LMFDB - Embedded newform 1206.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1206,4,Mod(1,1206)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1206, 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("1206.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1206 = 2 \cdot 3^{2} \cdot 67 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1206.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:	$71.1563034669$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 402)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1206.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +7.58579 q^{5} -7.34315 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} +7.58579 q^{5} -7.34315 q^{7} -8.00000 q^{8} -15.1716 q^{10} +49.3137 q^{11} -85.8823 q^{13} +14.6863 q^{14} +16.0000 q^{16} -38.2254 q^{17} -19.5147 q^{19} +30.3431 q^{20} -98.6274 q^{22} +110.497 q^{23} -67.4558 q^{25} +171.765 q^{26} -29.3726 q^{28} -48.2498 q^{29} +245.357 q^{31} -32.0000 q^{32} +76.4508 q^{34} -55.7035 q^{35} +23.2843 q^{37} +39.0294 q^{38} -60.6863 q^{40} +129.762 q^{41} -4.13708 q^{43} +197.255 q^{44} -220.995 q^{46} +362.098 q^{47} -289.078 q^{49} +134.912 q^{50} -343.529 q^{52} +87.2914 q^{53} +374.083 q^{55} +58.7452 q^{56} +96.4996 q^{58} +122.095 q^{59} +432.804 q^{61} -490.715 q^{62} +64.0000 q^{64} -651.484 q^{65} +67.0000 q^{67} -152.902 q^{68} +111.407 q^{70} -459.637 q^{71} +245.128 q^{73} -46.5685 q^{74} -78.0589 q^{76} -362.118 q^{77} +509.019 q^{79} +121.373 q^{80} -259.525 q^{82} +1435.72 q^{83} -289.970 q^{85} +8.27417 q^{86} -394.510 q^{88} +130.686 q^{89} +630.646 q^{91} +441.990 q^{92} -724.195 q^{94} -148.034 q^{95} -120.045 q^{97} +578.156 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} + 18 q^{5} - 26 q^{7} - 16 q^{8}+O(q^{10}) $ 2 * q - 4 * q^2 + 8 * q^4 + 18 * q^5 - 26 * q^7 - 16 * q^8	$ 2 q - 4 q^{2} + 8 q^{4} + 18 q^{5} - 26 q^{7} - 16 q^{8} - 36 q^{10} + 76 q^{11} - 36 q^{13} + 52 q^{14} + 32 q^{16} + 48 q^{17} - 56 q^{19} + 72 q^{20} - 152 q^{22} + 122 q^{23} - 84 q^{25} + 72 q^{26} - 104 q^{28} + 192 q^{29} - 58 q^{31} - 64 q^{32} - 96 q^{34} - 250 q^{35} - 10 q^{37} + 112 q^{38} - 144 q^{40} + 466 q^{41} + 218 q^{43} + 304 q^{44} - 244 q^{46} + 68 q^{47} - 284 q^{49} + 168 q^{50} - 144 q^{52} - 162 q^{53} + 652 q^{55} + 208 q^{56} - 384 q^{58} + 66 q^{59} + 1024 q^{61} + 116 q^{62} + 128 q^{64} - 132 q^{65} + 134 q^{67} + 192 q^{68} + 500 q^{70} - 116 q^{71} + 1022 q^{73} + 20 q^{74} - 224 q^{76} - 860 q^{77} + 656 q^{79} + 288 q^{80} - 932 q^{82} + 142 q^{83} + 608 q^{85} - 436 q^{86} - 608 q^{88} + 284 q^{89} - 300 q^{91} + 488 q^{92} - 136 q^{94} - 528 q^{95} - 868 q^{97} + 568 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 + 18 * q^5 - 26 * q^7 - 16 * q^8 - 36 * q^10 + 76 * q^11 - 36 * q^13 + 52 * q^14 + 32 * q^16 + 48 * q^17 - 56 * q^19 + 72 * q^20 - 152 * q^22 + 122 * q^23 - 84 * q^25 + 72 * q^26 - 104 * q^28 + 192 * q^29 - 58 * q^31 - 64 * q^32 - 96 * q^34 - 250 * q^35 - 10 * q^37 + 112 * q^38 - 144 * q^40 + 466 * q^41 + 218 * q^43 + 304 * q^44 - 244 * q^46 + 68 * q^47 - 284 * q^49 + 168 * q^50 - 144 * q^52 - 162 * q^53 + 652 * q^55 + 208 * q^56 - 384 * q^58 + 66 * q^59 + 1024 * q^61 + 116 * q^62 + 128 * q^64 - 132 * q^65 + 134 * q^67 + 192 * q^68 + 500 * q^70 - 116 * q^71 + 1022 * q^73 + 20 * q^74 - 224 * q^76 - 860 * q^77 + 656 * q^79 + 288 * q^80 - 932 * q^82 + 142 * q^83 + 608 * q^85 - 436 * q^86 - 608 * q^88 + 284 * q^89 - 300 * q^91 + 488 * q^92 - 136 * q^94 - 528 * q^95 - 868 * q^97 + 568 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	7.58579	0.678493	0.339247	−	0.940697i	$-0.389828\pi$
$5$	7.58579	0.678493	0.339247	+	0.940697i	$0.389828\pi$
$6$	0	0
$7$	−7.34315	−0.396493	−0.198246	−	0.980152i	$-0.563525\pi$
$7$	−7.34315	−0.396493	−0.198246	+	0.980152i	$0.563525\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−15.1716	−0.479767
$11$	49.3137	1.35169	0.675847	−	0.737042i	$-0.263778\pi$
$11$	49.3137	1.35169	0.675847	+	0.737042i	$0.263778\pi$
$12$	0	0
$13$	−85.8823	−1.83227	−0.916133	−	0.400875i	$-0.868706\pi$
$13$	−85.8823	−1.83227	−0.916133	+	0.400875i	$0.868706\pi$
$14$	14.6863	0.280363
$15$	0	0
$16$	16.0000	0.250000
$17$	−38.2254	−0.545354	−0.272677	−	0.962106i	$-0.587909\pi$
$17$	−38.2254	−0.545354	−0.272677	+	0.962106i	$0.587909\pi$
$18$	0	0
$19$	−19.5147	−0.235631	−0.117815	−	0.993036i	$-0.537589\pi$
$19$	−19.5147	−0.235631	−0.117815	+	0.993036i	$0.537589\pi$
$20$	30.3431	0.339247
$21$	0	0
$22$	−98.6274	−0.955793
$23$	110.497	1.00175	0.500876	−	0.865519i	$-0.333011\pi$
$23$	110.497	1.00175	0.500876	+	0.865519i	$0.333011\pi$
$24$	0	0
$25$	−67.4558	−0.539647
$26$	171.765	1.29561
$27$	0	0
$28$	−29.3726	−0.198246
$29$	−48.2498	−0.308957	−0.154479	−	0.987996i	$-0.549370\pi$
$29$	−48.2498	−0.308957	−0.154479	+	0.987996i	$0.549370\pi$
$30$	0	0
$31$	245.357	1.42153	0.710766	−	0.703428i	$-0.248348\pi$
$31$	245.357	1.42153	0.710766	+	0.703428i	$0.248348\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	76.4508	0.385624
$35$	−55.7035	−0.269018
$36$	0	0
$37$	23.2843	0.103457	0.0517285	−	0.998661i	$-0.483527\pi$
$37$	23.2843	0.103457	0.0517285	+	0.998661i	$0.483527\pi$
$38$	39.0294	0.166616
$39$	0	0
$40$	−60.6863	−0.239884
$41$	129.762	0.494280	0.247140	−	0.968980i	$-0.420509\pi$
$41$	129.762	0.494280	0.247140	+	0.968980i	$0.420509\pi$
$42$	0	0
$43$	−4.13708	−0.0146721	−0.00733604	−	0.999973i	$-0.502335\pi$
$43$	−4.13708	−0.0146721	−0.00733604	+	0.999973i	$0.502335\pi$
$44$	197.255	0.675847
$45$	0	0
$46$	−220.995	−0.708346
$47$	362.098	1.12377	0.561887	−	0.827214i	$-0.310076\pi$
$47$	362.098	1.12377	0.561887	+	0.827214i	$0.310076\pi$
$48$	0	0
$49$	−289.078	−0.842794
$50$	134.912	0.381588
$51$	0	0
$52$	−343.529	−0.916133
$53$	87.2914	0.226234	0.113117	−	0.993582i	$-0.463917\pi$
$53$	87.2914	0.226234	0.113117	+	0.993582i	$0.463917\pi$
$54$	0	0
$55$	374.083	0.917116
$56$	58.7452	0.140181
$57$	0	0
$58$	96.4996	0.218466
$59$	122.095	0.269415	0.134707	−	0.990885i	$-0.456991\pi$
$59$	122.095	0.269415	0.134707	+	0.990885i	$0.456991\pi$
$60$	0	0
$61$	432.804	0.908441	0.454220	−	0.890889i	$-0.349918\pi$
$61$	432.804	0.908441	0.454220	+	0.890889i	$0.349918\pi$
$62$	−490.715	−1.00517
$63$	0	0
$64$	64.0000	0.125000
$65$	−651.484	−1.24318
$66$	0	0
$67$	67.0000	0.122169
$67$	67.0000	0.122169
$68$	−152.902	−0.272677
$69$	0	0
$70$	111.407	0.190224
$71$	−459.637	−0.768293	−0.384147	−	0.923272i	$-0.625504\pi$
$71$	−459.637	−0.768293	−0.384147	+	0.923272i	$0.625504\pi$
$72$	0	0
$73$	245.128	0.393014	0.196507	−	0.980502i	$-0.437040\pi$
$73$	245.128	0.393014	0.196507	+	0.980502i	$0.437040\pi$
$74$	−46.5685	−0.0731552
$75$	0	0
$76$	−78.0589	−0.117815
$77$	−362.118	−0.535937
$78$	0	0
$79$	509.019	0.724926	0.362463	−	0.931998i	$-0.381936\pi$
$79$	509.019	0.724926	0.362463	+	0.931998i	$0.381936\pi$
$80$	121.373	0.169623
$81$	0	0
$82$	−259.525	−0.349509
$83$	1435.72	1.89868	0.949339	−	0.314253i	$-0.101754\pi$
$83$	1435.72	1.89868	0.949339	+	0.314253i	$0.101754\pi$
$84$	0	0
$85$	−289.970	−0.370019
$86$	8.27417	0.0103747
$87$	0	0
$88$	−394.510	−0.477896
$89$	130.686	0.155649	0.0778243	−	0.996967i	$-0.475203\pi$
$89$	130.686	0.155649	0.0778243	+	0.996967i	$0.475203\pi$
$90$	0	0
$91$	630.646	0.726480
$92$	441.990	0.500876
$93$	0	0
$94$	−724.195	−0.794628
$95$	−148.034	−0.159874
$96$	0	0
$97$	−120.045	−0.125657	−0.0628283	−	0.998024i	$-0.520012\pi$
$97$	−120.045	−0.125657	−0.0628283	+	0.998024i	$0.520012\pi$
$98$	578.156	0.595945
$99$	0	0
$100$	−269.823	−0.269823
$101$	−804.827	−0.792903	−0.396452	−	0.918056i	$-0.629759\pi$
$101$	−804.827	−0.792903	−0.396452	+	0.918056i	$0.629759\pi$
$102$	0	0
$103$	−290.014	−0.277436	−0.138718	−	0.990332i	$-0.544298\pi$
$103$	−290.014	−0.277436	−0.138718	+	0.990332i	$0.544298\pi$
$104$	687.058	0.647804
$105$	0	0
$106$	−174.583	−0.159972
$107$	−456.518	−0.412460	−0.206230	−	0.978504i	$-0.566120\pi$
$107$	−456.518	−0.412460	−0.206230	+	0.978504i	$0.566120\pi$
$108$	0	0
$109$	−410.926	−0.361097	−0.180548	−	0.983566i	$-0.557787\pi$
$109$	−410.926	−0.361097	−0.180548	+	0.983566i	$0.557787\pi$
$110$	−748.167	−0.648499
$111$	0	0
$112$	−117.490	−0.0991232
$113$	819.856	0.682527	0.341264	−	0.939968i	$-0.389145\pi$
$113$	819.856	0.682527	0.341264	+	0.939968i	$0.389145\pi$
$114$	0	0
$115$	838.210	0.679683
$116$	−192.999	−0.154479
$117$	0	0
$118$	−244.191	−0.190505
$119$	280.695	0.216229
$120$	0	0
$121$	1100.84	0.827079
$122$	−865.608	−0.642365
$123$	0	0
$124$	981.430	0.710766
$125$	−1459.93	−1.04464
$126$	0	0
$127$	−657.765	−0.459584	−0.229792	−	0.973240i	$-0.573805\pi$
$127$	−657.765	−0.459584	−0.229792	+	0.973240i	$0.573805\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	1302.97	0.879061
$131$	2103.42	1.40287	0.701437	−	0.712732i	$-0.252542\pi$
$131$	2103.42	1.40287	0.701437	+	0.712732i	$0.252542\pi$
$132$	0	0
$133$	143.299	0.0934258
$134$	−134.000	−0.0863868
$135$	0	0
$136$	305.803	0.192812
$137$	−944.471	−0.588990	−0.294495	−	0.955653i	$-0.595151\pi$
$137$	−944.471	−0.588990	−0.294495	+	0.955653i	$0.595151\pi$
$138$	0	0
$139$	−1055.58	−0.644125	−0.322062	−	0.946718i	$-0.604376\pi$
$139$	−1055.58	−0.644125	−0.322062	+	0.946718i	$0.604376\pi$
$140$	−222.814	−0.134509
$141$	0	0
$142$	919.273	0.543266
$143$	−4235.17	−2.47666
$144$	0	0
$145$	−366.013	−0.209625
$146$	−490.256	−0.277903
$147$	0	0
$148$	93.1371	0.0517285
$149$	1715.84	0.943401	0.471701	−	0.881759i	$-0.343640\pi$
$149$	1715.84	0.943401	0.471701	+	0.881759i	$0.343640\pi$
$150$	0	0
$151$	2545.65	1.37193	0.685966	−	0.727634i	$-0.259380\pi$
$151$	2545.65	1.37193	0.685966	+	0.727634i	$0.259380\pi$
$152$	156.118	0.0833080
$153$	0	0
$154$	724.235	0.378965
$155$	1861.23	0.964500
$156$	0	0
$157$	−1897.10	−0.964363	−0.482182	−	0.876071i	$-0.660156\pi$
$157$	−1897.10	−0.964363	−0.482182	+	0.876071i	$0.660156\pi$
$158$	−1018.04	−0.512600
$159$	0	0
$160$	−242.745	−0.119942
$161$	−811.399	−0.397188
$162$	0	0
$163$	2201.01	1.05764	0.528822	−	0.848733i	$-0.322634\pi$
$163$	2201.01	1.05764	0.528822	+	0.848733i	$0.322634\pi$
$164$	519.050	0.247140
$165$	0	0
$166$	−2871.43	−1.34257
$167$	−3423.45	−1.58632	−0.793158	−	0.609016i	$-0.791564\pi$
$167$	−3423.45	−1.58632	−0.793158	+	0.609016i	$0.791564\pi$
$168$	0	0
$169$	5178.76	2.35720
$170$	579.939	0.261643
$171$	0	0
$172$	−16.5483	−0.00733604
$173$	56.9075	0.0250092	0.0125046	−	0.999922i	$-0.496020\pi$
$173$	56.9075	0.0250092	0.0125046	+	0.999922i	$0.496020\pi$
$174$	0	0
$175$	495.338	0.213966
$176$	789.019	0.337924
$177$	0	0
$178$	−261.373	−0.110060
$179$	2929.89	1.22341	0.611705	−	0.791086i	$-0.290484\pi$
$179$	2929.89	1.22341	0.611705	+	0.791086i	$0.290484\pi$
$180$	0	0
$181$	3437.32	1.41157	0.705785	−	0.708426i	$-0.250595\pi$
$181$	3437.32	1.41157	0.705785	+	0.708426i	$0.250595\pi$
$182$	−1261.29	−0.513699
$183$	0	0
$184$	−883.980	−0.354173
$185$	176.630	0.0701949
$186$	0	0
$187$	−1885.04	−0.737152
$188$	1448.39	0.561887
$189$	0	0
$190$	296.069	0.113048
$191$	169.996	0.0644003	0.0322002	−	0.999481i	$-0.489749\pi$
$191$	169.996	0.0644003	0.0322002	+	0.999481i	$0.489749\pi$
$192$	0	0
$193$	1707.54	0.636846	0.318423	−	0.947949i	$-0.396847\pi$
$193$	1707.54	0.636846	0.318423	+	0.947949i	$0.396847\pi$
$194$	240.089	0.0888526
$195$	0	0
$196$	−1156.31	−0.421397
$197$	1309.06	0.473436	0.236718	−	0.971578i	$-0.423928\pi$
$197$	1309.06	0.473436	0.236718	+	0.971578i	$0.423928\pi$
$198$	0	0
$199$	2704.01	0.963229	0.481614	−	0.876383i	$-0.340051\pi$
$199$	2704.01	0.963229	0.481614	+	0.876383i	$0.340051\pi$
$200$	539.647	0.190794
$201$	0	0
$202$	1609.65	0.560667
$203$	354.305	0.122499
$204$	0	0
$205$	984.350	0.335366
$206$	580.029	0.196177
$207$	0	0
$208$	−1374.12	−0.458066
$209$	−962.343	−0.318501
$210$	0	0
$211$	−912.630	−0.297763	−0.148882	−	0.988855i	$-0.547567\pi$
$211$	−912.630	−0.297763	−0.148882	+	0.988855i	$0.547567\pi$
$212$	349.166	0.113117
$213$	0	0
$214$	913.036	0.291654
$215$	−31.3830	−0.00995491
$216$	0	0
$217$	−1801.70	−0.563627
$218$	821.852	0.255334
$219$	0	0
$220$	1496.33	0.458558
$221$	3282.88	0.999234
$222$	0	0
$223$	979.086	0.294011	0.147005	−	0.989136i	$-0.453037\pi$
$223$	979.086	0.294011	0.147005	+	0.989136i	$0.453037\pi$
$224$	234.981	0.0700907
$225$	0	0
$226$	−1639.71	−0.482620
$227$	1021.83	0.298771	0.149386	−	0.988779i	$-0.452270\pi$
$227$	1021.83	0.298771	0.149386	+	0.988779i	$0.452270\pi$
$228$	0	0
$229$	6305.79	1.81964	0.909822	−	0.414999i	$-0.136218\pi$
$229$	6305.79	1.81964	0.909822	+	0.414999i	$0.136218\pi$
$230$	−1676.42	−0.480608
$231$	0	0
$232$	385.998	0.109233
$233$	4758.43	1.33792	0.668960	−	0.743299i	$-0.266740\pi$
$233$	4758.43	1.33792	0.668960	+	0.743299i	$0.266740\pi$
$234$	0	0
$235$	2746.79	0.762473
$236$	488.382	0.134707
$237$	0	0
$238$	−561.389	−0.152897
$239$	3542.42	0.958744	0.479372	−	0.877612i	$-0.340864\pi$
$239$	3542.42	0.958744	0.479372	+	0.877612i	$0.340864\pi$
$240$	0	0
$241$	2926.20	0.782129	0.391065	−	0.920363i	$-0.372107\pi$
$241$	2926.20	0.782129	0.391065	+	0.920363i	$0.372107\pi$
$242$	−2201.68	−0.584833
$243$	0	0
$244$	1731.22	0.454220
$245$	−2192.89	−0.571830
$246$	0	0
$247$	1675.97	0.431738
$248$	−1962.86	−0.502587
$249$	0	0
$250$	2919.86	0.738672
$251$	3267.88	0.821779	0.410889	−	0.911685i	$-0.365218\pi$
$251$	3267.88	0.821779	0.410889	+	0.911685i	$0.365218\pi$
$252$	0	0
$253$	5449.04	1.35406
$254$	1315.53	0.324975
$255$	0	0
$256$	256.000	0.0625000
$257$	3290.22	0.798593	0.399297	−	0.916822i	$-0.369254\pi$
$257$	3290.22	0.798593	0.399297	+	0.916822i	$0.369254\pi$
$258$	0	0
$259$	−170.980	−0.0410200
$260$	−2605.94	−0.621590
$261$	0	0
$262$	−4206.84	−0.991981
$263$	1573.04	0.368813	0.184407	−	0.982850i	$-0.440964\pi$
$263$	1573.04	0.368813	0.184407	+	0.982850i	$0.440964\pi$
$264$	0	0
$265$	662.174	0.153498
$266$	−286.599	−0.0660620
$267$	0	0
$268$	268.000	0.0610847
$269$	6568.74	1.48886	0.744430	−	0.667701i	$-0.232721\pi$
$269$	6568.74	1.48886	0.744430	+	0.667701i	$0.232721\pi$
$270$	0	0
$271$	623.524	0.139765	0.0698826	−	0.997555i	$-0.477738\pi$
$271$	623.524	0.139765	0.0698826	+	0.997555i	$0.477738\pi$
$272$	−611.606	−0.136339
$273$	0	0
$274$	1888.94	0.416479
$275$	−3326.50	−0.729438
$276$	0	0
$277$	−7358.89	−1.59622	−0.798110	−	0.602512i	$-0.794167\pi$
$277$	−7358.89	−1.59622	−0.798110	+	0.602512i	$0.794167\pi$
$278$	2111.17	0.455465
$279$	0	0
$280$	445.628	0.0951121
$281$	4795.86	1.01814	0.509070	−	0.860725i	$-0.329989\pi$
$281$	4795.86	1.01814	0.509070	+	0.860725i	$0.329989\pi$
$282$	0	0
$283$	−1406.81	−0.295499	−0.147749	−	0.989025i	$-0.547203\pi$
$283$	−1406.81	−0.295499	−0.147749	+	0.989025i	$0.547203\pi$
$284$	−1838.55	−0.384147
$285$	0	0
$286$	8470.34	1.75127
$287$	−952.864	−0.195978
$288$	0	0
$289$	−3451.82	−0.702589
$290$	732.025	0.148228
$291$	0	0
$292$	980.511	0.196507
$293$	5850.21	1.16646	0.583231	−	0.812307i	$-0.301788\pi$
$293$	5850.21	1.16646	0.583231	+	0.812307i	$0.301788\pi$
$294$	0	0
$295$	926.190	0.182796
$296$	−186.274	−0.0365776
$297$	0	0
$298$	−3431.67	−0.667085
$299$	−9489.77	−1.83548
$300$	0	0
$301$	30.3792	0.00581737
$302$	−5091.29	−0.970102
$303$	0	0
$304$	−312.235	−0.0589077
$305$	3283.16	0.616371
$306$	0	0
$307$	7193.11	1.33724	0.668619	−	0.743605i	$-0.266886\pi$
$307$	7193.11	1.33724	0.668619	+	0.743605i	$0.266886\pi$
$308$	−1448.47	−0.267968
$309$	0	0
$310$	−3722.46	−0.682005
$311$	−6744.21	−1.22968	−0.614838	−	0.788653i	$-0.710779\pi$
$311$	−6744.21	−1.22968	−0.614838	+	0.788653i	$0.710779\pi$
$312$	0	0
$313$	−7456.91	−1.34661	−0.673306	−	0.739364i	$-0.735126\pi$
$313$	−7456.91	−1.34661	−0.673306	+	0.739364i	$0.735126\pi$
$314$	3794.20	0.681908
$315$	0	0
$316$	2036.08	0.362463
$317$	7782.36	1.37887	0.689433	−	0.724349i	$-0.257860\pi$
$317$	7782.36	1.37887	0.689433	+	0.724349i	$0.257860\pi$
$318$	0	0
$319$	−2379.38	−0.417616
$320$	485.490	0.0848117
$321$	0	0
$322$	1622.80	0.280854
$323$	745.958	0.128502
$324$	0	0
$325$	5793.26	0.988776
$326$	−4402.01	−0.747867
$327$	0	0
$328$	−1038.10	−0.174754
$329$	−2658.94	−0.445568
$330$	0	0
$331$	−7079.55	−1.17561	−0.587805	−	0.809002i	$-0.700008\pi$
$331$	−7079.55	−1.17561	−0.587805	+	0.809002i	$0.700008\pi$
$332$	5742.86	0.949339
$333$	0	0
$334$	6846.91	1.12169
$335$	508.248	0.0828912
$336$	0	0
$337$	−1259.65	−0.203613	−0.101806	−	0.994804i	$-0.532462\pi$
$337$	−1259.65	−0.203613	−0.101806	+	0.994804i	$0.532462\pi$
$338$	−10357.5	−1.66679
$339$	0	0
$340$	−1159.88	−0.185010
$341$	12099.5	1.92148
$342$	0	0
$343$	4641.44	0.730654
$344$	33.0967	0.00518736
$345$	0	0
$346$	−113.815	−0.0176842
$347$	−5571.15	−0.861888	−0.430944	−	0.902379i	$-0.641819\pi$
$347$	−5571.15	−0.861888	−0.430944	+	0.902379i	$0.641819\pi$
$348$	0	0
$349$	−5949.15	−0.912467	−0.456233	−	0.889860i	$-0.650802\pi$
$349$	−5949.15	−0.912467	−0.456233	+	0.889860i	$0.650802\pi$
$350$	−990.676	−0.151297
$351$	0	0
$352$	−1578.04	−0.238948
$353$	−6709.95	−1.01171	−0.505856	−	0.862618i	$-0.668823\pi$
$353$	−6709.95	−1.01171	−0.505856	+	0.862618i	$0.668823\pi$
$354$	0	0
$355$	−3486.71	−0.521282
$356$	522.745	0.0778243
$357$	0	0
$358$	−5859.78	−0.865082
$359$	−4616.63	−0.678709	−0.339355	−	0.940659i	$-0.610209\pi$
$359$	−4616.63	−0.678709	−0.339355	+	0.940659i	$0.610209\pi$
$360$	0	0
$361$	−6478.18	−0.944478
$362$	−6874.64	−0.998130
$363$	0	0
$364$	2522.58	0.363240
$365$	1859.49	0.266658
$366$	0	0
$367$	−12424.2	−1.76713	−0.883565	−	0.468308i	$-0.844864\pi$
$367$	−12424.2	−1.76713	−0.883565	+	0.468308i	$0.844864\pi$
$368$	1767.96	0.250438
$369$	0	0
$370$	−353.259	−0.0496353
$371$	−640.994	−0.0897001
$372$	0	0
$373$	2319.28	0.321952	0.160976	−	0.986958i	$-0.448536\pi$
$373$	2319.28	0.321952	0.160976	+	0.986958i	$0.448536\pi$
$374$	3770.07	0.521245
$375$	0	0
$376$	−2896.78	−0.397314
$377$	4143.80	0.566092
$378$	0	0
$379$	5341.33	0.723920	0.361960	−	0.932194i	$-0.382108\pi$
$379$	5341.33	0.723920	0.361960	+	0.932194i	$0.382108\pi$
$380$	−592.138	−0.0799369
$381$	0	0
$382$	−339.992	−0.0455379
$383$	8463.73	1.12918	0.564591	−	0.825371i	$-0.309034\pi$
$383$	8463.73	1.12918	0.564591	+	0.825371i	$0.309034\pi$
$384$	0	0
$385$	−2746.95	−0.363630
$386$	−3415.08	−0.450318
$387$	0	0
$388$	−480.178	−0.0628283
$389$	4747.20	0.618747	0.309374	−	0.950941i	$-0.399881\pi$
$389$	4747.20	0.618747	0.309374	+	0.950941i	$0.399881\pi$
$390$	0	0
$391$	−4223.81	−0.546310
$392$	2312.63	0.297973
$393$	0	0
$394$	−2618.12	−0.334769
$395$	3861.31	0.491857
$396$	0	0
$397$	−12501.0	−1.58037	−0.790187	−	0.612866i	$-0.790016\pi$
$397$	−12501.0	−1.58037	−0.790187	+	0.612866i	$0.790016\pi$
$398$	−5408.03	−0.681105
$399$	0	0
$400$	−1079.29	−0.134912
$401$	12820.6	1.59658	0.798292	−	0.602270i	$-0.205737\pi$
$401$	12820.6	1.59658	0.798292	+	0.602270i	$0.205737\pi$
$402$	0	0
$403$	−21071.8	−2.60462
$404$	−3219.31	−0.396452
$405$	0	0
$406$	−708.610	−0.0866201
$407$	1148.23	0.139842
$408$	0	0
$409$	−5571.01	−0.673518	−0.336759	−	0.941591i	$-0.609331\pi$
$409$	−5571.01	−0.673518	−0.336759	+	0.941591i	$0.609331\pi$
$410$	−1968.70	−0.237139
$411$	0	0
$412$	−1160.06	−0.138718
$413$	−896.565	−0.106821
$414$	0	0
$415$	10891.0	1.28824
$416$	2748.23	0.323902
$417$	0	0
$418$	1924.69	0.225214
$419$	3557.91	0.414834	0.207417	−	0.978253i	$-0.433494\pi$
$419$	3557.91	0.414834	0.207417	+	0.978253i	$0.433494\pi$
$420$	0	0
$421$	−515.408	−0.0596662	−0.0298331	−	0.999555i	$-0.509498\pi$
$421$	−515.408	−0.0596662	−0.0298331	+	0.999555i	$0.509498\pi$
$422$	1825.26	0.210550
$423$	0	0
$424$	−698.331	−0.0799858
$425$	2578.53	0.294299
$426$	0	0
$427$	−3178.14	−0.360190
$428$	−1826.07	−0.206230
$429$	0	0
$430$	62.7661	0.00703918
$431$	8536.99	0.954089	0.477044	−	0.878879i	$-0.341708\pi$
$431$	8536.99	0.954089	0.477044	+	0.878879i	$0.341708\pi$
$432$	0	0
$433$	11219.4	1.24520	0.622598	−	0.782542i	$-0.286077\pi$
$433$	11219.4	1.24520	0.622598	+	0.782542i	$0.286077\pi$
$434$	3603.39	0.398544
$435$	0	0
$436$	−1643.70	−0.180548
$437$	−2156.33	−0.236044
$438$	0	0
$439$	12665.0	1.37692	0.688462	−	0.725273i	$-0.258286\pi$
$439$	12665.0	1.37692	0.688462	+	0.725273i	$0.258286\pi$
$440$	−2992.67	−0.324249
$441$	0	0
$442$	−6565.77	−0.706565
$443$	−10382.6	−1.11352	−0.556762	−	0.830672i	$-0.687956\pi$
$443$	−10382.6	−1.11352	−0.556762	+	0.830672i	$0.687956\pi$
$444$	0	0
$445$	991.358	0.105606
$446$	−1958.17	−0.207897
$447$	0	0
$448$	−469.961	−0.0495616
$449$	15434.7	1.62229	0.811147	−	0.584843i	$-0.198844\pi$
$449$	15434.7	1.62229	0.811147	+	0.584843i	$0.198844\pi$
$450$	0	0
$451$	6399.07	0.668116
$452$	3279.42	0.341264
$453$	0	0
$454$	−2043.66	−0.211263
$455$	4783.95	0.492912
$456$	0	0
$457$	−354.364	−0.0362723	−0.0181361	−	0.999836i	$-0.505773\pi$
$457$	−354.364	−0.0362723	−0.0181361	+	0.999836i	$0.505773\pi$
$458$	−12611.6	−1.28668
$459$	0	0
$460$	3352.84	0.339841
$461$	367.463	0.0371247	0.0185623	−	0.999828i	$-0.494091\pi$
$461$	367.463	0.0371247	0.0185623	+	0.999828i	$0.494091\pi$
$462$	0	0
$463$	−4909.26	−0.492770	−0.246385	−	0.969172i	$-0.579243\pi$
$463$	−4909.26	−0.492770	−0.246385	+	0.969172i	$0.579243\pi$
$464$	−771.997	−0.0772393
$465$	0	0
$466$	−9516.86	−0.946052
$467$	−16158.5	−1.60113	−0.800565	−	0.599246i	$-0.795467\pi$
$467$	−16158.5	−1.60113	−0.800565	+	0.599246i	$0.795467\pi$
$468$	0	0
$469$	−491.991	−0.0484393
$470$	−5493.59	−0.539150
$471$	0	0
$472$	−976.764	−0.0952525
$473$	−204.015	−0.0198322
$474$	0	0
$475$	1316.38	0.127157
$476$	1122.78	0.108114
$477$	0	0
$478$	−7084.83	−0.677934
$479$	−18030.3	−1.71988	−0.859941	−	0.510393i	$-0.829500\pi$
$479$	−18030.3	−1.71988	−0.859941	+	0.510393i	$0.829500\pi$
$480$	0	0
$481$	−1999.71	−0.189561
$482$	−5852.40	−0.553049
$483$	0	0
$484$	4403.37	0.413539
$485$	−910.633	−0.0852571
$486$	0	0
$487$	13975.5	1.30039	0.650196	−	0.759767i	$-0.274687\pi$
$487$	13975.5	1.30039	0.650196	+	0.759767i	$0.274687\pi$
$488$	−3462.43	−0.321182
$489$	0	0
$490$	4385.77	0.404345
$491$	−1749.26	−0.160780	−0.0803898	−	0.996764i	$-0.525617\pi$
$491$	−1749.26	−0.160780	−0.0803898	+	0.996764i	$0.525617\pi$
$492$	0	0
$493$	1844.37	0.168491
$494$	−3351.94	−0.305285
$495$	0	0
$496$	3925.72	0.355383
$497$	3375.18	0.304623
$498$	0	0
$499$	−1966.06	−0.176378	−0.0881891	−	0.996104i	$-0.528108\pi$
$499$	−1966.06	−0.176378	−0.0881891	+	0.996104i	$0.528108\pi$
$500$	−5839.72	−0.522320
$501$	0	0
$502$	−6535.75	−0.581085
$503$	12092.8	1.07195	0.535975	−	0.844234i	$-0.319944\pi$
$503$	12092.8	1.07195	0.535975	+	0.844234i	$0.319944\pi$
$504$	0	0
$505$	−6105.24	−0.537980
$506$	−10898.1	−0.957468
$507$	0	0
$508$	−2631.06	−0.229792
$509$	13218.4	1.15107	0.575535	−	0.817777i	$-0.304794\pi$
$509$	13218.4	1.15107	0.575535	+	0.817777i	$0.304794\pi$
$510$	0	0
$511$	−1800.01	−0.155827
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−6580.45	−0.564691
$515$	−2199.99	−0.188239
$516$	0	0
$517$	17856.4	1.51900
$518$	341.960	0.0290055
$519$	0	0
$520$	5211.88	0.439530
$521$	−12583.5	−1.05814	−0.529071	−	0.848578i	$-0.677459\pi$
$521$	−12583.5	−1.05814	−0.529071	+	0.848578i	$0.677459\pi$
$522$	0	0
$523$	−16948.7	−1.41704	−0.708521	−	0.705690i	$-0.750637\pi$
$523$	−16948.7	−1.41704	−0.708521	+	0.705690i	$0.750637\pi$
$524$	8413.67	0.701437
$525$	0	0
$526$	−3146.08	−0.260790
$527$	−9378.89	−0.775238
$528$	0	0
$529$	42.6919	0.00350883
$530$	−1324.35	−0.108540
$531$	0	0
$532$	573.198	0.0467129
$533$	−11144.3	−0.905652
$534$	0	0
$535$	−3463.05	−0.279852
$536$	−536.000	−0.0431934
$537$	0	0
$538$	−13137.5	−1.05278
$539$	−14255.5	−1.13920
$540$	0	0
$541$	−2701.35	−0.214677	−0.107338	−	0.994223i	$-0.534233\pi$
$541$	−2701.35	−0.214677	−0.107338	+	0.994223i	$0.534233\pi$
$542$	−1247.05	−0.0988290
$543$	0	0
$544$	1223.21	0.0964059
$545$	−3117.20	−0.245002
$546$	0	0
$547$	−3787.38	−0.296045	−0.148023	−	0.988984i	$-0.547291\pi$
$547$	−3787.38	−0.296045	−0.148023	+	0.988984i	$0.547291\pi$
$548$	−3777.89	−0.294495
$549$	0	0
$550$	6653.00	0.515790
$551$	941.581	0.0727998
$552$	0	0
$553$	−3737.80	−0.287428
$554$	14717.8	1.12870
$555$	0	0
$556$	−4222.33	−0.322062
$557$	−16331.9	−1.24238	−0.621189	−	0.783661i	$-0.713350\pi$
$557$	−16331.9	−1.24238	−0.621189	+	0.783661i	$0.713350\pi$
$558$	0	0
$559$	355.302	0.0268831
$560$	−891.257	−0.0672544
$561$	0	0
$562$	−9591.73	−0.719934
$563$	−358.864	−0.0268638	−0.0134319	−	0.999910i	$-0.504276\pi$
$563$	−358.864	−0.0268638	−0.0134319	+	0.999910i	$0.504276\pi$
$564$	0	0
$565$	6219.25	0.463090
$566$	2813.62	0.208949
$567$	0	0
$568$	3677.09	0.271633
$569$	−4110.83	−0.302874	−0.151437	−	0.988467i	$-0.548390\pi$
$569$	−4110.83	−0.302874	−0.151437	+	0.988467i	$0.548390\pi$
$570$	0	0
$571$	−17546.5	−1.28598	−0.642992	−	0.765873i	$-0.722307\pi$
$571$	−17546.5	−1.28598	−0.642992	+	0.765873i	$0.722307\pi$
$572$	−16940.7	−1.23833
$573$	0	0
$574$	1905.73	0.138578
$575$	−7453.70	−0.540593
$576$	0	0
$577$	20266.8	1.46225	0.731124	−	0.682245i	$-0.238996\pi$
$577$	20266.8	1.46225	0.731124	+	0.682245i	$0.238996\pi$
$578$	6903.64	0.496805
$579$	0	0
$580$	−1464.05	−0.104813
$581$	−10542.7	−0.752812
$582$	0	0
$583$	4304.66	0.305799
$584$	−1961.02	−0.138952
$585$	0	0
$586$	−11700.4	−0.824813
$587$	11811.6	0.830522	0.415261	−	0.909702i	$-0.363690\pi$
$587$	11811.6	0.830522	0.415261	+	0.909702i	$0.363690\pi$
$588$	0	0
$589$	−4788.08	−0.334957
$590$	−1852.38	−0.129256
$591$	0	0
$592$	372.548	0.0258643
$593$	−2288.09	−0.158449	−0.0792247	−	0.996857i	$-0.525244\pi$
$593$	−2288.09	−0.158449	−0.0792247	+	0.996857i	$0.525244\pi$
$594$	0	0
$595$	2129.29	0.146710
$596$	6863.34	0.471701
$597$	0	0
$598$	18979.5	1.29788
$599$	−7487.78	−0.510755	−0.255378	−	0.966841i	$-0.582200\pi$
$599$	−7487.78	−0.510755	−0.255378	+	0.966841i	$0.582200\pi$
$600$	0	0
$601$	−25440.1	−1.72666	−0.863330	−	0.504639i	$-0.831626\pi$
$601$	−25440.1	−1.72666	−0.863330	+	0.504639i	$0.831626\pi$
$602$	−60.7584	−0.00411350
$603$	0	0
$604$	10182.6	0.685966
$605$	8350.75	0.561167
$606$	0	0
$607$	6300.94	0.421330	0.210665	−	0.977558i	$-0.432437\pi$
$607$	6300.94	0.421330	0.210665	+	0.977558i	$0.432437\pi$
$608$	624.471	0.0416540
$609$	0	0
$610$	−6566.32	−0.435840
$611$	−31097.8	−2.05905
$612$	0	0
$613$	−25317.6	−1.66814	−0.834070	−	0.551659i	$-0.813995\pi$
$613$	−25317.6	−1.66814	−0.834070	+	0.551659i	$0.813995\pi$
$614$	−14386.2	−0.945571
$615$	0	0
$616$	2896.94	0.189482
$617$	−2752.37	−0.179589	−0.0897943	−	0.995960i	$-0.528621\pi$
$617$	−2752.37	−0.179589	−0.0897943	+	0.995960i	$0.528621\pi$
$618$	0	0
$619$	14353.3	0.932002	0.466001	−	0.884784i	$-0.345694\pi$
$619$	14353.3	0.932002	0.466001	+	0.884784i	$0.345694\pi$
$620$	7444.92	0.482250
$621$	0	0
$622$	13488.4	0.869512
$623$	−959.648	−0.0617135
$624$	0	0
$625$	−2642.73	−0.169135
$626$	14913.8	0.952198
$627$	0	0
$628$	−7588.40	−0.482182
$629$	−890.051	−0.0564207
$630$	0	0
$631$	20483.1	1.29226	0.646132	−	0.763226i	$-0.276386\pi$
$631$	20483.1	1.29226	0.646132	+	0.763226i	$0.276386\pi$
$632$	−4072.15	−0.256300
$633$	0	0
$634$	−15564.7	−0.975006
$635$	−4989.66	−0.311825
$636$	0	0
$637$	24826.7	1.54422
$638$	4758.75	0.295299
$639$	0	0
$640$	−970.981	−0.0599709
$641$	12312.3	0.758670	0.379335	−	0.925259i	$-0.376153\pi$
$641$	12312.3	0.758670	0.379335	+	0.925259i	$0.376153\pi$
$642$	0	0
$643$	−151.612	−0.00929860	−0.00464930	−	0.999989i	$-0.501480\pi$
$643$	−151.612	−0.00929860	−0.00464930	+	0.999989i	$0.501480\pi$
$644$	−3245.60	−0.198594
$645$	0	0
$646$	−1491.92	−0.0908648
$647$	−19148.5	−1.16353	−0.581765	−	0.813357i	$-0.697638\pi$
$647$	−19148.5	−1.16353	−0.581765	+	0.813357i	$0.697638\pi$
$648$	0	0
$649$	6020.98	0.364167
$650$	−11586.5	−0.699170
$651$	0	0
$652$	8804.02	0.528822
$653$	−3586.14	−0.214910	−0.107455	−	0.994210i	$-0.534270\pi$
$653$	−3586.14	−0.214910	−0.107455	+	0.994210i	$0.534270\pi$
$654$	0	0
$655$	15956.1	0.951840
$656$	2076.20	0.123570
$657$	0	0
$658$	5317.87	0.315064
$659$	−2557.93	−0.151203	−0.0756016	−	0.997138i	$-0.524088\pi$
$659$	−2557.93	−0.151203	−0.0756016	+	0.997138i	$0.524088\pi$
$660$	0	0
$661$	3819.81	0.224771	0.112385	−	0.993665i	$-0.464151\pi$
$661$	3819.81	0.224771	0.112385	+	0.993665i	$0.464151\pi$
$662$	14159.1	0.831282
$663$	0	0
$664$	−11485.7	−0.671284
$665$	1087.04	0.0633888
$666$	0	0
$667$	−5331.48	−0.309499
$668$	−13693.8	−0.793158
$669$	0	0
$670$	−1016.50	−0.0586129
$671$	21343.2	1.22793
$672$	0	0
$673$	−33525.5	−1.92023	−0.960114	−	0.279610i	$-0.909795\pi$
$673$	−33525.5	−1.92023	−0.960114	+	0.279610i	$0.909795\pi$
$674$	2519.30	0.143976
$675$	0	0
$676$	20715.0	1.17860
$677$	−28910.7	−1.64125	−0.820626	−	0.571466i	$-0.806375\pi$
$677$	−28910.7	−1.64125	−0.820626	+	0.571466i	$0.806375\pi$
$678$	0	0
$679$	881.505	0.0498219
$680$	2319.76	0.130822
$681$	0	0
$682$	−24199.0	−1.35869
$683$	25271.7	1.41580	0.707902	−	0.706310i	$-0.249642\pi$
$683$	25271.7	1.41580	0.707902	+	0.706310i	$0.249642\pi$
$684$	0	0
$685$	−7164.56	−0.399626
$686$	−9282.88	−0.516650
$687$	0	0
$688$	−66.1934	−0.00366802
$689$	−7496.78	−0.414521
$690$	0	0
$691$	29074.6	1.60065	0.800326	−	0.599564i	$-0.204660\pi$
$691$	29074.6	1.60065	0.800326	+	0.599564i	$0.204660\pi$
$692$	227.630	0.0125046
$693$	0	0
$694$	11142.3	0.609447
$695$	−8007.43	−0.437034
$696$	0	0
$697$	−4960.22	−0.269558
$698$	11898.3	0.645212
$699$	0	0
$700$	1981.35	0.106983
$701$	−33065.0	−1.78152	−0.890762	−	0.454469i	$-0.849829\pi$
$701$	−33065.0	−1.78152	−0.890762	+	0.454469i	$0.849829\pi$
$702$	0	0
$703$	−454.386	−0.0243777
$704$	3156.08	0.168962
$705$	0	0
$706$	13419.9	0.715388
$707$	5909.96	0.314380
$708$	0	0
$709$	−29313.8	−1.55276	−0.776378	−	0.630267i	$-0.782945\pi$
$709$	−29313.8	−1.55276	−0.776378	+	0.630267i	$0.782945\pi$
$710$	6973.41	0.368602
$711$	0	0
$712$	−1045.49	−0.0550301
$713$	27111.4	1.42402
$714$	0	0
$715$	−32127.1	−1.68040
$716$	11719.6	0.611705
$717$	0	0
$718$	9233.27	0.479920
$719$	9724.83	0.504416	0.252208	−	0.967673i	$-0.418843\pi$
$719$	9724.83	0.504416	0.252208	+	0.967673i	$0.418843\pi$
$720$	0	0
$721$	2129.62	0.110002
$722$	12956.4	0.667847
$723$	0	0
$724$	13749.3	0.705785
$725$	3254.73	0.166728
$726$	0	0
$727$	15422.0	0.786753	0.393376	−	0.919378i	$-0.371307\pi$
$727$	15422.0	0.786753	0.393376	+	0.919378i	$0.371307\pi$
$728$	−5045.17	−0.256849
$729$	0	0
$730$	−3718.98	−0.188555
$731$	158.142	0.00800148
$732$	0	0
$733$	3814.22	0.192198	0.0960991	−	0.995372i	$-0.469363\pi$
$733$	3814.22	0.192198	0.0960991	+	0.995372i	$0.469363\pi$
$734$	24848.4	1.24955
$735$	0	0
$736$	−3535.92	−0.177087
$737$	3304.02	0.165136
$738$	0	0
$739$	8246.03	0.410467	0.205234	−	0.978713i	$-0.434205\pi$
$739$	8246.03	0.410467	0.205234	+	0.978713i	$0.434205\pi$
$740$	706.518	0.0350975
$741$	0	0
$742$	1281.99	0.0634275
$743$	−30927.7	−1.52709	−0.763545	−	0.645754i	$-0.776543\pi$
$743$	−30927.7	−1.52709	−0.763545	+	0.645754i	$0.776543\pi$
$744$	0	0
$745$	13016.0	0.640091
$746$	−4638.57	−0.227654
$747$	0	0
$748$	−7540.14	−0.368576
$749$	3352.28	0.163538
$750$	0	0
$751$	39387.1	1.91379	0.956895	−	0.290434i	$-0.0937996\pi$
$751$	39387.1	1.91379	0.956895	+	0.290434i	$0.0937996\pi$
$752$	5793.56	0.280943
$753$	0	0
$754$	−8287.60	−0.400287
$755$	19310.7	0.930846
$756$	0	0
$757$	−13593.9	−0.652679	−0.326340	−	0.945253i	$-0.605815\pi$
$757$	−13593.9	−0.652679	−0.326340	+	0.945253i	$0.605815\pi$
$758$	−10682.7	−0.511889
$759$	0	0
$760$	1184.28	0.0565240
$761$	−32192.5	−1.53348	−0.766740	−	0.641958i	$-0.778122\pi$
$761$	−32192.5	−1.53348	−0.766740	+	0.641958i	$0.778122\pi$
$762$	0	0
$763$	3017.49	0.143172
$764$	679.983	0.0322002
$765$	0	0
$766$	−16927.5	−0.798452
$767$	−10485.8	−0.493639
$768$	0	0
$769$	−6182.46	−0.289916	−0.144958	−	0.989438i	$-0.546305\pi$
$769$	−6182.46	−0.289916	−0.144958	+	0.989438i	$0.546305\pi$
$770$	5493.90	0.257125
$771$	0	0
$772$	6830.15	0.318423
$773$	6656.15	0.309709	0.154855	−	0.987937i	$-0.450509\pi$
$773$	6656.15	0.309709	0.154855	+	0.987937i	$0.450509\pi$
$774$	0	0
$775$	−16550.8	−0.767125
$776$	960.357	0.0444263
$777$	0	0
$778$	−9494.40	−0.437520
$779$	−2532.28	−0.116468
$780$	0	0
$781$	−22666.4	−1.03850
$782$	8447.62	0.386300
$783$	0	0
$784$	−4625.25	−0.210698
$785$	−14391.0	−0.654314
$786$	0	0
$787$	28307.2	1.28214	0.641069	−	0.767483i	$-0.278491\pi$
$787$	28307.2	1.28214	0.641069	+	0.767483i	$0.278491\pi$
$788$	5236.25	0.236718
$789$	0	0
$790$	−7722.62	−0.347796
$791$	−6020.32	−0.270617
$792$	0	0
$793$	−37170.2	−1.66450
$794$	25002.0	1.11749
$795$	0	0
$796$	10816.1	0.481614
$797$	40495.2	1.79977	0.899883	−	0.436131i	$-0.143652\pi$
$797$	40495.2	1.79977	0.899883	+	0.436131i	$0.143652\pi$
$798$	0	0
$799$	−13841.3	−0.612855
$800$	2158.59	0.0953970
$801$	0	0
$802$	−25641.2	−1.12896
$803$	12088.2	0.531235
$804$	0	0
$805$	−6155.10	−0.269489
$806$	42143.7	1.84175
$807$	0	0
$808$	6438.61	0.280334
$809$	−11327.0	−0.492259	−0.246129	−	0.969237i	$-0.579159\pi$
$809$	−11327.0	−0.492259	−0.246129	+	0.969237i	$0.579159\pi$
$810$	0	0
$811$	−35693.4	−1.54546	−0.772728	−	0.634738i	$-0.781108\pi$
$811$	−35693.4	−1.54546	−0.772728	+	0.634738i	$0.781108\pi$
$812$	1417.22	0.0612496
$813$	0	0
$814$	−2296.47	−0.0988835
$815$	16696.4	0.717605
$816$	0	0
$817$	80.7340	0.00345719
$818$	11142.0	0.476249
$819$	0	0
$820$	3937.40	0.167683
$821$	29092.0	1.23668	0.618342	−	0.785909i	$-0.287805\pi$
$821$	29092.0	1.23668	0.618342	+	0.785909i	$0.287805\pi$
$822$	0	0
$823$	−24127.3	−1.02190	−0.510951	−	0.859610i	$-0.670707\pi$
$823$	−24127.3	−1.02190	−0.510951	+	0.859610i	$0.670707\pi$
$824$	2320.11	0.0980886
$825$	0	0
$826$	1793.13	0.0755338
$827$	−35157.3	−1.47828	−0.739141	−	0.673551i	$-0.764768\pi$
$827$	−35157.3	−1.47828	−0.739141	+	0.673551i	$0.764768\pi$
$828$	0	0
$829$	−4301.14	−0.180199	−0.0900994	−	0.995933i	$-0.528718\pi$
$829$	−4301.14	−0.180199	−0.0900994	+	0.995933i	$0.528718\pi$
$830$	−21782.1	−0.910924
$831$	0	0
$832$	−5496.46	−0.229033
$833$	11050.1	0.459621
$834$	0	0
$835$	−25969.6	−1.07630
$836$	−3849.37	−0.159250
$837$	0	0
$838$	−7115.83	−0.293332
$839$	23434.4	0.964297	0.482148	−	0.876090i	$-0.339857\pi$
$839$	23434.4	0.964297	0.482148	+	0.876090i	$0.339857\pi$
$840$	0	0
$841$	−22061.0	−0.904545
$842$	1030.82	0.0421904
$843$	0	0
$844$	−3650.52	−0.148882
$845$	39285.0	1.59934
$846$	0	0
$847$	−8083.64	−0.327931
$848$	1396.66	0.0565585
$849$	0	0
$850$	−5157.05	−0.208101
$851$	2572.85	0.103638
$852$	0	0
$853$	43169.4	1.73282	0.866409	−	0.499335i	$-0.166422\pi$
$853$	43169.4	1.73282	0.866409	+	0.499335i	$0.166422\pi$
$854$	6356.29	0.254693
$855$	0	0
$856$	3652.14	0.145827
$857$	−25259.7	−1.00683	−0.503417	−	0.864044i	$-0.667924\pi$
$857$	−25259.7	−1.00683	−0.503417	+	0.864044i	$0.667924\pi$
$858$	0	0
$859$	−14689.7	−0.583477	−0.291738	−	0.956498i	$-0.594234\pi$
$859$	−14689.7	−0.583477	−0.291738	+	0.956498i	$0.594234\pi$
$860$	−125.532	−0.00497746
$861$	0	0
$862$	−17074.0	−0.674643
$863$	−1606.89	−0.0633825	−0.0316912	−	0.999498i	$-0.510089\pi$
$863$	−1606.89	−0.0633825	−0.0316912	+	0.999498i	$0.510089\pi$
$864$	0	0
$865$	431.688	0.0169686
$866$	−22438.8	−0.880486
$867$	0	0
$868$	−7206.78	−0.281813
$869$	25101.6	0.979879
$870$	0	0
$871$	−5754.11	−0.223847
$872$	3287.41	0.127667
$873$	0	0
$874$	4312.65	0.166908
$875$	10720.5	0.414192
$876$	0	0
$877$	25512.6	0.982327	0.491163	−	0.871067i	$-0.336572\pi$
$877$	25512.6	0.982327	0.491163	+	0.871067i	$0.336572\pi$
$878$	−25330.1	−0.973632
$879$	0	0
$880$	5985.33	0.229279
$881$	10289.8	0.393500	0.196750	−	0.980454i	$-0.436961\pi$
$881$	10289.8	0.393500	0.196750	+	0.980454i	$0.436961\pi$
$882$	0	0
$883$	46838.2	1.78509	0.892543	−	0.450962i	$-0.148919\pi$
$883$	46838.2	1.78509	0.892543	+	0.450962i	$0.148919\pi$
$884$	13131.5	0.499617
$885$	0	0
$886$	20765.1	0.787380
$887$	−5464.28	−0.206846	−0.103423	−	0.994637i	$-0.532980\pi$
$887$	−5464.28	−0.206846	−0.103423	+	0.994637i	$0.532980\pi$
$888$	0	0
$889$	4830.06	0.182222
$890$	−1982.72	−0.0746751
$891$	0	0
$892$	3916.34	0.147005
$893$	−7066.23	−0.264796
$894$	0	0
$895$	22225.5	0.830076
$896$	939.923	0.0350453
$897$	0	0
$898$	−30869.4	−1.14713
$899$	−11838.4	−0.439193
$900$	0	0
$901$	−3336.75	−0.123378
$902$	−12798.1	−0.472429
$903$	0	0
$904$	−6558.85	−0.241310
$905$	26074.8	0.957741
$906$	0	0
$907$	−53161.2	−1.94618	−0.973092	−	0.230418i	$-0.925991\pi$
$907$	−53161.2	−1.94618	−0.973092	+	0.230418i	$0.925991\pi$
$908$	4087.31	0.149386
$909$	0	0
$910$	−9567.89	−0.348541
$911$	34785.9	1.26510	0.632551	−	0.774518i	$-0.282008\pi$
$911$	34785.9	1.26510	0.632551	+	0.774518i	$0.282008\pi$
$912$	0	0
$913$	70800.5	2.56643
$914$	708.727	0.0256484
$915$	0	0
$916$	25223.2	0.909822
$917$	−15445.7	−0.556229
$918$	0	0
$919$	41997.2	1.50747	0.753733	−	0.657181i	$-0.228251\pi$
$919$	41997.2	1.50747	0.753733	+	0.657181i	$0.228251\pi$
$920$	−6705.68	−0.240304
$921$	0	0
$922$	−734.927	−0.0262511
$923$	39474.6	1.40772
$924$	0	0
$925$	−1570.66	−0.0558303
$926$	9818.51	0.348441
$927$	0	0
$928$	1543.99	0.0546164
$929$	−9612.89	−0.339493	−0.169746	−	0.985488i	$-0.554295\pi$
$929$	−9612.89	−0.339493	−0.169746	+	0.985488i	$0.554295\pi$
$930$	0	0
$931$	5641.28	0.198588
$932$	19033.7	0.668960
$933$	0	0
$934$	32317.1	1.13217
$935$	−14299.5	−0.500153
$936$	0	0
$937$	55264.0	1.92678	0.963391	−	0.268099i	$-0.0863955\pi$
$937$	55264.0	1.92678	0.963391	+	0.268099i	$0.0863955\pi$
$938$	983.982	0.0342517
$939$	0	0
$940$	10987.2	0.381236
$941$	18748.5	0.649503	0.324752	−	0.945799i	$-0.394719\pi$
$941$	18748.5	0.649503	0.324752	+	0.945799i	$0.394719\pi$
$942$	0	0
$943$	14338.4	0.495147
$944$	1953.53	0.0673537
$945$	0	0
$946$	408.030	0.0140235
$947$	2918.76	0.100155	0.0500776	−	0.998745i	$-0.484053\pi$
$947$	2918.76	0.100155	0.0500776	+	0.998745i	$0.484053\pi$
$948$	0	0
$949$	−21052.1	−0.720107
$950$	−2632.76	−0.0899138
$951$	0	0
$952$	−2245.56	−0.0764485
$953$	30385.2	1.03282	0.516408	−	0.856343i	$-0.327269\pi$
$953$	30385.2	1.03282	0.516408	+	0.856343i	$0.327269\pi$
$954$	0	0
$955$	1289.55	0.0436952
$956$	14169.7	0.479372
$957$	0	0
$958$	36060.5	1.21614
$959$	6935.39	0.233530
$960$	0	0
$961$	30409.3	1.02075
$962$	3999.41	0.134040
$963$	0	0
$964$	11704.8	0.391065
$965$	12953.0	0.432096
$966$	0	0
$967$	−3072.95	−0.102192	−0.0510958	−	0.998694i	$-0.516271\pi$
$967$	−3072.95	−0.102192	−0.0510958	+	0.998694i	$0.516271\pi$
$968$	−8806.73	−0.292417
$969$	0	0
$970$	1821.27	0.0602859
$971$	−3272.57	−0.108158	−0.0540792	−	0.998537i	$-0.517222\pi$
$971$	−3272.57	−0.108158	−0.0540792	+	0.998537i	$0.517222\pi$
$972$	0	0
$973$	7751.30	0.255391
$974$	−27951.0	−0.919516
$975$	0	0
$976$	6924.86	0.227110
$977$	−17470.0	−0.572073	−0.286036	−	0.958219i	$-0.592338\pi$
$977$	−17470.0	−0.572073	−0.286036	+	0.958219i	$0.592338\pi$
$978$	0	0
$979$	6444.63	0.210389
$980$	−8771.54	−0.285915
$981$	0	0
$982$	3498.51	0.113688
$983$	−15870.9	−0.514957	−0.257478	−	0.966284i	$-0.582892\pi$
$983$	−15870.9	−0.514957	−0.257478	+	0.966284i	$0.582892\pi$
$984$	0	0
$985$	9930.26	0.321223
$986$	−3688.73	−0.119141
$987$	0	0
$988$	6703.87	0.215869
$989$	−457.137	−0.0146978
$990$	0	0
$991$	56377.2	1.80715	0.903573	−	0.428434i	$-0.140934\pi$
$991$	56377.2	1.80715	0.903573	+	0.428434i	$0.140934\pi$
$992$	−7851.44	−0.251294
$993$	0	0
$994$	−6750.36	−0.215401
$995$	20512.1	0.653544
$996$	0	0
$997$	−3589.00	−0.114007	−0.0570033	−	0.998374i	$-0.518155\pi$
$997$	−3589.00	−0.114007	−0.0570033	+	0.998374i	$0.518155\pi$
$998$	3932.11	0.124718
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1206.4.a.d.1.1		2
3.2	odd	2		402.4.a.b.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.4.a.b.1.2	✓	2	3.2	odd	2
1206.4.a.d.1.1		2	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:	\( N \)	\(=\)	\( 1206 = 2 \cdot 3^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1206.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +7.58579 q^{5} -7.34315 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} +7.58579 q^{5} -7.34315 q^{7} -8.00000 q^{8} -15.1716 q^{10} +49.3137 q^{11} -85.8823 q^{13} +14.6863 q^{14} +16.0000 q^{16} -38.2254 q^{17} -19.5147 q^{19} +30.3431 q^{20} -98.6274 q^{22} +110.497 q^{23} -67.4558 q^{25} +171.765 q^{26} -29.3726 q^{28} -48.2498 q^{29} +245.357 q^{31} -32.0000 q^{32} +76.4508 q^{34} -55.7035 q^{35} +23.2843 q^{37} +39.0294 q^{38} -60.6863 q^{40} +129.762 q^{41} -4.13708 q^{43} +197.255 q^{44} -220.995 q^{46} +362.098 q^{47} -289.078 q^{49} +134.912 q^{50} -343.529 q^{52} +87.2914 q^{53} +374.083 q^{55} +58.7452 q^{56} +96.4996 q^{58} +122.095 q^{59} +432.804 q^{61} -490.715 q^{62} +64.0000 q^{64} -651.484 q^{65} +67.0000 q^{67} -152.902 q^{68} +111.407 q^{70} -459.637 q^{71} +245.128 q^{73} -46.5685 q^{74} -78.0589 q^{76} -362.118 q^{77} +509.019 q^{79} +121.373 q^{80} -259.525 q^{82} +1435.72 q^{83} -289.970 q^{85} +8.27417 q^{86} -394.510 q^{88} +130.686 q^{89} +630.646 q^{91} +441.990 q^{92} -724.195 q^{94} -148.034 q^{95} -120.045 q^{97} +578.156 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} + 18 q^{5} - 26 q^{7} - 16 q^{8}+O(q^{10}) \) 2 * q - 4 * q^2 + 8 * q^4 + 18 * q^5 - 26 * q^7 - 16 * q^8	\( 2 q - 4 q^{2} + 8 q^{4} + 18 q^{5} - 26 q^{7} - 16 q^{8} - 36 q^{10} + 76 q^{11} - 36 q^{13} + 52 q^{14} + 32 q^{16} + 48 q^{17} - 56 q^{19} + 72 q^{20} - 152 q^{22} + 122 q^{23} - 84 q^{25} + 72 q^{26} - 104 q^{28} + 192 q^{29} - 58 q^{31} - 64 q^{32} - 96 q^{34} - 250 q^{35} - 10 q^{37} + 112 q^{38} - 144 q^{40} + 466 q^{41} + 218 q^{43} + 304 q^{44} - 244 q^{46} + 68 q^{47} - 284 q^{49} + 168 q^{50} - 144 q^{52} - 162 q^{53} + 652 q^{55} + 208 q^{56} - 384 q^{58} + 66 q^{59} + 1024 q^{61} + 116 q^{62} + 128 q^{64} - 132 q^{65} + 134 q^{67} + 192 q^{68} + 500 q^{70} - 116 q^{71} + 1022 q^{73} + 20 q^{74} - 224 q^{76} - 860 q^{77} + 656 q^{79} + 288 q^{80} - 932 q^{82} + 142 q^{83} + 608 q^{85} - 436 q^{86} - 608 q^{88} + 284 q^{89} - 300 q^{91} + 488 q^{92} - 136 q^{94} - 528 q^{95} - 868 q^{97} + 568 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 + 18 * q^5 - 26 * q^7 - 16 * q^8 - 36 * q^10 + 76 * q^11 - 36 * q^13 + 52 * q^14 + 32 * q^16 + 48 * q^17 - 56 * q^19 + 72 * q^20 - 152 * q^22 + 122 * q^23 - 84 * q^25 + 72 * q^26 - 104 * q^28 + 192 * q^29 - 58 * q^31 - 64 * q^32 - 96 * q^34 - 250 * q^35 - 10 * q^37 + 112 * q^38 - 144 * q^40 + 466 * q^41 + 218 * q^43 + 304 * q^44 - 244 * q^46 + 68 * q^47 - 284 * q^49 + 168 * q^50 - 144 * q^52 - 162 * q^53 + 652 * q^55 + 208 * q^56 - 384 * q^58 + 66 * q^59 + 1024 * q^61 + 116 * q^62 + 128 * q^64 - 132 * q^65 + 134 * q^67 + 192 * q^68 + 500 * q^70 - 116 * q^71 + 1022 * q^73 + 20 * q^74 - 224 * q^76 - 860 * q^77 + 656 * q^79 + 288 * q^80 - 932 * q^82 + 142 * q^83 + 608 * q^85 - 436 * q^86 - 608 * q^88 + 284 * q^89 - 300 * q^91 + 488 * q^92 - 136 * q^94 - 528 * q^95 - 868 * q^97 + 568 * q^98

Introduction

L-functions

Modular forms

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Database

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