LMFDB - Embedded newform 2254.4.a.u.1.10

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)

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);

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")

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:	$0$
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			1.10
Root			$8.71952$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +8.71952 q^{3} +4.00000 q^{4} -10.2018 q^{5} -17.4390 q^{6} -8.00000 q^{8} +49.0300 q^{9} +20.4036 q^{10} -47.6338 q^{11} +34.8781 q^{12} +44.4484 q^{13} -88.9548 q^{15} +16.0000 q^{16} -24.4794 q^{17} -98.0600 q^{18} -116.475 q^{19} -40.8072 q^{20} +95.2675 q^{22} +23.0000 q^{23} -69.7561 q^{24} -20.9233 q^{25} -88.8968 q^{26} +192.091 q^{27} -57.1589 q^{29} +177.910 q^{30} -156.511 q^{31} -32.0000 q^{32} -415.343 q^{33} +48.9587 q^{34} +196.120 q^{36} +131.743 q^{37} +232.951 q^{38} +387.569 q^{39} +81.6144 q^{40} +391.272 q^{41} +144.126 q^{43} -190.535 q^{44} -500.194 q^{45} -46.0000 q^{46} +330.386 q^{47} +139.512 q^{48} +41.8466 q^{50} -213.448 q^{51} +177.794 q^{52} +222.623 q^{53} -384.182 q^{54} +485.950 q^{55} -1015.61 q^{57} +114.318 q^{58} +869.735 q^{59} -355.819 q^{60} -182.244 q^{61} +313.022 q^{62} +64.0000 q^{64} -453.454 q^{65} +830.687 q^{66} +492.053 q^{67} -97.9174 q^{68} +200.549 q^{69} +1085.88 q^{71} -392.240 q^{72} +1104.26 q^{73} -263.485 q^{74} -182.441 q^{75} -465.901 q^{76} -775.137 q^{78} -58.3125 q^{79} -163.229 q^{80} +351.130 q^{81} -782.545 q^{82} -1178.21 q^{83} +249.733 q^{85} -288.251 q^{86} -498.398 q^{87} +381.070 q^{88} +49.9905 q^{89} +1000.39 q^{90} +92.0000 q^{92} -1364.70 q^{93} -660.771 q^{94} +1188.26 q^{95} -279.025 q^{96} -548.910 q^{97} -2335.48 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	8.71952	1.67807	0.839036	−	0.544076i	$-0.183120\pi$
$3$	8.71952	1.67807	0.839036	+	0.544076i	$0.183120\pi$
$4$	4.00000	0.500000
$5$	−10.2018	−0.912477	−0.456238	−	0.889858i	$-0.650804\pi$
$5$	−10.2018	−0.912477	−0.456238	+	0.889858i	$0.650804\pi$
$6$	−17.4390	−1.18658
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	49.0300	1.81593
$10$	20.4036	0.645218
$11$	−47.6338	−1.30565	−0.652824	−	0.757510i	$-0.726416\pi$
$11$	−47.6338	−1.30565	−0.652824	+	0.757510i	$0.726416\pi$
$12$	34.8781	0.839036
$13$	44.4484	0.948290	0.474145	−	0.880447i	$-0.342757\pi$
$13$	44.4484	0.948290	0.474145	+	0.880447i	$0.342757\pi$
$14$	0	0
$15$	−88.9548	−1.53120
$16$	16.0000	0.250000
$17$	−24.4794	−0.349242	−0.174621	−	0.984636i	$-0.555870\pi$
$17$	−24.4794	−0.349242	−0.174621	+	0.984636i	$0.555870\pi$
$18$	−98.0600	−1.28405
$19$	−116.475	−1.40638	−0.703191	−	0.711001i	$-0.748242\pi$
$19$	−116.475	−1.40638	−0.703191	+	0.711001i	$0.748242\pi$
$20$	−40.8072	−0.456238
$21$	0	0
$22$	95.2675	0.923232
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−69.7561	−0.593288
$25$	−20.9233	−0.167386
$26$	−88.8968	−0.670542
$27$	192.091	1.36918
$28$	0	0
$29$	−57.1589	−0.366005	−0.183003	−	0.983112i	$-0.558582\pi$
$29$	−57.1589	−0.366005	−0.183003	+	0.983112i	$0.558582\pi$
$30$	177.910	1.08272
$31$	−156.511	−0.906780	−0.453390	−	0.891312i	$-0.649785\pi$
$31$	−156.511	−0.906780	−0.453390	+	0.891312i	$0.649785\pi$
$32$	−32.0000	−0.176777
$33$	−415.343	−2.19097
$34$	48.9587	0.246951
$35$	0	0
$36$	196.120	0.907963
$37$	131.743	0.585360	0.292680	−	0.956210i	$-0.405453\pi$
$37$	131.743	0.585360	0.292680	+	0.956210i	$0.405453\pi$
$38$	232.951	0.994463
$39$	387.569	1.59130
$40$	81.6144	0.322609
$41$	391.272	1.49040	0.745201	−	0.666840i	$-0.232353\pi$
$41$	391.272	1.49040	0.745201	+	0.666840i	$0.232353\pi$
$42$	0	0
$43$	144.126	0.511138	0.255569	−	0.966791i	$-0.417737\pi$
$43$	144.126	0.511138	0.255569	+	0.966791i	$0.417737\pi$
$44$	−190.535	−0.652824
$45$	−500.194	−1.65699
$46$	−46.0000	−0.147442
$47$	330.386	1.02536	0.512678	−	0.858581i	$-0.328654\pi$
$47$	330.386	1.02536	0.512678	+	0.858581i	$0.328654\pi$
$48$	139.512	0.419518
$49$	0	0
$50$	41.8466	0.118360
$51$	−213.448	−0.586053
$52$	177.794	0.474145
$53$	222.623	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$53$	222.623	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$54$	−384.182	−0.968158
$55$	485.950	1.19137
$56$	0	0
$57$	−1015.61	−2.36001
$58$	114.318	0.258805
$59$	869.735	1.91915	0.959575	−	0.281454i	$-0.0908167\pi$
$59$	869.735	1.91915	0.959575	+	0.281454i	$0.0908167\pi$
$60$	−355.819	−0.765601
$61$	−182.244	−0.382523	−0.191261	−	0.981539i	$-0.561258\pi$
$61$	−182.244	−0.382523	−0.191261	+	0.981539i	$0.561258\pi$
$62$	313.022	0.641190
$63$	0	0
$64$	64.0000	0.125000
$65$	−453.454	−0.865293
$66$	830.687	1.54925
$67$	492.053	0.897221	0.448610	−	0.893727i	$-0.351919\pi$
$67$	492.053	0.897221	0.448610	+	0.893727i	$0.351919\pi$
$68$	−97.9174	−0.174621
$69$	200.549	0.349902
$70$	0	0
$71$	1085.88	1.81507	0.907533	−	0.419980i	$-0.137963\pi$
$71$	1085.88	1.81507	0.907533	+	0.419980i	$0.137963\pi$
$72$	−392.240	−0.642027
$73$	1104.26	1.77046	0.885229	−	0.465156i	$-0.154002\pi$
$73$	1104.26	1.77046	0.885229	+	0.465156i	$0.154002\pi$
$74$	−263.485	−0.413912
$75$	−182.441	−0.280886
$76$	−465.901	−0.703191
$77$	0	0
$78$	−775.137	−1.12522
$79$	−58.3125	−0.0830464	−0.0415232	−	0.999138i	$-0.513221\pi$
$79$	−58.3125	−0.0830464	−0.0415232	+	0.999138i	$0.513221\pi$
$80$	−163.229	−0.228119
$81$	351.130	0.481660
$82$	−782.545	−1.05387
$83$	−1178.21	−1.55814	−0.779069	−	0.626939i	$-0.784308\pi$
$83$	−1178.21	−1.55814	−0.779069	+	0.626939i	$0.784308\pi$
$84$	0	0
$85$	249.733	0.318675
$86$	−288.251	−0.361429
$87$	−498.398	−0.614183
$88$	381.070	0.461616
$89$	49.9905	0.0595391	0.0297695	−	0.999557i	$-0.490523\pi$
$89$	49.9905	0.0595391	0.0297695	+	0.999557i	$0.490523\pi$
$90$	1000.39	1.17167
$91$	0	0
$92$	92.0000	0.104257
$93$	−1364.70	−1.52164
$94$	−660.771	−0.725036
$95$	1188.26	1.28329
$96$	−279.025	−0.296644
$97$	−548.910	−0.574570	−0.287285	−	0.957845i	$-0.592753\pi$
$97$	−548.910	−0.574570	−0.287285	+	0.957845i	$0.592753\pi$
$98$	0	0
$99$	−2335.48	−2.37096
$100$	−83.6932	−0.0836932
$101$	1709.02	1.68370	0.841850	−	0.539712i	$-0.181467\pi$
$101$	1709.02	1.68370	0.841850	+	0.539712i	$0.181467\pi$
$102$	426.896	0.414402
$103$	−1189.67	−1.13807	−0.569037	−	0.822312i	$-0.692684\pi$
$103$	−1189.67	−1.13807	−0.569037	+	0.822312i	$0.692684\pi$
$104$	−355.587	−0.335271
$105$	0	0
$106$	−445.246	−0.407983
$107$	−834.807	−0.754241	−0.377121	−	0.926164i	$-0.623086\pi$
$107$	−834.807	−0.754241	−0.377121	+	0.926164i	$0.623086\pi$
$108$	768.363	0.684591
$109$	−604.155	−0.530895	−0.265448	−	0.964125i	$-0.585520\pi$
$109$	−604.155	−0.530895	−0.265448	+	0.964125i	$0.585520\pi$
$110$	−971.900	−0.842428
$111$	1148.73	0.982277
$112$	0	0
$113$	695.157	0.578716	0.289358	−	0.957221i	$-0.406558\pi$
$113$	695.157	0.578716	0.289358	+	0.957221i	$0.406558\pi$
$114$	2031.22	1.66878
$115$	−234.641	−0.190265
$116$	−228.636	−0.183003
$117$	2179.31	1.72202
$118$	−1739.47	−1.35704
$119$	0	0
$120$	711.638	0.541361
$121$	937.975	0.704715
$122$	364.487	0.270484
$123$	3411.71	2.50100
$124$	−626.043	−0.453390
$125$	1488.68	1.06521
$126$	0	0
$127$	1227.11	0.857390	0.428695	−	0.903449i	$-0.358974\pi$
$127$	1227.11	0.857390	0.428695	+	0.903449i	$0.358974\pi$
$128$	−128.000	−0.0883883
$129$	1256.71	0.857727
$130$	906.908	0.611854
$131$	−1943.67	−1.29633	−0.648166	−	0.761499i	$-0.724464\pi$
$131$	−1943.67	−1.29633	−0.648166	+	0.761499i	$0.724464\pi$
$132$	−1661.37	−1.09549
$133$	0	0
$134$	−984.105	−0.634431
$135$	−1959.67	−1.24935
$136$	195.835	0.123476
$137$	3002.30	1.87229	0.936145	−	0.351615i	$-0.114368\pi$
$137$	3002.30	1.87229	0.936145	+	0.351615i	$0.114368\pi$
$138$	−401.098	−0.247418
$139$	−1599.49	−0.976023	−0.488012	−	0.872837i	$-0.662278\pi$
$139$	−1599.49	−0.976023	−0.488012	+	0.872837i	$0.662278\pi$
$140$	0	0
$141$	2880.80	1.72062
$142$	−2171.75	−1.28345
$143$	−2117.25	−1.23813
$144$	784.480	0.453981
$145$	583.124	0.333971
$146$	−2208.51	−1.25190
$147$	0	0
$148$	526.970	0.292680
$149$	373.153	0.205167	0.102583	−	0.994724i	$-0.467289\pi$
$149$	373.153	0.205167	0.102583	+	0.994724i	$0.467289\pi$
$150$	364.882	0.198617
$151$	276.916	0.149239	0.0746196	−	0.997212i	$-0.476226\pi$
$151$	276.916	0.149239	0.0746196	+	0.997212i	$0.476226\pi$
$152$	931.803	0.497231
$153$	−1200.22	−0.634198
$154$	0	0
$155$	1596.69	0.827415
$156$	1550.27	0.795650
$157$	−2195.24	−1.11592	−0.557960	−	0.829868i	$-0.688416\pi$
$157$	−2195.24	−1.11592	−0.557960	+	0.829868i	$0.688416\pi$
$158$	116.625	0.0587227
$159$	1941.17	0.968205
$160$	326.458	0.161305
$161$	0	0
$162$	−702.260	−0.340585
$163$	509.827	0.244986	0.122493	−	0.992469i	$-0.460911\pi$
$163$	509.827	0.244986	0.122493	+	0.992469i	$0.460911\pi$
$164$	1565.09	0.745201
$165$	4237.25	1.99921
$166$	2356.42	1.10177
$167$	2612.83	1.21070	0.605350	−	0.795960i	$-0.293033\pi$
$167$	2612.83	1.21070	0.605350	+	0.795960i	$0.293033\pi$
$168$	0	0
$169$	−221.339	−0.100746
$170$	−499.467	−0.225337
$171$	−5710.78	−2.55389
$172$	576.502	0.255569
$173$	1989.26	0.874222	0.437111	−	0.899407i	$-0.356002\pi$
$173$	1989.26	0.874222	0.437111	+	0.899407i	$0.356002\pi$
$174$	996.797	0.434293
$175$	0	0
$176$	−762.140	−0.326412
$177$	7583.67	3.22047
$178$	−99.9809	−0.0421005
$179$	−10.6376	−0.00444187	−0.00222093	−	0.999998i	$-0.500707\pi$
$179$	−10.6376	−0.00444187	−0.00222093	+	0.999998i	$0.500707\pi$
$180$	−2000.78	−0.828495
$181$	−310.208	−0.127390	−0.0636949	−	0.997969i	$-0.520288\pi$
$181$	−310.208	−0.127390	−0.0636949	+	0.997969i	$0.520288\pi$
$182$	0	0
$183$	−1589.08	−0.641901
$184$	−184.000	−0.0737210
$185$	−1344.01	−0.534128
$186$	2729.40	1.07596
$187$	1166.04	0.455987
$188$	1321.54	0.512678
$189$	0	0
$190$	−2376.52	−0.907424
$191$	−52.8909	−0.0200369	−0.0100185	−	0.999950i	$-0.503189\pi$
$191$	−52.8909	−0.0200369	−0.0100185	+	0.999950i	$0.503189\pi$
$192$	558.049	0.209759
$193$	−441.361	−0.164611	−0.0823054	−	0.996607i	$-0.526228\pi$
$193$	−441.361	−0.164611	−0.0823054	+	0.996607i	$0.526228\pi$
$194$	1097.82	0.406283
$195$	−3953.90	−1.45202
$196$	0	0
$197$	5481.13	1.98231	0.991153	−	0.132726i	$-0.0423730\pi$
$197$	5481.13	1.98231	0.991153	+	0.132726i	$0.0423730\pi$
$198$	4670.97	1.67652
$199$	1718.02	0.611996	0.305998	−	0.952032i	$-0.401010\pi$
$199$	1718.02	0.611996	0.305998	+	0.952032i	$0.401010\pi$
$200$	167.386	0.0591800
$201$	4290.46	1.50560
$202$	−3418.04	−1.19056
$203$	0	0
$204$	−853.793	−0.293027
$205$	−3991.68	−1.35996
$206$	2379.34	0.804739
$207$	1127.69	0.378647
$208$	711.175	0.237073
$209$	5548.16	1.83624
$210$	0	0
$211$	675.408	0.220365	0.110182	−	0.993911i	$-0.464856\pi$
$211$	675.408	0.220365	0.110182	+	0.993911i	$0.464856\pi$
$212$	890.493	0.288487
$213$	9468.31	3.04581
$214$	1669.61	0.533329
$215$	−1470.34	−0.466402
$216$	−1536.73	−0.484079
$217$	0	0
$218$	1208.31	0.375400
$219$	9628.58	2.97095
$220$	1943.80	0.595686
$221$	−1088.07	−0.331183
$222$	−2297.46	−0.694575
$223$	1948.32	0.585063	0.292532	−	0.956256i	$-0.405502\pi$
$223$	1948.32	0.585063	0.292532	+	0.956256i	$0.405502\pi$
$224$	0	0
$225$	−1025.87	−0.303961
$226$	−1390.31	−0.409214
$227$	−1606.14	−0.469617	−0.234808	−	0.972042i	$-0.575446\pi$
$227$	−1606.14	−0.469617	−0.234808	+	0.972042i	$0.575446\pi$
$228$	−4062.43	−1.18001
$229$	3301.37	0.952665	0.476333	−	0.879265i	$-0.341966\pi$
$229$	3301.37	0.952665	0.476333	+	0.879265i	$0.341966\pi$
$230$	469.283	0.134537
$231$	0	0
$232$	457.271	0.129402
$233$	2193.56	0.616760	0.308380	−	0.951263i	$-0.400213\pi$
$233$	2193.56	0.616760	0.308380	+	0.951263i	$0.400213\pi$
$234$	−4358.61	−1.21766
$235$	−3370.53	−0.935613
$236$	3478.94	0.959575
$237$	−508.457	−0.139358
$238$	0	0
$239$	−3553.43	−0.961725	−0.480862	−	0.876796i	$-0.659676\pi$
$239$	−3553.43	−0.961725	−0.480862	+	0.876796i	$0.659676\pi$
$240$	−1423.28	−0.382800
$241$	3204.36	0.856478	0.428239	−	0.903666i	$-0.359134\pi$
$241$	3204.36	0.856478	0.428239	+	0.903666i	$0.359134\pi$
$242$	−1875.95	−0.498309
$243$	−2124.77	−0.560922
$244$	−728.974	−0.191261
$245$	0	0
$246$	−6823.41	−1.76848
$247$	−5177.14	−1.33366
$248$	1252.09	0.320595
$249$	−10273.4	−2.61467
$250$	−2977.36	−0.753219
$251$	7610.49	1.91382	0.956912	−	0.290378i	$-0.0937810\pi$
$251$	7610.49	1.91382	0.956912	+	0.290378i	$0.0937810\pi$
$252$	0	0
$253$	−1095.58	−0.272246
$254$	−2454.22	−0.606266
$255$	2177.55	0.534760
$256$	256.000	0.0625000
$257$	−3868.87	−0.939040	−0.469520	−	0.882922i	$-0.655573\pi$
$257$	−3868.87	−0.939040	−0.469520	+	0.882922i	$0.655573\pi$
$258$	−2513.41	−0.606504
$259$	0	0
$260$	−1813.82	−0.432646
$261$	−2802.50	−0.664638
$262$	3887.35	0.916645
$263$	1455.83	0.341331	0.170666	−	0.985329i	$-0.445408\pi$
$263$	1455.83	0.341331	0.170666	+	0.985329i	$0.445408\pi$
$264$	3322.75	0.774625
$265$	−2271.16	−0.526476
$266$	0	0
$267$	435.893	0.0999109
$268$	1968.21	0.448610
$269$	−5926.71	−1.34334	−0.671669	−	0.740851i	$-0.734422\pi$
$269$	−5926.71	−1.34334	−0.671669	+	0.740851i	$0.734422\pi$
$270$	3919.34	0.883421
$271$	−7649.64	−1.71470	−0.857348	−	0.514737i	$-0.827890\pi$
$271$	−7649.64	−1.71470	−0.857348	+	0.514737i	$0.827890\pi$
$272$	−391.670	−0.0873105
$273$	0	0
$274$	−6004.60	−1.32391
$275$	996.656	0.218548
$276$	802.196	0.174951
$277$	−7120.13	−1.54443	−0.772215	−	0.635361i	$-0.780851\pi$
$277$	−7120.13	−1.54443	−0.772215	+	0.635361i	$0.780851\pi$
$278$	3198.99	0.690153
$279$	−7673.72	−1.64664
$280$	0	0
$281$	−96.0293	−0.0203866	−0.0101933	−	0.999948i	$-0.503245\pi$
$281$	−96.0293	−0.0203866	−0.0101933	+	0.999948i	$0.503245\pi$
$282$	−5761.61	−1.21666
$283$	−1899.83	−0.399057	−0.199528	−	0.979892i	$-0.563941\pi$
$283$	−1899.83	−0.399057	−0.199528	+	0.979892i	$0.563941\pi$
$284$	4343.50	0.907533
$285$	10361.0	2.15346
$286$	4234.49	0.875492
$287$	0	0
$288$	−1568.96	−0.321013
$289$	−4313.76	−0.878030
$290$	−1166.25	−0.236153
$291$	−4786.23	−0.964170
$292$	4417.02	0.885229
$293$	8109.02	1.61684	0.808420	−	0.588607i	$-0.200323\pi$
$293$	8109.02	1.61684	0.808420	+	0.588607i	$0.200323\pi$
$294$	0	0
$295$	−8872.86	−1.75118
$296$	−1053.94	−0.206956
$297$	−9150.01	−1.78767
$298$	−746.306	−0.145075
$299$	1022.31	0.197732
$300$	−729.764	−0.140443
$301$	0	0
$302$	−553.832	−0.105528
$303$	14901.8	2.82537
$304$	−1863.61	−0.351596
$305$	1859.21	0.349043
$306$	2400.44	0.448445
$307$	−421.150	−0.0782941	−0.0391470	−	0.999233i	$-0.512464\pi$
$307$	−421.150	−0.0782941	−0.0391470	+	0.999233i	$0.512464\pi$
$308$	0	0
$309$	−10373.3	−1.90977
$310$	−3193.38	−0.585071
$311$	1223.90	0.223154	0.111577	−	0.993756i	$-0.464410\pi$
$311$	1223.90	0.223154	0.111577	+	0.993756i	$0.464410\pi$
$312$	−3100.55	−0.562609
$313$	6678.19	1.20599	0.602993	−	0.797747i	$-0.293975\pi$
$313$	6678.19	1.20599	0.602993	+	0.797747i	$0.293975\pi$
$314$	4390.49	0.789075
$315$	0	0
$316$	−233.250	−0.0415232
$317$	−52.0616	−0.00922420	−0.00461210	−	0.999989i	$-0.501468\pi$
$317$	−52.0616	−0.00922420	−0.00461210	+	0.999989i	$0.501468\pi$
$318$	−3882.33	−0.684624
$319$	2722.70	0.477874
$320$	−652.915	−0.114060
$321$	−7279.11	−1.26567
$322$	0	0
$323$	2851.24	0.491168
$324$	1404.52	0.240830
$325$	−930.008	−0.158731
$326$	−1019.65	−0.173231
$327$	−5267.94	−0.890880
$328$	−3130.18	−0.526937
$329$	0	0
$330$	−8474.50	−1.41365
$331$	−7564.09	−1.25607	−0.628036	−	0.778184i	$-0.716141\pi$
$331$	−7564.09	−1.25607	−0.628036	+	0.778184i	$0.716141\pi$
$332$	−4712.84	−0.779069
$333$	6459.33	1.06297
$334$	−5225.66	−0.856094
$335$	−5019.82	−0.818693
$336$	0	0
$337$	9037.89	1.46091	0.730453	−	0.682963i	$-0.239309\pi$
$337$	9037.89	1.46091	0.730453	+	0.682963i	$0.239309\pi$
$338$	442.677	0.0712381
$339$	6061.43	0.971127
$340$	998.934	0.159338
$341$	7455.20	1.18393
$342$	11421.6	1.80587
$343$	0	0
$344$	−1153.00	−0.180715
$345$	−2045.96	−0.319278
$346$	−3978.52	−0.618169
$347$	1593.80	0.246570	0.123285	−	0.992371i	$-0.460657\pi$
$347$	1593.80	0.246570	0.123285	+	0.992371i	$0.460657\pi$
$348$	−1993.59	−0.307091
$349$	−7630.96	−1.17042	−0.585209	−	0.810882i	$-0.698988\pi$
$349$	−7630.96	−1.17042	−0.585209	+	0.810882i	$0.698988\pi$
$350$	0	0
$351$	8538.13	1.29838
$352$	1524.28	0.230808
$353$	9282.20	1.39955	0.699776	−	0.714362i	$-0.253283\pi$
$353$	9282.20	1.39955	0.699776	+	0.714362i	$0.253283\pi$
$354$	−15167.3	−2.27722
$355$	−11077.9	−1.65621
$356$	199.962	0.0297695
$357$	0	0
$358$	21.2753	0.00314087
$359$	−5980.57	−0.879226	−0.439613	−	0.898187i	$-0.644884\pi$
$359$	−5980.57	−0.879226	−0.439613	+	0.898187i	$0.644884\pi$
$360$	4001.55	0.585834
$361$	6707.50	0.977912
$362$	620.415	0.0900782
$363$	8178.69	1.18256
$364$	0	0
$365$	−11265.4	−1.61550
$366$	3178.15	0.453892
$367$	6369.05	0.905890	0.452945	−	0.891538i	$-0.350373\pi$
$367$	6369.05	0.905890	0.452945	+	0.891538i	$0.350373\pi$
$368$	368.000	0.0521286
$369$	19184.1	2.70646
$370$	2688.02	0.377685
$371$	0	0
$372$	−5458.79	−0.760821
$373$	2692.60	0.373774	0.186887	−	0.982381i	$-0.440160\pi$
$373$	2692.60	0.373774	0.186887	+	0.982381i	$0.440160\pi$
$374$	−2332.09	−0.322431
$375$	12980.6	1.78750
$376$	−2643.08	−0.362518
$377$	−2540.62	−0.347079
$378$	0	0
$379$	1084.67	0.147008	0.0735038	−	0.997295i	$-0.476582\pi$
$379$	1084.67	0.147008	0.0735038	+	0.997295i	$0.476582\pi$
$380$	4753.03	0.641646
$381$	10699.8	1.43876
$382$	105.782	0.0141683
$383$	95.3045	0.0127150	0.00635748	−	0.999980i	$-0.497976\pi$
$383$	95.3045	0.0127150	0.00635748	+	0.999980i	$0.497976\pi$
$384$	−1116.10	−0.148322
$385$	0	0
$386$	882.722	0.116397
$387$	7066.48	0.928189
$388$	−2195.64	−0.287285
$389$	3766.29	0.490895	0.245448	−	0.969410i	$-0.421065\pi$
$389$	3766.29	0.490895	0.245448	+	0.969410i	$0.421065\pi$
$390$	7907.80	1.02674
$391$	−563.025	−0.0728220
$392$	0	0
$393$	−16947.9	−2.17534
$394$	−10962.3	−1.40170
$395$	594.892	0.0757779
$396$	−9341.93	−1.18548
$397$	11403.4	1.44161	0.720807	−	0.693136i	$-0.243772\pi$
$397$	11403.4	1.44161	0.720807	+	0.693136i	$0.243772\pi$
$398$	−3436.04	−0.432746
$399$	0	0
$400$	−334.773	−0.0418466
$401$	−11444.0	−1.42515	−0.712575	−	0.701596i	$-0.752471\pi$
$401$	−11444.0	−1.42515	−0.712575	+	0.701596i	$0.752471\pi$
$402$	−8580.92	−1.06462
$403$	−6956.66	−0.859890
$404$	6836.07	0.841850
$405$	−3582.16	−0.439503
$406$	0	0
$407$	−6275.39	−0.764274
$408$	1707.59	0.207201
$409$	9949.99	1.20292	0.601461	−	0.798902i	$-0.294585\pi$
$409$	9949.99	1.20292	0.601461	+	0.798902i	$0.294585\pi$
$410$	7983.37	0.961635
$411$	26178.6	3.14184
$412$	−4758.68	−0.569037
$413$	0	0
$414$	−2255.38	−0.267744
$415$	12019.9	1.42176
$416$	−1422.35	−0.167636
$417$	−13946.8	−1.63784
$418$	−11096.3	−1.29842
$419$	−4875.16	−0.568418	−0.284209	−	0.958762i	$-0.591731\pi$
$419$	−4875.16	−0.568418	−0.284209	+	0.958762i	$0.591731\pi$
$420$	0	0
$421$	−2216.12	−0.256549	−0.128275	−	0.991739i	$-0.540944\pi$
$421$	−2216.12	−0.256549	−0.128275	+	0.991739i	$0.540944\pi$
$422$	−1350.82	−0.155821
$423$	16198.8	1.86197
$424$	−1780.99	−0.203991
$425$	512.189	0.0584584
$426$	−18936.6	−2.15371
$427$	0	0
$428$	−3339.23	−0.377121
$429$	−18461.4	−2.07768
$430$	2940.68	0.329796
$431$	1336.31	0.149346	0.0746729	−	0.997208i	$-0.476209\pi$
$431$	1336.31	0.149346	0.0746729	+	0.997208i	$0.476209\pi$
$432$	3073.45	0.342295
$433$	14882.9	1.65179	0.825897	−	0.563821i	$-0.190669\pi$
$433$	14882.9	1.65179	0.825897	+	0.563821i	$0.190669\pi$
$434$	0	0
$435$	5084.56	0.560428
$436$	−2416.62	−0.265448
$437$	−2678.93	−0.293251
$438$	−19257.2	−2.10078
$439$	−16084.7	−1.74870	−0.874351	−	0.485293i	$-0.838713\pi$
$439$	−16084.7	−1.74870	−0.874351	+	0.485293i	$0.838713\pi$
$440$	−3887.60	−0.421214
$441$	0	0
$442$	2176.14	0.234182
$443$	12470.5	1.33746	0.668728	−	0.743507i	$-0.266839\pi$
$443$	12470.5	1.33746	0.668728	+	0.743507i	$0.266839\pi$
$444$	4594.92	0.491138
$445$	−509.993	−0.0543280
$446$	−3896.64	−0.413702
$447$	3253.71	0.344285
$448$	0	0
$449$	−11606.7	−1.21994	−0.609971	−	0.792424i	$-0.708819\pi$
$449$	−11606.7	−1.21994	−0.609971	+	0.792424i	$0.708819\pi$
$450$	2051.74	0.214933
$451$	−18637.8	−1.94594
$452$	2780.63	0.289358
$453$	2414.57	0.250434
$454$	3212.27	0.332069
$455$	0	0
$456$	8124.87	0.834390
$457$	17684.6	1.81018	0.905091	−	0.425218i	$-0.139803\pi$
$457$	17684.6	1.81018	0.905091	+	0.425218i	$0.139803\pi$
$458$	−6602.73	−0.673636
$459$	−4702.26	−0.478176
$460$	−938.565	−0.0951323
$461$	1063.02	0.107396	0.0536982	−	0.998557i	$-0.482899\pi$
$461$	1063.02	0.107396	0.0536982	+	0.998557i	$0.482899\pi$
$462$	0	0
$463$	−9453.24	−0.948876	−0.474438	−	0.880289i	$-0.657349\pi$
$463$	−9453.24	−0.948876	−0.474438	+	0.880289i	$0.657349\pi$
$464$	−914.543	−0.0915013
$465$	13922.4	1.38846
$466$	−4387.12	−0.436115
$467$	17792.1	1.76300	0.881500	−	0.472183i	$-0.156534\pi$
$467$	17792.1	1.76300	0.881500	+	0.472183i	$0.156534\pi$
$468$	8717.22	0.861012
$469$	0	0
$470$	6741.05	0.661578
$471$	−19141.5	−1.87259
$472$	−6957.88	−0.678522
$473$	−6865.24	−0.667366
$474$	1016.91	0.0985409
$475$	2437.05	0.235409
$476$	0	0
$477$	10915.2	1.04774
$478$	7106.86	0.680042
$479$	1685.46	0.160774	0.0803870	−	0.996764i	$-0.474384\pi$
$479$	1685.46	0.160774	0.0803870	+	0.996764i	$0.474384\pi$
$480$	2846.55	0.270681
$481$	5855.75	0.555091
$482$	−6408.73	−0.605621
$483$	0	0
$484$	3751.90	0.352357
$485$	5599.86	0.524282
$486$	4249.54	0.396632
$487$	6246.48	0.581221	0.290611	−	0.956841i	$-0.406142\pi$
$487$	6246.48	0.581221	0.290611	+	0.956841i	$0.406142\pi$
$488$	1457.95	0.135242
$489$	4445.45	0.411105
$490$	0	0
$491$	−3481.97	−0.320039	−0.160020	−	0.987114i	$-0.551156\pi$
$491$	−3481.97	−0.320039	−0.160020	+	0.987114i	$0.551156\pi$
$492$	13646.8	1.25050
$493$	1399.21	0.127824
$494$	10354.3	0.943039
$495$	23826.1	2.16344
$496$	−2504.17	−0.226695
$497$	0	0
$498$	20546.9	1.84885
$499$	4686.08	0.420396	0.210198	−	0.977659i	$-0.432589\pi$
$499$	4686.08	0.420396	0.210198	+	0.977659i	$0.432589\pi$
$500$	5954.72	0.532606
$501$	22782.6	2.03164
$502$	−15221.0	−1.35328
$503$	14176.5	1.25665	0.628327	−	0.777949i	$-0.283740\pi$
$503$	14176.5	1.25665	0.628327	+	0.777949i	$0.283740\pi$
$504$	0	0
$505$	−17435.1	−1.53634
$506$	2191.15	0.192507
$507$	−1929.97	−0.169059
$508$	4908.44	0.428695
$509$	−5433.63	−0.473166	−0.236583	−	0.971611i	$-0.576028\pi$
$509$	−5433.63	−0.473166	−0.236583	+	0.971611i	$0.576028\pi$
$510$	−4355.11	−0.378132
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−22373.8	−1.92559
$514$	7737.73	0.664002
$515$	12136.8	1.03847
$516$	5026.82	0.428863
$517$	−15737.5	−1.33875
$518$	0	0
$519$	17345.4	1.46701
$520$	3627.63	0.305927
$521$	11942.6	1.00425	0.502124	−	0.864796i	$-0.332552\pi$
$521$	11942.6	1.00425	0.502124	+	0.864796i	$0.332552\pi$
$522$	5605.00	0.469970
$523$	8321.41	0.695736	0.347868	−	0.937544i	$-0.386906\pi$
$523$	8321.41	0.695736	0.347868	+	0.937544i	$0.386906\pi$
$524$	−7774.69	−0.648166
$525$	0	0
$526$	−2911.65	−0.241358
$527$	3831.28	0.316686
$528$	−6645.49	−0.547743
$529$	529.000	0.0434783
$530$	4542.31	0.372275
$531$	42643.1	3.48503
$532$	0	0
$533$	17391.4	1.41333
$534$	−871.786	−0.0706477
$535$	8516.53	0.688228
$536$	−3936.42	−0.317215
$537$	−92.7550	−0.00745377
$538$	11853.4	0.949883
$539$	0	0
$540$	−7838.69	−0.624673
$541$	−8200.26	−0.651676	−0.325838	−	0.945426i	$-0.605646\pi$
$541$	−8200.26	−0.651676	−0.325838	+	0.945426i	$0.605646\pi$
$542$	15299.3	1.21247
$543$	−2704.86	−0.213769
$544$	783.339	0.0617379
$545$	6163.47	0.484429
$546$	0	0
$547$	2098.90	0.164063	0.0820317	−	0.996630i	$-0.473859\pi$
$547$	2098.90	0.164063	0.0820317	+	0.996630i	$0.473859\pi$
$548$	12009.2	0.936145
$549$	−8935.40	−0.694633
$550$	−1993.31	−0.154537
$551$	6657.61	0.514743
$552$	−1604.39	−0.123709
$553$	0	0
$554$	14240.3	1.09208
$555$	−11719.1	−0.896305
$556$	−6397.97	−0.488012
$557$	−7430.71	−0.565259	−0.282630	−	0.959229i	$-0.591207\pi$
$557$	−7430.71	−0.565259	−0.282630	+	0.959229i	$0.591207\pi$
$558$	15347.4	1.16435
$559$	6406.15	0.484707
$560$	0	0
$561$	10167.3	0.765179
$562$	192.059	0.0144155
$563$	−813.440	−0.0608924	−0.0304462	−	0.999536i	$-0.509693\pi$
$563$	−813.440	−0.0608924	−0.0304462	+	0.999536i	$0.509693\pi$
$564$	11523.2	0.860310
$565$	−7091.85	−0.528065
$566$	3799.66	0.282176
$567$	0	0
$568$	−8687.01	−0.641723
$569$	−13513.3	−0.995622	−0.497811	−	0.867286i	$-0.665863\pi$
$569$	−13513.3	−0.995622	−0.497811	+	0.867286i	$0.665863\pi$
$570$	−20722.1	−1.52272
$571$	3982.03	0.291844	0.145922	−	0.989296i	$-0.453385\pi$
$571$	3982.03	0.291844	0.145922	+	0.989296i	$0.453385\pi$
$572$	−8468.98	−0.619066
$573$	−461.183	−0.0336234
$574$	0	0
$575$	−481.236	−0.0349025
$576$	3137.92	0.226991
$577$	−6101.18	−0.440200	−0.220100	−	0.975477i	$-0.570638\pi$
$577$	−6101.18	−0.440200	−0.220100	+	0.975477i	$0.570638\pi$
$578$	8627.52	0.620861
$579$	−3848.46	−0.276229
$580$	2332.50	0.166986
$581$	0	0
$582$	9572.45	0.681771
$583$	−10604.4	−0.753325
$584$	−8834.05	−0.625951
$585$	−22232.8	−1.57131
$586$	−16218.0	−1.14328
$587$	−20382.8	−1.43320	−0.716600	−	0.697485i	$-0.754303\pi$
$587$	−20382.8	−1.43320	−0.716600	+	0.697485i	$0.754303\pi$
$588$	0	0
$589$	18229.6	1.27528
$590$	17745.7	1.23827
$591$	47792.8	3.32645
$592$	2107.88	0.146340
$593$	9515.15	0.658922	0.329461	−	0.944169i	$-0.393133\pi$
$593$	9515.15	0.658922	0.329461	+	0.944169i	$0.393133\pi$
$594$	18300.0	1.26407
$595$	0	0
$596$	1492.61	0.102583
$597$	14980.3	1.02697
$598$	−2044.63	−0.139818
$599$	−1109.19	−0.0756598	−0.0378299	−	0.999284i	$-0.512045\pi$
$599$	−1109.19	−0.0756598	−0.0378299	+	0.999284i	$0.512045\pi$
$600$	1459.53	0.0993084
$601$	17645.1	1.19760	0.598802	−	0.800897i	$-0.295644\pi$
$601$	17645.1	1.19760	0.598802	+	0.800897i	$0.295644\pi$
$602$	0	0
$603$	24125.3	1.62929
$604$	1107.66	0.0746196
$605$	−9569.03	−0.643036
$606$	−29803.6	−1.99784
$607$	−22071.9	−1.47590	−0.737950	−	0.674856i	$-0.764206\pi$
$607$	−22071.9	−1.47590	−0.737950	+	0.674856i	$0.764206\pi$
$608$	3727.21	0.248616
$609$	0	0
$610$	−3718.42	−0.246811
$611$	14685.1	0.972334
$612$	−4800.89	−0.317099
$613$	−23736.1	−1.56393	−0.781967	−	0.623320i	$-0.785783\pi$
$613$	−23736.1	−1.56393	−0.781967	+	0.623320i	$0.785783\pi$
$614$	842.299	0.0553623
$615$	−34805.6	−2.28211
$616$	0	0
$617$	−27074.5	−1.76658	−0.883290	−	0.468826i	$-0.844677\pi$
$617$	−27074.5	−1.76658	−0.883290	+	0.468826i	$0.844677\pi$
$618$	20746.7	1.35041
$619$	26991.9	1.75266	0.876331	−	0.481710i	$-0.159984\pi$
$619$	26991.9	1.75266	0.876331	+	0.481710i	$0.159984\pi$
$620$	6386.77	0.413708
$621$	4418.09	0.285494
$622$	−2447.79	−0.157793
$623$	0	0
$624$	6201.10	0.397825
$625$	−12571.8	−0.804595
$626$	−13356.4	−0.852760
$627$	48377.3	3.08134
$628$	−8780.97	−0.557960
$629$	−3224.97	−0.204432
$630$	0	0
$631$	−10438.6	−0.658562	−0.329281	−	0.944232i	$-0.606806\pi$
$631$	−10438.6	−0.658562	−0.329281	+	0.944232i	$0.606806\pi$
$632$	466.500	0.0293613
$633$	5889.23	0.369788
$634$	104.123	0.00652250
$635$	−12518.7	−0.782348
$636$	7764.67	0.484102
$637$	0	0
$638$	−5445.39	−0.337908
$639$	53240.5	3.29603
$640$	1305.83	0.0806523
$641$	−5231.20	−0.322340	−0.161170	−	0.986927i	$-0.551527\pi$
$641$	−5231.20	−0.322340	−0.161170	+	0.986927i	$0.551527\pi$
$642$	14558.2	0.894965
$643$	23379.7	1.43391	0.716957	−	0.697117i	$-0.245534\pi$
$643$	23379.7	1.43391	0.716957	+	0.697117i	$0.245534\pi$
$644$	0	0
$645$	−12820.7	−0.782656
$646$	−5702.48	−0.347308
$647$	4299.72	0.261266	0.130633	−	0.991431i	$-0.458299\pi$
$647$	4299.72	0.261266	0.130633	+	0.991431i	$0.458299\pi$
$648$	−2809.04	−0.170292
$649$	−41428.7	−2.50573
$650$	1860.02	0.112240
$651$	0	0
$652$	2039.31	0.122493
$653$	−21893.3	−1.31202	−0.656011	−	0.754752i	$-0.727757\pi$
$653$	−21893.3	−1.31202	−0.656011	+	0.754752i	$0.727757\pi$
$654$	10535.9	0.629948
$655$	19829.0	1.18287
$656$	6260.36	0.372601
$657$	54141.7	3.21502
$658$	0	0
$659$	15102.7	0.892745	0.446372	−	0.894847i	$-0.352716\pi$
$659$	15102.7	0.892745	0.446372	+	0.894847i	$0.352716\pi$
$660$	16949.0	0.999604
$661$	96.2346	0.00566277	0.00283139	−	0.999996i	$-0.499099\pi$
$661$	96.2346	0.00566277	0.00283139	+	0.999996i	$0.499099\pi$
$662$	15128.2	0.888178
$663$	−9487.43	−0.555749
$664$	9425.68	0.550885
$665$	0	0
$666$	−12918.7	−0.751634
$667$	−1314.66	−0.0763173
$668$	10451.3	0.605350
$669$	16988.4	0.981778
$670$	10039.6	0.578903
$671$	8680.94	0.499440
$672$	0	0
$673$	23038.4	1.31956	0.659780	−	0.751459i	$-0.270649\pi$
$673$	23038.4	1.31956	0.659780	+	0.751459i	$0.270649\pi$
$674$	−18075.8	−1.03302
$675$	−4019.18	−0.229182
$676$	−885.354	−0.0503729
$677$	−22225.8	−1.26175	−0.630875	−	0.775884i	$-0.717304\pi$
$677$	−22225.8	−1.26175	−0.630875	+	0.775884i	$0.717304\pi$
$678$	−12122.9	−0.686690
$679$	0	0
$680$	−1997.87	−0.112669
$681$	−14004.7	−0.788051
$682$	−14910.4	−0.837168
$683$	7187.69	0.402679	0.201339	−	0.979522i	$-0.435471\pi$
$683$	7187.69	0.402679	0.201339	+	0.979522i	$0.435471\pi$
$684$	−22843.1	−1.27694
$685$	−30628.8	−1.70842
$686$	0	0
$687$	28786.3	1.59864
$688$	2306.01	0.127785
$689$	9895.25	0.547139
$690$	4091.92	0.225763
$691$	−4050.60	−0.222999	−0.111499	−	0.993765i	$-0.535565\pi$
$691$	−4050.60	−0.222999	−0.111499	+	0.993765i	$0.535565\pi$
$692$	7957.03	0.437111
$693$	0	0
$694$	−3187.60	−0.174351
$695$	16317.7	0.890598
$696$	3987.19	0.217146
$697$	−9578.10	−0.520511
$698$	15261.9	0.827611
$699$	19126.8	1.03497
$700$	0	0
$701$	−12995.1	−0.700168	−0.350084	−	0.936718i	$-0.613847\pi$
$701$	−12995.1	−0.700168	−0.350084	+	0.936718i	$0.613847\pi$
$702$	−17076.3	−0.918094
$703$	−15344.8	−0.823241
$704$	−3048.56	−0.163206
$705$	−29389.4	−1.57003
$706$	−18564.4	−0.989633
$707$	0	0
$708$	30334.7	1.61024
$709$	21281.5	1.12728	0.563640	−	0.826020i	$-0.309400\pi$
$709$	21281.5	1.12728	0.563640	+	0.826020i	$0.309400\pi$
$710$	22155.8	1.17111
$711$	−2859.06	−0.150806
$712$	−399.924	−0.0210502
$713$	−3599.75	−0.189077
$714$	0	0
$715$	21599.7	1.12977
$716$	−42.5505	−0.00222093
$717$	−30984.2	−1.61384
$718$	11961.1	0.621707
$719$	19425.2	1.00756	0.503781	−	0.863831i	$-0.331942\pi$
$719$	19425.2	1.00756	0.503781	+	0.863831i	$0.331942\pi$
$720$	−8003.11	−0.414247
$721$	0	0
$722$	−13415.0	−0.691488
$723$	27940.5	1.43723
$724$	−1240.83	−0.0636949
$725$	1195.95	0.0612643
$726$	−16357.4	−0.836198
$727$	16613.0	0.847515	0.423758	−	0.905776i	$-0.360711\pi$
$727$	16613.0	0.847515	0.423758	+	0.905776i	$0.360711\pi$
$728$	0	0
$729$	−28007.5	−1.42293
$730$	22530.8	1.14233
$731$	−3528.10	−0.178511
$732$	−6356.30	−0.320950
$733$	28860.0	1.45426	0.727128	−	0.686501i	$-0.240854\pi$
$733$	28860.0	1.45426	0.727128	+	0.686501i	$0.240854\pi$
$734$	−12738.1	−0.640561
$735$	0	0
$736$	−736.000	−0.0368605
$737$	−23438.3	−1.17145
$738$	−38368.2	−1.91376
$739$	4377.48	0.217900	0.108950	−	0.994047i	$-0.465251\pi$
$739$	4377.48	0.217900	0.108950	+	0.994047i	$0.465251\pi$
$740$	−5376.04	−0.267064
$741$	−45142.2	−2.23798
$742$	0	0
$743$	13092.1	0.646436	0.323218	−	0.946325i	$-0.395235\pi$
$743$	13092.1	0.646436	0.323218	+	0.946325i	$0.395235\pi$
$744$	10917.6	0.537981
$745$	−3806.83	−0.187210
$746$	−5385.20	−0.264298
$747$	−57767.6	−2.82946
$748$	4664.17	0.227993
$749$	0	0
$750$	−25961.1	−1.26396
$751$	−29469.3	−1.43189	−0.715944	−	0.698157i	$-0.754004\pi$
$751$	−29469.3	−1.43189	−0.715944	+	0.698157i	$0.754004\pi$
$752$	5286.17	0.256339
$753$	66359.8	3.21153
$754$	5081.25	0.245422
$755$	−2825.04	−0.136177
$756$	0	0
$757$	25951.8	1.24602	0.623009	−	0.782214i	$-0.285910\pi$
$757$	25951.8	1.24602	0.623009	+	0.782214i	$0.285910\pi$
$758$	−2169.34	−0.103950
$759$	−9552.90	−0.456849
$760$	−9506.06	−0.453712
$761$	9234.94	0.439903	0.219952	−	0.975511i	$-0.429410\pi$
$761$	9234.94	0.439903	0.219952	+	0.975511i	$0.429410\pi$
$762$	−21399.6	−1.01736
$763$	0	0
$764$	−211.564	−0.0100185
$765$	12244.4	0.578690
$766$	−190.609	−0.00899084
$767$	38658.3	1.81991
$768$	2232.20	0.104879
$769$	−13479.5	−0.632099	−0.316050	−	0.948743i	$-0.602357\pi$
$769$	−13479.5	−0.632099	−0.316050	+	0.948743i	$0.602357\pi$
$770$	0	0
$771$	−33734.7	−1.57578
$772$	−1765.44	−0.0823054
$773$	1615.52	0.0751697	0.0375849	−	0.999293i	$-0.488034\pi$
$773$	1615.52	0.0751697	0.0375849	+	0.999293i	$0.488034\pi$
$774$	−14133.0	−0.656329
$775$	3274.72	0.151783
$776$	4391.28	0.203141
$777$	0	0
$778$	−7532.57	−0.347115
$779$	−45573.6	−2.09608
$780$	−15815.6	−0.726012
$781$	−51724.4	−2.36984
$782$	1126.05	0.0514929
$783$	−10979.7	−0.501127
$784$	0	0
$785$	22395.4	1.01825
$786$	33895.8	1.53820
$787$	−20500.9	−0.928563	−0.464281	−	0.885688i	$-0.653687\pi$
$787$	−20500.9	−0.928563	−0.464281	+	0.885688i	$0.653687\pi$
$788$	21924.5	0.991153
$789$	12694.1	0.572778
$790$	−1189.78	−0.0535831
$791$	0	0
$792$	18683.9	0.838260
$793$	−8100.44	−0.362743
$794$	−22806.8	−1.01937
$795$	−19803.4	−0.883464
$796$	6872.08	0.305998
$797$	−30024.8	−1.33442	−0.667209	−	0.744870i	$-0.732511\pi$
$797$	−30024.8	−1.33442	−0.667209	+	0.744870i	$0.732511\pi$
$798$	0	0
$799$	−8087.62	−0.358097
$800$	669.546	0.0295900
$801$	2451.03	0.108119
$802$	22888.0	1.00773
$803$	−52599.9	−2.31159
$804$	17161.8	0.752801
$805$	0	0
$806$	13913.3	0.608034
$807$	−51678.0	−2.25422
$808$	−13672.1	−0.595278
$809$	−35008.7	−1.52143	−0.760716	−	0.649084i	$-0.775152\pi$
$809$	−35008.7	−1.52143	−0.760716	+	0.649084i	$0.775152\pi$
$810$	7164.32	0.310776
$811$	35468.2	1.53570	0.767852	−	0.640628i	$-0.221326\pi$
$811$	35468.2	1.53570	0.767852	+	0.640628i	$0.221326\pi$
$812$	0	0
$813$	−66701.2	−2.87738
$814$	12550.8	0.540423
$815$	−5201.16	−0.223544
$816$	−3415.17	−0.146513
$817$	−16787.1	−0.718856
$818$	−19900.0	−0.850595
$819$	0	0
$820$	−15966.7	−0.679979
$821$	−742.947	−0.0315823	−0.0157911	−	0.999875i	$-0.505027\pi$
$821$	−742.947	−0.0315823	−0.0157911	+	0.999875i	$0.505027\pi$
$822$	−52357.2	−2.22161
$823$	−43289.9	−1.83353	−0.916763	−	0.399431i	$-0.869208\pi$
$823$	−43289.9	−1.83353	−0.916763	+	0.399431i	$0.869208\pi$
$824$	9517.35	0.402370
$825$	8690.36	0.366739
$826$	0	0
$827$	−40732.0	−1.71269	−0.856343	−	0.516408i	$-0.827269\pi$
$827$	−40732.0	−1.71269	−0.856343	+	0.516408i	$0.827269\pi$
$828$	4510.76	0.189323
$829$	18816.7	0.788336	0.394168	−	0.919038i	$-0.371033\pi$
$829$	18816.7	0.788336	0.394168	+	0.919038i	$0.371033\pi$
$830$	−24039.7	−1.00534
$831$	−62084.1	−2.59166
$832$	2844.70	0.118536
$833$	0	0
$834$	27893.6	1.15813
$835$	−26655.6	−1.10473
$836$	22192.6	0.918120
$837$	−30064.3	−1.24155
$838$	9750.32	0.401932
$839$	43333.9	1.78314	0.891568	−	0.452886i	$-0.149606\pi$
$839$	43333.9	1.78314	0.891568	+	0.452886i	$0.149606\pi$
$840$	0	0
$841$	−21121.9	−0.866040
$842$	4432.24	0.181408
$843$	−837.329	−0.0342101
$844$	2701.63	0.110182
$845$	2258.05	0.0919282
$846$	−32397.6	−1.31661
$847$	0	0
$848$	3561.97	0.144244
$849$	−16565.6	−0.669646
$850$	−1024.38	−0.0413363
$851$	3030.08	0.122056
$852$	37873.3	1.52291
$853$	−29347.3	−1.17800	−0.588999	−	0.808134i	$-0.700478\pi$
$853$	−29347.3	−1.17800	−0.588999	+	0.808134i	$0.700478\pi$
$854$	0	0
$855$	58260.3	2.33036
$856$	6678.46	0.266665
$857$	37377.1	1.48982	0.744910	−	0.667164i	$-0.232492\pi$
$857$	37377.1	1.48982	0.744910	+	0.667164i	$0.232492\pi$
$858$	36922.7	1.46914
$859$	13909.6	0.552491	0.276246	−	0.961087i	$-0.410910\pi$
$859$	13909.6	0.552491	0.276246	+	0.961087i	$0.410910\pi$
$860$	−5881.36	−0.233201
$861$	0	0
$862$	−2672.63	−0.105603
$863$	−44798.7	−1.76705	−0.883525	−	0.468384i	$-0.844837\pi$
$863$	−44798.7	−1.76705	−0.883525	+	0.468384i	$0.844837\pi$
$864$	−6146.91	−0.242039
$865$	−20294.0	−0.797707
$866$	−29765.8	−1.16799
$867$	−37613.9	−1.47340
$868$	0	0
$869$	2777.64	0.108429
$870$	−10169.1	−0.396282
$871$	21871.0	0.850826
$872$	4833.24	0.187700
$873$	−26913.0	−1.04338
$874$	5357.86	0.207360
$875$	0	0
$876$	38514.3	1.48548
$877$	30660.0	1.18052	0.590259	−	0.807214i	$-0.299025\pi$
$877$	30660.0	1.18052	0.590259	+	0.807214i	$0.299025\pi$
$878$	32169.4	1.23652
$879$	70706.7	2.71317
$880$	7775.20	0.297843
$881$	−37429.6	−1.43137	−0.715683	−	0.698425i	$-0.753885\pi$
$881$	−37429.6	−1.43137	−0.715683	+	0.698425i	$0.753885\pi$
$882$	0	0
$883$	5281.76	0.201297	0.100649	−	0.994922i	$-0.467908\pi$
$883$	5281.76	0.201297	0.100649	+	0.994922i	$0.467908\pi$
$884$	−4352.27	−0.165591
$885$	−77367.0	−2.93860
$886$	−24941.1	−0.945725
$887$	8822.51	0.333969	0.166985	−	0.985959i	$-0.446597\pi$
$887$	8822.51	0.333969	0.166985	+	0.985959i	$0.446597\pi$
$888$	−9189.85	−0.347287
$889$	0	0
$890$	1019.99	0.0384157
$891$	−16725.6	−0.628878
$892$	7793.28	0.292532
$893$	−38481.8	−1.44204
$894$	−6507.43	−0.243446
$895$	108.523	0.00405310
$896$	0	0
$897$	8914.08	0.331809
$898$	23213.4	0.862629
$899$	8945.99	0.331886
$900$	−4103.48	−0.151981
$901$	−5449.67	−0.201504
$902$	37275.6	1.37599
$903$	0	0
$904$	−5561.26	−0.204607
$905$	3164.68	0.116240
$906$	−4829.15	−0.177084
$907$	−16922.0	−0.619498	−0.309749	−	0.950818i	$-0.600245\pi$
$907$	−16922.0	−0.619498	−0.309749	+	0.950818i	$0.600245\pi$
$908$	−6424.55	−0.234808
$909$	83793.1	3.05747
$910$	0	0
$911$	−4095.71	−0.148954	−0.0744769	−	0.997223i	$-0.523729\pi$
$911$	−4095.71	−0.148954	−0.0744769	+	0.997223i	$0.523729\pi$
$912$	−16249.7	−0.590003
$913$	56122.6	2.03438
$914$	−35369.3	−1.27999
$915$	16211.4	0.585719
$916$	13205.5	0.476333
$917$	0	0
$918$	9404.52	0.338121
$919$	−50949.8	−1.82881	−0.914407	−	0.404797i	$-0.867342\pi$
$919$	−50949.8	−1.82881	−0.914407	+	0.404797i	$0.867342\pi$
$920$	1877.13	0.0672687
$921$	−3672.22	−0.131383
$922$	−2126.04	−0.0759408
$923$	48265.5	1.72121
$924$	0	0
$925$	−2756.49	−0.0979814
$926$	18906.5	0.670957
$927$	−58329.5	−2.06666
$928$	1829.09	0.0647012
$929$	29361.2	1.03693	0.518465	−	0.855099i	$-0.326504\pi$
$929$	29361.2	1.03693	0.518465	+	0.855099i	$0.326504\pi$
$930$	−27844.8	−0.981791
$931$	0	0
$932$	8774.25	0.308380
$933$	10671.8	0.374468
$934$	−35584.2	−1.24663
$935$	−11895.7	−0.416077
$936$	−17434.4	−0.608828
$937$	−4856.38	−0.169318	−0.0846589	−	0.996410i	$-0.526980\pi$
$937$	−4856.38	−0.169318	−0.0846589	+	0.996410i	$0.526980\pi$
$938$	0	0
$939$	58230.6	2.02373
$940$	−13482.1	−0.467806
$941$	−17659.3	−0.611772	−0.305886	−	0.952068i	$-0.598953\pi$
$941$	−17659.3	−0.611772	−0.305886	+	0.952068i	$0.598953\pi$
$942$	38282.9	1.32412
$943$	8999.27	0.310770
$944$	13915.8	0.479787
$945$	0	0
$946$	13730.5	0.471899
$947$	2362.58	0.0810702	0.0405351	−	0.999178i	$-0.487094\pi$
$947$	2362.58	0.0810702	0.0405351	+	0.999178i	$0.487094\pi$
$948$	−2033.83	−0.0696789
$949$	49082.4	1.67891
$950$	−4874.10	−0.166460
$951$	−453.952	−0.0154789
$952$	0	0
$953$	51826.5	1.76162	0.880811	−	0.473469i	$-0.156998\pi$
$953$	51826.5	1.76162	0.880811	+	0.473469i	$0.156998\pi$
$954$	−21830.4	−0.740866
$955$	539.583	0.0182832
$956$	−14213.7	−0.480862
$957$	23740.6	0.801906
$958$	−3370.92	−0.113684
$959$	0	0
$960$	−5693.10	−0.191400
$961$	−5295.37	−0.177751
$962$	−11711.5	−0.392509
$963$	−40930.6	−1.36965
$964$	12817.5	0.428239
$965$	4502.68	0.150203
$966$	0	0
$967$	9508.48	0.316207	0.158103	−	0.987423i	$-0.449462\pi$
$967$	9508.48	0.316207	0.158103	+	0.987423i	$0.449462\pi$
$968$	−7503.80	−0.249154
$969$	24861.4	0.824215
$970$	−11199.7	−0.370723
$971$	−46010.2	−1.52064	−0.760318	−	0.649551i	$-0.774957\pi$
$971$	−46010.2	−1.52064	−0.760318	+	0.649551i	$0.774957\pi$
$972$	−8499.07	−0.280461
$973$	0	0
$974$	−12493.0	−0.410986
$975$	−8109.22	−0.266362
$976$	−2915.90	−0.0956307
$977$	24820.7	0.812778	0.406389	−	0.913700i	$-0.366788\pi$
$977$	24820.7	0.812778	0.406389	+	0.913700i	$0.366788\pi$
$978$	−8890.90	−0.290695
$979$	−2381.23	−0.0777371
$980$	0	0
$981$	−29621.7	−0.964066
$982$	6963.95	0.226302
$983$	59896.5	1.94344	0.971720	−	0.236137i	$-0.0758814\pi$
$983$	59896.5	1.94344	0.971720	+	0.236137i	$0.0758814\pi$
$984$	−27293.7	−0.884238
$985$	−55917.4	−1.80881
$986$	−2798.43	−0.0903855
$987$	0	0
$988$	−20708.6	−0.666829
$989$	3314.89	0.106580
$990$	−47652.3	−1.52979
$991$	−665.739	−0.0213399	−0.0106700	−	0.999943i	$-0.503396\pi$
$991$	−665.739	−0.0213399	−0.0106700	+	0.999943i	$0.503396\pi$
$992$	5008.35	0.160298
$993$	−65955.2	−2.10778
$994$	0	0
$995$	−17526.9	−0.558432
$996$	−41093.7	−1.30733
$997$	−16179.2	−0.513941	−0.256971	−	0.966419i	$-0.582724\pi$
$997$	−16179.2	−0.513941	−0.256971	+	0.966419i	$0.582724\pi$
$998$	−9372.16	−0.297265
$999$	25306.5	0.801465

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.u.1.10		11
7.3	odd	6		322.4.e.d.93.10	✓	22
7.5	odd	6		322.4.e.d.277.10	yes	22
7.6	odd	2		2254.4.a.r.1.2		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.d.93.10	✓	22	7.3	odd	6
322.4.e.d.277.10	yes	22	7.5	odd	6
2254.4.a.r.1.2		11	7.6	odd	2
2254.4.a.u.1.10		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +8.71952 q^{3} +4.00000 q^{4} -10.2018 q^{5} -17.4390 q^{6} -8.00000 q^{8} +49.0300 q^{9} +20.4036 q^{10} -47.6338 q^{11} +34.8781 q^{12} +44.4484 q^{13} -88.9548 q^{15} +16.0000 q^{16} -24.4794 q^{17} -98.0600 q^{18} -116.475 q^{19} -40.8072 q^{20} +95.2675 q^{22} +23.0000 q^{23} -69.7561 q^{24} -20.9233 q^{25} -88.8968 q^{26} +192.091 q^{27} -57.1589 q^{29} +177.910 q^{30} -156.511 q^{31} -32.0000 q^{32} -415.343 q^{33} +48.9587 q^{34} +196.120 q^{36} +131.743 q^{37} +232.951 q^{38} +387.569 q^{39} +81.6144 q^{40} +391.272 q^{41} +144.126 q^{43} -190.535 q^{44} -500.194 q^{45} -46.0000 q^{46} +330.386 q^{47} +139.512 q^{48} +41.8466 q^{50} -213.448 q^{51} +177.794 q^{52} +222.623 q^{53} -384.182 q^{54} +485.950 q^{55} -1015.61 q^{57} +114.318 q^{58} +869.735 q^{59} -355.819 q^{60} -182.244 q^{61} +313.022 q^{62} +64.0000 q^{64} -453.454 q^{65} +830.687 q^{66} +492.053 q^{67} -97.9174 q^{68} +200.549 q^{69} +1085.88 q^{71} -392.240 q^{72} +1104.26 q^{73} -263.485 q^{74} -182.441 q^{75} -465.901 q^{76} -775.137 q^{78} -58.3125 q^{79} -163.229 q^{80} +351.130 q^{81} -782.545 q^{82} -1178.21 q^{83} +249.733 q^{85} -288.251 q^{86} -498.398 q^{87} +381.070 q^{88} +49.9905 q^{89} +1000.39 q^{90} +92.0000 q^{92} -1364.70 q^{93} -660.771 q^{94} +1188.26 q^{95} -279.025 q^{96} -548.910 q^{97} -2335.48 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 22 q^{2} + 44 q^{4} + 23 q^{5} - 88 q^{8} + 171 q^{9} - 46 q^{10} - 48 q^{11} + 77 q^{13} + 104 q^{15} + 176 q^{16} + 97 q^{17} - 342 q^{18} + 138 q^{19} + 92 q^{20} + 96 q^{22} + 253 q^{23} + 30 q^{25}+ \cdots - 2545 q^{99}+O(q^{100}) \) 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

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