LMFDB - Embedded newform 1150.4.a.q.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(1,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1150.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$67.8521965066$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 107x^{3} - 3x^{2} + 2151x - 2916 $ x^5 - 107x^3 - 3x^2 + 2151*x - 2916
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.88892$ of defining polynomial
Character	$\chi$	$=$	1150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +2.88892 q^{3} +4.00000 q^{4} -5.77785 q^{6} +8.21839 q^{7} -8.00000 q^{8} -18.6541 q^{9} -71.5466 q^{11} +11.5557 q^{12} +66.1071 q^{13} -16.4368 q^{14} +16.0000 q^{16} +27.7650 q^{17} +37.3082 q^{18} +13.7670 q^{19} +23.7423 q^{21} +143.093 q^{22} -23.0000 q^{23} -23.1114 q^{24} -132.214 q^{26} -131.891 q^{27} +32.8736 q^{28} +274.558 q^{29} +265.877 q^{31} -32.0000 q^{32} -206.693 q^{33} -55.5300 q^{34} -74.6165 q^{36} -204.865 q^{37} -27.5341 q^{38} +190.978 q^{39} -69.5895 q^{41} -47.4846 q^{42} -187.919 q^{43} -286.186 q^{44} +46.0000 q^{46} -135.207 q^{47} +46.2228 q^{48} -275.458 q^{49} +80.2110 q^{51} +264.429 q^{52} -502.564 q^{53} +263.782 q^{54} -65.7472 q^{56} +39.7719 q^{57} -549.116 q^{58} -161.961 q^{59} -103.522 q^{61} -531.754 q^{62} -153.307 q^{63} +64.0000 q^{64} +413.385 q^{66} +985.315 q^{67} +111.060 q^{68} -66.4452 q^{69} -817.825 q^{71} +149.233 q^{72} -1119.51 q^{73} +409.729 q^{74} +55.0681 q^{76} -587.998 q^{77} -381.957 q^{78} +400.301 q^{79} +122.638 q^{81} +139.179 q^{82} -1113.02 q^{83} +94.9692 q^{84} +375.838 q^{86} +793.177 q^{87} +572.373 q^{88} -485.051 q^{89} +543.295 q^{91} -92.0000 q^{92} +768.098 q^{93} +270.413 q^{94} -92.4455 q^{96} +1036.55 q^{97} +550.916 q^{98} +1334.64 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} - 5 q^{3} + 20 q^{4} + 10 q^{6} + 3 q^{7} - 40 q^{8} + 84 q^{9} - 26 q^{11} - 20 q^{12} + 61 q^{13} - 6 q^{14} + 80 q^{16} - 231 q^{17} - 168 q^{18} + 74 q^{19} - 88 q^{21} + 52 q^{22} - 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) $ 5 * q - 10 * q^2 - 5 * q^3 + 20 * q^4 + 10 * q^6 + 3 * q^7 - 40 * q^8 + 84 * q^9 - 26 * q^11 - 20 * q^12 + 61 * q^13 - 6 * q^14 + 80 * q^16 - 231 * q^17 - 168 * q^18 + 74 * q^19 - 88 * q^21 + 52 * q^22 - 115 * q^23 + 40 * q^24 - 122 * q^26 - 368 * q^27 + 12 * q^28 + 37 * q^29 + 220 * q^31 - 160 * q^32 - 391 * q^33 + 462 * q^34 + 336 * q^36 - 220 * q^37 - 148 * q^38 + 195 * q^39 + 112 * q^41 + 176 * q^42 - 97 * q^43 - 104 * q^44 + 230 * q^46 - 370 * q^47 - 80 * q^48 + 182 * q^49 + 563 * q^51 + 244 * q^52 - 1224 * q^53 + 736 * q^54 - 24 * q^56 + 1009 * q^57 - 74 * q^58 - 181 * q^59 + 1574 * q^61 - 440 * q^62 - 1829 * q^63 + 320 * q^64 + 782 * q^66 + 435 * q^67 - 924 * q^68 + 115 * q^69 - 30 * q^71 - 672 * q^72 - 2180 * q^73 + 440 * q^74 + 296 * q^76 + 1401 * q^77 - 390 * q^78 + 609 * q^79 + 1453 * q^81 - 224 * q^82 - 2018 * q^83 - 352 * q^84 + 194 * q^86 + 1485 * q^87 + 208 * q^88 - 1663 * q^89 + 361 * q^91 - 460 * q^92 - 2741 * q^93 + 740 * q^94 + 160 * q^96 + 1274 * q^97 - 364 * q^98 + 2397 * 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
$3$	2.88892	0.555973	0.277987	−	0.960585i	$-0.410333\pi$
$3$	2.88892	0.555973	0.277987	+	0.960585i	$0.410333\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−5.77785	−0.393133
$7$	8.21839	0.443752	0.221876	−	0.975075i	$-0.428782\pi$
$7$	8.21839	0.443752	0.221876	+	0.975075i	$0.428782\pi$
$8$	−8.00000	−0.353553
$9$	−18.6541	−0.690894
$10$	0	0
$11$	−71.5466	−1.96110	−0.980551	−	0.196265i	$-0.937119\pi$
$11$	−71.5466	−1.96110	−0.980551	+	0.196265i	$0.937119\pi$
$12$	11.5557	0.277987
$13$	66.1071	1.41037	0.705185	−	0.709023i	$-0.250864\pi$
$13$	66.1071	1.41037	0.705185	+	0.709023i	$0.250864\pi$
$14$	−16.4368	−0.313780
$15$	0	0
$16$	16.0000	0.250000
$17$	27.7650	0.396118	0.198059	−	0.980190i	$-0.436536\pi$
$17$	27.7650	0.396118	0.198059	+	0.980190i	$0.436536\pi$
$18$	37.3082	0.488535
$19$	13.7670	0.166230	0.0831151	−	0.996540i	$-0.473513\pi$
$19$	13.7670	0.166230	0.0831151	+	0.996540i	$0.473513\pi$
$20$	0	0
$21$	23.7423	0.246714
$22$	143.093	1.38671
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−23.1114	−0.196566
$25$	0	0
$26$	−132.214	−0.997283
$27$	−131.891	−0.940092
$28$	32.8736	0.221876
$29$	274.558	1.75807	0.879037	−	0.476753i	$-0.158186\pi$
$29$	274.558	1.75807	0.879037	+	0.476753i	$0.158186\pi$
$30$	0	0
$31$	265.877	1.54042	0.770208	−	0.637792i	$-0.220152\pi$
$31$	265.877	1.54042	0.770208	+	0.637792i	$0.220152\pi$
$32$	−32.0000	−0.176777
$33$	−206.693	−1.09032
$34$	−55.5300	−0.280098
$35$	0	0
$36$	−74.6165	−0.345447
$37$	−204.865	−0.910258	−0.455129	−	0.890425i	$-0.650407\pi$
$37$	−204.865	−0.910258	−0.455129	+	0.890425i	$0.650407\pi$
$38$	−27.5341	−0.117542
$39$	190.978	0.784129
$40$	0	0
$41$	−69.5895	−0.265075	−0.132537	−	0.991178i	$-0.542312\pi$
$41$	−69.5895	−0.265075	−0.132537	+	0.991178i	$0.542312\pi$
$42$	−47.4846	−0.174453
$43$	−187.919	−0.666452	−0.333226	−	0.942847i	$-0.608137\pi$
$43$	−187.919	−0.666452	−0.333226	+	0.942847i	$0.608137\pi$
$44$	−286.186	−0.980551
$45$	0	0
$46$	46.0000	0.147442
$47$	−135.207	−0.419615	−0.209808	−	0.977743i	$-0.567284\pi$
$47$	−135.207	−0.419615	−0.209808	+	0.977743i	$0.567284\pi$
$48$	46.2228	0.138993
$49$	−275.458	−0.803085
$50$	0	0
$51$	80.2110	0.220231
$52$	264.429	0.705185
$53$	−502.564	−1.30250	−0.651249	−	0.758864i	$-0.725755\pi$
$53$	−502.564	−1.30250	−0.651249	+	0.758864i	$0.725755\pi$
$54$	263.782	0.664745
$55$	0	0
$56$	−65.7472	−0.156890
$57$	39.7719	0.0924195
$58$	−549.116	−1.24315
$59$	−161.961	−0.357382	−0.178691	−	0.983905i	$-0.557186\pi$
$59$	−161.961	−0.357382	−0.178691	+	0.983905i	$0.557186\pi$
$60$	0	0
$61$	−103.522	−0.217290	−0.108645	−	0.994081i	$-0.534651\pi$
$61$	−103.522	−0.217290	−0.108645	+	0.994081i	$0.534651\pi$
$62$	−531.754	−1.08924
$63$	−153.307	−0.306585
$64$	64.0000	0.125000
$65$	0	0
$66$	413.385	0.770973
$67$	985.315	1.79665	0.898324	−	0.439334i	$-0.144785\pi$
$67$	985.315	1.79665	0.898324	+	0.439334i	$0.144785\pi$
$68$	111.060	0.198059
$69$	−66.4452	−0.115928
$70$	0	0
$71$	−817.825	−1.36701	−0.683507	−	0.729944i	$-0.739546\pi$
$71$	−817.825	−1.36701	−0.683507	+	0.729944i	$0.739546\pi$
$72$	149.233	0.244268
$73$	−1119.51	−1.79492	−0.897460	−	0.441096i	$-0.854590\pi$
$73$	−1119.51	−1.79492	−0.897460	+	0.441096i	$0.854590\pi$
$74$	409.729	0.643650
$75$	0	0
$76$	55.0681	0.0831151
$77$	−587.998	−0.870242
$78$	−381.957	−0.554463
$79$	400.301	0.570094	0.285047	−	0.958514i	$-0.407991\pi$
$79$	400.301	0.570094	0.285047	+	0.958514i	$0.407991\pi$
$80$	0	0
$81$	122.638	0.168227
$82$	139.179	0.187436
$83$	−1113.02	−1.47193	−0.735964	−	0.677020i	$-0.763271\pi$
$83$	−1113.02	−1.47193	−0.735964	+	0.677020i	$0.763271\pi$
$84$	94.9692	0.123357
$85$	0	0
$86$	375.838	0.471252
$87$	793.177	0.977443
$88$	572.373	0.693354
$89$	−485.051	−0.577700	−0.288850	−	0.957374i	$-0.593273\pi$
$89$	−485.051	−0.577700	−0.288850	+	0.957374i	$0.593273\pi$
$90$	0	0
$91$	543.295	0.625854
$92$	−92.0000	−0.104257
$93$	768.098	0.856431
$94$	270.413	0.296713
$95$	0	0
$96$	−92.4455	−0.0982832
$97$	1036.55	1.08500	0.542501	−	0.840055i	$-0.317477\pi$
$97$	1036.55	1.08500	0.542501	+	0.840055i	$0.317477\pi$
$98$	550.916	0.567867
$99$	1334.64	1.35491
$100$	0	0
$101$	−1137.44	−1.12059	−0.560295	−	0.828293i	$-0.689312\pi$
$101$	−1137.44	−1.12059	−0.560295	+	0.828293i	$0.689312\pi$
$102$	−160.422	−0.155727
$103$	−102.870	−0.0984082	−0.0492041	−	0.998789i	$-0.515668\pi$
$103$	−102.870	−0.0984082	−0.0492041	+	0.998789i	$0.515668\pi$
$104$	−528.857	−0.498641
$105$	0	0
$106$	1005.13	0.921006
$107$	607.332	0.548720	0.274360	−	0.961627i	$-0.411534\pi$
$107$	607.332	0.548720	0.274360	+	0.961627i	$0.411534\pi$
$108$	−527.565	−0.470046
$109$	−2143.66	−1.88372	−0.941861	−	0.336002i	$-0.890925\pi$
$109$	−2143.66	−1.88372	−0.941861	+	0.336002i	$0.890925\pi$
$110$	0	0
$111$	−591.838	−0.506079
$112$	131.494	0.110938
$113$	−642.814	−0.535140	−0.267570	−	0.963538i	$-0.586221\pi$
$113$	−642.814	−0.535140	−0.267570	+	0.963538i	$0.586221\pi$
$114$	−79.5438	−0.0653505
$115$	0	0
$116$	1098.23	0.879037
$117$	−1233.17	−0.974416
$118$	323.923	0.252708
$119$	228.184	0.175778
$120$	0	0
$121$	3787.92	2.84592
$122$	207.045	0.153647
$123$	−201.039	−0.147374
$124$	1063.51	0.770208
$125$	0	0
$126$	306.614	0.216788
$127$	−819.689	−0.572722	−0.286361	−	0.958122i	$-0.592446\pi$
$127$	−819.689	−0.572722	−0.286361	+	0.958122i	$0.592446\pi$
$128$	−128.000	−0.0883883
$129$	−542.884	−0.370529
$130$	0	0
$131$	1730.93	1.15445	0.577223	−	0.816587i	$-0.304136\pi$
$131$	1730.93	1.15445	0.577223	+	0.816587i	$0.304136\pi$
$132$	−826.771	−0.545160
$133$	113.143	0.0737649
$134$	−1970.63	−1.27042
$135$	0	0
$136$	−222.120	−0.140049
$137$	−2385.91	−1.48790	−0.743950	−	0.668235i	$-0.767050\pi$
$137$	−2385.91	−1.48790	−0.743950	+	0.668235i	$0.767050\pi$
$138$	132.890	0.0819738
$139$	416.788	0.254327	0.127164	−	0.991882i	$-0.459413\pi$
$139$	416.788	0.254327	0.127164	+	0.991882i	$0.459413\pi$
$140$	0	0
$141$	−390.602	−0.233295
$142$	1635.65	0.966625
$143$	−4729.74	−2.76588
$144$	−298.466	−0.172723
$145$	0	0
$146$	2239.03	1.26920
$147$	−795.777	−0.446494
$148$	−819.459	−0.455129
$149$	−3026.53	−1.66405	−0.832025	−	0.554739i	$-0.812818\pi$
$149$	−3026.53	−1.66405	−0.832025	+	0.554739i	$0.812818\pi$
$150$	0	0
$151$	−734.096	−0.395628	−0.197814	−	0.980240i	$-0.563384\pi$
$151$	−734.096	−0.395628	−0.197814	+	0.980240i	$0.563384\pi$
$152$	−110.136	−0.0587712
$153$	−517.932	−0.273675
$154$	1176.00	0.615354
$155$	0	0
$156$	763.914	0.392064
$157$	−501.455	−0.254908	−0.127454	−	0.991845i	$-0.540680\pi$
$157$	−501.455	−0.254908	−0.127454	+	0.991845i	$0.540680\pi$
$158$	−800.603	−0.403117
$159$	−1451.87	−0.724155
$160$	0	0
$161$	−189.023	−0.0925286
$162$	−245.275	−0.118955
$163$	−976.374	−0.469175	−0.234588	−	0.972095i	$-0.575374\pi$
$163$	−976.374	−0.469175	−0.234588	+	0.972095i	$0.575374\pi$
$164$	−278.358	−0.132537
$165$	0	0
$166$	2226.04	1.04081
$167$	2431.80	1.12681	0.563407	−	0.826179i	$-0.309490\pi$
$167$	2431.80	1.12681	0.563407	+	0.826179i	$0.309490\pi$
$168$	−189.938	−0.0872266
$169$	2173.15	0.989146
$170$	0	0
$171$	−256.812	−0.114847
$172$	−751.677	−0.333226
$173$	847.107	0.372280	0.186140	−	0.982523i	$-0.440402\pi$
$173$	847.107	0.372280	0.186140	+	0.982523i	$0.440402\pi$
$174$	−1586.35	−0.691156
$175$	0	0
$176$	−1144.75	−0.490275
$177$	−467.894	−0.198695
$178$	970.102	0.408496
$179$	4060.04	1.69532	0.847658	−	0.530544i	$-0.178012\pi$
$179$	4060.04	1.69532	0.847658	+	0.530544i	$0.178012\pi$
$180$	0	0
$181$	−25.2071	−0.0103516	−0.00517578	−	0.999987i	$-0.501648\pi$
$181$	−25.2071	−0.0103516	−0.00517578	+	0.999987i	$0.501648\pi$
$182$	−1086.59	−0.442546
$183$	−299.068	−0.120807
$184$	184.000	0.0737210
$185$	0	0
$186$	−1536.20	−0.605588
$187$	−1986.49	−0.776828
$188$	−540.827	−0.209808
$189$	−1083.93	−0.417167
$190$	0	0
$191$	−1603.82	−0.607585	−0.303792	−	0.952738i	$-0.598253\pi$
$191$	−1603.82	−0.607585	−0.303792	+	0.952738i	$0.598253\pi$
$192$	184.891	0.0694967
$193$	2754.90	1.02747	0.513735	−	0.857949i	$-0.328261\pi$
$193$	2754.90	1.02747	0.513735	+	0.857949i	$0.328261\pi$
$194$	−2073.09	−0.767212
$195$	0	0
$196$	−1101.83	−0.401542
$197$	−2406.44	−0.870314	−0.435157	−	0.900355i	$-0.643307\pi$
$197$	−2406.44	−0.870314	−0.435157	+	0.900355i	$0.643307\pi$
$198$	−2669.28	−0.958068
$199$	−5565.61	−1.98259	−0.991295	−	0.131660i	$-0.957969\pi$
$199$	−5565.61	−1.98259	−0.991295	+	0.131660i	$0.957969\pi$
$200$	0	0
$201$	2846.50	0.998888
$202$	2274.88	0.792377
$203$	2256.43	0.780148
$204$	320.844	0.110116
$205$	0	0
$206$	205.739	0.0695851
$207$	429.045	0.144061
$208$	1057.71	0.352593
$209$	−984.984	−0.325994
$210$	0	0
$211$	−1477.89	−0.482189	−0.241095	−	0.970502i	$-0.577506\pi$
$211$	−1477.89	−0.482189	−0.241095	+	0.970502i	$0.577506\pi$
$212$	−2010.25	−0.651249
$213$	−2362.63	−0.760024
$214$	−1214.66	−0.388004
$215$	0	0
$216$	1055.13	0.332373
$217$	2185.08	0.683562
$218$	4287.33	1.33199
$219$	−3234.19	−0.997928
$220$	0	0
$221$	1835.47	0.558673
$222$	1183.68	0.357852
$223$	−6265.46	−1.88146	−0.940732	−	0.339152i	$-0.889860\pi$
$223$	−6265.46	−1.88146	−0.940732	+	0.339152i	$0.889860\pi$
$224$	−262.989	−0.0784449
$225$	0	0
$226$	1285.63	0.378401
$227$	−5108.12	−1.49356	−0.746780	−	0.665072i	$-0.768401\pi$
$227$	−5108.12	−1.49356	−0.746780	+	0.665072i	$0.768401\pi$
$228$	159.088	0.0462098
$229$	4598.10	1.32686	0.663430	−	0.748239i	$-0.269100\pi$
$229$	4598.10	1.32686	0.663430	+	0.748239i	$0.269100\pi$
$230$	0	0
$231$	−1698.68	−0.483831
$232$	−2196.46	−0.621573
$233$	−3746.09	−1.05328	−0.526640	−	0.850088i	$-0.676548\pi$
$233$	−3746.09	−1.05328	−0.526640	+	0.850088i	$0.676548\pi$
$234$	2466.34	0.689016
$235$	0	0
$236$	−647.845	−0.178691
$237$	1156.44	0.316957
$238$	−456.368	−0.124294
$239$	4409.95	1.19354	0.596770	−	0.802413i	$-0.296451\pi$
$239$	4409.95	1.19354	0.596770	+	0.802413i	$0.296451\pi$
$240$	0	0
$241$	−2747.38	−0.734334	−0.367167	−	0.930155i	$-0.619672\pi$
$241$	−2747.38	−0.734334	−0.367167	+	0.930155i	$0.619672\pi$
$242$	−7575.84	−2.01237
$243$	3915.35	1.03362
$244$	−414.090	−0.108645
$245$	0	0
$246$	402.078	0.104209
$247$	910.099	0.234446
$248$	−2127.02	−0.544620
$249$	−3215.44	−0.818353
$250$	0	0
$251$	3985.25	1.00218	0.501090	−	0.865395i	$-0.332933\pi$
$251$	3985.25	1.00218	0.501090	+	0.865395i	$0.332933\pi$
$252$	−613.228	−0.153293
$253$	1645.57	0.408918
$254$	1639.38	0.404975
$255$	0	0
$256$	256.000	0.0625000
$257$	7998.42	1.94135	0.970677	−	0.240388i	$-0.0772746\pi$
$257$	7998.42	1.94135	0.970677	+	0.240388i	$0.0772746\pi$
$258$	1085.77	0.262004
$259$	−1683.66	−0.403929
$260$	0	0
$261$	−5121.64	−1.21464
$262$	−3461.87	−0.816317
$263$	−3988.79	−0.935207	−0.467603	−	0.883938i	$-0.654882\pi$
$263$	−3988.79	−0.935207	−0.467603	+	0.883938i	$0.654882\pi$
$264$	1653.54	0.385487
$265$	0	0
$266$	−226.286	−0.0521597
$267$	−1401.28	−0.321186
$268$	3941.26	0.898324
$269$	−5546.77	−1.25722	−0.628611	−	0.777720i	$-0.716376\pi$
$269$	−5546.77	−1.25722	−0.628611	+	0.777720i	$0.716376\pi$
$270$	0	0
$271$	2660.23	0.596300	0.298150	−	0.954519i	$-0.403630\pi$
$271$	2660.23	0.596300	0.298150	+	0.954519i	$0.403630\pi$
$272$	444.240	0.0990295
$273$	1569.54	0.347958
$274$	4771.83	1.05210
$275$	0	0
$276$	−265.781	−0.0579642
$277$	−1756.71	−0.381048	−0.190524	−	0.981683i	$-0.561019\pi$
$277$	−1756.71	−0.381048	−0.190524	+	0.981683i	$0.561019\pi$
$278$	−833.576	−0.179836
$279$	−4959.70	−1.06426
$280$	0	0
$281$	−3137.33	−0.666041	−0.333020	−	0.942920i	$-0.608068\pi$
$281$	−3137.33	−0.666041	−0.333020	+	0.942920i	$0.608068\pi$
$282$	781.203	0.164964
$283$	−8279.78	−1.73916	−0.869579	−	0.493793i	$-0.835610\pi$
$283$	−8279.78	−1.73916	−0.869579	+	0.493793i	$0.835610\pi$
$284$	−3271.30	−0.683507
$285$	0	0
$286$	9459.48	1.95577
$287$	−571.914	−0.117627
$288$	596.932	0.122134
$289$	−4142.10	−0.843091
$290$	0	0
$291$	2994.50	0.603232
$292$	−4478.06	−0.897460
$293$	−1677.69	−0.334510	−0.167255	−	0.985914i	$-0.553490\pi$
$293$	−1677.69	−0.334510	−0.167255	+	0.985914i	$0.553490\pi$
$294$	1591.55	0.315719
$295$	0	0
$296$	1638.92	0.321825
$297$	9436.37	1.84362
$298$	6053.07	1.17666
$299$	−1520.46	−0.294083
$300$	0	0
$301$	−1544.39	−0.295739
$302$	1468.19	0.279752
$303$	−3285.98	−0.623018
$304$	220.272	0.0415575
$305$	0	0
$306$	1035.86	0.193518
$307$	−1021.48	−0.189899	−0.0949497	−	0.995482i	$-0.530269\pi$
$307$	−1021.48	−0.189899	−0.0949497	+	0.995482i	$0.530269\pi$
$308$	−2351.99	−0.435121
$309$	−297.183	−0.0547124
$310$	0	0
$311$	−492.850	−0.0898617	−0.0449308	−	0.998990i	$-0.514307\pi$
$311$	−492.850	−0.0898617	−0.0449308	+	0.998990i	$0.514307\pi$
$312$	−1527.83	−0.277231
$313$	−1711.97	−0.309157	−0.154579	−	0.987980i	$-0.549402\pi$
$313$	−1711.97	−0.309157	−0.154579	+	0.987980i	$0.549402\pi$
$314$	1002.91	0.180247
$315$	0	0
$316$	1601.21	0.285047
$317$	509.170	0.0902140	0.0451070	−	0.998982i	$-0.485637\pi$
$317$	509.170	0.0902140	0.0451070	+	0.998982i	$0.485637\pi$
$318$	2903.74	0.512055
$319$	−19643.7	−3.44776
$320$	0	0
$321$	1754.54	0.305074
$322$	378.046	0.0654276
$323$	382.242	0.0658467
$324$	490.551	0.0841137
$325$	0	0
$326$	1952.75	0.331757
$327$	−6192.88	−1.04730
$328$	556.716	0.0937180
$329$	−1111.18	−0.186205
$330$	0	0
$331$	436.915	0.0725529	0.0362764	−	0.999342i	$-0.488450\pi$
$331$	436.915	0.0725529	0.0362764	+	0.999342i	$0.488450\pi$
$332$	−4452.09	−0.735964
$333$	3821.57	0.628891
$334$	−4863.59	−0.796778
$335$	0	0
$336$	379.877	0.0616785
$337$	−10300.6	−1.66502	−0.832508	−	0.554013i	$-0.813096\pi$
$337$	−10300.6	−1.66502	−0.832508	+	0.554013i	$0.813096\pi$
$338$	−4346.31	−0.699432
$339$	−1857.04	−0.297524
$340$	0	0
$341$	−19022.6	−3.02091
$342$	513.624	0.0812093
$343$	−5082.73	−0.800122
$344$	1503.35	0.235626
$345$	0	0
$346$	−1694.21	−0.263241
$347$	−54.1055	−0.00837042	−0.00418521	−	0.999991i	$-0.501332\pi$
$347$	−54.1055	−0.00837042	−0.00418521	+	0.999991i	$0.501332\pi$
$348$	3172.71	0.488721
$349$	9862.75	1.51272	0.756362	−	0.654153i	$-0.226975\pi$
$349$	9862.75	1.51272	0.756362	+	0.654153i	$0.226975\pi$
$350$	0	0
$351$	−8718.95	−1.32588
$352$	2289.49	0.346677
$353$	−7397.97	−1.11545	−0.557726	−	0.830025i	$-0.688326\pi$
$353$	−7397.97	−1.11545	−0.557726	+	0.830025i	$0.688326\pi$
$354$	935.787	0.140499
$355$	0	0
$356$	−1940.20	−0.288850
$357$	659.206	0.0977279
$358$	−8120.07	−1.19877
$359$	7112.06	1.04557	0.522786	−	0.852464i	$-0.324893\pi$
$359$	7112.06	1.04557	0.522786	+	0.852464i	$0.324893\pi$
$360$	0	0
$361$	−6669.47	−0.972368
$362$	50.4143	0.00731966
$363$	10943.0	1.58226
$364$	2173.18	0.312927
$365$	0	0
$366$	598.136	0.0854237
$367$	10728.0	1.52588	0.762939	−	0.646470i	$-0.223755\pi$
$367$	10728.0	1.52588	0.762939	+	0.646470i	$0.223755\pi$
$368$	−368.000	−0.0521286
$369$	1298.13	0.183138
$370$	0	0
$371$	−4130.27	−0.577986
$372$	3072.39	0.428215
$373$	2519.27	0.349713	0.174856	−	0.984594i	$-0.444054\pi$
$373$	2519.27	0.349713	0.174856	+	0.984594i	$0.444054\pi$
$374$	3972.99	0.549300
$375$	0	0
$376$	1081.65	0.148356
$377$	18150.2	2.47954
$378$	2167.87	0.294982
$379$	2479.13	0.336001	0.168001	−	0.985787i	$-0.446269\pi$
$379$	2479.13	0.336001	0.168001	+	0.985787i	$0.446269\pi$
$380$	0	0
$381$	−2368.02	−0.318418
$382$	3207.65	0.429627
$383$	1310.10	0.174786	0.0873928	−	0.996174i	$-0.472146\pi$
$383$	1310.10	0.174786	0.0873928	+	0.996174i	$0.472146\pi$
$384$	−369.782	−0.0491416
$385$	0	0
$386$	−5509.79	−0.726531
$387$	3505.47	0.460447
$388$	4146.18	0.542501
$389$	11197.1	1.45942	0.729712	−	0.683755i	$-0.239654\pi$
$389$	11197.1	1.45942	0.729712	+	0.683755i	$0.239654\pi$
$390$	0	0
$391$	−638.595	−0.0825963
$392$	2203.66	0.283933
$393$	5000.54	0.641841
$394$	4812.88	0.615405
$395$	0	0
$396$	5338.56	0.677456
$397$	−8185.02	−1.03475	−0.517373	−	0.855760i	$-0.673090\pi$
$397$	−8185.02	−1.03475	−0.517373	+	0.855760i	$0.673090\pi$
$398$	11131.2	1.40190
$399$	326.861	0.0410113
$400$	0	0
$401$	5779.36	0.719719	0.359860	−	0.933006i	$-0.382825\pi$
$401$	5779.36	0.719719	0.359860	+	0.933006i	$0.382825\pi$
$402$	−5693.00	−0.706321
$403$	17576.4	2.17256
$404$	−4549.76	−0.560295
$405$	0	0
$406$	−4512.85	−0.551648
$407$	14657.4	1.78511
$408$	−641.688	−0.0778635
$409$	11007.3	1.33074	0.665372	−	0.746512i	$-0.268273\pi$
$409$	11007.3	1.33074	0.665372	+	0.746512i	$0.268273\pi$
$410$	0	0
$411$	−6892.72	−0.827233
$412$	−411.479	−0.0492041
$413$	−1331.06	−0.158589
$414$	−858.090	−0.101867
$415$	0	0
$416$	−2115.43	−0.249321
$417$	1204.07	0.141399
$418$	1969.97	0.230513
$419$	−3827.76	−0.446297	−0.223148	−	0.974784i	$-0.571633\pi$
$419$	−3827.76	−0.446297	−0.223148	+	0.974784i	$0.571633\pi$
$420$	0	0
$421$	6671.63	0.772341	0.386170	−	0.922428i	$-0.373798\pi$
$421$	6671.63	0.772341	0.386170	+	0.922428i	$0.373798\pi$
$422$	2955.77	0.340959
$423$	2522.16	0.289909
$424$	4020.51	0.460503
$425$	0	0
$426$	4725.27	0.537418
$427$	−850.788	−0.0964227
$428$	2429.33	0.274360
$429$	−13663.9	−1.53776
$430$	0	0
$431$	2813.78	0.314466	0.157233	−	0.987562i	$-0.449743\pi$
$431$	2813.78	0.314466	0.157233	+	0.987562i	$0.449743\pi$
$432$	−2110.26	−0.235023
$433$	14651.4	1.62610	0.813051	−	0.582193i	$-0.197805\pi$
$433$	14651.4	1.62610	0.813051	+	0.582193i	$0.197805\pi$
$434$	−4370.16	−0.483352
$435$	0	0
$436$	−8574.65	−0.941861
$437$	−316.642	−0.0346614
$438$	6468.38	0.705642
$439$	15021.5	1.63311	0.816557	−	0.577265i	$-0.195880\pi$
$439$	15021.5	1.63311	0.816557	+	0.577265i	$0.195880\pi$
$440$	0	0
$441$	5138.43	0.554846
$442$	−3670.93	−0.395042
$443$	−14722.2	−1.57895	−0.789473	−	0.613785i	$-0.789646\pi$
$443$	−14722.2	−1.57895	−0.789473	+	0.613785i	$0.789646\pi$
$444$	−2367.35	−0.253040
$445$	0	0
$446$	12530.9	1.33040
$447$	−8743.42	−0.925167
$448$	525.977	0.0554690
$449$	−6954.45	−0.730960	−0.365480	−	0.930819i	$-0.619095\pi$
$449$	−6954.45	−0.730960	−0.365480	+	0.930819i	$0.619095\pi$
$450$	0	0
$451$	4978.90	0.519838
$452$	−2571.26	−0.267570
$453$	−2120.75	−0.219959
$454$	10216.2	1.05611
$455$	0	0
$456$	−318.175	−0.0326752
$457$	911.330	0.0932828	0.0466414	−	0.998912i	$-0.485148\pi$
$457$	911.330	0.0932828	0.0466414	+	0.998912i	$0.485148\pi$
$458$	−9196.19	−0.938231
$459$	−3661.96	−0.372387
$460$	0	0
$461$	−5709.58	−0.576836	−0.288418	−	0.957505i	$-0.593129\pi$
$461$	−5709.58	−0.576836	−0.288418	+	0.957505i	$0.593129\pi$
$462$	3397.36	0.342121
$463$	−5728.63	−0.575015	−0.287508	−	0.957778i	$-0.592827\pi$
$463$	−5728.63	−0.575015	−0.287508	+	0.957778i	$0.592827\pi$
$464$	4392.93	0.439519
$465$	0	0
$466$	7492.17	0.744781
$467$	4467.88	0.442717	0.221358	−	0.975193i	$-0.428951\pi$
$467$	4467.88	0.442717	0.221358	+	0.975193i	$0.428951\pi$
$468$	−4932.68	−0.487208
$469$	8097.71	0.797265
$470$	0	0
$471$	−1448.67	−0.141722
$472$	1295.69	0.126354
$473$	13445.0	1.30698
$474$	−2312.88	−0.224123
$475$	0	0
$476$	912.736	0.0878890
$477$	9374.88	0.899888
$478$	−8819.90	−0.843960
$479$	7246.30	0.691215	0.345607	−	0.938379i	$-0.387673\pi$
$479$	7246.30	0.691215	0.345607	+	0.938379i	$0.387673\pi$
$480$	0	0
$481$	−13543.0	−1.28380
$482$	5494.77	0.519253
$483$	−546.073	−0.0514434
$484$	15151.7	1.42296
$485$	0	0
$486$	−7830.71	−0.730881
$487$	9362.52	0.871162	0.435581	−	0.900149i	$-0.356543\pi$
$487$	9362.52	0.871162	0.435581	+	0.900149i	$0.356543\pi$
$488$	828.179	0.0768236
$489$	−2820.67	−0.260849
$490$	0	0
$491$	10220.5	0.939401	0.469700	−	0.882826i	$-0.344362\pi$
$491$	10220.5	0.939401	0.469700	+	0.882826i	$0.344362\pi$
$492$	−804.155	−0.0736872
$493$	7623.11	0.696405
$494$	−1820.20	−0.165778
$495$	0	0
$496$	4254.03	0.385104
$497$	−6721.21	−0.606615
$498$	6430.87	0.578663
$499$	10463.5	0.938698	0.469349	−	0.883013i	$-0.344489\pi$
$499$	10463.5	0.938698	0.469349	+	0.883013i	$0.344489\pi$
$500$	0	0
$501$	7025.27	0.626479
$502$	−7970.51	−0.708648
$503$	−2129.28	−0.188748	−0.0943738	−	0.995537i	$-0.530085\pi$
$503$	−2129.28	−0.188748	−0.0943738	+	0.995537i	$0.530085\pi$
$504$	1226.46	0.108394
$505$	0	0
$506$	−3291.14	−0.289149
$507$	6278.07	0.549939
$508$	−3278.76	−0.286361
$509$	2890.57	0.251713	0.125857	−	0.992048i	$-0.459832\pi$
$509$	2890.57	0.251713	0.125857	+	0.992048i	$0.459832\pi$
$510$	0	0
$511$	−9200.61	−0.796499
$512$	−512.000	−0.0441942
$513$	−1815.75	−0.156272
$514$	−15996.8	−1.37274
$515$	0	0
$516$	−2171.54	−0.185265
$517$	9673.58	0.822908
$518$	3367.32	0.285621
$519$	2447.23	0.206978
$520$	0	0
$521$	14109.0	1.18642	0.593212	−	0.805046i	$-0.297860\pi$
$521$	14109.0	1.18642	0.593212	+	0.805046i	$0.297860\pi$
$522$	10243.3	0.858882
$523$	−7407.28	−0.619307	−0.309654	−	0.950849i	$-0.600213\pi$
$523$	−7407.28	−0.619307	−0.309654	+	0.950849i	$0.600213\pi$
$524$	6923.74	0.577223
$525$	0	0
$526$	7977.58	0.661291
$527$	7382.08	0.610187
$528$	−3307.08	−0.272580
$529$	529.000	0.0434783
$530$	0	0
$531$	3021.25	0.246913
$532$	452.571	0.0368824
$533$	−4600.36	−0.373853
$534$	2802.55	0.227113
$535$	0	0
$536$	−7882.52	−0.635211
$537$	11729.1	0.942550
$538$	11093.5	0.888990
$539$	19708.1	1.57493
$540$	0	0
$541$	−2546.42	−0.202365	−0.101182	−	0.994868i	$-0.532263\pi$
$541$	−2546.42	−0.202365	−0.101182	+	0.994868i	$0.532263\pi$
$542$	−5320.45	−0.421648
$543$	−72.8215	−0.00575519
$544$	−888.481	−0.0700244
$545$	0	0
$546$	−3139.07	−0.246044
$547$	−10521.1	−0.822397	−0.411199	−	0.911546i	$-0.634890\pi$
$547$	−10521.1	−0.822397	−0.411199	+	0.911546i	$0.634890\pi$
$548$	−9543.66	−0.743950
$549$	1931.12	0.150124
$550$	0	0
$551$	3779.85	0.292245
$552$	531.562	0.0409869
$553$	3289.83	0.252980
$554$	3513.41	0.269442
$555$	0	0
$556$	1667.15	0.127164
$557$	6773.86	0.515292	0.257646	−	0.966239i	$-0.417053\pi$
$557$	6773.86	0.515292	0.257646	+	0.966239i	$0.417053\pi$
$558$	9919.41	0.752548
$559$	−12422.8	−0.939944
$560$	0	0
$561$	−5738.83	−0.431896
$562$	6274.66	0.470962
$563$	1102.09	0.0825004	0.0412502	−	0.999149i	$-0.486866\pi$
$563$	1102.09	0.0825004	0.0412502	+	0.999149i	$0.486866\pi$
$564$	−1562.41	−0.116647
$565$	0	0
$566$	16559.6	1.22977
$567$	1007.89	0.0746512
$568$	6542.60	0.483312
$569$	11348.7	0.836137	0.418069	−	0.908415i	$-0.362707\pi$
$569$	11348.7	0.836137	0.418069	+	0.908415i	$0.362707\pi$
$570$	0	0
$571$	−5324.14	−0.390207	−0.195104	−	0.980783i	$-0.562504\pi$
$571$	−5324.14	−0.390207	−0.195104	+	0.980783i	$0.562504\pi$
$572$	−18919.0	−1.38294
$573$	−4633.33	−0.337801
$574$	1143.83	0.0831750
$575$	0	0
$576$	−1193.86	−0.0863617
$577$	−3382.91	−0.244077	−0.122039	−	0.992525i	$-0.538943\pi$
$577$	−3382.91	−0.244077	−0.122039	+	0.992525i	$0.538943\pi$
$578$	8284.21	0.596155
$579$	7958.68	0.571246
$580$	0	0
$581$	−9147.26	−0.653171
$582$	−5989.00	−0.426550
$583$	35956.7	2.55433
$584$	8956.11	0.634600
$585$	0	0
$586$	3355.37	0.236534
$587$	−5792.93	−0.407325	−0.203662	−	0.979041i	$-0.565285\pi$
$587$	−5792.93	−0.407325	−0.203662	+	0.979041i	$0.565285\pi$
$588$	−3183.11	−0.223247
$589$	3660.34	0.256064
$590$	0	0
$591$	−6952.02	−0.483871
$592$	−3277.84	−0.227565
$593$	−8483.73	−0.587496	−0.293748	−	0.955883i	$-0.594903\pi$
$593$	−8483.73	−0.587496	−0.293748	+	0.955883i	$0.594903\pi$
$594$	−18872.7	−1.30363
$595$	0	0
$596$	−12106.1	−0.832025
$597$	−16078.6	−1.10227
$598$	3040.93	0.207948
$599$	−10700.3	−0.729888	−0.364944	−	0.931030i	$-0.618912\pi$
$599$	−10700.3	−0.729888	−0.364944	+	0.931030i	$0.618912\pi$
$600$	0	0
$601$	−23988.6	−1.62815	−0.814074	−	0.580761i	$-0.802755\pi$
$601$	−23988.6	−1.62815	−0.814074	+	0.580761i	$0.802755\pi$
$602$	3088.79	0.209119
$603$	−18380.2	−1.24129
$604$	−2936.38	−0.197814
$605$	0	0
$606$	6571.96	0.440541
$607$	10243.8	0.684978	0.342489	−	0.939522i	$-0.388730\pi$
$607$	10243.8	0.684978	0.342489	+	0.939522i	$0.388730\pi$
$608$	−440.545	−0.0293856
$609$	6518.64	0.433742
$610$	0	0
$611$	−8938.12	−0.591813
$612$	−2071.73	−0.136838
$613$	−6397.72	−0.421536	−0.210768	−	0.977536i	$-0.567596\pi$
$613$	−6397.72	−0.421536	−0.210768	+	0.977536i	$0.567596\pi$
$614$	2042.97	0.134279
$615$	0	0
$616$	4703.99	0.307677
$617$	−12688.7	−0.827923	−0.413962	−	0.910294i	$-0.635855\pi$
$617$	−12688.7	−0.827923	−0.413962	+	0.910294i	$0.635855\pi$
$618$	594.365	0.0386875
$619$	11955.6	0.776310	0.388155	−	0.921594i	$-0.373113\pi$
$619$	11955.6	0.776310	0.388155	+	0.921594i	$0.373113\pi$
$620$	0	0
$621$	3033.50	0.196023
$622$	985.701	0.0635418
$623$	−3986.34	−0.256355
$624$	3055.65	0.196032
$625$	0	0
$626$	3423.94	0.218607
$627$	−2845.54	−0.181244
$628$	−2005.82	−0.127454
$629$	−5688.07	−0.360570
$630$	0	0
$631$	5947.90	0.375249	0.187625	−	0.982241i	$-0.439921\pi$
$631$	5947.90	0.375249	0.187625	+	0.982241i	$0.439921\pi$
$632$	−3202.41	−0.201559
$633$	−4269.50	−0.268084
$634$	−1018.34	−0.0637909
$635$	0	0
$636$	−5807.47	−0.362077
$637$	−18209.7	−1.13265
$638$	39287.4	2.43794
$639$	15255.8	0.944461
$640$	0	0
$641$	−27171.2	−1.67425	−0.837127	−	0.547009i	$-0.815766\pi$
$641$	−27171.2	−1.67425	−0.837127	+	0.547009i	$0.815766\pi$
$642$	−3509.07	−0.215720
$643$	17656.8	1.08292	0.541460	−	0.840726i	$-0.317872\pi$
$643$	17656.8	1.08292	0.541460	+	0.840726i	$0.317872\pi$
$644$	−756.092	−0.0462643
$645$	0	0
$646$	−764.483	−0.0465607
$647$	−8260.17	−0.501917	−0.250959	−	0.967998i	$-0.580746\pi$
$647$	−8260.17	−0.501917	−0.250959	+	0.967998i	$0.580746\pi$
$648$	−981.102	−0.0594773
$649$	11587.8	0.700863
$650$	0	0
$651$	6312.53	0.380043
$652$	−3905.50	−0.234588
$653$	25187.0	1.50941	0.754705	−	0.656065i	$-0.227780\pi$
$653$	25187.0	1.50941	0.754705	+	0.656065i	$0.227780\pi$
$654$	12385.8	0.740553
$655$	0	0
$656$	−1113.43	−0.0662686
$657$	20883.5	1.24010
$658$	2222.36	0.131667
$659$	15212.2	0.899218	0.449609	−	0.893225i	$-0.351563\pi$
$659$	15212.2	0.899218	0.449609	+	0.893225i	$0.351563\pi$
$660$	0	0
$661$	3311.16	0.194840	0.0974201	−	0.995243i	$-0.468941\pi$
$661$	3311.16	0.194840	0.0974201	+	0.995243i	$0.468941\pi$
$662$	−873.830	−0.0513026
$663$	5302.52	0.310608
$664$	8904.18	0.520405
$665$	0	0
$666$	−7643.14	−0.444693
$667$	−6314.84	−0.366584
$668$	9727.19	0.563407
$669$	−18100.4	−1.04604
$670$	0	0
$671$	7406.68	0.426128
$672$	−759.754	−0.0436133
$673$	6194.51	0.354801	0.177400	−	0.984139i	$-0.443231\pi$
$673$	6194.51	0.354801	0.177400	+	0.984139i	$0.443231\pi$
$674$	20601.2	1.17734
$675$	0	0
$676$	8692.61	0.494573
$677$	−16181.4	−0.918613	−0.459307	−	0.888278i	$-0.651902\pi$
$677$	−16181.4	−0.918613	−0.459307	+	0.888278i	$0.651902\pi$
$678$	3714.08	0.210381
$679$	8518.74	0.481471
$680$	0	0
$681$	−14757.0	−0.830379
$682$	38045.2	2.13611
$683$	3372.91	0.188962	0.0944810	−	0.995527i	$-0.469881\pi$
$683$	3372.91	0.188962	0.0944810	+	0.995527i	$0.469881\pi$
$684$	−1027.25	−0.0574237
$685$	0	0
$686$	10165.5	0.565771
$687$	13283.5	0.737699
$688$	−3006.71	−0.166613
$689$	−33223.0	−1.83701
$690$	0	0
$691$	32121.7	1.76841	0.884203	−	0.467102i	$-0.154702\pi$
$691$	32121.7	1.76841	0.884203	+	0.467102i	$0.154702\pi$
$692$	3388.43	0.186140
$693$	10968.6	0.601245
$694$	108.211	0.00591878
$695$	0	0
$696$	−6345.42	−0.345578
$697$	−1932.15	−0.105001
$698$	−19725.5	−1.06966
$699$	−10822.2	−0.585596
$700$	0	0
$701$	−29151.7	−1.57068	−0.785339	−	0.619066i	$-0.787511\pi$
$701$	−29151.7	−1.57068	−0.785339	+	0.619066i	$0.787511\pi$
$702$	17437.9	0.937537
$703$	−2820.38	−0.151312
$704$	−4578.98	−0.245138
$705$	0	0
$706$	14795.9	0.788743
$707$	−9347.94	−0.497264
$708$	−1871.57	−0.0993476
$709$	−12848.7	−0.680594	−0.340297	−	0.940318i	$-0.610528\pi$
$709$	−12848.7	−0.680594	−0.340297	+	0.940318i	$0.610528\pi$
$710$	0	0
$711$	−7467.27	−0.393874
$712$	3880.41	0.204248
$713$	−6115.17	−0.321199
$714$	−1318.41	−0.0691041
$715$	0	0
$716$	16240.1	0.847658
$717$	12740.0	0.663576
$718$	−14224.1	−0.739331
$719$	24086.1	1.24932	0.624659	−	0.780897i	$-0.285238\pi$
$719$	24086.1	1.24932	0.624659	+	0.780897i	$0.285238\pi$
$720$	0	0
$721$	−845.423	−0.0436688
$722$	13338.9	0.687568
$723$	−7936.98	−0.408270
$724$	−100.829	−0.00517578
$725$	0	0
$726$	−21886.0	−1.11882
$727$	23213.1	1.18422	0.592110	−	0.805857i	$-0.298295\pi$
$727$	23213.1	1.18422	0.592110	+	0.805857i	$0.298295\pi$
$728$	−4346.36	−0.221273
$729$	7999.94	0.406439
$730$	0	0
$731$	−5217.58	−0.263993
$732$	−1196.27	−0.0604037
$733$	9332.21	0.470250	0.235125	−	0.971965i	$-0.424450\pi$
$733$	9332.21	0.470250	0.235125	+	0.971965i	$0.424450\pi$
$734$	−21456.0	−1.07896
$735$	0	0
$736$	736.000	0.0368605
$737$	−70496.0	−3.52341
$738$	−2596.26	−0.129498
$739$	−8372.49	−0.416762	−0.208381	−	0.978048i	$-0.566819\pi$
$739$	−8372.49	−0.416762	−0.208381	+	0.978048i	$0.566819\pi$
$740$	0	0
$741$	2629.20	0.130346
$742$	8260.53	0.408698
$743$	−37771.7	−1.86502	−0.932509	−	0.361146i	$-0.882386\pi$
$743$	−37771.7	−1.86502	−0.932509	+	0.361146i	$0.882386\pi$
$744$	−6144.78	−0.302794
$745$	0	0
$746$	−5038.54	−0.247284
$747$	20762.5	1.01695
$748$	−7945.97	−0.388414
$749$	4991.30	0.243495
$750$	0	0
$751$	−16002.6	−0.777556	−0.388778	−	0.921331i	$-0.627103\pi$
$751$	−16002.6	−0.777556	−0.388778	+	0.921331i	$0.627103\pi$
$752$	−2163.31	−0.104904
$753$	11513.1	0.557185
$754$	−36300.5	−1.75330
$755$	0	0
$756$	−4335.74	−0.208584
$757$	−5717.38	−0.274507	−0.137254	−	0.990536i	$-0.543828\pi$
$757$	−5717.38	−0.274507	−0.137254	+	0.990536i	$0.543828\pi$
$758$	−4958.26	−0.237589
$759$	4753.93	0.227348
$760$	0	0
$761$	−2454.12	−0.116901	−0.0584507	−	0.998290i	$-0.518616\pi$
$761$	−2454.12	−0.116901	−0.0584507	+	0.998290i	$0.518616\pi$
$762$	4736.04	0.225156
$763$	−17617.5	−0.835905
$764$	−6415.30	−0.303792
$765$	0	0
$766$	−2620.20	−0.123592
$767$	−10706.8	−0.504042
$768$	739.564	0.0347483
$769$	−22439.3	−1.05225	−0.526125	−	0.850407i	$-0.676356\pi$
$769$	−22439.3	−1.05225	−0.526125	+	0.850407i	$0.676356\pi$
$770$	0	0
$771$	23106.8	1.07934
$772$	11019.6	0.513735
$773$	7376.09	0.343208	0.171604	−	0.985166i	$-0.445105\pi$
$773$	7376.09	0.343208	0.171604	+	0.985166i	$0.445105\pi$
$774$	−7010.94	−0.325585
$775$	0	0
$776$	−8292.36	−0.383606
$777$	−4863.96	−0.224574
$778$	−22394.2	−1.03197
$779$	−958.041	−0.0440634
$780$	0	0
$781$	58512.6	2.68085
$782$	1277.19	0.0584044
$783$	−36211.8	−1.65275
$784$	−4407.33	−0.200771
$785$	0	0
$786$	−10001.1	−0.453850
$787$	32539.0	1.47381	0.736906	−	0.675996i	$-0.236286\pi$
$787$	32539.0	1.47381	0.736906	+	0.675996i	$0.236286\pi$
$788$	−9625.76	−0.435157
$789$	−11523.3	−0.519950
$790$	0	0
$791$	−5282.90	−0.237469
$792$	−10677.1	−0.479034
$793$	−6843.57	−0.306459
$794$	16370.0	0.731676
$795$	0	0
$796$	−22262.4	−0.991295
$797$	9137.39	0.406101	0.203051	−	0.979168i	$-0.434914\pi$
$797$	9137.39	0.406101	0.203051	+	0.979168i	$0.434914\pi$
$798$	−653.722	−0.0289994
$799$	−3754.01	−0.166217
$800$	0	0
$801$	9048.20	0.399129
$802$	−11558.7	−0.508918
$803$	80097.4	3.52002
$804$	11386.0	0.499444
$805$	0	0
$806$	−35152.7	−1.53623
$807$	−16024.2	−0.698982
$808$	9099.53	0.396188
$809$	−38671.2	−1.68060	−0.840302	−	0.542119i	$-0.817622\pi$
$809$	−38671.2	−1.68060	−0.840302	+	0.542119i	$0.817622\pi$
$810$	0	0
$811$	3072.50	0.133033	0.0665166	−	0.997785i	$-0.478811\pi$
$811$	3072.50	0.133033	0.0665166	+	0.997785i	$0.478811\pi$
$812$	9025.71	0.390074
$813$	7685.19	0.331527
$814$	−29314.8	−1.26226
$815$	0	0
$816$	1283.38	0.0550578
$817$	−2587.09	−0.110784
$818$	−22014.5	−0.940979
$819$	−10134.7	−0.432399
$820$	0	0
$821$	−33710.0	−1.43299	−0.716497	−	0.697590i	$-0.754256\pi$
$821$	−33710.0	−1.43299	−0.716497	+	0.697590i	$0.754256\pi$
$822$	13785.4	0.584942
$823$	−23281.5	−0.986078	−0.493039	−	0.870007i	$-0.664114\pi$
$823$	−23281.5	−0.986078	−0.493039	+	0.870007i	$0.664114\pi$
$824$	822.957	0.0347926
$825$	0	0
$826$	2662.12	0.112139
$827$	−4175.78	−0.175582	−0.0877908	−	0.996139i	$-0.527981\pi$
$827$	−4175.78	−0.175582	−0.0877908	+	0.996139i	$0.527981\pi$
$828$	1716.18	0.0720306
$829$	−12532.6	−0.525061	−0.262530	−	0.964924i	$-0.584557\pi$
$829$	−12532.6	−0.525061	−0.262530	+	0.964924i	$0.584557\pi$
$830$	0	0
$831$	−5074.99	−0.211853
$832$	4230.86	0.176296
$833$	−7648.10	−0.318116
$834$	−2408.14	−0.0999843
$835$	0	0
$836$	−3939.94	−0.162997
$837$	−35066.8	−1.44813
$838$	7655.52	0.315579
$839$	−11496.2	−0.473055	−0.236527	−	0.971625i	$-0.576009\pi$
$839$	−11496.2	−0.473055	−0.236527	+	0.971625i	$0.576009\pi$
$840$	0	0
$841$	50993.1	2.09083
$842$	−13343.3	−0.546127
$843$	−9063.50	−0.370301
$844$	−5911.55	−0.241095
$845$	0	0
$846$	−5044.32	−0.204997
$847$	31130.6	1.26288
$848$	−8041.02	−0.325625
$849$	−23919.6	−0.966926
$850$	0	0
$851$	4711.89	0.189802
$852$	−9450.54	−0.380012
$853$	18885.9	0.758080	0.379040	−	0.925380i	$-0.376254\pi$
$853$	18885.9	0.758080	0.379040	+	0.925380i	$0.376254\pi$
$854$	1701.58	0.0681812
$855$	0	0
$856$	−4858.66	−0.194002
$857$	16127.6	0.642834	0.321417	−	0.946938i	$-0.395841\pi$
$857$	16127.6	0.642834	0.321417	+	0.946938i	$0.395841\pi$
$858$	27327.7	1.08736
$859$	27156.2	1.07865	0.539324	−	0.842098i	$-0.318680\pi$
$859$	27156.2	1.07865	0.539324	+	0.842098i	$0.318680\pi$
$860$	0	0
$861$	−1652.22	−0.0653976
$862$	−5627.56	−0.222361
$863$	−335.975	−0.0132523	−0.00662614	−	0.999978i	$-0.502109\pi$
$863$	−335.975	−0.0132523	−0.00662614	+	0.999978i	$0.502109\pi$
$864$	4220.52	0.166186
$865$	0	0
$866$	−29302.8	−1.14983
$867$	−11966.2	−0.468736
$868$	8740.33	0.341781
$869$	−28640.2	−1.11801
$870$	0	0
$871$	65136.4	2.53394
$872$	17149.3	0.665996
$873$	−19335.8	−0.749621
$874$	633.283	0.0245093
$875$	0	0
$876$	−12936.8	−0.498964
$877$	−28305.8	−1.08988	−0.544938	−	0.838477i	$-0.683447\pi$
$877$	−28305.8	−1.08988	−0.544938	+	0.838477i	$0.683447\pi$
$878$	−30043.0	−1.15479
$879$	−4846.71	−0.185979
$880$	0	0
$881$	35994.4	1.37648	0.688242	−	0.725482i	$-0.258383\pi$
$881$	35994.4	1.37648	0.688242	+	0.725482i	$0.258383\pi$
$882$	−10276.9	−0.392335
$883$	−3444.45	−0.131274	−0.0656370	−	0.997844i	$-0.520908\pi$
$883$	−3444.45	−0.131274	−0.0656370	+	0.997844i	$0.520908\pi$
$884$	7341.86	0.279337
$885$	0	0
$886$	29444.4	1.11648
$887$	−14360.6	−0.543608	−0.271804	−	0.962353i	$-0.587620\pi$
$887$	−14360.6	−0.543608	−0.271804	+	0.962353i	$0.587620\pi$
$888$	4734.71	0.178926
$889$	−6736.53	−0.254146
$890$	0	0
$891$	−8774.32	−0.329911
$892$	−25061.8	−0.940732
$893$	−1861.39	−0.0697527
$894$	17486.8	0.654192
$895$	0	0
$896$	−1051.95	−0.0392225
$897$	−4392.50	−0.163502
$898$	13908.9	0.516867
$899$	72998.7	2.70817
$900$	0	0
$901$	−13953.7	−0.515943
$902$	−9957.79	−0.367581
$903$	−4461.64	−0.164423
$904$	5142.51	0.189201
$905$	0	0
$906$	4241.49	0.155534
$907$	29262.6	1.07128	0.535638	−	0.844447i	$-0.320071\pi$
$907$	29262.6	1.07128	0.535638	+	0.844447i	$0.320071\pi$
$908$	−20432.5	−0.746780
$909$	21218.0	0.774208
$910$	0	0
$911$	−17719.2	−0.644415	−0.322207	−	0.946669i	$-0.604425\pi$
$911$	−17719.2	−0.644415	−0.322207	+	0.946669i	$0.604425\pi$
$912$	636.350	0.0231049
$913$	79633.0	2.88660
$914$	−1822.66	−0.0659609
$915$	0	0
$916$	18392.4	0.663430
$917$	14225.5	0.512287
$918$	7323.93	0.263318
$919$	−19281.9	−0.692112	−0.346056	−	0.938214i	$-0.612479\pi$
$919$	−19281.9	−0.692112	−0.346056	+	0.938214i	$0.612479\pi$
$920$	0	0
$921$	−2950.98	−0.105579
$922$	11419.2	0.407885
$923$	−54064.1	−1.92800
$924$	−6794.73	−0.241916
$925$	0	0
$926$	11457.3	0.406597
$927$	1918.94	0.0679896
$928$	−8785.86	−0.310787
$929$	16260.8	0.574274	0.287137	−	0.957889i	$-0.407296\pi$
$929$	16260.8	0.574274	0.287137	+	0.957889i	$0.407296\pi$
$930$	0	0
$931$	−3792.24	−0.133497
$932$	−14984.3	−0.526640
$933$	−1423.81	−0.0499607
$934$	−8935.76	−0.313048
$935$	0	0
$936$	9865.37	0.344508
$937$	26752.5	0.932729	0.466365	−	0.884593i	$-0.345563\pi$
$937$	26752.5	0.932729	0.466365	+	0.884593i	$0.345563\pi$
$938$	−16195.4	−0.563752
$939$	−4945.75	−0.171883
$940$	0	0
$941$	−11735.8	−0.406563	−0.203281	−	0.979120i	$-0.565161\pi$
$941$	−11735.8	−0.406563	−0.203281	+	0.979120i	$0.565161\pi$
$942$	2897.33	0.100212
$943$	1600.56	0.0552719
$944$	−2591.38	−0.0893456
$945$	0	0
$946$	−26890.0	−0.924174
$947$	−28051.3	−0.962559	−0.481279	−	0.876567i	$-0.659828\pi$
$947$	−28051.3	−0.962559	−0.481279	+	0.876567i	$0.659828\pi$
$948$	4625.76	0.158479
$949$	−74007.8	−2.53150
$950$	0	0
$951$	1470.95	0.0501566
$952$	−1825.47	−0.0621469
$953$	3824.07	0.129983	0.0649915	−	0.997886i	$-0.479298\pi$
$953$	3824.07	0.129983	0.0649915	+	0.997886i	$0.479298\pi$
$954$	−18749.8	−0.636317
$955$	0	0
$956$	17639.8	0.596770
$957$	−56749.1	−1.91686
$958$	−14492.6	−0.488763
$959$	−19608.4	−0.660258
$960$	0	0
$961$	40899.6	1.37288
$962$	27086.0	0.907785
$963$	−11329.3	−0.379107
$964$	−10989.5	−0.367167
$965$	0	0
$966$	1092.15	0.0363760
$967$	−23862.2	−0.793545	−0.396773	−	0.917917i	$-0.629870\pi$
$967$	−23862.2	−0.793545	−0.396773	+	0.917917i	$0.629870\pi$
$968$	−30303.4	−1.00618
$969$	1104.27	0.0366090
$970$	0	0
$971$	9788.89	0.323523	0.161761	−	0.986830i	$-0.448283\pi$
$971$	9788.89	0.323523	0.161761	+	0.986830i	$0.448283\pi$
$972$	15661.4	0.516811
$973$	3425.33	0.112858
$974$	−18725.0	−0.616005
$975$	0	0
$976$	−1656.36	−0.0543225
$977$	315.225	0.0103224	0.00516118	−	0.999987i	$-0.498357\pi$
$977$	315.225	0.0103224	0.00516118	+	0.999987i	$0.498357\pi$
$978$	5641.34	0.184448
$979$	34703.8	1.13293
$980$	0	0
$981$	39988.2	1.30145
$982$	−20441.0	−0.664257
$983$	5713.93	0.185398	0.0926988	−	0.995694i	$-0.470451\pi$
$983$	5713.93	0.185398	0.0926988	+	0.995694i	$0.470451\pi$
$984$	1608.31	0.0521047
$985$	0	0
$986$	−15246.2	−0.492433
$987$	−3210.12	−0.103525
$988$	3640.39	0.117223
$989$	4322.14	0.138965
$990$	0	0
$991$	−32110.3	−1.02928	−0.514641	−	0.857406i	$-0.672075\pi$
$991$	−32110.3	−1.02928	−0.514641	+	0.857406i	$0.672075\pi$
$992$	−8508.06	−0.272310
$993$	1262.21	0.0403375
$994$	13442.4	0.428941
$995$	0	0
$996$	−12861.7	−0.409177
$997$	−28704.5	−0.911817	−0.455909	−	0.890027i	$-0.650686\pi$
$997$	−28704.5	−0.911817	−0.455909	+	0.890027i	$0.650686\pi$
$998$	−20927.0	−0.663760
$999$	27019.9	0.855726

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.q.1.4	✓	5
5.2	odd	4		1150.4.b.r.599.2		10
5.3	odd	4		1150.4.b.r.599.9		10
5.4	even	2		1150.4.a.v.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.q.1.4	✓	5	1.1	even	1	trivial
1150.4.a.v.1.2	yes	5	5.4	even	2
1150.4.b.r.599.2		10	5.2	odd	4
1150.4.b.r.599.9		10	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +2.88892 q^{3} +4.00000 q^{4} -5.77785 q^{6} +8.21839 q^{7} -8.00000 q^{8} -18.6541 q^{9} -71.5466 q^{11} +11.5557 q^{12} +66.1071 q^{13} -16.4368 q^{14} +16.0000 q^{16} +27.7650 q^{17} +37.3082 q^{18} +13.7670 q^{19} +23.7423 q^{21} +143.093 q^{22} -23.0000 q^{23} -23.1114 q^{24} -132.214 q^{26} -131.891 q^{27} +32.8736 q^{28} +274.558 q^{29} +265.877 q^{31} -32.0000 q^{32} -206.693 q^{33} -55.5300 q^{34} -74.6165 q^{36} -204.865 q^{37} -27.5341 q^{38} +190.978 q^{39} -69.5895 q^{41} -47.4846 q^{42} -187.919 q^{43} -286.186 q^{44} +46.0000 q^{46} -135.207 q^{47} +46.2228 q^{48} -275.458 q^{49} +80.2110 q^{51} +264.429 q^{52} -502.564 q^{53} +263.782 q^{54} -65.7472 q^{56} +39.7719 q^{57} -549.116 q^{58} -161.961 q^{59} -103.522 q^{61} -531.754 q^{62} -153.307 q^{63} +64.0000 q^{64} +413.385 q^{66} +985.315 q^{67} +111.060 q^{68} -66.4452 q^{69} -817.825 q^{71} +149.233 q^{72} -1119.51 q^{73} +409.729 q^{74} +55.0681 q^{76} -587.998 q^{77} -381.957 q^{78} +400.301 q^{79} +122.638 q^{81} +139.179 q^{82} -1113.02 q^{83} +94.9692 q^{84} +375.838 q^{86} +793.177 q^{87} +572.373 q^{88} -485.051 q^{89} +543.295 q^{91} -92.0000 q^{92} +768.098 q^{93} +270.413 q^{94} -92.4455 q^{96} +1036.55 q^{97} +550.916 q^{98} +1334.64 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} - 5 q^{3} + 20 q^{4} + 10 q^{6} + 3 q^{7} - 40 q^{8} + 84 q^{9} - 26 q^{11} - 20 q^{12} + 61 q^{13} - 6 q^{14} + 80 q^{16} - 231 q^{17} - 168 q^{18} + 74 q^{19} - 88 q^{21} + 52 q^{22} - 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 - 5 * q^3 + 20 * q^4 + 10 * q^6 + 3 * q^7 - 40 * q^8 + 84 * q^9 - 26 * q^11 - 20 * q^12 + 61 * q^13 - 6 * q^14 + 80 * q^16 - 231 * q^17 - 168 * q^18 + 74 * q^19 - 88 * q^21 + 52 * q^22 - 115 * q^23 + 40 * q^24 - 122 * q^26 - 368 * q^27 + 12 * q^28 + 37 * q^29 + 220 * q^31 - 160 * q^32 - 391 * q^33 + 462 * q^34 + 336 * q^36 - 220 * q^37 - 148 * q^38 + 195 * q^39 + 112 * q^41 + 176 * q^42 - 97 * q^43 - 104 * q^44 + 230 * q^46 - 370 * q^47 - 80 * q^48 + 182 * q^49 + 563 * q^51 + 244 * q^52 - 1224 * q^53 + 736 * q^54 - 24 * q^56 + 1009 * q^57 - 74 * q^58 - 181 * q^59 + 1574 * q^61 - 440 * q^62 - 1829 * q^63 + 320 * q^64 + 782 * q^66 + 435 * q^67 - 924 * q^68 + 115 * q^69 - 30 * q^71 - 672 * q^72 - 2180 * q^73 + 440 * q^74 + 296 * q^76 + 1401 * q^77 - 390 * q^78 + 609 * q^79 + 1453 * q^81 - 224 * q^82 - 2018 * q^83 - 352 * q^84 + 194 * q^86 + 1485 * q^87 + 208 * q^88 - 1663 * q^89 + 361 * q^91 - 460 * q^92 - 2741 * q^93 + 740 * q^94 + 160 * q^96 + 1274 * q^97 - 364 * q^98 + 2397 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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