LMFDB - Embedded newform 384.4.f.i.191.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(191,384)]

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([1, 1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("384.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.f (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:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.77720518656.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} + 119x^{4} - 300x^{2} + 625 $ x^8 - 12x^6 + 119x^4 - 300*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.5
Root			$-1.41864 - 0.819051i$ of defining polynomial
Character	$\chi$	$=$	384.191
Dual form			384.4.f.i.191.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.89898 - 1.73205i) q^{3} -16.2481 q^{5} +28.1425i q^{7} +(21.0000 - 16.9706i) q^{9} +O(q^{10})$	$q+(4.89898 - 1.73205i) q^{3} -16.2481 q^{5} +28.1425i q^{7} +(21.0000 - 16.9706i) q^{9} -24.2487i q^{11} -45.9565i q^{13} +(-79.5990 + 28.1425i) q^{15} -79.1960i q^{17} -9.79796 q^{19} +(48.7442 + 137.870i) q^{21} +159.198 q^{23} +139.000 q^{25} +(73.4847 - 119.512i) q^{27} +113.737 q^{29} -196.997i q^{31} +(-42.0000 - 118.794i) q^{33} -457.261i q^{35} -321.696i q^{37} +(-79.5990 - 225.140i) q^{39} +316.784i q^{41} +480.100 q^{43} +(-341.210 + 275.739i) q^{45} -449.000 q^{49} +(-137.171 - 387.979i) q^{51} -341.210 q^{53} +393.995i q^{55} +(-48.0000 + 16.9706i) q^{57} -682.428i q^{59} -45.9565i q^{61} +(477.594 + 590.992i) q^{63} +746.705i q^{65} -342.929 q^{67} +(779.908 - 275.739i) q^{69} -1114.39 q^{71} -98.0000 q^{73} +(680.958 - 240.755i) q^{75} +682.419 q^{77} +140.712i q^{79} +(153.000 - 712.764i) q^{81} -924.915i q^{83} +1286.78i q^{85} +(557.193 - 196.997i) q^{87} +395.980i q^{89} +1293.33 q^{91} +(-341.210 - 965.087i) q^{93} +159.198 q^{95} +994.000 q^{97} +(-411.514 - 509.223i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 168 q^{9}+O(q^{10}) $ 8 * q + 168 * q^9	$ 8 q + 168 q^{9} + 1112 q^{25} - 336 q^{33} - 3592 q^{49} - 384 q^{57} - 784 q^{73} + 1224 q^{81} + 7952 q^{97}+O(q^{100}) $ 8 * q + 168 * q^9 + 1112 * q^25 - 336 * q^33 - 3592 * q^49 - 384 * q^57 - 784 * q^73 + 1224 * q^81 + 7952 * 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
$3$	4.89898	−	1.73205i	0.942809	−	0.333333i
$3$	4.89898	−	1.73205i	0.942809	−	0.333333i
$4$		0			0
$5$	−16.2481			−1.45327			−0.726636	−	0.687023i	$-0.758917\pi$
$5$	−16.2481			−1.45327			−0.726636	+	0.687023i	$0.758917\pi$
$6$		0			0
$7$			28.1425i			1.51955i	0.650185	+	0.759776i	$0.274691\pi$
$7$			28.1425i			1.51955i	−0.650185	+	0.759776i	$0.725309\pi$
$8$		0			0
$9$	21.0000	−	16.9706i	0.777778	−	0.628539i
$10$		0			0
$11$		−	24.2487i		−	0.664660i	−0.943163	−	0.332330i	$-0.892165\pi$
$11$		−	24.2487i		−	0.664660i	0.943163	−	0.332330i	$-0.107835\pi$
$12$		0			0
$13$		−	45.9565i		−	0.980465i	−0.871592	−	0.490232i	$-0.836912\pi$
$13$		−	45.9565i		−	0.980465i	0.871592	−	0.490232i	$-0.163088\pi$
$14$		0			0
$15$	−79.5990	+	28.1425i	−1.37016	+	0.484424i
$16$		0			0
$17$		−	79.1960i		−	1.12987i	−0.825134	−	0.564937i	$-0.808901\pi$
$17$		−	79.1960i		−	1.12987i	0.825134	−	0.564937i	$-0.191099\pi$
$18$		0			0
$19$	−9.79796			−0.118306			−0.0591528	−	0.998249i	$-0.518840\pi$
$19$	−9.79796			−0.118306			−0.0591528	+	0.998249i	$0.518840\pi$
$20$		0			0
$21$	48.7442	+	137.870i	0.506517	+	1.43265i
$22$		0			0
$23$	159.198			1.44326			0.721632	−	0.692277i	$-0.243392\pi$
$23$	159.198			1.44326			0.721632	+	0.692277i	$0.243392\pi$
$24$		0			0
$25$	139.000			1.11200
$26$		0			0
$27$	73.4847	−	119.512i	0.523783	−	0.851852i
$28$		0			0
$29$	113.737			0.728288			0.364144	−	0.931343i	$-0.381362\pi$
$29$	113.737			0.728288			0.364144	+	0.931343i	$0.381362\pi$
$30$		0			0
$31$		−	196.997i		−	1.14135i	−0.821177	−	0.570674i	$-0.806682\pi$
$31$		−	196.997i		−	1.14135i	0.821177	−	0.570674i	$-0.193318\pi$
$32$		0			0
$33$	−42.0000	−	118.794i	−0.221553	−	0.626648i
$34$		0			0
$35$		−	457.261i		−	2.20832i
$36$		0			0
$37$		−	321.696i		−	1.42936i	−0.699450	−	0.714681i	$-0.746572\pi$
$37$		−	321.696i		−	1.42936i	0.699450	−	0.714681i	$-0.253428\pi$
$38$		0			0
$39$	−79.5990	−	225.140i	−0.326822	−	0.924391i
$40$		0			0
$41$			316.784i			1.20667i	0.797489	+	0.603333i	$0.206161\pi$
$41$			316.784i			1.20667i	−0.797489	+	0.603333i	$0.793839\pi$
$42$		0			0
$43$	480.100			1.70266			0.851332	−	0.524627i	$-0.175795\pi$
$43$	480.100			1.70266			0.851332	+	0.524627i	$0.175795\pi$
$44$		0			0
$45$	−341.210	+	275.739i	−1.13032	+	0.913439i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−449.000			−1.30904
$50$		0			0
$51$	−137.171	−	387.979i	−0.376624	−	1.06525i
$52$		0			0
$53$	−341.210			−0.884316			−0.442158	−	0.896937i	$-0.645787\pi$
$53$	−341.210			−0.884316			−0.442158	+	0.896937i	$0.645787\pi$
$54$		0			0
$55$			393.995i			0.965932i
$56$		0			0
$57$	−48.0000	+	16.9706i	−0.111540	+	0.0394352i
$58$		0			0
$59$		−	682.428i		−	1.50584i	−0.658112	−	0.752920i	$-0.728645\pi$
$59$		−	682.428i		−	1.50584i	0.658112	−	0.752920i	$-0.271355\pi$
$60$		0			0
$61$		−	45.9565i		−	0.0964611i	−0.998836	−	0.0482305i	$-0.984642\pi$
$61$		−	45.9565i		−	0.0964611i	0.998836	−	0.0482305i	$-0.0153582\pi$
$62$		0			0
$63$	477.594	+	590.992i	0.955098	+	1.18187i
$64$		0			0
$65$			746.705i			1.42488i
$66$		0			0
$67$	−342.929			−0.625304			−0.312652	−	0.949868i	$-0.601217\pi$
$67$	−342.929			−0.625304			−0.312652	+	0.949868i	$0.601217\pi$
$68$		0			0
$69$	779.908	−	275.739i	1.36072	−	0.481088i
$70$		0			0
$71$	−1114.39			−1.86272			−0.931361	−	0.364097i	$-0.881378\pi$
$71$	−1114.39			−1.86272			−0.931361	+	0.364097i	$0.881378\pi$
$72$		0			0
$73$	−98.0000			−0.157124			−0.0785619	−	0.996909i	$-0.525033\pi$
$73$	−98.0000			−0.157124			−0.0785619	+	0.996909i	$0.525033\pi$
$74$		0			0
$75$	680.958	−	240.755i	1.04840	−	0.370667i
$76$		0			0
$77$	682.419			1.00999
$78$		0			0
$79$			140.712i			0.200397i	0.994967	+	0.100199i	$0.0319478\pi$
$79$			140.712i			0.200397i	−0.994967	+	0.100199i	$0.968052\pi$
$80$		0			0
$81$	153.000	−	712.764i	0.209877	−	0.977728i
$82$		0			0
$83$		−	924.915i		−	1.22316i	−0.791181	−	0.611582i	$-0.790533\pi$
$83$		−	924.915i		−	1.22316i	0.791181	−	0.611582i	$-0.209467\pi$
$84$		0			0
$85$			1286.78i			1.64201i
$86$		0			0
$87$	557.193	−	196.997i	0.686636	−	0.242763i
$88$		0			0
$89$			395.980i			0.471615i	0.971800	+	0.235808i	$0.0757736\pi$
$89$			395.980i			0.471615i	−0.971800	+	0.235808i	$0.924226\pi$
$90$		0			0
$91$	1293.33			1.48987
$92$		0			0
$93$	−341.210	−	965.087i	−0.380449	−	1.07607i
$94$		0			0
$95$	159.198			0.171930
$96$		0			0
$97$	994.000			1.04047			0.520234	−	0.854024i	$-0.325845\pi$
$97$	994.000			1.04047			0.520234	+	0.854024i	$0.325845\pi$
$98$		0			0
$99$	−411.514	−	509.223i	−0.417765	−	0.516958i
$100$		0			0
$101$	−926.140			−0.912420			−0.456210	−	0.889872i	$-0.650793\pi$
$101$	−926.140			−0.912420			−0.456210	+	0.889872i	$0.650793\pi$
$102$		0			0
$103$		−	984.987i		−	0.942269i	−0.882061	−	0.471134i	$-0.843845\pi$
$103$		−	984.987i		−	0.942269i	0.882061	−	0.471134i	$-0.156155\pi$
$104$		0			0
$105$	−792.000	−	2240.11i	−0.736107	−	2.08203i
$106$		0			0
$107$			897.202i			0.810615i	0.914180	+	0.405308i	$0.132836\pi$
$107$			897.202i			0.810615i	−0.914180	+	0.405308i	$0.867164\pi$
$108$		0			0
$109$			321.696i			0.282687i	0.989961	+	0.141343i	$0.0451421\pi$
$109$			321.696i			0.282687i	−0.989961	+	0.141343i	$0.954858\pi$
$110$		0			0
$111$	−557.193	−	1575.98i	−0.476454	−	1.34762i
$112$		0			0
$113$			520.431i			0.433257i	0.976254	+	0.216628i	$0.0695060\pi$
$113$			520.431i			0.433257i	−0.976254	+	0.216628i	$0.930494\pi$
$114$		0			0
$115$	−2586.66			−2.09746
$116$		0			0
$117$	−779.908	−	965.087i	−0.616261	−	0.762584i
$118$		0			0
$119$	2228.77			1.71690
$120$		0			0
$121$	743.000			0.558227
$122$		0			0
$123$	548.686	+	1551.92i	0.402222	+	1.13766i
$124$		0			0
$125$	−227.473			−0.162766
$126$		0			0
$127$		−	196.997i		−	0.137643i	−0.997629	−	0.0688216i	$-0.978076\pi$
$127$		−	196.997i		−	0.137643i	0.997629	−	0.0688216i	$-0.0219239\pi$
$128$		0			0
$129$	2352.00	−	831.558i	1.60529	−	0.567555i
$130$		0			0
$131$			1215.90i			0.810944i	0.914107	+	0.405472i	$0.132893\pi$
$131$			1215.90i			0.810944i	−0.914107	+	0.405472i	$0.867107\pi$
$132$		0			0
$133$		−	275.739i		−	0.179771i
$134$		0			0
$135$	−1193.98	+	1941.83i	−0.761199	+	1.23797i
$136$		0			0
$137$			791.960i			0.493881i	0.969031	+	0.246940i	$0.0794252\pi$
$137$			791.960i			0.493881i	−0.969031	+	0.246940i	$0.920575\pi$
$138$		0			0
$139$	1518.68			0.926713			0.463356	−	0.886172i	$-0.346645\pi$
$139$	1518.68			0.926713			0.463356	+	0.886172i	$0.346645\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1114.39			−0.651676
$144$		0			0
$145$	−1848.00			−1.05840
$146$		0			0
$147$	−2199.64	+	777.691i	−1.23417	+	0.436346i
$148$		0			0
$149$	2843.41			1.56337			0.781683	−	0.623676i	$-0.214362\pi$
$149$	2843.41			1.56337			0.781683	+	0.623676i	$0.214362\pi$
$150$		0			0
$151$			590.992i			0.318505i	0.987238	+	0.159253i	$0.0509084\pi$
$151$			590.992i			0.318505i	−0.987238	+	0.159253i	$0.949092\pi$
$152$		0			0
$153$	−1344.00	−	1663.12i	−0.710170	−	0.878790i
$154$		0			0
$155$			3200.83i			1.65869i
$156$		0			0
$157$			1976.13i			1.00454i	0.864712	+	0.502268i	$0.167501\pi$
$157$			1976.13i			1.00454i	−0.864712	+	0.502268i	$0.832499\pi$
$158$		0			0
$159$	−1671.58	+	590.992i	−0.833741	+	0.294772i
$160$		0			0
$161$			4480.23i			2.19311i
$162$		0			0
$163$	−205.757			−0.0988720			−0.0494360	−	0.998777i	$-0.515742\pi$
$163$	−205.757			−0.0988720			−0.0494360	+	0.998777i	$0.515742\pi$
$164$		0			0
$165$	682.419	+	1930.17i	0.321977	+	0.910690i
$166$		0			0
$167$	1114.39			0.516370			0.258185	−	0.966096i	$-0.416876\pi$
$167$	1114.39			0.516370			0.258185	+	0.966096i	$0.416876\pi$
$168$		0			0
$169$	85.0000			0.0386891
$170$		0			0
$171$	−205.757	+	166.277i	−0.0920154	+	0.0743597i
$172$		0			0
$173$	−1576.06			−0.692635			−0.346318	−	0.938117i	$-0.612568\pi$
$173$	−1576.06			−0.692635			−0.346318	+	0.938117i	$0.612568\pi$
$174$		0			0
$175$			3911.81i			1.68974i
$176$		0			0
$177$	−1182.00	−	3343.20i	−0.501947	−	1.41972i
$178$		0			0
$179$			1964.15i			0.820152i	0.912051	+	0.410076i	$0.134498\pi$
$179$			1964.15i			0.820152i	−0.912051	+	0.410076i	$0.865502\pi$
$180$		0			0
$181$		−	4549.69i		−	1.86838i	−0.356781	−	0.934188i	$-0.616126\pi$
$181$		−	4549.69i		−	1.86838i	0.356781	−	0.934188i	$-0.383874\pi$
$182$		0			0
$183$	−79.5990	−	225.140i	−0.0321537	−	0.0909444i
$184$		0			0
$185$			5226.93i			2.07725i
$186$		0			0
$187$	−1920.40			−0.750982
$188$		0			0
$189$	3363.35	+	2068.04i	1.29443	+	0.795915i
$190$		0			0
$191$	2228.77			0.844337			0.422168	−	0.906517i	$-0.361269\pi$
$191$	2228.77			0.844337			0.422168	+	0.906517i	$0.361269\pi$
$192$		0			0
$193$	−2618.00			−0.976413			−0.488207	−	0.872728i	$-0.662349\pi$
$193$	−2618.00			−0.976413			−0.488207	+	0.872728i	$0.662349\pi$
$194$		0			0
$195$	1293.33	+	3658.09i	0.474961	+	1.34339i
$196$		0			0
$197$	−2160.99			−0.781545			−0.390773	−	0.920487i	$-0.627792\pi$
$197$	−2160.99			−0.781545			−0.390773	+	0.920487i	$0.627792\pi$
$198$		0			0
$199$		−	2560.97i		−	0.912272i	−0.889910	−	0.456136i	$-0.849233\pi$
$199$		−	2560.97i		−	0.912272i	0.889910	−	0.456136i	$-0.150767\pi$
$200$		0			0
$201$	−1680.00	+	593.970i	−0.589543	+	0.208435i
$202$		0			0
$203$			3200.83i			1.10667i
$204$		0			0
$205$		−	5147.13i		−	1.75361i
$206$		0			0
$207$	3343.16	−	2701.68i	1.12254	−	0.907148i
$208$		0			0
$209$			237.588i			0.0786330i
$210$		0			0
$211$	−3909.39			−1.27551			−0.637756	−	0.770238i	$-0.720137\pi$
$211$	−3909.39			−1.27551			−0.637756	+	0.770238i	$0.720137\pi$
$212$		0			0
$213$	−5459.35	+	1930.17i	−1.75619	+	0.620907i
$214$		0			0
$215$	−7800.70			−2.47443
$216$		0			0
$217$	5544.00			1.73434
$218$		0			0
$219$	−480.100	+	169.741i	−0.148138	+	0.0523746i
$220$		0			0
$221$	−3639.57			−1.10780
$222$		0			0
$223$		−	5712.93i		−	1.71554i	−0.514032	−	0.857771i	$-0.671849\pi$
$223$		−	5712.93i		−	1.71554i	0.514032	−	0.857771i	$-0.328151\pi$
$224$		0			0
$225$	2919.00	−	2358.91i	0.864889	−	0.698936i
$226$		0			0
$227$			1208.97i			0.353490i	0.984257	+	0.176745i	$0.0565568\pi$
$227$			1208.97i			0.353490i	−0.984257	+	0.176745i	$0.943443\pi$
$228$		0			0
$229$			5193.08i			1.49855i	0.662257	+	0.749277i	$0.269599\pi$
$229$			5193.08i			1.49855i	−0.662257	+	0.749277i	$0.730401\pi$
$230$		0			0
$231$	3343.16	−	1181.98i	0.952224	−	0.336662i
$232$		0			0
$233$			1346.33i			0.378545i	0.981925	+	0.189273i	$0.0606130\pi$
$233$			1346.33i			0.378545i	−0.981925	+	0.189273i	$0.939387\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	243.721	+	689.348i	0.0667991	+	0.188936i
$238$		0			0
$239$	1273.58			0.344692			0.172346	−	0.985037i	$-0.444865\pi$
$239$	1273.58			0.344692			0.172346	+	0.985037i	$0.444865\pi$
$240$		0			0
$241$	−5642.00			−1.50802			−0.754011	−	0.656862i	$-0.771883\pi$
$241$	−5642.00			−1.50802			−0.754011	+	0.656862i	$0.771883\pi$
$242$		0			0
$243$	−484.999	−	3756.82i	−0.128036	−	0.991770i
$244$		0			0
$245$	7295.39			1.90239
$246$		0			0
$247$			450.280i			0.115994i
$248$		0			0
$249$	−1602.00	−	4531.14i	−0.407721	−	1.15321i
$250$		0			0
$251$		−	4264.31i		−	1.07235i	−0.844106	−	0.536177i	$-0.819868\pi$
$251$		−	4264.31i		−	1.07235i	0.844106	−	0.536177i	$-0.180132\pi$
$252$		0			0
$253$		−	3860.35i		−	0.959280i
$254$		0			0
$255$	2228.77	+	6303.92i	0.547338	+	1.54810i
$256$		0			0
$257$		−	2059.09i		−	0.499777i	−0.968275	−	0.249889i	$-0.919606\pi$
$257$		−	2059.09i		−	0.499777i	0.968275	−	0.249889i	$-0.0803940\pi$
$258$		0			0
$259$	9053.31			2.17199
$260$		0			0
$261$	2388.47	−	1930.17i	0.566446	−	0.457758i
$262$		0			0
$263$	3343.16			0.783832			0.391916	−	0.920001i	$-0.371812\pi$
$263$	3343.16			0.783832			0.391916	+	0.920001i	$0.371812\pi$
$264$		0			0
$265$	5544.00			1.28515
$266$		0			0
$267$	685.857	+	1939.90i	0.157205	+	0.444643i
$268$		0			0
$269$	5248.13			1.18953			0.594766	−	0.803899i	$-0.297245\pi$
$269$	5248.13			1.18953			0.594766	+	0.803899i	$0.297245\pi$
$270$		0			0
$271$			2166.97i			0.485735i	0.970059	+	0.242867i	$0.0780880\pi$
$271$			2166.97i			0.485735i	−0.970059	+	0.242867i	$0.921912\pi$
$272$		0			0
$273$	6336.00	−	2240.11i	1.40466	−	0.496622i
$274$		0			0
$275$		−	3370.57i		−	0.739102i
$276$		0			0
$277$			965.087i			0.209337i	0.994507	+	0.104669i	$0.0333782\pi$
$277$			965.087i			0.209337i	−0.994507	+	0.104669i	$0.966622\pi$
$278$		0			0
$279$	−3343.16	−	4136.95i	−0.717382	−	0.887715i
$280$		0			0
$281$		−	1029.55i		−	0.218568i	−0.994011	−	0.109284i	$-0.965144\pi$
$281$		−	1029.55i		−	0.218568i	0.994011	−	0.109284i	$-0.0348558\pi$
$282$		0			0
$283$	−4791.20			−1.00639			−0.503193	−	0.864174i	$-0.667842\pi$
$283$	−4791.20			−1.00639			−0.503193	+	0.864174i	$0.667842\pi$
$284$		0			0
$285$	779.908	−	275.739i	0.162097	−	0.0573101i
$286$		0			0
$287$	−8915.09			−1.83359
$288$		0			0
$289$	−1359.00			−0.276613
$290$		0			0
$291$	4869.59	−	1721.66i	0.980963	−	0.346823i
$292$		0			0
$293$	−4760.69			−0.949223			−0.474611	−	0.880195i	$-0.657411\pi$
$293$	−4760.69			−0.949223			−0.474611	+	0.880195i	$0.657411\pi$
$294$		0			0
$295$			11088.1i			2.18840i
$296$		0			0
$297$	−2898.00	−	1781.91i	−0.566192	−	0.348138i
$298$		0			0
$299$		−	7316.18i		−	1.41507i
$300$		0			0
$301$			13511.2i			2.58729i
$302$		0			0
$303$	−4537.14	+	1604.12i	−0.860238	+	0.304140i
$304$		0			0
$305$			746.705i			0.140184i
$306$		0			0
$307$	1773.43			0.329691			0.164845	−	0.986319i	$-0.447288\pi$
$307$	1773.43			0.329691			0.164845	+	0.986319i	$0.447288\pi$
$308$		0			0
$309$	−1706.05	−	4825.43i	−0.314090	−	0.888380i
$310$		0			0
$311$	5571.93			1.01593			0.507966	−	0.861377i	$-0.330397\pi$
$311$	5571.93			1.01593			0.507966	+	0.861377i	$0.330397\pi$
$312$		0			0
$313$	3374.00			0.609296			0.304648	−	0.952465i	$-0.401461\pi$
$313$	3374.00			0.609296			0.304648	+	0.952465i	$0.401461\pi$
$314$		0			0
$315$	−7759.98	−	9602.49i	−1.38802	−	1.71758i
$316$		0			0
$317$	−4890.67			−0.866522			−0.433261	−	0.901269i	$-0.642637\pi$
$317$	−4890.67			−0.866522			−0.433261	+	0.901269i	$0.642637\pi$
$318$		0			0
$319$		−	2757.96i		−	0.484064i
$320$		0			0
$321$	1554.00	+	4395.38i	0.270205	+	0.764255i
$322$		0			0
$323$			775.959i			0.133670i
$324$		0			0
$325$		−	6387.95i		−	1.09028i
$326$		0			0
$327$	557.193	+	1575.98i	0.0942289	+	0.266519i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	617.271			0.102502			0.0512512	−	0.998686i	$-0.483679\pi$
$331$	617.271			0.102502			0.0512512	+	0.998686i	$0.483679\pi$
$332$		0			0
$333$	−5459.35	−	6755.61i	−0.898411	−	1.11173i
$334$		0			0
$335$	5571.93			0.908737
$336$		0			0
$337$	1526.00			0.246666			0.123333	−	0.992365i	$-0.460642\pi$
$337$	1526.00			0.246666			0.123333	+	0.992365i	$0.460642\pi$
$338$		0			0
$339$	901.412	+	2549.58i	0.144419	+	0.408478i
$340$		0			0
$341$	−4776.93			−0.758609
$342$		0			0
$343$		−	2983.10i		−	0.469599i
$344$		0			0
$345$	−12672.0	+	4480.23i	−1.97750	+	0.699152i
$346$		0			0
$347$		−	11081.7i		−	1.71439i	−0.514989	−	0.857197i	$-0.672204\pi$
$347$		−	11081.7i		−	1.71439i	0.514989	−	0.857197i	$-0.327796\pi$
$348$		0			0
$349$			689.348i			0.105730i	0.998602	+	0.0528652i	$0.0168354\pi$
$349$			689.348i			0.105730i	−0.998602	+	0.0528652i	$0.983165\pi$
$350$		0			0
$351$	−5492.33	−	3377.10i	−0.835211	−	0.513551i
$352$		0			0
$353$			316.784i			0.0477640i	0.999715	+	0.0238820i	$0.00760260\pi$
$353$			316.784i			0.0477640i	−0.999715	+	0.0238820i	$0.992397\pi$
$354$		0			0
$355$	18106.6			2.70704
$356$		0			0
$357$	10918.7	−	3860.35i	1.61871	−	0.572300i
$358$		0			0
$359$	6527.12			0.959577			0.479788	−	0.877384i	$-0.340713\pi$
$359$	6527.12			0.959577			0.479788	+	0.877384i	$0.340713\pi$
$360$		0			0
$361$	−6763.00			−0.986004
$362$		0			0
$363$	3639.94	−	1286.91i	0.526301	−	0.186076i
$364$		0			0
$365$	1592.31			0.228344
$366$		0			0
$367$		−	9652.88i		−	1.37296i	−0.727149	−	0.686480i	$-0.759155\pi$
$367$		−	9652.88i		−	1.37296i	0.727149	−	0.686480i	$-0.240845\pi$
$368$		0			0
$369$	5376.00	+	6652.46i	0.758437	+	0.938518i
$370$		0			0
$371$		−	9602.49i		−	1.34376i
$372$		0			0
$373$			12546.1i			1.74159i	0.491645	+	0.870796i	$0.336396\pi$
$373$			12546.1i			1.74159i	−0.491645	+	0.870796i	$0.663604\pi$
$374$		0			0
$375$	−1114.39	+	393.995i	−0.153458	+	0.0542555i
$376$		0			0
$377$		−	5226.93i		−	0.714060i
$378$		0			0
$379$	−11042.3			−1.49658			−0.748291	−	0.663370i	$-0.769125\pi$
$379$	−11042.3			−1.49658			−0.748291	+	0.663370i	$0.769125\pi$
$380$		0			0
$381$	−341.210	−	965.087i	−0.0458811	−	0.129771i
$382$		0			0
$383$	6686.32			0.892049			0.446024	−	0.895021i	$-0.352839\pi$
$383$	6686.32			0.892049			0.446024	+	0.895021i	$0.352839\pi$
$384$		0			0
$385$	−11088.0			−1.46778
$386$		0			0
$387$	10082.1	−	8147.57i	1.32429	−	1.07019i
$388$		0			0
$389$	−341.210			−0.0444730			−0.0222365	−	0.999753i	$-0.507079\pi$
$389$	−341.210			−0.0444730			−0.0222365	+	0.999753i	$0.507079\pi$
$390$		0			0
$391$		−	12607.8i		−	1.63071i
$392$		0			0
$393$	2106.00	+	5956.67i	0.270315	+	0.764565i
$394$		0			0
$395$		−	2286.31i		−	0.291232i
$396$		0			0
$397$		−	4457.78i		−	0.563551i	−0.959480	−	0.281775i	$-0.909077\pi$
$397$		−	4457.78i		−	0.563551i	0.959480	−	0.281775i	$-0.0909233\pi$
$398$		0			0
$399$	−477.594	−	1350.84i	−0.0599238	−	0.169490i
$400$		0			0
$401$			11121.4i			1.38498i	0.721430	+	0.692488i	$0.243485\pi$
$401$			11121.4i			1.38498i	−0.721430	+	0.692488i	$0.756515\pi$
$402$		0			0
$403$	−9053.31			−1.11905
$404$		0			0
$405$	−2485.96	+	11581.0i	−0.305008	+	1.42090i
$406$		0			0
$407$	−7800.70			−0.950040
$408$		0			0
$409$	2618.00			0.316508			0.158254	−	0.987398i	$-0.449414\pi$
$409$	2618.00			0.316508			0.158254	+	0.987398i	$0.449414\pi$
$410$		0			0
$411$	1371.71	+	3879.79i	0.164627	+	0.465635i
$412$		0			0
$413$	19205.2			2.28820
$414$		0			0
$415$			15028.1i			1.77759i
$416$		0			0
$417$	7440.00	−	2630.44i	0.873713	−	0.308904i
$418$		0			0
$419$			6640.68i			0.774269i	0.922023	+	0.387134i	$0.126535\pi$
$419$			6640.68i			0.774269i	−0.922023	+	0.387134i	$0.873465\pi$
$420$		0			0
$421$		−	5468.82i		−	0.633098i	−0.948576	−	0.316549i	$-0.897476\pi$
$421$		−	5468.82i		−	0.633098i	0.948576	−	0.316549i	$-0.102524\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	11008.2i		−	1.25642i
$426$		0			0
$427$	1293.33			0.146578
$428$		0			0
$429$	−5459.35	+	1930.17i	−0.614406	+	0.217225i
$430$		0			0
$431$	−1273.58			−0.142335			−0.0711675	−	0.997464i	$-0.522672\pi$
$431$	−1273.58			−0.142335			−0.0711675	+	0.997464i	$0.522672\pi$
$432$		0			0
$433$	2674.00			0.296777			0.148388	−	0.988929i	$-0.452591\pi$
$433$	2674.00			0.296777			0.148388	+	0.988929i	$0.452591\pi$
$434$		0			0
$435$	−9053.31	+	3200.83i	−0.997869	+	0.352800i
$436$		0			0
$437$	−1559.82			−0.170746
$438$		0			0
$439$			6894.91i			0.749604i	0.927105	+	0.374802i	$0.122289\pi$
$439$			6894.91i			0.749604i	−0.927105	+	0.374802i	$0.877711\pi$
$440$		0			0
$441$	−9429.00	+	7619.78i	−1.01814	+	0.822782i
$442$		0			0
$443$			11809.1i			1.26652i	0.773939	+	0.633260i	$0.218284\pi$
$443$			11809.1i			1.26652i	−0.773939	+	0.633260i	$0.781716\pi$
$444$		0			0
$445$		−	6433.91i		−	0.685386i
$446$		0			0
$447$	13929.8	−	4924.94i	1.47396	−	0.521122i
$448$		0			0
$449$		−	18135.9i		−	1.90620i	−0.302652	−	0.953101i	$-0.597872\pi$
$449$		−	18135.9i		−	1.90620i	0.302652	−	0.953101i	$-0.402128\pi$
$450$		0			0
$451$	7681.60			0.802023
$452$		0			0
$453$	1023.63	+	2895.26i	0.106168	+	0.300290i
$454$		0			0
$455$	−21014.1			−2.16518
$456$		0			0
$457$	15722.0			1.60929			0.804643	−	0.593758i	$-0.202356\pi$
$457$	15722.0			1.60929			0.804643	+	0.593758i	$0.202356\pi$
$458$		0			0
$459$	−9464.83	−	5819.69i	−0.962484	−	0.591808i
$460$		0			0
$461$	−3655.82			−0.369346			−0.184673	−	0.982800i	$-0.559123\pi$
$461$	−3655.82			−0.369346			−0.184673	+	0.982800i	$0.559123\pi$
$462$		0			0
$463$			13536.5i			1.35874i	0.733796	+	0.679370i	$0.237747\pi$
$463$			13536.5i			1.35874i	−0.733796	+	0.679370i	$0.762253\pi$
$464$		0			0
$465$	5544.00	+	15680.8i	0.552896	+	1.56383i
$466$		0			0
$467$		−	1700.87i		−	0.168538i	−0.996443	−	0.0842688i	$-0.973145\pi$
$467$		−	1700.87i		−	0.168538i	0.996443	−	0.0842688i	$-0.0268554\pi$
$468$		0			0
$469$		−	9650.87i		−	0.950182i
$470$		0			0
$471$	3422.76	+	9681.02i	0.334846	+	0.947087i
$472$		0			0
$473$		−	11641.8i		−	1.13169i
$474$		0			0
$475$	−1361.92			−0.131556
$476$		0			0
$477$	−7165.40	+	5790.52i	−0.687801	+	0.555827i
$478$		0			0
$479$	−4457.54			−0.425199			−0.212600	−	0.977139i	$-0.568193\pi$
$479$	−4457.54			−0.425199			−0.212600	+	0.977139i	$0.568193\pi$
$480$		0			0
$481$	−14784.0			−1.40144
$482$		0			0
$483$	7759.98	+	21948.5i	0.731038	+	2.06769i
$484$		0			0
$485$	−16150.6			−1.51208
$486$		0			0
$487$		−	3348.96i		−	0.311613i	−0.987788	−	0.155807i	$-0.950202\pi$
$487$		−	3348.96i		−	0.311613i	0.987788	−	0.155807i	$-0.0497977\pi$
$488$		0			0
$489$	−1008.00	+	356.382i	−0.0932175	+	0.0329573i
$490$		0			0
$491$			5940.93i			0.546050i	0.962007	+	0.273025i	$0.0880242\pi$
$491$			5940.93i			0.546050i	−0.962007	+	0.273025i	$0.911976\pi$
$492$		0			0
$493$		−	9007.47i		−	0.822873i
$494$		0			0
$495$	6686.32	+	8273.89i	0.607126	+	0.751281i
$496$		0			0
$497$		−	31361.6i		−	2.83050i
$498$		0			0
$499$	−13237.0			−1.18752			−0.593759	−	0.804643i	$-0.702357\pi$
$499$	−13237.0			−1.18752			−0.593759	+	0.804643i	$0.702357\pi$
$500$		0			0
$501$	5459.35	−	1930.17i	0.486838	−	0.172123i
$502$		0			0
$503$	14487.0			1.28418			0.642092	−	0.766628i	$-0.278067\pi$
$503$	14487.0			1.28418			0.642092	+	0.766628i	$0.278067\pi$
$504$		0			0
$505$	15048.0			1.32599
$506$		0			0
$507$	416.413	−	147.224i	0.0364765	−	0.0128964i
$508$		0			0
$509$	−13404.7			−1.16729			−0.583646	−	0.812008i	$-0.698374\pi$
$509$	−13404.7			−1.16729			−0.583646	+	0.812008i	$0.698374\pi$
$510$		0			0
$511$		−	2757.96i		−	0.238758i
$512$		0			0
$513$	−720.000	+	1170.97i	−0.0619664	+	0.100779i
$514$		0			0
$515$			16004.1i			1.36937i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−7721.10	+	2729.82i	−0.653023	+	0.230878i
$520$		0			0
$521$			13146.5i			1.10549i	0.833351	+	0.552745i	$0.186420\pi$
$521$			13146.5i			1.10549i	−0.833351	+	0.552745i	$0.813580\pi$
$522$		0			0
$523$	2322.12			0.194147			0.0970737	−	0.995277i	$-0.469052\pi$
$523$	2322.12			0.194147			0.0970737	+	0.995277i	$0.469052\pi$
$524$		0			0
$525$	6775.45	+	19163.9i	0.563247	+	1.59310i
$526$		0			0
$527$	−15601.4			−1.28958
$528$		0			0
$529$	13177.0			1.08301
$530$		0			0
$531$	−11581.2	−	14331.0i	−0.946480	−	1.17121i
$532$		0			0
$533$	14558.3			1.18309
$534$		0			0
$535$		−	14577.8i		−	1.17804i
$536$		0			0
$537$	3402.00	+	9622.31i	0.273384	+	0.773246i
$538$		0			0
$539$			10887.7i			0.870065i
$540$		0			0
$541$			5468.82i			0.434608i	0.976104	+	0.217304i	$0.0697263\pi$
$541$			5468.82i			0.434608i	−0.976104	+	0.217304i	$0.930274\pi$
$542$		0			0
$543$	−7880.30	−	22288.9i	−0.622792	−	1.76152i
$544$		0			0
$545$		−	5226.93i		−	0.410821i
$546$		0			0
$547$	7750.19			0.605803			0.302901	−	0.953022i	$-0.402045\pi$
$547$	7750.19			0.605803			0.302901	+	0.953022i	$0.402045\pi$
$548$		0			0
$549$	−779.908	−	965.087i	−0.0606296	−	0.0750253i
$550$		0			0
$551$	−1114.39			−0.0861605
$552$		0			0
$553$	−3960.00			−0.304514
$554$		0			0
$555$	9053.31	+	25606.6i	0.692418	+	1.95845i
$556$		0			0
$557$	6028.04			0.458557			0.229278	−	0.973361i	$-0.426363\pi$
$557$	6028.04			0.458557			0.229278	+	0.973361i	$0.426363\pi$
$558$		0			0
$559$		−	22063.7i		−	1.66940i
$560$		0			0
$561$	−9408.00	+	3326.23i	−0.708032	+	0.250327i
$562$		0			0
$563$			15425.6i			1.15473i	0.816486	+	0.577365i	$0.195919\pi$
$563$			15425.6i			1.15473i	−0.816486	+	0.577365i	$0.804081\pi$
$564$		0			0
$565$		−	8456.00i		−	0.629640i
$566$		0			0
$567$	20058.9	+	4305.80i	1.48571	+	0.318918i
$568$		0			0
$569$		−	15092.5i		−	1.11197i	−0.831193	−	0.555984i	$-0.812342\pi$
$569$		−	15092.5i		−	1.11197i	0.831193	−	0.555984i	$-0.187658\pi$
$570$		0			0
$571$	−1577.47			−0.115613			−0.0578066	−	0.998328i	$-0.518411\pi$
$571$	−1577.47			−0.115613			−0.0578066	+	0.998328i	$0.518411\pi$
$572$		0			0
$573$	10918.7	−	3860.35i	0.796048	−	0.281446i
$574$		0			0
$575$	22128.5			1.60491
$576$		0			0
$577$	−20314.0			−1.46565			−0.732827	−	0.680415i	$-0.761800\pi$
$577$	−20314.0			−1.46565			−0.732827	+	0.680415i	$0.761800\pi$
$578$		0			0
$579$	−12825.5	+	4534.51i	−0.920571	+	0.325471i
$580$		0			0
$581$	26029.4			1.85866
$582$		0			0
$583$			8273.89i			0.587770i
$584$		0			0
$585$	12672.0	+	15680.8i	0.895594	+	1.10824i
$586$		0			0
$587$			11206.4i			0.787967i	0.919118	+	0.393983i	$0.128903\pi$
$587$			11206.4i			0.787967i	−0.919118	+	0.393983i	$0.871097\pi$
$588$		0			0
$589$			1930.17i			0.135028i
$590$		0			0
$591$	−10586.7	+	3742.95i	−0.736848	+	0.260515i
$592$		0			0
$593$			11721.0i			0.811676i	0.913945	+	0.405838i	$0.133020\pi$
$593$			11721.0i			0.811676i	−0.913945	+	0.405838i	$0.866980\pi$
$594$		0			0
$595$	−36213.3			−2.49512
$596$		0			0
$597$	−4435.72	−	12546.1i	−0.304091	−	0.860098i
$598$		0			0
$599$	21173.3			1.44427			0.722136	−	0.691751i	$-0.243160\pi$
$599$	21173.3			1.44427			0.722136	+	0.691751i	$0.243160\pi$
$600$		0			0
$601$	−5138.00			−0.348724			−0.174362	−	0.984682i	$-0.555786\pi$
$601$	−5138.00			−0.348724			−0.174362	+	0.984682i	$0.555786\pi$
$602$		0			0
$603$	−7201.50	+	5819.69i	−0.486348	+	0.393028i
$604$		0			0
$605$	−12072.3			−0.811256
$606$		0			0
$607$			17138.8i			1.14603i	0.819544	+	0.573016i	$0.194227\pi$
$607$			17138.8i			1.14603i	−0.819544	+	0.573016i	$0.805773\pi$
$608$		0			0
$609$	5544.00	+	15680.8i	0.368890	+	1.04338i
$610$		0			0
$611$		0			0
$612$		0			0
$613$			24127.2i			1.58970i	0.606805	+	0.794851i	$0.292451\pi$
$613$			24127.2i			1.58970i	−0.606805	+	0.794851i	$0.707549\pi$
$614$		0			0
$615$	−8915.09	−	25215.7i	−0.584538	−	1.65332i
$616$		0			0
$617$		−	1301.08i		−	0.0848936i	−0.999099	−	0.0424468i	$-0.986485\pi$
$617$		−	1301.08i		−	0.0848936i	0.999099	−	0.0424468i	$-0.0135153\pi$
$618$		0			0
$619$	16999.5			1.10382			0.551911	−	0.833903i	$-0.313899\pi$
$619$	16999.5			1.10382			0.551911	+	0.833903i	$0.313899\pi$
$620$		0			0
$621$	11698.6	−	19026.0i	0.755957	−	1.22945i
$622$		0			0
$623$	−11143.9			−0.716644
$624$		0			0
$625$	−13679.0			−0.875456
$626$		0			0
$627$	411.514	+	1163.94i	0.0262110	+	0.0741359i
$628$		0			0
$629$	−25477.0			−1.61500
$630$		0			0
$631$			10384.6i			0.655156i	0.944824	+	0.327578i	$0.106232\pi$
$631$			10384.6i			0.655156i	−0.944824	+	0.327578i	$0.893768\pi$
$632$		0			0
$633$	−19152.0	+	6771.25i	−1.20257	+	0.425171i
$634$		0			0
$635$			3200.83i			0.200033i
$636$		0			0
$637$			20634.5i			1.28347i
$638$		0			0
$639$	−23402.1	+	18911.8i	−1.44878	+	1.17079i
$640$		0			0
$641$		−	8428.71i		−	0.519367i	−0.965694	−	0.259683i	$-0.916382\pi$
$641$		−	8428.71i		−	0.519367i	0.965694	−	0.259683i	$-0.0836182\pi$
$642$		0			0
$643$	−14824.3			−0.909197			−0.454599	−	0.890696i	$-0.650217\pi$
$643$	−14824.3			−0.909197			−0.454599	+	0.890696i	$0.650217\pi$
$644$		0			0
$645$	−38215.5	+	13511.2i	−2.33292	+	0.824812i
$646$		0			0
$647$	10029.5			0.609427			0.304714	−	0.952444i	$-0.401439\pi$
$647$	10029.5			0.609427			0.304714	+	0.952444i	$0.401439\pi$
$648$		0			0
$649$	−16548.0			−1.00087
$650$		0			0
$651$	27159.9	−	9602.49i	1.63515	−	0.578113i
$652$		0			0
$653$	−7165.40			−0.429409			−0.214704	−	0.976679i	$-0.568879\pi$
$653$	−7165.40			−0.429409			−0.214704	+	0.976679i	$0.568879\pi$
$654$		0			0
$655$		−	19756.0i		−	1.17852i
$656$		0			0
$657$	−2058.00	+	1663.12i	−0.122207	+	0.0987585i
$658$		0			0
$659$		−	14331.0i		−	0.847126i	−0.905867	−	0.423563i	$-0.860779\pi$
$659$		−	14331.0i		−	0.847126i	0.905867	−	0.423563i	$-0.139221\pi$
$660$		0			0
$661$		−	5836.48i		−	0.343438i	−0.985146	−	0.171719i	$-0.945068\pi$
$661$		−	5836.48i		−	0.343438i	0.985146	−	0.171719i	$-0.0549321\pi$
$662$		0			0
$663$	−17830.2	+	6303.92i	−1.04444	+	0.369267i
$664$		0			0
$665$			4480.23i			0.261257i
$666$		0			0
$667$	18106.6			1.05111
$668$		0			0
$669$	−9895.08	−	27987.5i	−0.571847	−	1.61743i
$670$		0			0
$671$	−1114.39			−0.0641138
$672$		0			0
$673$	18562.0			1.06317			0.531584	−	0.847005i	$-0.321597\pi$
$673$	18562.0			1.06317			0.531584	+	0.847005i	$0.321597\pi$
$674$		0			0
$675$	10214.4	−	16612.1i	0.582446	−	0.947259i
$676$		0			0
$677$	16621.8			0.943614			0.471807	−	0.881702i	$-0.343602\pi$
$677$	16621.8			0.943614			0.471807	+	0.881702i	$0.343602\pi$
$678$		0			0
$679$			27973.6i			1.58105i
$680$		0			0
$681$	2094.00	+	5922.73i	0.117830	+	0.333274i
$682$		0			0
$683$			15882.9i			0.889813i	0.895577	+	0.444907i	$0.146763\pi$
$683$			15882.9i			0.889813i	−0.895577	+	0.444907i	$0.853237\pi$
$684$		0			0
$685$		−	12867.8i		−	0.717743i
$686$		0			0
$687$	8994.69	+	25440.8i	0.499518	+	1.41285i
$688$		0			0
$689$			15680.8i			0.867040i
$690$		0			0
$691$	−7965.74			−0.438540			−0.219270	−	0.975664i	$-0.570368\pi$
$691$	−7965.74			−0.438540			−0.219270	+	0.975664i	$0.570368\pi$
$692$		0			0
$693$	14330.8	−	11581.0i	0.785544	−	0.634816i
$694$		0			0
$695$	−24675.7			−1.34677
$696$		0			0
$697$	25088.0			1.36338
$698$		0			0
$699$	2331.91	+	6595.65i	0.126182	+	0.356896i
$700$		0			0
$701$	8757.71			0.471861			0.235930	−	0.971770i	$-0.424186\pi$
$701$	8757.71			0.471861			0.235930	+	0.971770i	$0.424186\pi$
$702$		0			0
$703$			3151.96i			0.169102i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	26063.9i		−	1.38647i
$708$		0			0
$709$			15119.7i			0.800891i	0.916321	+	0.400446i	$0.131145\pi$
$709$			15119.7i			0.800891i	−0.916321	+	0.400446i	$0.868855\pi$
$710$		0			0
$711$	2387.97	+	2954.96i	0.125958	+	0.155865i
$712$		0			0
$713$		−	31361.6i		−	1.64727i
$714$		0			0
$715$	18106.6			0.947062
$716$		0			0
$717$	6239.26	−	2205.91i	0.324978	−	0.114897i
$718$		0			0
$719$	33431.6			1.73406			0.867029	−	0.498258i	$-0.166027\pi$
$719$	33431.6			1.73406			0.867029	+	0.498258i	$0.166027\pi$
$720$		0			0
$721$	27720.0			1.43183
$722$		0			0
$723$	−27640.0	+	9772.23i	−1.42178	+	0.502674i
$724$		0			0
$725$	15809.4			0.809856
$726$		0			0
$727$		−	2560.97i		−	0.130648i	−0.997864	−	0.0653239i	$-0.979192\pi$
$727$		−	2560.97i		−	0.130648i	0.997864	−	0.0653239i	$-0.0208081\pi$
$728$		0			0
$729$	−8883.00	−	17564.5i	−0.451303	−	0.892371i
$730$		0			0
$731$		−	38022.0i		−	1.92379i
$732$		0			0
$733$			689.348i			0.0347362i	0.999849	+	0.0173681i	$0.00552872\pi$
$733$			689.348i			0.0347362i	−0.999849	+	0.0173681i	$0.994471\pi$
$734$		0			0
$735$	35739.9	−	12636.0i	1.79359	−	0.634129i
$736$		0			0
$737$			8315.58i			0.415615i
$738$		0			0
$739$	18038.0			0.897889			0.448945	−	0.893560i	$-0.351800\pi$
$739$	18038.0			0.897889			0.448945	+	0.893560i	$0.351800\pi$
$740$		0			0
$741$	779.908	+	2205.91i	0.0386648	+	0.109361i
$742$		0			0
$743$	−10984.7			−0.542380			−0.271190	−	0.962526i	$-0.587417\pi$
$743$	−10984.7			−0.542380			−0.271190	+	0.962526i	$0.587417\pi$
$744$		0			0
$745$	−46200.0			−2.27200
$746$		0			0
$747$	−15696.3	−	19423.2i	−0.768807	−	0.951350i
$748$		0			0
$749$	−25249.5			−1.23177
$750$		0			0
$751$		−	7288.91i		−	0.354162i	−0.984196	−	0.177081i	$-0.943334\pi$
$751$		−	7288.91i		−	0.354162i	0.984196	−	0.177081i	$-0.0566655\pi$
$752$		0			0
$753$	−7386.00	−	20890.8i	−0.357451	−	1.01102i
$754$		0			0
$755$		−	9602.49i		−	0.462875i
$756$		0			0
$757$			24127.2i			1.15841i	0.815182	+	0.579205i	$0.196637\pi$
$757$			24127.2i			1.15841i	−0.815182	+	0.579205i	$0.803363\pi$
$758$		0			0
$759$	−6686.32	−	18911.8i	−0.319760	−	0.904418i
$760$		0			0
$761$			20274.2i			0.965753i	0.875689	+	0.482876i	$0.160408\pi$
$761$			20274.2i			0.965753i	−0.875689	+	0.482876i	$0.839592\pi$
$762$		0			0
$763$	−9053.31			−0.429557
$764$		0			0
$765$	21837.4	+	27022.4i	1.03207	+	1.27712i
$766$		0			0
$767$	−31362.0			−1.47642
$768$		0			0
$769$	26950.0			1.26377			0.631887	−	0.775061i	$-0.282281\pi$
$769$	26950.0			1.26377			0.631887	+	0.775061i	$0.282281\pi$
$770$		0			0
$771$	−3566.46	−	10087.5i	−0.166592	−	0.471195i
$772$		0			0
$773$	2713.43			0.126255			0.0631276	−	0.998005i	$-0.479892\pi$
$773$	2713.43			0.126255			0.0631276	+	0.998005i	$0.479892\pi$
$774$		0			0
$775$		−	27382.6i		−	1.26918i
$776$		0			0
$777$	44352.0	−	15680.8i	2.04777	−	0.723997i
$778$		0			0
$779$		−	3103.84i		−	0.142755i
$780$		0			0
$781$			27022.4i			1.23808i
$782$		0			0
$783$	8357.89	−	13592.8i	0.381465	−	0.620393i
$784$		0			0
$785$		−	32108.3i		−	1.45987i
$786$		0			0
$787$	−2322.12			−0.105177			−0.0525886	−	0.998616i	$-0.516747\pi$
$787$	−2322.12			−0.105177			−0.0525886	+	0.998616i	$0.516747\pi$
$788$		0			0
$789$	16378.1	−	5790.52i	0.739004	−	0.261277i
$790$		0			0
$791$	−14646.2			−0.658356
$792$		0			0
$793$	−2112.00			−0.0945767
$794$		0			0
$795$	27159.9	−	9602.49i	1.21165	−	0.428384i
$796$		0			0
$797$	14997.0			0.666525			0.333262	−	0.942834i	$-0.391851\pi$
$797$	14997.0			0.666525			0.333262	+	0.942834i	$0.391851\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	6720.00	+	8315.58i	0.296429	+	0.366812i
$802$		0			0
$803$			2376.37i			0.104434i
$804$		0			0
$805$		−	72795.1i		−	3.18719i
$806$		0			0
$807$	25710.5	−	9090.03i	1.12150	−	0.396511i
$808$		0			0
$809$		−	19369.1i		−	0.841756i	−0.907117	−	0.420878i	$-0.861722\pi$
$809$		−	19369.1i		−	0.841756i	0.907117	−	0.420878i	$-0.138278\pi$
$810$		0			0
$811$	31559.2			1.36645			0.683227	−	0.730206i	$-0.260576\pi$
$811$	31559.2			1.36645			0.683227	+	0.730206i	$0.260576\pi$
$812$		0			0
$813$	3753.31	+	10616.0i	0.161912	+	0.457955i
$814$		0			0
$815$	3343.16			0.143688
$816$		0			0
$817$	−4704.00			−0.201435
$818$		0			0
$819$	27159.9	−	21948.5i	1.15879	−	0.936440i
$820$		0			0
$821$	−24453.4			−1.03950			−0.519749	−	0.854319i	$-0.673975\pi$
$821$	−24453.4			−1.03950			−0.519749	+	0.854319i	$0.673975\pi$
$822$		0			0
$823$		−	18658.5i		−	0.790272i	−0.918623	−	0.395136i	$-0.870698\pi$
$823$		−	18658.5i		−	0.790272i	0.918623	−	0.395136i	$-0.129302\pi$
$824$		0			0
$825$	−5838.00	−	16512.4i	−0.246367	−	0.696832i
$826$		0			0
$827$		−	29171.2i		−	1.22658i	−0.789858	−	0.613290i	$-0.789846\pi$
$827$		−	29171.2i		−	1.22658i	0.789858	−	0.613290i	$-0.210154\pi$
$828$		0			0
$829$		−	6479.87i		−	0.271478i	−0.990745	−	0.135739i	$-0.956659\pi$
$829$		−	6479.87i		−	0.271478i	0.990745	−	0.135739i	$-0.0433408\pi$
$830$		0			0
$831$	1671.58	+	4727.94i	0.0697791	+	0.197365i
$832$		0			0
$833$			35559.0i			1.47905i
$834$		0			0
$835$	−18106.6			−0.750426
$836$		0			0
$837$	−23543.5	−	14476.3i	−0.972259	−	0.597818i
$838$		0			0
$839$	16715.8			0.687834			0.343917	−	0.939000i	$-0.388246\pi$
$839$	16715.8			0.687834			0.343917	+	0.939000i	$0.388246\pi$
$840$		0			0
$841$	−11453.0			−0.469597
$842$		0			0
$843$	−1783.23	−	5043.73i	−0.0728561	−	0.206068i
$844$		0			0
$845$	−1381.09			−0.0562258
$846$		0			0
$847$			20909.9i			0.848255i
$848$		0			0
$849$	−23472.0	+	8298.61i	−0.948830	+	0.335462i
$850$		0			0
$851$		−	51213.3i		−	2.06295i
$852$		0			0
$853$		−	5101.17i		−	0.204761i	−0.994745	−	0.102380i	$-0.967354\pi$
$853$		−	5101.17i		−	0.204761i	0.994745	−	0.102380i	$-0.0326459\pi$
$854$		0			0
$855$	3343.16	−	2701.68i	0.133723	−	0.108065i
$856$		0			0
$857$			13146.5i			0.524010i	0.965066	+	0.262005i	$0.0843838\pi$
$857$			13146.5i			0.524010i	−0.965066	+	0.262005i	$0.915616\pi$
$858$		0			0
$859$	19057.0			0.756947			0.378474	−	0.925612i	$-0.376449\pi$
$859$	19057.0			0.756947			0.378474	+	0.925612i	$0.376449\pi$
$860$		0			0
$861$	−43674.8	+	15441.4i	−1.72873	+	0.611197i
$862$		0			0
$863$	−2228.77			−0.0879123			−0.0439561	−	0.999033i	$-0.513996\pi$
$863$	−2228.77			−0.0879123			−0.0439561	+	0.999033i	$0.513996\pi$
$864$		0			0
$865$	25608.0			1.00659
$866$		0			0
$867$	−6657.71	+	2353.86i	−0.260793	+	0.0922044i
$868$		0			0
$869$	3412.10			0.133196
$870$		0			0
$871$			15759.8i			0.613089i
$872$		0			0
$873$	20874.0	−	16868.7i	0.809253	−	0.653975i
$874$		0			0
$875$		−	6401.66i		−	0.247332i
$876$		0			0
$877$		−	44715.7i		−	1.72171i	−0.508849	−	0.860856i	$-0.669929\pi$
$877$		−	44715.7i		−	1.72171i	0.508849	−	0.860856i	$-0.330071\pi$
$878$		0			0
$879$	−23322.5	+	8245.75i	−0.894936	+	0.316408i
$880$		0			0
$881$			16947.9i			0.648116i	0.946037	+	0.324058i	$0.105047\pi$
$881$			16947.9i			0.648116i	−0.946037	+	0.324058i	$0.894953\pi$
$882$		0			0
$883$	32989.7			1.25730			0.628648	−	0.777690i	$-0.283609\pi$
$883$	32989.7			1.25730			0.628648	+	0.777690i	$0.283609\pi$
$884$		0			0
$885$	19205.2	+	54320.6i	0.729465	+	2.06324i
$886$		0			0
$887$	−25630.9			−0.970237			−0.485119	−	0.874448i	$-0.661224\pi$
$887$	−25630.9			−0.970237			−0.485119	+	0.874448i	$0.661224\pi$
$888$		0			0
$889$	5544.00			0.209156
$890$		0			0
$891$	−17283.6	−	3710.05i	−0.649857	−	0.139497i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	31913.6i		−	1.19190i
$896$		0			0
$897$	−12672.0	−	35841.8i	−0.471690	−	1.33414i
$898$		0			0
$899$		−	22405.8i		−	0.831230i
$900$		0			0
$901$			27022.4i			0.999165i
$902$		0			0
$903$	23402.1	+	66191.1i	0.862429	+	2.43932i
$904$		0			0
$905$			73923.8i			2.71526i
$906$		0			0
$907$	−36281.8			−1.32825			−0.664123	−	0.747623i	$-0.731195\pi$
$907$	−36281.8			−1.32825			−0.664123	+	0.747623i	$0.731195\pi$
$908$		0			0
$909$	−19448.9	+	15717.1i	−0.709660	+	0.573492i
$910$		0			0
$911$	−5731.13			−0.208431			−0.104216	−	0.994555i	$-0.533233\pi$
$911$	−5731.13			−0.208431			−0.104216	+	0.994555i	$0.533233\pi$
$912$		0			0
$913$	−22428.0			−0.812988
$914$		0			0
$915$	1293.33	+	3658.09i	0.0467281	+	0.132167i
$916$		0			0
$917$	−34218.4			−1.23227
$918$		0			0
$919$			30872.3i			1.10814i	0.832469	+	0.554072i	$0.186927\pi$
$919$			30872.3i			1.10814i	−0.832469	+	0.554072i	$0.813073\pi$
$920$		0			0
$921$	8688.00	−	3071.67i	0.310835	−	0.109897i
$922$		0			0
$923$			51213.3i			1.82633i
$924$		0			0
$925$		−	44715.7i		−	1.58945i
$926$		0			0
$927$	−16715.8	−	20684.7i	−0.592253	−	0.732876i
$928$		0			0
$929$			8949.14i			0.316052i	0.987435	+	0.158026i	$0.0505129\pi$
$929$			8949.14i			0.316052i	−0.987435	+	0.158026i	$0.949487\pi$
$930$		0			0
$931$	4399.28			0.154866
$932$		0			0
$933$	27296.8	−	9650.87i	0.957831	−	0.338644i
$934$		0			0
$935$	31202.8			1.09138
$936$		0			0
$937$	−43862.0			−1.52925			−0.764626	−	0.644474i	$-0.777076\pi$
$937$	−43862.0			−1.52925			−0.764626	+	0.644474i	$0.777076\pi$
$938$		0			0
$939$	16529.2	−	5843.94i	0.574450	−	0.203099i
$940$		0			0
$941$	24550.8			0.850515			0.425258	−	0.905072i	$-0.360184\pi$
$941$	24550.8			0.850515			0.425258	+	0.905072i	$0.360184\pi$
$942$		0			0
$943$			50431.4i			1.74154i
$944$		0			0
$945$	−54648.0	−	33601.7i	−1.88116	−	1.15668i
$946$		0			0
$947$			39986.1i			1.37210i	0.727557	+	0.686048i	$0.240656\pi$
$947$			39986.1i			1.37210i	−0.727557	+	0.686048i	$0.759344\pi$
$948$		0			0
$949$			4503.74i			0.154054i
$950$		0			0
$951$	−23959.3	+	8470.89i	−0.816965	+	0.288841i
$952$		0			0
$953$			46408.8i			1.57747i	0.614733	+	0.788735i	$0.289264\pi$
$953$			46408.8i			1.57747i	−0.614733	+	0.788735i	$0.710736\pi$
$954$		0			0
$955$	−36213.3			−1.22705
$956$		0			0
$957$	−4776.93	−	13511.2i	−0.161355	−	0.456380i
$958$		0			0
$959$	−22287.7			−0.750478
$960$		0			0
$961$	−9017.00			−0.302675
$962$		0			0
$963$	15226.0	+	18841.2i	0.509504	+	0.630478i
$964$		0			0
$965$	42537.5			1.41899
$966$		0			0
$967$		−	8076.90i		−	0.268599i	−0.990941	−	0.134300i	$-0.957122\pi$
$967$		−	8076.90i		−	0.268599i	0.990941	−	0.134300i	$-0.0428785\pi$
$968$		0			0
$969$	1344.00	+	3801.41i	0.0445568	+	0.126026i
$970$		0			0
$971$		−	41371.8i		−	1.36734i	−0.729793	−	0.683668i	$-0.760384\pi$
$971$		−	41371.8i		−	1.36734i	0.729793	−	0.683668i	$-0.239616\pi$
$972$		0			0
$973$			42739.5i			1.40819i
$974$		0			0
$975$	−11064.3	−	31294.5i	−0.363426	−	1.02792i
$976$		0			0
$977$			21778.9i			0.713171i	0.934263	+	0.356586i	$0.116059\pi$
$977$			21778.9i			0.713171i	−0.934263	+	0.356586i	$0.883941\pi$
$978$		0			0
$979$	9602.00			0.313464
$980$		0			0
$981$	5459.35	+	6755.61i	0.177680	+	0.219867i
$982$		0			0
$983$	41232.3			1.33785			0.668924	−	0.743330i	$-0.266755\pi$
$983$	41232.3			1.33785			0.668924	+	0.743330i	$0.266755\pi$
$984$		0			0
$985$	35112.0			1.13580
$986$		0			0
$987$		0			0
$988$		0			0
$989$	76431.0			2.45739
$990$		0			0
$991$			15225.1i			0.488033i	0.969771	+	0.244017i	$0.0784651\pi$
$991$			15225.1i			0.488033i	−0.969771	+	0.244017i	$0.921535\pi$
$992$		0			0
$993$	3024.00	−	1069.15i	0.0966402	−	0.0341675i
$994$		0			0
$995$			41610.8i			1.32578i
$996$		0			0
$997$			19899.2i			0.632109i	0.948741	+	0.316055i	$0.102358\pi$
$997$			19899.2i			0.632109i	−0.948741	+	0.316055i	$0.897642\pi$
$998$		0			0
$999$	−38446.3	−	23639.7i	−1.21761	−	0.748676i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.f.i.191.5	yes	8
3.2	odd	2	inner	384.4.f.i.191.8	yes	8
4.3	odd	2	inner	384.4.f.i.191.3	yes	8
8.3	odd	2	inner	384.4.f.i.191.6	yes	8
8.5	even	2	inner	384.4.f.i.191.4	yes	8
12.11	even	2	inner	384.4.f.i.191.2	yes	8
16.3	odd	4		768.4.c.s.767.2		8
16.5	even	4		768.4.c.s.767.1		8
16.11	odd	4		768.4.c.s.767.7		8
16.13	even	4		768.4.c.s.767.8		8
24.5	odd	2	inner	384.4.f.i.191.1	✓	8
24.11	even	2	inner	384.4.f.i.191.7	yes	8
48.5	odd	4		768.4.c.s.767.6		8
48.11	even	4		768.4.c.s.767.4		8
48.29	odd	4		768.4.c.s.767.3		8
48.35	even	4		768.4.c.s.767.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.f.i.191.1	✓	8	24.5	odd	2	inner
384.4.f.i.191.2	yes	8	12.11	even	2	inner
384.4.f.i.191.3	yes	8	4.3	odd	2	inner
384.4.f.i.191.4	yes	8	8.5	even	2	inner
384.4.f.i.191.5	yes	8	1.1	even	1	trivial
384.4.f.i.191.6	yes	8	8.3	odd	2	inner
384.4.f.i.191.7	yes	8	24.11	even	2	inner
384.4.f.i.191.8	yes	8	3.2	odd	2	inner
768.4.c.s.767.1		8	16.5	even	4
768.4.c.s.767.2		8	16.3	odd	4
768.4.c.s.767.3		8	48.29	odd	4
768.4.c.s.767.4		8	48.11	even	4
768.4.c.s.767.5		8	48.35	even	4
768.4.c.s.767.6		8	48.5	odd	4
768.4.c.s.767.7		8	16.11	odd	4
768.4.c.s.767.8		8	16.13	even	4

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(4.89898 - 1.73205i) q^{3} -16.2481 q^{5} +28.1425i q^{7} +(21.0000 - 16.9706i) q^{9} +O(q^{10})\)	\(q+(4.89898 - 1.73205i) q^{3} -16.2481 q^{5} +28.1425i q^{7} +(21.0000 - 16.9706i) q^{9} -24.2487i q^{11} -45.9565i q^{13} +(-79.5990 + 28.1425i) q^{15} -79.1960i q^{17} -9.79796 q^{19} +(48.7442 + 137.870i) q^{21} +159.198 q^{23} +139.000 q^{25} +(73.4847 - 119.512i) q^{27} +113.737 q^{29} -196.997i q^{31} +(-42.0000 - 118.794i) q^{33} -457.261i q^{35} -321.696i q^{37} +(-79.5990 - 225.140i) q^{39} +316.784i q^{41} +480.100 q^{43} +(-341.210 + 275.739i) q^{45} -449.000 q^{49} +(-137.171 - 387.979i) q^{51} -341.210 q^{53} +393.995i q^{55} +(-48.0000 + 16.9706i) q^{57} -682.428i q^{59} -45.9565i q^{61} +(477.594 + 590.992i) q^{63} +746.705i q^{65} -342.929 q^{67} +(779.908 - 275.739i) q^{69} -1114.39 q^{71} -98.0000 q^{73} +(680.958 - 240.755i) q^{75} +682.419 q^{77} +140.712i q^{79} +(153.000 - 712.764i) q^{81} -924.915i q^{83} +1286.78i q^{85} +(557.193 - 196.997i) q^{87} +395.980i q^{89} +1293.33 q^{91} +(-341.210 - 965.087i) q^{93} +159.198 q^{95} +994.000 q^{97} +(-411.514 - 509.223i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 168 q^{9}+O(q^{10}) \) 8 * q + 168 * q^9	\( 8 q + 168 q^{9} + 1112 q^{25} - 336 q^{33} - 3592 q^{49} - 384 q^{57} - 784 q^{73} + 1224 q^{81} + 7952 q^{97}+O(q^{100}) \) 8 * q + 168 * q^9 + 1112 * q^25 - 336 * q^33 - 3592 * q^49 - 384 * q^57 - 784 * q^73 + 1224 * q^81 + 7952 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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