	1.90246	0.951229	−	0.308485i	$-0.0998220\pi$
$193$	26.4298	1.90246	0.951229	+	0.308485i	$0.0998220\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.0718	−0.788833	−0.394417	−	0.918932i	$-0.629053\pi$
$197$	−11.0718	−0.788833	−0.394417	+	0.918932i	$0.629053\pi$
$198$	0	0
$199$	−16.5548	−1.17354	−0.586768	−	0.809755i	$-0.699600\pi$
$199$	−16.5548	−1.17354	−0.586768	+	0.809755i	$0.699600\pi$
$200$	0	0
$201$	3.17036	0.223620
$202$	0	0
$203$	7.23907	0.508083
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.05101	0.559584
$208$	0	0
$209$	−37.5796	−2.59944
$210$	0	0
$211$	−4.10714	−0.282747	−0.141373	−	0.989956i	$-0.545152\pi$
$211$	−4.10714	−0.282747	−0.141373	+	0.989956i	$0.545152\pi$
$212$	0	0
$213$	6.50896	0.445987
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.97021	−0.676822
$218$	0	0
$219$	−0.367806	−0.0248540
$220$	0	0
$221$	4.72118	0.317581
$222$	0	0
$223$	−6.26147	−0.419299	−0.209650	−	0.977777i	$-0.567232\pi$
$223$	−6.26147	−0.419299	−0.209650	+	0.977777i	$0.567232\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.4817	−0.695697	−0.347849	−	0.937551i	$-0.613088\pi$
$227$	−10.4817	−0.695697	−0.347849	+	0.937551i	$0.613088\pi$
$228$	0	0
$229$	7.56679	0.500028	0.250014	−	0.968242i	$-0.419565\pi$
$229$	7.56679	0.500028	0.250014	+	0.968242i	$0.419565\pi$
$230$	0	0
$231$	6.58345	0.433159
$232$	0	0
$233$	−15.9832	−1.04709	−0.523547	−	0.851997i	$-0.675392\pi$
$233$	−15.9832	−1.04709	−0.523547	+	0.851997i	$0.675392\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.32009	−0.215663
$238$	0	0
$239$	−2.91998	−0.188877	−0.0944387	−	0.995531i	$-0.530106\pi$
$239$	−2.91998	−0.188877	−0.0944387	+	0.995531i	$0.530106\pi$
$240$	0	0
$241$	−23.0477	−1.48463	−0.742315	−	0.670051i	$-0.766272\pi$
$241$	−23.0477	−1.48463	−0.742315	+	0.670051i	$0.766272\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.01315	0.382607
$248$	0	0
$249$	13.8579	0.878206
$250$	0	0
$251$	−9.79130	−0.618021	−0.309011	−	0.951059i	$-0.599998\pi$
$251$	−9.79130	−0.618021	−0.309011	+	0.951059i	$0.599998\pi$
$252$	0	0
$253$	50.6321	3.18321
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.3147	−1.39195	−0.695975	−	0.718066i	$-0.745027\pi$
$257$	−22.3147	−1.39195	−0.695975	+	0.718066i	$0.745027\pi$
$258$	0	0
$259$	8.05845	0.500728
$260$	0	0
$261$	6.91519	0.428039
$262$	0	0
$263$	0.729154	0.0449615	0.0224808	−	0.999747i	$-0.492844\pi$
$263$	0.729154	0.0449615	0.0224808	+	0.999747i	$0.492844\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.50645	−0.153392
$268$	0	0
$269$	−1.85922	−0.113359	−0.0566794	−	0.998392i	$-0.518051\pi$
$269$	−1.85922	−0.113359	−0.0566794	+	0.998392i	$0.518051\pi$
$270$	0	0
$271$	19.5534	1.18778	0.593891	−	0.804545i	$-0.297591\pi$
$271$	19.5534	1.18778	0.593891	+	0.804545i	$0.297591\pi$
$272$	0	0
$273$	−1.05342	−0.0637561
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−16.9612	−1.01910	−0.509551	−	0.860441i	$-0.670188\pi$
$277$	−16.9612	−1.01910	−0.509551	+	0.860441i	$0.670188\pi$
$278$	0	0
$279$	−9.52414	−0.570195
$280$	0	0
$281$	−26.0879	−1.55627	−0.778135	−	0.628096i	$-0.783834\pi$
$281$	−26.0879	−1.55627	−0.778135	+	0.628096i	$0.783834\pi$
$282$	0	0
$283$	25.6093	1.52232	0.761159	−	0.648565i	$-0.224631\pi$
$283$	25.6093	1.52232	0.761159	+	0.648565i	$0.224631\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.64282	0.0969724
$288$	0	0
$289$	5.01158	0.294799
$290$	0	0
$291$	0.182689	0.0107094
$292$	0	0
$293$	27.7845	1.62319	0.811595	−	0.584220i	$-0.198600\pi$
$293$	27.7845	1.62319	0.811595	+	0.584220i	$0.198600\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.28891	0.364919
$298$	0	0
$299$	−8.10169	−0.468533
$300$	0	0
$301$	10.4149	0.600307
$302$	0	0
$303$	−5.96970	−0.342951
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.1422	0.921283	0.460642	−	0.887586i	$-0.347619\pi$
$307$	16.1422	0.921283	0.460642	+	0.887586i	$0.347619\pi$
$308$	0	0
$309$	−8.95864	−0.509639
$310$	0	0
$311$	28.7381	1.62959	0.814795	−	0.579749i	$-0.196850\pi$
$311$	28.7381	1.62959	0.814795	+	0.579749i	$0.196850\pi$
$312$	0	0
$313$	26.4864	1.49710	0.748550	−	0.663079i	$-0.230751\pi$
$313$	26.4864	1.49710	0.748550	+	0.663079i	$0.230751\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.2980	1.47705	0.738523	−	0.674229i	$-0.235524\pi$
$317$	26.2980	1.47705	0.738523	+	0.674229i	$0.235524\pi$
$318$	0	0
$319$	43.4890	2.43491
$320$	0	0
$321$	−2.69495	−0.150417
$322$	0	0
$323$	28.0351	1.55992
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.2999	0.569585
$328$	0	0
$329$	−5.07358	−0.279716
$330$	0	0
$331$	−31.2634	−1.71839	−0.859195	−	0.511649i	$-0.829035\pi$
$331$	−31.2634	−1.71839	−0.859195	+	0.511649i	$0.829035\pi$
$332$	0	0
$333$	7.69791	0.421843
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.90526	0.485101	0.242550	−	0.970139i	$-0.422016\pi$
$337$	8.90526	0.485101	0.242550	+	0.970139i	$0.422016\pi$
$338$	0	0
$339$	−3.38070	−0.183614
$340$	0	0
$341$	−59.8964	−3.24357
$342$	0	0
$343$	−13.5085	−0.729391
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.5530	1.15702	0.578512	−	0.815674i	$-0.303634\pi$
$347$	21.5530	1.15702	0.578512	+	0.815674i	$0.303634\pi$
$348$	0	0
$349$	17.3958	0.931178	0.465589	−	0.885001i	$-0.345842\pi$
$349$	17.3958	0.931178	0.465589	+	0.885001i	$0.345842\pi$
$350$	0	0
$351$	−1.00629	−0.0537120
$352$	0	0
$353$	17.8613	0.950661	0.475331	−	0.879807i	$-0.342328\pi$
$353$	17.8613	0.950661	0.475331	+	0.879807i	$0.342328\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.91139	−0.259938
$358$	0	0
$359$	3.65870	0.193099	0.0965493	−	0.995328i	$-0.469219\pi$
$359$	3.65870	0.193099	0.0965493	+	0.995328i	$0.469219\pi$
$360$	0	0
$361$	16.7071	0.879319
$362$	0	0
$363$	28.5503	1.49850
$364$	0	0
$365$	0	0
$366$	0	0
$367$	26.0423	1.35940	0.679698	−	0.733492i	$-0.262111\pi$
$367$	26.0423	1.35940	0.679698	+	0.733492i	$0.262111\pi$
$368$	0	0
$369$	1.56932	0.0816954
$370$	0	0
$371$	2.95872	0.153609
$372$	0	0
$373$	−29.2609	−1.51507	−0.757537	−	0.652792i	$-0.773597\pi$
$373$	−29.2609	−1.51507	−0.757537	+	0.652792i	$0.773597\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.95871	−0.358392
$378$	0	0
$379$	−10.2669	−0.527374	−0.263687	−	0.964608i	$-0.584939\pi$
$379$	−10.2669	−0.527374	−0.263687	+	0.964608i	$0.584939\pi$
$380$	0	0
$381$	−5.97421	−0.306068
$382$	0	0
$383$	−25.6479	−1.31055	−0.655273	−	0.755392i	$-0.727446\pi$
$383$	−25.6479	−1.31055	−0.655273	+	0.755392i	$0.727446\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.94897	0.505735
$388$	0	0
$389$	−9.58813	−0.486137	−0.243069	−	0.970009i	$-0.578154\pi$
$389$	−9.58813	−0.486137	−0.243069	+	0.970009i	$0.578154\pi$
$390$	0	0
$391$	−37.7725	−1.91024
$392$	0	0
$393$	6.20733	0.313118
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−12.3309	−0.618869	−0.309435	−	0.950921i	$-0.600140\pi$
$397$	−12.3309	−0.618869	−0.309435	+	0.950921i	$0.600140\pi$
$398$	0	0
$399$	−6.25541	−0.313162
$400$	0	0
$401$	18.9779	0.947709	0.473855	−	0.880603i	$-0.342862\pi$
$401$	18.9779	0.947709	0.473855	+	0.880603i	$0.342862\pi$
$402$	0	0
$403$	9.58408	0.477417
$404$	0	0
$405$	0	0
$406$	0	0
$407$	48.4115	2.39967
$408$	0	0
$409$	19.5102	0.964719	0.482359	−	0.875973i	$-0.339780\pi$
$409$	19.5102	0.964719	0.482359	+	0.875973i	$0.339780\pi$
$410$	0	0
$411$	12.9777	0.640144
$412$	0	0
$413$	4.27328	0.210274
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.99948	0.195855
$418$	0	0
$419$	1.84915	0.0903370	0.0451685	−	0.998979i	$-0.485618\pi$
$419$	1.84915	0.0903370	0.0451685	+	0.998979i	$0.485618\pi$
$420$	0	0
$421$	−6.67752	−0.325443	−0.162721	−	0.986672i	$-0.552027\pi$
$421$	−6.67752	−0.325443	−0.162721	+	0.986672i	$0.552027\pi$
$422$	0	0
$423$	−4.84659	−0.235649
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.30856	0.160112
$428$	0	0
$429$	−6.32849	−0.305542
$430$	0	0
$431$	29.6459	1.42799	0.713996	−	0.700150i	$-0.246884\pi$
$431$	29.6459	1.42799	0.713996	+	0.700150i	$0.246884\pi$
$432$	0	0
$433$	28.0178	1.34645	0.673225	−	0.739438i	$-0.264909\pi$
$433$	28.0178	1.34645	0.673225	+	0.739438i	$0.264909\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−48.1091	−2.30137
$438$	0	0
$439$	10.2248	0.488004	0.244002	−	0.969775i	$-0.421540\pi$
$439$	10.2248	0.488004	0.244002	+	0.969775i	$0.421540\pi$
$440$	0	0
$441$	−5.90413	−0.281149
$442$	0	0
$443$	26.5115	1.25960	0.629800	−	0.776757i	$-0.283137\pi$
$443$	26.5115	1.25960	0.629800	+	0.776757i	$0.283137\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.68762	0.269015
$448$	0	0
$449$	−6.39692	−0.301889	−0.150945	−	0.988542i	$-0.548232\pi$
$449$	−6.39692	−0.301889	−0.150945	+	0.988542i	$0.548232\pi$
$450$	0	0
$451$	9.86929	0.464727
$452$	0	0
$453$	−5.51150	−0.258953
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−14.9784	−0.700661	−0.350330	−	0.936626i	$-0.613931\pi$
$457$	−14.9784	−0.700661	−0.350330	+	0.936626i	$0.613931\pi$
$458$	0	0
$459$	−4.69165	−0.218987
$460$	0	0
$461$	−12.1832	−0.567429	−0.283715	−	0.958909i	$-0.591567\pi$
$461$	−12.1832	−0.567429	−0.283715	+	0.958909i	$0.591567\pi$
$462$	0	0
$463$	8.31956	0.386643	0.193321	−	0.981135i	$-0.438074\pi$
$463$	8.31956	0.386643	0.193321	+	0.981135i	$0.438074\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.30463	−0.384293	−0.192146	−	0.981366i	$-0.561545\pi$
$467$	−8.30463	−0.384293	−0.192146	+	0.981366i	$0.561545\pi$
$468$	0	0
$469$	3.31884	0.153250
$470$	0	0
$471$	17.0217	0.784321
$472$	0	0
$473$	62.5681	2.87689
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.82635	0.129410
$478$	0	0
$479$	−3.05639	−0.139650	−0.0698249	−	0.997559i	$-0.522244\pi$
$479$	−3.05639	−0.139650	−0.0698249	+	0.997559i	$0.522244\pi$
$480$	0	0
$481$	−7.74636	−0.353204
$482$	0	0
$483$	8.42809	0.383491
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−11.7187	−0.531026	−0.265513	−	0.964107i	$-0.585541\pi$
$487$	−11.7187	−0.531026	−0.265513	+	0.964107i	$0.585541\pi$
$488$	0	0
$489$	12.2958	0.556033
$490$	0	0
$491$	16.9523	0.765048	0.382524	−	0.923946i	$-0.375055\pi$
$491$	16.9523	0.765048	0.382524	+	0.923946i	$0.375055\pi$
$492$	0	0
$493$	−32.4436	−1.46119
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.81381	0.305641
$498$	0	0
$499$	14.0674	0.629742	0.314871	−	0.949135i	$-0.398039\pi$
$499$	14.0674	0.629742	0.314871	+	0.949135i	$0.398039\pi$
$500$	0	0
$501$	−5.42858	−0.242531
$502$	0	0
$503$	5.60755	0.250028	0.125014	−	0.992155i	$-0.460102\pi$
$503$	5.60755	0.250028	0.125014	+	0.992155i	$0.460102\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−11.9874	−0.532378
$508$	0	0
$509$	−11.4650	−0.508176	−0.254088	−	0.967181i	$-0.581775\pi$
$509$	−11.4650	−0.508176	−0.254088	+	0.967181i	$0.581775\pi$
$510$	0	0
$511$	−0.385033	−0.0170328
$512$	0	0
$513$	−5.97554	−0.263826
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−30.4797	−1.34050
$518$	0	0
$519$	10.7456	0.471680
$520$	0	0
$521$	17.9012	0.784267	0.392133	−	0.919908i	$-0.371737\pi$
$521$	17.9012	0.784267	0.392133	+	0.919908i	$0.371737\pi$
$522$	0	0
$523$	1.34677	0.0588899	0.0294450	−	0.999566i	$-0.490626\pi$
$523$	1.34677	0.0588899	0.0294450	+	0.999566i	$0.490626\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	44.6839	1.94646
$528$	0	0
$529$	41.8188	1.81821
$530$	0	0
$531$	4.08209	0.177148
$532$	0	0
$533$	−1.57919	−0.0684025
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−19.3062	−0.833126
$538$	0	0
$539$	−37.1305	−1.59933
$540$	0	0
$541$	11.2977	0.485725	0.242862	−	0.970061i	$-0.421914\pi$
$541$	11.2977	0.485725	0.242862	+	0.970061i	$0.421914\pi$
$542$	0	0
$543$	16.7512	0.718864
$544$	0	0
$545$	0	0
$546$	0	0
$547$	25.8147	1.10376	0.551879	−	0.833924i	$-0.313911\pi$
$547$	25.8147	1.10376	0.551879	+	0.833924i	$0.313911\pi$
$548$	0	0
$549$	3.16053	0.134888
$550$	0	0
$551$	−41.3220	−1.76037
$552$	0	0
$553$	−3.47559	−0.147797
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−35.3575	−1.49814	−0.749072	−	0.662489i	$-0.769500\pi$
$557$	−35.3575	−1.49814	−0.749072	+	0.662489i	$0.769500\pi$
$558$	0	0
$559$	−10.0116	−0.423445
$560$	0	0
$561$	−29.5053	−1.24572
$562$	0	0
$563$	−4.55427	−0.191940	−0.0959698	−	0.995384i	$-0.530595\pi$
$563$	−4.55427	−0.191940	−0.0959698	+	0.995384i	$0.530595\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.04684	0.0439630
$568$	0	0
$569$	−37.1409	−1.55703	−0.778514	−	0.627627i	$-0.784026\pi$
$569$	−37.1409	−1.55703	−0.778514	+	0.627627i	$0.784026\pi$
$570$	0	0
$571$	−4.30015	−0.179956	−0.0899778	−	0.995944i	$-0.528680\pi$
$571$	−4.30015	−0.179956	−0.0899778	+	0.995944i	$0.528680\pi$
$572$	0	0
$573$	−11.9454	−0.499025
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0.467179	0.0194489	0.00972445	−	0.999953i	$-0.496905\pi$
$577$	0.467179	0.0194489	0.00972445	+	0.999953i	$0.496905\pi$
$578$	0	0
$579$	26.4298	1.09838
$580$	0	0
$581$	14.5069	0.601848
$582$	0	0
$583$	17.7746	0.736150
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.9548	1.36019	0.680095	−	0.733124i	$-0.261938\pi$
$587$	32.9548	1.36019	0.680095	+	0.733124i	$0.261938\pi$
$588$	0	0
$589$	56.9119	2.34501
$590$	0	0
$591$	−11.0718	−0.455433
$592$	0	0
$593$	−15.5285	−0.637680	−0.318840	−	0.947809i	$-0.603293\pi$
$593$	−15.5285	−0.637680	−0.318840	+	0.947809i	$0.603293\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.5548	−0.677542
$598$	0	0
$599$	−36.0116	−1.47140	−0.735698	−	0.677310i	$-0.763146\pi$
$599$	−36.0116	−1.47140	−0.735698	+	0.677310i	$0.763146\pi$
$600$	0	0
$601$	−24.1503	−0.985112	−0.492556	−	0.870281i	$-0.663937\pi$
$601$	−24.1503	−0.985112	−0.492556	+	0.870281i	$0.663937\pi$
$602$	0	0
$603$	3.17036	0.129107
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−29.8098	−1.20994	−0.604970	−	0.796248i	$-0.706815\pi$
$607$	−29.8098	−1.20994	−0.604970	+	0.796248i	$0.706815\pi$
$608$	0	0
$609$	7.23907	0.293342
$610$	0	0
$611$	4.87709	0.197306
$612$	0	0
$613$	−32.1330	−1.29784	−0.648919	−	0.760857i	$-0.724779\pi$
$613$	−32.1330	−1.29784	−0.648919	+	0.760857i	$0.724779\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17.2509	−0.694493	−0.347247	−	0.937774i	$-0.612883\pi$
$617$	−17.2509	−0.694493	−0.347247	+	0.937774i	$0.612883\pi$
$618$	0	0
$619$	7.06017	0.283772	0.141886	−	0.989883i	$-0.454683\pi$
$619$	7.06017	0.283772	0.141886	+	0.989883i	$0.454683\pi$
$620$	0	0
$621$	8.05101	0.323076
$622$	0	0
$623$	−2.62384	−0.105122
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−37.5796	−1.50078
$628$	0	0
$629$	−36.1159	−1.44004
$630$	0	0
$631$	−9.78513	−0.389540	−0.194770	−	0.980849i	$-0.562396\pi$
$631$	−9.78513	−0.389540	−0.194770	+	0.980849i	$0.562396\pi$
$632$	0	0
$633$	−4.10714	−0.163244
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.94129	0.235403
$638$	0	0
$639$	6.50896	0.257491
$640$	0	0
$641$	15.6701	0.618930	0.309465	−	0.950911i	$-0.399850\pi$
$641$	15.6701	0.618930	0.309465	+	0.950911i	$0.399850\pi$
$642$	0	0
$643$	0.766747	0.0302375	0.0151188	−	0.999886i	$-0.495187\pi$
$643$	0.766747	0.0302375	0.0151188	+	0.999886i	$0.495187\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.1728	−0.517878	−0.258939	−	0.965894i	$-0.583373\pi$
$647$	−13.1728	−0.517878	−0.258939	+	0.965894i	$0.583373\pi$
$648$	0	0
$649$	25.6719	1.00771
$650$	0	0
$651$	−9.97021	−0.390763
$652$	0	0
$653$	13.0921	0.512335	0.256167	−	0.966632i	$-0.417540\pi$
$653$	13.0921	0.512335	0.256167	+	0.966632i	$0.417540\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−0.367806	−0.0143495
$658$	0	0
$659$	−9.46731	−0.368794	−0.184397	−	0.982852i	$-0.559033\pi$
$659$	−9.46731	−0.368794	−0.184397	+	0.982852i	$0.559033\pi$
$660$	0	0
$661$	5.55884	0.216214	0.108107	−	0.994139i	$-0.465521\pi$
$661$	5.55884	0.216214	0.108107	+	0.994139i	$0.465521\pi$
$662$	0	0
$663$	4.72118	0.183355
$664$	0	0
$665$	0	0
$666$	0	0
$667$	55.6743	2.15572
$668$	0	0
$669$	−6.26147	−0.242083
$670$	0	0
$671$	19.8763	0.767316
$672$	0	0
$673$	−0.304023	−0.0117192	−0.00585962	−	0.999983i	$-0.501865\pi$
$673$	−0.304023	−0.0117192	−0.00585962	+	0.999983i	$0.501865\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−39.5278	−1.51918	−0.759589	−	0.650404i	$-0.774600\pi$
$677$	−39.5278	−1.51918	−0.759589	+	0.650404i	$0.774600\pi$
$678$	0	0
$679$	0.191246	0.00733934
$680$	0	0
$681$	−10.4817	−0.401661
$682$	0	0
$683$	−31.5480	−1.20715	−0.603576	−	0.797305i	$-0.706258\pi$
$683$	−31.5480	−1.20715	−0.603576	+	0.797305i	$0.706258\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.56679	0.288691
$688$	0	0
$689$	−2.84414	−0.108353
$690$	0	0
$691$	9.38385	0.356978	0.178489	−	0.983942i	$-0.442879\pi$
$691$	9.38385	0.356978	0.178489	+	0.983942i	$0.442879\pi$
$692$	0	0
$693$	6.58345	0.250085
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−7.36269	−0.278882
$698$	0	0
$699$	−15.9832	−0.604540
$700$	0	0
$701$	−20.4347	−0.771809	−0.385904	−	0.922539i	$-0.626111\pi$
$701$	−20.4347	−0.771809	−0.385904	+	0.922539i	$0.626111\pi$
$702$	0	0
$703$	−45.9992	−1.73489
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.24930	−0.235029
$708$	0	0
$709$	−5.91798	−0.222254	−0.111127	−	0.993806i	$-0.535446\pi$
$709$	−5.91798	−0.222254	−0.111127	+	0.993806i	$0.535446\pi$
$710$	0	0
$711$	−3.32009	−0.124513
$712$	0	0
$713$	−76.6790	−2.87165
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.91998	−0.109048
$718$	0	0
$719$	−8.00127	−0.298397	−0.149199	−	0.988807i	$-0.547669\pi$
$719$	−8.00127	−0.298397	−0.149199	+	0.988807i	$0.547669\pi$
$720$	0	0
$721$	−9.37822	−0.349263
$722$	0	0
$723$	−23.0477	−0.857152
$724$	0	0
$725$	0	0
$726$	0	0
$727$	40.2787	1.49385	0.746926	−	0.664907i	$-0.231529\pi$
$727$	40.2787	1.49385	0.746926	+	0.664907i	$0.231529\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−46.6771	−1.72641
$732$	0	0
$733$	−10.5973	−0.391420	−0.195710	−	0.980662i	$-0.562701\pi$
$733$	−10.5973	−0.391420	−0.195710	+	0.980662i	$0.562701\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.9381	0.734428
$738$	0	0
$739$	−45.5094	−1.67409	−0.837044	−	0.547135i	$-0.815718\pi$
$739$	−45.5094	−1.67409	−0.837044	+	0.547135i	$0.815718\pi$
$740$	0	0
$741$	6.01315	0.220898
$742$	0	0
$743$	8.64344	0.317097	0.158549	−	0.987351i	$-0.449319\pi$
$743$	8.64344	0.317097	0.158549	+	0.987351i	$0.449319\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.8579	0.507032
$748$	0	0
$749$	−2.82117	−0.103083
$750$	0	0
$751$	−54.4382	−1.98648	−0.993239	−	0.116086i	$-0.962965\pi$
$751$	−54.4382	−1.98648	−0.993239	+	0.116086i	$0.962965\pi$
$752$	0	0
$753$	−9.79130	−0.356815
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−34.1378	−1.24076	−0.620380	−	0.784302i	$-0.713022\pi$
$757$	−34.1378	−1.24076	−0.620380	+	0.784302i	$0.713022\pi$
$758$	0	0
$759$	50.6321	1.83783
$760$	0	0
$761$	3.12475	0.113272	0.0566361	−	0.998395i	$-0.481963\pi$
$761$	3.12475	0.113272	0.0566361	+	0.998395i	$0.481963\pi$
$762$	0	0
$763$	10.7823	0.390345
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.10778	−0.148323
$768$	0	0
$769$	−53.3949	−1.92547	−0.962735	−	0.270446i	$-0.912829\pi$
$769$	−53.3949	−1.92547	−0.962735	+	0.270446i	$0.912829\pi$
$770$	0	0
$771$	−22.3147	−0.803642
$772$	0	0
$773$	−0.797218	−0.0286739	−0.0143370	−	0.999897i	$-0.504564\pi$
$773$	−0.797218	−0.0286739	−0.0143370	+	0.999897i	$0.504564\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.05845	0.289095
$778$	0	0
$779$	−9.37751	−0.335984
$780$	0	0
$781$	40.9342	1.46474
$782$	0	0
$783$	6.91519	0.247129
$784$	0	0
$785$	0	0
$786$	0	0
$787$	41.7571	1.48848	0.744239	−	0.667913i	$-0.232812\pi$
$787$	41.7571	1.48848	0.744239	+	0.667913i	$0.232812\pi$
$788$	0	0
$789$	0.729154	0.0259586
$790$	0	0
$791$	−3.53904	−0.125834
$792$	0	0
$793$	−3.18042	−0.112940
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.6513	0.979458	0.489729	−	0.871875i	$-0.337096\pi$
$797$	27.6513	0.979458	0.489729	+	0.871875i	$0.337096\pi$
$798$	0	0
$799$	22.7385	0.804430
$800$	0	0
$801$	−2.50645	−0.0885611
$802$	0	0
$803$	−2.31310	−0.0816275
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.85922	−0.0654477
$808$	0	0
$809$	34.9805	1.22985	0.614924	−	0.788586i	$-0.289187\pi$
$809$	34.9805	1.22985	0.614924	+	0.788586i	$0.289187\pi$
$810$	0	0
$811$	9.15152	0.321353	0.160677	−	0.987007i	$-0.448632\pi$
$811$	9.15152	0.321353	0.160677	+	0.987007i	$0.448632\pi$
$812$	0	0
$813$	19.5534	0.685767
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−59.4504	−2.07991
$818$	0	0
$819$	−1.05342	−0.0368096
$820$	0	0
$821$	−3.23251	−0.112815	−0.0564077	−	0.998408i	$-0.517965\pi$
$821$	−3.23251	−0.112815	−0.0564077	+	0.998408i	$0.517965\pi$
$822$	0	0
$823$	45.2046	1.57573	0.787867	−	0.615846i	$-0.211186\pi$
$823$	45.2046	1.57573	0.787867	+	0.615846i	$0.211186\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.6568	0.822629	0.411314	−	0.911493i	$-0.365070\pi$
$827$	23.6568	0.822629	0.411314	+	0.911493i	$0.365070\pi$
$828$	0	0
$829$	12.7644	0.443326	0.221663	−	0.975123i	$-0.428852\pi$
$829$	12.7644	0.443326	0.221663	+	0.975123i	$0.428852\pi$
$830$	0	0
$831$	−16.9612	−0.588379
$832$	0	0
$833$	27.7001	0.959753
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9.52414	−0.329202
$838$	0	0
$839$	−52.0352	−1.79646	−0.898228	−	0.439530i	$-0.855145\pi$
$839$	−52.0352	−1.79646	−0.898228	+	0.439530i	$0.855145\pi$
$840$	0	0
$841$	18.8198	0.648959
$842$	0	0
$843$	−26.0879	−0.898513
$844$	0	0
$845$	0	0
$846$	0	0
$847$	29.8875	1.02695
$848$	0	0
$849$	25.6093	0.878911
$850$	0	0
$851$	61.9760	2.12451
$852$	0	0
$853$	−35.6260	−1.21981	−0.609905	−	0.792474i	$-0.708792\pi$
$853$	−35.6260	−1.21981	−0.609905	+	0.792474i	$0.708792\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.6745	0.842866	0.421433	−	0.906860i	$-0.361527\pi$
$857$	24.6745	0.842866	0.421433	+	0.906860i	$0.361527\pi$
$858$	0	0
$859$	21.9223	0.747978	0.373989	−	0.927433i	$-0.377990\pi$
$859$	21.9223	0.747978	0.373989	+	0.927433i	$0.377990\pi$
$860$	0	0
$861$	1.64282	0.0559871
$862$	0	0
$863$	−27.5761	−0.938702	−0.469351	−	0.883012i	$-0.655512\pi$
$863$	−27.5761	−0.938702	−0.469351	+	0.883012i	$0.655512\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.01158	0.170202
$868$	0	0
$869$	−20.8797	−0.708296
$870$	0	0
$871$	−3.19031	−0.108100
$872$	0	0
$873$	0.182689	0.00618310
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.6879	0.698579	0.349290	−	0.937015i	$-0.386423\pi$
$877$	20.6879	0.698579	0.349290	+	0.937015i	$0.386423\pi$
$878$	0	0
$879$	27.7845	0.937149
$880$	0	0
$881$	−37.8861	−1.27641	−0.638207	−	0.769865i	$-0.720324\pi$
$881$	−37.8861	−1.27641	−0.638207	+	0.769865i	$0.720324\pi$
$882$	0	0
$883$	39.3444	1.32404	0.662022	−	0.749484i	$-0.269698\pi$
$883$	39.3444	1.32404	0.662022	+	0.749484i	$0.269698\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.2764	−0.815121	−0.407560	−	0.913178i	$-0.633620\pi$
$887$	−24.2764	−0.815121	−0.407560	+	0.913178i	$0.633620\pi$
$888$	0	0
$889$	−6.25402	−0.209753
$890$	0	0
$891$	6.28891	0.210686
$892$	0	0
$893$	28.9610	0.969142
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−8.10169	−0.270507
$898$	0	0
$899$	−65.8612	−2.19659
$900$	0	0
$901$	−13.2602	−0.441762
$902$	0	0
$903$	10.4149	0.346587
$904$	0	0
$905$	0	0
$906$	0	0
$907$	13.1961	0.438169	0.219085	−	0.975706i	$-0.429693\pi$
$907$	13.1961	0.438169	0.219085	+	0.975706i	$0.429693\pi$
$908$	0	0
$909$	−5.96970	−0.198003
$910$	0	0
$911$	−27.6036	−0.914547	−0.457274	−	0.889326i	$-0.651174\pi$
$911$	−27.6036	−0.914547	−0.457274	+	0.889326i	$0.651174\pi$
$912$	0	0
$913$	87.1508	2.88427
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.49805	0.214585
$918$	0	0
$919$	12.1492	0.400764	0.200382	−	0.979718i	$-0.435782\pi$
$919$	12.1492	0.400764	0.200382	+	0.979718i	$0.435782\pi$
$920$	0	0
$921$	16.1422	0.531903
$922$	0	0
$923$	−6.54993	−0.215593
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−8.95864	−0.294240
$928$	0	0
$929$	35.2346	1.15601	0.578005	−	0.816033i	$-0.303831\pi$
$929$	35.2346	1.15601	0.578005	+	0.816033i	$0.303831\pi$
$930$	0	0
$931$	35.2804	1.15627
$932$	0	0
$933$	28.7381	0.940844
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−6.81965	−0.222788	−0.111394	−	0.993776i	$-0.535532\pi$
$937$	−6.81965	−0.222788	−0.111394	+	0.993776i	$0.535532\pi$
$938$	0	0
$939$	26.4864	0.864351
$940$	0	0
$941$	31.2034	1.01720	0.508601	−	0.861002i	$-0.330163\pi$
$941$	31.2034	1.01720	0.508601	+	0.861002i	$0.330163\pi$
$942$	0	0
$943$	12.6346	0.411439
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.3989	−0.922842	−0.461421	−	0.887181i	$-0.652660\pi$
$947$	−28.3989	−0.922842	−0.461421	+	0.887181i	$0.652660\pi$
$948$	0	0
$949$	0.370121	0.0120146
$950$	0	0
$951$	26.2980	0.852772
$952$	0	0
$953$	11.7015	0.379049	0.189525	−	0.981876i	$-0.439305\pi$
$953$	11.7015	0.379049	0.189525	+	0.981876i	$0.439305\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	43.4890	1.40580
$958$	0	0
$959$	13.5855	0.438700
$960$	0	0
$961$	59.7092	1.92610
$962$	0	0
$963$	−2.69495	−0.0868435
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−16.9318	−0.544490	−0.272245	−	0.962228i	$-0.587766\pi$
$967$	−16.9318	−0.544490	−0.272245	+	0.962228i	$0.587766\pi$
$968$	0	0
$969$	28.0351	0.900618
$970$	0	0
$971$	−12.3229	−0.395460	−0.197730	−	0.980257i	$-0.563357\pi$
$971$	−12.3229	−0.395460	−0.197730	+	0.980257i	$0.563357\pi$
$972$	0	0
$973$	4.18679	0.134222
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.15644	0.132976	0.0664882	−	0.997787i	$-0.478821\pi$
$977$	4.15644	0.132976	0.0664882	+	0.997787i	$0.478821\pi$
$978$	0	0
$979$	−15.7628	−0.503782
$980$	0	0
$981$	10.2999	0.328850
$982$	0	0
$983$	−13.6896	−0.436632	−0.218316	−	0.975878i	$-0.570056\pi$
$983$	−13.6896	−0.436632	−0.218316	+	0.975878i	$0.570056\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.07358	−0.161494
$988$	0	0
$989$	80.0993	2.54701
$990$	0	0
$991$	−7.65679	−0.243226	−0.121613	−	0.992578i	$-0.538807\pi$
$991$	−7.65679	−0.243226	−0.121613	+	0.992578i	$0.538807\pi$
$992$	0	0
$993$	−31.2634	−0.992113
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−51.5223	−1.63173	−0.815864	−	0.578244i	$-0.803738\pi$
$997$	−51.5223	−1.63173	−0.815864	+	0.578244i	$0.803738\pi$
$998$	0	0
$999$	7.69791	0.243551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.a.n.1.7		12
5.2	odd	4		7500.2.d.g.1249.7		24
5.3	odd	4		7500.2.d.g.1249.18		24
5.4	even	2		7500.2.a.m.1.6		12
25.3	odd	20		300.2.o.a.109.3	✓	24
25.4	even	10		1500.2.m.d.1201.3		24
25.6	even	5		1500.2.m.c.301.4		24
25.8	odd	20		1500.2.o.c.949.4		24
25.17	odd	20		300.2.o.a.289.3	yes	24
25.19	even	10		1500.2.m.d.301.3		24
25.21	even	5		1500.2.m.c.1201.4		24
25.22	odd	20		1500.2.o.c.49.4		24
75.17	even	20		900.2.w.c.289.2		24
75.53	even	20		900.2.w.c.109.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.109.3	✓	24	25.3	odd	20
300.2.o.a.289.3	yes	24	25.17	odd	20
900.2.w.c.109.2		24	75.53	even	20
900.2.w.c.289.2		24	75.17	even	20
1500.2.m.c.301.4		24	25.6	even	5
1500.2.m.c.1201.4		24	25.21	even	5
1500.2.m.d.301.3		24	25.19	even	10
1500.2.m.d.1201.3		24	25.4	even	10
1500.2.o.c.49.4		24	25.22	odd	20
1500.2.o.c.949.4		24	25.8	odd	20
7500.2.a.m.1.6		12	5.4	even	2
7500.2.a.n.1.7		12	1.1	even	1	trivial
7500.2.d.g.1249.7		24	5.2	odd	4
7500.2.d.g.1249.18		24	5.3	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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.04684 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.04684 q^{7} +1.00000 q^{9} +6.28891 q^{11} -1.00629 q^{13} -4.69165 q^{17} -5.97554 q^{19} +1.04684 q^{21} +8.05101 q^{23} +1.00000 q^{27} +6.91519 q^{29} -9.52414 q^{31} +6.28891 q^{33} +7.69791 q^{37} -1.00629 q^{39} +1.56932 q^{41} +9.94897 q^{43} -4.84659 q^{47} -5.90413 q^{49} -4.69165 q^{51} +2.82635 q^{53} -5.97554 q^{57} +4.08209 q^{59} +3.16053 q^{61} +1.04684 q^{63} +3.17036 q^{67} +8.05101 q^{69} +6.50896 q^{71} -0.367806 q^{73} +6.58345 q^{77} -3.32009 q^{79} +1.00000 q^{81} +13.8579 q^{83} +6.91519 q^{87} -2.50645 q^{89} -1.05342 q^{91} -9.52414 q^{93} +0.182689 q^{97} +6.28891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9	\( 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9} + 2 q^{11} + 8 q^{17} + 10 q^{19} + 8 q^{21} + 18 q^{23} + 12 q^{27} + 8 q^{29} - 2 q^{31} + 2 q^{33} + 4 q^{37} + 10 q^{41} + 28 q^{43} + 22 q^{47} + 28 q^{49} + 8 q^{51} + 16 q^{53} + 10 q^{57} - 2 q^{59} + 34 q^{61} + 8 q^{63} + 32 q^{67} + 18 q^{69} + 24 q^{73} + 18 q^{77} + 6 q^{79} + 12 q^{81} + 28 q^{83} + 8 q^{87} + 10 q^{89} + 20 q^{91} - 2 q^{93} + 16 q^{97} + 2 q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 8 * q^7 + 12 * q^9 + 2 * q^11 + 8 * q^17 + 10 * q^19 + 8 * q^21 + 18 * q^23 + 12 * q^27 + 8 * q^29 - 2 * q^31 + 2 * q^33 + 4 * q^37 + 10 * q^41 + 28 * q^43 + 22 * q^47 + 28 * q^49 + 8 * q^51 + 16 * q^53 + 10 * q^57 - 2 * q^59 + 34 * q^61 + 8 * q^63 + 32 * q^67 + 18 * q^69 + 24 * q^73 + 18 * q^77 + 6 * q^79 + 12 * q^81 + 28 * q^83 + 8 * q^87 + 10 * q^89 + 20 * q^91 - 2 * q^93 + 16 * q^97 + 2 * q^99

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