LMFDB - Embedded newform 2254.4.a.r.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2254,4,Mod(1,2254)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2254, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2254.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,-22,0,44,-23] 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:	$132.990305153$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 234 x^{9} - 105 x^{8} + 18997 x^{7} + 16513 x^{6} - 621598 x^{5} - 743169 x^{4} + \cdots - 12103441 $ x^11 - 234x^9 - 105x^8 + 18997x^7 + 16513x^6 - 621598x^5 - 743169x^4 + 7229916x^3 + 7324254x^2 - 26794327*x - 12103441
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$7.30492$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -7.30492 q^{3} +4.00000 q^{4} -7.55338 q^{5} +14.6098 q^{6} -8.00000 q^{8} +26.3619 q^{9} +15.1068 q^{10} +56.3122 q^{11} -29.2197 q^{12} -65.8379 q^{13} +55.1768 q^{15} +16.0000 q^{16} +108.835 q^{17} -52.7238 q^{18} -92.3659 q^{19} -30.2135 q^{20} -112.624 q^{22} +23.0000 q^{23} +58.4394 q^{24} -67.9465 q^{25} +131.676 q^{26} +4.66115 q^{27} +58.1564 q^{29} -110.354 q^{30} -234.881 q^{31} -32.0000 q^{32} -411.356 q^{33} -217.669 q^{34} +105.448 q^{36} -122.911 q^{37} +184.732 q^{38} +480.941 q^{39} +60.4270 q^{40} -359.331 q^{41} -33.0320 q^{43} +225.249 q^{44} -199.121 q^{45} -46.0000 q^{46} +147.849 q^{47} -116.879 q^{48} +135.893 q^{50} -795.028 q^{51} -263.352 q^{52} +640.721 q^{53} -9.32230 q^{54} -425.347 q^{55} +674.726 q^{57} -116.313 q^{58} +402.029 q^{59} +220.707 q^{60} +240.394 q^{61} +469.762 q^{62} +64.0000 q^{64} +497.298 q^{65} +822.713 q^{66} +112.885 q^{67} +435.338 q^{68} -168.013 q^{69} +171.363 q^{71} -210.895 q^{72} -361.719 q^{73} +245.823 q^{74} +496.344 q^{75} -369.464 q^{76} -961.881 q^{78} -842.316 q^{79} -120.854 q^{80} -745.821 q^{81} +718.661 q^{82} -1374.29 q^{83} -822.069 q^{85} +66.0640 q^{86} -424.828 q^{87} -450.498 q^{88} +1241.11 q^{89} +398.243 q^{90} +92.0000 q^{92} +1715.79 q^{93} -295.699 q^{94} +697.674 q^{95} +233.758 q^{96} +997.907 q^{97} +1484.50 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 22 q^{2} + 44 q^{4} - 23 q^{5} - 88 q^{8} + 171 q^{9} + 46 q^{10} - 48 q^{11} - 77 q^{13} + 104 q^{15} + 176 q^{16} - 97 q^{17} - 342 q^{18} - 138 q^{19} - 92 q^{20} + 96 q^{22} + 253 q^{23} + 30 q^{25}+ \cdots - 2545 q^{99}+O(q^{100}) $ 11 * q - 22 * q^2 + 44 * q^4 - 23 * q^5 - 88 * q^8 + 171 * q^9 + 46 * q^10 - 48 * q^11 - 77 * q^13 + 104 * q^15 + 176 * q^16 - 97 * q^17 - 342 * q^18 - 138 * q^19 - 92 * q^20 + 96 * q^22 + 253 * q^23 + 30 * q^25 + 154 * q^26 - 315 * q^27 + 137 * q^29 - 208 * q^30 - 234 * q^31 - 352 * q^32 - 28 * q^33 + 194 * q^34 + 684 * q^36 + 87 * q^37 + 276 * q^38 + 129 * q^39 + 184 * q^40 - 612 * q^41 + 187 * q^43 - 192 * q^44 - 597 * q^45 - 506 * q^46 - 793 * q^47 - 60 * q^50 - 1257 * q^51 - 308 * q^52 + 575 * q^53 + 630 * q^54 + 549 * q^55 + 39 * q^57 - 274 * q^58 - 1241 * q^59 + 416 * q^60 - 187 * q^61 + 468 * q^62 + 704 * q^64 - 937 * q^65 + 56 * q^66 + 2275 * q^67 - 388 * q^68 + 115 * q^71 - 1368 * q^72 - 2385 * q^73 - 174 * q^74 - 196 * q^75 - 552 * q^76 - 258 * q^78 - 1225 * q^79 - 368 * q^80 + 3635 * q^81 + 1224 * q^82 + 597 * q^83 - 2493 * q^85 - 374 * q^86 - 1428 * q^87 + 384 * q^88 - 602 * q^89 + 1194 * q^90 + 1012 * q^92 - 1360 * q^93 + 1586 * q^94 + 4835 * q^95 + 1039 * q^97 - 2545 * q^99

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−7.30492	−1.40583	−0.702917	−	0.711272i	$-0.748119\pi$
$3$	−7.30492	−1.40583	−0.702917	+	0.711272i	$0.748119\pi$
$4$	4.00000	0.500000
$5$	−7.55338	−0.675595	−0.337797	−	0.941219i	$-0.609682\pi$
$5$	−7.55338	−0.675595	−0.337797	+	0.941219i	$0.609682\pi$
$6$	14.6098	0.994074
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	26.3619	0.976367
$10$	15.1068	0.477717
$11$	56.3122	1.54352	0.771762	−	0.635911i	$-0.219376\pi$
$11$	56.3122	1.54352	0.771762	+	0.635911i	$0.219376\pi$
$12$	−29.2197	−0.702917
$13$	−65.8379	−1.40463	−0.702313	−	0.711868i	$-0.747849\pi$
$13$	−65.8379	−1.40463	−0.702313	+	0.711868i	$0.747849\pi$
$14$	0	0
$15$	55.1768	0.949773
$16$	16.0000	0.250000
$17$	108.835	1.55272	0.776361	−	0.630289i	$-0.217064\pi$
$17$	108.835	1.55272	0.776361	+	0.630289i	$0.217064\pi$
$18$	−52.7238	−0.690396
$19$	−92.3659	−1.11527	−0.557637	−	0.830085i	$-0.688292\pi$
$19$	−92.3659	−1.11527	−0.557637	+	0.830085i	$0.688292\pi$
$20$	−30.2135	−0.337797
$21$	0	0
$22$	−112.624	−1.09144
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	58.4394	0.497037
$25$	−67.9465	−0.543572
$26$	131.676	0.993221
$27$	4.66115	0.0332236
$28$	0	0
$29$	58.1564	0.372392	0.186196	−	0.982513i	$-0.440384\pi$
$29$	58.1564	0.372392	0.186196	+	0.982513i	$0.440384\pi$
$30$	−110.354	−0.671591
$31$	−234.881	−1.36083	−0.680417	−	0.732825i	$-0.738201\pi$
$31$	−234.881	−1.36083	−0.680417	+	0.732825i	$0.738201\pi$
$32$	−32.0000	−0.176777
$33$	−411.356	−2.16994
$34$	−217.669	−1.09794
$35$	0	0
$36$	105.448	0.488184
$37$	−122.911	−0.546122	−0.273061	−	0.961997i	$-0.588036\pi$
$37$	−122.911	−0.546122	−0.273061	+	0.961997i	$0.588036\pi$
$38$	184.732	0.788617
$39$	480.941	1.97467
$40$	60.4270	0.238859
$41$	−359.331	−1.36873	−0.684366	−	0.729139i	$-0.739921\pi$
$41$	−359.331	−1.36873	−0.684366	+	0.729139i	$0.739921\pi$
$42$	0	0
$43$	−33.0320	−0.117147	−0.0585736	−	0.998283i	$-0.518655\pi$
$43$	−33.0320	−0.117147	−0.0585736	+	0.998283i	$0.518655\pi$
$44$	225.249	0.771762
$45$	−199.121	−0.659628
$46$	−46.0000	−0.147442
$47$	147.849	0.458852	0.229426	−	0.973326i	$-0.426315\pi$
$47$	147.849	0.458852	0.229426	+	0.973326i	$0.426315\pi$
$48$	−116.879	−0.351458
$49$	0	0
$50$	135.893	0.384363
$51$	−795.028	−2.18287
$52$	−263.352	−0.702313
$53$	640.721	1.66056	0.830281	−	0.557345i	$-0.188180\pi$
$53$	640.721	1.66056	0.830281	+	0.557345i	$0.188180\pi$
$54$	−9.32230	−0.0234927
$55$	−425.347	−1.04280
$56$	0	0
$57$	674.726	1.56789
$58$	−116.313	−0.263321
$59$	402.029	0.887115	0.443557	−	0.896246i	$-0.353716\pi$
$59$	402.029	0.887115	0.443557	+	0.896246i	$0.353716\pi$
$60$	220.707	0.474887
$61$	240.394	0.504580	0.252290	−	0.967652i	$-0.418816\pi$
$61$	240.394	0.504580	0.252290	+	0.967652i	$0.418816\pi$
$62$	469.762	0.962255
$63$	0	0
$64$	64.0000	0.125000
$65$	497.298	0.948958
$66$	822.713	1.53438
$67$	112.885	0.205838	0.102919	−	0.994690i	$-0.467182\pi$
$67$	112.885	0.205838	0.102919	+	0.994690i	$0.467182\pi$
$68$	435.338	0.776361
$69$	−168.013	−0.293137
$70$	0	0
$71$	171.363	0.286437	0.143219	−	0.989691i	$-0.454255\pi$
$71$	171.363	0.286437	0.143219	+	0.989691i	$0.454255\pi$
$72$	−210.895	−0.345198
$73$	−361.719	−0.579946	−0.289973	−	0.957035i	$-0.593646\pi$
$73$	−361.719	−0.579946	−0.289973	+	0.957035i	$0.593646\pi$
$74$	245.823	0.386166
$75$	496.344	0.764172
$76$	−369.464	−0.557637
$77$	0	0
$78$	−961.881	−1.39630
$79$	−842.316	−1.19959	−0.599797	−	0.800152i	$-0.704752\pi$
$79$	−842.316	−1.19959	−0.599797	+	0.800152i	$0.704752\pi$
$80$	−120.854	−0.168899
$81$	−745.821	−1.02307
$82$	718.661	0.967840
$83$	−1374.29	−1.81744	−0.908721	−	0.417404i	$-0.862940\pi$
$83$	−1374.29	−1.81744	−0.908721	+	0.417404i	$0.862940\pi$
$84$	0	0
$85$	−822.069	−1.04901
$86$	66.0640	0.0828356
$87$	−424.828	−0.523521
$88$	−450.498	−0.545718
$89$	1241.11	1.47818	0.739088	−	0.673608i	$-0.235257\pi$
$89$	1241.11	1.47818	0.739088	+	0.673608i	$0.235257\pi$
$90$	398.243	0.466428
$91$	0	0
$92$	92.0000	0.104257
$93$	1715.79	1.91311
$94$	−295.699	−0.324457
$95$	697.674	0.753472
$96$	233.758	0.248519
$97$	997.907	1.04456	0.522279	−	0.852775i	$-0.325082\pi$
$97$	997.907	1.04456	0.522279	+	0.852775i	$0.325082\pi$
$98$	0	0
$99$	1484.50	1.50705
$100$	−271.786	−0.271786
$101$	946.648	0.932624	0.466312	−	0.884620i	$-0.345582\pi$
$101$	946.648	0.932624	0.466312	+	0.884620i	$0.345582\pi$
$102$	1590.06	1.54352
$103$	422.213	0.403902	0.201951	−	0.979396i	$-0.435272\pi$
$103$	422.213	0.403902	0.201951	+	0.979396i	$0.435272\pi$
$104$	526.703	0.496610
$105$	0	0
$106$	−1281.44	−1.17419
$107$	588.846	0.532018	0.266009	−	0.963971i	$-0.414295\pi$
$107$	588.846	0.532018	0.266009	+	0.963971i	$0.414295\pi$
$108$	18.6446	0.0166118
$109$	106.854	0.0938969	0.0469484	−	0.998897i	$-0.485050\pi$
$109$	106.854	0.0938969	0.0469484	+	0.998897i	$0.485050\pi$
$110$	850.694	0.737368
$111$	897.858	0.767756
$112$	0	0
$113$	21.1869	0.0176380	0.00881901	−	0.999961i	$-0.497193\pi$
$113$	21.1869	0.0176380	0.00881901	+	0.999961i	$0.497193\pi$
$114$	−1349.45	−1.10866
$115$	−173.728	−0.140871
$116$	232.625	0.186196
$117$	−1735.61	−1.37143
$118$	−804.059	−0.627285
$119$	0	0
$120$	−441.415	−0.335796
$121$	1840.06	1.38247
$122$	−480.789	−0.356792
$123$	2624.88	1.92421
$124$	−939.524	−0.680417
$125$	1457.40	1.04283
$126$	0	0
$127$	2173.95	1.51895	0.759475	−	0.650536i	$-0.225456\pi$
$127$	2173.95	1.51895	0.759475	+	0.650536i	$0.225456\pi$
$128$	−128.000	−0.0883883
$129$	241.296	0.164689
$130$	−994.597	−0.671015
$131$	−2455.77	−1.63788	−0.818939	−	0.573881i	$-0.805437\pi$
$131$	−2455.77	−1.63788	−0.818939	+	0.573881i	$0.805437\pi$
$132$	−1645.43	−1.08497
$133$	0	0
$134$	−225.770	−0.145549
$135$	−35.2074	−0.0224457
$136$	−870.677	−0.548970
$137$	1021.29	0.636896	0.318448	−	0.947940i	$-0.396838\pi$
$137$	1021.29	0.636896	0.318448	+	0.947940i	$0.396838\pi$
$138$	336.027	0.207279
$139$	418.904	0.255619	0.127809	−	0.991799i	$-0.459205\pi$
$139$	418.904	0.255619	0.127809	+	0.991799i	$0.459205\pi$
$140$	0	0
$141$	−1080.03	−0.645069
$142$	−342.726	−0.202542
$143$	−3707.48	−2.16807
$144$	421.791	0.244092
$145$	−439.277	−0.251586
$146$	723.438	0.410083
$147$	0	0
$148$	−491.645	−0.273061
$149$	984.319	0.541199	0.270599	−	0.962692i	$-0.412778\pi$
$149$	984.319	0.541199	0.270599	+	0.962692i	$0.412778\pi$
$150$	−992.688	−0.540351
$151$	2004.99	1.08056	0.540278	−	0.841486i	$-0.318319\pi$
$151$	2004.99	1.08056	0.540278	+	0.841486i	$0.318319\pi$
$152$	738.927	0.394309
$153$	2869.09	1.51603
$154$	0	0
$155$	1774.14	0.919372
$156$	1923.76	0.987335
$157$	−385.868	−0.196150	−0.0980752	−	0.995179i	$-0.531269\pi$
$157$	−385.868	−0.196150	−0.0980752	+	0.995179i	$0.531269\pi$
$158$	1684.63	0.848242
$159$	−4680.42	−2.33447
$160$	241.708	0.119429
$161$	0	0
$162$	1491.64	0.723423
$163$	−2397.38	−1.15201	−0.576003	−	0.817448i	$-0.695388\pi$
$163$	−2397.38	−1.15201	−0.576003	+	0.817448i	$0.695388\pi$
$164$	−1437.32	−0.684366
$165$	3107.13	1.46600
$166$	2748.58	1.28513
$167$	3438.93	1.59349	0.796743	−	0.604318i	$-0.206554\pi$
$167$	3438.93	1.59349	0.796743	+	0.604318i	$0.206554\pi$
$168$	0	0
$169$	2137.63	0.972975
$170$	1644.14	0.741762
$171$	−2434.94	−1.08892
$172$	−132.128	−0.0585736
$173$	2008.99	0.882894	0.441447	−	0.897287i	$-0.354465\pi$
$173$	2008.99	0.882894	0.441447	+	0.897287i	$0.354465\pi$
$174$	849.656	0.370185
$175$	0	0
$176$	900.995	0.385881
$177$	−2936.79	−1.24714
$178$	−2482.23	−1.04523
$179$	4012.23	1.67535	0.837677	−	0.546166i	$-0.183913\pi$
$179$	4012.23	1.67535	0.837677	+	0.546166i	$0.183913\pi$
$180$	−796.486	−0.329814
$181$	2231.44	0.916361	0.458181	−	0.888859i	$-0.348501\pi$
$181$	2231.44	0.916361	0.458181	+	0.888859i	$0.348501\pi$
$182$	0	0
$183$	−1756.06	−0.709355
$184$	−184.000	−0.0737210
$185$	928.395	0.368957
$186$	−3431.57	−1.35277
$187$	6128.71	2.39666
$188$	591.397	0.229426
$189$	0	0
$190$	−1395.35	−0.532785
$191$	130.667	0.0495011	0.0247505	−	0.999694i	$-0.492121\pi$
$191$	130.667	0.0495011	0.0247505	+	0.999694i	$0.492121\pi$
$192$	−467.515	−0.175729
$193$	−3443.32	−1.28423	−0.642113	−	0.766610i	$-0.721942\pi$
$193$	−3443.32	−1.28423	−0.642113	+	0.766610i	$0.721942\pi$
$194$	−1995.81	−0.738614
$195$	−3632.73	−1.33408
$196$	0	0
$197$	−2976.33	−1.07642	−0.538211	−	0.842810i	$-0.680900\pi$
$197$	−2976.33	−1.07642	−0.538211	+	0.842810i	$0.680900\pi$
$198$	−2968.99	−1.06564
$199$	−902.959	−0.321653	−0.160827	−	0.986983i	$-0.551416\pi$
$199$	−902.959	−0.321653	−0.160827	+	0.986983i	$0.551416\pi$
$200$	543.572	0.192182
$201$	−824.618	−0.289374
$202$	−1893.30	−0.659465
$203$	0	0
$204$	−3180.11	−1.09143
$205$	2714.16	0.924708
$206$	−844.426	−0.285602
$207$	606.324	0.203587
$208$	−1053.41	−0.351157
$209$	−5201.33	−1.72145
$210$	0	0
$211$	−2439.95	−0.796081	−0.398041	−	0.917368i	$-0.630310\pi$
$211$	−2439.95	−0.796081	−0.398041	+	0.917368i	$0.630310\pi$
$212$	2562.88	0.830281
$213$	−1251.79	−0.402683
$214$	−1177.69	−0.376194
$215$	249.503	0.0791440
$216$	−37.2892	−0.0117463
$217$	0	0
$218$	−213.708	−0.0663951
$219$	2642.33	0.815307
$220$	−1701.39	−0.521398
$221$	−7165.44	−2.18099
$222$	−1795.72	−0.542885
$223$	−5414.35	−1.62588	−0.812941	−	0.582346i	$-0.802135\pi$
$223$	−5414.35	−1.62588	−0.812941	+	0.582346i	$0.802135\pi$
$224$	0	0
$225$	−1791.20	−0.530726
$226$	−42.3738	−0.0124720
$227$	4479.54	1.30977	0.654884	−	0.755729i	$-0.272717\pi$
$227$	4479.54	1.30977	0.654884	+	0.755729i	$0.272717\pi$
$228$	2698.90	0.783944
$229$	−3627.89	−1.04689	−0.523445	−	0.852059i	$-0.675354\pi$
$229$	−3627.89	−1.04689	−0.523445	+	0.852059i	$0.675354\pi$
$230$	347.455	0.0996110
$231$	0	0
$232$	−465.251	−0.131660
$233$	−3682.73	−1.03547	−0.517734	−	0.855542i	$-0.673224\pi$
$233$	−3682.73	−1.03547	−0.517734	+	0.855542i	$0.673224\pi$
$234$	3471.23	0.969748
$235$	−1116.76	−0.309998
$236$	1608.12	0.443557
$237$	6153.06	1.68643
$238$	0	0
$239$	1289.59	0.349025	0.174512	−	0.984655i	$-0.444165\pi$
$239$	1289.59	0.349025	0.174512	+	0.984655i	$0.444165\pi$
$240$	882.829	0.237443
$241$	−664.153	−0.177518	−0.0887590	−	0.996053i	$-0.528290\pi$
$241$	−664.153	−0.177518	−0.0887590	+	0.996053i	$0.528290\pi$
$242$	−3680.13	−0.977552
$243$	5322.32	1.40505
$244$	961.578	0.252290
$245$	0	0
$246$	−5249.77	−1.36062
$247$	6081.17	1.56654
$248$	1879.05	0.481128
$249$	10039.1	2.55502
$250$	−2914.80	−0.737391
$251$	4771.08	1.19979	0.599896	−	0.800078i	$-0.295209\pi$
$251$	4771.08	1.19979	0.599896	+	0.800078i	$0.295209\pi$
$252$	0	0
$253$	1295.18	0.321847
$254$	−4347.90	−1.07406
$255$	6005.15	1.47473
$256$	256.000	0.0625000
$257$	−2134.29	−0.518028	−0.259014	−	0.965874i	$-0.583398\pi$
$257$	−2134.29	−0.518028	−0.259014	+	0.965874i	$0.583398\pi$
$258$	−482.592	−0.116453
$259$	0	0
$260$	1989.19	0.474479
$261$	1533.11	0.363591
$262$	4911.55	1.15815
$263$	6810.24	1.59672	0.798360	−	0.602180i	$-0.205701\pi$
$263$	6810.24	1.59672	0.798360	+	0.602180i	$0.205701\pi$
$264$	3290.85	0.767189
$265$	−4839.61	−1.12187
$266$	0	0
$267$	−9066.24	−2.07807
$268$	451.541	0.102919
$269$	−6693.13	−1.51705	−0.758526	−	0.651643i	$-0.774080\pi$
$269$	−6693.13	−1.51705	−0.758526	+	0.651643i	$0.774080\pi$
$270$	70.4148	0.0158715
$271$	−8641.37	−1.93699	−0.968497	−	0.249024i	$-0.919890\pi$
$271$	−8641.37	−1.93699	−0.968497	+	0.249024i	$0.919890\pi$
$272$	1741.35	0.388180
$273$	0	0
$274$	−2042.58	−0.450353
$275$	−3826.22	−0.839017
$276$	−672.053	−0.146568
$277$	−7493.58	−1.62544	−0.812718	−	0.582658i	$-0.802013\pi$
$277$	−7493.58	−1.62544	−0.812718	+	0.582658i	$0.802013\pi$
$278$	−837.808	−0.180750
$279$	−6191.91	−1.32867
$280$	0	0
$281$	1976.73	0.419651	0.209825	−	0.977739i	$-0.432710\pi$
$281$	1976.73	0.419651	0.209825	+	0.977739i	$0.432710\pi$
$282$	2160.06	0.456133
$283$	6154.61	1.29277	0.646384	−	0.763012i	$-0.276280\pi$
$283$	6154.61	1.29277	0.646384	+	0.763012i	$0.276280\pi$
$284$	685.452	0.143219
$285$	−5096.46	−1.05926
$286$	7414.95	1.53306
$287$	0	0
$288$	−843.581	−0.172599
$289$	6931.97	1.41094
$290$	878.554	0.177898
$291$	−7289.63	−1.46847
$292$	−1446.88	−0.289973
$293$	−1905.58	−0.379950	−0.189975	−	0.981789i	$-0.560841\pi$
$293$	−1905.58	−0.379950	−0.189975	+	0.981789i	$0.560841\pi$
$294$	0	0
$295$	−3036.68	−0.599330
$296$	983.291	0.193083
$297$	262.479	0.0512815
$298$	−1968.64	−0.382685
$299$	−1514.27	−0.292885
$300$	1985.38	0.382086
$301$	0	0
$302$	−4009.99	−0.764069
$303$	−6915.20	−1.31111
$304$	−1477.85	−0.278818
$305$	−1815.79	−0.340891
$306$	−5738.18	−1.07199
$307$	643.667	0.119661	0.0598307	−	0.998209i	$-0.480944\pi$
$307$	643.667	0.119661	0.0598307	+	0.998209i	$0.480944\pi$
$308$	0	0
$309$	−3084.23	−0.567818
$310$	−3548.29	−0.650094
$311$	−5131.05	−0.935547	−0.467773	−	0.883848i	$-0.654944\pi$
$311$	−5131.05	−0.935547	−0.467773	+	0.883848i	$0.654944\pi$
$312$	−3847.53	−0.698151
$313$	4845.93	0.875106	0.437553	−	0.899193i	$-0.355845\pi$
$313$	4845.93	0.875106	0.437553	+	0.899193i	$0.355845\pi$
$314$	771.736	0.138699
$315$	0	0
$316$	−3369.27	−0.599797
$317$	2149.07	0.380769	0.190384	−	0.981710i	$-0.439027\pi$
$317$	2149.07	0.380769	0.190384	+	0.981710i	$0.439027\pi$
$318$	9360.84	1.65072
$319$	3274.91	0.574796
$320$	−483.416	−0.0844493
$321$	−4301.48	−0.747929
$322$	0	0
$323$	−10052.6	−1.73171
$324$	−2983.28	−0.511537
$325$	4473.45	0.763516
$326$	4794.75	0.814591
$327$	−780.561	−0.132003
$328$	2874.64	0.483920
$329$	0	0
$330$	−6214.26	−1.03662
$331$	733.344	0.121777	0.0608886	−	0.998145i	$-0.480607\pi$
$331$	733.344	0.121777	0.0608886	+	0.998145i	$0.480607\pi$
$332$	−5497.15	−0.908721
$333$	−3240.18	−0.533215
$334$	−6877.86	−1.12677
$335$	−852.665	−0.139063
$336$	0	0
$337$	−1922.41	−0.310743	−0.155372	−	0.987856i	$-0.549657\pi$
$337$	−1922.41	−0.310743	−0.155372	+	0.987856i	$0.549657\pi$
$338$	−4275.25	−0.687997
$339$	−154.769	−0.0247961
$340$	−3288.27	−0.524505
$341$	−13226.7	−2.10048
$342$	4869.88	0.769980
$343$	0	0
$344$	264.256	0.0414178
$345$	1269.07	0.198041
$346$	−4017.98	−0.624300
$347$	11309.5	1.74964	0.874821	−	0.484446i	$-0.160979\pi$
$347$	11309.5	1.74964	0.874821	+	0.484446i	$0.160979\pi$
$348$	−1699.31	−0.261760
$349$	5202.37	0.797927	0.398963	−	0.916967i	$-0.369370\pi$
$349$	5202.37	0.797927	0.398963	+	0.916967i	$0.369370\pi$
$350$	0	0
$351$	−306.880	−0.0466668
$352$	−1801.99	−0.272859
$353$	6302.68	0.950306	0.475153	−	0.879903i	$-0.342393\pi$
$353$	6302.68	0.950306	0.475153	+	0.879903i	$0.342393\pi$
$354$	5873.59	0.881858
$355$	−1294.37	−0.193515
$356$	4964.45	0.739088
$357$	0	0
$358$	−8024.47	−1.18465
$359$	−12797.9	−1.88147	−0.940733	−	0.339148i	$-0.889861\pi$
$359$	−12797.9	−1.88147	−0.940733	+	0.339148i	$0.889861\pi$
$360$	1592.97	0.233214
$361$	1672.46	0.243834
$362$	−4462.87	−0.647965
$363$	−13441.5	−1.94352
$364$	0	0
$365$	2732.20	0.391808
$366$	3512.13	0.501590
$367$	−7794.73	−1.10867	−0.554335	−	0.832294i	$-0.687027\pi$
$367$	−7794.73	−1.10867	−0.554335	+	0.832294i	$0.687027\pi$
$368$	368.000	0.0521286
$369$	−9472.64	−1.33639
$370$	−1856.79	−0.260892
$371$	0	0
$372$	6863.15	0.956553
$373$	9352.02	1.29820	0.649101	−	0.760703i	$-0.275145\pi$
$373$	9352.02	1.29820	0.649101	+	0.760703i	$0.275145\pi$
$374$	−12257.4	−1.69470
$375$	−10646.2	−1.46604
$376$	−1182.79	−0.162229
$377$	−3828.89	−0.523072
$378$	0	0
$379$	−671.508	−0.0910107	−0.0455054	−	0.998964i	$-0.514490\pi$
$379$	−671.508	−0.0910107	−0.0455054	+	0.998964i	$0.514490\pi$
$380$	2790.70	0.376736
$381$	−15880.5	−2.13539
$382$	−261.333	−0.0350026
$383$	5923.13	0.790230	0.395115	−	0.918632i	$-0.370705\pi$
$383$	5923.13	0.790230	0.395115	+	0.918632i	$0.370705\pi$
$384$	935.030	0.124259
$385$	0	0
$386$	6886.64	0.908085
$387$	−870.787	−0.114379
$388$	3991.63	0.522279
$389$	−3452.45	−0.449991	−0.224995	−	0.974360i	$-0.572237\pi$
$389$	−3452.45	−0.449991	−0.224995	+	0.974360i	$0.572237\pi$
$390$	7265.45	0.943335
$391$	2503.20	0.323765
$392$	0	0
$393$	17939.2	2.30258
$394$	5952.67	0.761145
$395$	6362.33	0.810440
$396$	5937.99	0.753523
$397$	−2462.71	−0.311334	−0.155667	−	0.987810i	$-0.549753\pi$
$397$	−2462.71	−0.311334	−0.155667	+	0.987810i	$0.549753\pi$
$398$	1805.92	0.227443
$399$	0	0
$400$	−1087.14	−0.135893
$401$	−12628.0	−1.57260	−0.786301	−	0.617843i	$-0.788007\pi$
$401$	−12628.0	−1.57260	−0.786301	+	0.617843i	$0.788007\pi$
$402$	1649.24	0.204618
$403$	15464.1	1.91146
$404$	3786.59	0.466312
$405$	5633.47	0.691183
$406$	0	0
$407$	−6921.41	−0.842952
$408$	6360.23	0.771760
$409$	9857.74	1.19177	0.595885	−	0.803070i	$-0.296801\pi$
$409$	9857.74	1.19177	0.595885	+	0.803070i	$0.296801\pi$
$410$	−5428.32	−0.653867
$411$	−7460.45	−0.895370
$412$	1688.85	0.201951
$413$	0	0
$414$	−1212.65	−0.143958
$415$	10380.5	1.22785
$416$	2106.81	0.248305
$417$	−3060.06	−0.359357
$418$	10402.7	1.21725
$419$	−15594.0	−1.81818	−0.909090	−	0.416599i	$-0.863222\pi$
$419$	−15594.0	−1.81818	−0.909090	+	0.416599i	$0.863222\pi$
$420$	0	0
$421$	−2429.98	−0.281306	−0.140653	−	0.990059i	$-0.544920\pi$
$421$	−2429.98	−0.281306	−0.140653	+	0.990059i	$0.544920\pi$
$422$	4879.90	0.562914
$423$	3897.59	0.448008
$424$	−5125.77	−0.587097
$425$	−7394.93	−0.844016
$426$	2503.59	0.284740
$427$	0	0
$428$	2355.39	0.266009
$429$	27082.8	3.04795
$430$	−499.006	−0.0559633
$431$	−4540.29	−0.507420	−0.253710	−	0.967280i	$-0.581651\pi$
$431$	−4540.29	−0.507420	−0.253710	+	0.967280i	$0.581651\pi$
$432$	74.5784	0.00830591
$433$	−15322.6	−1.70060	−0.850299	−	0.526300i	$-0.823579\pi$
$433$	−15322.6	−1.70060	−0.850299	+	0.526300i	$0.823579\pi$
$434$	0	0
$435$	3208.88	0.353688
$436$	427.416	0.0469484
$437$	−2124.42	−0.232551
$438$	−5284.66	−0.576509
$439$	−5141.79	−0.559008	−0.279504	−	0.960145i	$-0.590170\pi$
$439$	−5141.79	−0.559008	−0.279504	+	0.960145i	$0.590170\pi$
$440$	3402.78	0.368684
$441$	0	0
$442$	14330.9	1.54220
$443$	−7538.33	−0.808480	−0.404240	−	0.914653i	$-0.632464\pi$
$443$	−7538.33	−0.808480	−0.404240	+	0.914653i	$0.632464\pi$
$444$	3591.43	0.383878
$445$	−9374.60	−0.998648
$446$	10828.7	1.14967
$447$	−7190.38	−0.760835
$448$	0	0
$449$	−7253.14	−0.762354	−0.381177	−	0.924502i	$-0.624481\pi$
$449$	−7253.14	−0.762354	−0.381177	+	0.924502i	$0.624481\pi$
$450$	3582.40	0.375280
$451$	−20234.7	−2.11267
$452$	84.7476	0.00881901
$453$	−14646.3	−1.51908
$454$	−8959.08	−0.926146
$455$	0	0
$456$	−5397.81	−0.554332
$457$	−14113.4	−1.44464	−0.722319	−	0.691561i	$-0.756924\pi$
$457$	−14113.4	−1.44464	−0.722319	+	0.691561i	$0.756924\pi$
$458$	7255.79	0.740264
$459$	507.294	0.0515871
$460$	−694.911	−0.0704356
$461$	1996.06	0.201662	0.100831	−	0.994904i	$-0.467850\pi$
$461$	1996.06	0.201662	0.100831	+	0.994904i	$0.467850\pi$
$462$	0	0
$463$	8301.53	0.833271	0.416636	−	0.909074i	$-0.363209\pi$
$463$	8301.53	0.833271	0.416636	+	0.909074i	$0.363209\pi$
$464$	930.502	0.0930980
$465$	−12960.0	−1.29248
$466$	7365.47	0.732186
$467$	−4517.84	−0.447668	−0.223834	−	0.974627i	$-0.571857\pi$
$467$	−4517.84	−0.447668	−0.223834	+	0.974627i	$0.571857\pi$
$468$	−6942.45	−0.685716
$469$	0	0
$470$	2233.52	0.219202
$471$	2818.74	0.275755
$472$	−3216.24	−0.313642
$473$	−1860.10	−0.180820
$474$	−12306.1	−1.19249
$475$	6275.94	0.606231
$476$	0	0
$477$	16890.6	1.62132
$478$	−2579.19	−0.246798
$479$	−16515.6	−1.57540	−0.787702	−	0.616056i	$-0.788729\pi$
$479$	−16515.6	−1.57540	−0.787702	+	0.616056i	$0.788729\pi$
$480$	−1765.66	−0.167898
$481$	8092.22	0.767097
$482$	1328.31	0.125524
$483$	0	0
$484$	7360.25	0.691233
$485$	−7537.57	−0.705697
$486$	−10644.6	−0.993519
$487$	9053.08	0.842370	0.421185	−	0.906975i	$-0.361614\pi$
$487$	9053.08	0.842370	0.421185	+	0.906975i	$0.361614\pi$
$488$	−1923.16	−0.178396
$489$	17512.6	1.61953
$490$	0	0
$491$	−7398.88	−0.680055	−0.340027	−	0.940416i	$-0.610436\pi$
$491$	−7398.88	−0.680055	−0.340027	+	0.940416i	$0.610436\pi$
$492$	10499.5	0.962105
$493$	6329.42	0.578221
$494$	−12162.3	−1.10771
$495$	−11213.0	−1.01815
$496$	−3758.09	−0.340209
$497$	0	0
$498$	−20078.1	−1.80667
$499$	−717.821	−0.0643970	−0.0321985	−	0.999481i	$-0.510251\pi$
$499$	−717.821	−0.0643970	−0.0321985	+	0.999481i	$0.510251\pi$
$500$	5829.59	0.521414
$501$	−25121.1	−2.24018
$502$	−9542.15	−0.848381
$503$	−1927.68	−0.170877	−0.0854385	−	0.996343i	$-0.527229\pi$
$503$	−1927.68	−0.170877	−0.0854385	+	0.996343i	$0.527229\pi$
$504$	0	0
$505$	−7150.39	−0.630076
$506$	−2590.36	−0.227580
$507$	−15615.2	−1.36784
$508$	8695.79	0.759475
$509$	−7804.87	−0.679656	−0.339828	−	0.940488i	$-0.610369\pi$
$509$	−7804.87	−0.679656	−0.339828	+	0.940488i	$0.610369\pi$
$510$	−12010.3	−1.04279
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−430.531	−0.0370534
$514$	4268.58	0.366301
$515$	−3189.13	−0.272874
$516$	965.185	0.0823447
$517$	8325.72	0.708249
$518$	0	0
$519$	−14675.5	−1.24120
$520$	−3978.39	−0.335507
$521$	−21406.8	−1.80010	−0.900049	−	0.435789i	$-0.856469\pi$
$521$	−21406.8	−1.80010	−0.900049	+	0.435789i	$0.856469\pi$
$522$	−3066.23	−0.257098
$523$	11327.8	0.947098	0.473549	−	0.880768i	$-0.342973\pi$
$523$	11327.8	0.947098	0.473549	+	0.880768i	$0.342973\pi$
$524$	−9823.10	−0.818939
$525$	0	0
$526$	−13620.5	−1.12905
$527$	−25563.2	−2.11300
$528$	−6581.70	−0.542484
$529$	529.000	0.0434783
$530$	9679.21	0.793279
$531$	10598.3	0.866150
$532$	0	0
$533$	23657.6	1.92256
$534$	18132.5	1.46942
$535$	−4447.78	−0.359428
$536$	−903.082	−0.0727746
$537$	−29309.1	−2.35527
$538$	13386.3	1.07272
$539$	0	0
$540$	−140.830	−0.0112229
$541$	−8508.86	−0.676201	−0.338100	−	0.941110i	$-0.609784\pi$
$541$	−8508.86	−0.676201	−0.338100	+	0.941110i	$0.609784\pi$
$542$	17282.7	1.36966
$543$	−16300.5	−1.28825
$544$	−3482.71	−0.274485
$545$	−807.109	−0.0634362
$546$	0	0
$547$	7097.86	0.554813	0.277406	−	0.960753i	$-0.410525\pi$
$547$	7097.86	0.554813	0.277406	+	0.960753i	$0.410525\pi$
$548$	4085.16	0.318448
$549$	6337.26	0.492655
$550$	7652.43	0.593274
$551$	−5371.66	−0.415319
$552$	1344.11	0.103639
$553$	0	0
$554$	14987.2	1.14936
$555$	−6781.86	−0.518692
$556$	1675.62	0.127809
$557$	269.957	0.0205358	0.0102679	−	0.999947i	$-0.496732\pi$
$557$	269.957	0.0205358	0.0102679	+	0.999947i	$0.496732\pi$
$558$	12383.8	0.939514
$559$	2174.76	0.164548
$560$	0	0
$561$	−44769.8	−3.36931
$562$	−3953.46	−0.296738
$563$	−22749.5	−1.70298	−0.851491	−	0.524370i	$-0.824301\pi$
$563$	−22749.5	−1.70298	−0.851491	+	0.524370i	$0.824301\pi$
$564$	−4320.11	−0.322535
$565$	−160.033	−0.0119161
$566$	−12309.2	−0.914126
$567$	0	0
$568$	−1370.90	−0.101271
$569$	−93.7196	−0.00690498	−0.00345249	−	0.999994i	$-0.501099\pi$
$569$	−93.7196	−0.00690498	−0.00345249	+	0.999994i	$0.501099\pi$
$570$	10192.9	0.749007
$571$	3511.92	0.257389	0.128695	−	0.991684i	$-0.458921\pi$
$571$	3511.92	0.257389	0.128695	+	0.991684i	$0.458921\pi$
$572$	−14829.9	−1.08404
$573$	−954.510	−0.0695903
$574$	0	0
$575$	−1562.77	−0.113343
$576$	1687.16	0.122046
$577$	−3643.87	−0.262905	−0.131453	−	0.991322i	$-0.541964\pi$
$577$	−3643.87	−0.262905	−0.131453	+	0.991322i	$0.541964\pi$
$578$	−13863.9	−0.997688
$579$	25153.2	1.80541
$580$	−1757.11	−0.125793
$581$	0	0
$582$	14579.3	1.03837
$583$	36080.4	2.56312
$584$	2893.75	0.205042
$585$	13109.7	0.926531
$586$	3811.16	0.268665
$587$	12611.0	0.886730	0.443365	−	0.896341i	$-0.353784\pi$
$587$	12611.0	0.886730	0.443365	+	0.896341i	$0.353784\pi$
$588$	0	0
$589$	21695.0	1.51770
$590$	6073.36	0.423790
$591$	21741.9	1.51327
$592$	−1966.58	−0.136530
$593$	−516.853	−0.0357919	−0.0178959	−	0.999840i	$-0.505697\pi$
$593$	−516.853	−0.0357919	−0.0178959	+	0.999840i	$0.505697\pi$
$594$	−524.959	−0.0362615
$595$	0	0
$596$	3937.28	0.270599
$597$	6596.04	0.452191
$598$	3028.54	0.207101
$599$	−6239.98	−0.425640	−0.212820	−	0.977091i	$-0.568265\pi$
$599$	−6239.98	−0.425640	−0.212820	+	0.977091i	$0.568265\pi$
$600$	−3970.75	−0.270176
$601$	−22085.1	−1.49895	−0.749477	−	0.662031i	$-0.769695\pi$
$601$	−22085.1	−1.49895	−0.749477	+	0.662031i	$0.769695\pi$
$602$	0	0
$603$	2975.87	0.200973
$604$	8019.97	0.540278
$605$	−13898.7	−0.933987
$606$	13830.4	0.927098
$607$	23388.3	1.56393	0.781963	−	0.623325i	$-0.214218\pi$
$607$	23388.3	1.56393	0.781963	+	0.623325i	$0.214218\pi$
$608$	2955.71	0.197154
$609$	0	0
$610$	3631.58	0.241046
$611$	−9734.09	−0.644516
$612$	11476.4	0.758013
$613$	23195.2	1.52830	0.764149	−	0.645039i	$-0.223159\pi$
$613$	23195.2	1.52830	0.764149	+	0.645039i	$0.223159\pi$
$614$	−1287.33	−0.0846134
$615$	−19826.7	−1.29999
$616$	0	0
$617$	−4370.58	−0.285175	−0.142588	−	0.989782i	$-0.545542\pi$
$617$	−4370.58	−0.285175	−0.142588	+	0.989782i	$0.545542\pi$
$618$	6168.47	0.401508
$619$	5754.61	0.373663	0.186831	−	0.982392i	$-0.440178\pi$
$619$	5754.61	0.373663	0.186831	+	0.982392i	$0.440178\pi$
$620$	7096.58	0.459686
$621$	107.206	0.00692761
$622$	10262.1	0.661531
$623$	0	0
$624$	7695.05	0.493668
$625$	−2514.96	−0.160957
$626$	−9691.86	−0.618793
$627$	37995.3	2.42007
$628$	−1543.47	−0.0980752
$629$	−13377.0	−0.847975
$630$	0	0
$631$	−10487.6	−0.661658	−0.330829	−	0.943691i	$-0.607328\pi$
$631$	−10487.6	−0.661658	−0.330829	+	0.943691i	$0.607328\pi$
$632$	6738.53	0.424121
$633$	17823.7	1.11916
$634$	−4298.14	−0.269244
$635$	−16420.7	−1.02619
$636$	−18721.7	−1.16724
$637$	0	0
$638$	−6549.82	−0.406442
$639$	4517.46	0.279668
$640$	966.832	0.0597147
$641$	22643.5	1.39526	0.697631	−	0.716457i	$-0.254237\pi$
$641$	22643.5	1.39526	0.697631	+	0.716457i	$0.254237\pi$
$642$	8602.96	0.528865
$643$	−8247.48	−0.505830	−0.252915	−	0.967488i	$-0.581389\pi$
$643$	−8247.48	−0.505830	−0.252915	+	0.967488i	$0.581389\pi$
$644$	0	0
$645$	−1822.60	−0.111263
$646$	20105.2	1.22450
$647$	3358.18	0.204055	0.102027	−	0.994782i	$-0.467467\pi$
$647$	3358.18	0.204055	0.102027	+	0.994782i	$0.467467\pi$
$648$	5966.57	0.361711
$649$	22639.2	1.36928
$650$	−8946.91	−0.539887
$651$	0	0
$652$	−9589.50	−0.576003
$653$	−11243.5	−0.673802	−0.336901	−	0.941540i	$-0.609379\pi$
$653$	−11243.5	−0.673802	−0.336901	+	0.941540i	$0.609379\pi$
$654$	1561.12	0.0933405
$655$	18549.4	1.10654
$656$	−5749.29	−0.342183
$657$	−9535.61	−0.566240
$658$	0	0
$659$	28375.7	1.67733	0.838664	−	0.544650i	$-0.183337\pi$
$659$	28375.7	1.67733	0.838664	+	0.544650i	$0.183337\pi$
$660$	12428.5	0.732999
$661$	−26521.0	−1.56058	−0.780292	−	0.625415i	$-0.784930\pi$
$661$	−26521.0	−1.56058	−0.780292	+	0.625415i	$0.784930\pi$
$662$	−1466.69	−0.0861094
$663$	52343.0	3.06611
$664$	10994.3	0.642563
$665$	0	0
$666$	6480.36	0.377040
$667$	1337.60	0.0776491
$668$	13755.7	0.796743
$669$	39551.4	2.28572
$670$	1705.33	0.0983323
$671$	13537.1	0.778831
$672$	0	0
$673$	−11574.0	−0.662921	−0.331460	−	0.943469i	$-0.607541\pi$
$673$	−11574.0	−0.662921	−0.331460	+	0.943469i	$0.607541\pi$
$674$	3844.82	0.219729
$675$	−316.709	−0.0180594
$676$	8550.51	0.486488
$677$	19964.6	1.13338	0.566692	−	0.823930i	$-0.308223\pi$
$677$	19964.6	1.13338	0.566692	+	0.823930i	$0.308223\pi$
$678$	309.537	0.0175335
$679$	0	0
$680$	6576.55	0.370881
$681$	−32722.7	−1.84132
$682$	26453.3	1.48526
$683$	30955.7	1.73424	0.867120	−	0.498099i	$-0.165968\pi$
$683$	30955.7	1.73424	0.867120	+	0.498099i	$0.165968\pi$
$684$	−9739.77	−0.544458
$685$	−7714.19	−0.430283
$686$	0	0
$687$	26501.5	1.47175
$688$	−528.512	−0.0292868
$689$	−42183.7	−2.33247
$690$	−2538.13	−0.140036
$691$	−10356.0	−0.570133	−0.285066	−	0.958508i	$-0.592016\pi$
$691$	−10356.0	−0.570133	−0.285066	+	0.958508i	$0.592016\pi$
$692$	8035.96	0.441447
$693$	0	0
$694$	−22619.0	−1.23718
$695$	−3164.14	−0.172694
$696$	3398.62	0.185093
$697$	−39107.6	−2.12526
$698$	−10404.7	−0.564220
$699$	26902.1	1.45569
$700$	0	0
$701$	−736.621	−0.0396887	−0.0198444	−	0.999803i	$-0.506317\pi$
$701$	−736.621	−0.0396887	−0.0198444	+	0.999803i	$0.506317\pi$
$702$	613.760	0.0329984
$703$	11352.8	0.609075
$704$	3603.98	0.192941
$705$	8157.86	0.435805
$706$	−12605.4	−0.671968
$707$	0	0
$708$	−11747.2	−0.623568
$709$	−32277.6	−1.70975	−0.854873	−	0.518838i	$-0.826365\pi$
$709$	−32277.6	−1.70975	−0.854873	+	0.518838i	$0.826365\pi$
$710$	2588.74	0.136836
$711$	−22205.1	−1.17125
$712$	−9928.91	−0.522614
$713$	−5402.26	−0.283754
$714$	0	0
$715$	28004.0	1.46474
$716$	16048.9	0.837677
$717$	−9420.39	−0.490671
$718$	25595.8	1.33040
$719$	34790.7	1.80455	0.902277	−	0.431156i	$-0.141894\pi$
$719$	34790.7	1.80455	0.902277	+	0.431156i	$0.141894\pi$
$720$	−3185.94	−0.164907
$721$	0	0
$722$	−3344.92	−0.172417
$723$	4851.59	0.249561
$724$	8925.75	0.458181
$725$	−3951.52	−0.202422
$726$	26883.0	1.37427
$727$	21693.3	1.10668	0.553341	−	0.832955i	$-0.313353\pi$
$727$	21693.3	1.10668	0.553341	+	0.832955i	$0.313353\pi$
$728$	0	0
$729$	−18741.9	−0.952189
$730$	−5464.40	−0.277050
$731$	−3595.02	−0.181897
$732$	−7024.25	−0.354677
$733$	18264.6	0.920352	0.460176	−	0.887828i	$-0.347786\pi$
$733$	18264.6	0.920352	0.460176	+	0.887828i	$0.347786\pi$
$734$	15589.5	0.783948
$735$	0	0
$736$	−736.000	−0.0368605
$737$	6356.82	0.317715
$738$	18945.3	0.944967
$739$	10487.5	0.522040	0.261020	−	0.965333i	$-0.415941\pi$
$739$	10487.5	0.522040	0.261020	+	0.965333i	$0.415941\pi$
$740$	3713.58	0.184478
$741$	−44422.5	−2.20230
$742$	0	0
$743$	6519.14	0.321889	0.160945	−	0.986963i	$-0.448546\pi$
$743$	6519.14	0.321889	0.160945	+	0.986963i	$0.448546\pi$
$744$	−13726.3	−0.676385
$745$	−7434.93	−0.365631
$746$	−18704.0	−0.917967
$747$	−36228.9	−1.77449
$748$	24514.9	1.19833
$749$	0	0
$750$	21292.4	1.03665
$751$	−24410.9	−1.18611	−0.593053	−	0.805164i	$-0.702077\pi$
$751$	−24410.9	−1.18611	−0.593053	+	0.805164i	$0.702077\pi$
$752$	2365.59	0.114713
$753$	−34852.4	−1.68671
$754$	7657.78	0.369867
$755$	−15144.5	−0.730018
$756$	0	0
$757$	−36222.9	−1.73916	−0.869580	−	0.493793i	$-0.835610\pi$
$757$	−36222.9	−1.73916	−0.869580	+	0.493793i	$0.835610\pi$
$758$	1343.02	0.0643543
$759$	−9461.19	−0.452463
$760$	−5581.39	−0.266393
$761$	10363.5	0.493660	0.246830	−	0.969059i	$-0.420611\pi$
$761$	10363.5	0.493660	0.246830	+	0.969059i	$0.420611\pi$
$762$	31761.1	1.50995
$763$	0	0
$764$	522.667	0.0247505
$765$	−21671.3	−1.02422
$766$	−11846.3	−0.558777
$767$	−26468.8	−1.24607
$768$	−1870.06	−0.0878646
$769$	−3915.55	−0.183613	−0.0918065	−	0.995777i	$-0.529264\pi$
$769$	−3915.55	−0.183613	−0.0918065	+	0.995777i	$0.529264\pi$
$770$	0	0
$771$	15590.8	0.728262
$772$	−13773.3	−0.642113
$773$	28820.3	1.34100	0.670500	−	0.741910i	$-0.266080\pi$
$773$	28820.3	1.34100	0.670500	+	0.741910i	$0.266080\pi$
$774$	1741.57	0.0808780
$775$	15959.3	0.739711
$776$	−7983.25	−0.369307
$777$	0	0
$778$	6904.91	0.318191
$779$	33189.9	1.52651
$780$	−14530.9	−0.667038
$781$	9649.82	0.442123
$782$	−5006.39	−0.228936
$783$	271.075	0.0123722
$784$	0	0
$785$	2914.61	0.132518
$786$	−35878.5	−1.62817
$787$	23146.5	1.04839	0.524196	−	0.851598i	$-0.324366\pi$
$787$	23146.5	1.04839	0.524196	+	0.851598i	$0.324366\pi$
$788$	−11905.3	−0.538211
$789$	−49748.3	−2.24472
$790$	−12724.7	−0.573067
$791$	0	0
$792$	−11876.0	−0.532821
$793$	−15827.1	−0.708746
$794$	4925.42	0.220147
$795$	35353.0	1.57716
$796$	−3611.83	−0.160827
$797$	8804.30	0.391298	0.195649	−	0.980674i	$-0.437319\pi$
$797$	8804.30	0.391298	0.195649	+	0.980674i	$0.437319\pi$
$798$	0	0
$799$	16091.1	0.712469
$800$	2174.29	0.0960909
$801$	32718.1	1.44324
$802$	25256.1	1.11200
$803$	−20369.2	−0.895160
$804$	−3298.47	−0.144687
$805$	0	0
$806$	−30928.1	−1.35161
$807$	48892.8	2.13272
$808$	−7573.19	−0.329732
$809$	−16308.6	−0.708752	−0.354376	−	0.935103i	$-0.615307\pi$
$809$	−16308.6	−0.708752	−0.354376	+	0.935103i	$0.615307\pi$
$810$	−11266.9	−0.488740
$811$	8308.25	0.359731	0.179866	−	0.983691i	$-0.442434\pi$
$811$	8308.25	0.359731	0.179866	+	0.983691i	$0.442434\pi$
$812$	0	0
$813$	63124.5	2.72309
$814$	13842.8	0.596057
$815$	18108.3	0.778289
$816$	−12720.5	−0.545717
$817$	3051.03	0.130651
$818$	−19715.5	−0.842709
$819$	0	0
$820$	10856.6	0.462354
$821$	28618.8	1.21657	0.608284	−	0.793720i	$-0.291858\pi$
$821$	28618.8	1.21657	0.608284	+	0.793720i	$0.291858\pi$
$822$	14920.9	0.633122
$823$	1464.41	0.0620244	0.0310122	−	0.999519i	$-0.490127\pi$
$823$	1464.41	0.0620244	0.0310122	+	0.999519i	$0.490127\pi$
$824$	−3377.70	−0.142801
$825$	27950.2	1.17952
$826$	0	0
$827$	−38045.8	−1.59974	−0.799869	−	0.600175i	$-0.795098\pi$
$827$	−38045.8	−1.59974	−0.799869	+	0.600175i	$0.795098\pi$
$828$	2425.30	0.101793
$829$	23247.3	0.973961	0.486980	−	0.873413i	$-0.338098\pi$
$829$	23247.3	0.973961	0.486980	+	0.873413i	$0.338098\pi$
$830$	−20761.0	−0.868224
$831$	54740.0	2.28509
$832$	−4213.62	−0.175578
$833$	0	0
$834$	6120.12	0.254104
$835$	−25975.5	−1.07655
$836$	−20805.3	−0.860725
$837$	−1094.81	−0.0452119
$838$	31188.0	1.28565
$839$	10511.6	0.432539	0.216270	−	0.976334i	$-0.430611\pi$
$839$	10511.6	0.432539	0.216270	+	0.976334i	$0.430611\pi$
$840$	0	0
$841$	−21006.8	−0.861324
$842$	4859.96	0.198914
$843$	−14439.9	−0.589959
$844$	−9759.80	−0.398041
$845$	−16146.3	−0.657337
$846$	−7795.18	−0.316790
$847$	0	0
$848$	10251.5	0.415141
$849$	−44959.0	−1.81742
$850$	14789.9	0.596809
$851$	−2826.96	−0.113874
$852$	−5007.17	−0.201341
$853$	10457.1	0.419748	0.209874	−	0.977728i	$-0.432695\pi$
$853$	10457.1	0.419748	0.209874	+	0.977728i	$0.432695\pi$
$854$	0	0
$855$	18392.0	0.735666
$856$	−4710.77	−0.188097
$857$	−17615.4	−0.702135	−0.351067	−	0.936350i	$-0.614181\pi$
$857$	−17615.4	−0.702135	−0.351067	+	0.936350i	$0.614181\pi$
$858$	−54165.7	−2.15523
$859$	−16600.7	−0.659381	−0.329691	−	0.944089i	$-0.606944\pi$
$859$	−16600.7	−0.659381	−0.329691	+	0.944089i	$0.606944\pi$
$860$	998.012	0.0395720
$861$	0	0
$862$	9080.57	0.358800
$863$	−11013.9	−0.434436	−0.217218	−	0.976123i	$-0.569698\pi$
$863$	−11013.9	−0.434436	−0.217218	+	0.976123i	$0.569698\pi$
$864$	−149.157	−0.00587317
$865$	−15174.7	−0.596478
$866$	30645.3	1.20250
$867$	−50637.5	−1.98355
$868$	0	0
$869$	−47432.7	−1.85160
$870$	−6417.77	−0.250095
$871$	−7432.13	−0.289125
$872$	−854.832	−0.0331976
$873$	26306.7	1.01987
$874$	4248.83	0.164438
$875$	0	0
$876$	10569.3	0.407653
$877$	22076.0	0.850003	0.425002	−	0.905193i	$-0.360274\pi$
$877$	22076.0	0.850003	0.425002	+	0.905193i	$0.360274\pi$
$878$	10283.6	0.395278
$879$	13920.1	0.534146
$880$	−6805.55	−0.260699
$881$	9443.65	0.361140	0.180570	−	0.983562i	$-0.442206\pi$
$881$	9443.65	0.361140	0.180570	+	0.983562i	$0.442206\pi$
$882$	0	0
$883$	31677.0	1.20727	0.603633	−	0.797262i	$-0.293719\pi$
$883$	31677.0	1.20727	0.603633	+	0.797262i	$0.293719\pi$
$884$	−28661.8	−1.09050
$885$	22182.7	0.842558
$886$	15076.7	0.571682
$887$	−96.4492	−0.00365101	−0.00182551	−	0.999998i	$-0.500581\pi$
$887$	−96.4492	−0.00365101	−0.00182551	+	0.999998i	$0.500581\pi$
$888$	−7182.86	−0.271443
$889$	0	0
$890$	18749.2	0.706151
$891$	−41998.8	−1.57914
$892$	−21657.4	−0.812941
$893$	−13656.2	−0.511745
$894$	14380.8	0.537992
$895$	−30305.9	−1.13186
$896$	0	0
$897$	11061.6	0.411747
$898$	14506.3	0.539065
$899$	−13659.8	−0.506764
$900$	−7164.80	−0.265363
$901$	69732.6	2.57839
$902$	40469.4	1.49388
$903$	0	0
$904$	−169.495	−0.00623598
$905$	−16854.9	−0.619088
$906$	29292.6	1.07415
$907$	−34721.3	−1.27112	−0.635559	−	0.772053i	$-0.719230\pi$
$907$	−34721.3	−1.27112	−0.635559	+	0.772053i	$0.719230\pi$
$908$	17918.2	0.654884
$909$	24955.5	0.910584
$910$	0	0
$911$	23562.8	0.856938	0.428469	−	0.903557i	$-0.359053\pi$
$911$	23562.8	0.856938	0.428469	+	0.903557i	$0.359053\pi$
$912$	10795.6	0.391972
$913$	−77389.2	−2.80527
$914$	28226.9	1.02151
$915$	13264.2	0.479236
$916$	−14511.6	−0.523445
$917$	0	0
$918$	−1014.59	−0.0364776
$919$	−26368.7	−0.946487	−0.473244	−	0.880932i	$-0.656917\pi$
$919$	−26368.7	−0.946487	−0.473244	+	0.880932i	$0.656917\pi$
$920$	1389.82	0.0498055
$921$	−4701.94	−0.168224
$922$	−3992.13	−0.142596
$923$	−11282.2	−0.402337
$924$	0	0
$925$	8351.39	0.296856
$926$	−16603.1	−0.589212
$927$	11130.3	0.394356
$928$	−1861.00	−0.0658302
$929$	−36036.3	−1.27267	−0.636336	−	0.771412i	$-0.719551\pi$
$929$	−36036.3	−1.27267	−0.636336	+	0.771412i	$0.719551\pi$
$930$	25920.0	0.913924
$931$	0	0
$932$	−14730.9	−0.517734
$933$	37481.9	1.31522
$934$	9035.68	0.316549
$935$	−46292.5	−1.61917
$936$	13884.9	0.484874
$937$	13236.3	0.461483	0.230742	−	0.973015i	$-0.425885\pi$
$937$	13236.3	0.461483	0.230742	+	0.973015i	$0.425885\pi$
$938$	0	0
$939$	−35399.2	−1.23025
$940$	−4467.05	−0.154999
$941$	−14141.9	−0.489918	−0.244959	−	0.969533i	$-0.578775\pi$
$941$	−14141.9	−0.489918	−0.244959	+	0.969533i	$0.578775\pi$
$942$	−5637.47	−0.194988
$943$	−8264.60	−0.285400
$944$	6432.47	0.221779
$945$	0	0
$946$	3720.21	0.127859
$947$	48138.7	1.65185	0.825923	−	0.563782i	$-0.190654\pi$
$947$	48138.7	1.65185	0.825923	+	0.563782i	$0.190654\pi$
$948$	24612.2	0.843215
$949$	23814.8	0.814607
$950$	−12551.9	−0.428670
$951$	−15698.8	−0.535298
$952$	0	0
$953$	−35613.3	−1.21052	−0.605262	−	0.796027i	$-0.706932\pi$
$953$	−35613.3	−1.21052	−0.605262	+	0.796027i	$0.706932\pi$
$954$	−33781.3	−1.14645
$955$	−986.974	−0.0334427
$956$	5158.38	0.174512
$957$	−23923.0	−0.808067
$958$	33031.3	1.11398
$959$	0	0
$960$	3531.32	0.118722
$961$	25378.0	0.851869
$962$	−16184.4	−0.542419
$963$	15523.1	0.519445
$964$	−2656.61	−0.0887590
$965$	26008.7	0.867616
$966$	0	0
$967$	−19501.2	−0.648518	−0.324259	−	0.945968i	$-0.605115\pi$
$967$	−19501.2	−0.648518	−0.324259	+	0.945968i	$0.605115\pi$
$968$	−14720.5	−0.488776
$969$	73433.5	2.43449
$970$	15075.1	0.499003
$971$	−8769.76	−0.289840	−0.144920	−	0.989443i	$-0.546293\pi$
$971$	−8769.76	−0.289840	−0.144920	+	0.989443i	$0.546293\pi$
$972$	21289.3	0.702524
$973$	0	0
$974$	−18106.2	−0.595646
$975$	−32678.2	−1.07338
$976$	3846.31	0.126145
$977$	−4744.61	−0.155367	−0.0776835	−	0.996978i	$-0.524752\pi$
$977$	−4744.61	−0.155367	−0.0776835	+	0.996978i	$0.524752\pi$
$978$	−35025.3	−1.14518
$979$	69889.8	2.28160
$980$	0	0
$981$	2816.88	0.0916779
$982$	14797.8	0.480871
$983$	−55524.2	−1.80157	−0.900787	−	0.434262i	$-0.857009\pi$
$983$	−55524.2	−1.80157	−0.900787	+	0.434262i	$0.857009\pi$
$984$	−20999.1	−0.680311
$985$	22481.4	0.727225
$986$	−12658.8	−0.408864
$987$	0	0
$988$	24324.7	0.783271
$989$	−759.736	−0.0244269
$990$	22425.9	0.719942
$991$	−16769.6	−0.537543	−0.268771	−	0.963204i	$-0.586618\pi$
$991$	−16769.6	−0.537543	−0.268771	+	0.963204i	$0.586618\pi$
$992$	7516.19	0.240564
$993$	−5357.02	−0.171198
$994$	0	0
$995$	6820.39	0.217307
$996$	40156.3	1.27751
$997$	14619.4	0.464394	0.232197	−	0.972669i	$-0.425409\pi$
$997$	14619.4	0.464394	0.232197	+	0.972669i	$0.425409\pi$
$998$	1435.64	0.0455356
$999$	−572.908	−0.0181441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.r.1.3		11
7.2	even	3		322.4.e.d.277.9	yes	22
7.4	even	3		322.4.e.d.93.9	✓	22
7.6	odd	2		2254.4.a.u.1.9		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.d.93.9	✓	22	7.4	even	3
322.4.e.d.277.9	yes	22	7.2	even	3
2254.4.a.r.1.3		11	1.1	even	1	trivial
2254.4.a.u.1.9		11	7.6	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:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -7.30492 q^{3} +4.00000 q^{4} -7.55338 q^{5} +14.6098 q^{6} -8.00000 q^{8} +26.3619 q^{9} +15.1068 q^{10} +56.3122 q^{11} -29.2197 q^{12} -65.8379 q^{13} +55.1768 q^{15} +16.0000 q^{16} +108.835 q^{17} -52.7238 q^{18} -92.3659 q^{19} -30.2135 q^{20} -112.624 q^{22} +23.0000 q^{23} +58.4394 q^{24} -67.9465 q^{25} +131.676 q^{26} +4.66115 q^{27} +58.1564 q^{29} -110.354 q^{30} -234.881 q^{31} -32.0000 q^{32} -411.356 q^{33} -217.669 q^{34} +105.448 q^{36} -122.911 q^{37} +184.732 q^{38} +480.941 q^{39} +60.4270 q^{40} -359.331 q^{41} -33.0320 q^{43} +225.249 q^{44} -199.121 q^{45} -46.0000 q^{46} +147.849 q^{47} -116.879 q^{48} +135.893 q^{50} -795.028 q^{51} -263.352 q^{52} +640.721 q^{53} -9.32230 q^{54} -425.347 q^{55} +674.726 q^{57} -116.313 q^{58} +402.029 q^{59} +220.707 q^{60} +240.394 q^{61} +469.762 q^{62} +64.0000 q^{64} +497.298 q^{65} +822.713 q^{66} +112.885 q^{67} +435.338 q^{68} -168.013 q^{69} +171.363 q^{71} -210.895 q^{72} -361.719 q^{73} +245.823 q^{74} +496.344 q^{75} -369.464 q^{76} -961.881 q^{78} -842.316 q^{79} -120.854 q^{80} -745.821 q^{81} +718.661 q^{82} -1374.29 q^{83} -822.069 q^{85} +66.0640 q^{86} -424.828 q^{87} -450.498 q^{88} +1241.11 q^{89} +398.243 q^{90} +92.0000 q^{92} +1715.79 q^{93} -295.699 q^{94} +697.674 q^{95} +233.758 q^{96} +997.907 q^{97} +1484.50 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 22 q^{2} + 44 q^{4} - 23 q^{5} - 88 q^{8} + 171 q^{9} + 46 q^{10} - 48 q^{11} - 77 q^{13} + 104 q^{15} + 176 q^{16} - 97 q^{17} - 342 q^{18} - 138 q^{19} - 92 q^{20} + 96 q^{22} + 253 q^{23} + 30 q^{25}+ \cdots - 2545 q^{99}+O(q^{100}) \) 11 * q - 22 * q^2 + 44 * q^4 - 23 * q^5 - 88 * q^8 + 171 * q^9 + 46 * q^10 - 48 * q^11 - 77 * q^13 + 104 * q^15 + 176 * q^16 - 97 * q^17 - 342 * q^18 - 138 * q^19 - 92 * q^20 + 96 * q^22 + 253 * q^23 + 30 * q^25 + 154 * q^26 - 315 * q^27 + 137 * q^29 - 208 * q^30 - 234 * q^31 - 352 * q^32 - 28 * q^33 + 194 * q^34 + 684 * q^36 + 87 * q^37 + 276 * q^38 + 129 * q^39 + 184 * q^40 - 612 * q^41 + 187 * q^43 - 192 * q^44 - 597 * q^45 - 506 * q^46 - 793 * q^47 - 60 * q^50 - 1257 * q^51 - 308 * q^52 + 575 * q^53 + 630 * q^54 + 549 * q^55 + 39 * q^57 - 274 * q^58 - 1241 * q^59 + 416 * q^60 - 187 * q^61 + 468 * q^62 + 704 * q^64 - 937 * q^65 + 56 * q^66 + 2275 * q^67 - 388 * q^68 + 115 * q^71 - 1368 * q^72 - 2385 * q^73 - 174 * q^74 - 196 * q^75 - 552 * q^76 - 258 * q^78 - 1225 * q^79 - 368 * q^80 + 3635 * q^81 + 1224 * q^82 + 597 * q^83 - 2493 * q^85 - 374 * q^86 - 1428 * q^87 + 384 * q^88 - 602 * q^89 + 1194 * q^90 + 1012 * q^92 - 1360 * q^93 + 1586 * q^94 + 4835 * q^95 + 1039 * q^97 - 2545 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more