LMFDB - Embedded newform 1449.4.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,4,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1449.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1449.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:	$85.4937675983$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 34x^{5} + 7x^{4} + 295x^{3} + 84x^{2} - 524x - 288 $ x^7 - x^6 - 34x^5 + 7x^4 + 295x^3 + 84x^2 - 524*x - 288
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-3.01159$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.01159 q^{2} +8.09283 q^{4} -9.29066 q^{5} +7.00000 q^{7} +0.372383 q^{8} -37.2703 q^{10} +26.0262 q^{11} +0.0530670 q^{13} +28.0811 q^{14} -63.2488 q^{16} +107.068 q^{17} -37.1814 q^{19} -75.1877 q^{20} +104.406 q^{22} -23.0000 q^{23} -38.6836 q^{25} +0.212883 q^{26} +56.6498 q^{28} +270.783 q^{29} -215.781 q^{31} -256.707 q^{32} +429.513 q^{34} -65.0346 q^{35} +186.628 q^{37} -149.157 q^{38} -3.45969 q^{40} -10.8910 q^{41} +205.043 q^{43} +210.626 q^{44} -92.2665 q^{46} +471.928 q^{47} +49.0000 q^{49} -155.183 q^{50} +0.429462 q^{52} -267.226 q^{53} -241.801 q^{55} +2.60668 q^{56} +1086.27 q^{58} +30.0157 q^{59} +539.671 q^{61} -865.624 q^{62} -523.812 q^{64} -0.493028 q^{65} +679.592 q^{67} +866.484 q^{68} -260.892 q^{70} +1111.20 q^{71} -182.981 q^{73} +748.673 q^{74} -300.903 q^{76} +182.184 q^{77} +162.187 q^{79} +587.623 q^{80} -43.6901 q^{82} +740.972 q^{83} -994.733 q^{85} +822.549 q^{86} +9.69174 q^{88} +378.808 q^{89} +0.371469 q^{91} -186.135 q^{92} +1893.18 q^{94} +345.440 q^{95} -1398.66 q^{97} +196.568 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 6 q^{2} + 18 q^{4} + 41 q^{5} + 49 q^{7} + 33 q^{8} + q^{10} + 126 q^{11} - 87 q^{13} + 42 q^{14} + 2 q^{16} + 204 q^{17} - 286 q^{19} + 418 q^{20} + 329 q^{22} - 161 q^{23} + 440 q^{25} + 360 q^{26}+ \cdots + 294 q^{98}+O(q^{100}) $ 7 * q + 6 * q^2 + 18 * q^4 + 41 * q^5 + 49 * q^7 + 33 * q^8 + q^10 + 126 * q^11 - 87 * q^13 + 42 * q^14 + 2 * q^16 + 204 * q^17 - 286 * q^19 + 418 * q^20 + 329 * q^22 - 161 * q^23 + 440 * q^25 + 360 * q^26 + 126 * q^28 + 329 * q^29 + 296 * q^31 + 270 * q^32 - 919 * q^34 + 287 * q^35 - 691 * q^37 + 367 * q^38 + 138 * q^40 + 343 * q^41 - 171 * q^43 + 1279 * q^44 - 138 * q^46 + 1403 * q^47 + 343 * q^49 - 230 * q^50 + 2157 * q^52 + 1024 * q^53 - 158 * q^55 + 231 * q^56 + 2608 * q^58 + 1388 * q^59 - 52 * q^61 + 309 * q^62 - 187 * q^64 + 1067 * q^65 - 1148 * q^67 - 1293 * q^68 + 7 * q^70 + 1590 * q^71 - 802 * q^73 + 878 * q^74 - 2505 * q^76 + 882 * q^77 + 618 * q^79 - 195 * q^80 - 1040 * q^82 + 1818 * q^83 - 1526 * q^85 - 1188 * q^86 + 1664 * q^88 + 354 * q^89 - 609 * q^91 - 414 * q^92 + 663 * q^94 - 78 * q^95 - 1575 * q^97 + 294 * q^98

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 $

$2$	4.01159	1.41831	0.709155	−	0.705053i	$-0.249077\pi$
$2$	4.01159	1.41831	0.709155	+	0.705053i	$0.249077\pi$
$3$	0	0
$3$	0	0
$4$	8.09283	1.01160
$5$	−9.29066	−0.830982	−0.415491	−	0.909597i	$-0.636390\pi$
$5$	−9.29066	−0.830982	−0.415491	+	0.909597i	$0.636390\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0.372383	0.0164572
$9$	0	0
$10$	−37.2703	−1.17859
$11$	26.0262	0.713382	0.356691	−	0.934222i	$-0.383905\pi$
$11$	26.0262	0.713382	0.356691	+	0.934222i	$0.383905\pi$
$12$	0	0
$13$	0.0530670	0.00113217	0.000566083	−	1.00000i	$-0.499820\pi$
$13$	0.0530670	0.00113217	0.000566083		1.00000i	$0.499820\pi$
$14$	28.0811	0.536071
$15$	0	0
$16$	−63.2488	−0.988262
$17$	107.068	1.52752	0.763760	−	0.645501i	$-0.223351\pi$
$17$	107.068	1.52752	0.763760	+	0.645501i	$0.223351\pi$
$18$	0	0
$19$	−37.1814	−0.448948	−0.224474	−	0.974480i	$-0.572066\pi$
$19$	−37.1814	−0.448948	−0.224474	+	0.974480i	$0.572066\pi$
$20$	−75.1877	−0.840624
$21$	0	0
$22$	104.406	1.01180
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−38.6836	−0.309469
$26$	0.212883	0.00160576
$27$	0	0
$28$	56.6498	0.382350
$29$	270.783	1.73390	0.866952	−	0.498392i	$-0.166076\pi$
$29$	270.783	1.73390	0.866952	+	0.498392i	$0.166076\pi$
$30$	0	0
$31$	−215.781	−1.25017	−0.625087	−	0.780555i	$-0.714937\pi$
$31$	−215.781	−1.25017	−0.625087	+	0.780555i	$0.714937\pi$
$32$	−256.707	−1.41812
$33$	0	0
$34$	429.513	2.16650
$35$	−65.0346	−0.314082
$36$	0	0
$37$	186.628	0.829227	0.414614	−	0.909998i	$-0.363917\pi$
$37$	186.628	0.829227	0.414614	+	0.909998i	$0.363917\pi$
$38$	−149.157	−0.636747
$39$	0	0
$40$	−3.45969	−0.0136756
$41$	−10.8910	−0.0414850	−0.0207425	−	0.999785i	$-0.506603\pi$
$41$	−10.8910	−0.0414850	−0.0207425	+	0.999785i	$0.506603\pi$
$42$	0	0
$43$	205.043	0.727182	0.363591	−	0.931559i	$-0.381551\pi$
$43$	205.043	0.727182	0.363591	+	0.931559i	$0.381551\pi$
$44$	210.626	0.721660
$45$	0	0
$46$	−92.2665	−0.295738
$47$	471.928	1.46463	0.732316	−	0.680965i	$-0.238439\pi$
$47$	471.928	1.46463	0.732316	+	0.680965i	$0.238439\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−155.183	−0.438923
$51$	0	0
$52$	0.429462	0.00114530
$53$	−267.226	−0.692571	−0.346286	−	0.938129i	$-0.612557\pi$
$53$	−267.226	−0.692571	−0.346286	+	0.938129i	$0.612557\pi$
$54$	0	0
$55$	−241.801	−0.592808
$56$	2.60668	0.00622023
$57$	0	0
$58$	1086.27	2.45921
$59$	30.0157	0.0662324	0.0331162	−	0.999452i	$-0.489457\pi$
$59$	30.0157	0.0662324	0.0331162	+	0.999452i	$0.489457\pi$
$60$	0	0
$61$	539.671	1.13275	0.566375	−	0.824147i	$-0.308345\pi$
$61$	539.671	1.13275	0.566375	+	0.824147i	$0.308345\pi$
$62$	−865.624	−1.77314
$63$	0	0
$64$	−523.812	−1.02307
$65$	−0.493028	−0.000940809 0
$66$	0	0
$67$	679.592	1.23918	0.619592	−	0.784924i	$-0.287298\pi$
$67$	679.592	1.23918	0.619592	+	0.784924i	$0.287298\pi$
$68$	866.484	1.54524
$69$	0	0
$70$	−260.892	−0.445465
$71$	1111.20	1.85739	0.928697	−	0.370840i	$-0.120930\pi$
$71$	1111.20	1.85739	0.928697	+	0.370840i	$0.120930\pi$
$72$	0	0
$73$	−182.981	−0.293374	−0.146687	−	0.989183i	$-0.546861\pi$
$73$	−182.981	−0.293374	−0.146687	+	0.989183i	$0.546861\pi$
$74$	748.673	1.17610
$75$	0	0
$76$	−300.903	−0.454157
$77$	182.184	0.269633
$78$	0	0
$79$	162.187	0.230980	0.115490	−	0.993309i	$-0.463156\pi$
$79$	162.187	0.230980	0.115490	+	0.993309i	$0.463156\pi$
$80$	587.623	0.821228
$81$	0	0
$82$	−43.6901	−0.0588386
$83$	740.972	0.979907	0.489954	−	0.871749i	$-0.337014\pi$
$83$	740.972	0.979907	0.489954	+	0.871749i	$0.337014\pi$
$84$	0	0
$85$	−994.733	−1.26934
$86$	822.549	1.03137
$87$	0	0
$88$	9.69174	0.0117403
$89$	378.808	0.451164	0.225582	−	0.974224i	$-0.427572\pi$
$89$	378.808	0.451164	0.225582	+	0.974224i	$0.427572\pi$
$90$	0	0
$91$	0.371469	0.000427918 0
$92$	−186.135	−0.210934
$93$	0	0
$94$	1893.18	2.07730
$95$	345.440	0.373068
$96$	0	0
$97$	−1398.66	−1.46404	−0.732021	−	0.681282i	$-0.761423\pi$
$97$	−1398.66	−1.46404	−0.732021	+	0.681282i	$0.761423\pi$
$98$	196.568	0.202616
$99$	0	0
$100$	−313.060	−0.313060
$101$	1080.66	1.06465	0.532324	−	0.846541i	$-0.321319\pi$
$101$	1080.66	1.06465	0.532324	+	0.846541i	$0.321319\pi$
$102$	0	0
$103$	370.703	0.354626	0.177313	−	0.984155i	$-0.443259\pi$
$103$	370.703	0.354626	0.177313	+	0.984155i	$0.443259\pi$
$104$	0.0197613	1.86322e−5 0
$105$	0	0
$106$	−1072.00	−0.982281
$107$	557.976	0.504127	0.252064	−	0.967711i	$-0.418891\pi$
$107$	557.976	0.504127	0.252064	+	0.967711i	$0.418891\pi$
$108$	0	0
$109$	1602.50	1.40818	0.704092	−	0.710109i	$-0.251354\pi$
$109$	1602.50	1.40818	0.704092	+	0.710109i	$0.251354\pi$
$110$	−970.005	−0.840785
$111$	0	0
$112$	−442.741	−0.373528
$113$	1506.65	1.25428	0.627139	−	0.778907i	$-0.284226\pi$
$113$	1506.65	1.25428	0.627139	+	0.778907i	$0.284226\pi$
$114$	0	0
$115$	213.685	0.173272
$116$	2191.40	1.75402
$117$	0	0
$118$	120.411	0.0939381
$119$	749.477	0.577348
$120$	0	0
$121$	−653.636	−0.491086
$122$	2164.94	1.60659
$123$	0	0
$124$	−1746.28	−1.26468
$125$	1520.73	1.08815
$126$	0	0
$127$	−827.281	−0.578026	−0.289013	−	0.957325i	$-0.593327\pi$
$127$	−827.281	−0.578026	−0.289013	+	0.957325i	$0.593327\pi$
$128$	−47.6619	−0.0329121
$129$	0	0
$130$	−1.97782	−0.00133436
$131$	−1303.11	−0.869107	−0.434553	−	0.900646i	$-0.643094\pi$
$131$	−1303.11	−0.869107	−0.434553	+	0.900646i	$0.643094\pi$
$132$	0	0
$133$	−260.270	−0.169686
$134$	2726.24	1.75755
$135$	0	0
$136$	39.8704	0.0251387
$137$	−247.574	−0.154392	−0.0771958	−	0.997016i	$-0.524597\pi$
$137$	−247.574	−0.154392	−0.0771958	+	0.997016i	$0.524597\pi$
$138$	0	0
$139$	560.415	0.341969	0.170985	−	0.985274i	$-0.445305\pi$
$139$	560.415	0.341969	0.170985	+	0.985274i	$0.445305\pi$
$140$	−526.314	−0.317726
$141$	0	0
$142$	4457.67	2.63436
$143$	1.38113	0.000807666 0
$144$	0	0
$145$	−2515.76	−1.44084
$146$	−734.045	−0.416096
$147$	0	0
$148$	1510.35	0.838849
$149$	−2650.36	−1.45722	−0.728612	−	0.684927i	$-0.759834\pi$
$149$	−2650.36	−1.45722	−0.728612	+	0.684927i	$0.759834\pi$
$150$	0	0
$151$	−734.446	−0.395817	−0.197908	−	0.980220i	$-0.563415\pi$
$151$	−734.446	−0.395817	−0.197908	+	0.980220i	$0.563415\pi$
$152$	−13.8458	−0.00738841
$153$	0	0
$154$	730.845	0.382423
$155$	2004.75	1.03887
$156$	0	0
$157$	−2448.16	−1.24449	−0.622243	−	0.782824i	$-0.713778\pi$
$157$	−2448.16	−1.24449	−0.622243	+	0.782824i	$0.713778\pi$
$158$	650.627	0.327602
$159$	0	0
$160$	2384.98	1.17843
$161$	−161.000	−0.0788110
$162$	0	0
$163$	3698.40	1.77718	0.888592	−	0.458699i	$-0.151684\pi$
$163$	3698.40	1.77718	0.888592	+	0.458699i	$0.151684\pi$
$164$	−88.1388	−0.0419664
$165$	0	0
$166$	2972.47	1.38981
$167$	−3599.66	−1.66796	−0.833981	−	0.551793i	$-0.813944\pi$
$167$	−3599.66	−1.66796	−0.833981	+	0.551793i	$0.813944\pi$
$168$	0	0
$169$	−2197.00	−0.999999
$170$	−3990.46	−1.80032
$171$	0	0
$172$	1659.38	0.735619
$173$	−1644.62	−0.722764	−0.361382	−	0.932418i	$-0.617695\pi$
$173$	−1644.62	−0.722764	−0.361382	+	0.932418i	$0.617695\pi$
$174$	0	0
$175$	−270.785	−0.116968
$176$	−1646.13	−0.705008
$177$	0	0
$178$	1519.62	0.639891
$179$	4760.11	1.98764	0.993819	−	0.111008i	$-0.0354080\pi$
$179$	4760.11	1.98764	0.993819	+	0.111008i	$0.0354080\pi$
$180$	0	0
$181$	799.206	0.328202	0.164101	−	0.986444i	$-0.447528\pi$
$181$	799.206	0.328202	0.164101	+	0.986444i	$0.447528\pi$
$182$	1.49018	0.000606921 0
$183$	0	0
$184$	−8.56482	−0.00343156
$185$	−1733.89	−0.689073
$186$	0	0
$187$	2786.58	1.08970
$188$	3819.23	1.48163
$189$	0	0
$190$	1385.76	0.529125
$191$	2733.54	1.03556	0.517781	−	0.855513i	$-0.326758\pi$
$191$	2733.54	1.03556	0.517781	+	0.855513i	$0.326758\pi$
$192$	0	0
$193$	2516.43	0.938533	0.469266	−	0.883057i	$-0.344518\pi$
$193$	2516.43	0.938533	0.469266	+	0.883057i	$0.344518\pi$
$194$	−5610.83	−2.07647
$195$	0	0
$196$	396.549	0.144515
$197$	−2839.10	−1.02679	−0.513395	−	0.858152i	$-0.671613\pi$
$197$	−2839.10	−1.02679	−0.513395	+	0.858152i	$0.671613\pi$
$198$	0	0
$199$	11.6659	0.00415566	0.00207783	−	0.999998i	$-0.499339\pi$
$199$	11.6659	0.00415566	0.00207783	+	0.999998i	$0.499339\pi$
$200$	−14.4051	−0.00509299
$201$	0	0
$202$	4335.15	1.51000
$203$	1895.48	0.655354
$204$	0	0
$205$	101.184	0.0344733
$206$	1487.11	0.502970
$207$	0	0
$208$	−3.35643	−0.00111888
$209$	−967.693	−0.320271
$210$	0	0
$211$	−3294.35	−1.07484	−0.537422	−	0.843313i	$-0.680602\pi$
$211$	−3294.35	−1.07484	−0.537422	+	0.843313i	$0.680602\pi$
$212$	−2162.61	−0.700607
$213$	0	0
$214$	2238.37	0.715009
$215$	−1904.99	−0.604275
$216$	0	0
$217$	−1510.47	−0.472522
$218$	6428.58	1.99724
$219$	0	0
$220$	−1956.85	−0.599686
$221$	5.68179	0.00172940
$222$	0	0
$223$	1709.66	0.513397	0.256699	−	0.966491i	$-0.417365\pi$
$223$	1709.66	0.513397	0.256699	+	0.966491i	$0.417365\pi$
$224$	−1796.95	−0.535999
$225$	0	0
$226$	6044.05	1.77896
$227$	−4910.88	−1.43589	−0.717944	−	0.696101i	$-0.754917\pi$
$227$	−4910.88	−1.43589	−0.717944	+	0.696101i	$0.754917\pi$
$228$	0	0
$229$	657.414	0.189708	0.0948540	−	0.995491i	$-0.469762\pi$
$229$	657.414	0.189708	0.0948540	+	0.995491i	$0.469762\pi$
$230$	857.217	0.245753
$231$	0	0
$232$	100.835	0.0285352
$233$	4042.94	1.13675	0.568373	−	0.822771i	$-0.307573\pi$
$233$	4042.94	1.13675	0.568373	+	0.822771i	$0.307573\pi$
$234$	0	0
$235$	−4384.52	−1.21708
$236$	242.912	0.0670010
$237$	0	0
$238$	3006.59	0.818858
$239$	−5849.57	−1.58317	−0.791584	−	0.611060i	$-0.790743\pi$
$239$	−5849.57	−1.58317	−0.791584	+	0.611060i	$0.790743\pi$
$240$	0	0
$241$	−1520.34	−0.406364	−0.203182	−	0.979141i	$-0.565128\pi$
$241$	−1520.34	−0.406364	−0.203182	+	0.979141i	$0.565128\pi$
$242$	−2622.12	−0.696512
$243$	0	0
$244$	4367.46	1.14589
$245$	−455.242	−0.118712
$246$	0	0
$247$	−1.97311	−0.000508283 0
$248$	−80.3533	−0.0205743
$249$	0	0
$250$	6100.54	1.54333
$251$	−2027.33	−0.509817	−0.254909	−	0.966965i	$-0.582045\pi$
$251$	−2027.33	−0.509817	−0.254909	+	0.966965i	$0.582045\pi$
$252$	0	0
$253$	−598.603	−0.148750
$254$	−3318.71	−0.819820
$255$	0	0
$256$	3999.30	0.976391
$257$	−460.382	−0.111742	−0.0558712	−	0.998438i	$-0.517794\pi$
$257$	−460.382	−0.111742	−0.0558712	+	0.998438i	$0.517794\pi$
$258$	0	0
$259$	1306.39	0.313418
$260$	−3.98999	−0.000951726 0
$261$	0	0
$262$	−5227.53	−1.23266
$263$	−855.717	−0.200630	−0.100315	−	0.994956i	$-0.531985\pi$
$263$	−855.717	−0.200630	−0.100315	+	0.994956i	$0.531985\pi$
$264$	0	0
$265$	2482.70	0.575514
$266$	−1044.10	−0.240668
$267$	0	0
$268$	5499.82	1.25356
$269$	−4915.03	−1.11403	−0.557016	−	0.830502i	$-0.688054\pi$
$269$	−4915.03	−1.11403	−0.557016	+	0.830502i	$0.688054\pi$
$270$	0	0
$271$	−4904.12	−1.09928	−0.549638	−	0.835403i	$-0.685234\pi$
$271$	−4904.12	−1.09928	−0.549638	+	0.835403i	$0.685234\pi$
$272$	−6771.92	−1.50959
$273$	0	0
$274$	−993.163	−0.218975
$275$	−1006.79	−0.220770
$276$	0	0
$277$	−1522.94	−0.330341	−0.165171	−	0.986265i	$-0.552818\pi$
$277$	−1522.94	−0.330341	−0.165171	+	0.986265i	$0.552818\pi$
$278$	2248.15	0.485019
$279$	0	0
$280$	−24.2178	−0.00516890
$281$	7847.92	1.66608	0.833038	−	0.553215i	$-0.186599\pi$
$281$	7847.92	1.66608	0.833038	+	0.553215i	$0.186599\pi$
$282$	0	0
$283$	−8315.72	−1.74671	−0.873354	−	0.487086i	$-0.838060\pi$
$283$	−8315.72	−1.74671	−0.873354	+	0.487086i	$0.838060\pi$
$284$	8992.73	1.87895
$285$	0	0
$286$	5.54054	0.00114552
$287$	−76.2369	−0.0156799
$288$	0	0
$289$	6550.58	1.33331
$290$	−10092.2	−2.04356
$291$	0	0
$292$	−1480.83	−0.296778
$293$	3652.35	0.728235	0.364117	−	0.931353i	$-0.381371\pi$
$293$	3652.35	0.728235	0.364117	+	0.931353i	$0.381371\pi$
$294$	0	0
$295$	−278.866	−0.0550380
$296$	69.4971	0.0136467
$297$	0	0
$298$	−10632.2	−2.06679
$299$	−1.22054	−0.000236073 0
$300$	0	0
$301$	1435.30	0.274849
$302$	−2946.29	−0.561391
$303$	0	0
$304$	2351.68	0.443678
$305$	−5013.90	−0.941295
$306$	0	0
$307$	−3056.40	−0.568201	−0.284101	−	0.958794i	$-0.591695\pi$
$307$	−3056.40	−0.568201	−0.284101	+	0.958794i	$0.591695\pi$
$308$	1474.38	0.272762
$309$	0	0
$310$	8042.22	1.47344
$311$	−3317.56	−0.604893	−0.302446	−	0.953166i	$-0.597803\pi$
$311$	−3317.56	−0.604893	−0.302446	+	0.953166i	$0.597803\pi$
$312$	0	0
$313$	5004.23	0.903693	0.451846	−	0.892096i	$-0.350766\pi$
$313$	5004.23	0.903693	0.451846	+	0.892096i	$0.350766\pi$
$314$	−9820.99	−1.76507
$315$	0	0
$316$	1312.55	0.233661
$317$	8828.01	1.56413	0.782066	−	0.623195i	$-0.214166\pi$
$317$	8828.01	1.56413	0.782066	+	0.623195i	$0.214166\pi$
$318$	0	0
$319$	7047.47	1.23694
$320$	4866.56	0.850153
$321$	0	0
$322$	−645.865	−0.111778
$323$	−3980.95	−0.685776
$324$	0	0
$325$	−2.05283	−0.000350370 0
$326$	14836.4	2.52060
$327$	0	0
$328$	−4.05562	−0.000682726 0
$329$	3303.49	0.553579
$330$	0	0
$331$	−312.884	−0.0519567	−0.0259784	−	0.999663i	$-0.508270\pi$
$331$	−312.884	−0.0519567	−0.0259784	+	0.999663i	$0.508270\pi$
$332$	5996.56	0.991277
$333$	0	0
$334$	−14440.3	−2.36569
$335$	−6313.86	−1.02974
$336$	0	0
$337$	−6915.33	−1.11781	−0.558905	−	0.829232i	$-0.688778\pi$
$337$	−6915.33	−1.11781	−0.558905	+	0.829232i	$0.688778\pi$
$338$	−8813.44	−1.41831
$339$	0	0
$340$	−8050.20	−1.28407
$341$	−5615.97	−0.891852
$342$	0	0
$343$	343.000	0.0539949
$344$	76.3547	0.0119674
$345$	0	0
$346$	−6597.53	−1.02510
$347$	8630.76	1.33523	0.667613	−	0.744508i	$-0.267316\pi$
$347$	8630.76	1.33523	0.667613	+	0.744508i	$0.267316\pi$
$348$	0	0
$349$	1398.48	0.214495	0.107248	−	0.994232i	$-0.465796\pi$
$349$	1398.48	0.214495	0.107248	+	0.994232i	$0.465796\pi$
$350$	−1086.28	−0.165897
$351$	0	0
$352$	−6681.11	−1.01166
$353$	7272.15	1.09648	0.548240	−	0.836321i	$-0.315298\pi$
$353$	7272.15	1.09648	0.548240	+	0.836321i	$0.315298\pi$
$354$	0	0
$355$	−10323.8	−1.54346
$356$	3065.63	0.456399
$357$	0	0
$358$	19095.6	2.81909
$359$	−3938.05	−0.578948	−0.289474	−	0.957186i	$-0.593480\pi$
$359$	−3938.05	−0.578948	−0.289474	+	0.957186i	$0.593480\pi$
$360$	0	0
$361$	−5476.54	−0.798446
$362$	3206.09	0.465492
$363$	0	0
$364$	3.00624	0.000432884 0
$365$	1700.02	0.243789
$366$	0	0
$367$	4038.29	0.574379	0.287190	−	0.957874i	$-0.407279\pi$
$367$	4038.29	0.574379	0.287190	+	0.957874i	$0.407279\pi$
$368$	1454.72	0.206067
$369$	0	0
$370$	−6955.67	−0.977319
$371$	−1870.58	−0.261767
$372$	0	0
$373$	2809.11	0.389947	0.194973	−	0.980809i	$-0.437538\pi$
$373$	2809.11	0.389947	0.194973	+	0.980809i	$0.437538\pi$
$374$	11178.6	1.54554
$375$	0	0
$376$	175.738	0.0241037
$377$	14.3697	0.00196307
$378$	0	0
$379$	4091.32	0.554504	0.277252	−	0.960797i	$-0.410576\pi$
$379$	4091.32	0.554504	0.277252	+	0.960797i	$0.410576\pi$
$380$	2795.59	0.377396
$381$	0	0
$382$	10965.8	1.46875
$383$	−4096.95	−0.546591	−0.273295	−	0.961930i	$-0.588114\pi$
$383$	−4096.95	−0.546591	−0.273295	+	0.961930i	$0.588114\pi$
$384$	0	0
$385$	−1692.61	−0.224060
$386$	10094.9	1.33113
$387$	0	0
$388$	−11319.1	−1.48103
$389$	54.3187	0.00707986	0.00353993	−	0.999994i	$-0.498873\pi$
$389$	54.3187	0.00707986	0.00353993	+	0.999994i	$0.498873\pi$
$390$	0	0
$391$	−2462.57	−0.318510
$392$	18.2468	0.00235103
$393$	0	0
$394$	−11389.3	−1.45631
$395$	−1506.82	−0.191941
$396$	0	0
$397$	9273.44	1.17234	0.586172	−	0.810187i	$-0.300634\pi$
$397$	9273.44	1.17234	0.586172	+	0.810187i	$0.300634\pi$
$398$	46.7989	0.00589402
$399$	0	0
$400$	2446.69	0.305836
$401$	−13718.2	−1.70837	−0.854185	−	0.519969i	$-0.825944\pi$
$401$	−13718.2	−1.70837	−0.854185	+	0.519969i	$0.825944\pi$
$402$	0	0
$403$	−11.4509	−0.00141540
$404$	8745.58	1.07700
$405$	0	0
$406$	7603.90	0.929495
$407$	4857.22	0.591556
$408$	0	0
$409$	−2397.58	−0.289860	−0.144930	−	0.989442i	$-0.546296\pi$
$409$	−2397.58	−0.289860	−0.144930	+	0.989442i	$0.546296\pi$
$410$	405.910	0.0488938
$411$	0	0
$412$	3000.04	0.358741
$413$	210.110	0.0250335
$414$	0	0
$415$	−6884.12	−0.814285
$416$	−13.6227	−0.00160555
$417$	0	0
$418$	−3881.98	−0.454244
$419$	−1423.39	−0.165960	−0.0829799	−	0.996551i	$-0.526444\pi$
$419$	−1423.39	−0.165960	−0.0829799	+	0.996551i	$0.526444\pi$
$420$	0	0
$421$	−3069.10	−0.355294	−0.177647	−	0.984094i	$-0.556849\pi$
$421$	−3069.10	−0.355294	−0.177647	+	0.984094i	$0.556849\pi$
$422$	−13215.6	−1.52446
$423$	0	0
$424$	−99.5104	−0.0113978
$425$	−4141.78	−0.472720
$426$	0	0
$427$	3777.70	0.428139
$428$	4515.61	0.509977
$429$	0	0
$430$	−7642.02	−0.857049
$431$	6600.69	0.737690	0.368845	−	0.929491i	$-0.379753\pi$
$431$	6600.69	0.737690	0.368845	+	0.929491i	$0.379753\pi$
$432$	0	0
$433$	−8663.43	−0.961519	−0.480760	−	0.876852i	$-0.659639\pi$
$433$	−8663.43	−0.961519	−0.480760	+	0.876852i	$0.659639\pi$
$434$	−6059.37	−0.670182
$435$	0	0
$436$	12968.8	1.42452
$437$	855.173	0.0936121
$438$	0	0
$439$	6948.87	0.755471	0.377735	−	0.925914i	$-0.376703\pi$
$439$	6948.87	0.755471	0.377735	+	0.925914i	$0.376703\pi$
$440$	−90.0426	−0.00975594
$441$	0	0
$442$	22.7930	0.00245283
$443$	−3400.22	−0.364672	−0.182336	−	0.983236i	$-0.558366\pi$
$443$	−3400.22	−0.364672	−0.182336	+	0.983236i	$0.558366\pi$
$444$	0	0
$445$	−3519.38	−0.374909
$446$	6858.47	0.728157
$447$	0	0
$448$	−3666.68	−0.386684
$449$	11407.2	1.19897	0.599487	−	0.800384i	$-0.295371\pi$
$449$	11407.2	1.19897	0.599487	+	0.800384i	$0.295371\pi$
$450$	0	0
$451$	−283.451	−0.0295947
$452$	12193.0	1.26883
$453$	0	0
$454$	−19700.4	−2.03654
$455$	−3.45120	−0.000355592 0
$456$	0	0
$457$	9912.76	1.01466	0.507330	−	0.861752i	$-0.330633\pi$
$457$	9912.76	1.01466	0.507330	+	0.861752i	$0.330633\pi$
$458$	2637.27	0.269065
$459$	0	0
$460$	1729.32	0.175282
$461$	−10159.1	−1.02637	−0.513185	−	0.858278i	$-0.671535\pi$
$461$	−10159.1	−1.02637	−0.513185	+	0.858278i	$0.671535\pi$
$462$	0	0
$463$	−9046.25	−0.908024	−0.454012	−	0.890996i	$-0.650008\pi$
$463$	−9046.25	−0.908024	−0.454012	+	0.890996i	$0.650008\pi$
$464$	−17126.7	−1.71355
$465$	0	0
$466$	16218.6	1.61226
$467$	3959.37	0.392329	0.196165	−	0.980571i	$-0.437151\pi$
$467$	3959.37	0.392329	0.196165	+	0.980571i	$0.437151\pi$
$468$	0	0
$469$	4757.14	0.468368
$470$	−17588.9	−1.72620
$471$	0	0
$472$	11.1774	0.00109000
$473$	5336.50	0.518758
$474$	0	0
$475$	1438.31	0.138935
$476$	6065.38	0.584047
$477$	0	0
$478$	−23466.1	−2.24542
$479$	−16222.6	−1.54745	−0.773726	−	0.633520i	$-0.781609\pi$
$479$	−16222.6	−1.54745	−0.773726	+	0.633520i	$0.781609\pi$
$480$	0	0
$481$	9.90378	0.000938823 0
$482$	−6098.97	−0.576350
$483$	0	0
$484$	−5289.76	−0.496784
$485$	12994.4	1.21659
$486$	0	0
$487$	6774.14	0.630320	0.315160	−	0.949039i	$-0.397942\pi$
$487$	6774.14	0.630320	0.315160	+	0.949039i	$0.397942\pi$
$488$	200.965	0.0186419
$489$	0	0
$490$	−1826.24	−0.168370
$491$	−14659.7	−1.34742	−0.673710	−	0.738996i	$-0.735300\pi$
$491$	−14659.7	−1.34742	−0.673710	+	0.738996i	$0.735300\pi$
$492$	0	0
$493$	28992.3	2.64857
$494$	−7.91530	−0.000720903 0
$495$	0	0
$496$	13647.9	1.23550
$497$	7778.38	0.702029
$498$	0	0
$499$	12484.8	1.12003	0.560015	−	0.828483i	$-0.310795\pi$
$499$	12484.8	1.12003	0.560015	+	0.828483i	$0.310795\pi$
$500$	12307.0	1.10077
$501$	0	0
$502$	−8132.82	−0.723079
$503$	17938.9	1.59017	0.795085	−	0.606498i	$-0.207426\pi$
$503$	17938.9	1.59017	0.795085	+	0.606498i	$0.207426\pi$
$504$	0	0
$505$	−10040.0	−0.884703
$506$	−2401.35	−0.210974
$507$	0	0
$508$	−6695.04	−0.584733
$509$	10505.0	0.914785	0.457392	−	0.889265i	$-0.348783\pi$
$509$	10505.0	0.914785	0.457392	+	0.889265i	$0.348783\pi$
$510$	0	0
$511$	−1280.87	−0.110885
$512$	16424.8	1.41774
$513$	0	0
$514$	−1846.86	−0.158485
$515$	−3444.08	−0.294688
$516$	0	0
$517$	12282.5	1.04484
$518$	5240.71	0.444525
$519$	0	0
$520$	−0.183595	−1.54831e−5 0
$521$	−3758.58	−0.316058	−0.158029	−	0.987434i	$-0.550514\pi$
$521$	−3758.58	−0.316058	−0.158029	+	0.987434i	$0.550514\pi$
$522$	0	0
$523$	−20595.1	−1.72191	−0.860955	−	0.508681i	$-0.830133\pi$
$523$	−20595.1	−1.72191	−0.860955	+	0.508681i	$0.830133\pi$
$524$	−10545.8	−0.879191
$525$	0	0
$526$	−3432.78	−0.284556
$527$	−23103.3	−1.90967
$528$	0	0
$529$	529.000	0.0434783
$530$	9959.58	0.816258
$531$	0	0
$532$	−2106.32	−0.171655
$533$	−0.577952	−4.69679e−5 0
$534$	0	0
$535$	−5183.97	−0.418921
$536$	253.069	0.0203935
$537$	0	0
$538$	−19717.1	−1.58004
$539$	1275.29	0.101912
$540$	0	0
$541$	6685.37	0.531288	0.265644	−	0.964071i	$-0.414415\pi$
$541$	6685.37	0.531288	0.265644	+	0.964071i	$0.414415\pi$
$542$	−19673.3	−1.55912
$543$	0	0
$544$	−27485.1	−2.16620
$545$	−14888.3	−1.17017
$546$	0	0
$547$	12675.1	0.990763	0.495381	−	0.868676i	$-0.335028\pi$
$547$	12675.1	0.990763	0.495381	+	0.868676i	$0.335028\pi$
$548$	−2003.57	−0.156183
$549$	0	0
$550$	−4038.82	−0.313120
$551$	−10068.1	−0.778432
$552$	0	0
$553$	1135.31	0.0873024
$554$	−6109.41	−0.468527
$555$	0	0
$556$	4535.34	0.345937
$557$	−12756.8	−0.970418	−0.485209	−	0.874398i	$-0.661256\pi$
$557$	−12756.8	−0.970418	−0.485209	+	0.874398i	$0.661256\pi$
$558$	0	0
$559$	10.8810	0.000823290 0
$560$	4113.36	0.310395
$561$	0	0
$562$	31482.6	2.36301
$563$	−11448.9	−0.857039	−0.428519	−	0.903533i	$-0.640965\pi$
$563$	−11448.9	−0.857039	−0.428519	+	0.903533i	$0.640965\pi$
$564$	0	0
$565$	−13997.7	−1.04228
$566$	−33359.2	−2.47737
$567$	0	0
$568$	413.792	0.0305675
$569$	−5508.53	−0.405852	−0.202926	−	0.979194i	$-0.565045\pi$
$569$	−5508.53	−0.405852	−0.202926	+	0.979194i	$0.565045\pi$
$570$	0	0
$571$	743.477	0.0544895	0.0272448	−	0.999629i	$-0.491327\pi$
$571$	743.477	0.0544895	0.0272448	+	0.999629i	$0.491327\pi$
$572$	11.1773	0.000817038 0
$573$	0	0
$574$	−305.831	−0.0222389
$575$	889.723	0.0645287
$576$	0	0
$577$	−17770.7	−1.28215	−0.641077	−	0.767477i	$-0.721512\pi$
$577$	−17770.7	−1.28215	−0.641077	+	0.767477i	$0.721512\pi$
$578$	26278.2	1.89105
$579$	0	0
$580$	−20359.6	−1.45756
$581$	5186.81	0.370370
$582$	0	0
$583$	−6954.88	−0.494068
$584$	−68.1391	−0.00482811
$585$	0	0
$586$	14651.7	1.03286
$587$	15758.0	1.10801	0.554007	−	0.832512i	$-0.313098\pi$
$587$	15758.0	1.10801	0.554007	+	0.832512i	$0.313098\pi$
$588$	0	0
$589$	8023.05	0.561263
$590$	−1118.69	−0.0780609
$591$	0	0
$592$	−11804.0	−0.819494
$593$	6098.60	0.422326	0.211163	−	0.977451i	$-0.432275\pi$
$593$	6098.60	0.422326	0.211163	+	0.977451i	$0.432275\pi$
$594$	0	0
$595$	−6963.13	−0.479766
$596$	−21448.9	−1.47413
$597$	0	0
$598$	−4.89631	−0.000334824 0
$599$	−13811.2	−0.942087	−0.471044	−	0.882110i	$-0.656123\pi$
$599$	−13811.2	−0.942087	−0.471044	+	0.882110i	$0.656123\pi$
$600$	0	0
$601$	24341.1	1.65207	0.826034	−	0.563620i	$-0.190592\pi$
$601$	24341.1	1.65207	0.826034	+	0.563620i	$0.190592\pi$
$602$	5757.84	0.389821
$603$	0	0
$604$	−5943.74	−0.400410
$605$	6072.71	0.408084
$606$	0	0
$607$	18529.0	1.23899	0.619497	−	0.784999i	$-0.287337\pi$
$607$	18529.0	1.23899	0.619497	+	0.784999i	$0.287337\pi$
$608$	9544.74	0.636661
$609$	0	0
$610$	−20113.7	−1.33505
$611$	25.0438	0.00165821
$612$	0	0
$613$	−9236.65	−0.608589	−0.304294	−	0.952578i	$-0.598421\pi$
$613$	−9236.65	−0.608589	−0.304294	+	0.952578i	$0.598421\pi$
$614$	−12261.0	−0.805886
$615$	0	0
$616$	67.8422	0.00443740
$617$	19494.7	1.27201	0.636004	−	0.771686i	$-0.280586\pi$
$617$	19494.7	1.27201	0.636004	+	0.771686i	$0.280586\pi$
$618$	0	0
$619$	−17261.4	−1.12083	−0.560417	−	0.828211i	$-0.689359\pi$
$619$	−17261.4	−1.12083	−0.560417	+	0.828211i	$0.689359\pi$
$620$	16224.1	1.05093
$621$	0	0
$622$	−13308.7	−0.857926
$623$	2651.66	0.170524
$624$	0	0
$625$	−9293.12	−0.594760
$626$	20074.9	1.28172
$627$	0	0
$628$	−19812.5	−1.25893
$629$	19981.9	1.26666
$630$	0	0
$631$	3497.01	0.220624	0.110312	−	0.993897i	$-0.464815\pi$
$631$	3497.01	0.220624	0.110312	+	0.993897i	$0.464815\pi$
$632$	60.3957	0.00380129
$633$	0	0
$634$	35414.3	2.21843
$635$	7685.99	0.480329
$636$	0	0
$637$	2.60029	0.000161738 0
$638$	28271.5	1.75436
$639$	0	0
$640$	442.810	0.0273494
$641$	−16199.9	−0.998219	−0.499110	−	0.866539i	$-0.666340\pi$
$641$	−16199.9	−0.998219	−0.499110	+	0.866539i	$0.666340\pi$
$642$	0	0
$643$	17595.3	1.07915	0.539574	−	0.841938i	$-0.318585\pi$
$643$	17595.3	1.07915	0.539574	+	0.841938i	$0.318585\pi$
$644$	−1302.95	−0.0797255
$645$	0	0
$646$	−15969.9	−0.972644
$647$	−1534.38	−0.0932343	−0.0466172	−	0.998913i	$-0.514844\pi$
$647$	−1534.38	−0.0932343	−0.0466172	+	0.998913i	$0.514844\pi$
$648$	0	0
$649$	781.196	0.0472490
$650$	−8.23509	−0.000496933 0
$651$	0	0
$652$	29930.5	1.79780
$653$	9932.96	0.595263	0.297631	−	0.954681i	$-0.403803\pi$
$653$	9932.96	0.595263	0.297631	+	0.954681i	$0.403803\pi$
$654$	0	0
$655$	12106.7	0.722212
$656$	688.841	0.0409981
$657$	0	0
$658$	13252.2	0.785147
$659$	−24631.9	−1.45603	−0.728013	−	0.685564i	$-0.759556\pi$
$659$	−24631.9	−1.45603	−0.728013	+	0.685564i	$0.759556\pi$
$660$	0	0
$661$	−24286.5	−1.42910	−0.714549	−	0.699585i	$-0.753368\pi$
$661$	−24286.5	−1.42910	−0.714549	+	0.699585i	$0.753368\pi$
$662$	−1255.16	−0.0736907
$663$	0	0
$664$	275.926	0.0161265
$665$	2418.08	0.141006
$666$	0	0
$667$	−6228.02	−0.361544
$668$	−29131.4	−1.68732
$669$	0	0
$670$	−25328.6	−1.46049
$671$	14045.6	0.808084
$672$	0	0
$673$	12874.3	0.737394	0.368697	−	0.929550i	$-0.379804\pi$
$673$	12874.3	0.737394	0.368697	+	0.929550i	$0.379804\pi$
$674$	−27741.4	−1.58540
$675$	0	0
$676$	−17779.9	−1.01160
$677$	17431.5	0.989584	0.494792	−	0.869012i	$-0.335244\pi$
$677$	17431.5	0.989584	0.494792	+	0.869012i	$0.335244\pi$
$678$	0	0
$679$	−9790.60	−0.553356
$680$	−370.422	−0.0208898
$681$	0	0
$682$	−22528.9	−1.26492
$683$	−6488.50	−0.363507	−0.181754	−	0.983344i	$-0.558177\pi$
$683$	−6488.50	−0.363507	−0.181754	+	0.983344i	$0.558177\pi$
$684$	0	0
$685$	2300.12	0.128297
$686$	1375.97	0.0765815
$687$	0	0
$688$	−12968.7	−0.718646
$689$	−14.1809	−0.000784105 0
$690$	0	0
$691$	34226.6	1.88428	0.942142	−	0.335215i	$-0.108809\pi$
$691$	34226.6	1.88428	0.942142	+	0.335215i	$0.108809\pi$
$692$	−13309.6	−0.731150
$693$	0	0
$694$	34623.0	1.89376
$695$	−5206.62	−0.284170
$696$	0	0
$697$	−1166.08	−0.0633692
$698$	5610.12	0.304221
$699$	0	0
$700$	−2191.42	−0.118326
$701$	−27477.7	−1.48048	−0.740242	−	0.672341i	$-0.765289\pi$
$701$	−27477.7	−1.48048	−0.740242	+	0.672341i	$0.765289\pi$
$702$	0	0
$703$	−6939.09	−0.372280
$704$	−13632.9	−0.729840
$705$	0	0
$706$	29172.8	1.55515
$707$	7564.60	0.402399
$708$	0	0
$709$	−7049.88	−0.373433	−0.186716	−	0.982414i	$-0.559785\pi$
$709$	−7049.88	−0.373433	−0.186716	+	0.982414i	$0.559785\pi$
$710$	−41414.7	−2.18911
$711$	0	0
$712$	141.062	0.00742489
$713$	4962.96	0.260679
$714$	0	0
$715$	−12.8317	−0.000671156 0
$716$	38522.8	2.01070
$717$	0	0
$718$	−15797.8	−0.821128
$719$	26340.1	1.36623	0.683114	−	0.730311i	$-0.260625\pi$
$719$	26340.1	1.36623	0.683114	+	0.730311i	$0.260625\pi$
$720$	0	0
$721$	2594.92	0.134036
$722$	−21969.6	−1.13244
$723$	0	0
$724$	6467.84	0.332010
$725$	−10474.9	−0.536589
$726$	0	0
$727$	−16913.3	−0.862833	−0.431416	−	0.902153i	$-0.641986\pi$
$727$	−16913.3	−0.862833	−0.431416	+	0.902153i	$0.641986\pi$
$728$	0.138329	7.04233e−6 0
$729$	0	0
$730$	6819.76	0.345768
$731$	21953.6	1.11078
$732$	0	0
$733$	−21070.8	−1.06176	−0.530878	−	0.847448i	$-0.678138\pi$
$733$	−21070.8	−1.06176	−0.530878	+	0.847448i	$0.678138\pi$
$734$	16200.0	0.814648
$735$	0	0
$736$	5904.26	0.295698
$737$	17687.2	0.884012
$738$	0	0
$739$	32778.5	1.63163	0.815816	−	0.578311i	$-0.196288\pi$
$739$	32778.5	1.63163	0.815816	+	0.578311i	$0.196288\pi$
$740$	−14032.1	−0.697069
$741$	0	0
$742$	−7503.99	−0.371267
$743$	24391.3	1.20435	0.602173	−	0.798365i	$-0.294302\pi$
$743$	24391.3	1.20435	0.602173	+	0.798365i	$0.294302\pi$
$744$	0	0
$745$	24623.6	1.21093
$746$	11269.0	0.553065
$747$	0	0
$748$	22551.3	1.10235
$749$	3905.83	0.190542
$750$	0	0
$751$	33092.5	1.60794	0.803969	−	0.594671i	$-0.202718\pi$
$751$	33092.5	1.60794	0.803969	+	0.594671i	$0.202718\pi$
$752$	−29848.8	−1.44744
$753$	0	0
$754$	57.6452	0.00278424
$755$	6823.49	0.328917
$756$	0	0
$757$	−7638.71	−0.366755	−0.183378	−	0.983043i	$-0.558703\pi$
$757$	−7638.71	−0.366755	−0.183378	+	0.983043i	$0.558703\pi$
$758$	16412.7	0.786459
$759$	0	0
$760$	128.636	0.00613964
$761$	−2717.83	−0.129463	−0.0647314	−	0.997903i	$-0.520619\pi$
$761$	−2717.83	−0.129463	−0.0647314	+	0.997903i	$0.520619\pi$
$762$	0	0
$763$	11217.5	0.532243
$764$	22122.1	1.04758
$765$	0	0
$766$	−16435.3	−0.775235
$767$	1.59285	7.49861e−5 0
$768$	0	0
$769$	−32383.5	−1.51857	−0.759285	−	0.650758i	$-0.774451\pi$
$769$	−32383.5	−1.51857	−0.759285	+	0.650758i	$0.774451\pi$
$770$	−6790.03	−0.317787
$771$	0	0
$772$	20365.1	0.949423
$773$	18976.7	0.882979	0.441490	−	0.897266i	$-0.354450\pi$
$773$	18976.7	0.882979	0.441490	+	0.897266i	$0.354450\pi$
$774$	0	0
$775$	8347.19	0.386890
$776$	−520.837	−0.0240940
$777$	0	0
$778$	217.904	0.0100414
$779$	404.943	0.0186246
$780$	0	0
$781$	28920.3	1.32503
$782$	−9878.80	−0.451746
$783$	0	0
$784$	−3099.19	−0.141180
$785$	22745.0	1.03414
$786$	0	0
$787$	−25315.0	−1.14661	−0.573304	−	0.819342i	$-0.694339\pi$
$787$	−25315.0	−1.14661	−0.573304	+	0.819342i	$0.694339\pi$
$788$	−22976.4	−1.03870
$789$	0	0
$790$	−6044.75	−0.272231
$791$	10546.5	0.474073
$792$	0	0
$793$	28.6387	0.00128246
$794$	37201.2	1.66275
$795$	0	0
$796$	94.4104	0.00420388
$797$	38523.5	1.71214	0.856068	−	0.516863i	$-0.172900\pi$
$797$	38523.5	1.71214	0.856068	+	0.516863i	$0.172900\pi$
$798$	0	0
$799$	50528.4	2.23725
$800$	9930.36	0.438864
$801$	0	0
$802$	−55031.9	−2.42300
$803$	−4762.31	−0.209288
$804$	0	0
$805$	1495.80	0.0654906
$806$	−45.9361	−0.00200748
$807$	0	0
$808$	402.419	0.0175211
$809$	27469.5	1.19379	0.596895	−	0.802319i	$-0.296401\pi$
$809$	27469.5	1.19379	0.596895	+	0.802319i	$0.296401\pi$
$810$	0	0
$811$	−24345.5	−1.05411	−0.527056	−	0.849830i	$-0.676704\pi$
$811$	−24345.5	−1.05411	−0.527056	+	0.849830i	$0.676704\pi$
$812$	15339.8	0.662958
$813$	0	0
$814$	19485.1	0.839010
$815$	−34360.6	−1.47681
$816$	0	0
$817$	−7623.81	−0.326467
$818$	−9618.11	−0.411111
$819$	0	0
$820$	818.868	0.0348733
$821$	−27134.4	−1.15347	−0.576733	−	0.816933i	$-0.695673\pi$
$821$	−27134.4	−1.15347	−0.576733	+	0.816933i	$0.695673\pi$
$822$	0	0
$823$	−15641.7	−0.662496	−0.331248	−	0.943544i	$-0.607470\pi$
$823$	−15641.7	−0.662496	−0.331248	+	0.943544i	$0.607470\pi$
$824$	138.044	0.00583615
$825$	0	0
$826$	842.875	0.0355053
$827$	−37881.7	−1.59284	−0.796418	−	0.604746i	$-0.793275\pi$
$827$	−37881.7	−1.59284	−0.796418	+	0.604746i	$0.793275\pi$
$828$	0	0
$829$	−23483.1	−0.983840	−0.491920	−	0.870640i	$-0.663705\pi$
$829$	−23483.1	−0.983840	−0.491920	+	0.870640i	$0.663705\pi$
$830$	−27616.3	−1.15491
$831$	0	0
$832$	−27.7972	−0.00115829
$833$	5246.34	0.218217
$834$	0	0
$835$	33443.2	1.38605
$836$	−7831.37	−0.323988
$837$	0	0
$838$	−5710.05	−0.235382
$839$	33381.1	1.37359	0.686796	−	0.726850i	$-0.259016\pi$
$839$	33381.1	1.37359	0.686796	+	0.726850i	$0.259016\pi$
$840$	0	0
$841$	48934.6	2.00642
$842$	−12312.0	−0.503917
$843$	0	0
$844$	−26660.6	−1.08732
$845$	20411.6	0.830981
$846$	0	0
$847$	−4575.45	−0.185613
$848$	16901.7	0.684442
$849$	0	0
$850$	−16615.1	−0.670463
$851$	−4292.44	−0.172906
$852$	0	0
$853$	−26043.1	−1.04537	−0.522683	−	0.852527i	$-0.675069\pi$
$853$	−26043.1	−1.04537	−0.522683	+	0.852527i	$0.675069\pi$
$854$	15154.6	0.607234
$855$	0	0
$856$	207.781	0.00829651
$857$	−45096.7	−1.79752	−0.898759	−	0.438443i	$-0.855530\pi$
$857$	−45096.7	−1.79752	−0.898759	+	0.438443i	$0.855530\pi$
$858$	0	0
$859$	19006.0	0.754921	0.377460	−	0.926026i	$-0.376797\pi$
$859$	19006.0	0.754921	0.377460	+	0.926026i	$0.376797\pi$
$860$	−15416.7	−0.611286
$861$	0	0
$862$	26479.3	1.04627
$863$	36333.2	1.43314	0.716568	−	0.697518i	$-0.245712\pi$
$863$	36333.2	1.43314	0.716568	+	0.697518i	$0.245712\pi$
$864$	0	0
$865$	15279.6	0.600603
$866$	−34754.1	−1.36373
$867$	0	0
$868$	−12223.9	−0.478004
$869$	4221.11	0.164777
$870$	0	0
$871$	36.0639	0.00140296
$872$	596.746	0.0231747
$873$	0	0
$874$	3430.60	0.132771
$875$	10645.1	0.411280
$876$	0	0
$877$	−20443.5	−0.787145	−0.393573	−	0.919294i	$-0.628761\pi$
$877$	−20443.5	−0.787145	−0.393573	+	0.919294i	$0.628761\pi$
$878$	27876.0	1.07149
$879$	0	0
$880$	15293.6	0.585849
$881$	−26429.0	−1.01069	−0.505345	−	0.862918i	$-0.668635\pi$
$881$	−26429.0	−1.01069	−0.505345	+	0.862918i	$0.668635\pi$
$882$	0	0
$883$	16823.1	0.641156	0.320578	−	0.947222i	$-0.396123\pi$
$883$	16823.1	0.641156	0.320578	+	0.947222i	$0.396123\pi$
$884$	45.9817	0.00174947
$885$	0	0
$886$	−13640.3	−0.517217
$887$	−37841.3	−1.43245	−0.716226	−	0.697868i	$-0.754132\pi$
$887$	−37841.3	−1.43245	−0.716226	+	0.697868i	$0.754132\pi$
$888$	0	0
$889$	−5790.97	−0.218473
$890$	−14118.3	−0.531738
$891$	0	0
$892$	13836.0	0.519354
$893$	−17546.9	−0.657543
$894$	0	0
$895$	−44224.6	−1.65169
$896$	−333.633	−0.0124396
$897$	0	0
$898$	45761.0	1.70052
$899$	−58429.9	−2.16768
$900$	0	0
$901$	−28611.3	−1.05792
$902$	−1137.09	−0.0419744
$903$	0	0
$904$	561.051	0.0206419
$905$	−7425.16	−0.272730
$906$	0	0
$907$	19199.0	0.702858	0.351429	−	0.936214i	$-0.385696\pi$
$907$	19199.0	0.702858	0.351429	+	0.936214i	$0.385696\pi$
$908$	−39742.9	−1.45255
$909$	0	0
$910$	−13.8448	−0.000504340 0
$911$	13080.8	0.475727	0.237863	−	0.971299i	$-0.423553\pi$
$911$	13080.8	0.475727	0.237863	+	0.971299i	$0.423553\pi$
$912$	0	0
$913$	19284.7	0.699048
$914$	39765.9	1.43910
$915$	0	0
$916$	5320.34	0.191909
$917$	−9121.75	−0.328491
$918$	0	0
$919$	301.836	0.0108342	0.00541712	−	0.999985i	$-0.498276\pi$
$919$	301.836	0.0108342	0.00541712	+	0.999985i	$0.498276\pi$
$920$	79.5728	0.00285156
$921$	0	0
$922$	−40754.1	−1.45571
$923$	58.9680	0.00210288
$924$	0	0
$925$	−7219.44	−0.256620
$926$	−36289.8	−1.28786
$927$	0	0
$928$	−69512.0	−2.45888
$929$	−49106.4	−1.73426	−0.867130	−	0.498082i	$-0.834038\pi$
$929$	−49106.4	−1.73426	−0.867130	+	0.498082i	$0.834038\pi$
$930$	0	0
$931$	−1821.89	−0.0641354
$932$	32718.8	1.14994
$933$	0	0
$934$	15883.4	0.556444
$935$	−25889.2	−0.905525
$936$	0	0
$937$	−16868.0	−0.588104	−0.294052	−	0.955789i	$-0.595004\pi$
$937$	−16868.0	−0.588104	−0.294052	+	0.955789i	$0.595004\pi$
$938$	19083.7	0.664291
$939$	0	0
$940$	−35483.2	−1.23121
$941$	15141.3	0.524541	0.262271	−	0.964994i	$-0.415529\pi$
$941$	15141.3	0.524541	0.262271	+	0.964994i	$0.415529\pi$
$942$	0	0
$943$	250.493	0.00865023
$944$	−1898.46	−0.0654550
$945$	0	0
$946$	21407.8	0.735760
$947$	−41117.0	−1.41090	−0.705450	−	0.708760i	$-0.749255\pi$
$947$	−41117.0	−1.41090	−0.705450	+	0.708760i	$0.749255\pi$
$948$	0	0
$949$	−9.71027	−0.000332148 0
$950$	5769.92	0.197054
$951$	0	0
$952$	279.093	0.00950152
$953$	304.023	0.0103340	0.00516698	−	0.999987i	$-0.498355\pi$
$953$	304.023	0.0103340	0.00516698	+	0.999987i	$0.498355\pi$
$954$	0	0
$955$	−25396.4	−0.860533
$956$	−47339.6	−1.60154
$957$	0	0
$958$	−65078.4	−2.19477
$959$	−1733.02	−0.0583545
$960$	0	0
$961$	16770.4	0.562937
$962$	39.7299	0.00133154
$963$	0	0
$964$	−12303.8	−0.411079
$965$	−23379.3	−0.779904
$966$	0	0
$967$	26528.3	0.882204	0.441102	−	0.897457i	$-0.354588\pi$
$967$	26528.3	0.882204	0.441102	+	0.897457i	$0.354588\pi$
$968$	−243.403	−0.00808189
$969$	0	0
$970$	52128.3	1.72551
$971$	−43836.5	−1.44879	−0.724397	−	0.689383i	$-0.757882\pi$
$971$	−43836.5	−1.44879	−0.724397	+	0.689383i	$0.757882\pi$
$972$	0	0
$973$	3922.90	0.129252
$974$	27175.1	0.893989
$975$	0	0
$976$	−34133.5	−1.11945
$977$	−26128.0	−0.855588	−0.427794	−	0.903876i	$-0.640709\pi$
$977$	−26128.0	−0.855588	−0.427794	+	0.903876i	$0.640709\pi$
$978$	0	0
$979$	9858.95	0.321852
$980$	−3684.20	−0.120089
$981$	0	0
$982$	−58808.7	−1.91106
$983$	−22600.8	−0.733321	−0.366661	−	0.930355i	$-0.619499\pi$
$983$	−22600.8	−0.733321	−0.366661	+	0.930355i	$0.619499\pi$
$984$	0	0
$985$	26377.1	0.853244
$986$	116305.	3.75650
$987$	0	0
$988$	−15.9680	−0.000514181 0
$989$	−4716.00	−0.151628
$990$	0	0
$991$	1642.25	0.0526417	0.0263208	−	0.999654i	$-0.491621\pi$
$991$	1642.25	0.0526417	0.0263208	+	0.999654i	$0.491621\pi$
$992$	55392.5	1.77290
$993$	0	0
$994$	31203.7	0.995694
$995$	−108.384	−0.00345328
$996$	0	0
$997$	8390.31	0.266523	0.133262	−	0.991081i	$-0.457455\pi$
$997$	8390.31	0.266523	0.133262	+	0.991081i	$0.457455\pi$
$998$	50083.7	1.58855
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.4.a.h.1.6		7
3.2	odd	2		483.4.a.c.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.4.a.c.1.2	✓	7	3.2	odd	2
1449.4.a.h.1.6		7	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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q+4.01159 q^{2} +8.09283 q^{4} -9.29066 q^{5} +7.00000 q^{7} +0.372383 q^{8} -37.2703 q^{10} +26.0262 q^{11} +0.0530670 q^{13} +28.0811 q^{14} -63.2488 q^{16} +107.068 q^{17} -37.1814 q^{19} -75.1877 q^{20} +104.406 q^{22} -23.0000 q^{23} -38.6836 q^{25} +0.212883 q^{26} +56.6498 q^{28} +270.783 q^{29} -215.781 q^{31} -256.707 q^{32} +429.513 q^{34} -65.0346 q^{35} +186.628 q^{37} -149.157 q^{38} -3.45969 q^{40} -10.8910 q^{41} +205.043 q^{43} +210.626 q^{44} -92.2665 q^{46} +471.928 q^{47} +49.0000 q^{49} -155.183 q^{50} +0.429462 q^{52} -267.226 q^{53} -241.801 q^{55} +2.60668 q^{56} +1086.27 q^{58} +30.0157 q^{59} +539.671 q^{61} -865.624 q^{62} -523.812 q^{64} -0.493028 q^{65} +679.592 q^{67} +866.484 q^{68} -260.892 q^{70} +1111.20 q^{71} -182.981 q^{73} +748.673 q^{74} -300.903 q^{76} +182.184 q^{77} +162.187 q^{79} +587.623 q^{80} -43.6901 q^{82} +740.972 q^{83} -994.733 q^{85} +822.549 q^{86} +9.69174 q^{88} +378.808 q^{89} +0.371469 q^{91} -186.135 q^{92} +1893.18 q^{94} +345.440 q^{95} -1398.66 q^{97} +196.568 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 6 q^{2} + 18 q^{4} + 41 q^{5} + 49 q^{7} + 33 q^{8} + q^{10} + 126 q^{11} - 87 q^{13} + 42 q^{14} + 2 q^{16} + 204 q^{17} - 286 q^{19} + 418 q^{20} + 329 q^{22} - 161 q^{23} + 440 q^{25} + 360 q^{26}+ \cdots + 294 q^{98}+O(q^{100}) \) 7 * q + 6 * q^2 + 18 * q^4 + 41 * q^5 + 49 * q^7 + 33 * q^8 + q^10 + 126 * q^11 - 87 * q^13 + 42 * q^14 + 2 * q^16 + 204 * q^17 - 286 * q^19 + 418 * q^20 + 329 * q^22 - 161 * q^23 + 440 * q^25 + 360 * q^26 + 126 * q^28 + 329 * q^29 + 296 * q^31 + 270 * q^32 - 919 * q^34 + 287 * q^35 - 691 * q^37 + 367 * q^38 + 138 * q^40 + 343 * q^41 - 171 * q^43 + 1279 * q^44 - 138 * q^46 + 1403 * q^47 + 343 * q^49 - 230 * q^50 + 2157 * q^52 + 1024 * q^53 - 158 * q^55 + 231 * q^56 + 2608 * q^58 + 1388 * q^59 - 52 * q^61 + 309 * q^62 - 187 * q^64 + 1067 * q^65 - 1148 * q^67 - 1293 * q^68 + 7 * q^70 + 1590 * q^71 - 802 * q^73 + 878 * q^74 - 2505 * q^76 + 882 * q^77 + 618 * q^79 - 195 * q^80 - 1040 * q^82 + 1818 * q^83 - 1526 * q^85 - 1188 * q^86 + 1664 * q^88 + 354 * q^89 - 609 * q^91 - 414 * q^92 + 663 * q^94 - 78 * q^95 - 1575 * q^97 + 294 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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