LMFDB - Embedded newform 384.5.h.h.65.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(65,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("384.65");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	384.h (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$39.6940658242$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.4
Character	$\chi$	$=$	384.65
Dual form			384.5.h.h.65.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.73496 + 2.16806i) q^{3} +14.5048 q^{5} +70.6373 q^{7} +(71.5990 - 37.8759i) q^{9} +O(q^{10})$	\(q+(-8.73496 + 2.16806i) q^{3} +14.5048 q^{5} +70.6373 q^{7} +(71.5990 - 37.8759i) q^{9} +163.140 q^{11} -293.353i q^{13} +(-126.699 + 31.4472i) q^{15} +533.139i q^{17} -314.108i q^{19} +(-617.014 + 153.146i) q^{21} -547.458i q^{23} -414.612 q^{25} +(-543.297 + 486.075i) q^{27} -493.989 q^{29} +1042.74 q^{31} +(-1425.02 + 353.697i) q^{33} +1024.58 q^{35} +550.113i q^{37} +(636.007 + 2562.42i) q^{39} -339.288i q^{41} -2585.68i q^{43} +(1038.53 - 549.380i) q^{45} -100.167i q^{47} +2588.62 q^{49} +(-1155.88 - 4656.95i) q^{51} -1409.01 q^{53} +2366.30 q^{55} +(681.006 + 2743.72i) q^{57} +1439.43 q^{59} -4990.69i q^{61} +(5057.56 - 2675.45i) q^{63} -4255.01i q^{65} -168.946i q^{67} +(1186.92 + 4782.02i) q^{69} +5261.70i q^{71} +1244.91 q^{73} +(3621.62 - 898.904i) q^{75} +11523.7 q^{77} +7421.94 q^{79} +(3691.84 - 5423.75i) q^{81} -116.567 q^{83} +7733.06i q^{85} +(4314.98 - 1071.00i) q^{87} -7446.53i q^{89} -20721.6i q^{91} +(-9108.26 + 2260.72i) q^{93} -4556.07i q^{95} -8270.40 q^{97} +(11680.6 - 6179.05i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 224 q^{9}+O(q^{10}) $ 32 * q + 224 * q^9	$ 32 q + 224 q^{9} + 5472 q^{25} - 3712 q^{33} + 13664 q^{49} - 17344 q^{57} - 17472 q^{73} - 10976 q^{81} - 39488 q^{97}+O(q^{100}) $ 32 * q + 224 * q^9 + 5472 * q^25 - 3712 * q^33 + 13664 * q^49 - 17344 * q^57 - 17472 * q^73 - 10976 * q^81 - 39488 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−8.73496	+	2.16806i	−0.970551	+	0.240896i
$3$	−8.73496	+	2.16806i	−0.970551	+	0.240896i
$4$		0			0
$5$	14.5048			0.580191			0.290095	−	0.956998i	$-0.406313\pi$
$5$	14.5048			0.580191			0.290095	+	0.956998i	$0.406313\pi$
$6$		0			0
$7$	70.6373			1.44158			0.720788	−	0.693155i	$-0.243780\pi$
$7$	70.6373			1.44158			0.720788	+	0.693155i	$0.243780\pi$
$8$		0			0
$9$	71.5990	−	37.8759i	0.883938	−	0.467603i
$10$		0			0
$11$	163.140			1.34826			0.674131	−	0.738612i	$-0.264519\pi$
$11$	163.140			1.34826			0.674131	+	0.738612i	$0.264519\pi$
$12$		0			0
$13$		−	293.353i		−	1.73582i	−0.496725	−	0.867908i	$-0.665464\pi$
$13$		−	293.353i		−	1.73582i	0.496725	−	0.867908i	$-0.334536\pi$
$14$		0			0
$15$	−126.699	+	31.4472i	−0.563105	+	0.139765i
$16$		0			0
$17$			533.139i			1.84477i	0.386269	+	0.922386i	$0.373764\pi$
$17$			533.139i			1.84477i	−0.386269	+	0.922386i	$0.626236\pi$
$18$		0			0
$19$		−	314.108i		−	0.870106i	−0.900405	−	0.435053i	$-0.856730\pi$
$19$		−	314.108i		−	0.870106i	0.900405	−	0.435053i	$-0.143270\pi$
$20$		0			0
$21$	−617.014	+	153.146i	−1.39912	+	0.347270i
$22$		0			0
$23$		−	547.458i		−	1.03489i	−0.855716	−	0.517446i	$-0.826883\pi$
$23$		−	547.458i		−	1.03489i	0.855716	−	0.517446i	$-0.173117\pi$
$24$		0			0
$25$	−414.612			−0.663379
$26$		0			0
$27$	−543.297	+	486.075i	−0.745264	+	0.666770i
$28$		0			0
$29$	−493.989			−0.587383			−0.293692	−	0.955900i	$-0.594884\pi$
$29$	−493.989			−0.587383			−0.293692	+	0.955900i	$0.594884\pi$
$30$		0			0
$31$	1042.74			1.08505			0.542527	−	0.840039i	$-0.317468\pi$
$31$	1042.74			1.08505			0.542527	+	0.840039i	$0.317468\pi$
$32$		0			0
$33$	−1425.02	+	353.697i	−1.30856	+	0.324790i
$34$		0			0
$35$	1024.58			0.836389
$36$		0			0
$37$			550.113i			0.401835i	0.979608	+	0.200918i	$0.0643924\pi$
$37$			550.113i			0.401835i	−0.979608	+	0.200918i	$0.935608\pi$
$38$		0			0
$39$	636.007	+	2562.42i	0.418151	+	1.68470i
$40$		0			0
$41$		−	339.288i		−	0.201837i	−0.994895	−	0.100918i	$-0.967822\pi$
$41$		−	339.288i		−	0.201837i	0.994895	−	0.100918i	$-0.0321781\pi$
$42$		0			0
$43$		−	2585.68i		−	1.39842i	−0.714915	−	0.699211i	$-0.753535\pi$
$43$		−	2585.68i		−	1.39842i	0.714915	−	0.699211i	$-0.246465\pi$
$44$		0			0
$45$	1038.53	−	549.380i	0.512853	−	0.271299i
$46$		0			0
$47$		−	100.167i		−	0.0453449i	−0.999743	−	0.0226725i	$-0.992783\pi$
$47$		−	100.167i		−	0.0453449i	0.999743	−	0.0226725i	$-0.00721749\pi$
$48$		0			0
$49$	2588.62			1.07814
$50$		0			0
$51$	−1155.88	−	4656.95i	−0.444398	−	1.79045i
$52$		0			0
$53$	−1409.01			−0.501605			−0.250802	−	0.968038i	$-0.580694\pi$
$53$	−1409.01			−0.501605			−0.250802	+	0.968038i	$0.580694\pi$
$54$		0			0
$55$	2366.30			0.782248
$56$		0			0
$57$	681.006	+	2743.72i	0.209605	+	0.844482i
$58$		0			0
$59$	1439.43			0.413511			0.206756	−	0.978393i	$-0.433710\pi$
$59$	1439.43			0.413511			0.206756	+	0.978393i	$0.433710\pi$
$60$		0			0
$61$		−	4990.69i		−	1.34122i	−0.741809	−	0.670611i	$-0.766032\pi$
$61$		−	4990.69i		−	1.34122i	0.741809	−	0.670611i	$-0.233968\pi$
$62$		0			0
$63$	5057.56	−	2675.45i	1.27426	−	0.674086i
$64$		0			0
$65$		−	4255.01i		−	1.00710i
$66$		0			0
$67$		−	168.946i		−	0.0376355i	−0.999823	−	0.0188178i	$-0.994010\pi$
$67$		−	168.946i		−	0.0376355i	0.999823	−	0.0188178i	$-0.00599024\pi$
$68$		0			0
$69$	1186.92	+	4782.02i	0.249301	+	1.00442i
$70$		0			0
$71$			5261.70i			1.04378i	0.853012	+	0.521890i	$0.174773\pi$
$71$			5261.70i			1.04378i	−0.853012	+	0.521890i	$0.825227\pi$
$72$		0			0
$73$	1244.91			0.233610			0.116805	−	0.993155i	$-0.462735\pi$
$73$	1244.91			0.233610			0.116805	+	0.993155i	$0.462735\pi$
$74$		0			0
$75$	3621.62	−	898.904i	0.643843	−	0.159805i
$76$		0			0
$77$	11523.7			1.94362
$78$		0			0
$79$	7421.94			1.18922			0.594611	−	0.804013i	$-0.297306\pi$
$79$	7421.94			1.18922			0.594611	+	0.804013i	$0.297306\pi$
$80$		0			0
$81$	3691.84	−	5423.75i	0.562694	−	0.826665i
$82$		0			0
$83$	−116.567			−0.0169207			−0.00846037	−	0.999964i	$-0.502693\pi$
$83$	−116.567			−0.0169207			−0.00846037	+	0.999964i	$0.502693\pi$
$84$		0			0
$85$			7733.06i			1.07032i
$86$		0			0
$87$	4314.98	−	1071.00i	0.570085	−	0.141498i
$88$		0			0
$89$		−	7446.53i		−	0.940100i	−0.882640	−	0.470050i	$-0.844236\pi$
$89$		−	7446.53i		−	0.940100i	0.882640	−	0.470050i	$-0.155764\pi$
$90$		0			0
$91$		−	20721.6i		−	2.50231i
$92$		0			0
$93$	−9108.26	+	2260.72i	−1.05310	+	0.261385i
$94$		0			0
$95$		−	4556.07i		−	0.504827i
$96$		0			0
$97$	−8270.40			−0.878988			−0.439494	−	0.898245i	$-0.644842\pi$
$97$	−8270.40			−0.878988			−0.439494	+	0.898245i	$0.644842\pi$
$98$		0			0
$99$	11680.6	−	6179.05i	1.19178	−	0.630451i
$100$		0			0
$101$	6550.90			0.642182			0.321091	−	0.947048i	$-0.395950\pi$
$101$	6550.90			0.642182			0.321091	+	0.947048i	$0.395950\pi$
$102$		0			0
$103$	14142.4			1.33306			0.666529	−	0.745479i	$-0.267779\pi$
$103$	14142.4			1.33306			0.666529	+	0.745479i	$0.267779\pi$
$104$		0			0
$105$	−8949.64	+	2221.35i	−0.811758	+	0.201483i
$106$		0			0
$107$	−3657.73			−0.319481			−0.159740	−	0.987159i	$-0.551066\pi$
$107$	−3657.73			−0.319481			−0.159740	+	0.987159i	$0.551066\pi$
$108$		0			0
$109$			8706.79i			0.732833i	0.930451	+	0.366416i	$0.119415\pi$
$109$			8706.79i			0.732833i	−0.930451	+	0.366416i	$0.880585\pi$
$110$		0			0
$111$	−1192.68	−	4805.21i	−0.0968004	−	0.390002i
$112$		0			0
$113$			15404.2i			1.20637i	0.797600	+	0.603187i	$0.206103\pi$
$113$			15404.2i			1.20637i	−0.797600	+	0.603187i	$0.793897\pi$
$114$		0			0
$115$		−	7940.75i		−	0.600435i
$116$		0			0
$117$	−11111.0	−	21003.8i	−0.811673	−	1.53435i
$118$		0			0
$119$			37659.5i			2.65938i
$120$		0			0
$121$	11973.5			0.817808
$122$		0			0
$123$	735.597	+	2963.67i	0.0486217	+	0.195893i
$124$		0			0
$125$	−15079.3			−0.965077
$126$		0			0
$127$	22620.8			1.40249			0.701247	−	0.712918i	$-0.252627\pi$
$127$	22620.8			1.40249			0.701247	+	0.712918i	$0.252627\pi$
$128$		0			0
$129$	5605.92	+	22585.8i	0.336874	+	1.35724i
$130$		0			0
$131$	−9335.23			−0.543980			−0.271990	−	0.962300i	$-0.587682\pi$
$131$	−9335.23			−0.543980			−0.271990	+	0.962300i	$0.587682\pi$
$132$		0			0
$133$		−	22187.7i		−	1.25432i
$134$		0			0
$135$	−7880.40	+	7050.41i	−0.432395	+	0.386854i
$136$		0			0
$137$		−	18160.9i		−	0.967602i	−0.875178	−	0.483801i	$-0.839256\pi$
$137$		−	18160.9i		−	0.967602i	0.875178	−	0.483801i	$-0.160744\pi$
$138$		0			0
$139$			9492.75i			0.491318i	0.969356	+	0.245659i	$0.0790043\pi$
$139$			9492.75i			0.491318i	−0.969356	+	0.245659i	$0.920996\pi$
$140$		0			0
$141$	217.168	+	874.954i	0.0109234	+	0.0440096i
$142$		0			0
$143$		−	47857.5i		−	2.34033i
$144$		0			0
$145$	−7165.20			−0.340794
$146$		0			0
$147$	−22611.5	+	5612.29i	−1.04639	+	0.259720i
$148$		0			0
$149$	19050.9			0.858112			0.429056	−	0.903278i	$-0.358846\pi$
$149$	19050.9			0.858112			0.429056	+	0.903278i	$0.358846\pi$
$150$		0			0
$151$	−39824.5			−1.74661			−0.873305	−	0.487174i	$-0.838028\pi$
$151$	−39824.5			−1.74661			−0.873305	+	0.487174i	$0.838028\pi$
$152$		0			0
$153$	20193.1	+	38172.2i	0.862621	+	1.63066i
$154$		0			0
$155$	15124.6			0.629538
$156$		0			0
$157$			1276.46i			0.0517855i	0.999665	+	0.0258928i	$0.00824284\pi$
$157$			1276.46i			0.0517855i	−0.999665	+	0.0258928i	$0.991757\pi$
$158$		0			0
$159$	12307.6	−	3054.81i	0.486833	−	0.120834i
$160$		0			0
$161$		−	38670.9i		−	1.49188i
$162$		0			0
$163$		−	12463.5i		−	0.469098i	−0.972104	−	0.234549i	$-0.924639\pi$
$163$		−	12463.5i		−	0.469098i	0.972104	−	0.234549i	$-0.0753614\pi$
$164$		0			0
$165$	−20669.5	+	5130.29i	−0.759212	+	0.188440i
$166$		0			0
$167$			18549.8i			0.665129i	0.943081	+	0.332564i	$0.107914\pi$
$167$			18549.8i			0.665129i	−0.943081	+	0.332564i	$0.892086\pi$
$168$		0			0
$169$	−57494.9			−2.01306
$170$		0			0
$171$	−11897.1	−	22489.8i	−0.406864	−	0.769120i
$172$		0			0
$173$	−2994.42			−0.100051			−0.0500254	−	0.998748i	$-0.515930\pi$
$173$	−2994.42			−0.100051			−0.0500254	+	0.998748i	$0.515930\pi$
$174$		0			0
$175$	−29287.0			−0.956311
$176$		0			0
$177$	−12573.4	+	3120.78i	−0.401334	+	0.0996131i
$178$		0			0
$179$	49036.1			1.53042			0.765208	−	0.643783i	$-0.222636\pi$
$179$	49036.1			1.53042			0.765208	+	0.643783i	$0.222636\pi$
$180$		0			0
$181$			28051.5i			0.856247i	0.903720	+	0.428124i	$0.140825\pi$
$181$			28051.5i			0.856247i	−0.903720	+	0.428124i	$0.859175\pi$
$182$		0			0
$183$	10820.1	+	43593.4i	0.323095	+	1.30172i
$184$		0			0
$185$			7979.25i			0.233141i
$186$		0			0
$187$			86976.1i			2.48723i
$188$		0			0
$189$	−38377.0	+	34335.0i	−1.07435	+	0.961200i
$190$		0			0
$191$		−	50035.7i		−	1.37156i	−0.727811	−	0.685778i	$-0.759462\pi$
$191$		−	50035.7i		−	1.37156i	0.727811	−	0.685778i	$-0.240538\pi$
$192$		0			0
$193$	30906.3			0.829722			0.414861	−	0.909885i	$-0.363830\pi$
$193$	30906.3			0.829722			0.414861	+	0.909885i	$0.363830\pi$
$194$		0			0
$195$	9225.13	+	37167.4i	0.242607	+	0.977446i
$196$		0			0
$197$	40327.4			1.03912			0.519562	−	0.854432i	$-0.326095\pi$
$197$	40327.4			1.03912			0.519562	+	0.854432i	$0.326095\pi$
$198$		0			0
$199$	706.369			0.0178372			0.00891858	−	0.999960i	$-0.497161\pi$
$199$	706.369			0.0178372			0.00891858	+	0.999960i	$0.497161\pi$
$200$		0			0
$201$	366.285	+	1475.74i	0.00906624	+	0.0365272i
$202$		0			0
$203$	−34894.0			−0.846758
$204$		0			0
$205$		−	4921.29i		−	0.117104i
$206$		0			0
$207$	−20735.4	−	39197.4i	−0.483919	−	0.914781i
$208$		0			0
$209$		−	51243.5i		−	1.17313i
$210$		0			0
$211$			33541.1i			0.753376i	0.926340	+	0.376688i	$0.122937\pi$
$211$			33541.1i			0.753376i	−0.926340	+	0.376688i	$0.877063\pi$
$212$		0			0
$213$	−11407.7	−	45960.7i	−0.251442	−	1.01304i
$214$		0			0
$215$		−	37504.7i		−	0.811351i
$216$		0			0
$217$	73656.0			1.56419
$218$		0			0
$219$	−10874.2	+	2699.04i	−0.226731	+	0.0562758i
$220$		0			0
$221$	156398.			3.20218
$222$		0			0
$223$	−58954.5			−1.18552			−0.592758	−	0.805381i	$-0.701961\pi$
$223$	−58954.5			−1.18552			−0.592758	+	0.805381i	$0.701961\pi$
$224$		0			0
$225$	−29685.8	+	15703.8i	−0.586386	+	0.310198i
$226$		0			0
$227$	−72396.6			−1.40497			−0.702484	−	0.711700i	$-0.747926\pi$
$227$	−72396.6			−1.40497			−0.702484	+	0.711700i	$0.747926\pi$
$228$		0			0
$229$			20382.8i			0.388681i	0.980934	+	0.194340i	$0.0622566\pi$
$229$			20382.8i			0.388681i	−0.980934	+	0.194340i	$0.937743\pi$
$230$		0			0
$231$	−100659.	+	24984.2i	−1.88638	+	0.468210i
$232$		0			0
$233$		−	77399.0i		−	1.42569i	−0.701324	−	0.712843i	$-0.747407\pi$
$233$		−	77399.0i		−	1.42569i	0.701324	−	0.712843i	$-0.252593\pi$
$234$		0			0
$235$		−	1452.90i		−	0.0263087i
$236$		0			0
$237$	−64830.3	+	16091.2i	−1.15420	+	0.286479i
$238$		0			0
$239$			31052.7i			0.543630i	0.962349	+	0.271815i	$0.0876239\pi$
$239$			31052.7i			0.543630i	−0.962349	+	0.271815i	$0.912376\pi$
$240$		0			0
$241$	61434.1			1.05773			0.528866	−	0.848706i	$-0.322618\pi$
$241$	61434.1			1.05773			0.528866	+	0.848706i	$0.322618\pi$
$242$		0			0
$243$	−20489.0	+	55380.4i	−0.346984	+	0.937871i
$244$		0			0
$245$	37547.3			0.625528
$246$		0			0
$247$	−92144.5			−1.51034
$248$		0			0
$249$	1018.21	−	252.724i	0.0164224	−	0.00407613i
$250$		0			0
$251$	58318.0			0.925668			0.462834	−	0.886445i	$-0.346833\pi$
$251$	58318.0			0.925668			0.462834	+	0.886445i	$0.346833\pi$
$252$		0			0
$253$		−	89312.0i		−	1.39530i
$254$		0			0
$255$	−16765.7	−	67547.9i	−0.257835	−	1.03880i
$256$		0			0
$257$		−	19572.9i		−	0.296339i	−0.988962	−	0.148170i	$-0.952662\pi$
$257$		−	19572.9i		−	0.296339i	0.988962	−	0.148170i	$-0.0473381\pi$
$258$		0			0
$259$			38858.4i			0.579276i
$260$		0			0
$261$	−35369.1	+	18710.3i	−0.519211	+	0.274662i
$262$		0			0
$263$			111228.i			1.60807i	0.594585	+	0.804033i	$0.297317\pi$
$263$			111228.i			1.60807i	−0.594585	+	0.804033i	$0.702683\pi$
$264$		0			0
$265$	−20437.3			−0.291026
$266$		0			0
$267$	16144.5	+	65045.1i	0.226466	+	0.912415i
$268$		0			0
$269$	−38148.0			−0.527190			−0.263595	−	0.964633i	$-0.584908\pi$
$269$	−38148.0			−0.527190			−0.263595	+	0.964633i	$0.584908\pi$
$270$		0			0
$271$	73859.9			1.00570			0.502852	−	0.864373i	$-0.332284\pi$
$271$	73859.9			1.00570			0.502852	+	0.864373i	$0.332284\pi$
$272$		0			0
$273$	44925.8	+	181003.i	0.602796	+	2.42862i
$274$		0			0
$275$	−67639.6			−0.894408
$276$		0			0
$277$			92295.7i			1.20288i	0.798918	+	0.601440i	$0.205406\pi$
$277$			92295.7i			1.20288i	−0.798918	+	0.601440i	$0.794594\pi$
$278$		0			0
$279$	74658.9	−	39494.5i	0.959120	−	0.507374i
$280$		0			0
$281$		−	24602.7i		−	0.311580i	−0.987790	−	0.155790i	$-0.950208\pi$
$281$		−	24602.7i		−	0.311580i	0.987790	−	0.155790i	$-0.0497923\pi$
$282$		0			0
$283$		−	23094.4i		−	0.288359i	−0.989552	−	0.144179i	$-0.953946\pi$
$283$		−	23094.4i		−	0.288359i	0.989552	−	0.144179i	$-0.0460542\pi$
$284$		0			0
$285$	9877.83	+	39797.0i	0.121611	+	0.489961i
$286$		0			0
$287$		−	23966.4i		−	0.290963i
$288$		0			0
$289$	−200716.			−2.40318
$290$		0			0
$291$	72241.6	−	17930.7i	0.853103	−	0.211745i
$292$		0			0
$293$	−94473.1			−1.10046			−0.550228	−	0.835014i	$-0.685459\pi$
$293$	−94473.1			−1.10046			−0.550228	+	0.835014i	$0.685459\pi$
$294$		0			0
$295$	20878.6			0.239915
$296$		0			0
$297$	−88633.3	+	79298.1i	−1.00481	+	0.898980i
$298$		0			0
$299$	−160598.			−1.79638
$300$		0			0
$301$		−	182646.i		−	2.01593i
$302$		0			0
$303$	−57221.8	+	14202.8i	−0.623270	+	0.154699i
$304$		0			0
$305$		−	72388.7i		−	0.778164i
$306$		0			0
$307$			21965.4i			0.233057i	0.993187	+	0.116529i	$0.0371767\pi$
$307$			21965.4i			0.233057i	−0.993187	+	0.116529i	$0.962823\pi$
$308$		0			0
$309$	−123533.	+	30661.6i	−1.29380	+	0.321128i
$310$		0			0
$311$		−	161.545i		−	0.00167022i	−1.00000		0.000835109i	$-0.999734\pi$
$311$		−	161.545i		−	0.00167022i	1.00000		0.000835109i	$-0.000265823\pi$
$312$		0			0
$313$	22811.8			0.232847			0.116424	−	0.993200i	$-0.462857\pi$
$313$	22811.8			0.232847			0.116424	+	0.993200i	$0.462857\pi$
$314$		0			0
$315$	73358.7	−	38806.7i	0.739317	−	0.391098i
$316$		0			0
$317$	11173.0			0.111187			0.0555933	−	0.998453i	$-0.482295\pi$
$317$	11173.0			0.111187			0.0555933	+	0.998453i	$0.482295\pi$
$318$		0			0
$319$	−80589.2			−0.791946
$320$		0			0
$321$	31950.2	−	7930.19i	0.310072	−	0.0769615i
$322$		0			0
$323$	167463.			1.60515
$324$		0			0
$325$			121628.i			1.15150i
$326$		0			0
$327$	−18876.9	−	76053.4i	−0.176536	−	0.711252i
$328$		0			0
$329$		−	7075.52i		−	0.0653682i
$330$		0			0
$331$		−	20196.8i		−	0.184343i	−0.995743	−	0.0921715i	$-0.970619\pi$
$331$		−	20196.8i		−	0.184343i	0.995743	−	0.0921715i	$-0.0293808\pi$
$332$		0			0
$333$	20836.0	+	39387.5i	0.187900	+	0.355198i
$334$		0			0
$335$		−	2450.52i		−	0.0218358i
$336$		0			0
$337$	69854.6			0.615085			0.307543	−	0.951534i	$-0.400493\pi$
$337$	69854.6			0.615085			0.307543	+	0.951534i	$0.400493\pi$
$338$		0			0
$339$	−33397.2	−	134555.i	−0.290610	−	1.17085i
$340$		0			0
$341$	170112.			1.46293
$342$		0			0
$343$	13253.1			0.112649
$344$		0			0
$345$	17216.0	+	69362.1i	0.144642	+	0.582752i
$346$		0			0
$347$	118904.			0.987503			0.493751	−	0.869603i	$-0.335625\pi$
$347$	118904.			0.987503			0.493751	+	0.869603i	$0.335625\pi$
$348$		0			0
$349$			32493.5i			0.266775i	0.991064	+	0.133388i	$0.0425856\pi$
$349$			32493.5i			0.266775i	−0.991064	+	0.133388i	$0.957414\pi$
$350$		0			0
$351$	142592.	+	159378.i	1.15739	+	1.29364i
$352$		0			0
$353$		−	15417.5i		−	0.123727i	−0.998085	−	0.0618636i	$-0.980296\pi$
$353$		−	15417.5i		−	0.123727i	0.998085	−	0.0618636i	$-0.0197044\pi$
$354$		0			0
$355$			76319.7i			0.605592i
$356$		0			0
$357$	−81648.1	−	328954.i	−0.640633	−	2.58106i
$358$		0			0
$359$		−	237534.i		−	1.84305i	−0.388321	−	0.921524i	$-0.626945\pi$
$359$		−	237534.i		−	1.84305i	0.388321	−	0.921524i	$-0.373055\pi$
$360$		0			0
$361$	31657.0			0.242916
$362$		0			0
$363$	−104588.	+	25959.3i	−0.793724	+	0.197006i
$364$		0			0
$365$	18057.1			0.135539
$366$		0			0
$367$	84181.0			0.625003			0.312502	−	0.949917i	$-0.398833\pi$
$367$	84181.0			0.625003			0.312502	+	0.949917i	$0.398833\pi$
$368$		0			0
$369$	−12850.8	−	24292.7i	−0.0943796	−	0.178411i
$370$		0			0
$371$	−99528.4			−0.723101
$372$		0			0
$373$		−	79258.9i		−	0.569679i	−0.958575	−	0.284840i	$-0.908060\pi$
$373$		−	79258.9i		−	0.569679i	0.958575	−	0.284840i	$-0.0919404\pi$
$374$		0			0
$375$	131717.	−	32692.9i	0.936656	−	0.232483i
$376$		0			0
$377$			144913.i			1.01959i
$378$		0			0
$379$		−	115025.i		−	0.800778i	−0.916345	−	0.400389i	$-0.868875\pi$
$379$		−	115025.i		−	0.800778i	0.916345	−	0.400389i	$-0.131125\pi$
$380$		0			0
$381$	−197592.	+	49043.4i	−1.36119	+	0.337855i
$382$		0			0
$383$			190876.i			1.30123i	0.759408	+	0.650615i	$0.225489\pi$
$383$			190876.i			1.30123i	−0.759408	+	0.650615i	$0.774511\pi$
$384$		0			0
$385$	167149.			1.12767
$386$		0			0
$387$	−97935.0	−	185132.i	−0.653907	−	1.23612i
$388$		0			0
$389$	190095.			1.25624			0.628119	−	0.778118i	$-0.283825\pi$
$389$	190095.			1.25624			0.628119	+	0.778118i	$0.283825\pi$
$390$		0			0
$391$	291871.			1.90914
$392$		0			0
$393$	81542.9	−	20239.4i	0.527960	−	0.131042i
$394$		0			0
$395$	107653.			0.689976
$396$		0			0
$397$			73036.2i			0.463401i	0.972787	+	0.231700i	$0.0744289\pi$
$397$			73036.2i			0.463401i	−0.972787	+	0.231700i	$0.925571\pi$
$398$		0			0
$399$	48104.4	+	193809.i	0.302161	+	1.21739i
$400$		0			0
$401$		−	141470.i		−	0.879782i	−0.898051	−	0.439891i	$-0.855017\pi$
$401$		−	141470.i		−	0.879782i	0.898051	−	0.439891i	$-0.144983\pi$
$402$		0			0
$403$		−	305890.i		−	1.88345i
$404$		0			0
$405$	53549.2	−	78670.2i	0.326470	−	0.479623i
$406$		0			0
$407$			89745.1i			0.541779i
$408$		0			0
$409$	−172765.			−1.03279			−0.516393	−	0.856352i	$-0.672725\pi$
$409$	−172765.			−1.03279			−0.516393	+	0.856352i	$0.672725\pi$
$410$		0			0
$411$	39374.0	+	158635.i	0.233091	+	0.939107i
$412$		0			0
$413$	101678.			0.596108
$414$		0			0
$415$	−1690.78			−0.00981725
$416$		0			0
$417$	−20580.9	−	82918.8i	−0.118356	−	0.476849i
$418$		0			0
$419$	192120.			1.09432			0.547159	−	0.837029i	$-0.315709\pi$
$419$	192120.			1.09432			0.547159	+	0.837029i	$0.315709\pi$
$420$		0			0
$421$			307033.i			1.73229i	0.499793	+	0.866145i	$0.333409\pi$
$421$			307033.i			1.73229i	−0.499793	+	0.866145i	$0.666591\pi$
$422$		0			0
$423$	−3793.91	−	7171.85i	−0.0212034	−	0.0400821i
$424$		0			0
$425$		−	221046.i		−	1.22378i
$426$		0			0
$427$		−	352528.i		−	1.93347i
$428$		0			0
$429$	103758.	+	418033.i	0.563776	+	2.27141i
$430$		0			0
$431$			106621.i			0.573969i	0.957935	+	0.286984i	$0.0926528\pi$
$431$			106621.i			0.573969i	−0.957935	+	0.286984i	$0.907347\pi$
$432$		0			0
$433$	−174550.			−0.930990			−0.465495	−	0.885051i	$-0.654124\pi$
$433$	−174550.			−0.930990			−0.465495	+	0.885051i	$0.654124\pi$
$434$		0			0
$435$	62587.7	−	15534.6i	0.330758	−	0.0820959i
$436$		0			0
$437$	−171961.			−0.900466
$438$		0			0
$439$	149048.			0.773389			0.386694	−	0.922208i	$-0.373617\pi$
$439$	149048.			0.773389			0.386694	+	0.922208i	$0.373617\pi$
$440$		0			0
$441$	185343.	−	98046.3i	0.953012	−	0.504143i
$442$		0			0
$443$	135015.			0.687979			0.343989	−	0.938974i	$-0.388222\pi$
$443$	135015.			0.687979			0.343989	+	0.938974i	$0.388222\pi$
$444$		0			0
$445$		−	108010.i		−	0.545437i
$446$		0			0
$447$	−166409.	+	41303.6i	−0.832842	+	0.206716i
$448$		0			0
$449$			94873.2i			0.470599i	0.971923	+	0.235299i	$0.0756071\pi$
$449$			94873.2i			0.470599i	−0.971923	+	0.235299i	$0.924393\pi$
$450$		0			0
$451$		−	55351.3i		−	0.272129i
$452$		0			0
$453$	347865.	−	86341.9i	1.69517	−	0.420751i
$454$		0			0
$455$		−	300562.i		−	1.45182i
$456$		0			0
$457$	−67995.0			−0.325570			−0.162785	−	0.986662i	$-0.552048\pi$
$457$	−67995.0			−0.325570			−0.162785	+	0.986662i	$0.552048\pi$
$458$		0			0
$459$	−259146.	−	289653.i	−1.23004	−	1.37484i
$460$		0			0
$461$	102453.			0.482086			0.241043	−	0.970514i	$-0.422511\pi$
$461$	102453.			0.482086			0.241043	+	0.970514i	$0.422511\pi$
$462$		0			0
$463$	256628.			1.19713			0.598567	−	0.801073i	$-0.295737\pi$
$463$	256628.			1.19713			0.598567	+	0.801073i	$0.295737\pi$
$464$		0			0
$465$	−132113.	+	32791.2i	−0.610998	+	0.151653i
$466$		0			0
$467$	−161298.			−0.739597			−0.369799	−	0.929112i	$-0.620573\pi$
$467$	−161298.			−0.739597			−0.369799	+	0.929112i	$0.620573\pi$
$468$		0			0
$469$		−	11933.9i		−	0.0542545i
$470$		0			0
$471$	−2767.45	−	11149.8i	−0.0124749	−	0.0502605i
$472$		0			0
$473$		−	421827.i		−	1.88544i
$474$		0			0
$475$			130233.i			0.577210i
$476$		0			0
$477$	−100884.	+	53367.4i	−0.443388	+	0.234552i
$478$		0			0
$479$			179193.i			0.780999i	0.920603	+	0.390499i	$0.127698\pi$
$479$			179193.i			0.780999i	−0.920603	+	0.390499i	$0.872302\pi$
$480$		0			0
$481$	161377.			0.697512
$482$		0			0
$483$	83840.9	+	337789.i	0.359387	+	1.44794i
$484$		0			0
$485$	−119960.			−0.509981
$486$		0			0
$487$	−13645.2			−0.0575339			−0.0287669	−	0.999586i	$-0.509158\pi$
$487$	−13645.2			−0.0575339			−0.0287669	+	0.999586i	$0.509158\pi$
$488$		0			0
$489$	27021.6	+	108868.i	0.113004	+	0.455284i
$490$		0			0
$491$	−425902.			−1.76664			−0.883318	−	0.468774i	$-0.844696\pi$
$491$	−425902.			−1.76664			−0.883318	+	0.468774i	$0.844696\pi$
$492$		0			0
$493$		−	263365.i		−	1.08359i
$494$		0			0
$495$	169425.	−	89625.7i	0.691459	−	0.365782i
$496$		0			0
$497$			371672.i			1.50469i
$498$		0			0
$499$			116410.i			0.467507i	0.972296	+	0.233753i	$0.0751008\pi$
$499$			116410.i			0.467507i	−0.972296	+	0.233753i	$0.924899\pi$
$500$		0			0
$501$	−40217.0	−	162031.i	−0.160227	−	0.645541i
$502$		0			0
$503$		−	395157.i		−	1.56183i	−0.624637	−	0.780915i	$-0.714753\pi$
$503$		−	395157.i		−	1.56183i	0.624637	−	0.780915i	$-0.285247\pi$
$504$		0			0
$505$	95019.3			0.372588
$506$		0			0
$507$	502215.	−	124652.i	1.95377	−	0.484936i
$508$		0			0
$509$	−492582.			−1.90126			−0.950632	−	0.310319i	$-0.899564\pi$
$509$	−492582.			−1.90126			−0.950632	+	0.310319i	$0.899564\pi$
$510$		0			0
$511$	87937.0			0.336767
$512$		0			0
$513$	152680.	+	170654.i	0.580160	+	0.648458i
$514$		0			0
$515$	205132.			0.773427
$516$		0			0
$517$		−	16341.2i		−	0.0611368i
$518$		0			0
$519$	26156.2	−	6492.09i	0.0971045	−	0.0241018i
$520$		0			0
$521$			498910.i			1.83801i	0.394249	+	0.919003i	$0.371005\pi$
$521$			498910.i			1.83801i	−0.394249	+	0.919003i	$0.628995\pi$
$522$		0			0
$523$			174522.i			0.638040i	0.947748	+	0.319020i	$0.103354\pi$
$523$			174522.i			0.638040i	−0.947748	+	0.319020i	$0.896646\pi$
$524$		0			0
$525$	255821.	−	63496.1i	0.928149	−	0.230371i
$526$		0			0
$527$			555923.i			2.00168i
$528$		0			0
$529$	−19869.1			−0.0710014
$530$		0			0
$531$	103062.	−	54519.7i	0.365518	−	0.193359i
$532$		0			0
$533$	−99531.0			−0.350352
$534$		0			0
$535$	−53054.6			−0.185360
$536$		0			0
$537$	−428328.	+	106313.i	−1.48535	+	0.368671i
$538$		0			0
$539$	422307.			1.45362
$540$		0			0
$541$		−	185893.i		−	0.635139i	−0.948235	−	0.317569i	$-0.897133\pi$
$541$		−	185893.i		−	0.635139i	0.948235	−	0.317569i	$-0.102867\pi$
$542$		0			0
$543$	−60817.4	−	245029.i	−0.206266	−	0.831032i
$544$		0			0
$545$			126290.i			0.425183i
$546$		0			0
$547$		−	378740.i		−	1.26580i	−0.774232	−	0.632902i	$-0.781864\pi$
$547$		−	378740.i		−	1.26580i	0.774232	−	0.632902i	$-0.218136\pi$
$548$		0			0
$549$	−189027.	−	357328.i	−0.627160	−	1.18556i
$550$		0			0
$551$			155166.i			0.511086i
$552$		0			0
$553$	524265.			1.71436
$554$		0			0
$555$	−17299.5	−	69698.5i	−0.0561627	−	0.226275i
$556$		0			0
$557$	−159324.			−0.513537			−0.256768	−	0.966473i	$-0.582658\pi$
$557$	−159324.			−0.513537			−0.256768	+	0.966473i	$0.582658\pi$
$558$		0			0
$559$	−758517.			−2.42740
$560$		0			0
$561$	−188570.	−	759733.i	−0.599164	−	2.41399i
$562$		0			0
$563$	−437248.			−1.37947			−0.689733	−	0.724063i	$-0.742272\pi$
$563$	−437248.			−1.37947			−0.689733	+	0.724063i	$0.742272\pi$
$564$		0			0
$565$			223434.i			0.699926i
$566$		0			0
$567$	260781.	−	383119.i	0.811167	−	1.19170i
$568$		0			0
$569$			300156.i			0.927092i	0.886073	+	0.463546i	$0.153423\pi$
$569$			300156.i			0.927092i	−0.886073	+	0.463546i	$0.846577\pi$
$570$		0			0
$571$			202047.i			0.619698i	0.950786	+	0.309849i	$0.100279\pi$
$571$			202047.i			0.619698i	−0.950786	+	0.309849i	$0.899721\pi$
$572$		0			0
$573$	108481.	+	437060.i	0.330402	+	1.33117i
$574$		0			0
$575$			226982.i			0.686525i
$576$		0			0
$577$	−207377.			−0.622887			−0.311443	−	0.950265i	$-0.600812\pi$
$577$	−207377.			−0.622887			−0.311443	+	0.950265i	$0.600812\pi$
$578$		0			0
$579$	−269965.	+	67006.8i	−0.805288	+	0.199877i
$580$		0			0
$581$	−8233.97			−0.0243925
$582$		0			0
$583$	−229865.			−0.676294
$584$		0			0
$585$	−161162.	−	304655.i	−0.470925	−	0.890218i
$586$		0			0
$587$	−545839.			−1.58412			−0.792060	−	0.610443i	$-0.790991\pi$
$587$	−545839.			−1.58412			−0.792060	+	0.610443i	$0.790991\pi$
$588$		0			0
$589$		−	327532.i		−	0.944111i
$590$		0			0
$591$	−352258.	+	87432.3i	−1.00852	+	0.250321i
$592$		0			0
$593$			124034.i			0.352720i	0.984326	+	0.176360i	$0.0564323\pi$
$593$			124034.i			0.352720i	−0.984326	+	0.176360i	$0.943568\pi$
$594$		0			0
$595$			546242.i			1.54295i
$596$		0			0
$597$	−6170.11	+	1531.45i	−0.0173119	+	0.00429690i
$598$		0			0
$599$		−	216086.i		−	0.602243i	−0.953586	−	0.301122i	$-0.902639\pi$
$599$		−	216086.i		−	0.602243i	0.953586	−	0.301122i	$-0.0973610\pi$
$600$		0			0
$601$	−590209.			−1.63402			−0.817009	−	0.576625i	$-0.804369\pi$
$601$	−590209.			−1.63402			−0.817009	+	0.576625i	$0.804369\pi$
$602$		0			0
$603$	−6398.97	−	12096.4i	−0.0175985	−	0.0332675i
$604$		0			0
$605$	173673.			0.474484
$606$		0			0
$607$	−27397.4			−0.0743587			−0.0371794	−	0.999309i	$-0.511837\pi$
$607$	−27397.4			−0.0743587			−0.0371794	+	0.999309i	$0.511837\pi$
$608$		0			0
$609$	304798.	−	75652.4i	0.821822	−	0.203980i
$610$		0			0
$611$	−29384.3			−0.0787104
$612$		0			0
$613$			594961.i			1.58331i	0.610965	+	0.791657i	$0.290781\pi$
$613$			594961.i			1.58331i	−0.610965	+	0.791657i	$0.709219\pi$
$614$		0			0
$615$	10669.7	+	42987.3i	0.0282098	+	0.113655i
$616$		0			0
$617$			614701.i			1.61471i	0.590068	+	0.807353i	$0.299101\pi$
$617$			614701.i			1.61471i	−0.590068	+	0.807353i	$0.700899\pi$
$618$		0			0
$619$			676244.i			1.76491i	0.470399	+	0.882454i	$0.344110\pi$
$619$			676244.i			1.76491i	−0.470399	+	0.882454i	$0.655890\pi$
$620$		0			0
$621$	266106.	+	297432.i	0.690035	+	0.771267i
$622$		0			0
$623$		−	526002.i		−	1.35523i
$624$		0			0
$625$	40410.3			0.103450
$626$		0			0
$627$	111099.	+	447610.i	0.282602	+	1.13858i
$628$		0			0
$629$	−293287.			−0.741295
$630$		0			0
$631$	−176191.			−0.442513			−0.221256	−	0.975216i	$-0.571016\pi$
$631$	−176191.			−0.442513			−0.221256	+	0.975216i	$0.571016\pi$
$632$		0			0
$633$	−72719.1	−	292980.i	−0.181485	−	0.731190i
$634$		0			0
$635$	328110.			0.813714
$636$		0			0
$637$		−	759379.i		−	1.87146i
$638$		0			0
$639$	199291.	+	376733.i	0.488075	+	0.922638i
$640$		0			0
$641$		−	191375.i		−	0.465767i	−0.972505	−	0.232883i	$-0.925184\pi$
$641$		−	191375.i		−	0.465767i	0.972505	−	0.232883i	$-0.0748161\pi$
$642$		0			0
$643$			323396.i			0.782191i	0.920350	+	0.391095i	$0.127904\pi$
$643$			323396.i			0.782191i	−0.920350	+	0.391095i	$0.872096\pi$
$644$		0			0
$645$	81312.6	+	327602.i	0.195451	+	0.787458i
$646$		0			0
$647$			451814.i			1.07932i	0.841882	+	0.539662i	$0.181448\pi$
$647$			451814.i			1.07932i	−0.841882	+	0.539662i	$0.818552\pi$
$648$		0			0
$649$	234828.			0.557521
$650$		0			0
$651$	−643382.	+	159691.i	−1.51812	+	0.376806i
$652$		0			0
$653$	−467328.			−1.09596			−0.547980	−	0.836491i	$-0.684603\pi$
$653$	−467328.			−1.09596			−0.547980	+	0.836491i	$0.684603\pi$
$654$		0			0
$655$	−135405.			−0.315612
$656$		0			0
$657$	89134.3	−	47152.0i	0.206497	−	0.109237i
$658$		0			0
$659$	163218.			0.375835			0.187918	−	0.982185i	$-0.439826\pi$
$659$	163218.			0.375835			0.187918	+	0.982185i	$0.439826\pi$
$660$		0			0
$661$			698278.i			1.59818i	0.601212	+	0.799090i	$0.294685\pi$
$661$			698278.i			1.59818i	−0.601212	+	0.799090i	$0.705315\pi$
$662$		0			0
$663$	−1.36613e6	+	339080.i	−3.10788	+	0.771392i
$664$		0			0
$665$		−	321828.i		−	0.727747i
$666$		0			0
$667$			270438.i			0.607878i
$668$		0			0
$669$	514965.	−	127817.i	1.15060	−	0.285586i
$670$		0			0
$671$		−	814178.i		−	1.80832i
$672$		0			0
$673$	−397119.			−0.876779			−0.438389	−	0.898785i	$-0.644451\pi$
$673$	−397119.			−0.876779			−0.438389	+	0.898785i	$0.644451\pi$
$674$		0			0
$675$	225257.	−	201533.i	0.494392	−	0.442321i
$676$		0			0
$677$	−495877.			−1.08192			−0.540962	−	0.841047i	$-0.681940\pi$
$677$	−495877.			−1.08192			−0.540962	+	0.841047i	$0.681940\pi$
$678$		0			0
$679$	−584198.			−1.26713
$680$		0			0
$681$	632381.	−	156960.i	1.36359	−	0.338451i
$682$		0			0
$683$	−713524.			−1.52956			−0.764781	−	0.644290i	$-0.777153\pi$
$683$	−713524.			−1.52956			−0.764781	+	0.644290i	$0.777153\pi$
$684$		0			0
$685$		−	263420.i		−	0.561394i
$686$		0			0
$687$	−44191.2	−	178043.i	−0.0936315	−	0.377234i
$688$		0			0
$689$			413336.i			0.870693i
$690$		0			0
$691$		−	801221.i		−	1.67802i	−0.544118	−	0.839009i	$-0.683136\pi$
$691$		−	801221.i		−	1.67802i	0.544118	−	0.839009i	$-0.316864\pi$
$692$		0			0
$693$	825088.	−	436471.i	1.71804	−	0.908844i
$694$		0			0
$695$			137690.i			0.285058i
$696$		0			0
$697$	180888.			0.372343
$698$		0			0
$699$	167806.	+	676077.i	0.343442	+	1.38370i
$700$		0			0
$701$	572005.			1.16403			0.582015	−	0.813178i	$-0.302265\pi$
$701$	572005.			1.16403			0.582015	+	0.813178i	$0.302265\pi$
$702$		0			0
$703$	172795.			0.349639
$704$		0			0
$705$	3149.97	+	12691.0i	0.00633765	+	0.0255339i
$706$		0			0
$707$	462738.			0.925755
$708$		0			0
$709$		−	447501.i		−	0.890228i	−0.895474	−	0.445114i	$-0.853163\pi$
$709$		−	447501.i		−	0.890228i	0.895474	−	0.445114i	$-0.146837\pi$
$710$		0			0
$711$	531404.	−	281112.i	1.05120	−	0.556084i
$712$		0			0
$713$		−	570854.i		−	1.12291i
$714$		0			0
$715$		−	694161.i		−	1.35784i
$716$		0			0
$717$	−67324.2	−	271244.i	−0.130958	−	0.527621i
$718$		0			0
$719$		−	614041.i		−	1.18779i	−0.804542	−	0.593895i	$-0.797589\pi$
$719$		−	614041.i		−	1.18779i	0.804542	−	0.593895i	$-0.202411\pi$
$720$		0			0
$721$	998981.			1.92170
$722$		0			0
$723$	−536624.	+	133193.i	−1.02658	+	0.254803i
$724$		0			0
$725$	204814.			0.389658
$726$		0			0
$727$	503006.			0.951709			0.475855	−	0.879524i	$-0.342139\pi$
$727$	503006.			0.951709			0.475855	+	0.879524i	$0.342139\pi$
$728$		0			0
$729$	58902.8	−	528167.i	0.110836	−	0.993839i
$730$		0			0
$731$	1.37853e6			2.57977
$732$		0			0
$733$			558326.i			1.03915i	0.854424	+	0.519577i	$0.173910\pi$
$733$			558326.i			1.03915i	−0.854424	+	0.519577i	$0.826090\pi$
$734$		0			0
$735$	−327975.	+	81405.0i	−0.607107	+	0.150687i
$736$		0			0
$737$		−	27561.8i		−	0.0507425i
$738$		0			0
$739$			897002.i			1.64250i	0.570571	+	0.821248i	$0.306722\pi$
$739$			897002.i			1.64250i	−0.570571	+	0.821248i	$0.693278\pi$
$740$		0			0
$741$	804879.	−	199775.i	1.46587	−	0.363835i
$742$		0			0
$743$			28448.2i			0.0515319i	0.999668	+	0.0257660i	$0.00820247\pi$
$743$			28448.2i			0.0515319i	−0.999668	+	0.0257660i	$0.991798\pi$
$744$		0			0
$745$	276330.			0.497869
$746$		0			0
$747$	−8346.08	+	4415.08i	−0.0149569	+	0.00791219i
$748$		0			0
$749$	−258372.			−0.460556
$750$		0			0
$751$	−157854.			−0.279883			−0.139941	−	0.990160i	$-0.544691\pi$
$751$	−157854.			−0.279883			−0.139941	+	0.990160i	$0.544691\pi$
$752$		0			0
$753$	−509405.	+	126437.i	−0.898408	+	0.222990i
$754$		0			0
$755$	−577644.			−1.01337
$756$		0			0
$757$			485489.i			0.847202i	0.905849	+	0.423601i	$0.139234\pi$
$757$			485489.i			0.847202i	−0.905849	+	0.423601i	$0.860766\pi$
$758$		0			0
$759$	193634.	+	780137.i	0.336123	+	1.35421i
$760$		0			0
$761$			496163.i			0.856752i	0.903601	+	0.428376i	$0.140914\pi$
$761$			496163.i			0.856752i	−0.903601	+	0.428376i	$0.859086\pi$
$762$		0			0
$763$			615024.i			1.05643i
$764$		0			0
$765$	292896.	+	553679.i	0.500485	+	0.946096i
$766$		0			0
$767$		−	422261.i		−	0.717779i
$768$		0			0
$769$	61547.3			0.104077			0.0520386	−	0.998645i	$-0.483428\pi$
$769$	61547.3			0.104077			0.0520386	+	0.998645i	$0.483428\pi$
$770$		0			0
$771$	42435.3	+	170968.i	0.0713868	+	0.287612i
$772$		0			0
$773$	−1.00051e6			−1.67441			−0.837205	−	0.546889i	$-0.815812\pi$
$773$	−1.00051e6			−1.67441			−0.837205	+	0.546889i	$0.815812\pi$
$774$		0			0
$775$	−432331.			−0.719801
$776$		0			0
$777$	−84247.5	−	339427.i	−0.139545	−	0.562217i
$778$		0			0
$779$	−106573.			−0.175619
$780$		0			0
$781$			858392.i			1.40729i
$782$		0			0
$783$	268383.	−	240116.i	0.437755	−	0.391649i
$784$		0			0
$785$			18514.8i			0.0300455i
$786$		0			0
$787$			574494.i			0.927548i	0.885954	+	0.463774i	$0.153505\pi$
$787$			574494.i			0.927548i	−0.885954	+	0.463774i	$0.846495\pi$
$788$		0			0
$789$	−241150.	−	971575.i	−0.387376	−	1.56071i
$790$		0			0
$791$			1.08811e6i			1.73908i
$792$		0			0
$793$	−1.46403e6			−2.32811
$794$		0			0
$795$	178519.	−	44309.4i	0.282456	−	0.0701070i
$796$		0			0
$797$	−677843.			−1.06712			−0.533559	−	0.845763i	$-0.679146\pi$
$797$	−677843.			−1.06712			−0.533559	+	0.845763i	$0.679146\pi$
$798$		0			0
$799$	53402.9			0.0836511
$800$		0			0
$801$	−282044.	−	533164.i	−0.439594	−	0.830990i
$802$		0			0
$803$	203094.			0.314968
$804$		0			0
$805$		−	560913.i		−	0.865572i
$806$		0			0
$807$	333221.	−	82707.3i	0.511665	−	0.126998i
$808$		0			0
$809$			1.06520e6i			1.62754i	0.581184	+	0.813772i	$0.302590\pi$
$809$			1.06520e6i			1.62754i	−0.581184	+	0.813772i	$0.697410\pi$
$810$		0			0
$811$		−	1.26372e6i		−	1.92137i	−0.277643	−	0.960684i	$-0.589553\pi$
$811$		−	1.26372e6i		−	1.92137i	0.277643	−	0.960684i	$-0.410447\pi$
$812$		0			0
$813$	−645163.	+	160133.i	−0.976087	+	0.242270i
$814$		0			0
$815$		−	180780.i		−	0.272166i
$816$		0			0
$817$	−812184.			−1.21678
$818$		0			0
$819$	−784850.	−	1.48365e6i	−1.17009	−	2.21189i
$820$		0			0
$821$	302546.			0.448854			0.224427	−	0.974491i	$-0.427949\pi$
$821$	302546.			0.448854			0.224427	+	0.974491i	$0.427949\pi$
$822$		0			0
$823$	−279454.			−0.412583			−0.206292	−	0.978491i	$-0.566140\pi$
$823$	−279454.			−0.412583			−0.206292	+	0.978491i	$0.566140\pi$
$824$		0			0
$825$	590829.	−	146647.i	0.868069	−	0.215459i
$826$		0			0
$827$	−853114.			−1.24737			−0.623686	−	0.781675i	$-0.714366\pi$
$827$	−853114.			−1.24737			−0.623686	+	0.781675i	$0.714366\pi$
$828$		0			0
$829$		−	99511.5i		−	0.144799i	−0.997376	−	0.0723993i	$-0.976934\pi$
$829$		−	99511.5i		−	0.144799i	0.997376	−	0.0723993i	$-0.0230656\pi$
$830$		0			0
$831$	−200103.	−	806199.i	−0.289768	−	1.16746i
$832$		0			0
$833$			1.38010e6i			1.98893i
$834$		0			0
$835$			269060.i			0.385901i
$836$		0			0
$837$	−566516.	+	506848.i	−0.808651	+	0.723481i
$838$		0			0
$839$		−	239237.i		−	0.339864i	−0.985456	−	0.169932i	$-0.945645\pi$
$839$		−	239237.i		−	0.339864i	0.985456	−	0.169932i	$-0.0543548\pi$
$840$		0			0
$841$	−463256.			−0.654981
$842$		0			0
$843$	53340.1	+	214903.i	0.0750583	+	0.302404i
$844$		0			0
$845$	−833950.			−1.16796
$846$		0			0
$847$	845777.			1.17893
$848$		0			0
$849$	50070.0	+	201728.i	0.0694644	+	0.279867i
$850$		0			0
$851$	301163.			0.415856
$852$		0			0
$853$		−	224778.i		−	0.308927i	−0.987999	−	0.154463i	$-0.950635\pi$
$853$		−	224778.i		−	0.308927i	0.987999	−	0.154463i	$-0.0493649\pi$
$854$		0			0
$855$	−172565.	−	326210.i	−0.236059	−	0.446236i
$856$		0			0
$857$			648711.i			0.883262i	0.897197	+	0.441631i	$0.145600\pi$
$857$			648711.i			0.883262i	−0.897197	+	0.441631i	$0.854400\pi$
$858$		0			0
$859$		−	761094.i		−	1.03146i	−0.856752	−	0.515729i	$-0.827521\pi$
$859$		−	761094.i		−	1.03146i	0.856752	−	0.515729i	$-0.172479\pi$
$860$		0			0
$861$	51960.6	+	209345.i	0.0700918	+	0.282395i
$862$		0			0
$863$			883951.i			1.18688i	0.804879	+	0.593440i	$0.202230\pi$
$863$			883951.i			1.18688i	−0.804879	+	0.593440i	$0.797770\pi$
$864$		0			0
$865$	−43433.4			−0.0580486
$866$		0			0
$867$	1.75325e6	−	435165.i	2.33241	−	0.578917i
$868$		0			0
$869$	1.21081e6			1.60338
$870$		0			0
$871$	−49560.8			−0.0653284
$872$		0			0
$873$	−592153.	+	313249.i	−0.776972	+	0.411018i
$874$		0			0
$875$	−1.06516e6			−1.39123
$876$		0			0
$877$		−	763734.i		−	0.992985i	−0.868041	−	0.496493i	$-0.834621\pi$
$877$		−	763734.i		−	0.992985i	0.868041	−	0.496493i	$-0.165379\pi$
$878$		0			0
$879$	825219.	−	204823.i	1.06805	−	0.265095i
$880$		0			0
$881$		−	1.42245e6i		−	1.83267i	−0.400408	−	0.916337i	$-0.631132\pi$
$881$		−	1.42245e6i		−	1.83267i	0.400408	−	0.916337i	$-0.368868\pi$
$882$		0			0
$883$		−	526843.i		−	0.675709i	−0.941198	−	0.337854i	$-0.890299\pi$
$883$		−	526843.i		−	0.675709i	0.941198	−	0.337854i	$-0.109701\pi$
$884$		0			0
$885$	−182374.	+	45266.1i	−0.232850	+	0.0577946i
$886$		0			0
$887$			1.22321e6i			1.55473i	0.629050	+	0.777365i	$0.283444\pi$
$887$			1.22321e6i			1.55473i	−0.629050	+	0.777365i	$0.716556\pi$
$888$		0			0
$889$	1.59787e6			2.02180
$890$		0			0
$891$	602285.	−	884828.i	0.758659	−	1.11456i
$892$		0			0
$893$	−31463.3			−0.0394549
$894$		0			0
$895$	711257.			0.887933
$896$		0			0
$897$	1.40282e6	−	348187.i	1.74348	−	0.432741i
$898$		0			0
$899$	−515100.			−0.637342
$900$		0			0
$901$		−	751197.i		−	0.925346i
$902$		0			0
$903$	395987.	+	1.59540e6i	0.485630	+	1.95657i
$904$		0			0
$905$			406881.i			0.496787i
$906$		0			0
$907$		−	58073.7i		−	0.0705935i	−0.999377	−	0.0352968i	$-0.988762\pi$
$907$		−	58073.7i		−	0.0705935i	0.999377	−	0.0352968i	$-0.0112376\pi$
$908$		0			0
$909$	469038.	−	248121.i	0.567649	−	0.300286i
$910$		0			0
$911$		−	101962.i		−	0.122858i	−0.998111	−	0.0614288i	$-0.980434\pi$
$911$		−	101962.i		−	0.122858i	0.998111	−	0.0614288i	$-0.0195657\pi$
$912$		0			0
$913$	−19016.7			−0.0228136
$914$		0			0
$915$	156943.	+	632313.i	0.187456	+	0.755248i
$916$		0			0
$917$	−659415.			−0.784188
$918$		0			0
$919$	1.66618e6			1.97284			0.986420	−	0.164240i	$-0.0525172\pi$
$919$	1.66618e6			1.97284			0.986420	+	0.164240i	$0.0525172\pi$
$920$		0			0
$921$	−47622.3	−	191867.i	−0.0561425	−	0.226194i
$922$		0			0
$923$	1.54353e6			1.81181
$924$		0			0
$925$		−	228083.i		−	0.266569i
$926$		0			0
$927$	1.01258e6	−	535656.i	1.17834	−	0.623342i
$928$		0			0
$929$		−	545932.i		−	0.632568i	−0.948665	−	0.316284i	$-0.897565\pi$
$929$		−	545932.i		−	0.632568i	0.948665	−	0.316284i	$-0.102435\pi$
$930$		0			0
$931$		−	813107.i		−	0.938099i
$932$		0			0
$933$	350.240	+	1411.09i	0.000402348	+	0.00162103i
$934$		0			0
$935$			1.26157e6i			1.44307i
$936$		0			0
$937$	−69359.6			−0.0790000			−0.0395000	−	0.999220i	$-0.512577\pi$
$937$	−69359.6			−0.0790000			−0.0395000	+	0.999220i	$0.512577\pi$
$938$		0			0
$939$	−199260.	+	49457.4i	−0.225990	+	0.0560919i
$940$		0			0
$941$	625701.			0.706622			0.353311	−	0.935506i	$-0.385056\pi$
$941$	625701.			0.706622			0.353311	+	0.935506i	$0.385056\pi$
$942$		0			0
$943$	−185746.			−0.208879
$944$		0			0
$945$	−556650.	+	498021.i	−0.623330	+	0.557679i
$946$		0			0
$947$	−341934.			−0.381278			−0.190639	−	0.981660i	$-0.561056\pi$
$947$	−341934.			−0.381278			−0.190639	+	0.981660i	$0.561056\pi$
$948$		0			0
$949$		−	365198.i		−	0.405505i
$950$		0			0
$951$	−97595.9	+	24223.8i	−0.107912	+	0.0267844i
$952$		0			0
$953$			178830.i			0.196904i	0.995142	+	0.0984518i	$0.0313890\pi$
$953$			178830.i			0.196904i	−0.995142	+	0.0984518i	$0.968611\pi$
$954$		0			0
$955$		−	725757.i		−	0.795764i
$956$		0			0
$957$	703943.	−	174722.i	0.768624	−	0.190776i
$958$		0			0
$959$		−	1.28284e6i		−	1.39487i
$960$		0			0
$961$	163777.			0.177340
$962$		0			0
$963$	−261890.	+	138540.i	−0.282401	+	0.149390i
$964$		0			0
$965$	448289.			0.481397
$966$		0			0
$967$	406495.			0.434712			0.217356	−	0.976092i	$-0.430257\pi$
$967$	406495.			0.434712			0.217356	+	0.976092i	$0.430257\pi$
$968$		0			0
$969$	−1.46279e6	+	363071.i	−1.55788	+	0.386673i
$970$		0			0
$971$	−8629.24			−0.00915239			−0.00457619	−	0.999990i	$-0.501457\pi$
$971$	−8629.24			−0.00915239			−0.00457619	+	0.999990i	$0.501457\pi$
$972$		0			0
$973$			670542.i			0.708273i
$974$		0			0
$975$	−263696.	−	1.06241e6i	−0.277392	−	1.11759i
$976$		0			0
$977$			455792.i			0.477505i	0.971080	+	0.238753i	$0.0767385\pi$
$977$			455792.i			0.477505i	−0.971080	+	0.238753i	$0.923262\pi$
$978$		0			0
$979$		−	1.21482e6i		−	1.26750i
$980$		0			0
$981$	329777.	+	623397.i	0.342675	+	0.647779i
$982$		0			0
$983$		−	1.67749e6i		−	1.73602i	−0.496549	−	0.868009i	$-0.665400\pi$
$983$		−	1.67749e6i		−	1.73602i	0.496549	−	0.868009i	$-0.334600\pi$
$984$		0			0
$985$	584939.			0.602890
$986$		0			0
$987$	15340.2	+	61804.4i	0.0157469	+	0.0634432i
$988$		0			0
$989$	−1.41555e6			−1.44722
$990$		0			0
$991$	1.02074e6			1.03937			0.519684	−	0.854358i	$-0.326050\pi$
$991$	1.02074e6			1.03937			0.519684	+	0.854358i	$0.326050\pi$
$992$		0			0
$993$	43787.9	+	176418.i	0.0444075	+	0.178914i
$994$		0			0
$995$	10245.7			0.0103490
$996$		0			0
$997$			1.21614e6i			1.22347i	0.791064	+	0.611733i	$0.209527\pi$
$997$			1.21614e6i			1.22347i	−0.791064	+	0.611733i	$0.790473\pi$
$998$		0			0
$999$	−267396.	−	298875.i	−0.267932	−	0.299473i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.h.h.65.4	yes	32
3.2	odd	2	inner	384.5.h.h.65.31	yes	32
4.3	odd	2	inner	384.5.h.h.65.30	yes	32
8.3	odd	2	inner	384.5.h.h.65.3	yes	32
8.5	even	2	inner	384.5.h.h.65.29	yes	32
12.11	even	2	inner	384.5.h.h.65.1	✓	32
24.5	odd	2	inner	384.5.h.h.65.2	yes	32
24.11	even	2	inner	384.5.h.h.65.32	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.h.h.65.1	✓	32	12.11	even	2	inner
384.5.h.h.65.2	yes	32	24.5	odd	2	inner
384.5.h.h.65.3	yes	32	8.3	odd	2	inner
384.5.h.h.65.4	yes	32	1.1	even	1	trivial
384.5.h.h.65.29	yes	32	8.5	even	2	inner
384.5.h.h.65.30	yes	32	4.3	odd	2	inner
384.5.h.h.65.31	yes	32	3.2	odd	2	inner
384.5.h.h.65.32	yes	32	24.11	even	2	inner

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	384.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-8.73496 + 2.16806i) q^{3} +14.5048 q^{5} +70.6373 q^{7} +(71.5990 - 37.8759i) q^{9} +O(q^{10})\)	\(q+(-8.73496 + 2.16806i) q^{3} +14.5048 q^{5} +70.6373 q^{7} +(71.5990 - 37.8759i) q^{9} +163.140 q^{11} -293.353i q^{13} +(-126.699 + 31.4472i) q^{15} +533.139i q^{17} -314.108i q^{19} +(-617.014 + 153.146i) q^{21} -547.458i q^{23} -414.612 q^{25} +(-543.297 + 486.075i) q^{27} -493.989 q^{29} +1042.74 q^{31} +(-1425.02 + 353.697i) q^{33} +1024.58 q^{35} +550.113i q^{37} +(636.007 + 2562.42i) q^{39} -339.288i q^{41} -2585.68i q^{43} +(1038.53 - 549.380i) q^{45} -100.167i q^{47} +2588.62 q^{49} +(-1155.88 - 4656.95i) q^{51} -1409.01 q^{53} +2366.30 q^{55} +(681.006 + 2743.72i) q^{57} +1439.43 q^{59} -4990.69i q^{61} +(5057.56 - 2675.45i) q^{63} -4255.01i q^{65} -168.946i q^{67} +(1186.92 + 4782.02i) q^{69} +5261.70i q^{71} +1244.91 q^{73} +(3621.62 - 898.904i) q^{75} +11523.7 q^{77} +7421.94 q^{79} +(3691.84 - 5423.75i) q^{81} -116.567 q^{83} +7733.06i q^{85} +(4314.98 - 1071.00i) q^{87} -7446.53i q^{89} -20721.6i q^{91} +(-9108.26 + 2260.72i) q^{93} -4556.07i q^{95} -8270.40 q^{97} +(11680.6 - 6179.05i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 224 q^{9}+O(q^{10}) \) 32 * q + 224 * q^9	\( 32 q + 224 q^{9} + 5472 q^{25} - 3712 q^{33} + 13664 q^{49} - 17344 q^{57} - 17472 q^{73} - 10976 q^{81} - 39488 q^{97}+O(q^{100}) \) 32 * q + 224 * q^9 + 5472 * q^25 - 3712 * q^33 + 13664 * q^49 - 17344 * q^57 - 17472 * q^73 - 10976 * q^81 - 39488 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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