LMFDB - Embedded newform 2254.4.a.y.1.11

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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4 x^{10} - 212 x^{9} + 487 x^{8} + 16315 x^{7} - 9025 x^{6} - 516068 x^{5} - 504693 x^{4} + \cdots - 11394027 $ x^11 - 4x^10 - 212x^9 + 487x^8 + 16315x^7 - 9025x^6 - 516068x^5 - 504693x^4 + 5020566x^3 + 8901846x^2 - 4796415x - 11394027
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-8.28452$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +10.2845 q^{3} +4.00000 q^{4} -16.3087 q^{5} +20.5690 q^{6} +8.00000 q^{8} +78.7713 q^{9} -32.6174 q^{10} -27.6343 q^{11} +41.1381 q^{12} -18.8028 q^{13} -167.727 q^{15} +16.0000 q^{16} -32.0518 q^{17} +157.543 q^{18} +57.0546 q^{19} -65.2348 q^{20} -55.2687 q^{22} +23.0000 q^{23} +82.2761 q^{24} +140.974 q^{25} -37.6056 q^{26} +532.442 q^{27} +104.726 q^{29} -335.454 q^{30} +225.677 q^{31} +32.0000 q^{32} -284.206 q^{33} -64.1036 q^{34} +315.085 q^{36} -1.51721 q^{37} +114.109 q^{38} -193.377 q^{39} -130.470 q^{40} +418.186 q^{41} -488.129 q^{43} -110.537 q^{44} -1284.66 q^{45} +46.0000 q^{46} +548.023 q^{47} +164.552 q^{48} +281.947 q^{50} -329.637 q^{51} -75.2111 q^{52} +451.501 q^{53} +1064.88 q^{54} +450.680 q^{55} +586.779 q^{57} +209.452 q^{58} -396.484 q^{59} -670.908 q^{60} +851.200 q^{61} +451.353 q^{62} +64.0000 q^{64} +306.649 q^{65} -568.412 q^{66} +291.392 q^{67} -128.207 q^{68} +236.544 q^{69} +504.181 q^{71} +630.170 q^{72} -18.6661 q^{73} -3.03442 q^{74} +1449.85 q^{75} +228.218 q^{76} -386.755 q^{78} +438.759 q^{79} -260.939 q^{80} +3349.09 q^{81} +836.373 q^{82} +83.2225 q^{83} +522.723 q^{85} -976.257 q^{86} +1077.06 q^{87} -221.075 q^{88} -425.962 q^{89} -2569.31 q^{90} +92.0000 q^{92} +2320.98 q^{93} +1096.05 q^{94} -930.487 q^{95} +329.104 q^{96} +784.072 q^{97} -2176.79 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 22 q^{2} + 18 q^{3} + 44 q^{4} + 33 q^{5} + 36 q^{6} + 88 q^{8} + 171 q^{9} + 66 q^{10} + 8 q^{11} + 72 q^{12} + 185 q^{13} - 186 q^{15} + 176 q^{16} + 107 q^{17} + 342 q^{18} + 114 q^{19} + 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) $ 11 * q + 22 * q^2 + 18 * q^3 + 44 * q^4 + 33 * q^5 + 36 * q^6 + 88 * q^8 + 171 * q^9 + 66 * q^10 + 8 * q^11 + 72 * q^12 + 185 * q^13 - 186 * q^15 + 176 * q^16 + 107 * q^17 + 342 * q^18 + 114 * q^19 + 132 * q^20 + 16 * q^22 + 253 * q^23 + 144 * q^24 + 494 * q^25 + 370 * q^26 + 561 * q^27 - 79 * q^29 - 372 * q^30 + 488 * q^31 + 352 * q^32 + 304 * q^33 + 214 * q^34 + 684 * q^36 - 41 * q^37 + 228 * q^38 + 425 * q^39 + 264 * q^40 + 812 * q^41 + 219 * q^43 + 32 * q^44 + 619 * q^45 + 506 * q^46 + 1141 * q^47 + 288 * q^48 + 988 * q^50 + 83 * q^51 + 740 * q^52 + 579 * q^53 + 1122 * q^54 + 951 * q^55 + 383 * q^57 - 158 * q^58 + 1695 * q^59 - 744 * q^60 + 1105 * q^61 + 976 * q^62 + 704 * q^64 - 347 * q^65 + 608 * q^66 + 695 * q^67 + 428 * q^68 + 414 * q^69 + 65 * q^71 + 1368 * q^72 + 1935 * q^73 - 82 * q^74 + 1712 * q^75 + 456 * q^76 + 850 * q^78 + 447 * q^79 + 528 * q^80 + 2203 * q^81 + 1624 * q^82 + 1389 * q^83 + 153 * q^85 + 438 * q^86 + 3354 * q^87 + 64 * q^88 + 2698 * q^89 + 1238 * q^90 + 1012 * q^92 - 686 * q^93 + 2282 * q^94 + 827 * q^95 + 576 * q^96 + 1589 * q^97 - 1729 * 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
$3$	10.2845	1.97926	0.989628	−	0.143654i	$-0.0458853\pi$
$3$	10.2845	1.97926	0.989628	+	0.143654i	$0.0458853\pi$
$4$	4.00000	0.500000
$5$	−16.3087	−1.45869	−0.729347	−	0.684144i	$-0.760176\pi$
$5$	−16.3087	−1.45869	−0.729347	+	0.684144i	$0.760176\pi$
$6$	20.5690	1.39955
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	78.7713	2.91745
$10$	−32.6174	−1.03145
$11$	−27.6343	−0.757461	−0.378730	−	0.925507i	$-0.623639\pi$
$11$	−27.6343	−0.757461	−0.378730	+	0.925507i	$0.623639\pi$
$12$	41.1381	0.989628
$13$	−18.8028	−0.401150	−0.200575	−	0.979678i	$-0.564281\pi$
$13$	−18.8028	−0.401150	−0.200575	+	0.979678i	$0.564281\pi$
$14$	0	0
$15$	−167.727	−2.88713
$16$	16.0000	0.250000
$17$	−32.0518	−0.457277	−0.228638	−	0.973511i	$-0.573427\pi$
$17$	−32.0518	−0.457277	−0.228638	+	0.973511i	$0.573427\pi$
$18$	157.543	2.06295
$19$	57.0546	0.688907	0.344453	−	0.938803i	$-0.388064\pi$
$19$	57.0546	0.688907	0.344453	+	0.938803i	$0.388064\pi$
$20$	−65.2348	−0.729347
$21$	0	0
$22$	−55.2687	−0.535606
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	82.2761	0.699773
$25$	140.974	1.12779
$26$	−37.6056	−0.283656
$27$	532.442	3.79513
$28$	0	0
$29$	104.726	0.670591	0.335296	−	0.942113i	$-0.391164\pi$
$29$	104.726	0.670591	0.335296	+	0.942113i	$0.391164\pi$
$30$	−335.454	−2.04151
$31$	225.677	1.30751	0.653754	−	0.756707i	$-0.273193\pi$
$31$	225.677	1.30751	0.653754	+	0.756707i	$0.273193\pi$
$32$	32.0000	0.176777
$33$	−284.206	−1.49921
$34$	−64.1036	−0.323343
$35$	0	0
$36$	315.085	1.45873
$37$	−1.51721	−0.00674130	−0.00337065	−	0.999994i	$-0.501073\pi$
$37$	−1.51721	−0.00674130	−0.00337065	+	0.999994i	$0.501073\pi$
$38$	114.109	0.487131
$39$	−193.377	−0.793979
$40$	−130.470	−0.515726
$41$	418.186	1.59292	0.796460	−	0.604691i	$-0.206703\pi$
$41$	418.186	1.59292	0.796460	+	0.604691i	$0.206703\pi$
$42$	0	0
$43$	−488.129	−1.73114	−0.865569	−	0.500790i	$-0.833043\pi$
$43$	−488.129	−1.73114	−0.865569	+	0.500790i	$0.833043\pi$
$44$	−110.537	−0.378730
$45$	−1284.66	−4.25567
$46$	46.0000	0.147442
$47$	548.023	1.70080	0.850398	−	0.526140i	$-0.176361\pi$
$47$	548.023	1.70080	0.850398	+	0.526140i	$0.176361\pi$
$48$	164.552	0.494814
$49$	0	0
$50$	281.947	0.797468
$51$	−329.637	−0.905068
$52$	−75.2111	−0.200575
$53$	451.501	1.17016	0.585080	−	0.810976i	$-0.301063\pi$
$53$	451.501	1.17016	0.585080	+	0.810976i	$0.301063\pi$
$54$	1064.88	2.68356
$55$	450.680	1.10490
$56$	0	0
$57$	586.779	1.36352
$58$	209.452	0.474180
$59$	−396.484	−0.874878	−0.437439	−	0.899248i	$-0.644114\pi$
$59$	−396.484	−0.874878	−0.437439	+	0.899248i	$0.644114\pi$
$60$	−670.908	−1.44356
$61$	851.200	1.78664	0.893319	−	0.449422i	$-0.148370\pi$
$61$	851.200	1.78664	0.893319	+	0.449422i	$0.148370\pi$
$62$	451.353	0.924547
$63$	0	0
$64$	64.0000	0.125000
$65$	306.649	0.585156
$66$	−568.412	−1.06010
$67$	291.392	0.531331	0.265665	−	0.964065i	$-0.414408\pi$
$67$	291.392	0.531331	0.265665	+	0.964065i	$0.414408\pi$
$68$	−128.207	−0.228638
$69$	236.544	0.412703
$70$	0	0
$71$	504.181	0.842751	0.421375	−	0.906886i	$-0.361548\pi$
$71$	504.181	0.842751	0.421375	+	0.906886i	$0.361548\pi$
$72$	630.170	1.03148
$73$	−18.6661	−0.0299274	−0.0149637	−	0.999888i	$-0.504763\pi$
$73$	−18.6661	−0.0299274	−0.0149637	+	0.999888i	$0.504763\pi$
$74$	−3.03442	−0.00476682
$75$	1449.85	2.23218
$76$	228.218	0.344453
$77$	0	0
$78$	−386.755	−0.561428
$79$	438.759	0.624864	0.312432	−	0.949940i	$-0.398856\pi$
$79$	438.759	0.624864	0.312432	+	0.949940i	$0.398856\pi$
$80$	−260.939	−0.364674
$81$	3349.09	4.59408
$82$	836.373	1.12636
$83$	83.2225	0.110059	0.0550293	−	0.998485i	$-0.482475\pi$
$83$	83.2225	0.110059	0.0550293	+	0.998485i	$0.482475\pi$
$84$	0	0
$85$	522.723	0.667027
$86$	−976.257	−1.22410
$87$	1077.06	1.32727
$88$	−221.075	−0.267803
$89$	−425.962	−0.507325	−0.253662	−	0.967293i	$-0.581635\pi$
$89$	−425.962	−0.507325	−0.253662	+	0.967293i	$0.581635\pi$
$90$	−2569.31	−3.00922
$91$	0	0
$92$	92.0000	0.104257
$93$	2320.98	2.58789
$94$	1096.05	1.20264
$95$	−930.487	−1.00490
$96$	329.104	0.349886
$97$	784.072	0.820727	0.410363	−	0.911922i	$-0.365402\pi$
$97$	784.072	0.820727	0.410363	+	0.911922i	$0.365402\pi$
$98$	0	0
$99$	−2176.79	−2.20986
$100$	563.895	0.563895
$101$	−1266.56	−1.24779	−0.623897	−	0.781506i	$-0.714452\pi$
$101$	−1266.56	−1.24779	−0.623897	+	0.781506i	$0.714452\pi$
$102$	−659.275	−0.639979
$103$	−484.177	−0.463179	−0.231589	−	0.972814i	$-0.574393\pi$
$103$	−484.177	−0.463179	−0.231589	+	0.972814i	$0.574393\pi$
$104$	−150.422	−0.141828
$105$	0	0
$106$	903.002	0.827427
$107$	−425.648	−0.384570	−0.192285	−	0.981339i	$-0.561590\pi$
$107$	−425.648	−0.384570	−0.192285	+	0.981339i	$0.561590\pi$
$108$	2129.77	1.89757
$109$	−2088.02	−1.83482	−0.917412	−	0.397938i	$-0.869726\pi$
$109$	−2088.02	−1.83482	−0.917412	+	0.397938i	$0.869726\pi$
$110$	901.360	0.781285
$111$	−15.6038	−0.0133428
$112$	0	0
$113$	−580.792	−0.483508	−0.241754	−	0.970338i	$-0.577723\pi$
$113$	−580.792	−0.483508	−0.241754	+	0.970338i	$0.577723\pi$
$114$	1173.56	0.964156
$115$	−375.100	−0.304159
$116$	418.904	0.335296
$117$	−1481.12	−1.17034
$118$	−792.968	−0.618632
$119$	0	0
$120$	−1341.82	−1.02075
$121$	−567.343	−0.426253
$122$	1702.40	1.26334
$123$	4300.84	3.15280
$124$	902.707	0.653754
$125$	−260.511	−0.186406
$126$	0	0
$127$	257.544	0.179948	0.0899738	−	0.995944i	$-0.471322\pi$
$127$	257.544	0.179948	0.0899738	+	0.995944i	$0.471322\pi$
$128$	128.000	0.0883883
$129$	−5020.17	−3.42637
$130$	613.298	0.413767
$131$	881.324	0.587799	0.293900	−	0.955836i	$-0.405047\pi$
$131$	881.324	0.587799	0.293900	+	0.955836i	$0.405047\pi$
$132$	−1136.82	−0.749604
$133$	0	0
$134$	582.783	0.375707
$135$	−8683.44	−5.53594
$136$	−256.414	−0.161672
$137$	−643.403	−0.401238	−0.200619	−	0.979669i	$-0.564295\pi$
$137$	−643.403	−0.401238	−0.200619	+	0.979669i	$0.564295\pi$
$138$	473.088	0.291825
$139$	−1723.42	−1.05164	−0.525822	−	0.850595i	$-0.676242\pi$
$139$	−1723.42	−1.05164	−0.525822	+	0.850595i	$0.676242\pi$
$140$	0	0
$141$	5636.15	3.36631
$142$	1008.36	0.595915
$143$	519.602	0.303855
$144$	1260.34	0.729363
$145$	−1707.95	−0.978188
$146$	−37.3322	−0.0211619
$147$	0	0
$148$	−6.06885	−0.00337065
$149$	2402.05	1.32069	0.660347	−	0.750960i	$-0.270409\pi$
$149$	2402.05	1.32069	0.660347	+	0.750960i	$0.270409\pi$
$150$	2899.69	1.57839
$151$	−1059.86	−0.571195	−0.285597	−	0.958350i	$-0.592192\pi$
$151$	−1059.86	−0.571195	−0.285597	+	0.958350i	$0.592192\pi$
$152$	456.437	0.243565
$153$	−2524.76	−1.33408
$154$	0	0
$155$	−3680.49	−1.90725
$156$	−773.510	−0.396989
$157$	702.454	0.357082	0.178541	−	0.983932i	$-0.442862\pi$
$157$	702.454	0.357082	0.178541	+	0.983932i	$0.442862\pi$
$158$	877.518	0.441845
$159$	4643.47	2.31604
$160$	−521.878	−0.257863
$161$	0	0
$162$	6698.17	3.24851
$163$	2947.52	1.41637	0.708184	−	0.706028i	$-0.249515\pi$
$163$	2947.52	1.41637	0.708184	+	0.706028i	$0.249515\pi$
$164$	1672.75	0.796460
$165$	4635.03	2.18689
$166$	166.445	0.0778231
$167$	−566.737	−0.262608	−0.131304	−	0.991342i	$-0.541916\pi$
$167$	−566.737	−0.262608	−0.131304	+	0.991342i	$0.541916\pi$
$168$	0	0
$169$	−1843.46	−0.839079
$170$	1045.45	0.471659
$171$	4494.26	2.00985
$172$	−1952.51	−0.865569
$173$	−1057.60	−0.464787	−0.232394	−	0.972622i	$-0.574656\pi$
$173$	−1057.60	−0.464787	−0.232394	+	0.972622i	$0.574656\pi$
$174$	2154.11	0.938523
$175$	0	0
$176$	−442.149	−0.189365
$177$	−4077.65	−1.73161
$178$	−851.925	−0.358733
$179$	951.374	0.397257	0.198629	−	0.980075i	$-0.436351\pi$
$179$	951.374	0.397257	0.198629	+	0.980075i	$0.436351\pi$
$180$	−5138.63	−2.12784
$181$	1978.85	0.812636	0.406318	−	0.913732i	$-0.366813\pi$
$181$	1978.85	0.812636	0.406318	+	0.913732i	$0.366813\pi$
$182$	0	0
$183$	8754.18	3.53622
$184$	184.000	0.0737210
$185$	24.7438	0.00983350
$186$	4641.95	1.82992
$187$	885.730	0.346369
$188$	2192.09	0.850398
$189$	0	0
$190$	−1860.97	−0.710575
$191$	3164.81	1.19894	0.599471	−	0.800396i	$-0.295378\pi$
$191$	3164.81	1.19894	0.599471	+	0.800396i	$0.295378\pi$
$192$	658.209	0.247407
$193$	−5119.57	−1.90940	−0.954701	−	0.297568i	$-0.903825\pi$
$193$	−5119.57	−1.90940	−0.954701	+	0.297568i	$0.903825\pi$
$194$	1568.14	0.580341
$195$	3153.73	1.15817
$196$	0	0
$197$	−3056.05	−1.10525	−0.552626	−	0.833429i	$-0.686374\pi$
$197$	−3056.05	−1.10525	−0.552626	+	0.833429i	$0.686374\pi$
$198$	−4353.58	−1.56260
$199$	−2448.58	−0.872237	−0.436118	−	0.899889i	$-0.643647\pi$
$199$	−2448.58	−0.872237	−0.436118	+	0.899889i	$0.643647\pi$
$200$	1127.79	0.398734
$201$	2996.82	1.05164
$202$	−2533.12	−0.882324
$203$	0	0
$204$	−1318.55	−0.452534
$205$	−6820.08	−2.32358
$206$	−968.355	−0.327517
$207$	1811.74	0.608331
$208$	−300.844	−0.100288
$209$	−1576.67	−0.521820
$210$	0	0
$211$	−2369.26	−0.773017	−0.386509	−	0.922286i	$-0.626319\pi$
$211$	−2369.26	−0.773017	−0.386509	+	0.922286i	$0.626319\pi$
$212$	1806.00	0.585080
$213$	5185.26	1.66802
$214$	−851.297	−0.271932
$215$	7960.75	2.52520
$216$	4259.54	1.34178
$217$	0	0
$218$	−4176.04	−1.29742
$219$	−191.972	−0.0592341
$220$	1802.72	0.552452
$221$	602.663	0.183437
$222$	−31.2076	−0.00943475
$223$	435.066	0.130647	0.0653233	−	0.997864i	$-0.479192\pi$
$223$	435.066	0.130647	0.0653233	+	0.997864i	$0.479192\pi$
$224$	0	0
$225$	11104.7	3.29027
$226$	−1161.58	−0.341891
$227$	2979.41	0.871146	0.435573	−	0.900153i	$-0.356546\pi$
$227$	2979.41	0.871146	0.435573	+	0.900153i	$0.356546\pi$
$228$	2347.12	0.681761
$229$	4647.35	1.34107	0.670537	−	0.741877i	$-0.266064\pi$
$229$	4647.35	1.34107	0.670537	+	0.741877i	$0.266064\pi$
$230$	−750.200	−0.215073
$231$	0	0
$232$	837.809	0.237090
$233$	−1651.85	−0.464449	−0.232224	−	0.972662i	$-0.574600\pi$
$233$	−1651.85	−0.464449	−0.232224	+	0.972662i	$0.574600\pi$
$234$	−2962.24	−0.827553
$235$	−8937.55	−2.48094
$236$	−1585.94	−0.437439
$237$	4512.42	1.23677
$238$	0	0
$239$	−4737.07	−1.28207	−0.641036	−	0.767511i	$-0.721495\pi$
$239$	−4737.07	−1.28207	−0.641036	+	0.767511i	$0.721495\pi$
$240$	−2683.63	−0.721782
$241$	1374.90	0.367491	0.183746	−	0.982974i	$-0.441178\pi$
$241$	1374.90	0.367491	0.183746	+	0.982974i	$0.441178\pi$
$242$	−1134.69	−0.301407
$243$	20067.8	5.29773
$244$	3404.80	0.893319
$245$	0	0
$246$	8601.69	2.22936
$247$	−1072.79	−0.276355
$248$	1805.41	0.462274
$249$	855.903	0.217834
$250$	−521.022	−0.131809
$251$	3368.32	0.847037	0.423518	−	0.905888i	$-0.360795\pi$
$251$	3368.32	0.847037	0.423518	+	0.905888i	$0.360795\pi$
$252$	0	0
$253$	−635.590	−0.157941
$254$	515.088	0.127242
$255$	5375.96	1.32022
$256$	256.000	0.0625000
$257$	2705.83	0.656752	0.328376	−	0.944547i	$-0.393499\pi$
$257$	2705.83	0.656752	0.328376	+	0.944547i	$0.393499\pi$
$258$	−10040.3	−2.42281
$259$	0	0
$260$	1226.60	0.292578
$261$	8249.40	1.95642
$262$	1762.65	0.415637
$263$	4051.59	0.949930	0.474965	−	0.880005i	$-0.342461\pi$
$263$	4051.59	0.949930	0.474965	+	0.880005i	$0.342461\pi$
$264$	−2273.65	−0.530050
$265$	−7363.39	−1.70690
$266$	0	0
$267$	−4380.82	−1.00413
$268$	1165.57	0.265665
$269$	3373.56	0.764645	0.382323	−	0.924029i	$-0.375124\pi$
$269$	3373.56	0.764645	0.382323	+	0.924029i	$0.375124\pi$
$270$	−17366.9	−3.91450
$271$	−2094.48	−0.469485	−0.234742	−	0.972058i	$-0.575425\pi$
$271$	−2094.48	−0.469485	−0.234742	+	0.972058i	$0.575425\pi$
$272$	−512.829	−0.114319
$273$	0	0
$274$	−1286.81	−0.283718
$275$	−3895.72	−0.854256
$276$	946.175	0.206352
$277$	−8241.37	−1.78764	−0.893819	−	0.448428i	$-0.851984\pi$
$277$	−8241.37	−1.78764	−0.893819	+	0.448428i	$0.851984\pi$
$278$	−3446.84	−0.743625
$279$	17776.8	3.81459
$280$	0	0
$281$	−5436.99	−1.15425	−0.577124	−	0.816656i	$-0.695825\pi$
$281$	−5436.99	−1.15425	−0.577124	+	0.816656i	$0.695825\pi$
$282$	11272.3	2.38034
$283$	1020.20	0.214291	0.107146	−	0.994243i	$-0.465829\pi$
$283$	1020.20	0.214291	0.107146	+	0.994243i	$0.465829\pi$
$284$	2016.72	0.421375
$285$	−9569.60	−1.98896
$286$	1039.20	0.214858
$287$	0	0
$288$	2520.68	0.515738
$289$	−3885.68	−0.790898
$290$	−3415.89	−0.691683
$291$	8063.80	1.62443
$292$	−74.6645	−0.0149637
$293$	4610.02	0.919181	0.459590	−	0.888131i	$-0.347996\pi$
$293$	4610.02	0.919181	0.459590	+	0.888131i	$0.347996\pi$
$294$	0	0
$295$	6466.14	1.27618
$296$	−12.1377	−0.00238341
$297$	−14713.7	−2.87466
$298$	4804.10	0.933872
$299$	−432.464	−0.0836456
$300$	5799.39	1.11609
$301$	0	0
$302$	−2119.73	−0.403896
$303$	−13025.9	−2.46970
$304$	912.874	0.172227
$305$	−13882.0	−2.60616
$306$	−5049.52	−0.943340
$307$	6959.27	1.29377	0.646883	−	0.762589i	$-0.276072\pi$
$307$	6959.27	1.29377	0.646883	+	0.762589i	$0.276072\pi$
$308$	0	0
$309$	−4979.53	−0.916749
$310$	−7360.99	−1.34863
$311$	−7781.70	−1.41884	−0.709421	−	0.704785i	$-0.751044\pi$
$311$	−7781.70	−1.41884	−0.709421	+	0.704785i	$0.751044\pi$
$312$	−1547.02	−0.280714
$313$	−7958.66	−1.43722	−0.718610	−	0.695413i	$-0.755221\pi$
$313$	−7958.66	−1.43722	−0.718610	+	0.695413i	$0.755221\pi$
$314$	1404.91	0.252495
$315$	0	0
$316$	1755.04	0.312432
$317$	5277.43	0.935048	0.467524	−	0.883980i	$-0.345146\pi$
$317$	5277.43	0.935048	0.467524	+	0.883980i	$0.345146\pi$
$318$	9286.94	1.63769
$319$	−2894.04	−0.507946
$320$	−1043.76	−0.182337
$321$	−4377.59	−0.761162
$322$	0	0
$323$	−1828.70	−0.315021
$324$	13396.3	2.29704
$325$	−2650.70	−0.452413
$326$	5895.05	1.00152
$327$	−21474.3	−3.63159
$328$	3345.49	0.563182
$329$	0	0
$330$	9270.05	1.54636
$331$	9512.16	1.57956	0.789782	−	0.613387i	$-0.210194\pi$
$331$	9512.16	1.57956	0.789782	+	0.613387i	$0.210194\pi$
$332$	332.890	0.0550293
$333$	−119.513	−0.0196674
$334$	−1133.47	−0.185692
$335$	−4752.22	−0.775049
$336$	0	0
$337$	11801.0	1.90754	0.953770	−	0.300539i	$-0.0971665\pi$
$337$	11801.0	1.90754	0.953770	+	0.300539i	$0.0971665\pi$
$338$	−3686.91	−0.593318
$339$	−5973.17	−0.956985
$340$	2090.89	0.333514
$341$	−6236.43	−0.990385
$342$	8988.53	1.42118
$343$	0	0
$344$	−3905.03	−0.612050
$345$	−3857.72	−0.602008
$346$	−2115.21	−0.328654
$347$	−1896.21	−0.293354	−0.146677	−	0.989184i	$-0.546858\pi$
$347$	−1896.21	−0.293354	−0.146677	+	0.989184i	$0.546858\pi$
$348$	4308.23	0.663636
$349$	1645.13	0.252326	0.126163	−	0.992010i	$-0.459734\pi$
$349$	1645.13	0.252326	0.126163	+	0.992010i	$0.459734\pi$
$350$	0	0
$351$	−10011.4	−1.52242
$352$	−884.299	−0.133901
$353$	436.545	0.0658214	0.0329107	−	0.999458i	$-0.489522\pi$
$353$	436.545	0.0658214	0.0329107	+	0.999458i	$0.489522\pi$
$354$	−8155.29	−1.22443
$355$	−8222.54	−1.22932
$356$	−1703.85	−0.253662
$357$	0	0
$358$	1902.75	0.280903
$359$	1760.32	0.258792	0.129396	−	0.991593i	$-0.458696\pi$
$359$	1760.32	0.258792	0.129396	+	0.991593i	$0.458696\pi$
$360$	−10277.3	−1.50461
$361$	−3603.77	−0.525408
$362$	3957.71	0.574620
$363$	−5834.85	−0.843665
$364$	0	0
$365$	304.420	0.0436550
$366$	17508.4	2.50048
$367$	1332.61	0.189541	0.0947706	−	0.995499i	$-0.469788\pi$
$367$	1332.61	0.189541	0.0947706	+	0.995499i	$0.469788\pi$
$368$	368.000	0.0521286
$369$	32941.1	4.64727
$370$	49.4875	0.00695333
$371$	0	0
$372$	9283.90	1.29395
$373$	8386.95	1.16424	0.582118	−	0.813104i	$-0.302224\pi$
$373$	8386.95	1.16424	0.582118	+	0.813104i	$0.302224\pi$
$374$	1771.46	0.244920
$375$	−2679.23	−0.368946
$376$	4384.19	0.601322
$377$	−1969.14	−0.269008
$378$	0	0
$379$	−7783.45	−1.05490	−0.527452	−	0.849585i	$-0.676853\pi$
$379$	−7783.45	−1.05490	−0.527452	+	0.849585i	$0.676853\pi$
$380$	−3721.95	−0.502452
$381$	2648.72	0.356162
$382$	6329.63	0.847780
$383$	−1362.28	−0.181748	−0.0908739	−	0.995862i	$-0.528966\pi$
$383$	−1362.28	−0.181748	−0.0908739	+	0.995862i	$0.528966\pi$
$384$	1316.42	0.174943
$385$	0	0
$386$	−10239.1	−1.35015
$387$	−38450.5	−5.05052
$388$	3136.29	0.410363
$389$	340.973	0.0444421	0.0222211	−	0.999753i	$-0.492926\pi$
$389$	340.973	0.0444421	0.0222211	+	0.999753i	$0.492926\pi$
$390$	6307.47	0.818952
$391$	−737.191	−0.0953488
$392$	0	0
$393$	9063.99	1.16340
$394$	−6112.10	−0.781531
$395$	−7155.59	−0.911486
$396$	−8707.17	−1.10493
$397$	9217.51	1.16527	0.582637	−	0.812733i	$-0.302021\pi$
$397$	9217.51	1.16527	0.582637	+	0.812733i	$0.302021\pi$
$398$	−4897.16	−0.616764
$399$	0	0
$400$	2255.58	0.281947
$401$	−4101.87	−0.510817	−0.255408	−	0.966833i	$-0.582210\pi$
$401$	−4101.87	−0.510817	−0.255408	+	0.966833i	$0.582210\pi$
$402$	5993.64	0.743621
$403$	−4243.35	−0.524507
$404$	−5066.23	−0.623897
$405$	−54619.3	−6.70136
$406$	0	0
$407$	41.9271	0.00510627
$408$	−2637.10	−0.319990
$409$	3555.48	0.429846	0.214923	−	0.976631i	$-0.431050\pi$
$409$	3555.48	0.429846	0.214923	+	0.976631i	$0.431050\pi$
$410$	−13640.2	−1.64302
$411$	−6617.09	−0.794153
$412$	−1936.71	−0.231589
$413$	0	0
$414$	3623.48	0.430155
$415$	−1357.25	−0.160542
$416$	−601.689	−0.0709140
$417$	−17724.5	−2.08147
$418$	−3153.33	−0.368982
$419$	5901.09	0.688036	0.344018	−	0.938963i	$-0.388212\pi$
$419$	5901.09	0.688036	0.344018	+	0.938963i	$0.388212\pi$
$420$	0	0
$421$	−13920.0	−1.61145	−0.805726	−	0.592288i	$-0.798225\pi$
$421$	−13920.0	−1.61145	−0.805726	+	0.592288i	$0.798225\pi$
$422$	−4738.52	−0.546606
$423$	43168.5	4.96199
$424$	3612.01	0.413714
$425$	−4518.46	−0.515712
$426$	10370.5	1.17947
$427$	0	0
$428$	−1702.59	−0.192285
$429$	5343.86	0.601408
$430$	15921.5	1.78559
$431$	−5217.18	−0.583068	−0.291534	−	0.956560i	$-0.594166\pi$
$431$	−5217.18	−0.583068	−0.291534	+	0.956560i	$0.594166\pi$
$432$	8519.08	0.948783
$433$	−16199.5	−1.79792	−0.898959	−	0.438034i	$-0.855675\pi$
$433$	−16199.5	−1.79792	−0.898959	+	0.438034i	$0.855675\pi$
$434$	0	0
$435$	−17565.4	−1.93608
$436$	−8352.07	−0.917412
$437$	1312.26	0.143647
$438$	−383.944	−0.0418848
$439$	−11622.1	−1.26354	−0.631770	−	0.775156i	$-0.717671\pi$
$439$	−11622.1	−1.26354	−0.631770	+	0.775156i	$0.717671\pi$
$440$	3605.44	0.390642
$441$	0	0
$442$	1205.33	0.129709
$443$	907.320	0.0973094	0.0486547	−	0.998816i	$-0.484507\pi$
$443$	907.320	0.0973094	0.0486547	+	0.998816i	$0.484507\pi$
$444$	−62.4151	−0.00667138
$445$	6946.89	0.740032
$446$	870.133	0.0923811
$447$	24703.9	2.61399
$448$	0	0
$449$	1283.93	0.134950	0.0674750	−	0.997721i	$-0.478506\pi$
$449$	1283.93	0.134950	0.0674750	+	0.997721i	$0.478506\pi$
$450$	22209.4	2.32658
$451$	−11556.3	−1.20657
$452$	−2323.17	−0.241754
$453$	−10900.2	−1.13054
$454$	5958.81	0.615993
$455$	0	0
$456$	4694.23	0.482078
$457$	−712.015	−0.0728811	−0.0364406	−	0.999336i	$-0.511602\pi$
$457$	−712.015	−0.0728811	−0.0364406	+	0.999336i	$0.511602\pi$
$458$	9294.71	0.948282
$459$	−17065.7	−1.73543
$460$	−1500.40	−0.152079
$461$	−9315.85	−0.941177	−0.470588	−	0.882353i	$-0.655958\pi$
$461$	−9315.85	−0.941177	−0.470588	+	0.882353i	$0.655958\pi$
$462$	0	0
$463$	−12792.9	−1.28409	−0.642046	−	0.766666i	$-0.721914\pi$
$463$	−12792.9	−1.28409	−0.642046	+	0.766666i	$0.721914\pi$
$464$	1675.62	0.167648
$465$	−37852.1	−3.77494
$466$	−3303.71	−0.328415
$467$	247.428	0.0245174	0.0122587	−	0.999925i	$-0.496098\pi$
$467$	247.428	0.0245174	0.0122587	+	0.999925i	$0.496098\pi$
$468$	−5924.47	−0.585168
$469$	0	0
$470$	−17875.1	−1.75429
$471$	7224.40	0.706757
$472$	−3171.87	−0.309316
$473$	13489.1	1.31127
$474$	9024.85	0.874525
$475$	8043.20	0.776942
$476$	0	0
$477$	35565.3	3.41389
$478$	−9474.13	−0.906562
$479$	−1802.17	−0.171907	−0.0859533	−	0.996299i	$-0.527394\pi$
$479$	−1802.17	−0.171907	−0.0859533	+	0.996299i	$0.527394\pi$
$480$	−5367.27	−0.510377
$481$	28.5278	0.00270427
$482$	2749.81	0.259855
$483$	0	0
$484$	−2269.37	−0.213127
$485$	−12787.2	−1.19719
$486$	40135.6	3.74606
$487$	8106.29	0.754274	0.377137	−	0.926158i	$-0.376909\pi$
$487$	8106.29	0.754274	0.377137	+	0.926158i	$0.376909\pi$
$488$	6809.60	0.631672
$489$	30313.9	2.80335
$490$	0	0
$491$	15619.0	1.43560	0.717798	−	0.696251i	$-0.245150\pi$
$491$	15619.0	1.43560	0.717798	+	0.696251i	$0.245150\pi$
$492$	17203.4	1.57640
$493$	−3356.66	−0.306646
$494$	−2145.57	−0.195412
$495$	35500.6	3.22351
$496$	3610.83	0.326877
$497$	0	0
$498$	1711.81	0.154032
$499$	−21443.4	−1.92372	−0.961862	−	0.273536i	$-0.911807\pi$
$499$	−21443.4	−1.92372	−0.961862	+	0.273536i	$0.911807\pi$
$500$	−1042.04	−0.0932032
$501$	−5828.62	−0.519767
$502$	6736.63	0.598945
$503$	1097.74	0.0973073	0.0486537	−	0.998816i	$-0.484507\pi$
$503$	1097.74	0.0973073	0.0486537	+	0.998816i	$0.484507\pi$
$504$	0	0
$505$	20655.9	1.82015
$506$	−1271.18	−0.111681
$507$	−18959.0	−1.66075
$508$	1030.18	0.0899738
$509$	17686.9	1.54019	0.770094	−	0.637930i	$-0.220209\pi$
$509$	17686.9	1.54019	0.770094	+	0.637930i	$0.220209\pi$
$510$	10751.9	0.933535
$511$	0	0
$512$	512.000	0.0441942
$513$	30378.3	2.61449
$514$	5411.67	0.464394
$515$	7896.30	0.675636
$516$	−20080.7	−1.71318
$517$	−15144.3	−1.28829
$518$	0	0
$519$	−10876.9	−0.919933
$520$	2453.19	0.206884
$521$	8771.02	0.737554	0.368777	−	0.929518i	$-0.379777\pi$
$521$	8771.02	0.737554	0.368777	+	0.929518i	$0.379777\pi$
$522$	16498.8	1.38340
$523$	−15519.9	−1.29759	−0.648795	−	0.760963i	$-0.724727\pi$
$523$	−15519.9	−1.29759	−0.648795	+	0.760963i	$0.724727\pi$
$524$	3525.30	0.293900
$525$	0	0
$526$	8103.17	0.671702
$527$	−7233.35	−0.597893
$528$	−4547.29	−0.374802
$529$	529.000	0.0434783
$530$	−14726.8	−1.20696
$531$	−31231.5	−2.55242
$532$	0	0
$533$	−7863.06	−0.639000
$534$	−8761.63	−0.710024
$535$	6941.77	0.560970
$536$	2331.13	0.187854
$537$	9784.42	0.786274
$538$	6747.12	0.540686
$539$	0	0
$540$	−34733.8	−2.76797
$541$	−5504.03	−0.437406	−0.218703	−	0.975791i	$-0.570183\pi$
$541$	−5504.03	−0.437406	−0.218703	+	0.975791i	$0.570183\pi$
$542$	−4188.95	−0.331976
$543$	20351.6	1.60841
$544$	−1025.66	−0.0808359
$545$	34052.9	2.67645
$546$	0	0
$547$	−4995.13	−0.390450	−0.195225	−	0.980758i	$-0.562544\pi$
$547$	−4995.13	−0.390450	−0.195225	+	0.980758i	$0.562544\pi$
$548$	−2573.61	−0.200619
$549$	67050.1	5.21244
$550$	−7791.43	−0.604050
$551$	5975.11	0.461975
$552$	1892.35	0.145913
$553$	0	0
$554$	−16482.7	−1.26405
$555$	254.478	0.0194630
$556$	−6893.68	−0.525822
$557$	8868.66	0.674645	0.337322	−	0.941389i	$-0.390479\pi$
$557$	8868.66	0.674645	0.337322	+	0.941389i	$0.390479\pi$
$558$	35553.7	2.69732
$559$	9178.17	0.694446
$560$	0	0
$561$	9109.31	0.685553
$562$	−10874.0	−0.816177
$563$	−15661.6	−1.17240	−0.586199	−	0.810167i	$-0.699376\pi$
$563$	−15661.6	−1.17240	−0.586199	+	0.810167i	$0.699376\pi$
$564$	22544.6	1.68315
$565$	9471.97	0.705290
$566$	2040.40	0.151527
$567$	0	0
$568$	4033.45	0.297957
$569$	8385.22	0.617798	0.308899	−	0.951095i	$-0.400040\pi$
$569$	8385.22	0.617798	0.308899	+	0.951095i	$0.400040\pi$
$570$	−19139.2	−1.40641
$571$	−20243.9	−1.48368	−0.741840	−	0.670577i	$-0.766046\pi$
$571$	−20243.9	−1.48368	−0.741840	+	0.670577i	$0.766046\pi$
$572$	2078.41	0.151928
$573$	32548.6	2.37301
$574$	0	0
$575$	3242.40	0.235160
$576$	5041.36	0.364682
$577$	5780.04	0.417030	0.208515	−	0.978019i	$-0.433137\pi$
$577$	5780.04	0.417030	0.208515	+	0.978019i	$0.433137\pi$
$578$	−7771.36	−0.559249
$579$	−52652.3	−3.77919
$580$	−6831.78	−0.489094
$581$	0	0
$582$	16127.6	1.14864
$583$	−12476.9	−0.886349
$584$	−149.329	−0.0105810
$585$	24155.1	1.70716
$586$	9220.03	0.649959
$587$	−9242.05	−0.649847	−0.324924	−	0.945740i	$-0.605339\pi$
$587$	−9242.05	−0.649847	−0.324924	+	0.945740i	$0.605339\pi$
$588$	0	0
$589$	12875.9	0.900751
$590$	12932.3	0.902396
$591$	−31430.0	−2.18758
$592$	−24.2754	−0.00168532
$593$	−21131.7	−1.46336	−0.731681	−	0.681647i	$-0.761264\pi$
$593$	−21131.7	−1.46336	−0.731681	+	0.681647i	$0.761264\pi$
$594$	−29427.4	−2.03269
$595$	0	0
$596$	9608.20	0.660347
$597$	−25182.4	−1.72638
$598$	−864.928	−0.0591464
$599$	13238.3	0.903012	0.451506	−	0.892268i	$-0.350887\pi$
$599$	13238.3	0.903012	0.451506	+	0.892268i	$0.350887\pi$
$600$	11598.8	0.789196
$601$	−23291.5	−1.58083	−0.790415	−	0.612571i	$-0.790135\pi$
$601$	−23291.5	−1.58083	−0.790415	+	0.612571i	$0.790135\pi$
$602$	0	0
$603$	22953.3	1.55013
$604$	−4239.45	−0.285597
$605$	9252.63	0.621774
$606$	−26051.9	−1.74635
$607$	10032.1	0.670824	0.335412	−	0.942072i	$-0.391124\pi$
$607$	10032.1	0.670824	0.335412	+	0.942072i	$0.391124\pi$
$608$	1825.75	0.121783
$609$	0	0
$610$	−27763.9	−1.84283
$611$	−10304.4	−0.682274
$612$	−10099.0	−0.667042
$613$	4707.08	0.310142	0.155071	−	0.987903i	$-0.450439\pi$
$613$	4707.08	0.310142	0.155071	+	0.987903i	$0.450439\pi$
$614$	13918.5	0.914831
$615$	−70141.2	−4.59897
$616$	0	0
$617$	−25595.4	−1.67007	−0.835033	−	0.550200i	$-0.814552\pi$
$617$	−25595.4	−1.67007	−0.835033	+	0.550200i	$0.814552\pi$
$618$	−9959.06	−0.648240
$619$	11681.1	0.758489	0.379244	−	0.925297i	$-0.376184\pi$
$619$	11681.1	0.758489	0.379244	+	0.925297i	$0.376184\pi$
$620$	−14722.0	−0.953627
$621$	12246.2	0.791340
$622$	−15563.4	−1.00327
$623$	0	0
$624$	−3094.04	−0.198495
$625$	−13373.1	−0.855880
$626$	−15917.3	−1.01627
$627$	−16215.3	−1.03281
$628$	2809.81	0.178541
$629$	48.6294	0.00308264
$630$	0	0
$631$	6134.76	0.387038	0.193519	−	0.981097i	$-0.438010\pi$
$631$	6134.76	0.387038	0.193519	+	0.981097i	$0.438010\pi$
$632$	3510.07	0.220923
$633$	−24366.7	−1.53000
$634$	10554.9	0.661178
$635$	−4200.21	−0.262489
$636$	18573.9	1.15802
$637$	0	0
$638$	−5788.07	−0.359172
$639$	39715.0	2.45869
$640$	−2087.51	−0.128932
$641$	17461.6	1.07596	0.537981	−	0.842957i	$-0.319187\pi$
$641$	17461.6	1.07596	0.537981	+	0.842957i	$0.319187\pi$
$642$	−8755.17	−0.538223
$643$	−11315.5	−0.693997	−0.346998	−	0.937866i	$-0.612799\pi$
$643$	−11315.5	−0.693997	−0.346998	+	0.937866i	$0.612799\pi$
$644$	0	0
$645$	81872.4	4.99802
$646$	−3657.41	−0.222753
$647$	−10354.5	−0.629177	−0.314588	−	0.949228i	$-0.601866\pi$
$647$	−10354.5	−0.629177	−0.314588	+	0.949228i	$0.601866\pi$
$648$	26792.7	1.62425
$649$	10956.6	0.662686
$650$	−5301.39	−0.319904
$651$	0	0
$652$	11790.1	0.708184
$653$	21004.8	1.25877	0.629387	−	0.777092i	$-0.283306\pi$
$653$	21004.8	1.25877	0.629387	+	0.777092i	$0.283306\pi$
$654$	−42948.5	−2.56792
$655$	−14373.3	−0.857419
$656$	6690.98	0.398230
$657$	−1470.35	−0.0873120
$658$	0	0
$659$	−11472.6	−0.678161	−0.339080	−	0.940757i	$-0.610116\pi$
$659$	−11472.6	−0.678161	−0.339080	+	0.940757i	$0.610116\pi$
$660$	18540.1	1.09344
$661$	19392.9	1.14114	0.570571	−	0.821248i	$-0.306722\pi$
$661$	19392.9	1.14114	0.570571	+	0.821248i	$0.306722\pi$
$662$	19024.3	1.11692
$663$	6198.10	0.363068
$664$	665.780	0.0389116
$665$	0	0
$666$	−239.025	−0.0139070
$667$	2408.70	0.139828
$668$	−2266.95	−0.131304
$669$	4474.45	0.258583
$670$	−9504.44	−0.548042
$671$	−23522.3	−1.35331
$672$	0	0
$673$	4238.83	0.242786	0.121393	−	0.992605i	$-0.461264\pi$
$673$	4238.83	0.242786	0.121393	+	0.992605i	$0.461264\pi$
$674$	23602.0	1.34883
$675$	75060.4	4.28011
$676$	−7373.82	−0.419539
$677$	4205.95	0.238771	0.119385	−	0.992848i	$-0.461908\pi$
$677$	4205.95	0.238771	0.119385	+	0.992848i	$0.461908\pi$
$678$	−11946.3	−0.676691
$679$	0	0
$680$	4181.79	0.235830
$681$	30641.8	1.72422
$682$	−12472.9	−0.700308
$683$	−12328.9	−0.690708	−0.345354	−	0.938472i	$-0.612241\pi$
$683$	−12328.9	−0.690708	−0.345354	+	0.938472i	$0.612241\pi$
$684$	17977.1	1.00493
$685$	10493.1	0.585284
$686$	0	0
$687$	47795.8	2.65433
$688$	−7810.06	−0.432784
$689$	−8489.47	−0.469409
$690$	−7715.45	−0.425684
$691$	15725.1	0.865720	0.432860	−	0.901461i	$-0.357504\pi$
$691$	15725.1	0.865720	0.432860	+	0.901461i	$0.357504\pi$
$692$	−4230.42	−0.232394
$693$	0	0
$694$	−3792.41	−0.207432
$695$	28106.7	1.53403
$696$	8616.45	0.469261
$697$	−13403.6	−0.728405
$698$	3290.26	0.178422
$699$	−16988.5	−0.919263
$700$	0	0
$701$	35899.1	1.93422	0.967112	−	0.254349i	$-0.0818613\pi$
$701$	35899.1	1.93422	0.967112	+	0.254349i	$0.0818613\pi$
$702$	−20022.8	−1.07651
$703$	−86.5639	−0.00464413
$704$	−1768.60	−0.0946826
$705$	−91918.3	−4.91042
$706$	873.090	0.0465427
$707$	0	0
$708$	−16310.6	−0.865804
$709$	12087.3	0.640264	0.320132	−	0.947373i	$-0.396273\pi$
$709$	12087.3	0.640264	0.320132	+	0.947373i	$0.396273\pi$
$710$	−16445.1	−0.869257
$711$	34561.6	1.82301
$712$	−3407.70	−0.179366
$713$	5190.56	0.272634
$714$	0	0
$715$	−8474.04	−0.443232
$716$	3805.50	0.198629
$717$	−48718.4	−2.53755
$718$	3520.64	0.182993
$719$	20736.2	1.07556	0.537780	−	0.843085i	$-0.319263\pi$
$719$	20736.2	1.07556	0.537780	+	0.843085i	$0.319263\pi$
$720$	−20554.5	−1.06392
$721$	0	0
$722$	−7207.54	−0.371519
$723$	14140.2	0.727359
$724$	7915.42	0.406318
$725$	14763.6	0.756286
$726$	−11669.7	−0.596561
$727$	−7267.27	−0.370740	−0.185370	−	0.982669i	$-0.559348\pi$
$727$	−7267.27	−0.370740	−0.185370	+	0.982669i	$0.559348\pi$
$728$	0	0
$729$	115962.	5.89149
$730$	608.840	0.0308688
$731$	15645.4	0.791609
$732$	35016.7	1.76811
$733$	33570.0	1.69159	0.845796	−	0.533506i	$-0.179126\pi$
$733$	33570.0	1.69159	0.845796	+	0.533506i	$0.179126\pi$
$734$	2665.22	0.134026
$735$	0	0
$736$	736.000	0.0368605
$737$	−8052.41	−0.402462
$738$	65882.1	3.28612
$739$	−19700.5	−0.980644	−0.490322	−	0.871541i	$-0.663121\pi$
$739$	−19700.5	−0.980644	−0.490322	+	0.871541i	$0.663121\pi$
$740$	98.9750	0.00491675
$741$	−11033.1	−0.546977
$742$	0	0
$743$	−68.4168	−0.00337816	−0.00168908	−	0.999999i	$-0.500538\pi$
$743$	−68.4168	−0.00337816	−0.00168908	+	0.999999i	$0.500538\pi$
$744$	18567.8	0.914958
$745$	−39174.3	−1.92649
$746$	16773.9	0.823239
$747$	6555.54	0.321091
$748$	3542.92	0.173185
$749$	0	0
$750$	−5358.45	−0.260884
$751$	−27894.6	−1.35538	−0.677688	−	0.735349i	$-0.737018\pi$
$751$	−27894.6	−1.35538	−0.677688	+	0.735349i	$0.737018\pi$
$752$	8768.37	0.425199
$753$	34641.5	1.67650
$754$	−3938.28	−0.190217
$755$	17285.0	0.833199
$756$	0	0
$757$	−37170.0	−1.78463	−0.892317	−	0.451409i	$-0.850922\pi$
$757$	−37170.0	−1.78463	−0.892317	+	0.451409i	$0.850922\pi$
$758$	−15566.9	−0.745930
$759$	−6536.73	−0.312607
$760$	−7443.89	−0.355287
$761$	4006.34	0.190841	0.0954203	−	0.995437i	$-0.469580\pi$
$761$	4006.34	0.190841	0.0954203	+	0.995437i	$0.469580\pi$
$762$	5297.43	0.251845
$763$	0	0
$764$	12659.3	0.599471
$765$	41175.6	1.94602
$766$	−2724.57	−0.128515
$767$	7455.00	0.350957
$768$	2632.84	0.123703
$769$	10828.0	0.507759	0.253879	−	0.967236i	$-0.418293\pi$
$769$	10828.0	0.507759	0.253879	+	0.967236i	$0.418293\pi$
$770$	0	0
$771$	27828.2	1.29988
$772$	−20478.3	−0.954701
$773$	5766.47	0.268312	0.134156	−	0.990960i	$-0.457168\pi$
$773$	5766.47	0.268312	0.134156	+	0.990960i	$0.457168\pi$
$774$	−76901.0	−3.57125
$775$	31814.5	1.47459
$776$	6272.58	0.290171
$777$	0	0
$778$	681.945	0.0314253
$779$	23859.5	1.09737
$780$	12614.9	0.579086
$781$	−13932.7	−0.638350
$782$	−1474.38	−0.0674218
$783$	55760.6	2.54498
$784$	0	0
$785$	−11456.1	−0.520874
$786$	18128.0	0.822651
$787$	−24553.5	−1.11212	−0.556059	−	0.831143i	$-0.687687\pi$
$787$	−24553.5	−1.11212	−0.556059	+	0.831143i	$0.687687\pi$
$788$	−12224.2	−0.552626
$789$	41668.6	1.88015
$790$	−14311.2	−0.644518
$791$	0	0
$792$	−17414.3	−0.781302
$793$	−16004.9	−0.716710
$794$	18435.0	0.823973
$795$	−75728.9	−3.37840
$796$	−9794.31	−0.436118
$797$	7978.02	0.354575	0.177287	−	0.984159i	$-0.443268\pi$
$797$	7978.02	0.354575	0.177287	+	0.984159i	$0.443268\pi$
$798$	0	0
$799$	−17565.1	−0.777734
$800$	4511.16	0.199367
$801$	−33553.6	−1.48010
$802$	−8203.74	−0.361202
$803$	515.826	0.0226689
$804$	11987.3	0.525820
$805$	0	0
$806$	−8486.70	−0.370882
$807$	34695.4	1.51343
$808$	−10132.5	−0.441162
$809$	21020.9	0.913543	0.456771	−	0.889584i	$-0.349006\pi$
$809$	21020.9	0.913543	0.456771	+	0.889584i	$0.349006\pi$
$810$	−109239.	−4.73858
$811$	17741.2	0.768158	0.384079	−	0.923300i	$-0.374519\pi$
$811$	17741.2	0.768158	0.384079	+	0.923300i	$0.374519\pi$
$812$	0	0
$813$	−21540.7	−0.929230
$814$	83.8543	0.00361068
$815$	−48070.3	−2.06605
$816$	−5274.20	−0.226267
$817$	−27850.0	−1.19259
$818$	7110.96	0.303947
$819$	0	0
$820$	−27280.3	−1.16179
$821$	−85.8127	−0.00364785	−0.00182392	−	0.999998i	$-0.500581\pi$
$821$	−85.8127	−0.00364785	−0.00182392	+	0.999998i	$0.500581\pi$
$822$	−13234.2	−0.561551
$823$	−12447.6	−0.527212	−0.263606	−	0.964630i	$-0.584912\pi$
$823$	−12447.6	−0.527212	−0.263606	+	0.964630i	$0.584912\pi$
$824$	−3873.42	−0.163758
$825$	−40065.5	−1.69079
$826$	0	0
$827$	3949.72	0.166076	0.0830382	−	0.996546i	$-0.473538\pi$
$827$	3949.72	0.166076	0.0830382	+	0.996546i	$0.473538\pi$
$828$	7246.96	0.304166
$829$	−1354.04	−0.0567282	−0.0283641	−	0.999598i	$-0.509030\pi$
$829$	−1354.04	−0.0567282	−0.0283641	+	0.999598i	$0.509030\pi$
$830$	−2714.50	−0.113520
$831$	−84758.5	−3.53819
$832$	−1203.38	−0.0501438
$833$	0	0
$834$	−35449.1	−1.47182
$835$	9242.75	0.383064
$836$	−6306.67	−0.260910
$837$	120160.	4.96216
$838$	11802.2	0.486515
$839$	31163.1	1.28233	0.641163	−	0.767405i	$-0.278452\pi$
$839$	31163.1	1.28233	0.641163	+	0.767405i	$0.278452\pi$
$840$	0	0
$841$	−13421.5	−0.550308
$842$	−27840.1	−1.13947
$843$	−55916.8	−2.28455
$844$	−9477.04	−0.386509
$845$	30064.4	1.22396
$846$	86336.9	3.50866
$847$	0	0
$848$	7224.02	0.292540
$849$	10492.2	0.424138
$850$	−9036.92	−0.364663
$851$	−34.8959	−0.00140566
$852$	20741.0	0.834010
$853$	13541.1	0.543538	0.271769	−	0.962363i	$-0.412391\pi$
$853$	13541.1	0.543538	0.271769	+	0.962363i	$0.412391\pi$
$854$	0	0
$855$	−73295.6	−2.93176
$856$	−3405.19	−0.135966
$857$	−19892.0	−0.792880	−0.396440	−	0.918061i	$-0.629755\pi$
$857$	−19892.0	−0.792880	−0.396440	+	0.918061i	$0.629755\pi$
$858$	10687.7	0.425259
$859$	8754.07	0.347712	0.173856	−	0.984771i	$-0.444377\pi$
$859$	8754.07	0.347712	0.173856	+	0.984771i	$0.444377\pi$
$860$	31843.0	1.26260
$861$	0	0
$862$	−10434.4	−0.412292
$863$	3090.34	0.121896	0.0609482	−	0.998141i	$-0.480588\pi$
$863$	3090.34	0.121896	0.0609482	+	0.998141i	$0.480588\pi$
$864$	17038.2	0.670891
$865$	17248.2	0.677983
$866$	−32399.0	−1.27132
$867$	−39962.4	−1.56539
$868$	0	0
$869$	−12124.8	−0.473310
$870$	−35130.8	−1.36902
$871$	−5478.97	−0.213143
$872$	−16704.1	−0.648708
$873$	61762.4	2.39443
$874$	2624.51	0.101574
$875$	0	0
$876$	−767.888	−0.0296170
$877$	−15764.8	−0.606999	−0.303500	−	0.952832i	$-0.598155\pi$
$877$	−15764.8	−0.606999	−0.303500	+	0.952832i	$0.598155\pi$
$878$	−23244.3	−0.893457
$879$	47411.8	1.81929
$880$	7210.88	0.276226
$881$	−44203.5	−1.69041	−0.845207	−	0.534440i	$-0.820523\pi$
$881$	−44203.5	−1.69041	−0.845207	+	0.534440i	$0.820523\pi$
$882$	0	0
$883$	−1534.78	−0.0584932	−0.0292466	−	0.999572i	$-0.509311\pi$
$883$	−1534.78	−0.0584932	−0.0292466	+	0.999572i	$0.509311\pi$
$884$	2410.65	0.0917183
$885$	66501.1	2.52589
$886$	1814.64	0.0688081
$887$	−33510.6	−1.26852	−0.634260	−	0.773120i	$-0.718695\pi$
$887$	−33510.6	−1.26852	−0.634260	+	0.773120i	$0.718695\pi$
$888$	−124.830	−0.00471738
$889$	0	0
$890$	13893.8	0.523282
$891$	−92549.8	−3.47984
$892$	1740.27	0.0653233
$893$	31267.2	1.17169
$894$	49407.8	1.84837
$895$	−15515.7	−0.579477
$896$	0	0
$897$	−4447.68	−0.165556
$898$	2567.86	0.0954240
$899$	23634.2	0.876803
$900$	44418.7	1.64514
$901$	−14471.4	−0.535086
$902$	−23112.6	−0.853177
$903$	0	0
$904$	−4646.34	−0.170946
$905$	−32272.5	−1.18539
$906$	−21800.4	−0.799413
$907$	28338.0	1.03743	0.518715	−	0.854947i	$-0.326411\pi$
$907$	28338.0	1.03743	0.518715	+	0.854947i	$0.326411\pi$
$908$	11917.6	0.435573
$909$	−99768.4	−3.64038
$910$	0	0
$911$	−47263.6	−1.71889	−0.859447	−	0.511224i	$-0.829192\pi$
$911$	−47263.6	−1.71889	−0.859447	+	0.511224i	$0.829192\pi$
$912$	9388.46	0.340881
$913$	−2299.80	−0.0833650
$914$	−1424.03	−0.0515347
$915$	−142769.	−5.15826
$916$	18589.4	0.670537
$917$	0	0
$918$	−34131.5	−1.22713
$919$	12273.4	0.440547	0.220273	−	0.975438i	$-0.429305\pi$
$919$	12273.4	0.440547	0.220273	+	0.975438i	$0.429305\pi$
$920$	−3000.80	−0.107536
$921$	71572.7	2.56069
$922$	−18631.7	−0.665513
$923$	−9480.00	−0.338070
$924$	0	0
$925$	−213.887	−0.00760277
$926$	−25585.7	−0.907990
$927$	−38139.3	−1.35130
$928$	3351.23	0.118545
$929$	−7433.58	−0.262527	−0.131264	−	0.991348i	$-0.541903\pi$
$929$	−7433.58	−0.262527	−0.131264	+	0.991348i	$0.541903\pi$
$930$	−75704.2	−2.66929
$931$	0	0
$932$	−6607.42	−0.232224
$933$	−80031.1	−2.80825
$934$	494.856	0.0173364
$935$	−14445.1	−0.505247
$936$	−11848.9	−0.413777
$937$	4558.85	0.158945	0.0794724	−	0.996837i	$-0.474676\pi$
$937$	4558.85	0.158945	0.0794724	+	0.996837i	$0.474676\pi$
$938$	0	0
$939$	−81851.0	−2.84463
$940$	−35750.2	−1.24047
$941$	−29098.7	−1.00806	−0.504032	−	0.863685i	$-0.668151\pi$
$941$	−29098.7	−1.00806	−0.504032	+	0.863685i	$0.668151\pi$
$942$	14448.8	0.499753
$943$	9618.29	0.332147
$944$	−6343.74	−0.218720
$945$	0	0
$946$	26978.2	0.927207
$947$	23915.7	0.820649	0.410325	−	0.911940i	$-0.365415\pi$
$947$	23915.7	0.820649	0.410325	+	0.911940i	$0.365415\pi$
$948$	18049.7	0.618383
$949$	350.975	0.0120054
$950$	16086.4	0.549381
$951$	54275.8	1.85070
$952$	0	0
$953$	−41112.8	−1.39745	−0.698727	−	0.715388i	$-0.746250\pi$
$953$	−41112.8	−1.39745	−0.698727	+	0.715388i	$0.746250\pi$
$954$	71130.6	2.41398
$955$	−51614.0	−1.74889
$956$	−18948.3	−0.641036
$957$	−29763.8	−1.00536
$958$	−3604.34	−0.121556
$959$	0	0
$960$	−10734.5	−0.360891
$961$	21139.0	0.709576
$962$	57.0556	0.00191221
$963$	−33528.9	−1.12197
$964$	5499.62	0.183746
$965$	83493.5	2.78523
$966$	0	0
$967$	5731.82	0.190613	0.0953065	−	0.995448i	$-0.469617\pi$
$967$	5731.82	0.190613	0.0953065	+	0.995448i	$0.469617\pi$
$968$	−4538.75	−0.150703
$969$	−18807.3	−0.623507
$970$	−25574.4	−0.846541
$971$	23001.0	0.760182	0.380091	−	0.924949i	$-0.375893\pi$
$971$	23001.0	0.760182	0.380091	+	0.924949i	$0.375893\pi$
$972$	80271.2	2.64887
$973$	0	0
$974$	16212.6	0.533352
$975$	−27261.1	−0.895441
$976$	13619.2	0.446660
$977$	19142.1	0.626826	0.313413	−	0.949617i	$-0.398528\pi$
$977$	19142.1	0.626826	0.313413	+	0.949617i	$0.398528\pi$
$978$	60627.7	1.98227
$979$	11771.2	0.384279
$980$	0	0
$981$	−164476.	−5.35302
$982$	31238.1	1.01512
$983$	−10104.1	−0.327843	−0.163922	−	0.986473i	$-0.552414\pi$
$983$	−10104.1	−0.327843	−0.163922	+	0.986473i	$0.552414\pi$
$984$	34406.8	1.11468
$985$	49840.2	1.61223
$986$	−6713.32	−0.216831
$987$	0	0
$988$	−4291.14	−0.138177
$989$	−11227.0	−0.360967
$990$	71001.3	2.27936
$991$	−43901.2	−1.40723	−0.703617	−	0.710579i	$-0.748433\pi$
$991$	−43901.2	−1.40723	−0.703617	+	0.710579i	$0.748433\pi$
$992$	7221.65	0.231137
$993$	97828.0	3.12636
$994$	0	0
$995$	39933.1	1.27233
$996$	3423.61	0.108917
$997$	49131.6	1.56070	0.780348	−	0.625345i	$-0.215042\pi$
$997$	49131.6	1.56070	0.780348	+	0.625345i	$0.215042\pi$
$998$	−42886.8	−1.36028
$999$	−807.828	−0.0255841

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.y.1.11		11
7.2	even	3		322.4.e.a.277.1	yes	22
7.4	even	3		322.4.e.a.93.1	✓	22
7.6	odd	2		2254.4.a.v.1.1		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.a.93.1	✓	22	7.4	even	3
322.4.e.a.277.1	yes	22	7.2	even	3
2254.4.a.v.1.1		11	7.6	odd	2
2254.4.a.y.1.11		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +10.2845 q^{3} +4.00000 q^{4} -16.3087 q^{5} +20.5690 q^{6} +8.00000 q^{8} +78.7713 q^{9} -32.6174 q^{10} -27.6343 q^{11} +41.1381 q^{12} -18.8028 q^{13} -167.727 q^{15} +16.0000 q^{16} -32.0518 q^{17} +157.543 q^{18} +57.0546 q^{19} -65.2348 q^{20} -55.2687 q^{22} +23.0000 q^{23} +82.2761 q^{24} +140.974 q^{25} -37.6056 q^{26} +532.442 q^{27} +104.726 q^{29} -335.454 q^{30} +225.677 q^{31} +32.0000 q^{32} -284.206 q^{33} -64.1036 q^{34} +315.085 q^{36} -1.51721 q^{37} +114.109 q^{38} -193.377 q^{39} -130.470 q^{40} +418.186 q^{41} -488.129 q^{43} -110.537 q^{44} -1284.66 q^{45} +46.0000 q^{46} +548.023 q^{47} +164.552 q^{48} +281.947 q^{50} -329.637 q^{51} -75.2111 q^{52} +451.501 q^{53} +1064.88 q^{54} +450.680 q^{55} +586.779 q^{57} +209.452 q^{58} -396.484 q^{59} -670.908 q^{60} +851.200 q^{61} +451.353 q^{62} +64.0000 q^{64} +306.649 q^{65} -568.412 q^{66} +291.392 q^{67} -128.207 q^{68} +236.544 q^{69} +504.181 q^{71} +630.170 q^{72} -18.6661 q^{73} -3.03442 q^{74} +1449.85 q^{75} +228.218 q^{76} -386.755 q^{78} +438.759 q^{79} -260.939 q^{80} +3349.09 q^{81} +836.373 q^{82} +83.2225 q^{83} +522.723 q^{85} -976.257 q^{86} +1077.06 q^{87} -221.075 q^{88} -425.962 q^{89} -2569.31 q^{90} +92.0000 q^{92} +2320.98 q^{93} +1096.05 q^{94} -930.487 q^{95} +329.104 q^{96} +784.072 q^{97} -2176.79 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 18 q^{3} + 44 q^{4} + 33 q^{5} + 36 q^{6} + 88 q^{8} + 171 q^{9} + 66 q^{10} + 8 q^{11} + 72 q^{12} + 185 q^{13} - 186 q^{15} + 176 q^{16} + 107 q^{17} + 342 q^{18} + 114 q^{19} + 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) \) 11 * q + 22 * q^2 + 18 * q^3 + 44 * q^4 + 33 * q^5 + 36 * q^6 + 88 * q^8 + 171 * q^9 + 66 * q^10 + 8 * q^11 + 72 * q^12 + 185 * q^13 - 186 * q^15 + 176 * q^16 + 107 * q^17 + 342 * q^18 + 114 * q^19 + 132 * q^20 + 16 * q^22 + 253 * q^23 + 144 * q^24 + 494 * q^25 + 370 * q^26 + 561 * q^27 - 79 * q^29 - 372 * q^30 + 488 * q^31 + 352 * q^32 + 304 * q^33 + 214 * q^34 + 684 * q^36 - 41 * q^37 + 228 * q^38 + 425 * q^39 + 264 * q^40 + 812 * q^41 + 219 * q^43 + 32 * q^44 + 619 * q^45 + 506 * q^46 + 1141 * q^47 + 288 * q^48 + 988 * q^50 + 83 * q^51 + 740 * q^52 + 579 * q^53 + 1122 * q^54 + 951 * q^55 + 383 * q^57 - 158 * q^58 + 1695 * q^59 - 744 * q^60 + 1105 * q^61 + 976 * q^62 + 704 * q^64 - 347 * q^65 + 608 * q^66 + 695 * q^67 + 428 * q^68 + 414 * q^69 + 65 * q^71 + 1368 * q^72 + 1935 * q^73 - 82 * q^74 + 1712 * q^75 + 456 * q^76 + 850 * q^78 + 447 * q^79 + 528 * q^80 + 2203 * q^81 + 1624 * q^82 + 1389 * q^83 + 153 * q^85 + 438 * q^86 + 3354 * q^87 + 64 * q^88 + 2698 * q^89 + 1238 * q^90 + 1012 * q^92 - 686 * q^93 + 2282 * q^94 + 827 * q^95 + 576 * q^96 + 1589 * q^97 - 1729 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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