LMFDB - Embedded newform 378.4.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, 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("378.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{23}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 23 $ x^2 - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.79583$ of defining polynomial
Character	$\chi$	$=$	378.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -18.3875 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} -18.3875 q^{5} -7.00000 q^{7} +8.00000 q^{8} -36.7750 q^{10} +53.1625 q^{11} +7.22501 q^{13} -14.0000 q^{14} +16.0000 q^{16} +88.0000 q^{17} -134.100 q^{19} -73.5500 q^{20} +106.325 q^{22} +155.162 q^{23} +213.100 q^{25} +14.4500 q^{26} -28.0000 q^{28} +119.550 q^{29} +321.975 q^{31} +32.0000 q^{32} +176.000 q^{34} +128.712 q^{35} +87.7750 q^{37} -268.200 q^{38} -147.100 q^{40} -226.587 q^{41} -62.7750 q^{43} +212.650 q^{44} +310.325 q^{46} +382.525 q^{47} +49.0000 q^{49} +426.200 q^{50} +28.9000 q^{52} -220.400 q^{53} -977.525 q^{55} -56.0000 q^{56} +239.100 q^{58} +800.975 q^{59} -813.050 q^{61} +643.950 q^{62} +64.0000 q^{64} -132.850 q^{65} -262.575 q^{67} +352.000 q^{68} +257.425 q^{70} +105.088 q^{71} -455.925 q^{73} +175.550 q^{74} -536.400 q^{76} -372.137 q^{77} +1164.72 q^{79} -294.200 q^{80} -453.175 q^{82} -414.650 q^{83} -1618.10 q^{85} -125.550 q^{86} +425.300 q^{88} -17.9375 q^{89} -50.5751 q^{91} +620.650 q^{92} +765.050 q^{94} +2465.76 q^{95} +1493.45 q^{97} +98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} - 8 q^{5} - 14 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 - 8 * q^5 - 14 * q^7 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} - 8 q^{5} - 14 q^{7} + 16 q^{8} - 16 q^{10} + 20 q^{11} + 72 q^{13} - 28 q^{14} + 32 q^{16} + 176 q^{17} - 38 q^{19} - 32 q^{20} + 40 q^{22} + 224 q^{23} + 196 q^{25} + 144 q^{26} - 56 q^{28} + 124 q^{29} + 126 q^{31} + 64 q^{32} + 352 q^{34} + 56 q^{35} + 118 q^{37} - 76 q^{38} - 64 q^{40} + 36 q^{41} - 68 q^{43} + 80 q^{44} + 448 q^{46} + 132 q^{47} + 98 q^{49} + 392 q^{50} + 288 q^{52} + 480 q^{53} - 1322 q^{55} - 112 q^{56} + 248 q^{58} + 1084 q^{59} - 360 q^{61} + 252 q^{62} + 128 q^{64} + 540 q^{65} - 928 q^{67} + 704 q^{68} + 112 q^{70} + 872 q^{71} - 1660 q^{73} + 236 q^{74} - 152 q^{76} - 140 q^{77} + 1236 q^{79} - 128 q^{80} + 72 q^{82} - 484 q^{83} - 704 q^{85} - 136 q^{86} + 160 q^{88} + 108 q^{89} - 504 q^{91} + 896 q^{92} + 264 q^{94} + 3464 q^{95} + 800 q^{97} + 196 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 - 8 * q^5 - 14 * q^7 + 16 * q^8 - 16 * q^10 + 20 * q^11 + 72 * q^13 - 28 * q^14 + 32 * q^16 + 176 * q^17 - 38 * q^19 - 32 * q^20 + 40 * q^22 + 224 * q^23 + 196 * q^25 + 144 * q^26 - 56 * q^28 + 124 * q^29 + 126 * q^31 + 64 * q^32 + 352 * q^34 + 56 * q^35 + 118 * q^37 - 76 * q^38 - 64 * q^40 + 36 * q^41 - 68 * q^43 + 80 * q^44 + 448 * q^46 + 132 * q^47 + 98 * q^49 + 392 * q^50 + 288 * q^52 + 480 * q^53 - 1322 * q^55 - 112 * q^56 + 248 * q^58 + 1084 * q^59 - 360 * q^61 + 252 * q^62 + 128 * q^64 + 540 * q^65 - 928 * q^67 + 704 * q^68 + 112 * q^70 + 872 * q^71 - 1660 * q^73 + 236 * q^74 - 152 * q^76 - 140 * q^77 + 1236 * q^79 - 128 * q^80 + 72 * q^82 - 484 * q^83 - 704 * q^85 - 136 * q^86 + 160 * q^88 + 108 * q^89 - 504 * q^91 + 896 * q^92 + 264 * q^94 + 3464 * q^95 + 800 * q^97 + 196 * 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
$5$	−18.3875	−1.64463	−0.822314	−	0.569034i	$-0.807317\pi$
$5$	−18.3875	−1.64463	−0.822314	+	0.569034i	$0.807317\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	−36.7750	−1.16293
$11$	53.1625	1.45719	0.728595	−	0.684945i	$-0.240174\pi$
$11$	53.1625	1.45719	0.728595	+	0.684945i	$0.240174\pi$
$12$	0	0
$13$	7.22501	0.154143	0.0770714	−	0.997026i	$-0.475443\pi$
$13$	7.22501	0.154143	0.0770714	+	0.997026i	$0.475443\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	88.0000	1.25548	0.627739	−	0.778424i	$-0.283980\pi$
$17$	88.0000	1.25548	0.627739	+	0.778424i	$0.283980\pi$
$18$	0	0
$19$	−134.100	−1.61919	−0.809596	−	0.586988i	$-0.800314\pi$
$19$	−134.100	−1.61919	−0.809596	+	0.586988i	$0.800314\pi$
$20$	−73.5500	−0.822314
$21$	0	0
$22$	106.325	1.03039
$23$	155.162	1.40668	0.703339	−	0.710854i	$-0.251691\pi$
$23$	155.162	1.40668	0.703339	+	0.710854i	$0.251691\pi$
$24$	0	0
$25$	213.100	1.70480
$26$	14.4500	0.108995
$27$	0	0
$28$	−28.0000	−0.188982
$29$	119.550	0.765513	0.382756	−	0.923849i	$-0.374975\pi$
$29$	119.550	0.765513	0.382756	+	0.923849i	$0.374975\pi$
$30$	0	0
$31$	321.975	1.86543	0.932716	−	0.360611i	$-0.117432\pi$
$31$	321.975	1.86543	0.932716	+	0.360611i	$0.117432\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	176.000	0.887757
$35$	128.712	0.621611
$36$	0	0
$37$	87.7750	0.390003	0.195002	−	0.980803i	$-0.437529\pi$
$37$	87.7750	0.390003	0.195002	+	0.980803i	$0.437529\pi$
$38$	−268.200	−1.14494
$39$	0	0
$40$	−147.100	−0.581464
$41$	−226.587	−0.863098	−0.431549	−	0.902090i	$-0.642033\pi$
$41$	−226.587	−0.863098	−0.431549	+	0.902090i	$0.642033\pi$
$42$	0	0
$43$	−62.7750	−0.222630	−0.111315	−	0.993785i	$-0.535506\pi$
$43$	−62.7750	−0.222630	−0.111315	+	0.993785i	$0.535506\pi$
$44$	212.650	0.728595
$45$	0	0
$46$	310.325	0.994672
$47$	382.525	1.18717	0.593585	−	0.804771i	$-0.297712\pi$
$47$	382.525	1.18717	0.593585	+	0.804771i	$0.297712\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	426.200	1.20548
$51$	0	0
$52$	28.9000	0.0770714
$53$	−220.400	−0.571212	−0.285606	−	0.958347i	$-0.592195\pi$
$53$	−220.400	−0.571212	−0.285606	+	0.958347i	$0.592195\pi$
$54$	0	0
$55$	−977.525	−2.39654
$56$	−56.0000	−0.133631
$57$	0	0
$58$	239.100	0.541299
$59$	800.975	1.76742	0.883712	−	0.468031i	$-0.155036\pi$
$59$	800.975	1.76742	0.883712	+	0.468031i	$0.155036\pi$
$60$	0	0
$61$	−813.050	−1.70656	−0.853282	−	0.521450i	$-0.825391\pi$
$61$	−813.050	−1.70656	−0.853282	+	0.521450i	$0.825391\pi$
$62$	643.950	1.31906
$63$	0	0
$64$	64.0000	0.125000
$65$	−132.850	−0.253508
$66$	0	0
$67$	−262.575	−0.478786	−0.239393	−	0.970923i	$-0.576948\pi$
$67$	−262.575	−0.478786	−0.239393	+	0.970923i	$0.576948\pi$
$68$	352.000	0.627739
$69$	0	0
$70$	257.425	0.439545
$71$	105.088	0.175656	0.0878282	−	0.996136i	$-0.472007\pi$
$71$	105.088	0.175656	0.0878282	+	0.996136i	$0.472007\pi$
$72$	0	0
$73$	−455.925	−0.730986	−0.365493	−	0.930814i	$-0.619100\pi$
$73$	−455.925	−0.730986	−0.365493	+	0.930814i	$0.619100\pi$
$74$	175.550	0.275774
$75$	0	0
$76$	−536.400	−0.809596
$77$	−372.137	−0.550766
$78$	0	0
$79$	1164.72	1.65876	0.829378	−	0.558687i	$-0.188695\pi$
$79$	1164.72	1.65876	0.829378	+	0.558687i	$0.188695\pi$
$80$	−294.200	−0.411157
$81$	0	0
$82$	−453.175	−0.610302
$83$	−414.650	−0.548358	−0.274179	−	0.961679i	$-0.588406\pi$
$83$	−414.650	−0.548358	−0.274179	+	0.961679i	$0.588406\pi$
$84$	0	0
$85$	−1618.10	−2.06479
$86$	−125.550	−0.157423
$87$	0	0
$88$	425.300	0.515195
$89$	−17.9375	−0.0213637	−0.0106818	−	0.999943i	$-0.503400\pi$
$89$	−17.9375	−0.0213637	−0.0106818	+	0.999943i	$0.503400\pi$
$90$	0	0
$91$	−50.5751	−0.0582605
$92$	620.650	0.703339
$93$	0	0
$94$	765.050	0.839456
$95$	2465.76	2.66297
$96$	0	0
$97$	1493.45	1.56327	0.781633	−	0.623738i	$-0.214387\pi$
$97$	1493.45	1.56327	0.781633	+	0.623738i	$0.214387\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	852.400	0.852400
$101$	590.100	0.581357	0.290679	−	0.956821i	$-0.406119\pi$
$101$	590.100	0.581357	0.290679	+	0.956821i	$0.406119\pi$
$102$	0	0
$103$	−1639.97	−1.56885	−0.784425	−	0.620224i	$-0.787042\pi$
$103$	−1639.97	−1.56885	−0.784425	+	0.620224i	$0.787042\pi$
$104$	57.8001	0.0544977
$105$	0	0
$106$	−440.800	−0.403908
$107$	−696.375	−0.629169	−0.314585	−	0.949229i	$-0.601865\pi$
$107$	−696.375	−0.629169	−0.314585	+	0.949229i	$0.601865\pi$
$108$	0	0
$109$	308.575	0.271157	0.135578	−	0.990767i	$-0.456711\pi$
$109$	308.575	0.271157	0.135578	+	0.990767i	$0.456711\pi$
$110$	−1955.05	−1.69461
$111$	0	0
$112$	−112.000	−0.0944911
$113$	−70.4747	−0.0586699	−0.0293350	−	0.999570i	$-0.509339\pi$
$113$	−70.4747	−0.0586699	−0.0293350	+	0.999570i	$0.509339\pi$
$114$	0	0
$115$	−2853.05	−2.31346
$116$	478.200	0.382756
$117$	0	0
$118$	1601.95	1.24976
$119$	−616.000	−0.474526
$120$	0	0
$121$	1495.25	1.12340
$122$	−1626.10	−1.20672
$123$	0	0
$124$	1287.90	0.932716
$125$	−1619.94	−1.15913
$126$	0	0
$127$	891.524	0.622913	0.311457	−	0.950260i	$-0.399183\pi$
$127$	891.524	0.622913	0.311457	+	0.950260i	$0.399183\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−265.700	−0.179257
$131$	477.225	0.318285	0.159143	−	0.987256i	$-0.449127\pi$
$131$	477.225	0.318285	0.159143	+	0.987256i	$0.449127\pi$
$132$	0	0
$133$	938.700	0.611997
$134$	−525.150	−0.338553
$135$	0	0
$136$	704.000	0.443879
$137$	−1505.10	−0.938609	−0.469304	−	0.883037i	$-0.655495\pi$
$137$	−1505.10	−0.938609	−0.469304	+	0.883037i	$0.655495\pi$
$138$	0	0
$139$	942.950	0.575396	0.287698	−	0.957721i	$-0.407110\pi$
$139$	942.950	0.575396	0.287698	+	0.957721i	$0.407110\pi$
$140$	514.850	0.310805
$141$	0	0
$142$	210.175	0.124208
$143$	384.100	0.224615
$144$	0	0
$145$	−2198.22	−1.25898
$146$	−911.850	−0.516885
$147$	0	0
$148$	351.100	0.195002
$149$	2020.27	1.11079	0.555394	−	0.831587i	$-0.312568\pi$
$149$	2020.27	1.11079	0.555394	+	0.831587i	$0.312568\pi$
$150$	0	0
$151$	−1693.25	−0.912548	−0.456274	−	0.889839i	$-0.650816\pi$
$151$	−1693.25	−0.912548	−0.456274	+	0.889839i	$0.650816\pi$
$152$	−1072.80	−0.572471
$153$	0	0
$154$	−744.275	−0.389450
$155$	−5920.31	−3.06794
$156$	0	0
$157$	1514.22	0.769734	0.384867	−	0.922972i	$-0.374247\pi$
$157$	1514.22	0.769734	0.384867	+	0.922972i	$0.374247\pi$
$158$	2329.45	1.17292
$159$	0	0
$160$	−588.400	−0.290732
$161$	−1086.14	−0.531675
$162$	0	0
$163$	−2745.22	−1.31916	−0.659578	−	0.751636i	$-0.729265\pi$
$163$	−2745.22	−1.31916	−0.659578	+	0.751636i	$0.729265\pi$
$164$	−906.350	−0.431549
$165$	0	0
$166$	−829.300	−0.387748
$167$	1492.17	0.691425	0.345712	−	0.938341i	$-0.387637\pi$
$167$	1492.17	0.691425	0.345712	+	0.938341i	$0.387637\pi$
$168$	0	0
$169$	−2144.80	−0.976240
$170$	−3236.20	−1.46003
$171$	0	0
$172$	−251.100	−0.111315
$173$	578.912	0.254416	0.127208	−	0.991876i	$-0.459398\pi$
$173$	578.912	0.254416	0.127208	+	0.991876i	$0.459398\pi$
$174$	0	0
$175$	−1491.70	−0.644354
$176$	850.600	0.364298
$177$	0	0
$178$	−35.8749	−0.0151064
$179$	−2311.42	−0.965162	−0.482581	−	0.875851i	$-0.660301\pi$
$179$	−2311.42	−0.965162	−0.482581	+	0.875851i	$0.660301\pi$
$180$	0	0
$181$	−534.449	−0.219477	−0.109738	−	0.993961i	$-0.535001\pi$
$181$	−534.449	−0.219477	−0.109738	+	0.993961i	$0.535001\pi$
$182$	−101.150	−0.0411964
$183$	0	0
$184$	1241.30	0.497336
$185$	−1613.96	−0.641410
$186$	0	0
$187$	4678.30	1.82947
$188$	1530.10	0.593585
$189$	0	0
$190$	4931.52	1.88300
$191$	−478.363	−0.181221	−0.0906103	−	0.995886i	$-0.528882\pi$
$191$	−478.363	−0.181221	−0.0906103	+	0.995886i	$0.528882\pi$
$192$	0	0
$193$	619.150	0.230919	0.115460	−	0.993312i	$-0.463166\pi$
$193$	619.150	0.230919	0.115460	+	0.993312i	$0.463166\pi$
$194$	2986.90	1.10540
$195$	0	0
$196$	196.000	0.0714286
$197$	4589.27	1.65976	0.829879	−	0.557944i	$-0.188410\pi$
$197$	4589.27	1.65976	0.829879	+	0.557944i	$0.188410\pi$
$198$	0	0
$199$	478.974	0.170621	0.0853106	−	0.996354i	$-0.472812\pi$
$199$	478.974	0.170621	0.0853106	+	0.996354i	$0.472812\pi$
$200$	1704.80	0.602738
$201$	0	0
$202$	1180.20	0.411082
$203$	−836.850	−0.289337
$204$	0	0
$205$	4166.37	1.41947
$206$	−3279.95	−1.10934
$207$	0	0
$208$	115.600	0.0385357
$209$	−7129.09	−2.35947
$210$	0	0
$211$	5354.40	1.74698	0.873488	−	0.486846i	$-0.161853\pi$
$211$	5354.40	1.74698	0.873488	+	0.486846i	$0.161853\pi$
$212$	−881.599	−0.285606
$213$	0	0
$214$	−1392.75	−0.444890
$215$	1154.27	0.366144
$216$	0	0
$217$	−2253.82	−0.705067
$218$	617.149	0.191737
$219$	0	0
$220$	−3910.10	−1.19827
$221$	635.801	0.193523
$222$	0	0
$223$	587.324	0.176368	0.0881841	−	0.996104i	$-0.471894\pi$
$223$	587.324	0.176368	0.0881841	+	0.996104i	$0.471894\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−140.949	−0.0414859
$227$	−1851.77	−0.541439	−0.270719	−	0.962658i	$-0.587262\pi$
$227$	−1851.77	−0.541439	−0.270719	+	0.962658i	$0.587262\pi$
$228$	0	0
$229$	3465.35	0.999986	0.499993	−	0.866030i	$-0.333336\pi$
$229$	3465.35	0.999986	0.499993	+	0.866030i	$0.333336\pi$
$230$	−5706.10	−1.63587
$231$	0	0
$232$	956.400	0.270650
$233$	2875.52	0.808506	0.404253	−	0.914647i	$-0.367532\pi$
$233$	2875.52	0.808506	0.404253	+	0.914647i	$0.367532\pi$
$234$	0	0
$235$	−7033.67	−1.95245
$236$	3203.90	0.883712
$237$	0	0
$238$	−1232.00	−0.335541
$239$	−4039.92	−1.09339	−0.546697	−	0.837331i	$-0.684115\pi$
$239$	−4039.92	−1.09339	−0.546697	+	0.837331i	$0.684115\pi$
$240$	0	0
$241$	−5236.72	−1.39970	−0.699849	−	0.714291i	$-0.746749\pi$
$241$	−5236.72	−1.39970	−0.699849	+	0.714291i	$0.746749\pi$
$242$	2990.50	0.794366
$243$	0	0
$244$	−3252.20	−0.853282
$245$	−900.987	−0.234947
$246$	0	0
$247$	−968.874	−0.249587
$248$	2575.80	0.659530
$249$	0	0
$250$	−3239.87	−0.819631
$251$	−5529.95	−1.39063	−0.695313	−	0.718707i	$-0.744734\pi$
$251$	−5529.95	−1.39063	−0.695313	+	0.718707i	$0.744734\pi$
$252$	0	0
$253$	8248.82	2.04980
$254$	1783.05	0.440466
$255$	0	0
$256$	256.000	0.0625000
$257$	1084.74	0.263284	0.131642	−	0.991297i	$-0.457975\pi$
$257$	1084.74	0.263284	0.131642	+	0.991297i	$0.457975\pi$
$258$	0	0
$259$	−614.425	−0.147407
$260$	−531.399	−0.126754
$261$	0	0
$262$	954.450	0.225062
$263$	6844.21	1.60469	0.802343	−	0.596864i	$-0.203587\pi$
$263$	6844.21	1.60469	0.802343	+	0.596864i	$0.203587\pi$
$264$	0	0
$265$	4052.60	0.939431
$266$	1877.40	0.432747
$267$	0	0
$268$	−1050.30	−0.239393
$269$	−5045.31	−1.14356	−0.571781	−	0.820406i	$-0.693747\pi$
$269$	−5045.31	−1.14356	−0.571781	+	0.820406i	$0.693747\pi$
$270$	0	0
$271$	−581.650	−0.130379	−0.0651896	−	0.997873i	$-0.520765\pi$
$271$	−581.650	−0.130379	−0.0651896	+	0.997873i	$0.520765\pi$
$272$	1408.00	0.313870
$273$	0	0
$274$	−3010.20	−0.663696
$275$	11328.9	2.48422
$276$	0	0
$277$	−6341.42	−1.37552	−0.687760	−	0.725938i	$-0.741406\pi$
$277$	−6341.42	−1.37552	−0.687760	+	0.725938i	$0.741406\pi$
$278$	1885.90	0.406866
$279$	0	0
$280$	1029.70	0.219773
$281$	−2422.48	−0.514280	−0.257140	−	0.966374i	$-0.582780\pi$
$281$	−2422.48	−0.514280	−0.257140	+	0.966374i	$0.582780\pi$
$282$	0	0
$283$	−6353.95	−1.33464	−0.667320	−	0.744771i	$-0.732559\pi$
$283$	−6353.95	−1.33464	−0.667320	+	0.744771i	$0.732559\pi$
$284$	420.350	0.0878282
$285$	0	0
$286$	768.199	0.158827
$287$	1586.11	0.326220
$288$	0	0
$289$	2831.00	0.576226
$290$	−4396.45	−0.890236
$291$	0	0
$292$	−1823.70	−0.365493
$293$	3726.55	0.743028	0.371514	−	0.928427i	$-0.378839\pi$
$293$	3726.55	0.743028	0.371514	+	0.928427i	$0.378839\pi$
$294$	0	0
$295$	−14727.9	−2.90676
$296$	702.200	0.137887
$297$	0	0
$298$	4040.55	0.785445
$299$	1121.05	0.216830
$300$	0	0
$301$	439.425	0.0841463
$302$	−3386.50	−0.645269
$303$	0	0
$304$	−2145.60	−0.404798
$305$	14949.9	2.80666
$306$	0	0
$307$	1644.20	0.305666	0.152833	−	0.988252i	$-0.451160\pi$
$307$	1644.20	0.305666	0.152833	+	0.988252i	$0.451160\pi$
$308$	−1488.55	−0.275383
$309$	0	0
$310$	−11840.6	−2.16936
$311$	−628.901	−0.114668	−0.0573339	−	0.998355i	$-0.518260\pi$
$311$	−628.901	−0.114668	−0.0573339	+	0.998355i	$0.518260\pi$
$312$	0	0
$313$	9479.12	1.71179	0.855897	−	0.517146i	$-0.173006\pi$
$313$	9479.12	1.71179	0.855897	+	0.517146i	$0.173006\pi$
$314$	3028.45	0.544284
$315$	0	0
$316$	4658.90	0.829378
$317$	−1259.17	−0.223098	−0.111549	−	0.993759i	$-0.535581\pi$
$317$	−1259.17	−0.223098	−0.111549	+	0.993759i	$0.535581\pi$
$318$	0	0
$319$	6355.57	1.11550
$320$	−1176.80	−0.205578
$321$	0	0
$322$	−2172.27	−0.375951
$323$	−11800.8	−2.03286
$324$	0	0
$325$	1539.65	0.262783
$326$	−5490.45	−0.932785
$327$	0	0
$328$	−1812.70	−0.305151
$329$	−2677.67	−0.448708
$330$	0	0
$331$	5103.05	0.847399	0.423699	−	0.905803i	$-0.360731\pi$
$331$	5103.05	0.847399	0.423699	+	0.905803i	$0.360731\pi$
$332$	−1658.60	−0.274179
$333$	0	0
$334$	2984.35	0.488911
$335$	4828.10	0.787424
$336$	0	0
$337$	−353.550	−0.0571486	−0.0285743	−	0.999592i	$-0.509097\pi$
$337$	−353.550	−0.0571486	−0.0285743	+	0.999592i	$0.509097\pi$
$338$	−4289.60	−0.690306
$339$	0	0
$340$	−6472.40	−1.03240
$341$	17117.0	2.71829
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−502.200	−0.0787116
$345$	0	0
$346$	1157.82	0.179899
$347$	10196.6	1.57748	0.788738	−	0.614729i	$-0.210735\pi$
$347$	10196.6	1.57748	0.788738	+	0.614729i	$0.210735\pi$
$348$	0	0
$349$	7361.55	1.12910	0.564548	−	0.825400i	$-0.309050\pi$
$349$	7361.55	1.12910	0.564548	+	0.825400i	$0.309050\pi$
$350$	−2983.40	−0.455627
$351$	0	0
$352$	1701.20	0.257597
$353$	8990.44	1.35556	0.677780	−	0.735265i	$-0.262942\pi$
$353$	8990.44	1.35556	0.677780	+	0.735265i	$0.262942\pi$
$354$	0	0
$355$	−1932.30	−0.288889
$356$	−71.7499	−0.0106818
$357$	0	0
$358$	−4622.85	−0.682472
$359$	2135.58	0.313959	0.156980	−	0.987602i	$-0.449824\pi$
$359$	2135.58	0.313959	0.156980	+	0.987602i	$0.449824\pi$
$360$	0	0
$361$	11123.8	1.62178
$362$	−1068.90	−0.155194
$363$	0	0
$364$	−202.300	−0.0291303
$365$	8383.32	1.20220
$366$	0	0
$367$	4367.92	0.621264	0.310632	−	0.950530i	$-0.399459\pi$
$367$	4367.92	0.621264	0.310632	+	0.950530i	$0.399459\pi$
$368$	2482.60	0.351670
$369$	0	0
$370$	−3227.92	−0.453545
$371$	1542.80	0.215898
$372$	0	0
$373$	1908.97	0.264995	0.132497	−	0.991183i	$-0.457700\pi$
$373$	1908.97	0.264995	0.132497	+	0.991183i	$0.457700\pi$
$374$	9356.60	1.29363
$375$	0	0
$376$	3060.20	0.419728
$377$	863.750	0.117998
$378$	0	0
$379$	−13762.9	−1.86531	−0.932653	−	0.360773i	$-0.882513\pi$
$379$	−13762.9	−1.86531	−0.932653	+	0.360773i	$0.882513\pi$
$380$	9863.05	1.33148
$381$	0	0
$382$	−956.726	−0.128142
$383$	−9759.32	−1.30203	−0.651016	−	0.759064i	$-0.725657\pi$
$383$	−9759.32	−1.30203	−0.651016	+	0.759064i	$0.725657\pi$
$384$	0	0
$385$	6842.67	0.905805
$386$	1238.30	0.163285
$387$	0	0
$388$	5973.80	0.781633
$389$	7673.10	1.00011	0.500054	−	0.865994i	$-0.333314\pi$
$389$	7673.10	1.00011	0.500054	+	0.865994i	$0.333314\pi$
$390$	0	0
$391$	13654.3	1.76606
$392$	392.000	0.0505076
$393$	0	0
$394$	9178.55	1.17363
$395$	−21416.4	−2.72804
$396$	0	0
$397$	−2959.80	−0.374176	−0.187088	−	0.982343i	$-0.559905\pi$
$397$	−2959.80	−0.374176	−0.187088	+	0.982343i	$0.559905\pi$
$398$	957.949	0.120647
$399$	0	0
$400$	3409.60	0.426200
$401$	−13432.8	−1.67283	−0.836414	−	0.548098i	$-0.815352\pi$
$401$	−13432.8	−1.67283	−0.836414	+	0.548098i	$0.815352\pi$
$402$	0	0
$403$	2326.27	0.287543
$404$	2360.40	0.290679
$405$	0	0
$406$	−1673.70	−0.204592
$407$	4666.34	0.568309
$408$	0	0
$409$	6747.38	0.815737	0.407868	−	0.913041i	$-0.366272\pi$
$409$	6747.38	0.815737	0.407868	+	0.913041i	$0.366272\pi$
$410$	8332.75	1.00372
$411$	0	0
$412$	−6559.90	−0.784425
$413$	−5606.82	−0.668024
$414$	0	0
$415$	7624.37	0.901845
$416$	231.200	0.0272489
$417$	0	0
$418$	−14258.2	−1.66840
$419$	−7894.93	−0.920507	−0.460253	−	0.887788i	$-0.652241\pi$
$419$	−7894.93	−0.920507	−0.460253	+	0.887788i	$0.652241\pi$
$420$	0	0
$421$	15412.8	1.78426	0.892132	−	0.451775i	$-0.149209\pi$
$421$	15412.8	1.78426	0.892132	+	0.451775i	$0.149209\pi$
$422$	10708.8	1.23530
$423$	0	0
$424$	−1763.20	−0.201954
$425$	18752.8	2.14034
$426$	0	0
$427$	5691.35	0.645020
$428$	−2785.50	−0.314585
$429$	0	0
$430$	2308.55	0.258903
$431$	5503.88	0.615111	0.307555	−	0.951530i	$-0.400489\pi$
$431$	5503.88	0.615111	0.307555	+	0.951530i	$0.400489\pi$
$432$	0	0
$433$	−13592.2	−1.50854	−0.754270	−	0.656564i	$-0.772009\pi$
$433$	−13592.2	−1.50854	−0.754270	+	0.656564i	$0.772009\pi$
$434$	−4507.65	−0.498558
$435$	0	0
$436$	1234.30	0.135578
$437$	−20807.3	−2.27768
$438$	0	0
$439$	−3179.60	−0.345681	−0.172840	−	0.984950i	$-0.555294\pi$
$439$	−3179.60	−0.345681	−0.172840	+	0.984950i	$0.555294\pi$
$440$	−7820.20	−0.847303
$441$	0	0
$442$	1271.60	0.136841
$443$	−2448.21	−0.262569	−0.131284	−	0.991345i	$-0.541910\pi$
$443$	−2448.21	−0.262569	−0.131284	+	0.991345i	$0.541910\pi$
$444$	0	0
$445$	329.825	0.0351353
$446$	1174.65	0.124711
$447$	0	0
$448$	−448.000	−0.0472456
$449$	1012.37	0.106407	0.0532036	−	0.998584i	$-0.483057\pi$
$449$	1012.37	0.106407	0.0532036	+	0.998584i	$0.483057\pi$
$450$	0	0
$451$	−12045.9	−1.25770
$452$	−281.899	−0.0293350
$453$	0	0
$454$	−3703.55	−0.382855
$455$	929.949	0.0958169
$456$	0	0
$457$	2642.60	0.270494	0.135247	−	0.990812i	$-0.456817\pi$
$457$	2642.60	0.270494	0.135247	+	0.990812i	$0.456817\pi$
$458$	6930.70	0.707097
$459$	0	0
$460$	−11412.2	−1.15673
$461$	−16767.3	−1.69399	−0.846996	−	0.531599i	$-0.821591\pi$
$461$	−16767.3	−1.69399	−0.846996	+	0.531599i	$0.821591\pi$
$462$	0	0
$463$	−11477.1	−1.15203	−0.576013	−	0.817440i	$-0.695392\pi$
$463$	−11477.1	−1.15203	−0.576013	+	0.817440i	$0.695392\pi$
$464$	1912.80	0.191378
$465$	0	0
$466$	5751.05	0.571700
$467$	6741.58	0.668015	0.334007	−	0.942570i	$-0.391599\pi$
$467$	6741.58	0.668015	0.334007	+	0.942570i	$0.391599\pi$
$468$	0	0
$469$	1838.03	0.180964
$470$	−14067.3	−1.38059
$471$	0	0
$472$	6407.80	0.624879
$473$	−3337.27	−0.324414
$474$	0	0
$475$	−28576.7	−2.76040
$476$	−2464.00	−0.237263
$477$	0	0
$478$	−8079.85	−0.773146
$479$	9145.20	0.872348	0.436174	−	0.899862i	$-0.356333\pi$
$479$	9145.20	0.872348	0.436174	+	0.899862i	$0.356333\pi$
$480$	0	0
$481$	634.175	0.0601162
$482$	−10473.4	−0.989736
$483$	0	0
$484$	5981.00	0.561702
$485$	−27460.8	−2.57099
$486$	0	0
$487$	−6828.32	−0.635361	−0.317681	−	0.948198i	$-0.602904\pi$
$487$	−6828.32	−0.635361	−0.317681	+	0.948198i	$0.602904\pi$
$488$	−6504.40	−0.603361
$489$	0	0
$490$	−1801.97	−0.166132
$491$	−18359.9	−1.68751	−0.843757	−	0.536726i	$-0.819661\pi$
$491$	−18359.9	−1.68751	−0.843757	+	0.536726i	$0.819661\pi$
$492$	0	0
$493$	10520.4	0.961085
$494$	−1937.75	−0.176485
$495$	0	0
$496$	5151.60	0.466358
$497$	−735.613	−0.0663919
$498$	0	0
$499$	−11674.0	−1.04730	−0.523648	−	0.851934i	$-0.675429\pi$
$499$	−11674.0	−1.04730	−0.523648	+	0.851934i	$0.675429\pi$
$500$	−6479.75	−0.579566
$501$	0	0
$502$	−11059.9	−0.983321
$503$	1256.30	0.111363	0.0556814	−	0.998449i	$-0.482267\pi$
$503$	1256.30	0.111363	0.0556814	+	0.998449i	$0.482267\pi$
$504$	0	0
$505$	−10850.5	−0.956116
$506$	16497.6	1.44943
$507$	0	0
$508$	3566.10	0.311457
$509$	−19052.4	−1.65910	−0.829549	−	0.558434i	$-0.811402\pi$
$509$	−19052.4	−1.65910	−0.829549	+	0.558434i	$0.811402\pi$
$510$	0	0
$511$	3191.48	0.276287
$512$	512.000	0.0441942
$513$	0	0
$514$	2169.48	0.186170
$515$	30155.0	2.58017
$516$	0	0
$517$	20336.0	1.72993
$518$	−1228.85	−0.104233
$519$	0	0
$520$	−1062.80	−0.0896285
$521$	−3395.14	−0.285497	−0.142748	−	0.989759i	$-0.545594\pi$
$521$	−3395.14	−0.285497	−0.142748	+	0.989759i	$0.545594\pi$
$522$	0	0
$523$	−11493.5	−0.960952	−0.480476	−	0.877008i	$-0.659536\pi$
$523$	−11493.5	−0.960952	−0.480476	+	0.877008i	$0.659536\pi$
$524$	1908.90	0.159143
$525$	0	0
$526$	13688.4	1.13468
$527$	28333.8	2.34201
$528$	0	0
$529$	11908.4	0.978745
$530$	8105.20	0.664278
$531$	0	0
$532$	3754.80	0.305998
$533$	−1637.10	−0.133040
$534$	0	0
$535$	12804.6	1.03475
$536$	−2100.60	−0.169276
$537$	0	0
$538$	−10090.6	−0.808620
$539$	2604.96	0.208170
$540$	0	0
$541$	−13248.6	−1.05287	−0.526433	−	0.850216i	$-0.676471\pi$
$541$	−13248.6	−1.05287	−0.526433	+	0.850216i	$0.676471\pi$
$542$	−1163.30	−0.0921920
$543$	0	0
$544$	2816.00	0.221939
$545$	−5673.91	−0.445952
$546$	0	0
$547$	9932.74	0.776405	0.388202	−	0.921574i	$-0.373096\pi$
$547$	9932.74	0.776405	0.388202	+	0.921574i	$0.373096\pi$
$548$	−6020.40	−0.469304
$549$	0	0
$550$	22657.8	1.75661
$551$	−16031.6	−1.23951
$552$	0	0
$553$	−8153.07	−0.626951
$554$	−12682.8	−0.972640
$555$	0	0
$556$	3771.80	0.287698
$557$	−8651.92	−0.658157	−0.329079	−	0.944303i	$-0.606738\pi$
$557$	−8651.92	−0.658157	−0.329079	+	0.944303i	$0.606738\pi$
$558$	0	0
$559$	−453.550	−0.0343169
$560$	2059.40	0.155403
$561$	0	0
$562$	−4844.95	−0.363651
$563$	−13554.0	−1.01462	−0.507310	−	0.861764i	$-0.669360\pi$
$563$	−13554.0	−1.01462	−0.507310	+	0.861764i	$0.669360\pi$
$564$	0	0
$565$	1295.85	0.0964902
$566$	−12707.9	−0.943733
$567$	0	0
$568$	840.701	0.0621039
$569$	−1204.72	−0.0887604	−0.0443802	−	0.999015i	$-0.514131\pi$
$569$	−1204.72	−0.0887604	−0.0443802	+	0.999015i	$0.514131\pi$
$570$	0	0
$571$	11598.8	0.850078	0.425039	−	0.905175i	$-0.360260\pi$
$571$	11598.8	0.850078	0.425039	+	0.905175i	$0.360260\pi$
$572$	1536.40	0.112308
$573$	0	0
$574$	3172.22	0.230673
$575$	33065.1	2.39811
$576$	0	0
$577$	11616.8	0.838148	0.419074	−	0.907952i	$-0.362355\pi$
$577$	11616.8	0.838148	0.419074	+	0.907952i	$0.362355\pi$
$578$	5662.00	0.407454
$579$	0	0
$580$	−8792.90	−0.629492
$581$	2902.55	0.207260
$582$	0	0
$583$	−11717.0	−0.832365
$584$	−3647.40	−0.258443
$585$	0	0
$586$	7453.10	0.525400
$587$	−20276.6	−1.42573	−0.712865	−	0.701301i	$-0.752603\pi$
$587$	−20276.6	−1.42573	−0.712865	+	0.701301i	$0.752603\pi$
$588$	0	0
$589$	−43176.8	−3.02049
$590$	−29455.8	−2.05539
$591$	0	0
$592$	1404.40	0.0975008
$593$	10819.3	0.749234	0.374617	−	0.927180i	$-0.377774\pi$
$593$	10819.3	0.749234	0.374617	+	0.927180i	$0.377774\pi$
$594$	0	0
$595$	11326.7	0.780419
$596$	8081.10	0.555394
$597$	0	0
$598$	2242.10	0.153322
$599$	7654.14	0.522103	0.261052	−	0.965325i	$-0.415931\pi$
$599$	7654.14	0.522103	0.261052	+	0.965325i	$0.415931\pi$
$600$	0	0
$601$	4349.23	0.295189	0.147595	−	0.989048i	$-0.452847\pi$
$601$	4349.23	0.295189	0.147595	+	0.989048i	$0.452847\pi$
$602$	878.850	0.0595004
$603$	0	0
$604$	−6773.00	−0.456274
$605$	−27493.9	−1.84758
$606$	0	0
$607$	5204.85	0.348037	0.174018	−	0.984742i	$-0.444325\pi$
$607$	5204.85	0.348037	0.174018	+	0.984742i	$0.444325\pi$
$608$	−4291.20	−0.286235
$609$	0	0
$610$	29899.9	1.98461
$611$	2763.75	0.182994
$612$	0	0
$613$	−23934.7	−1.57702	−0.788509	−	0.615023i	$-0.789147\pi$
$613$	−23934.7	−1.57702	−0.788509	+	0.615023i	$0.789147\pi$
$614$	3288.40	0.216138
$615$	0	0
$616$	−2977.10	−0.194725
$617$	−2840.68	−0.185351	−0.0926754	−	0.995696i	$-0.529542\pi$
$617$	−2840.68	−0.185351	−0.0926754	+	0.995696i	$0.529542\pi$
$618$	0	0
$619$	−23706.7	−1.53935	−0.769673	−	0.638439i	$-0.779581\pi$
$619$	−23706.7	−1.53935	−0.769673	+	0.638439i	$0.779581\pi$
$620$	−23681.2	−1.53397
$621$	0	0
$622$	−1257.80	−0.0810824
$623$	125.562	0.00807472
$624$	0	0
$625$	3149.10	0.201542
$626$	18958.2	1.21042
$627$	0	0
$628$	6056.90	0.384867
$629$	7724.20	0.489641
$630$	0	0
$631$	−31532.9	−1.98939	−0.994695	−	0.102865i	$-0.967199\pi$
$631$	−31532.9	−1.98939	−0.994695	+	0.102865i	$0.967199\pi$
$632$	9317.80	0.586459
$633$	0	0
$634$	−2518.35	−0.157754
$635$	−16392.9	−1.02446
$636$	0	0
$637$	354.026	0.0220204
$638$	12711.1	0.788776
$639$	0	0
$640$	−2353.60	−0.145366
$641$	−8246.47	−0.508137	−0.254069	−	0.967186i	$-0.581769\pi$
$641$	−8246.47	−0.508137	−0.254069	+	0.967186i	$0.581769\pi$
$642$	0	0
$643$	−15090.7	−0.925535	−0.462767	−	0.886480i	$-0.653143\pi$
$643$	−15090.7	−0.925535	−0.462767	+	0.886480i	$0.653143\pi$
$644$	−4344.55	−0.265837
$645$	0	0
$646$	−23601.6	−1.43745
$647$	18639.2	1.13259	0.566293	−	0.824204i	$-0.308377\pi$
$647$	18639.2	1.13259	0.566293	+	0.824204i	$0.308377\pi$
$648$	0	0
$649$	42581.8	2.57547
$650$	3079.30	0.185815
$651$	0	0
$652$	−10980.9	−0.659578
$653$	24107.7	1.44473	0.722364	−	0.691513i	$-0.243055\pi$
$653$	24107.7	1.44473	0.722364	+	0.691513i	$0.243055\pi$
$654$	0	0
$655$	−8774.97	−0.523460
$656$	−3625.40	−0.215774
$657$	0	0
$658$	−5355.35	−0.317285
$659$	10380.4	0.613598	0.306799	−	0.951774i	$-0.400742\pi$
$659$	10380.4	0.613598	0.306799	+	0.951774i	$0.400742\pi$
$660$	0	0
$661$	−5475.48	−0.322196	−0.161098	−	0.986938i	$-0.551503\pi$
$661$	−5475.48	−0.322196	−0.161098	+	0.986938i	$0.551503\pi$
$662$	10206.1	0.599201
$663$	0	0
$664$	−3317.20	−0.193874
$665$	−17260.3	−1.00651
$666$	0	0
$667$	18549.7	1.07683
$668$	5968.70	0.345712
$669$	0	0
$670$	9656.20	0.556793
$671$	−43223.7	−2.48679
$672$	0	0
$673$	3132.85	0.179439	0.0897194	−	0.995967i	$-0.471403\pi$
$673$	3132.85	0.179439	0.0897194	+	0.995967i	$0.471403\pi$
$674$	−707.099	−0.0404101
$675$	0	0
$676$	−8579.20	−0.488120
$677$	21992.1	1.24849	0.624244	−	0.781230i	$-0.285407\pi$
$677$	21992.1	1.24849	0.624244	+	0.781230i	$0.285407\pi$
$678$	0	0
$679$	−10454.1	−0.590859
$680$	−12944.8	−0.730015
$681$	0	0
$682$	34234.0	1.92212
$683$	1436.83	0.0804961	0.0402480	−	0.999190i	$-0.487185\pi$
$683$	1436.83	0.0804961	0.0402480	+	0.999190i	$0.487185\pi$
$684$	0	0
$685$	27675.0	1.54366
$686$	−686.000	−0.0381802
$687$	0	0
$688$	−1004.40	−0.0556575
$689$	−1592.39	−0.0880483
$690$	0	0
$691$	−3705.10	−0.203978	−0.101989	−	0.994786i	$-0.532521\pi$
$691$	−3705.10	−0.203978	−0.101989	+	0.994786i	$0.532521\pi$
$692$	2315.65	0.127208
$693$	0	0
$694$	20393.3	1.11544
$695$	−17338.5	−0.946312
$696$	0	0
$697$	−19939.7	−1.08360
$698$	14723.1	0.798392
$699$	0	0
$700$	−5966.80	−0.322177
$701$	−898.597	−0.0484159	−0.0242079	−	0.999707i	$-0.507706\pi$
$701$	−898.597	−0.0484159	−0.0242079	+	0.999707i	$0.507706\pi$
$702$	0	0
$703$	−11770.6	−0.631490
$704$	3402.40	0.182149
$705$	0	0
$706$	17980.9	0.958526
$707$	−4130.70	−0.219732
$708$	0	0
$709$	−7393.57	−0.391638	−0.195819	−	0.980640i	$-0.562737\pi$
$709$	−7393.57	−0.391638	−0.195819	+	0.980640i	$0.562737\pi$
$710$	−3864.60	−0.204276
$711$	0	0
$712$	−143.500	−0.00755320
$713$	49958.4	2.62406
$714$	0	0
$715$	−7062.63	−0.369409
$716$	−9245.70	−0.482581
$717$	0	0
$718$	4271.15	0.222003
$719$	3189.33	0.165427	0.0827135	−	0.996573i	$-0.473641\pi$
$719$	3189.33	0.165427	0.0827135	+	0.996573i	$0.473641\pi$
$720$	0	0
$721$	11479.8	0.592969
$722$	22247.6	1.14677
$723$	0	0
$724$	−2137.80	−0.109738
$725$	25476.1	1.30505
$726$	0	0
$727$	−1094.66	−0.0558439	−0.0279220	−	0.999610i	$-0.508889\pi$
$727$	−1094.66	−0.0558439	−0.0279220	+	0.999610i	$0.508889\pi$
$728$	−404.601	−0.0205982
$729$	0	0
$730$	16766.6	0.850084
$731$	−5524.20	−0.279507
$732$	0	0
$733$	24527.7	1.23595	0.617975	−	0.786197i	$-0.287953\pi$
$733$	24527.7	1.23595	0.617975	+	0.786197i	$0.287953\pi$
$734$	8735.85	0.439300
$735$	0	0
$736$	4965.20	0.248668
$737$	−13959.1	−0.697682
$738$	0	0
$739$	−3198.20	−0.159198	−0.0795992	−	0.996827i	$-0.525364\pi$
$739$	−3198.20	−0.159198	−0.0795992	+	0.996827i	$0.525364\pi$
$740$	−6455.85	−0.320705
$741$	0	0
$742$	3085.60	0.152663
$743$	−27708.1	−1.36812	−0.684059	−	0.729427i	$-0.739787\pi$
$743$	−27708.1	−1.36812	−0.684059	+	0.729427i	$0.739787\pi$
$744$	0	0
$745$	−37147.8	−1.82683
$746$	3817.95	0.187379
$747$	0	0
$748$	18713.2	0.914735
$749$	4874.63	0.237804
$750$	0	0
$751$	−14712.9	−0.714890	−0.357445	−	0.933934i	$-0.616352\pi$
$751$	−14712.9	−0.714890	−0.357445	+	0.933934i	$0.616352\pi$
$752$	6120.40	0.296792
$753$	0	0
$754$	1727.50	0.0834374
$755$	31134.6	1.50080
$756$	0	0
$757$	−1870.35	−0.0898005	−0.0449003	−	0.998991i	$-0.514297\pi$
$757$	−1870.35	−0.0898005	−0.0449003	+	0.998991i	$0.514297\pi$
$758$	−27525.7	−1.31897
$759$	0	0
$760$	19726.1	0.941501
$761$	−5683.10	−0.270713	−0.135356	−	0.990797i	$-0.543218\pi$
$761$	−5683.10	−0.270713	−0.135356	+	0.990797i	$0.543218\pi$
$762$	0	0
$763$	−2160.02	−0.102488
$764$	−1913.45	−0.0906103
$765$	0	0
$766$	−19518.6	−0.920675
$767$	5787.05	0.272436
$768$	0	0
$769$	8462.12	0.396816	0.198408	−	0.980119i	$-0.436423\pi$
$769$	8462.12	0.396816	0.198408	+	0.980119i	$0.436423\pi$
$770$	13685.3	0.640501
$771$	0	0
$772$	2476.60	0.115460
$773$	990.886	0.0461057	0.0230528	−	0.999734i	$-0.492661\pi$
$773$	990.886	0.0461057	0.0230528	+	0.999734i	$0.492661\pi$
$774$	0	0
$775$	68612.8	3.18019
$776$	11947.6	0.552698
$777$	0	0
$778$	15346.2	0.707183
$779$	30385.4	1.39752
$780$	0	0
$781$	5586.72	0.255965
$782$	27308.6	1.24879
$783$	0	0
$784$	784.000	0.0357143
$785$	−27842.8	−1.26593
$786$	0	0
$787$	4470.60	0.202490	0.101245	−	0.994862i	$-0.467717\pi$
$787$	4470.60	0.202490	0.101245	+	0.994862i	$0.467717\pi$
$788$	18357.1	0.829879
$789$	0	0
$790$	−42832.7	−1.92901
$791$	493.323	0.0221751
$792$	0	0
$793$	−5874.29	−0.263055
$794$	−5919.59	−0.264582
$795$	0	0
$796$	1915.90	0.0853106
$797$	37216.0	1.65403	0.827014	−	0.562182i	$-0.190038\pi$
$797$	37216.0	1.65403	0.827014	+	0.562182i	$0.190038\pi$
$798$	0	0
$799$	33662.2	1.49047
$800$	6819.20	0.301369
$801$	0	0
$802$	−26865.7	−1.18287
$803$	−24238.1	−1.06519
$804$	0	0
$805$	19971.3	0.874407
$806$	4652.54	0.203324
$807$	0	0
$808$	4720.80	0.205541
$809$	−35504.7	−1.54299	−0.771495	−	0.636236i	$-0.780491\pi$
$809$	−35504.7	−1.54299	−0.771495	+	0.636236i	$0.780491\pi$
$810$	0	0
$811$	−6036.65	−0.261375	−0.130688	−	0.991424i	$-0.541719\pi$
$811$	−6036.65	−0.261375	−0.130688	+	0.991424i	$0.541719\pi$
$812$	−3347.40	−0.144668
$813$	0	0
$814$	9332.67	0.401855
$815$	50477.8	2.16952
$816$	0	0
$817$	8418.12	0.360481
$818$	13494.8	0.576813
$819$	0	0
$820$	16665.5	0.709737
$821$	25063.6	1.06544	0.532720	−	0.846292i	$-0.321170\pi$
$821$	25063.6	1.06544	0.532720	+	0.846292i	$0.321170\pi$
$822$	0	0
$823$	8923.90	0.377968	0.188984	−	0.981980i	$-0.439481\pi$
$823$	8923.90	0.377968	0.188984	+	0.981980i	$0.439481\pi$
$824$	−13119.8	−0.554672
$825$	0	0
$826$	−11213.6	−0.472364
$827$	−27007.8	−1.13561	−0.567807	−	0.823162i	$-0.692208\pi$
$827$	−27007.8	−1.13561	−0.567807	+	0.823162i	$0.692208\pi$
$828$	0	0
$829$	−8558.07	−0.358545	−0.179273	−	0.983799i	$-0.557374\pi$
$829$	−8558.07	−0.358545	−0.179273	+	0.983799i	$0.557374\pi$
$830$	15248.7	0.637701
$831$	0	0
$832$	462.401	0.0192679
$833$	4312.00	0.179354
$834$	0	0
$835$	−27437.3	−1.13714
$836$	−28516.3	−1.17974
$837$	0	0
$838$	−15789.9	−0.650897
$839$	−19210.9	−0.790507	−0.395254	−	0.918572i	$-0.629343\pi$
$839$	−19210.9	−0.790507	−0.395254	+	0.918572i	$0.629343\pi$
$840$	0	0
$841$	−10096.8	−0.413990
$842$	30825.6	1.26167
$843$	0	0
$844$	21417.6	0.873488
$845$	39437.5	1.60555
$846$	0	0
$847$	−10466.7	−0.424607
$848$	−3526.40	−0.142803
$849$	0	0
$850$	37505.6	1.51345
$851$	13619.4	0.548609
$852$	0	0
$853$	−17851.8	−0.716571	−0.358286	−	0.933612i	$-0.616639\pi$
$853$	−17851.8	−0.716571	−0.358286	+	0.933612i	$0.616639\pi$
$854$	11382.7	0.456098
$855$	0	0
$856$	−5571.00	−0.222445
$857$	46372.9	1.84839	0.924193	−	0.381926i	$-0.124739\pi$
$857$	46372.9	1.84839	0.924193	+	0.381926i	$0.124739\pi$
$858$	0	0
$859$	13380.8	0.531486	0.265743	−	0.964044i	$-0.414383\pi$
$859$	13380.8	0.531486	0.265743	+	0.964044i	$0.414383\pi$
$860$	4617.10	0.183072
$861$	0	0
$862$	11007.8	0.434949
$863$	−15173.5	−0.598509	−0.299254	−	0.954173i	$-0.596738\pi$
$863$	−15173.5	−0.598509	−0.299254	+	0.954173i	$0.596738\pi$
$864$	0	0
$865$	−10644.7	−0.418419
$866$	−27184.3	−1.06670
$867$	0	0
$868$	−9015.30	−0.352534
$869$	61919.7	2.41712
$870$	0	0
$871$	−1897.11	−0.0738014
$872$	2468.60	0.0958684
$873$	0	0
$874$	−41614.6	−1.61056
$875$	11339.6	0.438111
$876$	0	0
$877$	−36273.9	−1.39667	−0.698337	−	0.715769i	$-0.746076\pi$
$877$	−36273.9	−1.39667	−0.698337	+	0.715769i	$0.746076\pi$
$878$	−6359.19	−0.244433
$879$	0	0
$880$	−15640.4	−0.599134
$881$	−5588.76	−0.213723	−0.106862	−	0.994274i	$-0.534080\pi$
$881$	−5588.76	−0.213723	−0.106862	+	0.994274i	$0.534080\pi$
$882$	0	0
$883$	−36347.5	−1.38527	−0.692634	−	0.721289i	$-0.743550\pi$
$883$	−36347.5	−1.38527	−0.692634	+	0.721289i	$0.743550\pi$
$884$	2543.20	0.0967615
$885$	0	0
$886$	−4896.42	−0.185664
$887$	−38630.6	−1.46233	−0.731166	−	0.682200i	$-0.761023\pi$
$887$	−38630.6	−1.46233	−0.731166	+	0.682200i	$0.761023\pi$
$888$	0	0
$889$	−6240.67	−0.235439
$890$	659.650	0.0248444
$891$	0	0
$892$	2349.30	0.0881841
$893$	−51296.6	−1.92226
$894$	0	0
$895$	42501.3	1.58733
$896$	−896.000	−0.0334077
$897$	0	0
$898$	2024.75	0.0752413
$899$	38492.1	1.42801
$900$	0	0
$901$	−19395.2	−0.717145
$902$	−24091.9	−0.889326
$903$	0	0
$904$	−563.797	−0.0207429
$905$	9827.18	0.360958
$906$	0	0
$907$	−45493.9	−1.66549	−0.832746	−	0.553655i	$-0.813233\pi$
$907$	−45493.9	−1.66549	−0.832746	+	0.553655i	$0.813233\pi$
$908$	−7407.10	−0.270719
$909$	0	0
$910$	1859.90	0.0677528
$911$	18693.0	0.679830	0.339915	−	0.940456i	$-0.389602\pi$
$911$	18693.0	0.679830	0.339915	+	0.940456i	$0.389602\pi$
$912$	0	0
$913$	−22043.8	−0.799062
$914$	5285.21	0.191268
$915$	0	0
$916$	13861.4	0.499993
$917$	−3340.58	−0.120300
$918$	0	0
$919$	−20904.0	−0.750338	−0.375169	−	0.926956i	$-0.622415\pi$
$919$	−20904.0	−0.750338	−0.375169	+	0.926956i	$0.622415\pi$
$920$	−22824.4	−0.817933
$921$	0	0
$922$	−33534.6	−1.19783
$923$	759.259	0.0270762
$924$	0	0
$925$	18704.8	0.664877
$926$	−22954.3	−0.814606
$927$	0	0
$928$	3825.60	0.135325
$929$	30322.5	1.07088	0.535441	−	0.844572i	$-0.320145\pi$
$929$	30322.5	1.07088	0.535441	+	0.844572i	$0.320145\pi$
$930$	0	0
$931$	−6570.90	−0.231313
$932$	11502.1	0.404253
$933$	0	0
$934$	13483.2	0.472358
$935$	−86022.2	−3.00880
$936$	0	0
$937$	29241.8	1.01952	0.509758	−	0.860318i	$-0.329735\pi$
$937$	29241.8	1.01952	0.509758	+	0.860318i	$0.329735\pi$
$938$	3676.05	0.127961
$939$	0	0
$940$	−28134.7	−0.976226
$941$	5814.51	0.201432	0.100716	−	0.994915i	$-0.467887\pi$
$941$	5814.51	0.201432	0.100716	+	0.994915i	$0.467887\pi$
$942$	0	0
$943$	−35157.9	−1.21410
$944$	12815.6	0.441856
$945$	0	0
$946$	−6674.55	−0.229396
$947$	−17776.4	−0.609984	−0.304992	−	0.952355i	$-0.598654\pi$
$947$	−17776.4	−0.609984	−0.304992	+	0.952355i	$0.598654\pi$
$948$	0	0
$949$	−3294.06	−0.112676
$950$	−57153.4	−1.95190
$951$	0	0
$952$	−4928.00	−0.167770
$953$	24642.9	0.837629	0.418814	−	0.908072i	$-0.362446\pi$
$953$	24642.9	0.837629	0.418814	+	0.908072i	$0.362446\pi$
$954$	0	0
$955$	8795.89	0.298040
$956$	−16159.7	−0.546697
$957$	0	0
$958$	18290.4	0.616843
$959$	10535.7	0.354761
$960$	0	0
$961$	73876.8	2.47984
$962$	1268.35	0.0425086
$963$	0	0
$964$	−20946.9	−0.699849
$965$	−11384.6	−0.379776
$966$	0	0
$967$	14393.1	0.478646	0.239323	−	0.970940i	$-0.423074\pi$
$967$	14393.1	0.478646	0.239323	+	0.970940i	$0.423074\pi$
$968$	11962.0	0.397183
$969$	0	0
$970$	−54921.6	−1.81797
$971$	41393.7	1.36806	0.684031	−	0.729453i	$-0.260225\pi$
$971$	41393.7	1.36806	0.684031	+	0.729453i	$0.260225\pi$
$972$	0	0
$973$	−6600.65	−0.217479
$974$	−13656.6	−0.449268
$975$	0	0
$976$	−13008.8	−0.426641
$977$	−15614.4	−0.511310	−0.255655	−	0.966768i	$-0.582291\pi$
$977$	−15614.4	−0.511310	−0.255655	+	0.966768i	$0.582291\pi$
$978$	0	0
$979$	−953.601	−0.0311310
$980$	−3603.95	−0.117473
$981$	0	0
$982$	−36719.7	−1.19325
$983$	−13392.9	−0.434556	−0.217278	−	0.976110i	$-0.569718\pi$
$983$	−13392.9	−0.434556	−0.217278	+	0.976110i	$0.569718\pi$
$984$	0	0
$985$	−84385.2	−2.72968
$986$	21040.8	0.679590
$987$	0	0
$988$	−3875.49	−0.124793
$989$	−9740.32	−0.313169
$990$	0	0
$991$	54945.5	1.76125	0.880627	−	0.473811i	$-0.157122\pi$
$991$	54945.5	1.76125	0.880627	+	0.473811i	$0.157122\pi$
$992$	10303.2	0.329765
$993$	0	0
$994$	−1471.23	−0.0469462
$995$	−8807.14	−0.280608
$996$	0	0
$997$	56308.0	1.78866	0.894329	−	0.447410i	$-0.147654\pi$
$997$	56308.0	1.78866	0.894329	+	0.447410i	$0.147654\pi$
$998$	−23348.1	−0.740551
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.a.q.1.1	yes	2
3.2	odd	2		378.4.a.n.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.a.n.1.2	✓	2	3.2	odd	2
378.4.a.q.1.1	yes	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -18.3875 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -18.3875 q^{5} -7.00000 q^{7} +8.00000 q^{8} -36.7750 q^{10} +53.1625 q^{11} +7.22501 q^{13} -14.0000 q^{14} +16.0000 q^{16} +88.0000 q^{17} -134.100 q^{19} -73.5500 q^{20} +106.325 q^{22} +155.162 q^{23} +213.100 q^{25} +14.4500 q^{26} -28.0000 q^{28} +119.550 q^{29} +321.975 q^{31} +32.0000 q^{32} +176.000 q^{34} +128.712 q^{35} +87.7750 q^{37} -268.200 q^{38} -147.100 q^{40} -226.587 q^{41} -62.7750 q^{43} +212.650 q^{44} +310.325 q^{46} +382.525 q^{47} +49.0000 q^{49} +426.200 q^{50} +28.9000 q^{52} -220.400 q^{53} -977.525 q^{55} -56.0000 q^{56} +239.100 q^{58} +800.975 q^{59} -813.050 q^{61} +643.950 q^{62} +64.0000 q^{64} -132.850 q^{65} -262.575 q^{67} +352.000 q^{68} +257.425 q^{70} +105.088 q^{71} -455.925 q^{73} +175.550 q^{74} -536.400 q^{76} -372.137 q^{77} +1164.72 q^{79} -294.200 q^{80} -453.175 q^{82} -414.650 q^{83} -1618.10 q^{85} -125.550 q^{86} +425.300 q^{88} -17.9375 q^{89} -50.5751 q^{91} +620.650 q^{92} +765.050 q^{94} +2465.76 q^{95} +1493.45 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} - 8 q^{5} - 14 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 - 8 * q^5 - 14 * q^7 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} - 8 q^{5} - 14 q^{7} + 16 q^{8} - 16 q^{10} + 20 q^{11} + 72 q^{13} - 28 q^{14} + 32 q^{16} + 176 q^{17} - 38 q^{19} - 32 q^{20} + 40 q^{22} + 224 q^{23} + 196 q^{25} + 144 q^{26} - 56 q^{28} + 124 q^{29} + 126 q^{31} + 64 q^{32} + 352 q^{34} + 56 q^{35} + 118 q^{37} - 76 q^{38} - 64 q^{40} + 36 q^{41} - 68 q^{43} + 80 q^{44} + 448 q^{46} + 132 q^{47} + 98 q^{49} + 392 q^{50} + 288 q^{52} + 480 q^{53} - 1322 q^{55} - 112 q^{56} + 248 q^{58} + 1084 q^{59} - 360 q^{61} + 252 q^{62} + 128 q^{64} + 540 q^{65} - 928 q^{67} + 704 q^{68} + 112 q^{70} + 872 q^{71} - 1660 q^{73} + 236 q^{74} - 152 q^{76} - 140 q^{77} + 1236 q^{79} - 128 q^{80} + 72 q^{82} - 484 q^{83} - 704 q^{85} - 136 q^{86} + 160 q^{88} + 108 q^{89} - 504 q^{91} + 896 q^{92} + 264 q^{94} + 3464 q^{95} + 800 q^{97} + 196 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 - 8 * q^5 - 14 * q^7 + 16 * q^8 - 16 * q^10 + 20 * q^11 + 72 * q^13 - 28 * q^14 + 32 * q^16 + 176 * q^17 - 38 * q^19 - 32 * q^20 + 40 * q^22 + 224 * q^23 + 196 * q^25 + 144 * q^26 - 56 * q^28 + 124 * q^29 + 126 * q^31 + 64 * q^32 + 352 * q^34 + 56 * q^35 + 118 * q^37 - 76 * q^38 - 64 * q^40 + 36 * q^41 - 68 * q^43 + 80 * q^44 + 448 * q^46 + 132 * q^47 + 98 * q^49 + 392 * q^50 + 288 * q^52 + 480 * q^53 - 1322 * q^55 - 112 * q^56 + 248 * q^58 + 1084 * q^59 - 360 * q^61 + 252 * q^62 + 128 * q^64 + 540 * q^65 - 928 * q^67 + 704 * q^68 + 112 * q^70 + 872 * q^71 - 1660 * q^73 + 236 * q^74 - 152 * q^76 - 140 * q^77 + 1236 * q^79 - 128 * q^80 + 72 * q^82 - 484 * q^83 - 704 * q^85 - 136 * q^86 + 160 * q^88 + 108 * q^89 - 504 * q^91 + 896 * q^92 + 264 * q^94 + 3464 * q^95 + 800 * q^97 + 196 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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