LMFDB - Embedded newform 8550.2.a.cj.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:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 950)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.91223$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +4.22547 q^{7} -1.00000 q^{8} +5.13770 q^{11} +3.16784 q^{13} -4.22547 q^{14} +1.00000 q^{16} +6.48108 q^{17} -1.00000 q^{19} -5.13770 q^{22} -7.56885 q^{23} -3.16784 q^{26} +4.22547 q^{28} -0.832162 q^{29} -4.51122 q^{31} -1.00000 q^{32} -6.48108 q^{34} +0.137699 q^{37} +1.00000 q^{38} +11.6489 q^{41} -2.51122 q^{43} +5.13770 q^{44} +7.56885 q^{46} +5.96216 q^{47} +10.8546 q^{49} +3.16784 q^{52} -0.225470 q^{53} -4.22547 q^{56} +0.832162 q^{58} -5.39331 q^{59} +14.4509 q^{61} +4.51122 q^{62} +1.00000 q^{64} -4.11021 q^{67} +6.48108 q^{68} -3.82446 q^{71} +4.70655 q^{73} -0.137699 q^{74} -1.00000 q^{76} +21.7092 q^{77} +10.6265 q^{79} -11.6489 q^{82} +12.0999 q^{83} +2.51122 q^{86} -5.13770 q^{88} +10.0000 q^{89} +13.3856 q^{91} -7.56885 q^{92} -5.96216 q^{94} +3.93972 q^{97} -10.8546 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 2 q^{7} - 3 q^{8} - 2 q^{11} - 2 q^{13} - 2 q^{14} + 3 q^{16} + 4 q^{17} - 3 q^{19} + 2 q^{22} - 14 q^{23} + 2 q^{26} + 2 q^{28} - 14 q^{29} - 4 q^{31} - 3 q^{32} - 4 q^{34} - 17 q^{37}+ \cdots - 25 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 2 * q^7 - 3 * q^8 - 2 * q^11 - 2 * q^13 - 2 * q^14 + 3 * q^16 + 4 * q^17 - 3 * q^19 + 2 * q^22 - 14 * q^23 + 2 * q^26 + 2 * q^28 - 14 * q^29 - 4 * q^31 - 3 * q^32 - 4 * q^34 - 17 * q^37 + 3 * q^38 + 8 * q^41 + 2 * q^43 - 2 * q^44 + 14 * q^46 - 13 * q^47 + 25 * q^49 - 2 * q^52 + 10 * q^53 - 2 * q^56 + 14 * q^58 + 6 * q^59 + 22 * q^61 + 4 * q^62 + 3 * q^64 + 4 * q^68 + 2 * q^71 - 12 * q^73 + 17 * q^74 - 3 * q^76 + 50 * q^77 + 24 * q^79 - 8 * q^82 - 12 * q^83 - 2 * q^86 + 2 * q^88 + 30 * q^89 - 7 * q^91 - 14 * q^92 + 13 * q^94 - 25 * q^98

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
$7$	4.22547	1.59708	0.798539	−	0.601943i	$-0.205607\pi$
$7$	4.22547	1.59708	0.798539	+	0.601943i	$0.205607\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	5.13770	1.54907	0.774537	−	0.632528i	$-0.217983\pi$
$11$	5.13770	1.54907	0.774537	+	0.632528i	$0.217983\pi$
$12$	0	0
$13$	3.16784	0.878600	0.439300	−	0.898340i	$-0.355226\pi$
$13$	3.16784	0.878600	0.439300	+	0.898340i	$0.355226\pi$
$14$	−4.22547	−1.12930
$15$	0	0
$16$	1.00000	0.250000
$17$	6.48108	1.57189	0.785946	−	0.618295i	$-0.212176\pi$
$17$	6.48108	1.57189	0.785946	+	0.618295i	$0.212176\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−5.13770	−1.09536
$23$	−7.56885	−1.57821	−0.789107	−	0.614256i	$-0.789456\pi$
$23$	−7.56885	−1.57821	−0.789107	+	0.614256i	$0.789456\pi$
$24$	0	0
$25$	0	0
$26$	−3.16784	−0.621264
$27$	0	0
$28$	4.22547	0.798539
$29$	−0.832162	−0.154529	−0.0772643	−	0.997011i	$-0.524619\pi$
$29$	−0.832162	−0.154529	−0.0772643	+	0.997011i	$0.524619\pi$
$30$	0	0
$31$	−4.51122	−0.810239	−0.405119	−	0.914264i	$-0.632770\pi$
$31$	−4.51122	−0.810239	−0.405119	+	0.914264i	$0.632770\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.48108	−1.11150
$35$	0	0
$36$	0	0
$37$	0.137699	0.0226376	0.0113188	−	0.999936i	$-0.496397\pi$
$37$	0.137699	0.0226376	0.0113188	+	0.999936i	$0.496397\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	11.6489	1.81926	0.909628	−	0.415425i	$-0.136367\pi$
$41$	11.6489	1.81926	0.909628	+	0.415425i	$0.136367\pi$
$42$	0	0
$43$	−2.51122	−0.382957	−0.191479	−	0.981497i	$-0.561328\pi$
$43$	−2.51122	−0.382957	−0.191479	+	0.981497i	$0.561328\pi$
$44$	5.13770	0.774537
$45$	0	0
$46$	7.56885	1.11597
$47$	5.96216	0.869670	0.434835	−	0.900510i	$-0.356807\pi$
$47$	5.96216	0.869670	0.434835	+	0.900510i	$0.356807\pi$
$48$	0	0
$49$	10.8546	1.55066
$50$	0	0
$51$	0	0
$52$	3.16784	0.439300
$53$	−0.225470	−0.0309707	−0.0154853	−	0.999880i	$-0.504929\pi$
$53$	−0.225470	−0.0309707	−0.0154853	+	0.999880i	$0.504929\pi$
$54$	0	0
$55$	0	0
$56$	−4.22547	−0.564652
$57$	0	0
$58$	0.832162	0.109268
$59$	−5.39331	−0.702149	−0.351074	−	0.936348i	$-0.614184\pi$
$59$	−5.39331	−0.702149	−0.351074	+	0.936348i	$0.614184\pi$
$60$	0	0
$61$	14.4509	1.85025	0.925127	−	0.379659i	$-0.123959\pi$
$61$	14.4509	1.85025	0.925127	+	0.379659i	$0.123959\pi$
$62$	4.51122	0.572925
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.11021	−0.502142	−0.251071	−	0.967969i	$-0.580783\pi$
$67$	−4.11021	−0.502142	−0.251071	+	0.967969i	$0.580783\pi$
$68$	6.48108	0.785946
$69$	0	0
$70$	0	0
$71$	−3.82446	−0.453880	−0.226940	−	0.973909i	$-0.572872\pi$
$71$	−3.82446	−0.453880	−0.226940	+	0.973909i	$0.572872\pi$
$72$	0	0
$73$	4.70655	0.550860	0.275430	−	0.961321i	$-0.411180\pi$
$73$	4.70655	0.550860	0.275430	+	0.961321i	$0.411180\pi$
$74$	−0.137699	−0.0160072
$75$	0	0
$76$	−1.00000	−0.114708
$77$	21.7092	2.47399
$78$	0	0
$79$	10.6265	1.19557	0.597786	−	0.801655i	$-0.296047\pi$
$79$	10.6265	1.19557	0.597786	+	0.801655i	$0.296047\pi$
$80$	0	0
$81$	0	0
$82$	−11.6489	−1.28641
$83$	12.0999	1.32813	0.664066	−	0.747674i	$-0.268829\pi$
$83$	12.0999	1.32813	0.664066	+	0.747674i	$0.268829\pi$
$84$	0	0
$85$	0	0
$86$	2.51122	0.270792
$87$	0	0
$88$	−5.13770	−0.547681
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	13.3856	1.40319
$92$	−7.56885	−0.789107
$93$	0	0
$94$	−5.96216	−0.614950
$95$	0	0
$96$	0	0
$97$	3.93972	0.400018	0.200009	−	0.979794i	$-0.435903\pi$
$97$	3.93972	0.400018	0.200009	+	0.979794i	$0.435903\pi$
$98$	−10.8546	−1.09648
$99$	0	0
$100$	0	0
$101$	−3.19798	−0.318211	−0.159105	−	0.987262i	$-0.550861\pi$
$101$	−3.19798	−0.318211	−0.159105	+	0.987262i	$0.550861\pi$
$102$	0	0
$103$	−10.6868	−1.05300	−0.526499	−	0.850176i	$-0.676496\pi$
$103$	−10.6868	−1.05300	−0.526499	+	0.850176i	$0.676496\pi$
$104$	−3.16784	−0.310632
$105$	0	0
$106$	0.225470	0.0218996
$107$	−10.8168	−1.04570	−0.522848	−	0.852426i	$-0.675130\pi$
$107$	−10.8168	−1.04570	−0.522848	+	0.852426i	$0.675130\pi$
$108$	0	0
$109$	9.24791	0.885789	0.442894	−	0.896574i	$-0.353952\pi$
$109$	9.24791	0.885789	0.442894	+	0.896574i	$0.353952\pi$
$110$	0	0
$111$	0	0
$112$	4.22547	0.399269
$113$	−17.6489	−1.66027	−0.830135	−	0.557562i	$-0.811737\pi$
$113$	−17.6489	−1.66027	−0.830135	+	0.557562i	$0.811737\pi$
$114$	0	0
$115$	0	0
$116$	−0.832162	−0.0772643
$117$	0	0
$118$	5.39331	0.496494
$119$	27.3856	2.51043
$120$	0	0
$121$	15.3960	1.39963
$122$	−14.4509	−1.30833
$123$	0	0
$124$	−4.51122	−0.405119
$125$	0	0
$126$	0	0
$127$	−12.7866	−1.13463	−0.567314	−	0.823501i	$-0.692018\pi$
$127$	−12.7866	−1.13463	−0.567314	+	0.823501i	$0.692018\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	2.11526	0.184811	0.0924057	−	0.995721i	$-0.470544\pi$
$131$	2.11526	0.184811	0.0924057	+	0.995721i	$0.470544\pi$
$132$	0	0
$133$	−4.22547	−0.366395
$134$	4.11021	0.355068
$135$	0	0
$136$	−6.48108	−0.555748
$137$	5.36317	0.458206	0.229103	−	0.973402i	$-0.426421\pi$
$137$	5.36317	0.458206	0.229103	+	0.973402i	$0.426421\pi$
$138$	0	0
$139$	−10.1601	−0.861771	−0.430886	−	0.902407i	$-0.641799\pi$
$139$	−10.1601	−0.861771	−0.430886	+	0.902407i	$0.641799\pi$
$140$	0	0
$141$	0	0
$142$	3.82446	0.320941
$143$	16.2754	1.36102
$144$	0	0
$145$	0	0
$146$	−4.70655	−0.389517
$147$	0	0
$148$	0.137699	0.0113188
$149$	5.93972	0.486601	0.243301	−	0.969951i	$-0.421770\pi$
$149$	5.93972	0.486601	0.243301	+	0.969951i	$0.421770\pi$
$150$	0	0
$151$	−15.2978	−1.24492	−0.622460	−	0.782652i	$-0.713867\pi$
$151$	−15.2978	−1.24492	−0.622460	+	0.782652i	$0.713867\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−21.7092	−1.74938
$155$	0	0
$156$	0	0
$157$	−12.7866	−1.02048	−0.510242	−	0.860031i	$-0.670444\pi$
$157$	−12.7866	−1.02048	−0.510242	+	0.860031i	$0.670444\pi$
$158$	−10.6265	−0.845397
$159$	0	0
$160$	0	0
$161$	−31.9819	−2.52053
$162$	0	0
$163$	−11.4734	−0.898664	−0.449332	−	0.893365i	$-0.648338\pi$
$163$	−11.4734	−0.898664	−0.449332	+	0.893365i	$0.648338\pi$
$164$	11.6489	0.909628
$165$	0	0
$166$	−12.0999	−0.939131
$167$	−19.4131	−1.50223	−0.751115	−	0.660171i	$-0.770484\pi$
$167$	−19.4131	−1.50223	−0.751115	+	0.660171i	$0.770484\pi$
$168$	0	0
$169$	−2.96480	−0.228062
$170$	0	0
$171$	0	0
$172$	−2.51122	−0.191479
$173$	−9.78662	−0.744063	−0.372031	−	0.928220i	$-0.621339\pi$
$173$	−9.78662	−0.744063	−0.372031	+	0.928220i	$0.621339\pi$
$174$	0	0
$175$	0	0
$176$	5.13770	0.387269
$177$	0	0
$178$	−10.0000	−0.749532
$179$	6.82446	0.510084	0.255042	−	0.966930i	$-0.417911\pi$
$179$	6.82446	0.510084	0.255042	+	0.966930i	$0.417911\pi$
$180$	0	0
$181$	−0.137699	−0.0102351	−0.00511755	−	0.999987i	$-0.501629\pi$
$181$	−0.137699	−0.0102351	−0.00511755	+	0.999987i	$0.501629\pi$
$182$	−13.3856	−0.992207
$183$	0	0
$184$	7.56885	0.557983
$185$	0	0
$186$	0	0
$187$	33.2978	2.43498
$188$	5.96216	0.434835
$189$	0	0
$190$	0	0
$191$	−19.3779	−1.40214	−0.701068	−	0.713095i	$-0.747293\pi$
$191$	−19.3779	−1.40214	−0.701068	+	0.713095i	$0.747293\pi$
$192$	0	0
$193$	−5.42851	−0.390752	−0.195376	−	0.980728i	$-0.562593\pi$
$193$	−5.42851	−0.390752	−0.195376	+	0.980728i	$0.562593\pi$
$194$	−3.93972	−0.282856
$195$	0	0
$196$	10.8546	0.775328
$197$	15.6489	1.11494	0.557470	−	0.830197i	$-0.311772\pi$
$197$	15.6489	1.11494	0.557470	+	0.830197i	$0.311772\pi$
$198$	0	0
$199$	18.0499	1.27953	0.639763	−	0.768572i	$-0.279033\pi$
$199$	18.0499	1.27953	0.639763	+	0.768572i	$0.279033\pi$
$200$	0	0
$201$	0	0
$202$	3.19798	0.225009
$203$	−3.51628	−0.246794
$204$	0	0
$205$	0	0
$206$	10.6868	0.744582
$207$	0	0
$208$	3.16784	0.219650
$209$	−5.13770	−0.355382
$210$	0	0
$211$	−7.50857	−0.516911	−0.258456	−	0.966023i	$-0.583214\pi$
$211$	−7.50857	−0.516911	−0.258456	+	0.966023i	$0.583214\pi$
$212$	−0.225470	−0.0154853
$213$	0	0
$214$	10.8168	0.739418
$215$	0	0
$216$	0	0
$217$	−19.0620	−1.29401
$218$	−9.24791	−0.626347
$219$	0	0
$220$	0	0
$221$	20.5310	1.38106
$222$	0	0
$223$	27.8091	1.86223	0.931116	−	0.364723i	$-0.118836\pi$
$223$	27.8091	1.86223	0.931116	+	0.364723i	$0.118836\pi$
$224$	−4.22547	−0.282326
$225$	0	0
$226$	17.6489	1.17399
$227$	−8.91223	−0.591525	−0.295763	−	0.955261i	$-0.595574\pi$
$227$	−8.91223	−0.591525	−0.295763	+	0.955261i	$0.595574\pi$
$228$	0	0
$229$	−13.6489	−0.901946	−0.450973	−	0.892538i	$-0.648923\pi$
$229$	−13.6489	−0.901946	−0.450973	+	0.892538i	$0.648923\pi$
$230$	0	0
$231$	0	0
$232$	0.832162	0.0546341
$233$	−23.2754	−1.52482	−0.762411	−	0.647093i	$-0.775985\pi$
$233$	−23.2754	−1.52482	−0.762411	+	0.647093i	$0.775985\pi$
$234$	0	0
$235$	0	0
$236$	−5.39331	−0.351074
$237$	0	0
$238$	−27.3856	−1.77515
$239$	−3.72898	−0.241208	−0.120604	−	0.992701i	$-0.538483\pi$
$239$	−3.72898	−0.241208	−0.120604	+	0.992701i	$0.538483\pi$
$240$	0	0
$241$	−1.48878	−0.0959009	−0.0479505	−	0.998850i	$-0.515269\pi$
$241$	−1.48878	−0.0959009	−0.0479505	+	0.998850i	$0.515269\pi$
$242$	−15.3960	−0.989689
$243$	0	0
$244$	14.4509	0.925127
$245$	0	0
$246$	0	0
$247$	−3.16784	−0.201565
$248$	4.51122	0.286463
$249$	0	0
$250$	0	0
$251$	8.78662	0.554606	0.277303	−	0.960782i	$-0.410559\pi$
$251$	8.78662	0.554606	0.277303	+	0.960782i	$0.410559\pi$
$252$	0	0
$253$	−38.8865	−2.44477
$254$	12.7866	0.802304
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.74175	−0.171025	−0.0855127	−	0.996337i	$-0.527253\pi$
$257$	−2.74175	−0.171025	−0.0855127	+	0.996337i	$0.527253\pi$
$258$	0	0
$259$	0.581844	0.0361540
$260$	0	0
$261$	0	0
$262$	−2.11526	−0.130681
$263$	−2.74704	−0.169390	−0.0846948	−	0.996407i	$-0.526992\pi$
$263$	−2.74704	−0.169390	−0.0846948	+	0.996407i	$0.526992\pi$
$264$	0	0
$265$	0	0
$266$	4.22547	0.259080
$267$	0	0
$268$	−4.11021	−0.251071
$269$	9.74175	0.593965	0.296982	−	0.954883i	$-0.404020\pi$
$269$	9.74175	0.593965	0.296982	+	0.954883i	$0.404020\pi$
$270$	0	0
$271$	26.3555	1.60098	0.800490	−	0.599346i	$-0.204573\pi$
$271$	26.3555	1.60098	0.800490	+	0.599346i	$0.204573\pi$
$272$	6.48108	0.392973
$273$	0	0
$274$	−5.36317	−0.324001
$275$	0	0
$276$	0	0
$277$	−0.962158	−0.0578104	−0.0289052	−	0.999582i	$-0.509202\pi$
$277$	−0.962158	−0.0578104	−0.0289052	+	0.999582i	$0.509202\pi$
$278$	10.1601	0.609364
$279$	0	0
$280$	0	0
$281$	14.6714	0.875219	0.437610	−	0.899165i	$-0.355825\pi$
$281$	14.6714	0.875219	0.437610	+	0.899165i	$0.355825\pi$
$282$	0	0
$283$	−26.1601	−1.55506	−0.777529	−	0.628847i	$-0.783527\pi$
$283$	−26.1601	−1.55506	−0.777529	+	0.628847i	$0.783527\pi$
$284$	−3.82446	−0.226940
$285$	0	0
$286$	−16.2754	−0.962384
$287$	49.2221	2.90549
$288$	0	0
$289$	25.0044	1.47085
$290$	0	0
$291$	0	0
$292$	4.70655	0.275430
$293$	22.4657	1.31246	0.656229	−	0.754562i	$-0.272150\pi$
$293$	22.4657	1.31246	0.656229	+	0.754562i	$0.272150\pi$
$294$	0	0
$295$	0	0
$296$	−0.137699	−0.00800360
$297$	0	0
$298$	−5.93972	−0.344079
$299$	−23.9769	−1.38662
$300$	0	0
$301$	−10.6111	−0.611612
$302$	15.2978	0.880291
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	17.2204	0.982821	0.491410	−	0.870928i	$-0.336482\pi$
$307$	17.2204	0.982821	0.491410	+	0.870928i	$0.336482\pi$
$308$	21.7092	1.23700
$309$	0	0
$310$	0	0
$311$	7.87439	0.446516	0.223258	−	0.974759i	$-0.428331\pi$
$311$	7.87439	0.446516	0.223258	+	0.974759i	$0.428331\pi$
$312$	0	0
$313$	−25.1678	−1.42257	−0.711285	−	0.702904i	$-0.751887\pi$
$313$	−25.1678	−1.42257	−0.711285	+	0.702904i	$0.751887\pi$
$314$	12.7866	0.721590
$315$	0	0
$316$	10.6265	0.597786
$317$	23.9045	1.34261	0.671306	−	0.741180i	$-0.265734\pi$
$317$	23.9045	1.34261	0.671306	+	0.741180i	$0.265734\pi$
$318$	0	0
$319$	−4.27540	−0.239376
$320$	0	0
$321$	0	0
$322$	31.9819	1.78228
$323$	−6.48108	−0.360617
$324$	0	0
$325$	0	0
$326$	11.4734	0.635451
$327$	0	0
$328$	−11.6489	−0.643204
$329$	25.1929	1.38893
$330$	0	0
$331$	30.7565	1.69053	0.845264	−	0.534348i	$-0.179443\pi$
$331$	30.7565	1.69053	0.845264	+	0.534348i	$0.179443\pi$
$332$	12.0999	0.664066
$333$	0	0
$334$	19.4131	1.06224
$335$	0	0
$336$	0	0
$337$	13.4734	0.733942	0.366971	−	0.930232i	$-0.380395\pi$
$337$	13.4734	0.733942	0.366971	+	0.930232i	$0.380395\pi$
$338$	2.96480	0.161264
$339$	0	0
$340$	0	0
$341$	−23.1773	−1.25512
$342$	0	0
$343$	16.2875	0.879441
$344$	2.51122	0.135396
$345$	0	0
$346$	9.78662	0.526132
$347$	−28.9468	−1.55394	−0.776971	−	0.629536i	$-0.783245\pi$
$347$	−28.9468	−1.55394	−0.776971	+	0.629536i	$0.783245\pi$
$348$	0	0
$349$	27.9243	1.49475	0.747377	−	0.664400i	$-0.231313\pi$
$349$	27.9243	1.49475	0.747377	+	0.664400i	$0.231313\pi$
$350$	0	0
$351$	0	0
$352$	−5.13770	−0.273840
$353$	−28.3099	−1.50678	−0.753392	−	0.657571i	$-0.771584\pi$
$353$	−28.3099	−1.50678	−0.753392	+	0.657571i	$0.771584\pi$
$354$	0	0
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	−6.82446	−0.360684
$359$	−2.60163	−0.137309	−0.0686545	−	0.997640i	$-0.521871\pi$
$359$	−2.60163	−0.137309	−0.0686545	+	0.997640i	$0.521871\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0.137699	0.00723731
$363$	0	0
$364$	13.3856	0.701596
$365$	0	0
$366$	0	0
$367$	11.6489	0.608069	0.304034	−	0.952661i	$-0.401666\pi$
$367$	11.6489	0.608069	0.304034	+	0.952661i	$0.401666\pi$
$368$	−7.56885	−0.394554
$369$	0	0
$370$	0	0
$371$	−0.952717	−0.0494626
$372$	0	0
$373$	−12.8064	−0.663091	−0.331545	−	0.943439i	$-0.607570\pi$
$373$	−12.8064	−0.663091	−0.331545	+	0.943439i	$0.607570\pi$
$374$	−33.2978	−1.72179
$375$	0	0
$376$	−5.96216	−0.307475
$377$	−2.63615	−0.135769
$378$	0	0
$379$	−20.9122	−1.07419	−0.537095	−	0.843522i	$-0.680478\pi$
$379$	−20.9122	−1.07419	−0.537095	+	0.843522i	$0.680478\pi$
$380$	0	0
$381$	0	0
$382$	19.3779	0.991460
$383$	−10.3511	−0.528916	−0.264458	−	0.964397i	$-0.585193\pi$
$383$	−10.3511	−0.528916	−0.264458	+	0.964397i	$0.585193\pi$
$384$	0	0
$385$	0	0
$386$	5.42851	0.276304
$387$	0	0
$388$	3.93972	0.200009
$389$	−6.56620	−0.332920	−0.166460	−	0.986048i	$-0.553234\pi$
$389$	−6.56620	−0.332920	−0.166460	+	0.986048i	$0.553234\pi$
$390$	0	0
$391$	−49.0543	−2.48078
$392$	−10.8546	−0.548240
$393$	0	0
$394$	−15.6489	−0.788381
$395$	0	0
$396$	0	0
$397$	23.5284	1.18085	0.590427	−	0.807091i	$-0.298959\pi$
$397$	23.5284	1.18085	0.590427	+	0.807091i	$0.298959\pi$
$398$	−18.0499	−0.904761
$399$	0	0
$400$	0	0
$401$	−6.22041	−0.310633	−0.155316	−	0.987865i	$-0.549640\pi$
$401$	−6.22041	−0.310633	−0.155316	+	0.987865i	$0.549640\pi$
$402$	0	0
$403$	−14.2908	−0.711876
$404$	−3.19798	−0.159105
$405$	0	0
$406$	3.51628	0.174510
$407$	0.707457	0.0350674
$408$	0	0
$409$	−34.4905	−1.70545	−0.852723	−	0.522363i	$-0.825051\pi$
$409$	−34.4905	−1.70545	−0.852723	+	0.522363i	$0.825051\pi$
$410$	0	0
$411$	0	0
$412$	−10.6868	−0.526499
$413$	−22.7893	−1.12139
$414$	0	0
$415$	0	0
$416$	−3.16784	−0.155316
$417$	0	0
$418$	5.13770	0.251293
$419$	10.6265	0.519138	0.259569	−	0.965725i	$-0.416420\pi$
$419$	10.6265	0.519138	0.259569	+	0.965725i	$0.416420\pi$
$420$	0	0
$421$	−1.98021	−0.0965095	−0.0482548	−	0.998835i	$-0.515366\pi$
$421$	−1.98021	−0.0965095	−0.0482548	+	0.998835i	$0.515366\pi$
$422$	7.50857	0.365512
$423$	0	0
$424$	0.225470	0.0109498
$425$	0	0
$426$	0	0
$427$	61.0620	2.95500
$428$	−10.8168	−0.522848
$429$	0	0
$430$	0	0
$431$	−15.0774	−0.726254	−0.363127	−	0.931740i	$-0.618291\pi$
$431$	−15.0774	−0.726254	−0.363127	+	0.931740i	$0.618291\pi$
$432$	0	0
$433$	13.5337	0.650386	0.325193	−	0.945648i	$-0.394571\pi$
$433$	13.5337	0.650386	0.325193	+	0.945648i	$0.394571\pi$
$434$	19.0620	0.915006
$435$	0	0
$436$	9.24791	0.442894
$437$	7.56885	0.362067
$438$	0	0
$439$	39.1773	1.86983	0.934915	−	0.354872i	$-0.115476\pi$
$439$	39.1773	1.86983	0.934915	+	0.354872i	$0.115476\pi$
$440$	0	0
$441$	0	0
$442$	−20.5310	−0.976560
$443$	−19.3132	−0.917600	−0.458800	−	0.888540i	$-0.651721\pi$
$443$	−19.3132	−0.917600	−0.458800	+	0.888540i	$0.651721\pi$
$444$	0	0
$445$	0	0
$446$	−27.8091	−1.31680
$447$	0	0
$448$	4.22547	0.199635
$449$	−41.4131	−1.95440	−0.977202	−	0.212310i	$-0.931901\pi$
$449$	−41.4131	−1.95440	−0.977202	+	0.212310i	$0.931901\pi$
$450$	0	0
$451$	59.8486	2.81816
$452$	−17.6489	−0.830135
$453$	0	0
$454$	8.91223	0.418272
$455$	0	0
$456$	0	0
$457$	−37.8392	−1.77004	−0.885021	−	0.465551i	$-0.845856\pi$
$457$	−37.8392	−1.77004	−0.885021	+	0.465551i	$0.845856\pi$
$458$	13.6489	0.637772
$459$	0	0
$460$	0	0
$461$	−1.58864	−0.0739903	−0.0369952	−	0.999315i	$-0.511779\pi$
$461$	−1.58864	−0.0739903	−0.0369952	+	0.999315i	$0.511779\pi$
$462$	0	0
$463$	40.3581	1.87560	0.937800	−	0.347175i	$-0.112859\pi$
$463$	40.3581	1.87560	0.937800	+	0.347175i	$0.112859\pi$
$464$	−0.832162	−0.0386322
$465$	0	0
$466$	23.2754	1.07821
$467$	39.5130	1.82844	0.914221	−	0.405217i	$-0.132804\pi$
$467$	39.5130	1.82844	0.914221	+	0.405217i	$0.132804\pi$
$468$	0	0
$469$	−17.3676	−0.801959
$470$	0	0
$471$	0	0
$472$	5.39331	0.248247
$473$	−12.9019	−0.593229
$474$	0	0
$475$	0	0
$476$	27.3856	1.25522
$477$	0	0
$478$	3.72898	0.170560
$479$	−7.61107	−0.347759	−0.173879	−	0.984767i	$-0.555630\pi$
$479$	−7.61107	−0.347759	−0.173879	+	0.984767i	$0.555630\pi$
$480$	0	0
$481$	0.436209	0.0198894
$482$	1.48878	0.0678122
$483$	0	0
$484$	15.3960	0.699816
$485$	0	0
$486$	0	0
$487$	−20.3907	−0.923989	−0.461995	−	0.886883i	$-0.652866\pi$
$487$	−20.3907	−0.923989	−0.461995	+	0.886883i	$0.652866\pi$
$488$	−14.4509	−0.654163
$489$	0	0
$490$	0	0
$491$	−25.0224	−1.12925	−0.564623	−	0.825349i	$-0.690979\pi$
$491$	−25.0224	−1.12925	−0.564623	+	0.825349i	$0.690979\pi$
$492$	0	0
$493$	−5.39331	−0.242902
$494$	3.16784	0.142528
$495$	0	0
$496$	−4.51122	−0.202560
$497$	−16.1601	−0.724881
$498$	0	0
$499$	22.6111	1.01221	0.506105	−	0.862472i	$-0.331085\pi$
$499$	22.6111	1.01221	0.506105	+	0.862472i	$0.331085\pi$
$500$	0	0
$501$	0	0
$502$	−8.78662	−0.392166
$503$	−11.5035	−0.512916	−0.256458	−	0.966555i	$-0.582556\pi$
$503$	−11.5035	−0.512916	−0.256458	+	0.966555i	$0.582556\pi$
$504$	0	0
$505$	0	0
$506$	38.8865	1.72871
$507$	0	0
$508$	−12.7866	−0.567314
$509$	−9.11526	−0.404027	−0.202013	−	0.979383i	$-0.564748\pi$
$509$	−9.11526	−0.404027	−0.202013	+	0.979383i	$0.564748\pi$
$510$	0	0
$511$	19.8874	0.879766
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	2.74175	0.120933
$515$	0	0
$516$	0	0
$517$	30.6318	1.34718
$518$	−0.581844	−0.0255648
$519$	0	0
$520$	0	0
$521$	−15.4888	−0.678576	−0.339288	−	0.940683i	$-0.610186\pi$
$521$	−15.4888	−0.678576	−0.339288	+	0.940683i	$0.610186\pi$
$522$	0	0
$523$	4.34073	0.189807	0.0949035	−	0.995486i	$-0.469746\pi$
$523$	4.34073	0.189807	0.0949035	+	0.995486i	$0.469746\pi$
$524$	2.11526	0.0924057
$525$	0	0
$526$	2.74704	0.119776
$527$	−29.2376	−1.27361
$528$	0	0
$529$	34.2875	1.49076
$530$	0	0
$531$	0	0
$532$	−4.22547	−0.183197
$533$	36.9019	1.59840
$534$	0	0
$535$	0	0
$536$	4.11021	0.177534
$537$	0	0
$538$	−9.74175	−0.419996
$539$	55.7677	2.40208
$540$	0	0
$541$	19.5491	0.840480	0.420240	−	0.907413i	$-0.361946\pi$
$541$	19.5491	0.840480	0.420240	+	0.907413i	$0.361946\pi$
$542$	−26.3555	−1.13206
$543$	0	0
$544$	−6.48108	−0.277874
$545$	0	0
$546$	0	0
$547$	−41.8038	−1.78740	−0.893700	−	0.448665i	$-0.851900\pi$
$547$	−41.8038	−1.78740	−0.893700	+	0.448665i	$0.851900\pi$
$548$	5.36317	0.229103
$549$	0	0
$550$	0	0
$551$	0.832162	0.0354513
$552$	0	0
$553$	44.9019	1.90942
$554$	0.962158	0.0408782
$555$	0	0
$556$	−10.1601	−0.430886
$557$	9.76418	0.413722	0.206861	−	0.978370i	$-0.433675\pi$
$557$	9.76418	0.413722	0.206861	+	0.978370i	$0.433675\pi$
$558$	0	0
$559$	−7.95513	−0.336466
$560$	0	0
$561$	0	0
$562$	−14.6714	−0.618874
$563$	−11.4509	−0.482600	−0.241300	−	0.970451i	$-0.577574\pi$
$563$	−11.4509	−0.482600	−0.241300	+	0.970451i	$0.577574\pi$
$564$	0	0
$565$	0	0
$566$	26.1601	1.09959
$567$	0	0
$568$	3.82446	0.160471
$569$	13.4338	0.563174	0.281587	−	0.959536i	$-0.409139\pi$
$569$	13.4338	0.563174	0.281587	+	0.959536i	$0.409139\pi$
$570$	0	0
$571$	37.6335	1.57491	0.787457	−	0.616370i	$-0.211397\pi$
$571$	37.6335	1.57491	0.787457	+	0.616370i	$0.211397\pi$
$572$	16.2754	0.680509
$573$	0	0
$574$	−49.2221	−2.05449
$575$	0	0
$576$	0	0
$577$	1.56885	0.0653121	0.0326560	−	0.999467i	$-0.489603\pi$
$577$	1.56885	0.0653121	0.0326560	+	0.999467i	$0.489603\pi$
$578$	−25.0044	−1.04005
$579$	0	0
$580$	0	0
$581$	51.1276	2.12113
$582$	0	0
$583$	−1.15840	−0.0479759
$584$	−4.70655	−0.194758
$585$	0	0
$586$	−22.4657	−0.928048
$587$	−25.8693	−1.06774	−0.533871	−	0.845566i	$-0.679263\pi$
$587$	−25.8693	−1.06774	−0.533871	+	0.845566i	$0.679263\pi$
$588$	0	0
$589$	4.51122	0.185881
$590$	0	0
$591$	0	0
$592$	0.137699	0.00565940
$593$	28.9243	1.18778	0.593890	−	0.804547i	$-0.297592\pi$
$593$	28.9243	1.18778	0.593890	+	0.804547i	$0.297592\pi$
$594$	0	0
$595$	0	0
$596$	5.93972	0.243301
$597$	0	0
$598$	23.9769	0.980488
$599$	−41.3581	−1.68985	−0.844923	−	0.534887i	$-0.820354\pi$
$599$	−41.3581	−1.68985	−0.844923	+	0.534887i	$0.820354\pi$
$600$	0	0
$601$	2.68147	0.109379	0.0546897	−	0.998503i	$-0.482583\pi$
$601$	2.68147	0.109379	0.0546897	+	0.998503i	$0.482583\pi$
$602$	10.6111	0.432475
$603$	0	0
$604$	−15.2978	−0.622460
$605$	0	0
$606$	0	0
$607$	3.31324	0.134480	0.0672401	−	0.997737i	$-0.478581\pi$
$607$	3.31324	0.134480	0.0672401	+	0.997737i	$0.478581\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	18.8871	0.764092
$612$	0	0
$613$	−10.0603	−0.406331	−0.203165	−	0.979144i	$-0.565123\pi$
$613$	−10.0603	−0.406331	−0.203165	+	0.979144i	$0.565123\pi$
$614$	−17.2204	−0.694959
$615$	0	0
$616$	−21.7092	−0.874688
$617$	27.6265	1.11220	0.556100	−	0.831115i	$-0.312297\pi$
$617$	27.6265	1.11220	0.556100	+	0.831115i	$0.312297\pi$
$618$	0	0
$619$	16.6714	0.670078	0.335039	−	0.942204i	$-0.391250\pi$
$619$	16.6714	0.670078	0.335039	+	0.942204i	$0.391250\pi$
$620$	0	0
$621$	0	0
$622$	−7.87439	−0.315734
$623$	42.2547	1.69290
$624$	0	0
$625$	0	0
$626$	25.1678	1.00591
$627$	0	0
$628$	−12.7866	−0.510242
$629$	0.892439	0.0355839
$630$	0	0
$631$	−0.709194	−0.0282326	−0.0141163	−	0.999900i	$-0.504494\pi$
$631$	−0.709194	−0.0282326	−0.0141163	+	0.999900i	$0.504494\pi$
$632$	−10.6265	−0.422699
$633$	0	0
$634$	−23.9045	−0.949370
$635$	0	0
$636$	0	0
$637$	34.3856	1.36241
$638$	4.27540	0.169265
$639$	0	0
$640$	0	0
$641$	2.43553	0.0961978	0.0480989	−	0.998843i	$-0.484684\pi$
$641$	2.43553	0.0961978	0.0480989	+	0.998843i	$0.484684\pi$
$642$	0	0
$643$	7.70390	0.303812	0.151906	−	0.988395i	$-0.451459\pi$
$643$	7.70390	0.303812	0.151906	+	0.988395i	$0.451459\pi$
$644$	−31.9819	−1.26027
$645$	0	0
$646$	6.48108	0.254995
$647$	10.0499	0.395103	0.197552	−	0.980292i	$-0.436701\pi$
$647$	10.0499	0.395103	0.197552	+	0.980292i	$0.436701\pi$
$648$	0	0
$649$	−27.7092	−1.08768
$650$	0	0
$651$	0	0
$652$	−11.4734	−0.449332
$653$	7.09283	0.277564	0.138782	−	0.990323i	$-0.455681\pi$
$653$	7.09283	0.277564	0.138782	+	0.990323i	$0.455681\pi$
$654$	0	0
$655$	0	0
$656$	11.6489	0.454814
$657$	0	0
$658$	−25.1929	−0.982122
$659$	1.16346	0.0453218	0.0226609	−	0.999743i	$-0.492786\pi$
$659$	1.16346	0.0453218	0.0226609	+	0.999743i	$0.492786\pi$
$660$	0	0
$661$	1.79432	0.0697909	0.0348955	−	0.999391i	$-0.488890\pi$
$661$	1.79432	0.0697909	0.0348955	+	0.999391i	$0.488890\pi$
$662$	−30.7565	−1.19538
$663$	0	0
$664$	−12.0999	−0.469566
$665$	0	0
$666$	0	0
$667$	6.29851	0.243879
$668$	−19.4131	−0.751115
$669$	0	0
$670$	0	0
$671$	74.2446	2.86618
$672$	0	0
$673$	−25.2824	−0.974566	−0.487283	−	0.873244i	$-0.662012\pi$
$673$	−25.2824	−0.974566	−0.487283	+	0.873244i	$0.662012\pi$
$674$	−13.4734	−0.518975
$675$	0	0
$676$	−2.96480	−0.114031
$677$	4.89682	0.188200	0.0941001	−	0.995563i	$-0.470003\pi$
$677$	4.89682	0.188200	0.0941001	+	0.995563i	$0.470003\pi$
$678$	0	0
$679$	16.6472	0.638860
$680$	0	0
$681$	0	0
$682$	23.1773	0.887504
$683$	−10.1980	−0.390215	−0.195107	−	0.980782i	$-0.562506\pi$
$683$	−10.1980	−0.390215	−0.195107	+	0.980782i	$0.562506\pi$
$684$	0	0
$685$	0	0
$686$	−16.2875	−0.621859
$687$	0	0
$688$	−2.51122	−0.0957393
$689$	−0.714253	−0.0272109
$690$	0	0
$691$	−39.9846	−1.52109	−0.760543	−	0.649288i	$-0.775067\pi$
$691$	−39.9846	−1.52109	−0.760543	+	0.649288i	$0.775067\pi$
$692$	−9.78662	−0.372031
$693$	0	0
$694$	28.9468	1.09880
$695$	0	0
$696$	0	0
$697$	75.4975	2.85967
$698$	−27.9243	−1.05695
$699$	0	0
$700$	0	0
$701$	−4.91729	−0.185723	−0.0928617	−	0.995679i	$-0.529601\pi$
$701$	−4.91729	−0.185723	−0.0928617	+	0.995679i	$0.529601\pi$
$702$	0	0
$703$	−0.137699	−0.00519342
$704$	5.13770	0.193634
$705$	0	0
$706$	28.3099	1.06546
$707$	−13.5130	−0.508207
$708$	0	0
$709$	−36.8865	−1.38530	−0.692650	−	0.721274i	$-0.743557\pi$
$709$	−36.8865	−1.38530	−0.692650	+	0.721274i	$0.743557\pi$
$710$	0	0
$711$	0	0
$712$	−10.0000	−0.374766
$713$	34.1447	1.27873
$714$	0	0
$715$	0	0
$716$	6.82446	0.255042
$717$	0	0
$718$	2.60163	0.0970921
$719$	23.8891	0.890914	0.445457	−	0.895303i	$-0.353041\pi$
$719$	23.8891	0.890914	0.445457	+	0.895303i	$0.353041\pi$
$720$	0	0
$721$	−45.1566	−1.68172
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−0.137699	−0.00511755
$725$	0	0
$726$	0	0
$727$	19.6894	0.730240	0.365120	−	0.930961i	$-0.381028\pi$
$727$	19.6894	0.730240	0.365120	+	0.930961i	$0.381028\pi$
$728$	−13.3856	−0.496104
$729$	0	0
$730$	0	0
$731$	−16.2754	−0.601967
$732$	0	0
$733$	1.76418	0.0651615	0.0325808	−	0.999469i	$-0.489627\pi$
$733$	1.76418	0.0651615	0.0325808	+	0.999469i	$0.489627\pi$
$734$	−11.6489	−0.429969
$735$	0	0
$736$	7.56885	0.278991
$737$	−21.1170	−0.777855
$738$	0	0
$739$	28.1755	1.03645	0.518227	−	0.855243i	$-0.326592\pi$
$739$	28.1755	1.03645	0.518227	+	0.855243i	$0.326592\pi$
$740$	0	0
$741$	0	0
$742$	0.952717	0.0349753
$743$	5.42851	0.199153	0.0995763	−	0.995030i	$-0.468251\pi$
$743$	5.42851	0.199153	0.0995763	+	0.995030i	$0.468251\pi$
$744$	0	0
$745$	0	0
$746$	12.8064	0.468876
$747$	0	0
$748$	33.2978	1.21749
$749$	−45.7059	−1.67006
$750$	0	0
$751$	25.8640	0.943792	0.471896	−	0.881654i	$-0.343570\pi$
$751$	25.8640	0.943792	0.471896	+	0.881654i	$0.343570\pi$
$752$	5.96216	0.217418
$753$	0	0
$754$	2.63615	0.0960031
$755$	0	0
$756$	0	0
$757$	−3.76947	−0.137004	−0.0685019	−	0.997651i	$-0.521822\pi$
$757$	−3.76947	−0.137004	−0.0685019	+	0.997651i	$0.521822\pi$
$758$	20.9122	0.759566
$759$	0	0
$760$	0	0
$761$	−18.3605	−0.665568	−0.332784	−	0.943003i	$-0.607988\pi$
$761$	−18.3605	−0.665568	−0.332784	+	0.943003i	$0.607988\pi$
$762$	0	0
$763$	39.0767	1.41467
$764$	−19.3779	−0.701068
$765$	0	0
$766$	10.3511	0.374000
$767$	−17.0851	−0.616908
$768$	0	0
$769$	−38.1300	−1.37500	−0.687501	−	0.726183i	$-0.741292\pi$
$769$	−38.1300	−1.37500	−0.687501	+	0.726183i	$0.741292\pi$
$770$	0	0
$771$	0	0
$772$	−5.42851	−0.195376
$773$	29.1575	1.04872	0.524361	−	0.851496i	$-0.324304\pi$
$773$	29.1575	1.04872	0.524361	+	0.851496i	$0.324304\pi$
$774$	0	0
$775$	0	0
$776$	−3.93972	−0.141428
$777$	0	0
$778$	6.56620	0.235410
$779$	−11.6489	−0.417366
$780$	0	0
$781$	−19.6489	−0.703094
$782$	49.0543	1.75418
$783$	0	0
$784$	10.8546	0.387664
$785$	0	0
$786$	0	0
$787$	−47.0165	−1.67596	−0.837978	−	0.545704i	$-0.816262\pi$
$787$	−47.0165	−1.67596	−0.837978	+	0.545704i	$0.816262\pi$
$788$	15.6489	0.557470
$789$	0	0
$790$	0	0
$791$	−74.5750	−2.65158
$792$	0	0
$793$	45.7782	1.62563
$794$	−23.5284	−0.834990
$795$	0	0
$796$	18.0499	0.639763
$797$	42.6359	1.51024	0.755121	−	0.655586i	$-0.227578\pi$
$797$	42.6359	1.51024	0.755121	+	0.655586i	$0.227578\pi$
$798$	0	0
$799$	38.6412	1.36703
$800$	0	0
$801$	0	0
$802$	6.22041	0.219650
$803$	24.1808	0.853323
$804$	0	0
$805$	0	0
$806$	14.2908	0.503372
$807$	0	0
$808$	3.19798	0.112504
$809$	−49.4630	−1.73903	−0.869514	−	0.493909i	$-0.835568\pi$
$809$	−49.4630	−1.73903	−0.869514	+	0.493909i	$0.835568\pi$
$810$	0	0
$811$	−16.7816	−0.589280	−0.294640	−	0.955608i	$-0.595200\pi$
$811$	−16.7816	−0.589280	−0.294640	+	0.955608i	$0.595200\pi$
$812$	−3.51628	−0.123397
$813$	0	0
$814$	−0.707457	−0.0247964
$815$	0	0
$816$	0	0
$817$	2.51122	0.0878564
$818$	34.4905	1.20593
$819$	0	0
$820$	0	0
$821$	−11.5337	−0.402527	−0.201264	−	0.979537i	$-0.564505\pi$
$821$	−11.5337	−0.402527	−0.201264	+	0.979537i	$0.564505\pi$
$822$	0	0
$823$	32.6309	1.13744	0.568720	−	0.822531i	$-0.307439\pi$
$823$	32.6309	1.13744	0.568720	+	0.822531i	$0.307439\pi$
$824$	10.6868	0.372291
$825$	0	0
$826$	22.7893	0.792940
$827$	−6.64650	−0.231122	−0.115561	−	0.993300i	$-0.536867\pi$
$827$	−6.64650	−0.231122	−0.115561	+	0.993300i	$0.536867\pi$
$828$	0	0
$829$	−30.9217	−1.07395	−0.536977	−	0.843597i	$-0.680434\pi$
$829$	−30.9217	−1.07395	−0.536977	+	0.843597i	$0.680434\pi$
$830$	0	0
$831$	0	0
$832$	3.16784	0.109825
$833$	70.3495	2.43747
$834$	0	0
$835$	0	0
$836$	−5.13770	−0.177691
$837$	0	0
$838$	−10.6265	−0.367086
$839$	48.7109	1.68169	0.840844	−	0.541277i	$-0.182059\pi$
$839$	48.7109	1.68169	0.840844	+	0.541277i	$0.182059\pi$
$840$	0	0
$841$	−28.3075	−0.976121
$842$	1.98021	0.0682426
$843$	0	0
$844$	−7.50857	−0.258456
$845$	0	0
$846$	0	0
$847$	65.0551	2.23532
$848$	−0.225470	−0.00774267
$849$	0	0
$850$	0	0
$851$	−1.04222	−0.0357270
$852$	0	0
$853$	46.1447	1.57997	0.789983	−	0.613129i	$-0.210089\pi$
$853$	46.1447	1.57997	0.789983	+	0.613129i	$0.210089\pi$
$854$	−61.0620	−2.08950
$855$	0	0
$856$	10.8168	0.369709
$857$	8.40607	0.287146	0.143573	−	0.989640i	$-0.454141\pi$
$857$	8.40607	0.287146	0.143573	+	0.989640i	$0.454141\pi$
$858$	0	0
$859$	−8.49581	−0.289873	−0.144937	−	0.989441i	$-0.546298\pi$
$859$	−8.49581	−0.289873	−0.144937	+	0.989441i	$0.546298\pi$
$860$	0	0
$861$	0	0
$862$	15.0774	0.513539
$863$	−3.37352	−0.114836	−0.0574179	−	0.998350i	$-0.518287\pi$
$863$	−3.37352	−0.114836	−0.0574179	+	0.998350i	$0.518287\pi$
$864$	0	0
$865$	0	0
$866$	−13.5337	−0.459892
$867$	0	0
$868$	−19.0620	−0.647007
$869$	54.5957	1.85203
$870$	0	0
$871$	−13.0205	−0.441182
$872$	−9.24791	−0.313174
$873$	0	0
$874$	−7.56885	−0.256020
$875$	0	0
$876$	0	0
$877$	−13.2780	−0.448368	−0.224184	−	0.974547i	$-0.571972\pi$
$877$	−13.2780	−0.448368	−0.224184	+	0.974547i	$0.571972\pi$
$878$	−39.1773	−1.32217
$879$	0	0
$880$	0	0
$881$	26.6489	0.897825	0.448912	−	0.893576i	$-0.351812\pi$
$881$	26.6489	0.897825	0.448912	+	0.893576i	$0.351812\pi$
$882$	0	0
$883$	−47.7884	−1.60821	−0.804103	−	0.594490i	$-0.797354\pi$
$883$	−47.7884	−1.60821	−0.804103	+	0.594490i	$0.797354\pi$
$884$	20.5310	0.690532
$885$	0	0
$886$	19.3132	0.648841
$887$	36.3304	1.21985	0.609927	−	0.792457i	$-0.291199\pi$
$887$	36.3304	1.21985	0.609927	+	0.792457i	$0.291199\pi$
$888$	0	0
$889$	−54.0295	−1.81209
$890$	0	0
$891$	0	0
$892$	27.8091	0.931116
$893$	−5.96216	−0.199516
$894$	0	0
$895$	0	0
$896$	−4.22547	−0.141163
$897$	0	0
$898$	41.4131	1.38197
$899$	3.75406	0.125205
$900$	0	0
$901$	−1.46129	−0.0486826
$902$	−59.8486	−1.99274
$903$	0	0
$904$	17.6489	0.586994
$905$	0	0
$906$	0	0
$907$	−39.3873	−1.30784	−0.653918	−	0.756566i	$-0.726876\pi$
$907$	−39.3873	−1.30784	−0.653918	+	0.756566i	$0.726876\pi$
$908$	−8.91223	−0.295763
$909$	0	0
$910$	0	0
$911$	20.5561	0.681054	0.340527	−	0.940235i	$-0.389395\pi$
$911$	20.5561	0.681054	0.340527	+	0.940235i	$0.389395\pi$
$912$	0	0
$913$	62.1654	2.05738
$914$	37.8392	1.25161
$915$	0	0
$916$	−13.6489	−0.450973
$917$	8.93799	0.295158
$918$	0	0
$919$	12.4054	0.409216	0.204608	−	0.978844i	$-0.434408\pi$
$919$	12.4054	0.409216	0.204608	+	0.978844i	$0.434408\pi$
$920$	0	0
$921$	0	0
$922$	1.58864	0.0523190
$923$	−12.1153	−0.398779
$924$	0	0
$925$	0	0
$926$	−40.3581	−1.32625
$927$	0	0
$928$	0.832162	0.0273171
$929$	31.3330	1.02800	0.514002	−	0.857789i	$-0.328162\pi$
$929$	31.3330	1.02800	0.514002	+	0.857789i	$0.328162\pi$
$930$	0	0
$931$	−10.8546	−0.355745
$932$	−23.2754	−0.762411
$933$	0	0
$934$	−39.5130	−1.29290
$935$	0	0
$936$	0	0
$937$	7.27299	0.237598	0.118799	−	0.992918i	$-0.462096\pi$
$937$	7.27299	0.237598	0.118799	+	0.992918i	$0.462096\pi$
$938$	17.3676	0.567071
$939$	0	0
$940$	0	0
$941$	−6.72128	−0.219107	−0.109554	−	0.993981i	$-0.534942\pi$
$941$	−6.72128	−0.219107	−0.109554	+	0.993981i	$0.534942\pi$
$942$	0	0
$943$	−88.1689	−2.87117
$944$	−5.39331	−0.175537
$945$	0	0
$946$	12.9019	0.419476
$947$	−10.0757	−0.327416	−0.163708	−	0.986509i	$-0.552345\pi$
$947$	−10.0757	−0.327416	−0.163708	+	0.986509i	$0.552345\pi$
$948$	0	0
$949$	14.9096	0.483986
$950$	0	0
$951$	0	0
$952$	−27.3856	−0.887573
$953$	8.14473	0.263834	0.131917	−	0.991261i	$-0.457887\pi$
$953$	8.14473	0.263834	0.131917	+	0.991261i	$0.457887\pi$
$954$	0	0
$955$	0	0
$956$	−3.72898	−0.120604
$957$	0	0
$958$	7.61107	0.245903
$959$	22.6619	0.731791
$960$	0	0
$961$	−10.6489	−0.343513
$962$	−0.436209	−0.0140639
$963$	0	0
$964$	−1.48878	−0.0479505
$965$	0	0
$966$	0	0
$967$	47.1773	1.51712	0.758560	−	0.651604i	$-0.225903\pi$
$967$	47.1773	1.51712	0.758560	+	0.651604i	$0.225903\pi$
$968$	−15.3960	−0.494845
$969$	0	0
$970$	0	0
$971$	−0.230528	−0.00739801	−0.00369901	−	0.999993i	$-0.501177\pi$
$971$	−0.230528	−0.00739801	−0.00369901	+	0.999993i	$0.501177\pi$
$972$	0	0
$973$	−42.9313	−1.37632
$974$	20.3907	0.653359
$975$	0	0
$976$	14.4509	0.462563
$977$	5.80905	0.185848	0.0929240	−	0.995673i	$-0.470379\pi$
$977$	5.80905	0.185848	0.0929240	+	0.995673i	$0.470379\pi$
$978$	0	0
$979$	51.3770	1.64202
$980$	0	0
$981$	0	0
$982$	25.0224	0.798498
$983$	22.3511	0.712889	0.356444	−	0.934317i	$-0.383989\pi$
$983$	22.3511	0.712889	0.356444	+	0.934317i	$0.383989\pi$
$984$	0	0
$985$	0	0
$986$	5.39331	0.171758
$987$	0	0
$988$	−3.16784	−0.100782
$989$	19.0070	0.604388
$990$	0	0
$991$	−34.9415	−1.10995	−0.554976	−	0.831866i	$-0.687273\pi$
$991$	−34.9415	−1.10995	−0.554976	+	0.831866i	$0.687273\pi$
$992$	4.51122	0.143231
$993$	0	0
$994$	16.1601	0.512568
$995$	0	0
$996$	0	0
$997$	−9.35282	−0.296207	−0.148103	−	0.988972i	$-0.547317\pi$
$997$	−9.35282	−0.296207	−0.148103	+	0.988972i	$0.547317\pi$
$998$	−22.6111	−0.715741
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cj.1.3		3
3.2	odd	2		950.2.a.m.1.2	yes	3
5.4	even	2		8550.2.a.co.1.1		3
12.11	even	2		7600.2.a.bm.1.2		3
15.2	even	4		950.2.b.g.799.5		6
15.8	even	4		950.2.b.g.799.2		6
15.14	odd	2		950.2.a.k.1.2	✓	3
60.59	even	2		7600.2.a.cb.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.a.k.1.2	✓	3	15.14	odd	2
950.2.a.m.1.2	yes	3	3.2	odd	2
950.2.b.g.799.2		6	15.8	even	4
950.2.b.g.799.5		6	15.2	even	4
7600.2.a.bm.1.2		3	12.11	even	2
7600.2.a.cb.1.2		3	60.59	even	2
8550.2.a.cj.1.3		3	1.1	even	1	trivial
8550.2.a.co.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.22547 q^{7} -1.00000 q^{8} +5.13770 q^{11} +3.16784 q^{13} -4.22547 q^{14} +1.00000 q^{16} +6.48108 q^{17} -1.00000 q^{19} -5.13770 q^{22} -7.56885 q^{23} -3.16784 q^{26} +4.22547 q^{28} -0.832162 q^{29} -4.51122 q^{31} -1.00000 q^{32} -6.48108 q^{34} +0.137699 q^{37} +1.00000 q^{38} +11.6489 q^{41} -2.51122 q^{43} +5.13770 q^{44} +7.56885 q^{46} +5.96216 q^{47} +10.8546 q^{49} +3.16784 q^{52} -0.225470 q^{53} -4.22547 q^{56} +0.832162 q^{58} -5.39331 q^{59} +14.4509 q^{61} +4.51122 q^{62} +1.00000 q^{64} -4.11021 q^{67} +6.48108 q^{68} -3.82446 q^{71} +4.70655 q^{73} -0.137699 q^{74} -1.00000 q^{76} +21.7092 q^{77} +10.6265 q^{79} -11.6489 q^{82} +12.0999 q^{83} +2.51122 q^{86} -5.13770 q^{88} +10.0000 q^{89} +13.3856 q^{91} -7.56885 q^{92} -5.96216 q^{94} +3.93972 q^{97} -10.8546 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 2 q^{7} - 3 q^{8} - 2 q^{11} - 2 q^{13} - 2 q^{14} + 3 q^{16} + 4 q^{17} - 3 q^{19} + 2 q^{22} - 14 q^{23} + 2 q^{26} + 2 q^{28} - 14 q^{29} - 4 q^{31} - 3 q^{32} - 4 q^{34} - 17 q^{37}+ \cdots - 25 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 2 * q^7 - 3 * q^8 - 2 * q^11 - 2 * q^13 - 2 * q^14 + 3 * q^16 + 4 * q^17 - 3 * q^19 + 2 * q^22 - 14 * q^23 + 2 * q^26 + 2 * q^28 - 14 * q^29 - 4 * q^31 - 3 * q^32 - 4 * q^34 - 17 * q^37 + 3 * q^38 + 8 * q^41 + 2 * q^43 - 2 * q^44 + 14 * q^46 - 13 * q^47 + 25 * q^49 - 2 * q^52 + 10 * q^53 - 2 * q^56 + 14 * q^58 + 6 * q^59 + 22 * q^61 + 4 * q^62 + 3 * q^64 + 4 * q^68 + 2 * q^71 - 12 * q^73 + 17 * q^74 - 3 * q^76 + 50 * q^77 + 24 * q^79 - 8 * q^82 - 12 * q^83 - 2 * q^86 + 2 * q^88 + 30 * q^89 - 7 * q^91 - 14 * q^92 + 13 * q^94 - 25 * q^98

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