LMFDB - Embedded newform 414.4.a.n.1.2

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			1.2
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	$-0.605570\pi$
$5$	−7.28088	−0.651222	−0.325611	+	0.945504i	$0.605570\pi$
$6$	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	$-0.690295\pi$
$11$	−41.0688	−1.12570	−0.562850	+	0.826559i	$0.690295\pi$
$12$	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	$-0.502605\pi$
$17$	−1.14743	−0.0163702	−0.00818510	+	0.999967i	$0.502605\pi$
$18$	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	$-0.158185\pi$
$29$	274.559	1.75808	0.879039	+	0.476749i	$0.158185\pi$
$30$	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	$-0.496228\pi$
$41$	6.22194	0.0237001	0.0118501	+	0.999930i	$0.496228\pi$
$42$	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	$-0.413241\pi$
$47$	173.480	0.538398	0.269199	+	0.963085i	$0.413241\pi$
$48$	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	$-0.295090\pi$
$53$	463.166	1.20039	0.600195	+	0.799854i	$0.295090\pi$
$54$	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	$-0.430584\pi$
$59$	196.095	0.432703	0.216351	+	0.976316i	$0.430584\pi$
$60$	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	$-0.708880\pi$
$71$	−730.020	−1.22024	−0.610122	+	0.792307i	$0.708880\pi$
$72$	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	$-0.789430\pi$
$83$	−1193.32	−1.57811	−0.789057	+	0.614321i	$0.789430\pi$
$84$	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	$-0.346198\pi$
$89$	780.180	0.929201	0.464601	+	0.885520i	$0.346198\pi$
$90$	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	$-0.849061\pi$
$101$	−1806.08	−1.77933	−0.889664	+	0.456616i	$0.849061\pi$
$102$	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	$-0.571438\pi$
$107$	−492.642	−0.445099	−0.222549	+	0.974921i	$0.571438\pi$
$108$	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	$-0.268260\pi$
$113$	1598.57	1.33081	0.665403	+	0.746484i	$0.268260\pi$
$114$	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	$-0.557041\pi$
$131$	−534.502	−0.356486	−0.178243	+	0.983986i	$0.557041\pi$
$132$	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	$-0.791733\pi$
$137$	−2544.76	−1.58696	−0.793481	+	0.608595i	$0.791733\pi$
$138$	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	$-0.773626\pi$
$149$	−2755.79	−1.51519	−0.757596	+	0.652724i	$0.773626\pi$
$150$	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	$-0.110944\pi$
$167$	4056.70	1.87974	0.939872	+	0.341528i	$0.110944\pi$
$168$	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	$-0.439625\pi$
$173$	858.019	0.377075	0.188537	+	0.982066i	$0.439625\pi$
$174$	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	$-0.257454\pi$
$179$	3306.61	1.38071	0.690356	+	0.723469i	$0.257454\pi$
$180$	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	$-0.881484\pi$
$191$	−4917.62	−1.86297	−0.931483	+	0.363786i	$0.881484\pi$
$192$	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	$-0.506098\pi$
$197$	−105.944	−0.0383156	−0.0191578	+	0.999816i	$0.506098\pi$
$198$	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	$-0.360292\pi$
$227$	2906.74	0.849898	0.424949	+	0.905217i	$0.360292\pi$
$228$	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	$-0.768650\pi$
$233$	−5315.68	−1.49460	−0.747300	+	0.664487i	$0.768650\pi$
$234$	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	$-0.570952\pi$
$239$	−1633.57	−0.442120	−0.221060	+	0.975260i	$0.570952\pi$
$240$	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	$-0.236045\pi$
$251$	5864.81	1.47483	0.737417	+	0.675437i	$0.236045\pi$
$252$	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	$-0.724932\pi$
$257$	−5350.14	−1.29857	−0.649285	+	0.760545i	$0.724932\pi$
$258$	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	$-0.302608\pi$
$263$	4957.27	1.16228	0.581138	+	0.813805i	$0.302608\pi$
$264$	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	$-0.508061\pi$
$269$	−223.436	−0.0506437	−0.0253218	+	0.999679i	$0.508061\pi$
$270$	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	$-0.628622\pi$
$281$	−3704.01	−0.786345	−0.393172	+	0.919465i	$0.628622\pi$
$282$	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	$-0.501657\pi$
$293$	−52.2109	−0.0104102	−0.00520510	+	0.999986i	$0.501657\pi$
$294$	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	$-0.813958\pi$
$311$	−9148.29	−1.66801	−0.834006	+	0.551755i	$0.813958\pi$
$312$	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	$-0.289704\pi$
$317$	6926.81	1.22728	0.613641	+	0.789585i	$0.289704\pi$
$318$	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	$-0.457999\pi$
$347$	1700.89	0.263137	0.131569	+	0.991307i	$0.457999\pi$
$348$	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	$-0.409588\pi$
$353$	3717.19	0.560471	0.280235	+	0.959931i	$0.409588\pi$
$354$	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	$-0.521738\pi$
$359$	−928.351	−0.136480	−0.0682402	+	0.997669i	$0.521738\pi$
$360$	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	$-0.401121\pi$
$383$	4582.21	0.611331	0.305665	+	0.952139i	$0.401121\pi$
$384$	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	$-0.697383\pi$
$389$	−8916.93	−1.16223	−0.581114	+	0.813822i	$0.697383\pi$
$390$	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	$-0.580428\pi$
$401$	−4014.90	−0.499987	−0.249993	+	0.968248i	$0.580428\pi$
$402$	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	$-0.522184\pi$
$419$	−1194.52	−0.139275	−0.0696376	+	0.997572i	$0.522184\pi$
$420$	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	$-0.414010\pi$
$431$	4775.85	0.533746	0.266873	+	0.963732i	$0.414010\pi$
$432$	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	$-0.482613\pi$
$443$	1018.10	0.109190	0.0545952	+	0.998509i	$0.482613\pi$
$444$	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	$-0.389646\pi$
$449$	6465.49	0.679567	0.339783	+	0.940504i	$0.389646\pi$
$450$	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	$-0.362550\pi$
$461$	8285.05	0.837035	0.418517	+	0.908209i	$0.362550\pi$
$462$	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	$-0.324747\pi$
$467$	10559.7	1.04635	0.523176	+	0.852225i	$0.324747\pi$
$468$	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	$-0.595165\pi$
$479$	−6175.49	−0.589071	−0.294536	+	0.955641i	$0.595165\pi$
$480$	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	$-0.471827\pi$
$491$	1923.41	0.176787	0.0883936	+	0.996086i	$0.471827\pi$
$492$	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	$-0.306207\pi$
$503$	12903.3	1.14380	0.571899	+	0.820324i	$0.306207\pi$
$504$	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	$-0.530662\pi$
$509$	−2208.97	−0.192360	−0.0961798	+	0.995364i	$0.530662\pi$
$510$	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	$-0.602471\pi$
$521$	−7525.04	−0.632779	−0.316390	+	0.948629i	$0.602471\pi$
$522$	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	$-0.371876\pi$
$557$	10299.2	0.783466	0.391733	+	0.920079i	$0.371876\pi$
$558$	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	$-0.289764\pi$
$563$	16390.8	1.22698	0.613492	+	0.789701i	$0.289764\pi$
$564$	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	$-0.742397\pi$
$569$	−18730.9	−1.38003	−0.690017	+	0.723793i	$0.742397\pi$
$570$	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	$-0.252174\pi$
$587$	19974.9	1.40452	0.702261	+	0.711920i	$0.252174\pi$
$588$	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	$-0.349670\pi$
$593$	13138.3	0.909827	0.454913	+	0.890536i	$0.349670\pi$
$594$	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	$-0.451557\pi$
$599$	4445.05	0.303205	0.151602	+	0.988442i	$0.451557\pi$
$600$	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	$-0.729273\pi$
$617$	−20217.9	−1.31919	−0.659597	+	0.751619i	$0.729273\pi$
$618$	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	$-0.807532\pi$
$641$	−26702.8	−1.64540	−0.822698	+	0.568479i	$0.807532\pi$
$642$	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	$-0.661384\pi$
$647$	−15981.9	−0.971116	−0.485558	+	0.874204i	$0.661384\pi$
$648$	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	$-0.118043\pi$
$653$	31104.7	1.86404	0.932022	+	0.362401i	$0.118043\pi$
$654$	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	$-0.523925\pi$
$659$	−2540.74	−0.150187	−0.0750933	+	0.997177i	$0.523925\pi$
$660$	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	$-0.724083\pi$
$677$	−22802.8	−1.29451	−0.647254	+	0.762274i	$0.724083\pi$
$678$	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	$-0.626092\pi$
$683$	−13774.7	−0.771703	−0.385852	+	0.922561i	$0.626092\pi$
$684$	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	$-0.447257\pi$
$701$	6122.50	0.329877	0.164938	+	0.986304i	$0.447257\pi$
$702$	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	$-0.368165\pi$
$719$	15517.2	0.804862	0.402431	+	0.915450i	$0.368165\pi$
$720$	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	$-0.681589\pi$
$743$	−21874.4	−1.08007	−0.540036	+	0.841642i	$0.681589\pi$
$744$	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	$-0.601991\pi$
$761$	−13223.9	−0.629917	−0.314959	+	0.949105i	$0.601991\pi$
$762$	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	$-0.365657\pi$
$773$	17607.4	0.819266	0.409633	+	0.912250i	$0.365657\pi$
$774$	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	$-0.487562\pi$
$797$	1758.02	0.0781331	0.0390666	+	0.999237i	$0.487562\pi$
$798$	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	$-0.454818\pi$
$809$	6510.49	0.282938	0.141469	+	0.989943i	$0.454818\pi$
$810$	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	$-0.766775\pi$
$821$	−34974.5	−1.48674	−0.743372	+	0.668878i	$0.766775\pi$
$822$	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	$-0.533241\pi$
$827$	−4958.24	−0.208482	−0.104241	+	0.994552i	$0.533241\pi$
$828$	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	$-0.486259\pi$
$839$	2097.47	0.0863082	0.0431541	+	0.999068i	$0.486259\pi$
$840$	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	$-0.414540\pi$
$857$	13310.2	0.530535	0.265267	+	0.964175i	$0.414540\pi$
$858$	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	$-0.338757\pi$
$863$	24600.4	0.970346	0.485173	+	0.874418i	$0.338757\pi$
$864$	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	$-0.409131\pi$
$881$	14728.1	0.563225	0.281612	+	0.959528i	$0.409131\pi$
$882$	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	$-0.494298\pi$
$887$	946.347	0.0358233	0.0179116	+	0.999840i	$0.494298\pi$
$888$	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	$-0.884955\pi$
$911$	−51440.1	−1.87079	−0.935394	+	0.353607i	$0.884955\pi$
$912$	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	$-0.548271\pi$
$929$	−8555.16	−0.302137	−0.151069	+	0.988523i	$0.548271\pi$
$930$	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	$-0.181756\pi$
$941$	48573.1	1.68272	0.841360	+	0.540475i	$0.181756\pi$
$942$	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	$-0.424566\pi$
$947$	13683.6	0.469542	0.234771	+	0.972051i	$0.424566\pi$
$948$	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	$-0.724379\pi$
$953$	−38125.9	−1.29593	−0.647964	+	0.761671i	$0.724379\pi$
$954$	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	$-0.838604\pi$
$971$	−52900.8	−1.74837	−0.874186	+	0.485592i	$0.838604\pi$
$972$	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	$-0.291131\pi$
$977$	37262.3	1.22019	0.610095	+	0.792329i	$0.291131\pi$
$978$	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	$-0.557831\pi$
$983$	−11137.2	−0.361366	−0.180683	+	0.983541i	$0.557831\pi$
$984$	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.n.1.2	yes	4
3.2	odd	2		414.4.a.m.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.4.a.m.1.3	✓	4	3.2	odd	2
414.4.a.n.1.2	yes	4	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:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
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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