LMFDB - Embedded newform 8550.2.a.ck.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.12489$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +4.12489 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +4.12489 q^{7} -1.00000 q^{8} +2.64002 q^{11} +2.51514 q^{13} -4.12489 q^{14} +1.00000 q^{16} +0.515138 q^{17} +1.00000 q^{19} -2.64002 q^{22} -3.09461 q^{23} -2.51514 q^{26} +4.12489 q^{28} +7.79518 q^{29} +3.67030 q^{31} -1.00000 q^{32} -0.515138 q^{34} +10.2498 q^{37} -1.00000 q^{38} -8.88979 q^{41} +8.64002 q^{43} +2.64002 q^{44} +3.09461 q^{46} +4.96972 q^{47} +10.0147 q^{49} +2.51514 q^{52} -5.48486 q^{53} -4.12489 q^{56} -7.79518 q^{58} -3.15516 q^{59} -12.6400 q^{61} -3.67030 q^{62} +1.00000 q^{64} -7.40493 q^{67} +0.515138 q^{68} -11.1396 q^{71} -2.70436 q^{73} -10.2498 q^{74} +1.00000 q^{76} +10.8898 q^{77} +16.7493 q^{79} +8.88979 q^{82} -3.28005 q^{83} -8.64002 q^{86} -2.64002 q^{88} +7.60975 q^{89} +10.3747 q^{91} -3.09461 q^{92} -4.96972 q^{94} +3.93945 q^{97} -10.0147 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 4 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 4 q^{7} - 3 q^{8} + 8 q^{13} - 4 q^{14} + 3 q^{16} + 2 q^{17} + 3 q^{19} - 8 q^{26} + 4 q^{28} + 8 q^{29} + 4 q^{31} - 3 q^{32} - 2 q^{34} + 14 q^{37} - 3 q^{38} - 2 q^{41} + 18 q^{43} + 14 q^{47} - 3 q^{49} + 8 q^{52} - 16 q^{53} - 4 q^{56} - 8 q^{58} - 2 q^{59} - 30 q^{61} - 4 q^{62} + 3 q^{64} + 2 q^{67} + 2 q^{68} + 8 q^{71} + 10 q^{73} - 14 q^{74} + 3 q^{76} + 8 q^{77} + 2 q^{82} + 6 q^{83} - 18 q^{86} + 14 q^{89} + 6 q^{91} - 14 q^{94} + 10 q^{97} + 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 4 * q^7 - 3 * q^8 + 8 * q^13 - 4 * q^14 + 3 * q^16 + 2 * q^17 + 3 * q^19 - 8 * q^26 + 4 * q^28 + 8 * q^29 + 4 * q^31 - 3 * q^32 - 2 * q^34 + 14 * q^37 - 3 * q^38 - 2 * q^41 + 18 * q^43 + 14 * q^47 - 3 * q^49 + 8 * q^52 - 16 * q^53 - 4 * q^56 - 8 * q^58 - 2 * q^59 - 30 * q^61 - 4 * q^62 + 3 * q^64 + 2 * q^67 + 2 * q^68 + 8 * q^71 + 10 * q^73 - 14 * q^74 + 3 * q^76 + 8 * q^77 + 2 * q^82 + 6 * q^83 - 18 * q^86 + 14 * q^89 + 6 * q^91 - 14 * q^94 + 10 * q^97 + 3 * 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
$7$	4.12489	1.55906	0.779530	−	0.626365i	$-0.215458\pi$
$7$	4.12489	1.55906	0.779530	+	0.626365i	$0.215458\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	2.64002	0.795997	0.397999	−	0.917386i	$-0.369705\pi$
$11$	2.64002	0.795997	0.397999	+	0.917386i	$0.369705\pi$
$12$	0	0
$13$	2.51514	0.697574	0.348787	−	0.937202i	$-0.386594\pi$
$13$	2.51514	0.697574	0.348787	+	0.937202i	$0.386594\pi$
$14$	−4.12489	−1.10242
$15$	0	0
$16$	1.00000	0.250000
$17$	0.515138	0.124939	0.0624697	−	0.998047i	$-0.480102\pi$
$17$	0.515138	0.124939	0.0624697	+	0.998047i	$0.480102\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−2.64002	−0.562855
$23$	−3.09461	−0.645271	−0.322635	−	0.946523i	$-0.604569\pi$
$23$	−3.09461	−0.645271	−0.322635	+	0.946523i	$0.604569\pi$
$24$	0	0
$25$	0	0
$26$	−2.51514	−0.493259
$27$	0	0
$28$	4.12489	0.779530
$29$	7.79518	1.44753	0.723765	−	0.690047i	$-0.242410\pi$
$29$	7.79518	1.44753	0.723765	+	0.690047i	$0.242410\pi$
$30$	0	0
$31$	3.67030	0.659205	0.329603	−	0.944120i	$-0.393085\pi$
$31$	3.67030	0.659205	0.329603	+	0.944120i	$0.393085\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.515138	−0.0883454
$35$	0	0
$36$	0	0
$37$	10.2498	1.68505	0.842526	−	0.538656i	$-0.181068\pi$
$37$	10.2498	1.68505	0.842526	+	0.538656i	$0.181068\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	−8.88979	−1.38835	−0.694176	−	0.719805i	$-0.744231\pi$
$41$	−8.88979	−1.38835	−0.694176	+	0.719805i	$0.744231\pi$
$42$	0	0
$43$	8.64002	1.31759	0.658796	−	0.752322i	$-0.271066\pi$
$43$	8.64002	1.31759	0.658796	+	0.752322i	$0.271066\pi$
$44$	2.64002	0.397999
$45$	0	0
$46$	3.09461	0.456275
$47$	4.96972	0.724909	0.362454	−	0.932002i	$-0.381939\pi$
$47$	4.96972	0.724909	0.362454	+	0.932002i	$0.381939\pi$
$48$	0	0
$49$	10.0147	1.43067
$50$	0	0
$51$	0	0
$52$	2.51514	0.348787
$53$	−5.48486	−0.753404	−0.376702	−	0.926335i	$-0.622942\pi$
$53$	−5.48486	−0.753404	−0.376702	+	0.926335i	$0.622942\pi$
$54$	0	0
$55$	0	0
$56$	−4.12489	−0.551211
$57$	0	0
$58$	−7.79518	−1.02356
$59$	−3.15516	−0.410767	−0.205384	−	0.978682i	$-0.565844\pi$
$59$	−3.15516	−0.410767	−0.205384	+	0.978682i	$0.565844\pi$
$60$	0	0
$61$	−12.6400	−1.61839	−0.809195	−	0.587541i	$-0.800096\pi$
$61$	−12.6400	−1.61839	−0.809195	+	0.587541i	$0.800096\pi$
$62$	−3.67030	−0.466129
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−7.40493	−0.904656	−0.452328	−	0.891852i	$-0.649406\pi$
$67$	−7.40493	−0.904656	−0.452328	+	0.891852i	$0.649406\pi$
$68$	0.515138	0.0624697
$69$	0	0
$70$	0	0
$71$	−11.1396	−1.32202	−0.661012	−	0.750376i	$-0.729873\pi$
$71$	−11.1396	−1.32202	−0.661012	+	0.750376i	$0.729873\pi$
$72$	0	0
$73$	−2.70436	−0.316521	−0.158261	−	0.987397i	$-0.550589\pi$
$73$	−2.70436	−0.316521	−0.158261	+	0.987397i	$0.550589\pi$
$74$	−10.2498	−1.19151
$75$	0	0
$76$	1.00000	0.114708
$77$	10.8898	1.24101
$78$	0	0
$79$	16.7493	1.88444	0.942222	−	0.334988i	$-0.108732\pi$
$79$	16.7493	1.88444	0.942222	+	0.334988i	$0.108732\pi$
$80$	0	0
$81$	0	0
$82$	8.88979	0.981714
$83$	−3.28005	−0.360032	−0.180016	−	0.983664i	$-0.557615\pi$
$83$	−3.28005	−0.360032	−0.180016	+	0.983664i	$0.557615\pi$
$84$	0	0
$85$	0	0
$86$	−8.64002	−0.931678
$87$	0	0
$88$	−2.64002	−0.281427
$89$	7.60975	0.806632	0.403316	−	0.915061i	$-0.367858\pi$
$89$	7.60975	0.806632	0.403316	+	0.915061i	$0.367858\pi$
$90$	0	0
$91$	10.3747	1.08756
$92$	−3.09461	−0.322635
$93$	0	0
$94$	−4.96972	−0.512588
$95$	0	0
$96$	0	0
$97$	3.93945	0.399990	0.199995	−	0.979797i	$-0.435907\pi$
$97$	3.93945	0.399990	0.199995	+	0.979797i	$0.435907\pi$
$98$	−10.0147	−1.01164
$99$	0	0
$100$	0	0
$101$	13.7990	1.37305	0.686524	−	0.727107i	$-0.259136\pi$
$101$	13.7990	1.37305	0.686524	+	0.727107i	$0.259136\pi$
$102$	0	0
$103$	4.57947	0.451229	0.225614	−	0.974217i	$-0.427561\pi$
$103$	4.57947	0.451229	0.225614	+	0.974217i	$0.427561\pi$
$104$	−2.51514	−0.246630
$105$	0	0
$106$	5.48486	0.532737
$107$	10.3747	1.00296	0.501478	−	0.865170i	$-0.332790\pi$
$107$	10.3747	1.00296	0.501478	+	0.865170i	$0.332790\pi$
$108$	0	0
$109$	3.01468	0.288754	0.144377	−	0.989523i	$-0.453882\pi$
$109$	3.01468	0.288754	0.144377	+	0.989523i	$0.453882\pi$
$110$	0	0
$111$	0	0
$112$	4.12489	0.389765
$113$	19.2001	1.80620	0.903098	−	0.429435i	$-0.141287\pi$
$113$	19.2001	1.80620	0.903098	+	0.429435i	$0.141287\pi$
$114$	0	0
$115$	0	0
$116$	7.79518	0.723765
$117$	0	0
$118$	3.15516	0.290456
$119$	2.12489	0.194788
$120$	0	0
$121$	−4.03028	−0.366389
$122$	12.6400	1.14437
$123$	0	0
$124$	3.67030	0.329603
$125$	0	0
$126$	0	0
$127$	14.3103	1.26984	0.634918	−	0.772580i	$-0.281034\pi$
$127$	14.3103	1.26984	0.634918	+	0.772580i	$0.281034\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	6.56009	0.573158	0.286579	−	0.958057i	$-0.407482\pi$
$131$	6.56009	0.573158	0.286579	+	0.958057i	$0.407482\pi$
$132$	0	0
$133$	4.12489	0.357673
$134$	7.40493	0.639689
$135$	0	0
$136$	−0.515138	−0.0441727
$137$	6.45459	0.551452	0.275726	−	0.961236i	$-0.411082\pi$
$137$	6.45459	0.551452	0.275726	+	0.961236i	$0.411082\pi$
$138$	0	0
$139$	−23.0596	−1.95589	−0.977946	−	0.208856i	$-0.933026\pi$
$139$	−23.0596	−1.95589	−0.977946	+	0.208856i	$0.933026\pi$
$140$	0	0
$141$	0	0
$142$	11.1396	0.934812
$143$	6.64002	0.555267
$144$	0	0
$145$	0	0
$146$	2.70436	0.223814
$147$	0	0
$148$	10.2498	0.842526
$149$	−16.0294	−1.31318	−0.656588	−	0.754249i	$-0.728001\pi$
$149$	−16.0294	−1.31318	−0.656588	+	0.754249i	$0.728001\pi$
$150$	0	0
$151$	−14.3103	−1.16456	−0.582279	−	0.812989i	$-0.697839\pi$
$151$	−14.3103	−1.16456	−0.582279	+	0.812989i	$0.697839\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−10.8898	−0.877525
$155$	0	0
$156$	0	0
$157$	−18.0294	−1.43890	−0.719450	−	0.694544i	$-0.755606\pi$
$157$	−18.0294	−1.43890	−0.719450	+	0.694544i	$0.755606\pi$
$158$	−16.7493	−1.33250
$159$	0	0
$160$	0	0
$161$	−12.7649	−1.00602
$162$	0	0
$163$	2.70058	0.211525	0.105763	−	0.994391i	$-0.466272\pi$
$163$	2.70058	0.211525	0.105763	+	0.994391i	$0.466272\pi$
$164$	−8.88979	−0.694176
$165$	0	0
$166$	3.28005	0.254581
$167$	−8.95035	−0.692599	−0.346299	−	0.938124i	$-0.612562\pi$
$167$	−8.95035	−0.692599	−0.346299	+	0.938124i	$0.612562\pi$
$168$	0	0
$169$	−6.67408	−0.513391
$170$	0	0
$171$	0	0
$172$	8.64002	0.658796
$173$	−0.310323	−0.0235934	−0.0117967	−	0.999930i	$-0.503755\pi$
$173$	−0.310323	−0.0235934	−0.0117967	+	0.999930i	$0.503755\pi$
$174$	0	0
$175$	0	0
$176$	2.64002	0.198999
$177$	0	0
$178$	−7.60975	−0.570375
$179$	1.52982	0.114344	0.0571720	−	0.998364i	$-0.481792\pi$
$179$	1.52982	0.114344	0.0571720	+	0.998364i	$0.481792\pi$
$180$	0	0
$181$	−14.7493	−1.09631	−0.548154	−	0.836377i	$-0.684669\pi$
$181$	−14.7493	−1.09631	−0.548154	+	0.836377i	$0.684669\pi$
$182$	−10.3747	−0.769021
$183$	0	0
$184$	3.09461	0.228138
$185$	0	0
$186$	0	0
$187$	1.35998	0.0994513
$188$	4.96972	0.362454
$189$	0	0
$190$	0	0
$191$	−18.1249	−1.31147	−0.655735	−	0.754991i	$-0.727641\pi$
$191$	−18.1249	−1.31147	−0.655735	+	0.754991i	$0.727641\pi$
$192$	0	0
$193$	−4.56009	−0.328243	−0.164121	−	0.986440i	$-0.552479\pi$
$193$	−4.56009	−0.328243	−0.164121	+	0.986440i	$0.552479\pi$
$194$	−3.93945	−0.282836
$195$	0	0
$196$	10.0147	0.715334
$197$	2.14048	0.152503	0.0762515	−	0.997089i	$-0.475705\pi$
$197$	2.14048	0.152503	0.0762515	+	0.997089i	$0.475705\pi$
$198$	0	0
$199$	16.5639	1.17418	0.587091	−	0.809521i	$-0.300273\pi$
$199$	16.5639	1.17418	0.587091	+	0.809521i	$0.300273\pi$
$200$	0	0
$201$	0	0
$202$	−13.7990	−0.970892
$203$	32.1542	2.25679
$204$	0	0
$205$	0	0
$206$	−4.57947	−0.319067
$207$	0	0
$208$	2.51514	0.174393
$209$	2.64002	0.182614
$210$	0	0
$211$	−15.2838	−1.05218	−0.526091	−	0.850428i	$-0.676343\pi$
$211$	−15.2838	−1.05218	−0.526091	+	0.850428i	$0.676343\pi$
$212$	−5.48486	−0.376702
$213$	0	0
$214$	−10.3747	−0.709197
$215$	0	0
$216$	0	0
$217$	15.1396	1.02774
$218$	−3.01468	−0.204180
$219$	0	0
$220$	0	0
$221$	1.29564	0.0871544
$222$	0	0
$223$	−3.75023	−0.251134	−0.125567	−	0.992085i	$-0.540075\pi$
$223$	−3.75023	−0.251134	−0.125567	+	0.992085i	$0.540075\pi$
$224$	−4.12489	−0.275606
$225$	0	0
$226$	−19.2001	−1.27717
$227$	24.1542	1.60317	0.801587	−	0.597878i	$-0.203989\pi$
$227$	24.1542	1.60317	0.801587	+	0.597878i	$0.203989\pi$
$228$	0	0
$229$	−13.0596	−0.863005	−0.431503	−	0.902112i	$-0.642016\pi$
$229$	−13.0596	−0.863005	−0.431503	+	0.902112i	$0.642016\pi$
$230$	0	0
$231$	0	0
$232$	−7.79518	−0.511779
$233$	6.43899	0.421832	0.210916	−	0.977504i	$-0.432355\pi$
$233$	6.43899	0.421832	0.210916	+	0.977504i	$0.432355\pi$
$234$	0	0
$235$	0	0
$236$	−3.15516	−0.205384
$237$	0	0
$238$	−2.12489	−0.137736
$239$	−22.8742	−1.47961	−0.739804	−	0.672822i	$-0.765082\pi$
$239$	−22.8742	−1.47961	−0.739804	+	0.672822i	$0.765082\pi$
$240$	0	0
$241$	4.96972	0.320128	0.160064	−	0.987107i	$-0.448830\pi$
$241$	4.96972	0.320128	0.160064	+	0.987107i	$0.448830\pi$
$242$	4.03028	0.259076
$243$	0	0
$244$	−12.6400	−0.809195
$245$	0	0
$246$	0	0
$247$	2.51514	0.160034
$248$	−3.67030	−0.233064
$249$	0	0
$250$	0	0
$251$	−15.9201	−1.00487	−0.502433	−	0.864616i	$-0.667562\pi$
$251$	−15.9201	−1.00487	−0.502433	+	0.864616i	$0.667562\pi$
$252$	0	0
$253$	−8.16984	−0.513634
$254$	−14.3103	−0.897910
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.04965	0.0654756	0.0327378	−	0.999464i	$-0.489577\pi$
$257$	1.04965	0.0654756	0.0327378	+	0.999464i	$0.489577\pi$
$258$	0	0
$259$	42.2791	2.62710
$260$	0	0
$261$	0	0
$262$	−6.56009	−0.405284
$263$	11.9394	0.736218	0.368109	−	0.929783i	$-0.380005\pi$
$263$	11.9394	0.736218	0.368109	+	0.929783i	$0.380005\pi$
$264$	0	0
$265$	0	0
$266$	−4.12489	−0.252913
$267$	0	0
$268$	−7.40493	−0.452328
$269$	15.9394	0.971845	0.485923	−	0.874002i	$-0.338484\pi$
$269$	15.9394	0.971845	0.485923	+	0.874002i	$0.338484\pi$
$270$	0	0
$271$	1.46548	0.0890218	0.0445109	−	0.999009i	$-0.485827\pi$
$271$	1.46548	0.0890218	0.0445109	+	0.999009i	$0.485827\pi$
$272$	0.515138	0.0312348
$273$	0	0
$274$	−6.45459	−0.389936
$275$	0	0
$276$	0	0
$277$	32.4995	1.95271	0.976354	−	0.216177i	$-0.0693590\pi$
$277$	32.4995	1.95271	0.976354	+	0.216177i	$0.0693590\pi$
$278$	23.0596	1.38303
$279$	0	0
$280$	0	0
$281$	−1.04965	−0.0626171	−0.0313085	−	0.999510i	$-0.509967\pi$
$281$	−1.04965	−0.0626171	−0.0313085	+	0.999510i	$0.509967\pi$
$282$	0	0
$283$	−9.34060	−0.555241	−0.277620	−	0.960691i	$-0.589546\pi$
$283$	−9.34060	−0.555241	−0.277620	+	0.960691i	$0.589546\pi$
$284$	−11.1396	−0.661012
$285$	0	0
$286$	−6.64002	−0.392633
$287$	−36.6694	−2.16453
$288$	0	0
$289$	−16.7346	−0.984390
$290$	0	0
$291$	0	0
$292$	−2.70436	−0.158261
$293$	−25.5748	−1.49409	−0.747047	−	0.664771i	$-0.768529\pi$
$293$	−25.5748	−1.49409	−0.747047	+	0.664771i	$0.768529\pi$
$294$	0	0
$295$	0	0
$296$	−10.2498	−0.595756
$297$	0	0
$298$	16.0294	0.928556
$299$	−7.78337	−0.450124
$300$	0	0
$301$	35.6391	2.05420
$302$	14.3103	0.823467
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	9.43991	0.538764	0.269382	−	0.963033i	$-0.413181\pi$
$307$	9.43991	0.538764	0.269382	+	0.963033i	$0.413181\pi$
$308$	10.8898	0.620504
$309$	0	0
$310$	0	0
$311$	17.4655	0.990377	0.495188	−	0.868786i	$-0.335099\pi$
$311$	17.4655	0.990377	0.495188	+	0.868786i	$0.335099\pi$
$312$	0	0
$313$	−10.4546	−0.590928	−0.295464	−	0.955354i	$-0.595474\pi$
$313$	−10.4546	−0.590928	−0.295464	+	0.955354i	$0.595474\pi$
$314$	18.0294	1.01746
$315$	0	0
$316$	16.7493	0.942222
$317$	−4.33348	−0.243393	−0.121696	−	0.992567i	$-0.538833\pi$
$317$	−4.33348	−0.243393	−0.121696	+	0.992567i	$0.538833\pi$
$318$	0	0
$319$	20.5795	1.15223
$320$	0	0
$321$	0	0
$322$	12.7649	0.711361
$323$	0.515138	0.0286630
$324$	0	0
$325$	0	0
$326$	−2.70058	−0.149571
$327$	0	0
$328$	8.88979	0.490857
$329$	20.4995	1.13018
$330$	0	0
$331$	−4.56387	−0.250853	−0.125427	−	0.992103i	$-0.540030\pi$
$331$	−4.56387	−0.250853	−0.125427	+	0.992103i	$0.540030\pi$
$332$	−3.28005	−0.180016
$333$	0	0
$334$	8.95035	0.489741
$335$	0	0
$336$	0	0
$337$	13.0109	0.708749	0.354374	−	0.935104i	$-0.384694\pi$
$337$	13.0109	0.708749	0.354374	+	0.935104i	$0.384694\pi$
$338$	6.67408	0.363022
$339$	0	0
$340$	0	0
$341$	9.68968	0.524725
$342$	0	0
$343$	12.4352	0.671438
$344$	−8.64002	−0.465839
$345$	0	0
$346$	0.310323	0.0166831
$347$	−13.2195	−0.709660	−0.354830	−	0.934931i	$-0.615461\pi$
$347$	−13.2195	−0.709660	−0.354830	+	0.934931i	$0.615461\pi$
$348$	0	0
$349$	−20.4390	−1.09407	−0.547037	−	0.837108i	$-0.684244\pi$
$349$	−20.4390	−1.09407	−0.547037	+	0.837108i	$0.684244\pi$
$350$	0	0
$351$	0	0
$352$	−2.64002	−0.140714
$353$	−10.7044	−0.569735	−0.284868	−	0.958567i	$-0.591950\pi$
$353$	−10.7044	−0.569735	−0.284868	+	0.958567i	$0.591950\pi$
$354$	0	0
$355$	0	0
$356$	7.60975	0.403316
$357$	0	0
$358$	−1.52982	−0.0808534
$359$	4.31410	0.227690	0.113845	−	0.993499i	$-0.463683\pi$
$359$	4.31410	0.227690	0.113845	+	0.993499i	$0.463683\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	14.7493	0.775207
$363$	0	0
$364$	10.3747	0.543780
$365$	0	0
$366$	0	0
$367$	−12.8099	−0.668669	−0.334335	−	0.942454i	$-0.608512\pi$
$367$	−12.8099	−0.668669	−0.334335	+	0.942454i	$0.608512\pi$
$368$	−3.09461	−0.161318
$369$	0	0
$370$	0	0
$371$	−22.6244	−1.17460
$372$	0	0
$373$	0.704357	0.0364702	0.0182351	−	0.999834i	$-0.494195\pi$
$373$	0.704357	0.0364702	0.0182351	+	0.999834i	$0.494195\pi$
$374$	−1.35998	−0.0703227
$375$	0	0
$376$	−4.96972	−0.256294
$377$	19.6060	1.00976
$378$	0	0
$379$	6.12489	0.314614	0.157307	−	0.987550i	$-0.449719\pi$
$379$	6.12489	0.314614	0.157307	+	0.987550i	$0.449719\pi$
$380$	0	0
$381$	0	0
$382$	18.1249	0.927350
$383$	−18.6400	−0.952461	−0.476230	−	0.879321i	$-0.657997\pi$
$383$	−18.6400	−0.952461	−0.476230	+	0.879321i	$0.657997\pi$
$384$	0	0
$385$	0	0
$386$	4.56009	0.232103
$387$	0	0
$388$	3.93945	0.199995
$389$	−13.9201	−0.705776	−0.352888	−	0.935666i	$-0.614800\pi$
$389$	−13.9201	−0.705776	−0.352888	+	0.935666i	$0.614800\pi$
$390$	0	0
$391$	−1.59415	−0.0806197
$392$	−10.0147	−0.505818
$393$	0	0
$394$	−2.14048	−0.107836
$395$	0	0
$396$	0	0
$397$	−28.3784	−1.42427	−0.712136	−	0.702041i	$-0.752272\pi$
$397$	−28.3784	−1.42427	−0.712136	+	0.702041i	$0.752272\pi$
$398$	−16.5639	−0.830272
$399$	0	0
$400$	0	0
$401$	−5.54920	−0.277114	−0.138557	−	0.990354i	$-0.544246\pi$
$401$	−5.54920	−0.277114	−0.138557	+	0.990354i	$0.544246\pi$
$402$	0	0
$403$	9.23131	0.459844
$404$	13.7990	0.686524
$405$	0	0
$406$	−32.1542	−1.59579
$407$	27.0596	1.34130
$408$	0	0
$409$	−5.01090	−0.247773	−0.123886	−	0.992296i	$-0.539536\pi$
$409$	−5.01090	−0.247773	−0.123886	+	0.992296i	$0.539536\pi$
$410$	0	0
$411$	0	0
$412$	4.57947	0.225614
$413$	−13.0147	−0.640411
$414$	0	0
$415$	0	0
$416$	−2.51514	−0.123315
$417$	0	0
$418$	−2.64002	−0.129128
$419$	15.8889	0.776222	0.388111	−	0.921613i	$-0.373128\pi$
$419$	15.8889	0.776222	0.388111	+	0.921613i	$0.373128\pi$
$420$	0	0
$421$	−2.38647	−0.116310	−0.0581548	−	0.998308i	$-0.518522\pi$
$421$	−2.38647	−0.116310	−0.0581548	+	0.998308i	$0.518522\pi$
$422$	15.2838	0.744005
$423$	0	0
$424$	5.48486	0.266368
$425$	0	0
$426$	0	0
$427$	−52.1386	−2.52317
$428$	10.3747	0.501478
$429$	0	0
$430$	0	0
$431$	2.35906	0.113632	0.0568160	−	0.998385i	$-0.481905\pi$
$431$	2.35906	0.113632	0.0568160	+	0.998385i	$0.481905\pi$
$432$	0	0
$433$	−0.640023	−0.0307576	−0.0153788	−	0.999882i	$-0.504895\pi$
$433$	−0.640023	−0.0307576	−0.0153788	+	0.999882i	$0.504895\pi$
$434$	−15.1396	−0.726722
$435$	0	0
$436$	3.01468	0.144377
$437$	−3.09461	−0.148035
$438$	0	0
$439$	−25.4499	−1.21466	−0.607328	−	0.794451i	$-0.707759\pi$
$439$	−25.4499	−1.21466	−0.607328	+	0.794451i	$0.707759\pi$
$440$	0	0
$441$	0	0
$442$	−1.29564	−0.0616275
$443$	35.6685	1.69466	0.847330	−	0.531067i	$-0.178209\pi$
$443$	35.6685	1.69466	0.847330	+	0.531067i	$0.178209\pi$
$444$	0	0
$445$	0	0
$446$	3.75023	0.177578
$447$	0	0
$448$	4.12489	0.194883
$449$	11.4399	0.539883	0.269941	−	0.962877i	$-0.412996\pi$
$449$	11.4399	0.539883	0.269941	+	0.962877i	$0.412996\pi$
$450$	0	0
$451$	−23.4693	−1.10512
$452$	19.2001	0.903098
$453$	0	0
$454$	−24.1542	−1.13361
$455$	0	0
$456$	0	0
$457$	19.4849	0.911463	0.455732	−	0.890117i	$-0.349378\pi$
$457$	19.4849	0.911463	0.455732	+	0.890117i	$0.349378\pi$
$458$	13.0596	0.610237
$459$	0	0
$460$	0	0
$461$	29.3893	1.36880	0.684399	−	0.729108i	$-0.260065\pi$
$461$	29.3893	1.36880	0.684399	+	0.729108i	$0.260065\pi$
$462$	0	0
$463$	12.1892	0.566481	0.283241	−	0.959049i	$-0.408591\pi$
$463$	12.1892	0.566481	0.283241	+	0.959049i	$0.408591\pi$
$464$	7.79518	0.361882
$465$	0	0
$466$	−6.43899	−0.298280
$467$	17.8889	0.827799	0.413899	−	0.910323i	$-0.364167\pi$
$467$	17.8889	0.827799	0.413899	+	0.910323i	$0.364167\pi$
$468$	0	0
$469$	−30.5445	−1.41041
$470$	0	0
$471$	0	0
$472$	3.15516	0.145228
$473$	22.8099	1.04880
$474$	0	0
$475$	0	0
$476$	2.12489	0.0973940
$477$	0	0
$478$	22.8742	1.04624
$479$	1.15894	0.0529534	0.0264767	−	0.999649i	$-0.491571\pi$
$479$	1.15894	0.0529534	0.0264767	+	0.999649i	$0.491571\pi$
$480$	0	0
$481$	25.7796	1.17545
$482$	−4.96972	−0.226365
$483$	0	0
$484$	−4.03028	−0.183194
$485$	0	0
$486$	0	0
$487$	30.6888	1.39064	0.695320	−	0.718700i	$-0.255263\pi$
$487$	30.6888	1.39064	0.695320	+	0.718700i	$0.255263\pi$
$488$	12.6400	0.572187
$489$	0	0
$490$	0	0
$491$	−3.67030	−0.165638	−0.0828192	−	0.996565i	$-0.526392\pi$
$491$	−3.67030	−0.165638	−0.0828192	+	0.996565i	$0.526392\pi$
$492$	0	0
$493$	4.01560	0.180853
$494$	−2.51514	−0.113161
$495$	0	0
$496$	3.67030	0.164801
$497$	−45.9494	−2.06111
$498$	0	0
$499$	31.3893	1.40518	0.702590	−	0.711595i	$-0.252027\pi$
$499$	31.3893	1.40518	0.702590	+	0.711595i	$0.252027\pi$
$500$	0	0
$501$	0	0
$502$	15.9201	0.710548
$503$	18.1542	0.809458	0.404729	−	0.914437i	$-0.367366\pi$
$503$	18.1542	0.809458	0.404729	+	0.914437i	$0.367366\pi$
$504$	0	0
$505$	0	0
$506$	8.16984	0.363194
$507$	0	0
$508$	14.3103	0.634918
$509$	−11.5298	−0.511050	−0.255525	−	0.966802i	$-0.582248\pi$
$509$	−11.5298	−0.511050	−0.255525	+	0.966802i	$0.582248\pi$
$510$	0	0
$511$	−11.1552	−0.493475
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−1.04965	−0.0462982
$515$	0	0
$516$	0	0
$517$	13.1202	0.577025
$518$	−42.2791	−1.85764
$519$	0	0
$520$	0	0
$521$	42.5895	1.86588	0.932939	−	0.360035i	$-0.117235\pi$
$521$	42.5895	1.86588	0.932939	+	0.360035i	$0.117235\pi$
$522$	0	0
$523$	1.21571	0.0531594	0.0265797	−	0.999647i	$-0.491538\pi$
$523$	1.21571	0.0531594	0.0265797	+	0.999647i	$0.491538\pi$
$524$	6.56009	0.286579
$525$	0	0
$526$	−11.9394	−0.520585
$527$	1.89071	0.0823607
$528$	0	0
$529$	−13.4234	−0.583626
$530$	0	0
$531$	0	0
$532$	4.12489	0.178836
$533$	−22.3591	−0.968478
$534$	0	0
$535$	0	0
$536$	7.40493	0.319844
$537$	0	0
$538$	−15.9394	−0.687198
$539$	26.4390	1.13881
$540$	0	0
$541$	−1.92007	−0.0825503	−0.0412751	−	0.999148i	$-0.513142\pi$
$541$	−1.92007	−0.0825503	−0.0412751	+	0.999148i	$0.513142\pi$
$542$	−1.46548	−0.0629480
$543$	0	0
$544$	−0.515138	−0.0220864
$545$	0	0
$546$	0	0
$547$	10.9697	0.469032	0.234516	−	0.972112i	$-0.424650\pi$
$547$	10.9697	0.469032	0.234516	+	0.972112i	$0.424650\pi$
$548$	6.45459	0.275726
$549$	0	0
$550$	0	0
$551$	7.79518	0.332086
$552$	0	0
$553$	69.0890	2.93796
$554$	−32.4995	−1.38077
$555$	0	0
$556$	−23.0596	−0.977946
$557$	−23.5005	−0.995746	−0.497873	−	0.867250i	$-0.665886\pi$
$557$	−23.5005	−0.995746	−0.497873	+	0.867250i	$0.665886\pi$
$558$	0	0
$559$	21.7309	0.919117
$560$	0	0
$561$	0	0
$562$	1.04965	0.0442770
$563$	7.87890	0.332056	0.166028	−	0.986121i	$-0.446906\pi$
$563$	7.87890	0.332056	0.166028	+	0.986121i	$0.446906\pi$
$564$	0	0
$565$	0	0
$566$	9.34060	0.392615
$567$	0	0
$568$	11.1396	0.467406
$569$	1.09083	0.0457299	0.0228650	−	0.999739i	$-0.492721\pi$
$569$	1.09083	0.0457299	0.0228650	+	0.999739i	$0.492721\pi$
$570$	0	0
$571$	21.4886	0.899272	0.449636	−	0.893212i	$-0.351554\pi$
$571$	21.4886	0.899272	0.449636	+	0.893212i	$0.351554\pi$
$572$	6.64002	0.277633
$573$	0	0
$574$	36.6694	1.53055
$575$	0	0
$576$	0	0
$577$	−16.1055	−0.670481	−0.335241	−	0.942133i	$-0.608818\pi$
$577$	−16.1055	−0.670481	−0.335241	+	0.942133i	$0.608818\pi$
$578$	16.7346	0.696069
$579$	0	0
$580$	0	0
$581$	−13.5298	−0.561311
$582$	0	0
$583$	−14.4802	−0.599707
$584$	2.70436	0.111907
$585$	0	0
$586$	25.5748	1.05648
$587$	−0.480164	−0.0198185	−0.00990925	−	0.999951i	$-0.503154\pi$
$587$	−0.480164	−0.0198185	−0.00990925	+	0.999951i	$0.503154\pi$
$588$	0	0
$589$	3.67030	0.151232
$590$	0	0
$591$	0	0
$592$	10.2498	0.421263
$593$	−40.4683	−1.66184	−0.830918	−	0.556395i	$-0.812184\pi$
$593$	−40.4683	−1.66184	−0.830918	+	0.556395i	$0.812184\pi$
$594$	0	0
$595$	0	0
$596$	−16.0294	−0.656588
$597$	0	0
$598$	7.78337	0.318286
$599$	36.1698	1.47786	0.738930	−	0.673782i	$-0.235331\pi$
$599$	36.1698	1.47786	0.738930	+	0.673782i	$0.235331\pi$
$600$	0	0
$601$	24.3903	0.994899	0.497450	−	0.867493i	$-0.334270\pi$
$601$	24.3903	0.994899	0.497450	+	0.867493i	$0.334270\pi$
$602$	−35.6391	−1.45254
$603$	0	0
$604$	−14.3103	−0.582279
$605$	0	0
$606$	0	0
$607$	21.6897	0.880357	0.440178	−	0.897910i	$-0.354915\pi$
$607$	21.6897	0.880357	0.440178	+	0.897910i	$0.354915\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	12.4995	0.505677
$612$	0	0
$613$	−26.2304	−1.05944	−0.529718	−	0.848174i	$-0.677702\pi$
$613$	−26.2304	−1.05944	−0.529718	+	0.848174i	$0.677702\pi$
$614$	−9.43991	−0.380964
$615$	0	0
$616$	−10.8898	−0.438762
$617$	22.7200	0.914671	0.457335	−	0.889294i	$-0.348804\pi$
$617$	22.7200	0.914671	0.457335	+	0.889294i	$0.348804\pi$
$618$	0	0
$619$	−32.9192	−1.32313	−0.661566	−	0.749887i	$-0.730108\pi$
$619$	−32.9192	−1.32313	−0.661566	+	0.749887i	$0.730108\pi$
$620$	0	0
$621$	0	0
$622$	−17.4655	−0.700302
$623$	31.3893	1.25759
$624$	0	0
$625$	0	0
$626$	10.4546	0.417849
$627$	0	0
$628$	−18.0294	−0.719450
$629$	5.28005	0.210529
$630$	0	0
$631$	17.2876	0.688209	0.344104	−	0.938931i	$-0.388183\pi$
$631$	17.2876	0.688209	0.344104	+	0.938931i	$0.388183\pi$
$632$	−16.7493	−0.666252
$633$	0	0
$634$	4.33348	0.172105
$635$	0	0
$636$	0	0
$637$	25.1883	0.997997
$638$	−20.5795	−0.814749
$639$	0	0
$640$	0	0
$641$	9.29942	0.367305	0.183653	−	0.982991i	$-0.441208\pi$
$641$	9.29942	0.367305	0.183653	+	0.982991i	$0.441208\pi$
$642$	0	0
$643$	−3.40115	−0.134128	−0.0670642	−	0.997749i	$-0.521363\pi$
$643$	−3.40115	−0.134128	−0.0670642	+	0.997749i	$0.521363\pi$
$644$	−12.7649	−0.503008
$645$	0	0
$646$	−0.515138	−0.0202678
$647$	−14.9348	−0.587146	−0.293573	−	0.955937i	$-0.594844\pi$
$647$	−14.9348	−0.587146	−0.293573	+	0.955937i	$0.594844\pi$
$648$	0	0
$649$	−8.32970	−0.326969
$650$	0	0
$651$	0	0
$652$	2.70058	0.105763
$653$	−21.1202	−0.826497	−0.413248	−	0.910618i	$-0.635606\pi$
$653$	−21.1202	−0.826497	−0.413248	+	0.910618i	$0.635606\pi$
$654$	0	0
$655$	0	0
$656$	−8.88979	−0.347088
$657$	0	0
$658$	−20.4995	−0.799155
$659$	−27.6547	−1.07727	−0.538637	−	0.842538i	$-0.681061\pi$
$659$	−27.6547	−1.07727	−0.538637	+	0.842538i	$0.681061\pi$
$660$	0	0
$661$	10.2342	0.398063	0.199032	−	0.979993i	$-0.436220\pi$
$661$	10.2342	0.398063	0.199032	+	0.979993i	$0.436220\pi$
$662$	4.56387	0.177380
$663$	0	0
$664$	3.28005	0.127291
$665$	0	0
$666$	0	0
$667$	−24.1231	−0.934048
$668$	−8.95035	−0.346299
$669$	0	0
$670$	0	0
$671$	−33.3700	−1.28823
$672$	0	0
$673$	−13.4693	−0.519202	−0.259601	−	0.965716i	$-0.583591\pi$
$673$	−13.4693	−0.519202	−0.259601	+	0.965716i	$0.583591\pi$
$674$	−13.0109	−0.501161
$675$	0	0
$676$	−6.67408	−0.256695
$677$	−15.1433	−0.582006	−0.291003	−	0.956722i	$-0.593989\pi$
$677$	−15.1433	−0.582006	−0.291003	+	0.956722i	$0.593989\pi$
$678$	0	0
$679$	16.2498	0.623609
$680$	0	0
$681$	0	0
$682$	−9.68968	−0.371037
$683$	−15.8789	−0.607589	−0.303795	−	0.952738i	$-0.598254\pi$
$683$	−15.8789	−0.607589	−0.303795	+	0.952738i	$0.598254\pi$
$684$	0	0
$685$	0	0
$686$	−12.4352	−0.474778
$687$	0	0
$688$	8.64002	0.329398
$689$	−13.7952	−0.525555
$690$	0	0
$691$	27.3094	1.03890	0.519449	−	0.854501i	$-0.326137\pi$
$691$	27.3094	1.03890	0.519449	+	0.854501i	$0.326137\pi$
$692$	−0.310323	−0.0117967
$693$	0	0
$694$	13.2195	0.501805
$695$	0	0
$696$	0	0
$697$	−4.57947	−0.173460
$698$	20.4390	0.773627
$699$	0	0
$700$	0	0
$701$	20.9697	0.792016	0.396008	−	0.918247i	$-0.370395\pi$
$701$	20.9697	0.792016	0.396008	+	0.918247i	$0.370395\pi$
$702$	0	0
$703$	10.2498	0.386577
$704$	2.64002	0.0994996
$705$	0	0
$706$	10.7044	0.402864
$707$	56.9192	2.14067
$708$	0	0
$709$	37.5592	1.41056	0.705282	−	0.708927i	$-0.250820\pi$
$709$	37.5592	1.41056	0.705282	+	0.708927i	$0.250820\pi$
$710$	0	0
$711$	0	0
$712$	−7.60975	−0.285187
$713$	−11.3581	−0.425366
$714$	0	0
$715$	0	0
$716$	1.52982	0.0571720
$717$	0	0
$718$	−4.31410	−0.161001
$719$	−3.94323	−0.147058	−0.0735288	−	0.997293i	$-0.523426\pi$
$719$	−3.94323	−0.147058	−0.0735288	+	0.997293i	$0.523426\pi$
$720$	0	0
$721$	18.8898	0.703493
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−14.7493	−0.548154
$725$	0	0
$726$	0	0
$727$	33.2139	1.23183	0.615917	−	0.787811i	$-0.288786\pi$
$727$	33.2139	1.23183	0.615917	+	0.787811i	$0.288786\pi$
$728$	−10.3747	−0.384510
$729$	0	0
$730$	0	0
$731$	4.45080	0.164619
$732$	0	0
$733$	−8.62065	−0.318411	−0.159205	−	0.987245i	$-0.550893\pi$
$733$	−8.62065	−0.318411	−0.159205	+	0.987245i	$0.550893\pi$
$734$	12.8099	0.472821
$735$	0	0
$736$	3.09461	0.114069
$737$	−19.5492	−0.720104
$738$	0	0
$739$	−45.1689	−1.66157	−0.830783	−	0.556597i	$-0.812107\pi$
$739$	−45.1689	−1.66157	−0.830783	+	0.556597i	$0.812107\pi$
$740$	0	0
$741$	0	0
$742$	22.6244	0.830569
$743$	24.7905	0.909475	0.454737	−	0.890626i	$-0.349733\pi$
$743$	24.7905	0.909475	0.454737	+	0.890626i	$0.349733\pi$
$744$	0	0
$745$	0	0
$746$	−0.704357	−0.0257883
$747$	0	0
$748$	1.35998	0.0497257
$749$	42.7943	1.56367
$750$	0	0
$751$	−6.76869	−0.246993	−0.123497	−	0.992345i	$-0.539411\pi$
$751$	−6.76869	−0.246993	−0.123497	+	0.992345i	$0.539411\pi$
$752$	4.96972	0.181227
$753$	0	0
$754$	−19.6060	−0.714007
$755$	0	0
$756$	0	0
$757$	45.2101	1.64319	0.821595	−	0.570072i	$-0.193085\pi$
$757$	45.2101	1.64319	0.821595	+	0.570072i	$0.193085\pi$
$758$	−6.12489	−0.222466
$759$	0	0
$760$	0	0
$761$	−19.8851	−0.720834	−0.360417	−	0.932791i	$-0.617366\pi$
$761$	−19.8851	−0.720834	−0.360417	+	0.932791i	$0.617366\pi$
$762$	0	0
$763$	12.4352	0.450185
$764$	−18.1249	−0.655735
$765$	0	0
$766$	18.6400	0.673491
$767$	−7.93567	−0.286540
$768$	0	0
$769$	8.07615	0.291233	0.145617	−	0.989341i	$-0.453483\pi$
$769$	8.07615	0.291233	0.145617	+	0.989341i	$0.453483\pi$
$770$	0	0
$771$	0	0
$772$	−4.56009	−0.164121
$773$	−45.9456	−1.65255	−0.826275	−	0.563267i	$-0.809544\pi$
$773$	−45.9456	−1.65255	−0.826275	+	0.563267i	$0.809544\pi$
$774$	0	0
$775$	0	0
$776$	−3.93945	−0.141418
$777$	0	0
$778$	13.9201	0.499059
$779$	−8.88979	−0.318510
$780$	0	0
$781$	−29.4087	−1.05233
$782$	1.59415	0.0570067
$783$	0	0
$784$	10.0147	0.357667
$785$	0	0
$786$	0	0
$787$	−17.8439	−0.636067	−0.318034	−	0.948079i	$-0.603022\pi$
$787$	−17.8439	−0.636067	−0.318034	+	0.948079i	$0.603022\pi$
$788$	2.14048	0.0762515
$789$	0	0
$790$	0	0
$791$	79.1983	2.81597
$792$	0	0
$793$	−31.7914	−1.12895
$794$	28.3784	1.00711
$795$	0	0
$796$	16.5639	0.587091
$797$	37.3326	1.32239	0.661194	−	0.750215i	$-0.270050\pi$
$797$	37.3326	1.32239	0.661194	+	0.750215i	$0.270050\pi$
$798$	0	0
$799$	2.56009	0.0905696
$800$	0	0
$801$	0	0
$802$	5.54920	0.195949
$803$	−7.13957	−0.251950
$804$	0	0
$805$	0	0
$806$	−9.23131	−0.325159
$807$	0	0
$808$	−13.7990	−0.485446
$809$	38.6950	1.36044	0.680221	−	0.733007i	$-0.261884\pi$
$809$	38.6950	1.36044	0.680221	+	0.733007i	$0.261884\pi$
$810$	0	0
$811$	13.3444	0.468585	0.234292	−	0.972166i	$-0.424723\pi$
$811$	13.3444	0.468585	0.234292	+	0.972166i	$0.424723\pi$
$812$	32.1542	1.12839
$813$	0	0
$814$	−27.0596	−0.948440
$815$	0	0
$816$	0	0
$817$	8.64002	0.302276
$818$	5.01090	0.175202
$819$	0	0
$820$	0	0
$821$	37.3482	1.30346	0.651730	−	0.758451i	$-0.274044\pi$
$821$	37.3482	1.30346	0.651730	+	0.758451i	$0.274044\pi$
$822$	0	0
$823$	−51.5630	−1.79737	−0.898686	−	0.438593i	$-0.855477\pi$
$823$	−51.5630	−1.79737	−0.898686	+	0.438593i	$0.855477\pi$
$824$	−4.57947	−0.159533
$825$	0	0
$826$	13.0147	0.452839
$827$	48.5639	1.68873	0.844366	−	0.535767i	$-0.179978\pi$
$827$	48.5639	1.68873	0.844366	+	0.535767i	$0.179978\pi$
$828$	0	0
$829$	0.325919	0.0113196	0.00565982	−	0.999984i	$-0.498198\pi$
$829$	0.325919	0.0113196	0.00565982	+	0.999984i	$0.498198\pi$
$830$	0	0
$831$	0	0
$832$	2.51514	0.0871967
$833$	5.15894	0.178747
$834$	0	0
$835$	0	0
$836$	2.64002	0.0913071
$837$	0	0
$838$	−15.8889	−0.548872
$839$	17.1202	0.591055	0.295527	−	0.955334i	$-0.404505\pi$
$839$	17.1202	0.591055	0.295527	+	0.955334i	$0.404505\pi$
$840$	0	0
$841$	31.7649	1.09534
$842$	2.38647	0.0822432
$843$	0	0
$844$	−15.2838	−0.526091
$845$	0	0
$846$	0	0
$847$	−16.6244	−0.571222
$848$	−5.48486	−0.188351
$849$	0	0
$850$	0	0
$851$	−31.7190	−1.08731
$852$	0	0
$853$	−24.9092	−0.852874	−0.426437	−	0.904517i	$-0.640231\pi$
$853$	−24.9092	−0.852874	−0.426437	+	0.904517i	$0.640231\pi$
$854$	52.1386	1.78415
$855$	0	0
$856$	−10.3747	−0.354598
$857$	11.6509	0.397988	0.198994	−	0.980001i	$-0.436233\pi$
$857$	11.6509	0.397988	0.198994	+	0.980001i	$0.436233\pi$
$858$	0	0
$859$	−5.35998	−0.182880	−0.0914400	−	0.995811i	$-0.529147\pi$
$859$	−5.35998	−0.182880	−0.0914400	+	0.995811i	$0.529147\pi$
$860$	0	0
$861$	0	0
$862$	−2.35906	−0.0803499
$863$	41.0790	1.39835	0.699173	−	0.714953i	$-0.253552\pi$
$863$	41.0790	1.39835	0.699173	+	0.714953i	$0.253552\pi$
$864$	0	0
$865$	0	0
$866$	0.640023	0.0217489
$867$	0	0
$868$	15.1396	0.513870
$869$	44.2186	1.50001
$870$	0	0
$871$	−18.6244	−0.631065
$872$	−3.01468	−0.102090
$873$	0	0
$874$	3.09461	0.104677
$875$	0	0
$876$	0	0
$877$	45.9532	1.55173	0.775865	−	0.630899i	$-0.217314\pi$
$877$	45.9532	1.55173	0.775865	+	0.630899i	$0.217314\pi$
$878$	25.4499	0.858892
$879$	0	0
$880$	0	0
$881$	6.90917	0.232776	0.116388	−	0.993204i	$-0.462868\pi$
$881$	6.90917	0.232776	0.116388	+	0.993204i	$0.462868\pi$
$882$	0	0
$883$	48.5213	1.63287	0.816437	−	0.577435i	$-0.195946\pi$
$883$	48.5213	1.63287	0.816437	+	0.577435i	$0.195946\pi$
$884$	1.29564	0.0435772
$885$	0	0
$886$	−35.6685	−1.19831
$887$	−23.7990	−0.799091	−0.399546	−	0.916713i	$-0.630832\pi$
$887$	−23.7990	−0.799091	−0.399546	+	0.916713i	$0.630832\pi$
$888$	0	0
$889$	59.0284	1.97975
$890$	0	0
$891$	0	0
$892$	−3.75023	−0.125567
$893$	4.96972	0.166305
$894$	0	0
$895$	0	0
$896$	−4.12489	−0.137803
$897$	0	0
$898$	−11.4399	−0.381755
$899$	28.6107	0.954219
$900$	0	0
$901$	−2.82546	−0.0941298
$902$	23.4693	0.781441
$903$	0	0
$904$	−19.2001	−0.638586
$905$	0	0
$906$	0	0
$907$	17.9726	0.596770	0.298385	−	0.954446i	$-0.403552\pi$
$907$	17.9726	0.596770	0.298385	+	0.954446i	$0.403552\pi$
$908$	24.1542	0.801587
$909$	0	0
$910$	0	0
$911$	−19.5592	−0.648024	−0.324012	−	0.946053i	$-0.605032\pi$
$911$	−19.5592	−0.648024	−0.324012	+	0.946053i	$0.605032\pi$
$912$	0	0
$913$	−8.65940	−0.286584
$914$	−19.4849	−0.644502
$915$	0	0
$916$	−13.0596	−0.431503
$917$	27.0596	0.893588
$918$	0	0
$919$	35.4948	1.17087	0.585433	−	0.810720i	$-0.300924\pi$
$919$	35.4948	1.17087	0.585433	+	0.810720i	$0.300924\pi$
$920$	0	0
$921$	0	0
$922$	−29.3893	−0.967886
$923$	−28.0175	−0.922209
$924$	0	0
$925$	0	0
$926$	−12.1892	−0.400563
$927$	0	0
$928$	−7.79518	−0.255889
$929$	−17.9532	−0.589026	−0.294513	−	0.955648i	$-0.595157\pi$
$929$	−17.9532	−0.589026	−0.294513	+	0.955648i	$0.595157\pi$
$930$	0	0
$931$	10.0147	0.328218
$932$	6.43899	0.210916
$933$	0	0
$934$	−17.8889	−0.585342
$935$	0	0
$936$	0	0
$937$	−5.66652	−0.185117	−0.0925585	−	0.995707i	$-0.529505\pi$
$937$	−5.66652	−0.185117	−0.0925585	+	0.995707i	$0.529505\pi$
$938$	30.5445	0.997313
$939$	0	0
$940$	0	0
$941$	24.7044	0.805339	0.402670	−	0.915345i	$-0.368082\pi$
$941$	24.7044	0.805339	0.402670	+	0.915345i	$0.368082\pi$
$942$	0	0
$943$	27.5104	0.895863
$944$	−3.15516	−0.102692
$945$	0	0
$946$	−22.8099	−0.741613
$947$	−13.5904	−0.441628	−0.220814	−	0.975316i	$-0.570871\pi$
$947$	−13.5904	−0.441628	−0.220814	+	0.975316i	$0.570871\pi$
$948$	0	0
$949$	−6.80183	−0.220797
$950$	0	0
$951$	0	0
$952$	−2.12489	−0.0688679
$953$	−24.7375	−0.801326	−0.400663	−	0.916225i	$-0.631220\pi$
$953$	−24.7375	−0.801326	−0.400663	+	0.916225i	$0.631220\pi$
$954$	0	0
$955$	0	0
$956$	−22.8742	−0.739804
$957$	0	0
$958$	−1.15894	−0.0374437
$959$	26.6244	0.859748
$960$	0	0
$961$	−17.5289	−0.565448
$962$	−25.7796	−0.831167
$963$	0	0
$964$	4.96972	0.160064
$965$	0	0
$966$	0	0
$967$	−35.6897	−1.14770	−0.573851	−	0.818960i	$-0.694551\pi$
$967$	−35.6897	−1.14770	−0.573851	+	0.818960i	$0.694551\pi$
$968$	4.03028	0.129538
$969$	0	0
$970$	0	0
$971$	−16.4995	−0.529495	−0.264748	−	0.964318i	$-0.585289\pi$
$971$	−16.4995	−0.529495	−0.264748	+	0.964318i	$0.585289\pi$
$972$	0	0
$973$	−95.1184	−3.04935
$974$	−30.6888	−0.983331
$975$	0	0
$976$	−12.6400	−0.404597
$977$	−49.8501	−1.59485	−0.797423	−	0.603420i	$-0.793804\pi$
$977$	−49.8501	−1.59485	−0.797423	+	0.603420i	$0.793804\pi$
$978$	0	0
$979$	20.0899	0.642076
$980$	0	0
$981$	0	0
$982$	3.67030	0.117124
$983$	5.19014	0.165540	0.0827698	−	0.996569i	$-0.473623\pi$
$983$	5.19014	0.165540	0.0827698	+	0.996569i	$0.473623\pi$
$984$	0	0
$985$	0	0
$986$	−4.01560	−0.127883
$987$	0	0
$988$	2.51514	0.0800172
$989$	−26.7375	−0.850203
$990$	0	0
$991$	−32.4272	−1.03008	−0.515042	−	0.857165i	$-0.672224\pi$
$991$	−32.4272	−1.03008	−0.515042	+	0.857165i	$0.672224\pi$
$992$	−3.67030	−0.116532
$993$	0	0
$994$	45.9494	1.45743
$995$	0	0
$996$	0	0
$997$	10.1992	0.323012	0.161506	−	0.986872i	$-0.448365\pi$
$997$	10.1992	0.323012	0.161506	+	0.986872i	$0.448365\pi$
$998$	−31.3893	−0.993612
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.ck.1.3		3
3.2	odd	2		950.2.a.n.1.1		3
5.2	odd	4		1710.2.d.d.1369.1		6
5.3	odd	4		1710.2.d.d.1369.4		6
5.4	even	2		8550.2.a.cl.1.1		3
12.11	even	2		7600.2.a.bi.1.3		3
15.2	even	4		190.2.b.b.39.6	yes	6
15.8	even	4		190.2.b.b.39.1	✓	6
15.14	odd	2		950.2.a.i.1.3		3
60.23	odd	4		1520.2.d.j.609.5		6
60.47	odd	4		1520.2.d.j.609.2		6
60.59	even	2		7600.2.a.cd.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.b.39.1	✓	6	15.8	even	4
190.2.b.b.39.6	yes	6	15.2	even	4
950.2.a.i.1.3		3	15.14	odd	2
950.2.a.n.1.1		3	3.2	odd	2
1520.2.d.j.609.2		6	60.47	odd	4
1520.2.d.j.609.5		6	60.23	odd	4
1710.2.d.d.1369.1		6	5.2	odd	4
1710.2.d.d.1369.4		6	5.3	odd	4
7600.2.a.bi.1.3		3	12.11	even	2
7600.2.a.cd.1.1		3	60.59	even	2
8550.2.a.ck.1.3		3	1.1	even	1	trivial
8550.2.a.cl.1.1		3	5.4	even	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.12489 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.12489 q^{7} -1.00000 q^{8} +2.64002 q^{11} +2.51514 q^{13} -4.12489 q^{14} +1.00000 q^{16} +0.515138 q^{17} +1.00000 q^{19} -2.64002 q^{22} -3.09461 q^{23} -2.51514 q^{26} +4.12489 q^{28} +7.79518 q^{29} +3.67030 q^{31} -1.00000 q^{32} -0.515138 q^{34} +10.2498 q^{37} -1.00000 q^{38} -8.88979 q^{41} +8.64002 q^{43} +2.64002 q^{44} +3.09461 q^{46} +4.96972 q^{47} +10.0147 q^{49} +2.51514 q^{52} -5.48486 q^{53} -4.12489 q^{56} -7.79518 q^{58} -3.15516 q^{59} -12.6400 q^{61} -3.67030 q^{62} +1.00000 q^{64} -7.40493 q^{67} +0.515138 q^{68} -11.1396 q^{71} -2.70436 q^{73} -10.2498 q^{74} +1.00000 q^{76} +10.8898 q^{77} +16.7493 q^{79} +8.88979 q^{82} -3.28005 q^{83} -8.64002 q^{86} -2.64002 q^{88} +7.60975 q^{89} +10.3747 q^{91} -3.09461 q^{92} -4.96972 q^{94} +3.93945 q^{97} -10.0147 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 4 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 4 q^{7} - 3 q^{8} + 8 q^{13} - 4 q^{14} + 3 q^{16} + 2 q^{17} + 3 q^{19} - 8 q^{26} + 4 q^{28} + 8 q^{29} + 4 q^{31} - 3 q^{32} - 2 q^{34} + 14 q^{37} - 3 q^{38} - 2 q^{41} + 18 q^{43} + 14 q^{47} - 3 q^{49} + 8 q^{52} - 16 q^{53} - 4 q^{56} - 8 q^{58} - 2 q^{59} - 30 q^{61} - 4 q^{62} + 3 q^{64} + 2 q^{67} + 2 q^{68} + 8 q^{71} + 10 q^{73} - 14 q^{74} + 3 q^{76} + 8 q^{77} + 2 q^{82} + 6 q^{83} - 18 q^{86} + 14 q^{89} + 6 q^{91} - 14 q^{94} + 10 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 4 * q^7 - 3 * q^8 + 8 * q^13 - 4 * q^14 + 3 * q^16 + 2 * q^17 + 3 * q^19 - 8 * q^26 + 4 * q^28 + 8 * q^29 + 4 * q^31 - 3 * q^32 - 2 * q^34 + 14 * q^37 - 3 * q^38 - 2 * q^41 + 18 * q^43 + 14 * q^47 - 3 * q^49 + 8 * q^52 - 16 * q^53 - 4 * q^56 - 8 * q^58 - 2 * q^59 - 30 * q^61 - 4 * q^62 + 3 * q^64 + 2 * q^67 + 2 * q^68 + 8 * q^71 + 10 * q^73 - 14 * q^74 + 3 * q^76 + 8 * q^77 + 2 * q^82 + 6 * q^83 - 18 * q^86 + 14 * q^89 + 6 * q^91 - 14 * q^94 + 10 * q^97 + 3 * q^98

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