LMFDB - Embedded newform 384.5.h.h.65.6

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			65.6
Character	$\chi$	$=$	384.65
Dual form			384.5.h.h.65.8

$q$-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	$-0.295662\pi$
$5$	29.9378			1.19751			0.598757	+	0.800931i	$0.295662\pi$
$6$		0			0
$7$	5.14305			0.104960			0.0524801	−	0.998622i	$-0.483287\pi$
$7$	5.14305			0.104960			0.0524801	+	0.998622i	$0.483287\pi$
$8$		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	$-0.841566\pi$
$13$		−	161.374i		−	0.954875i	0.878666	−	0.477438i	$-0.158434\pi$
$14$		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	$-0.897879\pi$
$23$		−	333.638i		−	0.630696i	0.948976	−	0.315348i	$-0.102121\pi$
$24$		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	$-0.313303\pi$
$29$	930.939			1.10694			0.553471	+	0.832868i	$0.313303\pi$
$30$		0			0
$31$	−128.177			−0.133379			−0.0666895	−	0.997774i	$-0.521244\pi$
$31$	−128.177			−0.133379			−0.0666895	+	0.997774i	$0.521244\pi$
$32$		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	$-0.657329\pi$
$37$		−	2410.31i		−	1.76064i	0.474383	−	0.880318i	$-0.342671\pi$
$38$		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	$0.353716\pi$
$47$			3959.61i			1.79249i	−0.443558	+	0.896246i	$0.646284\pi$
$48$		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	$-0.720464\pi$
$53$	−3587.36			−1.27709			−0.638547	+	0.769583i	$0.720464\pi$
$54$		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	$0.0697125\pi$
$61$			1616.86i			0.434523i	−0.976113	+	0.217262i	$0.930287\pi$
$62$		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	$-0.898701\pi$
$71$		−	3154.62i		−	0.625793i	0.949787	−	0.312897i	$-0.101299\pi$
$72$		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	$-0.789945\pi$
$79$	−9861.40			−1.58010			−0.790049	+	0.613043i	$0.789945\pi$
$80$		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	$-0.551402\pi$
$101$	−3280.32			−0.321568			−0.160784	+	0.986990i	$0.551402\pi$
$102$		0			0
$103$	−10235.3			−0.964773			−0.482386	−	0.875959i	$-0.660230\pi$
$103$	−10235.3			−0.964773			−0.482386	+	0.875959i	$0.660230\pi$
$104$		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	$0.0651966\pi$
$109$			4833.00i			0.406784i	−0.979097	+	0.203392i	$0.934803\pi$
$110$		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	$-0.219915\pi$
$127$	24860.7			1.54137			0.770683	+	0.637219i	$0.219915\pi$
$128$		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	$-0.419928\pi$
$149$	11052.1			0.497818			0.248909	+	0.968527i	$0.419928\pi$
$150$		0			0
$151$	−26646.9			−1.16867			−0.584336	−	0.811512i	$-0.698645\pi$
$151$	−26646.9			−1.16867			−0.584336	+	0.811512i	$0.698645\pi$
$152$		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	$-0.924382\pi$
$157$		−	11601.5i		−	0.470666i	0.971915	−	0.235333i	$-0.0756181\pi$
$158$		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	$-0.611943\pi$
$167$		−	52364.2i		−	1.87759i	0.344474	−	0.938796i	$-0.388057\pi$
$168$		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	$-0.822546\pi$
$173$	−50794.7			−1.69717			−0.848587	+	0.529056i	$0.822546\pi$
$174$		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	$0.101969\pi$
$181$			20632.5i			0.629788i	−0.949127	+	0.314894i	$0.898031\pi$
$182$		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	$0.0484032\pi$
$191$			11052.1i			0.302955i	−0.988461	+	0.151478i	$0.951597\pi$
$192$		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	$-0.693284\pi$
$197$	−44287.8			−1.14117			−0.570586	+	0.821238i	$0.693284\pi$
$198$		0			0
$199$	38718.7			0.977720			0.488860	−	0.872362i	$-0.337413\pi$
$199$	38718.7			0.977720			0.488860	+	0.872362i	$0.337413\pi$
$200$		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	$-0.280653\pi$
$223$	63239.5			1.27168			0.635841	+	0.771820i	$0.280653\pi$
$224$		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	$-0.707951\pi$
$229$		−	83285.2i		−	1.58817i	0.607807	−	0.794084i	$-0.292049\pi$
$230$		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	$0.202294\pi$
$239$			67814.2i			1.18720i	−0.804759	+	0.593601i	$0.797706\pi$
$240$		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	$0.0218679\pi$
$263$			9496.36i			0.137292i	−0.997641	+	0.0686461i	$0.978132\pi$
$264$		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	$-0.625330\pi$
$269$	−55521.4			−0.767283			−0.383642	+	0.923482i	$0.625330\pi$
$270$		0			0
$271$	123146.			1.67680			0.838400	−	0.545055i	$-0.183491\pi$
$271$	123146.			1.67680			0.838400	+	0.545055i	$0.183491\pi$
$272$		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	$-0.752420\pi$
$277$		−	107683.i		−	1.40342i	0.712462	−	0.701711i	$-0.247580\pi$
$278$		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	$-0.675231\pi$
$293$	−89818.1			−1.04623			−0.523117	+	0.852261i	$0.675231\pi$
$294$		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	$0.00822183\pi$
$311$			4995.98i			0.0516535i	−0.999666	+	0.0258268i	$0.991778\pi$
$312$		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	$-0.362544\pi$
$317$	84116.1			0.837067			0.418534	+	0.908201i	$0.362544\pi$
$318$		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	$0.224610\pi$
$349$			157980.i			1.29703i	−0.761202	+	0.648515i	$0.775390\pi$
$350$		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	$0.236639\pi$
$359$			174457.i			1.35363i	−0.736155	+	0.676813i	$0.763361\pi$
$360$		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	$-0.675325\pi$
$367$	−140984.			−1.04674			−0.523370	+	0.852106i	$0.675325\pi$
$368$		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	$-0.845883\pi$
$373$		−	129523.i		−	0.930953i	0.885060	−	0.465476i	$-0.154117\pi$
$374$		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	$0.0474412\pi$
$383$			43563.6i			0.296979i	−0.988914	+	0.148490i	$0.952559\pi$
$384$		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	$-0.522659\pi$
$389$	−21525.5			−0.142250			−0.0711252	+	0.997467i	$0.522659\pi$
$390$		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	$0.289120\pi$
$397$			248536.i			1.57692i	−0.615088	+	0.788458i	$0.710880\pi$
$398$		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	$0.237940\pi$
$421$			240982.i			1.35963i	−0.733383	+	0.679815i	$0.762060\pi$
$422$		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	$-0.976377\pi$
$431$		−	27547.2i		−	0.148294i	0.997247	−	0.0741470i	$-0.0236234\pi$
$432$		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	$-0.424705\pi$
$439$	90326.6			0.468691			0.234345	+	0.972153i	$0.424705\pi$
$440$		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	$-0.750209\pi$
$461$	−300747.			−1.41514			−0.707571	+	0.706643i	$0.750209\pi$
$462$		0			0
$463$	121032.			0.564597			0.282298	−	0.959327i	$-0.408903\pi$
$463$	121032.			0.564597			0.282298	+	0.959327i	$0.408903\pi$
$464$		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	$-0.822204\pi$
$479$		−	243193.i		−	1.05994i	0.848017	−	0.529969i	$-0.177796\pi$
$480$		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	$-0.613674\pi$
$487$	−165816.			−0.699149			−0.349574	+	0.936909i	$0.613674\pi$
$488$		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	$-0.920458\pi$
$503$		−	125137.i		−	0.494595i	0.968940	−	0.247297i	$-0.0795425\pi$
$504$		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	$-0.342616\pi$
$509$	245886.			0.949071			0.474536	+	0.880236i	$0.342616\pi$
$510$		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	$0.181516\pi$
$541$			316003.i			1.07968i	−0.841767	+	0.539842i	$0.818484\pi$
$542$		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	$-0.214876\pi$
$557$	484406.			1.56135			0.780673	+	0.624939i	$0.214876\pi$
$558$		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	$-0.983496\pi$
$599$		−	37189.5i		−	0.103649i	0.998656	−	0.0518247i	$-0.0165037\pi$
$600$		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	$-0.201558\pi$
$607$	594036.			1.61226			0.806130	+	0.591738i	$0.201558\pi$
$608$		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	$-0.942718\pi$
$613$		−	134516.i		−	0.357975i	0.983851	−	0.178987i	$-0.0572821\pi$
$614$		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	$-0.419491\pi$
$631$	199271.			0.500478			0.250239	+	0.968184i	$0.419491\pi$
$632$		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	$-0.932515\pi$
$647$		−	176172.i		−	0.420852i	0.977610	−	0.210426i	$-0.0674851\pi$
$648$		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	$-0.469316\pi$
$653$	82082.5			0.192497			0.0962486	+	0.995357i	$0.469316\pi$
$654$		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	$0.0340649\pi$
$661$			93338.4i			0.213628i	−0.994279	+	0.106814i	$0.965935\pi$
$662$		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	$-0.541308\pi$
$677$	−118624.			−0.258818			−0.129409	+	0.991591i	$0.541308\pi$
$678$		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	$-0.534253\pi$
$701$	−105553.			−0.214801			−0.107401	+	0.994216i	$0.534253\pi$
$702$		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	$-0.987328\pi$
$709$		−	40012.7i		−	0.0795985i	0.999208	−	0.0397993i	$-0.0126718\pi$
$710$		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	$0.0214708\pi$
$719$			69687.8i			0.134803i	−0.997726	+	0.0674015i	$0.978529\pi$
$720$		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	$-0.584845\pi$
$727$	−278432.			−0.526805			−0.263402	+	0.964686i	$0.584845\pi$
$728$		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	$0.0631176\pi$
$733$			211684.i			0.393986i	−0.980405	+	0.196993i	$0.936882\pi$
$734$		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	$-0.845831\pi$
$743$		−	514092.i		−	0.931244i	0.884984	−	0.465622i	$-0.154169\pi$
$744$		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	$-0.580803\pi$
$751$	−283278.			−0.502265			−0.251133	+	0.967953i	$0.580803\pi$
$752$		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	$0.0324383\pi$
$757$			116594.i			0.203463i	−0.994812	+	0.101732i	$0.967562\pi$
$758$		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	$-0.342108\pi$
$773$	568776.			0.951880			0.475940	+	0.879478i	$0.342108\pi$
$774$		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	$-0.345502\pi$
$797$	592697.			0.933074			0.466537	+	0.884502i	$0.345502\pi$
$798$		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	$-0.508557\pi$
$821$	−36236.4			−0.0537600			−0.0268800	+	0.999639i	$0.508557\pi$
$822$		0			0
$823$	−306097.			−0.451918			−0.225959	−	0.974137i	$-0.572552\pi$
$823$	−306097.			−0.451918			−0.225959	+	0.974137i	$0.572552\pi$
$824$		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	$-0.766244\pi$
$829$		−	921062.i		−	1.34023i	0.742257	−	0.670116i	$-0.233756\pi$
$830$		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	$-0.807090\pi$
$839$		−	801936.i		−	1.13924i	0.821908	−	0.569620i	$-0.192910\pi$
$840$		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	$0.0595525\pi$
$853$			270671.i			0.372000i	−0.982550	+	0.186000i	$0.940448\pi$
$854$		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	$0.333147\pi$
$863$			1.28954e6i			1.73147i	−0.500507	+	0.865733i	$0.666853\pi$
$864$		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	$-0.834914\pi$
$877$		−	762506.i		−	0.991389i	0.868497	−	0.495695i	$-0.165086\pi$
$878$		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	$0.00684734\pi$
$887$			33846.7i			0.0430198i	−0.999769	+	0.0215099i	$0.993153\pi$
$888$		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	$0.214508\pi$
$911$			1.03580e6i			1.24807i	−0.781395	+	0.624037i	$0.785492\pi$
$912$		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	$-0.573537\pi$
$919$	−386764.			−0.457947			−0.228974	+	0.973433i	$0.573537\pi$
$920$		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	$-0.322841\pi$
$941$	935546.			1.05654			0.528270	+	0.849076i	$0.322841\pi$
$942$		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	$-0.623827\pi$
$967$	−709312.			−0.758550			−0.379275	+	0.925284i	$0.623827\pi$
$968$		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	$-0.685087\pi$
$983$		−	1.61497e6i		−	1.67132i	0.549250	−	0.835658i	$-0.314913\pi$
$984$		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	$-0.621331\pi$
$991$	−730687.			−0.744019			−0.372010	+	0.928229i	$0.621331\pi$
$992$		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	$-0.898021\pi$
$997$		−	626076.i		−	0.629850i	0.949117	−	0.314925i	$-0.101979\pi$
$998$		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.65.6	yes	32
3.2	odd	2	inner	384.5.h.h.65.25	yes	32
4.3	odd	2	inner	384.5.h.h.65.28	yes	32
8.3	odd	2	inner	384.5.h.h.65.5	✓	32
8.5	even	2	inner	384.5.h.h.65.27	yes	32
12.11	even	2	inner	384.5.h.h.65.7	yes	32
24.5	odd	2	inner	384.5.h.h.65.8	yes	32
24.11	even	2	inner	384.5.h.h.65.26	yes	32

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