LMFDB - Embedded newform 9522.2.a.ce.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9522,2,Mod(1,9522)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9522, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9522.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.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:	$76.0335528047$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.819879542784.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 151x^{4} - 440x^{3} - 298x^{2} + 532x - 146 $ x^8 - 4x^7 - 22x^6 + 80x^5 + 151x^4 - 440x^3 - 298x^2 + 532*x - 146
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1058)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$4.30747$ of defining polynomial
Character	$\chi$	$=$	9522.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +4.07171 q^{5} +1.94542 q^{7} -1.00000 q^{8} -4.07171 q^{10} -2.44949 q^{11} +1.35643 q^{13} -1.94542 q^{14} +1.00000 q^{16} -5.09341 q^{17} +3.55407 q^{19} +4.07171 q^{20} +2.44949 q^{22} +11.5788 q^{25} -1.35643 q^{26} +1.94542 q^{28} +4.40183 q^{29} +3.46410 q^{31} -1.00000 q^{32} +5.09341 q^{34} +7.92118 q^{35} -1.94542 q^{37} -3.55407 q^{38} -4.07171 q^{40} +5.57177 q^{41} +1.63579 q^{43} -2.44949 q^{44} +2.29416 q^{47} -3.21534 q^{49} -11.5788 q^{50} +1.35643 q^{52} +8.84555 q^{53} -9.97360 q^{55} -1.94542 q^{56} -4.40183 q^{58} -14.3300 q^{59} -5.96285 q^{61} -3.46410 q^{62} +1.00000 q^{64} +5.52299 q^{65} +8.83199 q^{67} -5.09341 q^{68} -7.92118 q^{70} +13.9877 q^{71} -4.92118 q^{73} +1.94542 q^{74} +3.55407 q^{76} -4.76529 q^{77} +7.06114 q^{79} +4.07171 q^{80} -5.57177 q^{82} -4.96828 q^{83} -20.7389 q^{85} -1.63579 q^{86} +2.44949 q^{88} +16.0689 q^{89} +2.63883 q^{91} -2.29416 q^{94} +14.4711 q^{95} -8.59175 q^{97} +3.21534 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 12 q^{13} + 8 q^{16} + 32 q^{25} - 12 q^{26} - 8 q^{32} + 12 q^{35} - 12 q^{41} + 12 q^{47} + 32 q^{49} - 32 q^{50} + 12 q^{52} + 12 q^{55} - 24 q^{59} + 8 q^{64} - 12 q^{70}+ \cdots - 32 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 12 * q^13 + 8 * q^16 + 32 * q^25 - 12 * q^26 - 8 * q^32 + 12 * q^35 - 12 * q^41 + 12 * q^47 + 32 * q^49 - 32 * q^50 + 12 * q^52 + 12 * q^55 - 24 * q^59 + 8 * q^64 - 12 * q^70 + 12 * q^71 + 12 * q^73 + 12 * q^82 - 36 * q^85 - 12 * q^94 + 84 * q^95 - 32 * 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
$5$	4.07171	1.82092	0.910461	−	0.413594i	$-0.135727\pi$
$5$	4.07171	1.82092	0.910461	+	0.413594i	$0.135727\pi$
$6$	0	0
$7$	1.94542	0.735300	0.367650	−	0.929964i	$-0.380163\pi$
$7$	1.94542	0.735300	0.367650	+	0.929964i	$0.380163\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−4.07171	−1.28759
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	1.35643	0.376206	0.188103	−	0.982149i	$-0.439766\pi$
$13$	1.35643	0.376206	0.188103	+	0.982149i	$0.439766\pi$
$14$	−1.94542	−0.519935
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.09341	−1.23533	−0.617667	−	0.786440i	$-0.711922\pi$
$17$	−5.09341	−1.23533	−0.617667	+	0.786440i	$0.711922\pi$
$18$	0	0
$19$	3.55407	0.815359	0.407680	−	0.913125i	$-0.366338\pi$
$19$	3.55407	0.815359	0.407680	+	0.913125i	$0.366338\pi$
$20$	4.07171	0.910461
$21$	0	0
$22$	2.44949	0.522233
$23$	0	0
$23$	0	0
$24$	0	0
$25$	11.5788	2.31576
$26$	−1.35643	−0.266018
$27$	0	0
$28$	1.94542	0.367650
$29$	4.40183	0.817400	0.408700	−	0.912669i	$-0.365982\pi$
$29$	4.40183	0.817400	0.408700	+	0.912669i	$0.365982\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.09341	0.873513
$35$	7.92118	1.33892
$36$	0	0
$37$	−1.94542	−0.319825	−0.159913	−	0.987131i	$-0.551121\pi$
$37$	−1.94542	−0.319825	−0.159913	+	0.987131i	$0.551121\pi$
$38$	−3.55407	−0.576546
$39$	0	0
$40$	−4.07171	−0.643793
$41$	5.57177	0.870165	0.435082	−	0.900391i	$-0.356719\pi$
$41$	5.57177	0.870165	0.435082	+	0.900391i	$0.356719\pi$
$42$	0	0
$43$	1.63579	0.249455	0.124727	−	0.992191i	$-0.460194\pi$
$43$	1.63579	0.249455	0.124727	+	0.992191i	$0.460194\pi$
$44$	−2.44949	−0.369274
$45$	0	0
$46$	0	0
$47$	2.29416	0.334638	0.167319	−	0.985903i	$-0.446489\pi$
$47$	2.29416	0.334638	0.167319	+	0.985903i	$0.446489\pi$
$48$	0	0
$49$	−3.21534	−0.459334
$50$	−11.5788	−1.63749
$51$	0	0
$52$	1.35643	0.188103
$53$	8.84555	1.21503	0.607515	−	0.794308i	$-0.292166\pi$
$53$	8.84555	1.21503	0.607515	+	0.794308i	$0.292166\pi$
$54$	0	0
$55$	−9.97360	−1.34484
$56$	−1.94542	−0.259968
$57$	0	0
$58$	−4.40183	−0.577989
$59$	−14.3300	−1.86561	−0.932806	−	0.360379i	$-0.882647\pi$
$59$	−14.3300	−1.86561	−0.932806	+	0.360379i	$0.882647\pi$
$60$	0	0
$61$	−5.96285	−0.763465	−0.381733	−	0.924273i	$-0.624672\pi$
$61$	−5.96285	−0.763465	−0.381733	+	0.924273i	$0.624672\pi$
$62$	−3.46410	−0.439941
$63$	0	0
$64$	1.00000	0.125000
$65$	5.52299	0.685043
$66$	0	0
$67$	8.83199	1.07900	0.539499	−	0.841986i	$-0.318614\pi$
$67$	8.83199	1.07900	0.539499	+	0.841986i	$0.318614\pi$
$68$	−5.09341	−0.617667
$69$	0	0
$70$	−7.92118	−0.946762
$71$	13.9877	1.66003	0.830014	−	0.557742i	$-0.188332\pi$
$71$	13.9877	1.66003	0.830014	+	0.557742i	$0.188332\pi$
$72$	0	0
$73$	−4.92118	−0.575981	−0.287990	−	0.957633i	$-0.592987\pi$
$73$	−4.92118	−0.575981	−0.287990	+	0.957633i	$0.592987\pi$
$74$	1.94542	0.226150
$75$	0	0
$76$	3.55407	0.407680
$77$	−4.76529	−0.543055
$78$	0	0
$79$	7.06114	0.794440	0.397220	−	0.917723i	$-0.369975\pi$
$79$	7.06114	0.794440	0.397220	+	0.917723i	$0.369975\pi$
$80$	4.07171	0.455231
$81$	0	0
$82$	−5.57177	−0.615299
$83$	−4.96828	−0.545340	−0.272670	−	0.962108i	$-0.587907\pi$
$83$	−4.96828	−0.545340	−0.272670	+	0.962108i	$0.587907\pi$
$84$	0	0
$85$	−20.7389	−2.24945
$86$	−1.63579	−0.176391
$87$	0	0
$88$	2.44949	0.261116
$89$	16.0689	1.70330	0.851650	−	0.524112i	$-0.175603\pi$
$89$	16.0689	1.70330	0.851650	+	0.524112i	$0.175603\pi$
$90$	0	0
$91$	2.63883	0.276624
$92$	0	0
$93$	0	0
$94$	−2.29416	−0.236625
$95$	14.4711	1.48471
$96$	0	0
$97$	−8.59175	−0.872360	−0.436180	−	0.899859i	$-0.643669\pi$
$97$	−8.59175	−0.872360	−0.436180	+	0.899859i	$0.643669\pi$
$98$	3.21534	0.324798
$99$	0	0
$100$	11.5788	1.15788
$101$	−15.5858	−1.55085	−0.775423	−	0.631442i	$-0.782464\pi$
$101$	−15.5858	−1.55085	−0.775423	+	0.631442i	$0.782464\pi$
$102$	0	0
$103$	−9.82510	−0.968095	−0.484048	−	0.875042i	$-0.660834\pi$
$103$	−9.82510	−0.968095	−0.484048	+	0.875042i	$0.660834\pi$
$104$	−1.35643	−0.133009
$105$	0	0
$106$	−8.84555	−0.859156
$107$	2.10762	0.203751	0.101876	−	0.994797i	$-0.467516\pi$
$107$	2.10762	0.203751	0.101876	+	0.994797i	$0.467516\pi$
$108$	0	0
$109$	−11.1328	−1.06633	−0.533166	−	0.846010i	$-0.678998\pi$
$109$	−11.1328	−1.06633	−0.533166	+	0.846010i	$0.678998\pi$
$110$	9.97360	0.950946
$111$	0	0
$112$	1.94542	0.183825
$113$	11.6146	1.09261	0.546305	−	0.837586i	$-0.316034\pi$
$113$	11.6146	1.09261	0.546305	+	0.837586i	$0.316034\pi$
$114$	0	0
$115$	0	0
$116$	4.40183	0.408700
$117$	0	0
$118$	14.3300	1.31919
$119$	−9.90883	−0.908341
$120$	0	0
$121$	−5.00000	−0.454545
$122$	5.96285	0.539851
$123$	0	0
$124$	3.46410	0.311086
$125$	26.7869	2.39590
$126$	0	0
$127$	8.22939	0.730240	0.365120	−	0.930960i	$-0.381028\pi$
$127$	8.22939	0.730240	0.365120	+	0.930960i	$0.381028\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−5.52299	−0.484398
$131$	8.81351	0.770040	0.385020	−	0.922908i	$-0.374195\pi$
$131$	8.81351	0.770040	0.385020	+	0.922908i	$0.374195\pi$
$132$	0	0
$133$	6.91416	0.599533
$134$	−8.83199	−0.762967
$135$	0	0
$136$	5.09341	0.436757
$137$	3.87727	0.331258	0.165629	−	0.986188i	$-0.447035\pi$
$137$	3.87727	0.331258	0.165629	+	0.986188i	$0.447035\pi$
$138$	0	0
$139$	−0.533395	−0.0452420	−0.0226210	−	0.999744i	$-0.507201\pi$
$139$	−0.533395	−0.0452420	−0.0226210	+	0.999744i	$0.507201\pi$
$140$	7.92118	0.669462
$141$	0	0
$142$	−13.9877	−1.17382
$143$	−3.32256	−0.277847
$144$	0	0
$145$	17.9230	1.48842
$146$	4.92118	0.407280
$147$	0	0
$148$	−1.94542	−0.159913
$149$	−2.79256	−0.228775	−0.114388	−	0.993436i	$-0.536491\pi$
$149$	−2.79256	−0.228775	−0.114388	+	0.993436i	$0.536491\pi$
$150$	0	0
$151$	22.2030	1.80685	0.903427	−	0.428742i	$-0.141043\pi$
$151$	22.2030	1.80685	0.903427	+	0.428742i	$0.141043\pi$
$152$	−3.55407	−0.288273
$153$	0	0
$154$	4.76529	0.383998
$155$	14.1048	1.13293
$156$	0	0
$157$	12.4925	0.997012	0.498506	−	0.866886i	$-0.333882\pi$
$157$	12.4925	0.997012	0.498506	+	0.866886i	$0.333882\pi$
$158$	−7.06114	−0.561754
$159$	0	0
$160$	−4.07171	−0.321897
$161$	0	0
$162$	0	0
$163$	6.33254	0.496003	0.248001	−	0.968760i	$-0.420226\pi$
$163$	6.33254	0.496003	0.248001	+	0.968760i	$0.420226\pi$
$164$	5.57177	0.435082
$165$	0	0
$166$	4.96828	0.385613
$167$	20.6741	1.59981	0.799906	−	0.600126i	$-0.204883\pi$
$167$	20.6741	1.59981	0.799906	+	0.600126i	$0.204883\pi$
$168$	0	0
$169$	−11.1601	−0.858469
$170$	20.7389	1.59060
$171$	0	0
$172$	1.63579	0.124727
$173$	−1.59817	−0.121506	−0.0607532	−	0.998153i	$-0.519350\pi$
$173$	−1.59817	−0.121506	−0.0607532	+	0.998153i	$0.519350\pi$
$174$	0	0
$175$	22.5256	1.70278
$176$	−2.44949	−0.184637
$177$	0	0
$178$	−16.0689	−1.20441
$179$	−8.87107	−0.663055	−0.331528	−	0.943446i	$-0.607564\pi$
$179$	−8.87107	−0.663055	−0.331528	+	0.943446i	$0.607564\pi$
$180$	0	0
$181$	−4.26624	−0.317107	−0.158553	−	0.987350i	$-0.550683\pi$
$181$	−4.26624	−0.317107	−0.158553	+	0.987350i	$0.550683\pi$
$182$	−2.63883	−0.195603
$183$	0	0
$184$	0	0
$185$	−7.92118	−0.582377
$186$	0	0
$187$	12.4763	0.912355
$188$	2.29416	0.167319
$189$	0	0
$190$	−14.4711	−1.04985
$191$	3.58630	0.259496	0.129748	−	0.991547i	$-0.458583\pi$
$191$	3.58630	0.259496	0.129748	+	0.991547i	$0.458583\pi$
$192$	0	0
$193$	15.2870	1.10038	0.550190	−	0.835040i	$-0.314555\pi$
$193$	15.2870	1.10038	0.550190	+	0.835040i	$0.314555\pi$
$194$	8.59175	0.616852
$195$	0	0
$196$	−3.21534	−0.229667
$197$	7.79946	0.555689	0.277844	−	0.960626i	$-0.410380\pi$
$197$	7.79946	0.555689	0.277844	+	0.960626i	$0.410380\pi$
$198$	0	0
$199$	24.1394	1.71119	0.855597	−	0.517642i	$-0.173190\pi$
$199$	24.1394	1.71119	0.855597	+	0.517642i	$0.173190\pi$
$200$	−11.5788	−0.818744
$201$	0	0
$202$	15.5858	1.09661
$203$	8.56341	0.601034
$204$	0	0
$205$	22.6866	1.58450
$206$	9.82510	0.684547
$207$	0	0
$208$	1.35643	0.0940516
$209$	−8.70565	−0.602183
$210$	0	0
$211$	−0.100647	−0.00692882	−0.00346441	−	0.999994i	$-0.501103\pi$
$211$	−0.100647	−0.00692882	−0.00346441	+	0.999994i	$0.501103\pi$
$212$	8.84555	0.607515
$213$	0	0
$214$	−2.10762	−0.144074
$215$	6.66044	0.454238
$216$	0	0
$217$	6.73913	0.457482
$218$	11.1328	0.754011
$219$	0	0
$220$	−9.97360	−0.672420
$221$	−6.90887	−0.464741
$222$	0	0
$223$	6.11720	0.409638	0.204819	−	0.978800i	$-0.434339\pi$
$223$	6.11720	0.409638	0.204819	+	0.978800i	$0.434339\pi$
$224$	−1.94542	−0.129984
$225$	0	0
$226$	−11.6146	−0.772592
$227$	−24.5033	−1.62634	−0.813170	−	0.582027i	$-0.802260\pi$
$227$	−24.5033	−1.62634	−0.813170	+	0.582027i	$0.802260\pi$
$228$	0	0
$229$	13.9724	0.923322	0.461661	−	0.887056i	$-0.347254\pi$
$229$	13.9724	0.923322	0.461661	+	0.887056i	$0.347254\pi$
$230$	0	0
$231$	0	0
$232$	−4.40183	−0.288994
$233$	−4.85641	−0.318154	−0.159077	−	0.987266i	$-0.550852\pi$
$233$	−4.85641	−0.318154	−0.159077	+	0.987266i	$0.550852\pi$
$234$	0	0
$235$	9.34115	0.609350
$236$	−14.3300	−0.932806
$237$	0	0
$238$	9.90883	0.642294
$239$	−10.6601	−0.689543	−0.344771	−	0.938687i	$-0.612044\pi$
$239$	−10.6601	−0.689543	−0.344771	+	0.938687i	$0.612044\pi$
$240$	0	0
$241$	−13.3806	−0.861922	−0.430961	−	0.902371i	$-0.641825\pi$
$241$	−13.3806	−0.861922	−0.430961	+	0.902371i	$0.641825\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	−5.96285	−0.381733
$245$	−13.0919	−0.836412
$246$	0	0
$247$	4.82085	0.306743
$248$	−3.46410	−0.219971
$249$	0	0
$250$	−26.7869	−1.69415
$251$	3.44260	0.217295	0.108647	−	0.994080i	$-0.465348\pi$
$251$	3.44260	0.217295	0.108647	+	0.994080i	$0.465348\pi$
$252$	0	0
$253$	0	0
$254$	−8.22939	−0.516358
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.79150	0.548399	0.274199	−	0.961673i	$-0.411587\pi$
$257$	8.79150	0.548399	0.274199	+	0.961673i	$0.411587\pi$
$258$	0	0
$259$	−3.78466	−0.235167
$260$	5.52299	0.342521
$261$	0	0
$262$	−8.81351	−0.544500
$263$	17.0957	1.05417	0.527083	−	0.849814i	$-0.323286\pi$
$263$	17.0957	1.05417	0.527083	+	0.849814i	$0.323286\pi$
$264$	0	0
$265$	36.0165	2.21248
$266$	−6.91416	−0.423934
$267$	0	0
$268$	8.83199	0.539499
$269$	4.28181	0.261067	0.130533	−	0.991444i	$-0.458331\pi$
$269$	4.28181	0.261067	0.130533	+	0.991444i	$0.458331\pi$
$270$	0	0
$271$	−13.6337	−0.828190	−0.414095	−	0.910234i	$-0.635902\pi$
$271$	−13.6337	−0.828190	−0.414095	+	0.910234i	$0.635902\pi$
$272$	−5.09341	−0.308834
$273$	0	0
$274$	−3.87727	−0.234235
$275$	−28.3621	−1.71030
$276$	0	0
$277$	−30.3395	−1.82292	−0.911462	−	0.411384i	$-0.865046\pi$
$277$	−30.3395	−1.82292	−0.911462	+	0.411384i	$0.865046\pi$
$278$	0.533395	0.0319909
$279$	0	0
$280$	−7.92118	−0.473381
$281$	−9.25793	−0.552282	−0.276141	−	0.961117i	$-0.589056\pi$
$281$	−9.25793	−0.552282	−0.276141	+	0.961117i	$0.589056\pi$
$282$	0	0
$283$	14.7391	0.876149	0.438074	−	0.898939i	$-0.355661\pi$
$283$	14.7391	0.876149	0.438074	+	0.898939i	$0.355661\pi$
$284$	13.9877	0.830014
$285$	0	0
$286$	3.32256	0.196467
$287$	10.8394	0.639832
$288$	0	0
$289$	8.94287	0.526051
$290$	−17.9230	−1.05247
$291$	0	0
$292$	−4.92118	−0.287990
$293$	−15.7879	−0.922342	−0.461171	−	0.887311i	$-0.652571\pi$
$293$	−15.7879	−0.922342	−0.461171	+	0.887311i	$0.652571\pi$
$294$	0	0
$295$	−58.3477	−3.39713
$296$	1.94542	0.113075
$297$	0	0
$298$	2.79256	0.161769
$299$	0	0
$300$	0	0
$301$	3.18229	0.183424
$302$	−22.2030	−1.27764
$303$	0	0
$304$	3.55407	0.203840
$305$	−24.2790	−1.39021
$306$	0	0
$307$	32.9589	1.88107	0.940533	−	0.339703i	$-0.110326\pi$
$307$	32.9589	1.88107	0.940533	+	0.339703i	$0.110326\pi$
$308$	−4.76529	−0.271527
$309$	0	0
$310$	−14.1048	−0.801099
$311$	19.9212	1.12963	0.564813	−	0.825219i	$-0.308948\pi$
$311$	19.9212	1.12963	0.564813	+	0.825219i	$0.308948\pi$
$312$	0	0
$313$	−9.67273	−0.546735	−0.273368	−	0.961910i	$-0.588138\pi$
$313$	−9.67273	−0.546735	−0.273368	+	0.961910i	$0.588138\pi$
$314$	−12.4925	−0.704994
$315$	0	0
$316$	7.06114	0.397220
$317$	18.3490	1.03058	0.515292	−	0.857014i	$-0.327683\pi$
$317$	18.3490	1.03058	0.515292	+	0.857014i	$0.327683\pi$
$318$	0	0
$319$	−10.7822	−0.603690
$320$	4.07171	0.227615
$321$	0	0
$322$	0	0
$323$	−18.1023	−1.00724
$324$	0	0
$325$	15.7058	0.871203
$326$	−6.33254	−0.350727
$327$	0	0
$328$	−5.57177	−0.307650
$329$	4.46311	0.246059
$330$	0	0
$331$	−25.5384	−1.40371	−0.701857	−	0.712317i	$-0.747646\pi$
$331$	−25.5384	−1.40371	−0.701857	+	0.712317i	$0.747646\pi$
$332$	−4.96828	−0.272670
$333$	0	0
$334$	−20.6741	−1.13124
$335$	35.9613	1.96477
$336$	0	0
$337$	−32.6280	−1.77736	−0.888681	−	0.458526i	$-0.848377\pi$
$337$	−32.6280	−1.77736	−0.888681	+	0.458526i	$0.848377\pi$
$338$	11.1601	0.607029
$339$	0	0
$340$	−20.7389	−1.12472
$341$	−8.48528	−0.459504
$342$	0	0
$343$	−19.8731	−1.07305
$344$	−1.63579	−0.0881956
$345$	0	0
$346$	1.59817	0.0859181
$347$	−11.6909	−0.627598	−0.313799	−	0.949489i	$-0.601602\pi$
$347$	−11.6909	−0.627598	−0.313799	+	0.949489i	$0.601602\pi$
$348$	0	0
$349$	11.3012	0.604939	0.302469	−	0.953159i	$-0.402189\pi$
$349$	11.3012	0.604939	0.302469	+	0.953159i	$0.402189\pi$
$350$	−22.5256	−1.20405
$351$	0	0
$352$	2.44949	0.130558
$353$	25.1190	1.33695	0.668476	−	0.743734i	$-0.266947\pi$
$353$	25.1190	1.33695	0.668476	+	0.743734i	$0.266947\pi$
$354$	0	0
$355$	56.9536	3.02278
$356$	16.0689	0.851650
$357$	0	0
$358$	8.87107	0.468851
$359$	15.9761	0.843186	0.421593	−	0.906785i	$-0.361471\pi$
$359$	15.9761	0.843186	0.421593	+	0.906785i	$0.361471\pi$
$360$	0	0
$361$	−6.36860	−0.335189
$362$	4.26624	0.224228
$363$	0	0
$364$	2.63883	0.138312
$365$	−20.0376	−1.04882
$366$	0	0
$367$	−0.337877	−0.0176370	−0.00881851	−	0.999961i	$-0.502807\pi$
$367$	−0.337877	−0.0176370	−0.00881851	+	0.999961i	$0.502807\pi$
$368$	0	0
$369$	0	0
$370$	7.92118	0.411803
$371$	17.2083	0.893411
$372$	0	0
$373$	−2.17582	−0.112660	−0.0563298	−	0.998412i	$-0.517940\pi$
$373$	−2.17582	−0.112660	−0.0563298	+	0.998412i	$0.517940\pi$
$374$	−12.4763	−0.645132
$375$	0	0
$376$	−2.29416	−0.118312
$377$	5.97078	0.307511
$378$	0	0
$379$	−3.99367	−0.205141	−0.102571	−	0.994726i	$-0.532707\pi$
$379$	−3.99367	−0.205141	−0.102571	+	0.994726i	$0.532707\pi$
$380$	14.4711	0.742353
$381$	0	0
$382$	−3.58630	−0.183491
$383$	−16.2398	−0.829816	−0.414908	−	0.909863i	$-0.636186\pi$
$383$	−16.2398	−0.829816	−0.414908	+	0.909863i	$0.636186\pi$
$384$	0	0
$385$	−19.4028	−0.988861
$386$	−15.2870	−0.778085
$387$	0	0
$388$	−8.59175	−0.436180
$389$	3.96170	0.200866	0.100433	−	0.994944i	$-0.467977\pi$
$389$	3.96170	0.200866	0.100433	+	0.994944i	$0.467977\pi$
$390$	0	0
$391$	0	0
$392$	3.21534	0.162399
$393$	0	0
$394$	−7.79946	−0.392931
$395$	28.7509	1.44661
$396$	0	0
$397$	9.53339	0.478467	0.239234	−	0.970962i	$-0.423104\pi$
$397$	9.53339	0.478467	0.239234	+	0.970962i	$0.423104\pi$
$398$	−24.1394	−1.21000
$399$	0	0
$400$	11.5788	0.578940
$401$	−3.05120	−0.152370	−0.0761848	−	0.997094i	$-0.524274\pi$
$401$	−3.05120	−0.152370	−0.0761848	+	0.997094i	$0.524274\pi$
$402$	0	0
$403$	4.69882	0.234065
$404$	−15.5858	−0.775423
$405$	0	0
$406$	−8.56341	−0.424995
$407$	4.76529	0.236206
$408$	0	0
$409$	39.7028	1.96318	0.981589	−	0.191003i	$-0.0611742\pi$
$409$	39.7028	1.96318	0.981589	+	0.191003i	$0.0611742\pi$
$410$	−22.6866	−1.12041
$411$	0	0
$412$	−9.82510	−0.484048
$413$	−27.8779	−1.37178
$414$	0	0
$415$	−20.2294	−0.993022
$416$	−1.35643	−0.0665045
$417$	0	0
$418$	8.70565	0.425807
$419$	−30.8981	−1.50947	−0.754736	−	0.656028i	$-0.772235\pi$
$419$	−30.8981	−1.50947	−0.754736	+	0.656028i	$0.772235\pi$
$420$	0	0
$421$	2.14868	0.104720	0.0523602	−	0.998628i	$-0.483326\pi$
$421$	2.14868	0.104720	0.0523602	+	0.998628i	$0.483326\pi$
$422$	0.100647	0.00489942
$423$	0	0
$424$	−8.84555	−0.429578
$425$	−58.9756	−2.86074
$426$	0	0
$427$	−11.6003	−0.561376
$428$	2.10762	0.101876
$429$	0	0
$430$	−6.66044	−0.321195
$431$	−25.1005	−1.20905	−0.604524	−	0.796587i	$-0.706637\pi$
$431$	−25.1005	−1.20905	−0.604524	+	0.796587i	$0.706637\pi$
$432$	0	0
$433$	5.95292	0.286079	0.143040	−	0.989717i	$-0.454312\pi$
$433$	5.95292	0.286079	0.143040	+	0.989717i	$0.454312\pi$
$434$	−6.73913	−0.323489
$435$	0	0
$436$	−11.1328	−0.533166
$437$	0	0
$438$	0	0
$439$	24.1766	1.15389	0.576943	−	0.816784i	$-0.304245\pi$
$439$	24.1766	1.15389	0.576943	+	0.816784i	$0.304245\pi$
$440$	9.97360	0.475473
$441$	0	0
$442$	6.90887	0.328621
$443$	−6.51214	−0.309401	−0.154701	−	0.987961i	$-0.549441\pi$
$443$	−6.51214	−0.309401	−0.154701	+	0.987961i	$0.549441\pi$
$444$	0	0
$445$	65.4278	3.10158
$446$	−6.11720	−0.289658
$447$	0	0
$448$	1.94542	0.0919125
$449$	−29.7300	−1.40304	−0.701522	−	0.712647i	$-0.747496\pi$
$449$	−29.7300	−1.40304	−0.701522	+	0.712647i	$0.747496\pi$
$450$	0	0
$451$	−13.6480	−0.642659
$452$	11.6146	0.546305
$453$	0	0
$454$	24.5033	1.15000
$455$	10.7445	0.503712
$456$	0	0
$457$	4.37251	0.204538	0.102269	−	0.994757i	$-0.467390\pi$
$457$	4.37251	0.204538	0.102269	+	0.994757i	$0.467390\pi$
$458$	−13.9724	−0.652887
$459$	0	0
$460$	0	0
$461$	−7.86593	−0.366353	−0.183177	−	0.983080i	$-0.558638\pi$
$461$	−7.86593	−0.366353	−0.183177	+	0.983080i	$0.558638\pi$
$462$	0	0
$463$	−21.9472	−1.01997	−0.509987	−	0.860182i	$-0.670350\pi$
$463$	−21.9472	−1.01997	−0.509987	+	0.860182i	$0.670350\pi$
$464$	4.40183	0.204350
$465$	0	0
$466$	4.85641	0.224969
$467$	29.6404	1.37159	0.685797	−	0.727793i	$-0.259454\pi$
$467$	29.6404	1.37159	0.685797	+	0.727793i	$0.259454\pi$
$468$	0	0
$469$	17.1819	0.793387
$470$	−9.34115	−0.430875
$471$	0	0
$472$	14.3300	0.659593
$473$	−4.00684	−0.184235
$474$	0	0
$475$	41.1518	1.88818
$476$	−9.90883	−0.454171
$477$	0	0
$478$	10.6601	0.487580
$479$	−38.7485	−1.77046	−0.885232	−	0.465150i	$-0.846001\pi$
$479$	−38.7485	−1.77046	−0.885232	+	0.465150i	$0.846001\pi$
$480$	0	0
$481$	−2.63883	−0.120320
$482$	13.3806	0.609471
$483$	0	0
$484$	−5.00000	−0.227273
$485$	−34.9831	−1.58850
$486$	0	0
$487$	15.6777	0.710426	0.355213	−	0.934785i	$-0.384408\pi$
$487$	15.6777	0.710426	0.355213	+	0.934785i	$0.384408\pi$
$488$	5.96285	0.269926
$489$	0	0
$490$	13.0919	0.591433
$491$	16.8561	0.760705	0.380352	−	0.924842i	$-0.375803\pi$
$491$	16.8561	0.760705	0.380352	+	0.924842i	$0.375803\pi$
$492$	0	0
$493$	−22.4204	−1.00976
$494$	−4.82085	−0.216900
$495$	0	0
$496$	3.46410	0.155543
$497$	27.2119	1.22062
$498$	0	0
$499$	−16.9853	−0.760368	−0.380184	−	0.924911i	$-0.624139\pi$
$499$	−16.9853	−0.760368	−0.380184	+	0.924911i	$0.624139\pi$
$500$	26.7869	1.19795
$501$	0	0
$502$	−3.44260	−0.153651
$503$	−5.66325	−0.252512	−0.126256	−	0.991998i	$-0.540296\pi$
$503$	−5.66325	−0.252512	−0.126256	+	0.991998i	$0.540296\pi$
$504$	0	0
$505$	−63.4609	−2.82397
$506$	0	0
$507$	0	0
$508$	8.22939	0.365120
$509$	−10.4262	−0.462131	−0.231066	−	0.972938i	$-0.574221\pi$
$509$	−10.4262	−0.462131	−0.231066	+	0.972938i	$0.574221\pi$
$510$	0	0
$511$	−9.57376	−0.423518
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−8.79150	−0.387776
$515$	−40.0049	−1.76283
$516$	0	0
$517$	−5.61952	−0.247146
$518$	3.78466	0.166288
$519$	0	0
$520$	−5.52299	−0.242199
$521$	27.8011	1.21799	0.608995	−	0.793174i	$-0.291573\pi$
$521$	27.8011	1.21799	0.608995	+	0.793174i	$0.291573\pi$
$522$	0	0
$523$	−21.1154	−0.923312	−0.461656	−	0.887059i	$-0.652745\pi$
$523$	−21.1154	−0.923312	−0.461656	+	0.887059i	$0.652745\pi$
$524$	8.81351	0.385020
$525$	0	0
$526$	−17.0957	−0.745408
$527$	−17.6441	−0.768589
$528$	0	0
$529$	0	0
$530$	−36.0165	−1.56446
$531$	0	0
$532$	6.91416	0.299767
$533$	7.55773	0.327361
$534$	0	0
$535$	8.58162	0.371016
$536$	−8.83199	−0.381484
$537$	0	0
$538$	−4.28181	−0.184602
$539$	7.87594	0.339241
$540$	0	0
$541$	−31.1340	−1.33856	−0.669278	−	0.743012i	$-0.733396\pi$
$541$	−31.1340	−1.33856	−0.669278	+	0.743012i	$0.733396\pi$
$542$	13.6337	0.585619
$543$	0	0
$544$	5.09341	0.218378
$545$	−45.3297	−1.94171
$546$	0	0
$547$	23.8705	1.02063	0.510313	−	0.859988i	$-0.329529\pi$
$547$	23.8705	1.02063	0.510313	+	0.859988i	$0.329529\pi$
$548$	3.87727	0.165629
$549$	0	0
$550$	28.3621	1.20937
$551$	15.6444	0.666474
$552$	0	0
$553$	13.7369	0.584151
$554$	30.3395	1.28900
$555$	0	0
$556$	−0.533395	−0.0226210
$557$	46.8977	1.98712	0.993560	−	0.113305i	$-0.0361439\pi$
$557$	46.8977	1.98712	0.993560	+	0.113305i	$0.0361439\pi$
$558$	0	0
$559$	2.21883	0.0938465
$560$	7.92118	0.334731
$561$	0	0
$562$	9.25793	0.390522
$563$	6.76143	0.284960	0.142480	−	0.989798i	$-0.454492\pi$
$563$	6.76143	0.284960	0.142480	+	0.989798i	$0.454492\pi$
$564$	0	0
$565$	47.2913	1.98956
$566$	−14.7391	−0.619531
$567$	0	0
$568$	−13.9877	−0.586909
$569$	2.06692	0.0866497	0.0433248	−	0.999061i	$-0.486205\pi$
$569$	2.06692	0.0866497	0.0433248	+	0.999061i	$0.486205\pi$
$570$	0	0
$571$	17.8482	0.746924	0.373462	−	0.927645i	$-0.378171\pi$
$571$	17.8482	0.746924	0.373462	+	0.927645i	$0.378171\pi$
$572$	−3.32256	−0.138923
$573$	0	0
$574$	−10.8394	−0.452429
$575$	0	0
$576$	0	0
$577$	−14.8779	−0.619376	−0.309688	−	0.950838i	$-0.600225\pi$
$577$	−14.8779	−0.619376	−0.309688	+	0.950838i	$0.600225\pi$
$578$	−8.94287	−0.371974
$579$	0	0
$580$	17.9230	0.744211
$581$	−9.66540	−0.400988
$582$	0	0
$583$	−21.6671	−0.897359
$584$	4.92118	0.203640
$585$	0	0
$586$	15.7879	0.652194
$587$	−29.5524	−1.21976	−0.609879	−	0.792495i	$-0.708782\pi$
$587$	−29.5524	−1.21976	−0.609879	+	0.792495i	$0.708782\pi$
$588$	0	0
$589$	12.3117	0.507293
$590$	58.3477	2.40214
$591$	0	0
$592$	−1.94542	−0.0799563
$593$	−24.1392	−0.991276	−0.495638	−	0.868529i	$-0.665066\pi$
$593$	−24.1392	−0.991276	−0.495638	+	0.868529i	$0.665066\pi$
$594$	0	0
$595$	−40.3459	−1.65402
$596$	−2.79256	−0.114388
$597$	0	0
$598$	0	0
$599$	12.0404	0.491959	0.245980	−	0.969275i	$-0.420890\pi$
$599$	12.0404	0.491959	0.245980	+	0.969275i	$0.420890\pi$
$600$	0	0
$601$	32.6696	1.33262	0.666310	−	0.745675i	$-0.267873\pi$
$601$	32.6696	1.33262	0.666310	+	0.745675i	$0.267873\pi$
$602$	−3.18229	−0.129700
$603$	0	0
$604$	22.2030	0.903427
$605$	−20.3585	−0.827692
$606$	0	0
$607$	4.09287	0.166124	0.0830622	−	0.996544i	$-0.473530\pi$
$607$	4.09287	0.166124	0.0830622	+	0.996544i	$0.473530\pi$
$608$	−3.55407	−0.144137
$609$	0	0
$610$	24.2790	0.983028
$611$	3.11187	0.125893
$612$	0	0
$613$	−29.7094	−1.19995	−0.599975	−	0.800019i	$-0.704823\pi$
$613$	−29.7094	−1.19995	−0.599975	+	0.800019i	$0.704823\pi$
$614$	−32.9589	−1.33011
$615$	0	0
$616$	4.76529	0.191999
$617$	−20.8936	−0.841146	−0.420573	−	0.907259i	$-0.638171\pi$
$617$	−20.8936	−0.841146	−0.420573	+	0.907259i	$0.638171\pi$
$618$	0	0
$619$	−36.3294	−1.46020	−0.730102	−	0.683339i	$-0.760527\pi$
$619$	−36.3294	−1.46020	−0.730102	+	0.683339i	$0.760527\pi$
$620$	14.1048	0.566463
$621$	0	0
$622$	−19.9212	−0.798767
$623$	31.2607	1.25244
$624$	0	0
$625$	51.1745	2.04698
$626$	9.67273	0.386600
$627$	0	0
$628$	12.4925	0.498506
$629$	9.90883	0.395091
$630$	0	0
$631$	−1.77569	−0.0706890	−0.0353445	−	0.999375i	$-0.511253\pi$
$631$	−1.77569	−0.0706890	−0.0353445	+	0.999375i	$0.511253\pi$
$632$	−7.06114	−0.280877
$633$	0	0
$634$	−18.3490	−0.728733
$635$	33.5077	1.32971
$636$	0	0
$637$	−4.36139	−0.172805
$638$	10.7822	0.426873
$639$	0	0
$640$	−4.07171	−0.160948
$641$	33.0344	1.30478	0.652389	−	0.757884i	$-0.273767\pi$
$641$	33.0344	1.30478	0.652389	+	0.757884i	$0.273767\pi$
$642$	0	0
$643$	43.0895	1.69928	0.849642	−	0.527360i	$-0.176818\pi$
$643$	43.0895	1.69928	0.849642	+	0.527360i	$0.176818\pi$
$644$	0	0
$645$	0	0
$646$	18.1023	0.712227
$647$	0.0891034	0.00350302	0.00175151	−	0.999998i	$-0.499442\pi$
$647$	0.0891034	0.00350302	0.00175151	+	0.999998i	$0.499442\pi$
$648$	0	0
$649$	35.1013	1.37785
$650$	−15.7058	−0.616034
$651$	0	0
$652$	6.33254	0.248001
$653$	9.03587	0.353601	0.176801	−	0.984247i	$-0.443425\pi$
$653$	9.03587	0.353601	0.176801	+	0.984247i	$0.443425\pi$
$654$	0	0
$655$	35.8860	1.40218
$656$	5.57177	0.217541
$657$	0	0
$658$	−4.46311	−0.173990
$659$	−7.78944	−0.303433	−0.151717	−	0.988424i	$-0.548480\pi$
$659$	−7.78944	−0.303433	−0.151717	+	0.988424i	$0.548480\pi$
$660$	0	0
$661$	−27.8692	−1.08399	−0.541993	−	0.840383i	$-0.682330\pi$
$661$	−27.8692	−1.08399	−0.541993	+	0.840383i	$0.682330\pi$
$662$	25.5384	0.992576
$663$	0	0
$664$	4.96828	0.192807
$665$	28.1524	1.09170
$666$	0	0
$667$	0	0
$668$	20.6741	0.799906
$669$	0	0
$670$	−35.9613	−1.38930
$671$	14.6059	0.563856
$672$	0	0
$673$	0.885122	0.0341189	0.0170595	−	0.999854i	$-0.494570\pi$
$673$	0.885122	0.0341189	0.0170595	+	0.999854i	$0.494570\pi$
$674$	32.6280	1.25678
$675$	0	0
$676$	−11.1601	−0.429234
$677$	−45.3102	−1.74141	−0.870707	−	0.491802i	$-0.836338\pi$
$677$	−45.3102	−1.74141	−0.870707	+	0.491802i	$0.836338\pi$
$678$	0	0
$679$	−16.7146	−0.641446
$680$	20.7389	0.795300
$681$	0	0
$682$	8.48528	0.324918
$683$	51.8395	1.98358	0.991791	−	0.127866i	$-0.0408128\pi$
$683$	51.8395	1.98358	0.991791	+	0.127866i	$0.0408128\pi$
$684$	0	0
$685$	15.7871	0.603195
$686$	19.8731	0.758760
$687$	0	0
$688$	1.63579	0.0623637
$689$	11.9984	0.457102
$690$	0	0
$691$	31.3012	1.19075	0.595377	−	0.803447i	$-0.297003\pi$
$691$	31.3012	1.19075	0.595377	+	0.803447i	$0.297003\pi$
$692$	−1.59817	−0.0607532
$693$	0	0
$694$	11.6909	0.443779
$695$	−2.17183	−0.0823821
$696$	0	0
$697$	−28.3793	−1.07494
$698$	−11.3012	−0.427756
$699$	0	0
$700$	22.5256	0.851388
$701$	−12.3084	−0.464881	−0.232440	−	0.972611i	$-0.574671\pi$
$701$	−12.3084	−0.464881	−0.232440	+	0.972611i	$0.574671\pi$
$702$	0	0
$703$	−6.91416	−0.260772
$704$	−2.44949	−0.0923186
$705$	0	0
$706$	−25.1190	−0.945367
$707$	−30.3210	−1.14034
$708$	0	0
$709$	−19.3568	−0.726959	−0.363480	−	0.931602i	$-0.618411\pi$
$709$	−19.3568	−0.726959	−0.363480	+	0.931602i	$0.618411\pi$
$710$	−56.9536	−2.13743
$711$	0	0
$712$	−16.0689	−0.602207
$713$	0	0
$714$	0	0
$715$	−13.5285	−0.505937
$716$	−8.87107	−0.331528
$717$	0	0
$718$	−15.9761	−0.596222
$719$	−0.616416	−0.0229885	−0.0114942	−	0.999934i	$-0.503659\pi$
$719$	−0.616416	−0.0229885	−0.0114942	+	0.999934i	$0.503659\pi$
$720$	0	0
$721$	−19.1139	−0.711840
$722$	6.36860	0.237015
$723$	0	0
$724$	−4.26624	−0.158553
$725$	50.9679	1.89290
$726$	0	0
$727$	−45.3126	−1.68055	−0.840275	−	0.542160i	$-0.817607\pi$
$727$	−45.3126	−1.68055	−0.840275	+	0.542160i	$0.817607\pi$
$728$	−2.63883	−0.0978015
$729$	0	0
$730$	20.0376	0.741625
$731$	−8.33173	−0.308160
$732$	0	0
$733$	17.3915	0.642370	0.321185	−	0.947017i	$-0.395919\pi$
$733$	17.3915	0.642370	0.321185	+	0.947017i	$0.395919\pi$
$734$	0.337877	0.0124713
$735$	0	0
$736$	0	0
$737$	−21.6339	−0.796893
$738$	0	0
$739$	33.9686	1.24956	0.624778	−	0.780803i	$-0.285190\pi$
$739$	33.9686	1.24956	0.624778	+	0.780803i	$0.285190\pi$
$740$	−7.92118	−0.291188
$741$	0	0
$742$	−17.2083	−0.631737
$743$	25.9127	0.950643	0.475321	−	0.879812i	$-0.342332\pi$
$743$	25.9127	0.950643	0.475321	+	0.879812i	$0.342332\pi$
$744$	0	0
$745$	−11.3705	−0.416582
$746$	2.17582	0.0796624
$747$	0	0
$748$	12.4763	0.456177
$749$	4.10021	0.149818
$750$	0	0
$751$	−21.5515	−0.786427	−0.393213	−	0.919447i	$-0.628637\pi$
$751$	−21.5515	−0.786427	−0.393213	+	0.919447i	$0.628637\pi$
$752$	2.29416	0.0836595
$753$	0	0
$754$	−5.97078	−0.217443
$755$	90.4041	3.29014
$756$	0	0
$757$	30.5862	1.11167	0.555837	−	0.831292i	$-0.312398\pi$
$757$	30.5862	1.11167	0.555837	+	0.831292i	$0.312398\pi$
$758$	3.99367	0.145057
$759$	0	0
$760$	−14.4711	−0.524923
$761$	−22.0332	−0.798702	−0.399351	−	0.916798i	$-0.630765\pi$
$761$	−22.0332	−0.798702	−0.399351	+	0.916798i	$0.630765\pi$
$762$	0	0
$763$	−21.6581	−0.784074
$764$	3.58630	0.129748
$765$	0	0
$766$	16.2398	0.586769
$767$	−19.4377	−0.701855
$768$	0	0
$769$	42.3961	1.52884	0.764421	−	0.644717i	$-0.223025\pi$
$769$	42.3961	1.52884	0.764421	+	0.644717i	$0.223025\pi$
$770$	19.4028	0.699230
$771$	0	0
$772$	15.2870	0.550190
$773$	−10.2548	−0.368838	−0.184419	−	0.982848i	$-0.559040\pi$
$773$	−10.2548	−0.368838	−0.184419	+	0.982848i	$0.559040\pi$
$774$	0	0
$775$	40.1101	1.44080
$776$	8.59175	0.308426
$777$	0	0
$778$	−3.96170	−0.142034
$779$	19.8025	0.709497
$780$	0	0
$781$	−34.2626	−1.22601
$782$	0	0
$783$	0	0
$784$	−3.21534	−0.114834
$785$	50.8659	1.81548
$786$	0	0
$787$	48.6462	1.73405	0.867025	−	0.498265i	$-0.166030\pi$
$787$	48.6462	1.73405	0.867025	+	0.498265i	$0.166030\pi$
$788$	7.79946	0.277844
$789$	0	0
$790$	−28.7509	−1.02291
$791$	22.5953	0.803396
$792$	0	0
$793$	−8.08820	−0.287220
$794$	−9.53339	−0.338328
$795$	0	0
$796$	24.1394	0.855597
$797$	−46.1175	−1.63356	−0.816782	−	0.576946i	$-0.804244\pi$
$797$	−46.1175	−1.63356	−0.816782	+	0.576946i	$0.804244\pi$
$798$	0	0
$799$	−11.6851	−0.413390
$800$	−11.5788	−0.409372
$801$	0	0
$802$	3.05120	0.107742
$803$	12.0544	0.425390
$804$	0	0
$805$	0	0
$806$	−4.69882	−0.165509
$807$	0	0
$808$	15.5858	0.548307
$809$	−33.9809	−1.19470	−0.597352	−	0.801979i	$-0.703780\pi$
$809$	−33.9809	−1.19470	−0.597352	+	0.801979i	$0.703780\pi$
$810$	0	0
$811$	5.04290	0.177080	0.0885400	−	0.996073i	$-0.471780\pi$
$811$	5.04290	0.177080	0.0885400	+	0.996073i	$0.471780\pi$
$812$	8.56341	0.300517
$813$	0	0
$814$	−4.76529	−0.167023
$815$	25.7842	0.903182
$816$	0	0
$817$	5.81369	0.203395
$818$	−39.7028	−1.38818
$819$	0	0
$820$	22.6866	0.792251
$821$	−2.44191	−0.0852231	−0.0426116	−	0.999092i	$-0.513568\pi$
$821$	−2.44191	−0.0852231	−0.0426116	+	0.999092i	$0.513568\pi$
$822$	0	0
$823$	−2.84938	−0.0993232	−0.0496616	−	0.998766i	$-0.515814\pi$
$823$	−2.84938	−0.0993232	−0.0496616	+	0.998766i	$0.515814\pi$
$824$	9.82510	0.342273
$825$	0	0
$826$	27.8779	0.969997
$827$	−3.88626	−0.135139	−0.0675693	−	0.997715i	$-0.521524\pi$
$827$	−3.88626	−0.135139	−0.0675693	+	0.997715i	$0.521524\pi$
$828$	0	0
$829$	20.3631	0.707239	0.353620	−	0.935389i	$-0.384951\pi$
$829$	20.3631	0.707239	0.353620	+	0.935389i	$0.384951\pi$
$830$	20.2294	0.702172
$831$	0	0
$832$	1.35643	0.0470258
$833$	16.3771	0.567432
$834$	0	0
$835$	84.1789	2.91313
$836$	−8.70565	−0.301091
$837$	0	0
$838$	30.8981	1.06736
$839$	7.05733	0.243646	0.121823	−	0.992552i	$-0.461126\pi$
$839$	7.05733	0.243646	0.121823	+	0.992552i	$0.461126\pi$
$840$	0	0
$841$	−9.62388	−0.331858
$842$	−2.14868	−0.0740485
$843$	0	0
$844$	−0.100647	−0.00346441
$845$	−45.4406	−1.56321
$846$	0	0
$847$	−9.72710	−0.334227
$848$	8.84555	0.303758
$849$	0	0
$850$	58.9756	2.02285
$851$	0	0
$852$	0	0
$853$	−22.4781	−0.769635	−0.384818	−	0.922993i	$-0.625736\pi$
$853$	−22.4781	−0.769635	−0.384818	+	0.922993i	$0.625736\pi$
$854$	11.6003	0.396953
$855$	0	0
$856$	−2.10762	−0.0720370
$857$	−6.43977	−0.219978	−0.109989	−	0.993933i	$-0.535082\pi$
$857$	−6.43977	−0.219978	−0.109989	+	0.993933i	$0.535082\pi$
$858$	0	0
$859$	37.7008	1.28633	0.643167	−	0.765726i	$-0.277620\pi$
$859$	37.7008	1.28633	0.643167	+	0.765726i	$0.277620\pi$
$860$	6.66044	0.227119
$861$	0	0
$862$	25.1005	0.854927
$863$	−34.5566	−1.17632	−0.588159	−	0.808745i	$-0.700147\pi$
$863$	−34.5566	−1.17632	−0.588159	+	0.808745i	$0.700147\pi$
$864$	0	0
$865$	−6.50727	−0.221254
$866$	−5.95292	−0.202288
$867$	0	0
$868$	6.73913	0.228741
$869$	−17.2962	−0.586733
$870$	0	0
$871$	11.9800	0.405926
$872$	11.1328	0.377006
$873$	0	0
$874$	0	0
$875$	52.1118	1.76170
$876$	0	0
$877$	−7.17164	−0.242169	−0.121085	−	0.992642i	$-0.538637\pi$
$877$	−7.17164	−0.242169	−0.121085	+	0.992642i	$0.538637\pi$
$878$	−24.1766	−0.815921
$879$	0	0
$880$	−9.97360	−0.336210
$881$	−8.08174	−0.272281	−0.136140	−	0.990690i	$-0.543470\pi$
$881$	−8.08174	−0.272281	−0.136140	+	0.990690i	$0.543470\pi$
$882$	0	0
$883$	−41.0042	−1.37990	−0.689950	−	0.723857i	$-0.742367\pi$
$883$	−41.0042	−1.37990	−0.689950	+	0.723857i	$0.742367\pi$
$884$	−6.90887	−0.232370
$885$	0	0
$886$	6.51214	0.218780
$887$	37.8321	1.27028	0.635138	−	0.772398i	$-0.280943\pi$
$887$	37.8321	1.27028	0.635138	+	0.772398i	$0.280943\pi$
$888$	0	0
$889$	16.0096	0.536945
$890$	−65.4278	−2.19315
$891$	0	0
$892$	6.11720	0.204819
$893$	8.15361	0.272850
$894$	0	0
$895$	−36.1204	−1.20737
$896$	−1.94542	−0.0649919
$897$	0	0
$898$	29.7300	0.992102
$899$	15.2484	0.508562
$900$	0	0
$901$	−45.0541	−1.50097
$902$	13.6480	0.454429
$903$	0	0
$904$	−11.6146	−0.386296
$905$	−17.3709	−0.577427
$906$	0	0
$907$	−15.9611	−0.529978	−0.264989	−	0.964251i	$-0.585368\pi$
$907$	−15.9611	−0.529978	−0.264989	+	0.964251i	$0.585368\pi$
$908$	−24.5033	−0.813170
$909$	0	0
$910$	−10.7445	−0.356178
$911$	−37.5172	−1.24300	−0.621501	−	0.783414i	$-0.713477\pi$
$911$	−37.5172	−1.24300	−0.621501	+	0.783414i	$0.713477\pi$
$912$	0	0
$913$	12.1698	0.402760
$914$	−4.37251	−0.144630
$915$	0	0
$916$	13.9724	0.461661
$917$	17.1460	0.566210
$918$	0	0
$919$	26.7175	0.881330	0.440665	−	0.897672i	$-0.354743\pi$
$919$	26.7175	0.881330	0.440665	+	0.897672i	$0.354743\pi$
$920$	0	0
$921$	0	0
$922$	7.86593	0.259051
$923$	18.9733	0.624513
$924$	0	0
$925$	−22.5256	−0.740638
$926$	21.9472	0.721230
$927$	0	0
$928$	−4.40183	−0.144497
$929$	43.8177	1.43761	0.718805	−	0.695211i	$-0.244689\pi$
$929$	43.8177	1.43761	0.718805	+	0.695211i	$0.244689\pi$
$930$	0	0
$931$	−11.4275	−0.374523
$932$	−4.85641	−0.159077
$933$	0	0
$934$	−29.6404	−0.969864
$935$	50.7997	1.66133
$936$	0	0
$937$	−34.4360	−1.12498	−0.562488	−	0.826805i	$-0.690156\pi$
$937$	−34.4360	−1.12498	−0.562488	+	0.826805i	$0.690156\pi$
$938$	−17.1819	−0.561010
$939$	0	0
$940$	9.34115	0.304675
$941$	−35.8492	−1.16865	−0.584326	−	0.811519i	$-0.698641\pi$
$941$	−35.8492	−1.16865	−0.584326	+	0.811519i	$0.698641\pi$
$942$	0	0
$943$	0	0
$944$	−14.3300	−0.466403
$945$	0	0
$946$	4.00684	0.130274
$947$	−0.975923	−0.0317132	−0.0158566	−	0.999874i	$-0.505048\pi$
$947$	−0.975923	−0.0317132	−0.0158566	+	0.999874i	$0.505048\pi$
$948$	0	0
$949$	−6.67524	−0.216688
$950$	−41.1518	−1.33514
$951$	0	0
$952$	9.90883	0.321147
$953$	−16.6476	−0.539267	−0.269634	−	0.962963i	$-0.586903\pi$
$953$	−16.6476	−0.539267	−0.269634	+	0.962963i	$0.586903\pi$
$954$	0	0
$955$	14.6024	0.472522
$956$	−10.6601	−0.344771
$957$	0	0
$958$	38.7485	1.25191
$959$	7.54292	0.243574
$960$	0	0
$961$	−19.0000	−0.612903
$962$	2.63883	0.0850792
$963$	0	0
$964$	−13.3806	−0.430961
$965$	62.2440	2.00370
$966$	0	0
$967$	52.3940	1.68488	0.842439	−	0.538792i	$-0.181119\pi$
$967$	52.3940	1.68488	0.842439	+	0.538792i	$0.181119\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	34.9831	1.12324
$971$	51.8696	1.66457	0.832287	−	0.554344i	$-0.187031\pi$
$971$	51.8696	1.66457	0.832287	+	0.554344i	$0.187031\pi$
$972$	0	0
$973$	−1.03768	−0.0332664
$974$	−15.6777	−0.502347
$975$	0	0
$976$	−5.96285	−0.190866
$977$	−18.1728	−0.581399	−0.290699	−	0.956814i	$-0.593888\pi$
$977$	−18.1728	−0.581399	−0.290699	+	0.956814i	$0.593888\pi$
$978$	0	0
$979$	−39.3606	−1.25797
$980$	−13.0919	−0.418206
$981$	0	0
$982$	−16.8561	−0.537899
$983$	−23.6079	−0.752975	−0.376488	−	0.926422i	$-0.622868\pi$
$983$	−23.6079	−0.752975	−0.376488	+	0.926422i	$0.622868\pi$
$984$	0	0
$985$	31.7571	1.01187
$986$	22.4204	0.714010
$987$	0	0
$988$	4.82085	0.153372
$989$	0	0
$990$	0	0
$991$	3.77401	0.119885	0.0599427	−	0.998202i	$-0.480908\pi$
$991$	3.77401	0.119885	0.0599427	+	0.998202i	$0.480908\pi$
$992$	−3.46410	−0.109985
$993$	0	0
$994$	−27.2119	−0.863108
$995$	98.2884	3.11595
$996$	0	0
$997$	−14.6939	−0.465361	−0.232681	−	0.972553i	$-0.574750\pi$
$997$	−14.6939	−0.465361	−0.232681	+	0.972553i	$0.574750\pi$
$998$	16.9853	0.537661
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ce.1.8		8
3.2	odd	2		1058.2.a.n.1.5	✓	8
12.11	even	2		8464.2.a.cb.1.3		8
23.22	odd	2	inner	9522.2.a.ce.1.1		8
69.68	even	2		1058.2.a.n.1.6	yes	8
276.275	odd	2		8464.2.a.cb.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1058.2.a.n.1.5	✓	8	3.2	odd	2
1058.2.a.n.1.6	yes	8	69.68	even	2
8464.2.a.cb.1.3		8	12.11	even	2
8464.2.a.cb.1.4		8	276.275	odd	2
9522.2.a.ce.1.1		8	23.22	odd	2	inner
9522.2.a.ce.1.8		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.07171 q^{5} +1.94542 q^{7} -1.00000 q^{8} -4.07171 q^{10} -2.44949 q^{11} +1.35643 q^{13} -1.94542 q^{14} +1.00000 q^{16} -5.09341 q^{17} +3.55407 q^{19} +4.07171 q^{20} +2.44949 q^{22} +11.5788 q^{25} -1.35643 q^{26} +1.94542 q^{28} +4.40183 q^{29} +3.46410 q^{31} -1.00000 q^{32} +5.09341 q^{34} +7.92118 q^{35} -1.94542 q^{37} -3.55407 q^{38} -4.07171 q^{40} +5.57177 q^{41} +1.63579 q^{43} -2.44949 q^{44} +2.29416 q^{47} -3.21534 q^{49} -11.5788 q^{50} +1.35643 q^{52} +8.84555 q^{53} -9.97360 q^{55} -1.94542 q^{56} -4.40183 q^{58} -14.3300 q^{59} -5.96285 q^{61} -3.46410 q^{62} +1.00000 q^{64} +5.52299 q^{65} +8.83199 q^{67} -5.09341 q^{68} -7.92118 q^{70} +13.9877 q^{71} -4.92118 q^{73} +1.94542 q^{74} +3.55407 q^{76} -4.76529 q^{77} +7.06114 q^{79} +4.07171 q^{80} -5.57177 q^{82} -4.96828 q^{83} -20.7389 q^{85} -1.63579 q^{86} +2.44949 q^{88} +16.0689 q^{89} +2.63883 q^{91} -2.29416 q^{94} +14.4711 q^{95} -8.59175 q^{97} +3.21534 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 12 q^{13} + 8 q^{16} + 32 q^{25} - 12 q^{26} - 8 q^{32} + 12 q^{35} - 12 q^{41} + 12 q^{47} + 32 q^{49} - 32 q^{50} + 12 q^{52} + 12 q^{55} - 24 q^{59} + 8 q^{64} - 12 q^{70}+ \cdots - 32 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 12 * q^13 + 8 * q^16 + 32 * q^25 - 12 * q^26 - 8 * q^32 + 12 * q^35 - 12 * q^41 + 12 * q^47 + 32 * q^49 - 32 * q^50 + 12 * q^52 + 12 * q^55 - 24 * q^59 + 8 * q^64 - 12 * q^70 + 12 * q^71 + 12 * q^73 + 12 * q^82 - 36 * q^85 - 12 * q^94 + 84 * q^95 - 32 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more