LMFDB - Embedded newform 8550.2.a.ck.1.1

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			1.1
Root			$-1.76156$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.761557 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -0.761557 q^{7} -1.00000 q^{8} +0.864641 q^{11} +5.62620 q^{13} +0.761557 q^{14} +1.00000 q^{16} +3.62620 q^{17} +1.00000 q^{19} -0.864641 q^{22} +8.01395 q^{23} -5.62620 q^{26} -0.761557 q^{28} +7.35548 q^{29} +8.11704 q^{31} -1.00000 q^{32} -3.62620 q^{34} +0.476886 q^{37} -1.00000 q^{38} +2.65847 q^{41} +6.86464 q^{43} +0.864641 q^{44} -8.01395 q^{46} -1.25240 q^{47} -6.42003 q^{49} +5.62620 q^{52} -2.37380 q^{53} +0.761557 q^{56} -7.35548 q^{58} -4.49084 q^{59} -10.8646 q^{61} -8.11704 q^{62} +1.00000 q^{64} +1.03228 q^{67} +3.62620 q^{68} +10.1816 q^{71} +16.4017 q^{73} -0.476886 q^{74} +1.00000 q^{76} -0.658473 q^{77} -12.5693 q^{79} -2.65847 q^{82} +0.270718 q^{83} -6.86464 q^{86} -0.864641 q^{88} -0.387755 q^{89} -4.28467 q^{91} +8.01395 q^{92} +1.25240 q^{94} -8.50479 q^{97} +6.42003 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−0.761557	−0.287842	−0.143921	−	0.989589i	$-0.545971\pi$
$7$	−0.761557	−0.287842	−0.143921	+	0.989589i	$0.545971\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	0.864641	0.260699	0.130350	−	0.991468i	$-0.458390\pi$
$11$	0.864641	0.260699	0.130350	+	0.991468i	$0.458390\pi$
$12$	0	0
$13$	5.62620	1.56043	0.780213	−	0.625514i	$-0.215111\pi$
$13$	5.62620	1.56043	0.780213	+	0.625514i	$0.215111\pi$
$14$	0.761557	0.203535
$15$	0	0
$16$	1.00000	0.250000
$17$	3.62620	0.879482	0.439741	−	0.898125i	$-0.355070\pi$
$17$	3.62620	0.879482	0.439741	+	0.898125i	$0.355070\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−0.864641	−0.184342
$23$	8.01395	1.67102	0.835512	−	0.549472i	$-0.185171\pi$
$23$	8.01395	1.67102	0.835512	+	0.549472i	$0.185171\pi$
$24$	0	0
$25$	0	0
$26$	−5.62620	−1.10339
$27$	0	0
$28$	−0.761557	−0.143921
$29$	7.35548	1.36588	0.682939	−	0.730475i	$-0.260701\pi$
$29$	7.35548	1.36588	0.682939	+	0.730475i	$0.260701\pi$
$30$	0	0
$31$	8.11704	1.45786	0.728931	−	0.684587i	$-0.240017\pi$
$31$	8.11704	1.45786	0.728931	+	0.684587i	$0.240017\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.62620	−0.621888
$35$	0	0
$36$	0	0
$37$	0.476886	0.0783995	0.0391998	−	0.999231i	$-0.487519\pi$
$37$	0.476886	0.0783995	0.0391998	+	0.999231i	$0.487519\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	2.65847	0.415184	0.207592	−	0.978216i	$-0.433437\pi$
$41$	2.65847	0.415184	0.207592	+	0.978216i	$0.433437\pi$
$42$	0	0
$43$	6.86464	1.04685	0.523424	−	0.852072i	$-0.324654\pi$
$43$	6.86464	1.04685	0.523424	+	0.852072i	$0.324654\pi$
$44$	0.864641	0.130350
$45$	0	0
$46$	−8.01395	−1.18159
$47$	−1.25240	−0.182681	−0.0913404	−	0.995820i	$-0.529115\pi$
$47$	−1.25240	−0.182681	−0.0913404	+	0.995820i	$0.529115\pi$
$48$	0	0
$49$	−6.42003	−0.917147
$50$	0	0
$51$	0	0
$52$	5.62620	0.780213
$53$	−2.37380	−0.326067	−0.163033	−	0.986621i	$-0.552128\pi$
$53$	−2.37380	−0.326067	−0.163033	+	0.986621i	$0.552128\pi$
$54$	0	0
$55$	0	0
$56$	0.761557	0.101767
$57$	0	0
$58$	−7.35548	−0.965822
$59$	−4.49084	−0.584657	−0.292329	−	0.956318i	$-0.594430\pi$
$59$	−4.49084	−0.584657	−0.292329	+	0.956318i	$0.594430\pi$
$60$	0	0
$61$	−10.8646	−1.39107	−0.695537	−	0.718490i	$-0.744834\pi$
$61$	−10.8646	−1.39107	−0.695537	+	0.718490i	$0.744834\pi$
$62$	−8.11704	−1.03086
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	1.03228	0.126113	0.0630563	−	0.998010i	$-0.479915\pi$
$67$	1.03228	0.126113	0.0630563	+	0.998010i	$0.479915\pi$
$68$	3.62620	0.439741
$69$	0	0
$70$	0	0
$71$	10.1816	1.20833	0.604166	−	0.796858i	$-0.293506\pi$
$71$	10.1816	1.20833	0.604166	+	0.796858i	$0.293506\pi$
$72$	0	0
$73$	16.4017	1.91967	0.959837	−	0.280557i	$-0.0905191\pi$
$73$	16.4017	1.91967	0.959837	+	0.280557i	$0.0905191\pi$
$74$	−0.476886	−0.0554368
$75$	0	0
$76$	1.00000	0.114708
$77$	−0.658473	−0.0750400
$78$	0	0
$79$	−12.5693	−1.41416	−0.707081	−	0.707133i	$-0.749988\pi$
$79$	−12.5693	−1.41416	−0.707081	+	0.707133i	$0.749988\pi$
$80$	0	0
$81$	0	0
$82$	−2.65847	−0.293579
$83$	0.270718	0.0297152	0.0148576	−	0.999890i	$-0.495271\pi$
$83$	0.270718	0.0297152	0.0148576	+	0.999890i	$0.495271\pi$
$84$	0	0
$85$	0	0
$86$	−6.86464	−0.740233
$87$	0	0
$88$	−0.864641	−0.0921710
$89$	−0.387755	−0.0411020	−0.0205510	−	0.999789i	$-0.506542\pi$
$89$	−0.387755	−0.0411020	−0.0205510	+	0.999789i	$0.506542\pi$
$90$	0	0
$91$	−4.28467	−0.449156
$92$	8.01395	0.835512
$93$	0	0
$94$	1.25240	0.129175
$95$	0	0
$96$	0	0
$97$	−8.50479	−0.863531	−0.431765	−	0.901986i	$-0.642109\pi$
$97$	−8.50479	−0.863531	−0.431765	+	0.901986i	$0.642109\pi$
$98$	6.42003	0.648521
$99$	0	0
$100$	0	0
$101$	−16.4157	−1.63342	−0.816710	−	0.577049i	$-0.804204\pi$
$101$	−16.4157	−1.63342	−0.816710	+	0.577049i	$0.804204\pi$
$102$	0	0
$103$	−9.64015	−0.949872	−0.474936	−	0.880020i	$-0.657529\pi$
$103$	−9.64015	−0.949872	−0.474936	+	0.880020i	$0.657529\pi$
$104$	−5.62620	−0.551694
$105$	0	0
$106$	2.37380	0.230564
$107$	−4.28467	−0.414215	−0.207107	−	0.978318i	$-0.566405\pi$
$107$	−4.28467	−0.414215	−0.207107	+	0.978318i	$0.566405\pi$
$108$	0	0
$109$	−13.4200	−1.28541	−0.642703	−	0.766116i	$-0.722187\pi$
$109$	−13.4200	−1.28541	−0.642703	+	0.766116i	$0.722187\pi$
$110$	0	0
$111$	0	0
$112$	−0.761557	−0.0719604
$113$	10.3232	0.971125	0.485563	−	0.874202i	$-0.338615\pi$
$113$	10.3232	0.971125	0.485563	+	0.874202i	$0.338615\pi$
$114$	0	0
$115$	0	0
$116$	7.35548	0.682939
$117$	0	0
$118$	4.49084	0.413415
$119$	−2.76156	−0.253152
$120$	0	0
$121$	−10.2524	−0.932036
$122$	10.8646	0.983638
$123$	0	0
$124$	8.11704	0.728931
$125$	0	0
$126$	0	0
$127$	16.9817	1.50688	0.753440	−	0.657517i	$-0.228393\pi$
$127$	16.9817	1.50688	0.753440	+	0.657517i	$0.228393\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−0.541436	−0.0473055	−0.0236528	−	0.999720i	$-0.507530\pi$
$131$	−0.541436	−0.0473055	−0.0236528	+	0.999720i	$0.507530\pi$
$132$	0	0
$133$	−0.761557	−0.0660354
$134$	−1.03228	−0.0891750
$135$	0	0
$136$	−3.62620	−0.310944
$137$	−2.87859	−0.245935	−0.122967	−	0.992411i	$-0.539241\pi$
$137$	−2.87859	−0.245935	−0.122967	+	0.992411i	$0.539241\pi$
$138$	0	0
$139$	3.58767	0.304302	0.152151	−	0.988357i	$-0.451380\pi$
$139$	3.58767	0.304302	0.152151	+	0.988357i	$0.451380\pi$
$140$	0	0
$141$	0	0
$142$	−10.1816	−0.854420
$143$	4.86464	0.406802
$144$	0	0
$145$	0	0
$146$	−16.4017	−1.35742
$147$	0	0
$148$	0.476886	0.0391998
$149$	16.8401	1.37959	0.689796	−	0.724004i	$-0.257700\pi$
$149$	16.8401	1.37959	0.689796	+	0.724004i	$0.257700\pi$
$150$	0	0
$151$	−16.9817	−1.38195	−0.690975	−	0.722879i	$-0.742818\pi$
$151$	−16.9817	−1.38195	−0.690975	+	0.722879i	$0.742818\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	0.658473	0.0530613
$155$	0	0
$156$	0	0
$157$	14.8401	1.18437	0.592183	−	0.805804i	$-0.298266\pi$
$157$	14.8401	1.18437	0.592183	+	0.805804i	$0.298266\pi$
$158$	12.5693	0.999963
$159$	0	0
$160$	0	0
$161$	−6.10308	−0.480990
$162$	0	0
$163$	13.3694	1.04717	0.523587	−	0.851972i	$-0.324593\pi$
$163$	13.3694	1.04717	0.523587	+	0.851972i	$0.324593\pi$
$164$	2.65847	0.207592
$165$	0	0
$166$	−0.270718	−0.0210118
$167$	−9.84632	−0.761931	−0.380966	−	0.924589i	$-0.624408\pi$
$167$	−9.84632	−0.761931	−0.380966	+	0.924589i	$0.624408\pi$
$168$	0	0
$169$	18.6541	1.43493
$170$	0	0
$171$	0	0
$172$	6.86464	0.523424
$173$	−2.98168	−0.226693	−0.113346	−	0.993556i	$-0.536157\pi$
$173$	−2.98168	−0.226693	−0.113346	+	0.993556i	$0.536157\pi$
$174$	0	0
$175$	0	0
$176$	0.864641	0.0651748
$177$	0	0
$178$	0.387755	0.0290635
$179$	−11.7938	−0.881512	−0.440756	−	0.897627i	$-0.645290\pi$
$179$	−11.7938	−0.881512	−0.440756	+	0.897627i	$0.645290\pi$
$180$	0	0
$181$	14.5693	1.08293	0.541465	−	0.840723i	$-0.317870\pi$
$181$	14.5693	1.08293	0.541465	+	0.840723i	$0.317870\pi$
$182$	4.28467	0.317601
$183$	0	0
$184$	−8.01395	−0.590796
$185$	0	0
$186$	0	0
$187$	3.13536	0.229280
$188$	−1.25240	−0.0913404
$189$	0	0
$190$	0	0
$191$	−13.2384	−0.957900	−0.478950	−	0.877842i	$-0.658983\pi$
$191$	−13.2384	−0.957900	−0.478950	+	0.877842i	$0.658983\pi$
$192$	0	0
$193$	2.54144	0.182937	0.0914683	−	0.995808i	$-0.470844\pi$
$193$	2.54144	0.182937	0.0914683	+	0.995808i	$0.470844\pi$
$194$	8.50479	0.610609
$195$	0	0
$196$	−6.42003	−0.458574
$197$	19.9109	1.41859	0.709295	−	0.704911i	$-0.249013\pi$
$197$	19.9109	1.41859	0.709295	+	0.704911i	$0.249013\pi$
$198$	0	0
$199$	−20.3126	−1.43992	−0.719960	−	0.694015i	$-0.755840\pi$
$199$	−20.3126	−1.43992	−0.719960	+	0.694015i	$0.755840\pi$
$200$	0	0
$201$	0	0
$202$	16.4157	1.15500
$203$	−5.60162	−0.393157
$204$	0	0
$205$	0	0
$206$	9.64015	0.671661
$207$	0	0
$208$	5.62620	0.390107
$209$	0.864641	0.0598085
$210$	0	0
$211$	18.0419	1.24205	0.621026	−	0.783790i	$-0.286716\pi$
$211$	18.0419	1.24205	0.621026	+	0.783790i	$0.286716\pi$
$212$	−2.37380	−0.163033
$213$	0	0
$214$	4.28467	0.292894
$215$	0	0
$216$	0	0
$217$	−6.18159	−0.419634
$218$	13.4200	0.908919
$219$	0	0
$220$	0	0
$221$	20.4017	1.37237
$222$	0	0
$223$	−13.5231	−0.905575	−0.452787	−	0.891619i	$-0.649570\pi$
$223$	−13.5231	−0.905575	−0.452787	+	0.891619i	$0.649570\pi$
$224$	0.761557	0.0508837
$225$	0	0
$226$	−10.3232	−0.686689
$227$	−13.6016	−0.902771	−0.451386	−	0.892329i	$-0.649070\pi$
$227$	−13.6016	−0.902771	−0.451386	+	0.892329i	$0.649070\pi$
$228$	0	0
$229$	13.5877	0.897898	0.448949	−	0.893557i	$-0.351798\pi$
$229$	13.5877	0.897898	0.448949	+	0.893557i	$0.351798\pi$
$230$	0	0
$231$	0	0
$232$	−7.35548	−0.482911
$233$	−25.5510	−1.67390	−0.836952	−	0.547277i	$-0.815664\pi$
$233$	−25.5510	−1.67390	−0.836952	+	0.547277i	$0.815664\pi$
$234$	0	0
$235$	0	0
$236$	−4.49084	−0.292329
$237$	0	0
$238$	2.76156	0.179005
$239$	11.3309	0.732935	0.366468	−	0.930431i	$-0.380567\pi$
$239$	11.3309	0.732935	0.366468	+	0.930431i	$0.380567\pi$
$240$	0	0
$241$	−1.25240	−0.0806739	−0.0403370	−	0.999186i	$-0.512843\pi$
$241$	−1.25240	−0.0806739	−0.0403370	+	0.999186i	$0.512843\pi$
$242$	10.2524	0.659049
$243$	0	0
$244$	−10.8646	−0.695537
$245$	0	0
$246$	0	0
$247$	5.62620	0.357986
$248$	−8.11704	−0.515432
$249$	0	0
$250$	0	0
$251$	−10.5939	−0.668682	−0.334341	−	0.942452i	$-0.608514\pi$
$251$	−10.5939	−0.668682	−0.334341	+	0.942452i	$0.608514\pi$
$252$	0	0
$253$	6.92919	0.435635
$254$	−16.9817	−1.06553
$255$	0	0
$256$	1.00000	0.0625000
$257$	0.153681	0.00958637	0.00479319	−	0.999989i	$-0.498474\pi$
$257$	0.153681	0.00958637	0.00479319	+	0.999989i	$0.498474\pi$
$258$	0	0
$259$	−0.363176	−0.0225666
$260$	0	0
$261$	0	0
$262$	0.541436	0.0334501
$263$	−0.504792	−0.0311268	−0.0155634	−	0.999879i	$-0.504954\pi$
$263$	−0.504792	−0.0311268	−0.0155634	+	0.999879i	$0.504954\pi$
$264$	0	0
$265$	0	0
$266$	0.761557	0.0466941
$267$	0	0
$268$	1.03228	0.0630563
$269$	3.49521	0.213107	0.106553	−	0.994307i	$-0.466019\pi$
$269$	3.49521	0.213107	0.106553	+	0.994307i	$0.466019\pi$
$270$	0	0
$271$	5.47252	0.332432	0.166216	−	0.986089i	$-0.446845\pi$
$271$	5.47252	0.332432	0.166216	+	0.986089i	$0.446845\pi$
$272$	3.62620	0.219871
$273$	0	0
$274$	2.87859	0.173902
$275$	0	0
$276$	0	0
$277$	12.9538	0.778317	0.389158	−	0.921171i	$-0.372766\pi$
$277$	12.9538	0.778317	0.389158	+	0.921171i	$0.372766\pi$
$278$	−3.58767	−0.215174
$279$	0	0
$280$	0	0
$281$	−0.153681	−0.00916785	−0.00458393	−	0.999989i	$-0.501459\pi$
$281$	−0.153681	−0.00916785	−0.00458393	+	0.999989i	$0.501459\pi$
$282$	0	0
$283$	−18.2341	−1.08390	−0.541952	−	0.840410i	$-0.682314\pi$
$283$	−18.2341	−1.08390	−0.541952	+	0.840410i	$0.682314\pi$
$284$	10.1816	0.604166
$285$	0	0
$286$	−4.86464	−0.287652
$287$	−2.02458	−0.119507
$288$	0	0
$289$	−3.85069	−0.226511
$290$	0	0
$291$	0	0
$292$	16.4017	0.959837
$293$	−2.03853	−0.119092	−0.0595462	−	0.998226i	$-0.518965\pi$
$293$	−2.03853	−0.119092	−0.0595462	+	0.998226i	$0.518965\pi$
$294$	0	0
$295$	0	0
$296$	−0.476886	−0.0277184
$297$	0	0
$298$	−16.8401	−0.975519
$299$	45.0881	2.60751
$300$	0	0
$301$	−5.22782	−0.301326
$302$	16.9817	0.977186
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	16.5414	0.944070	0.472035	−	0.881580i	$-0.343520\pi$
$307$	16.5414	0.944070	0.472035	+	0.881580i	$0.343520\pi$
$308$	−0.658473	−0.0375200
$309$	0	0
$310$	0	0
$311$	21.4725	1.21759	0.608797	−	0.793326i	$-0.291652\pi$
$311$	21.4725	1.21759	0.608797	+	0.793326i	$0.291652\pi$
$312$	0	0
$313$	−1.12141	−0.0633856	−0.0316928	−	0.999498i	$-0.510090\pi$
$313$	−1.12141	−0.0633856	−0.0316928	+	0.999498i	$0.510090\pi$
$314$	−14.8401	−0.837473
$315$	0	0
$316$	−12.5693	−0.707081
$317$	29.8882	1.67869	0.839344	−	0.543601i	$-0.182940\pi$
$317$	29.8882	1.67869	0.839344	+	0.543601i	$0.182940\pi$
$318$	0	0
$319$	6.35985	0.356083
$320$	0	0
$321$	0	0
$322$	6.10308	0.340112
$323$	3.62620	0.201767
$324$	0	0
$325$	0	0
$326$	−13.3694	−0.740464
$327$	0	0
$328$	−2.65847	−0.146790
$329$	0.953771	0.0525831
$330$	0	0
$331$	32.3126	1.77606	0.888030	−	0.459786i	$-0.152074\pi$
$331$	32.3126	1.77606	0.888030	+	0.459786i	$0.152074\pi$
$332$	0.270718	0.0148576
$333$	0	0
$334$	9.84632	0.538767
$335$	0	0
$336$	0	0
$337$	26.3511	1.43544	0.717718	−	0.696334i	$-0.245187\pi$
$337$	26.3511	1.43544	0.717718	+	0.696334i	$0.245187\pi$
$338$	−18.6541	−1.01465
$339$	0	0
$340$	0	0
$341$	7.01832	0.380063
$342$	0	0
$343$	10.2201	0.551835
$344$	−6.86464	−0.370117
$345$	0	0
$346$	2.98168	0.160296
$347$	2.77551	0.148997	0.0744986	−	0.997221i	$-0.476264\pi$
$347$	2.77551	0.148997	0.0744986	+	0.997221i	$0.476264\pi$
$348$	0	0
$349$	11.5510	0.618312	0.309156	−	0.951011i	$-0.399953\pi$
$349$	11.5510	0.618312	0.309156	+	0.951011i	$0.399953\pi$
$350$	0	0
$351$	0	0
$352$	−0.864641	−0.0460855
$353$	8.40171	0.447178	0.223589	−	0.974684i	$-0.428223\pi$
$353$	8.40171	0.447178	0.223589	+	0.974684i	$0.428223\pi$
$354$	0	0
$355$	0	0
$356$	−0.387755	−0.0205510
$357$	0	0
$358$	11.7938	0.623323
$359$	−22.7895	−1.20278	−0.601391	−	0.798955i	$-0.705387\pi$
$359$	−22.7895	−1.20278	−0.601391	+	0.798955i	$0.705387\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−14.5693	−0.765748
$363$	0	0
$364$	−4.28467	−0.224578
$365$	0	0
$366$	0	0
$367$	4.06455	0.212168	0.106084	−	0.994357i	$-0.466169\pi$
$367$	4.06455	0.212168	0.106084	+	0.994357i	$0.466169\pi$
$368$	8.01395	0.417756
$369$	0	0
$370$	0	0
$371$	1.80779	0.0938556
$372$	0	0
$373$	−18.4017	−0.952804	−0.476402	−	0.879227i	$-0.658059\pi$
$373$	−18.4017	−0.952804	−0.476402	+	0.879227i	$0.658059\pi$
$374$	−3.13536	−0.162126
$375$	0	0
$376$	1.25240	0.0645874
$377$	41.3834	2.13135
$378$	0	0
$379$	1.23844	0.0636145	0.0318073	−	0.999494i	$-0.489874\pi$
$379$	1.23844	0.0636145	0.0318073	+	0.999494i	$0.489874\pi$
$380$	0	0
$381$	0	0
$382$	13.2384	0.677338
$383$	−16.8646	−0.861743	−0.430871	−	0.902413i	$-0.641794\pi$
$383$	−16.8646	−0.861743	−0.430871	+	0.902413i	$0.641794\pi$
$384$	0	0
$385$	0	0
$386$	−2.54144	−0.129356
$387$	0	0
$388$	−8.50479	−0.431765
$389$	−8.59392	−0.435729	−0.217865	−	0.975979i	$-0.569909\pi$
$389$	−8.59392	−0.435729	−0.217865	+	0.975979i	$0.569909\pi$
$390$	0	0
$391$	29.0602	1.46964
$392$	6.42003	0.324261
$393$	0	0
$394$	−19.9109	−1.00310
$395$	0	0
$396$	0	0
$397$	16.0558	0.805818	0.402909	−	0.915240i	$-0.367999\pi$
$397$	16.0558	0.805818	0.402909	+	0.915240i	$0.367999\pi$
$398$	20.3126	1.01818
$399$	0	0
$400$	0	0
$401$	14.8925	0.743698	0.371849	−	0.928293i	$-0.378724\pi$
$401$	14.8925	0.743698	0.371849	+	0.928293i	$0.378724\pi$
$402$	0	0
$403$	45.6681	2.27489
$404$	−16.4157	−0.816710
$405$	0	0
$406$	5.60162	0.278004
$407$	0.412335	0.0204387
$408$	0	0
$409$	−18.3511	−0.907404	−0.453702	−	0.891153i	$-0.649897\pi$
$409$	−18.3511	−0.907404	−0.453702	+	0.891153i	$0.649897\pi$
$410$	0	0
$411$	0	0
$412$	−9.64015	−0.474936
$413$	3.42003	0.168289
$414$	0	0
$415$	0	0
$416$	−5.62620	−0.275847
$417$	0	0
$418$	−0.864641	−0.0422910
$419$	−34.7509	−1.69769	−0.848847	−	0.528639i	$-0.822703\pi$
$419$	−34.7509	−1.69769	−0.848847	+	0.528639i	$0.822703\pi$
$420$	0	0
$421$	−40.1589	−1.95722	−0.978612	−	0.205713i	$-0.934049\pi$
$421$	−40.1589	−1.95722	−0.978612	+	0.205713i	$0.934049\pi$
$422$	−18.0419	−0.878264
$423$	0	0
$424$	2.37380	0.115282
$425$	0	0
$426$	0	0
$427$	8.27405	0.400409
$428$	−4.28467	−0.207107
$429$	0	0
$430$	0	0
$431$	−34.9571	−1.68382	−0.841912	−	0.539615i	$-0.818570\pi$
$431$	−34.9571	−1.68382	−0.841912	+	0.539615i	$0.818570\pi$
$432$	0	0
$433$	1.13536	0.0545619	0.0272809	−	0.999628i	$-0.491315\pi$
$433$	1.13536	0.0545619	0.0272809	+	0.999628i	$0.491315\pi$
$434$	6.18159	0.296726
$435$	0	0
$436$	−13.4200	−0.642703
$437$	8.01395	0.383359
$438$	0	0
$439$	−6.80009	−0.324551	−0.162275	−	0.986746i	$-0.551883\pi$
$439$	−6.80009	−0.324551	−0.162275	+	0.986746i	$0.551883\pi$
$440$	0	0
$441$	0	0
$442$	−20.4017	−0.970410
$443$	−38.0679	−1.80866	−0.904330	−	0.426835i	$-0.859629\pi$
$443$	−38.0679	−1.80866	−0.904330	+	0.426835i	$0.859629\pi$
$444$	0	0
$445$	0	0
$446$	13.5231	0.640338
$447$	0	0
$448$	−0.761557	−0.0359802
$449$	18.5414	0.875024	0.437512	−	0.899212i	$-0.355860\pi$
$449$	18.5414	0.875024	0.437512	+	0.899212i	$0.355860\pi$
$450$	0	0
$451$	2.29862	0.108238
$452$	10.3232	0.485563
$453$	0	0
$454$	13.6016	0.638356
$455$	0	0
$456$	0	0
$457$	16.3738	0.765934	0.382967	−	0.923762i	$-0.374902\pi$
$457$	16.3738	0.765934	0.382967	+	0.923762i	$0.374902\pi$
$458$	−13.5877	−0.634910
$459$	0	0
$460$	0	0
$461$	−1.70470	−0.0793959	−0.0396979	−	0.999212i	$-0.512640\pi$
$461$	−1.70470	−0.0793959	−0.0396979	+	0.999212i	$0.512640\pi$
$462$	0	0
$463$	−10.0279	−0.466036	−0.233018	−	0.972472i	$-0.574860\pi$
$463$	−10.0279	−0.466036	−0.233018	+	0.972472i	$0.574860\pi$
$464$	7.35548	0.341470
$465$	0	0
$466$	25.5510	1.18363
$467$	−32.7509	−1.51553	−0.757766	−	0.652526i	$-0.773709\pi$
$467$	−32.7509	−1.51553	−0.757766	+	0.652526i	$0.773709\pi$
$468$	0	0
$469$	−0.786137	−0.0363004
$470$	0	0
$471$	0	0
$472$	4.49084	0.206708
$473$	5.93545	0.272912
$474$	0	0
$475$	0	0
$476$	−2.76156	−0.126576
$477$	0	0
$478$	−11.3309	−0.518263
$479$	−27.2803	−1.24647	−0.623234	−	0.782035i	$-0.714182\pi$
$479$	−27.2803	−1.24647	−0.623234	+	0.782035i	$0.714182\pi$
$480$	0	0
$481$	2.68305	0.122337
$482$	1.25240	0.0570451
$483$	0	0
$484$	−10.2524	−0.466018
$485$	0	0
$486$	0	0
$487$	−11.0741	−0.501817	−0.250908	−	0.968011i	$-0.580729\pi$
$487$	−11.0741	−0.501817	−0.250908	+	0.968011i	$0.580729\pi$
$488$	10.8646	0.491819
$489$	0	0
$490$	0	0
$491$	−8.11704	−0.366317	−0.183158	−	0.983083i	$-0.558632\pi$
$491$	−8.11704	−0.366317	−0.183158	+	0.983083i	$0.558632\pi$
$492$	0	0
$493$	26.6724	1.20127
$494$	−5.62620	−0.253135
$495$	0	0
$496$	8.11704	0.364466
$497$	−7.75386	−0.347808
$498$	0	0
$499$	0.295298	0.0132193	0.00660967	−	0.999978i	$-0.497896\pi$
$499$	0.295298	0.0132193	0.00660967	+	0.999978i	$0.497896\pi$
$500$	0	0
$501$	0	0
$502$	10.5939	0.472830
$503$	−19.6016	−0.873993	−0.436996	−	0.899463i	$-0.643958\pi$
$503$	−19.6016	−0.873993	−0.436996	+	0.899463i	$0.643958\pi$
$504$	0	0
$505$	0	0
$506$	−6.92919	−0.308040
$507$	0	0
$508$	16.9817	0.753440
$509$	1.79383	0.0795102	0.0397551	−	0.999209i	$-0.487342\pi$
$509$	1.79383	0.0795102	0.0397551	+	0.999209i	$0.487342\pi$
$510$	0	0
$511$	−12.4908	−0.552562
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−0.153681	−0.00677859
$515$	0	0
$516$	0	0
$517$	−1.08287	−0.0476247
$518$	0.363176	0.0159570
$519$	0	0
$520$	0	0
$521$	2.61850	0.114719	0.0573593	−	0.998354i	$-0.481732\pi$
$521$	2.61850	0.114719	0.0573593	+	0.998354i	$0.481732\pi$
$522$	0	0
$523$	14.9956	0.655713	0.327857	−	0.944728i	$-0.393674\pi$
$523$	14.9956	0.655713	0.327857	+	0.944728i	$0.393674\pi$
$524$	−0.541436	−0.0236528
$525$	0	0
$526$	0.504792	0.0220100
$527$	29.4340	1.28216
$528$	0	0
$529$	41.2234	1.79232
$530$	0	0
$531$	0	0
$532$	−0.761557	−0.0330177
$533$	14.9571	0.647864
$534$	0	0
$535$	0	0
$536$	−1.03228	−0.0445875
$537$	0	0
$538$	−3.49521	−0.150689
$539$	−5.55102	−0.239099
$540$	0	0
$541$	3.40608	0.146439	0.0732194	−	0.997316i	$-0.476673\pi$
$541$	3.40608	0.146439	0.0732194	+	0.997316i	$0.476673\pi$
$542$	−5.47252	−0.235065
$543$	0	0
$544$	−3.62620	−0.155472
$545$	0	0
$546$	0	0
$547$	4.74760	0.202993	0.101496	−	0.994836i	$-0.467637\pi$
$547$	4.74760	0.202993	0.101496	+	0.994836i	$0.467637\pi$
$548$	−2.87859	−0.122967
$549$	0	0
$550$	0	0
$551$	7.35548	0.313354
$552$	0	0
$553$	9.57227	0.407054
$554$	−12.9538	−0.550353
$555$	0	0
$556$	3.58767	0.152151
$557$	−43.0462	−1.82393	−0.911964	−	0.410271i	$-0.865434\pi$
$557$	−43.0462	−1.82393	−0.911964	+	0.410271i	$0.865434\pi$
$558$	0	0
$559$	38.6218	1.63353
$560$	0	0
$561$	0	0
$562$	0.153681	0.00648265
$563$	−17.0096	−0.716869	−0.358434	−	0.933555i	$-0.616689\pi$
$563$	−17.0096	−0.716869	−0.358434	+	0.933555i	$0.616689\pi$
$564$	0	0
$565$	0	0
$566$	18.2341	0.766435
$567$	0	0
$568$	−10.1816	−0.427210
$569$	19.7572	0.828264	0.414132	−	0.910217i	$-0.364085\pi$
$569$	19.7572	0.828264	0.414132	+	0.910217i	$0.364085\pi$
$570$	0	0
$571$	−11.3973	−0.476964	−0.238482	−	0.971147i	$-0.576650\pi$
$571$	−11.3973	−0.476964	−0.238482	+	0.971147i	$0.576650\pi$
$572$	4.86464	0.203401
$573$	0	0
$574$	2.02458	0.0845043
$575$	0	0
$576$	0	0
$577$	−18.3372	−0.763386	−0.381693	−	0.924289i	$-0.624659\pi$
$577$	−18.3372	−0.763386	−0.381693	+	0.924289i	$0.624659\pi$
$578$	3.85069	0.160167
$579$	0	0
$580$	0	0
$581$	−0.206167	−0.00855327
$582$	0	0
$583$	−2.05249	−0.0850053
$584$	−16.4017	−0.678708
$585$	0	0
$586$	2.03853	0.0842110
$587$	11.9475	0.493127	0.246563	−	0.969127i	$-0.420699\pi$
$587$	11.9475	0.493127	0.246563	+	0.969127i	$0.420699\pi$
$588$	0	0
$589$	8.11704	0.334457
$590$	0	0
$591$	0	0
$592$	0.476886	0.0195999
$593$	24.3911	1.00162	0.500811	−	0.865557i	$-0.333035\pi$
$593$	24.3911	1.00162	0.500811	+	0.865557i	$0.333035\pi$
$594$	0	0
$595$	0	0
$596$	16.8401	0.689796
$597$	0	0
$598$	−45.0881	−1.84379
$599$	21.0708	0.860930	0.430465	−	0.902607i	$-0.358350\pi$
$599$	21.0708	0.860930	0.430465	+	0.902607i	$0.358350\pi$
$600$	0	0
$601$	32.3878	1.32112	0.660562	−	0.750771i	$-0.270318\pi$
$601$	32.3878	1.32112	0.660562	+	0.750771i	$0.270318\pi$
$602$	5.22782	0.213070
$603$	0	0
$604$	−16.9817	−0.690975
$605$	0	0
$606$	0	0
$607$	19.0183	0.771930	0.385965	−	0.922513i	$-0.373869\pi$
$607$	19.0183	0.771930	0.385965	+	0.922513i	$0.373869\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	−7.04623	−0.285060
$612$	0	0
$613$	−23.5756	−0.952210	−0.476105	−	0.879389i	$-0.657952\pi$
$613$	−23.5756	−0.952210	−0.476105	+	0.879389i	$0.657952\pi$
$614$	−16.5414	−0.667558
$615$	0	0
$616$	0.658473	0.0265307
$617$	26.2707	1.05762	0.528810	−	0.848740i	$-0.322639\pi$
$617$	26.2707	1.05762	0.528810	+	0.848740i	$0.322639\pi$
$618$	0	0
$619$	11.4985	0.462165	0.231083	−	0.972934i	$-0.425773\pi$
$619$	11.4985	0.462165	0.231083	+	0.972934i	$0.425773\pi$
$620$	0	0
$621$	0	0
$622$	−21.4725	−0.860969
$623$	0.295298	0.0118309
$624$	0	0
$625$	0	0
$626$	1.12141	0.0448204
$627$	0	0
$628$	14.8401	0.592183
$629$	1.72928	0.0689510
$630$	0	0
$631$	−45.8130	−1.82379	−0.911893	−	0.410427i	$-0.865380\pi$
$631$	−45.8130	−1.82379	−0.911893	+	0.410427i	$0.865380\pi$
$632$	12.5693	0.499982
$633$	0	0
$634$	−29.8882	−1.18701
$635$	0	0
$636$	0	0
$637$	−36.1204	−1.43114
$638$	−6.35985	−0.251789
$639$	0	0
$640$	0	0
$641$	−1.36943	−0.0540894	−0.0270447	−	0.999634i	$-0.508610\pi$
$641$	−1.36943	−0.0540894	−0.0270447	+	0.999634i	$0.508610\pi$
$642$	0	0
$643$	−24.7389	−0.975606	−0.487803	−	0.872954i	$-0.662201\pi$
$643$	−24.7389	−0.975606	−0.487803	+	0.872954i	$0.662201\pi$
$644$	−6.10308	−0.240495
$645$	0	0
$646$	−3.62620	−0.142671
$647$	6.82611	0.268362	0.134181	−	0.990957i	$-0.457160\pi$
$647$	6.82611	0.268362	0.134181	+	0.990957i	$0.457160\pi$
$648$	0	0
$649$	−3.88296	−0.152420
$650$	0	0
$651$	0	0
$652$	13.3694	0.523587
$653$	−6.91713	−0.270688	−0.135344	−	0.990799i	$-0.543214\pi$
$653$	−6.91713	−0.270688	−0.135344	+	0.990799i	$0.543214\pi$
$654$	0	0
$655$	0	0
$656$	2.65847	0.103796
$657$	0	0
$658$	−0.953771	−0.0371819
$659$	−9.44461	−0.367910	−0.183955	−	0.982935i	$-0.558890\pi$
$659$	−9.44461	−0.367910	−0.183955	+	0.982935i	$0.558890\pi$
$660$	0	0
$661$	−22.1955	−0.863306	−0.431653	−	0.902040i	$-0.642070\pi$
$661$	−22.1955	−0.863306	−0.431653	+	0.902040i	$0.642070\pi$
$662$	−32.3126	−1.25586
$663$	0	0
$664$	−0.270718	−0.0105059
$665$	0	0
$666$	0	0
$667$	58.9465	2.28242
$668$	−9.84632	−0.380966
$669$	0	0
$670$	0	0
$671$	−9.39401	−0.362652
$672$	0	0
$673$	12.2986	0.474077	0.237039	−	0.971500i	$-0.423823\pi$
$673$	12.2986	0.474077	0.237039	+	0.971500i	$0.423823\pi$
$674$	−26.3511	−1.01501
$675$	0	0
$676$	18.6541	0.717466
$677$	35.9527	1.38178	0.690888	−	0.722962i	$-0.257220\pi$
$677$	35.9527	1.38178	0.690888	+	0.722962i	$0.257220\pi$
$678$	0	0
$679$	6.47689	0.248560
$680$	0	0
$681$	0	0
$682$	−7.01832	−0.268745
$683$	9.00958	0.344742	0.172371	−	0.985032i	$-0.444857\pi$
$683$	9.00958	0.344742	0.172371	+	0.985032i	$0.444857\pi$
$684$	0	0
$685$	0	0
$686$	−10.2201	−0.390206
$687$	0	0
$688$	6.86464	0.261712
$689$	−13.3555	−0.508803
$690$	0	0
$691$	−9.11078	−0.346590	−0.173295	−	0.984870i	$-0.555441\pi$
$691$	−9.11078	−0.346590	−0.173295	+	0.984870i	$0.555441\pi$
$692$	−2.98168	−0.113346
$693$	0	0
$694$	−2.77551	−0.105357
$695$	0	0
$696$	0	0
$697$	9.64015	0.365147
$698$	−11.5510	−0.437213
$699$	0	0
$700$	0	0
$701$	14.7476	0.557009	0.278505	−	0.960435i	$-0.410161\pi$
$701$	14.7476	0.557009	0.278505	+	0.960435i	$0.410161\pi$
$702$	0	0
$703$	0.476886	0.0179861
$704$	0.864641	0.0325874
$705$	0	0
$706$	−8.40171	−0.316202
$707$	12.5015	0.470166
$708$	0	0
$709$	−8.63389	−0.324253	−0.162126	−	0.986770i	$-0.551835\pi$
$709$	−8.63389	−0.324253	−0.162126	+	0.986770i	$0.551835\pi$
$710$	0	0
$711$	0	0
$712$	0.387755	0.0145317
$713$	65.0496	2.43613
$714$	0	0
$715$	0	0
$716$	−11.7938	−0.440756
$717$	0	0
$718$	22.7895	0.850495
$719$	38.2759	1.42745	0.713726	−	0.700425i	$-0.247006\pi$
$719$	38.2759	1.42745	0.713726	+	0.700425i	$0.247006\pi$
$720$	0	0
$721$	7.34153	0.273413
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	14.5693	0.541465
$725$	0	0
$726$	0	0
$727$	−31.1893	−1.15675	−0.578373	−	0.815772i	$-0.696312\pi$
$727$	−31.1893	−1.15675	−0.578373	+	0.815772i	$0.696312\pi$
$728$	4.28467	0.158800
$729$	0	0
$730$	0	0
$731$	24.8925	0.920684
$732$	0	0
$733$	−13.9634	−0.515748	−0.257874	−	0.966179i	$-0.583022\pi$
$733$	−13.9634	−0.515748	−0.257874	+	0.966179i	$0.583022\pi$
$734$	−4.06455	−0.150025
$735$	0	0
$736$	−8.01395	−0.295398
$737$	0.892548	0.0328774
$738$	0	0
$739$	9.02165	0.331867	0.165933	−	0.986137i	$-0.446936\pi$
$739$	9.02165	0.331867	0.165933	+	0.986137i	$0.446936\pi$
$740$	0	0
$741$	0	0
$742$	−1.80779	−0.0663659
$743$	15.0342	0.551550	0.275775	−	0.961222i	$-0.411066\pi$
$743$	15.0342	0.551550	0.275775	+	0.961222i	$0.411066\pi$
$744$	0	0
$745$	0	0
$746$	18.4017	0.673734
$747$	0	0
$748$	3.13536	0.114640
$749$	3.26302	0.119228
$750$	0	0
$751$	29.6681	1.08260	0.541301	−	0.840829i	$-0.317932\pi$
$751$	29.6681	1.08260	0.541301	+	0.840829i	$0.317932\pi$
$752$	−1.25240	−0.0456702
$753$	0	0
$754$	−41.3834	−1.50709
$755$	0	0
$756$	0	0
$757$	10.5819	0.384604	0.192302	−	0.981336i	$-0.438405\pi$
$757$	10.5819	0.384604	0.192302	+	0.981336i	$0.438405\pi$
$758$	−1.23844	−0.0449823
$759$	0	0
$760$	0	0
$761$	0.979789	0.0355173	0.0177587	−	0.999842i	$-0.494347\pi$
$761$	0.979789	0.0355173	0.0177587	+	0.999842i	$0.494347\pi$
$762$	0	0
$763$	10.2201	0.369993
$764$	−13.2384	−0.478950
$765$	0	0
$766$	16.8646	0.609344
$767$	−25.2663	−0.912315
$768$	0	0
$769$	43.1772	1.55701	0.778505	−	0.627638i	$-0.215978\pi$
$769$	43.1772	1.55701	0.778505	+	0.627638i	$0.215978\pi$
$770$	0	0
$771$	0	0
$772$	2.54144	0.0914683
$773$	−37.5250	−1.34968	−0.674840	−	0.737964i	$-0.735787\pi$
$773$	−37.5250	−1.34968	−0.674840	+	0.737964i	$0.735787\pi$
$774$	0	0
$775$	0	0
$776$	8.50479	0.305304
$777$	0	0
$778$	8.59392	0.308107
$779$	2.65847	0.0952497
$780$	0	0
$781$	8.80342	0.315011
$782$	−29.0602	−1.03919
$783$	0	0
$784$	−6.42003	−0.229287
$785$	0	0
$786$	0	0
$787$	22.5833	0.805008	0.402504	−	0.915418i	$-0.368140\pi$
$787$	22.5833	0.805008	0.402504	+	0.915418i	$0.368140\pi$
$788$	19.9109	0.709295
$789$	0	0
$790$	0	0
$791$	−7.86171	−0.279530
$792$	0	0
$793$	−61.1266	−2.17067
$794$	−16.0558	−0.569799
$795$	0	0
$796$	−20.3126	−0.719960
$797$	−35.9806	−1.27450	−0.637250	−	0.770657i	$-0.719928\pi$
$797$	−35.9806	−1.27450	−0.637250	+	0.770657i	$0.719928\pi$
$798$	0	0
$799$	−4.54144	−0.160664
$800$	0	0
$801$	0	0
$802$	−14.8925	−0.525874
$803$	14.1816	0.500457
$804$	0	0
$805$	0	0
$806$	−45.6681	−1.60859
$807$	0	0
$808$	16.4157	0.577501
$809$	0.955660	0.0335992	0.0167996	−	0.999859i	$-0.494652\pi$
$809$	0.955660	0.0335992	0.0167996	+	0.999859i	$0.494652\pi$
$810$	0	0
$811$	−7.53707	−0.264662	−0.132331	−	0.991206i	$-0.542246\pi$
$811$	−7.53707	−0.264662	−0.132331	+	0.991206i	$0.542246\pi$
$812$	−5.60162	−0.196578
$813$	0	0
$814$	−0.412335	−0.0144523
$815$	0	0
$816$	0	0
$817$	6.86464	0.240163
$818$	18.3511	0.641632
$819$	0	0
$820$	0	0
$821$	−13.3082	−0.464460	−0.232230	−	0.972661i	$-0.574602\pi$
$821$	−13.3082	−0.464460	−0.232230	+	0.972661i	$0.574602\pi$
$822$	0	0
$823$	24.4050	0.850706	0.425353	−	0.905028i	$-0.360150\pi$
$823$	24.4050	0.850706	0.425353	+	0.905028i	$0.360150\pi$
$824$	9.64015	0.335831
$825$	0	0
$826$	−3.42003	−0.118998
$827$	11.6874	0.406411	0.203206	−	0.979136i	$-0.434864\pi$
$827$	11.6874	0.406411	0.203206	+	0.979136i	$0.434864\pi$
$828$	0	0
$829$	25.6541	0.891004	0.445502	−	0.895281i	$-0.353025\pi$
$829$	25.6541	0.891004	0.445502	+	0.895281i	$0.353025\pi$
$830$	0	0
$831$	0	0
$832$	5.62620	0.195053
$833$	−23.2803	−0.806615
$834$	0	0
$835$	0	0
$836$	0.864641	0.0299042
$837$	0	0
$838$	34.7509	1.20045
$839$	2.91713	0.100710	0.0503552	−	0.998731i	$-0.483965\pi$
$839$	2.91713	0.100710	0.0503552	+	0.998731i	$0.483965\pi$
$840$	0	0
$841$	25.1031	0.865624
$842$	40.1589	1.38397
$843$	0	0
$844$	18.0419	0.621026
$845$	0	0
$846$	0	0
$847$	7.80779	0.268279
$848$	−2.37380	−0.0815167
$849$	0	0
$850$	0	0
$851$	3.82174	0.131008
$852$	0	0
$853$	−6.24281	−0.213750	−0.106875	−	0.994272i	$-0.534084\pi$
$853$	−6.24281	−0.213750	−0.106875	+	0.994272i	$0.534084\pi$
$854$	−8.27405	−0.283132
$855$	0	0
$856$	4.28467	0.146447
$857$	23.2158	0.793035	0.396517	−	0.918027i	$-0.370219\pi$
$857$	23.2158	0.793035	0.396517	+	0.918027i	$0.370219\pi$
$858$	0	0
$859$	−7.13536	−0.243455	−0.121728	−	0.992564i	$-0.538843\pi$
$859$	−7.13536	−0.243455	−0.121728	+	0.992564i	$0.538843\pi$
$860$	0	0
$861$	0	0
$862$	34.9571	1.19064
$863$	7.31362	0.248959	0.124479	−	0.992222i	$-0.460274\pi$
$863$	7.31362	0.248959	0.124479	+	0.992222i	$0.460274\pi$
$864$	0	0
$865$	0	0
$866$	−1.13536	−0.0385811
$867$	0	0
$868$	−6.18159	−0.209817
$869$	−10.8680	−0.368671
$870$	0	0
$871$	5.80779	0.196789
$872$	13.4200	0.454460
$873$	0	0
$874$	−8.01395	−0.271076
$875$	0	0
$876$	0	0
$877$	−22.0173	−0.743471	−0.371735	−	0.928339i	$-0.621237\pi$
$877$	−22.0173	−0.743471	−0.371735	+	0.928339i	$0.621237\pi$
$878$	6.80009	0.229492
$879$	0	0
$880$	0	0
$881$	−11.7572	−0.396110	−0.198055	−	0.980191i	$-0.563462\pi$
$881$	−11.7572	−0.396110	−0.198055	+	0.980191i	$0.563462\pi$
$882$	0	0
$883$	55.6560	1.87297	0.936487	−	0.350703i	$-0.114057\pi$
$883$	55.6560	1.87297	0.936487	+	0.350703i	$0.114057\pi$
$884$	20.4017	0.686184
$885$	0	0
$886$	38.0679	1.27892
$887$	6.41566	0.215417	0.107708	−	0.994183i	$-0.465649\pi$
$887$	6.41566	0.215417	0.107708	+	0.994183i	$0.465649\pi$
$888$	0	0
$889$	−12.9325	−0.433743
$890$	0	0
$891$	0	0
$892$	−13.5231	−0.452787
$893$	−1.25240	−0.0419098
$894$	0	0
$895$	0	0
$896$	0.761557	0.0254418
$897$	0	0
$898$	−18.5414	−0.618736
$899$	59.7047	1.99126
$900$	0	0
$901$	−8.60788	−0.286770
$902$	−2.29862	−0.0765358
$903$	0	0
$904$	−10.3232	−0.343345
$905$	0	0
$906$	0	0
$907$	−57.1160	−1.89651	−0.948253	−	0.317516i	$-0.897151\pi$
$907$	−57.1160	−1.89651	−0.948253	+	0.317516i	$0.897151\pi$
$908$	−13.6016	−0.451386
$909$	0	0
$910$	0	0
$911$	26.6339	0.882420	0.441210	−	0.897404i	$-0.354549\pi$
$911$	26.6339	0.882420	0.441210	+	0.897404i	$0.354549\pi$
$912$	0	0
$913$	0.234074	0.00774672
$914$	−16.3738	−0.541597
$915$	0	0
$916$	13.5877	0.448949
$917$	0.412335	0.0136165
$918$	0	0
$919$	6.63246	0.218785	0.109392	−	0.993999i	$-0.465110\pi$
$919$	6.63246	0.218785	0.109392	+	0.993999i	$0.465110\pi$
$920$	0	0
$921$	0	0
$922$	1.70470	0.0561414
$923$	57.2836	1.88551
$924$	0	0
$925$	0	0
$926$	10.0279	0.329537
$927$	0	0
$928$	−7.35548	−0.241455
$929$	50.0173	1.64101	0.820507	−	0.571637i	$-0.193691\pi$
$929$	50.0173	1.64101	0.820507	+	0.571637i	$0.193691\pi$
$930$	0	0
$931$	−6.42003	−0.210408
$932$	−25.5510	−0.836952
$933$	0	0
$934$	32.7509	1.07164
$935$	0	0
$936$	0	0
$937$	−39.8882	−1.30309	−0.651545	−	0.758610i	$-0.725879\pi$
$937$	−39.8882	−1.30309	−0.651545	+	0.758610i	$0.725879\pi$
$938$	0.786137	0.0256683
$939$	0	0
$940$	0	0
$941$	5.59829	0.182499	0.0912495	−	0.995828i	$-0.470914\pi$
$941$	5.59829	0.182499	0.0912495	+	0.995828i	$0.470914\pi$
$942$	0	0
$943$	21.3049	0.693782
$944$	−4.49084	−0.146164
$945$	0	0
$946$	−5.93545	−0.192978
$947$	−12.7110	−0.413051	−0.206525	−	0.978441i	$-0.566216\pi$
$947$	−12.7110	−0.413051	−0.206525	+	0.978441i	$0.566216\pi$
$948$	0	0
$949$	92.2793	2.99551
$950$	0	0
$951$	0	0
$952$	2.76156	0.0895026
$953$	57.0129	1.84683	0.923415	−	0.383804i	$-0.125386\pi$
$953$	57.0129	1.84683	0.923415	+	0.383804i	$0.125386\pi$
$954$	0	0
$955$	0	0
$956$	11.3309	0.366468
$957$	0	0
$958$	27.2803	0.881387
$959$	2.19221	0.0707903
$960$	0	0
$961$	34.8863	1.12536
$962$	−2.68305	−0.0865051
$963$	0	0
$964$	−1.25240	−0.0403370
$965$	0	0
$966$	0	0
$967$	−33.0183	−1.06180	−0.530899	−	0.847435i	$-0.678146\pi$
$967$	−33.0183	−1.06180	−0.530899	+	0.847435i	$0.678146\pi$
$968$	10.2524	0.329524
$969$	0	0
$970$	0	0
$971$	3.04623	0.0977581	0.0488791	−	0.998805i	$-0.484435\pi$
$971$	3.04623	0.0977581	0.0488791	+	0.998805i	$0.484435\pi$
$972$	0	0
$973$	−2.73221	−0.0875907
$974$	11.0741	0.354838
$975$	0	0
$976$	−10.8646	−0.347769
$977$	−13.4465	−0.430192	−0.215096	−	0.976593i	$-0.569006\pi$
$977$	−13.4465	−0.430192	−0.215096	+	0.976593i	$0.569006\pi$
$978$	0	0
$979$	−0.335269	−0.0107152
$980$	0	0
$981$	0	0
$982$	8.11704	0.259025
$983$	22.0646	0.703750	0.351875	−	0.936047i	$-0.385544\pi$
$983$	22.0646	0.703750	0.351875	+	0.936047i	$0.385544\pi$
$984$	0	0
$985$	0	0
$986$	−26.6724	−0.849423
$987$	0	0
$988$	5.62620	0.178993
$989$	55.0129	1.74931
$990$	0	0
$991$	51.9946	1.65166	0.825831	−	0.563917i	$-0.190706\pi$
$991$	51.9946	1.65166	0.825831	+	0.563917i	$0.190706\pi$
$992$	−8.11704	−0.257716
$993$	0	0
$994$	7.75386	0.245938
$995$	0	0
$996$	0	0
$997$	−37.7693	−1.19616	−0.598082	−	0.801435i	$-0.704070\pi$
$997$	−37.7693	−1.19616	−0.598082	+	0.801435i	$0.704070\pi$
$998$	−0.295298	−0.00934749
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.ck.1.1		3
3.2	odd	2		950.2.a.n.1.3		3
5.2	odd	4		1710.2.d.d.1369.3		6
5.3	odd	4		1710.2.d.d.1369.6		6
5.4	even	2		8550.2.a.cl.1.3		3
12.11	even	2		7600.2.a.bi.1.1		3
15.2	even	4		190.2.b.b.39.4	yes	6
15.8	even	4		190.2.b.b.39.3	✓	6
15.14	odd	2		950.2.a.i.1.1		3
60.23	odd	4		1520.2.d.j.609.1		6
60.47	odd	4		1520.2.d.j.609.6		6
60.59	even	2		7600.2.a.cd.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.b.39.3	✓	6	15.8	even	4
190.2.b.b.39.4	yes	6	15.2	even	4
950.2.a.i.1.1		3	15.14	odd	2
950.2.a.n.1.3		3	3.2	odd	2
1520.2.d.j.609.1		6	60.23	odd	4
1520.2.d.j.609.6		6	60.47	odd	4
1710.2.d.d.1369.3		6	5.2	odd	4
1710.2.d.d.1369.6		6	5.3	odd	4
7600.2.a.bi.1.1		3	12.11	even	2
7600.2.a.cd.1.3		3	60.59	even	2
8550.2.a.ck.1.1		3	1.1	even	1	trivial
8550.2.a.cl.1.3		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.761557 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.761557 q^{7} -1.00000 q^{8} +0.864641 q^{11} +5.62620 q^{13} +0.761557 q^{14} +1.00000 q^{16} +3.62620 q^{17} +1.00000 q^{19} -0.864641 q^{22} +8.01395 q^{23} -5.62620 q^{26} -0.761557 q^{28} +7.35548 q^{29} +8.11704 q^{31} -1.00000 q^{32} -3.62620 q^{34} +0.476886 q^{37} -1.00000 q^{38} +2.65847 q^{41} +6.86464 q^{43} +0.864641 q^{44} -8.01395 q^{46} -1.25240 q^{47} -6.42003 q^{49} +5.62620 q^{52} -2.37380 q^{53} +0.761557 q^{56} -7.35548 q^{58} -4.49084 q^{59} -10.8646 q^{61} -8.11704 q^{62} +1.00000 q^{64} +1.03228 q^{67} +3.62620 q^{68} +10.1816 q^{71} +16.4017 q^{73} -0.476886 q^{74} +1.00000 q^{76} -0.658473 q^{77} -12.5693 q^{79} -2.65847 q^{82} +0.270718 q^{83} -6.86464 q^{86} -0.864641 q^{88} -0.387755 q^{89} -4.28467 q^{91} +8.01395 q^{92} +1.25240 q^{94} -8.50479 q^{97} +6.42003 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) 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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