LMFDB - Embedded newform 8464.2.a.cb.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-4,0,0,0,0,0,20,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.5853802708$
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			1.4
Root			$0.443768$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37562 q^{3} +4.07171 q^{5} +1.94542 q^{7} +2.64357 q^{9} +2.44949 q^{11} +1.35643 q^{13} -9.67283 q^{15} -5.09341 q^{17} +3.55407 q^{19} -4.62158 q^{21} +11.5788 q^{25} +0.846745 q^{27} -4.40183 q^{29} -3.46410 q^{31} -5.81906 q^{33} +7.92118 q^{35} +1.94542 q^{37} -3.22236 q^{39} -5.57177 q^{41} +1.63579 q^{43} +10.7638 q^{45} +2.29416 q^{47} -3.21534 q^{49} +12.1000 q^{51} +8.84555 q^{53} +9.97360 q^{55} -8.44312 q^{57} -14.3300 q^{59} +5.96285 q^{61} +5.14285 q^{63} +5.52299 q^{65} +8.83199 q^{67} +13.9877 q^{71} -4.92118 q^{73} -27.5068 q^{75} +4.76529 q^{77} +7.06114 q^{79} -9.94225 q^{81} +4.96828 q^{83} -20.7389 q^{85} +10.4571 q^{87} +16.0689 q^{89} +2.63883 q^{91} +8.22939 q^{93} +14.4711 q^{95} +8.59175 q^{97} +6.47539 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 20 q^{9} + 12 q^{13} + 32 q^{25} - 40 q^{27} + 12 q^{35} + 36 q^{39} + 12 q^{41} + 12 q^{47} + 32 q^{49} - 12 q^{55} - 24 q^{59} + 12 q^{71} + 12 q^{73} + 8 q^{75} - 16 q^{81} - 36 q^{85}+ \cdots + 84 q^{95}+O(q^{100}) $ 8 * q - 4 * q^3 + 20 * q^9 + 12 * q^13 + 32 * q^25 - 40 * q^27 + 12 * q^35 + 36 * q^39 + 12 * q^41 + 12 * q^47 + 32 * q^49 - 12 * q^55 - 24 * q^59 + 12 * q^71 + 12 * q^73 + 8 * q^75 - 16 * q^81 - 36 * q^85 + 60 * q^87 + 84 * q^95

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 $

$2$	0	0
$2$	0	0
$3$	−2.37562	−1.37156	−0.685782	−	0.727807i	$-0.740540\pi$
$3$	−2.37562	−1.37156	−0.685782	+	0.727807i	$0.740540\pi$
$4$	0	0
$5$	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	$0.380163\pi$
$8$	0	0
$9$	2.64357	0.881190
$10$	0	0
$11$	2.44949	0.738549	0.369274	−	0.929320i	$-0.379606\pi$
$11$	2.44949	0.738549	0.369274	+	0.929320i	$0.379606\pi$
$12$	0	0
$13$	1.35643	0.376206	0.188103	−	0.982149i	$-0.439766\pi$
$13$	1.35643	0.376206	0.188103	+	0.982149i	$0.439766\pi$
$14$	0	0
$15$	−9.67283	−2.49751
$16$	0	0
$17$	−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	$0.366338\pi$
$20$	0	0
$21$	−4.62158	−1.00851
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	11.5788	2.31576
$26$	0	0
$27$	0.846745	0.162956
$28$	0	0
$29$	−4.40183	−0.817400	−0.408700	−	0.912669i	$-0.634018\pi$
$29$	−4.40183	−0.817400	−0.408700	+	0.912669i	$0.634018\pi$
$30$	0	0
$31$	−3.46410	−0.622171	−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410	−0.622171	−0.311086	+	0.950382i	$0.600693\pi$
$32$	0	0
$33$	−5.81906	−1.01297
$34$	0	0
$35$	7.92118	1.33892
$36$	0	0
$37$	1.94542	0.319825	0.159913	−	0.987131i	$-0.448879\pi$
$37$	1.94542	0.319825	0.159913	+	0.987131i	$0.448879\pi$
$38$	0	0
$39$	−3.22236	−0.515991
$40$	0	0
$41$	−5.57177	−0.870165	−0.435082	−	0.900391i	$-0.643281\pi$
$41$	−5.57177	−0.870165	−0.435082	+	0.900391i	$0.643281\pi$
$42$	0	0
$43$	1.63579	0.249455	0.124727	−	0.992191i	$-0.460194\pi$
$43$	1.63579	0.249455	0.124727	+	0.992191i	$0.460194\pi$
$44$	0	0
$45$	10.7638	1.60458
$46$	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
$50$	0	0
$51$	12.1000	1.69434
$52$	0	0
$53$	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
$56$	0	0
$57$	−8.44312	−1.11832
$58$	0	0
$59$	−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	$-0.375328\pi$
$61$	5.96285	0.763465	0.381733	+	0.924273i	$0.375328\pi$
$62$	0	0
$63$	5.14285	0.647938
$64$	0	0
$65$	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	$0.318614\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	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	$0.592987\pi$
$74$	0	0
$75$	−27.5068	−3.17621
$76$	0	0
$77$	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	$0.369975\pi$
$80$	0	0
$81$	−9.94225	−1.10469
$82$	0	0
$83$	4.96828	0.545340	0.272670	−	0.962108i	$-0.412093\pi$
$83$	4.96828	0.545340	0.272670	+	0.962108i	$0.412093\pi$
$84$	0	0
$85$	−20.7389	−2.24945
$86$	0	0
$87$	10.4571	1.12112
$88$	0	0
$89$	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
$93$	8.22939	0.853348
$94$	0	0
$95$	14.4711	1.48471
$96$	0	0
$97$	8.59175	0.872360	0.436180	−	0.899859i	$-0.356331\pi$
$97$	8.59175	0.872360	0.436180	+	0.899859i	$0.356331\pi$
$98$	0	0
$99$	6.47539	0.650802
$100$	0	0
$101$	15.5858	1.55085	0.775423	−	0.631442i	$-0.217536\pi$
$101$	15.5858	1.55085	0.775423	+	0.631442i	$0.217536\pi$
$102$	0	0
$103$	−9.82510	−0.968095	−0.484048	−	0.875042i	$-0.660834\pi$
$103$	−9.82510	−0.968095	−0.484048	+	0.875042i	$0.660834\pi$
$104$	0	0
$105$	−18.8177	−1.83642
$106$	0	0
$107$	−2.10762	−0.203751	−0.101876	−	0.994797i	$-0.532484\pi$
$107$	−2.10762	−0.203751	−0.101876	+	0.994797i	$0.532484\pi$
$108$	0	0
$109$	11.1328	1.06633	0.533166	−	0.846010i	$-0.321002\pi$
$109$	11.1328	1.06633	0.533166	+	0.846010i	$0.321002\pi$
$110$	0	0
$111$	−4.62158	−0.438661
$112$	0	0
$113$	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
$116$	0	0
$117$	3.58582	0.331509
$118$	0	0
$119$	−9.90883	−0.908341
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	13.2364	1.19349
$124$	0	0
$125$	26.7869	2.39590
$126$	0	0
$127$	−8.22939	−0.730240	−0.365120	−	0.930960i	$-0.618972\pi$
$127$	−8.22939	−0.730240	−0.365120	+	0.930960i	$0.618972\pi$
$128$	0	0
$129$	−3.88600	−0.342144
$130$	0	0
$131$	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
$134$	0	0
$135$	3.44770	0.296730
$136$	0	0
$137$	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	$-0.492799\pi$
$139$	0.533395	0.0452420	0.0226210	+	0.999744i	$0.492799\pi$
$140$	0	0
$141$	−5.45005	−0.458977
$142$	0	0
$143$	3.32256	0.277847
$144$	0	0
$145$	−17.9230	−1.48842
$146$	0	0
$147$	7.63843	0.630007
$148$	0	0
$149$	−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	$-0.858957\pi$
$151$	−22.2030	−1.80685	−0.903427	+	0.428742i	$0.858957\pi$
$152$	0	0
$153$	−13.4648	−1.08856
$154$	0	0
$155$	−14.1048	−1.13293
$156$	0	0
$157$	−12.4925	−0.997012	−0.498506	−	0.866886i	$-0.666118\pi$
$157$	−12.4925	−0.997012	−0.498506	+	0.866886i	$0.666118\pi$
$158$	0	0
$159$	−21.0137	−1.66649
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.33254	−0.496003	−0.248001	−	0.968760i	$-0.579774\pi$
$163$	−6.33254	−0.496003	−0.248001	+	0.968760i	$0.579774\pi$
$164$	0	0
$165$	−23.6935	−1.84454
$166$	0	0
$167$	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
$170$	0	0
$171$	9.39543	0.718486
$172$	0	0
$173$	1.59817	0.121506	0.0607532	−	0.998153i	$-0.480650\pi$
$173$	1.59817	0.121506	0.0607532	+	0.998153i	$0.480650\pi$
$174$	0	0
$175$	22.5256	1.70278
$176$	0	0
$177$	34.0427	2.55881
$178$	0	0
$179$	−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	$-0.449317\pi$
$181$	4.26624	0.317107	0.158553	+	0.987350i	$0.449317\pi$
$182$	0	0
$183$	−14.1655	−1.04714
$184$	0	0
$185$	7.92118	0.582377
$186$	0	0
$187$	−12.4763	−0.912355
$188$	0	0
$189$	1.64727	0.119822
$190$	0	0
$191$	−3.58630	−0.259496	−0.129748	−	0.991547i	$-0.541417\pi$
$191$	−3.58630	−0.259496	−0.129748	+	0.991547i	$0.541417\pi$
$192$	0	0
$193$	15.2870	1.10038	0.550190	−	0.835040i	$-0.314555\pi$
$193$	15.2870	1.10038	0.550190	+	0.835040i	$0.314555\pi$
$194$	0	0
$195$	−13.1205	−0.939580
$196$	0	0
$197$	−7.79946	−0.555689	−0.277844	−	0.960626i	$-0.589620\pi$
$197$	−7.79946	−0.555689	−0.277844	+	0.960626i	$0.589620\pi$
$198$	0	0
$199$	24.1394	1.71119	0.855597	−	0.517642i	$-0.173190\pi$
$199$	24.1394	1.71119	0.855597	+	0.517642i	$0.173190\pi$
$200$	0	0
$201$	−20.9814	−1.47992
$202$	0	0
$203$	−8.56341	−0.601034
$204$	0	0
$205$	−22.6866	−1.58450
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.70565	0.602183
$210$	0	0
$211$	0.100647	0.00692882	0.00346441	−	0.999994i	$-0.498897\pi$
$211$	0.100647	0.00692882	0.00346441	+	0.999994i	$0.498897\pi$
$212$	0	0
$213$	−33.2293	−2.27684
$214$	0	0
$215$	6.66044	0.454238
$216$	0	0
$217$	−6.73913	−0.457482
$218$	0	0
$219$	11.6909	0.789995
$220$	0	0
$221$	−6.90887	−0.464741
$222$	0	0
$223$	−6.11720	−0.409638	−0.204819	−	0.978800i	$-0.565661\pi$
$223$	−6.11720	−0.409638	−0.204819	+	0.978800i	$0.565661\pi$
$224$	0	0
$225$	30.6093	2.04062
$226$	0	0
$227$	24.5033	1.62634	0.813170	−	0.582027i	$-0.197740\pi$
$227$	24.5033	1.62634	0.813170	+	0.582027i	$0.197740\pi$
$228$	0	0
$229$	−13.9724	−0.923322	−0.461661	−	0.887056i	$-0.652746\pi$
$229$	−13.9724	−0.923322	−0.461661	+	0.887056i	$0.652746\pi$
$230$	0	0
$231$	−11.3205	−0.744835
$232$	0	0
$233$	4.85641	0.318154	0.159077	−	0.987266i	$-0.449148\pi$
$233$	4.85641	0.318154	0.159077	+	0.987266i	$0.449148\pi$
$234$	0	0
$235$	9.34115	0.609350
$236$	0	0
$237$	−16.7746	−1.08963
$238$	0	0
$239$	−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	$-0.358175\pi$
$241$	13.3806	0.861922	0.430961	+	0.902371i	$0.358175\pi$
$242$	0	0
$243$	21.0788	1.35220
$244$	0	0
$245$	−13.0919	−0.836412
$246$	0	0
$247$	4.82085	0.306743
$248$	0	0
$249$	−11.8027	−0.747969
$250$	0	0
$251$	−3.44260	−0.217295	−0.108647	−	0.994080i	$-0.534652\pi$
$251$	−3.44260	−0.217295	−0.108647	+	0.994080i	$0.534652\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	49.2677	3.08526
$256$	0	0
$257$	−8.79150	−0.548399	−0.274199	−	0.961673i	$-0.588413\pi$
$257$	−8.79150	−0.548399	−0.274199	+	0.961673i	$0.588413\pi$
$258$	0	0
$259$	3.78466	0.235167
$260$	0	0
$261$	−11.6365	−0.720284
$262$	0	0
$263$	−17.0957	−1.05417	−0.527083	−	0.849814i	$-0.676714\pi$
$263$	−17.0957	−1.05417	−0.527083	+	0.849814i	$0.676714\pi$
$264$	0	0
$265$	36.0165	2.21248
$266$	0	0
$267$	−38.1736	−2.33618
$268$	0	0
$269$	−4.28181	−0.261067	−0.130533	−	0.991444i	$-0.541669\pi$
$269$	−4.28181	−0.261067	−0.130533	+	0.991444i	$0.541669\pi$
$270$	0	0
$271$	13.6337	0.828190	0.414095	−	0.910234i	$-0.364098\pi$
$271$	13.6337	0.828190	0.414095	+	0.910234i	$0.364098\pi$
$272$	0	0
$273$	−6.26885	−0.379408
$274$	0	0
$275$	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	0
$279$	−9.15759	−0.548251
$280$	0	0
$281$	−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	$0.355661\pi$
$284$	0	0
$285$	−34.3779	−2.03637
$286$	0	0
$287$	−10.8394	−0.639832
$288$	0	0
$289$	8.94287	0.526051
$290$	0	0
$291$	−20.4107	−1.19650
$292$	0	0
$293$	−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
$296$	0	0
$297$	2.07409	0.120351
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.18229	0.183424
$302$	0	0
$303$	−37.0260	−2.12709
$304$	0	0
$305$	24.2790	1.39021
$306$	0	0
$307$	−32.9589	−1.88107	−0.940533	−	0.339703i	$-0.889674\pi$
$307$	−32.9589	−1.88107	−0.940533	+	0.339703i	$0.889674\pi$
$308$	0	0
$309$	23.3407	1.32781
$310$	0	0
$311$	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	$-0.411862\pi$
$313$	9.67273	0.546735	0.273368	+	0.961910i	$0.411862\pi$
$314$	0	0
$315$	20.9402	1.17985
$316$	0	0
$317$	−18.3490	−1.03058	−0.515292	−	0.857014i	$-0.672317\pi$
$317$	−18.3490	−1.03058	−0.515292	+	0.857014i	$0.672317\pi$
$318$	0	0
$319$	−10.7822	−0.603690
$320$	0	0
$321$	5.00691	0.279458
$322$	0	0
$323$	−18.1023	−1.00724
$324$	0	0
$325$	15.7058	0.871203
$326$	0	0
$327$	−26.4474	−1.46254
$328$	0	0
$329$	4.46311	0.246059
$330$	0	0
$331$	25.5384	1.40371	0.701857	−	0.712317i	$-0.252354\pi$
$331$	25.5384	1.40371	0.701857	+	0.712317i	$0.252354\pi$
$332$	0	0
$333$	5.14285	0.281827
$334$	0	0
$335$	35.9613	1.96477
$336$	0	0
$337$	32.6280	1.77736	0.888681	−	0.458526i	$-0.151623\pi$
$337$	32.6280	1.77736	0.888681	+	0.458526i	$0.151623\pi$
$338$	0	0
$339$	−27.5919	−1.49859
$340$	0	0
$341$	−8.48528	−0.459504
$342$	0	0
$343$	−19.8731	−1.07305
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−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	$0.402189\pi$
$350$	0	0
$351$	1.14855	0.0613051
$352$	0	0
$353$	−25.1190	−1.33695	−0.668476	−	0.743734i	$-0.733053\pi$
$353$	−25.1190	−1.33695	−0.668476	+	0.743734i	$0.733053\pi$
$354$	0	0
$355$	56.9536	3.02278
$356$	0	0
$357$	23.5396	1.24585
$358$	0	0
$359$	−15.9761	−0.843186	−0.421593	−	0.906785i	$-0.638529\pi$
$359$	−15.9761	−0.843186	−0.421593	+	0.906785i	$0.638529\pi$
$360$	0	0
$361$	−6.36860	−0.335189
$362$	0	0
$363$	11.8781	0.623438
$364$	0	0
$365$	−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
$369$	−14.7294	−0.766780
$370$	0	0
$371$	17.2083	0.893411
$372$	0	0
$373$	2.17582	0.112660	0.0563298	−	0.998412i	$-0.482060\pi$
$373$	2.17582	0.112660	0.0563298	+	0.998412i	$0.482060\pi$
$374$	0	0
$375$	−63.6355	−3.28613
$376$	0	0
$377$	−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	$0.532707\pi$
$380$	0	0
$381$	19.5499	1.00157
$382$	0	0
$383$	16.2398	0.829816	0.414908	−	0.909863i	$-0.363814\pi$
$383$	16.2398	0.829816	0.414908	+	0.909863i	$0.363814\pi$
$384$	0	0
$385$	19.4028	0.988861
$386$	0	0
$387$	4.32431	0.219817
$388$	0	0
$389$	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
$392$	0	0
$393$	−20.9375	−1.05616
$394$	0	0
$395$	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	$0.423104\pi$
$398$	0	0
$399$	−16.4254	−0.822299
$400$	0	0
$401$	−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
$404$	0	0
$405$	−40.4819	−2.01156
$406$	0	0
$407$	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	$0.0611742\pi$
$410$	0	0
$411$	−9.21092	−0.454341
$412$	0	0
$413$	−27.8779	−1.37178
$414$	0	0
$415$	20.2294	0.993022
$416$	0	0
$417$	−1.26714	−0.0620523
$418$	0	0
$419$	30.8981	1.50947	0.754736	−	0.656028i	$-0.227765\pi$
$419$	30.8981	1.50947	0.754736	+	0.656028i	$0.227765\pi$
$420$	0	0
$421$	−2.14868	−0.104720	−0.0523602	−	0.998628i	$-0.516674\pi$
$421$	−2.14868	−0.104720	−0.0523602	+	0.998628i	$0.516674\pi$
$422$	0	0
$423$	6.06477	0.294879
$424$	0	0
$425$	−58.9756	−2.86074
$426$	0	0
$427$	11.6003	0.561376
$428$	0	0
$429$	−7.89315	−0.381085
$430$	0	0
$431$	25.1005	1.20905	0.604524	−	0.796587i	$-0.293363\pi$
$431$	25.1005	1.20905	0.604524	+	0.796587i	$0.293363\pi$
$432$	0	0
$433$	−5.95292	−0.286079	−0.143040	−	0.989717i	$-0.545688\pi$
$433$	−5.95292	−0.286079	−0.143040	+	0.989717i	$0.545688\pi$
$434$	0	0
$435$	42.5782	2.04147
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−24.1766	−1.15389	−0.576943	−	0.816784i	$-0.695755\pi$
$439$	−24.1766	−1.15389	−0.576943	+	0.816784i	$0.695755\pi$
$440$	0	0
$441$	−8.49998	−0.404761
$442$	0	0
$443$	−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
$446$	0	0
$447$	6.63406	0.313780
$448$	0	0
$449$	29.7300	1.40304	0.701522	−	0.712647i	$-0.252504\pi$
$449$	29.7300	1.40304	0.701522	+	0.712647i	$0.252504\pi$
$450$	0	0
$451$	−13.6480	−0.642659
$452$	0	0
$453$	52.7459	2.47822
$454$	0	0
$455$	10.7445	0.503712
$456$	0	0
$457$	−4.37251	−0.204538	−0.102269	−	0.994757i	$-0.532610\pi$
$457$	−4.37251	−0.204538	−0.102269	+	0.994757i	$0.532610\pi$
$458$	0	0
$459$	−4.31282	−0.201305
$460$	0	0
$461$	7.86593	0.366353	0.183177	−	0.983080i	$-0.441362\pi$
$461$	7.86593	0.366353	0.183177	+	0.983080i	$0.441362\pi$
$462$	0	0
$463$	21.9472	1.01997	0.509987	−	0.860182i	$-0.329650\pi$
$463$	21.9472	1.01997	0.509987	+	0.860182i	$0.329650\pi$
$464$	0	0
$465$	33.5077	1.55388
$466$	0	0
$467$	−29.6404	−1.37159	−0.685797	−	0.727793i	$-0.740546\pi$
$467$	−29.6404	−1.37159	−0.685797	+	0.727793i	$0.740546\pi$
$468$	0	0
$469$	17.1819	0.793387
$470$	0	0
$471$	29.6775	1.36747
$472$	0	0
$473$	4.00684	0.184235
$474$	0	0
$475$	41.1518	1.88818
$476$	0	0
$477$	23.3838	1.07067
$478$	0	0
$479$	38.7485	1.77046	0.885232	−	0.465150i	$-0.153999\pi$
$479$	38.7485	1.77046	0.885232	+	0.465150i	$0.153999\pi$
$480$	0	0
$481$	2.63883	0.120320
$482$	0	0
$483$	0	0
$484$	0	0
$485$	34.9831	1.58850
$486$	0	0
$487$	−15.6777	−0.710426	−0.355213	−	0.934785i	$-0.615592\pi$
$487$	−15.6777	−0.710426	−0.355213	+	0.934785i	$0.615592\pi$
$488$	0	0
$489$	15.0437	0.680300
$490$	0	0
$491$	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
$494$	0	0
$495$	26.3659	1.18506
$496$	0	0
$497$	27.2119	1.22062
$498$	0	0
$499$	16.9853	0.760368	0.380184	−	0.924911i	$-0.375861\pi$
$499$	16.9853	0.760368	0.380184	+	0.924911i	$0.375861\pi$
$500$	0	0
$501$	−49.1138	−2.19424
$502$	0	0
$503$	5.66325	0.252512	0.126256	−	0.991998i	$-0.459704\pi$
$503$	5.66325	0.252512	0.126256	+	0.991998i	$0.459704\pi$
$504$	0	0
$505$	63.4609	2.82397
$506$	0	0
$507$	26.5121	1.17745
$508$	0	0
$509$	10.4262	0.462131	0.231066	−	0.972938i	$-0.425779\pi$
$509$	10.4262	0.462131	0.231066	+	0.972938i	$0.425779\pi$
$510$	0	0
$511$	−9.57376	−0.423518
$512$	0	0
$513$	3.00939	0.132868
$514$	0	0
$515$	−40.0049	−1.76283
$516$	0	0
$517$	5.61952	0.247146
$518$	0	0
$519$	−3.79664	−0.166654
$520$	0	0
$521$	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	$0.652745\pi$
$524$	0	0
$525$	−53.5123	−2.33547
$526$	0	0
$527$	17.6441	0.768589
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−37.8824	−1.64396
$532$	0	0
$533$	−7.55773	−0.327361
$534$	0	0
$535$	−8.58162	−0.371016
$536$	0	0
$537$	21.0743	0.909423
$538$	0	0
$539$	−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	$0.733396\pi$
$542$	0	0
$543$	−10.1350	−0.434933
$544$	0	0
$545$	45.3297	1.94171
$546$	0	0
$547$	−23.8705	−1.02063	−0.510313	−	0.859988i	$-0.670471\pi$
$547$	−23.8705	−1.02063	−0.510313	+	0.859988i	$0.670471\pi$
$548$	0	0
$549$	15.7632	0.672758
$550$	0	0
$551$	−15.6444	−0.666474
$552$	0	0
$553$	13.7369	0.584151
$554$	0	0
$555$	−18.8177	−0.798767
$556$	0	0
$557$	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
$560$	0	0
$561$	29.6389	1.25135
$562$	0	0
$563$	−6.76143	−0.284960	−0.142480	−	0.989798i	$-0.545508\pi$
$563$	−6.76143	−0.284960	−0.142480	+	0.989798i	$0.545508\pi$
$564$	0	0
$565$	47.2913	1.98956
$566$	0	0
$567$	−19.3419	−0.812281
$568$	0	0
$569$	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	$0.378171\pi$
$572$	0	0
$573$	8.51969	0.355915
$574$	0	0
$575$	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	$0.600225\pi$
$578$	0	0
$579$	−36.3160	−1.50924
$580$	0	0
$581$	9.66540	0.400988
$582$	0	0
$583$	21.6671	0.897359
$584$	0	0
$585$	14.6004	0.603652
$586$	0	0
$587$	−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
$590$	0	0
$591$	18.5286	0.762163
$592$	0	0
$593$	24.1392	0.991276	0.495638	−	0.868529i	$-0.334934\pi$
$593$	24.1392	0.991276	0.495638	+	0.868529i	$0.334934\pi$
$594$	0	0
$595$	−40.3459	−1.65402
$596$	0	0
$597$	−57.3460	−2.34701
$598$	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	$0.267873\pi$
$602$	0	0
$603$	23.3480	0.950803
$604$	0	0
$605$	−20.3585	−0.827692
$606$	0	0
$607$	−4.09287	−0.166124	−0.0830622	−	0.996544i	$-0.526470\pi$
$607$	−4.09287	−0.166124	−0.0830622	+	0.996544i	$0.526470\pi$
$608$	0	0
$609$	20.3434	0.824357
$610$	0	0
$611$	3.11187	0.125893
$612$	0	0
$613$	29.7094	1.19995	0.599975	−	0.800019i	$-0.295177\pi$
$613$	29.7094	1.19995	0.599975	+	0.800019i	$0.295177\pi$
$614$	0	0
$615$	53.8948	2.17325
$616$	0	0
$617$	−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	$0.760527\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	31.2607	1.25244
$624$	0	0
$625$	51.1745	2.04698
$626$	0	0
$627$	−20.6813	−0.825933
$628$	0	0
$629$	−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	$0.511253\pi$
$632$	0	0
$633$	−0.239099	−0.00950333
$634$	0	0
$635$	−33.5077	−1.32971
$636$	0	0
$637$	−4.36139	−0.172805
$638$	0	0
$639$	36.9773	1.46280
$640$	0	0
$641$	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
$645$	−15.8227	−0.623017
$646$	0	0
$647$	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
$650$	0	0
$651$	16.0096	0.627466
$652$	0	0
$653$	−9.03587	−0.353601	−0.176801	−	0.984247i	$-0.556575\pi$
$653$	−9.03587	−0.353601	−0.176801	+	0.984247i	$0.556575\pi$
$654$	0	0
$655$	35.8860	1.40218
$656$	0	0
$657$	−13.0095	−0.507548
$658$	0	0
$659$	7.78944	0.303433	0.151717	−	0.988424i	$-0.451520\pi$
$659$	7.78944	0.303433	0.151717	+	0.988424i	$0.451520\pi$
$660$	0	0
$661$	27.8692	1.08399	0.541993	−	0.840383i	$-0.317670\pi$
$661$	27.8692	1.08399	0.541993	+	0.840383i	$0.317670\pi$
$662$	0	0
$663$	16.4128	0.637422
$664$	0	0
$665$	28.1524	1.09170
$666$	0	0
$667$	0	0
$668$	0	0
$669$	14.5321	0.561845
$670$	0	0
$671$	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	$0.494570\pi$
$674$	0	0
$675$	9.80428	0.377367
$676$	0	0
$677$	−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
$680$	0	0
$681$	−58.2105	−2.23063
$682$	0	0
$683$	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
$686$	0	0
$687$	33.1931	1.26640
$688$	0	0
$689$	11.9984	0.457102
$690$	0	0
$691$	−31.3012	−1.19075	−0.595377	−	0.803447i	$-0.702997\pi$
$691$	−31.3012	−1.19075	−0.595377	+	0.803447i	$0.702997\pi$
$692$	0	0
$693$	12.5974	0.478534
$694$	0	0
$695$	2.17183	0.0823821
$696$	0	0
$697$	28.3793	1.07494
$698$	0	0
$699$	−11.5370	−0.436368
$700$	0	0
$701$	−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
$704$	0	0
$705$	−22.1910	−0.835762
$706$	0	0
$707$	30.3210	1.14034
$708$	0	0
$709$	19.3568	0.726959	0.363480	−	0.931602i	$-0.381589\pi$
$709$	19.3568	0.726959	0.363480	+	0.931602i	$0.381589\pi$
$710$	0	0
$711$	18.6666	0.700052
$712$	0	0
$713$	0	0
$714$	0	0
$715$	13.5285	0.505937
$716$	0	0
$717$	25.3243	0.945752
$718$	0	0
$719$	−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
$722$	0	0
$723$	−31.7873	−1.18218
$724$	0	0
$725$	−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	$0.817607\pi$
$728$	0	0
$729$	−20.2484	−0.749940
$730$	0	0
$731$	−8.33173	−0.308160
$732$	0	0
$733$	−17.3915	−0.642370	−0.321185	−	0.947017i	$-0.604081\pi$
$733$	−17.3915	−0.642370	−0.321185	+	0.947017i	$0.604081\pi$
$734$	0	0
$735$	31.1014	1.14719
$736$	0	0
$737$	21.6339	0.796893
$738$	0	0
$739$	−33.9686	−1.24956	−0.624778	−	0.780803i	$-0.714810\pi$
$739$	−33.9686	−1.24956	−0.624778	+	0.780803i	$0.714810\pi$
$740$	0	0
$741$	−11.4525	−0.420718
$742$	0	0
$743$	−25.9127	−0.950643	−0.475321	−	0.879812i	$-0.657668\pi$
$743$	−25.9127	−0.950643	−0.475321	+	0.879812i	$0.657668\pi$
$744$	0	0
$745$	−11.3705	−0.416582
$746$	0	0
$747$	13.1340	0.480548
$748$	0	0
$749$	−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	$0.628637\pi$
$752$	0	0
$753$	8.17831	0.298034
$754$	0	0
$755$	−90.4041	−3.29014
$756$	0	0
$757$	−30.5862	−1.11167	−0.555837	−	0.831292i	$-0.687602\pi$
$757$	−30.5862	−1.11167	−0.555837	+	0.831292i	$0.687602\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.0332	0.798702	0.399351	−	0.916798i	$-0.369235\pi$
$761$	22.0332	0.798702	0.399351	+	0.916798i	$0.369235\pi$
$762$	0	0
$763$	21.6581	0.784074
$764$	0	0
$765$	−54.8247	−1.98219
$766$	0	0
$767$	−19.4377	−0.701855
$768$	0	0
$769$	−42.3961	−1.52884	−0.764421	−	0.644717i	$-0.776975\pi$
$769$	−42.3961	−1.52884	−0.764421	+	0.644717i	$0.776975\pi$
$770$	0	0
$771$	20.8853	0.752164
$772$	0	0
$773$	−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
$776$	0	0
$777$	−8.99091	−0.322547
$778$	0	0
$779$	−19.8025	−0.709497
$780$	0	0
$781$	34.2626	1.22601
$782$	0	0
$783$	−3.72723	−0.133200
$784$	0	0
$785$	−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	$0.166030\pi$
$788$	0	0
$789$	40.6129	1.44586
$790$	0	0
$791$	22.5953	0.803396
$792$	0	0
$793$	8.08820	0.287220
$794$	0	0
$795$	−85.5615	−3.03455
$796$	0	0
$797$	−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
$800$	0	0
$801$	42.4792	1.50093
$802$	0	0
$803$	−12.0544	−0.425390
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.1720	0.358070
$808$	0	0
$809$	33.9809	1.19470	0.597352	−	0.801979i	$-0.296220\pi$
$809$	33.9809	1.19470	0.597352	+	0.801979i	$0.296220\pi$
$810$	0	0
$811$	−5.04290	−0.177080	−0.0885400	−	0.996073i	$-0.528220\pi$
$811$	−5.04290	−0.177080	−0.0885400	+	0.996073i	$0.528220\pi$
$812$	0	0
$813$	−32.3885	−1.13592
$814$	0	0
$815$	−25.7842	−0.903182
$816$	0	0
$817$	5.81369	0.203395
$818$	0	0
$819$	6.97592	0.243759
$820$	0	0
$821$	2.44191	0.0852231	0.0426116	−	0.999092i	$-0.486432\pi$
$821$	2.44191	0.0852231	0.0426116	+	0.999092i	$0.486432\pi$
$822$	0	0
$823$	2.84938	0.0993232	0.0496616	−	0.998766i	$-0.484186\pi$
$823$	2.84938	0.0993232	0.0496616	+	0.998766i	$0.484186\pi$
$824$	0	0
$825$	−67.3777	−2.34579
$826$	0	0
$827$	3.88626	0.135139	0.0675693	−	0.997715i	$-0.478476\pi$
$827$	3.88626	0.135139	0.0675693	+	0.997715i	$0.478476\pi$
$828$	0	0
$829$	20.3631	0.707239	0.353620	−	0.935389i	$-0.384951\pi$
$829$	20.3631	0.707239	0.353620	+	0.935389i	$0.384951\pi$
$830$	0	0
$831$	72.0751	2.50026
$832$	0	0
$833$	16.3771	0.567432
$834$	0	0
$835$	84.1789	2.91313
$836$	0	0
$837$	−2.93321	−0.101387
$838$	0	0
$839$	−7.05733	−0.243646	−0.121823	−	0.992552i	$-0.538874\pi$
$839$	−7.05733	−0.243646	−0.121823	+	0.992552i	$0.538874\pi$
$840$	0	0
$841$	−9.62388	−0.331858
$842$	0	0
$843$	21.9933	0.757490
$844$	0	0
$845$	−45.4406	−1.56321
$846$	0	0
$847$	−9.72710	−0.334227
$848$	0	0
$849$	−35.0145	−1.20169
$850$	0	0
$851$	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	$0.625736\pi$
$854$	0	0
$855$	38.2554	1.30831
$856$	0	0
$857$	6.43977	0.219978	0.109989	−	0.993933i	$-0.464918\pi$
$857$	6.43977	0.219978	0.109989	+	0.993933i	$0.464918\pi$
$858$	0	0
$859$	−37.7008	−1.28633	−0.643167	−	0.765726i	$-0.722380\pi$
$859$	−37.7008	−1.28633	−0.643167	+	0.765726i	$0.722380\pi$
$860$	0	0
$861$	25.7504	0.877571
$862$	0	0
$863$	−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
$866$	0	0
$867$	−21.2449	−0.721513
$868$	0	0
$869$	17.2962	0.586733
$870$	0	0
$871$	11.9800	0.405926
$872$	0	0
$873$	22.7129	0.768715
$874$	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	$0.538637\pi$
$878$	0	0
$879$	37.5062	1.26505
$880$	0	0
$881$	−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	$-0.257633\pi$
$883$	41.0042	1.37990	0.689950	+	0.723857i	$0.257633\pi$
$884$	0	0
$885$	138.612	4.65939
$886$	0	0
$887$	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
$890$	0	0
$891$	−24.3534	−0.815871
$892$	0	0
$893$	8.15361	0.272850
$894$	0	0
$895$	−36.1204	−1.20737
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.2484	0.508562
$900$	0	0
$901$	−45.0541	−1.50097
$902$	0	0
$903$	−7.55991	−0.251578
$904$	0	0
$905$	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	$0.585368\pi$
$908$	0	0
$909$	41.2022	1.36659
$910$	0	0
$911$	37.5172	1.24300	0.621501	−	0.783414i	$-0.286523\pi$
$911$	37.5172	1.24300	0.621501	+	0.783414i	$0.286523\pi$
$912$	0	0
$913$	12.1698	0.402760
$914$	0	0
$915$	−57.6776	−1.90676
$916$	0	0
$917$	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
$921$	78.2979	2.58000
$922$	0	0
$923$	18.9733	0.624513
$924$	0	0
$925$	22.5256	0.740638
$926$	0	0
$927$	−25.9733	−0.853076
$928$	0	0
$929$	−43.8177	−1.43761	−0.718805	−	0.695211i	$-0.755311\pi$
$929$	−43.8177	−1.43761	−0.718805	+	0.695211i	$0.755311\pi$
$930$	0	0
$931$	−11.4275	−0.374523
$932$	0	0
$933$	−47.3251	−1.54936
$934$	0	0
$935$	−50.7997	−1.66133
$936$	0	0
$937$	34.4360	1.12498	0.562488	−	0.826805i	$-0.309844\pi$
$937$	34.4360	1.12498	0.562488	+	0.826805i	$0.309844\pi$
$938$	0	0
$939$	−22.9787	−0.749883
$940$	0	0
$941$	−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
$944$	0	0
$945$	6.70722	0.218186
$946$	0	0
$947$	−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
$950$	0	0
$951$	43.5903	1.41351
$952$	0	0
$953$	−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
$956$	0	0
$957$	25.6145	0.827999
$958$	0	0
$959$	7.54292	0.243574
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	−5.57164	−0.179544
$964$	0	0
$965$	62.2440	2.00370
$966$	0	0
$967$	−52.3940	−1.68488	−0.842439	−	0.538792i	$-0.818881\pi$
$967$	−52.3940	−1.68488	−0.842439	+	0.538792i	$0.818881\pi$
$968$	0	0
$969$	43.0043	1.38150
$970$	0	0
$971$	−51.8696	−1.66457	−0.832287	−	0.554344i	$-0.812969\pi$
$971$	−51.8696	−1.66457	−0.832287	+	0.554344i	$0.812969\pi$
$972$	0	0
$973$	1.03768	0.0332664
$974$	0	0
$975$	−37.3111	−1.19491
$976$	0	0
$977$	−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
$980$	0	0
$981$	29.4304	0.939641
$982$	0	0
$983$	23.6079	0.752975	0.376488	−	0.926422i	$-0.377132\pi$
$983$	23.6079	0.752975	0.376488	+	0.926422i	$0.377132\pi$
$984$	0	0
$985$	−31.7571	−1.01187
$986$	0	0
$987$	−10.6026	−0.337486
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−3.77401	−0.119885	−0.0599427	−	0.998202i	$-0.519092\pi$
$991$	−3.77401	−0.119885	−0.0599427	+	0.998202i	$0.519092\pi$
$992$	0	0
$993$	−60.6694	−1.92529
$994$	0	0
$995$	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	$0.574750\pi$
$998$	0	0
$999$	1.64727	0.0521175

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1058.2.a.n.1.5	✓	8	92.91	even	2
1058.2.a.n.1.6	yes	8	4.3	odd	2
8464.2.a.cb.1.3		8	23.22	odd	2	inner
8464.2.a.cb.1.4		8	1.1	even	1	trivial
9522.2.a.ce.1.1		8	12.11	even	2
9522.2.a.ce.1.8		8	276.275	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q-2.37562 q^{3} +4.07171 q^{5} +1.94542 q^{7} +2.64357 q^{9} +2.44949 q^{11} +1.35643 q^{13} -9.67283 q^{15} -5.09341 q^{17} +3.55407 q^{19} -4.62158 q^{21} +11.5788 q^{25} +0.846745 q^{27} -4.40183 q^{29} -3.46410 q^{31} -5.81906 q^{33} +7.92118 q^{35} +1.94542 q^{37} -3.22236 q^{39} -5.57177 q^{41} +1.63579 q^{43} +10.7638 q^{45} +2.29416 q^{47} -3.21534 q^{49} +12.1000 q^{51} +8.84555 q^{53} +9.97360 q^{55} -8.44312 q^{57} -14.3300 q^{59} +5.96285 q^{61} +5.14285 q^{63} +5.52299 q^{65} +8.83199 q^{67} +13.9877 q^{71} -4.92118 q^{73} -27.5068 q^{75} +4.76529 q^{77} +7.06114 q^{79} -9.94225 q^{81} +4.96828 q^{83} -20.7389 q^{85} +10.4571 q^{87} +16.0689 q^{89} +2.63883 q^{91} +8.22939 q^{93} +14.4711 q^{95} +8.59175 q^{97} +6.47539 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 20 q^{9} + 12 q^{13} + 32 q^{25} - 40 q^{27} + 12 q^{35} + 36 q^{39} + 12 q^{41} + 12 q^{47} + 32 q^{49} - 12 q^{55} - 24 q^{59} + 12 q^{71} + 12 q^{73} + 8 q^{75} - 16 q^{81} - 36 q^{85}+ \cdots + 84 q^{95}+O(q^{100}) \) 8 * q - 4 * q^3 + 20 * q^9 + 12 * q^13 + 32 * q^25 - 40 * q^27 + 12 * q^35 + 36 * q^39 + 12 * q^41 + 12 * q^47 + 32 * q^49 - 12 * q^55 - 24 * q^59 + 12 * q^71 + 12 * q^73 + 8 * q^75 - 16 * q^81 - 36 * q^85 + 60 * q^87 + 84 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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