LMFDB - Embedded newform 8550.2.a.cn.1.2

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.51414$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.193252 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -0.193252 q^{7} +1.00000 q^{8} +0.514137 q^{11} -3.12763 q^{13} -0.193252 q^{14} +1.00000 q^{16} +2.83502 q^{17} +1.00000 q^{19} +0.514137 q^{22} -2.32088 q^{23} -3.12763 q^{26} -0.193252 q^{28} -0.164979 q^{29} -9.05655 q^{31} +1.00000 q^{32} +2.83502 q^{34} -3.02827 q^{37} +1.00000 q^{38} -9.96265 q^{41} -5.51414 q^{43} +0.514137 q^{44} -2.32088 q^{46} -1.70739 q^{47} -6.96265 q^{49} -3.12763 q^{52} +2.90064 q^{53} -0.193252 q^{56} -0.164979 q^{58} -13.9572 q^{59} +5.92892 q^{61} -9.05655 q^{62} +1.00000 q^{64} -4.22153 q^{67} +2.83502 q^{68} -3.16498 q^{71} +8.37743 q^{73} -3.02827 q^{74} +1.00000 q^{76} -0.0993582 q^{77} +6.02827 q^{79} -9.96265 q^{82} +12.8633 q^{83} -5.51414 q^{86} +0.514137 q^{88} +9.01920 q^{89} +0.604422 q^{91} -2.32088 q^{92} -1.70739 q^{94} +8.89157 q^{97} -6.96265 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 5 q^{11} - 2 q^{14} + 3 q^{16} - 6 q^{17} + 3 q^{19} - 5 q^{22} + q^{23} - 2 q^{28} - 15 q^{29} - q^{31} + 3 q^{32} - 6 q^{34} + 4 q^{37} + 3 q^{38} - 6 q^{41} - 10 q^{43} - 5 q^{44} + q^{46} + 3 q^{49} + 5 q^{53} - 2 q^{56} - 15 q^{58} - 12 q^{59} + q^{61} - q^{62} + 3 q^{64} - q^{67} - 6 q^{68} - 24 q^{71} - 9 q^{73} + 4 q^{74} + 3 q^{76} - 4 q^{77} + 5 q^{79} - 6 q^{82} + 11 q^{83} - 10 q^{86} - 5 q^{88} - 23 q^{89} - 38 q^{91} + q^{92} - 14 q^{97} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 5 * q^11 - 2 * q^14 + 3 * q^16 - 6 * q^17 + 3 * q^19 - 5 * q^22 + q^23 - 2 * q^28 - 15 * q^29 - q^31 + 3 * q^32 - 6 * q^34 + 4 * q^37 + 3 * q^38 - 6 * q^41 - 10 * q^43 - 5 * q^44 + q^46 + 3 * q^49 + 5 * q^53 - 2 * q^56 - 15 * q^58 - 12 * q^59 + q^61 - q^62 + 3 * q^64 - q^67 - 6 * q^68 - 24 * q^71 - 9 * q^73 + 4 * q^74 + 3 * q^76 - 4 * q^77 + 5 * q^79 - 6 * q^82 + 11 * q^83 - 10 * q^86 - 5 * q^88 - 23 * q^89 - 38 * q^91 + q^92 - 14 * 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
$7$	−0.193252	−0.0730426	−0.0365213	−	0.999333i	$-0.511628\pi$
$7$	−0.193252	−0.0730426	−0.0365213	+	0.999333i	$0.511628\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	0.514137	0.155018	0.0775091	−	0.996992i	$-0.475303\pi$
$11$	0.514137	0.155018	0.0775091	+	0.996992i	$0.475303\pi$
$12$	0	0
$13$	−3.12763	−0.867449	−0.433725	−	0.901046i	$-0.642801\pi$
$13$	−3.12763	−0.867449	−0.433725	+	0.901046i	$0.642801\pi$
$14$	−0.193252	−0.0516489
$15$	0	0
$16$	1.00000	0.250000
$17$	2.83502	0.687594	0.343797	−	0.939044i	$-0.388287\pi$
$17$	2.83502	0.687594	0.343797	+	0.939044i	$0.388287\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0.514137	0.109614
$23$	−2.32088	−0.483938	−0.241969	−	0.970284i	$-0.577793\pi$
$23$	−2.32088	−0.483938	−0.241969	+	0.970284i	$0.577793\pi$
$24$	0	0
$25$	0	0
$26$	−3.12763	−0.613379
$27$	0	0
$28$	−0.193252	−0.0365213
$29$	−0.164979	−0.0306358	−0.0153179	−	0.999883i	$-0.504876\pi$
$29$	−0.164979	−0.0306358	−0.0153179	+	0.999883i	$0.504876\pi$
$30$	0	0
$31$	−9.05655	−1.62660	−0.813302	−	0.581842i	$-0.802332\pi$
$31$	−9.05655	−1.62660	−0.813302	+	0.581842i	$0.802332\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.83502	0.486202
$35$	0	0
$36$	0	0
$37$	−3.02827	−0.497845	−0.248923	−	0.968523i	$-0.580076\pi$
$37$	−3.02827	−0.497845	−0.248923	+	0.968523i	$0.580076\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	−9.96265	−1.55591	−0.777953	−	0.628323i	$-0.783742\pi$
$41$	−9.96265	−1.55591	−0.777953	+	0.628323i	$0.783742\pi$
$42$	0	0
$43$	−5.51414	−0.840898	−0.420449	−	0.907316i	$-0.638127\pi$
$43$	−5.51414	−0.840898	−0.420449	+	0.907316i	$0.638127\pi$
$44$	0.514137	0.0775091
$45$	0	0
$46$	−2.32088	−0.342196
$47$	−1.70739	−0.249048	−0.124524	−	0.992217i	$-0.539740\pi$
$47$	−1.70739	−0.249048	−0.124524	+	0.992217i	$0.539740\pi$
$48$	0	0
$49$	−6.96265	−0.994665
$50$	0	0
$51$	0	0
$52$	−3.12763	−0.433725
$53$	2.90064	0.398434	0.199217	−	0.979955i	$-0.436160\pi$
$53$	2.90064	0.398434	0.199217	+	0.979955i	$0.436160\pi$
$54$	0	0
$55$	0	0
$56$	−0.193252	−0.0258244
$57$	0	0
$58$	−0.164979	−0.0216627
$59$	−13.9572	−1.81707	−0.908536	−	0.417807i	$-0.862799\pi$
$59$	−13.9572	−1.81707	−0.908536	+	0.417807i	$0.862799\pi$
$60$	0	0
$61$	5.92892	0.759120	0.379560	−	0.925167i	$-0.376075\pi$
$61$	5.92892	0.759120	0.379560	+	0.925167i	$0.376075\pi$
$62$	−9.05655	−1.15018
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.22153	−0.515742	−0.257871	−	0.966179i	$-0.583021\pi$
$67$	−4.22153	−0.515742	−0.257871	+	0.966179i	$0.583021\pi$
$68$	2.83502	0.343797
$69$	0	0
$70$	0	0
$71$	−3.16498	−0.375614	−0.187807	−	0.982206i	$-0.560138\pi$
$71$	−3.16498	−0.375614	−0.187807	+	0.982206i	$0.560138\pi$
$72$	0	0
$73$	8.37743	0.980504	0.490252	−	0.871581i	$-0.336905\pi$
$73$	8.37743	0.980504	0.490252	+	0.871581i	$0.336905\pi$
$74$	−3.02827	−0.352030
$75$	0	0
$76$	1.00000	0.114708
$77$	−0.0993582	−0.0113229
$78$	0	0
$79$	6.02827	0.678234	0.339117	−	0.940744i	$-0.389872\pi$
$79$	6.02827	0.678234	0.339117	+	0.940744i	$0.389872\pi$
$80$	0	0
$81$	0	0
$82$	−9.96265	−1.10019
$83$	12.8633	1.41193	0.705965	−	0.708247i	$-0.250514\pi$
$83$	12.8633	1.41193	0.705965	+	0.708247i	$0.250514\pi$
$84$	0	0
$85$	0	0
$86$	−5.51414	−0.594605
$87$	0	0
$88$	0.514137	0.0548072
$89$	9.01920	0.956033	0.478017	−	0.878351i	$-0.341356\pi$
$89$	9.01920	0.956033	0.478017	+	0.878351i	$0.341356\pi$
$90$	0	0
$91$	0.604422	0.0633607
$92$	−2.32088	−0.241969
$93$	0	0
$94$	−1.70739	−0.176104
$95$	0	0
$96$	0	0
$97$	8.89157	0.902802	0.451401	−	0.892321i	$-0.350924\pi$
$97$	8.89157	0.902802	0.451401	+	0.892321i	$0.350924\pi$
$98$	−6.96265	−0.703334
$99$	0	0
$100$	0	0
$101$	2.15591	0.214521	0.107260	−	0.994231i	$-0.465792\pi$
$101$	2.15591	0.214521	0.107260	+	0.994231i	$0.465792\pi$
$102$	0	0
$103$	−10.3209	−1.01695	−0.508473	−	0.861078i	$-0.669790\pi$
$103$	−10.3209	−1.01695	−0.508473	+	0.861078i	$0.669790\pi$
$104$	−3.12763	−0.306690
$105$	0	0
$106$	2.90064	0.281735
$107$	5.57068	0.538538	0.269269	−	0.963065i	$-0.413218\pi$
$107$	5.57068	0.538538	0.269269	+	0.963065i	$0.413218\pi$
$108$	0	0
$109$	5.61350	0.537675	0.268838	−	0.963186i	$-0.413360\pi$
$109$	5.61350	0.537675	0.268838	+	0.963186i	$0.413360\pi$
$110$	0	0
$111$	0	0
$112$	−0.193252	−0.0182606
$113$	−6.96265	−0.654991	−0.327496	−	0.944853i	$-0.606205\pi$
$113$	−6.96265	−0.654991	−0.327496	+	0.944853i	$0.606205\pi$
$114$	0	0
$115$	0	0
$116$	−0.164979	−0.0153179
$117$	0	0
$118$	−13.9572	−1.28486
$119$	−0.547875	−0.0502236
$120$	0	0
$121$	−10.7357	−0.975969
$122$	5.92892	0.536779
$123$	0	0
$124$	−9.05655	−0.813302
$125$	0	0
$126$	0	0
$127$	13.6983	1.21553	0.607765	−	0.794117i	$-0.292066\pi$
$127$	13.6983	1.21553	0.607765	+	0.794117i	$0.292066\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−10.4768	−0.915361	−0.457681	−	0.889117i	$-0.651320\pi$
$131$	−10.4768	−0.915361	−0.457681	+	0.889117i	$0.651320\pi$
$132$	0	0
$133$	−0.193252	−0.0167571
$134$	−4.22153	−0.364684
$135$	0	0
$136$	2.83502	0.243101
$137$	−11.6700	−0.997039	−0.498520	−	0.866878i	$-0.666123\pi$
$137$	−11.6700	−0.997039	−0.498520	+	0.866878i	$0.666123\pi$
$138$	0	0
$139$	12.7411	1.08069	0.540344	−	0.841444i	$-0.318294\pi$
$139$	12.7411	1.08069	0.540344	+	0.841444i	$0.318294\pi$
$140$	0	0
$141$	0	0
$142$	−3.16498	−0.265599
$143$	−1.60803	−0.134470
$144$	0	0
$145$	0	0
$146$	8.37743	0.693321
$147$	0	0
$148$	−3.02827	−0.248923
$149$	−9.61350	−0.787568	−0.393784	−	0.919203i	$-0.628834\pi$
$149$	−9.61350	−0.787568	−0.393784	+	0.919203i	$0.628834\pi$
$150$	0	0
$151$	−9.12217	−0.742352	−0.371176	−	0.928563i	$-0.621045\pi$
$151$	−9.12217	−0.742352	−0.371176	+	0.928563i	$0.621045\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−0.0993582	−0.00800651
$155$	0	0
$156$	0	0
$157$	−4.93438	−0.393806	−0.196903	−	0.980423i	$-0.563088\pi$
$157$	−4.93438	−0.393806	−0.196903	+	0.980423i	$0.563088\pi$
$158$	6.02827	0.479584
$159$	0	0
$160$	0	0
$161$	0.448517	0.0353481
$162$	0	0
$163$	−11.2835	−0.883795	−0.441897	−	0.897066i	$-0.645694\pi$
$163$	−11.2835	−0.883795	−0.441897	+	0.897066i	$0.645694\pi$
$164$	−9.96265	−0.777953
$165$	0	0
$166$	12.8633	0.998385
$167$	−11.8688	−0.918432	−0.459216	−	0.888325i	$-0.651870\pi$
$167$	−11.8688	−0.918432	−0.459216	+	0.888325i	$0.651870\pi$
$168$	0	0
$169$	−3.21792	−0.247532
$170$	0	0
$171$	0	0
$172$	−5.51414	−0.420449
$173$	−4.27807	−0.325256	−0.162628	−	0.986687i	$-0.551997\pi$
$173$	−4.27807	−0.325256	−0.162628	+	0.986687i	$0.551997\pi$
$174$	0	0
$175$	0	0
$176$	0.514137	0.0387545
$177$	0	0
$178$	9.01920	0.676018
$179$	−22.3118	−1.66766	−0.833832	−	0.552019i	$-0.813858\pi$
$179$	−22.3118	−1.66766	−0.833832	+	0.552019i	$0.813858\pi$
$180$	0	0
$181$	−11.4576	−0.851636	−0.425818	−	0.904809i	$-0.640014\pi$
$181$	−11.4576	−0.851636	−0.425818	+	0.904809i	$0.640014\pi$
$182$	0.604422	0.0448028
$183$	0	0
$184$	−2.32088	−0.171098
$185$	0	0
$186$	0	0
$187$	1.45759	0.106589
$188$	−1.70739	−0.124524
$189$	0	0
$190$	0	0
$191$	−18.4148	−1.33245	−0.666223	−	0.745752i	$-0.732090\pi$
$191$	−18.4148	−1.33245	−0.666223	+	0.745752i	$0.732090\pi$
$192$	0	0
$193$	−0.193252	−0.0139106	−0.00695531	−	0.999976i	$-0.502214\pi$
$193$	−0.193252	−0.0139106	−0.00695531	+	0.999976i	$0.502214\pi$
$194$	8.89157	0.638377
$195$	0	0
$196$	−6.96265	−0.497332
$197$	−18.5990	−1.32512	−0.662560	−	0.749008i	$-0.730530\pi$
$197$	−18.5990	−1.32512	−0.662560	+	0.749008i	$0.730530\pi$
$198$	0	0
$199$	20.5671	1.45796	0.728981	−	0.684534i	$-0.239994\pi$
$199$	20.5671	1.45796	0.728981	+	0.684534i	$0.239994\pi$
$200$	0	0
$201$	0	0
$202$	2.15591	0.151689
$203$	0.0318825	0.00223771
$204$	0	0
$205$	0	0
$206$	−10.3209	−0.719090
$207$	0	0
$208$	−3.12763	−0.216862
$209$	0.514137	0.0355636
$210$	0	0
$211$	4.57068	0.314659	0.157329	−	0.987546i	$-0.449712\pi$
$211$	4.57068	0.314659	0.157329	+	0.987546i	$0.449712\pi$
$212$	2.90064	0.199217
$213$	0	0
$214$	5.57068	0.380804
$215$	0	0
$216$	0	0
$217$	1.75020	0.118811
$218$	5.61350	0.380194
$219$	0	0
$220$	0	0
$221$	−8.86690	−0.596453
$222$	0	0
$223$	3.93438	0.263466	0.131733	−	0.991285i	$-0.457946\pi$
$223$	3.93438	0.263466	0.131733	+	0.991285i	$0.457946\pi$
$224$	−0.193252	−0.0129122
$225$	0	0
$226$	−6.96265	−0.463149
$227$	−11.4713	−0.761379	−0.380689	−	0.924703i	$-0.624313\pi$
$227$	−11.4713	−0.761379	−0.380689	+	0.924703i	$0.624313\pi$
$228$	0	0
$229$	12.4768	0.824490	0.412245	−	0.911073i	$-0.364745\pi$
$229$	12.4768	0.824490	0.412245	+	0.911073i	$0.364745\pi$
$230$	0	0
$231$	0	0
$232$	−0.164979	−0.0108314
$233$	27.4021	1.79517	0.897586	−	0.440840i	$-0.145319\pi$
$233$	27.4021	1.79517	0.897586	+	0.440840i	$0.145319\pi$
$234$	0	0
$235$	0	0
$236$	−13.9572	−0.908536
$237$	0	0
$238$	−0.547875	−0.0355134
$239$	14.4057	0.931828	0.465914	−	0.884830i	$-0.345726\pi$
$239$	14.4057	0.931828	0.465914	+	0.884830i	$0.345726\pi$
$240$	0	0
$241$	−21.7886	−1.40353	−0.701764	−	0.712410i	$-0.747604\pi$
$241$	−21.7886	−1.40353	−0.701764	+	0.712410i	$0.747604\pi$
$242$	−10.7357	−0.690115
$243$	0	0
$244$	5.92892	0.379560
$245$	0	0
$246$	0	0
$247$	−3.12763	−0.199006
$248$	−9.05655	−0.575091
$249$	0	0
$250$	0	0
$251$	−21.9627	−1.38627	−0.693135	−	0.720808i	$-0.743771\pi$
$251$	−21.9627	−1.38627	−0.693135	+	0.720808i	$0.743771\pi$
$252$	0	0
$253$	−1.19325	−0.0750191
$254$	13.6983	0.859509
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.0656204	−0.00409329	−0.00204664	−	0.999998i	$-0.500651\pi$
$257$	−0.0656204	−0.00409329	−0.00204664	+	0.999998i	$0.500651\pi$
$258$	0	0
$259$	0.585221	0.0363639
$260$	0	0
$261$	0	0
$262$	−10.4768	−0.647258
$263$	24.8314	1.53117	0.765585	−	0.643334i	$-0.222450\pi$
$263$	24.8314	1.53117	0.765585	+	0.643334i	$0.222450\pi$
$264$	0	0
$265$	0	0
$266$	−0.193252	−0.0118491
$267$	0	0
$268$	−4.22153	−0.257871
$269$	−19.8578	−1.21075	−0.605377	−	0.795939i	$-0.706978\pi$
$269$	−19.8578	−1.21075	−0.605377	+	0.795939i	$0.706978\pi$
$270$	0	0
$271$	0.891569	0.0541590	0.0270795	−	0.999633i	$-0.491379\pi$
$271$	0.891569	0.0541590	0.0270795	+	0.999633i	$0.491379\pi$
$272$	2.83502	0.171898
$273$	0	0
$274$	−11.6700	−0.705013
$275$	0	0
$276$	0	0
$277$	6.76940	0.406734	0.203367	−	0.979103i	$-0.434812\pi$
$277$	6.76940	0.406734	0.203367	+	0.979103i	$0.434812\pi$
$278$	12.7411	0.764162
$279$	0	0
$280$	0	0
$281$	−17.2553	−1.02936	−0.514681	−	0.857382i	$-0.672090\pi$
$281$	−17.2553	−1.02936	−0.514681	+	0.857382i	$0.672090\pi$
$282$	0	0
$283$	−20.6983	−1.23039	−0.615194	−	0.788376i	$-0.710922\pi$
$283$	−20.6983	−1.23039	−0.615194	+	0.788376i	$0.710922\pi$
$284$	−3.16498	−0.187807
$285$	0	0
$286$	−1.60803	−0.0950849
$287$	1.92531	0.113647
$288$	0	0
$289$	−8.96265	−0.527215
$290$	0	0
$291$	0	0
$292$	8.37743	0.490252
$293$	0.712853	0.0416453	0.0208227	−	0.999783i	$-0.493371\pi$
$293$	0.712853	0.0416453	0.0208227	+	0.999783i	$0.493371\pi$
$294$	0	0
$295$	0	0
$296$	−3.02827	−0.176015
$297$	0	0
$298$	−9.61350	−0.556895
$299$	7.25887	0.419791
$300$	0	0
$301$	1.06562	0.0614213
$302$	−9.12217	−0.524922
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	−5.11856	−0.292132	−0.146066	−	0.989275i	$-0.546661\pi$
$307$	−5.11856	−0.292132	−0.146066	+	0.989275i	$0.546661\pi$
$308$	−0.0993582	−0.00566146
$309$	0	0
$310$	0	0
$311$	−3.26434	−0.185104	−0.0925518	−	0.995708i	$-0.529502\pi$
$311$	−3.26434	−0.185104	−0.0925518	+	0.995708i	$0.529502\pi$
$312$	0	0
$313$	−2.96265	−0.167459	−0.0837295	−	0.996489i	$-0.526683\pi$
$313$	−2.96265	−0.167459	−0.0837295	+	0.996489i	$0.526683\pi$
$314$	−4.93438	−0.278463
$315$	0	0
$316$	6.02827	0.339117
$317$	−12.1650	−0.683253	−0.341627	−	0.939836i	$-0.610978\pi$
$317$	−12.1650	−0.683253	−0.341627	+	0.939836i	$0.610978\pi$
$318$	0	0
$319$	−0.0848216	−0.00474910
$320$	0	0
$321$	0	0
$322$	0.448517	0.0249949
$323$	2.83502	0.157745
$324$	0	0
$325$	0	0
$326$	−11.2835	−0.624937
$327$	0	0
$328$	−9.96265	−0.550096
$329$	0.329957	0.0181911
$330$	0	0
$331$	6.67551	0.366919	0.183460	−	0.983027i	$-0.441270\pi$
$331$	6.67551	0.366919	0.183460	+	0.983027i	$0.441270\pi$
$332$	12.8633	0.705965
$333$	0	0
$334$	−11.8688	−0.649430
$335$	0	0
$336$	0	0
$337$	−9.72659	−0.529841	−0.264921	−	0.964270i	$-0.585346\pi$
$337$	−9.72659	−0.529841	−0.264921	+	0.964270i	$0.585346\pi$
$338$	−3.21792	−0.175032
$339$	0	0
$340$	0	0
$341$	−4.65631	−0.252153
$342$	0	0
$343$	2.69832	0.145695
$344$	−5.51414	−0.297302
$345$	0	0
$346$	−4.27807	−0.229991
$347$	9.08482	0.487699	0.243849	−	0.969813i	$-0.421590\pi$
$347$	9.08482	0.487699	0.243849	+	0.969813i	$0.421590\pi$
$348$	0	0
$349$	26.0047	1.39200	0.695999	−	0.718043i	$-0.254962\pi$
$349$	26.0047	1.39200	0.695999	+	0.718043i	$0.254962\pi$
$350$	0	0
$351$	0	0
$352$	0.514137	0.0274036
$353$	27.7321	1.47603	0.738014	−	0.674785i	$-0.235764\pi$
$353$	27.7321	1.47603	0.738014	+	0.674785i	$0.235764\pi$
$354$	0	0
$355$	0	0
$356$	9.01920	0.478017
$357$	0	0
$358$	−22.3118	−1.17922
$359$	0.746591	0.0394036	0.0197018	−	0.999806i	$-0.493728\pi$
$359$	0.746591	0.0394036	0.0197018	+	0.999806i	$0.493728\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−11.4576	−0.602198
$363$	0	0
$364$	0.604422	0.0316803
$365$	0	0
$366$	0	0
$367$	13.9945	0.730509	0.365254	−	0.930908i	$-0.380982\pi$
$367$	13.9945	0.730509	0.365254	+	0.930908i	$0.380982\pi$
$368$	−2.32088	−0.120984
$369$	0	0
$370$	0	0
$371$	−0.560556	−0.0291026
$372$	0	0
$373$	20.6099	1.06714	0.533570	−	0.845756i	$-0.320850\pi$
$373$	20.6099	1.06714	0.533570	+	0.845756i	$0.320850\pi$
$374$	1.45759	0.0753701
$375$	0	0
$376$	−1.70739	−0.0880519
$377$	0.515992	0.0265750
$378$	0	0
$379$	1.63164	0.0838117	0.0419059	−	0.999122i	$-0.486657\pi$
$379$	1.63164	0.0838117	0.0419059	+	0.999122i	$0.486657\pi$
$380$	0	0
$381$	0	0
$382$	−18.4148	−0.942182
$383$	−12.3173	−0.629383	−0.314692	−	0.949194i	$-0.601901\pi$
$383$	−12.3173	−0.629383	−0.314692	+	0.949194i	$0.601901\pi$
$384$	0	0
$385$	0	0
$386$	−0.193252	−0.00983629
$387$	0	0
$388$	8.89157	0.451401
$389$	−3.61350	−0.183211	−0.0916057	−	0.995795i	$-0.529200\pi$
$389$	−3.61350	−0.183211	−0.0916057	+	0.995795i	$0.529200\pi$
$390$	0	0
$391$	−6.57976	−0.332753
$392$	−6.96265	−0.351667
$393$	0	0
$394$	−18.5990	−0.937002
$395$	0	0
$396$	0	0
$397$	−2.88250	−0.144668	−0.0723342	−	0.997380i	$-0.523045\pi$
$397$	−2.88250	−0.144668	−0.0723342	+	0.997380i	$0.523045\pi$
$398$	20.5671	1.03093
$399$	0	0
$400$	0	0
$401$	−28.9253	−1.44446	−0.722230	−	0.691652i	$-0.756883\pi$
$401$	−28.9253	−1.44446	−0.722230	+	0.691652i	$0.756883\pi$
$402$	0	0
$403$	28.3255	1.41100
$404$	2.15591	0.107260
$405$	0	0
$406$	0.0318825	0.00158230
$407$	−1.55695	−0.0771750
$408$	0	0
$409$	21.8578	1.08080	0.540400	−	0.841408i	$-0.318273\pi$
$409$	21.8578	1.08080	0.540400	+	0.841408i	$0.318273\pi$
$410$	0	0
$411$	0	0
$412$	−10.3209	−0.508473
$413$	2.69726	0.132724
$414$	0	0
$415$	0	0
$416$	−3.12763	−0.153345
$417$	0	0
$418$	0.514137	0.0251473
$419$	8.38650	0.409708	0.204854	−	0.978793i	$-0.434328\pi$
$419$	8.38650	0.409708	0.204854	+	0.978793i	$0.434328\pi$
$420$	0	0
$421$	−13.1523	−0.641004	−0.320502	−	0.947248i	$-0.603852\pi$
$421$	−13.1523	−0.641004	−0.320502	+	0.947248i	$0.603852\pi$
$422$	4.57068	0.222497
$423$	0	0
$424$	2.90064	0.140868
$425$	0	0
$426$	0	0
$427$	−1.14578	−0.0554481
$428$	5.57068	0.269269
$429$	0	0
$430$	0	0
$431$	−16.1806	−0.779391	−0.389695	−	0.920944i	$-0.627420\pi$
$431$	−16.1806	−0.779391	−0.389695	+	0.920944i	$0.627420\pi$
$432$	0	0
$433$	13.9945	0.672534	0.336267	−	0.941767i	$-0.390835\pi$
$433$	13.9945	0.672534	0.336267	+	0.941767i	$0.390835\pi$
$434$	1.75020	0.0840123
$435$	0	0
$436$	5.61350	0.268838
$437$	−2.32088	−0.111023
$438$	0	0
$439$	−12.3401	−0.588960	−0.294480	−	0.955658i	$-0.595146\pi$
$439$	−12.3401	−0.588960	−0.294480	+	0.955658i	$0.595146\pi$
$440$	0	0
$441$	0	0
$442$	−8.86690	−0.421756
$443$	−14.5899	−0.693186	−0.346593	−	0.938016i	$-0.612662\pi$
$443$	−14.5899	−0.693186	−0.346593	+	0.938016i	$0.612662\pi$
$444$	0	0
$445$	0	0
$446$	3.93438	0.186298
$447$	0	0
$448$	−0.193252	−0.00913032
$449$	−3.74659	−0.176813	−0.0884063	−	0.996084i	$-0.528177\pi$
$449$	−3.74659	−0.176813	−0.0884063	+	0.996084i	$0.528177\pi$
$450$	0	0
$451$	−5.12217	−0.241193
$452$	−6.96265	−0.327496
$453$	0	0
$454$	−11.4713	−0.538376
$455$	0	0
$456$	0	0
$457$	18.2745	0.854843	0.427422	−	0.904052i	$-0.359422\pi$
$457$	18.2745	0.854843	0.427422	+	0.904052i	$0.359422\pi$
$458$	12.4768	0.583002
$459$	0	0
$460$	0	0
$461$	13.0957	0.609930	0.304965	−	0.952364i	$-0.401355\pi$
$461$	13.0957	0.609930	0.304965	+	0.952364i	$0.401355\pi$
$462$	0	0
$463$	−15.0283	−0.698423	−0.349212	−	0.937044i	$-0.613551\pi$
$463$	−15.0283	−0.698423	−0.349212	+	0.937044i	$0.613551\pi$
$464$	−0.164979	−0.00765894
$465$	0	0
$466$	27.4021	1.26938
$467$	−21.7302	−1.00555	−0.502777	−	0.864416i	$-0.667688\pi$
$467$	−21.7302	−1.00555	−0.502777	+	0.864416i	$0.667688\pi$
$468$	0	0
$469$	0.815820	0.0376711
$470$	0	0
$471$	0	0
$472$	−13.9572	−0.642432
$473$	−2.83502	−0.130354
$474$	0	0
$475$	0	0
$476$	−0.547875	−0.0251118
$477$	0	0
$478$	14.4057	0.658902
$479$	0.962653	0.0439848	0.0219924	−	0.999758i	$-0.492999\pi$
$479$	0.962653	0.0439848	0.0219924	+	0.999758i	$0.492999\pi$
$480$	0	0
$481$	9.47133	0.431855
$482$	−21.7886	−0.992444
$483$	0	0
$484$	−10.7357	−0.487985
$485$	0	0
$486$	0	0
$487$	42.5561	1.92840	0.964202	−	0.265170i	$-0.0854282\pi$
$487$	42.5561	1.92840	0.964202	+	0.265170i	$0.0854282\pi$
$488$	5.92892	0.268389
$489$	0	0
$490$	0	0
$491$	14.7549	0.665878	0.332939	−	0.942948i	$-0.391960\pi$
$491$	14.7549	0.665878	0.332939	+	0.942948i	$0.391960\pi$
$492$	0	0
$493$	−0.467718	−0.0210649
$494$	−3.12763	−0.140719
$495$	0	0
$496$	−9.05655	−0.406651
$497$	0.611640	0.0274358
$498$	0	0
$499$	−7.64538	−0.342254	−0.171127	−	0.985249i	$-0.554741\pi$
$499$	−7.64538	−0.342254	−0.171127	+	0.985249i	$0.554741\pi$
$500$	0	0
$501$	0	0
$502$	−21.9627	−0.980241
$503$	4.51053	0.201115	0.100557	−	0.994931i	$-0.467937\pi$
$503$	4.51053	0.201115	0.100557	+	0.994931i	$0.467937\pi$
$504$	0	0
$505$	0	0
$506$	−1.19325	−0.0530465
$507$	0	0
$508$	13.6983	0.607765
$509$	41.1943	1.82591	0.912953	−	0.408065i	$-0.133796\pi$
$509$	41.1943	1.82591	0.912953	+	0.408065i	$0.133796\pi$
$510$	0	0
$511$	−1.61896	−0.0716185
$512$	1.00000	0.0441942
$513$	0	0
$514$	−0.0656204	−0.00289439
$515$	0	0
$516$	0	0
$517$	−0.877832	−0.0386070
$518$	0.585221	0.0257131
$519$	0	0
$520$	0	0
$521$	−10.1595	−0.445096	−0.222548	−	0.974922i	$-0.571437\pi$
$521$	−10.1595	−0.445096	−0.222548	+	0.974922i	$0.571437\pi$
$522$	0	0
$523$	−14.6308	−0.639762	−0.319881	−	0.947458i	$-0.603643\pi$
$523$	−14.6308	−0.639762	−0.319881	+	0.947458i	$0.603643\pi$
$524$	−10.4768	−0.457681
$525$	0	0
$526$	24.8314	1.08270
$527$	−25.6755	−1.11844
$528$	0	0
$529$	−17.6135	−0.765804
$530$	0	0
$531$	0	0
$532$	−0.193252	−0.00837856
$533$	31.1595	1.34967
$534$	0	0
$535$	0	0
$536$	−4.22153	−0.182342
$537$	0	0
$538$	−19.8578	−0.856132
$539$	−3.57976	−0.154191
$540$	0	0
$541$	−13.0511	−0.561110	−0.280555	−	0.959838i	$-0.590518\pi$
$541$	−13.0511	−0.561110	−0.280555	+	0.959838i	$0.590518\pi$
$542$	0.891569	0.0382962
$543$	0	0
$544$	2.83502	0.121551
$545$	0	0
$546$	0	0
$547$	−10.0420	−0.429365	−0.214683	−	0.976684i	$-0.568872\pi$
$547$	−10.0420	−0.429365	−0.214683	+	0.976684i	$0.568872\pi$
$548$	−11.6700	−0.498520
$549$	0	0
$550$	0	0
$551$	−0.164979	−0.00702832
$552$	0	0
$553$	−1.16498	−0.0495399
$554$	6.76940	0.287604
$555$	0	0
$556$	12.7411	0.540344
$557$	6.03188	0.255579	0.127790	−	0.991801i	$-0.459212\pi$
$557$	6.03188	0.255579	0.127790	+	0.991801i	$0.459212\pi$
$558$	0	0
$559$	17.2462	0.729436
$560$	0	0
$561$	0	0
$562$	−17.2553	−0.727869
$563$	35.2407	1.48522	0.742610	−	0.669724i	$-0.233588\pi$
$563$	35.2407	1.48522	0.742610	+	0.669724i	$0.233588\pi$
$564$	0	0
$565$	0	0
$566$	−20.6983	−0.870015
$567$	0	0
$568$	−3.16498	−0.132800
$569$	−21.9144	−0.918699	−0.459349	−	0.888256i	$-0.651917\pi$
$569$	−21.9144	−0.918699	−0.459349	+	0.888256i	$0.651917\pi$
$570$	0	0
$571$	−21.2407	−0.888897	−0.444448	−	0.895804i	$-0.646600\pi$
$571$	−21.2407	−0.888897	−0.444448	+	0.895804i	$0.646600\pi$
$572$	−1.60803	−0.0672352
$573$	0	0
$574$	1.92531	0.0803608
$575$	0	0
$576$	0	0
$577$	32.9253	1.37070	0.685349	−	0.728215i	$-0.259650\pi$
$577$	32.9253	1.37070	0.685349	+	0.728215i	$0.259650\pi$
$578$	−8.96265	−0.372797
$579$	0	0
$580$	0	0
$581$	−2.48586	−0.103131
$582$	0	0
$583$	1.49133	0.0617645
$584$	8.37743	0.346661
$585$	0	0
$586$	0.712853	0.0294477
$587$	−21.5798	−0.890692	−0.445346	−	0.895359i	$-0.646919\pi$
$587$	−21.5798	−0.890692	−0.445346	+	0.895359i	$0.646919\pi$
$588$	0	0
$589$	−9.05655	−0.373169
$590$	0	0
$591$	0	0
$592$	−3.02827	−0.124461
$593$	1.98907	0.0816814	0.0408407	−	0.999166i	$-0.486996\pi$
$593$	1.98907	0.0816814	0.0408407	+	0.999166i	$0.486996\pi$
$594$	0	0
$595$	0	0
$596$	−9.61350	−0.393784
$597$	0	0
$598$	7.25887	0.296837
$599$	−33.7321	−1.37825	−0.689127	−	0.724640i	$-0.742006\pi$
$599$	−33.7321	−1.37825	−0.689127	+	0.724640i	$0.742006\pi$
$600$	0	0
$601$	25.7211	1.04919	0.524593	−	0.851353i	$-0.324217\pi$
$601$	25.7211	1.04919	0.524593	+	0.851353i	$0.324217\pi$
$602$	1.06562	0.0434314
$603$	0	0
$604$	−9.12217	−0.371176
$605$	0	0
$606$	0	0
$607$	37.8488	1.53623	0.768117	−	0.640310i	$-0.221194\pi$
$607$	37.8488	1.53623	0.768117	+	0.640310i	$0.221194\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	5.34009	0.216037
$612$	0	0
$613$	15.4713	0.624881	0.312440	−	0.949937i	$-0.398854\pi$
$613$	15.4713	0.624881	0.312440	+	0.949937i	$0.398854\pi$
$614$	−5.11856	−0.206568
$615$	0	0
$616$	−0.0993582	−0.00400326
$617$	−10.3118	−0.415138	−0.207569	−	0.978220i	$-0.566555\pi$
$617$	−10.3118	−0.415138	−0.207569	+	0.978220i	$0.566555\pi$
$618$	0	0
$619$	12.6099	0.506834	0.253417	−	0.967357i	$-0.418446\pi$
$619$	12.6099	0.506834	0.253417	+	0.967357i	$0.418446\pi$
$620$	0	0
$621$	0	0
$622$	−3.26434	−0.130888
$623$	−1.74298	−0.0698311
$624$	0	0
$625$	0	0
$626$	−2.96265	−0.118411
$627$	0	0
$628$	−4.93438	−0.196903
$629$	−8.58522	−0.342315
$630$	0	0
$631$	−9.34009	−0.371823	−0.185911	−	0.982566i	$-0.559524\pi$
$631$	−9.34009	−0.371823	−0.185911	+	0.982566i	$0.559524\pi$
$632$	6.02827	0.239792
$633$	0	0
$634$	−12.1650	−0.483133
$635$	0	0
$636$	0	0
$637$	21.7766	0.862821
$638$	−0.0848216	−0.00335812
$639$	0	0
$640$	0	0
$641$	−28.3310	−1.11901	−0.559504	−	0.828828i	$-0.689008\pi$
$641$	−28.3310	−1.11901	−0.559504	+	0.828828i	$0.689008\pi$
$642$	0	0
$643$	25.0848	0.989249	0.494624	−	0.869107i	$-0.335306\pi$
$643$	25.0848	0.989249	0.494624	+	0.869107i	$0.335306\pi$
$644$	0.448517	0.0176740
$645$	0	0
$646$	2.83502	0.111542
$647$	−19.4623	−0.765140	−0.382570	−	0.923926i	$-0.624961\pi$
$647$	−19.4623	−0.765140	−0.382570	+	0.923926i	$0.624961\pi$
$648$	0	0
$649$	−7.17591	−0.281679
$650$	0	0
$651$	0	0
$652$	−11.2835	−0.441897
$653$	−12.5671	−0.491788	−0.245894	−	0.969297i	$-0.579081\pi$
$653$	−12.5671	−0.491788	−0.245894	+	0.969297i	$0.579081\pi$
$654$	0	0
$655$	0	0
$656$	−9.96265	−0.388976
$657$	0	0
$658$	0.329957	0.0128631
$659$	8.22338	0.320337	0.160169	−	0.987090i	$-0.448796\pi$
$659$	8.22338	0.320337	0.160169	+	0.987090i	$0.448796\pi$
$660$	0	0
$661$	26.2371	1.02051	0.510253	−	0.860024i	$-0.329552\pi$
$661$	26.2371	1.02051	0.510253	+	0.860024i	$0.329552\pi$
$662$	6.67551	0.259451
$663$	0	0
$664$	12.8633	0.499193
$665$	0	0
$666$	0	0
$667$	0.382896	0.0148258
$668$	−11.8688	−0.459216
$669$	0	0
$670$	0	0
$671$	3.04827	0.117677
$672$	0	0
$673$	7.29622	0.281249	0.140624	−	0.990063i	$-0.455089\pi$
$673$	7.29622	0.281249	0.140624	+	0.990063i	$0.455089\pi$
$674$	−9.72659	−0.374654
$675$	0	0
$676$	−3.21792	−0.123766
$677$	−29.2234	−1.12315	−0.561573	−	0.827427i	$-0.689804\pi$
$677$	−29.2234	−1.12315	−0.561573	+	0.827427i	$0.689804\pi$
$678$	0	0
$679$	−1.71832	−0.0659430
$680$	0	0
$681$	0	0
$682$	−4.65631	−0.178299
$683$	−11.9006	−0.455365	−0.227683	−	0.973735i	$-0.573115\pi$
$683$	−11.9006	−0.455365	−0.227683	+	0.973735i	$0.573115\pi$
$684$	0	0
$685$	0	0
$686$	2.69832	0.103022
$687$	0	0
$688$	−5.51414	−0.210224
$689$	−9.07214	−0.345621
$690$	0	0
$691$	0.0137372	0.000522589 0	0.000261294	−	1.00000i	$-0.499917\pi$
$691$	0.0137372	0.000522589 0	0.000261294		1.00000i	$0.499917\pi$
$692$	−4.27807	−0.162628
$693$	0	0
$694$	9.08482	0.344855
$695$	0	0
$696$	0	0
$697$	−28.2443	−1.06983
$698$	26.0047	0.984291
$699$	0	0
$700$	0	0
$701$	−0.266192	−0.0100539	−0.00502697	−	0.999987i	$-0.501600\pi$
$701$	−0.266192	−0.0100539	−0.00502697	+	0.999987i	$0.501600\pi$
$702$	0	0
$703$	−3.02827	−0.114214
$704$	0.514137	0.0193773
$705$	0	0
$706$	27.7321	1.04371
$707$	−0.416634	−0.0156691
$708$	0	0
$709$	−9.50506	−0.356970	−0.178485	−	0.983943i	$-0.557120\pi$
$709$	−9.50506	−0.356970	−0.178485	+	0.983943i	$0.557120\pi$
$710$	0	0
$711$	0	0
$712$	9.01920	0.338009
$713$	21.0192	0.787175
$714$	0	0
$715$	0	0
$716$	−22.3118	−0.833832
$717$	0	0
$718$	0.746591	0.0278625
$719$	−10.4895	−0.391191	−0.195596	−	0.980685i	$-0.562664\pi$
$719$	−10.4895	−0.391191	−0.195596	+	0.980685i	$0.562664\pi$
$720$	0	0
$721$	1.99454	0.0742804
$722$	1.00000	0.0372161
$723$	0	0
$724$	−11.4576	−0.425818
$725$	0	0
$726$	0	0
$727$	−36.3502	−1.34815	−0.674077	−	0.738661i	$-0.735459\pi$
$727$	−36.3502	−1.34815	−0.674077	+	0.738661i	$0.735459\pi$
$728$	0.604422	0.0224014
$729$	0	0
$730$	0	0
$731$	−15.6327	−0.578196
$732$	0	0
$733$	22.0337	0.813835	0.406917	−	0.913465i	$-0.366604\pi$
$733$	22.0337	0.813835	0.406917	+	0.913465i	$0.366604\pi$
$734$	13.9945	0.516548
$735$	0	0
$736$	−2.32088	−0.0855489
$737$	−2.17044	−0.0799493
$738$	0	0
$739$	17.0848	0.628475	0.314238	−	0.949344i	$-0.398251\pi$
$739$	17.0848	0.628475	0.314238	+	0.949344i	$0.398251\pi$
$740$	0	0
$741$	0	0
$742$	−0.560556	−0.0205787
$743$	26.7549	0.981541	0.490770	−	0.871289i	$-0.336715\pi$
$743$	26.7549	0.981541	0.490770	+	0.871289i	$0.336715\pi$
$744$	0	0
$745$	0	0
$746$	20.6099	0.754582
$747$	0	0
$748$	1.45759	0.0532947
$749$	−1.07655	−0.0393362
$750$	0	0
$751$	−31.7002	−1.15676	−0.578378	−	0.815769i	$-0.696314\pi$
$751$	−31.7002	−1.15676	−0.578378	+	0.815769i	$0.696314\pi$
$752$	−1.70739	−0.0622621
$753$	0	0
$754$	0.515992	0.0187913
$755$	0	0
$756$	0	0
$757$	35.1933	1.27912	0.639560	−	0.768741i	$-0.279116\pi$
$757$	35.1933	1.27912	0.639560	+	0.768741i	$0.279116\pi$
$758$	1.63164	0.0592638
$759$	0	0
$760$	0	0
$761$	25.9764	0.941643	0.470822	−	0.882228i	$-0.343958\pi$
$761$	25.9764	0.941643	0.470822	+	0.882228i	$0.343958\pi$
$762$	0	0
$763$	−1.08482	−0.0392732
$764$	−18.4148	−0.666223
$765$	0	0
$766$	−12.3173	−0.445041
$767$	43.6530	1.57622
$768$	0	0
$769$	−44.6146	−1.60884	−0.804421	−	0.594060i	$-0.797524\pi$
$769$	−44.6146	−1.60884	−0.804421	+	0.594060i	$0.797524\pi$
$770$	0	0
$771$	0	0
$772$	−0.193252	−0.00695531
$773$	−51.5844	−1.85536	−0.927681	−	0.373373i	$-0.878201\pi$
$773$	−51.5844	−1.85536	−0.927681	+	0.373373i	$0.878201\pi$
$774$	0	0
$775$	0	0
$776$	8.89157	0.319189
$777$	0	0
$778$	−3.61350	−0.129550
$779$	−9.96265	−0.356949
$780$	0	0
$781$	−1.62723	−0.0582270
$782$	−6.57976	−0.235292
$783$	0	0
$784$	−6.96265	−0.248666
$785$	0	0
$786$	0	0
$787$	−15.8916	−0.566473	−0.283237	−	0.959050i	$-0.591408\pi$
$787$	−15.8916	−0.566473	−0.283237	+	0.959050i	$0.591408\pi$
$788$	−18.5990	−0.662560
$789$	0	0
$790$	0	0
$791$	1.34555	0.0478422
$792$	0	0
$793$	−18.5435	−0.658498
$794$	−2.88250	−0.102296
$795$	0	0
$796$	20.5671	0.728981
$797$	43.6146	1.54491	0.772453	−	0.635072i	$-0.219029\pi$
$797$	43.6146	1.54491	0.772453	+	0.635072i	$0.219029\pi$
$798$	0	0
$799$	−4.84049	−0.171244
$800$	0	0
$801$	0	0
$802$	−28.9253	−1.02139
$803$	4.30715	0.151996
$804$	0	0
$805$	0	0
$806$	28.3255	0.997725
$807$	0	0
$808$	2.15591	0.0758445
$809$	−21.4659	−0.754700	−0.377350	−	0.926071i	$-0.623165\pi$
$809$	−21.4659	−0.754700	−0.377350	+	0.926071i	$0.623165\pi$
$810$	0	0
$811$	−4.84409	−0.170099	−0.0850496	−	0.996377i	$-0.527105\pi$
$811$	−4.84409	−0.170099	−0.0850496	+	0.996377i	$0.527105\pi$
$812$	0.0318825	0.00111886
$813$	0	0
$814$	−1.55695	−0.0545710
$815$	0	0
$816$	0	0
$817$	−5.51414	−0.192915
$818$	21.8578	0.764241
$819$	0	0
$820$	0	0
$821$	−7.65631	−0.267207	−0.133603	−	0.991035i	$-0.542655\pi$
$821$	−7.65631	−0.267207	−0.133603	+	0.991035i	$0.542655\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	−10.3209	−0.359545
$825$	0	0
$826$	2.69726	0.0938497
$827$	13.7875	0.479440	0.239720	−	0.970842i	$-0.422944\pi$
$827$	13.7875	0.479440	0.239720	+	0.970842i	$0.422944\pi$
$828$	0	0
$829$	35.0529	1.21744	0.608719	−	0.793386i	$-0.291684\pi$
$829$	35.0529	1.21744	0.608719	+	0.793386i	$0.291684\pi$
$830$	0	0
$831$	0	0
$832$	−3.12763	−0.108431
$833$	−19.7393	−0.683925
$834$	0	0
$835$	0	0
$836$	0.514137	0.0177818
$837$	0	0
$838$	8.38650	0.289707
$839$	43.6464	1.50684	0.753421	−	0.657538i	$-0.228402\pi$
$839$	43.6464	1.50684	0.753421	+	0.657538i	$0.228402\pi$
$840$	0	0
$841$	−28.9728	−0.999061
$842$	−13.1523	−0.453258
$843$	0	0
$844$	4.57068	0.157329
$845$	0	0
$846$	0	0
$847$	2.07469	0.0712873
$848$	2.90064	0.0996084
$849$	0	0
$850$	0	0
$851$	7.02827	0.240926
$852$	0	0
$853$	44.4431	1.52170	0.760851	−	0.648927i	$-0.224782\pi$
$853$	44.4431	1.52170	0.760851	+	0.648927i	$0.224782\pi$
$854$	−1.14578	−0.0392077
$855$	0	0
$856$	5.57068	0.190402
$857$	−40.5935	−1.38665	−0.693324	−	0.720626i	$-0.743854\pi$
$857$	−40.5935	−1.38665	−0.693324	+	0.720626i	$0.743854\pi$
$858$	0	0
$859$	−6.58522	−0.224685	−0.112342	−	0.993670i	$-0.535835\pi$
$859$	−6.58522	−0.224685	−0.112342	+	0.993670i	$0.535835\pi$
$860$	0	0
$861$	0	0
$862$	−16.1806	−0.551112
$863$	7.19792	0.245020	0.122510	−	0.992467i	$-0.460906\pi$
$863$	7.19792	0.245020	0.122510	+	0.992467i	$0.460906\pi$
$864$	0	0
$865$	0	0
$866$	13.9945	0.475554
$867$	0	0
$868$	1.75020	0.0594057
$869$	3.09936	0.105139
$870$	0	0
$871$	13.2034	0.447379
$872$	5.61350	0.190097
$873$	0	0
$874$	−2.32088	−0.0785051
$875$	0	0
$876$	0	0
$877$	36.6236	1.23669	0.618346	−	0.785906i	$-0.287803\pi$
$877$	36.6236	1.23669	0.618346	+	0.785906i	$0.287803\pi$
$878$	−12.3401	−0.416458
$879$	0	0
$880$	0	0
$881$	−5.80128	−0.195450	−0.0977251	−	0.995213i	$-0.531157\pi$
$881$	−5.80128	−0.195450	−0.0977251	+	0.995213i	$0.531157\pi$
$882$	0	0
$883$	−5.84409	−0.196669	−0.0983347	−	0.995153i	$-0.531352\pi$
$883$	−5.84409	−0.196669	−0.0983347	+	0.995153i	$0.531352\pi$
$884$	−8.86690	−0.298226
$885$	0	0
$886$	−14.5899	−0.490157
$887$	7.68819	0.258144	0.129072	−	0.991635i	$-0.458800\pi$
$887$	7.68819	0.258144	0.129072	+	0.991635i	$0.458800\pi$
$888$	0	0
$889$	−2.64723	−0.0887853
$890$	0	0
$891$	0	0
$892$	3.93438	0.131733
$893$	−1.70739	−0.0571356
$894$	0	0
$895$	0	0
$896$	−0.193252	−0.00645611
$897$	0	0
$898$	−3.74659	−0.125025
$899$	1.49414	0.0498322
$900$	0	0
$901$	8.22338	0.273961
$902$	−5.12217	−0.170550
$903$	0	0
$904$	−6.96265	−0.231574
$905$	0	0
$906$	0	0
$907$	21.8880	0.726778	0.363389	−	0.931638i	$-0.381620\pi$
$907$	21.8880	0.726778	0.363389	+	0.931638i	$0.381620\pi$
$908$	−11.4713	−0.380689
$909$	0	0
$910$	0	0
$911$	−26.4249	−0.875496	−0.437748	−	0.899098i	$-0.644224\pi$
$911$	−26.4249	−0.875496	−0.437748	+	0.899098i	$0.644224\pi$
$912$	0	0
$913$	6.61350	0.218875
$914$	18.2745	0.604466
$915$	0	0
$916$	12.4768	0.412245
$917$	2.02467	0.0668603
$918$	0	0
$919$	12.5232	0.413103	0.206551	−	0.978436i	$-0.433776\pi$
$919$	12.5232	0.413103	0.206551	+	0.978436i	$0.433776\pi$
$920$	0	0
$921$	0	0
$922$	13.0957	0.431286
$923$	9.89889	0.325826
$924$	0	0
$925$	0	0
$926$	−15.0283	−0.493860
$927$	0	0
$928$	−0.164979	−0.00541569
$929$	49.9253	1.63800	0.818998	−	0.573796i	$-0.194530\pi$
$929$	49.9253	1.63800	0.818998	+	0.573796i	$0.194530\pi$
$930$	0	0
$931$	−6.96265	−0.228192
$932$	27.4021	0.897586
$933$	0	0
$934$	−21.7302	−0.711034
$935$	0	0
$936$	0	0
$937$	40.7658	1.33176	0.665880	−	0.746059i	$-0.268056\pi$
$937$	40.7658	1.33176	0.665880	+	0.746059i	$0.268056\pi$
$938$	0.815820	0.0266375
$939$	0	0
$940$	0	0
$941$	−13.2125	−0.430714	−0.215357	−	0.976535i	$-0.569091\pi$
$941$	−13.2125	−0.430714	−0.215357	+	0.976535i	$0.569091\pi$
$942$	0	0
$943$	23.1222	0.752961
$944$	−13.9572	−0.454268
$945$	0	0
$946$	−2.83502	−0.0921745
$947$	−12.4996	−0.406182	−0.203091	−	0.979160i	$-0.565099\pi$
$947$	−12.4996	−0.406182	−0.203091	+	0.979160i	$0.565099\pi$
$948$	0	0
$949$	−26.2015	−0.850538
$950$	0	0
$951$	0	0
$952$	−0.547875	−0.0177567
$953$	−19.8405	−0.642696	−0.321348	−	0.946961i	$-0.604136\pi$
$953$	−19.8405	−0.642696	−0.321348	+	0.946961i	$0.604136\pi$
$954$	0	0
$955$	0	0
$956$	14.4057	0.465914
$957$	0	0
$958$	0.962653	0.0311019
$959$	2.25526	0.0728263
$960$	0	0
$961$	51.0211	1.64584
$962$	9.47133	0.305368
$963$	0	0
$964$	−21.7886	−0.701764
$965$	0	0
$966$	0	0
$967$	21.9709	0.706537	0.353269	−	0.935522i	$-0.385070\pi$
$967$	21.9709	0.706537	0.353269	+	0.935522i	$0.385070\pi$
$968$	−10.7357	−0.345057
$969$	0	0
$970$	0	0
$971$	−14.3546	−0.460662	−0.230331	−	0.973112i	$-0.573981\pi$
$971$	−14.3546	−0.460662	−0.230331	+	0.973112i	$0.573981\pi$
$972$	0	0
$973$	−2.46225	−0.0789362
$974$	42.5561	1.36359
$975$	0	0
$976$	5.92892	0.189780
$977$	51.4340	1.64552	0.822759	−	0.568390i	$-0.192433\pi$
$977$	51.4340	1.64552	0.822759	+	0.568390i	$0.192433\pi$
$978$	0	0
$979$	4.63710	0.148202
$980$	0	0
$981$	0	0
$982$	14.7549	0.470847
$983$	14.8350	0.473164	0.236582	−	0.971612i	$-0.423973\pi$
$983$	14.8350	0.473164	0.236582	+	0.971612i	$0.423973\pi$
$984$	0	0
$985$	0	0
$986$	−0.467718	−0.0148952
$987$	0	0
$988$	−3.12763	−0.0995032
$989$	12.7977	0.406942
$990$	0	0
$991$	9.59349	0.304747	0.152374	−	0.988323i	$-0.451308\pi$
$991$	9.59349	0.304747	0.152374	+	0.988323i	$0.451308\pi$
$992$	−9.05655	−0.287546
$993$	0	0
$994$	0.611640	0.0194000
$995$	0	0
$996$	0	0
$997$	24.7694	0.784455	0.392227	−	0.919868i	$-0.371705\pi$
$997$	24.7694	0.784455	0.392227	+	0.919868i	$0.371705\pi$
$998$	−7.64538	−0.242010
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cn.1.2	yes	3
3.2	odd	2		8550.2.a.cd.1.2	✓	3
5.4	even	2		8550.2.a.cg.1.2	yes	3
15.14	odd	2		8550.2.a.cs.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8550.2.a.cd.1.2	✓	3	3.2	odd	2
8550.2.a.cg.1.2	yes	3	5.4	even	2
8550.2.a.cn.1.2	yes	3	1.1	even	1	trivial
8550.2.a.cs.1.2	yes	3	15.14	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.193252 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.193252 q^{7} +1.00000 q^{8} +0.514137 q^{11} -3.12763 q^{13} -0.193252 q^{14} +1.00000 q^{16} +2.83502 q^{17} +1.00000 q^{19} +0.514137 q^{22} -2.32088 q^{23} -3.12763 q^{26} -0.193252 q^{28} -0.164979 q^{29} -9.05655 q^{31} +1.00000 q^{32} +2.83502 q^{34} -3.02827 q^{37} +1.00000 q^{38} -9.96265 q^{41} -5.51414 q^{43} +0.514137 q^{44} -2.32088 q^{46} -1.70739 q^{47} -6.96265 q^{49} -3.12763 q^{52} +2.90064 q^{53} -0.193252 q^{56} -0.164979 q^{58} -13.9572 q^{59} +5.92892 q^{61} -9.05655 q^{62} +1.00000 q^{64} -4.22153 q^{67} +2.83502 q^{68} -3.16498 q^{71} +8.37743 q^{73} -3.02827 q^{74} +1.00000 q^{76} -0.0993582 q^{77} +6.02827 q^{79} -9.96265 q^{82} +12.8633 q^{83} -5.51414 q^{86} +0.514137 q^{88} +9.01920 q^{89} +0.604422 q^{91} -2.32088 q^{92} -1.70739 q^{94} +8.89157 q^{97} -6.96265 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 5 q^{11} - 2 q^{14} + 3 q^{16} - 6 q^{17} + 3 q^{19} - 5 q^{22} + q^{23} - 2 q^{28} - 15 q^{29} - q^{31} + 3 q^{32} - 6 q^{34} + 4 q^{37} + 3 q^{38} - 6 q^{41} - 10 q^{43} - 5 q^{44} + q^{46} + 3 q^{49} + 5 q^{53} - 2 q^{56} - 15 q^{58} - 12 q^{59} + q^{61} - q^{62} + 3 q^{64} - q^{67} - 6 q^{68} - 24 q^{71} - 9 q^{73} + 4 q^{74} + 3 q^{76} - 4 q^{77} + 5 q^{79} - 6 q^{82} + 11 q^{83} - 10 q^{86} - 5 q^{88} - 23 q^{89} - 38 q^{91} + q^{92} - 14 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 5 * q^11 - 2 * q^14 + 3 * q^16 - 6 * q^17 + 3 * q^19 - 5 * q^22 + q^23 - 2 * q^28 - 15 * q^29 - q^31 + 3 * q^32 - 6 * q^34 + 4 * q^37 + 3 * q^38 - 6 * q^41 - 10 * q^43 - 5 * q^44 + q^46 + 3 * q^49 + 5 * q^53 - 2 * q^56 - 15 * q^58 - 12 * q^59 + q^61 - q^62 + 3 * q^64 - q^67 - 6 * q^68 - 24 * q^71 - 9 * q^73 + 4 * q^74 + 3 * q^76 - 4 * q^77 + 5 * q^79 - 6 * q^82 + 11 * q^83 - 10 * q^86 - 5 * q^88 - 23 * q^89 - 38 * q^91 + q^92 - 14 * q^97 + 3 * q^98

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