LMFDB - Embedded newform 2254.4.a.l.1.4

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:	$5$
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} - 92x^{3} - 28x^{2} + 1593x - 1782 $ x^5 - 92x^3 - 28x^2 + 1593*x - 1782
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.52502$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +2.52502 q^{3} +4.00000 q^{4} +15.4206 q^{5} +5.05004 q^{6} +8.00000 q^{8} -20.6243 q^{9} +30.8412 q^{10} +16.2367 q^{11} +10.1001 q^{12} -50.2121 q^{13} +38.9372 q^{15} +16.0000 q^{16} -130.670 q^{17} -41.2486 q^{18} -85.0072 q^{19} +61.6823 q^{20} +32.4733 q^{22} +23.0000 q^{23} +20.2002 q^{24} +112.794 q^{25} -100.424 q^{26} -120.252 q^{27} -65.1081 q^{29} +77.8745 q^{30} +87.4054 q^{31} +32.0000 q^{32} +40.9979 q^{33} -261.341 q^{34} -82.4971 q^{36} -341.524 q^{37} -170.014 q^{38} -126.786 q^{39} +123.365 q^{40} -423.344 q^{41} +487.541 q^{43} +64.9467 q^{44} -318.038 q^{45} +46.0000 q^{46} +102.708 q^{47} +40.4003 q^{48} +225.588 q^{50} -329.945 q^{51} -200.848 q^{52} -565.138 q^{53} -240.504 q^{54} +250.379 q^{55} -214.645 q^{57} -130.216 q^{58} -344.174 q^{59} +155.749 q^{60} -203.883 q^{61} +174.811 q^{62} +64.0000 q^{64} -774.299 q^{65} +81.9958 q^{66} -118.878 q^{67} -522.682 q^{68} +58.0754 q^{69} +449.076 q^{71} -164.994 q^{72} -309.286 q^{73} -683.048 q^{74} +284.807 q^{75} -340.029 q^{76} -253.573 q^{78} +274.745 q^{79} +246.729 q^{80} +253.216 q^{81} -846.688 q^{82} -530.690 q^{83} -2015.01 q^{85} +975.082 q^{86} -164.399 q^{87} +129.893 q^{88} -775.186 q^{89} -636.077 q^{90} +92.0000 q^{92} +220.700 q^{93} +205.417 q^{94} -1310.86 q^{95} +80.8006 q^{96} +1070.49 q^{97} -334.870 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 10 q^{2} - 5 q^{3} + 20 q^{4} - 22 q^{5} - 10 q^{6} + 40 q^{8} + 54 q^{9} - 44 q^{10} + 42 q^{11} - 20 q^{12} - 107 q^{13} + 122 q^{15} + 80 q^{16} - 218 q^{17} + 108 q^{18} - 194 q^{19} - 88 q^{20}+ \cdots - 2616 q^{99}+O(q^{100}) $ 5 * q + 10 * q^2 - 5 * q^3 + 20 * q^4 - 22 * q^5 - 10 * q^6 + 40 * q^8 + 54 * q^9 - 44 * q^10 + 42 * q^11 - 20 * q^12 - 107 * q^13 + 122 * q^15 + 80 * q^16 - 218 * q^17 + 108 * q^18 - 194 * q^19 - 88 * q^20 + 84 * q^22 + 115 * q^23 - 40 * q^24 + 219 * q^25 - 214 * q^26 - 203 * q^27 + 33 * q^29 + 244 * q^30 - 19 * q^31 + 160 * q^32 - 240 * q^33 - 436 * q^34 + 216 * q^36 + 214 * q^37 - 388 * q^38 - 243 * q^39 - 176 * q^40 - 395 * q^41 - 284 * q^43 + 168 * q^44 - 470 * q^45 + 230 * q^46 + 493 * q^47 - 80 * q^48 + 438 * q^50 - 244 * q^51 - 428 * q^52 - 70 * q^53 - 406 * q^54 + 186 * q^55 + 22 * q^57 + 66 * q^58 - 226 * q^59 + 488 * q^60 - 1606 * q^61 - 38 * q^62 + 320 * q^64 + 362 * q^65 - 480 * q^66 + 298 * q^67 - 872 * q^68 - 115 * q^69 - 969 * q^71 + 432 * q^72 - 697 * q^73 + 428 * q^74 - 1065 * q^75 - 776 * q^76 - 486 * q^78 + 186 * q^79 - 352 * q^80 - 335 * q^81 - 790 * q^82 + 1104 * q^83 - 1344 * q^85 - 568 * q^86 + 1603 * q^87 + 336 * q^88 + 814 * q^89 - 940 * q^90 + 460 * q^92 - 3423 * q^93 + 986 * q^94 - 2812 * q^95 - 160 * q^96 + 256 * q^97 - 2616 * 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
$3$	2.52502	0.485940	0.242970	−	0.970034i	$-0.421878\pi$
$3$	2.52502	0.485940	0.242970	+	0.970034i	$0.421878\pi$
$4$	4.00000	0.500000
$5$	15.4206	1.37926	0.689629	−	0.724163i	$-0.257774\pi$
$5$	15.4206	1.37926	0.689629	+	0.724163i	$0.257774\pi$
$6$	5.05004	0.343612
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−20.6243	−0.763862
$10$	30.8412	0.975283
$11$	16.2367	0.445049	0.222525	−	0.974927i	$-0.428570\pi$
$11$	16.2367	0.445049	0.222525	+	0.974927i	$0.428570\pi$
$12$	10.1001	0.242970
$13$	−50.2121	−1.07126	−0.535628	−	0.844454i	$-0.679925\pi$
$13$	−50.2121	−1.07126	−0.535628	+	0.844454i	$0.679925\pi$
$14$	0	0
$15$	38.9372	0.670237
$16$	16.0000	0.250000
$17$	−130.670	−1.86425	−0.932124	−	0.362138i	$-0.882047\pi$
$17$	−130.670	−1.86425	−0.932124	+	0.362138i	$0.882047\pi$
$18$	−41.2486	−0.540132
$19$	−85.0072	−1.02642	−0.513210	−	0.858263i	$-0.671544\pi$
$19$	−85.0072	−1.02642	−0.513210	+	0.858263i	$0.671544\pi$
$20$	61.6823	0.689629
$21$	0	0
$22$	32.4733	0.314697
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	20.2002	0.171806
$25$	112.794	0.902354
$26$	−100.424	−0.757492
$27$	−120.252	−0.857131
$28$	0	0
$29$	−65.1081	−0.416906	−0.208453	−	0.978032i	$-0.566843\pi$
$29$	−65.1081	−0.416906	−0.208453	+	0.978032i	$0.566843\pi$
$30$	77.8745	0.473929
$31$	87.4054	0.506403	0.253201	−	0.967414i	$-0.418517\pi$
$31$	87.4054	0.506403	0.253201	+	0.967414i	$0.418517\pi$
$32$	32.0000	0.176777
$33$	40.9979	0.216267
$34$	−261.341	−1.31822
$35$	0	0
$36$	−82.4971	−0.381931
$37$	−341.524	−1.51747	−0.758733	−	0.651402i	$-0.774181\pi$
$37$	−341.524	−1.51747	−0.758733	+	0.651402i	$0.774181\pi$
$38$	−170.014	−0.725789
$39$	−126.786	−0.520566
$40$	123.365	0.487641
$41$	−423.344	−1.61257	−0.806283	−	0.591530i	$-0.798524\pi$
$41$	−423.344	−1.61257	−0.806283	+	0.591530i	$0.798524\pi$
$42$	0	0
$43$	487.541	1.72905	0.864526	−	0.502587i	$-0.167619\pi$
$43$	487.541	1.72905	0.864526	+	0.502587i	$0.167619\pi$
$44$	64.9467	0.222525
$45$	−318.038	−1.05356
$46$	46.0000	0.147442
$47$	102.708	0.318756	0.159378	−	0.987218i	$-0.449051\pi$
$47$	102.708	0.318756	0.159378	+	0.987218i	$0.449051\pi$
$48$	40.4003	0.121485
$49$	0	0
$50$	225.588	0.638060
$51$	−329.945	−0.905913
$52$	−200.848	−0.535628
$53$	−565.138	−1.46467	−0.732336	−	0.680943i	$-0.761570\pi$
$53$	−565.138	−1.46467	−0.732336	+	0.680943i	$0.761570\pi$
$54$	−240.504	−0.606083
$55$	250.379	0.613838
$56$	0	0
$57$	−214.645	−0.498779
$58$	−130.216	−0.294797
$59$	−344.174	−0.759452	−0.379726	−	0.925099i	$-0.623982\pi$
$59$	−344.174	−0.759452	−0.379726	+	0.925099i	$0.623982\pi$
$60$	155.749	0.335118
$61$	−203.883	−0.427944	−0.213972	−	0.976840i	$-0.568640\pi$
$61$	−203.883	−0.427944	−0.213972	+	0.976840i	$0.568640\pi$
$62$	174.811	0.358081
$63$	0	0
$64$	64.0000	0.125000
$65$	−774.299	−1.47754
$66$	81.9958	0.152924
$67$	−118.878	−0.216764	−0.108382	−	0.994109i	$-0.534567\pi$
$67$	−118.878	−0.216764	−0.108382	+	0.994109i	$0.534567\pi$
$68$	−522.682	−0.932124
$69$	58.0754	0.101326
$70$	0	0
$71$	449.076	0.750642	0.375321	−	0.926895i	$-0.377532\pi$
$71$	449.076	0.750642	0.375321	+	0.926895i	$0.377532\pi$
$72$	−164.994	−0.270066
$73$	−309.286	−0.495879	−0.247939	−	0.968776i	$-0.579753\pi$
$73$	−309.286	−0.495879	−0.247939	+	0.968776i	$0.579753\pi$
$74$	−683.048	−1.07301
$75$	284.807	0.438490
$76$	−340.029	−0.513210
$77$	0	0
$78$	−253.573	−0.368096
$79$	274.745	0.391281	0.195641	−	0.980676i	$-0.437321\pi$
$79$	274.745	0.391281	0.195641	+	0.980676i	$0.437321\pi$
$80$	246.729	0.344815
$81$	253.216	0.347348
$82$	−846.688	−1.14026
$83$	−530.690	−0.701817	−0.350908	−	0.936410i	$-0.614127\pi$
$83$	−530.690	−0.701817	−0.350908	+	0.936410i	$0.614127\pi$
$84$	0	0
$85$	−2015.01	−2.57128
$86$	975.082	1.22262
$87$	−164.399	−0.202591
$88$	129.893	0.157349
$89$	−775.186	−0.923253	−0.461627	−	0.887074i	$-0.652734\pi$
$89$	−775.186	−0.923253	−0.461627	+	0.887074i	$0.652734\pi$
$90$	−636.077	−0.744982
$91$	0	0
$92$	92.0000	0.104257
$93$	220.700	0.246081
$94$	205.417	0.225395
$95$	−1310.86	−1.41570
$96$	80.8006	0.0859029
$97$	1070.49	1.12054	0.560269	−	0.828311i	$-0.310698\pi$
$97$	1070.49	1.12054	0.560269	+	0.828311i	$0.310698\pi$
$98$	0	0
$99$	−334.870	−0.339956
$100$	451.177	0.451177
$101$	−891.235	−0.878031	−0.439016	−	0.898479i	$-0.644673\pi$
$101$	−891.235	−0.878031	−0.439016	+	0.898479i	$0.644673\pi$
$102$	−659.890	−0.640577
$103$	1457.76	1.39454	0.697269	−	0.716810i	$-0.254398\pi$
$103$	1457.76	1.39454	0.697269	+	0.716810i	$0.254398\pi$
$104$	−401.696	−0.378746
$105$	0	0
$106$	−1130.28	−1.03568
$107$	730.304	0.659824	0.329912	−	0.944012i	$-0.392981\pi$
$107$	730.304	0.659824	0.329912	+	0.944012i	$0.392981\pi$
$108$	−481.009	−0.428566
$109$	1655.91	1.45512	0.727558	−	0.686046i	$-0.240655\pi$
$109$	1655.91	1.45512	0.727558	+	0.686046i	$0.240655\pi$
$110$	500.758	0.434049
$111$	−862.355	−0.737397
$112$	0	0
$113$	625.622	0.520828	0.260414	−	0.965497i	$-0.416141\pi$
$113$	625.622	0.520828	0.260414	+	0.965497i	$0.416141\pi$
$114$	−429.289	−0.352690
$115$	354.673	0.287595
$116$	−260.432	−0.208453
$117$	1035.59	0.818291
$118$	−688.349	−0.537014
$119$	0	0
$120$	311.498	0.236965
$121$	−1067.37	−0.801931
$122$	−407.767	−0.302602
$123$	−1068.95	−0.783611
$124$	349.622	0.253201
$125$	−188.220	−0.134680
$126$	0	0
$127$	−530.024	−0.370331	−0.185165	−	0.982707i	$-0.559282\pi$
$127$	−530.024	−0.370331	−0.185165	+	0.982707i	$0.559282\pi$
$128$	128.000	0.0883883
$129$	1231.05	0.840216
$130$	−1548.60	−1.04478
$131$	1501.76	1.00160	0.500800	−	0.865563i	$-0.333039\pi$
$131$	1501.76	1.00160	0.500800	+	0.865563i	$0.333039\pi$
$132$	163.992	0.108134
$133$	0	0
$134$	−237.755	−0.153276
$135$	−1854.36	−1.18221
$136$	−1045.36	−0.659111
$137$	593.298	0.369992	0.184996	−	0.982739i	$-0.440773\pi$
$137$	593.298	0.369992	0.184996	+	0.982739i	$0.440773\pi$
$138$	116.151	0.0716480
$139$	2110.17	1.28764	0.643822	−	0.765175i	$-0.277348\pi$
$139$	2110.17	1.28764	0.643822	+	0.765175i	$0.277348\pi$
$140$	0	0
$141$	259.340	0.154897
$142$	898.153	0.530784
$143$	−815.277	−0.476761
$144$	−329.988	−0.190966
$145$	−1004.00	−0.575021
$146$	−618.571	−0.350639
$147$	0	0
$148$	−1366.10	−0.758733
$149$	2173.54	1.19506	0.597528	−	0.801848i	$-0.296150\pi$
$149$	2173.54	1.19506	0.597528	+	0.801848i	$0.296150\pi$
$150$	569.615	0.310059
$151$	2044.01	1.10158	0.550792	−	0.834643i	$-0.314326\pi$
$151$	2044.01	1.10158	0.550792	+	0.834643i	$0.314326\pi$
$152$	−680.057	−0.362894
$153$	2694.98	1.42403
$154$	0	0
$155$	1347.84	0.698460
$156$	−507.146	−0.260283
$157$	−3307.56	−1.68135	−0.840675	−	0.541540i	$-0.817841\pi$
$157$	−3307.56	−1.68135	−0.840675	+	0.541540i	$0.817841\pi$
$158$	549.490	0.276677
$159$	−1426.98	−0.711743
$160$	493.458	0.243821
$161$	0	0
$162$	506.433	0.245612
$163$	−1902.98	−0.914437	−0.457218	−	0.889354i	$-0.651154\pi$
$163$	−1902.98	−0.914437	−0.457218	+	0.889354i	$0.651154\pi$
$164$	−1693.38	−0.806283
$165$	632.211	0.298288
$166$	−1061.38	−0.496259
$167$	−511.817	−0.237159	−0.118580	−	0.992945i	$-0.537834\pi$
$167$	−511.817	−0.237159	−0.118580	+	0.992945i	$0.537834\pi$
$168$	0	0
$169$	324.250	0.147588
$170$	−4030.03	−1.81817
$171$	1753.21	0.784044
$172$	1950.16	0.864526
$173$	150.429	0.0661091	0.0330546	−	0.999454i	$-0.489476\pi$
$173$	150.429	0.0661091	0.0330546	+	0.999454i	$0.489476\pi$
$174$	−328.798	−0.143254
$175$	0	0
$176$	259.787	0.111262
$177$	−869.047	−0.369048
$178$	−1550.37	−0.652838
$179$	−2821.78	−1.17827	−0.589133	−	0.808036i	$-0.700530\pi$
$179$	−2821.78	−1.17827	−0.589133	+	0.808036i	$0.700530\pi$
$180$	−1272.15	−0.526782
$181$	607.565	0.249502	0.124751	−	0.992188i	$-0.460187\pi$
$181$	607.565	0.249502	0.124751	+	0.992188i	$0.460187\pi$
$182$	0	0
$183$	−514.809	−0.207955
$184$	184.000	0.0737210
$185$	−5266.50	−2.09298
$186$	441.401	0.174006
$187$	−2121.65	−0.829682
$188$	410.833	0.159378
$189$	0	0
$190$	−2621.72	−1.00105
$191$	1087.29	0.411904	0.205952	−	0.978562i	$-0.433971\pi$
$191$	1087.29	0.411904	0.205952	+	0.978562i	$0.433971\pi$
$192$	161.601	0.0607425
$193$	3217.91	1.20016	0.600078	−	0.799942i	$-0.295136\pi$
$193$	3217.91	1.20016	0.600078	+	0.799942i	$0.295136\pi$
$194$	2140.99	0.792340
$195$	−1955.12	−0.717995
$196$	0	0
$197$	−3303.35	−1.19469	−0.597345	−	0.801985i	$-0.703778\pi$
$197$	−3303.35	−1.19469	−0.597345	+	0.801985i	$0.703778\pi$
$198$	−669.739	−0.240385
$199$	2423.32	0.863240	0.431620	−	0.902056i	$-0.357942\pi$
$199$	2423.32	0.863240	0.431620	+	0.902056i	$0.357942\pi$
$200$	902.354	0.319030
$201$	−300.168	−0.105335
$202$	−1782.47	−0.620862
$203$	0	0
$204$	−1319.78	−0.452957
$205$	−6528.21	−2.22415
$206$	2915.52	0.986087
$207$	−474.358	−0.159276
$208$	−803.393	−0.267814
$209$	−1380.23	−0.456807
$210$	0	0
$211$	−2363.30	−0.771072	−0.385536	−	0.922693i	$-0.625983\pi$
$211$	−2363.30	−0.771072	−0.385536	+	0.922693i	$0.625983\pi$
$212$	−2260.55	−0.732336
$213$	1133.93	0.364767
$214$	1460.61	0.466566
$215$	7518.16	2.38481
$216$	−962.018	−0.303042
$217$	0	0
$218$	3311.82	1.02892
$219$	−780.952	−0.240967
$220$	1001.52	0.306919
$221$	6561.23	1.99709
$222$	−1724.71	−0.521419
$223$	4272.53	1.28300	0.641502	−	0.767121i	$-0.278312\pi$
$223$	4272.53	1.28300	0.641502	+	0.767121i	$0.278312\pi$
$224$	0	0
$225$	−2326.30	−0.689274
$226$	1251.24	0.368281
$227$	−2313.03	−0.676306	−0.338153	−	0.941091i	$-0.609802\pi$
$227$	−2313.03	−0.676306	−0.338153	+	0.941091i	$0.609802\pi$
$228$	−858.579	−0.249389
$229$	−6857.40	−1.97882	−0.989410	−	0.145150i	$-0.953634\pi$
$229$	−6857.40	−1.97882	−0.989410	+	0.145150i	$0.953634\pi$
$230$	709.347	0.203361
$231$	0	0
$232$	−520.865	−0.147398
$233$	5132.80	1.44318	0.721590	−	0.692321i	$-0.243412\pi$
$233$	5132.80	1.44318	0.721590	+	0.692321i	$0.243412\pi$
$234$	2071.17	0.578619
$235$	1583.82	0.439647
$236$	−1376.70	−0.379726
$237$	693.736	0.190139
$238$	0	0
$239$	−3025.18	−0.818756	−0.409378	−	0.912365i	$-0.634254\pi$
$239$	−3025.18	−0.818756	−0.409378	+	0.912365i	$0.634254\pi$
$240$	622.996	0.167559
$241$	3626.40	0.969282	0.484641	−	0.874713i	$-0.338950\pi$
$241$	3626.40	0.969282	0.484641	+	0.874713i	$0.338950\pi$
$242$	−2134.74	−0.567051
$243$	3886.19	1.02592
$244$	−815.533	−0.213972
$245$	0	0
$246$	−2137.90	−0.554096
$247$	4268.38	1.09956
$248$	699.243	0.179040
$249$	−1340.00	−0.341041
$250$	−376.441	−0.0952329
$251$	−3420.84	−0.860245	−0.430123	−	0.902771i	$-0.641530\pi$
$251$	−3420.84	−0.860245	−0.430123	+	0.902771i	$0.641530\pi$
$252$	0	0
$253$	373.444	0.0927992
$254$	−1060.05	−0.261863
$255$	−5087.95	−1.24949
$256$	256.000	0.0625000
$257$	3976.53	0.965173	0.482586	−	0.875848i	$-0.339697\pi$
$257$	3976.53	0.965173	0.482586	+	0.875848i	$0.339697\pi$
$258$	2462.10	0.594122
$259$	0	0
$260$	−3097.20	−0.738769
$261$	1342.81	0.318459
$262$	3003.52	0.708238
$263$	8125.09	1.90500	0.952499	−	0.304541i	$-0.0985030\pi$
$263$	8125.09	1.90500	0.952499	+	0.304541i	$0.0985030\pi$
$264$	327.983	0.0764620
$265$	−8714.75	−2.02016
$266$	0	0
$267$	−1957.36	−0.448646
$268$	−475.511	−0.108382
$269$	3052.55	0.691886	0.345943	−	0.938255i	$-0.387559\pi$
$269$	3052.55	0.691886	0.345943	+	0.938255i	$0.387559\pi$
$270$	−3708.72	−0.835946
$271$	7253.38	1.62587	0.812936	−	0.582352i	$-0.197868\pi$
$271$	7253.38	1.62587	0.812936	+	0.582352i	$0.197868\pi$
$272$	−2090.73	−0.466062
$273$	0	0
$274$	1186.60	0.261624
$275$	1831.40	0.401592
$276$	232.302	0.0506628
$277$	2128.80	0.461758	0.230879	−	0.972982i	$-0.425840\pi$
$277$	2128.80	0.461758	0.230879	+	0.972982i	$0.425840\pi$
$278$	4220.35	0.910502
$279$	−1802.67	−0.386822
$280$	0	0
$281$	−5832.58	−1.23823	−0.619115	−	0.785300i	$-0.712508\pi$
$281$	−5832.58	−1.23823	−0.619115	+	0.785300i	$0.712508\pi$
$282$	518.681	0.109528
$283$	−6614.39	−1.38935	−0.694673	−	0.719326i	$-0.744451\pi$
$283$	−6614.39	−1.38935	−0.694673	+	0.719326i	$0.744451\pi$
$284$	1796.31	0.375321
$285$	−3309.95	−0.687945
$286$	−1630.55	−0.337121
$287$	0	0
$288$	−659.977	−0.135033
$289$	12161.8	2.47542
$290$	−2008.01	−0.406601
$291$	2703.02	0.544514
$292$	−1237.14	−0.247939
$293$	−8352.00	−1.66529	−0.832643	−	0.553810i	$-0.813173\pi$
$293$	−8352.00	−1.66529	−0.832643	+	0.553810i	$0.813173\pi$
$294$	0	0
$295$	−5307.37	−1.04748
$296$	−2732.19	−0.536505
$297$	−1952.50	−0.381466
$298$	4347.08	0.845032
$299$	−1154.88	−0.223372
$300$	1139.23	0.219245
$301$	0	0
$302$	4088.02	0.778937
$303$	−2250.38	−0.426671
$304$	−1360.11	−0.256605
$305$	−3144.00	−0.590245
$306$	5389.97	1.00694
$307$	−4736.08	−0.880463	−0.440232	−	0.897884i	$-0.645104\pi$
$307$	−4736.08	−0.880463	−0.440232	+	0.897884i	$0.645104\pi$
$308$	0	0
$309$	3680.87	0.677662
$310$	2695.68	0.493886
$311$	1024.10	0.186725	0.0933623	−	0.995632i	$-0.470239\pi$
$311$	1024.10	0.186725	0.0933623	+	0.995632i	$0.470239\pi$
$312$	−1014.29	−0.184048
$313$	−9492.18	−1.71415	−0.857076	−	0.515190i	$-0.827721\pi$
$313$	−9492.18	−1.71415	−0.857076	+	0.515190i	$0.827721\pi$
$314$	−6615.12	−1.18889
$315$	0	0
$316$	1098.98	0.195641
$317$	−949.926	−0.168306	−0.0841532	−	0.996453i	$-0.526819\pi$
$317$	−949.926	−0.168306	−0.0841532	+	0.996453i	$0.526819\pi$
$318$	−2853.97	−0.503278
$319$	−1057.14	−0.185544
$320$	986.917	0.172407
$321$	1844.03	0.320635
$322$	0	0
$323$	11107.9	1.91350
$324$	1012.87	0.173674
$325$	−5663.63	−0.966651
$326$	−3805.97	−0.646605
$327$	4181.21	0.707099
$328$	−3386.75	−0.570128
$329$	0	0
$330$	1264.42	0.210922
$331$	−3601.89	−0.598120	−0.299060	−	0.954234i	$-0.596673\pi$
$331$	−3601.89	−0.598120	−0.299060	+	0.954234i	$0.596673\pi$
$332$	−2122.76	−0.350908
$333$	7043.69	1.15913
$334$	−1023.63	−0.167697
$335$	−1833.16	−0.298974
$336$	0	0
$337$	−2325.88	−0.375961	−0.187980	−	0.982173i	$-0.560194\pi$
$337$	−2325.88	−0.375961	−0.187980	+	0.982173i	$0.560194\pi$
$338$	648.501	0.104360
$339$	1579.71	0.253091
$340$	−8060.05	−1.28564
$341$	1419.17	0.225374
$342$	3506.42	0.554403
$343$	0	0
$344$	3900.33	0.611312
$345$	895.557	0.139754
$346$	300.857	0.0467462
$347$	−743.523	−0.115027	−0.0575136	−	0.998345i	$-0.518317\pi$
$347$	−743.523	−0.115027	−0.0575136	+	0.998345i	$0.518317\pi$
$348$	−657.596	−0.101296
$349$	−7785.32	−1.19409	−0.597047	−	0.802206i	$-0.703659\pi$
$349$	−7785.32	−1.19409	−0.597047	+	0.802206i	$0.703659\pi$
$350$	0	0
$351$	6038.11	0.918207
$352$	519.574	0.0786743
$353$	−4465.40	−0.673284	−0.336642	−	0.941633i	$-0.609291\pi$
$353$	−4465.40	−0.673284	−0.336642	+	0.941633i	$0.609291\pi$
$354$	−1738.09	−0.260957
$355$	6925.02	1.03533
$356$	−3100.74	−0.461627
$357$	0	0
$358$	−5643.55	−0.833159
$359$	1190.89	0.175077	0.0875385	−	0.996161i	$-0.472100\pi$
$359$	1190.89	0.175077	0.0875385	+	0.996161i	$0.472100\pi$
$360$	−2544.31	−0.372491
$361$	367.219	0.0535383
$362$	1215.13	0.176425
$363$	−2695.13	−0.389691
$364$	0	0
$365$	−4769.36	−0.683945
$366$	−1029.62	−0.147047
$367$	4712.30	0.670245	0.335122	−	0.942175i	$-0.391222\pi$
$367$	4712.30	0.670245	0.335122	+	0.942175i	$0.391222\pi$
$368$	368.000	0.0521286
$369$	8731.16	1.23178
$370$	−10533.0	−1.47996
$371$	0	0
$372$	882.801	0.123041
$373$	−2899.46	−0.402489	−0.201245	−	0.979541i	$-0.564499\pi$
$373$	−2899.46	−0.402489	−0.201245	+	0.979541i	$0.564499\pi$
$374$	−4243.31	−0.586674
$375$	−475.260	−0.0654462
$376$	821.667	0.112697
$377$	3269.21	0.446612
$378$	0	0
$379$	−8339.00	−1.13020	−0.565100	−	0.825022i	$-0.691162\pi$
$379$	−8339.00	−1.13020	−0.565100	+	0.825022i	$0.691162\pi$
$380$	−5243.44	−0.707849
$381$	−1338.32	−0.179958
$382$	2174.58	0.291260
$383$	3573.90	0.476809	0.238405	−	0.971166i	$-0.423376\pi$
$383$	3573.90	0.476809	0.238405	+	0.971166i	$0.423376\pi$
$384$	323.202	0.0429514
$385$	0	0
$386$	6435.81	0.848638
$387$	−10055.2	−1.32076
$388$	4281.98	0.560269
$389$	3082.12	0.401722	0.200861	−	0.979620i	$-0.435626\pi$
$389$	3082.12	0.401722	0.200861	+	0.979620i	$0.435626\pi$
$390$	−3910.24	−0.507699
$391$	−3005.42	−0.388723
$392$	0	0
$393$	3791.98	0.486717
$394$	−6606.70	−0.844773
$395$	4236.72	0.539678
$396$	−1339.48	−0.169978
$397$	5825.10	0.736406	0.368203	−	0.929745i	$-0.379973\pi$
$397$	5825.10	0.736406	0.368203	+	0.929745i	$0.379973\pi$
$398$	4846.64	0.610403
$399$	0	0
$400$	1804.71	0.225588
$401$	−9361.76	−1.16585	−0.582923	−	0.812528i	$-0.698091\pi$
$401$	−9361.76	−1.16585	−0.582923	+	0.812528i	$0.698091\pi$
$402$	−600.337	−0.0744828
$403$	−4388.81	−0.542486
$404$	−3564.94	−0.439016
$405$	3904.74	0.479082
$406$	0	0
$407$	−5545.22	−0.675347
$408$	−2639.56	−0.320289
$409$	8828.05	1.06728	0.533642	−	0.845711i	$-0.320823\pi$
$409$	8828.05	1.06728	0.533642	+	0.845711i	$0.320823\pi$
$410$	−13056.4	−1.57271
$411$	1498.09	0.179794
$412$	5831.04	0.697269
$413$	0	0
$414$	−948.717	−0.112625
$415$	−8183.55	−0.967987
$416$	−1606.79	−0.189373
$417$	5328.23	0.625718
$418$	−2760.47	−0.323012
$419$	−9440.13	−1.10067	−0.550335	−	0.834944i	$-0.685500\pi$
$419$	−9440.13	−1.10067	−0.550335	+	0.834944i	$0.685500\pi$
$420$	0	0
$421$	−8022.19	−0.928687	−0.464344	−	0.885655i	$-0.653710\pi$
$421$	−8022.19	−0.928687	−0.464344	+	0.885655i	$0.653710\pi$
$422$	−4726.60	−0.545230
$423$	−2118.29	−0.243486
$424$	−4521.10	−0.517840
$425$	−14738.9	−1.68221
$426$	2267.85	0.257929
$427$	0	0
$428$	2921.22	0.329912
$429$	−2058.59	−0.231677
$430$	15036.3	1.68632
$431$	−15910.7	−1.77817	−0.889083	−	0.457746i	$-0.848657\pi$
$431$	−15910.7	−1.77817	−0.889083	+	0.457746i	$0.848657\pi$
$432$	−1924.04	−0.214283
$433$	−4098.01	−0.454822	−0.227411	−	0.973799i	$-0.573026\pi$
$433$	−4098.01	−0.454822	−0.227411	+	0.973799i	$0.573026\pi$
$434$	0	0
$435$	−2535.13	−0.279426
$436$	6623.65	0.727558
$437$	−1955.16	−0.214023
$438$	−1561.90	−0.170390
$439$	8184.15	0.889768	0.444884	−	0.895588i	$-0.353245\pi$
$439$	8184.15	0.889768	0.444884	+	0.895588i	$0.353245\pi$
$440$	2003.03	0.217024
$441$	0	0
$442$	13122.5	1.41215
$443$	10534.5	1.12982	0.564910	−	0.825152i	$-0.308911\pi$
$443$	10534.5	1.12982	0.564910	+	0.825152i	$0.308911\pi$
$444$	−3449.42	−0.368699
$445$	−11953.8	−1.27340
$446$	8545.06	0.907221
$447$	5488.23	0.580725
$448$	0	0
$449$	−14963.9	−1.57281	−0.786403	−	0.617714i	$-0.788059\pi$
$449$	−14963.9	−1.57281	−0.786403	+	0.617714i	$0.788059\pi$
$450$	−4652.60	−0.487390
$451$	−6873.70	−0.717671
$452$	2502.49	0.260414
$453$	5161.16	0.535303
$454$	−4626.07	−0.478221
$455$	0	0
$456$	−1717.16	−0.176345
$457$	−8900.42	−0.911038	−0.455519	−	0.890226i	$-0.650546\pi$
$457$	−8900.42	−0.911038	−0.455519	+	0.890226i	$0.650546\pi$
$458$	−13714.8	−1.39924
$459$	15713.4	1.59791
$460$	1418.69	0.143798
$461$	−12896.8	−1.30296	−0.651480	−	0.758666i	$-0.725852\pi$
$461$	−12896.8	−1.30296	−0.651480	+	0.758666i	$0.725852\pi$
$462$	0	0
$463$	−13810.2	−1.38621	−0.693104	−	0.720838i	$-0.743757\pi$
$463$	−13810.2	−1.38621	−0.693104	+	0.720838i	$0.743757\pi$
$464$	−1041.73	−0.104226
$465$	3403.33	0.339410
$466$	10265.6	1.02048
$467$	−15279.1	−1.51399	−0.756995	−	0.653421i	$-0.773333\pi$
$467$	−15279.1	−1.51399	−0.756995	+	0.653421i	$0.773333\pi$
$468$	4142.35	0.409146
$469$	0	0
$470$	3167.64	0.310878
$471$	−8351.65	−0.817035
$472$	−2753.39	−0.268507
$473$	7916.04	0.769514
$474$	1387.47	0.134449
$475$	−9588.32	−0.926194
$476$	0	0
$477$	11655.6	1.11881
$478$	−6050.36	−0.578948
$479$	15499.2	1.47845	0.739225	−	0.673458i	$-0.235192\pi$
$479$	15499.2	1.47845	0.739225	+	0.673458i	$0.235192\pi$
$480$	1245.99	0.118482
$481$	17148.6	1.62559
$482$	7252.80	0.685386
$483$	0	0
$484$	−4269.48	−0.400966
$485$	16507.6	1.54551
$486$	7772.37	0.725436
$487$	−12482.8	−1.16150	−0.580751	−	0.814082i	$-0.697241\pi$
$487$	−12482.8	−1.16150	−0.580751	+	0.814082i	$0.697241\pi$
$488$	−1631.07	−0.151301
$489$	−4805.07	−0.444362
$490$	0	0
$491$	14831.3	1.36319	0.681596	−	0.731729i	$-0.261286\pi$
$491$	14831.3	1.36319	0.681596	+	0.731729i	$0.261286\pi$
$492$	−4275.81	−0.391805
$493$	8507.70	0.777216
$494$	8536.77	0.777505
$495$	−5163.88	−0.468888
$496$	1398.49	0.126601
$497$	0	0
$498$	−2680.00	−0.241152
$499$	2865.86	0.257101	0.128551	−	0.991703i	$-0.458968\pi$
$499$	2865.86	0.257101	0.128551	+	0.991703i	$0.458968\pi$
$500$	−752.882	−0.0673398
$501$	−1292.35	−0.115245
$502$	−6841.68	−0.608285
$503$	−9737.57	−0.863174	−0.431587	−	0.902071i	$-0.642046\pi$
$503$	−9737.57	−0.863174	−0.431587	+	0.902071i	$0.642046\pi$
$504$	0	0
$505$	−13743.4	−1.21103
$506$	746.887	0.0656189
$507$	818.739	0.0717188
$508$	−2120.09	−0.185165
$509$	14015.6	1.22049	0.610247	−	0.792211i	$-0.291070\pi$
$509$	14015.6	1.22049	0.610247	+	0.792211i	$0.291070\pi$
$510$	−10175.9	−0.883522
$511$	0	0
$512$	512.000	0.0441942
$513$	10222.3	0.879777
$514$	7953.07	0.682480
$515$	22479.5	1.92343
$516$	4924.20	0.420108
$517$	1667.64	0.141862
$518$	0	0
$519$	379.835	0.0321251
$520$	−6194.39	−0.522388
$521$	83.3495	0.00700884	0.00350442	−	0.999994i	$-0.498885\pi$
$521$	83.3495	0.00700884	0.00350442	+	0.999994i	$0.498885\pi$
$522$	2685.61	0.225184
$523$	14731.1	1.23164	0.615818	−	0.787888i	$-0.288826\pi$
$523$	14731.1	1.23164	0.615818	+	0.787888i	$0.288826\pi$
$524$	6007.05	0.500800
$525$	0	0
$526$	16250.2	1.34704
$527$	−11421.3	−0.944060
$528$	655.967	0.0540668
$529$	529.000	0.0434783
$530$	−17429.5	−1.42847
$531$	7098.35	0.580117
$532$	0	0
$533$	21257.0	1.72747
$534$	−3914.72	−0.317240
$535$	11261.7	0.910068
$536$	−951.021	−0.0766378
$537$	−7125.04	−0.572566
$538$	6105.11	0.489238
$539$	0	0
$540$	−7417.43	−0.591103
$541$	−3356.19	−0.266717	−0.133359	−	0.991068i	$-0.542576\pi$
$541$	−3356.19	−0.266717	−0.133359	+	0.991068i	$0.542576\pi$
$542$	14506.8	1.14967
$543$	1534.11	0.121243
$544$	−4181.45	−0.329556
$545$	25535.1	2.00698
$546$	0	0
$547$	−14151.2	−1.10615	−0.553073	−	0.833133i	$-0.686545\pi$
$547$	−14151.2	−1.10615	−0.553073	+	0.833133i	$0.686545\pi$
$548$	2373.19	0.184996
$549$	4204.95	0.326890
$550$	3662.81	0.283968
$551$	5534.65	0.427920
$552$	464.603	0.0358240
$553$	0	0
$554$	4257.59	0.326512
$555$	−13298.0	−1.01706
$556$	8440.69	0.643822
$557$	−1801.72	−0.137058	−0.0685292	−	0.997649i	$-0.521831\pi$
$557$	−1801.72	−0.137058	−0.0685292	+	0.997649i	$0.521831\pi$
$558$	−3605.35	−0.273524
$559$	−24480.4	−1.85226
$560$	0	0
$561$	−5357.21	−0.403176
$562$	−11665.2	−0.875561
$563$	4338.87	0.324799	0.162399	−	0.986725i	$-0.448077\pi$
$563$	4338.87	0.324799	0.162399	+	0.986725i	$0.448077\pi$
$564$	1037.36	0.0774483
$565$	9647.45	0.718356
$566$	−13228.8	−0.982415
$567$	0	0
$568$	3592.61	0.265392
$569$	8654.77	0.637657	0.318829	−	0.947812i	$-0.396711\pi$
$569$	8654.77	0.637657	0.318829	+	0.947812i	$0.396711\pi$
$570$	−6619.89	−0.486450
$571$	−25113.7	−1.84059	−0.920293	−	0.391230i	$-0.872050\pi$
$571$	−25113.7	−1.84059	−0.920293	+	0.391230i	$0.872050\pi$
$572$	−3261.11	−0.238381
$573$	2745.43	0.200161
$574$	0	0
$575$	2594.27	0.188154
$576$	−1319.95	−0.0954828
$577$	6299.88	0.454536	0.227268	−	0.973832i	$-0.427021\pi$
$577$	6299.88	0.454536	0.227268	+	0.973832i	$0.427021\pi$
$578$	24323.5	1.75039
$579$	8125.27	0.583204
$580$	−4016.02	−0.287510
$581$	0	0
$582$	5406.03	0.385030
$583$	−9175.96	−0.651851
$584$	−2474.28	−0.175320
$585$	15969.4	1.12864
$586$	−16704.0	−1.17754
$587$	−19473.7	−1.36928	−0.684639	−	0.728882i	$-0.740040\pi$
$587$	−19473.7	−1.36928	−0.684639	+	0.728882i	$0.740040\pi$
$588$	0	0
$589$	−7430.09	−0.519782
$590$	−10614.7	−0.740681
$591$	−8341.02	−0.580547
$592$	−5464.39	−0.379366
$593$	618.222	0.0428117	0.0214058	−	0.999771i	$-0.493186\pi$
$593$	618.222	0.0428117	0.0214058	+	0.999771i	$0.493186\pi$
$594$	−3904.99	−0.269737
$595$	0	0
$596$	8694.16	0.597528
$597$	6118.93	0.419483
$598$	−2309.75	−0.157948
$599$	−100.330	−0.00684368	−0.00342184	−	0.999994i	$-0.501089\pi$
$599$	−100.330	−0.00684368	−0.00342184	+	0.999994i	$0.501089\pi$
$600$	2278.46	0.155030
$601$	326.619	0.0221682	0.0110841	−	0.999939i	$-0.496472\pi$
$601$	326.619	0.0221682	0.0110841	+	0.999939i	$0.496472\pi$
$602$	0	0
$603$	2451.77	0.165578
$604$	8176.03	0.550792
$605$	−16459.5	−1.10607
$606$	−4500.77	−0.301702
$607$	3897.82	0.260639	0.130319	−	0.991472i	$-0.458400\pi$
$607$	3897.82	0.260639	0.130319	+	0.991472i	$0.458400\pi$
$608$	−2720.23	−0.181447
$609$	0	0
$610$	−6288.00	−0.417367
$611$	−5157.20	−0.341469
$612$	10779.9	0.712015
$613$	21469.4	1.41458	0.707292	−	0.706922i	$-0.249917\pi$
$613$	21469.4	1.41458	0.707292	+	0.706922i	$0.249917\pi$
$614$	−9472.16	−0.622582
$615$	−16483.8	−1.08080
$616$	0	0
$617$	−1312.99	−0.0856712	−0.0428356	−	0.999082i	$-0.513639\pi$
$617$	−1312.99	−0.0856712	−0.0428356	+	0.999082i	$0.513639\pi$
$618$	7361.74	0.479179
$619$	−24376.6	−1.58284	−0.791420	−	0.611273i	$-0.790658\pi$
$619$	−24376.6	−1.58284	−0.791420	+	0.611273i	$0.790658\pi$
$620$	5391.37	0.349230
$621$	−2765.80	−0.178724
$622$	2048.20	0.132034
$623$	0	0
$624$	−2028.58	−0.130141
$625$	−17001.7	−1.08811
$626$	−18984.4	−1.21209
$627$	−3485.12	−0.221981
$628$	−13230.2	−0.840675
$629$	44627.1	2.82893
$630$	0	0
$631$	−4304.58	−0.271573	−0.135787	−	0.990738i	$-0.543356\pi$
$631$	−4304.58	−0.271573	−0.135787	+	0.990738i	$0.543356\pi$
$632$	2197.96	0.138339
$633$	−5967.37	−0.374695
$634$	−1899.85	−0.119011
$635$	−8173.27	−0.510782
$636$	−5707.93	−0.355871
$637$	0	0
$638$	−2114.28	−0.131199
$639$	−9261.88	−0.573387
$640$	1973.83	0.121910
$641$	817.563	0.0503772	0.0251886	−	0.999683i	$-0.491981\pi$
$641$	817.563	0.0503772	0.0251886	+	0.999683i	$0.491981\pi$
$642$	3688.06	0.226723
$643$	25946.6	1.59134	0.795672	−	0.605728i	$-0.207118\pi$
$643$	25946.6	1.59134	0.795672	+	0.605728i	$0.207118\pi$
$644$	0	0
$645$	18983.5	1.15888
$646$	22215.8	1.35305
$647$	11020.7	0.669655	0.334827	−	0.942279i	$-0.391322\pi$
$647$	11020.7	0.669655	0.334827	+	0.942279i	$0.391322\pi$
$648$	2025.73	0.122806
$649$	−5588.25	−0.337994
$650$	−11327.3	−0.683526
$651$	0	0
$652$	−7611.94	−0.457218
$653$	13448.0	0.805914	0.402957	−	0.915219i	$-0.367982\pi$
$653$	13448.0	0.805914	0.402957	+	0.915219i	$0.367982\pi$
$654$	8362.42	0.499995
$655$	23158.0	1.38146
$656$	−6773.50	−0.403142
$657$	6378.79	0.378783
$658$	0	0
$659$	−25104.9	−1.48399	−0.741994	−	0.670407i	$-0.766120\pi$
$659$	−25104.9	−1.48399	−0.741994	+	0.670407i	$0.766120\pi$
$660$	2528.85	0.149144
$661$	−29107.5	−1.71278	−0.856391	−	0.516328i	$-0.827299\pi$
$661$	−29107.5	−1.71278	−0.856391	+	0.516328i	$0.827299\pi$
$662$	−7203.78	−0.422935
$663$	16567.2	0.970464
$664$	−4245.52	−0.248130
$665$	0	0
$666$	14087.4	0.819632
$667$	−1497.49	−0.0869309
$668$	−2047.27	−0.118580
$669$	10788.2	0.623463
$670$	−3666.32	−0.211407
$671$	−3310.39	−0.190456
$672$	0	0
$673$	29743.1	1.70358	0.851791	−	0.523882i	$-0.175517\pi$
$673$	29743.1	1.70358	0.851791	+	0.523882i	$0.175517\pi$
$674$	−4651.76	−0.265844
$675$	−13563.8	−0.773436
$676$	1297.00	0.0737939
$677$	5690.32	0.323038	0.161519	−	0.986870i	$-0.448361\pi$
$677$	5690.32	0.323038	0.161519	+	0.986870i	$0.448361\pi$
$678$	3159.41	0.178963
$679$	0	0
$680$	−16120.1	−0.909085
$681$	−5840.45	−0.328644
$682$	2838.35	0.159364
$683$	7310.12	0.409537	0.204769	−	0.978810i	$-0.434356\pi$
$683$	7310.12	0.409537	0.204769	+	0.978810i	$0.434356\pi$
$684$	7012.85	0.392022
$685$	9149.00	0.510314
$686$	0	0
$687$	−17315.1	−0.961588
$688$	7800.65	0.432263
$689$	28376.7	1.56904
$690$	1791.11	0.0988210
$691$	−13020.6	−0.716825	−0.358412	−	0.933563i	$-0.616682\pi$
$691$	−13020.6	−0.716825	−0.358412	+	0.933563i	$0.616682\pi$
$692$	601.715	0.0330546
$693$	0	0
$694$	−1487.05	−0.0813365
$695$	32540.1	1.77599
$696$	−1315.19	−0.0716268
$697$	55318.5	3.00622
$698$	−15570.6	−0.844352
$699$	12960.4	0.701299
$700$	0	0
$701$	23471.0	1.26460	0.632301	−	0.774723i	$-0.282111\pi$
$701$	23471.0	1.26460	0.632301	+	0.774723i	$0.282111\pi$
$702$	12076.2	0.649270
$703$	29032.0	1.55756
$704$	1039.15	0.0556312
$705$	3999.18	0.213642
$706$	−8930.80	−0.476084
$707$	0	0
$708$	−3476.19	−0.184524
$709$	−2970.92	−0.157370	−0.0786849	−	0.996900i	$-0.525072\pi$
$709$	−2970.92	−0.157370	−0.0786849	+	0.996900i	$0.525072\pi$
$710$	13850.0	0.732088
$711$	−5666.41	−0.298885
$712$	−6201.49	−0.326419
$713$	2010.32	0.105592
$714$	0	0
$715$	−12572.0	−0.657577
$716$	−11287.1	−0.589133
$717$	−7638.64	−0.397867
$718$	2381.77	0.123798
$719$	2702.14	0.140157	0.0700783	−	0.997541i	$-0.477675\pi$
$719$	2702.14	0.140157	0.0700783	+	0.997541i	$0.477675\pi$
$720$	−5088.61	−0.263391
$721$	0	0
$722$	734.439	0.0378573
$723$	9156.72	0.471013
$724$	2430.26	0.124751
$725$	−7343.81	−0.376196
$726$	−5390.26	−0.275553
$727$	15464.8	0.788938	0.394469	−	0.918909i	$-0.370929\pi$
$727$	15464.8	0.788938	0.394469	+	0.918909i	$0.370929\pi$
$728$	0	0
$729$	2975.85	0.151189
$730$	−9538.72	−0.483622
$731$	−63707.2	−3.22338
$732$	−2059.24	−0.103978
$733$	7652.95	0.385632	0.192816	−	0.981235i	$-0.438238\pi$
$733$	7652.95	0.385632	0.192816	+	0.981235i	$0.438238\pi$
$734$	9424.59	0.473935
$735$	0	0
$736$	736.000	0.0368605
$737$	−1930.18	−0.0964709
$738$	17462.3	0.870999
$739$	−37809.5	−1.88207	−0.941033	−	0.338315i	$-0.890143\pi$
$739$	−37809.5	−1.88207	−0.941033	+	0.338315i	$0.890143\pi$
$740$	−21066.0	−1.04649
$741$	10777.8	0.534319
$742$	0	0
$743$	20288.3	1.00176	0.500878	−	0.865518i	$-0.333010\pi$
$743$	20288.3	1.00176	0.500878	+	0.865518i	$0.333010\pi$
$744$	1765.60	0.0870029
$745$	33517.2	1.64829
$746$	−5798.92	−0.284603
$747$	10945.1	0.536091
$748$	−8486.61	−0.414841
$749$	0	0
$750$	−950.520	−0.0462775
$751$	7531.84	0.365966	0.182983	−	0.983116i	$-0.441425\pi$
$751$	7531.84	0.365966	0.182983	+	0.983116i	$0.441425\pi$
$752$	1643.33	0.0796891
$753$	−8637.69	−0.418028
$754$	6538.42	0.315803
$755$	31519.8	1.51937
$756$	0	0
$757$	22633.7	1.08671	0.543354	−	0.839504i	$-0.317154\pi$
$757$	22633.7	1.08671	0.543354	+	0.839504i	$0.317154\pi$
$758$	−16678.0	−0.799172
$759$	942.952	0.0450948
$760$	−10486.9	−0.500525
$761$	−30974.0	−1.47544	−0.737719	−	0.675108i	$-0.764097\pi$
$761$	−30974.0	−1.47544	−0.737719	+	0.675108i	$0.764097\pi$
$762$	−2676.64	−0.127250
$763$	0	0
$764$	4349.17	0.205952
$765$	41558.2	1.96410
$766$	7147.81	0.337155
$767$	17281.7	0.813567
$768$	646.405	0.0303713
$769$	3067.29	0.143835	0.0719177	−	0.997411i	$-0.477088\pi$
$769$	3067.29	0.143835	0.0719177	+	0.997411i	$0.477088\pi$
$770$	0	0
$771$	10040.8	0.469016
$772$	12871.6	0.600078
$773$	−18054.8	−0.840084	−0.420042	−	0.907505i	$-0.637985\pi$
$773$	−18054.8	−0.840084	−0.420042	+	0.907505i	$0.637985\pi$
$774$	−20110.4	−0.933917
$775$	9858.82	0.456954
$776$	8563.95	0.396170
$777$	0	0
$778$	6164.25	0.284060
$779$	35987.3	1.65517
$780$	−7820.48	−0.358997
$781$	7291.51	0.334073
$782$	−6010.84	−0.274868
$783$	7829.39	0.357343
$784$	0	0
$785$	−51004.5	−2.31902
$786$	7583.95	0.344161
$787$	40117.2	1.81706	0.908529	−	0.417822i	$-0.137206\pi$
$787$	40117.2	1.81706	0.908529	+	0.417822i	$0.137206\pi$
$788$	−13213.4	−0.597345
$789$	20516.0	0.925715
$790$	8473.45	0.381610
$791$	0	0
$792$	−2678.96	−0.120193
$793$	10237.4	0.458437
$794$	11650.2	0.520718
$795$	−22004.9	−0.981677
$796$	9693.28	0.431620
$797$	2326.75	0.103410	0.0517049	−	0.998662i	$-0.483534\pi$
$797$	2326.75	0.103410	0.0517049	+	0.998662i	$0.483534\pi$
$798$	0	0
$799$	−13420.9	−0.594241
$800$	3609.41	0.159515
$801$	15987.6	0.705238
$802$	−18723.5	−0.824377
$803$	−5021.77	−0.220690
$804$	−1200.67	−0.0526673
$805$	0	0
$806$	−8777.61	−0.383596
$807$	7707.75	0.336215
$808$	−7129.88	−0.310431
$809$	24592.8	1.06877	0.534386	−	0.845241i	$-0.320543\pi$
$809$	24592.8	1.06877	0.534386	+	0.845241i	$0.320543\pi$
$810$	7809.49	0.338762
$811$	−14087.1	−0.609944	−0.304972	−	0.952361i	$-0.598647\pi$
$811$	−14087.1	−0.609944	−0.304972	+	0.952361i	$0.598647\pi$
$812$	0	0
$813$	18314.9	0.790077
$814$	−11090.4	−0.477542
$815$	−29345.1	−1.26124
$816$	−5279.12	−0.226478
$817$	−41444.5	−1.77473
$818$	17656.1	0.754683
$819$	0	0
$820$	−26112.8	−1.11207
$821$	−42907.0	−1.82395	−0.911976	−	0.410244i	$-0.865443\pi$
$821$	−42907.0	−1.82395	−0.911976	+	0.410244i	$0.865443\pi$
$822$	2996.18	0.127133
$823$	−10009.2	−0.423936	−0.211968	−	0.977277i	$-0.567987\pi$
$823$	−10009.2	−0.423936	−0.211968	+	0.977277i	$0.567987\pi$
$824$	11662.1	0.493044
$825$	4624.33	0.195150
$826$	0	0
$827$	12764.8	0.536730	0.268365	−	0.963317i	$-0.413517\pi$
$827$	12764.8	0.536730	0.268365	+	0.963317i	$0.413517\pi$
$828$	−1897.43	−0.0796381
$829$	−15637.5	−0.655141	−0.327571	−	0.944827i	$-0.606230\pi$
$829$	−15637.5	−0.655141	−0.327571	+	0.944827i	$0.606230\pi$
$830$	−16367.1	−0.684470
$831$	5375.25	0.224387
$832$	−3213.57	−0.133907
$833$	0	0
$834$	10656.5	0.442450
$835$	−7892.51	−0.327104
$836$	−5520.94	−0.228404
$837$	−10510.7	−0.434053
$838$	−18880.3	−0.778291
$839$	4670.47	0.192184	0.0960920	−	0.995372i	$-0.469366\pi$
$839$	4670.47	0.192184	0.0960920	+	0.995372i	$0.469366\pi$
$840$	0	0
$841$	−20149.9	−0.826190
$842$	−16044.4	−0.656681
$843$	−14727.4	−0.601705
$844$	−9453.19	−0.385536
$845$	5000.13	0.203562
$846$	−4236.57	−0.172171
$847$	0	0
$848$	−9042.20	−0.366168
$849$	−16701.5	−0.675139
$850$	−29477.7	−1.18950
$851$	−7855.05	−0.316413
$852$	4535.71	0.182383
$853$	−31330.0	−1.25758	−0.628792	−	0.777574i	$-0.716450\pi$
$853$	−31330.0	−1.25758	−0.628792	+	0.777574i	$0.716450\pi$
$854$	0	0
$855$	27035.5	1.08140
$856$	5842.43	0.233283
$857$	−21364.6	−0.851575	−0.425787	−	0.904823i	$-0.640003\pi$
$857$	−21364.6	−0.851575	−0.425787	+	0.904823i	$0.640003\pi$
$858$	−4117.18	−0.163821
$859$	26075.6	1.03572	0.517862	−	0.855464i	$-0.326728\pi$
$859$	26075.6	1.03572	0.517862	+	0.855464i	$0.326728\pi$
$860$	30072.6	1.19241
$861$	0	0
$862$	−31821.3	−1.25735
$863$	3264.55	0.128768	0.0643839	−	0.997925i	$-0.479492\pi$
$863$	3264.55	0.128768	0.0643839	+	0.997925i	$0.479492\pi$
$864$	−3848.07	−0.151521
$865$	2319.70	0.0911815
$866$	−8196.02	−0.321607
$867$	30708.7	1.20291
$868$	0	0
$869$	4460.94	0.174139
$870$	−5070.26	−0.197584
$871$	5969.09	0.232210
$872$	13247.3	0.514461
$873$	−22078.2	−0.855937
$874$	−3910.33	−0.151337
$875$	0	0
$876$	−3123.81	−0.120484
$877$	18208.5	0.701091	0.350546	−	0.936546i	$-0.385996\pi$
$877$	18208.5	0.701091	0.350546	+	0.936546i	$0.385996\pi$
$878$	16368.3	0.629161
$879$	−21089.0	−0.809229
$880$	4006.06	0.153459
$881$	27778.9	1.06231	0.531154	−	0.847275i	$-0.321758\pi$
$881$	27778.9	1.06231	0.531154	+	0.847275i	$0.321758\pi$
$882$	0	0
$883$	3117.19	0.118801	0.0594007	−	0.998234i	$-0.481081\pi$
$883$	3117.19	0.118801	0.0594007	+	0.998234i	$0.481081\pi$
$884$	26244.9	0.998543
$885$	−13401.2	−0.509013
$886$	21069.1	0.798904
$887$	10267.6	0.388671	0.194335	−	0.980935i	$-0.437745\pi$
$887$	10267.6	0.388671	0.194335	+	0.980935i	$0.437745\pi$
$888$	−6898.84	−0.260709
$889$	0	0
$890$	−23907.6	−0.900433
$891$	4111.39	0.154587
$892$	17090.1	0.641502
$893$	−8730.95	−0.327178
$894$	10976.5	0.410635
$895$	−43513.4	−1.62513
$896$	0	0
$897$	−2916.09	−0.108545
$898$	−29927.8	−1.11214
$899$	−5690.80	−0.211122
$900$	−9305.20	−0.344637
$901$	73846.8	2.73051
$902$	−13747.4	−0.507470
$903$	0	0
$904$	5004.98	0.184141
$905$	9369.00	0.344128
$906$	10322.3	0.378517
$907$	−42982.9	−1.57356	−0.786782	−	0.617231i	$-0.788255\pi$
$907$	−42982.9	−1.57356	−0.786782	+	0.617231i	$0.788255\pi$
$908$	−9252.14	−0.338153
$909$	18381.1	0.670695
$910$	0	0
$911$	41425.0	1.50656	0.753278	−	0.657702i	$-0.228471\pi$
$911$	41425.0	1.50656	0.753278	+	0.657702i	$0.228471\pi$
$912$	−3434.32	−0.124695
$913$	−8616.64	−0.312343
$914$	−17800.8	−0.644201
$915$	−7938.66	−0.286824
$916$	−27429.6	−0.989410
$917$	0	0
$918$	31426.8	1.12989
$919$	40128.8	1.44040	0.720200	−	0.693766i	$-0.244050\pi$
$919$	40128.8	1.44040	0.720200	+	0.693766i	$0.244050\pi$
$920$	2837.39	0.101680
$921$	−11958.7	−0.427852
$922$	−25793.6	−0.921332
$923$	−22549.1	−0.804129
$924$	0	0
$925$	−38521.9	−1.36929
$926$	−27620.4	−0.980197
$927$	−30065.3	−1.06523
$928$	−2083.46	−0.0736992
$929$	−36983.1	−1.30611	−0.653055	−	0.757311i	$-0.726513\pi$
$929$	−36983.1	−1.30611	−0.653055	+	0.757311i	$0.726513\pi$
$930$	6806.65	0.239999
$931$	0	0
$932$	20531.2	0.721590
$933$	2585.87	0.0907370
$934$	−30558.2	−1.07055
$935$	−32717.1	−1.14435
$936$	8284.70	0.289310
$937$	41169.1	1.43536	0.717682	−	0.696371i	$-0.245203\pi$
$937$	41169.1	1.43536	0.717682	+	0.696371i	$0.245203\pi$
$938$	0	0
$939$	−23967.9	−0.832975
$940$	6335.29	0.219824
$941$	10.1220	0.000350658 0	0.000175329	−	1.00000i	$-0.499944\pi$
$941$	10.1220	0.000350658 0	0.000175329		1.00000i	$0.499944\pi$
$942$	−16703.3	−0.577731
$943$	−9736.91	−0.336243
$944$	−5506.79	−0.189863
$945$	0	0
$946$	15832.1	0.544128
$947$	−1271.29	−0.0436233	−0.0218117	−	0.999762i	$-0.506943\pi$
$947$	−1271.29	−0.0436233	−0.0218117	+	0.999762i	$0.506943\pi$
$948$	2774.94	0.0950696
$949$	15529.9	0.531212
$950$	−19176.6	−0.654918
$951$	−2398.58	−0.0817869
$952$	0	0
$953$	16604.4	0.564396	0.282198	−	0.959356i	$-0.408936\pi$
$953$	16604.4	0.564396	0.282198	+	0.959356i	$0.408936\pi$
$954$	23311.1	0.791116
$955$	16766.7	0.568122
$956$	−12100.7	−0.409378
$957$	−2669.29	−0.0901631
$958$	30998.5	1.04542
$959$	0	0
$960$	2491.98	0.0837796
$961$	−22151.3	−0.743557
$962$	34297.3	1.14947
$963$	−15062.0	−0.504015
$964$	14505.6	0.484641
$965$	49622.0	1.65532
$966$	0	0
$967$	−759.098	−0.0252440	−0.0126220	−	0.999920i	$-0.504018\pi$
$967$	−759.098	−0.0252440	−0.0126220	+	0.999920i	$0.504018\pi$
$968$	−8538.96	−0.283525
$969$	28047.7	0.929848
$970$	33015.3	1.09284
$971$	48924.8	1.61696	0.808481	−	0.588522i	$-0.200290\pi$
$971$	48924.8	1.61696	0.808481	+	0.588522i	$0.200290\pi$
$972$	15544.7	0.512961
$973$	0	0
$974$	−24965.7	−0.821305
$975$	−14300.8	−0.469735
$976$	−3262.13	−0.106986
$977$	−53353.5	−1.74711	−0.873556	−	0.486724i	$-0.838192\pi$
$977$	−53353.5	−1.74711	−0.873556	+	0.486724i	$0.838192\pi$
$978$	−9610.14	−0.314211
$979$	−12586.4	−0.410893
$980$	0	0
$981$	−34152.0	−1.11151
$982$	29662.6	0.963922
$983$	24632.2	0.799233	0.399617	−	0.916682i	$-0.369143\pi$
$983$	24632.2	0.799233	0.399617	+	0.916682i	$0.369143\pi$
$984$	−8551.61	−0.277048
$985$	−50939.5	−1.64779
$986$	17015.4	0.549575
$987$	0	0
$988$	17073.5	0.549779
$989$	11213.4	0.360532
$990$	−10327.8	−0.331554
$991$	42448.6	1.36067	0.680336	−	0.732901i	$-0.261834\pi$
$991$	42448.6	1.36067	0.680336	+	0.732901i	$0.261834\pi$
$992$	2796.97	0.0895202
$993$	−9094.84	−0.290651
$994$	0	0
$995$	37369.0	1.19063
$996$	−5360.01	−0.170520
$997$	7135.08	0.226650	0.113325	−	0.993558i	$-0.463850\pi$
$997$	7135.08	0.226650	0.113325	+	0.993558i	$0.463850\pi$
$998$	5731.72	0.181798
$999$	41069.0	1.30067

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.l.1.4		5
7.6	odd	2		322.4.a.h.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.a.h.1.2	✓	5	7.6	odd	2
2254.4.a.l.1.4		5	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +2.52502 q^{3} +4.00000 q^{4} +15.4206 q^{5} +5.05004 q^{6} +8.00000 q^{8} -20.6243 q^{9} +30.8412 q^{10} +16.2367 q^{11} +10.1001 q^{12} -50.2121 q^{13} +38.9372 q^{15} +16.0000 q^{16} -130.670 q^{17} -41.2486 q^{18} -85.0072 q^{19} +61.6823 q^{20} +32.4733 q^{22} +23.0000 q^{23} +20.2002 q^{24} +112.794 q^{25} -100.424 q^{26} -120.252 q^{27} -65.1081 q^{29} +77.8745 q^{30} +87.4054 q^{31} +32.0000 q^{32} +40.9979 q^{33} -261.341 q^{34} -82.4971 q^{36} -341.524 q^{37} -170.014 q^{38} -126.786 q^{39} +123.365 q^{40} -423.344 q^{41} +487.541 q^{43} +64.9467 q^{44} -318.038 q^{45} +46.0000 q^{46} +102.708 q^{47} +40.4003 q^{48} +225.588 q^{50} -329.945 q^{51} -200.848 q^{52} -565.138 q^{53} -240.504 q^{54} +250.379 q^{55} -214.645 q^{57} -130.216 q^{58} -344.174 q^{59} +155.749 q^{60} -203.883 q^{61} +174.811 q^{62} +64.0000 q^{64} -774.299 q^{65} +81.9958 q^{66} -118.878 q^{67} -522.682 q^{68} +58.0754 q^{69} +449.076 q^{71} -164.994 q^{72} -309.286 q^{73} -683.048 q^{74} +284.807 q^{75} -340.029 q^{76} -253.573 q^{78} +274.745 q^{79} +246.729 q^{80} +253.216 q^{81} -846.688 q^{82} -530.690 q^{83} -2015.01 q^{85} +975.082 q^{86} -164.399 q^{87} +129.893 q^{88} -775.186 q^{89} -636.077 q^{90} +92.0000 q^{92} +220.700 q^{93} +205.417 q^{94} -1310.86 q^{95} +80.8006 q^{96} +1070.49 q^{97} -334.870 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 10 q^{2} - 5 q^{3} + 20 q^{4} - 22 q^{5} - 10 q^{6} + 40 q^{8} + 54 q^{9} - 44 q^{10} + 42 q^{11} - 20 q^{12} - 107 q^{13} + 122 q^{15} + 80 q^{16} - 218 q^{17} + 108 q^{18} - 194 q^{19} - 88 q^{20}+ \cdots - 2616 q^{99}+O(q^{100}) \) 5 * q + 10 * q^2 - 5 * q^3 + 20 * q^4 - 22 * q^5 - 10 * q^6 + 40 * q^8 + 54 * q^9 - 44 * q^10 + 42 * q^11 - 20 * q^12 - 107 * q^13 + 122 * q^15 + 80 * q^16 - 218 * q^17 + 108 * q^18 - 194 * q^19 - 88 * q^20 + 84 * q^22 + 115 * q^23 - 40 * q^24 + 219 * q^25 - 214 * q^26 - 203 * q^27 + 33 * q^29 + 244 * q^30 - 19 * q^31 + 160 * q^32 - 240 * q^33 - 436 * q^34 + 216 * q^36 + 214 * q^37 - 388 * q^38 - 243 * q^39 - 176 * q^40 - 395 * q^41 - 284 * q^43 + 168 * q^44 - 470 * q^45 + 230 * q^46 + 493 * q^47 - 80 * q^48 + 438 * q^50 - 244 * q^51 - 428 * q^52 - 70 * q^53 - 406 * q^54 + 186 * q^55 + 22 * q^57 + 66 * q^58 - 226 * q^59 + 488 * q^60 - 1606 * q^61 - 38 * q^62 + 320 * q^64 + 362 * q^65 - 480 * q^66 + 298 * q^67 - 872 * q^68 - 115 * q^69 - 969 * q^71 + 432 * q^72 - 697 * q^73 + 428 * q^74 - 1065 * q^75 - 776 * q^76 - 486 * q^78 + 186 * q^79 - 352 * q^80 - 335 * q^81 - 790 * q^82 + 1104 * q^83 - 1344 * q^85 - 568 * q^86 + 1603 * q^87 + 336 * q^88 + 814 * q^89 - 940 * q^90 + 460 * q^92 - 3423 * q^93 + 986 * q^94 - 2812 * q^95 - 160 * q^96 + 256 * q^97 - 2616 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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