LMFDB - Embedded newform 1058.2.a.n.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1058,2,Mod(1,1058)] 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("1058.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1058, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1058 = 2 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1058.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,8,4,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.44817253385$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.443768$ of defining polynomial
Character	$\chi$	$=$	1058.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.37562 q^{3} +1.00000 q^{4} -4.07171 q^{5} +2.37562 q^{6} +1.94542 q^{7} +1.00000 q^{8} +2.64357 q^{9} -4.07171 q^{10} +2.44949 q^{11} +2.37562 q^{12} +1.35643 q^{13} +1.94542 q^{14} -9.67283 q^{15} +1.00000 q^{16} +5.09341 q^{17} +2.64357 q^{18} +3.55407 q^{19} -4.07171 q^{20} +4.62158 q^{21} +2.44949 q^{22} +2.37562 q^{24} +11.5788 q^{25} +1.35643 q^{26} -0.846745 q^{27} +1.94542 q^{28} -4.40183 q^{29} -9.67283 q^{30} +3.46410 q^{31} +1.00000 q^{32} +5.81906 q^{33} +5.09341 q^{34} -7.92118 q^{35} +2.64357 q^{36} -1.94542 q^{37} +3.55407 q^{38} +3.22236 q^{39} -4.07171 q^{40} -5.57177 q^{41} +4.62158 q^{42} +1.63579 q^{43} +2.44949 q^{44} -10.7638 q^{45} -2.29416 q^{47} +2.37562 q^{48} -3.21534 q^{49} +11.5788 q^{50} +12.1000 q^{51} +1.35643 q^{52} -8.84555 q^{53} -0.846745 q^{54} -9.97360 q^{55} +1.94542 q^{56} +8.44312 q^{57} -4.40183 q^{58} +14.3300 q^{59} -9.67283 q^{60} -5.96285 q^{61} +3.46410 q^{62} +5.14285 q^{63} +1.00000 q^{64} -5.52299 q^{65} +5.81906 q^{66} +8.83199 q^{67} +5.09341 q^{68} -7.92118 q^{70} -13.9877 q^{71} +2.64357 q^{72} -4.92118 q^{73} -1.94542 q^{74} +27.5068 q^{75} +3.55407 q^{76} +4.76529 q^{77} +3.22236 q^{78} +7.06114 q^{79} -4.07171 q^{80} -9.94225 q^{81} -5.57177 q^{82} +4.96828 q^{83} +4.62158 q^{84} -20.7389 q^{85} +1.63579 q^{86} -10.4571 q^{87} +2.44949 q^{88} -16.0689 q^{89} -10.7638 q^{90} +2.63883 q^{91} +8.22939 q^{93} -2.29416 q^{94} -14.4711 q^{95} +2.37562 q^{96} -8.59175 q^{97} -3.21534 q^{98} +6.47539 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{8} + 20 q^{9} + 4 q^{12} + 12 q^{13} + 8 q^{16} + 20 q^{18} + 4 q^{24} + 32 q^{25} + 12 q^{26} + 40 q^{27} + 8 q^{32} - 12 q^{35} + 20 q^{36} - 36 q^{39}+ \cdots + 32 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^8 + 20 * q^9 + 4 * q^12 + 12 * q^13 + 8 * q^16 + 20 * q^18 + 4 * q^24 + 32 * q^25 + 12 * q^26 + 40 * q^27 + 8 * q^32 - 12 * q^35 + 20 * q^36 - 36 * q^39 + 12 * q^41 - 12 * q^47 + 4 * q^48 + 32 * q^49 + 32 * q^50 + 12 * q^52 + 40 * q^54 + 12 * q^55 + 24 * q^59 + 8 * q^64 - 12 * q^70 - 12 * q^71 + 20 * q^72 + 12 * q^73 - 8 * q^75 - 36 * q^78 - 16 * q^81 + 12 * q^82 - 36 * q^85 - 60 * q^87 - 12 * q^94 - 84 * q^95 + 4 * q^96 + 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
$3$	2.37562	1.37156	0.685782	−	0.727807i	$-0.259460\pi$
$3$	2.37562	1.37156	0.685782	+	0.727807i	$0.259460\pi$
$4$	1.00000	0.500000
$5$	−4.07171	−1.82092	−0.910461	−	0.413594i	$-0.864273\pi$
$5$	−4.07171	−1.82092	−0.910461	+	0.413594i	$0.864273\pi$
$6$	2.37562	0.969843
$7$	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
$9$	2.64357	0.881190
$10$	−4.07171	−1.28759
$11$	2.44949	0.738549	0.369274	−	0.929320i	$-0.379606\pi$
$11$	2.44949	0.738549	0.369274	+	0.929320i	$0.379606\pi$
$12$	2.37562	0.685782
$13$	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
$15$	−9.67283	−2.49751
$16$	1.00000	0.250000
$17$	5.09341	1.23533	0.617667	−	0.786440i	$-0.288078\pi$
$17$	5.09341	1.23533	0.617667	+	0.786440i	$0.288078\pi$
$18$	2.64357	0.623095
$19$	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
$21$	4.62158	1.00851
$22$	2.44949	0.522233
$23$	0	0
$23$	0	0
$24$	2.37562	0.484921
$25$	11.5788	2.31576
$26$	1.35643	0.266018
$27$	−0.846745	−0.162956
$28$	1.94542	0.367650
$29$	−4.40183	−0.817400	−0.408700	−	0.912669i	$-0.634018\pi$
$29$	−4.40183	−0.817400	−0.408700	+	0.912669i	$0.634018\pi$
$30$	−9.67283	−1.76601
$31$	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
$33$	5.81906	1.01297
$34$	5.09341	0.873513
$35$	−7.92118	−1.33892
$36$	2.64357	0.440595
$37$	−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
$39$	3.22236	0.515991
$40$	−4.07171	−0.643793
$41$	−5.57177	−0.870165	−0.435082	−	0.900391i	$-0.643281\pi$
$41$	−5.57177	−0.870165	−0.435082	+	0.900391i	$0.643281\pi$
$42$	4.62158	0.713125
$43$	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
$45$	−10.7638	−1.60458
$46$	0	0
$47$	−2.29416	−0.334638	−0.167319	−	0.985903i	$-0.553511\pi$
$47$	−2.29416	−0.334638	−0.167319	+	0.985903i	$0.553511\pi$
$48$	2.37562	0.342891
$49$	−3.21534	−0.459334
$50$	11.5788	1.63749
$51$	12.1000	1.69434
$52$	1.35643	0.188103
$53$	−8.84555	−1.21503	−0.607515	−	0.794308i	$-0.707834\pi$
$53$	−8.84555	−1.21503	−0.607515	+	0.794308i	$0.707834\pi$
$54$	−0.846745	−0.115227
$55$	−9.97360	−1.34484
$56$	1.94542	0.259968
$57$	8.44312	1.11832
$58$	−4.40183	−0.577989
$59$	14.3300	1.86561	0.932806	−	0.360379i	$-0.117353\pi$
$59$	14.3300	1.86561	0.932806	+	0.360379i	$0.117353\pi$
$60$	−9.67283	−1.24876
$61$	−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
$63$	5.14285	0.647938
$64$	1.00000	0.125000
$65$	−5.52299	−0.685043
$66$	5.81906	0.716276
$67$	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	$-0.811668\pi$
$71$	−13.9877	−1.66003	−0.830014	+	0.557742i	$0.811668\pi$
$72$	2.64357	0.311548
$73$	−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
$75$	27.5068	3.17621
$76$	3.55407	0.407680
$77$	4.76529	0.543055
$78$	3.22236	0.364861
$79$	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
$81$	−9.94225	−1.10469
$82$	−5.57177	−0.615299
$83$	4.96828	0.545340	0.272670	−	0.962108i	$-0.412093\pi$
$83$	4.96828	0.545340	0.272670	+	0.962108i	$0.412093\pi$
$84$	4.62158	0.504256
$85$	−20.7389	−2.24945
$86$	1.63579	0.176391
$87$	−10.4571	−1.12112
$88$	2.44949	0.261116
$89$	−16.0689	−1.70330	−0.851650	−	0.524112i	$-0.824397\pi$
$89$	−16.0689	−1.70330	−0.851650	+	0.524112i	$0.824397\pi$
$90$	−10.7638	−1.13461
$91$	2.63883	0.276624
$92$	0	0
$93$	8.22939	0.853348
$94$	−2.29416	−0.236625
$95$	−14.4711	−1.48471
$96$	2.37562	0.242461
$97$	−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
$99$	6.47539	0.650802
$100$	11.5788	1.15788
$101$	15.5858	1.55085	0.775423	−	0.631442i	$-0.217536\pi$
$101$	15.5858	1.55085	0.775423	+	0.631442i	$0.217536\pi$
$102$	12.1000	1.19808
$103$	−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
$105$	−18.8177	−1.83642
$106$	−8.84555	−0.859156
$107$	−2.10762	−0.203751	−0.101876	−	0.994797i	$-0.532484\pi$
$107$	−2.10762	−0.203751	−0.101876	+	0.994797i	$0.532484\pi$
$108$	−0.846745	−0.0814781
$109$	−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
$111$	−4.62158	−0.438661
$112$	1.94542	0.183825
$113$	−11.6146	−1.09261	−0.546305	−	0.837586i	$-0.683966\pi$
$113$	−11.6146	−1.09261	−0.546305	+	0.837586i	$0.683966\pi$
$114$	8.44312	0.790770
$115$	0	0
$116$	−4.40183	−0.408700
$117$	3.58582	0.331509
$118$	14.3300	1.31919
$119$	9.90883	0.908341
$120$	−9.67283	−0.883004
$121$	−5.00000	−0.454545
$122$	−5.96285	−0.539851
$123$	−13.2364	−1.19349
$124$	3.46410	0.311086
$125$	−26.7869	−2.39590
$126$	5.14285	0.458162
$127$	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
$129$	3.88600	0.342144
$130$	−5.52299	−0.484398
$131$	−8.81351	−0.770040	−0.385020	−	0.922908i	$-0.625805\pi$
$131$	−8.81351	−0.770040	−0.385020	+	0.922908i	$0.625805\pi$
$132$	5.81906	0.506484
$133$	6.91416	0.599533
$134$	8.83199	0.762967
$135$	3.44770	0.296730
$136$	5.09341	0.436757
$137$	−3.87727	−0.331258	−0.165629	−	0.986188i	$-0.552965\pi$
$137$	−3.87727	−0.331258	−0.165629	+	0.986188i	$0.552965\pi$
$138$	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
$141$	−5.45005	−0.458977
$142$	−13.9877	−1.17382
$143$	3.32256	0.277847
$144$	2.64357	0.220297
$145$	17.9230	1.48842
$146$	−4.92118	−0.407280
$147$	−7.63843	−0.630007
$148$	−1.94542	−0.159913
$149$	2.79256	0.228775	0.114388	−	0.993436i	$-0.463509\pi$
$149$	2.79256	0.228775	0.114388	+	0.993436i	$0.463509\pi$
$150$	27.5068	2.24592
$151$	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
$153$	13.4648	1.08856
$154$	4.76529	0.383998
$155$	−14.1048	−1.13293
$156$	3.22236	0.257996
$157$	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
$159$	−21.0137	−1.66649
$160$	−4.07171	−0.321897
$161$	0	0
$162$	−9.94225	−0.781137
$163$	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
$165$	−23.6935	−1.84454
$166$	4.96828	0.385613
$167$	−20.6741	−1.59981	−0.799906	−	0.600126i	$-0.795117\pi$
$167$	−20.6741	−1.59981	−0.799906	+	0.600126i	$0.795117\pi$
$168$	4.62158	0.356562
$169$	−11.1601	−0.858469
$170$	−20.7389	−1.59060
$171$	9.39543	0.718486
$172$	1.63579	0.124727
$173$	1.59817	0.121506	0.0607532	−	0.998153i	$-0.480650\pi$
$173$	1.59817	0.121506	0.0607532	+	0.998153i	$0.480650\pi$
$174$	−10.4571	−0.792749
$175$	22.5256	1.70278
$176$	2.44949	0.184637
$177$	34.0427	2.55881
$178$	−16.0689	−1.20441
$179$	8.87107	0.663055	0.331528	−	0.943446i	$-0.392436\pi$
$179$	8.87107	0.663055	0.331528	+	0.943446i	$0.392436\pi$
$180$	−10.7638	−0.802289
$181$	−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
$183$	−14.1655	−1.04714
$184$	0	0
$185$	7.92118	0.582377
$186$	8.22939	0.603408
$187$	12.4763	0.912355
$188$	−2.29416	−0.167319
$189$	−1.64727	−0.119822
$190$	−14.4711	−1.04985
$191$	−3.58630	−0.259496	−0.129748	−	0.991547i	$-0.541417\pi$
$191$	−3.58630	−0.259496	−0.129748	+	0.991547i	$0.541417\pi$
$192$	2.37562	0.171446
$193$	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
$195$	−13.1205	−0.939580
$196$	−3.21534	−0.229667
$197$	−7.79946	−0.555689	−0.277844	−	0.960626i	$-0.589620\pi$
$197$	−7.79946	−0.555689	−0.277844	+	0.960626i	$0.589620\pi$
$198$	6.47539	0.460186
$199$	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
$201$	20.9814	1.47992
$202$	15.5858	1.09661
$203$	−8.56341	−0.601034
$204$	12.1000	0.847171
$205$	22.6866	1.58450
$206$	−9.82510	−0.684547
$207$	0	0
$208$	1.35643	0.0940516
$209$	8.70565	0.602183
$210$	−18.8177	−1.29855
$211$	−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
$213$	−33.2293	−2.27684
$214$	−2.10762	−0.144074
$215$	−6.66044	−0.454238
$216$	−0.846745	−0.0576137
$217$	6.73913	0.457482
$218$	−11.1328	−0.754011
$219$	−11.6909	−0.789995
$220$	−9.97360	−0.672420
$221$	6.90887	0.464741
$222$	−4.62158	−0.310180
$223$	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
$225$	30.6093	2.04062
$226$	−11.6146	−0.772592
$227$	24.5033	1.62634	0.813170	−	0.582027i	$-0.197740\pi$
$227$	24.5033	1.62634	0.813170	+	0.582027i	$0.197740\pi$
$228$	8.44312	0.559159
$229$	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
$231$	11.3205	0.744835
$232$	−4.40183	−0.288994
$233$	4.85641	0.318154	0.159077	−	0.987266i	$-0.449148\pi$
$233$	4.85641	0.318154	0.159077	+	0.987266i	$0.449148\pi$
$234$	3.58582	0.234412
$235$	9.34115	0.609350
$236$	14.3300	0.932806
$237$	16.7746	1.08963
$238$	9.90883	0.642294
$239$	10.6601	0.689543	0.344771	−	0.938687i	$-0.387956\pi$
$239$	10.6601	0.689543	0.344771	+	0.938687i	$0.387956\pi$
$240$	−9.67283	−0.624378
$241$	−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
$243$	−21.0788	−1.35220
$244$	−5.96285	−0.381733
$245$	13.0919	0.836412
$246$	−13.2364	−0.843923
$247$	4.82085	0.306743
$248$	3.46410	0.219971
$249$	11.8027	0.747969
$250$	−26.7869	−1.69415
$251$	−3.44260	−0.217295	−0.108647	−	0.994080i	$-0.534652\pi$
$251$	−3.44260	−0.217295	−0.108647	+	0.994080i	$0.534652\pi$
$252$	5.14285	0.323969
$253$	0	0
$254$	8.22939	0.516358
$255$	−49.2677	−3.08526
$256$	1.00000	0.0625000
$257$	−8.79150	−0.548399	−0.274199	−	0.961673i	$-0.588413\pi$
$257$	−8.79150	−0.548399	−0.274199	+	0.961673i	$0.588413\pi$
$258$	3.88600	0.241932
$259$	−3.78466	−0.235167
$260$	−5.52299	−0.342521
$261$	−11.6365	−0.720284
$262$	−8.81351	−0.544500
$263$	−17.0957	−1.05417	−0.527083	−	0.849814i	$-0.676714\pi$
$263$	−17.0957	−1.05417	−0.527083	+	0.849814i	$0.676714\pi$
$264$	5.81906	0.358138
$265$	36.0165	2.21248
$266$	6.91416	0.423934
$267$	−38.1736	−2.33618
$268$	8.83199	0.539499
$269$	−4.28181	−0.261067	−0.130533	−	0.991444i	$-0.541669\pi$
$269$	−4.28181	−0.261067	−0.130533	+	0.991444i	$0.541669\pi$
$270$	3.44770	0.209820
$271$	−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
$273$	6.26885	0.379408
$274$	−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
$279$	9.15759	0.548251
$280$	−7.92118	−0.473381
$281$	9.25793	0.552282	0.276141	−	0.961117i	$-0.410944\pi$
$281$	9.25793	0.552282	0.276141	+	0.961117i	$0.410944\pi$
$282$	−5.45005	−0.324546
$283$	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
$285$	−34.3779	−2.03637
$286$	3.32256	0.196467
$287$	−10.8394	−0.639832
$288$	2.64357	0.155774
$289$	8.94287	0.526051
$290$	17.9230	1.05247
$291$	−20.4107	−1.19650
$292$	−4.92118	−0.287990
$293$	15.7879	0.922342	0.461171	−	0.887311i	$-0.347429\pi$
$293$	15.7879	0.922342	0.461171	+	0.887311i	$0.347429\pi$
$294$	−7.63843	−0.445482
$295$	−58.3477	−3.39713
$296$	−1.94542	−0.113075
$297$	−2.07409	−0.120351
$298$	2.79256	0.161769
$299$	0	0
$300$	27.5068	1.58811
$301$	3.18229	0.183424
$302$	22.2030	1.27764
$303$	37.0260	2.12709
$304$	3.55407	0.203840
$305$	24.2790	1.39021
$306$	13.4648	0.769731
$307$	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
$309$	−23.3407	−1.32781
$310$	−14.1048	−0.801099
$311$	−19.9212	−1.12963	−0.564813	−	0.825219i	$-0.691052\pi$
$311$	−19.9212	−1.12963	−0.564813	+	0.825219i	$0.691052\pi$
$312$	3.22236	0.182430
$313$	−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
$315$	−20.9402	−1.17985
$316$	7.06114	0.397220
$317$	−18.3490	−1.03058	−0.515292	−	0.857014i	$-0.672317\pi$
$317$	−18.3490	−1.03058	−0.515292	+	0.857014i	$0.672317\pi$
$318$	−21.0137	−1.17839
$319$	−10.7822	−0.603690
$320$	−4.07171	−0.227615
$321$	−5.00691	−0.279458
$322$	0	0
$323$	18.1023	1.00724
$324$	−9.94225	−0.552347
$325$	15.7058	0.871203
$326$	6.33254	0.350727
$327$	−26.4474	−1.46254
$328$	−5.57177	−0.307650
$329$	−4.46311	−0.246059
$330$	−23.6935	−1.30428
$331$	−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
$333$	−5.14285	−0.281827
$334$	−20.6741	−1.13124
$335$	−35.9613	−1.96477
$336$	4.62158	0.252128
$337$	−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
$339$	−27.5919	−1.49859
$340$	−20.7389	−1.12472
$341$	8.48528	0.459504
$342$	9.39543	0.508046
$343$	−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	$-0.398398\pi$
$347$	11.6909	0.627598	0.313799	+	0.949489i	$0.398398\pi$
$348$	−10.4571	−0.560558
$349$	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
$351$	−1.14855	−0.0613051
$352$	2.44949	0.130558
$353$	−25.1190	−1.33695	−0.668476	−	0.743734i	$-0.733053\pi$
$353$	−25.1190	−1.33695	−0.668476	+	0.743734i	$0.733053\pi$
$354$	34.0427	1.80935
$355$	56.9536	3.02278
$356$	−16.0689	−0.851650
$357$	23.5396	1.24585
$358$	8.87107	0.468851
$359$	−15.9761	−0.843186	−0.421593	−	0.906785i	$-0.638529\pi$
$359$	−15.9761	−0.843186	−0.421593	+	0.906785i	$0.638529\pi$
$360$	−10.7638	−0.567304
$361$	−6.36860	−0.335189
$362$	−4.26624	−0.224228
$363$	−11.8781	−0.623438
$364$	2.63883	0.138312
$365$	20.0376	1.04882
$366$	−14.1655	−0.740441
$367$	−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
$370$	7.92118	0.411803
$371$	−17.2083	−0.893411
$372$	8.22939	0.426674
$373$	−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
$375$	−63.6355	−3.28613
$376$	−2.29416	−0.118312
$377$	−5.97078	−0.307511
$378$	−1.64727	−0.0847266
$379$	−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
$381$	19.5499	1.00157
$382$	−3.58630	−0.183491
$383$	16.2398	0.829816	0.414908	−	0.909863i	$-0.363814\pi$
$383$	16.2398	0.829816	0.414908	+	0.909863i	$0.363814\pi$
$384$	2.37562	0.121230
$385$	−19.4028	−0.988861
$386$	15.2870	0.778085
$387$	4.32431	0.219817
$388$	−8.59175	−0.436180
$389$	−3.96170	−0.200866	−0.100433	−	0.994944i	$-0.532023\pi$
$389$	−3.96170	−0.200866	−0.100433	+	0.994944i	$0.532023\pi$
$390$	−13.1205	−0.664384
$391$	0	0
$392$	−3.21534	−0.162399
$393$	−20.9375	−1.05616
$394$	−7.79946	−0.392931
$395$	−28.7509	−1.44661
$396$	6.47539	0.325401
$397$	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
$399$	16.4254	0.822299
$400$	11.5788	0.578940
$401$	3.05120	0.152370	0.0761848	−	0.997094i	$-0.475726\pi$
$401$	3.05120	0.152370	0.0761848	+	0.997094i	$0.475726\pi$
$402$	20.9814	1.04646
$403$	4.69882	0.234065
$404$	15.5858	0.775423
$405$	40.4819	2.01156
$406$	−8.56341	−0.424995
$407$	−4.76529	−0.236206
$408$	12.1000	0.599040
$409$	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
$411$	−9.21092	−0.454341
$412$	−9.82510	−0.484048
$413$	27.8779	1.37178
$414$	0	0
$415$	−20.2294	−0.993022
$416$	1.35643	0.0665045
$417$	−1.26714	−0.0620523
$418$	8.70565	0.425807
$419$	30.8981	1.50947	0.754736	−	0.656028i	$-0.227765\pi$
$419$	30.8981	1.50947	0.754736	+	0.656028i	$0.227765\pi$
$420$	−18.8177	−0.918210
$421$	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
$423$	−6.06477	−0.294879
$424$	−8.84555	−0.429578
$425$	58.9756	2.86074
$426$	−33.2293	−1.60997
$427$	−11.6003	−0.561376
$428$	−2.10762	−0.101876
$429$	7.89315	0.381085
$430$	−6.66044	−0.321195
$431$	25.1005	1.20905	0.604524	−	0.796587i	$-0.293363\pi$
$431$	25.1005	1.20905	0.604524	+	0.796587i	$0.293363\pi$
$432$	−0.846745	−0.0407390
$433$	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
$435$	42.5782	2.04147
$436$	−11.1328	−0.533166
$437$	0	0
$438$	−11.6909	−0.558610
$439$	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
$441$	−8.49998	−0.404761
$442$	6.90887	0.328621
$443$	6.51214	0.309401	0.154701	−	0.987961i	$-0.450559\pi$
$443$	6.51214	0.309401	0.154701	+	0.987961i	$0.450559\pi$
$444$	−4.62158	−0.219330
$445$	65.4278	3.10158
$446$	6.11720	0.289658
$447$	6.63406	0.313780
$448$	1.94542	0.0919125
$449$	29.7300	1.40304	0.701522	−	0.712647i	$-0.252504\pi$
$449$	29.7300	1.40304	0.701522	+	0.712647i	$0.252504\pi$
$450$	30.6093	1.44294
$451$	−13.6480	−0.642659
$452$	−11.6146	−0.546305
$453$	52.7459	2.47822
$454$	24.5033	1.15000
$455$	−10.7445	−0.503712
$456$	8.44312	0.395385
$457$	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
$459$	−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	$0.441362\pi$
$462$	11.3205	0.526678
$463$	−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
$465$	−33.5077	−1.55388
$466$	4.85641	0.224969
$467$	−29.6404	−1.37159	−0.685797	−	0.727793i	$-0.740546\pi$
$467$	−29.6404	−1.37159	−0.685797	+	0.727793i	$0.740546\pi$
$468$	3.58582	0.165755
$469$	17.1819	0.793387
$470$	9.34115	0.430875
$471$	29.6775	1.36747
$472$	14.3300	0.659593
$473$	4.00684	0.184235
$474$	16.7746	0.770482
$475$	41.1518	1.88818
$476$	9.90883	0.454171
$477$	−23.3838	−1.07067
$478$	10.6601	0.487580
$479$	38.7485	1.77046	0.885232	−	0.465150i	$-0.153999\pi$
$479$	38.7485	1.77046	0.885232	+	0.465150i	$0.153999\pi$
$480$	−9.67283	−0.441502
$481$	−2.63883	−0.120320
$482$	−13.3806	−0.609471
$483$	0	0
$484$	−5.00000	−0.227273
$485$	34.9831	1.58850
$486$	−21.0788	−0.956152
$487$	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
$489$	15.0437	0.680300
$490$	13.0919	0.591433
$491$	−16.8561	−0.760705	−0.380352	−	0.924842i	$-0.624197\pi$
$491$	−16.8561	−0.760705	−0.380352	+	0.924842i	$0.624197\pi$
$492$	−13.2364	−0.596744
$493$	−22.4204	−1.00976
$494$	4.82085	0.216900
$495$	−26.3659	−1.18506
$496$	3.46410	0.155543
$497$	−27.2119	−1.22062
$498$	11.8027	0.528894
$499$	−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
$501$	−49.1138	−2.19424
$502$	−3.44260	−0.153651
$503$	5.66325	0.252512	0.126256	−	0.991998i	$-0.459704\pi$
$503$	5.66325	0.252512	0.126256	+	0.991998i	$0.459704\pi$
$504$	5.14285	0.229081
$505$	−63.4609	−2.82397
$506$	0	0
$507$	−26.5121	−1.17745
$508$	8.22939	0.365120
$509$	10.4262	0.462131	0.231066	−	0.972938i	$-0.425779\pi$
$509$	10.4262	0.462131	0.231066	+	0.972938i	$0.425779\pi$
$510$	−49.2677	−2.18161
$511$	−9.57376	−0.423518
$512$	1.00000	0.0441942
$513$	−3.00939	−0.132868
$514$	−8.79150	−0.387776
$515$	40.0049	1.76283
$516$	3.88600	0.171072
$517$	−5.61952	−0.247146
$518$	−3.78466	−0.166288
$519$	3.79664	0.166654
$520$	−5.52299	−0.242199
$521$	−27.8011	−1.21799	−0.608995	−	0.793174i	$-0.708427\pi$
$521$	−27.8011	−1.21799	−0.608995	+	0.793174i	$0.708427\pi$
$522$	−11.6365	−0.509318
$523$	−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
$525$	53.5123	2.33547
$526$	−17.0957	−0.745408
$527$	17.6441	0.768589
$528$	5.81906	0.253242
$529$	0	0
$530$	36.0165	1.56446
$531$	37.8824	1.64396
$532$	6.91416	0.299767
$533$	−7.55773	−0.327361
$534$	−38.1736	−1.65193
$535$	8.58162	0.371016
$536$	8.83199	0.381484
$537$	21.0743	0.909423
$538$	−4.28181	−0.184602
$539$	−7.87594	−0.339241
$540$	3.44770	0.148365
$541$	−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
$543$	−10.1350	−0.434933
$544$	5.09341	0.218378
$545$	45.3297	1.94171
$546$	6.26885	0.268282
$547$	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
$549$	−15.7632	−0.672758
$550$	28.3621	1.20937
$551$	−15.6444	−0.666474
$552$	0	0
$553$	13.7369	0.584151
$554$	−30.3395	−1.28900
$555$	18.8177	0.798767
$556$	−0.533395	−0.0226210
$557$	−46.8977	−1.98712	−0.993560	−	0.113305i	$-0.963856\pi$
$557$	−46.8977	−1.98712	−0.993560	+	0.113305i	$0.963856\pi$
$558$	9.15759	0.387672
$559$	2.21883	0.0938465
$560$	−7.92118	−0.334731
$561$	29.6389	1.25135
$562$	9.25793	0.390522
$563$	−6.76143	−0.284960	−0.142480	−	0.989798i	$-0.545508\pi$
$563$	−6.76143	−0.284960	−0.142480	+	0.989798i	$0.545508\pi$
$564$	−5.45005	−0.229489
$565$	47.2913	1.98956
$566$	14.7391	0.619531
$567$	−19.3419	−0.812281
$568$	−13.9877	−0.586909
$569$	−2.06692	−0.0866497	−0.0433248	−	0.999061i	$-0.513795\pi$
$569$	−2.06692	−0.0866497	−0.0433248	+	0.999061i	$0.513795\pi$
$570$	−34.3779	−1.43993
$571$	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
$573$	−8.51969	−0.355915
$574$	−10.8394	−0.452429
$575$	0	0
$576$	2.64357	0.110149
$577$	−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
$579$	36.3160	1.50924
$580$	17.9230	0.744211
$581$	9.66540	0.400988
$582$	−20.4107	−0.846052
$583$	−21.6671	−0.897359
$584$	−4.92118	−0.203640
$585$	−14.6004	−0.603652
$586$	15.7879	0.652194
$587$	29.5524	1.21976	0.609879	−	0.792495i	$-0.291218\pi$
$587$	29.5524	1.21976	0.609879	+	0.792495i	$0.291218\pi$
$588$	−7.63843	−0.315003
$589$	12.3117	0.507293
$590$	−58.3477	−2.40214
$591$	−18.5286	−0.762163
$592$	−1.94542	−0.0799563
$593$	24.1392	0.991276	0.495638	−	0.868529i	$-0.334934\pi$
$593$	24.1392	0.991276	0.495638	+	0.868529i	$0.334934\pi$
$594$	−2.07409	−0.0851011
$595$	−40.3459	−1.65402
$596$	2.79256	0.114388
$597$	57.3460	2.34701
$598$	0	0
$599$	−12.0404	−0.491959	−0.245980	−	0.969275i	$-0.579110\pi$
$599$	−12.0404	−0.491959	−0.245980	+	0.969275i	$0.579110\pi$
$600$	27.5068	1.12296
$601$	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
$603$	23.3480	0.950803
$604$	22.2030	0.903427
$605$	20.3585	0.827692
$606$	37.0260	1.50408
$607$	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
$609$	−20.3434	−0.824357
$610$	24.2790	0.983028
$611$	−3.11187	−0.125893
$612$	13.4648	0.544282
$613$	−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
$615$	53.8948	2.17325
$616$	4.76529	0.191999
$617$	20.8936	0.841146	0.420573	−	0.907259i	$-0.361829\pi$
$617$	20.8936	0.841146	0.420573	+	0.907259i	$0.361829\pi$
$618$	−23.3407	−0.938900
$619$	−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
$624$	3.22236	0.128998
$625$	51.1745	2.04698
$626$	−9.67273	−0.386600
$627$	20.6813	0.825933
$628$	12.4925	0.498506
$629$	−9.90883	−0.395091
$630$	−20.9402	−0.834277
$631$	−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
$633$	−0.239099	−0.00950333
$634$	−18.3490	−0.728733
$635$	−33.5077	−1.32971
$636$	−21.0137	−0.833246
$637$	−4.36139	−0.172805
$638$	−10.7822	−0.426873
$639$	−36.9773	−1.46280
$640$	−4.07171	−0.160948
$641$	−33.0344	−1.30478	−0.652389	−	0.757884i	$-0.726233\pi$
$641$	−33.0344	−1.30478	−0.652389	+	0.757884i	$0.726233\pi$
$642$	−5.00691	−0.197607
$643$	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
$646$	18.1023	0.712227
$647$	−0.0891034	−0.00350302	−0.00175151	−	0.999998i	$-0.500558\pi$
$647$	−0.0891034	−0.00350302	−0.00175151	+	0.999998i	$0.500558\pi$
$648$	−9.94225	−0.390568
$649$	35.1013	1.37785
$650$	15.7058	0.616034
$651$	16.0096	0.627466
$652$	6.33254	0.248001
$653$	−9.03587	−0.353601	−0.176801	−	0.984247i	$-0.556575\pi$
$653$	−9.03587	−0.353601	−0.176801	+	0.984247i	$0.556575\pi$
$654$	−26.4474	−1.03418
$655$	35.8860	1.40218
$656$	−5.57177	−0.217541
$657$	−13.0095	−0.507548
$658$	−4.46311	−0.173990
$659$	7.78944	0.303433	0.151717	−	0.988424i	$-0.451520\pi$
$659$	7.78944	0.303433	0.151717	+	0.988424i	$0.451520\pi$
$660$	−23.6935	−0.922268
$661$	−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
$663$	16.4128	0.637422
$664$	4.96828	0.192807
$665$	−28.1524	−1.09170
$666$	−5.14285	−0.199281
$667$	0	0
$668$	−20.6741	−0.799906
$669$	14.5321	0.561845
$670$	−35.9613	−1.38930
$671$	−14.6059	−0.563856
$672$	4.62158	0.178281
$673$	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
$675$	−9.80428	−0.377367
$676$	−11.1601	−0.429234
$677$	45.3102	1.74141	0.870707	−	0.491802i	$-0.163662\pi$
$677$	45.3102	1.74141	0.870707	+	0.491802i	$0.163662\pi$
$678$	−27.5919	−1.05966
$679$	−16.7146	−0.641446
$680$	−20.7389	−0.795300
$681$	58.2105	2.23063
$682$	8.48528	0.324918
$683$	−51.8395	−1.98358	−0.991791	−	0.127866i	$-0.959187\pi$
$683$	−51.8395	−1.98358	−0.991791	+	0.127866i	$0.959187\pi$
$684$	9.39543	0.359243
$685$	15.7871	0.603195
$686$	−19.8731	−0.758760
$687$	33.1931	1.26640
$688$	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
$693$	12.5974	0.478534
$694$	11.6909	0.443779
$695$	2.17183	0.0823821
$696$	−10.4571	−0.396375
$697$	−28.3793	−1.07494
$698$	11.3012	0.427756
$699$	11.5370	0.436368
$700$	22.5256	0.851388
$701$	12.3084	0.464881	0.232440	−	0.972611i	$-0.425329\pi$
$701$	12.3084	0.464881	0.232440	+	0.972611i	$0.425329\pi$
$702$	−1.14855	−0.0433493
$703$	−6.91416	−0.260772
$704$	2.44949	0.0923186
$705$	22.1910	0.835762
$706$	−25.1190	−0.945367
$707$	30.3210	1.14034
$708$	34.0427	1.27940
$709$	−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
$711$	18.6666	0.700052
$712$	−16.0689	−0.602207
$713$	0	0
$714$	23.5396	0.880948
$715$	−13.5285	−0.505937
$716$	8.87107	0.331528
$717$	25.3243	0.945752
$718$	−15.9761	−0.596222
$719$	0.616416	0.0229885	0.0114942	−	0.999934i	$-0.496341\pi$
$719$	0.616416	0.0229885	0.0114942	+	0.999934i	$0.496341\pi$
$720$	−10.7638	−0.401145
$721$	−19.1139	−0.711840
$722$	−6.36860	−0.237015
$723$	−31.7873	−1.18218
$724$	−4.26624	−0.158553
$725$	−50.9679	−1.89290
$726$	−11.8781	−0.440838
$727$	−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
$729$	−20.2484	−0.749940
$730$	20.0376	0.741625
$731$	8.33173	0.308160
$732$	−14.1655	−0.523571
$733$	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
$735$	31.1014	1.14719
$736$	0	0
$737$	21.6339	0.796893
$738$	−14.7294	−0.542195
$739$	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
$741$	11.4525	0.420718
$742$	−17.2083	−0.631737
$743$	−25.9127	−0.950643	−0.475321	−	0.879812i	$-0.657668\pi$
$743$	−25.9127	−0.950643	−0.475321	+	0.879812i	$0.657668\pi$
$744$	8.22939	0.301704
$745$	−11.3705	−0.416582
$746$	−2.17582	−0.0796624
$747$	13.1340	0.480548
$748$	12.4763	0.456177
$749$	−4.10021	−0.149818
$750$	−63.6355	−2.32364
$751$	−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
$753$	−8.17831	−0.298034
$754$	−5.97078	−0.217443
$755$	−90.4041	−3.29014
$756$	−1.64727	−0.0599108
$757$	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	$-0.369235\pi$
$761$	22.0332	0.798702	0.399351	+	0.916798i	$0.369235\pi$
$762$	19.5499	0.708218
$763$	−21.6581	−0.784074
$764$	−3.58630	−0.129748
$765$	−54.8247	−1.98219
$766$	16.2398	0.586769
$767$	19.4377	0.701855
$768$	2.37562	0.0857228
$769$	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
$771$	−20.8853	−0.752164
$772$	15.2870	0.550190
$773$	10.2548	0.368838	0.184419	−	0.982848i	$-0.440960\pi$
$773$	10.2548	0.368838	0.184419	+	0.982848i	$0.440960\pi$
$774$	4.32431	0.155434
$775$	40.1101	1.44080
$776$	−8.59175	−0.308426
$777$	−8.99091	−0.322547
$778$	−3.96170	−0.142034
$779$	−19.8025	−0.709497
$780$	−13.1205	−0.469790
$781$	−34.2626	−1.22601
$782$	0	0
$783$	3.72723	0.133200
$784$	−3.21534	−0.114834
$785$	−50.8659	−1.81548
$786$	−20.9375	−0.746818
$787$	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
$789$	−40.6129	−1.44586
$790$	−28.7509	−1.02291
$791$	−22.5953	−0.803396
$792$	6.47539	0.230093
$793$	−8.08820	−0.287220
$794$	9.53339	0.338328
$795$	85.5615	3.03455
$796$	24.1394	0.855597
$797$	46.1175	1.63356	0.816782	−	0.576946i	$-0.195756\pi$
$797$	46.1175	1.63356	0.816782	+	0.576946i	$0.195756\pi$
$798$	16.4254	0.581453
$799$	−11.6851	−0.413390
$800$	11.5788	0.409372
$801$	−42.4792	−1.50093
$802$	3.05120	0.107742
$803$	−12.0544	−0.425390
$804$	20.9814	0.739958
$805$	0	0
$806$	4.69882	0.165509
$807$	−10.1720	−0.358070
$808$	15.5858	0.548307
$809$	33.9809	1.19470	0.597352	−	0.801979i	$-0.296220\pi$
$809$	33.9809	1.19470	0.597352	+	0.801979i	$0.296220\pi$
$810$	40.4819	1.42239
$811$	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
$813$	−32.3885	−1.13592
$814$	−4.76529	−0.167023
$815$	−25.7842	−0.903182
$816$	12.1000	0.423585
$817$	5.81369	0.203395
$818$	39.7028	1.38818
$819$	6.97592	0.243759
$820$	22.6866	0.792251
$821$	2.44191	0.0852231	0.0426116	−	0.999092i	$-0.486432\pi$
$821$	2.44191	0.0852231	0.0426116	+	0.999092i	$0.486432\pi$
$822$	−9.21092	−0.321268
$823$	−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
$825$	67.3777	2.34579
$826$	27.8779	0.969997
$827$	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	$0.384951\pi$
$830$	−20.2294	−0.702172
$831$	−72.0751	−2.50026
$832$	1.35643	0.0470258
$833$	−16.3771	−0.567432
$834$	−1.26714	−0.0438776
$835$	84.1789	2.91313
$836$	8.70565	0.301091
$837$	−2.93321	−0.101387
$838$	30.8981	1.06736
$839$	−7.05733	−0.243646	−0.121823	−	0.992552i	$-0.538874\pi$
$839$	−7.05733	−0.243646	−0.121823	+	0.992552i	$0.538874\pi$
$840$	−18.8177	−0.649273
$841$	−9.62388	−0.331858
$842$	2.14868	0.0740485
$843$	21.9933	0.757490
$844$	−0.100647	−0.00346441
$845$	45.4406	1.56321
$846$	−6.06477	−0.208511
$847$	−9.72710	−0.334227
$848$	−8.84555	−0.303758
$849$	35.0145	1.20169
$850$	58.9756	2.02285
$851$	0	0
$852$	−33.2293	−1.13842
$853$	−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
$855$	−38.2554	−1.30831
$856$	−2.10762	−0.0720370
$857$	6.43977	0.219978	0.109989	−	0.993933i	$-0.464918\pi$
$857$	6.43977	0.219978	0.109989	+	0.993933i	$0.464918\pi$
$858$	7.89315	0.269468
$859$	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
$861$	−25.7504	−0.877571
$862$	25.1005	0.854927
$863$	34.5566	1.17632	0.588159	−	0.808745i	$-0.299853\pi$
$863$	34.5566	1.17632	0.588159	+	0.808745i	$0.299853\pi$
$864$	−0.846745	−0.0288068
$865$	−6.50727	−0.221254
$866$	5.95292	0.202288
$867$	21.2449	0.721513
$868$	6.73913	0.228741
$869$	17.2962	0.586733
$870$	42.5782	1.44353
$871$	11.9800	0.405926
$872$	−11.1328	−0.377006
$873$	−22.7129	−0.768715
$874$	0	0
$875$	−52.1118	−1.76170
$876$	−11.6909	−0.394997
$877$	−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
$879$	37.5062	1.26505
$880$	−9.97360	−0.336210
$881$	8.08174	0.272281	0.136140	−	0.990690i	$-0.456530\pi$
$881$	8.08174	0.272281	0.136140	+	0.990690i	$0.456530\pi$
$882$	−8.49998	−0.286209
$883$	−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
$885$	−138.612	−4.65939
$886$	6.51214	0.218780
$887$	−37.8321	−1.27028	−0.635138	−	0.772398i	$-0.719057\pi$
$887$	−37.8321	−1.27028	−0.635138	+	0.772398i	$0.719057\pi$
$888$	−4.62158	−0.155090
$889$	16.0096	0.536945
$890$	65.4278	2.19315
$891$	−24.3534	−0.815871
$892$	6.11720	0.204819
$893$	−8.15361	−0.272850
$894$	6.63406	0.221876
$895$	−36.1204	−1.20737
$896$	1.94542	0.0649919
$897$	0	0
$898$	29.7300	0.992102
$899$	−15.2484	−0.508562
$900$	30.6093	1.02031
$901$	−45.0541	−1.50097
$902$	−13.6480	−0.454429
$903$	7.55991	0.251578
$904$	−11.6146	−0.386296
$905$	17.3709	0.577427
$906$	52.7459	1.75236
$907$	−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
$909$	41.2022	1.36659
$910$	−10.7445	−0.356178
$911$	37.5172	1.24300	0.621501	−	0.783414i	$-0.286523\pi$
$911$	37.5172	1.24300	0.621501	+	0.783414i	$0.286523\pi$
$912$	8.44312	0.279579
$913$	12.1698	0.402760
$914$	4.37251	0.144630
$915$	57.6776	1.90676
$916$	13.9724	0.461661
$917$	−17.1460	−0.566210
$918$	−4.31282	−0.142344
$919$	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
$922$	7.86593	0.259051
$923$	−18.9733	−0.624513
$924$	11.3205	0.372417
$925$	−22.5256	−0.740638
$926$	−21.9472	−0.721230
$927$	−25.9733	−0.853076
$928$	−4.40183	−0.144497
$929$	−43.8177	−1.43761	−0.718805	−	0.695211i	$-0.755311\pi$
$929$	−43.8177	−1.43761	−0.718805	+	0.695211i	$0.755311\pi$
$930$	−33.5077	−1.09876
$931$	−11.4275	−0.374523
$932$	4.85641	0.159077
$933$	−47.3251	−1.54936
$934$	−29.6404	−0.969864
$935$	−50.7997	−1.66133
$936$	3.58582	0.117206
$937$	−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
$939$	−22.9787	−0.749883
$940$	9.34115	0.304675
$941$	35.8492	1.16865	0.584326	−	0.811519i	$-0.301359\pi$
$941$	35.8492	1.16865	0.584326	+	0.811519i	$0.301359\pi$
$942$	29.6775	0.966945
$943$	0	0
$944$	14.3300	0.466403
$945$	6.70722	0.218186
$946$	4.00684	0.130274
$947$	0.975923	0.0317132	0.0158566	−	0.999874i	$-0.494952\pi$
$947$	0.975923	0.0317132	0.0158566	+	0.999874i	$0.494952\pi$
$948$	16.7746	0.544813
$949$	−6.67524	−0.216688
$950$	41.1518	1.33514
$951$	−43.5903	−1.41351
$952$	9.90883	0.321147
$953$	16.6476	0.539267	0.269634	−	0.962963i	$-0.413097\pi$
$953$	16.6476	0.539267	0.269634	+	0.962963i	$0.413097\pi$
$954$	−23.3838	−0.757079
$955$	14.6024	0.472522
$956$	10.6601	0.344771
$957$	−25.6145	−0.827999
$958$	38.7485	1.25191
$959$	−7.54292	−0.243574
$960$	−9.67283	−0.312189
$961$	−19.0000	−0.612903
$962$	−2.63883	−0.0850792
$963$	−5.57164	−0.179544
$964$	−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
$969$	43.0043	1.38150
$970$	34.9831	1.12324
$971$	−51.8696	−1.66457	−0.832287	−	0.554344i	$-0.812969\pi$
$971$	−51.8696	−1.66457	−0.832287	+	0.554344i	$0.812969\pi$
$972$	−21.0788	−0.676102
$973$	−1.03768	−0.0332664
$974$	15.6777	0.502347
$975$	37.3111	1.19491
$976$	−5.96285	−0.190866
$977$	18.1728	0.581399	0.290699	−	0.956814i	$-0.406112\pi$
$977$	18.1728	0.581399	0.290699	+	0.956814i	$0.406112\pi$
$978$	15.0437	0.481045
$979$	−39.3606	−1.25797
$980$	13.0919	0.418206
$981$	−29.4304	−0.939641
$982$	−16.8561	−0.537899
$983$	23.6079	0.752975	0.376488	−	0.926422i	$-0.377132\pi$
$983$	23.6079	0.752975	0.376488	+	0.926422i	$0.377132\pi$
$984$	−13.2364	−0.421961
$985$	31.7571	1.01187
$986$	−22.4204	−0.714010
$987$	−10.6026	−0.337486
$988$	4.82085	0.153372
$989$	0	0
$990$	−26.3659	−0.837964
$991$	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
$993$	−60.6694	−1.92529
$994$	−27.2119	−0.863108
$995$	−98.2884	−3.11595
$996$	11.8027	0.373984
$997$	−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
$999$	1.64727	0.0521175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1058.2.a.n.1.5	✓	8
3.2	odd	2		9522.2.a.ce.1.8		8
4.3	odd	2		8464.2.a.cb.1.3		8
23.22	odd	2	inner	1058.2.a.n.1.6	yes	8
69.68	even	2		9522.2.a.ce.1.1		8
92.91	even	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	1.1	even	1	trivial
1058.2.a.n.1.6	yes	8	23.22	odd	2	inner
8464.2.a.cb.1.3		8	4.3	odd	2
8464.2.a.cb.1.4		8	92.91	even	2
9522.2.a.ce.1.1		8	69.68	even	2
9522.2.a.ce.1.8		8	3.2	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:	\( N \)	\(=\)	\( 1058 = 2 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1058.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.37562 q^{3} +1.00000 q^{4} -4.07171 q^{5} +2.37562 q^{6} +1.94542 q^{7} +1.00000 q^{8} +2.64357 q^{9} -4.07171 q^{10} +2.44949 q^{11} +2.37562 q^{12} +1.35643 q^{13} +1.94542 q^{14} -9.67283 q^{15} +1.00000 q^{16} +5.09341 q^{17} +2.64357 q^{18} +3.55407 q^{19} -4.07171 q^{20} +4.62158 q^{21} +2.44949 q^{22} +2.37562 q^{24} +11.5788 q^{25} +1.35643 q^{26} -0.846745 q^{27} +1.94542 q^{28} -4.40183 q^{29} -9.67283 q^{30} +3.46410 q^{31} +1.00000 q^{32} +5.81906 q^{33} +5.09341 q^{34} -7.92118 q^{35} +2.64357 q^{36} -1.94542 q^{37} +3.55407 q^{38} +3.22236 q^{39} -4.07171 q^{40} -5.57177 q^{41} +4.62158 q^{42} +1.63579 q^{43} +2.44949 q^{44} -10.7638 q^{45} -2.29416 q^{47} +2.37562 q^{48} -3.21534 q^{49} +11.5788 q^{50} +12.1000 q^{51} +1.35643 q^{52} -8.84555 q^{53} -0.846745 q^{54} -9.97360 q^{55} +1.94542 q^{56} +8.44312 q^{57} -4.40183 q^{58} +14.3300 q^{59} -9.67283 q^{60} -5.96285 q^{61} +3.46410 q^{62} +5.14285 q^{63} +1.00000 q^{64} -5.52299 q^{65} +5.81906 q^{66} +8.83199 q^{67} +5.09341 q^{68} -7.92118 q^{70} -13.9877 q^{71} +2.64357 q^{72} -4.92118 q^{73} -1.94542 q^{74} +27.5068 q^{75} +3.55407 q^{76} +4.76529 q^{77} +3.22236 q^{78} +7.06114 q^{79} -4.07171 q^{80} -9.94225 q^{81} -5.57177 q^{82} +4.96828 q^{83} +4.62158 q^{84} -20.7389 q^{85} +1.63579 q^{86} -10.4571 q^{87} +2.44949 q^{88} -16.0689 q^{89} -10.7638 q^{90} +2.63883 q^{91} +8.22939 q^{93} -2.29416 q^{94} -14.4711 q^{95} +2.37562 q^{96} -8.59175 q^{97} -3.21534 q^{98} +6.47539 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 4 q^{6} + 8 q^{8} + 20 q^{9} + 4 q^{12} + 12 q^{13} + 8 q^{16} + 20 q^{18} + 4 q^{24} + 32 q^{25} + 12 q^{26} + 40 q^{27} + 8 q^{32} - 12 q^{35} + 20 q^{36} - 36 q^{39}+ \cdots + 32 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 4 * q^6 + 8 * q^8 + 20 * q^9 + 4 * q^12 + 12 * q^13 + 8 * q^16 + 20 * q^18 + 4 * q^24 + 32 * q^25 + 12 * q^26 + 40 * q^27 + 8 * q^32 - 12 * q^35 + 20 * q^36 - 36 * q^39 + 12 * q^41 - 12 * q^47 + 4 * q^48 + 32 * q^49 + 32 * q^50 + 12 * q^52 + 40 * q^54 + 12 * q^55 + 24 * q^59 + 8 * q^64 - 12 * q^70 - 12 * q^71 + 20 * q^72 + 12 * q^73 - 8 * q^75 - 36 * q^78 - 16 * q^81 + 12 * q^82 - 36 * q^85 - 60 * q^87 - 12 * q^94 - 84 * q^95 + 4 * q^96 + 32 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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