LMFDB - Embedded newform 261.4.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [261,4,Mod(1,261)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(261, 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("261.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 261 = 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	261.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:	$15.3994985115$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.13458092.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 $ x^5 - x^4 - 14x^3 + 18x^2 + 20*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.03898$ of defining polynomial
Character	$\chi$	$=$	261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.49488 q^{2} +22.1937 q^{4} -2.14270 q^{5} +20.3573 q^{7} +77.9928 q^{8} +O(q^{10})$	\(q+5.49488 q^{2} +22.1937 q^{4} -2.14270 q^{5} +20.3573 q^{7} +77.9928 q^{8} -11.7739 q^{10} -52.0703 q^{11} +7.04574 q^{13} +111.861 q^{14} +251.011 q^{16} -28.7724 q^{17} +76.4208 q^{19} -47.5545 q^{20} -286.120 q^{22} -59.7251 q^{23} -120.409 q^{25} +38.7155 q^{26} +451.804 q^{28} +29.0000 q^{29} -3.25229 q^{31} +755.335 q^{32} -158.101 q^{34} -43.6196 q^{35} +150.673 q^{37} +419.923 q^{38} -167.115 q^{40} +92.3254 q^{41} -100.703 q^{43} -1155.63 q^{44} -328.182 q^{46} -324.003 q^{47} +71.4197 q^{49} -661.632 q^{50} +156.371 q^{52} -374.774 q^{53} +111.571 q^{55} +1587.72 q^{56} +159.352 q^{58} -489.567 q^{59} +221.508 q^{61} -17.8709 q^{62} +2142.39 q^{64} -15.0969 q^{65} -427.538 q^{67} -638.567 q^{68} -239.685 q^{70} +898.999 q^{71} -1087.35 q^{73} +827.929 q^{74} +1696.06 q^{76} -1060.01 q^{77} -798.018 q^{79} -537.842 q^{80} +507.317 q^{82} +436.713 q^{83} +61.6507 q^{85} -553.353 q^{86} -4061.11 q^{88} -456.763 q^{89} +143.432 q^{91} -1325.52 q^{92} -1780.36 q^{94} -163.747 q^{95} +803.714 q^{97} +392.442 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8}+O(q^{10}) $ 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8	$ 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8} - 64 q^{10} - 12 q^{11} + 14 q^{13} + 192 q^{14} + 146 q^{16} - 66 q^{17} + 214 q^{19} - 6 q^{20} - 98 q^{22} - 164 q^{23} + 207 q^{25} - 56 q^{26} + 540 q^{28} + 145 q^{29} + 420 q^{31} + 652 q^{32} + 204 q^{34} + 52 q^{35} + 378 q^{37} + 496 q^{38} - 80 q^{40} + 1158 q^{41} - 204 q^{43} - 784 q^{44} + 580 q^{46} - 248 q^{47} - 283 q^{49} - 908 q^{50} + 1482 q^{52} + 554 q^{53} + 546 q^{55} + 608 q^{56} - 440 q^{59} + 618 q^{61} - 1250 q^{62} + 2594 q^{64} + 1656 q^{65} + 1164 q^{67} - 356 q^{68} - 2184 q^{70} + 692 q^{71} - 1950 q^{73} + 1832 q^{74} + 1376 q^{76} + 1616 q^{77} + 272 q^{79} + 890 q^{80} + 92 q^{82} - 512 q^{83} - 1628 q^{85} - 2446 q^{86} - 6954 q^{88} - 866 q^{89} + 2580 q^{91} - 3468 q^{92} - 5942 q^{94} - 2244 q^{95} + 1562 q^{97} + 3408 q^{98}+O(q^{100}) $ 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8 - 64 * q^10 - 12 * q^11 + 14 * q^13 + 192 * q^14 + 146 * q^16 - 66 * q^17 + 214 * q^19 - 6 * q^20 - 98 * q^22 - 164 * q^23 + 207 * q^25 - 56 * q^26 + 540 * q^28 + 145 * q^29 + 420 * q^31 + 652 * q^32 + 204 * q^34 + 52 * q^35 + 378 * q^37 + 496 * q^38 - 80 * q^40 + 1158 * q^41 - 204 * q^43 - 784 * q^44 + 580 * q^46 - 248 * q^47 - 283 * q^49 - 908 * q^50 + 1482 * q^52 + 554 * q^53 + 546 * q^55 + 608 * q^56 - 440 * q^59 + 618 * q^61 - 1250 * q^62 + 2594 * q^64 + 1656 * q^65 + 1164 * q^67 - 356 * q^68 - 2184 * q^70 + 692 * q^71 - 1950 * q^73 + 1832 * q^74 + 1376 * q^76 + 1616 * q^77 + 272 * q^79 + 890 * q^80 + 92 * q^82 - 512 * q^83 - 1628 * q^85 - 2446 * q^86 - 6954 * q^88 - 866 * q^89 + 2580 * q^91 - 3468 * q^92 - 5942 * q^94 - 2244 * q^95 + 1562 * q^97 + 3408 * 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 $

$2$	5.49488	1.94273	0.971367	−	0.237584i	$-0.0763557\pi$
$2$	5.49488	1.94273	0.971367	+	0.237584i	$0.0763557\pi$
$3$	0	0
$3$	0	0
$4$	22.1937	2.77421
$5$	−2.14270	−0.191649	−0.0958246	−	0.995398i	$-0.530549\pi$
$5$	−2.14270	−0.191649	−0.0958246	+	0.995398i	$0.530549\pi$
$6$	0	0
$7$	20.3573	1.09919	0.549595	−	0.835431i	$-0.314782\pi$
$7$	20.3573	1.09919	0.549595	+	0.835431i	$0.314782\pi$
$8$	77.9928	3.44683
$9$	0	0
$10$	−11.7739	−0.372323
$11$	−52.0703	−1.42725	−0.713627	−	0.700526i	$-0.752949\pi$
$11$	−52.0703	−1.42725	−0.713627	+	0.700526i	$0.752949\pi$
$12$	0	0
$13$	7.04574	0.150318	0.0751591	−	0.997172i	$-0.476054\pi$
$13$	7.04574	0.150318	0.0751591	+	0.997172i	$0.476054\pi$
$14$	111.861	2.13544
$15$	0	0
$16$	251.011	3.92205
$17$	−28.7724	−0.410490	−0.205245	−	0.978711i	$-0.565799\pi$
$17$	−28.7724	−0.410490	−0.205245	+	0.978711i	$0.565799\pi$
$18$	0	0
$19$	76.4208	0.922744	0.461372	−	0.887207i	$-0.347357\pi$
$19$	76.4208	0.922744	0.461372	+	0.887207i	$0.347357\pi$
$20$	−47.5545	−0.531676
$21$	0	0
$22$	−286.120	−2.77277
$23$	−59.7251	−0.541458	−0.270729	−	0.962656i	$-0.587265\pi$
$23$	−59.7251	−0.541458	−0.270729	+	0.962656i	$0.587265\pi$
$24$	0	0
$25$	−120.409	−0.963271
$26$	38.7155	0.292028
$27$	0	0
$28$	451.804	3.04939
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−3.25229	−0.0188428	−0.00942142	−	0.999956i	$-0.502999\pi$
$31$	−3.25229	−0.0188428	−0.00942142	+	0.999956i	$0.502999\pi$
$32$	755.335	4.17268
$33$	0	0
$34$	−158.101	−0.797473
$35$	−43.6196	−0.210659
$36$	0	0
$37$	150.673	0.669471	0.334736	−	0.942312i	$-0.391353\pi$
$37$	150.673	0.669471	0.334736	+	0.942312i	$0.391353\pi$
$38$	419.923	1.79265
$39$	0	0
$40$	−167.115	−0.660581
$41$	92.3254	0.351678	0.175839	−	0.984419i	$-0.443736\pi$
$41$	92.3254	0.351678	0.175839	+	0.984419i	$0.443736\pi$
$42$	0	0
$43$	−100.703	−0.357142	−0.178571	−	0.983927i	$-0.557148\pi$
$43$	−100.703	−0.357142	−0.178571	+	0.983927i	$0.557148\pi$
$44$	−1155.63	−3.95951
$45$	0	0
$46$	−328.182	−1.05191
$47$	−324.003	−1.00555	−0.502774	−	0.864418i	$-0.667687\pi$
$47$	−324.003	−1.00555	−0.502774	+	0.864418i	$0.667687\pi$
$48$	0	0
$49$	71.4197	0.208221
$50$	−661.632	−1.87138
$51$	0	0
$52$	156.371	0.417015
$53$	−374.774	−0.971305	−0.485653	−	0.874152i	$-0.661418\pi$
$53$	−374.774	−0.971305	−0.485653	+	0.874152i	$0.661418\pi$
$54$	0	0
$55$	111.571	0.273532
$56$	1587.72	3.78872
$57$	0	0
$58$	159.352	0.360757
$59$	−489.567	−1.08027	−0.540137	−	0.841577i	$-0.681628\pi$
$59$	−489.567	−1.08027	−0.540137	+	0.841577i	$0.681628\pi$
$60$	0	0
$61$	221.508	0.464937	0.232468	−	0.972604i	$-0.425320\pi$
$61$	221.508	0.464937	0.232468	+	0.972604i	$0.425320\pi$
$62$	−17.8709	−0.0366066
$63$	0	0
$64$	2142.39	4.18435
$65$	−15.0969	−0.0288083
$66$	0	0
$67$	−427.538	−0.779584	−0.389792	−	0.920903i	$-0.627453\pi$
$67$	−427.538	−0.779584	−0.389792	+	0.920903i	$0.627453\pi$
$68$	−638.567	−1.13879
$69$	0	0
$70$	−239.685	−0.409254
$71$	898.999	1.50270	0.751349	−	0.659905i	$-0.229404\pi$
$71$	898.999	1.50270	0.751349	+	0.659905i	$0.229404\pi$
$72$	0	0
$73$	−1087.35	−1.74335	−0.871673	−	0.490087i	$-0.836965\pi$
$73$	−1087.35	−1.74335	−0.871673	+	0.490087i	$0.836965\pi$
$74$	827.929	1.30060
$75$	0	0
$76$	1696.06	2.55989
$77$	−1060.01	−1.56882
$78$	0	0
$79$	−798.018	−1.13651	−0.568253	−	0.822854i	$-0.692381\pi$
$79$	−798.018	−1.13651	−0.568253	+	0.822854i	$0.692381\pi$
$80$	−537.842	−0.751658
$81$	0	0
$82$	507.317	0.683217
$83$	436.713	0.577536	0.288768	−	0.957399i	$-0.406754\pi$
$83$	436.713	0.577536	0.288768	+	0.957399i	$0.406754\pi$
$84$	0	0
$85$	61.6507	0.0786701
$86$	−553.353	−0.693833
$87$	0	0
$88$	−4061.11	−4.91950
$89$	−456.763	−0.544009	−0.272004	−	0.962296i	$-0.587687\pi$
$89$	−456.763	−0.544009	−0.272004	+	0.962296i	$0.587687\pi$
$90$	0	0
$91$	143.432	0.165228
$92$	−1325.52	−1.50212
$93$	0	0
$94$	−1780.36	−1.95351
$95$	−163.747	−0.176843
$96$	0	0
$97$	803.714	0.841287	0.420643	−	0.907226i	$-0.361804\pi$
$97$	803.714	0.841287	0.420643	+	0.907226i	$0.361804\pi$
$98$	392.442	0.404517
$99$	0	0
$100$	−2672.32	−2.67232
$101$	738.800	0.727855	0.363928	−	0.931427i	$-0.381436\pi$
$101$	738.800	0.727855	0.363928	+	0.931427i	$0.381436\pi$
$102$	0	0
$103$	2031.60	1.94349	0.971743	−	0.236042i	$-0.0758501\pi$
$103$	2031.60	1.94349	0.971743	+	0.236042i	$0.0758501\pi$
$104$	549.517	0.518121
$105$	0	0
$106$	−2059.34	−1.88699
$107$	−1594.33	−1.44047	−0.720233	−	0.693732i	$-0.755965\pi$
$107$	−1594.33	−1.44047	−0.720233	+	0.693732i	$0.755965\pi$
$108$	0	0
$109$	−229.668	−0.201818	−0.100909	−	0.994896i	$-0.532175\pi$
$109$	−229.668	−0.201818	−0.100909	+	0.994896i	$0.532175\pi$
$110$	613.071	0.531400
$111$	0	0
$112$	5109.91	4.31108
$113$	1584.73	1.31928	0.659640	−	0.751582i	$-0.270709\pi$
$113$	1584.73	1.31928	0.659640	+	0.751582i	$0.270709\pi$
$114$	0	0
$115$	127.973	0.103770
$116$	643.618	0.515159
$117$	0	0
$118$	−2690.11	−2.09869
$119$	−585.728	−0.451207
$120$	0	0
$121$	1380.32	1.03705
$122$	1217.16	0.903248
$123$	0	0
$124$	−72.1803	−0.0522741
$125$	525.838	0.376259
$126$	0	0
$127$	621.184	0.434025	0.217013	−	0.976169i	$-0.430369\pi$
$127$	621.184	0.434025	0.217013	+	0.976169i	$0.430369\pi$
$128$	5729.47	3.95639
$129$	0	0
$130$	−82.9558	−0.0559669
$131$	1504.34	1.00332	0.501659	−	0.865066i	$-0.332723\pi$
$131$	1504.34	1.00332	0.501659	+	0.865066i	$0.332723\pi$
$132$	0	0
$133$	1555.72	1.01427
$134$	−2349.27	−1.51452
$135$	0	0
$136$	−2244.04	−1.41489
$137$	29.3812	0.0183227	0.00916135	−	0.999958i	$-0.497084\pi$
$137$	29.3812	0.0183227	0.00916135	+	0.999958i	$0.497084\pi$
$138$	0	0
$139$	1262.60	0.770450	0.385225	−	0.922823i	$-0.374124\pi$
$139$	1262.60	0.770450	0.385225	+	0.922823i	$0.374124\pi$
$140$	−968.082	−0.584413
$141$	0	0
$142$	4939.89	2.91934
$143$	−366.874	−0.214542
$144$	0	0
$145$	−62.1384	−0.0355883
$146$	−5974.84	−3.38686
$147$	0	0
$148$	3343.99	1.85726
$149$	−826.526	−0.454441	−0.227220	−	0.973843i	$-0.572964\pi$
$149$	−826.526	−0.454441	−0.227220	+	0.973843i	$0.572964\pi$
$150$	0	0
$151$	−2632.03	−1.41849	−0.709243	−	0.704964i	$-0.750963\pi$
$151$	−2632.03	−1.41849	−0.709243	+	0.704964i	$0.750963\pi$
$152$	5960.27	3.18054
$153$	0	0
$154$	−5824.64	−3.04781
$155$	6.96868	0.00361121
$156$	0	0
$157$	234.540	0.119225	0.0596124	−	0.998222i	$-0.481014\pi$
$157$	234.540	0.119225	0.0596124	+	0.998222i	$0.481014\pi$
$158$	−4385.01	−2.20793
$159$	0	0
$160$	−1618.46	−0.799689
$161$	−1215.84	−0.595166
$162$	0	0
$163$	−1650.78	−0.793246	−0.396623	−	0.917982i	$-0.629818\pi$
$163$	−1650.78	−0.793246	−0.396623	+	0.917982i	$0.629818\pi$
$164$	2049.04	0.975631
$165$	0	0
$166$	2399.69	1.12200
$167$	3684.28	1.70717	0.853587	−	0.520950i	$-0.174422\pi$
$167$	3684.28	1.70717	0.853587	+	0.520950i	$0.174422\pi$
$168$	0	0
$169$	−2147.36	−0.977404
$170$	338.763	0.152835
$171$	0	0
$172$	−2234.98	−0.990790
$173$	3073.58	1.35075	0.675375	−	0.737474i	$-0.263982\pi$
$173$	3073.58	1.35075	0.675375	+	0.737474i	$0.263982\pi$
$174$	0	0
$175$	−2451.20	−1.05882
$176$	−13070.2	−5.59776
$177$	0	0
$178$	−2509.86	−1.05686
$179$	−2980.70	−1.24463	−0.622313	−	0.782769i	$-0.713807\pi$
$179$	−2980.70	−1.24463	−0.622313	+	0.782769i	$0.713807\pi$
$180$	0	0
$181$	3145.07	1.29155	0.645777	−	0.763526i	$-0.276533\pi$
$181$	3145.07	1.29155	0.645777	+	0.763526i	$0.276533\pi$
$182$	788.143	0.320995
$183$	0	0
$184$	−4658.13	−1.86631
$185$	−322.847	−0.128304
$186$	0	0
$187$	1498.19	0.585874
$188$	−7190.84	−2.78960
$189$	0	0
$190$	−899.771	−0.343559
$191$	2125.56	0.805236	0.402618	−	0.915368i	$-0.368100\pi$
$191$	2125.56	0.805236	0.402618	+	0.915368i	$0.368100\pi$
$192$	0	0
$193$	3085.29	1.15069	0.575347	−	0.817909i	$-0.304867\pi$
$193$	3085.29	1.15069	0.575347	+	0.817909i	$0.304867\pi$
$194$	4416.31	1.63440
$195$	0	0
$196$	1585.07	0.577649
$197$	2484.45	0.898526	0.449263	−	0.893400i	$-0.351687\pi$
$197$	2484.45	0.898526	0.449263	+	0.893400i	$0.351687\pi$
$198$	0	0
$199$	5489.57	1.95550	0.977752	−	0.209764i	$-0.0672696\pi$
$199$	5489.57	1.95550	0.977752	+	0.209764i	$0.0672696\pi$
$200$	−9391.02	−3.32023
$201$	0	0
$202$	4059.62	1.41403
$203$	590.362	0.204115
$204$	0	0
$205$	−197.826	−0.0673988
$206$	11163.4	3.77568
$207$	0	0
$208$	1768.56	0.589556
$209$	−3979.26	−1.31699
$210$	0	0
$211$	−1884.64	−0.614902	−0.307451	−	0.951564i	$-0.599476\pi$
$211$	−1884.64	−0.614902	−0.307451	+	0.951564i	$0.599476\pi$
$212$	−8317.63	−2.69461
$213$	0	0
$214$	−8760.67	−2.79844
$215$	215.777	0.0684460
$216$	0	0
$217$	−66.2078	−0.0207119
$218$	−1262.00	−0.392079
$219$	0	0
$220$	2476.18	0.758836
$221$	−202.723	−0.0617041
$222$	0	0
$223$	−5208.50	−1.56407	−0.782033	−	0.623237i	$-0.785817\pi$
$223$	−5208.50	−1.56407	−0.782033	+	0.623237i	$0.785817\pi$
$224$	15376.6	4.58657
$225$	0	0
$226$	8707.89	2.56301
$227$	5243.29	1.53308	0.766540	−	0.642196i	$-0.221977\pi$
$227$	5243.29	1.53308	0.766540	+	0.642196i	$0.221977\pi$
$228$	0	0
$229$	−846.348	−0.244228	−0.122114	−	0.992516i	$-0.538967\pi$
$229$	−846.348	−0.244228	−0.122114	+	0.992516i	$0.538967\pi$
$230$	703.197	0.201597
$231$	0	0
$232$	2261.79	0.640060
$233$	−823.620	−0.231576	−0.115788	−	0.993274i	$-0.536939\pi$
$233$	−823.620	−0.231576	−0.115788	+	0.993274i	$0.536939\pi$
$234$	0	0
$235$	694.243	0.192712
$236$	−10865.3	−2.99691
$237$	0	0
$238$	−3218.51	−0.876575
$239$	2471.50	0.668903	0.334452	−	0.942413i	$-0.391449\pi$
$239$	2471.50	0.668903	0.334452	+	0.942413i	$0.391449\pi$
$240$	0	0
$241$	−1148.79	−0.307055	−0.153527	−	0.988144i	$-0.549063\pi$
$241$	−1148.79	−0.307055	−0.153527	+	0.988144i	$0.549063\pi$
$242$	7584.69	2.01472
$243$	0	0
$244$	4916.08	1.28983
$245$	−153.031	−0.0399053
$246$	0	0
$247$	538.441	0.138705
$248$	−253.655	−0.0649480
$249$	0	0
$250$	2889.42	0.730971
$251$	1686.20	0.424032	0.212016	−	0.977266i	$-0.431997\pi$
$251$	1686.20	0.424032	0.212016	+	0.977266i	$0.431997\pi$
$252$	0	0
$253$	3109.91	0.772799
$254$	3413.33	0.843195
$255$	0	0
$256$	14343.7	3.50188
$257$	−3593.97	−0.872318	−0.436159	−	0.899870i	$-0.643662\pi$
$257$	−3593.97	−0.872318	−0.436159	+	0.899870i	$0.643662\pi$
$258$	0	0
$259$	3067.29	0.735877
$260$	−335.057	−0.0799205
$261$	0	0
$262$	8266.15	1.94918
$263$	2321.87	0.544382	0.272191	−	0.962243i	$-0.412252\pi$
$263$	2321.87	0.544382	0.272191	+	0.962243i	$0.412252\pi$
$264$	0	0
$265$	803.029	0.186150
$266$	8548.51	1.97046
$267$	0	0
$268$	−9488.66	−2.16273
$269$	−2365.41	−0.536140	−0.268070	−	0.963399i	$-0.586386\pi$
$269$	−2365.41	−0.536140	−0.268070	+	0.963399i	$0.586386\pi$
$270$	0	0
$271$	1732.51	0.388348	0.194174	−	0.980967i	$-0.437797\pi$
$271$	1732.51	0.388348	0.194174	+	0.980967i	$0.437797\pi$
$272$	−7222.20	−1.60996
$273$	0	0
$274$	161.446	0.0355961
$275$	6269.73	1.37483
$276$	0	0
$277$	−688.616	−0.149368	−0.0746839	−	0.997207i	$-0.523795\pi$
$277$	−688.616	−0.149368	−0.0746839	+	0.997207i	$0.523795\pi$
$278$	6937.85	1.49678
$279$	0	0
$280$	−3402.02	−0.726105
$281$	−2224.44	−0.472238	−0.236119	−	0.971724i	$-0.575875\pi$
$281$	−2224.44	−0.472238	−0.236119	+	0.971724i	$0.575875\pi$
$282$	0	0
$283$	5037.52	1.05813	0.529063	−	0.848582i	$-0.322543\pi$
$283$	5037.52	1.05813	0.529063	+	0.848582i	$0.322543\pi$
$284$	19952.1	4.16881
$285$	0	0
$286$	−2015.93	−0.416798
$287$	1879.50	0.386561
$288$	0	0
$289$	−4085.15	−0.831498
$290$	−341.443	−0.0691387
$291$	0	0
$292$	−24132.3	−4.83642
$293$	6761.01	1.34806	0.674032	−	0.738702i	$-0.264561\pi$
$293$	6761.01	1.34806	0.674032	+	0.738702i	$0.264561\pi$
$294$	0	0
$295$	1049.00	0.207034
$296$	11751.4	2.30755
$297$	0	0
$298$	−4541.66	−0.882857
$299$	−420.807	−0.0813910
$300$	0	0
$301$	−2050.05	−0.392568
$302$	−14462.7	−2.75574
$303$	0	0
$304$	19182.5	3.61905
$305$	−474.625	−0.0891047
$306$	0	0
$307$	8860.99	1.64731	0.823653	−	0.567093i	$-0.191932\pi$
$307$	8860.99	1.64731	0.823653	+	0.567093i	$0.191932\pi$
$308$	−23525.6	−4.35226
$309$	0	0
$310$	38.2921	0.00701563
$311$	−4127.77	−0.752619	−0.376309	−	0.926494i	$-0.622807\pi$
$311$	−4127.77	−0.752619	−0.376309	+	0.926494i	$0.622807\pi$
$312$	0	0
$313$	−5384.53	−0.972369	−0.486184	−	0.873856i	$-0.661612\pi$
$313$	−5384.53	−0.972369	−0.486184	+	0.873856i	$0.661612\pi$
$314$	1288.77	0.231622
$315$	0	0
$316$	−17711.0	−3.15291
$317$	5379.49	0.953129	0.476565	−	0.879139i	$-0.341882\pi$
$317$	5379.49	0.953129	0.476565	+	0.879139i	$0.341882\pi$
$318$	0	0
$319$	−1510.04	−0.265034
$320$	−4590.49	−0.801926
$321$	0	0
$322$	−6680.90	−1.15625
$323$	−2198.81	−0.378778
$324$	0	0
$325$	−848.369	−0.144797
$326$	−9070.84	−1.54106
$327$	0	0
$328$	7200.71	1.21217
$329$	−6595.83	−1.10529
$330$	0	0
$331$	5825.09	0.967298	0.483649	−	0.875262i	$-0.339311\pi$
$331$	5825.09	0.967298	0.483649	+	0.875262i	$0.339311\pi$
$332$	9692.29	1.60221
$333$	0	0
$334$	20244.7	3.31658
$335$	916.087	0.149407
$336$	0	0
$337$	−3627.26	−0.586318	−0.293159	−	0.956064i	$-0.594707\pi$
$337$	−3627.26	−0.586318	−0.293159	+	0.956064i	$0.594707\pi$
$338$	−11799.5	−1.89884
$339$	0	0
$340$	1368.26	0.218248
$341$	169.348	0.0268935
$342$	0	0
$343$	−5528.64	−0.870317
$344$	−7854.14	−1.23101
$345$	0	0
$346$	16889.0	2.62415
$347$	−1172.68	−0.181421	−0.0907104	−	0.995877i	$-0.528914\pi$
$347$	−1172.68	−0.181421	−0.0907104	+	0.995877i	$0.528914\pi$
$348$	0	0
$349$	−173.078	−0.0265463	−0.0132731	−	0.999912i	$-0.504225\pi$
$349$	−173.078	−0.0265463	−0.0132731	+	0.999912i	$0.504225\pi$
$350$	−13469.0	−2.05700
$351$	0	0
$352$	−39330.5	−5.95547
$353$	−13008.4	−1.96137	−0.980687	−	0.195582i	$-0.937340\pi$
$353$	−13008.4	−1.96137	−0.980687	+	0.195582i	$0.937340\pi$
$354$	0	0
$355$	−1926.29	−0.287991
$356$	−10137.3	−1.50920
$357$	0	0
$358$	−16378.6	−2.41798
$359$	4487.40	0.659710	0.329855	−	0.944032i	$-0.393000\pi$
$359$	4487.40	0.659710	0.329855	+	0.944032i	$0.393000\pi$
$360$	0	0
$361$	−1018.85	−0.148543
$362$	17281.8	2.50915
$363$	0	0
$364$	3183.29	0.458379
$365$	2329.86	0.334111
$366$	0	0
$367$	1674.11	0.238114	0.119057	−	0.992887i	$-0.462013\pi$
$367$	1674.11	0.238114	0.119057	+	0.992887i	$0.462013\pi$
$368$	−14991.7	−2.12363
$369$	0	0
$370$	−1774.00	−0.249260
$371$	−7629.39	−1.06765
$372$	0	0
$373$	−6800.89	−0.944066	−0.472033	−	0.881581i	$-0.656480\pi$
$373$	−6800.89	−0.944066	−0.472033	+	0.881581i	$0.656480\pi$
$374$	8232.37	1.13820
$375$	0	0
$376$	−25269.9	−3.46595
$377$	204.326	0.0279134
$378$	0	0
$379$	10216.7	1.38468	0.692341	−	0.721571i	$-0.256579\pi$
$379$	10216.7	1.38468	0.692341	+	0.721571i	$0.256579\pi$
$380$	−3634.16	−0.490601
$381$	0	0
$382$	11679.7	1.56436
$383$	7184.47	0.958509	0.479255	−	0.877676i	$-0.340907\pi$
$383$	7184.47	0.958509	0.479255	+	0.877676i	$0.340907\pi$
$384$	0	0
$385$	2271.29	0.300664
$386$	16953.3	2.23549
$387$	0	0
$388$	17837.4	2.33391
$389$	3848.67	0.501633	0.250817	−	0.968035i	$-0.419301\pi$
$389$	3848.67	0.501633	0.250817	+	0.968035i	$0.419301\pi$
$390$	0	0
$391$	1718.43	0.222263
$392$	5570.22	0.717700
$393$	0	0
$394$	13651.7	1.74560
$395$	1709.91	0.217810
$396$	0	0
$397$	−1622.51	−0.205117	−0.102559	−	0.994727i	$-0.532703\pi$
$397$	−1622.51	−0.205117	−0.102559	+	0.994727i	$0.532703\pi$
$398$	30164.5	3.79902
$399$	0	0
$400$	−30224.0	−3.77800
$401$	−13842.9	−1.72389	−0.861947	−	0.506999i	$-0.830755\pi$
$401$	−13842.9	−1.72389	−0.861947	+	0.506999i	$0.830755\pi$
$402$	0	0
$403$	−22.9148	−0.00283242
$404$	16396.7	2.01923
$405$	0	0
$406$	3243.97	0.396540
$407$	−7845.58	−0.955506
$408$	0	0
$409$	3357.24	0.405880	0.202940	−	0.979191i	$-0.434950\pi$
$409$	3357.24	0.405880	0.202940	+	0.979191i	$0.434950\pi$
$410$	−1087.03	−0.130938
$411$	0	0
$412$	45088.7	5.39165
$413$	−9966.26	−1.18743
$414$	0	0
$415$	−935.746	−0.110684
$416$	5321.89	0.627229
$417$	0	0
$418$	−21865.6	−2.55856
$419$	−1652.11	−0.192627	−0.0963134	−	0.995351i	$-0.530705\pi$
$419$	−1652.11	−0.192627	−0.0963134	+	0.995351i	$0.530705\pi$
$420$	0	0
$421$	−6405.13	−0.741490	−0.370745	−	0.928735i	$-0.620898\pi$
$421$	−6405.13	−0.741490	−0.370745	+	0.928735i	$0.620898\pi$
$422$	−10355.9	−1.19459
$423$	0	0
$424$	−29229.7	−3.34792
$425$	3464.45	0.395413
$426$	0	0
$427$	4509.30	0.511054
$428$	−35384.2	−3.99616
$429$	0	0
$430$	1185.67	0.132972
$431$	−252.536	−0.0282233	−0.0141116	−	0.999900i	$-0.504492\pi$
$431$	−252.536	−0.0282233	−0.0141116	+	0.999900i	$0.504492\pi$
$432$	0	0
$433$	−12779.1	−1.41831	−0.709153	−	0.705055i	$-0.750922\pi$
$433$	−12779.1	−1.41831	−0.709153	+	0.705055i	$0.750922\pi$
$434$	−363.804	−0.0402377
$435$	0	0
$436$	−5097.18	−0.559887
$437$	−4564.24	−0.499628
$438$	0	0
$439$	−3760.84	−0.408873	−0.204437	−	0.978880i	$-0.565536\pi$
$439$	−3760.84	−0.408873	−0.204437	+	0.978880i	$0.565536\pi$
$440$	8701.75	0.942817
$441$	0	0
$442$	−1113.94	−0.119875
$443$	−3713.77	−0.398299	−0.199149	−	0.979969i	$-0.563818\pi$
$443$	−3713.77	−0.398299	−0.199149	+	0.979969i	$0.563818\pi$
$444$	0	0
$445$	978.707	0.104259
$446$	−28620.1	−3.03856
$447$	0	0
$448$	43613.2	4.59939
$449$	12043.9	1.26589	0.632947	−	0.774195i	$-0.281845\pi$
$449$	12043.9	1.26589	0.632947	+	0.774195i	$0.281845\pi$
$450$	0	0
$451$	−4807.41	−0.501934
$452$	35171.0	3.65997
$453$	0	0
$454$	28811.2	2.97837
$455$	−307.333	−0.0316659
$456$	0	0
$457$	−12995.7	−1.33023	−0.665113	−	0.746743i	$-0.731617\pi$
$457$	−12995.7	−1.33023	−0.665113	+	0.746743i	$0.731617\pi$
$458$	−4650.58	−0.474470
$459$	0	0
$460$	2840.20	0.287880
$461$	−7874.30	−0.795538	−0.397769	−	0.917486i	$-0.630215\pi$
$461$	−7874.30	−0.795538	−0.397769	+	0.917486i	$0.630215\pi$
$462$	0	0
$463$	−3466.08	−0.347910	−0.173955	−	0.984754i	$-0.555655\pi$
$463$	−3466.08	−0.347910	−0.173955	+	0.984754i	$0.555655\pi$
$464$	7279.33	0.728307
$465$	0	0
$466$	−4525.69	−0.449890
$467$	−14835.9	−1.47007	−0.735034	−	0.678030i	$-0.762834\pi$
$467$	−14835.9	−1.47007	−0.735034	+	0.678030i	$0.762834\pi$
$468$	0	0
$469$	−8703.53	−0.856912
$470$	3814.78	0.374389
$471$	0	0
$472$	−38182.7	−3.72352
$473$	5243.66	0.509733
$474$	0	0
$475$	−9201.74	−0.888853
$476$	−12999.5	−1.25175
$477$	0	0
$478$	13580.6	1.29950
$479$	−8119.86	−0.774542	−0.387271	−	0.921966i	$-0.626582\pi$
$479$	−8119.86	−0.774542	−0.387271	+	0.921966i	$0.626582\pi$
$480$	0	0
$481$	1061.60	0.100634
$482$	−6312.48	−0.596526
$483$	0	0
$484$	30634.4	2.87701
$485$	−1722.12	−0.161232
$486$	0	0
$487$	9698.92	0.902464	0.451232	−	0.892407i	$-0.350985\pi$
$487$	9698.92	0.902464	0.451232	+	0.892407i	$0.350985\pi$
$488$	17276.0	1.60256
$489$	0	0
$490$	−840.887	−0.0775253
$491$	6844.25	0.629076	0.314538	−	0.949245i	$-0.398150\pi$
$491$	6844.25	0.629076	0.314538	+	0.949245i	$0.398150\pi$
$492$	0	0
$493$	−834.400	−0.0762261
$494$	2958.67	0.269467
$495$	0	0
$496$	−816.361	−0.0739026
$497$	18301.2	1.65175
$498$	0	0
$499$	−18603.4	−1.66895	−0.834473	−	0.551049i	$-0.814228\pi$
$499$	−18603.4	−1.66895	−0.834473	+	0.551049i	$0.814228\pi$
$500$	11670.3	1.04382
$501$	0	0
$502$	9265.47	0.823781
$503$	−10624.8	−0.941822	−0.470911	−	0.882181i	$-0.656075\pi$
$503$	−10624.8	−0.941822	−0.470911	+	0.882181i	$0.656075\pi$
$504$	0	0
$505$	−1583.03	−0.139493
$506$	17088.6	1.50134
$507$	0	0
$508$	13786.4	1.20408
$509$	7931.22	0.690658	0.345329	−	0.938482i	$-0.387767\pi$
$509$	7931.22	0.690658	0.345329	+	0.938482i	$0.387767\pi$
$510$	0	0
$511$	−22135.4	−1.91627
$512$	32981.1	2.84682
$513$	0	0
$514$	−19748.4	−1.69468
$515$	−4353.10	−0.372467
$516$	0	0
$517$	16871.0	1.43517
$518$	16854.4	1.42961
$519$	0	0
$520$	−1177.45	−0.0992974
$521$	−13771.1	−1.15801	−0.579005	−	0.815324i	$-0.696559\pi$
$521$	−13771.1	−1.15801	−0.579005	+	0.815324i	$0.696559\pi$
$522$	0	0
$523$	20397.1	1.70536	0.852680	−	0.522433i	$-0.174976\pi$
$523$	20397.1	1.70536	0.852680	+	0.522433i	$0.174976\pi$
$524$	33386.8	2.78342
$525$	0	0
$526$	12758.4	1.05759
$527$	93.5761	0.00773480
$528$	0	0
$529$	−8599.91	−0.706823
$530$	4412.55	0.361640
$531$	0	0
$532$	34527.3	2.81381
$533$	650.501	0.0528636
$534$	0	0
$535$	3416.18	0.276064
$536$	−33344.9	−2.68709
$537$	0	0
$538$	−12997.7	−1.04158
$539$	−3718.85	−0.297184
$540$	0	0
$541$	1883.72	0.149699	0.0748496	−	0.997195i	$-0.476152\pi$
$541$	1883.72	0.149699	0.0748496	+	0.997195i	$0.476152\pi$
$542$	9519.92	0.754457
$543$	0	0
$544$	−21732.8	−1.71284
$545$	492.110	0.0386783
$546$	0	0
$547$	−9079.06	−0.709676	−0.354838	−	0.934928i	$-0.615464\pi$
$547$	−9079.06	−0.709676	−0.354838	+	0.934928i	$0.615464\pi$
$548$	652.079	0.0508311
$549$	0	0
$550$	34451.4	2.67093
$551$	2216.20	0.171349
$552$	0	0
$553$	−16245.5	−1.24924
$554$	−3783.86	−0.290182
$555$	0	0
$556$	28021.8	2.13739
$557$	5982.38	0.455084	0.227542	−	0.973768i	$-0.426931\pi$
$557$	5982.38	0.455084	0.227542	+	0.973768i	$0.426931\pi$
$558$	0	0
$559$	−709.530	−0.0536850
$560$	−10949.0	−0.826215
$561$	0	0
$562$	−12223.0	−0.917432
$563$	7837.88	0.586727	0.293363	−	0.956001i	$-0.405225\pi$
$563$	7837.88	0.586727	0.293363	+	0.956001i	$0.405225\pi$
$564$	0	0
$565$	−3395.60	−0.252839
$566$	27680.6	2.05566
$567$	0	0
$568$	70115.4	5.17954
$569$	−10019.2	−0.738182	−0.369091	−	0.929393i	$-0.620331\pi$
$569$	−10019.2	−0.738182	−0.369091	+	0.929393i	$0.620331\pi$
$570$	0	0
$571$	8832.38	0.647327	0.323663	−	0.946172i	$-0.395085\pi$
$571$	8832.38	0.647327	0.323663	+	0.946172i	$0.395085\pi$
$572$	−8142.30	−0.595186
$573$	0	0
$574$	10327.6	0.750986
$575$	7191.43	0.521571
$576$	0	0
$577$	−17583.3	−1.26863	−0.634317	−	0.773073i	$-0.718719\pi$
$577$	−17583.3	−1.26863	−0.634317	+	0.773073i	$0.718719\pi$
$578$	−22447.4	−1.61538
$579$	0	0
$580$	−1379.08	−0.0987297
$581$	8890.30	0.634823
$582$	0	0
$583$	19514.6	1.38630
$584$	−84805.2	−6.00901
$585$	0	0
$586$	37151.0	2.61893
$587$	10120.7	0.711630	0.355815	−	0.934556i	$-0.384203\pi$
$587$	10120.7	0.711630	0.355815	+	0.934556i	$0.384203\pi$
$588$	0	0
$589$	−248.543	−0.0173871
$590$	5764.11	0.402211
$591$	0	0
$592$	37820.6	2.62570
$593$	−3597.06	−0.249095	−0.124548	−	0.992214i	$-0.539748\pi$
$593$	−3597.06	−0.249095	−0.124548	+	0.992214i	$0.539748\pi$
$594$	0	0
$595$	1255.04	0.0864734
$596$	−18343.7	−1.26072
$597$	0	0
$598$	−2312.29	−0.158121
$599$	7987.90	0.544870	0.272435	−	0.962174i	$-0.412171\pi$
$599$	7987.90	0.544870	0.272435	+	0.962174i	$0.412171\pi$
$600$	0	0
$601$	2646.60	0.179629	0.0898147	−	0.995958i	$-0.471373\pi$
$601$	2646.60	0.179629	0.0898147	+	0.995958i	$0.471373\pi$
$602$	−11264.8	−0.762655
$603$	0	0
$604$	−58414.5	−3.93518
$605$	−2957.61	−0.198751
$606$	0	0
$607$	−15181.8	−1.01517	−0.507587	−	0.861600i	$-0.669463\pi$
$607$	−15181.8	−1.01517	−0.507587	+	0.861600i	$0.669463\pi$
$608$	57723.3	3.85031
$609$	0	0
$610$	−2608.01	−0.173107
$611$	−2282.84	−0.151152
$612$	0	0
$613$	−9721.86	−0.640558	−0.320279	−	0.947323i	$-0.603777\pi$
$613$	−9721.86	−0.640558	−0.320279	+	0.947323i	$0.603777\pi$
$614$	48690.1	3.20028
$615$	0	0
$616$	−82673.2	−5.40747
$617$	14150.2	0.923282	0.461641	−	0.887067i	$-0.347261\pi$
$617$	14150.2	0.923282	0.461641	+	0.887067i	$0.347261\pi$
$618$	0	0
$619$	−10804.4	−0.701560	−0.350780	−	0.936458i	$-0.614083\pi$
$619$	−10804.4	−0.701560	−0.350780	+	0.936458i	$0.614083\pi$
$620$	154.661	0.0100183
$621$	0	0
$622$	−22681.6	−1.46214
$623$	−9298.46	−0.597970
$624$	0	0
$625$	13924.4	0.891161
$626$	−29587.3	−1.88905
$627$	0	0
$628$	5205.30	0.330755
$629$	−4335.22	−0.274811
$630$	0	0
$631$	20692.1	1.30545	0.652725	−	0.757595i	$-0.273626\pi$
$631$	20692.1	1.30545	0.652725	+	0.757595i	$0.273626\pi$
$632$	−62239.6	−3.91734
$633$	0	0
$634$	29559.6	1.85168
$635$	−1331.01	−0.0831805
$636$	0	0
$637$	503.204	0.0312993
$638$	−8297.49	−0.514891
$639$	0	0
$640$	−12276.6	−0.758239
$641$	1995.59	0.122966	0.0614828	−	0.998108i	$-0.480417\pi$
$641$	1995.59	0.122966	0.0614828	+	0.998108i	$0.480417\pi$
$642$	0	0
$643$	10911.2	0.669203	0.334602	−	0.942360i	$-0.391398\pi$
$643$	10911.2	0.669203	0.334602	+	0.942360i	$0.391398\pi$
$644$	−26984.0	−1.65112
$645$	0	0
$646$	−12082.2	−0.735864
$647$	−20046.3	−1.21808	−0.609042	−	0.793138i	$-0.708446\pi$
$647$	−20046.3	−1.21808	−0.609042	+	0.793138i	$0.708446\pi$
$648$	0	0
$649$	25491.9	1.54183
$650$	−4661.69	−0.281302
$651$	0	0
$652$	−36636.9	−2.20063
$653$	−20205.6	−1.21088	−0.605441	−	0.795890i	$-0.707003\pi$
$653$	−20205.6	−1.21088	−0.605441	+	0.795890i	$0.707003\pi$
$654$	0	0
$655$	−3223.35	−0.192285
$656$	23174.7	1.37930
$657$	0	0
$658$	−36243.3	−2.14728
$659$	9268.73	0.547888	0.273944	−	0.961746i	$-0.411672\pi$
$659$	9268.73	0.547888	0.273944	+	0.961746i	$0.411672\pi$
$660$	0	0
$661$	−18209.9	−1.07153	−0.535765	−	0.844367i	$-0.679977\pi$
$661$	−18209.9	−1.07153	−0.535765	+	0.844367i	$0.679977\pi$
$662$	32008.1	1.87920
$663$	0	0
$664$	34060.5	1.99067
$665$	−3333.45	−0.194384
$666$	0	0
$667$	−1732.03	−0.100546
$668$	81767.8	4.73607
$669$	0	0
$670$	5033.79	0.290257
$671$	−11534.0	−0.663583
$672$	0	0
$673$	−63.9154	−0.00366086	−0.00183043	−	0.999998i	$-0.500583\pi$
$673$	−63.9154	−0.00366086	−0.00183043	+	0.999998i	$0.500583\pi$
$674$	−19931.3	−1.13906
$675$	0	0
$676$	−47657.8	−2.71153
$677$	−11436.2	−0.649231	−0.324616	−	0.945846i	$-0.605235\pi$
$677$	−11436.2	−0.649231	−0.324616	+	0.945846i	$0.605235\pi$
$678$	0	0
$679$	16361.5	0.924735
$680$	4808.31	0.271162
$681$	0	0
$682$	930.545	0.0522470
$683$	22719.5	1.27282	0.636410	−	0.771351i	$-0.280419\pi$
$683$	22719.5	1.27282	0.636410	+	0.771351i	$0.280419\pi$
$684$	0	0
$685$	−62.9553	−0.00351153
$686$	−30379.2	−1.69079
$687$	0	0
$688$	−25277.7	−1.40073
$689$	−2640.56	−0.146005
$690$	0	0
$691$	10495.0	0.577783	0.288891	−	0.957362i	$-0.406713\pi$
$691$	10495.0	0.577783	0.288891	+	0.957362i	$0.406713\pi$
$692$	68214.1	3.74727
$693$	0	0
$694$	−6443.76	−0.352452
$695$	−2705.38	−0.147656
$696$	0	0
$697$	−2656.42	−0.144360
$698$	−951.043	−0.0515723
$699$	0	0
$700$	−54401.2	−2.93739
$701$	6530.49	0.351859	0.175930	−	0.984403i	$-0.443707\pi$
$701$	6530.49	0.351859	0.175930	+	0.984403i	$0.443707\pi$
$702$	0	0
$703$	11514.5	0.617751
$704$	−111555.	−5.97212
$705$	0	0
$706$	−71479.4	−3.81043
$707$	15040.0	0.800052
$708$	0	0
$709$	−16597.7	−0.879183	−0.439591	−	0.898198i	$-0.644877\pi$
$709$	−16597.7	−0.879183	−0.439591	+	0.898198i	$0.644877\pi$
$710$	−10584.7	−0.559489
$711$	0	0
$712$	−35624.2	−1.87510
$713$	194.243	0.0102026
$714$	0	0
$715$	786.102	0.0411168
$716$	−66152.8	−3.45286
$717$	0	0
$718$	24657.7	1.28164
$719$	−6525.25	−0.338457	−0.169229	−	0.985577i	$-0.554128\pi$
$719$	−6525.25	−0.338457	−0.169229	+	0.985577i	$0.554128\pi$
$720$	0	0
$721$	41357.8	2.13626
$722$	−5598.48	−0.288579
$723$	0	0
$724$	69800.8	3.58305
$725$	−3491.86	−0.178875
$726$	0	0
$727$	1066.84	0.0544249	0.0272125	−	0.999630i	$-0.491337\pi$
$727$	1066.84	0.0544249	0.0272125	+	0.999630i	$0.491337\pi$
$728$	11186.7	0.569514
$729$	0	0
$730$	12802.3	0.649088
$731$	2897.48	0.146603
$732$	0	0
$733$	−6226.75	−0.313765	−0.156883	−	0.987617i	$-0.550144\pi$
$733$	−6226.75	−0.313765	−0.156883	+	0.987617i	$0.550144\pi$
$734$	9199.02	0.462592
$735$	0	0
$736$	−45112.4	−2.25933
$737$	22262.1	1.11266
$738$	0	0
$739$	31226.3	1.55437	0.777183	−	0.629274i	$-0.216648\pi$
$739$	31226.3	1.55437	0.777183	+	0.629274i	$0.216648\pi$
$740$	−7165.17	−0.355942
$741$	0	0
$742$	−41922.6	−2.07416
$743$	30790.4	1.52031	0.760156	−	0.649741i	$-0.225123\pi$
$743$	30790.4	1.52031	0.760156	+	0.649741i	$0.225123\pi$
$744$	0	0
$745$	1771.00	0.0870932
$746$	−37370.1	−1.83407
$747$	0	0
$748$	33250.4	1.62534
$749$	−32456.3	−1.58335
$750$	0	0
$751$	11908.0	0.578598	0.289299	−	0.957239i	$-0.406578\pi$
$751$	11908.0	0.578598	0.289299	+	0.957239i	$0.406578\pi$
$752$	−81328.5	−3.94381
$753$	0	0
$754$	1122.75	0.0542283
$755$	5639.65	0.271852
$756$	0	0
$757$	−7197.74	−0.345583	−0.172792	−	0.984958i	$-0.555279\pi$
$757$	−7197.74	−0.345583	−0.172792	+	0.984958i	$0.555279\pi$
$758$	56139.3	2.69007
$759$	0	0
$760$	−12771.1	−0.609548
$761$	−9185.20	−0.437534	−0.218767	−	0.975777i	$-0.570203\pi$
$761$	−9185.20	−0.437534	−0.218767	+	0.975777i	$0.570203\pi$
$762$	0	0
$763$	−4675.42	−0.221837
$764$	47174.1	2.23390
$765$	0	0
$766$	39477.8	1.86213
$767$	−3449.36	−0.162385
$768$	0	0
$769$	32791.1	1.53768	0.768840	−	0.639441i	$-0.220834\pi$
$769$	32791.1	1.53768	0.768840	+	0.639441i	$0.220834\pi$
$770$	12480.5	0.584110
$771$	0	0
$772$	68474.0	3.19227
$773$	28237.1	1.31386	0.656932	−	0.753950i	$-0.271854\pi$
$773$	28237.1	1.31386	0.656932	+	0.753950i	$0.271854\pi$
$774$	0	0
$775$	391.604	0.0181508
$776$	62683.9	2.89977
$777$	0	0
$778$	21148.0	0.974540
$779$	7055.58	0.324509
$780$	0	0
$781$	−46811.2	−2.14473
$782$	9442.59	0.431798
$783$	0	0
$784$	17927.1	0.816652
$785$	−502.548	−0.0228493
$786$	0	0
$787$	12082.3	0.547254	0.273627	−	0.961836i	$-0.411777\pi$
$787$	12082.3	0.547254	0.273627	+	0.961836i	$0.411777\pi$
$788$	55139.1	2.49270
$789$	0	0
$790$	9395.77	0.423148
$791$	32260.8	1.45014
$792$	0	0
$793$	1560.68	0.0698884
$794$	−8915.52	−0.398489
$795$	0	0
$796$	121834.	5.42499
$797$	1248.33	0.0554807	0.0277404	−	0.999615i	$-0.491169\pi$
$797$	1248.33	0.0554807	0.0277404	+	0.999615i	$0.491169\pi$
$798$	0	0
$799$	9322.35	0.412767
$800$	−90949.0	−4.01942
$801$	0	0
$802$	−76065.1	−3.34907
$803$	56618.5	2.48820
$804$	0	0
$805$	2605.19	0.114063
$806$	−125.914	−0.00550264
$807$	0	0
$808$	57621.1	2.50879
$809$	23512.6	1.02183	0.510914	−	0.859632i	$-0.329307\pi$
$809$	23512.6	1.02183	0.510914	+	0.859632i	$0.329307\pi$
$810$	0	0
$811$	23248.4	1.00661	0.503306	−	0.864108i	$-0.332117\pi$
$811$	23248.4	1.00661	0.503306	+	0.864108i	$0.332117\pi$
$812$	13102.3	0.566258
$813$	0	0
$814$	−43110.5	−1.85629
$815$	3537.13	0.152025
$816$	0	0
$817$	−7695.84	−0.329551
$818$	18447.6	0.788516
$819$	0	0
$820$	−4390.49	−0.186979
$821$	−26898.0	−1.14342	−0.571708	−	0.820457i	$-0.693719\pi$
$821$	−26898.0	−1.14342	−0.571708	+	0.820457i	$0.693719\pi$
$822$	0	0
$823$	12422.7	0.526158	0.263079	−	0.964774i	$-0.415262\pi$
$823$	12422.7	0.526158	0.263079	+	0.964774i	$0.415262\pi$
$824$	158450.	6.69886
$825$	0	0
$826$	−54763.4	−2.30686
$827$	4915.14	0.206670	0.103335	−	0.994647i	$-0.467049\pi$
$827$	4915.14	0.206670	0.103335	+	0.994647i	$0.467049\pi$
$828$	0	0
$829$	−45226.4	−1.89478	−0.947392	−	0.320074i	$-0.896292\pi$
$829$	−45226.4	−1.89478	−0.947392	+	0.320074i	$0.896292\pi$
$830$	−5141.81	−0.215030
$831$	0	0
$832$	15094.7	0.628983
$833$	−2054.92	−0.0854725
$834$	0	0
$835$	−7894.31	−0.327178
$836$	−88314.5	−3.65362
$837$	0	0
$838$	−9078.13	−0.374223
$839$	−3982.56	−0.163877	−0.0819387	−	0.996637i	$-0.526111\pi$
$839$	−3982.56	−0.163877	−0.0819387	+	0.996637i	$0.526111\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	−35195.5	−1.44052
$843$	0	0
$844$	−41827.3	−1.70587
$845$	4601.15	0.187319
$846$	0	0
$847$	28099.6	1.13992
$848$	−94072.5	−3.80951
$849$	0	0
$850$	19036.7	0.768182
$851$	−8998.94	−0.362491
$852$	0	0
$853$	−37142.6	−1.49090	−0.745450	−	0.666562i	$-0.767765\pi$
$853$	−37142.6	−1.49090	−0.745450	+	0.666562i	$0.767765\pi$
$854$	24778.0	0.992842
$855$	0	0
$856$	−124346.	−4.96504
$857$	−18690.5	−0.744990	−0.372495	−	0.928034i	$-0.621498\pi$
$857$	−18690.5	−0.744990	−0.372495	+	0.928034i	$0.621498\pi$
$858$	0	0
$859$	−22852.1	−0.907687	−0.453843	−	0.891081i	$-0.649947\pi$
$859$	−22852.1	−0.907687	−0.453843	+	0.891081i	$0.649947\pi$
$860$	4788.90	0.189884
$861$	0	0
$862$	−1387.65	−0.0548303
$863$	−41976.4	−1.65573	−0.827865	−	0.560928i	$-0.810445\pi$
$863$	−41976.4	−1.65573	−0.827865	+	0.560928i	$0.810445\pi$
$864$	0	0
$865$	−6585.77	−0.258870
$866$	−70219.9	−2.75539
$867$	0	0
$868$	−1469.40	−0.0574592
$869$	41553.0	1.62208
$870$	0	0
$871$	−3012.32	−0.117186
$872$	−17912.4	−0.695633
$873$	0	0
$874$	−25080.0	−0.970644
$875$	10704.6	0.413581
$876$	0	0
$877$	44394.0	1.70932	0.854662	−	0.519184i	$-0.173764\pi$
$877$	44394.0	1.70932	0.854662	+	0.519184i	$0.173764\pi$
$878$	−20665.4	−0.794331
$879$	0	0
$880$	28005.6	1.07281
$881$	6337.13	0.242342	0.121171	−	0.992632i	$-0.461335\pi$
$881$	6337.13	0.242342	0.121171	+	0.992632i	$0.461335\pi$
$882$	0	0
$883$	−2834.24	−0.108018	−0.0540090	−	0.998540i	$-0.517200\pi$
$883$	−2834.24	−0.108018	−0.0540090	+	0.998540i	$0.517200\pi$
$884$	−4499.17	−0.171181
$885$	0	0
$886$	−20406.7	−0.773788
$887$	−76.3532	−0.00289029	−0.00144515	−	0.999999i	$-0.500460\pi$
$887$	−76.3532	−0.00289029	−0.00144515	+	0.999999i	$0.500460\pi$
$888$	0	0
$889$	12645.6	0.477077
$890$	5377.88	0.202547
$891$	0	0
$892$	−115596.	−4.33905
$893$	−24760.6	−0.927863
$894$	0	0
$895$	6386.75	0.238531
$896$	116637.	4.34883
$897$	0	0
$898$	66179.7	2.45929
$899$	−94.3163	−0.00349903
$900$	0	0
$901$	10783.2	0.398711
$902$	−26416.2	−0.975124
$903$	0	0
$904$	123597.	4.54733
$905$	−6738.95	−0.247525
$906$	0	0
$907$	18209.7	0.666641	0.333320	−	0.942814i	$-0.391831\pi$
$907$	18209.7	0.666641	0.333320	+	0.942814i	$0.391831\pi$
$908$	116368.	4.25309
$909$	0	0
$910$	−1688.76	−0.0615184
$911$	−33348.8	−1.21284	−0.606419	−	0.795145i	$-0.707395\pi$
$911$	−33348.8	−1.21284	−0.606419	+	0.795145i	$0.707395\pi$
$912$	0	0
$913$	−22739.8	−0.824291
$914$	−71409.8	−2.58427
$915$	0	0
$916$	−18783.6	−0.677541
$917$	30624.2	1.10284
$918$	0	0
$919$	20190.6	0.724730	0.362365	−	0.932036i	$-0.381969\pi$
$919$	20190.6	0.724730	0.362365	+	0.932036i	$0.381969\pi$
$920$	9980.98	0.357677
$921$	0	0
$922$	−43268.4	−1.54552
$923$	6334.11	0.225883
$924$	0	0
$925$	−18142.3	−0.644882
$926$	−19045.7	−0.675897
$927$	0	0
$928$	21904.7	0.774846
$929$	−9089.58	−0.321011	−0.160506	−	0.987035i	$-0.551312\pi$
$929$	−9089.58	−0.321011	−0.160506	+	0.987035i	$0.551312\pi$
$930$	0	0
$931$	5457.95	0.192134
$932$	−18279.2	−0.642440
$933$	0	0
$934$	−81521.3	−2.85595
$935$	−3210.17	−0.112282
$936$	0	0
$937$	39646.5	1.38228	0.691140	−	0.722721i	$-0.257109\pi$
$937$	39646.5	1.38228	0.691140	+	0.722721i	$0.257109\pi$
$938$	−47824.8	−1.66475
$939$	0	0
$940$	15407.8	0.534625
$941$	51013.4	1.76726	0.883629	−	0.468188i	$-0.155093\pi$
$941$	51013.4	1.76726	0.883629	+	0.468188i	$0.155093\pi$
$942$	0	0
$943$	−5514.14	−0.190419
$944$	−122887.	−4.23689
$945$	0	0
$946$	28813.3	0.990276
$947$	−43867.1	−1.50527	−0.752634	−	0.658439i	$-0.771217\pi$
$947$	−43867.1	−1.50527	−0.752634	+	0.658439i	$0.771217\pi$
$948$	0	0
$949$	−7661.16	−0.262057
$950$	−50562.5	−1.72680
$951$	0	0
$952$	−45682.6	−1.55523
$953$	4167.10	0.141643	0.0708214	−	0.997489i	$-0.477438\pi$
$953$	4167.10	0.141643	0.0708214	+	0.997489i	$0.477438\pi$
$954$	0	0
$955$	−4554.44	−0.154323
$956$	54851.7	1.85568
$957$	0	0
$958$	−44617.6	−1.50473
$959$	598.123	0.0201401
$960$	0	0
$961$	−29780.4	−0.999645
$962$	5833.37	0.195505
$963$	0	0
$964$	−25496.0	−0.851836
$965$	−6610.85	−0.220529
$966$	0	0
$967$	−21935.0	−0.729454	−0.364727	−	0.931115i	$-0.618838\pi$
$967$	−21935.0	−0.729454	−0.364727	+	0.931115i	$0.618838\pi$
$968$	107655.	3.57455
$969$	0	0
$970$	−9462.85	−0.313231
$971$	24169.6	0.798805	0.399403	−	0.916776i	$-0.369218\pi$
$971$	24169.6	0.798805	0.399403	+	0.916776i	$0.369218\pi$
$972$	0	0
$973$	25703.2	0.846871
$974$	53294.4	1.75325
$975$	0	0
$976$	55600.9	1.82351
$977$	−16106.2	−0.527415	−0.263707	−	0.964603i	$-0.584945\pi$
$977$	−16106.2	−0.527415	−0.263707	+	0.964603i	$0.584945\pi$
$978$	0	0
$979$	23783.8	0.776439
$980$	−3396.33	−0.110706
$981$	0	0
$982$	37608.3	1.22213
$983$	22527.2	0.730933	0.365466	−	0.930825i	$-0.380910\pi$
$983$	22527.2	0.730933	0.365466	+	0.930825i	$0.380910\pi$
$984$	0	0
$985$	−5323.43	−0.172202
$986$	−4584.93	−0.148087
$987$	0	0
$988$	11950.0	0.384798
$989$	6014.52	0.193378
$990$	0	0
$991$	8337.94	0.267269	0.133634	−	0.991031i	$-0.457335\pi$
$991$	8337.94	0.267269	0.133634	+	0.991031i	$0.457335\pi$
$992$	−2456.57	−0.0786251
$993$	0	0
$994$	100563.	3.20891
$995$	−11762.5	−0.374771
$996$	0	0
$997$	−18827.2	−0.598058	−0.299029	−	0.954244i	$-0.596663\pi$
$997$	−18827.2	−0.598058	−0.299029	+	0.954244i	$0.596663\pi$
$998$	−102224.	−3.24232
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.4.a.f.1.5		5
3.2	odd	2		29.4.a.b.1.1	✓	5
12.11	even	2		464.4.a.l.1.5		5
15.14	odd	2		725.4.a.c.1.5		5
21.20	even	2		1421.4.a.e.1.1		5
24.5	odd	2		1856.4.a.y.1.5		5
24.11	even	2		1856.4.a.bb.1.1		5
87.86	odd	2		841.4.a.b.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.4.a.b.1.1	✓	5	3.2	odd	2
261.4.a.f.1.5		5	1.1	even	1	trivial
464.4.a.l.1.5		5	12.11	even	2
725.4.a.c.1.5		5	15.14	odd	2
841.4.a.b.1.5		5	87.86	odd	2
1421.4.a.e.1.1		5	21.20	even	2
1856.4.a.y.1.5		5	24.5	odd	2
1856.4.a.bb.1.1		5	24.11	even	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:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)

\(f(q)\)	\(=\)	\(q+5.49488 q^{2} +22.1937 q^{4} -2.14270 q^{5} +20.3573 q^{7} +77.9928 q^{8} +O(q^{10})\)	\(q+5.49488 q^{2} +22.1937 q^{4} -2.14270 q^{5} +20.3573 q^{7} +77.9928 q^{8} -11.7739 q^{10} -52.0703 q^{11} +7.04574 q^{13} +111.861 q^{14} +251.011 q^{16} -28.7724 q^{17} +76.4208 q^{19} -47.5545 q^{20} -286.120 q^{22} -59.7251 q^{23} -120.409 q^{25} +38.7155 q^{26} +451.804 q^{28} +29.0000 q^{29} -3.25229 q^{31} +755.335 q^{32} -158.101 q^{34} -43.6196 q^{35} +150.673 q^{37} +419.923 q^{38} -167.115 q^{40} +92.3254 q^{41} -100.703 q^{43} -1155.63 q^{44} -328.182 q^{46} -324.003 q^{47} +71.4197 q^{49} -661.632 q^{50} +156.371 q^{52} -374.774 q^{53} +111.571 q^{55} +1587.72 q^{56} +159.352 q^{58} -489.567 q^{59} +221.508 q^{61} -17.8709 q^{62} +2142.39 q^{64} -15.0969 q^{65} -427.538 q^{67} -638.567 q^{68} -239.685 q^{70} +898.999 q^{71} -1087.35 q^{73} +827.929 q^{74} +1696.06 q^{76} -1060.01 q^{77} -798.018 q^{79} -537.842 q^{80} +507.317 q^{82} +436.713 q^{83} +61.6507 q^{85} -553.353 q^{86} -4061.11 q^{88} -456.763 q^{89} +143.432 q^{91} -1325.52 q^{92} -1780.36 q^{94} -163.747 q^{95} +803.714 q^{97} +392.442 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8}+O(q^{10}) \) 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8	\( 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8} - 64 q^{10} - 12 q^{11} + 14 q^{13} + 192 q^{14} + 146 q^{16} - 66 q^{17} + 214 q^{19} - 6 q^{20} - 98 q^{22} - 164 q^{23} + 207 q^{25} - 56 q^{26} + 540 q^{28} + 145 q^{29} + 420 q^{31} + 652 q^{32} + 204 q^{34} + 52 q^{35} + 378 q^{37} + 496 q^{38} - 80 q^{40} + 1158 q^{41} - 204 q^{43} - 784 q^{44} + 580 q^{46} - 248 q^{47} - 283 q^{49} - 908 q^{50} + 1482 q^{52} + 554 q^{53} + 546 q^{55} + 608 q^{56} - 440 q^{59} + 618 q^{61} - 1250 q^{62} + 2594 q^{64} + 1656 q^{65} + 1164 q^{67} - 356 q^{68} - 2184 q^{70} + 692 q^{71} - 1950 q^{73} + 1832 q^{74} + 1376 q^{76} + 1616 q^{77} + 272 q^{79} + 890 q^{80} + 92 q^{82} - 512 q^{83} - 1628 q^{85} - 2446 q^{86} - 6954 q^{88} - 866 q^{89} + 2580 q^{91} - 3468 q^{92} - 5942 q^{94} - 2244 q^{95} + 1562 q^{97} + 3408 q^{98}+O(q^{100}) \) 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8 - 64 * q^10 - 12 * q^11 + 14 * q^13 + 192 * q^14 + 146 * q^16 - 66 * q^17 + 214 * q^19 - 6 * q^20 - 98 * q^22 - 164 * q^23 + 207 * q^25 - 56 * q^26 + 540 * q^28 + 145 * q^29 + 420 * q^31 + 652 * q^32 + 204 * q^34 + 52 * q^35 + 378 * q^37 + 496 * q^38 - 80 * q^40 + 1158 * q^41 - 204 * q^43 - 784 * q^44 + 580 * q^46 - 248 * q^47 - 283 * q^49 - 908 * q^50 + 1482 * q^52 + 554 * q^53 + 546 * q^55 + 608 * q^56 - 440 * q^59 + 618 * q^61 - 1250 * q^62 + 2594 * q^64 + 1656 * q^65 + 1164 * q^67 - 356 * q^68 - 2184 * q^70 + 692 * q^71 - 1950 * q^73 + 1832 * q^74 + 1376 * q^76 + 1616 * q^77 + 272 * q^79 + 890 * q^80 + 92 * q^82 - 512 * q^83 - 1628 * q^85 - 2446 * q^86 - 6954 * q^88 - 866 * q^89 + 2580 * q^91 - 3468 * q^92 - 5942 * q^94 - 2244 * q^95 + 1562 * q^97 + 3408 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more