LMFDB - Embedded newform 384.5.h.h.65.5

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			65.5
Character	$\chi$	$=$	384.65
Dual form			384.5.h.h.65.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.10449 - 3.91372i) q^{3} -29.9378 q^{5} -5.14305 q^{7} +(50.3656 + 63.4374i) q^{9} +O(q^{10})$	\(q+(-8.10449 - 3.91372i) q^{3} -29.9378 q^{5} -5.14305 q^{7} +(50.3656 + 63.4374i) q^{9} -170.063 q^{11} +161.374i q^{13} +(242.631 + 117.168i) q^{15} +32.6278i q^{17} +299.419i q^{19} +(41.6818 + 20.1284i) q^{21} +333.638i q^{23} +271.273 q^{25} +(-159.911 - 711.245i) q^{27} -930.939 q^{29} +128.177 q^{31} +(1378.28 + 665.580i) q^{33} +153.972 q^{35} +2410.31i q^{37} +(631.572 - 1307.85i) q^{39} +1190.24i q^{41} -2969.61i q^{43} +(-1507.84 - 1899.18i) q^{45} -3959.61i q^{47} -2374.55 q^{49} +(127.696 - 264.432i) q^{51} +3587.36 q^{53} +5091.33 q^{55} +(1171.84 - 2426.64i) q^{57} -1882.24 q^{59} -1616.86i q^{61} +(-259.033 - 326.262i) q^{63} -4831.18i q^{65} -8281.38i q^{67} +(1305.77 - 2703.97i) q^{69} +3154.62i q^{71} -5570.00 q^{73} +(-2198.53 - 1061.69i) q^{75} +874.643 q^{77} +9861.40 q^{79} +(-1487.61 + 6390.13i) q^{81} -8992.50 q^{83} -976.806i q^{85} +(7544.79 + 3643.43i) q^{87} -1596.14i q^{89} -829.953i q^{91} +(-1038.81 - 501.649i) q^{93} -8963.97i q^{95} +2547.74 q^{97} +(-8565.34 - 10788.4i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−8.10449	−	3.91372i	−0.900499	−	0.434858i
$3$	−8.10449	−	3.91372i	−0.900499	−	0.434858i
$4$		0			0
$5$	−29.9378			−1.19751			−0.598757	−	0.800931i	$-0.704338\pi$
$5$	−29.9378			−1.19751			−0.598757	+	0.800931i	$0.704338\pi$
$6$		0			0
$7$	−5.14305			−0.104960			−0.0524801	−	0.998622i	$-0.516713\pi$
$7$	−5.14305			−0.104960			−0.0524801	+	0.998622i	$0.516713\pi$
$8$		0			0
$9$	50.3656	+	63.4374i	0.621797	+	0.783178i
$10$		0			0
$11$	−170.063			−1.40548			−0.702741	−	0.711446i	$-0.748041\pi$
$11$	−170.063			−1.40548			−0.702741	+	0.711446i	$0.748041\pi$
$12$		0			0
$13$			161.374i			0.954875i	0.878666	+	0.477438i	$0.158434\pi$
$13$			161.374i			0.954875i	−0.878666	+	0.477438i	$0.841566\pi$
$14$		0			0
$15$	242.631	+	117.168i	1.07836	+	0.520748i
$16$		0			0
$17$			32.6278i			0.112899i	0.998405	+	0.0564495i	$0.0179780\pi$
$17$			32.6278i			0.112899i	−0.998405	+	0.0564495i	$0.982022\pi$
$18$		0			0
$19$			299.419i			0.829417i	0.909954	+	0.414708i	$0.136116\pi$
$19$			299.419i			0.829417i	−0.909954	+	0.414708i	$0.863884\pi$
$20$		0			0
$21$	41.6818	+	20.1284i	0.0945165	+	0.0456427i
$22$		0			0
$23$			333.638i			0.630696i	0.948976	+	0.315348i	$0.102121\pi$
$23$			333.638i			0.630696i	−0.948976	+	0.315348i	$0.897879\pi$
$24$		0			0
$25$	271.273			0.434038
$26$		0			0
$27$	−159.911	−	711.245i	−0.219357	−	0.975645i
$28$		0			0
$29$	−930.939			−1.10694			−0.553471	−	0.832868i	$-0.686697\pi$
$29$	−930.939			−1.10694			−0.553471	+	0.832868i	$0.686697\pi$
$30$		0			0
$31$	128.177			0.133379			0.0666895	−	0.997774i	$-0.478756\pi$
$31$	128.177			0.133379			0.0666895	+	0.997774i	$0.478756\pi$
$32$		0			0
$33$	1378.28	+	665.580i	1.26564	+	0.611185i
$34$		0			0
$35$	153.972			0.125691
$36$		0			0
$37$			2410.31i			1.76064i	0.474383	+	0.880318i	$0.342671\pi$
$37$			2410.31i			1.76064i	−0.474383	+	0.880318i	$0.657329\pi$
$38$		0			0
$39$	631.572	−	1307.85i	0.415235	−	0.859864i
$40$		0			0
$41$			1190.24i			0.708057i	0.935235	+	0.354028i	$0.115188\pi$
$41$			1190.24i			0.708057i	−0.935235	+	0.354028i	$0.884812\pi$
$42$		0			0
$43$		−	2969.61i		−	1.60606i	−0.595937	−	0.803031i	$-0.703219\pi$
$43$		−	2969.61i		−	1.60606i	0.595937	−	0.803031i	$-0.296781\pi$
$44$		0			0
$45$	−1507.84	−	1899.18i	−0.744611	−	0.937866i
$46$		0			0
$47$		−	3959.61i		−	1.79249i	−0.443558	−	0.896246i	$-0.646284\pi$
$47$		−	3959.61i		−	1.79249i	0.443558	−	0.896246i	$-0.353716\pi$
$48$		0			0
$49$	−2374.55			−0.988983
$50$		0			0
$51$	127.696	−	264.432i	0.0490950	−	0.101666i
$52$		0			0
$53$	3587.36			1.27709			0.638547	−	0.769583i	$-0.279536\pi$
$53$	3587.36			1.27709			0.638547	+	0.769583i	$0.279536\pi$
$54$		0			0
$55$	5091.33			1.68308
$56$		0			0
$57$	1171.84	−	2426.64i	0.360678	−	0.746889i
$58$		0			0
$59$	−1882.24			−0.540719			−0.270359	−	0.962759i	$-0.587143\pi$
$59$	−1882.24			−0.540719			−0.270359	+	0.962759i	$0.587143\pi$
$60$		0			0
$61$		−	1616.86i		−	0.434523i	−0.976113	−	0.217262i	$-0.930287\pi$
$61$		−	1616.86i		−	0.434523i	0.976113	−	0.217262i	$-0.0697125\pi$
$62$		0			0
$63$	−259.033	−	326.262i	−0.0652639	−	0.0822025i
$64$		0			0
$65$		−	4831.18i		−	1.14348i
$66$		0			0
$67$		−	8281.38i		−	1.84482i	−0.386216	−	0.922408i	$-0.626218\pi$
$67$		−	8281.38i		−	1.84482i	0.386216	−	0.922408i	$-0.373782\pi$
$68$		0			0
$69$	1305.77	−	2703.97i	0.274263	−	0.567941i
$70$		0			0
$71$			3154.62i			0.625793i	0.949787	+	0.312897i	$0.101299\pi$
$71$			3154.62i			0.625793i	−0.949787	+	0.312897i	$0.898701\pi$
$72$		0			0
$73$	−5570.00			−1.04522			−0.522612	−	0.852571i	$-0.675042\pi$
$73$	−5570.00			−1.04522			−0.522612	+	0.852571i	$0.675042\pi$
$74$		0			0
$75$	−2198.53	−	1061.69i	−0.390850	−	0.188745i
$76$		0			0
$77$	874.643			0.147520
$78$		0			0
$79$	9861.40			1.58010			0.790049	−	0.613043i	$-0.210055\pi$
$79$	9861.40			1.58010			0.790049	+	0.613043i	$0.210055\pi$
$80$		0			0
$81$	−1487.61	+	6390.13i	−0.226736	+	0.973956i
$82$		0			0
$83$	−8992.50			−1.30534			−0.652671	−	0.757641i	$-0.726352\pi$
$83$	−8992.50			−1.30534			−0.652671	+	0.757641i	$0.726352\pi$
$84$		0			0
$85$		−	976.806i		−	0.135198i
$86$		0			0
$87$	7544.79	+	3643.43i	0.996801	+	0.481363i
$88$		0			0
$89$		−	1596.14i		−	0.201507i	−0.994911	−	0.100754i	$-0.967875\pi$
$89$		−	1596.14i		−	0.201507i	0.994911	−	0.100754i	$-0.0321254\pi$
$90$		0			0
$91$		−	829.953i		−	0.100224i
$92$		0			0
$93$	−1038.81	−	501.649i	−0.120108	−	0.0580009i
$94$		0			0
$95$		−	8963.97i		−	0.993237i
$96$		0			0
$97$	2547.74			0.270777			0.135389	−	0.990793i	$-0.456772\pi$
$97$	2547.74			0.270777			0.135389	+	0.990793i	$0.456772\pi$
$98$		0			0
$99$	−8565.34	−	10788.4i	−0.873925	−	1.10074i
$100$		0			0
$101$	3280.32			0.321568			0.160784	−	0.986990i	$-0.448598\pi$
$101$	3280.32			0.321568			0.160784	+	0.986990i	$0.448598\pi$
$102$		0			0
$103$	10235.3			0.964773			0.482386	−	0.875959i	$-0.339770\pi$
$103$	10235.3			0.964773			0.482386	+	0.875959i	$0.339770\pi$
$104$		0			0
$105$	−1247.86	−	602.602i	−0.113185	−	0.0546577i
$106$		0			0
$107$	−1304.53			−0.113943			−0.0569714	−	0.998376i	$-0.518144\pi$
$107$	−1304.53			−0.113943			−0.0569714	+	0.998376i	$0.518144\pi$
$108$		0			0
$109$		−	4833.00i		−	0.406784i	−0.979097	−	0.203392i	$-0.934803\pi$
$109$		−	4833.00i		−	0.406784i	0.979097	−	0.203392i	$-0.0651966\pi$
$110$		0			0
$111$	9433.28	−	19534.4i	0.765627	−	1.58545i
$112$		0			0
$113$		−	17488.4i		−	1.36960i	−0.728732	−	0.684799i	$-0.759890\pi$
$113$		−	17488.4i		−	1.36960i	0.728732	−	0.684799i	$-0.240110\pi$
$114$		0			0
$115$		−	9988.40i		−	0.755266i
$116$		0			0
$117$	−10237.1	+	8127.69i	−0.747837	+	0.593739i
$118$		0			0
$119$		−	167.806i		−	0.0118499i
$120$		0			0
$121$	14280.5			0.975380
$122$		0			0
$123$	4658.28	−	9646.32i	0.307904	−	0.637605i
$124$		0			0
$125$	10589.8			0.677747
$126$		0			0
$127$	−24860.7			−1.54137			−0.770683	−	0.637219i	$-0.780085\pi$
$127$	−24860.7			−1.54137			−0.770683	+	0.637219i	$0.780085\pi$
$128$		0			0
$129$	−11622.2	+	24067.2i	−0.698409	+	1.44626i
$130$		0			0
$131$	13359.8			0.778497			0.389248	−	0.921133i	$-0.372735\pi$
$131$	13359.8			0.778497			0.389248	+	0.921133i	$0.372735\pi$
$132$		0			0
$133$		−	1539.93i		−	0.0870557i
$134$		0			0
$135$	4787.40	+	21293.1i	0.262683	+	1.16835i
$136$		0			0
$137$			19181.4i			1.02197i	0.859589	+	0.510987i	$0.170720\pi$
$137$			19181.4i			1.02197i	−0.859589	+	0.510987i	$0.829280\pi$
$138$		0			0
$139$		−	10860.0i		−	0.562083i	−0.959696	−	0.281042i	$-0.909320\pi$
$139$		−	10860.0i		−	0.562083i	0.959696	−	0.281042i	$-0.0906799\pi$
$140$		0			0
$141$	−15496.8	+	32090.7i	−0.779479	+	1.61414i
$142$		0			0
$143$		−	27443.8i		−	1.34206i
$144$		0			0
$145$	27870.3			1.32558
$146$		0			0
$147$	19244.5	+	9293.32i	0.890579	+	0.430067i
$148$		0			0
$149$	−11052.1			−0.497818			−0.248909	−	0.968527i	$-0.580072\pi$
$149$	−11052.1			−0.497818			−0.248909	+	0.968527i	$0.580072\pi$
$150$		0			0
$151$	26646.9			1.16867			0.584336	−	0.811512i	$-0.301355\pi$
$151$	26646.9			1.16867			0.584336	+	0.811512i	$0.301355\pi$
$152$		0			0
$153$	−2069.83	+	1643.32i	−0.0884201	+	0.0702004i
$154$		0			0
$155$	−3837.35			−0.159723
$156$		0			0
$157$			11601.5i			0.470666i	0.971915	+	0.235333i	$0.0756181\pi$
$157$			11601.5i			0.470666i	−0.971915	+	0.235333i	$0.924382\pi$
$158$		0			0
$159$	−29073.7	−	14039.9i	−1.15002	−	0.555354i
$160$		0			0
$161$		−	1715.92i		−	0.0661979i
$162$		0			0
$163$			32130.6i			1.20932i	0.796482	+	0.604662i	$0.206692\pi$
$163$			32130.6i			1.20932i	−0.796482	+	0.604662i	$0.793308\pi$
$164$		0			0
$165$	−41262.6	−	19926.0i	−1.51561	−	0.731902i
$166$		0			0
$167$			52364.2i			1.87759i	0.344474	+	0.938796i	$0.388057\pi$
$167$			52364.2i			1.87759i	−0.344474	+	0.938796i	$0.611943\pi$
$168$		0			0
$169$	2519.47			0.0882135
$170$		0			0
$171$	−18994.4	+	15080.4i	−0.649581	+	0.515729i
$172$		0			0
$173$	50794.7			1.69717			0.848587	−	0.529056i	$-0.177454\pi$
$173$	50794.7			1.69717			0.848587	+	0.529056i	$0.177454\pi$
$174$		0			0
$175$	−1395.17			−0.0455566
$176$		0			0
$177$	15254.6	+	7366.57i	0.486917	+	0.235136i
$178$		0			0
$179$	−16928.7			−0.528344			−0.264172	−	0.964476i	$-0.585099\pi$
$179$	−16928.7			−0.528344			−0.264172	+	0.964476i	$0.585099\pi$
$180$		0			0
$181$		−	20632.5i		−	0.629788i	−0.949127	−	0.314894i	$-0.898031\pi$
$181$		−	20632.5i		−	0.629788i	0.949127	−	0.314894i	$-0.101969\pi$
$182$		0			0
$183$	−6327.94	+	13103.8i	−0.188956	+	0.391288i
$184$		0			0
$185$		−	72159.5i		−	2.10839i
$186$		0			0
$187$		−	5548.80i		−	0.158678i
$188$		0			0
$189$	822.431	+	3657.96i	0.0230237	+	0.102404i
$190$		0			0
$191$		−	11052.1i		−	0.302955i	−0.988461	−	0.151478i	$-0.951597\pi$
$191$		−	11052.1i		−	0.302955i	0.988461	−	0.151478i	$-0.0484032\pi$
$192$		0			0
$193$	−54333.7			−1.45866			−0.729330	−	0.684162i	$-0.760168\pi$
$193$	−54333.7			−1.45866			−0.729330	+	0.684162i	$0.760168\pi$
$194$		0			0
$195$	−18907.9	+	39154.3i	−0.497249	+	1.02970i
$196$		0			0
$197$	44287.8			1.14117			0.570586	−	0.821238i	$-0.306716\pi$
$197$	44287.8			1.14117			0.570586	+	0.821238i	$0.306716\pi$
$198$		0			0
$199$	−38718.7			−0.977720			−0.488860	−	0.872362i	$-0.662587\pi$
$199$	−38718.7			−0.977720			−0.488860	+	0.872362i	$0.662587\pi$
$200$		0			0
$201$	−32411.0	+	67116.4i	−0.802233	+	1.66126i
$202$		0			0
$203$	4787.86			0.116185
$204$		0			0
$205$		−	35633.3i		−	0.847907i
$206$		0			0
$207$	−21165.1	+	16803.9i	−0.493947	+	0.392165i
$208$		0			0
$209$		−	50920.3i		−	1.16573i
$210$		0			0
$211$			33390.2i			0.749987i	0.927027	+	0.374993i	$0.122355\pi$
$211$			33390.2i			0.749987i	−0.927027	+	0.374993i	$0.877645\pi$
$212$		0			0
$213$	12346.3	−	25566.6i	0.272131	−	0.563526i
$214$		0			0
$215$			88903.7i			1.92328i
$216$		0			0
$217$	−659.221			−0.0139995
$218$		0			0
$219$	45142.0	+	21799.4i	0.941223	+	0.454524i
$220$		0			0
$221$	−5265.28			−0.107805
$222$		0			0
$223$	−63239.5			−1.27168			−0.635841	−	0.771820i	$-0.719347\pi$
$223$	−63239.5			−1.27168			−0.635841	+	0.771820i	$0.719347\pi$
$224$		0			0
$225$	13662.9	+	17208.9i	0.269883	+	0.339929i
$226$		0			0
$227$	67121.7			1.30260			0.651300	−	0.758820i	$-0.274224\pi$
$227$	67121.7			1.30260			0.651300	+	0.758820i	$0.274224\pi$
$228$		0			0
$229$			83285.2i			1.58817i	0.607807	+	0.794084i	$0.292049\pi$
$229$			83285.2i			1.58817i	−0.607807	+	0.794084i	$0.707951\pi$
$230$		0			0
$231$	−7088.54	−	3423.11i	−0.132841	−	0.0641500i
$232$		0			0
$233$		−	43135.7i		−	0.794558i	−0.917698	−	0.397279i	$-0.869955\pi$
$233$		−	43135.7i		−	0.794558i	0.917698	−	0.397279i	$-0.130045\pi$
$234$		0			0
$235$			118542.i			2.14653i
$236$		0			0
$237$	−79921.6	−	38594.7i	−1.42288	−	0.687118i
$238$		0			0
$239$		−	67814.2i		−	1.18720i	−0.804759	−	0.593601i	$-0.797706\pi$
$239$		−	67814.2i		−	1.18720i	0.804759	−	0.593601i	$-0.202294\pi$
$240$		0			0
$241$	−25824.5			−0.444630			−0.222315	−	0.974975i	$-0.571361\pi$
$241$	−25824.5			−0.444630			−0.222315	+	0.974975i	$0.571361\pi$
$242$		0			0
$243$	37065.5	−	45966.6i	0.627708	−	0.778449i
$244$		0			0
$245$	71088.8			1.18432
$246$		0			0
$247$	−48318.5			−0.791989
$248$		0			0
$249$	72879.7	+	35194.1i	1.17546	+	0.567638i
$250$		0			0
$251$	75750.6			1.20237			0.601186	−	0.799109i	$-0.294695\pi$
$251$	75750.6			1.20237			0.601186	+	0.799109i	$0.294695\pi$
$252$		0			0
$253$		−	56739.6i		−	0.886431i
$254$		0			0
$255$	−3822.95	+	7916.52i	−0.0587919	+	0.121746i
$256$		0			0
$257$			3320.52i			0.0502735i	0.999684	+	0.0251368i	$0.00800212\pi$
$257$			3320.52i			0.0502735i	−0.999684	+	0.0251368i	$0.991998\pi$
$258$		0			0
$259$		−	12396.3i		−	0.184797i
$260$		0			0
$261$	−46887.3	−	59056.4i	−0.688294	−	0.866933i
$262$		0			0
$263$		−	9496.36i		−	0.137292i	−0.997641	−	0.0686461i	$-0.978132\pi$
$263$		−	9496.36i		−	0.137292i	0.997641	−	0.0686461i	$-0.0218679\pi$
$264$		0			0
$265$	−107398.			−1.52934
$266$		0			0
$267$	−6246.84	+	12935.9i	−0.0876270	+	0.181457i
$268$		0			0
$269$	55521.4			0.767283			0.383642	−	0.923482i	$-0.374670\pi$
$269$	55521.4			0.767283			0.383642	+	0.923482i	$0.374670\pi$
$270$		0			0
$271$	−123146.			−1.67680			−0.838400	−	0.545055i	$-0.816509\pi$
$271$	−123146.			−1.67680			−0.838400	+	0.545055i	$0.816509\pi$
$272$		0			0
$273$	−3248.20	+	6726.35i	−0.0435831	+	0.0902514i
$274$		0			0
$275$	−46133.7			−0.610032
$276$		0			0
$277$			107683.i			1.40342i	0.712462	+	0.701711i	$0.247580\pi$
$277$			107683.i			1.40342i	−0.712462	+	0.701711i	$0.752420\pi$
$278$		0			0
$279$	6455.72	+	8131.23i	0.0829347	+	0.104459i
$280$		0			0
$281$		−	99208.5i		−	1.25642i	−0.778042	−	0.628212i	$-0.783787\pi$
$281$		−	99208.5i		−	1.25642i	0.778042	−	0.628212i	$-0.216213\pi$
$282$		0			0
$283$			18221.6i			0.227517i	0.993508	+	0.113759i	$0.0362890\pi$
$283$			18221.6i			0.227517i	−0.993508	+	0.113759i	$0.963711\pi$
$284$		0			0
$285$	−35082.5	+	72648.4i	−0.431917	+	0.894409i
$286$		0			0
$287$		−	6121.48i		−	0.0743177i
$288$		0			0
$289$	82456.4			0.987254
$290$		0			0
$291$	−20648.2	−	9971.15i	−0.243835	−	0.117750i
$292$		0			0
$293$	89818.1			1.04623			0.523117	−	0.852261i	$-0.324769\pi$
$293$	89818.1			1.04623			0.523117	+	0.852261i	$0.324769\pi$
$294$		0			0
$295$	56350.2			0.647518
$296$		0			0
$297$	27195.0	+	120957.i	0.308302	+	1.37125i
$298$		0			0
$299$	−53840.5			−0.602236
$300$		0			0
$301$			15272.8i			0.168572i
$302$		0			0
$303$	−26585.3	−	12838.3i	−0.289572	−	0.139837i
$304$		0			0
$305$			48405.3i			0.520348i
$306$		0			0
$307$			23652.4i			0.250957i	0.992096	+	0.125478i	$0.0400465\pi$
$307$			23652.4i			0.250957i	−0.992096	+	0.125478i	$0.959953\pi$
$308$		0			0
$309$	−82951.7	−	40058.0i	−0.868777	−	0.419539i
$310$		0			0
$311$		−	4995.98i		−	0.0516535i	−0.999666	−	0.0258268i	$-0.991778\pi$
$311$		−	4995.98i		−	0.0516535i	0.999666	−	0.0258268i	$-0.00822183\pi$
$312$		0			0
$313$	123447.			1.26007			0.630033	−	0.776569i	$-0.283041\pi$
$313$	123447.			1.26007			0.630033	+	0.776569i	$0.283041\pi$
$314$		0			0
$315$	7754.87	+	9767.56i	0.0781544	+	0.0984385i
$316$		0			0
$317$	−84116.1			−0.837067			−0.418534	−	0.908201i	$-0.637456\pi$
$317$	−84116.1			−0.837067			−0.418534	+	0.908201i	$0.637456\pi$
$318$		0			0
$319$	158319.			1.55579
$320$		0			0
$321$	10572.6	+	5105.57i	0.102605	+	0.0495489i
$322$		0			0
$323$	−9769.41			−0.0936404
$324$		0			0
$325$			43776.5i			0.414452i
$326$		0			0
$327$	−18915.0	+	39169.0i	−0.176893	+	0.366309i
$328$		0			0
$329$			20364.5i			0.188140i
$330$		0			0
$331$		−	67717.3i		−	0.618079i	−0.951049	−	0.309039i	$-0.899992\pi$
$331$		−	67717.3i		−	0.618079i	0.951049	−	0.309039i	$-0.100008\pi$
$332$		0			0
$333$	−152904.	+	121397.i	−1.37889	+	1.09476i
$334$		0			0
$335$			247927.i			2.20919i
$336$		0			0
$337$	11076.2			0.0975282			0.0487641	−	0.998810i	$-0.484472\pi$
$337$	11076.2			0.0975282			0.0487641	+	0.998810i	$0.484472\pi$
$338$		0			0
$339$	−68444.7	+	141735.i	−0.595580	+	1.23332i
$340$		0			0
$341$	−21798.2			−0.187462
$342$		0			0
$343$	24560.9			0.208764
$344$		0			0
$345$	−39091.8	+	80950.9i	−0.328433	+	0.680117i
$346$		0			0
$347$	33822.4			0.280896			0.140448	−	0.990088i	$-0.455146\pi$
$347$	33822.4			0.280896			0.140448	+	0.990088i	$0.455146\pi$
$348$		0			0
$349$		−	157980.i		−	1.29703i	−0.761202	−	0.648515i	$-0.775390\pi$
$349$		−	157980.i		−	1.29703i	0.761202	−	0.648515i	$-0.224610\pi$
$350$		0			0
$351$	114776.	−	25805.5i	0.931619	−	0.209459i
$352$		0			0
$353$		−	40542.5i		−	0.325358i	−0.986679	−	0.162679i	$-0.947987\pi$
$353$		−	40542.5i		−	0.325358i	0.986679	−	0.162679i	$-0.0520134\pi$
$354$		0			0
$355$		−	94442.6i		−	0.749395i
$356$		0			0
$357$	−656.747	+	1359.99i	−0.00515302	+	0.0106708i
$358$		0			0
$359$		−	174457.i		−	1.35363i	−0.736155	−	0.676813i	$-0.763361\pi$
$359$		−	174457.i		−	1.35363i	0.736155	−	0.676813i	$-0.236639\pi$
$360$		0			0
$361$	40669.0			0.312068
$362$		0			0
$363$	−115736.	−	55890.0i	−0.878328	−	0.424151i
$364$		0			0
$365$	166754.			1.25167
$366$		0			0
$367$	140984.			1.04674			0.523370	−	0.852106i	$-0.324675\pi$
$367$	140984.			1.04674			0.523370	+	0.852106i	$0.324675\pi$
$368$		0			0
$369$	−75506.0	+	59947.3i	−0.554535	+	0.440268i
$370$		0			0
$371$	−18449.9			−0.134044
$372$		0			0
$373$			129523.i			0.930953i	0.885060	+	0.465476i	$0.154117\pi$
$373$			129523.i			0.930953i	−0.885060	+	0.465476i	$0.845883\pi$
$374$		0			0
$375$	−85825.0	−	41445.5i	−0.610311	−	0.294724i
$376$		0			0
$377$		−	150229.i		−	1.05699i
$378$		0			0
$379$		−	282353.i		−	1.96569i	−0.184442	−	0.982843i	$-0.559048\pi$
$379$		−	282353.i		−	1.96569i	0.184442	−	0.982843i	$-0.440952\pi$
$380$		0			0
$381$	201483.	+	97297.7i	1.38800	+	0.670275i
$382$		0			0
$383$		−	43563.6i		−	0.296979i	−0.988914	−	0.148490i	$-0.952559\pi$
$383$		−	43563.6i		−	0.296979i	0.988914	−	0.148490i	$-0.0474412\pi$
$384$		0			0
$385$	−26184.9			−0.176657
$386$		0			0
$387$	188384.	−	149566.i	1.25783	−	0.998646i
$388$		0			0
$389$	21525.5			0.142250			0.0711252	−	0.997467i	$-0.477341\pi$
$389$	21525.5			0.142250			0.0711252	+	0.997467i	$0.477341\pi$
$390$		0			0
$391$	−10885.9			−0.0712049
$392$		0			0
$393$	−108274.	−	52286.5i	−0.701036	−	0.338535i
$394$		0			0
$395$	−295229.			−1.89219
$396$		0			0
$397$		−	248536.i		−	1.57692i	−0.615088	−	0.788458i	$-0.710880\pi$
$397$		−	248536.i		−	1.57692i	0.615088	−	0.788458i	$-0.289120\pi$
$398$		0			0
$399$	−6026.84	+	12480.3i	−0.0378568	+	0.0783935i
$400$		0			0
$401$		−	169418.i		−	1.05359i	−0.849993	−	0.526794i	$-0.823394\pi$
$401$		−	169418.i		−	1.05359i	0.849993	−	0.526794i	$-0.176606\pi$
$402$		0			0
$403$			20684.4i			0.127360i
$404$		0			0
$405$	44535.9	−	191307.i	0.271519	−	1.16633i
$406$		0			0
$407$		−	409906.i		−	2.47454i
$408$		0			0
$409$	−124151.			−0.742169			−0.371085	−	0.928599i	$-0.621014\pi$
$409$	−124151.			−0.742169			−0.371085	+	0.928599i	$0.621014\pi$
$410$		0			0
$411$	75070.7	−	155456.i	0.444413	−	0.920286i
$412$		0			0
$413$	9680.46			0.0567539
$414$		0			0
$415$	269216.			1.56316
$416$		0			0
$417$	−42503.0	+	88014.8i	−0.244426	+	0.506155i
$418$		0			0
$419$	37995.6			0.216424			0.108212	−	0.994128i	$-0.465488\pi$
$419$	37995.6			0.216424			0.108212	+	0.994128i	$0.465488\pi$
$420$		0			0
$421$		−	240982.i		−	1.35963i	−0.733383	−	0.679815i	$-0.762060\pi$
$421$		−	240982.i		−	1.35963i	0.733383	−	0.679815i	$-0.237940\pi$
$422$		0			0
$423$	251188.	−	199428.i	1.40384	−	1.11457i
$424$		0			0
$425$			8851.07i			0.0490024i
$426$		0			0
$427$			8315.59i			0.0456076i
$428$		0			0
$429$	−107407.	+	222418.i	−0.583605	+	1.20852i
$430$		0			0
$431$			27547.2i			0.148294i	0.997247	+	0.0741470i	$0.0236234\pi$
$431$			27547.2i			0.148294i	−0.997247	+	0.0741470i	$0.976377\pi$
$432$		0			0
$433$	113306.			0.604333			0.302167	−	0.953255i	$-0.402290\pi$
$433$	113306.			0.604333			0.302167	+	0.953255i	$0.402290\pi$
$434$		0			0
$435$	−225875.	−	109077.i	−1.19368	−	0.576438i
$436$		0			0
$437$	−99897.7			−0.523109
$438$		0			0
$439$	−90326.6			−0.468691			−0.234345	−	0.972153i	$-0.575295\pi$
$439$	−90326.6			−0.468691			−0.234345	+	0.972153i	$0.575295\pi$
$440$		0			0
$441$	−119596.	−	150635.i	−0.614947	−	0.774550i
$442$		0			0
$443$	293915.			1.49766			0.748832	−	0.662760i	$-0.230615\pi$
$443$	293915.			1.49766			0.748832	+	0.662760i	$0.230615\pi$
$444$		0			0
$445$			47784.9i			0.241308i
$446$		0			0
$447$	89571.3	+	43254.7i	0.448285	+	0.216480i
$448$		0			0
$449$		−	56159.4i		−	0.278567i	−0.990253	−	0.139284i	$-0.955520\pi$
$449$		−	56159.4i		−	0.278567i	0.990253	−	0.139284i	$-0.0444800\pi$
$450$		0			0
$451$		−	202417.i		−	0.995161i
$452$		0			0
$453$	−215960.	−	104289.i	−1.05239	−	0.508206i
$454$		0			0
$455$			24847.0i			0.120019i
$456$		0			0
$457$	218485.			1.04614			0.523069	−	0.852290i	$-0.324787\pi$
$457$	218485.			1.04614			0.523069	+	0.852290i	$0.324787\pi$
$458$		0			0
$459$	23206.4	−	5217.56i	0.110149	−	0.0247652i
$460$		0			0
$461$	300747.			1.41514			0.707571	−	0.706643i	$-0.249791\pi$
$461$	300747.			1.41514			0.707571	+	0.706643i	$0.249791\pi$
$462$		0			0
$463$	−121032.			−0.564597			−0.282298	−	0.959327i	$-0.591097\pi$
$463$	−121032.			−0.564597			−0.282298	+	0.959327i	$0.591097\pi$
$464$		0			0
$465$	31099.7	+	15018.3i	0.143830	+	0.0694568i
$466$		0			0
$467$	−63141.2			−0.289520			−0.144760	−	0.989467i	$-0.546241\pi$
$467$	−63141.2			−0.289520			−0.144760	+	0.989467i	$0.546241\pi$
$468$		0			0
$469$			42591.5i			0.193632i
$470$		0			0
$471$	45404.8	−	94023.9i	0.204673	−	0.423835i
$472$		0			0
$473$			505022.i			2.25729i
$474$		0			0
$475$			81224.5i			0.359998i
$476$		0			0
$477$	180679.	+	227573.i	0.794094	+	1.00019i
$478$		0			0
$479$			243193.i			1.05994i	0.848017	+	0.529969i	$0.177796\pi$
$479$			243193.i			1.05994i	−0.848017	+	0.529969i	$0.822204\pi$
$480$		0			0
$481$	−388961.			−1.68119
$482$		0			0
$483$	−6715.61	+	13906.6i	−0.0287867	+	0.0596111i
$484$		0			0
$485$	−76273.9			−0.324259
$486$		0			0
$487$	165816.			0.699149			0.349574	−	0.936909i	$-0.386326\pi$
$487$	165816.			0.699149			0.349574	+	0.936909i	$0.386326\pi$
$488$		0			0
$489$	125750.	−	260402.i	0.525884	−	1.08900i
$490$		0			0
$491$	−291207.			−1.20792			−0.603961	−	0.797014i	$-0.706412\pi$
$491$	−291207.			−1.20792			−0.603961	+	0.797014i	$0.706412\pi$
$492$		0			0
$493$		−	30374.5i		−	0.124973i
$494$		0			0
$495$	256428.	+	322981.i	1.04654	+	1.31815i
$496$		0			0
$497$		−	16224.4i		−	0.0656833i
$498$		0			0
$499$		−	92369.0i		−	0.370959i	−0.982648	−	0.185479i	$-0.940616\pi$
$499$		−	92369.0i		−	0.370959i	0.982648	−	0.185479i	$-0.0593838\pi$
$500$		0			0
$501$	204939.	−	424385.i	0.816485	−	1.69077i
$502$		0			0
$503$			125137.i			0.494595i	0.968940	+	0.247297i	$0.0795425\pi$
$503$			125137.i			0.494595i	−0.968940	+	0.247297i	$0.920458\pi$
$504$		0			0
$505$	−98205.6			−0.385082
$506$		0			0
$507$	−20419.0	−	9860.48i	−0.0794362	−	0.0383603i
$508$		0			0
$509$	−245886.			−0.949071			−0.474536	−	0.880236i	$-0.657384\pi$
$509$	−245886.			−0.949071			−0.474536	+	0.880236i	$0.657384\pi$
$510$		0			0
$511$	28646.8			0.109707
$512$		0			0
$513$	212961.	−	47880.5i	0.809216	−	0.181938i
$514$		0			0
$515$	−306422.			−1.15533
$516$		0			0
$517$			673385.i			2.51931i
$518$		0			0
$519$	−411665.	−	198796.i	−1.52830	−	0.738029i
$520$		0			0
$521$		−	145433.i		−	0.535781i	−0.963449	−	0.267890i	$-0.913673\pi$
$521$		−	145433.i		−	0.535781i	0.963449	−	0.267890i	$-0.0863265\pi$
$522$		0			0
$523$			203742.i			0.744865i	0.928059	+	0.372433i	$0.121476\pi$
$523$			203742.i			0.744865i	−0.928059	+	0.372433i	$0.878524\pi$
$524$		0			0
$525$	11307.2	+	5460.31i	0.0410237	+	0.0198107i
$526$		0			0
$527$			4182.14i			0.0150584i
$528$		0			0
$529$	168527.			0.602223
$530$		0			0
$531$	−94800.2	−	119405.i	−0.336218	−	0.423479i
$532$		0			0
$533$	−192074.			−0.676106
$534$		0			0
$535$	39054.8			0.136448
$536$		0			0
$537$	137198.	+	66254.1i	0.475773	+	0.229755i
$538$		0			0
$539$	403824.			1.39000
$540$		0			0
$541$		−	316003.i		−	1.07968i	−0.841767	−	0.539842i	$-0.818484\pi$
$541$		−	316003.i		−	1.07968i	0.841767	−	0.539842i	$-0.181516\pi$
$542$		0			0
$543$	−80749.8	+	167216.i	−0.273868	+	0.567124i
$544$		0			0
$545$			144689.i			0.487129i
$546$		0			0
$547$			76410.8i			0.255376i	0.991814	+	0.127688i	$0.0407556\pi$
$547$			76410.8i			0.255376i	−0.991814	+	0.127688i	$0.959244\pi$
$548$		0			0
$549$	102570.	−	81434.2i	0.340309	−	0.270186i
$550$		0			0
$551$		−	278741.i		−	0.918117i
$552$		0			0
$553$	−50717.6			−0.165847
$554$		0			0
$555$	−282412.	+	584816.i	−0.916848	+	1.89860i
$556$		0			0
$557$	−484406.			−1.56135			−0.780673	−	0.624939i	$-0.785124\pi$
$557$	−484406.			−1.56135			−0.780673	+	0.624939i	$0.785124\pi$
$558$		0			0
$559$	479217.			1.53359
$560$		0			0
$561$	−21716.4	+	44970.2i	−0.0690022	+	0.142889i
$562$		0			0
$563$	91818.6			0.289677			0.144838	−	0.989455i	$-0.453734\pi$
$563$	91818.6			0.289677			0.144838	+	0.989455i	$0.453734\pi$
$564$		0			0
$565$			523565.i			1.64011i
$566$		0			0
$567$	7650.86	−	32864.7i	0.0237982	−	0.102227i
$568$		0			0
$569$			430154.i			1.32862i	0.747459	+	0.664308i	$0.231274\pi$
$569$			430154.i			1.32862i	−0.747459	+	0.664308i	$0.768726\pi$
$570$		0			0
$571$		−	120296.i		−	0.368960i	−0.982836	−	0.184480i	$-0.940940\pi$
$571$		−	120296.i		−	0.368960i	0.982836	−	0.184480i	$-0.0590601\pi$
$572$		0			0
$573$	−43254.9	+	89571.8i	−0.131743	+	0.272811i
$574$		0			0
$575$			90507.1i			0.273746i
$576$		0			0
$577$	493660.			1.48278			0.741389	−	0.671076i	$-0.234167\pi$
$577$	493660.			1.48278			0.741389	+	0.671076i	$0.234167\pi$
$578$		0			0
$579$	440347.	+	212647.i	1.31352	+	0.634310i
$580$		0			0
$581$	46248.8			0.137009
$582$		0			0
$583$	−610078.			−1.79493
$584$		0			0
$585$	306478.	−	243325.i	0.895545	−	0.711010i
$586$		0			0
$587$	−365807.			−1.06164			−0.530819	−	0.847485i	$-0.678116\pi$
$587$	−365807.			−1.06164			−0.530819	+	0.847485i	$0.678116\pi$
$588$		0			0
$589$			38378.7i			0.110627i
$590$		0			0
$591$	−358930.	−	173330.i	−1.02763	−	0.496248i
$592$		0			0
$593$			490452.i			1.39472i	0.716721	+	0.697360i	$0.245642\pi$
$593$			490452.i			1.39472i	−0.716721	+	0.697360i	$0.754358\pi$
$594$		0			0
$595$			5023.76i			0.0141904i
$596$		0			0
$597$	313795.	+	151534.i	0.880436	+	0.425169i
$598$		0			0
$599$			37189.5i			0.103649i	0.998656	+	0.0518247i	$0.0165037\pi$
$599$			37189.5i			0.103649i	−0.998656	+	0.0518247i	$0.983496\pi$
$600$		0			0
$601$	−461210.			−1.27688			−0.638440	−	0.769672i	$-0.720420\pi$
$601$	−461210.			−1.27688			−0.638440	+	0.769672i	$0.720420\pi$
$602$		0			0
$603$	525350.	−	417097.i	1.44482	−	1.14710i
$604$		0			0
$605$	−427528.			−1.16803
$606$		0			0
$607$	−594036.			−1.61226			−0.806130	−	0.591738i	$-0.798442\pi$
$607$	−594036.			−1.61226			−0.806130	+	0.591738i	$0.798442\pi$
$608$		0			0
$609$	−38803.2	−	18738.3i	−0.104624	−	0.0505239i
$610$		0			0
$611$	638978.			1.71161
$612$		0			0
$613$			134516.i			0.357975i	0.983851	+	0.178987i	$0.0572821\pi$
$613$			134516.i			0.357975i	−0.983851	+	0.178987i	$0.942718\pi$
$614$		0			0
$615$	−139459.	+	288790.i	−0.368719	+	0.763540i
$616$		0			0
$617$			97225.4i			0.255393i	0.991813	+	0.127697i	$0.0407584\pi$
$617$			97225.4i			0.255393i	−0.991813	+	0.127697i	$0.959242\pi$
$618$		0			0
$619$			241624.i			0.630608i	0.948991	+	0.315304i	$0.102106\pi$
$619$			241624.i			0.630608i	−0.948991	+	0.315304i	$0.897894\pi$
$620$		0			0
$621$	237298.	−	53352.5i	0.615335	−	0.138348i
$622$		0			0
$623$			8209.02i			0.0211502i
$624$		0			0
$625$	−486582.			−1.24565
$626$		0			0
$627$	−199288.	+	412683.i	−0.506927	+	1.04974i
$628$		0			0
$629$	−78643.2			−0.198774
$630$		0			0
$631$	−199271.			−0.500478			−0.250239	−	0.968184i	$-0.580509\pi$
$631$	−199271.			−0.500478			−0.250239	+	0.968184i	$0.580509\pi$
$632$		0			0
$633$	130680.	−	270610.i	0.326138	−	0.675363i
$634$		0			0
$635$	744275.			1.84580
$636$		0			0
$637$		−	383190.i		−	0.944356i
$638$		0			0
$639$	−200121.	+	158884.i	−0.490107	+	0.389117i
$640$		0			0
$641$		−	648365.i		−	1.57799i	−0.614401	−	0.788994i	$-0.710602\pi$
$641$		−	648365.i		−	1.57799i	0.614401	−	0.788994i	$-0.289398\pi$
$642$		0			0
$643$			701137.i			1.69582i	0.530137	+	0.847912i	$0.322140\pi$
$643$			701137.i			1.69582i	−0.530137	+	0.847912i	$0.677860\pi$
$644$		0			0
$645$	347944.	−	720519.i	0.836354	−	1.73191i
$646$		0			0
$647$			176172.i			0.420852i	0.977610	+	0.210426i	$0.0674851\pi$
$647$			176172.i			0.420852i	−0.977610	+	0.210426i	$0.932515\pi$
$648$		0			0
$649$	320100.			0.759970
$650$		0			0
$651$	5342.65	+	2580.01i	0.0126065	+	0.00608778i
$652$		0			0
$653$	−82082.5			−0.192497			−0.0962486	−	0.995357i	$-0.530684\pi$
$653$	−82082.5			−0.192497			−0.0962486	+	0.995357i	$0.530684\pi$
$654$		0			0
$655$	−399963.			−0.932260
$656$		0			0
$657$	−280536.	−	353346.i	−0.649918	−	0.818596i
$658$		0			0
$659$	90850.5			0.209198			0.104599	−	0.994515i	$-0.466644\pi$
$659$	90850.5			0.209198			0.104599	+	0.994515i	$0.466644\pi$
$660$		0			0
$661$		−	93338.4i		−	0.213628i	−0.994279	−	0.106814i	$-0.965935\pi$
$661$		−	93338.4i		−	0.213628i	0.994279	−	0.106814i	$-0.0340649\pi$
$662$		0			0
$663$	42672.4	+	20606.8i	0.0970779	+	0.0468796i
$664$		0			0
$665$			46102.1i			0.104250i
$666$		0			0
$667$		−	310597.i		−	0.698144i
$668$		0			0
$669$	512524.	+	247502.i	1.14515	+	0.553001i
$670$		0			0
$671$			274969.i			0.610715i
$672$		0			0
$673$	−193401.			−0.427002			−0.213501	−	0.976943i	$-0.568487\pi$
$673$	−193401.			−0.427002			−0.213501	+	0.976943i	$0.568487\pi$
$674$		0			0
$675$	−43379.7	−	192942.i	−0.0952092	−	0.423466i
$676$		0			0
$677$	118624.			0.258818			0.129409	−	0.991591i	$-0.458692\pi$
$677$	118624.			0.258818			0.129409	+	0.991591i	$0.458692\pi$
$678$		0			0
$679$	−13103.2			−0.0284208
$680$		0			0
$681$	−543987.	−	262695.i	−1.17299	−	0.566446i
$682$		0			0
$683$	−578837.			−1.24084			−0.620419	−	0.784270i	$-0.713038\pi$
$683$	−578837.			−1.24084			−0.620419	+	0.784270i	$0.713038\pi$
$684$		0			0
$685$		−	574250.i		−	1.22383i
$686$		0			0
$687$	325955.	−	674984.i	0.690628	−	1.43014i
$688$		0			0
$689$			578906.i			1.21947i
$690$		0			0
$691$			825321.i			1.72849i	0.503071	+	0.864245i	$0.332203\pi$
$691$			825321.i			1.72849i	−0.503071	+	0.864245i	$0.667797\pi$
$692$		0			0
$693$	44051.9	+	55485.1i	0.0917273	+	0.115534i
$694$		0			0
$695$			325125.i			0.673102i
$696$		0			0
$697$	−38835.1			−0.0799390
$698$		0			0
$699$	−168821.	+	349593.i	−0.345520	+	0.715498i
$700$		0			0
$701$	105553.			0.214801			0.107401	−	0.994216i	$-0.465747\pi$
$701$	105553.			0.214801			0.107401	+	0.994216i	$0.465747\pi$
$702$		0			0
$703$	−721694.			−1.46030
$704$		0			0
$705$	463941.	−	960725.i	0.933436	−	1.93295i
$706$		0			0
$707$	−16870.8			−0.0337519
$708$		0			0
$709$			40012.7i			0.0795985i	0.999208	+	0.0397993i	$0.0126718\pi$
$709$			40012.7i			0.0795985i	−0.999208	+	0.0397993i	$0.987328\pi$
$710$		0			0
$711$	496675.	+	625582.i	0.982501	+	1.23750i
$712$		0			0
$713$			42764.8i			0.0841215i
$714$		0			0
$715$			821607.i			1.60713i
$716$		0			0
$717$	−265406.	+	549600.i	−0.516264	+	1.06907i
$718$		0			0
$719$		−	69687.8i		−	0.134803i	−0.997726	−	0.0674015i	$-0.978529\pi$
$719$		−	69687.8i		−	0.134803i	0.997726	−	0.0674015i	$-0.0214708\pi$
$720$		0			0
$721$	−52640.5			−0.101263
$722$		0			0
$723$	209295.	+	101070.i	0.400389	+	0.193351i
$724$		0			0
$725$	−252539.			−0.480455
$726$		0			0
$727$	278432.			0.526805			0.263402	−	0.964686i	$-0.415155\pi$
$727$	278432.			0.526805			0.263402	+	0.964686i	$0.415155\pi$
$728$		0			0
$729$	−480298.	+	227472.i	−0.903765	+	0.428029i
$730$		0			0
$731$	96891.9			0.181323
$732$		0			0
$733$		−	211684.i		−	0.393986i	−0.980405	−	0.196993i	$-0.936882\pi$
$733$		−	211684.i		−	0.393986i	0.980405	−	0.196993i	$-0.0631176\pi$
$734$		0			0
$735$	−576139.	−	278222.i	−1.06648	−	0.515011i
$736$		0			0
$737$			1.40836e6i			2.59286i
$738$		0			0
$739$		−	319029.i		−	0.584172i	−0.956392	−	0.292086i	$-0.905651\pi$
$739$		−	319029.i		−	0.584172i	0.956392	−	0.292086i	$-0.0943494\pi$
$740$		0			0
$741$	391597.	+	189105.i	0.713186	+	0.344403i
$742$		0			0
$743$			514092.i			0.931244i	0.884984	+	0.465622i	$0.154169\pi$
$743$			514092.i			0.931244i	−0.884984	+	0.465622i	$0.845831\pi$
$744$		0			0
$745$	330875.			0.596144
$746$		0			0
$747$	−452913.	−	570461.i	−0.811659	−	1.02232i
$748$		0			0
$749$	6709.27			0.0119595
$750$		0			0
$751$	283278.			0.502265			0.251133	−	0.967953i	$-0.419197\pi$
$751$	283278.			0.502265			0.251133	+	0.967953i	$0.419197\pi$
$752$		0			0
$753$	−613920.	−	296467.i	−1.08273	−	0.522861i
$754$		0			0
$755$	−797750.			−1.39950
$756$		0			0
$757$		−	116594.i		−	0.203463i	−0.994812	−	0.101732i	$-0.967562\pi$
$757$		−	116594.i		−	0.203463i	0.994812	−	0.101732i	$-0.0324383\pi$
$758$		0			0
$759$	−222063.	+	459845.i	−0.385471	+	0.798231i
$760$		0			0
$761$		−	191031.i		−	0.329863i	−0.986305	−	0.164932i	$-0.947260\pi$
$761$		−	191031.i		−	0.329863i	0.986305	−	0.164932i	$-0.0527403\pi$
$762$		0			0
$763$			24856.3i			0.0426961i
$764$		0			0
$765$	61966.1	−	49197.4i	0.105884	−	0.0840658i
$766$		0			0
$767$		−	303745.i		−	0.516319i
$768$		0			0
$769$	389168.			0.658088			0.329044	−	0.944315i	$-0.393273\pi$
$769$	389168.			0.658088			0.329044	+	0.944315i	$0.393273\pi$
$770$		0			0
$771$	12995.6	−	26911.1i	0.0218618	−	0.0452713i
$772$		0			0
$773$	−568776.			−0.951880			−0.475940	−	0.879478i	$-0.657892\pi$
$773$	−568776.			−0.951880			−0.475940	+	0.879478i	$0.657892\pi$
$774$		0			0
$775$	34771.1			0.0578915
$776$		0			0
$777$	−48515.8	+	100466.i	−0.0803602	+	0.166409i
$778$		0			0
$779$	−356382.			−0.587274
$780$		0			0
$781$		−	536486.i		−	0.879541i
$782$		0			0
$783$	148868.	+	662126.i	0.242816	+	1.07998i
$784$		0			0
$785$		−	347322.i		−	0.563629i
$786$		0			0
$787$		−	467885.i		−	0.755422i	−0.925923	−	0.377711i	$-0.876711\pi$
$787$		−	467885.i		−	0.755422i	0.925923	−	0.377711i	$-0.123289\pi$
$788$		0			0
$789$	−37166.1	+	76963.2i	−0.0597025	+	0.123631i
$790$		0			0
$791$			89943.6i			0.143753i
$792$		0			0
$793$	260919.			0.414916
$794$		0			0
$795$	870404.	+	420325.i	1.37717	+	0.665044i
$796$		0			0
$797$	−592697.			−0.933074			−0.466537	−	0.884502i	$-0.654498\pi$
$797$	−592697.			−0.933074			−0.466537	+	0.884502i	$0.654498\pi$
$798$		0			0
$799$	129194.			0.202371
$800$		0			0
$801$	101255.	−	80390.5i	0.157816	−	0.125297i
$802$		0			0
$803$	947252.			1.46904
$804$		0			0
$805$			51370.8i			0.0792728i
$806$		0			0
$807$	−449973.	−	217295.i	−0.690938	−	0.333659i
$808$		0			0
$809$		−	491874.i		−	0.751548i	−0.926711	−	0.375774i	$-0.877377\pi$
$809$		−	491874.i		−	0.751548i	0.926711	−	0.375774i	$-0.122623\pi$
$810$		0			0
$811$			140444.i			0.213531i	0.994284	+	0.106766i	$0.0340494\pi$
$811$			140444.i			0.213531i	−0.994284	+	0.106766i	$0.965951\pi$
$812$		0			0
$813$	998035.	+	481958.i	1.50996	+	0.729170i
$814$		0			0
$815$		−	961919.i		−	1.44818i
$816$		0			0
$817$	889159.			1.33209
$818$		0			0
$819$	52650.1	−	41801.1i	0.0784931	−	0.0623189i
$820$		0			0
$821$	36236.4			0.0537600			0.0268800	−	0.999639i	$-0.491443\pi$
$821$	36236.4			0.0537600			0.0268800	+	0.999639i	$0.491443\pi$
$822$		0			0
$823$	306097.			0.451918			0.225959	−	0.974137i	$-0.427448\pi$
$823$	306097.			0.451918			0.225959	+	0.974137i	$0.427448\pi$
$824$		0			0
$825$	373890.	+	180554.i	0.549333	+	0.265277i
$826$		0			0
$827$	1.01391e6			1.48247			0.741236	−	0.671244i	$-0.234240\pi$
$827$	1.01391e6			1.48247			0.741236	+	0.671244i	$0.234240\pi$
$828$		0			0
$829$			921062.i			1.34023i	0.742257	+	0.670116i	$0.233756\pi$
$829$			921062.i			1.34023i	−0.742257	+	0.670116i	$0.766244\pi$
$830$		0			0
$831$	421442.	−	872717.i	0.610289	−	1.26378i
$832$		0			0
$833$		−	77476.4i		−	0.111655i
$834$		0			0
$835$		−	1.56767e6i		−	2.24844i
$836$		0			0
$837$	−20497.0	−	91165.4i	−0.0292576	−	0.130130i
$838$		0			0
$839$			801936.i			1.13924i	0.821908	+	0.569620i	$0.192910\pi$
$839$			801936.i			1.13924i	−0.821908	+	0.569620i	$0.807090\pi$
$840$		0			0
$841$	159366.			0.225323
$842$		0			0
$843$	−388274.	+	804034.i	−0.546366	+	1.13141i
$844$		0			0
$845$	−75427.3			−0.105637
$846$		0			0
$847$	−73445.4			−0.102376
$848$		0			0
$849$	71314.3	−	147677.i	0.0989376	−	0.204879i
$850$		0			0
$851$	−804171.			−1.11043
$852$		0			0
$853$		−	270671.i		−	0.372000i	−0.982550	−	0.186000i	$-0.940448\pi$
$853$		−	270671.i		−	0.372000i	0.982550	−	0.186000i	$-0.0595525\pi$
$854$		0			0
$855$	568651.	−	451476.i	0.777882	−	0.617592i
$856$		0			0
$857$			872199.i			1.18755i	0.804629	+	0.593777i	$0.202364\pi$
$857$			872199.i			1.18755i	−0.804629	+	0.593777i	$0.797636\pi$
$858$		0			0
$859$		−	989515.i		−	1.34102i	−0.741900	−	0.670511i	$-0.766075\pi$
$859$		−	989515.i		−	1.34102i	0.741900	−	0.670511i	$-0.233925\pi$
$860$		0			0
$861$	−23957.7	+	49611.5i	−0.0323176	+	0.0669230i
$862$		0			0
$863$		−	1.28954e6i		−	1.73147i	−0.500507	−	0.865733i	$-0.666853\pi$
$863$		−	1.28954e6i		−	1.73147i	0.500507	−	0.865733i	$-0.333147\pi$
$864$		0			0
$865$	−1.52068e6			−2.03239
$866$		0			0
$867$	−668267.	−	322711.i	−0.889021	−	0.429315i
$868$		0			0
$869$	−1.67706e6			−2.22080
$870$		0			0
$871$	1.33640e6			1.76157
$872$		0			0
$873$	128319.	+	161622.i	0.168369	+	0.212067i
$874$		0			0
$875$	−54463.8			−0.0711364
$876$		0			0
$877$			762506.i			0.991389i	0.868497	+	0.495695i	$0.165086\pi$
$877$			762506.i			0.991389i	−0.868497	+	0.495695i	$0.834914\pi$
$878$		0			0
$879$	−727930.	−	351523.i	−0.942132	−	0.454963i
$880$		0			0
$881$			1.07039e6i			1.37908i	0.724247	+	0.689540i	$0.242187\pi$
$881$			1.07039e6i			1.37908i	−0.724247	+	0.689540i	$0.757813\pi$
$882$		0			0
$883$			379958.i			0.487320i	0.969861	+	0.243660i	$0.0783480\pi$
$883$			379958.i			0.487320i	−0.969861	+	0.243660i	$0.921652\pi$
$884$		0			0
$885$	−456690.	−	220539.i	−0.583089	−	0.281578i
$886$		0			0
$887$		−	33846.7i		−	0.0430198i	−0.999769	−	0.0215099i	$-0.993153\pi$
$887$		−	33846.7i		−	0.0430198i	0.999769	−	0.0215099i	$-0.00684734\pi$
$888$		0			0
$889$	127860.			0.161782
$890$		0			0
$891$	252988.	−	1.08673e6i	0.318673	−	1.36888i
$892$		0			0
$893$	1.18559e6			1.48672
$894$		0			0
$895$	506808.			0.632699
$896$		0			0
$897$	436350.	+	210716.i	0.542313	+	0.261887i
$898$		0			0
$899$	−119325.			−0.147643
$900$		0			0
$901$			117048.i			0.144183i
$902$		0			0
$903$	59773.6	−	123779.i	0.0733051	−	0.151799i
$904$		0			0
$905$			617692.i			0.754180i
$906$		0			0
$907$		−	650249.i		−	0.790433i	−0.918588	−	0.395217i	$-0.870670\pi$
$907$		−	650249.i		−	0.790433i	0.918588	−	0.395217i	$-0.129330\pi$
$908$		0			0
$909$	165215.	+	208095.i	0.199950	+	0.251845i
$910$		0			0
$911$		−	1.03580e6i		−	1.24807i	−0.781395	−	0.624037i	$-0.785492\pi$
$911$		−	1.03580e6i		−	1.24807i	0.781395	−	0.624037i	$-0.214508\pi$
$912$		0			0
$913$	1.52929e6			1.83463
$914$		0			0
$915$	189445.	−	392301.i	0.226277	−	0.468573i
$916$		0			0
$917$	−68710.0			−0.0817111
$918$		0			0
$919$	386764.			0.457947			0.228974	−	0.973433i	$-0.426463\pi$
$919$	386764.			0.457947			0.228974	+	0.973433i	$0.426463\pi$
$920$		0			0
$921$	92568.9	−	191691.i	0.109130	−	0.225986i
$922$		0			0
$923$	−509074.			−0.597554
$924$		0			0
$925$			653854.i			0.764182i
$926$		0			0
$927$	515506.	+	649299.i	0.599893	+	0.755589i
$928$		0			0
$929$		−	258141.i		−	0.299106i	−0.988754	−	0.149553i	$-0.952216\pi$
$929$		−	258141.i		−	0.299106i	0.988754	−	0.149553i	$-0.0477835\pi$
$930$		0			0
$931$		−	710986.i		−	0.820279i
$932$		0			0
$933$	−19552.9	+	40489.9i	−0.0224619	+	0.0465140i
$934$		0			0
$935$			166119.i			0.190019i
$936$		0			0
$937$	−294387.			−0.335305			−0.167652	−	0.985846i	$-0.553619\pi$
$937$	−294387.			−0.335305			−0.167652	+	0.985846i	$0.553619\pi$
$938$		0			0
$939$	−1.00048e6	−	483138.i	−1.13469	−	0.547949i
$940$		0			0
$941$	−935546.			−1.05654			−0.528270	−	0.849076i	$-0.677159\pi$
$941$	−935546.			−1.05654			−0.528270	+	0.849076i	$0.677159\pi$
$942$		0			0
$943$	−397110.			−0.446568
$944$		0			0
$945$	−24621.8	−	109512.i	−0.0275712	−	0.122630i
$946$		0			0
$947$	−762646.			−0.850400			−0.425200	−	0.905099i	$-0.639796\pi$
$947$	−762646.			−0.850400			−0.425200	+	0.905099i	$0.639796\pi$
$948$		0			0
$949$		−	898852.i		−	0.998058i
$950$		0			0
$951$	681718.	+	329207.i	0.753778	+	0.364005i
$952$		0			0
$953$		−	1.66347e6i		−	1.83159i	−0.401646	−	0.915795i	$-0.631562\pi$
$953$		−	1.66347e6i		−	1.83159i	0.401646	−	0.915795i	$-0.368438\pi$
$954$		0			0
$955$			330876.i			0.362793i
$956$		0			0
$957$	−1.28309e6	−	619615.i	−1.40099	−	0.676547i
$958$		0			0
$959$		−	98650.9i		−	0.107266i
$960$		0			0
$961$	−907092.			−0.982210
$962$		0			0
$963$	−65703.5	−	82756.1i	−0.0708494	−	0.0892376i
$964$		0			0
$965$	1.62663e6			1.74677
$966$		0			0
$967$	709312.			0.758550			0.379275	−	0.925284i	$-0.376173\pi$
$967$	709312.			0.758550			0.379275	+	0.925284i	$0.376173\pi$
$968$		0			0
$969$	79176.1	+	38234.7i	0.0843231	+	0.0407202i
$970$		0			0
$971$	−1.18218e6			−1.25385			−0.626924	−	0.779081i	$-0.715686\pi$
$971$	−1.18218e6			−1.25385			−0.626924	+	0.779081i	$0.715686\pi$
$972$		0			0
$973$			55853.5i			0.0589963i
$974$		0			0
$975$	171329.	−	354786.i	0.180228	−	0.373213i
$976$		0			0
$977$		−	1.20925e6i		−	1.26685i	−0.773804	−	0.633425i	$-0.781648\pi$
$977$		−	1.20925e6i		−	1.26685i	0.773804	−	0.633425i	$-0.218352\pi$
$978$		0			0
$979$			271445.i			0.283215i
$980$		0			0
$981$	306593.	−	243417.i	0.318584	−	0.252937i
$982$		0			0
$983$			1.61497e6i			1.67132i	0.549250	+	0.835658i	$0.314913\pi$
$983$			1.61497e6i			1.67132i	−0.549250	+	0.835658i	$0.685087\pi$
$984$		0			0
$985$	−1.32588e6			−1.36657
$986$		0			0
$987$	79700.8	−	165044.i	0.0818142	−	0.169420i
$988$		0			0
$989$	990774.			1.01294
$990$		0			0
$991$	730687.			0.744019			0.372010	−	0.928229i	$-0.378669\pi$
$991$	730687.			0.744019			0.372010	+	0.928229i	$0.378669\pi$
$992$		0			0
$993$	−265027.	+	548815.i	−0.268776	+	0.556580i
$994$		0			0
$995$	1.15915e6			1.17083
$996$		0			0
$997$			626076.i			0.629850i	0.949117	+	0.314925i	$0.101979\pi$
$997$			626076.i			0.629850i	−0.949117	+	0.314925i	$0.898021\pi$
$998$		0			0
$999$	1.71432e6	−	385436.i	1.71776	−	0.386208i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.h.h.65.5	✓	32
3.2	odd	2	inner	384.5.h.h.65.26	yes	32
4.3	odd	2	inner	384.5.h.h.65.27	yes	32
8.3	odd	2	inner	384.5.h.h.65.6	yes	32
8.5	even	2	inner	384.5.h.h.65.28	yes	32
12.11	even	2	inner	384.5.h.h.65.8	yes	32
24.5	odd	2	inner	384.5.h.h.65.7	yes	32
24.11	even	2	inner	384.5.h.h.65.25	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.h.h.65.5	✓	32	1.1	even	1	trivial
384.5.h.h.65.6	yes	32	8.3	odd	2	inner
384.5.h.h.65.7	yes	32	24.5	odd	2	inner
384.5.h.h.65.8	yes	32	12.11	even	2	inner
384.5.h.h.65.25	yes	32	24.11	even	2	inner
384.5.h.h.65.26	yes	32	3.2	odd	2	inner
384.5.h.h.65.27	yes	32	4.3	odd	2	inner
384.5.h.h.65.28	yes	32	8.5	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)

\(f(q)\)	\(=\)	\(q+(-8.10449 - 3.91372i) q^{3} -29.9378 q^{5} -5.14305 q^{7} +(50.3656 + 63.4374i) q^{9} +O(q^{10})\)	\(q+(-8.10449 - 3.91372i) q^{3} -29.9378 q^{5} -5.14305 q^{7} +(50.3656 + 63.4374i) q^{9} -170.063 q^{11} +161.374i q^{13} +(242.631 + 117.168i) q^{15} +32.6278i q^{17} +299.419i q^{19} +(41.6818 + 20.1284i) q^{21} +333.638i q^{23} +271.273 q^{25} +(-159.911 - 711.245i) q^{27} -930.939 q^{29} +128.177 q^{31} +(1378.28 + 665.580i) q^{33} +153.972 q^{35} +2410.31i q^{37} +(631.572 - 1307.85i) q^{39} +1190.24i q^{41} -2969.61i q^{43} +(-1507.84 - 1899.18i) q^{45} -3959.61i q^{47} -2374.55 q^{49} +(127.696 - 264.432i) q^{51} +3587.36 q^{53} +5091.33 q^{55} +(1171.84 - 2426.64i) q^{57} -1882.24 q^{59} -1616.86i q^{61} +(-259.033 - 326.262i) q^{63} -4831.18i q^{65} -8281.38i q^{67} +(1305.77 - 2703.97i) q^{69} +3154.62i q^{71} -5570.00 q^{73} +(-2198.53 - 1061.69i) q^{75} +874.643 q^{77} +9861.40 q^{79} +(-1487.61 + 6390.13i) q^{81} -8992.50 q^{83} -976.806i q^{85} +(7544.79 + 3643.43i) q^{87} -1596.14i q^{89} -829.953i q^{91} +(-1038.81 - 501.649i) q^{93} -8963.97i q^{95} +2547.74 q^{97} +(-8565.34 - 10788.4i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 224 q^{9}+O(q^{10}) \) 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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