LMFDB - Embedded newform 9450.2.a.eh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9450,2,Mod(1,9450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9450, 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("9450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9450.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:	$75.4586299101$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.16228$ of defining polynomial
Character	$\chi$	$=$	9450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{11} +4.16228 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.32456 q^{17} +3.00000 q^{19} +2.00000 q^{22} -1.16228 q^{23} -4.16228 q^{26} +1.00000 q^{28} -1.83772 q^{29} -6.32456 q^{31} -1.00000 q^{32} +7.32456 q^{34} +7.48683 q^{37} -3.00000 q^{38} -4.00000 q^{41} +3.16228 q^{43} -2.00000 q^{44} +1.16228 q^{46} +10.4868 q^{47} +1.00000 q^{49} +4.16228 q^{52} +0.162278 q^{53} -1.00000 q^{56} +1.83772 q^{58} +0.324555 q^{59} -3.83772 q^{61} +6.32456 q^{62} +1.00000 q^{64} -3.48683 q^{67} -7.32456 q^{68} +10.3246 q^{71} -4.32456 q^{73} -7.48683 q^{74} +3.00000 q^{76} -2.00000 q^{77} -14.4868 q^{79} +4.00000 q^{82} -3.16228 q^{83} -3.16228 q^{86} +2.00000 q^{88} -5.00000 q^{89} +4.16228 q^{91} -1.16228 q^{92} -10.4868 q^{94} -11.4868 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 4 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} + 6 q^{19} + 4 q^{22} + 4 q^{23} - 2 q^{26} + 2 q^{28} - 10 q^{29} - 2 q^{32} + 2 q^{34} - 4 q^{37} - 6 q^{38} - 8 q^{41} - 4 q^{44} - 4 q^{46} + 2 q^{47} + 2 q^{49} + 2 q^{52} - 6 q^{53} - 2 q^{56} + 10 q^{58} - 12 q^{59} - 14 q^{61} + 2 q^{64} + 12 q^{67} - 2 q^{68} + 8 q^{71} + 4 q^{73} + 4 q^{74} + 6 q^{76} - 4 q^{77} - 10 q^{79} + 8 q^{82} + 4 q^{88} - 10 q^{89} + 2 q^{91} + 4 q^{92} - 2 q^{94} - 4 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 4 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 + 6 * q^19 + 4 * q^22 + 4 * q^23 - 2 * q^26 + 2 * q^28 - 10 * q^29 - 2 * q^32 + 2 * q^34 - 4 * q^37 - 6 * q^38 - 8 * q^41 - 4 * q^44 - 4 * q^46 + 2 * q^47 + 2 * q^49 + 2 * q^52 - 6 * q^53 - 2 * q^56 + 10 * q^58 - 12 * q^59 - 14 * q^61 + 2 * q^64 + 12 * q^67 - 2 * q^68 + 8 * q^71 + 4 * q^73 + 4 * q^74 + 6 * q^76 - 4 * q^77 - 10 * q^79 + 8 * q^82 + 4 * q^88 - 10 * q^89 + 2 * q^91 + 4 * q^92 - 2 * q^94 - 4 * q^97 - 2 * 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
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	4.16228	1.15441	0.577204	−	0.816600i	$-0.304144\pi$
$13$	4.16228	1.15441	0.577204	+	0.816600i	$0.304144\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.32456	−1.77647	−0.888233	−	0.459394i	$-0.848067\pi$
$17$	−7.32456	−1.77647	−0.888233	+	0.459394i	$0.848067\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	−1.16228	−0.242352	−0.121176	−	0.992631i	$-0.538666\pi$
$23$	−1.16228	−0.242352	−0.121176	+	0.992631i	$0.538666\pi$
$24$	0	0
$25$	0	0
$26$	−4.16228	−0.816290
$27$	0	0
$28$	1.00000	0.188982
$29$	−1.83772	−0.341256	−0.170628	−	0.985335i	$-0.554580\pi$
$29$	−1.83772	−0.341256	−0.170628	+	0.985335i	$0.554580\pi$
$30$	0	0
$31$	−6.32456	−1.13592	−0.567962	−	0.823055i	$-0.692268\pi$
$31$	−6.32456	−1.13592	−0.567962	+	0.823055i	$0.692268\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.32456	1.25615
$35$	0	0
$36$	0	0
$37$	7.48683	1.23083	0.615414	−	0.788204i	$-0.288989\pi$
$37$	7.48683	1.23083	0.615414	+	0.788204i	$0.288989\pi$
$38$	−3.00000	−0.486664
$39$	0	0
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	3.16228	0.482243	0.241121	−	0.970495i	$-0.422485\pi$
$43$	3.16228	0.482243	0.241121	+	0.970495i	$0.422485\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	1.16228	0.171368
$47$	10.4868	1.52966	0.764831	−	0.644231i	$-0.222822\pi$
$47$	10.4868	1.52966	0.764831	+	0.644231i	$0.222822\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	4.16228	0.577204
$53$	0.162278	0.0222906	0.0111453	−	0.999938i	$-0.496452\pi$
$53$	0.162278	0.0222906	0.0111453	+	0.999938i	$0.496452\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	1.83772	0.241305
$59$	0.324555	0.0422535	0.0211268	−	0.999777i	$-0.493275\pi$
$59$	0.324555	0.0422535	0.0211268	+	0.999777i	$0.493275\pi$
$60$	0	0
$61$	−3.83772	−0.491370	−0.245685	−	0.969350i	$-0.579013\pi$
$61$	−3.83772	−0.491370	−0.245685	+	0.969350i	$0.579013\pi$
$62$	6.32456	0.803219
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−3.48683	−0.425984	−0.212992	−	0.977054i	$-0.568321\pi$
$67$	−3.48683	−0.425984	−0.212992	+	0.977054i	$0.568321\pi$
$68$	−7.32456	−0.888233
$69$	0	0
$70$	0	0
$71$	10.3246	1.22530	0.612650	−	0.790355i	$-0.290104\pi$
$71$	10.3246	1.22530	0.612650	+	0.790355i	$0.290104\pi$
$72$	0	0
$73$	−4.32456	−0.506151	−0.253075	−	0.967447i	$-0.581442\pi$
$73$	−4.32456	−0.506151	−0.253075	+	0.967447i	$0.581442\pi$
$74$	−7.48683	−0.870327
$75$	0	0
$76$	3.00000	0.344124
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−14.4868	−1.62990	−0.814948	−	0.579534i	$-0.803235\pi$
$79$	−14.4868	−1.62990	−0.814948	+	0.579534i	$0.803235\pi$
$80$	0	0
$81$	0	0
$82$	4.00000	0.441726
$83$	−3.16228	−0.347105	−0.173553	−	0.984825i	$-0.555525\pi$
$83$	−3.16228	−0.347105	−0.173553	+	0.984825i	$0.555525\pi$
$84$	0	0
$85$	0	0
$86$	−3.16228	−0.340997
$87$	0	0
$88$	2.00000	0.213201
$89$	−5.00000	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$89$	−5.00000	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$90$	0	0
$91$	4.16228	0.436325
$92$	−1.16228	−0.121176
$93$	0	0
$94$	−10.4868	−1.08163
$95$	0	0
$96$	0	0
$97$	−11.4868	−1.16631	−0.583156	−	0.812360i	$-0.698182\pi$
$97$	−11.4868	−1.16631	−0.583156	+	0.812360i	$0.698182\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	0	0
$101$	−15.4868	−1.54100	−0.770499	−	0.637442i	$-0.779993\pi$
$101$	−15.4868	−1.54100	−0.770499	+	0.637442i	$0.779993\pi$
$102$	0	0
$103$	−6.32456	−0.623177	−0.311588	−	0.950217i	$-0.600861\pi$
$103$	−6.32456	−0.623177	−0.311588	+	0.950217i	$0.600861\pi$
$104$	−4.16228	−0.408145
$105$	0	0
$106$	−0.162278	−0.0157618
$107$	−8.32456	−0.804765	−0.402383	−	0.915472i	$-0.631818\pi$
$107$	−8.32456	−0.804765	−0.402383	+	0.915472i	$0.631818\pi$
$108$	0	0
$109$	15.4868	1.48337	0.741685	−	0.670749i	$-0.234027\pi$
$109$	15.4868	1.48337	0.741685	+	0.670749i	$0.234027\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	14.6491	1.37807	0.689036	−	0.724727i	$-0.258034\pi$
$113$	14.6491	1.37807	0.689036	+	0.724727i	$0.258034\pi$
$114$	0	0
$115$	0	0
$116$	−1.83772	−0.170628
$117$	0	0
$118$	−0.324555	−0.0298777
$119$	−7.32456	−0.671441
$120$	0	0
$121$	−7.00000	−0.636364
$122$	3.83772	0.347451
$123$	0	0
$124$	−6.32456	−0.567962
$125$	0	0
$126$	0	0
$127$	−4.48683	−0.398142	−0.199071	−	0.979985i	$-0.563792\pi$
$127$	−4.48683	−0.398142	−0.199071	+	0.979985i	$0.563792\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	5.48683	0.479387	0.239693	−	0.970849i	$-0.422953\pi$
$131$	5.48683	0.479387	0.239693	+	0.970849i	$0.422953\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	3.48683	0.301216
$135$	0	0
$136$	7.32456	0.628075
$137$	2.51317	0.214714	0.107357	−	0.994221i	$-0.465761\pi$
$137$	2.51317	0.214714	0.107357	+	0.994221i	$0.465761\pi$
$138$	0	0
$139$	12.3246	1.04536	0.522678	−	0.852530i	$-0.324933\pi$
$139$	12.3246	1.04536	0.522678	+	0.852530i	$0.324933\pi$
$140$	0	0
$141$	0	0
$142$	−10.3246	−0.866417
$143$	−8.32456	−0.696134
$144$	0	0
$145$	0	0
$146$	4.32456	0.357903
$147$	0	0
$148$	7.48683	0.615414
$149$	−18.4868	−1.51450	−0.757250	−	0.653125i	$-0.773458\pi$
$149$	−18.4868	−1.51450	−0.757250	+	0.653125i	$0.773458\pi$
$150$	0	0
$151$	−8.16228	−0.664237	−0.332118	−	0.943238i	$-0.607763\pi$
$151$	−8.16228	−0.664237	−0.332118	+	0.943238i	$0.607763\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	2.00000	0.161165
$155$	0	0
$156$	0	0
$157$	24.1623	1.92836	0.964180	−	0.265249i	$-0.0854543\pi$
$157$	24.1623	1.92836	0.964180	+	0.265249i	$0.0854543\pi$
$158$	14.4868	1.15251
$159$	0	0
$160$	0	0
$161$	−1.16228	−0.0916003
$162$	0	0
$163$	0.324555	0.0254211	0.0127106	−	0.999919i	$-0.495954\pi$
$163$	0.324555	0.0254211	0.0127106	+	0.999919i	$0.495954\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	3.16228	0.245440
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	4.32456	0.332658
$170$	0	0
$171$	0	0
$172$	3.16228	0.241121
$173$	−12.8377	−0.976034	−0.488017	−	0.872834i	$-0.662280\pi$
$173$	−12.8377	−0.976034	−0.488017	+	0.872834i	$0.662280\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	5.00000	0.374766
$179$	−12.6754	−0.947407	−0.473704	−	0.880684i	$-0.657083\pi$
$179$	−12.6754	−0.947407	−0.473704	+	0.880684i	$0.657083\pi$
$180$	0	0
$181$	−9.83772	−0.731232	−0.365616	−	0.930766i	$-0.619142\pi$
$181$	−9.83772	−0.731232	−0.365616	+	0.930766i	$0.619142\pi$
$182$	−4.16228	−0.308529
$183$	0	0
$184$	1.16228	0.0856842
$185$	0	0
$186$	0	0
$187$	14.6491	1.07125
$188$	10.4868	0.764831
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−14.6754	−1.05636	−0.528181	−	0.849132i	$-0.677126\pi$
$193$	−14.6754	−1.05636	−0.528181	+	0.849132i	$0.677126\pi$
$194$	11.4868	0.824707
$195$	0	0
$196$	1.00000	0.0714286
$197$	12.8114	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$197$	12.8114	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$198$	0	0
$199$	−24.1359	−1.71095	−0.855476	−	0.517843i	$-0.826735\pi$
$199$	−24.1359	−1.71095	−0.855476	+	0.517843i	$0.826735\pi$
$200$	0	0
$201$	0	0
$202$	15.4868	1.08965
$203$	−1.83772	−0.128983
$204$	0	0
$205$	0	0
$206$	6.32456	0.440653
$207$	0	0
$208$	4.16228	0.288602
$209$	−6.00000	−0.415029
$210$	0	0
$211$	0.324555	0.0223433	0.0111717	−	0.999938i	$-0.496444\pi$
$211$	0.324555	0.0223433	0.0111717	+	0.999938i	$0.496444\pi$
$212$	0.162278	0.0111453
$213$	0	0
$214$	8.32456	0.569055
$215$	0	0
$216$	0	0
$217$	−6.32456	−0.429339
$218$	−15.4868	−1.04890
$219$	0	0
$220$	0	0
$221$	−30.4868	−2.05077
$222$	0	0
$223$	0.837722	0.0560980	0.0280490	−	0.999607i	$-0.491071\pi$
$223$	0.837722	0.0560980	0.0280490	+	0.999607i	$0.491071\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−14.6491	−0.974444
$227$	9.81139	0.651205	0.325602	−	0.945507i	$-0.394433\pi$
$227$	9.81139	0.651205	0.325602	+	0.945507i	$0.394433\pi$
$228$	0	0
$229$	17.8377	1.17875	0.589375	−	0.807860i	$-0.299374\pi$
$229$	17.8377	1.17875	0.589375	+	0.807860i	$0.299374\pi$
$230$	0	0
$231$	0	0
$232$	1.83772	0.120652
$233$	15.4868	1.01458	0.507288	−	0.861777i	$-0.330648\pi$
$233$	15.4868	1.01458	0.507288	+	0.861777i	$0.330648\pi$
$234$	0	0
$235$	0	0
$236$	0.324555	0.0211268
$237$	0	0
$238$	7.32456	0.474780
$239$	−25.8114	−1.66960	−0.834800	−	0.550553i	$-0.814417\pi$
$239$	−25.8114	−1.66960	−0.834800	+	0.550553i	$0.814417\pi$
$240$	0	0
$241$	−23.4868	−1.51292	−0.756460	−	0.654040i	$-0.773073\pi$
$241$	−23.4868	−1.51292	−0.756460	+	0.654040i	$0.773073\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−3.83772	−0.245685
$245$	0	0
$246$	0	0
$247$	12.4868	0.794518
$248$	6.32456	0.401610
$249$	0	0
$250$	0	0
$251$	3.67544	0.231992	0.115996	−	0.993250i	$-0.462994\pi$
$251$	3.67544	0.231992	0.115996	+	0.993250i	$0.462994\pi$
$252$	0	0
$253$	2.32456	0.146144
$254$	4.48683	0.281529
$255$	0	0
$256$	1.00000	0.0625000
$257$	19.0000	1.18519	0.592594	−	0.805502i	$-0.298104\pi$
$257$	19.0000	1.18519	0.592594	+	0.805502i	$0.298104\pi$
$258$	0	0
$259$	7.48683	0.465209
$260$	0	0
$261$	0	0
$262$	−5.48683	−0.338978
$263$	−16.3246	−1.00662	−0.503308	−	0.864107i	$-0.667884\pi$
$263$	−16.3246	−1.00662	−0.503308	+	0.864107i	$0.667884\pi$
$264$	0	0
$265$	0	0
$266$	−3.00000	−0.183942
$267$	0	0
$268$	−3.48683	−0.212992
$269$	−22.6491	−1.38094	−0.690470	−	0.723361i	$-0.742596\pi$
$269$	−22.6491	−1.38094	−0.690470	+	0.723361i	$0.742596\pi$
$270$	0	0
$271$	−2.32456	−0.141207	−0.0706033	−	0.997504i	$-0.522492\pi$
$271$	−2.32456	−0.141207	−0.0706033	+	0.997504i	$0.522492\pi$
$272$	−7.32456	−0.444116
$273$	0	0
$274$	−2.51317	−0.151826
$275$	0	0
$276$	0	0
$277$	8.32456	0.500174	0.250087	−	0.968223i	$-0.419541\pi$
$277$	8.32456	0.500174	0.250087	+	0.968223i	$0.419541\pi$
$278$	−12.3246	−0.739178
$279$	0	0
$280$	0	0
$281$	−8.83772	−0.527214	−0.263607	−	0.964630i	$-0.584912\pi$
$281$	−8.83772	−0.527214	−0.263607	+	0.964630i	$0.584912\pi$
$282$	0	0
$283$	−21.6491	−1.28691	−0.643453	−	0.765486i	$-0.722499\pi$
$283$	−21.6491	−1.28691	−0.643453	+	0.765486i	$0.722499\pi$
$284$	10.3246	0.612650
$285$	0	0
$286$	8.32456	0.492241
$287$	−4.00000	−0.236113
$288$	0	0
$289$	36.6491	2.15583
$290$	0	0
$291$	0	0
$292$	−4.32456	−0.253075
$293$	−24.1359	−1.41004	−0.705018	−	0.709189i	$-0.749061\pi$
$293$	−24.1359	−1.41004	−0.705018	+	0.709189i	$0.749061\pi$
$294$	0	0
$295$	0	0
$296$	−7.48683	−0.435163
$297$	0	0
$298$	18.4868	1.07091
$299$	−4.83772	−0.279773
$300$	0	0
$301$	3.16228	0.182271
$302$	8.16228	0.469686
$303$	0	0
$304$	3.00000	0.172062
$305$	0	0
$306$	0	0
$307$	6.64911	0.379485	0.189742	−	0.981834i	$-0.439235\pi$
$307$	6.64911	0.379485	0.189742	+	0.981834i	$0.439235\pi$
$308$	−2.00000	−0.113961
$309$	0	0
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	5.16228	0.291789	0.145895	−	0.989300i	$-0.453394\pi$
$313$	5.16228	0.291789	0.145895	+	0.989300i	$0.453394\pi$
$314$	−24.1623	−1.36356
$315$	0	0
$316$	−14.4868	−0.814948
$317$	21.8377	1.22653	0.613264	−	0.789878i	$-0.289856\pi$
$317$	21.8377	1.22653	0.613264	+	0.789878i	$0.289856\pi$
$318$	0	0
$319$	3.67544	0.205785
$320$	0	0
$321$	0	0
$322$	1.16228	0.0647712
$323$	−21.9737	−1.22265
$324$	0	0
$325$	0	0
$326$	−0.324555	−0.0179755
$327$	0	0
$328$	4.00000	0.220863
$329$	10.4868	0.578158
$330$	0	0
$331$	−2.18861	−0.120297	−0.0601485	−	0.998189i	$-0.519157\pi$
$331$	−2.18861	−0.120297	−0.0601485	+	0.998189i	$0.519157\pi$
$332$	−3.16228	−0.173553
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	13.3246	0.725835	0.362917	−	0.931821i	$-0.381781\pi$
$337$	13.3246	0.725835	0.362917	+	0.931821i	$0.381781\pi$
$338$	−4.32456	−0.235225
$339$	0	0
$340$	0	0
$341$	12.6491	0.684988
$342$	0	0
$343$	1.00000	0.0539949
$344$	−3.16228	−0.170499
$345$	0	0
$346$	12.8377	0.690160
$347$	23.0000	1.23470	0.617352	−	0.786687i	$-0.288205\pi$
$347$	23.0000	1.23470	0.617352	+	0.786687i	$0.288205\pi$
$348$	0	0
$349$	−10.8114	−0.578720	−0.289360	−	0.957220i	$-0.593443\pi$
$349$	−10.8114	−0.578720	−0.289360	+	0.957220i	$0.593443\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	0	0
$356$	−5.00000	−0.264999
$357$	0	0
$358$	12.6754	0.669918
$359$	−22.0000	−1.16112	−0.580558	−	0.814219i	$-0.697165\pi$
$359$	−22.0000	−1.16112	−0.580558	+	0.814219i	$0.697165\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	9.83772	0.517059
$363$	0	0
$364$	4.16228	0.218163
$365$	0	0
$366$	0	0
$367$	30.4605	1.59003	0.795013	−	0.606593i	$-0.207464\pi$
$367$	30.4605	1.59003	0.795013	+	0.606593i	$0.207464\pi$
$368$	−1.16228	−0.0605879
$369$	0	0
$370$	0	0
$371$	0.162278	0.00842504
$372$	0	0
$373$	−4.32456	−0.223917	−0.111958	−	0.993713i	$-0.535712\pi$
$373$	−4.32456	−0.223917	−0.111958	+	0.993713i	$0.535712\pi$
$374$	−14.6491	−0.757487
$375$	0	0
$376$	−10.4868	−0.540817
$377$	−7.64911	−0.393949
$378$	0	0
$379$	−4.83772	−0.248497	−0.124249	−	0.992251i	$-0.539652\pi$
$379$	−4.83772	−0.248497	−0.124249	+	0.992251i	$0.539652\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−26.3246	−1.34512	−0.672561	−	0.740042i	$-0.734806\pi$
$383$	−26.3246	−1.34512	−0.672561	+	0.740042i	$0.734806\pi$
$384$	0	0
$385$	0	0
$386$	14.6754	0.746960
$387$	0	0
$388$	−11.4868	−0.583156
$389$	26.4868	1.34294	0.671468	−	0.741034i	$-0.265664\pi$
$389$	26.4868	1.34294	0.671468	+	0.741034i	$0.265664\pi$
$390$	0	0
$391$	8.51317	0.430529
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−12.8114	−0.645428
$395$	0	0
$396$	0	0
$397$	−18.8114	−0.944117	−0.472058	−	0.881567i	$-0.656489\pi$
$397$	−18.8114	−0.944117	−0.472058	+	0.881567i	$0.656489\pi$
$398$	24.1359	1.20983
$399$	0	0
$400$	0	0
$401$	−9.16228	−0.457542	−0.228771	−	0.973480i	$-0.573471\pi$
$401$	−9.16228	−0.457542	−0.228771	+	0.973480i	$0.573471\pi$
$402$	0	0
$403$	−26.3246	−1.31132
$404$	−15.4868	−0.770499
$405$	0	0
$406$	1.83772	0.0912046
$407$	−14.9737	−0.742217
$408$	0	0
$409$	20.1359	0.995658	0.497829	−	0.867275i	$-0.334131\pi$
$409$	20.1359	0.995658	0.497829	+	0.867275i	$0.334131\pi$
$410$	0	0
$411$	0	0
$412$	−6.32456	−0.311588
$413$	0.324555	0.0159703
$414$	0	0
$415$	0	0
$416$	−4.16228	−0.204072
$417$	0	0
$418$	6.00000	0.293470
$419$	20.1359	0.983705	0.491853	−	0.870678i	$-0.336320\pi$
$419$	20.1359	0.983705	0.491853	+	0.870678i	$0.336320\pi$
$420$	0	0
$421$	−5.35089	−0.260786	−0.130393	−	0.991462i	$-0.541624\pi$
$421$	−5.35089	−0.260786	−0.130393	+	0.991462i	$0.541624\pi$
$422$	−0.324555	−0.0157991
$423$	0	0
$424$	−0.162278	−0.00788090
$425$	0	0
$426$	0	0
$427$	−3.83772	−0.185720
$428$	−8.32456	−0.402383
$429$	0	0
$430$	0	0
$431$	−30.9737	−1.49195	−0.745974	−	0.665975i	$-0.768016\pi$
$431$	−30.9737	−1.49195	−0.745974	+	0.665975i	$0.768016\pi$
$432$	0	0
$433$	30.3246	1.45731	0.728653	−	0.684884i	$-0.240147\pi$
$433$	30.3246	1.45731	0.728653	+	0.684884i	$0.240147\pi$
$434$	6.32456	0.303588
$435$	0	0
$436$	15.4868	0.741685
$437$	−3.48683	−0.166798
$438$	0	0
$439$	33.8114	1.61373	0.806865	−	0.590736i	$-0.201163\pi$
$439$	33.8114	1.61373	0.806865	+	0.590736i	$0.201163\pi$
$440$	0	0
$441$	0	0
$442$	30.4868	1.45011
$443$	28.6228	1.35991	0.679955	−	0.733254i	$-0.261999\pi$
$443$	28.6228	1.35991	0.679955	+	0.733254i	$0.261999\pi$
$444$	0	0
$445$	0	0
$446$	−0.837722	−0.0396673
$447$	0	0
$448$	1.00000	0.0472456
$449$	−25.1623	−1.18748	−0.593741	−	0.804656i	$-0.702349\pi$
$449$	−25.1623	−1.18748	−0.593741	+	0.804656i	$0.702349\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	14.6491	0.689036
$453$	0	0
$454$	−9.81139	−0.460471
$455$	0	0
$456$	0	0
$457$	−30.6228	−1.43247	−0.716237	−	0.697858i	$-0.754137\pi$
$457$	−30.6228	−1.43247	−0.716237	+	0.697858i	$0.754137\pi$
$458$	−17.8377	−0.833502
$459$	0	0
$460$	0	0
$461$	5.16228	0.240431	0.120216	−	0.992748i	$-0.461641\pi$
$461$	5.16228	0.240431	0.120216	+	0.992748i	$0.461641\pi$
$462$	0	0
$463$	−20.4868	−0.952104	−0.476052	−	0.879417i	$-0.657933\pi$
$463$	−20.4868	−0.952104	−0.476052	+	0.879417i	$0.657933\pi$
$464$	−1.83772	−0.0853141
$465$	0	0
$466$	−15.4868	−0.717414
$467$	−10.3246	−0.477763	−0.238882	−	0.971049i	$-0.576781\pi$
$467$	−10.3246	−0.477763	−0.238882	+	0.971049i	$0.576781\pi$
$468$	0	0
$469$	−3.48683	−0.161007
$470$	0	0
$471$	0	0
$472$	−0.324555	−0.0149389
$473$	−6.32456	−0.290803
$474$	0	0
$475$	0	0
$476$	−7.32456	−0.335720
$477$	0	0
$478$	25.8114	1.18059
$479$	4.16228	0.190179	0.0950897	−	0.995469i	$-0.469686\pi$
$479$	4.16228	0.190179	0.0950897	+	0.995469i	$0.469686\pi$
$480$	0	0
$481$	31.1623	1.42088
$482$	23.4868	1.06980
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	0	0
$487$	−26.9737	−1.22229	−0.611147	−	0.791517i	$-0.709291\pi$
$487$	−26.9737	−1.22229	−0.611147	+	0.791517i	$0.709291\pi$
$488$	3.83772	0.173726
$489$	0	0
$490$	0	0
$491$	17.3246	0.781846	0.390923	−	0.920423i	$-0.372156\pi$
$491$	17.3246	0.781846	0.390923	+	0.920423i	$0.372156\pi$
$492$	0	0
$493$	13.4605	0.606230
$494$	−12.4868	−0.561809
$495$	0	0
$496$	−6.32456	−0.283981
$497$	10.3246	0.463120
$498$	0	0
$499$	4.83772	0.216566	0.108283	−	0.994120i	$-0.465465\pi$
$499$	4.83772	0.216566	0.108283	+	0.994120i	$0.465465\pi$
$500$	0	0
$501$	0	0
$502$	−3.67544	−0.164043
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	0	0
$505$	0	0
$506$	−2.32456	−0.103339
$507$	0	0
$508$	−4.48683	−0.199071
$509$	19.4868	0.863739	0.431869	−	0.901936i	$-0.357854\pi$
$509$	19.4868	0.863739	0.431869	+	0.901936i	$0.357854\pi$
$510$	0	0
$511$	−4.32456	−0.191307
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−19.0000	−0.838054
$515$	0	0
$516$	0	0
$517$	−20.9737	−0.922421
$518$	−7.48683	−0.328953
$519$	0	0
$520$	0	0
$521$	31.3246	1.37235	0.686177	−	0.727435i	$-0.259288\pi$
$521$	31.3246	1.37235	0.686177	+	0.727435i	$0.259288\pi$
$522$	0	0
$523$	−27.3246	−1.19482	−0.597410	−	0.801936i	$-0.703803\pi$
$523$	−27.3246	−1.19482	−0.597410	+	0.801936i	$0.703803\pi$
$524$	5.48683	0.239693
$525$	0	0
$526$	16.3246	0.711784
$527$	46.3246	2.01793
$528$	0	0
$529$	−21.6491	−0.941266
$530$	0	0
$531$	0	0
$532$	3.00000	0.130066
$533$	−16.6491	−0.721153
$534$	0	0
$535$	0	0
$536$	3.48683	0.150608
$537$	0	0
$538$	22.6491	0.976472
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−17.8114	−0.765771	−0.382886	−	0.923796i	$-0.625070\pi$
$541$	−17.8114	−0.765771	−0.382886	+	0.923796i	$0.625070\pi$
$542$	2.32456	0.0998482
$543$	0	0
$544$	7.32456	0.314038
$545$	0	0
$546$	0	0
$547$	−33.2982	−1.42373	−0.711865	−	0.702317i	$-0.752149\pi$
$547$	−33.2982	−1.42373	−0.711865	+	0.702317i	$0.752149\pi$
$548$	2.51317	0.107357
$549$	0	0
$550$	0	0
$551$	−5.51317	−0.234869
$552$	0	0
$553$	−14.4868	−0.616043
$554$	−8.32456	−0.353676
$555$	0	0
$556$	12.3246	0.522678
$557$	−1.18861	−0.0503631	−0.0251815	−	0.999683i	$-0.508016\pi$
$557$	−1.18861	−0.0503631	−0.0251815	+	0.999683i	$0.508016\pi$
$558$	0	0
$559$	13.1623	0.556705
$560$	0	0
$561$	0	0
$562$	8.83772	0.372797
$563$	14.4605	0.609437	0.304719	−	0.952442i	$-0.401438\pi$
$563$	14.4605	0.609437	0.304719	+	0.952442i	$0.401438\pi$
$564$	0	0
$565$	0	0
$566$	21.6491	0.909980
$567$	0	0
$568$	−10.3246	−0.433209
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	39.2982	1.64458	0.822290	−	0.569069i	$-0.192696\pi$
$571$	39.2982	1.64458	0.822290	+	0.569069i	$0.192696\pi$
$572$	−8.32456	−0.348067
$573$	0	0
$574$	4.00000	0.166957
$575$	0	0
$576$	0	0
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	−36.6491	−1.52440
$579$	0	0
$580$	0	0
$581$	−3.16228	−0.131193
$582$	0	0
$583$	−0.324555	−0.0134417
$584$	4.32456	0.178951
$585$	0	0
$586$	24.1359	0.997047
$587$	−18.8377	−0.777516	−0.388758	−	0.921340i	$-0.627096\pi$
$587$	−18.8377	−0.777516	−0.388758	+	0.921340i	$0.627096\pi$
$588$	0	0
$589$	−18.9737	−0.781796
$590$	0	0
$591$	0	0
$592$	7.48683	0.307707
$593$	−0.675445	−0.0277372	−0.0138686	−	0.999904i	$-0.504415\pi$
$593$	−0.675445	−0.0277372	−0.0138686	+	0.999904i	$0.504415\pi$
$594$	0	0
$595$	0	0
$596$	−18.4868	−0.757250
$597$	0	0
$598$	4.83772	0.197829
$599$	−20.1359	−0.822732	−0.411366	−	0.911470i	$-0.634948\pi$
$599$	−20.1359	−0.822732	−0.411366	+	0.911470i	$0.634948\pi$
$600$	0	0
$601$	−21.6754	−0.884160	−0.442080	−	0.896976i	$-0.645759\pi$
$601$	−21.6754	−0.884160	−0.442080	+	0.896976i	$0.645759\pi$
$602$	−3.16228	−0.128885
$603$	0	0
$604$	−8.16228	−0.332118
$605$	0	0
$606$	0	0
$607$	−20.8377	−0.845777	−0.422889	−	0.906182i	$-0.638984\pi$
$607$	−20.8377	−0.845777	−0.422889	+	0.906182i	$0.638984\pi$
$608$	−3.00000	−0.121666
$609$	0	0
$610$	0	0
$611$	43.6491	1.76585
$612$	0	0
$613$	−11.8114	−0.477057	−0.238529	−	0.971135i	$-0.576665\pi$
$613$	−11.8114	−0.477057	−0.238529	+	0.971135i	$0.576665\pi$
$614$	−6.64911	−0.268336
$615$	0	0
$616$	2.00000	0.0805823
$617$	37.4868	1.50916	0.754582	−	0.656206i	$-0.227840\pi$
$617$	37.4868	1.50916	0.754582	+	0.656206i	$0.227840\pi$
$618$	0	0
$619$	8.97367	0.360682	0.180341	−	0.983604i	$-0.442280\pi$
$619$	8.97367	0.360682	0.180341	+	0.983604i	$0.442280\pi$
$620$	0	0
$621$	0	0
$622$	18.0000	0.721734
$623$	−5.00000	−0.200321
$624$	0	0
$625$	0	0
$626$	−5.16228	−0.206326
$627$	0	0
$628$	24.1623	0.964180
$629$	−54.8377	−2.18652
$630$	0	0
$631$	−23.8377	−0.948965	−0.474482	−	0.880265i	$-0.657365\pi$
$631$	−23.8377	−0.948965	−0.474482	+	0.880265i	$0.657365\pi$
$632$	14.4868	0.576255
$633$	0	0
$634$	−21.8377	−0.867287
$635$	0	0
$636$	0	0
$637$	4.16228	0.164915
$638$	−3.67544	−0.145512
$639$	0	0
$640$	0	0
$641$	10.3246	0.407795	0.203898	−	0.978992i	$-0.434639\pi$
$641$	10.3246	0.407795	0.203898	+	0.978992i	$0.434639\pi$
$642$	0	0
$643$	−2.35089	−0.0927100	−0.0463550	−	0.998925i	$-0.514761\pi$
$643$	−2.35089	−0.0927100	−0.0463550	+	0.998925i	$0.514761\pi$
$644$	−1.16228	−0.0458002
$645$	0	0
$646$	21.9737	0.864542
$647$	1.13594	0.0446586	0.0223293	−	0.999751i	$-0.492892\pi$
$647$	1.13594	0.0446586	0.0223293	+	0.999751i	$0.492892\pi$
$648$	0	0
$649$	−0.649111	−0.0254798
$650$	0	0
$651$	0	0
$652$	0.324555	0.0127106
$653$	21.5132	0.841875	0.420938	−	0.907090i	$-0.361701\pi$
$653$	21.5132	0.841875	0.420938	+	0.907090i	$0.361701\pi$
$654$	0	0
$655$	0	0
$656$	−4.00000	−0.156174
$657$	0	0
$658$	−10.4868	−0.408819
$659$	−6.35089	−0.247396	−0.123698	−	0.992320i	$-0.539475\pi$
$659$	−6.35089	−0.247396	−0.123698	+	0.992320i	$0.539475\pi$
$660$	0	0
$661$	2.64911	0.103038	0.0515192	−	0.998672i	$-0.483594\pi$
$661$	2.64911	0.103038	0.0515192	+	0.998672i	$0.483594\pi$
$662$	2.18861	0.0850628
$663$	0	0
$664$	3.16228	0.122720
$665$	0	0
$666$	0	0
$667$	2.13594	0.0827041
$668$	12.0000	0.464294
$669$	0	0
$670$	0	0
$671$	7.67544	0.296307
$672$	0	0
$673$	−34.9737	−1.34814	−0.674068	−	0.738669i	$-0.735454\pi$
$673$	−34.9737	−1.34814	−0.674068	+	0.738669i	$0.735454\pi$
$674$	−13.3246	−0.513243
$675$	0	0
$676$	4.32456	0.166329
$677$	3.67544	0.141259	0.0706294	−	0.997503i	$-0.477499\pi$
$677$	3.67544	0.141259	0.0706294	+	0.997503i	$0.477499\pi$
$678$	0	0
$679$	−11.4868	−0.440824
$680$	0	0
$681$	0	0
$682$	−12.6491	−0.484359
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	3.16228	0.120561
$689$	0.675445	0.0257324
$690$	0	0
$691$	−23.2982	−0.886306	−0.443153	−	0.896446i	$-0.646140\pi$
$691$	−23.2982	−0.886306	−0.443153	+	0.896446i	$0.646140\pi$
$692$	−12.8377	−0.488017
$693$	0	0
$694$	−23.0000	−0.873068
$695$	0	0
$696$	0	0
$697$	29.2982	1.10975
$698$	10.8114	0.409217
$699$	0	0
$700$	0	0
$701$	−16.3246	−0.616570	−0.308285	−	0.951294i	$-0.599755\pi$
$701$	−16.3246	−0.616570	−0.308285	+	0.951294i	$0.599755\pi$
$702$	0	0
$703$	22.4605	0.847114
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	24.0000	0.903252
$707$	−15.4868	−0.582442
$708$	0	0
$709$	6.32456	0.237524	0.118762	−	0.992923i	$-0.462108\pi$
$709$	6.32456	0.237524	0.118762	+	0.992923i	$0.462108\pi$
$710$	0	0
$711$	0	0
$712$	5.00000	0.187383
$713$	7.35089	0.275293
$714$	0	0
$715$	0	0
$716$	−12.6754	−0.473704
$717$	0	0
$718$	22.0000	0.821033
$719$	1.83772	0.0685355	0.0342677	−	0.999413i	$-0.489090\pi$
$719$	1.83772	0.0685355	0.0342677	+	0.999413i	$0.489090\pi$
$720$	0	0
$721$	−6.32456	−0.235539
$722$	10.0000	0.372161
$723$	0	0
$724$	−9.83772	−0.365616
$725$	0	0
$726$	0	0
$727$	−25.1623	−0.933217	−0.466609	−	0.884464i	$-0.654524\pi$
$727$	−25.1623	−0.933217	−0.466609	+	0.884464i	$0.654524\pi$
$728$	−4.16228	−0.154264
$729$	0	0
$730$	0	0
$731$	−23.1623	−0.856688
$732$	0	0
$733$	25.1359	0.928417	0.464209	−	0.885726i	$-0.346339\pi$
$733$	25.1359	0.928417	0.464209	+	0.885726i	$0.346339\pi$
$734$	−30.4605	−1.12432
$735$	0	0
$736$	1.16228	0.0428421
$737$	6.97367	0.256878
$738$	0	0
$739$	−6.64911	−0.244591	−0.122296	−	0.992494i	$-0.539026\pi$
$739$	−6.64911	−0.244591	−0.122296	+	0.992494i	$0.539026\pi$
$740$	0	0
$741$	0	0
$742$	−0.162278	−0.00595740
$743$	39.2982	1.44171	0.720856	−	0.693085i	$-0.243749\pi$
$743$	39.2982	1.44171	0.720856	+	0.693085i	$0.243749\pi$
$744$	0	0
$745$	0	0
$746$	4.32456	0.158333
$747$	0	0
$748$	14.6491	0.535625
$749$	−8.32456	−0.304173
$750$	0	0
$751$	−29.9473	−1.09279	−0.546397	−	0.837526i	$-0.684001\pi$
$751$	−29.9473	−1.09279	−0.546397	+	0.837526i	$0.684001\pi$
$752$	10.4868	0.382415
$753$	0	0
$754$	7.64911	0.278564
$755$	0	0
$756$	0	0
$757$	53.1096	1.93030	0.965151	−	0.261694i	$-0.0842812\pi$
$757$	53.1096	1.93030	0.965151	+	0.261694i	$0.0842812\pi$
$758$	4.83772	0.175714
$759$	0	0
$760$	0	0
$761$	−40.9473	−1.48434	−0.742170	−	0.670212i	$-0.766203\pi$
$761$	−40.9473	−1.48434	−0.742170	+	0.670212i	$0.766203\pi$
$762$	0	0
$763$	15.4868	0.560661
$764$	0	0
$765$	0	0
$766$	26.3246	0.951145
$767$	1.35089	0.0487778
$768$	0	0
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	0	0
$772$	−14.6754	−0.528181
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	0	0
$775$	0	0
$776$	11.4868	0.412353
$777$	0	0
$778$	−26.4868	−0.949599
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−20.6491	−0.738883
$782$	−8.51317	−0.304430
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	−16.9737	−0.605046	−0.302523	−	0.953142i	$-0.597829\pi$
$787$	−16.9737	−0.605046	−0.302523	+	0.953142i	$0.597829\pi$
$788$	12.8114	0.456387
$789$	0	0
$790$	0	0
$791$	14.6491	0.520862
$792$	0	0
$793$	−15.9737	−0.567242
$794$	18.8114	0.667591
$795$	0	0
$796$	−24.1359	−0.855476
$797$	2.51317	0.0890209	0.0445105	−	0.999009i	$-0.485827\pi$
$797$	2.51317	0.0890209	0.0445105	+	0.999009i	$0.485827\pi$
$798$	0	0
$799$	−76.8114	−2.71739
$800$	0	0
$801$	0	0
$802$	9.16228	0.323531
$803$	8.64911	0.305220
$804$	0	0
$805$	0	0
$806$	26.3246	0.927243
$807$	0	0
$808$	15.4868	0.544825
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	16.3246	0.573233	0.286616	−	0.958045i	$-0.407470\pi$
$811$	16.3246	0.573233	0.286616	+	0.958045i	$0.407470\pi$
$812$	−1.83772	−0.0644914
$813$	0	0
$814$	14.9737	0.524827
$815$	0	0
$816$	0	0
$817$	9.48683	0.331902
$818$	−20.1359	−0.704037
$819$	0	0
$820$	0	0
$821$	43.4605	1.51678	0.758391	−	0.651800i	$-0.225986\pi$
$821$	43.4605	1.51678	0.758391	+	0.651800i	$0.225986\pi$
$822$	0	0
$823$	−0.701779	−0.0244625	−0.0122312	−	0.999925i	$-0.503893\pi$
$823$	−0.701779	−0.0244625	−0.0122312	+	0.999925i	$0.503893\pi$
$824$	6.32456	0.220326
$825$	0	0
$826$	−0.324555	−0.0112927
$827$	−52.6228	−1.82987	−0.914937	−	0.403598i	$-0.867760\pi$
$827$	−52.6228	−1.82987	−0.914937	+	0.403598i	$0.867760\pi$
$828$	0	0
$829$	32.1096	1.11521	0.557606	−	0.830105i	$-0.311720\pi$
$829$	32.1096	1.11521	0.557606	+	0.830105i	$0.311720\pi$
$830$	0	0
$831$	0	0
$832$	4.16228	0.144301
$833$	−7.32456	−0.253781
$834$	0	0
$835$	0	0
$836$	−6.00000	−0.207514
$837$	0	0
$838$	−20.1359	−0.695585
$839$	−49.7851	−1.71877	−0.859385	−	0.511328i	$-0.829154\pi$
$839$	−49.7851	−1.71877	−0.859385	+	0.511328i	$0.829154\pi$
$840$	0	0
$841$	−25.6228	−0.883544
$842$	5.35089	0.184404
$843$	0	0
$844$	0.324555	0.0111717
$845$	0	0
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0.162278	0.00557264
$849$	0	0
$850$	0	0
$851$	−8.70178	−0.298293
$852$	0	0
$853$	−43.6228	−1.49362	−0.746808	−	0.665040i	$-0.768414\pi$
$853$	−43.6228	−1.49362	−0.746808	+	0.665040i	$0.768414\pi$
$854$	3.83772	0.131324
$855$	0	0
$856$	8.32456	0.284527
$857$	−57.5964	−1.96746	−0.983728	−	0.179661i	$-0.942500\pi$
$857$	−57.5964	−1.96746	−0.983728	+	0.179661i	$0.942500\pi$
$858$	0	0
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	0	0
$861$	0	0
$862$	30.9737	1.05497
$863$	8.18861	0.278744	0.139372	−	0.990240i	$-0.455492\pi$
$863$	8.18861	0.278744	0.139372	+	0.990240i	$0.455492\pi$
$864$	0	0
$865$	0	0
$866$	−30.3246	−1.03047
$867$	0	0
$868$	−6.32456	−0.214669
$869$	28.9737	0.982864
$870$	0	0
$871$	−14.5132	−0.491760
$872$	−15.4868	−0.524450
$873$	0	0
$874$	3.48683	0.117944
$875$	0	0
$876$	0	0
$877$	−14.9737	−0.505625	−0.252812	−	0.967515i	$-0.581356\pi$
$877$	−14.9737	−0.505625	−0.252812	+	0.967515i	$0.581356\pi$
$878$	−33.8114	−1.14108
$879$	0	0
$880$	0	0
$881$	10.0263	0.337796	0.168898	−	0.985634i	$-0.445979\pi$
$881$	10.0263	0.337796	0.168898	+	0.985634i	$0.445979\pi$
$882$	0	0
$883$	−54.9737	−1.85001	−0.925006	−	0.379954i	$-0.875940\pi$
$883$	−54.9737	−1.85001	−0.925006	+	0.379954i	$0.875940\pi$
$884$	−30.4868	−1.02538
$885$	0	0
$886$	−28.6228	−0.961601
$887$	30.3246	1.01820	0.509099	−	0.860708i	$-0.329979\pi$
$887$	30.3246	1.01820	0.509099	+	0.860708i	$0.329979\pi$
$888$	0	0
$889$	−4.48683	−0.150484
$890$	0	0
$891$	0	0
$892$	0.837722	0.0280490
$893$	31.4605	1.05279
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	25.1623	0.839676
$899$	11.6228	0.387641
$900$	0	0
$901$	−1.18861	−0.0395984
$902$	−8.00000	−0.266371
$903$	0	0
$904$	−14.6491	−0.487222
$905$	0	0
$906$	0	0
$907$	22.9737	0.762828	0.381414	−	0.924404i	$-0.375437\pi$
$907$	22.9737	0.762828	0.381414	+	0.924404i	$0.375437\pi$
$908$	9.81139	0.325602
$909$	0	0
$910$	0	0
$911$	−45.2982	−1.50080	−0.750399	−	0.660986i	$-0.770138\pi$
$911$	−45.2982	−1.50080	−0.750399	+	0.660986i	$0.770138\pi$
$912$	0	0
$913$	6.32456	0.209312
$914$	30.6228	1.01291
$915$	0	0
$916$	17.8377	0.589375
$917$	5.48683	0.181191
$918$	0	0
$919$	23.2982	0.768537	0.384269	−	0.923221i	$-0.374454\pi$
$919$	23.2982	0.768537	0.384269	+	0.923221i	$0.374454\pi$
$920$	0	0
$921$	0	0
$922$	−5.16228	−0.170011
$923$	42.9737	1.41450
$924$	0	0
$925$	0	0
$926$	20.4868	0.673239
$927$	0	0
$928$	1.83772	0.0603262
$929$	−14.2982	−0.469109	−0.234555	−	0.972103i	$-0.575363\pi$
$929$	−14.2982	−0.469109	−0.234555	+	0.972103i	$0.575363\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	15.4868	0.507288
$933$	0	0
$934$	10.3246	0.337830
$935$	0	0
$936$	0	0
$937$	12.1359	0.396464	0.198232	−	0.980155i	$-0.436480\pi$
$937$	12.1359	0.396464	0.198232	+	0.980155i	$0.436480\pi$
$938$	3.48683	0.113849
$939$	0	0
$940$	0	0
$941$	14.7851	0.481979	0.240989	−	0.970528i	$-0.422528\pi$
$941$	14.7851	0.481979	0.240989	+	0.970528i	$0.422528\pi$
$942$	0	0
$943$	4.64911	0.151396
$944$	0.324555	0.0105634
$945$	0	0
$946$	6.32456	0.205629
$947$	46.0000	1.49480	0.747400	−	0.664375i	$-0.231302\pi$
$947$	46.0000	1.49480	0.747400	+	0.664375i	$0.231302\pi$
$948$	0	0
$949$	−18.0000	−0.584305
$950$	0	0
$951$	0	0
$952$	7.32456	0.237390
$953$	24.7851	0.802867	0.401433	−	0.915888i	$-0.368512\pi$
$953$	24.7851	0.802867	0.401433	+	0.915888i	$0.368512\pi$
$954$	0	0
$955$	0	0
$956$	−25.8114	−0.834800
$957$	0	0
$958$	−4.16228	−0.134477
$959$	2.51317	0.0811544
$960$	0	0
$961$	9.00000	0.290323
$962$	−31.1623	−1.00471
$963$	0	0
$964$	−23.4868	−0.756460
$965$	0	0
$966$	0	0
$967$	−22.4868	−0.723128	−0.361564	−	0.932347i	$-0.617757\pi$
$967$	−22.4868	−0.723128	−0.361564	+	0.932347i	$0.617757\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	0	0
$971$	−46.4605	−1.49099	−0.745494	−	0.666512i	$-0.767786\pi$
$971$	−46.4605	−1.49099	−0.745494	+	0.666512i	$0.767786\pi$
$972$	0	0
$973$	12.3246	0.395107
$974$	26.9737	0.864292
$975$	0	0
$976$	−3.83772	−0.122842
$977$	40.4605	1.29445	0.647223	−	0.762301i	$-0.275930\pi$
$977$	40.4605	1.29445	0.647223	+	0.762301i	$0.275930\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	0	0
$981$	0	0
$982$	−17.3246	−0.552849
$983$	−58.4342	−1.86376	−0.931880	−	0.362766i	$-0.881832\pi$
$983$	−58.4342	−1.86376	−0.931880	+	0.362766i	$0.881832\pi$
$984$	0	0
$985$	0	0
$986$	−13.4605	−0.428670
$987$	0	0
$988$	12.4868	0.397259
$989$	−3.67544	−0.116872
$990$	0	0
$991$	58.4868	1.85790	0.928948	−	0.370211i	$-0.120715\pi$
$991$	58.4868	1.85790	0.928948	+	0.370211i	$0.120715\pi$
$992$	6.32456	0.200805
$993$	0	0
$994$	−10.3246	−0.327475
$995$	0	0
$996$	0	0
$997$	−8.97367	−0.284199	−0.142099	−	0.989852i	$-0.545385\pi$
$997$	−8.97367	−0.284199	−0.142099	+	0.989852i	$0.545385\pi$
$998$	−4.83772	−0.153135
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9450.2.a.eh.1.2	yes	2
3.2	odd	2		9450.2.a.ex.1.2	yes	2
5.4	even	2		9450.2.a.em.1.1	yes	2
15.14	odd	2		9450.2.a.ee.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9450.2.a.ee.1.1	✓	2	15.14	odd	2
9450.2.a.eh.1.2	yes	2	1.1	even	1	trivial
9450.2.a.em.1.1	yes	2	5.4	even	2
9450.2.a.ex.1.2	yes	2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{11} +4.16228 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.32456 q^{17} +3.00000 q^{19} +2.00000 q^{22} -1.16228 q^{23} -4.16228 q^{26} +1.00000 q^{28} -1.83772 q^{29} -6.32456 q^{31} -1.00000 q^{32} +7.32456 q^{34} +7.48683 q^{37} -3.00000 q^{38} -4.00000 q^{41} +3.16228 q^{43} -2.00000 q^{44} +1.16228 q^{46} +10.4868 q^{47} +1.00000 q^{49} +4.16228 q^{52} +0.162278 q^{53} -1.00000 q^{56} +1.83772 q^{58} +0.324555 q^{59} -3.83772 q^{61} +6.32456 q^{62} +1.00000 q^{64} -3.48683 q^{67} -7.32456 q^{68} +10.3246 q^{71} -4.32456 q^{73} -7.48683 q^{74} +3.00000 q^{76} -2.00000 q^{77} -14.4868 q^{79} +4.00000 q^{82} -3.16228 q^{83} -3.16228 q^{86} +2.00000 q^{88} -5.00000 q^{89} +4.16228 q^{91} -1.16228 q^{92} -10.4868 q^{94} -11.4868 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 4 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} + 6 q^{19} + 4 q^{22} + 4 q^{23} - 2 q^{26} + 2 q^{28} - 10 q^{29} - 2 q^{32} + 2 q^{34} - 4 q^{37} - 6 q^{38} - 8 q^{41} - 4 q^{44} - 4 q^{46} + 2 q^{47} + 2 q^{49} + 2 q^{52} - 6 q^{53} - 2 q^{56} + 10 q^{58} - 12 q^{59} - 14 q^{61} + 2 q^{64} + 12 q^{67} - 2 q^{68} + 8 q^{71} + 4 q^{73} + 4 q^{74} + 6 q^{76} - 4 q^{77} - 10 q^{79} + 8 q^{82} + 4 q^{88} - 10 q^{89} + 2 q^{91} + 4 q^{92} - 2 q^{94} - 4 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 4 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 + 6 * q^19 + 4 * q^22 + 4 * q^23 - 2 * q^26 + 2 * q^28 - 10 * q^29 - 2 * q^32 + 2 * q^34 - 4 * q^37 - 6 * q^38 - 8 * q^41 - 4 * q^44 - 4 * q^46 + 2 * q^47 + 2 * q^49 + 2 * q^52 - 6 * q^53 - 2 * q^56 + 10 * q^58 - 12 * q^59 - 14 * q^61 + 2 * q^64 + 12 * q^67 - 2 * q^68 + 8 * q^71 + 4 * q^73 + 4 * q^74 + 6 * q^76 - 4 * q^77 - 10 * q^79 + 8 * q^82 + 4 * q^88 - 10 * q^89 + 2 * q^91 + 4 * q^92 - 2 * q^94 - 4 * q^97 - 2 * q^98

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