LMFDB - Embedded newform 2254.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

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

lf = mfeigenbasis(mf)

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")

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

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

Self dual:	yes
Analytic conductor:	$132.990305153$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.27492$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +3.27492 q^{3} +4.00000 q^{4} +1.45017 q^{5} +6.54983 q^{6} +8.00000 q^{8} -16.2749 q^{9} +2.90033 q^{10} -24.3505 q^{11} +13.0997 q^{12} +9.62541 q^{13} +4.74917 q^{15} +16.0000 q^{16} -55.8488 q^{17} -32.5498 q^{18} +150.797 q^{19} +5.80066 q^{20} -48.7010 q^{22} +23.0000 q^{23} +26.1993 q^{24} -122.897 q^{25} +19.2508 q^{26} -141.722 q^{27} -129.124 q^{29} +9.49834 q^{30} -3.07558 q^{31} +32.0000 q^{32} -79.7459 q^{33} -111.698 q^{34} -65.0997 q^{36} -342.096 q^{37} +301.595 q^{38} +31.5224 q^{39} +11.6013 q^{40} +253.571 q^{41} -256.797 q^{43} -97.4020 q^{44} -23.6013 q^{45} +46.0000 q^{46} -70.1686 q^{47} +52.3987 q^{48} -245.794 q^{50} -182.900 q^{51} +38.5017 q^{52} -20.1030 q^{53} -283.444 q^{54} -35.3123 q^{55} +493.849 q^{57} -258.248 q^{58} +414.640 q^{59} +18.9967 q^{60} -42.0548 q^{61} -6.15116 q^{62} +64.0000 q^{64} +13.9584 q^{65} -159.492 q^{66} +510.234 q^{67} -223.395 q^{68} +75.3231 q^{69} -692.815 q^{71} -130.199 q^{72} -1001.87 q^{73} -684.193 q^{74} -402.478 q^{75} +603.189 q^{76} +63.0449 q^{78} +66.1379 q^{79} +23.2026 q^{80} -24.7043 q^{81} +507.141 q^{82} -78.8455 q^{83} -80.9901 q^{85} -513.595 q^{86} -422.870 q^{87} -194.804 q^{88} -34.8323 q^{89} -47.2026 q^{90} +92.0000 q^{92} -10.0723 q^{93} -140.337 q^{94} +218.681 q^{95} +104.797 q^{96} -1265.54 q^{97} +396.302 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - q^{3} + 8 q^{4} + 18 q^{5} - 2 q^{6} + 16 q^{8} - 25 q^{9} + 36 q^{10} - 94 q^{11} - 4 q^{12} + 57 q^{13} - 66 q^{15} + 32 q^{16} - 6 q^{17} - 50 q^{18} + 60 q^{19} + 72 q^{20} - 188 q^{22}+ \cdots + 1004 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 - q^3 + 8 * q^4 + 18 * q^5 - 2 * q^6 + 16 * q^8 - 25 * q^9 + 36 * q^10 - 94 * q^11 - 4 * q^12 + 57 * q^13 - 66 * q^15 + 32 * q^16 - 6 * q^17 - 50 * q^18 + 60 * q^19 + 72 * q^20 - 188 * q^22 + 46 * q^23 - 8 * q^24 + 26 * q^25 + 114 * q^26 + 11 * q^27 - 145 * q^29 - 132 * q^30 - 59 * q^31 + 64 * q^32 + 218 * q^33 - 12 * q^34 - 100 * q^36 - 352 * q^37 + 120 * q^38 - 171 * q^39 + 144 * q^40 + 107 * q^41 - 272 * q^43 - 376 * q^44 - 168 * q^45 + 92 * q^46 + 441 * q^47 - 16 * q^48 + 52 * q^50 - 396 * q^51 + 228 * q^52 - 312 * q^53 + 22 * q^54 - 1188 * q^55 + 882 * q^57 - 290 * q^58 - 122 * q^59 - 264 * q^60 - 522 * q^61 - 118 * q^62 + 128 * q^64 + 798 * q^65 + 436 * q^66 - 414 * q^67 - 24 * q^68 - 23 * q^69 - 457 * q^71 - 200 * q^72 - 1815 * q^73 - 704 * q^74 - 1039 * q^75 + 240 * q^76 - 342 * q^78 - 970 * q^79 + 288 * q^80 - 442 * q^81 + 214 * q^82 + 250 * q^83 + 744 * q^85 - 544 * q^86 - 355 * q^87 - 752 * q^88 + 1546 * q^89 - 336 * q^90 + 184 * q^92 + 229 * q^93 + 882 * q^94 - 1284 * q^95 - 32 * q^96 - 1610 * q^97 + 1004 * 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
$3$	3.27492	0.630258	0.315129	−	0.949049i	$-0.397952\pi$
$3$	3.27492	0.630258	0.315129	+	0.949049i	$0.397952\pi$
$4$	4.00000	0.500000
$5$	1.45017	0.129707	0.0648534	−	0.997895i	$-0.479342\pi$
$5$	1.45017	0.129707	0.0648534	+	0.997895i	$0.479342\pi$
$6$	6.54983	0.445660
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−16.2749	−0.602775
$10$	2.90033	0.0917165
$11$	−24.3505	−0.667450	−0.333725	−	0.942670i	$-0.608306\pi$
$11$	−24.3505	−0.667450	−0.333725	+	0.942670i	$0.608306\pi$
$12$	13.0997	0.315129
$13$	9.62541	0.205355	0.102677	−	0.994715i	$-0.467259\pi$
$13$	9.62541	0.205355	0.102677	+	0.994715i	$0.467259\pi$
$14$	0	0
$15$	4.74917	0.0817487
$16$	16.0000	0.250000
$17$	−55.8488	−0.796784	−0.398392	−	0.917215i	$-0.630432\pi$
$17$	−55.8488	−0.796784	−0.398392	+	0.917215i	$0.630432\pi$
$18$	−32.5498	−0.426226
$19$	150.797	1.82080	0.910402	−	0.413724i	$-0.135772\pi$
$19$	150.797	1.82080	0.910402	+	0.413724i	$0.135772\pi$
$20$	5.80066	0.0648534
$21$	0	0
$22$	−48.7010	−0.471958
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	26.1993	0.222830
$25$	−122.897	−0.983176
$26$	19.2508	0.145208
$27$	−141.722	−1.01016
$28$	0	0
$29$	−129.124	−0.826817	−0.413408	−	0.910546i	$-0.635662\pi$
$29$	−129.124	−0.826817	−0.413408	+	0.910546i	$0.635662\pi$
$30$	9.49834	0.0578051
$31$	−3.07558	−0.0178190	−0.00890952	−	0.999960i	$-0.502836\pi$
$31$	−3.07558	−0.0178190	−0.00890952	+	0.999960i	$0.502836\pi$
$32$	32.0000	0.176777
$33$	−79.7459	−0.420666
$34$	−111.698	−0.563412
$35$	0	0
$36$	−65.0997	−0.301387
$37$	−342.096	−1.52001	−0.760004	−	0.649918i	$-0.774803\pi$
$37$	−342.096	−1.52001	−0.760004	+	0.649918i	$0.774803\pi$
$38$	301.595	1.28750
$39$	31.5224	0.129426
$40$	11.6013	0.0458583
$41$	253.571	0.965880	0.482940	−	0.875653i	$-0.339569\pi$
$41$	253.571	0.965880	0.482940	+	0.875653i	$0.339569\pi$
$42$	0	0
$43$	−256.797	−0.910726	−0.455363	−	0.890306i	$-0.650491\pi$
$43$	−256.797	−0.910726	−0.455363	+	0.890306i	$0.650491\pi$
$44$	−97.4020	−0.333725
$45$	−23.6013	−0.0781839
$46$	46.0000	0.147442
$47$	−70.1686	−0.217769	−0.108885	−	0.994054i	$-0.534728\pi$
$47$	−70.1686	−0.217769	−0.108885	+	0.994054i	$0.534728\pi$
$48$	52.3987	0.157565
$49$	0	0
$50$	−245.794	−0.695211
$51$	−182.900	−0.502180
$52$	38.5017	0.102677
$53$	−20.1030	−0.0521011	−0.0260505	−	0.999661i	$-0.508293\pi$
$53$	−20.1030	−0.0521011	−0.0260505	+	0.999661i	$0.508293\pi$
$54$	−283.444	−0.714292
$55$	−35.3123	−0.0865728
$56$	0	0
$57$	493.849	1.14758
$58$	−258.248	−0.584648
$59$	414.640	0.914940	0.457470	−	0.889225i	$-0.348756\pi$
$59$	414.640	0.914940	0.457470	+	0.889225i	$0.348756\pi$
$60$	18.9967	0.0408744
$61$	−42.0548	−0.0882716	−0.0441358	−	0.999026i	$-0.514053\pi$
$61$	−42.0548	−0.0882716	−0.0441358	+	0.999026i	$0.514053\pi$
$62$	−6.15116	−0.0126000
$63$	0	0
$64$	64.0000	0.125000
$65$	13.9584	0.0266359
$66$	−159.492	−0.297456
$67$	510.234	0.930374	0.465187	−	0.885213i	$-0.345987\pi$
$67$	510.234	0.930374	0.465187	+	0.885213i	$0.345987\pi$
$68$	−223.395	−0.398392
$69$	75.3231	0.131418
$70$	0	0
$71$	−692.815	−1.15806	−0.579028	−	0.815308i	$-0.696568\pi$
$71$	−692.815	−1.15806	−0.579028	+	0.815308i	$0.696568\pi$
$72$	−130.199	−0.213113
$73$	−1001.87	−1.60631	−0.803153	−	0.595773i	$-0.796846\pi$
$73$	−1001.87	−1.60631	−0.803153	+	0.595773i	$0.796846\pi$
$74$	−684.193	−1.07481
$75$	−402.478	−0.619655
$76$	603.189	0.910402
$77$	0	0
$78$	63.0449	0.0915183
$79$	66.1379	0.0941911	0.0470955	−	0.998890i	$-0.485003\pi$
$79$	66.1379	0.0941911	0.0470955	+	0.998890i	$0.485003\pi$
$80$	23.2026	0.0324267
$81$	−24.7043	−0.0338879
$82$	507.141	0.682980
$83$	−78.8455	−0.104270	−0.0521351	−	0.998640i	$-0.516603\pi$
$83$	−78.8455	−0.104270	−0.0521351	+	0.998640i	$0.516603\pi$
$84$	0	0
$85$	−80.9901	−0.103348
$86$	−513.595	−0.643981
$87$	−422.870	−0.521108
$88$	−194.804	−0.235979
$89$	−34.8323	−0.0414856	−0.0207428	−	0.999785i	$-0.506603\pi$
$89$	−34.8323	−0.0414856	−0.0207428	+	0.999785i	$0.506603\pi$
$90$	−47.2026	−0.0552844
$91$	0	0
$92$	92.0000	0.104257
$93$	−10.0723	−0.0112306
$94$	−140.337	−0.153986
$95$	218.681	0.236171
$96$	104.797	0.111415
$97$	−1265.54	−1.32470	−0.662351	−	0.749194i	$-0.730441\pi$
$97$	−1265.54	−1.32470	−0.662351	+	0.749194i	$0.730441\pi$
$98$	0	0
$99$	396.302	0.402322
$100$	−491.588	−0.491588
$101$	−1037.75	−1.02238	−0.511189	−	0.859468i	$-0.670795\pi$
$101$	−1037.75	−1.02238	−0.511189	+	0.859468i	$0.670795\pi$
$102$	−365.801	−0.355095
$103$	−1657.58	−1.58569	−0.792846	−	0.609422i	$-0.791402\pi$
$103$	−1657.58	−1.58569	−0.792846	+	0.609422i	$0.791402\pi$
$104$	77.0033	0.0726038
$105$	0	0
$106$	−40.2060	−0.0368410
$107$	1237.82	1.11836	0.559181	−	0.829045i	$-0.311116\pi$
$107$	1237.82	1.11836	0.559181	+	0.829045i	$0.311116\pi$
$108$	−566.887	−0.505081
$109$	−1402.43	−1.23237	−0.616187	−	0.787600i	$-0.711323\pi$
$109$	−1402.43	−1.23237	−0.616187	+	0.787600i	$0.711323\pi$
$110$	−70.6245	−0.0612162
$111$	−1120.34	−0.957997
$112$	0	0
$113$	−1449.94	−1.20707	−0.603534	−	0.797337i	$-0.706241\pi$
$113$	−1449.94	−1.20707	−0.603534	+	0.797337i	$0.706241\pi$
$114$	987.698	0.811459
$115$	33.3538	0.0270457
$116$	−516.495	−0.413408
$117$	−156.653	−0.123783
$118$	829.279	0.646960
$119$	0	0
$120$	37.9934	0.0289025
$121$	−738.053	−0.554510
$122$	−84.1096	−0.0624174
$123$	830.423	0.608754
$124$	−12.3023	−0.00890952
$125$	−359.492	−0.257231
$126$	0	0
$127$	3.94760	0.00275821	0.00137911	−	0.999999i	$-0.499561\pi$
$127$	3.94760	0.00275821	0.00137911	+	0.999999i	$0.499561\pi$
$128$	128.000	0.0883883
$129$	−840.990	−0.573993
$130$	27.9169	0.0188344
$131$	−1101.10	−0.734376	−0.367188	−	0.930147i	$-0.619680\pi$
$131$	−1101.10	−0.734376	−0.367188	+	0.930147i	$0.619680\pi$
$132$	−318.983	−0.210333
$133$	0	0
$134$	1020.47	0.657874
$135$	−205.520	−0.131025
$136$	−446.791	−0.281706
$137$	381.855	0.238132	0.119066	−	0.992886i	$-0.462010\pi$
$137$	381.855	0.238132	0.119066	+	0.992886i	$0.462010\pi$
$138$	150.646	0.0929265
$139$	753.303	0.459672	0.229836	−	0.973229i	$-0.426181\pi$
$139$	753.303	0.459672	0.229836	+	0.973229i	$0.426181\pi$
$140$	0	0
$141$	−229.796	−0.137251
$142$	−1385.63	−0.818869
$143$	−234.384	−0.137064
$144$	−260.399	−0.150694
$145$	−187.251	−0.107244
$146$	−2003.75	−1.13583
$147$	0	0
$148$	−1368.39	−0.760004
$149$	−2924.17	−1.60777	−0.803884	−	0.594785i	$-0.797237\pi$
$149$	−2924.17	−1.60777	−0.803884	+	0.594785i	$0.797237\pi$
$150$	−804.955	−0.438162
$151$	2924.73	1.57623	0.788116	−	0.615527i	$-0.211057\pi$
$151$	2924.73	1.57623	0.788116	+	0.615527i	$0.211057\pi$
$152$	1206.38	0.643752
$153$	908.935	0.480281
$154$	0	0
$155$	−4.46010	−0.00231125
$156$	126.090	0.0647132
$157$	−1442.17	−0.733108	−0.366554	−	0.930397i	$-0.619462\pi$
$157$	−1442.17	−0.733108	−0.366554	+	0.930397i	$0.619462\pi$
$158$	132.276	0.0666032
$159$	−65.8356	−0.0328371
$160$	46.4053	0.0229291
$161$	0	0
$162$	−49.4086	−0.0239624
$163$	1464.27	0.703621	0.351811	−	0.936071i	$-0.385566\pi$
$163$	1464.27	0.703621	0.351811	+	0.936071i	$0.385566\pi$
$164$	1014.28	0.482940
$165$	−115.645	−0.0545632
$166$	−157.691	−0.0737301
$167$	1996.24	0.924990	0.462495	−	0.886622i	$-0.346954\pi$
$167$	1996.24	0.924990	0.462495	+	0.886622i	$0.346954\pi$
$168$	0	0
$169$	−2104.35	−0.957829
$170$	−161.980	−0.0730783
$171$	−2454.21	−1.09753
$172$	−1027.19	−0.455363
$173$	2175.24	0.955955	0.477978	−	0.878372i	$-0.341370\pi$
$173$	2175.24	0.955955	0.477978	+	0.878372i	$0.341370\pi$
$174$	−845.739	−0.368479
$175$	0	0
$176$	−389.608	−0.166863
$177$	1357.91	0.576649
$178$	−69.6646	−0.0293347
$179$	651.934	0.272223	0.136111	−	0.990694i	$-0.456540\pi$
$179$	651.934	0.272223	0.136111	+	0.990694i	$0.456540\pi$
$180$	−94.4053	−0.0390920
$181$	2760.83	1.13376	0.566881	−	0.823799i	$-0.308150\pi$
$181$	2760.83	1.13376	0.566881	+	0.823799i	$0.308150\pi$
$182$	0	0
$183$	−137.726	−0.0556339
$184$	184.000	0.0737210
$185$	−496.096	−0.197155
$186$	−20.1445	−0.00794123
$187$	1359.95	0.531814
$188$	−280.675	−0.108885
$189$	0	0
$190$	437.362	0.166998
$191$	813.038	0.308007	0.154004	−	0.988070i	$-0.450783\pi$
$191$	813.038	0.308007	0.154004	+	0.988070i	$0.450783\pi$
$192$	209.595	0.0787823
$193$	4552.05	1.69774	0.848869	−	0.528603i	$-0.177284\pi$
$193$	4552.05	1.69774	0.848869	+	0.528603i	$0.177284\pi$
$194$	−2531.08	−0.936706
$195$	45.7127	0.0167875
$196$	0	0
$197$	−1971.84	−0.713138	−0.356569	−	0.934269i	$-0.616053\pi$
$197$	−1971.84	−0.713138	−0.356569	+	0.934269i	$0.616053\pi$
$198$	792.605	0.284485
$199$	1053.92	0.375429	0.187714	−	0.982224i	$-0.439892\pi$
$199$	1053.92	0.375429	0.187714	+	0.982224i	$0.439892\pi$
$200$	−983.176	−0.347605
$201$	1670.98	0.586376
$202$	−2075.50	−0.722931
$203$	0	0
$204$	−731.601	−0.251090
$205$	367.719	0.125281
$206$	−3315.16	−1.12125
$207$	−374.323	−0.125687
$208$	154.007	0.0513386
$209$	−3671.99	−1.21530
$210$	0	0
$211$	3624.90	1.18269	0.591347	−	0.806418i	$-0.298597\pi$
$211$	3624.90	1.18269	0.591347	+	0.806418i	$0.298597\pi$
$212$	−80.4119	−0.0260505
$213$	−2268.91	−0.729874
$214$	2475.64	0.790802
$215$	−372.399	−0.118127
$216$	−1133.77	−0.357146
$217$	0	0
$218$	−2804.87	−0.871420
$219$	−3281.05	−1.01239
$220$	−141.249	−0.0432864
$221$	−537.568	−0.163623
$222$	−2240.67	−0.677406
$223$	−4175.81	−1.25396	−0.626979	−	0.779036i	$-0.715709\pi$
$223$	−4175.81	−1.25396	−0.626979	+	0.779036i	$0.715709\pi$
$224$	0	0
$225$	2000.14	0.592634
$226$	−2899.88	−0.853526
$227$	−2267.05	−0.662861	−0.331431	−	0.943480i	$-0.607531\pi$
$227$	−2267.05	−0.662861	−0.331431	+	0.943480i	$0.607531\pi$
$228$	1975.40	0.573788
$229$	1473.52	0.425211	0.212605	−	0.977138i	$-0.431805\pi$
$229$	1473.52	0.425211	0.212605	+	0.977138i	$0.431805\pi$
$230$	66.7076	0.0191242
$231$	0	0
$232$	−1032.99	−0.292324
$233$	−4406.55	−1.23898	−0.619490	−	0.785004i	$-0.712661\pi$
$233$	−4406.55	−1.23898	−0.619490	+	0.785004i	$0.712661\pi$
$234$	−313.306	−0.0875275
$235$	−101.756	−0.0282461
$236$	1658.56	0.457470
$237$	216.596	0.0593647
$238$	0	0
$239$	−5928.34	−1.60449	−0.802243	−	0.596997i	$-0.796360\pi$
$239$	−5928.34	−1.60449	−0.802243	+	0.596997i	$0.796360\pi$
$240$	75.9868	0.0204372
$241$	3062.74	0.818624	0.409312	−	0.912394i	$-0.365769\pi$
$241$	3062.74	0.818624	0.409312	+	0.912394i	$0.365769\pi$
$242$	−1476.11	−0.392098
$243$	3745.58	0.988804
$244$	−168.219	−0.0441358
$245$	0	0
$246$	1660.85	0.430454
$247$	1451.49	0.373911
$248$	−24.6046	−0.00629998
$249$	−258.213	−0.0657171
$250$	−718.983	−0.181890
$251$	−5907.69	−1.48562	−0.742808	−	0.669504i	$-0.766507\pi$
$251$	−5907.69	−1.48562	−0.742808	+	0.669504i	$0.766507\pi$
$252$	0	0
$253$	−560.061	−0.139173
$254$	7.89520	0.00195035
$255$	−265.236	−0.0651361
$256$	256.000	0.0625000
$257$	2450.88	0.594869	0.297435	−	0.954742i	$-0.403869\pi$
$257$	2450.88	0.594869	0.297435	+	0.954742i	$0.403869\pi$
$258$	−1681.98	−0.405874
$259$	0	0
$260$	55.8338	0.0133179
$261$	2101.48	0.498384
$262$	−2202.19	−0.519283
$263$	5522.13	1.29471	0.647356	−	0.762188i	$-0.275875\pi$
$263$	5522.13	1.29471	0.647356	+	0.762188i	$0.275875\pi$
$264$	−637.967	−0.148728
$265$	−29.1526	−0.00675786
$266$	0	0
$267$	−114.073	−0.0261466
$268$	2040.94	0.465187
$269$	568.714	0.128904	0.0644518	−	0.997921i	$-0.479470\pi$
$269$	568.714	0.128904	0.0644518	+	0.997921i	$0.479470\pi$
$270$	−411.040	−0.0926485
$271$	6090.70	1.36525	0.682626	−	0.730767i	$-0.260838\pi$
$271$	6090.70	1.36525	0.682626	+	0.730767i	$0.260838\pi$
$272$	−893.581	−0.199196
$273$	0	0
$274$	763.711	0.168385
$275$	2992.60	0.656221
$276$	301.292	0.0657090
$277$	−4228.13	−0.917126	−0.458563	−	0.888662i	$-0.651636\pi$
$277$	−4228.13	−0.917126	−0.458563	+	0.888662i	$0.651636\pi$
$278$	1506.61	0.325037
$279$	50.0548	0.0107409
$280$	0	0
$281$	−8083.10	−1.71600	−0.858002	−	0.513646i	$-0.828294\pi$
$281$	−8083.10	−1.71600	−0.858002	+	0.513646i	$0.828294\pi$
$282$	−459.593	−0.0970509
$283$	8328.24	1.74934	0.874669	−	0.484721i	$-0.161079\pi$
$283$	8328.24	1.74934	0.874669	+	0.484721i	$0.161079\pi$
$284$	−2771.26	−0.579028
$285$	716.163	0.148848
$286$	−468.767	−0.0969188
$287$	0	0
$288$	−520.797	−0.106557
$289$	−1793.91	−0.365135
$290$	−374.502	−0.0758327
$291$	−4144.54	−0.834904
$292$	−4007.49	−0.803153
$293$	6045.17	1.20533	0.602667	−	0.797993i	$-0.294105\pi$
$293$	6045.17	1.20533	0.602667	+	0.797993i	$0.294105\pi$
$294$	0	0
$295$	601.296	0.118674
$296$	−2736.77	−0.537404
$297$	3451.00	0.674233
$298$	−5848.35	−1.13686
$299$	221.385	0.0428194
$300$	−1609.91	−0.309827
$301$	0	0
$302$	5849.45	1.11456
$303$	−3398.55	−0.644362
$304$	2412.76	0.455201
$305$	−60.9864	−0.0114494
$306$	1817.87	0.339610
$307$	−7839.97	−1.45749	−0.728747	−	0.684783i	$-0.759897\pi$
$307$	−7839.97	−1.45749	−0.728747	+	0.684783i	$0.759897\pi$
$308$	0	0
$309$	−5428.44	−0.999396
$310$	−8.92020	−0.00163430
$311$	−6388.45	−1.16481	−0.582405	−	0.812899i	$-0.697888\pi$
$311$	−6388.45	−1.16481	−0.582405	+	0.812899i	$0.697888\pi$
$312$	252.179	0.0457591
$313$	8381.32	1.51355	0.756774	−	0.653677i	$-0.226775\pi$
$313$	8381.32	1.51355	0.756774	+	0.653677i	$0.226775\pi$
$314$	−2884.35	−0.518385
$315$	0	0
$316$	264.552	0.0470955
$317$	5150.54	0.912565	0.456283	−	0.889835i	$-0.349181\pi$
$317$	5150.54	0.912565	0.456283	+	0.889835i	$0.349181\pi$
$318$	−131.671	−0.0232194
$319$	3144.23	0.551859
$320$	92.8106	0.0162133
$321$	4053.77	0.704857
$322$	0	0
$323$	−8421.86	−1.45079
$324$	−98.8172	−0.0169440
$325$	−1182.93	−0.201900
$326$	2928.53	0.497535
$327$	−4592.85	−0.776714
$328$	2028.56	0.341490
$329$	0	0
$330$	−231.289	−0.0385820
$331$	−1447.51	−0.240370	−0.120185	−	0.992751i	$-0.538349\pi$
$331$	−1447.51	−0.240370	−0.120185	+	0.992751i	$0.538349\pi$
$332$	−315.382	−0.0521351
$333$	5567.59	0.916222
$334$	3992.47	0.654067
$335$	739.924	0.120676
$336$	0	0
$337$	−8514.92	−1.37637	−0.688186	−	0.725535i	$-0.741593\pi$
$337$	−8514.92	−1.37637	−0.688186	+	0.725535i	$0.741593\pi$
$338$	−4208.70	−0.677288
$339$	−4748.43	−0.760765
$340$	−323.960	−0.0516742
$341$	74.8919	0.0118933
$342$	−4908.43	−0.776074
$343$	0	0
$344$	−2054.38	−0.321990
$345$	109.231	0.0170458
$346$	4350.48	0.675962
$347$	−6331.50	−0.979519	−0.489759	−	0.871858i	$-0.662915\pi$
$347$	−6331.50	−0.979519	−0.489759	+	0.871858i	$0.662915\pi$
$348$	−1691.48	−0.260554
$349$	−6613.34	−1.01434	−0.507169	−	0.861847i	$-0.669308\pi$
$349$	−6613.34	−1.01434	−0.507169	+	0.861847i	$0.669308\pi$
$350$	0	0
$351$	−1364.13	−0.207441
$352$	−779.216	−0.117990
$353$	2449.80	0.369376	0.184688	−	0.982797i	$-0.440873\pi$
$353$	2449.80	0.369376	0.184688	+	0.982797i	$0.440873\pi$
$354$	2715.82	0.407752
$355$	−1004.70	−0.150208
$356$	−139.329	−0.0207428
$357$	0	0
$358$	1303.87	0.192491
$359$	−5606.60	−0.824248	−0.412124	−	0.911128i	$-0.635213\pi$
$359$	−5606.60	−0.824248	−0.412124	+	0.911128i	$0.635213\pi$
$360$	−188.811	−0.0276422
$361$	15880.8	2.31533
$362$	5521.66	0.801691
$363$	−2417.06	−0.349485
$364$	0	0
$365$	−1452.88	−0.208349
$366$	−275.452	−0.0393391
$367$	8049.92	1.14497	0.572483	−	0.819917i	$-0.305980\pi$
$367$	8049.92	1.14497	0.572483	+	0.819917i	$0.305980\pi$
$368$	368.000	0.0521286
$369$	−4126.84	−0.582208
$370$	−992.193	−0.139410
$371$	0	0
$372$	−40.2891	−0.00561530
$373$	−3108.79	−0.431548	−0.215774	−	0.976443i	$-0.569227\pi$
$373$	−3108.79	−0.431548	−0.215774	+	0.976443i	$0.569227\pi$
$374$	2719.89	0.376049
$375$	−1177.31	−0.162122
$376$	−561.349	−0.0769930
$377$	−1242.87	−0.169791
$378$	0	0
$379$	142.458	0.0193076	0.00965381	−	0.999953i	$-0.496927\pi$
$379$	142.458	0.0193076	0.00965381	+	0.999953i	$0.496927\pi$
$380$	874.725	0.118085
$381$	12.9281	0.00173838
$382$	1626.08	0.217794
$383$	6562.68	0.875554	0.437777	−	0.899084i	$-0.355766\pi$
$383$	6562.68	0.875554	0.437777	+	0.899084i	$0.355766\pi$
$384$	419.189	0.0557075
$385$	0	0
$386$	9104.09	1.20048
$387$	4179.36	0.548963
$388$	−5062.16	−0.662351
$389$	−9507.95	−1.23926	−0.619630	−	0.784894i	$-0.712717\pi$
$389$	−9507.95	−1.23926	−0.619630	+	0.784894i	$0.712717\pi$
$390$	91.4255	0.0118705
$391$	−1284.52	−0.166141
$392$	0	0
$393$	−3606.00	−0.462847
$394$	−3943.69	−0.504264
$395$	95.9109	0.0122172
$396$	1585.21	0.201161
$397$	−3317.24	−0.419365	−0.209682	−	0.977770i	$-0.567243\pi$
$397$	−3317.24	−0.419365	−0.209682	+	0.977770i	$0.567243\pi$
$398$	2107.84	0.265468
$399$	0	0
$400$	−1966.35	−0.245794
$401$	7841.22	0.976489	0.488244	−	0.872707i	$-0.337637\pi$
$401$	7841.22	0.976489	0.488244	+	0.872707i	$0.337637\pi$
$402$	3341.95	0.414630
$403$	−29.6037	−0.00365922
$404$	−4151.01	−0.511189
$405$	−35.8253	−0.00439549
$406$	0	0
$407$	8330.22	1.01453
$408$	−1463.20	−0.177547
$409$	−12419.7	−1.50151	−0.750753	−	0.660583i	$-0.770309\pi$
$409$	−12419.7	−1.50151	−0.750753	+	0.660583i	$0.770309\pi$
$410$	735.439	0.0885871
$411$	1250.55	0.150085
$412$	−6630.33	−0.792846
$413$	0	0
$414$	−748.646	−0.0888743
$415$	−114.339	−0.0135245
$416$	308.013	0.0363019
$417$	2467.01	0.289712
$418$	−7343.98	−0.859344
$419$	−6439.02	−0.750756	−0.375378	−	0.926872i	$-0.622487\pi$
$419$	−6439.02	−0.750756	−0.375378	+	0.926872i	$0.622487\pi$
$420$	0	0
$421$	1995.73	0.231035	0.115518	−	0.993305i	$-0.463147\pi$
$421$	1995.73	0.231035	0.115518	+	0.993305i	$0.463147\pi$
$422$	7249.79	0.836290
$423$	1141.99	0.131266
$424$	−160.824	−0.0184205
$425$	6863.66	0.783379
$426$	−4537.82	−0.516099
$427$	0	0
$428$	4951.29	0.559181
$429$	−767.587	−0.0863857
$430$	−744.797	−0.0835286
$431$	9633.35	1.07662	0.538309	−	0.842748i	$-0.319063\pi$
$431$	9633.35	1.07662	0.538309	+	0.842748i	$0.319063\pi$
$432$	−2267.55	−0.252540
$433$	−10609.6	−1.17751	−0.588757	−	0.808310i	$-0.700383\pi$
$433$	−10609.6	−1.17751	−0.588757	+	0.808310i	$0.700383\pi$
$434$	0	0
$435$	−613.231	−0.0675912
$436$	−5609.73	−0.616187
$437$	3468.34	0.379664
$438$	−6562.10	−0.715866
$439$	−5715.27	−0.621356	−0.310678	−	0.950515i	$-0.600556\pi$
$439$	−5715.27	−0.621356	−0.310678	+	0.950515i	$0.600556\pi$
$440$	−282.498	−0.0306081
$441$	0	0
$442$	−1075.14	−0.115699
$443$	−8164.69	−0.875657	−0.437828	−	0.899059i	$-0.644252\pi$
$443$	−8164.69	−0.875657	−0.437828	+	0.899059i	$0.644252\pi$
$444$	−4481.35	−0.478999
$445$	−50.5126	−0.00538096
$446$	−8351.61	−0.886683
$447$	−9576.42	−1.01331
$448$	0	0
$449$	4139.67	0.435107	0.217554	−	0.976048i	$-0.430192\pi$
$449$	4139.67	0.435107	0.217554	+	0.976048i	$0.430192\pi$
$450$	4000.28	0.419055
$451$	−6174.57	−0.644677
$452$	−5799.75	−0.603534
$453$	9578.24	0.993432
$454$	−4534.10	−0.468714
$455$	0	0
$456$	3950.79	0.405730
$457$	11418.7	1.16880	0.584401	−	0.811465i	$-0.301329\pi$
$457$	11418.7	1.16880	0.584401	+	0.811465i	$0.301329\pi$
$458$	2947.05	0.300669
$459$	7915.00	0.804881
$460$	133.415	0.0135229
$461$	2502.18	0.252794	0.126397	−	0.991980i	$-0.459659\pi$
$461$	2502.18	0.252794	0.126397	+	0.991980i	$0.459659\pi$
$462$	0	0
$463$	9916.65	0.995391	0.497695	−	0.867352i	$-0.334180\pi$
$463$	9916.65	0.995391	0.497695	+	0.867352i	$0.334180\pi$
$464$	−2065.98	−0.206704
$465$	−14.6065	−0.00145668
$466$	−8813.09	−0.876092
$467$	15914.2	1.57691	0.788457	−	0.615089i	$-0.210880\pi$
$467$	15914.2	1.57691	0.788457	+	0.615089i	$0.210880\pi$
$468$	−626.611	−0.0618913
$469$	0	0
$470$	−203.512	−0.0199730
$471$	−4723.00	−0.462047
$472$	3317.12	0.323480
$473$	6253.14	0.607864
$474$	433.192	0.0419772
$475$	−18532.5	−1.79017
$476$	0	0
$477$	327.174	0.0314052
$478$	−11856.7	−1.13454
$479$	−5914.27	−0.564154	−0.282077	−	0.959392i	$-0.591023\pi$
$479$	−5914.27	−0.564154	−0.282077	+	0.959392i	$0.591023\pi$
$480$	151.974	0.0144513
$481$	−3292.82	−0.312141
$482$	6125.48	0.578855
$483$	0	0
$484$	−2952.21	−0.277255
$485$	−1835.24	−0.171823
$486$	7491.17	0.699190
$487$	6116.19	0.569099	0.284549	−	0.958661i	$-0.408156\pi$
$487$	6116.19	0.569099	0.284549	+	0.958661i	$0.408156\pi$
$488$	−336.438	−0.0312087
$489$	4795.35	0.443463
$490$	0	0
$491$	−1284.68	−0.118079	−0.0590394	−	0.998256i	$-0.518804\pi$
$491$	−1284.68	−0.118079	−0.0590394	+	0.998256i	$0.518804\pi$
$492$	3321.69	0.304377
$493$	7211.41	0.658794
$494$	2902.97	0.264395
$495$	574.704	0.0521839
$496$	−49.2093	−0.00445476
$497$	0	0
$498$	−516.425	−0.0464690
$499$	14278.2	1.28093	0.640463	−	0.767989i	$-0.278743\pi$
$499$	14278.2	1.28093	0.640463	+	0.767989i	$0.278743\pi$
$500$	−1437.97	−0.128616
$501$	6537.51	0.582983
$502$	−11815.4	−1.05049
$503$	1329.55	0.117856	0.0589281	−	0.998262i	$-0.481232\pi$
$503$	1329.55	0.117856	0.0589281	+	0.998262i	$0.481232\pi$
$504$	0	0
$505$	−1504.91	−0.132609
$506$	−1120.12	−0.0984101
$507$	−6891.58	−0.603680
$508$	15.7904	0.00137911
$509$	13566.0	1.18134	0.590670	−	0.806913i	$-0.298863\pi$
$509$	13566.0	1.18134	0.590670	+	0.806913i	$0.298863\pi$
$510$	−530.472	−0.0460582
$511$	0	0
$512$	512.000	0.0441942
$513$	−21371.3	−1.83931
$514$	4901.75	0.420636
$515$	−2403.77	−0.205675
$516$	−3363.96	−0.286996
$517$	1708.64	0.145350
$518$	0	0
$519$	7123.72	0.602499
$520$	111.668	0.00941720
$521$	1749.47	0.147113	0.0735563	−	0.997291i	$-0.476565\pi$
$521$	1749.47	0.147113	0.0735563	+	0.997291i	$0.476565\pi$
$522$	4202.96	0.352411
$523$	14240.4	1.19061	0.595305	−	0.803500i	$-0.297031\pi$
$523$	14240.4	1.19061	0.595305	+	0.803500i	$0.297031\pi$
$524$	−4404.39	−0.367188
$525$	0	0
$526$	11044.3	0.915499
$527$	171.768	0.0141979
$528$	−1275.93	−0.105166
$529$	529.000	0.0434783
$530$	−58.3053	−0.00477853
$531$	−6748.22	−0.551503
$532$	0	0
$533$	2440.72	0.198348
$534$	−228.146	−0.0184884
$535$	1795.05	0.145059
$536$	4081.87	0.328937
$537$	2135.03	0.171571
$538$	1137.43	0.0911486
$539$	0	0
$540$	−822.080	−0.0655124
$541$	−11435.1	−0.908753	−0.454376	−	0.890810i	$-0.650138\pi$
$541$	−11435.1	−0.908753	−0.454376	+	0.890810i	$0.650138\pi$
$542$	12181.4	0.965380
$543$	9041.50	0.714563
$544$	−1787.16	−0.140853
$545$	−2033.76	−0.159847
$546$	0	0
$547$	13676.4	1.06903	0.534517	−	0.845158i	$-0.320494\pi$
$547$	13676.4	1.06903	0.534517	+	0.845158i	$0.320494\pi$
$548$	1527.42	0.119066
$549$	684.438	0.0532079
$550$	5985.21	0.464018
$551$	−19471.5	−1.50547
$552$	602.585	0.0464632
$553$	0	0
$554$	−8456.27	−0.648506
$555$	−1624.67	−0.124259
$556$	3013.21	0.229836
$557$	5590.35	0.425262	0.212631	−	0.977133i	$-0.431797\pi$
$557$	5590.35	0.425262	0.212631	+	0.977133i	$0.431797\pi$
$558$	100.110	0.00759494
$559$	−2471.78	−0.187022
$560$	0	0
$561$	4453.71	0.335180
$562$	−16166.2	−1.21340
$563$	18355.8	1.37408	0.687040	−	0.726620i	$-0.258910\pi$
$563$	18355.8	1.37408	0.687040	+	0.726620i	$0.258910\pi$
$564$	−919.186	−0.0686254
$565$	−2102.65	−0.156565
$566$	16656.5	1.23697
$567$	0	0
$568$	−5542.52	−0.409435
$569$	−21808.4	−1.60677	−0.803387	−	0.595458i	$-0.796971\pi$
$569$	−21808.4	−1.60677	−0.803387	+	0.595458i	$0.796971\pi$
$570$	1432.33	0.105252
$571$	10537.7	0.772311	0.386155	−	0.922434i	$-0.373803\pi$
$571$	10537.7	0.772311	0.386155	+	0.922434i	$0.373803\pi$
$572$	−937.534	−0.0685320
$573$	2662.63	0.194124
$574$	0	0
$575$	−2826.63	−0.205006
$576$	−1041.59	−0.0753468
$577$	−8270.43	−0.596711	−0.298356	−	0.954455i	$-0.596438\pi$
$577$	−8270.43	−0.596711	−0.298356	+	0.954455i	$0.596438\pi$
$578$	−3587.81	−0.258189
$579$	14907.6	1.07001
$580$	−749.003	−0.0536218
$581$	0	0
$582$	−8289.08	−0.590366
$583$	489.518	0.0347749
$584$	−8014.98	−0.567915
$585$	−227.173	−0.0160554
$586$	12090.3	0.852300
$587$	6251.52	0.439570	0.219785	−	0.975548i	$-0.429464\pi$
$587$	6251.52	0.439570	0.219785	+	0.975548i	$0.429464\pi$
$588$	0	0
$589$	−463.789	−0.0324450
$590$	1202.59	0.0839151
$591$	−6457.63	−0.449461
$592$	−5473.54	−0.380002
$593$	−16683.7	−1.15534	−0.577671	−	0.816270i	$-0.696038\pi$
$593$	−16683.7	−1.15534	−0.577671	+	0.816270i	$0.696038\pi$
$594$	6901.99	0.476754
$595$	0	0
$596$	−11696.7	−0.803884
$597$	3451.50	0.236617
$598$	442.769	0.0302779
$599$	4617.49	0.314967	0.157484	−	0.987522i	$-0.449662\pi$
$599$	4617.49	0.314967	0.157484	+	0.987522i	$0.449662\pi$
$600$	−3219.82	−0.219081
$601$	−22063.2	−1.49746	−0.748732	−	0.662872i	$-0.769337\pi$
$601$	−22063.2	−1.49746	−0.748732	+	0.662872i	$0.769337\pi$
$602$	0	0
$603$	−8304.02	−0.560806
$604$	11698.9	0.788116
$605$	−1070.30	−0.0719237
$606$	−6797.11	−0.455633
$607$	−228.182	−0.0152580	−0.00762901	−	0.999971i	$-0.502428\pi$
$607$	−228.182	−0.0152580	−0.00762901	+	0.999971i	$0.502428\pi$
$608$	4825.52	0.321876
$609$	0	0
$610$	−121.973	−0.00809596
$611$	−675.402	−0.0447199
$612$	3635.74	0.240141
$613$	21872.4	1.44114	0.720569	−	0.693384i	$-0.243881\pi$
$613$	21872.4	1.44114	0.720569	+	0.693384i	$0.243881\pi$
$614$	−15679.9	−1.03060
$615$	1204.25	0.0789595
$616$	0	0
$617$	15631.6	1.01994	0.509971	−	0.860191i	$-0.329656\pi$
$617$	15631.6	1.01994	0.509971	+	0.860191i	$0.329656\pi$
$618$	−10856.9	−0.706679
$619$	44.1753	0.00286843	0.00143421	−	0.999999i	$-0.499543\pi$
$619$	44.1753	0.00286843	0.00143421	+	0.999999i	$0.499543\pi$
$620$	−17.8404	−0.00115563
$621$	−3259.60	−0.210633
$622$	−12776.9	−0.823645
$623$	0	0
$624$	504.359	0.0323566
$625$	14840.8	0.949812
$626$	16762.6	1.07024
$627$	−12025.5	−0.765950
$628$	−5768.69	−0.366554
$629$	19105.7	1.21112
$630$	0	0
$631$	−12440.7	−0.784876	−0.392438	−	0.919778i	$-0.628368\pi$
$631$	−12440.7	−0.784876	−0.392438	+	0.919778i	$0.628368\pi$
$632$	529.103	0.0333016
$633$	11871.2	0.745402
$634$	10301.1	0.645281
$635$	5.72467	0.000357759 0
$636$	−263.342	−0.0164186
$637$	0	0
$638$	6288.46	0.390223
$639$	11275.5	0.698047
$640$	185.621	0.0114646
$641$	−23632.0	−1.45618	−0.728088	−	0.685484i	$-0.759591\pi$
$641$	−23632.0	−1.45618	−0.728088	+	0.685484i	$0.759591\pi$
$642$	8107.53	0.498409
$643$	16232.8	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$643$	16232.8	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$644$	0	0
$645$	−1219.57	−0.0744507
$646$	−16843.7	−1.02586
$647$	−10225.3	−0.621324	−0.310662	−	0.950520i	$-0.600551\pi$
$647$	−10225.3	−0.621324	−0.310662	+	0.950520i	$0.600551\pi$
$648$	−197.634	−0.0119812
$649$	−10096.7	−0.610677
$650$	−2365.87	−0.142765
$651$	0	0
$652$	5857.07	0.351811
$653$	−13678.0	−0.819694	−0.409847	−	0.912154i	$-0.634418\pi$
$653$	−13678.0	−0.819694	−0.409847	+	0.912154i	$0.634418\pi$
$654$	−9185.71	−0.549220
$655$	−1596.77	−0.0952536
$656$	4057.13	0.241470
$657$	16305.4	0.968241
$658$	0	0
$659$	−18598.7	−1.09940	−0.549699	−	0.835363i	$-0.685258\pi$
$659$	−18598.7	−1.09940	−0.549699	+	0.835363i	$0.685258\pi$
$660$	−462.579	−0.0272816
$661$	12146.3	0.714729	0.357364	−	0.933965i	$-0.383675\pi$
$661$	12146.3	0.714729	0.357364	+	0.933965i	$0.383675\pi$
$662$	−2895.03	−0.169967
$663$	−1760.49	−0.103125
$664$	−630.764	−0.0368651
$665$	0	0
$666$	11135.2	0.647867
$667$	−2969.85	−0.172403
$668$	7984.94	0.462495
$669$	−13675.4	−0.790317
$670$	1479.85	0.0853306
$671$	1024.06	0.0589169
$672$	0	0
$673$	−18740.9	−1.07342	−0.536708	−	0.843768i	$-0.680332\pi$
$673$	−18740.9	−1.07342	−0.536708	+	0.843768i	$0.680332\pi$
$674$	−17029.8	−0.973242
$675$	17417.2	0.993167
$676$	−8417.41	−0.478915
$677$	−9897.01	−0.561851	−0.280925	−	0.959730i	$-0.590641\pi$
$677$	−9897.01	−0.561851	−0.280925	+	0.959730i	$0.590641\pi$
$678$	−9496.86	−0.537942
$679$	0	0
$680$	−647.921	−0.0365391
$681$	−7424.41	−0.417774
$682$	149.784	0.00840985
$683$	−6884.66	−0.385702	−0.192851	−	0.981228i	$-0.561773\pi$
$683$	−6884.66	−0.385702	−0.192851	+	0.981228i	$0.561773\pi$
$684$	−9816.86	−0.548767
$685$	553.754	0.0308874
$686$	0	0
$687$	4825.67	0.267993
$688$	−4108.76	−0.227682
$689$	−193.500	−0.0106992
$690$	218.462	0.0120532
$691$	14078.3	0.775057	0.387529	−	0.921858i	$-0.373329\pi$
$691$	14078.3	0.775057	0.387529	+	0.921858i	$0.373329\pi$
$692$	8700.95	0.477978
$693$	0	0
$694$	−12663.0	−0.692624
$695$	1092.41	0.0596225
$696$	−3382.96	−0.184239
$697$	−14161.6	−0.769598
$698$	−13226.7	−0.717245
$699$	−14431.1	−0.780878
$700$	0	0
$701$	1417.54	0.0763763	0.0381882	−	0.999271i	$-0.487841\pi$
$701$	1417.54	0.0763763	0.0381882	+	0.999271i	$0.487841\pi$
$702$	−2728.26	−0.146683
$703$	−51587.2	−2.76764
$704$	−1558.43	−0.0834313
$705$	−333.243	−0.0178023
$706$	4899.60	0.261188
$707$	0	0
$708$	5431.64	0.288324
$709$	13220.9	0.700313	0.350157	−	0.936691i	$-0.386128\pi$
$709$	13220.9	0.700313	0.350157	+	0.936691i	$0.386128\pi$
$710$	−2009.39	−0.106213
$711$	−1076.39	−0.0567760
$712$	−278.658	−0.0146674
$713$	−70.7383	−0.00371553
$714$	0	0
$715$	−339.895	−0.0177781
$716$	2607.74	0.136111
$717$	−19414.8	−1.01124
$718$	−11213.2	−0.582831
$719$	31598.3	1.63897	0.819484	−	0.573102i	$-0.194260\pi$
$719$	31598.3	1.63897	0.819484	+	0.573102i	$0.194260\pi$
$720$	−377.621	−0.0195460
$721$	0	0
$722$	31761.7	1.63718
$723$	10030.2	0.515945
$724$	11043.3	0.566881
$725$	15868.9	0.812906
$726$	−4834.13	−0.247123
$727$	16092.8	0.820975	0.410488	−	0.911866i	$-0.365358\pi$
$727$	16092.8	0.820975	0.410488	+	0.911866i	$0.365358\pi$
$728$	0	0
$729$	12933.5	0.657089
$730$	−2905.76	−0.147325
$731$	14341.8	0.725652
$732$	−550.904	−0.0278169
$733$	22882.5	1.15305	0.576525	−	0.817079i	$-0.304408\pi$
$733$	22882.5	1.15305	0.576525	+	0.817079i	$0.304408\pi$
$734$	16099.8	0.809613
$735$	0	0
$736$	736.000	0.0368605
$737$	−12424.5	−0.620978
$738$	−8253.68	−0.411683
$739$	12333.1	0.613912	0.306956	−	0.951724i	$-0.400690\pi$
$739$	12333.1	0.613912	0.306956	+	0.951724i	$0.400690\pi$
$740$	−1984.39	−0.0985776
$741$	4753.50	0.235660
$742$	0	0
$743$	−20823.4	−1.02818	−0.514088	−	0.857737i	$-0.671870\pi$
$743$	−20823.4	−1.02818	−0.514088	+	0.857737i	$0.671870\pi$
$744$	−80.5781	−0.00397062
$745$	−4240.53	−0.208538
$746$	−6217.59	−0.305150
$747$	1283.20	0.0628514
$748$	5439.79	0.265907
$749$	0	0
$750$	−2354.61	−0.114638
$751$	2968.78	0.144251	0.0721254	−	0.997396i	$-0.477022\pi$
$751$	2968.78	0.144251	0.0721254	+	0.997396i	$0.477022\pi$
$752$	−1122.70	−0.0544423
$753$	−19347.2	−0.936322
$754$	−2485.74	−0.120060
$755$	4241.34	0.204448
$756$	0	0
$757$	−25476.1	−1.22318	−0.611588	−	0.791176i	$-0.709469\pi$
$757$	−25476.1	−1.22318	−0.611588	+	0.791176i	$0.709469\pi$
$758$	284.917	0.0136526
$759$	−1834.15	−0.0877149
$760$	1749.45	0.0834989
$761$	26646.7	1.26931	0.634654	−	0.772796i	$-0.281142\pi$
$761$	26646.7	1.26931	0.634654	+	0.772796i	$0.281142\pi$
$762$	25.8561	0.00122922
$763$	0	0
$764$	3252.15	0.154004
$765$	1318.11	0.0622957
$766$	13125.4	0.619110
$767$	3991.08	0.187887
$768$	838.379	0.0393911
$769$	−8988.61	−0.421506	−0.210753	−	0.977539i	$-0.567591\pi$
$769$	−8988.61	−0.421506	−0.210753	+	0.977539i	$0.567591\pi$
$770$	0	0
$771$	8026.42	0.374921
$772$	18208.2	0.848869
$773$	1658.67	0.0771773	0.0385887	−	0.999255i	$-0.487714\pi$
$773$	1658.67	0.0771773	0.0385887	+	0.999255i	$0.487714\pi$
$774$	8358.71	0.388175
$775$	377.980	0.0175193
$776$	−10124.3	−0.468353
$777$	0	0
$778$	−19015.9	−0.876289
$779$	38237.8	1.75868
$780$	182.851	0.00839374
$781$	16870.4	0.772945
$782$	−2569.05	−0.117479
$783$	18299.6	0.835218
$784$	0	0
$785$	−2091.39	−0.0950890
$786$	−7212.00	−0.327282
$787$	−1458.02	−0.0660392	−0.0330196	−	0.999455i	$-0.510512\pi$
$787$	−1458.02	−0.0660392	−0.0330196	+	0.999455i	$0.510512\pi$
$788$	−7887.38	−0.356569
$789$	18084.5	0.816002
$790$	191.822	0.00863888
$791$	0	0
$792$	3170.42	0.142242
$793$	−404.795	−0.0181270
$794$	−6634.49	−0.296536
$795$	−95.4725	−0.00425920
$796$	4215.67	0.187714
$797$	10538.6	0.468378	0.234189	−	0.972191i	$-0.424757\pi$
$797$	10538.6	0.468378	0.234189	+	0.972191i	$0.424757\pi$
$798$	0	0
$799$	3918.84	0.173515
$800$	−3932.70	−0.173803
$801$	566.893	0.0250064
$802$	15682.4	0.690482
$803$	24396.1	1.07213
$804$	6683.90	0.293188
$805$	0	0
$806$	−59.2075	−0.00258746
$807$	1862.49	0.0812426
$808$	−8302.02	−0.361465
$809$	−10372.3	−0.450769	−0.225384	−	0.974270i	$-0.572364\pi$
$809$	−10372.3	−0.450769	−0.225384	+	0.974270i	$0.572364\pi$
$810$	−71.6507	−0.00310808
$811$	14500.3	0.627837	0.313919	−	0.949450i	$-0.398358\pi$
$811$	14500.3	0.627837	0.313919	+	0.949450i	$0.398358\pi$
$812$	0	0
$813$	19946.5	0.860462
$814$	16660.4	0.717381
$815$	2123.43	0.0912644
$816$	−2926.41	−0.125545
$817$	−38724.4	−1.65825
$818$	−24839.4	−1.06172
$819$	0	0
$820$	1470.88	0.0626406
$821$	31142.8	1.32386	0.661930	−	0.749565i	$-0.269737\pi$
$821$	31142.8	1.32386	0.661930	+	0.749565i	$0.269737\pi$
$822$	2501.09	0.106126
$823$	−11026.8	−0.467037	−0.233518	−	0.972352i	$-0.575024\pi$
$823$	−11026.8	−0.467037	−0.233518	+	0.972352i	$0.575024\pi$
$824$	−13260.7	−0.560627
$825$	9800.53	0.413589
$826$	0	0
$827$	−10587.6	−0.445185	−0.222592	−	0.974912i	$-0.571452\pi$
$827$	−10587.6	−0.445185	−0.222592	+	0.974912i	$0.571452\pi$
$828$	−1497.29	−0.0628436
$829$	−13342.7	−0.559002	−0.279501	−	0.960145i	$-0.590169\pi$
$829$	−13342.7	−0.559002	−0.279501	+	0.960145i	$0.590169\pi$
$830$	−228.678	−0.00956329
$831$	−13846.8	−0.578026
$832$	616.026	0.0256693
$833$	0	0
$834$	4934.01	0.204857
$835$	2894.87	0.119977
$836$	−14688.0	−0.607648
$837$	435.877	0.0180001
$838$	−12878.0	−0.530865
$839$	1269.25	0.0522280	0.0261140	−	0.999659i	$-0.491687\pi$
$839$	1269.25	0.0522280	0.0261140	+	0.999659i	$0.491687\pi$
$840$	0	0
$841$	−7716.06	−0.316374
$842$	3991.46	0.163367
$843$	−26471.5	−1.08153
$844$	14499.6	0.591347
$845$	−3051.66	−0.124237
$846$	2283.98	0.0928189
$847$	0	0
$848$	−321.648	−0.0130253
$849$	27274.3	1.10253
$850$	13727.3	0.553933
$851$	−7868.22	−0.316944
$852$	−9075.64	−0.364937
$853$	45943.8	1.84418	0.922089	−	0.386977i	$-0.126481\pi$
$853$	45943.8	1.84418	0.922089	+	0.386977i	$0.126481\pi$
$854$	0	0
$855$	−3559.02	−0.142358
$856$	9902.58	0.395401
$857$	16704.6	0.665832	0.332916	−	0.942956i	$-0.391967\pi$
$857$	16704.6	0.665832	0.332916	+	0.942956i	$0.391967\pi$
$858$	−1535.17	−0.0610839
$859$	−9613.50	−0.381849	−0.190924	−	0.981605i	$-0.561149\pi$
$859$	−9613.50	−0.381849	−0.190924	+	0.981605i	$0.561149\pi$
$860$	−1489.59	−0.0590637
$861$	0	0
$862$	19266.7	0.761283
$863$	29112.3	1.14831	0.574157	−	0.818745i	$-0.305330\pi$
$863$	29112.3	1.14831	0.574157	+	0.818745i	$0.305330\pi$
$864$	−4535.10	−0.178573
$865$	3154.45	0.123994
$866$	−21219.1	−0.832628
$867$	−5874.90	−0.230129
$868$	0	0
$869$	−1610.49	−0.0628679
$870$	−1226.46	−0.0477942
$871$	4911.22	0.191056
$872$	−11219.5	−0.435710
$873$	20596.6	0.798497
$874$	6936.68	0.268463
$875$	0	0
$876$	−13124.2	−0.506194
$877$	16002.4	0.616147	0.308074	−	0.951362i	$-0.400316\pi$
$877$	16002.4	0.616147	0.308074	+	0.951362i	$0.400316\pi$
$878$	−11430.5	−0.439365
$879$	19797.4	0.759671
$880$	−564.996	−0.0216432
$881$	24886.3	0.951691	0.475846	−	0.879529i	$-0.342142\pi$
$881$	24886.3	0.951691	0.475846	+	0.879529i	$0.342142\pi$
$882$	0	0
$883$	−5564.81	−0.212085	−0.106042	−	0.994362i	$-0.533818\pi$
$883$	−5564.81	−0.212085	−0.106042	+	0.994362i	$0.533818\pi$
$884$	−2150.27	−0.0818117
$885$	1969.19	0.0747952
$886$	−16329.4	−0.619183
$887$	−31281.3	−1.18413	−0.592065	−	0.805890i	$-0.701687\pi$
$887$	−31281.3	−1.18413	−0.592065	+	0.805890i	$0.701687\pi$
$888$	−8962.70	−0.338703
$889$	0	0
$890$	−101.025	−0.00380491
$891$	601.562	0.0226185
$892$	−16703.2	−0.626979
$893$	−10581.2	−0.396515
$894$	−19152.8	−0.716518
$895$	945.413	0.0353091
$896$	0	0
$897$	725.016	0.0269873
$898$	8279.34	0.307667
$899$	397.130	0.0147331
$900$	8000.56	0.296317
$901$	1122.73	0.0415133
$902$	−12349.1	−0.455855
$903$	0	0
$904$	−11599.5	−0.426763
$905$	4003.66	0.147057
$906$	19156.5	0.702463
$907$	22174.4	0.811784	0.405892	−	0.913921i	$-0.366961\pi$
$907$	22174.4	0.811784	0.405892	+	0.913921i	$0.366961\pi$
$908$	−9068.21	−0.331431
$909$	16889.3	0.616264
$910$	0	0
$911$	−44322.7	−1.61194	−0.805970	−	0.591956i	$-0.798356\pi$
$911$	−44322.7	−1.61194	−0.805970	+	0.591956i	$0.798356\pi$
$912$	7901.58	0.286894
$913$	1919.93	0.0695951
$914$	22837.3	0.826468
$915$	−199.725	−0.00721609
$916$	5894.10	0.212605
$917$	0	0
$918$	15830.0	0.569137
$919$	19890.9	0.713971	0.356986	−	0.934110i	$-0.383804\pi$
$919$	19890.9	0.713971	0.356986	+	0.934110i	$0.383804\pi$
$920$	266.830	0.00956211
$921$	−25675.2	−0.918597
$922$	5004.36	0.178753
$923$	−6668.63	−0.237812
$924$	0	0
$925$	42042.6	1.49444
$926$	19833.3	0.703848
$927$	26977.0	0.955815
$928$	−4131.96	−0.146162
$929$	−34030.5	−1.20183	−0.600917	−	0.799311i	$-0.705198\pi$
$929$	−34030.5	−1.20183	−0.600917	+	0.799311i	$0.705198\pi$
$930$	−29.2129	−0.00103003
$931$	0	0
$932$	−17626.2	−0.619490
$933$	−20921.7	−0.734131
$934$	31828.3	1.11505
$935$	1972.15	0.0689798
$936$	−1253.22	−0.0437637
$937$	17488.1	0.609725	0.304863	−	0.952396i	$-0.401389\pi$
$937$	17488.1	0.609725	0.304863	+	0.952396i	$0.401389\pi$
$938$	0	0
$939$	27448.1	0.953925
$940$	−407.024	−0.0141231
$941$	−6832.13	−0.236685	−0.118343	−	0.992973i	$-0.537758\pi$
$941$	−6832.13	−0.236685	−0.118343	+	0.992973i	$0.537758\pi$
$942$	−9445.99	−0.326717
$943$	5832.12	0.201400
$944$	6634.23	0.228735
$945$	0	0
$946$	12506.3	0.429825
$947$	−12310.5	−0.422425	−0.211213	−	0.977440i	$-0.567741\pi$
$947$	−12310.5	−0.422425	−0.211213	+	0.977440i	$0.567741\pi$
$948$	866.385	0.0296824
$949$	−9643.44	−0.329862
$950$	−37065.1	−1.26584
$951$	16867.6	0.575152
$952$	0	0
$953$	31238.8	1.06183	0.530914	−	0.847425i	$-0.321849\pi$
$953$	31238.8	1.06183	0.530914	+	0.847425i	$0.321849\pi$
$954$	654.349	0.0222068
$955$	1179.04	0.0399506
$956$	−23713.3	−0.802243
$957$	10297.1	0.347813
$958$	−11828.5	−0.398917
$959$	0	0
$960$	303.947	0.0102186
$961$	−29781.5	−0.999682
$962$	−6585.64	−0.220717
$963$	−20145.5	−0.674121
$964$	12251.0	0.409312
$965$	6601.22	0.220208
$966$	0	0
$967$	41989.0	1.39636	0.698178	−	0.715924i	$-0.253994\pi$
$967$	41989.0	1.39636	0.698178	+	0.715924i	$0.253994\pi$
$968$	−5904.43	−0.196049
$969$	−27580.9	−0.914371
$970$	−3670.48	−0.121497
$971$	22454.0	0.742103	0.371052	−	0.928612i	$-0.378997\pi$
$971$	22454.0	0.742103	0.371052	+	0.928612i	$0.378997\pi$
$972$	14982.3	0.494402
$973$	0	0
$974$	12232.4	0.402414
$975$	−3874.01	−0.127249
$976$	−672.877	−0.0220679
$977$	2197.25	0.0719512	0.0359756	−	0.999353i	$-0.488546\pi$
$977$	2197.25	0.0719512	0.0359756	+	0.999353i	$0.488546\pi$
$978$	9590.71	0.313576
$979$	848.183	0.0276895
$980$	0	0
$981$	22824.5	0.742844
$982$	−2569.36	−0.0834943
$983$	35114.3	1.13934	0.569670	−	0.821873i	$-0.307071\pi$
$983$	35114.3	1.13934	0.569670	+	0.821873i	$0.307071\pi$
$984$	6643.38	0.215227
$985$	−2859.50	−0.0924988
$986$	14422.8	0.465838
$987$	0	0
$988$	5805.95	0.186955
$989$	−5906.34	−0.189900
$990$	1149.41	0.0368996
$991$	−33767.5	−1.08240	−0.541201	−	0.840893i	$-0.682030\pi$
$991$	−33767.5	−1.08240	−0.541201	+	0.840893i	$0.682030\pi$
$992$	−98.4185	−0.00314999
$993$	−4740.49	−0.151495
$994$	0	0
$995$	1528.36	0.0486956
$996$	−1032.85	−0.0328585
$997$	−34616.5	−1.09961	−0.549807	−	0.835292i	$-0.685299\pi$
$997$	−34616.5	−1.09961	−0.549807	+	0.835292i	$0.685299\pi$
$998$	28556.5	0.905751
$999$	48482.5	1.53545

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.g.1.2		2
7.6	odd	2		322.4.a.c.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.a.c.1.1	✓	2	7.6	odd	2
2254.4.a.g.1.2		2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.27492 q^{3} +4.00000 q^{4} +1.45017 q^{5} +6.54983 q^{6} +8.00000 q^{8} -16.2749 q^{9} +2.90033 q^{10} -24.3505 q^{11} +13.0997 q^{12} +9.62541 q^{13} +4.74917 q^{15} +16.0000 q^{16} -55.8488 q^{17} -32.5498 q^{18} +150.797 q^{19} +5.80066 q^{20} -48.7010 q^{22} +23.0000 q^{23} +26.1993 q^{24} -122.897 q^{25} +19.2508 q^{26} -141.722 q^{27} -129.124 q^{29} +9.49834 q^{30} -3.07558 q^{31} +32.0000 q^{32} -79.7459 q^{33} -111.698 q^{34} -65.0997 q^{36} -342.096 q^{37} +301.595 q^{38} +31.5224 q^{39} +11.6013 q^{40} +253.571 q^{41} -256.797 q^{43} -97.4020 q^{44} -23.6013 q^{45} +46.0000 q^{46} -70.1686 q^{47} +52.3987 q^{48} -245.794 q^{50} -182.900 q^{51} +38.5017 q^{52} -20.1030 q^{53} -283.444 q^{54} -35.3123 q^{55} +493.849 q^{57} -258.248 q^{58} +414.640 q^{59} +18.9967 q^{60} -42.0548 q^{61} -6.15116 q^{62} +64.0000 q^{64} +13.9584 q^{65} -159.492 q^{66} +510.234 q^{67} -223.395 q^{68} +75.3231 q^{69} -692.815 q^{71} -130.199 q^{72} -1001.87 q^{73} -684.193 q^{74} -402.478 q^{75} +603.189 q^{76} +63.0449 q^{78} +66.1379 q^{79} +23.2026 q^{80} -24.7043 q^{81} +507.141 q^{82} -78.8455 q^{83} -80.9901 q^{85} -513.595 q^{86} -422.870 q^{87} -194.804 q^{88} -34.8323 q^{89} -47.2026 q^{90} +92.0000 q^{92} -10.0723 q^{93} -140.337 q^{94} +218.681 q^{95} +104.797 q^{96} -1265.54 q^{97} +396.302 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - q^{3} + 8 q^{4} + 18 q^{5} - 2 q^{6} + 16 q^{8} - 25 q^{9} + 36 q^{10} - 94 q^{11} - 4 q^{12} + 57 q^{13} - 66 q^{15} + 32 q^{16} - 6 q^{17} - 50 q^{18} + 60 q^{19} + 72 q^{20} - 188 q^{22}+ \cdots + 1004 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - q^3 + 8 * q^4 + 18 * q^5 - 2 * q^6 + 16 * q^8 - 25 * q^9 + 36 * q^10 - 94 * q^11 - 4 * q^12 + 57 * q^13 - 66 * q^15 + 32 * q^16 - 6 * q^17 - 50 * q^18 + 60 * q^19 + 72 * q^20 - 188 * q^22 + 46 * q^23 - 8 * q^24 + 26 * q^25 + 114 * q^26 + 11 * q^27 - 145 * q^29 - 132 * q^30 - 59 * q^31 + 64 * q^32 + 218 * q^33 - 12 * q^34 - 100 * q^36 - 352 * q^37 + 120 * q^38 - 171 * q^39 + 144 * q^40 + 107 * q^41 - 272 * q^43 - 376 * q^44 - 168 * q^45 + 92 * q^46 + 441 * q^47 - 16 * q^48 + 52 * q^50 - 396 * q^51 + 228 * q^52 - 312 * q^53 + 22 * q^54 - 1188 * q^55 + 882 * q^57 - 290 * q^58 - 122 * q^59 - 264 * q^60 - 522 * q^61 - 118 * q^62 + 128 * q^64 + 798 * q^65 + 436 * q^66 - 414 * q^67 - 24 * q^68 - 23 * q^69 - 457 * q^71 - 200 * q^72 - 1815 * q^73 - 704 * q^74 - 1039 * q^75 + 240 * q^76 - 342 * q^78 - 970 * q^79 + 288 * q^80 - 442 * q^81 + 214 * q^82 + 250 * q^83 + 744 * q^85 - 544 * q^86 - 355 * q^87 - 752 * q^88 + 1546 * q^89 - 336 * q^90 + 184 * q^92 + 229 * q^93 + 882 * q^94 - 1284 * q^95 - 32 * q^96 - 1610 * q^97 + 1004 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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