LMFDB - Embedded newform 2151.4.a.g.1.40

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.40
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.92673 q^{2} -4.28770 q^{4} +4.54299 q^{5} -14.1149 q^{7} -23.6751 q^{8} +O(q^{10})$	\(q+1.92673 q^{2} -4.28770 q^{4} +4.54299 q^{5} -14.1149 q^{7} -23.6751 q^{8} +8.75313 q^{10} +28.0304 q^{11} -31.9364 q^{13} -27.1957 q^{14} -11.3141 q^{16} +71.9548 q^{17} +110.437 q^{19} -19.4790 q^{20} +54.0072 q^{22} -55.0861 q^{23} -104.361 q^{25} -61.5328 q^{26} +60.5204 q^{28} -12.1306 q^{29} +19.8759 q^{31} +167.602 q^{32} +138.638 q^{34} -64.1239 q^{35} -243.779 q^{37} +212.783 q^{38} -107.556 q^{40} +349.249 q^{41} -243.760 q^{43} -120.186 q^{44} -106.136 q^{46} +96.5623 q^{47} -143.769 q^{49} -201.076 q^{50} +136.933 q^{52} +728.153 q^{53} +127.342 q^{55} +334.172 q^{56} -23.3724 q^{58} -259.682 q^{59} +616.386 q^{61} +38.2956 q^{62} +413.436 q^{64} -145.087 q^{65} -34.5550 q^{67} -308.520 q^{68} -123.550 q^{70} -110.119 q^{71} -908.850 q^{73} -469.697 q^{74} -473.522 q^{76} -395.647 q^{77} +82.5884 q^{79} -51.3996 q^{80} +672.910 q^{82} +616.389 q^{83} +326.890 q^{85} -469.661 q^{86} -663.624 q^{88} -700.181 q^{89} +450.779 q^{91} +236.193 q^{92} +186.050 q^{94} +501.716 q^{95} -1734.66 q^{97} -277.005 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.92673	0.681203	0.340602	−	0.940208i	$-0.389369\pi$
$2$	1.92673	0.681203	0.340602	+	0.940208i	$0.389369\pi$
$3$	0	0
$3$	0	0
$4$	−4.28770	−0.535962
$5$	4.54299	0.406337	0.203169	−	0.979144i	$-0.434876\pi$
$5$	4.54299	0.406337	0.203169	+	0.979144i	$0.434876\pi$
$6$	0	0
$7$	−14.1149	−0.762133	−0.381067	−	0.924548i	$-0.624443\pi$
$7$	−14.1149	−0.762133	−0.381067	+	0.924548i	$0.624443\pi$
$8$	−23.6751	−1.04630
$9$	0	0
$10$	8.75313	0.276798
$11$	28.0304	0.768318	0.384159	−	0.923267i	$-0.374491\pi$
$11$	28.0304	0.768318	0.384159	+	0.923267i	$0.374491\pi$
$12$	0	0
$13$	−31.9364	−0.681350	−0.340675	−	0.940181i	$-0.610656\pi$
$13$	−31.9364	−0.681350	−0.340675	+	0.940181i	$0.610656\pi$
$14$	−27.1957	−0.519167
$15$	0	0
$16$	−11.3141	−0.176782
$17$	71.9548	1.02656	0.513282	−	0.858220i	$-0.328429\pi$
$17$	71.9548	1.02656	0.513282	+	0.858220i	$0.328429\pi$
$18$	0	0
$19$	110.437	1.33348	0.666739	−	0.745292i	$-0.267690\pi$
$19$	110.437	1.33348	0.666739	+	0.745292i	$0.267690\pi$
$20$	−19.4790	−0.217781
$21$	0	0
$22$	54.0072	0.523381
$23$	−55.0861	−0.499402	−0.249701	−	0.968323i	$-0.580332\pi$
$23$	−55.0861	−0.499402	−0.249701	+	0.968323i	$0.580332\pi$
$24$	0	0
$25$	−104.361	−0.834890
$26$	−61.5328	−0.464138
$27$	0	0
$28$	60.5204	0.408475
$29$	−12.1306	−0.0776755	−0.0388378	−	0.999246i	$-0.512366\pi$
$29$	−12.1306	−0.0776755	−0.0388378	+	0.999246i	$0.512366\pi$
$30$	0	0
$31$	19.8759	0.115156	0.0575778	−	0.998341i	$-0.481662\pi$
$31$	19.8759	0.115156	0.0575778	+	0.998341i	$0.481662\pi$
$32$	167.602	0.925878
$33$	0	0
$34$	138.638	0.699299
$35$	−64.1239	−0.309683
$36$	0	0
$37$	−243.779	−1.08316	−0.541581	−	0.840649i	$-0.682174\pi$
$37$	−243.779	−1.08316	−0.541581	+	0.840649i	$0.682174\pi$
$38$	212.783	0.908369
$39$	0	0
$40$	−107.556	−0.425152
$41$	349.249	1.33033	0.665165	−	0.746697i	$-0.268361\pi$
$41$	349.249	1.33033	0.665165	+	0.746697i	$0.268361\pi$
$42$	0	0
$43$	−243.760	−0.864490	−0.432245	−	0.901756i	$-0.642278\pi$
$43$	−243.760	−0.864490	−0.432245	+	0.901756i	$0.642278\pi$
$44$	−120.186	−0.411789
$45$	0	0
$46$	−106.136	−0.340194
$47$	96.5623	0.299682	0.149841	−	0.988710i	$-0.452124\pi$
$47$	96.5623	0.299682	0.149841	+	0.988710i	$0.452124\pi$
$48$	0	0
$49$	−143.769	−0.419153
$50$	−201.076	−0.568730
$51$	0	0
$52$	136.933	0.365178
$53$	728.153	1.88716	0.943580	−	0.331145i	$-0.107435\pi$
$53$	728.153	1.88716	0.943580	+	0.331145i	$0.107435\pi$
$54$	0	0
$55$	127.342	0.312196
$56$	334.172	0.797422
$57$	0	0
$58$	−23.3724	−0.0529128
$59$	−259.682	−0.573013	−0.286506	−	0.958078i	$-0.592494\pi$
$59$	−259.682	−0.573013	−0.286506	+	0.958078i	$0.592494\pi$
$60$	0	0
$61$	616.386	1.29377	0.646886	−	0.762586i	$-0.276071\pi$
$61$	616.386	1.29377	0.646886	+	0.762586i	$0.276071\pi$
$62$	38.2956	0.0784444
$63$	0	0
$64$	413.436	0.807493
$65$	−145.087	−0.276858
$66$	0	0
$67$	−34.5550	−0.0630084	−0.0315042	−	0.999504i	$-0.510030\pi$
$67$	−34.5550	−0.0630084	−0.0315042	+	0.999504i	$0.510030\pi$
$68$	−308.520	−0.550200
$69$	0	0
$70$	−123.550	−0.210957
$71$	−110.119	−0.184067	−0.0920335	−	0.995756i	$-0.529337\pi$
$71$	−110.119	−0.184067	−0.0920335	+	0.995756i	$0.529337\pi$
$72$	0	0
$73$	−908.850	−1.45716	−0.728581	−	0.684959i	$-0.759820\pi$
$73$	−908.850	−1.45716	−0.728581	+	0.684959i	$0.759820\pi$
$74$	−469.697	−0.737853
$75$	0	0
$76$	−473.522	−0.714694
$77$	−395.647	−0.585561
$78$	0	0
$79$	82.5884	0.117619	0.0588097	−	0.998269i	$-0.481269\pi$
$79$	82.5884	0.117619	0.0588097	+	0.998269i	$0.481269\pi$
$80$	−51.3996	−0.0718332
$81$	0	0
$82$	672.910	0.906225
$83$	616.389	0.815150	0.407575	−	0.913172i	$-0.366375\pi$
$83$	616.389	0.815150	0.407575	+	0.913172i	$0.366375\pi$
$84$	0	0
$85$	326.890	0.417131
$86$	−469.661	−0.588893
$87$	0	0
$88$	−663.624	−0.803893
$89$	−700.181	−0.833922	−0.416961	−	0.908924i	$-0.636905\pi$
$89$	−700.181	−0.833922	−0.416961	+	0.908924i	$0.636905\pi$
$90$	0	0
$91$	450.779	0.519279
$92$	236.193	0.267661
$93$	0	0
$94$	186.050	0.204144
$95$	501.716	0.541842
$96$	0	0
$97$	−1734.66	−1.81576	−0.907879	−	0.419233i	$-0.862299\pi$
$97$	−1734.66	−1.81576	−0.907879	+	0.419233i	$0.862299\pi$
$98$	−277.005	−0.285528
$99$	0	0
$100$	447.470	0.447470
$101$	−124.059	−0.122221	−0.0611104	−	0.998131i	$-0.519464\pi$
$101$	−124.059	−0.122221	−0.0611104	+	0.998131i	$0.519464\pi$
$102$	0	0
$103$	−1711.43	−1.63721	−0.818604	−	0.574358i	$-0.805252\pi$
$103$	−1711.43	−1.63721	−0.818604	+	0.574358i	$0.805252\pi$
$104$	756.097	0.712898
$105$	0	0
$106$	1402.96	1.28554
$107$	−1312.09	−1.18547	−0.592733	−	0.805399i	$-0.701951\pi$
$107$	−1312.09	−1.18547	−0.592733	+	0.805399i	$0.701951\pi$
$108$	0	0
$109$	−1547.57	−1.35991	−0.679955	−	0.733254i	$-0.738001\pi$
$109$	−1547.57	−1.35991	−0.679955	+	0.733254i	$0.738001\pi$
$110$	245.354	0.212669
$111$	0	0
$112$	159.697	0.134732
$113$	1304.61	1.08608	0.543040	−	0.839707i	$-0.317273\pi$
$113$	1304.61	1.08608	0.543040	+	0.839707i	$0.317273\pi$
$114$	0	0
$115$	−250.256	−0.202926
$116$	52.0122	0.0416312
$117$	0	0
$118$	−500.339	−0.390338
$119$	−1015.63	−0.782379
$120$	0	0
$121$	−545.294	−0.409688
$122$	1187.61	0.881322
$123$	0	0
$124$	−85.2221	−0.0617191
$125$	−1041.99	−0.745584
$126$	0	0
$127$	479.592	0.335094	0.167547	−	0.985864i	$-0.446415\pi$
$127$	479.592	0.335094	0.167547	+	0.985864i	$0.446415\pi$
$128$	−544.233	−0.375811
$129$	0	0
$130$	−279.543	−0.188596
$131$	−1823.07	−1.21590	−0.607949	−	0.793976i	$-0.708008\pi$
$131$	−1823.07	−1.21590	−0.607949	+	0.793976i	$0.708008\pi$
$132$	0	0
$133$	−1558.81	−1.01629
$134$	−66.5782	−0.0429215
$135$	0	0
$136$	−1703.54	−1.07410
$137$	1887.11	1.17684	0.588419	−	0.808556i	$-0.299751\pi$
$137$	1887.11	1.17684	0.588419	+	0.808556i	$0.299751\pi$
$138$	0	0
$139$	465.291	0.283924	0.141962	−	0.989872i	$-0.454659\pi$
$139$	465.291	0.283924	0.141962	+	0.989872i	$0.454659\pi$
$140$	274.944	0.165978
$141$	0	0
$142$	−212.170	−0.125387
$143$	−895.190	−0.523493
$144$	0	0
$145$	−55.1091	−0.0315625
$146$	−1751.11	−0.992624
$147$	0	0
$148$	1045.25	0.580534
$149$	−2487.89	−1.36789	−0.683947	−	0.729532i	$-0.739738\pi$
$149$	−2487.89	−1.36789	−0.683947	+	0.729532i	$0.739738\pi$
$150$	0	0
$151$	−3323.76	−1.79129	−0.895643	−	0.444774i	$-0.853284\pi$
$151$	−3323.76	−1.79129	−0.895643	+	0.444774i	$0.853284\pi$
$152$	−2614.62	−1.39522
$153$	0	0
$154$	−762.306	−0.398886
$155$	90.2962	0.0467920
$156$	0	0
$157$	−720.418	−0.366214	−0.183107	−	0.983093i	$-0.558615\pi$
$157$	−720.418	−0.366214	−0.183107	+	0.983093i	$0.558615\pi$
$158$	159.126	0.0801226
$159$	0	0
$160$	761.413	0.376219
$161$	777.535	0.380611
$162$	0	0
$163$	1108.25	0.532545	0.266272	−	0.963898i	$-0.414208\pi$
$163$	1108.25	0.532545	0.266272	+	0.963898i	$0.414208\pi$
$164$	−1497.47	−0.713007
$165$	0	0
$166$	1187.62	0.555283
$167$	1676.79	0.776968	0.388484	−	0.921456i	$-0.372999\pi$
$167$	1676.79	0.776968	0.388484	+	0.921456i	$0.372999\pi$
$168$	0	0
$169$	−1177.07	−0.535762
$170$	629.829	0.284151
$171$	0	0
$172$	1045.17	0.463334
$173$	3900.26	1.71405	0.857026	−	0.515274i	$-0.172310\pi$
$173$	3900.26	1.71405	0.857026	+	0.515274i	$0.172310\pi$
$174$	0	0
$175$	1473.05	0.636297
$176$	−317.138	−0.135825
$177$	0	0
$178$	−1349.06	−0.568070
$179$	−663.778	−0.277168	−0.138584	−	0.990351i	$-0.544255\pi$
$179$	−663.778	−0.277168	−0.138584	+	0.990351i	$0.544255\pi$
$180$	0	0
$181$	2098.72	0.861859	0.430929	−	0.902386i	$-0.358186\pi$
$181$	2098.72	0.861859	0.430929	+	0.902386i	$0.358186\pi$
$182$	868.530	0.353735
$183$	0	0
$184$	1304.17	0.522526
$185$	−1107.48	−0.440129
$186$	0	0
$187$	2016.92	0.788728
$188$	−414.030	−0.160618
$189$	0	0
$190$	966.673	0.369104
$191$	−302.401	−0.114560	−0.0572801	−	0.998358i	$-0.518243\pi$
$191$	−302.401	−0.114560	−0.0572801	+	0.998358i	$0.518243\pi$
$192$	0	0
$193$	−3757.87	−1.40154	−0.700770	−	0.713387i	$-0.747160\pi$
$193$	−3757.87	−1.40154	−0.700770	+	0.713387i	$0.747160\pi$
$194$	−3342.24	−1.23690
$195$	0	0
$196$	616.440	0.224650
$197$	−2992.66	−1.08233	−0.541163	−	0.840918i	$-0.682016\pi$
$197$	−2992.66	−1.08233	−0.541163	+	0.840918i	$0.682016\pi$
$198$	0	0
$199$	−574.375	−0.204605	−0.102302	−	0.994753i	$-0.532621\pi$
$199$	−574.375	−0.204605	−0.102302	+	0.994753i	$0.532621\pi$
$200$	2470.76	0.873547
$201$	0	0
$202$	−239.028	−0.0832572
$203$	171.222	0.0591991
$204$	0	0
$205$	1586.63	0.540563
$206$	−3297.47	−1.11527
$207$	0	0
$208$	361.330	0.120451
$209$	3095.61	1.02453
$210$	0	0
$211$	−1130.97	−0.369002	−0.184501	−	0.982832i	$-0.559067\pi$
$211$	−1130.97	−0.369002	−0.184501	+	0.982832i	$0.559067\pi$
$212$	−3122.10	−1.01145
$213$	0	0
$214$	−2528.05	−0.807543
$215$	−1107.40	−0.351274
$216$	0	0
$217$	−280.547	−0.0877639
$218$	−2981.75	−0.926375
$219$	0	0
$220$	−546.004	−0.167325
$221$	−2297.97	−0.699450
$222$	0	0
$223$	4758.26	1.42886	0.714432	−	0.699705i	$-0.246685\pi$
$223$	4758.26	1.42886	0.714432	+	0.699705i	$0.246685\pi$
$224$	−2365.68	−0.705642
$225$	0	0
$226$	2513.63	0.739841
$227$	−4921.32	−1.43894	−0.719470	−	0.694523i	$-0.755615\pi$
$227$	−4921.32	−1.43894	−0.719470	+	0.694523i	$0.755615\pi$
$228$	0	0
$229$	−5415.04	−1.56260	−0.781301	−	0.624154i	$-0.785444\pi$
$229$	−5415.04	−1.56260	−0.781301	+	0.624154i	$0.785444\pi$
$230$	−482.176	−0.138234
$231$	0	0
$232$	287.193	0.0812721
$233$	−4829.08	−1.35778	−0.678892	−	0.734238i	$-0.737540\pi$
$233$	−4829.08	−1.35778	−0.678892	+	0.734238i	$0.737540\pi$
$234$	0	0
$235$	438.682	0.121772
$236$	1113.44	0.307113
$237$	0	0
$238$	−1956.86	−0.532959
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	5048.84	1.34948	0.674739	−	0.738056i	$-0.264256\pi$
$241$	5048.84	1.34948	0.674739	+	0.738056i	$0.264256\pi$
$242$	−1050.64	−0.279080
$243$	0	0
$244$	−2642.88	−0.693413
$245$	−653.143	−0.170318
$246$	0	0
$247$	−3526.97	−0.908565
$248$	−470.565	−0.120488
$249$	0	0
$250$	−2007.63	−0.507894
$251$	−6977.00	−1.75452	−0.877259	−	0.480017i	$-0.840630\pi$
$251$	−6977.00	−1.75452	−0.877259	+	0.480017i	$0.840630\pi$
$252$	0	0
$253$	−1544.09	−0.383700
$254$	924.046	0.228267
$255$	0	0
$256$	−4356.08	−1.06350
$257$	7599.68	1.84457	0.922286	−	0.386507i	$-0.126319\pi$
$257$	7599.68	1.84457	0.922286	+	0.386507i	$0.126319\pi$
$258$	0	0
$259$	3440.91	0.825514
$260$	622.087	0.148385
$261$	0	0
$262$	−3512.58	−0.828274
$263$	−1590.80	−0.372977	−0.186489	−	0.982457i	$-0.559711\pi$
$263$	−1590.80	−0.372977	−0.186489	+	0.982457i	$0.559711\pi$
$264$	0	0
$265$	3307.99	0.766823
$266$	−3003.42	−0.692298
$267$	0	0
$268$	148.161	0.0337701
$269$	4255.95	0.964645	0.482323	−	0.875994i	$-0.339793\pi$
$269$	4255.95	0.964645	0.482323	+	0.875994i	$0.339793\pi$
$270$	0	0
$271$	8311.36	1.86302	0.931511	−	0.363713i	$-0.118491\pi$
$271$	8311.36	1.86302	0.931511	+	0.363713i	$0.118491\pi$
$272$	−814.100	−0.181478
$273$	0	0
$274$	3635.96	0.801665
$275$	−2925.29	−0.641461
$276$	0	0
$277$	3888.34	0.843421	0.421710	−	0.906731i	$-0.361430\pi$
$277$	3888.34	0.843421	0.421710	+	0.906731i	$0.361430\pi$
$278$	896.492	0.193410
$279$	0	0
$280$	1518.14	0.324022
$281$	−2351.52	−0.499217	−0.249609	−	0.968347i	$-0.580302\pi$
$281$	−2351.52	−0.499217	−0.249609	+	0.968347i	$0.580302\pi$
$282$	0	0
$283$	−1830.05	−0.384400	−0.192200	−	0.981356i	$-0.561562\pi$
$283$	−1830.05	−0.384400	−0.192200	+	0.981356i	$0.561562\pi$
$284$	472.158	0.0986529
$285$	0	0
$286$	−1724.79	−0.356605
$287$	−4929.61	−1.01389
$288$	0	0
$289$	264.488	0.0538344
$290$	−106.180	−0.0215005
$291$	0	0
$292$	3896.88	0.780984
$293$	−4939.16	−0.984808	−0.492404	−	0.870367i	$-0.663882\pi$
$293$	−4939.16	−0.984808	−0.492404	+	0.870367i	$0.663882\pi$
$294$	0	0
$295$	−1179.73	−0.232837
$296$	5771.49	1.13331
$297$	0	0
$298$	−4793.51	−0.931814
$299$	1759.25	0.340268
$300$	0	0
$301$	3440.65	0.658856
$302$	−6404.01	−1.22023
$303$	0	0
$304$	−1249.49	−0.235735
$305$	2800.24	0.525708
$306$	0	0
$307$	−5082.19	−0.944808	−0.472404	−	0.881382i	$-0.656614\pi$
$307$	−5082.19	−0.944808	−0.472404	+	0.881382i	$0.656614\pi$
$308$	1696.42	0.313838
$309$	0	0
$310$	173.977	0.0318749
$311$	249.779	0.0455424	0.0227712	−	0.999741i	$-0.492751\pi$
$311$	249.779	0.0455424	0.0227712	+	0.999741i	$0.492751\pi$
$312$	0	0
$313$	1335.72	0.241213	0.120606	−	0.992700i	$-0.461516\pi$
$313$	1335.72	0.241213	0.120606	+	0.992700i	$0.461516\pi$
$314$	−1388.05	−0.249466
$315$	0	0
$316$	−354.114	−0.0630395
$317$	−4099.50	−0.726344	−0.363172	−	0.931722i	$-0.618306\pi$
$317$	−4099.50	−0.726344	−0.363172	+	0.931722i	$0.618306\pi$
$318$	0	0
$319$	−340.025	−0.0596795
$320$	1878.24	0.328115
$321$	0	0
$322$	1498.10	0.259273
$323$	7946.50	1.36890
$324$	0	0
$325$	3332.92	0.568852
$326$	2135.30	0.362771
$327$	0	0
$328$	−8268.51	−1.39193
$329$	−1362.97	−0.228398
$330$	0	0
$331$	−704.646	−0.117012	−0.0585058	−	0.998287i	$-0.518634\pi$
$331$	−704.646	−0.117012	−0.0585058	+	0.998287i	$0.518634\pi$
$332$	−2642.89	−0.436890
$333$	0	0
$334$	3230.72	0.529273
$335$	−156.983	−0.0256027
$336$	0	0
$337$	−3937.55	−0.636474	−0.318237	−	0.948011i	$-0.603091\pi$
$337$	−3937.55	−0.636474	−0.318237	+	0.948011i	$0.603091\pi$
$338$	−2267.90	−0.364963
$339$	0	0
$340$	−1401.60	−0.223567
$341$	557.132	0.0884761
$342$	0	0
$343$	6870.70	1.08158
$344$	5771.05	0.904518
$345$	0	0
$346$	7514.75	1.16762
$347$	−4188.64	−0.648006	−0.324003	−	0.946056i	$-0.605029\pi$
$347$	−4188.64	−0.648006	−0.324003	+	0.946056i	$0.605029\pi$
$348$	0	0
$349$	−5622.54	−0.862372	−0.431186	−	0.902263i	$-0.641905\pi$
$349$	−5622.54	−0.862372	−0.431186	+	0.902263i	$0.641905\pi$
$350$	2838.17	0.433448
$351$	0	0
$352$	4697.95	0.711368
$353$	10770.7	1.62399	0.811994	−	0.583665i	$-0.198382\pi$
$353$	10770.7	1.62399	0.811994	+	0.583665i	$0.198382\pi$
$354$	0	0
$355$	−500.271	−0.0747933
$356$	3002.17	0.446951
$357$	0	0
$358$	−1278.92	−0.188808
$359$	−4653.73	−0.684163	−0.342081	−	0.939670i	$-0.611132\pi$
$359$	−4653.73	−0.684163	−0.342081	+	0.939670i	$0.611132\pi$
$360$	0	0
$361$	5337.41	0.778162
$362$	4043.67	0.587101
$363$	0	0
$364$	−1932.80	−0.278314
$365$	−4128.90	−0.592099
$366$	0	0
$367$	−12201.2	−1.73541	−0.867706	−	0.497077i	$-0.834407\pi$
$367$	−12201.2	−1.73541	−0.867706	+	0.497077i	$0.834407\pi$
$368$	623.247	0.0882854
$369$	0	0
$370$	−2133.83	−0.299817
$371$	−10277.8	−1.43827
$372$	0	0
$373$	5045.08	0.700333	0.350167	−	0.936687i	$-0.386125\pi$
$373$	5045.08	0.700333	0.350167	+	0.936687i	$0.386125\pi$
$374$	3886.07	0.537284
$375$	0	0
$376$	−2286.12	−0.313558
$377$	387.406	0.0529242
$378$	0	0
$379$	−4999.72	−0.677621	−0.338810	−	0.940855i	$-0.610024\pi$
$379$	−4999.72	−0.677621	−0.338810	+	0.940855i	$0.610024\pi$
$380$	−2151.21	−0.290407
$381$	0	0
$382$	−582.647	−0.0780387
$383$	−10546.4	−1.40704	−0.703521	−	0.710674i	$-0.748390\pi$
$383$	−10546.4	−1.40704	−0.703521	+	0.710674i	$0.748390\pi$
$384$	0	0
$385$	−1797.42	−0.237935
$386$	−7240.41	−0.954734
$387$	0	0
$388$	7437.72	0.973177
$389$	−11923.9	−1.55415	−0.777075	−	0.629408i	$-0.783298\pi$
$389$	−11923.9	−1.55415	−0.777075	+	0.629408i	$0.783298\pi$
$390$	0	0
$391$	−3963.71	−0.512668
$392$	3403.76	0.438561
$393$	0	0
$394$	−5766.06	−0.737284
$395$	375.198	0.0477931
$396$	0	0
$397$	4067.23	0.514178	0.257089	−	0.966388i	$-0.417237\pi$
$397$	4067.23	0.514178	0.257089	+	0.966388i	$0.417237\pi$
$398$	−1106.67	−0.139377
$399$	0	0
$400$	1180.75	0.147594
$401$	−14266.9	−1.77670	−0.888350	−	0.459168i	$-0.848148\pi$
$401$	−14266.9	−1.77670	−0.888350	+	0.459168i	$0.848148\pi$
$402$	0	0
$403$	−634.765	−0.0784613
$404$	531.926	0.0655057
$405$	0	0
$406$	329.899	0.0403266
$407$	−6833.23	−0.832213
$408$	0	0
$409$	8198.77	0.991205	0.495603	−	0.868549i	$-0.334947\pi$
$409$	8198.77	0.991205	0.495603	+	0.868549i	$0.334947\pi$
$410$	3057.02	0.368233
$411$	0	0
$412$	7338.11	0.877482
$413$	3665.39	0.436712
$414$	0	0
$415$	2800.25	0.331226
$416$	−5352.59	−0.630847
$417$	0	0
$418$	5964.41	0.697916
$419$	−3509.69	−0.409212	−0.204606	−	0.978844i	$-0.565591\pi$
$419$	−3509.69	−0.409212	−0.204606	+	0.978844i	$0.565591\pi$
$420$	0	0
$421$	−1228.24	−0.142188	−0.0710938	−	0.997470i	$-0.522649\pi$
$421$	−1228.24	−0.142188	−0.0710938	+	0.997470i	$0.522649\pi$
$422$	−2179.09	−0.251365
$423$	0	0
$424$	−17239.1	−1.97454
$425$	−7509.29	−0.857068
$426$	0	0
$427$	−8700.23	−0.986027
$428$	5625.86	0.635365
$429$	0	0
$430$	−2133.66	−0.239289
$431$	−15746.8	−1.75985	−0.879926	−	0.475111i	$-0.842408\pi$
$431$	−15746.8	−1.75985	−0.879926	+	0.475111i	$0.842408\pi$
$432$	0	0
$433$	15416.5	1.71101	0.855507	−	0.517792i	$-0.173246\pi$
$433$	15416.5	1.71101	0.855507	+	0.517792i	$0.173246\pi$
$434$	−540.539	−0.0597851
$435$	0	0
$436$	6635.51	0.728860
$437$	−6083.57	−0.665941
$438$	0	0
$439$	−13029.2	−1.41651	−0.708256	−	0.705956i	$-0.750518\pi$
$439$	−13029.2	−1.41651	−0.708256	+	0.705956i	$0.750518\pi$
$440$	−3014.84	−0.326652
$441$	0	0
$442$	−4427.58	−0.476467
$443$	8891.43	0.953599	0.476800	−	0.879012i	$-0.341797\pi$
$443$	8891.43	0.953599	0.476800	+	0.879012i	$0.341797\pi$
$444$	0	0
$445$	−3180.92	−0.338854
$446$	9167.90	0.973346
$447$	0	0
$448$	−5835.61	−0.615417
$449$	−12338.8	−1.29689	−0.648447	−	0.761260i	$-0.724581\pi$
$449$	−12338.8	−1.29689	−0.648447	+	0.761260i	$0.724581\pi$
$450$	0	0
$451$	9789.60	1.02212
$452$	−5593.76	−0.582098
$453$	0	0
$454$	−9482.07	−0.980211
$455$	2047.88	0.211003
$456$	0	0
$457$	−14357.2	−1.46958	−0.734791	−	0.678293i	$-0.762720\pi$
$457$	−14357.2	−1.46958	−0.734791	+	0.678293i	$0.762720\pi$
$458$	−10433.3	−1.06445
$459$	0	0
$460$	1073.02	0.108761
$461$	13873.5	1.40164	0.700818	−	0.713341i	$-0.252819\pi$
$461$	13873.5	1.40164	0.700818	+	0.713341i	$0.252819\pi$
$462$	0	0
$463$	6720.52	0.674577	0.337288	−	0.941401i	$-0.390490\pi$
$463$	6720.52	0.674577	0.337288	+	0.941401i	$0.390490\pi$
$464$	137.246	0.0137316
$465$	0	0
$466$	−9304.36	−0.924927
$467$	13853.0	1.37267	0.686337	−	0.727284i	$-0.259218\pi$
$467$	13853.0	1.37267	0.686337	+	0.727284i	$0.259218\pi$
$468$	0	0
$469$	487.740	0.0480208
$470$	845.222	0.0829515
$471$	0	0
$472$	6148.01	0.599545
$473$	−6832.70	−0.664203
$474$	0	0
$475$	−11525.4	−1.11331
$476$	4354.73	0.419325
$477$	0	0
$478$	460.489	0.0440634
$479$	6813.41	0.649922	0.324961	−	0.945727i	$-0.394649\pi$
$479$	6813.41	0.649922	0.324961	+	0.945727i	$0.394649\pi$
$480$	0	0
$481$	7785.41	0.738012
$482$	9727.76	0.919269
$483$	0	0
$484$	2338.06	0.219577
$485$	−7880.56	−0.737810
$486$	0	0
$487$	6464.04	0.601465	0.300733	−	0.953708i	$-0.402769\pi$
$487$	6464.04	0.601465	0.300733	+	0.953708i	$0.402769\pi$
$488$	−14593.0	−1.35368
$489$	0	0
$490$	−1258.43	−0.116021
$491$	4063.25	0.373467	0.186733	−	0.982411i	$-0.440210\pi$
$491$	4063.25	0.373467	0.186733	+	0.982411i	$0.440210\pi$
$492$	0	0
$493$	−872.852	−0.0797389
$494$	−6795.53	−0.618917
$495$	0	0
$496$	−224.878	−0.0203575
$497$	1554.32	0.140284
$498$	0	0
$499$	15417.9	1.38317	0.691583	−	0.722297i	$-0.256913\pi$
$499$	15417.9	1.38317	0.691583	+	0.722297i	$0.256913\pi$
$500$	4467.72	0.399605
$501$	0	0
$502$	−13442.8	−1.19518
$503$	10985.5	0.973792	0.486896	−	0.873460i	$-0.338129\pi$
$503$	10985.5	0.973792	0.486896	+	0.873460i	$0.338129\pi$
$504$	0	0
$505$	−563.597	−0.0496629
$506$	−2975.05	−0.261377
$507$	0	0
$508$	−2056.35	−0.179598
$509$	−12010.2	−1.04586	−0.522929	−	0.852376i	$-0.675161\pi$
$509$	−12010.2	−1.04586	−0.522929	+	0.852376i	$0.675161\pi$
$510$	0	0
$511$	12828.3	1.11055
$512$	−4039.15	−0.348646
$513$	0	0
$514$	14642.6	1.25653
$515$	−7775.02	−0.665259
$516$	0	0
$517$	2706.68	0.230251
$518$	6629.73	0.562343
$519$	0	0
$520$	3434.94	0.289677
$521$	6895.43	0.579836	0.289918	−	0.957052i	$-0.406372\pi$
$521$	6895.43	0.579836	0.289918	+	0.957052i	$0.406372\pi$
$522$	0	0
$523$	16667.0	1.39349	0.696747	−	0.717317i	$-0.254630\pi$
$523$	16667.0	1.39349	0.696747	+	0.717317i	$0.254630\pi$
$524$	7816.79	0.651676
$525$	0	0
$526$	−3065.05	−0.254073
$527$	1430.17	0.118215
$528$	0	0
$529$	−9132.52	−0.750598
$530$	6373.62	0.522363
$531$	0	0
$532$	6683.72	0.544692
$533$	−11153.7	−0.906420
$534$	0	0
$535$	−5960.83	−0.481699
$536$	818.093	0.0659258
$537$	0	0
$538$	8200.07	0.657119
$539$	−4029.92	−0.322043
$540$	0	0
$541$	5011.21	0.398242	0.199121	−	0.979975i	$-0.436191\pi$
$541$	5011.21	0.398242	0.199121	+	0.979975i	$0.436191\pi$
$542$	16013.8	1.26910
$543$	0	0
$544$	12059.7	0.950473
$545$	−7030.59	−0.552582
$546$	0	0
$547$	22306.8	1.74364	0.871819	−	0.489828i	$-0.162940\pi$
$547$	22306.8	1.74364	0.871819	+	0.489828i	$0.162940\pi$
$548$	−8091.36	−0.630740
$549$	0	0
$550$	−5636.26	−0.436965
$551$	−1339.67	−0.103579
$552$	0	0
$553$	−1165.73	−0.0896416
$554$	7491.79	0.574541
$555$	0	0
$556$	−1995.03	−0.152173
$557$	8349.63	0.635162	0.317581	−	0.948231i	$-0.397129\pi$
$557$	8349.63	0.635162	0.317581	+	0.948231i	$0.397129\pi$
$558$	0	0
$559$	7784.81	0.589020
$560$	725.501	0.0547464
$561$	0	0
$562$	−4530.75	−0.340068
$563$	10064.6	0.753417	0.376708	−	0.926332i	$-0.377056\pi$
$563$	10064.6	0.753417	0.376708	+	0.926332i	$0.377056\pi$
$564$	0	0
$565$	5926.81	0.441315
$566$	−3526.02	−0.261854
$567$	0	0
$568$	2607.09	0.192590
$569$	−1651.40	−0.121670	−0.0608350	−	0.998148i	$-0.519376\pi$
$569$	−1651.40	−0.121670	−0.0608350	+	0.998148i	$0.519376\pi$
$570$	0	0
$571$	23898.3	1.75151	0.875757	−	0.482752i	$-0.160363\pi$
$571$	23898.3	1.75151	0.875757	+	0.482752i	$0.160363\pi$
$572$	3838.31	0.280573
$573$	0	0
$574$	−9498.05	−0.690664
$575$	5748.86	0.416946
$576$	0	0
$577$	10181.4	0.734586	0.367293	−	0.930105i	$-0.380285\pi$
$577$	10181.4	0.734586	0.367293	+	0.930105i	$0.380285\pi$
$578$	509.598	0.0366721
$579$	0	0
$580$	236.291	0.0169163
$581$	−8700.27	−0.621253
$582$	0	0
$583$	20410.4	1.44994
$584$	21517.1	1.52463
$585$	0	0
$586$	−9516.44	−0.670854
$587$	−9971.33	−0.701126	−0.350563	−	0.936539i	$-0.614010\pi$
$587$	−9971.33	−0.701126	−0.350563	+	0.936539i	$0.614010\pi$
$588$	0	0
$589$	2195.05	0.153557
$590$	−2273.03	−0.158609
$591$	0	0
$592$	2758.13	0.191484
$593$	16799.7	1.16337	0.581686	−	0.813414i	$-0.302393\pi$
$593$	16799.7	1.16337	0.581686	+	0.813414i	$0.302393\pi$
$594$	0	0
$595$	−4614.02	−0.317910
$596$	10667.3	0.733140
$597$	0	0
$598$	3389.60	0.231791
$599$	−4059.29	−0.276892	−0.138446	−	0.990370i	$-0.544211\pi$
$599$	−4059.29	−0.276892	−0.138446	+	0.990370i	$0.544211\pi$
$600$	0	0
$601$	−2424.31	−0.164542	−0.0822711	−	0.996610i	$-0.526217\pi$
$601$	−2424.31	−0.164542	−0.0822711	+	0.996610i	$0.526217\pi$
$602$	6629.21	0.448815
$603$	0	0
$604$	14251.3	0.960062
$605$	−2477.27	−0.166471
$606$	0	0
$607$	17901.3	1.19702	0.598510	−	0.801115i	$-0.295760\pi$
$607$	17901.3	1.19702	0.598510	+	0.801115i	$0.295760\pi$
$608$	18509.5	1.23464
$609$	0	0
$610$	5395.31	0.358114
$611$	−3083.85	−0.204188
$612$	0	0
$613$	3978.84	0.262160	0.131080	−	0.991372i	$-0.458156\pi$
$613$	3978.84	0.262160	0.131080	+	0.991372i	$0.458156\pi$
$614$	−9792.03	−0.643606
$615$	0	0
$616$	9366.99	0.612673
$617$	−14969.5	−0.976741	−0.488371	−	0.872636i	$-0.662409\pi$
$617$	−14969.5	−0.976741	−0.488371	+	0.872636i	$0.662409\pi$
$618$	0	0
$619$	−19925.6	−1.29383	−0.646913	−	0.762564i	$-0.723940\pi$
$619$	−19925.6	−1.29383	−0.646913	+	0.762564i	$0.723940\pi$
$620$	−387.163	−0.0250788
$621$	0	0
$622$	481.258	0.0310236
$623$	9882.99	0.635560
$624$	0	0
$625$	8311.43	0.531931
$626$	2573.58	0.164315
$627$	0	0
$628$	3088.93	0.196277
$629$	−17541.0	−1.11194
$630$	0	0
$631$	−14340.9	−0.904757	−0.452379	−	0.891826i	$-0.649424\pi$
$631$	−14340.9	−0.904757	−0.452379	+	0.891826i	$0.649424\pi$
$632$	−1955.29	−0.123065
$633$	0	0
$634$	−7898.65	−0.494788
$635$	2178.78	0.136161
$636$	0	0
$637$	4591.47	0.285590
$638$	−655.138	−0.0406539
$639$	0	0
$640$	−2472.44	−0.152706
$641$	3101.58	0.191116	0.0955578	−	0.995424i	$-0.469537\pi$
$641$	3101.58	0.191116	0.0955578	+	0.995424i	$0.469537\pi$
$642$	0	0
$643$	−24212.2	−1.48497	−0.742486	−	0.669861i	$-0.766354\pi$
$643$	−24212.2	−1.48497	−0.742486	+	0.669861i	$0.766354\pi$
$644$	−3333.84	−0.203993
$645$	0	0
$646$	15310.8	0.932499
$647$	−14136.6	−0.858992	−0.429496	−	0.903069i	$-0.641309\pi$
$647$	−14136.6	−0.858992	−0.429496	+	0.903069i	$0.641309\pi$
$648$	0	0
$649$	−7279.01	−0.440256
$650$	6421.64	0.387504
$651$	0	0
$652$	−4751.84	−0.285424
$653$	8407.94	0.503872	0.251936	−	0.967744i	$-0.418933\pi$
$653$	8407.94	0.503872	0.251936	+	0.967744i	$0.418933\pi$
$654$	0	0
$655$	−8282.21	−0.494065
$656$	−3951.42	−0.235178
$657$	0	0
$658$	−2626.08	−0.155585
$659$	−406.791	−0.0240460	−0.0120230	−	0.999928i	$-0.503827\pi$
$659$	−406.791	−0.0240460	−0.0120230	+	0.999928i	$0.503827\pi$
$660$	0	0
$661$	−22192.4	−1.30588	−0.652939	−	0.757410i	$-0.726464\pi$
$661$	−22192.4	−1.30588	−0.652939	+	0.757410i	$0.726464\pi$
$662$	−1357.66	−0.0797086
$663$	0	0
$664$	−14593.1	−0.852893
$665$	−7081.67	−0.412955
$666$	0	0
$667$	668.226	0.0387913
$668$	−7189.55	−0.416425
$669$	0	0
$670$	−302.464	−0.0174406
$671$	17277.6	0.994029
$672$	0	0
$673$	−17824.1	−1.02090	−0.510452	−	0.859906i	$-0.670522\pi$
$673$	−17824.1	−1.02090	−0.510452	+	0.859906i	$0.670522\pi$
$674$	−7586.60	−0.433568
$675$	0	0
$676$	5046.92	0.287148
$677$	27332.0	1.55163	0.775815	−	0.630961i	$-0.217339\pi$
$677$	27332.0	1.55163	0.775815	+	0.630961i	$0.217339\pi$
$678$	0	0
$679$	24484.6	1.38385
$680$	−7739.15	−0.436446
$681$	0	0
$682$	1073.44	0.0602702
$683$	31853.8	1.78456	0.892278	−	0.451486i	$-0.149106\pi$
$683$	31853.8	1.78456	0.892278	+	0.451486i	$0.149106\pi$
$684$	0	0
$685$	8573.12	0.478193
$686$	13238.0	0.736778
$687$	0	0
$688$	2757.91	0.152826
$689$	−23254.5	−1.28582
$690$	0	0
$691$	24069.9	1.32513	0.662564	−	0.749006i	$-0.269468\pi$
$691$	24069.9	1.32513	0.662564	+	0.749006i	$0.269468\pi$
$692$	−16723.1	−0.918667
$693$	0	0
$694$	−8070.40	−0.441424
$695$	2113.81	0.115369
$696$	0	0
$697$	25130.1	1.36567
$698$	−10833.1	−0.587450
$699$	0	0
$700$	−6315.99	−0.341031
$701$	−610.991	−0.0329199	−0.0164599	−	0.999865i	$-0.505240\pi$
$701$	−610.991	−0.0329199	−0.0164599	+	0.999865i	$0.505240\pi$
$702$	0	0
$703$	−26922.3	−1.44437
$704$	11588.8	0.620411
$705$	0	0
$706$	20752.3	1.10627
$707$	1751.08	0.0931485
$708$	0	0
$709$	−15273.0	−0.809014	−0.404507	−	0.914535i	$-0.632557\pi$
$709$	−15273.0	−0.809014	−0.404507	+	0.914535i	$0.632557\pi$
$710$	−963.888	−0.0509494
$711$	0	0
$712$	16576.9	0.872535
$713$	−1094.89	−0.0575090
$714$	0	0
$715$	−4066.84	−0.212715
$716$	2846.08	0.148552
$717$	0	0
$718$	−8966.49	−0.466054
$719$	27319.3	1.41702	0.708510	−	0.705700i	$-0.249368\pi$
$719$	27319.3	1.41702	0.708510	+	0.705700i	$0.249368\pi$
$720$	0	0
$721$	24156.7	1.24777
$722$	10283.8	0.530087
$723$	0	0
$724$	−8998.67	−0.461924
$725$	1265.96	0.0648505
$726$	0	0
$727$	15530.0	0.792262	0.396131	−	0.918194i	$-0.370353\pi$
$727$	15530.0	0.792262	0.396131	+	0.918194i	$0.370353\pi$
$728$	−10672.2	−0.543323
$729$	0	0
$730$	−7955.28	−0.403340
$731$	−17539.7	−0.887454
$732$	0	0
$733$	18585.2	0.936507	0.468254	−	0.883594i	$-0.344883\pi$
$733$	18585.2	0.936507	0.468254	+	0.883594i	$0.344883\pi$
$734$	−23508.4	−1.18217
$735$	0	0
$736$	−9232.53	−0.462385
$737$	−968.591	−0.0484105
$738$	0	0
$739$	−27867.1	−1.38715	−0.693577	−	0.720382i	$-0.743966\pi$
$739$	−27867.1	−1.38715	−0.693577	+	0.720382i	$0.743966\pi$
$740$	4748.56	0.235893
$741$	0	0
$742$	−19802.6	−0.979752
$743$	26465.7	1.30677	0.653387	−	0.757024i	$-0.273347\pi$
$743$	26465.7	1.30677	0.653387	+	0.757024i	$0.273347\pi$
$744$	0	0
$745$	−11302.5	−0.555826
$746$	9720.52	0.477069
$747$	0	0
$748$	−8647.96	−0.422728
$749$	18520.1	0.903483
$750$	0	0
$751$	−17010.6	−0.826533	−0.413267	−	0.910610i	$-0.635612\pi$
$751$	−17010.6	−0.826533	−0.413267	+	0.910610i	$0.635612\pi$
$752$	−1092.51	−0.0529784
$753$	0	0
$754$	746.428	0.0360522
$755$	−15099.8	−0.727866
$756$	0	0
$757$	−16202.7	−0.777934	−0.388967	−	0.921252i	$-0.627168\pi$
$757$	−16202.7	−0.777934	−0.388967	+	0.921252i	$0.627168\pi$
$758$	−9633.12	−0.461597
$759$	0	0
$760$	−11878.2	−0.566930
$761$	−13718.1	−0.653457	−0.326729	−	0.945118i	$-0.605946\pi$
$761$	−13718.1	−0.653457	−0.326729	+	0.945118i	$0.605946\pi$
$762$	0	0
$763$	21843.8	1.03643
$764$	1296.61	0.0613999
$765$	0	0
$766$	−20320.2	−0.958482
$767$	8293.31	0.390422
$768$	0	0
$769$	−19139.3	−0.897505	−0.448752	−	0.893656i	$-0.648131\pi$
$769$	−19139.3	−0.897505	−0.448752	+	0.893656i	$0.648131\pi$
$770$	−3463.15	−0.162082
$771$	0	0
$772$	16112.6	0.751173
$773$	−1454.06	−0.0676572	−0.0338286	−	0.999428i	$-0.510770\pi$
$773$	−1454.06	−0.0676572	−0.0338286	+	0.999428i	$0.510770\pi$
$774$	0	0
$775$	−2074.28	−0.0961423
$776$	41068.4	1.89983
$777$	0	0
$778$	−22974.1	−1.05869
$779$	38570.1	1.77396
$780$	0	0
$781$	−3086.69	−0.141422
$782$	−7637.01	−0.349231
$783$	0	0
$784$	1626.62	0.0740988
$785$	−3272.85	−0.148806
$786$	0	0
$787$	−34703.9	−1.57187	−0.785934	−	0.618311i	$-0.787817\pi$
$787$	−34703.9	−1.57187	−0.785934	+	0.618311i	$0.787817\pi$
$788$	12831.6	0.580086
$789$	0	0
$790$	722.907	0.0325568
$791$	−18414.4	−0.827738
$792$	0	0
$793$	−19685.1	−0.881512
$794$	7836.47	0.350260
$795$	0	0
$796$	2462.75	0.109660
$797$	5327.34	0.236768	0.118384	−	0.992968i	$-0.462229\pi$
$797$	5327.34	0.236768	0.118384	+	0.992968i	$0.462229\pi$
$798$	0	0
$799$	6948.12	0.307643
$800$	−17491.1	−0.773006
$801$	0	0
$802$	−27488.6	−1.21029
$803$	−25475.5	−1.11956
$804$	0	0
$805$	3532.33	0.154656
$806$	−1223.02	−0.0534481
$807$	0	0
$808$	2937.10	0.127880
$809$	−23677.3	−1.02899	−0.514494	−	0.857494i	$-0.672020\pi$
$809$	−23677.3	−1.02899	−0.514494	+	0.857494i	$0.672020\pi$
$810$	0	0
$811$	−24380.5	−1.05563	−0.527814	−	0.849360i	$-0.676988\pi$
$811$	−24380.5	−1.05563	−0.527814	+	0.849360i	$0.676988\pi$
$812$	−734.148	−0.0317285
$813$	0	0
$814$	−13165.8	−0.566906
$815$	5034.76	0.216393
$816$	0	0
$817$	−26920.2	−1.15278
$818$	15796.8	0.675212
$819$	0	0
$820$	−6803.01	−0.289721
$821$	−21296.0	−0.905283	−0.452641	−	0.891693i	$-0.649518\pi$
$821$	−21296.0	−0.905283	−0.452641	+	0.891693i	$0.649518\pi$
$822$	0	0
$823$	4908.88	0.207913	0.103957	−	0.994582i	$-0.466850\pi$
$823$	4908.88	0.207913	0.103957	+	0.994582i	$0.466850\pi$
$824$	40518.4	1.71301
$825$	0	0
$826$	7062.23	0.297490
$827$	−38572.6	−1.62189	−0.810944	−	0.585124i	$-0.801046\pi$
$827$	−38572.6	−1.62189	−0.810944	+	0.585124i	$0.801046\pi$
$828$	0	0
$829$	30118.0	1.26181	0.630905	−	0.775860i	$-0.282684\pi$
$829$	30118.0	1.26181	0.630905	+	0.775860i	$0.282684\pi$
$830$	5395.33	0.225632
$831$	0	0
$832$	−13203.7	−0.550185
$833$	−10344.9	−0.430288
$834$	0	0
$835$	7617.62	0.315711
$836$	−13273.0	−0.549112
$837$	0	0
$838$	−6762.25	−0.278756
$839$	24615.7	1.01291	0.506454	−	0.862267i	$-0.330956\pi$
$839$	24615.7	1.01291	0.506454	+	0.862267i	$0.330956\pi$
$840$	0	0
$841$	−24241.8	−0.993967
$842$	−2366.50	−0.0968586
$843$	0	0
$844$	4849.28	0.197771
$845$	−5347.41	−0.217700
$846$	0	0
$847$	7696.78	0.312237
$848$	−8238.36	−0.333616
$849$	0	0
$850$	−14468.4	−0.583838
$851$	13428.8	0.540933
$852$	0	0
$853$	24728.8	0.992614	0.496307	−	0.868147i	$-0.334689\pi$
$853$	24728.8	0.992614	0.496307	+	0.868147i	$0.334689\pi$
$854$	−16763.0	−0.671685
$855$	0	0
$856$	31064.0	1.24036
$857$	−1605.12	−0.0639788	−0.0319894	−	0.999488i	$-0.510184\pi$
$857$	−1605.12	−0.0639788	−0.0319894	+	0.999488i	$0.510184\pi$
$858$	0	0
$859$	−43424.7	−1.72483	−0.862417	−	0.506199i	$-0.831050\pi$
$859$	−43424.7	−1.72483	−0.862417	+	0.506199i	$0.831050\pi$
$860$	4748.19	0.188270
$861$	0	0
$862$	−30339.9	−1.19882
$863$	−28312.7	−1.11678	−0.558388	−	0.829580i	$-0.688580\pi$
$863$	−28312.7	−1.11678	−0.558388	+	0.829580i	$0.688580\pi$
$864$	0	0
$865$	17718.8	0.696483
$866$	29703.4	1.16555
$867$	0	0
$868$	1202.90	0.0470382
$869$	2314.99	0.0903690
$870$	0	0
$871$	1103.56	0.0429308
$872$	36638.9	1.42288
$873$	0	0
$874$	−11721.4	−0.453641
$875$	14707.5	0.568234
$876$	0	0
$877$	16723.1	0.643897	0.321948	−	0.946757i	$-0.395662\pi$
$877$	16723.1	0.643897	0.321948	+	0.946757i	$0.395662\pi$
$878$	−25103.7	−0.964932
$879$	0	0
$880$	−1440.75	−0.0551907
$881$	986.914	0.0377412	0.0188706	−	0.999822i	$-0.493993\pi$
$881$	986.914	0.0377412	0.0188706	+	0.999822i	$0.493993\pi$
$882$	0	0
$883$	21840.2	0.832368	0.416184	−	0.909280i	$-0.363367\pi$
$883$	21840.2	0.832368	0.416184	+	0.909280i	$0.363367\pi$
$884$	9853.01	0.374879
$885$	0	0
$886$	17131.4	0.649595
$887$	629.867	0.0238431	0.0119216	−	0.999929i	$-0.496205\pi$
$887$	629.867	0.0238431	0.0119216	+	0.999929i	$0.496205\pi$
$888$	0	0
$889$	−6769.40	−0.255386
$890$	−6128.78	−0.230828
$891$	0	0
$892$	−20402.0	−0.765817
$893$	10664.1	0.399619
$894$	0	0
$895$	−3015.54	−0.112624
$896$	7681.79	0.286418
$897$	0	0
$898$	−23773.6	−0.883449
$899$	−241.107	−0.00894478
$900$	0	0
$901$	52394.1	1.93729
$902$	18862.0	0.696269
$903$	0	0
$904$	−30886.7	−1.13637
$905$	9534.45	0.350205
$906$	0	0
$907$	34384.5	1.25879	0.629393	−	0.777088i	$-0.283304\pi$
$907$	34384.5	1.25879	0.629393	+	0.777088i	$0.283304\pi$
$908$	21101.1	0.771218
$909$	0	0
$910$	3945.72	0.143736
$911$	33862.9	1.23153	0.615767	−	0.787928i	$-0.288846\pi$
$911$	33862.9	1.23153	0.615767	+	0.787928i	$0.288846\pi$
$912$	0	0
$913$	17277.6	0.626294
$914$	−27662.4	−1.00108
$915$	0	0
$916$	23218.1	0.837496
$917$	25732.5	0.926677
$918$	0	0
$919$	−26656.8	−0.956830	−0.478415	−	0.878134i	$-0.658788\pi$
$919$	−26656.8	−0.956830	−0.478415	+	0.878134i	$0.658788\pi$
$920$	5924.83	0.212322
$921$	0	0
$922$	26730.6	0.954798
$923$	3516.81	0.125414
$924$	0	0
$925$	25441.1	0.904321
$926$	12948.7	0.459524
$927$	0	0
$928$	−2033.11	−0.0719181
$929$	−12246.8	−0.432514	−0.216257	−	0.976336i	$-0.569385\pi$
$929$	−12246.8	−0.432514	−0.216257	+	0.976336i	$0.569385\pi$
$930$	0	0
$931$	−15877.5	−0.558931
$932$	20705.7	0.727721
$933$	0	0
$934$	26691.0	0.935070
$935$	9162.86	0.320490
$936$	0	0
$937$	−11620.7	−0.405155	−0.202577	−	0.979266i	$-0.564932\pi$
$937$	−11620.7	−0.405155	−0.202577	+	0.979266i	$0.564932\pi$
$938$	939.745	0.0327119
$939$	0	0
$940$	−1880.93	−0.0652652
$941$	−50209.2	−1.73940	−0.869698	−	0.493584i	$-0.835687\pi$
$941$	−50209.2	−1.73940	−0.869698	+	0.493584i	$0.835687\pi$
$942$	0	0
$943$	−19238.8	−0.664369
$944$	2938.06	0.101298
$945$	0	0
$946$	−13164.8	−0.452457
$947$	−28445.8	−0.976097	−0.488049	−	0.872816i	$-0.662291\pi$
$947$	−28445.8	−0.976097	−0.488049	+	0.872816i	$0.662291\pi$
$948$	0	0
$949$	29025.4	0.992838
$950$	−22206.3	−0.758388
$951$	0	0
$952$	24045.3	0.818605
$953$	−38782.3	−1.31824	−0.659119	−	0.752038i	$-0.729071\pi$
$953$	−38782.3	−1.31824	−0.659119	+	0.752038i	$0.729071\pi$
$954$	0	0
$955$	−1373.81	−0.0465501
$956$	−1024.76	−0.0346685
$957$	0	0
$958$	13127.6	0.442729
$959$	−26636.4	−0.896907
$960$	0	0
$961$	−29395.9	−0.986739
$962$	15000.4	0.502736
$963$	0	0
$964$	−21647.9	−0.723269
$965$	−17072.0	−0.569498
$966$	0	0
$967$	54398.6	1.80904	0.904520	−	0.426432i	$-0.140230\pi$
$967$	54398.6	1.80904	0.904520	+	0.426432i	$0.140230\pi$
$968$	12909.9	0.428657
$969$	0	0
$970$	−15183.7	−0.502598
$971$	−2476.35	−0.0818432	−0.0409216	−	0.999162i	$-0.513029\pi$
$971$	−2476.35	−0.0818432	−0.0409216	+	0.999162i	$0.513029\pi$
$972$	0	0
$973$	−6567.54	−0.216388
$974$	12454.5	0.409720
$975$	0	0
$976$	−6973.83	−0.228716
$977$	48041.8	1.57318	0.786588	−	0.617478i	$-0.211846\pi$
$977$	48041.8	1.57318	0.786588	+	0.617478i	$0.211846\pi$
$978$	0	0
$979$	−19626.4	−0.640717
$980$	2800.48	0.0912838
$981$	0	0
$982$	7828.81	0.254407
$983$	3188.75	0.103464	0.0517322	−	0.998661i	$-0.483526\pi$
$983$	3188.75	0.103464	0.0517322	+	0.998661i	$0.483526\pi$
$984$	0	0
$985$	−13595.6	−0.439789
$986$	−1681.75	−0.0543184
$987$	0	0
$988$	15122.6	0.486957
$989$	13427.8	0.431728
$990$	0	0
$991$	−48105.8	−1.54201	−0.771005	−	0.636829i	$-0.780246\pi$
$991$	−48105.8	−1.54201	−0.771005	+	0.636829i	$0.780246\pi$
$992$	3331.24	0.106620
$993$	0	0
$994$	2994.77	0.0955616
$995$	−2609.38	−0.0831385
$996$	0	0
$997$	54934.0	1.74501	0.872507	−	0.488602i	$-0.162493\pi$
$997$	54934.0	1.74501	0.872507	+	0.488602i	$0.162493\pi$
$998$	29706.2	0.942218
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.40	✓	59
3.2	odd	2		2151.4.a.h.1.20	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.40	✓	59	1.1	even	1	trivial
2151.4.a.h.1.20	yes	59	3.2	odd	2

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+1.92673 q^{2} -4.28770 q^{4} +4.54299 q^{5} -14.1149 q^{7} -23.6751 q^{8} +O(q^{10})\)	\(q+1.92673 q^{2} -4.28770 q^{4} +4.54299 q^{5} -14.1149 q^{7} -23.6751 q^{8} +8.75313 q^{10} +28.0304 q^{11} -31.9364 q^{13} -27.1957 q^{14} -11.3141 q^{16} +71.9548 q^{17} +110.437 q^{19} -19.4790 q^{20} +54.0072 q^{22} -55.0861 q^{23} -104.361 q^{25} -61.5328 q^{26} +60.5204 q^{28} -12.1306 q^{29} +19.8759 q^{31} +167.602 q^{32} +138.638 q^{34} -64.1239 q^{35} -243.779 q^{37} +212.783 q^{38} -107.556 q^{40} +349.249 q^{41} -243.760 q^{43} -120.186 q^{44} -106.136 q^{46} +96.5623 q^{47} -143.769 q^{49} -201.076 q^{50} +136.933 q^{52} +728.153 q^{53} +127.342 q^{55} +334.172 q^{56} -23.3724 q^{58} -259.682 q^{59} +616.386 q^{61} +38.2956 q^{62} +413.436 q^{64} -145.087 q^{65} -34.5550 q^{67} -308.520 q^{68} -123.550 q^{70} -110.119 q^{71} -908.850 q^{73} -469.697 q^{74} -473.522 q^{76} -395.647 q^{77} +82.5884 q^{79} -51.3996 q^{80} +672.910 q^{82} +616.389 q^{83} +326.890 q^{85} -469.661 q^{86} -663.624 q^{88} -700.181 q^{89} +450.779 q^{91} +236.193 q^{92} +186.050 q^{94} +501.716 q^{95} -1734.66 q^{97} -277.005 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more