LMFDB - Embedded newform 414.4.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,4,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, 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("414.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	414.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:	$24.4267907424$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 219x^{2} - 468x + 3240 $ x^4 - 2x^3 - 219x^2 - 468*x + 3240
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.90028$ of defining polynomial
Character	$\chi$	$=$	414.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} +7.28088 q^{5} +35.9873 q^{7} -8.00000 q^{8} -14.5618 q^{10} +41.0688 q^{11} +28.7612 q^{13} -71.9747 q^{14} +16.0000 q^{16} +1.14743 q^{17} +118.845 q^{19} +29.1235 q^{20} -82.1376 q^{22} -23.0000 q^{23} -71.9888 q^{25} -57.5224 q^{26} +143.949 q^{28} -274.559 q^{29} -93.9916 q^{31} -32.0000 q^{32} -2.29487 q^{34} +262.020 q^{35} +15.1419 q^{37} -237.691 q^{38} -58.2471 q^{40} -6.22194 q^{41} +188.731 q^{43} +164.275 q^{44} +46.0000 q^{46} -173.480 q^{47} +952.089 q^{49} +143.978 q^{50} +115.045 q^{52} -463.166 q^{53} +299.017 q^{55} -287.899 q^{56} +549.118 q^{58} -196.095 q^{59} -341.380 q^{61} +187.983 q^{62} +64.0000 q^{64} +209.407 q^{65} +463.772 q^{67} +4.58973 q^{68} -524.039 q^{70} +730.020 q^{71} +389.517 q^{73} -30.2838 q^{74} +475.382 q^{76} +1477.96 q^{77} -568.720 q^{79} +116.494 q^{80} +12.4439 q^{82} +1193.32 q^{83} +8.35433 q^{85} -377.461 q^{86} -328.550 q^{88} -780.180 q^{89} +1035.04 q^{91} -92.0000 q^{92} +346.960 q^{94} +865.300 q^{95} -343.593 q^{97} -1904.18 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 16 q^{4} + 18 q^{7} - 32 q^{8} - 42 q^{11} + 108 q^{13} - 36 q^{14} + 64 q^{16} - 26 q^{17} + 132 q^{19} + 84 q^{22} - 92 q^{23} + 260 q^{25} - 216 q^{26} + 72 q^{28} - 252 q^{29} + 428 q^{31}+ \cdots - 2152 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 16 * q^4 + 18 * q^7 - 32 * q^8 - 42 * q^11 + 108 * q^13 - 36 * q^14 + 64 * q^16 - 26 * q^17 + 132 * q^19 + 84 * q^22 - 92 * q^23 + 260 * q^25 - 216 * q^26 + 72 * q^28 - 252 * q^29 + 428 * q^31 - 128 * q^32 + 52 * q^34 - 348 * q^35 + 278 * q^37 - 264 * q^38 + 4 * q^41 + 216 * q^43 - 168 * q^44 + 184 * q^46 + 4 * q^47 + 1076 * q^49 - 520 * q^50 + 432 * q^52 + 460 * q^53 + 512 * q^55 - 144 * q^56 + 504 * q^58 + 504 * q^59 + 1086 * q^61 - 856 * q^62 + 256 * q^64 + 1420 * q^65 + 1484 * q^67 - 104 * q^68 + 696 * q^70 + 876 * q^71 + 1072 * q^73 - 556 * q^74 + 528 * q^76 + 1580 * q^77 + 1634 * q^79 - 8 * q^82 + 2246 * q^83 + 872 * q^85 - 432 * q^86 + 336 * q^88 + 474 * q^89 + 40 * q^91 - 368 * q^92 - 8 * q^94 + 2780 * q^95 + 1044 * q^97 - 2152 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	7.28088	0.651222	0.325611	−	0.945504i	$-0.394430\pi$
$5$	7.28088	0.651222	0.325611	+	0.945504i	$0.394430\pi$
$6$	0	0
$7$	35.9873	1.94313	0.971567	−	0.236764i	$-0.0760870\pi$
$7$	35.9873	1.94313	0.971567	+	0.236764i	$0.0760870\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−14.5618	−0.460483
$11$	41.0688	1.12570	0.562850	−	0.826559i	$-0.309705\pi$
$11$	41.0688	1.12570	0.562850	+	0.826559i	$0.309705\pi$
$12$	0	0
$13$	28.7612	0.613609	0.306805	−	0.951772i	$-0.400740\pi$
$13$	28.7612	0.613609	0.306805	+	0.951772i	$0.400740\pi$
$14$	−71.9747	−1.37400
$15$	0	0
$16$	16.0000	0.250000
$17$	1.14743	0.0163702	0.00818510	−	0.999967i	$-0.497395\pi$
$17$	1.14743	0.0163702	0.00818510	+	0.999967i	$0.497395\pi$
$18$	0	0
$19$	118.845	1.43500	0.717500	−	0.696558i	$-0.245286\pi$
$19$	118.845	1.43500	0.717500	+	0.696558i	$0.245286\pi$
$20$	29.1235	0.325611
$21$	0	0
$22$	−82.1376	−0.795991
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−71.9888	−0.575910
$26$	−57.5224	−0.433887
$27$	0	0
$28$	143.949	0.971567
$29$	−274.559	−1.75808	−0.879039	−	0.476749i	$-0.841815\pi$
$29$	−274.559	−1.75808	−0.879039	+	0.476749i	$0.841815\pi$
$30$	0	0
$31$	−93.9916	−0.544561	−0.272280	−	0.962218i	$-0.587778\pi$
$31$	−93.9916	−0.544561	−0.272280	+	0.962218i	$0.587778\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−2.29487	−0.0115755
$35$	262.020	1.26541
$36$	0	0
$37$	15.1419	0.0672787	0.0336394	−	0.999434i	$-0.489290\pi$
$37$	15.1419	0.0672787	0.0336394	+	0.999434i	$0.489290\pi$
$38$	−237.691	−1.01470
$39$	0	0
$40$	−58.2471	−0.230242
$41$	−6.22194	−0.0237001	−0.0118501	−	0.999930i	$-0.503772\pi$
$41$	−6.22194	−0.0237001	−0.0118501	+	0.999930i	$0.503772\pi$
$42$	0	0
$43$	188.731	0.669329	0.334664	−	0.942337i	$-0.391377\pi$
$43$	188.731	0.669329	0.334664	+	0.942337i	$0.391377\pi$
$44$	164.275	0.562850
$45$	0	0
$46$	46.0000	0.147442
$47$	−173.480	−0.538398	−0.269199	−	0.963085i	$-0.586759\pi$
$47$	−173.480	−0.538398	−0.269199	+	0.963085i	$0.586759\pi$
$48$	0	0
$49$	952.089	2.77577
$50$	143.978	0.407230
$51$	0	0
$52$	115.045	0.306805
$53$	−463.166	−1.20039	−0.600195	−	0.799854i	$-0.704910\pi$
$53$	−463.166	−1.20039	−0.600195	+	0.799854i	$0.704910\pi$
$54$	0	0
$55$	299.017	0.733081
$56$	−287.899	−0.687002
$57$	0	0
$58$	549.118	1.24315
$59$	−196.095	−0.432703	−0.216351	−	0.976316i	$-0.569416\pi$
$59$	−196.095	−0.432703	−0.216351	+	0.976316i	$0.569416\pi$
$60$	0	0
$61$	−341.380	−0.716545	−0.358273	−	0.933617i	$-0.616634\pi$
$61$	−341.380	−0.716545	−0.358273	+	0.933617i	$0.616634\pi$
$62$	187.983	0.385063
$63$	0	0
$64$	64.0000	0.125000
$65$	209.407	0.399596
$66$	0	0
$67$	463.772	0.845654	0.422827	−	0.906210i	$-0.361038\pi$
$67$	463.772	0.845654	0.422827	+	0.906210i	$0.361038\pi$
$68$	4.58973	0.00818510
$69$	0	0
$70$	−524.039	−0.894781
$71$	730.020	1.22024	0.610122	−	0.792307i	$-0.291120\pi$
$71$	730.020	1.22024	0.610122	+	0.792307i	$0.291120\pi$
$72$	0	0
$73$	389.517	0.624514	0.312257	−	0.949998i	$-0.398915\pi$
$73$	389.517	0.624514	0.312257	+	0.949998i	$0.398915\pi$
$74$	−30.2838	−0.0475732
$75$	0	0
$76$	475.382	0.717500
$77$	1477.96	2.18739
$78$	0	0
$79$	−568.720	−0.809949	−0.404975	−	0.914328i	$-0.632720\pi$
$79$	−568.720	−0.809949	−0.404975	+	0.914328i	$0.632720\pi$
$80$	116.494	0.162805
$81$	0	0
$82$	12.4439	0.0167585
$83$	1193.32	1.57811	0.789057	−	0.614321i	$-0.210570\pi$
$83$	1193.32	1.57811	0.789057	+	0.614321i	$0.210570\pi$
$84$	0	0
$85$	8.35433	0.0106606
$86$	−377.461	−0.473287
$87$	0	0
$88$	−328.550	−0.397995
$89$	−780.180	−0.929201	−0.464601	−	0.885520i	$-0.653802\pi$
$89$	−780.180	−0.929201	−0.464601	+	0.885520i	$0.653802\pi$
$90$	0	0
$91$	1035.04	1.19233
$92$	−92.0000	−0.104257
$93$	0	0
$94$	346.960	0.380705
$95$	865.300	0.934504
$96$	0	0
$97$	−343.593	−0.359655	−0.179828	−	0.983698i	$-0.557554\pi$
$97$	−343.593	−0.359655	−0.179828	+	0.983698i	$0.557554\pi$
$98$	−1904.18	−1.96277
$99$	0	0
$100$	−287.955	−0.287955
$101$	1806.08	1.77933	0.889664	−	0.456616i	$-0.150939\pi$
$101$	1806.08	1.77933	0.889664	+	0.456616i	$0.150939\pi$
$102$	0	0
$103$	−1247.98	−1.19386	−0.596928	−	0.802295i	$-0.703612\pi$
$103$	−1247.98	−1.19386	−0.596928	+	0.802295i	$0.703612\pi$
$104$	−230.090	−0.216944
$105$	0	0
$106$	926.331	0.848804
$107$	492.642	0.445099	0.222549	−	0.974921i	$-0.428562\pi$
$107$	492.642	0.445099	0.222549	+	0.974921i	$0.428562\pi$
$108$	0	0
$109$	374.270	0.328886	0.164443	−	0.986387i	$-0.447417\pi$
$109$	374.270	0.328886	0.164443	+	0.986387i	$0.447417\pi$
$110$	−598.034	−0.518366
$111$	0	0
$112$	575.798	0.485784
$113$	−1598.57	−1.33081	−0.665403	−	0.746484i	$-0.731740\pi$
$113$	−1598.57	−1.33081	−0.665403	+	0.746484i	$0.731740\pi$
$114$	0	0
$115$	−167.460	−0.135789
$116$	−1098.24	−0.879039
$117$	0	0
$118$	392.191	0.305967
$119$	41.2931	0.0318095
$120$	0	0
$121$	355.646	0.267202
$122$	682.761	0.506674
$123$	0	0
$124$	−375.966	−0.272280
$125$	−1434.25	−1.02627
$126$	0	0
$127$	1320.51	0.922650	0.461325	−	0.887231i	$-0.347374\pi$
$127$	1320.51	0.922650	0.461325	+	0.887231i	$0.347374\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−418.814	−0.282557
$131$	534.502	0.356486	0.178243	−	0.983986i	$-0.442959\pi$
$131$	534.502	0.356486	0.178243	+	0.983986i	$0.442959\pi$
$132$	0	0
$133$	4276.93	2.78840
$134$	−927.545	−0.597968
$135$	0	0
$136$	−9.17947	−0.00578774
$137$	2544.76	1.58696	0.793481	−	0.608595i	$-0.208267\pi$
$137$	2544.76	1.58696	0.793481	+	0.608595i	$0.208267\pi$
$138$	0	0
$139$	−1987.49	−1.21278	−0.606392	−	0.795166i	$-0.707384\pi$
$139$	−1987.49	−1.21278	−0.606392	+	0.795166i	$0.707384\pi$
$140$	1048.08	0.632706
$141$	0	0
$142$	−1460.04	−0.862843
$143$	1181.19	0.690741
$144$	0	0
$145$	−1999.03	−1.14490
$146$	−779.034	−0.441598
$147$	0	0
$148$	60.5676	0.0336394
$149$	2755.79	1.51519	0.757596	−	0.652724i	$-0.226374\pi$
$149$	2755.79	1.51519	0.757596	+	0.652724i	$0.226374\pi$
$150$	0	0
$151$	−2507.96	−1.35162	−0.675812	−	0.737074i	$-0.736207\pi$
$151$	−2507.96	−1.35162	−0.675812	+	0.737074i	$0.736207\pi$
$152$	−950.764	−0.507349
$153$	0	0
$154$	−2955.91	−1.54672
$155$	−684.341	−0.354630
$156$	0	0
$157$	−2109.91	−1.07254	−0.536270	−	0.844046i	$-0.680167\pi$
$157$	−2109.91	−1.07254	−0.536270	+	0.844046i	$0.680167\pi$
$158$	1137.44	0.572720
$159$	0	0
$160$	−232.988	−0.115121
$161$	−827.709	−0.405171
$162$	0	0
$163$	−2258.04	−1.08505	−0.542526	−	0.840039i	$-0.682532\pi$
$163$	−2258.04	−1.08505	−0.542526	+	0.840039i	$0.682532\pi$
$164$	−24.8878	−0.0118501
$165$	0	0
$166$	−2386.63	−1.11589
$167$	−4056.70	−1.87974	−0.939872	−	0.341528i	$-0.889056\pi$
$167$	−4056.70	−1.87974	−0.939872	+	0.341528i	$0.889056\pi$
$168$	0	0
$169$	−1369.79	−0.623483
$170$	−16.7087	−0.00753821
$171$	0	0
$172$	754.922	0.334664
$173$	−858.019	−0.377075	−0.188537	−	0.982066i	$-0.560375\pi$
$173$	−858.019	−0.377075	−0.188537	+	0.982066i	$0.560375\pi$
$174$	0	0
$175$	−2590.68	−1.11907
$176$	657.101	0.281425
$177$	0	0
$178$	1560.36	0.657044
$179$	−3306.61	−1.38071	−0.690356	−	0.723469i	$-0.742546\pi$
$179$	−3306.61	−1.38071	−0.690356	+	0.723469i	$0.742546\pi$
$180$	0	0
$181$	−262.105	−0.107636	−0.0538181	−	0.998551i	$-0.517139\pi$
$181$	−262.105	−0.107636	−0.0538181	+	0.998551i	$0.517139\pi$
$182$	−2070.08	−0.843101
$183$	0	0
$184$	184.000	0.0737210
$185$	110.246	0.0438134
$186$	0	0
$187$	47.1237	0.0184279
$188$	−693.921	−0.269199
$189$	0	0
$190$	−1730.60	−0.660794
$191$	4917.62	1.86297	0.931483	−	0.363786i	$-0.118516\pi$
$191$	4917.62	1.86297	0.931483	+	0.363786i	$0.118516\pi$
$192$	0	0
$193$	462.411	0.172462	0.0862308	−	0.996275i	$-0.472518\pi$
$193$	462.411	0.172462	0.0862308	+	0.996275i	$0.472518\pi$
$194$	687.186	0.254315
$195$	0	0
$196$	3808.36	1.38789
$197$	105.944	0.0383156	0.0191578	−	0.999816i	$-0.493902\pi$
$197$	105.944	0.0383156	0.0191578	+	0.999816i	$0.493902\pi$
$198$	0	0
$199$	−687.512	−0.244907	−0.122453	−	0.992474i	$-0.539076\pi$
$199$	−687.512	−0.244907	−0.122453	+	0.992474i	$0.539076\pi$
$200$	575.910	0.203615
$201$	0	0
$202$	−3612.17	−1.25817
$203$	−9880.64	−3.41618
$204$	0	0
$205$	−45.3012	−0.0154340
$206$	2495.96	0.844184
$207$	0	0
$208$	460.179	0.153402
$209$	4880.84	1.61538
$210$	0	0
$211$	4204.41	1.37177	0.685884	−	0.727711i	$-0.259416\pi$
$211$	4204.41	1.37177	0.685884	+	0.727711i	$0.259416\pi$
$212$	−1852.66	−0.600195
$213$	0	0
$214$	−985.285	−0.314732
$215$	1374.12	0.435882
$216$	0	0
$217$	−3382.51	−1.05815
$218$	−748.540	−0.232557
$219$	0	0
$220$	1196.07	0.366540
$221$	33.0016	0.0100449
$222$	0	0
$223$	3439.63	1.03289	0.516445	−	0.856320i	$-0.327255\pi$
$223$	3439.63	1.03289	0.516445	+	0.856320i	$0.327255\pi$
$224$	−1151.60	−0.343501
$225$	0	0
$226$	3197.15	0.941022
$227$	−2906.74	−0.849898	−0.424949	−	0.905217i	$-0.639708\pi$
$227$	−2906.74	−0.849898	−0.424949	+	0.905217i	$0.639708\pi$
$228$	0	0
$229$	−6220.56	−1.79505	−0.897524	−	0.440965i	$-0.854636\pi$
$229$	−6220.56	−1.79505	−0.897524	+	0.440965i	$0.854636\pi$
$230$	334.921	0.0960174
$231$	0	0
$232$	2196.47	0.621575
$233$	5315.68	1.49460	0.747300	−	0.664487i	$-0.231350\pi$
$233$	5315.68	1.49460	0.747300	+	0.664487i	$0.231350\pi$
$234$	0	0
$235$	−1263.09	−0.350616
$236$	−784.382	−0.216351
$237$	0	0
$238$	−82.5862	−0.0224927
$239$	1633.57	0.442120	0.221060	−	0.975260i	$-0.429048\pi$
$239$	1633.57	0.442120	0.221060	+	0.975260i	$0.429048\pi$
$240$	0	0
$241$	30.5253	0.00815896	0.00407948	−	0.999992i	$-0.498701\pi$
$241$	30.5253	0.00815896	0.00407948	+	0.999992i	$0.498701\pi$
$242$	−711.291	−0.188940
$243$	0	0
$244$	−1365.52	−0.358273
$245$	6932.05	1.80764
$246$	0	0
$247$	3418.14	0.880530
$248$	751.932	0.192531
$249$	0	0
$250$	2868.50	0.725680
$251$	−5864.81	−1.47483	−0.737417	−	0.675437i	$-0.763955\pi$
$251$	−5864.81	−1.47483	−0.737417	+	0.675437i	$0.763955\pi$
$252$	0	0
$253$	−944.582	−0.234725
$254$	−2641.02	−0.652412
$255$	0	0
$256$	256.000	0.0625000
$257$	5350.14	1.29857	0.649285	−	0.760545i	$-0.275068\pi$
$257$	5350.14	1.29857	0.649285	+	0.760545i	$0.275068\pi$
$258$	0	0
$259$	544.917	0.130732
$260$	837.628	0.199798
$261$	0	0
$262$	−1069.00	−0.252074
$263$	−4957.27	−1.16228	−0.581138	−	0.813805i	$-0.697392\pi$
$263$	−4957.27	−1.16228	−0.581138	+	0.813805i	$0.697392\pi$
$264$	0	0
$265$	−3372.25	−0.781720
$266$	−8553.87	−1.97170
$267$	0	0
$268$	1855.09	0.422827
$269$	223.436	0.0506437	0.0253218	−	0.999679i	$-0.491939\pi$
$269$	223.436	0.0506437	0.0253218	+	0.999679i	$0.491939\pi$
$270$	0	0
$271$	−5153.96	−1.15528	−0.577640	−	0.816292i	$-0.696026\pi$
$271$	−5153.96	−1.15528	−0.577640	+	0.816292i	$0.696026\pi$
$272$	18.3589	0.00409255
$273$	0	0
$274$	−5089.52	−1.12215
$275$	−2956.49	−0.648302
$276$	0	0
$277$	6182.51	1.34105	0.670526	−	0.741886i	$-0.266069\pi$
$277$	6182.51	1.34105	0.670526	+	0.741886i	$0.266069\pi$
$278$	3974.99	0.857568
$279$	0	0
$280$	−2096.16	−0.447391
$281$	3704.01	0.786345	0.393172	−	0.919465i	$-0.371378\pi$
$281$	3704.01	0.786345	0.393172	+	0.919465i	$0.371378\pi$
$282$	0	0
$283$	1947.10	0.408986	0.204493	−	0.978868i	$-0.434445\pi$
$283$	1947.10	0.408986	0.204493	+	0.978868i	$0.434445\pi$
$284$	2920.08	0.610122
$285$	0	0
$286$	−2362.38	−0.488427
$287$	−223.911	−0.0460525
$288$	0	0
$289$	−4911.68	−0.999732
$290$	3998.06	0.809566
$291$	0	0
$292$	1558.07	0.312257
$293$	52.2109	0.0104102	0.00520510	−	0.999986i	$-0.498343\pi$
$293$	52.2109	0.0104102	0.00520510	+	0.999986i	$0.498343\pi$
$294$	0	0
$295$	−1427.75	−0.281785
$296$	−121.135	−0.0237866
$297$	0	0
$298$	−5511.59	−1.07140
$299$	−661.508	−0.127946
$300$	0	0
$301$	6791.91	1.30060
$302$	5015.93	0.955742
$303$	0	0
$304$	1901.53	0.358750
$305$	−2485.55	−0.466630
$306$	0	0
$307$	9584.44	1.78180	0.890900	−	0.454199i	$-0.150074\pi$
$307$	9584.44	1.78180	0.890900	+	0.454199i	$0.150074\pi$
$308$	5911.83	1.09369
$309$	0	0
$310$	1368.68	0.250761
$311$	9148.29	1.66801	0.834006	−	0.551755i	$-0.186042\pi$
$311$	9148.29	1.66801	0.834006	+	0.551755i	$0.186042\pi$
$312$	0	0
$313$	443.616	0.0801107	0.0400553	−	0.999197i	$-0.487247\pi$
$313$	443.616	0.0801107	0.0400553	+	0.999197i	$0.487247\pi$
$314$	4219.81	0.758401
$315$	0	0
$316$	−2274.88	−0.404975
$317$	−6926.81	−1.22728	−0.613641	−	0.789585i	$-0.710296\pi$
$317$	−6926.81	−1.22728	−0.613641	+	0.789585i	$0.710296\pi$
$318$	0	0
$319$	−11275.8	−1.97907
$320$	465.976	0.0814027
$321$	0	0
$322$	1655.42	0.286500
$323$	136.367	0.0234913
$324$	0	0
$325$	−2070.48	−0.353384
$326$	4516.08	0.767247
$327$	0	0
$328$	49.7756	0.00837925
$329$	−6243.09	−1.04618
$330$	0	0
$331$	677.095	0.112437	0.0562183	−	0.998419i	$-0.482096\pi$
$331$	677.095	0.112437	0.0562183	+	0.998419i	$0.482096\pi$
$332$	4773.26	0.789057
$333$	0	0
$334$	8113.41	1.32918
$335$	3376.67	0.550708
$336$	0	0
$337$	−3143.44	−0.508113	−0.254056	−	0.967189i	$-0.581765\pi$
$337$	−3143.44	−0.508113	−0.254056	+	0.967189i	$0.581765\pi$
$338$	2739.59	0.440869
$339$	0	0
$340$	33.4173	0.00533032
$341$	−3860.12	−0.613012
$342$	0	0
$343$	21919.5	3.45056
$344$	−1509.84	−0.236643
$345$	0	0
$346$	1716.04	0.266632
$347$	−1700.89	−0.263137	−0.131569	−	0.991307i	$-0.542001\pi$
$347$	−1700.89	−0.263137	−0.131569	+	0.991307i	$0.542001\pi$
$348$	0	0
$349$	11022.8	1.69065	0.845324	−	0.534254i	$-0.179407\pi$
$349$	11022.8	1.69065	0.845324	+	0.534254i	$0.179407\pi$
$350$	5181.37	0.791302
$351$	0	0
$352$	−1314.20	−0.198998
$353$	−3717.19	−0.560471	−0.280235	−	0.959931i	$-0.590412\pi$
$353$	−3717.19	−0.560471	−0.280235	+	0.959931i	$0.590412\pi$
$354$	0	0
$355$	5315.19	0.794650
$356$	−3120.72	−0.464601
$357$	0	0
$358$	6613.22	0.976311
$359$	928.351	0.136480	0.0682402	−	0.997669i	$-0.478262\pi$
$359$	928.351	0.136480	0.0682402	+	0.997669i	$0.478262\pi$
$360$	0	0
$361$	7265.24	1.05923
$362$	524.211	0.0761102
$363$	0	0
$364$	4140.16	0.596163
$365$	2836.03	0.406697
$366$	0	0
$367$	−2808.73	−0.399494	−0.199747	−	0.979848i	$-0.564012\pi$
$367$	−2808.73	−0.399494	−0.199747	+	0.979848i	$0.564012\pi$
$368$	−368.000	−0.0521286
$369$	0	0
$370$	−220.493	−0.0309807
$371$	−16668.1	−2.33252
$372$	0	0
$373$	−4512.14	−0.626353	−0.313177	−	0.949695i	$-0.601393\pi$
$373$	−4512.14	−0.626353	−0.313177	+	0.949695i	$0.601393\pi$
$374$	−94.2474	−0.0130305
$375$	0	0
$376$	1387.84	0.190352
$377$	−7896.64	−1.07877
$378$	0	0
$379$	18.9605	0.00256975	0.00128487	−	0.999999i	$-0.499591\pi$
$379$	18.9605	0.00256975	0.00128487	+	0.999999i	$0.499591\pi$
$380$	3461.20	0.467252
$381$	0	0
$382$	−9835.24	−1.31732
$383$	−4582.21	−0.611331	−0.305665	−	0.952139i	$-0.598879\pi$
$383$	−4582.21	−0.611331	−0.305665	+	0.952139i	$0.598879\pi$
$384$	0	0
$385$	10760.8	1.42447
$386$	−924.822	−0.121949
$387$	0	0
$388$	−1374.37	−0.179828
$389$	8916.93	1.16223	0.581114	−	0.813822i	$-0.302617\pi$
$389$	8916.93	1.16223	0.581114	+	0.813822i	$0.302617\pi$
$390$	0	0
$391$	−26.3910	−0.00341342
$392$	−7616.71	−0.981383
$393$	0	0
$394$	−211.887	−0.0270932
$395$	−4140.78	−0.527457
$396$	0	0
$397$	3271.65	0.413600	0.206800	−	0.978383i	$-0.433695\pi$
$397$	3271.65	0.413600	0.206800	+	0.978383i	$0.433695\pi$
$398$	1375.02	0.173175
$399$	0	0
$400$	−1151.82	−0.143978
$401$	4014.90	0.499987	0.249993	−	0.968248i	$-0.419572\pi$
$401$	4014.90	0.499987	0.249993	+	0.968248i	$0.419572\pi$
$402$	0	0
$403$	−2703.31	−0.334148
$404$	7224.34	0.889664
$405$	0	0
$406$	19761.3	2.41561
$407$	621.860	0.0757357
$408$	0	0
$409$	6139.64	0.742263	0.371131	−	0.928580i	$-0.378970\pi$
$409$	6139.64	0.742263	0.371131	+	0.928580i	$0.378970\pi$
$410$	90.6025	0.0109135
$411$	0	0
$412$	−4991.92	−0.596928
$413$	−7056.95	−0.840799
$414$	0	0
$415$	8688.39	1.02770
$416$	−920.359	−0.108472
$417$	0	0
$418$	−9761.68	−1.14225
$419$	1194.52	0.139275	0.0696376	−	0.997572i	$-0.477816\pi$
$419$	1194.52	0.139275	0.0696376	+	0.997572i	$0.477816\pi$
$420$	0	0
$421$	−9138.18	−1.05788	−0.528940	−	0.848659i	$-0.677410\pi$
$421$	−9138.18	−1.05788	−0.528940	+	0.848659i	$0.677410\pi$
$422$	−8408.81	−0.969987
$423$	0	0
$424$	3705.32	0.424402
$425$	−82.6023	−0.00942776
$426$	0	0
$427$	−12285.4	−1.39234
$428$	1970.57	0.222549
$429$	0	0
$430$	−2748.25	−0.308215
$431$	−4775.85	−0.533746	−0.266873	−	0.963732i	$-0.585990\pi$
$431$	−4775.85	−0.533746	−0.266873	+	0.963732i	$0.585990\pi$
$432$	0	0
$433$	−6005.84	−0.666564	−0.333282	−	0.942827i	$-0.608156\pi$
$433$	−6005.84	−0.666564	−0.333282	+	0.942827i	$0.608156\pi$
$434$	6765.01	0.748228
$435$	0	0
$436$	1497.08	0.164443
$437$	−2733.45	−0.299218
$438$	0	0
$439$	16077.0	1.74786	0.873932	−	0.486048i	$-0.161562\pi$
$439$	16077.0	1.74786	0.873932	+	0.486048i	$0.161562\pi$
$440$	−2392.14	−0.259183
$441$	0	0
$442$	−66.0031	−0.00710283
$443$	−1018.10	−0.109190	−0.0545952	−	0.998509i	$-0.517387\pi$
$443$	−1018.10	−0.109190	−0.0545952	+	0.998509i	$0.517387\pi$
$444$	0	0
$445$	−5680.40	−0.605116
$446$	−6879.26	−0.730364
$447$	0	0
$448$	2303.19	0.242892
$449$	−6465.49	−0.679567	−0.339783	−	0.940504i	$-0.610354\pi$
$449$	−6465.49	−0.679567	−0.339783	+	0.940504i	$0.610354\pi$
$450$	0	0
$451$	−255.528	−0.0266792
$452$	−6394.29	−0.665403
$453$	0	0
$454$	5813.47	0.600969
$455$	7536.00	0.776469
$456$	0	0
$457$	−6983.59	−0.714833	−0.357416	−	0.933945i	$-0.616342\pi$
$457$	−6983.59	−0.714833	−0.357416	+	0.933945i	$0.616342\pi$
$458$	12441.1	1.26929
$459$	0	0
$460$	−669.841	−0.0678946
$461$	−8285.05	−0.837035	−0.418517	−	0.908209i	$-0.637450\pi$
$461$	−8285.05	−0.837035	−0.418517	+	0.908209i	$0.637450\pi$
$462$	0	0
$463$	−17934.5	−1.80019	−0.900093	−	0.435697i	$-0.856502\pi$
$463$	−17934.5	−1.80019	−0.900093	+	0.435697i	$0.856502\pi$
$464$	−4392.94	−0.439520
$465$	0	0
$466$	−10631.4	−1.05684
$467$	−10559.7	−1.04635	−0.523176	−	0.852225i	$-0.675253\pi$
$467$	−10559.7	−1.04635	−0.523176	+	0.852225i	$0.675253\pi$
$468$	0	0
$469$	16689.9	1.64322
$470$	2526.18	0.247923
$471$	0	0
$472$	1568.76	0.152983
$473$	7750.93	0.753464
$474$	0	0
$475$	−8555.54	−0.826431
$476$	165.172	0.0159048
$477$	0	0
$478$	−3267.14	−0.312626
$479$	6175.49	0.589071	0.294536	−	0.955641i	$-0.404835\pi$
$479$	6175.49	0.589071	0.294536	+	0.955641i	$0.404835\pi$
$480$	0	0
$481$	435.499	0.0412829
$482$	−61.0506	−0.00576925
$483$	0	0
$484$	1422.58	0.133601
$485$	−2501.66	−0.234215
$486$	0	0
$487$	−16616.4	−1.54612	−0.773062	−	0.634331i	$-0.781276\pi$
$487$	−16616.4	−1.54612	−0.773062	+	0.634331i	$0.781276\pi$
$488$	2731.04	0.253337
$489$	0	0
$490$	−13864.1	−1.27820
$491$	−1923.41	−0.176787	−0.0883936	−	0.996086i	$-0.528173\pi$
$491$	−1923.41	−0.176787	−0.0883936	+	0.996086i	$0.528173\pi$
$492$	0	0
$493$	−315.038	−0.0287801
$494$	−6836.28	−0.622629
$495$	0	0
$496$	−1503.86	−0.136140
$497$	26271.5	2.37110
$498$	0	0
$499$	−2001.76	−0.179582	−0.0897908	−	0.995961i	$-0.528620\pi$
$499$	−2001.76	−0.179582	−0.0897908	+	0.995961i	$0.528620\pi$
$500$	−5737.01	−0.513134
$501$	0	0
$502$	11729.6	1.04287
$503$	−12903.3	−1.14380	−0.571899	−	0.820324i	$-0.693793\pi$
$503$	−12903.3	−1.14380	−0.571899	+	0.820324i	$0.693793\pi$
$504$	0	0
$505$	13149.9	1.15874
$506$	1889.16	0.165975
$507$	0	0
$508$	5282.05	0.461325
$509$	2208.97	0.192360	0.0961798	−	0.995364i	$-0.469338\pi$
$509$	2208.97	0.192360	0.0961798	+	0.995364i	$0.469338\pi$
$510$	0	0
$511$	14017.7	1.21351
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−10700.3	−0.918227
$515$	−9086.40	−0.777465
$516$	0	0
$517$	−7124.62	−0.606075
$518$	−1089.83	−0.0924412
$519$	0	0
$520$	−1675.26	−0.141278
$521$	7525.04	0.632779	0.316390	−	0.948629i	$-0.397529\pi$
$521$	7525.04	0.632779	0.316390	+	0.948629i	$0.397529\pi$
$522$	0	0
$523$	−5782.76	−0.483485	−0.241742	−	0.970340i	$-0.577719\pi$
$523$	−5782.76	−0.483485	−0.241742	+	0.970340i	$0.577719\pi$
$524$	2138.01	0.178243
$525$	0	0
$526$	9914.55	0.821853
$527$	−107.849	−0.00891457
$528$	0	0
$529$	529.000	0.0434783
$530$	6744.51	0.552760
$531$	0	0
$532$	17107.7	1.39420
$533$	−178.951	−0.0145426
$534$	0	0
$535$	3586.87	0.289858
$536$	−3710.18	−0.298984
$537$	0	0
$538$	−446.872	−0.0358105
$539$	39101.2	3.12469
$540$	0	0
$541$	13452.7	1.06909	0.534546	−	0.845139i	$-0.320483\pi$
$541$	13452.7	1.06909	0.534546	+	0.845139i	$0.320483\pi$
$542$	10307.9	0.816906
$543$	0	0
$544$	−36.7179	−0.00289387
$545$	2725.02	0.214178
$546$	0	0
$547$	−9628.07	−0.752589	−0.376295	−	0.926500i	$-0.622802\pi$
$547$	−9628.07	−0.752589	−0.376295	+	0.926500i	$0.622802\pi$
$548$	10179.0	0.793481
$549$	0	0
$550$	5912.98	0.458419
$551$	−32630.1	−2.52284
$552$	0	0
$553$	−20466.7	−1.57384
$554$	−12365.0	−0.948266
$555$	0	0
$556$	−7949.98	−0.606392
$557$	−10299.2	−0.783466	−0.391733	−	0.920079i	$-0.628124\pi$
$557$	−10299.2	−0.783466	−0.391733	+	0.920079i	$0.628124\pi$
$558$	0	0
$559$	5428.12	0.410706
$560$	4192.31	0.316353
$561$	0	0
$562$	−7408.02	−0.556030
$563$	−16390.8	−1.22698	−0.613492	−	0.789701i	$-0.710236\pi$
$563$	−16390.8	−1.22698	−0.613492	+	0.789701i	$0.710236\pi$
$564$	0	0
$565$	−11639.0	−0.866650
$566$	−3894.20	−0.289197
$567$	0	0
$568$	−5840.16	−0.431422
$569$	18730.9	1.38003	0.690017	−	0.723793i	$-0.257603\pi$
$569$	18730.9	1.38003	0.690017	+	0.723793i	$0.257603\pi$
$570$	0	0
$571$	−23266.5	−1.70520	−0.852602	−	0.522561i	$-0.824977\pi$
$571$	−23266.5	−1.70520	−0.852602	+	0.522561i	$0.824977\pi$
$572$	4724.75	0.345370
$573$	0	0
$574$	447.823	0.0325640
$575$	1655.74	0.120086
$576$	0	0
$577$	−22221.0	−1.60324	−0.801622	−	0.597831i	$-0.796029\pi$
$577$	−22221.0	−1.60324	−0.801622	+	0.597831i	$0.796029\pi$
$578$	9823.37	0.706917
$579$	0	0
$580$	−7996.12	−0.572450
$581$	42944.3	3.06649
$582$	0	0
$583$	−19021.7	−1.35128
$584$	−3116.13	−0.220799
$585$	0	0
$586$	−104.422	−0.00736113
$587$	−19974.9	−1.40452	−0.702261	−	0.711920i	$-0.747826\pi$
$587$	−19974.9	−1.40452	−0.702261	+	0.711920i	$0.747826\pi$
$588$	0	0
$589$	−11170.5	−0.781445
$590$	2855.50	0.199252
$591$	0	0
$592$	242.270	0.0168197
$593$	−13138.3	−0.909827	−0.454913	−	0.890536i	$-0.650330\pi$
$593$	−13138.3	−0.909827	−0.454913	+	0.890536i	$0.650330\pi$
$594$	0	0
$595$	300.650	0.0207150
$596$	11023.2	0.757596
$597$	0	0
$598$	1323.02	0.0904718
$599$	−4445.05	−0.303205	−0.151602	−	0.988442i	$-0.548443\pi$
$599$	−4445.05	−0.303205	−0.151602	+	0.988442i	$0.548443\pi$
$600$	0	0
$601$	9950.36	0.675347	0.337674	−	0.941263i	$-0.390360\pi$
$601$	9950.36	0.675347	0.337674	+	0.941263i	$0.390360\pi$
$602$	−13583.8	−0.919660
$603$	0	0
$604$	−10031.9	−0.675812
$605$	2589.41	0.174008
$606$	0	0
$607$	21986.0	1.47015	0.735076	−	0.677985i	$-0.237146\pi$
$607$	21986.0	1.47015	0.735076	+	0.677985i	$0.237146\pi$
$608$	−3803.05	−0.253675
$609$	0	0
$610$	4971.10	0.329957
$611$	−4989.50	−0.330366
$612$	0	0
$613$	−17727.9	−1.16807	−0.584033	−	0.811730i	$-0.698526\pi$
$613$	−17727.9	−1.16807	−0.584033	+	0.811730i	$0.698526\pi$
$614$	−19168.9	−1.25992
$615$	0	0
$616$	−11823.7	−0.773358
$617$	20217.9	1.31919	0.659597	−	0.751619i	$-0.270727\pi$
$617$	20217.9	1.31919	0.659597	+	0.751619i	$0.270727\pi$
$618$	0	0
$619$	26698.5	1.73361	0.866803	−	0.498651i	$-0.166171\pi$
$619$	26698.5	1.73361	0.866803	+	0.498651i	$0.166171\pi$
$620$	−2737.37	−0.177315
$621$	0	0
$622$	−18296.6	−1.17946
$623$	−28076.6	−1.80556
$624$	0	0
$625$	−1444.03	−0.0924177
$626$	−887.231	−0.0566468
$627$	0	0
$628$	−8439.62	−0.536270
$629$	17.3743	0.00110137
$630$	0	0
$631$	−21906.3	−1.38205	−0.691027	−	0.722829i	$-0.742842\pi$
$631$	−21906.3	−1.38205	−0.691027	+	0.722829i	$0.742842\pi$
$632$	4549.76	0.286360
$633$	0	0
$634$	13853.6	0.867819
$635$	9614.49	0.600850
$636$	0	0
$637$	27383.2	1.70324
$638$	22551.6	1.39941
$639$	0	0
$640$	−931.953	−0.0575604
$641$	26702.8	1.64540	0.822698	−	0.568479i	$-0.192468\pi$
$641$	26702.8	1.64540	0.822698	+	0.568479i	$0.192468\pi$
$642$	0	0
$643$	−25413.0	−1.55862	−0.779310	−	0.626639i	$-0.784430\pi$
$643$	−25413.0	−1.55862	−0.779310	+	0.626639i	$0.784430\pi$
$644$	−3310.84	−0.202586
$645$	0	0
$646$	−272.734	−0.0166108
$647$	15981.9	0.971116	0.485558	−	0.874204i	$-0.338616\pi$
$647$	15981.9	0.971116	0.485558	+	0.874204i	$0.338616\pi$
$648$	0	0
$649$	−8053.40	−0.487094
$650$	4140.97	0.249880
$651$	0	0
$652$	−9032.16	−0.542526
$653$	−31104.7	−1.86404	−0.932022	−	0.362401i	$-0.881957\pi$
$653$	−31104.7	−1.86404	−0.932022	+	0.362401i	$0.881957\pi$
$654$	0	0
$655$	3891.65	0.232152
$656$	−99.5511	−0.00592503
$657$	0	0
$658$	12486.2	0.739760
$659$	2540.74	0.150187	0.0750933	−	0.997177i	$-0.476075\pi$
$659$	2540.74	0.150187	0.0750933	+	0.997177i	$0.476075\pi$
$660$	0	0
$661$	11474.7	0.675211	0.337605	−	0.941288i	$-0.390383\pi$
$661$	11474.7	0.675211	0.337605	+	0.941288i	$0.390383\pi$
$662$	−1354.19	−0.0795046
$663$	0	0
$664$	−9546.52	−0.557947
$665$	31139.8	1.81587
$666$	0	0
$667$	6314.85	0.366585
$668$	−16226.8	−0.939872
$669$	0	0
$670$	−6753.34	−0.389410
$671$	−14020.1	−0.806616
$672$	0	0
$673$	−3530.79	−0.202232	−0.101116	−	0.994875i	$-0.532241\pi$
$673$	−3530.79	−0.202232	−0.101116	+	0.994875i	$0.532241\pi$
$674$	6286.88	0.359290
$675$	0	0
$676$	−5479.17	−0.311742
$677$	22802.8	1.29451	0.647254	−	0.762274i	$-0.275917\pi$
$677$	22802.8	1.29451	0.647254	+	0.762274i	$0.275917\pi$
$678$	0	0
$679$	−12365.0	−0.698858
$680$	−66.8346	−0.00376910
$681$	0	0
$682$	7720.24	0.433465
$683$	13774.7	0.771703	0.385852	−	0.922561i	$-0.373908\pi$
$683$	13774.7	0.771703	0.385852	+	0.922561i	$0.373908\pi$
$684$	0	0
$685$	18528.1	1.03346
$686$	−43839.0	−2.43991
$687$	0	0
$688$	3019.69	0.167332
$689$	−13321.2	−0.736571
$690$	0	0
$691$	9020.06	0.496584	0.248292	−	0.968685i	$-0.420131\pi$
$691$	9020.06	0.496584	0.248292	+	0.968685i	$0.420131\pi$
$692$	−3432.07	−0.188537
$693$	0	0
$694$	3401.78	0.186066
$695$	−14470.7	−0.789792
$696$	0	0
$697$	−7.13927	−0.000387976 0
$698$	−22045.6	−1.19547
$699$	0	0
$700$	−10362.7	−0.559535
$701$	−6122.50	−0.329877	−0.164938	−	0.986304i	$-0.552743\pi$
$701$	−6122.50	−0.329877	−0.164938	+	0.986304i	$0.552743\pi$
$702$	0	0
$703$	1799.55	0.0965450
$704$	2628.40	0.140713
$705$	0	0
$706$	7434.38	0.396313
$707$	64996.2	3.45747
$708$	0	0
$709$	18858.6	0.998940	0.499470	−	0.866331i	$-0.333528\pi$
$709$	18858.6	0.998940	0.499470	+	0.866331i	$0.333528\pi$
$710$	−10630.4	−0.561903
$711$	0	0
$712$	6241.44	0.328522
$713$	2161.81	0.113549
$714$	0	0
$715$	8600.09	0.449825
$716$	−13226.4	−0.690356
$717$	0	0
$718$	−1856.70	−0.0965063
$719$	−15517.2	−0.804862	−0.402431	−	0.915450i	$-0.631835\pi$
$719$	−15517.2	−0.804862	−0.402431	+	0.915450i	$0.631835\pi$
$720$	0	0
$721$	−44911.5	−2.31982
$722$	−14530.5	−0.748987
$723$	0	0
$724$	−1048.42	−0.0538181
$725$	19765.1	1.01250
$726$	0	0
$727$	32390.6	1.65241	0.826203	−	0.563372i	$-0.190496\pi$
$727$	32390.6	1.65241	0.826203	+	0.563372i	$0.190496\pi$
$728$	−8280.32	−0.421551
$729$	0	0
$730$	−5672.05	−0.287578
$731$	216.556	0.0109570
$732$	0	0
$733$	−30356.1	−1.52964	−0.764821	−	0.644243i	$-0.777173\pi$
$733$	−30356.1	−1.52964	−0.764821	+	0.644243i	$0.777173\pi$
$734$	5617.45	0.282485
$735$	0	0
$736$	736.000	0.0368605
$737$	19046.6	0.951953
$738$	0	0
$739$	20619.4	1.02638	0.513190	−	0.858275i	$-0.328464\pi$
$739$	20619.4	1.02638	0.513190	+	0.858275i	$0.328464\pi$
$740$	440.986	0.0219067
$741$	0	0
$742$	33336.2	1.64934
$743$	21874.4	1.08007	0.540036	−	0.841642i	$-0.318411\pi$
$743$	21874.4	1.08007	0.540036	+	0.841642i	$0.318411\pi$
$744$	0	0
$745$	20064.6	0.986726
$746$	9024.28	0.442898
$747$	0	0
$748$	188.495	0.00921397
$749$	17728.9	0.864886
$750$	0	0
$751$	−6510.13	−0.316322	−0.158161	−	0.987413i	$-0.550557\pi$
$751$	−6510.13	−0.316322	−0.158161	+	0.987413i	$0.550557\pi$
$752$	−2775.68	−0.134599
$753$	0	0
$754$	15793.3	0.762808
$755$	−18260.2	−0.880207
$756$	0	0
$757$	−9314.75	−0.447226	−0.223613	−	0.974678i	$-0.571785\pi$
$757$	−9314.75	−0.447226	−0.223613	+	0.974678i	$0.571785\pi$
$758$	−37.9209	−0.00181709
$759$	0	0
$760$	−6922.40	−0.330397
$761$	13223.9	0.629917	0.314959	−	0.949105i	$-0.398009\pi$
$761$	13223.9	0.629917	0.314959	+	0.949105i	$0.398009\pi$
$762$	0	0
$763$	13469.0	0.639069
$764$	19670.5	0.931483
$765$	0	0
$766$	9164.41	0.432276
$767$	−5639.94	−0.265510
$768$	0	0
$769$	−6700.61	−0.314214	−0.157107	−	0.987582i	$-0.550217\pi$
$769$	−6700.61	−0.314214	−0.157107	+	0.987582i	$0.550217\pi$
$770$	−21521.7	−1.00726
$771$	0	0
$772$	1849.64	0.0862308
$773$	−17607.4	−0.819266	−0.409633	−	0.912250i	$-0.634343\pi$
$773$	−17607.4	−0.819266	−0.409633	+	0.912250i	$0.634343\pi$
$774$	0	0
$775$	6766.33	0.313618
$776$	2748.74	0.127157
$777$	0	0
$778$	−17833.9	−0.821819
$779$	−739.450	−0.0340097
$780$	0	0
$781$	29981.0	1.37363
$782$	52.7819	0.00241365
$783$	0	0
$784$	15233.4	0.693943
$785$	−15362.0	−0.698462
$786$	0	0
$787$	−40685.3	−1.84279	−0.921393	−	0.388631i	$-0.872948\pi$
$787$	−40685.3	−1.84279	−0.921393	+	0.388631i	$0.872948\pi$
$788$	423.774	0.0191578
$789$	0	0
$790$	8281.56	0.372968
$791$	−57528.4	−2.58594
$792$	0	0
$793$	−9818.51	−0.439679
$794$	−6543.29	−0.292459
$795$	0	0
$796$	−2750.05	−0.122453
$797$	−1758.02	−0.0781331	−0.0390666	−	0.999237i	$-0.512438\pi$
$797$	−1758.02	−0.0781331	−0.0390666	+	0.999237i	$0.512438\pi$
$798$	0	0
$799$	−199.057	−0.00881368
$800$	2303.64	0.101807
$801$	0	0
$802$	−8029.81	−0.353544
$803$	15997.0	0.703015
$804$	0	0
$805$	−6026.45	−0.263857
$806$	5406.62	0.236278
$807$	0	0
$808$	−14448.7	−0.629087
$809$	−6510.49	−0.282938	−0.141469	−	0.989943i	$-0.545182\pi$
$809$	−6510.49	−0.282938	−0.141469	+	0.989943i	$0.545182\pi$
$810$	0	0
$811$	−14997.6	−0.649369	−0.324684	−	0.945822i	$-0.605258\pi$
$811$	−14997.6	−0.649369	−0.324684	+	0.945822i	$0.605258\pi$
$812$	−39522.6	−1.70809
$813$	0	0
$814$	−1243.72	−0.0535532
$815$	−16440.5	−0.706609
$816$	0	0
$817$	22429.8	0.960487
$818$	−12279.3	−0.524859
$819$	0	0
$820$	−181.205	−0.00771702
$821$	34974.5	1.48674	0.743372	−	0.668878i	$-0.233225\pi$
$821$	34974.5	1.48674	0.743372	+	0.668878i	$0.233225\pi$
$822$	0	0
$823$	−14794.5	−0.626616	−0.313308	−	0.949652i	$-0.601437\pi$
$823$	−14794.5	−0.626616	−0.313308	+	0.949652i	$0.601437\pi$
$824$	9983.85	0.422092
$825$	0	0
$826$	14113.9	0.594535
$827$	4958.24	0.208482	0.104241	−	0.994552i	$-0.466759\pi$
$827$	4958.24	0.208482	0.104241	+	0.994552i	$0.466759\pi$
$828$	0	0
$829$	−7537.16	−0.315774	−0.157887	−	0.987457i	$-0.550468\pi$
$829$	−7537.16	−0.315774	−0.157887	+	0.987457i	$0.550468\pi$
$830$	−17376.8	−0.726695
$831$	0	0
$832$	1840.72	0.0767012
$833$	1092.46	0.0454399
$834$	0	0
$835$	−29536.4	−1.22413
$836$	19523.4	0.807691
$837$	0	0
$838$	−2389.05	−0.0984825
$839$	−2097.47	−0.0863082	−0.0431541	−	0.999068i	$-0.513741\pi$
$839$	−2097.47	−0.0863082	−0.0431541	+	0.999068i	$0.513741\pi$
$840$	0	0
$841$	50993.5	2.09084
$842$	18276.4	0.748034
$843$	0	0
$844$	16817.6	0.685884
$845$	−9973.30	−0.406026
$846$	0	0
$847$	12798.7	0.519209
$848$	−7410.65	−0.300098
$849$	0	0
$850$	165.205	0.00666644
$851$	−348.264	−0.0140286
$852$	0	0
$853$	−28566.3	−1.14665	−0.573324	−	0.819328i	$-0.694347\pi$
$853$	−28566.3	−1.14665	−0.573324	+	0.819328i	$0.694347\pi$
$854$	24570.7	0.984536
$855$	0	0
$856$	−3941.14	−0.157366
$857$	−13310.2	−0.530535	−0.265267	−	0.964175i	$-0.585460\pi$
$857$	−13310.2	−0.530535	−0.265267	+	0.964175i	$0.585460\pi$
$858$	0	0
$859$	4372.20	0.173664	0.0868322	−	0.996223i	$-0.472326\pi$
$859$	4372.20	0.173664	0.0868322	+	0.996223i	$0.472326\pi$
$860$	5496.50	0.217941
$861$	0	0
$862$	9551.70	0.377416
$863$	−24600.4	−0.970346	−0.485173	−	0.874418i	$-0.661243\pi$
$863$	−24600.4	−0.970346	−0.485173	+	0.874418i	$0.661243\pi$
$864$	0	0
$865$	−6247.13	−0.245559
$866$	12011.7	0.471332
$867$	0	0
$868$	−13530.0	−0.529077
$869$	−23356.6	−0.911760
$870$	0	0
$871$	13338.7	0.518901
$872$	−2994.16	−0.116279
$873$	0	0
$874$	5466.89	0.211579
$875$	−51614.9	−1.99417
$876$	0	0
$877$	24776.6	0.953988	0.476994	−	0.878907i	$-0.341726\pi$
$877$	24776.6	0.953988	0.476994	+	0.878907i	$0.341726\pi$
$878$	−32154.0	−1.23593
$879$	0	0
$880$	4784.27	0.183270
$881$	−14728.1	−0.563225	−0.281612	−	0.959528i	$-0.590869\pi$
$881$	−14728.1	−0.563225	−0.281612	+	0.959528i	$0.590869\pi$
$882$	0	0
$883$	15491.8	0.590420	0.295210	−	0.955432i	$-0.404610\pi$
$883$	15491.8	0.590420	0.295210	+	0.955432i	$0.404610\pi$
$884$	132.006	0.00502246
$885$	0	0
$886$	2036.20	0.0772093
$887$	−946.347	−0.0358233	−0.0179116	−	0.999840i	$-0.505702\pi$
$887$	−946.347	−0.0358233	−0.0179116	+	0.999840i	$0.505702\pi$
$888$	0	0
$889$	47521.7	1.79283
$890$	11360.8	0.427882
$891$	0	0
$892$	13758.5	0.516445
$893$	−20617.3	−0.772601
$894$	0	0
$895$	−24075.0	−0.899151
$896$	−4606.38	−0.171750
$897$	0	0
$898$	12931.0	0.480526
$899$	25806.2	0.957381
$900$	0	0
$901$	−531.452	−0.0196506
$902$	511.055	0.0188651
$903$	0	0
$904$	12788.6	0.470511
$905$	−1908.36	−0.0700950
$906$	0	0
$907$	36233.2	1.32647	0.663233	−	0.748413i	$-0.269184\pi$
$907$	36233.2	1.32647	0.663233	+	0.748413i	$0.269184\pi$
$908$	−11626.9	−0.424949
$909$	0	0
$910$	−15072.0	−0.549046
$911$	51440.1	1.87079	0.935394	−	0.353607i	$-0.115045\pi$
$911$	51440.1	1.87079	0.935394	+	0.353607i	$0.115045\pi$
$912$	0	0
$913$	49008.0	1.77648
$914$	13967.2	0.505463
$915$	0	0
$916$	−24882.2	−0.897524
$917$	19235.3	0.692701
$918$	0	0
$919$	−2379.26	−0.0854020	−0.0427010	−	0.999088i	$-0.513596\pi$
$919$	−2379.26	−0.0854020	−0.0427010	+	0.999088i	$0.513596\pi$
$920$	1339.68	0.0480087
$921$	0	0
$922$	16570.1	0.591873
$923$	20996.2	0.748754
$924$	0	0
$925$	−1090.05	−0.0387465
$926$	35869.0	1.27292
$927$	0	0
$928$	8785.88	0.310787
$929$	8555.16	0.302137	0.151069	−	0.988523i	$-0.451729\pi$
$929$	8555.16	0.302137	0.151069	+	0.988523i	$0.451729\pi$
$930$	0	0
$931$	113151.	3.98323
$932$	21262.7	0.747300
$933$	0	0
$934$	21119.5	0.739882
$935$	343.102	0.0120007
$936$	0	0
$937$	33864.3	1.18068	0.590340	−	0.807155i	$-0.298994\pi$
$937$	33864.3	1.18068	0.590340	+	0.807155i	$0.298994\pi$
$938$	−33379.9	−1.16193
$939$	0	0
$940$	−5052.36	−0.175308
$941$	−48573.1	−1.68272	−0.841360	−	0.540475i	$-0.818244\pi$
$941$	−48573.1	−1.68272	−0.841360	+	0.540475i	$0.818244\pi$
$942$	0	0
$943$	143.105	0.00494181
$944$	−3137.53	−0.108176
$945$	0	0
$946$	−15501.9	−0.532779
$947$	−13683.6	−0.469542	−0.234771	−	0.972051i	$-0.575434\pi$
$947$	−13683.6	−0.469542	−0.234771	+	0.972051i	$0.575434\pi$
$948$	0	0
$949$	11203.0	0.383207
$950$	17111.1	0.584375
$951$	0	0
$952$	−330.345	−0.0112464
$953$	38125.9	1.29593	0.647964	−	0.761671i	$-0.275621\pi$
$953$	38125.9	1.29593	0.647964	+	0.761671i	$0.275621\pi$
$954$	0	0
$955$	35804.6	1.21320
$956$	6534.27	0.221060
$957$	0	0
$958$	−12351.0	−0.416536
$959$	91579.2	3.08368
$960$	0	0
$961$	−20956.6	−0.703454
$962$	−870.999	−0.0291914
$963$	0	0
$964$	122.101	0.00407948
$965$	3366.76	0.112311
$966$	0	0
$967$	24301.7	0.808159	0.404079	−	0.914724i	$-0.367592\pi$
$967$	24301.7	0.808159	0.404079	+	0.914724i	$0.367592\pi$
$968$	−2845.17	−0.0944701
$969$	0	0
$970$	5003.32	0.165615
$971$	52900.8	1.74837	0.874186	−	0.485592i	$-0.161396\pi$
$971$	52900.8	1.74837	0.874186	+	0.485592i	$0.161396\pi$
$972$	0	0
$973$	−71524.6	−2.35660
$974$	33232.9	1.09327
$975$	0	0
$976$	−5462.09	−0.179136
$977$	−37262.3	−1.22019	−0.610095	−	0.792329i	$-0.708869\pi$
$977$	−37262.3	−1.22019	−0.610095	+	0.792329i	$0.708869\pi$
$978$	0	0
$979$	−32041.0	−1.04600
$980$	27728.2	0.903821
$981$	0	0
$982$	3846.83	0.125007
$983$	11137.2	0.361366	0.180683	−	0.983541i	$-0.442169\pi$
$983$	11137.2	0.361366	0.180683	+	0.983541i	$0.442169\pi$
$984$	0	0
$985$	771.363	0.0249519
$986$	630.076	0.0203506
$987$	0	0
$988$	13672.6	0.440265
$989$	−4340.80	−0.139565
$990$	0	0
$991$	24618.4	0.789130	0.394565	−	0.918868i	$-0.370895\pi$
$991$	24618.4	0.789130	0.394565	+	0.918868i	$0.370895\pi$
$992$	3007.73	0.0962656
$993$	0	0
$994$	−52542.9	−1.67662
$995$	−5005.69	−0.159489
$996$	0	0
$997$	−29265.4	−0.929635	−0.464817	−	0.885407i	$-0.653880\pi$
$997$	−29265.4	−0.929635	−0.464817	+	0.885407i	$0.653880\pi$
$998$	4003.53	0.126983
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.4.a.m.1.3	✓	4
3.2	odd	2		414.4.a.n.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.4.a.m.1.3	✓	4	1.1	even	1	trivial
414.4.a.n.1.2	yes	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +7.28088 q^{5} +35.9873 q^{7} -8.00000 q^{8} -14.5618 q^{10} +41.0688 q^{11} +28.7612 q^{13} -71.9747 q^{14} +16.0000 q^{16} +1.14743 q^{17} +118.845 q^{19} +29.1235 q^{20} -82.1376 q^{22} -23.0000 q^{23} -71.9888 q^{25} -57.5224 q^{26} +143.949 q^{28} -274.559 q^{29} -93.9916 q^{31} -32.0000 q^{32} -2.29487 q^{34} +262.020 q^{35} +15.1419 q^{37} -237.691 q^{38} -58.2471 q^{40} -6.22194 q^{41} +188.731 q^{43} +164.275 q^{44} +46.0000 q^{46} -173.480 q^{47} +952.089 q^{49} +143.978 q^{50} +115.045 q^{52} -463.166 q^{53} +299.017 q^{55} -287.899 q^{56} +549.118 q^{58} -196.095 q^{59} -341.380 q^{61} +187.983 q^{62} +64.0000 q^{64} +209.407 q^{65} +463.772 q^{67} +4.58973 q^{68} -524.039 q^{70} +730.020 q^{71} +389.517 q^{73} -30.2838 q^{74} +475.382 q^{76} +1477.96 q^{77} -568.720 q^{79} +116.494 q^{80} +12.4439 q^{82} +1193.32 q^{83} +8.35433 q^{85} -377.461 q^{86} -328.550 q^{88} -780.180 q^{89} +1035.04 q^{91} -92.0000 q^{92} +346.960 q^{94} +865.300 q^{95} -343.593 q^{97} -1904.18 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 16 q^{4} + 18 q^{7} - 32 q^{8} - 42 q^{11} + 108 q^{13} - 36 q^{14} + 64 q^{16} - 26 q^{17} + 132 q^{19} + 84 q^{22} - 92 q^{23} + 260 q^{25} - 216 q^{26} + 72 q^{28} - 252 q^{29} + 428 q^{31}+ \cdots - 2152 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 16 * q^4 + 18 * q^7 - 32 * q^8 - 42 * q^11 + 108 * q^13 - 36 * q^14 + 64 * q^16 - 26 * q^17 + 132 * q^19 + 84 * q^22 - 92 * q^23 + 260 * q^25 - 216 * q^26 + 72 * q^28 - 252 * q^29 + 428 * q^31 - 128 * q^32 + 52 * q^34 - 348 * q^35 + 278 * q^37 - 264 * q^38 + 4 * q^41 + 216 * q^43 - 168 * q^44 + 184 * q^46 + 4 * q^47 + 1076 * q^49 - 520 * q^50 + 432 * q^52 + 460 * q^53 + 512 * q^55 - 144 * q^56 + 504 * q^58 + 504 * q^59 + 1086 * q^61 - 856 * q^62 + 256 * q^64 + 1420 * q^65 + 1484 * q^67 - 104 * q^68 + 696 * q^70 + 876 * q^71 + 1072 * q^73 - 556 * q^74 + 528 * q^76 + 1580 * q^77 + 1634 * q^79 - 8 * q^82 + 2246 * q^83 + 872 * q^85 - 432 * q^86 + 336 * q^88 + 474 * q^89 + 40 * q^91 - 368 * q^92 - 8 * q^94 + 2780 * q^95 + 1044 * q^97 - 2152 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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