LMFDB - Embedded newform 384.6.d.i.193.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,6,Mod(193,384)]

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, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	384.d (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:	$61.5873868082$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.110166016.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 $ x^8 + 10x^6 + 19x^4 + 10*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.7
Root			$-2.77462i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.6.d.i.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} +42.1845i q^{5} -139.463 q^{7} -81.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} +42.1845i q^{5} -139.463 q^{7} -81.0000 q^{9} -42.1564i q^{11} -424.060i q^{13} -379.661 q^{15} -100.940 q^{17} -879.411i q^{19} -1255.17i q^{21} -2752.09 q^{23} +1345.47 q^{25} -729.000i q^{27} -5489.12i q^{29} +5488.74 q^{31} +379.408 q^{33} -5883.17i q^{35} +10308.5i q^{37} +3816.54 q^{39} +9170.82 q^{41} -6667.63i q^{43} -3416.94i q^{45} +2547.92 q^{47} +2642.87 q^{49} -908.462i q^{51} +3499.70i q^{53} +1778.35 q^{55} +7914.70 q^{57} +2438.50i q^{59} +30827.7i q^{61} +11296.5 q^{63} +17888.8 q^{65} -9600.25i q^{67} -24768.8i q^{69} -76573.8 q^{71} +78348.0 q^{73} +12109.2i q^{75} +5879.25i q^{77} -29359.4 q^{79} +6561.00 q^{81} +18384.9i q^{83} -4258.11i q^{85} +49402.1 q^{87} +117560. q^{89} +59140.6i q^{91} +49398.7i q^{93} +37097.5 q^{95} +92221.1 q^{97} +3414.67i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 648 q^{9}+O(q^{10}) $ 8 * q - 648 * q^9	$ 8 q - 648 q^{9} + 4080 q^{17} - 5528 q^{25} - 18144 q^{33} + 58704 q^{41} - 44024 q^{49} + 38880 q^{57} - 203904 q^{65} + 248816 q^{73} + 52488 q^{81} - 46800 q^{89} - 262544 q^{97}+O(q^{100}) $ 8 * q - 648 * q^9 + 4080 * q^17 - 5528 * q^25 - 18144 * q^33 + 58704 * q^41 - 44024 * q^49 + 38880 * q^57 - 203904 * q^65 + 248816 * q^73 + 52488 * q^81 - 46800 * q^89 - 262544 * 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
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	0	0
$5$	42.1845i	0.754619i	0.926087	+	0.377310i	$0.123151\pi$
$5$	42.1845i	0.754619i	−0.926087	+	0.377310i	$0.876849\pi$
$6$	0	0
$7$	−139.463	−1.07575	−0.537877	−	0.843023i	$-0.680774\pi$
$7$	−139.463	−1.07575	−0.537877	+	0.843023i	$0.680774\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	− 42.1564i	− 0.105047i	−0.998620	−	0.0525233i	$-0.983274\pi$
$11$	− 42.1564i	− 0.105047i	0.998620	−	0.0525233i	$-0.0167264\pi$
$12$	0	0
$13$	− 424.060i	− 0.695935i	−0.937507	−	0.347968i	$-0.886872\pi$
$13$	− 424.060i	− 0.695935i	0.937507	−	0.347968i	$-0.113128\pi$
$14$	0	0
$15$	−379.661	−0.435680
$16$	0	0
$17$	−100.940	−0.0847114	−0.0423557	−	0.999103i	$-0.513486\pi$
$17$	−100.940	−0.0847114	−0.0423557	+	0.999103i	$0.513486\pi$
$18$	0	0
$19$	− 879.411i	− 0.558866i	−0.960165	−	0.279433i	$-0.909853\pi$
$19$	− 879.411i	− 0.558866i	0.960165	−	0.279433i	$-0.0901466\pi$
$20$	0	0
$21$	− 1255.17i	− 0.621087i
$22$	0	0
$23$	−2752.09	−1.08479	−0.542393	−	0.840125i	$-0.682481\pi$
$23$	−2752.09	−1.08479	−0.542393	+	0.840125i	$0.682481\pi$
$24$	0	0
$25$	1345.47	0.430550
$26$	0	0
$27$	− 729.000i	− 0.192450i
$28$	0	0
$29$	− 5489.12i	− 1.21201i	−0.795459	−	0.606007i	$-0.792770\pi$
$29$	− 5489.12i	− 1.21201i	0.795459	−	0.606007i	$-0.207230\pi$
$30$	0	0
$31$	5488.74	1.02581	0.512907	−	0.858444i	$-0.328569\pi$
$31$	5488.74	1.02581	0.512907	+	0.858444i	$0.328569\pi$
$32$	0	0
$33$	379.408	0.0606487
$34$	0	0
$35$	− 5883.17i	− 0.811785i
$36$	0	0
$37$	10308.5i	1.23792i	0.785423	+	0.618959i	$0.212445\pi$
$37$	10308.5i	1.23792i	−0.785423	+	0.618959i	$0.787555\pi$
$38$	0	0
$39$	3816.54	0.401798
$40$	0	0
$41$	9170.82	0.852018	0.426009	−	0.904719i	$-0.359919\pi$
$41$	9170.82	0.852018	0.426009	+	0.904719i	$0.359919\pi$
$42$	0	0
$43$	− 6667.63i	− 0.549921i	−0.961456	−	0.274961i	$-0.911335\pi$
$43$	− 6667.63i	− 0.549921i	0.961456	−	0.274961i	$-0.0886648\pi$
$44$	0	0
$45$	− 3416.94i	− 0.251540i
$46$	0	0
$47$	2547.92	0.168244	0.0841222	−	0.996455i	$-0.473191\pi$
$47$	2547.92	0.168244	0.0841222	+	0.996455i	$0.473191\pi$
$48$	0	0
$49$	2642.87	0.157248
$50$	0	0
$51$	− 908.462i	− 0.0489082i
$52$	0	0
$53$	3499.70i	0.171136i	0.996332	+	0.0855680i	$0.0272705\pi$
$53$	3499.70i	0.171136i	−0.996332	+	0.0855680i	$0.972730\pi$
$54$	0	0
$55$	1778.35	0.0792702
$56$	0	0
$57$	7914.70	0.322662
$58$	0	0
$59$	2438.50i	0.0911997i	0.998960	+	0.0455998i	$0.0145199\pi$
$59$	2438.50i	0.0911997i	−0.998960	+	0.0455998i	$0.985480\pi$
$60$	0	0
$61$	30827.7i	1.06076i	0.847760	+	0.530379i	$0.177951\pi$
$61$	30827.7i	1.06076i	−0.847760	+	0.530379i	$0.822049\pi$
$62$	0	0
$63$	11296.5	0.358585
$64$	0	0
$65$	17888.8	0.525166
$66$	0	0
$67$	− 9600.25i	− 0.261274i	−0.991430	−	0.130637i	$-0.958298\pi$
$67$	− 9600.25i	− 0.261274i	0.991430	−	0.130637i	$-0.0417022\pi$
$68$	0	0
$69$	− 24768.8i	− 0.626301i
$70$	0	0
$71$	−76573.8	−1.80275	−0.901373	−	0.433044i	$-0.857440\pi$
$71$	−76573.8	−1.80275	−0.901373	+	0.433044i	$0.857440\pi$
$72$	0	0
$73$	78348.0	1.72076	0.860381	−	0.509651i	$-0.170225\pi$
$73$	78348.0	1.72076	0.860381	+	0.509651i	$0.170225\pi$
$74$	0	0
$75$	12109.2i	0.248578i
$76$	0	0
$77$	5879.25i	0.113004i
$78$	0	0
$79$	−29359.4	−0.529273	−0.264637	−	0.964348i	$-0.585252\pi$
$79$	−29359.4	−0.529273	−0.264637	+	0.964348i	$0.585252\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	18384.9i	0.292931i	0.989216	+	0.146466i	$0.0467898\pi$
$83$	18384.9i	0.292931i	−0.989216	+	0.146466i	$0.953210\pi$
$84$	0	0
$85$	− 4258.11i	− 0.0639249i
$86$	0	0
$87$	49402.1	0.699757
$88$	0	0
$89$	117560.	1.57320	0.786601	−	0.617462i	$-0.211839\pi$
$89$	117560.	1.57320	0.786601	+	0.617462i	$0.211839\pi$
$90$	0	0
$91$	59140.6i	0.748656i
$92$	0	0
$93$	49398.7i	0.592254i
$94$	0	0
$95$	37097.5	0.421731
$96$	0	0
$97$	92221.1	0.995179	0.497589	−	0.867413i	$-0.334219\pi$
$97$	92221.1	0.995179	0.497589	+	0.867413i	$0.334219\pi$
$98$	0	0
$99$	3414.67i	0.0350155i
$100$	0	0
$101$	− 89728.6i	− 0.875241i	−0.899160	−	0.437621i	$-0.855821\pi$
$101$	− 89728.6i	− 0.875241i	0.899160	−	0.437621i	$-0.144179\pi$
$102$	0	0
$103$	150476.	1.39757	0.698787	−	0.715330i	$-0.253724\pi$
$103$	150476.	1.39757	0.698787	+	0.715330i	$0.253724\pi$
$104$	0	0
$105$	52948.5	0.468684
$106$	0	0
$107$	122672.i	1.03582i	0.855435	+	0.517910i	$0.173290\pi$
$107$	122672.i	1.03582i	−0.855435	+	0.517910i	$0.826710\pi$
$108$	0	0
$109$	− 238672.i	− 1.92413i	−0.272813	−	0.962067i	$-0.587954\pi$
$109$	− 238672.i	− 1.92413i	0.272813	−	0.962067i	$-0.412046\pi$
$110$	0	0
$111$	−92776.7	−0.714713
$112$	0	0
$113$	140424.	1.03454	0.517268	−	0.855823i	$-0.326949\pi$
$113$	140424.	1.03454	0.517268	+	0.855823i	$0.326949\pi$
$114$	0	0
$115$	− 116096.i	− 0.818600i
$116$	0	0
$117$	34348.9i	0.231978i
$118$	0	0
$119$	14077.4	0.0911287
$120$	0	0
$121$	159274.	0.988965
$122$	0	0
$123$	82537.4i	0.491913i
$124$	0	0
$125$	188584.i	1.07952i
$126$	0	0
$127$	−21900.2	−0.120487	−0.0602434	−	0.998184i	$-0.519188\pi$
$127$	−21900.2	−0.120487	−0.0602434	+	0.998184i	$0.519188\pi$
$128$	0	0
$129$	60008.7	0.317497
$130$	0	0
$131$	− 134987.i	− 0.687249i	−0.939107	−	0.343624i	$-0.888345\pi$
$131$	− 134987.i	− 0.687249i	0.939107	−	0.343624i	$-0.111655\pi$
$132$	0	0
$133$	122645.i	0.601203i
$134$	0	0
$135$	30752.5	0.145227
$136$	0	0
$137$	208753.	0.950233	0.475117	−	0.879923i	$-0.342406\pi$
$137$	208753.	0.950233	0.475117	+	0.879923i	$0.342406\pi$
$138$	0	0
$139$	− 39611.3i	− 0.173893i	−0.996213	−	0.0869466i	$-0.972289\pi$
$139$	− 39611.3i	− 0.173893i	0.996213	−	0.0869466i	$-0.0277110\pi$
$140$	0	0
$141$	22931.3i	0.0971360i
$142$	0	0
$143$	−17876.8	−0.0731056
$144$	0	0
$145$	231556.	0.914609
$146$	0	0
$147$	23785.8i	0.0907873i
$148$	0	0
$149$	214820.i	0.792700i	0.918100	+	0.396350i	$0.129723\pi$
$149$	214820.i	0.792700i	−0.918100	+	0.396350i	$0.870277\pi$
$150$	0	0
$151$	−299050.	−1.06734	−0.533668	−	0.845694i	$-0.679187\pi$
$151$	−299050.	−1.06734	−0.533668	+	0.845694i	$0.679187\pi$
$152$	0	0
$153$	8176.16	0.0282371
$154$	0	0
$155$	231540.i	0.774099i
$156$	0	0
$157$	− 108444.i	− 0.351121i	−0.984469	−	0.175560i	$-0.943826\pi$
$157$	− 108444.i	− 0.351121i	0.984469	−	0.175560i	$-0.0561737\pi$
$158$	0	0
$159$	−31497.3	−0.0988054
$160$	0	0
$161$	383815.	1.16696
$162$	0	0
$163$	− 616203.i	− 1.81658i	−0.418340	−	0.908291i	$-0.637388\pi$
$163$	− 616203.i	− 1.81658i	0.418340	−	0.908291i	$-0.362612\pi$
$164$	0	0
$165$	16005.1i	0.0457667i
$166$	0	0
$167$	124713.	0.346037	0.173018	−	0.984919i	$-0.444648\pi$
$167$	124713.	0.346037	0.173018	+	0.984919i	$0.444648\pi$
$168$	0	0
$169$	191466.	0.515674
$170$	0	0
$171$	71232.3i	0.186289i
$172$	0	0
$173$	− 168525.i	− 0.428103i	−0.976822	−	0.214052i	$-0.931334\pi$
$173$	− 168525.i	− 0.428103i	0.976822	−	0.214052i	$-0.0686661\pi$
$174$	0	0
$175$	−187643.	−0.463166
$176$	0	0
$177$	−21946.5	−0.0526541
$178$	0	0
$179$	− 349648.i	− 0.815639i	−0.913063	−	0.407819i	$-0.866289\pi$
$179$	− 349648.i	− 0.815639i	0.913063	−	0.407819i	$-0.133711\pi$
$180$	0	0
$181$	219153.i	0.497223i	0.968603	+	0.248612i	$0.0799743\pi$
$181$	219153.i	0.497223i	−0.968603	+	0.248612i	$0.920026\pi$
$182$	0	0
$183$	−277449.	−0.612429
$184$	0	0
$185$	−434860.	−0.934157
$186$	0	0
$187$	4255.28i	0.00889865i
$188$	0	0
$189$	101668.i	0.207029i
$190$	0	0
$191$	769923.	1.52709	0.763543	−	0.645757i	$-0.223458\pi$
$191$	769923.	1.52709	0.763543	+	0.645757i	$0.223458\pi$
$192$	0	0
$193$	99052.3	0.191413	0.0957065	−	0.995410i	$-0.469489\pi$
$193$	99052.3	0.191413	0.0957065	+	0.995410i	$0.469489\pi$
$194$	0	0
$195$	160999.i	0.303205i
$196$	0	0
$197$	− 671217.i	− 1.23225i	−0.787650	−	0.616123i	$-0.788703\pi$
$197$	− 671217.i	− 1.23225i	0.787650	−	0.616123i	$-0.211297\pi$
$198$	0	0
$199$	83208.3	0.148948	0.0744739	−	0.997223i	$-0.476272\pi$
$199$	83208.3	0.148948	0.0744739	+	0.997223i	$0.476272\pi$
$200$	0	0
$201$	86402.3	0.150846
$202$	0	0
$203$	765528.i	1.30383i
$204$	0	0
$205$	386867.i	0.642949i
$206$	0	0
$207$	222920.	0.361595
$208$	0	0
$209$	−37072.8	−0.0587070
$210$	0	0
$211$	− 1.19938e6i	− 1.85460i	−0.374318	−	0.927300i	$-0.622123\pi$
$211$	− 1.19938e6i	− 1.85460i	0.374318	−	0.927300i	$-0.377877\pi$
$212$	0	0
$213$	− 689164.i	− 1.04082i
$214$	0	0
$215$	281271.	0.414981
$216$	0	0
$217$	−765475.	−1.10352
$218$	0	0
$219$	705132.i	0.993483i
$220$	0	0
$221$	42804.7i	0.0589537i
$222$	0	0
$223$	480791.	0.647432	0.323716	−	0.946154i	$-0.395068\pi$
$223$	480791.	0.647432	0.323716	+	0.946154i	$0.395068\pi$
$224$	0	0
$225$	−108983.	−0.143517
$226$	0	0
$227$	45370.4i	0.0584397i	0.999573	+	0.0292198i	$0.00930229\pi$
$227$	45370.4i	0.0584397i	−0.999573	+	0.0292198i	$0.990698\pi$
$228$	0	0
$229$	121331.i	0.152891i	0.997074	+	0.0764455i	$0.0243571\pi$
$229$	121331.i	0.152891i	−0.997074	+	0.0764455i	$0.975643\pi$
$230$	0	0
$231$	−52913.3	−0.0652431
$232$	0	0
$233$	−458391.	−0.553155	−0.276577	−	0.960992i	$-0.589200\pi$
$233$	−458391.	−0.553155	−0.276577	+	0.960992i	$0.589200\pi$
$234$	0	0
$235$	107483.i	0.126961i
$236$	0	0
$237$	− 264235.i	− 0.305576i
$238$	0	0
$239$	−1.55181e6	−1.75729	−0.878643	−	0.477479i	$-0.841551\pi$
$239$	−1.55181e6	−1.75729	−0.878643	+	0.477479i	$0.841551\pi$
$240$	0	0
$241$	−915057.	−1.01486	−0.507430	−	0.861693i	$-0.669404\pi$
$241$	−915057.	−1.01486	−0.507430	+	0.861693i	$0.669404\pi$
$242$	0	0
$243$	59049.0i	0.0641500i
$244$	0	0
$245$	111488.i	0.118663i
$246$	0	0
$247$	−372923.	−0.388935
$248$	0	0
$249$	−165464.	−0.169124
$250$	0	0
$251$	1.73684e6i	1.74011i	0.492955	+	0.870055i	$0.335917\pi$
$251$	1.73684e6i	1.74011i	−0.492955	+	0.870055i	$0.664083\pi$
$252$	0	0
$253$	116018.i	0.113953i
$254$	0	0
$255$	38323.0	0.0369071
$256$	0	0
$257$	−536538.	−0.506720	−0.253360	−	0.967372i	$-0.581536\pi$
$257$	−536538.	−0.506720	−0.253360	+	0.967372i	$0.581536\pi$
$258$	0	0
$259$	− 1.43766e6i	− 1.33170i
$260$	0	0
$261$	444619.i	0.404005i
$262$	0	0
$263$	−113645.	−0.101312	−0.0506558	−	0.998716i	$-0.516131\pi$
$263$	−113645.	−0.101312	−0.0506558	+	0.998716i	$0.516131\pi$
$264$	0	0
$265$	−147633.	−0.129143
$266$	0	0
$267$	1.05804e6i	0.908288i
$268$	0	0
$269$	− 121806.i	− 0.102633i	−0.998682	−	0.0513166i	$-0.983658\pi$
$269$	− 121806.i	− 0.102633i	0.998682	−	0.0513166i	$-0.0163418\pi$
$270$	0	0
$271$	627118.	0.518712	0.259356	−	0.965782i	$-0.416490\pi$
$271$	627118.	0.518712	0.259356	+	0.965782i	$0.416490\pi$
$272$	0	0
$273$	−532265.	−0.432237
$274$	0	0
$275$	− 56720.1i	− 0.0452278i
$276$	0	0
$277$	− 663979.i	− 0.519942i	−0.965616	−	0.259971i	$-0.916287\pi$
$277$	− 663979.i	− 0.519942i	0.965616	−	0.259971i	$-0.0837130\pi$
$278$	0	0
$279$	−444588.	−0.341938
$280$	0	0
$281$	−2.04008e6	−1.54128	−0.770638	−	0.637273i	$-0.780062\pi$
$281$	−2.04008e6	−1.54128	−0.770638	+	0.637273i	$0.780062\pi$
$282$	0	0
$283$	− 161692.i	− 0.120012i	−0.998198	−	0.0600058i	$-0.980888\pi$
$283$	− 161692.i	− 0.120012i	0.998198	−	0.0600058i	$-0.0191119\pi$
$284$	0	0
$285$	333878.i	0.243487i
$286$	0	0
$287$	−1.27899e6	−0.916562
$288$	0	0
$289$	−1.40967e6	−0.992824
$290$	0	0
$291$	829990.i	0.574567i
$292$	0	0
$293$	1.97921e6i	1.34686i	0.739252	+	0.673429i	$0.235179\pi$
$293$	1.97921e6i	1.34686i	−0.739252	+	0.673429i	$0.764821\pi$
$294$	0	0
$295$	−102867.	−0.0688210
$296$	0	0
$297$	−30732.0	−0.0202162
$298$	0	0
$299$	1.16705e6i	0.754940i
$300$	0	0
$301$	929886.i	0.591580i
$302$	0	0
$303$	807558.	0.505321
$304$	0	0
$305$	−1.30045e6	−0.800469
$306$	0	0
$307$	− 2.99375e6i	− 1.81288i	−0.422334	−	0.906440i	$-0.638789\pi$
$307$	− 2.99375e6i	− 1.81288i	0.422334	−	0.906440i	$-0.361211\pi$
$308$	0	0
$309$	1.35429e6i	0.806890i
$310$	0	0
$311$	1.67394e6	0.981386	0.490693	−	0.871333i	$-0.336744\pi$
$311$	1.67394e6	0.981386	0.490693	+	0.871333i	$0.336744\pi$
$312$	0	0
$313$	−996412.	−0.574881	−0.287440	−	0.957798i	$-0.592804\pi$
$313$	−996412.	−0.574881	−0.287440	+	0.957798i	$0.592804\pi$
$314$	0	0
$315$	476537.i	0.270595i
$316$	0	0
$317$	− 1.45256e6i	− 0.811867i	−0.913903	−	0.405933i	$-0.866946\pi$
$317$	− 1.45256e6i	− 0.811867i	0.913903	−	0.405933i	$-0.133054\pi$
$318$	0	0
$319$	−231402.	−0.127318
$320$	0	0
$321$	−1.10404e6	−0.598031
$322$	0	0
$323$	88768.0i	0.0473424i
$324$	0	0
$325$	− 570559.i	− 0.299635i
$326$	0	0
$327$	2.14805e6	1.11090
$328$	0	0
$329$	−355340.	−0.180990
$330$	0	0
$331$	− 3.29087e6i	− 1.65097i	−0.564421	−	0.825487i	$-0.690900\pi$
$331$	− 3.29087e6i	− 1.65097i	0.564421	−	0.825487i	$-0.309100\pi$
$332$	0	0
$333$	− 834990.i	− 0.412640i
$334$	0	0
$335$	404982.	0.197162
$336$	0	0
$337$	−2.66888e6	−1.28013	−0.640066	−	0.768320i	$-0.721093\pi$
$337$	−2.66888e6	−1.28013	−0.640066	+	0.768320i	$0.721093\pi$
$338$	0	0
$339$	1.26382e6i	0.597290i
$340$	0	0
$341$	− 231386.i	− 0.107758i
$342$	0	0
$343$	1.97537e6	0.906594
$344$	0	0
$345$	1.04486e6	0.472619
$346$	0	0
$347$	2.26305e6i	1.00895i	0.863426	+	0.504476i	$0.168314\pi$
$347$	2.26305e6i	1.00895i	−0.863426	+	0.504476i	$0.831686\pi$
$348$	0	0
$349$	− 3.63602e6i	− 1.59795i	−0.601364	−	0.798975i	$-0.705376\pi$
$349$	− 3.63602e6i	− 1.59795i	0.601364	−	0.798975i	$-0.294624\pi$
$350$	0	0
$351$	−309140.	−0.133933
$352$	0	0
$353$	884847.	0.377947	0.188974	−	0.981982i	$-0.439484\pi$
$353$	884847.	0.377947	0.188974	+	0.981982i	$0.439484\pi$
$354$	0	0
$355$	− 3.23023e6i	− 1.36039i
$356$	0	0
$357$	126697.i	0.0526132i
$358$	0	0
$359$	−3.89334e6	−1.59436	−0.797180	−	0.603741i	$-0.793676\pi$
$359$	−3.89334e6	−1.59436	−0.797180	+	0.603741i	$0.793676\pi$
$360$	0	0
$361$	1.70273e6	0.687668
$362$	0	0
$363$	1.43346e6i	0.570979i
$364$	0	0
$365$	3.30507e6i	1.29852i
$366$	0	0
$367$	−1.88925e6	−0.732192	−0.366096	−	0.930577i	$-0.619306\pi$
$367$	−1.88925e6	−0.732192	−0.366096	+	0.930577i	$0.619306\pi$
$368$	0	0
$369$	−742836.	−0.284006
$370$	0	0
$371$	− 488078.i	− 0.184100i
$372$	0	0
$373$	− 3.27290e6i	− 1.21804i	−0.793156	−	0.609018i	$-0.791564\pi$
$373$	− 3.27290e6i	− 1.21804i	0.793156	−	0.609018i	$-0.208436\pi$
$374$	0	0
$375$	−1.69726e6	−0.623261
$376$	0	0
$377$	−2.32772e6	−0.843483
$378$	0	0
$379$	− 1.91569e6i	− 0.685057i	−0.939507	−	0.342529i	$-0.888717\pi$
$379$	− 1.91569e6i	− 0.685057i	0.939507	−	0.342529i	$-0.111283\pi$
$380$	0	0
$381$	− 197102.i	− 0.0695630i
$382$	0	0
$383$	1.89465e6	0.659984	0.329992	−	0.943984i	$-0.392954\pi$
$383$	1.89465e6	0.659984	0.329992	+	0.943984i	$0.392954\pi$
$384$	0	0
$385$	−248013.	−0.0852753
$386$	0	0
$387$	540078.i	0.183307i
$388$	0	0
$389$	− 3.55031e6i	− 1.18958i	−0.803882	−	0.594789i	$-0.797236\pi$
$389$	− 3.55031e6i	− 1.18958i	0.803882	−	0.594789i	$-0.202764\pi$
$390$	0	0
$391$	277797.	0.0918937
$392$	0	0
$393$	1.21488e6	0.396783
$394$	0	0
$395$	− 1.23851e6i	− 0.399400i
$396$	0	0
$397$	4.13433e6i	1.31653i	0.752788	+	0.658263i	$0.228708\pi$
$397$	4.13433e6i	1.31653i	−0.752788	+	0.658263i	$0.771292\pi$
$398$	0	0
$399$	−1.10381e6	−0.347105
$400$	0	0
$401$	2.77048e6	0.860387	0.430194	−	0.902737i	$-0.358445\pi$
$401$	2.77048e6	0.860387	0.430194	+	0.902737i	$0.358445\pi$
$402$	0	0
$403$	− 2.32755e6i	− 0.713900i
$404$	0	0
$405$	276773.i	0.0838466i
$406$	0	0
$407$	434570.	0.130039
$408$	0	0
$409$	1.94195e6	0.574022	0.287011	−	0.957927i	$-0.407338\pi$
$409$	1.94195e6	0.574022	0.287011	+	0.957927i	$0.407338\pi$
$410$	0	0
$411$	1.87877e6i	0.548618i
$412$	0	0
$413$	− 340080.i	− 0.0981085i
$414$	0	0
$415$	−775557.	−0.221051
$416$	0	0
$417$	356502.	0.100397
$418$	0	0
$419$	4.22070e6i	1.17449i	0.809410	+	0.587244i	$0.199787\pi$
$419$	4.22070e6i	1.17449i	−0.809410	+	0.587244i	$0.800213\pi$
$420$	0	0
$421$	3.18554e6i	0.875947i	0.898988	+	0.437973i	$0.144304\pi$
$421$	3.18554e6i	0.875947i	−0.898988	+	0.437973i	$0.855696\pi$
$422$	0	0
$423$	−206381.	−0.0560815
$424$	0	0
$425$	−135812.	−0.0364725
$426$	0	0
$427$	− 4.29932e6i	− 1.14112i
$428$	0	0
$429$	− 160892.i	− 0.0422076i
$430$	0	0
$431$	4.44734e6	1.15321	0.576603	−	0.817025i	$-0.304378\pi$
$431$	4.44734e6	1.15321	0.576603	+	0.817025i	$0.304378\pi$
$432$	0	0
$433$	1.82838e6	0.468648	0.234324	−	0.972159i	$-0.424712\pi$
$433$	1.82838e6	0.468648	0.234324	+	0.972159i	$0.424712\pi$
$434$	0	0
$435$	2.08400e6i	0.528050i
$436$	0	0
$437$	2.42022e6i	0.606250i
$438$	0	0
$439$	−5.16183e6	−1.27833	−0.639164	−	0.769070i	$-0.720720\pi$
$439$	−5.16183e6	−1.27833	−0.639164	+	0.769070i	$0.720720\pi$
$440$	0	0
$441$	−214072.	−0.0524161
$442$	0	0
$443$	3.23701e6i	0.783674i	0.920035	+	0.391837i	$0.128160\pi$
$443$	3.23701e6i	0.783674i	−0.920035	+	0.391837i	$0.871840\pi$
$444$	0	0
$445$	4.95921e6i	1.18717i
$446$	0	0
$447$	−1.93338e6	−0.457666
$448$	0	0
$449$	5.74960e6	1.34593	0.672964	−	0.739676i	$-0.265021\pi$
$449$	5.74960e6	1.34593	0.672964	+	0.739676i	$0.265021\pi$
$450$	0	0
$451$	− 386609.i	− 0.0895016i
$452$	0	0
$453$	− 2.69145e6i	− 0.616227i
$454$	0	0
$455$	−2.49482e6	−0.564950
$456$	0	0
$457$	4.00591e6	0.897244	0.448622	−	0.893722i	$-0.351915\pi$
$457$	4.00591e6	0.897244	0.448622	+	0.893722i	$0.351915\pi$
$458$	0	0
$459$	73585.4i	0.0163027i
$460$	0	0
$461$	− 8.39187e6i	− 1.83910i	−0.392967	−	0.919552i	$-0.628551\pi$
$461$	− 8.39187e6i	− 1.83910i	0.392967	−	0.919552i	$-0.371449\pi$
$462$	0	0
$463$	7.20365e6	1.56171	0.780854	−	0.624713i	$-0.214784\pi$
$463$	7.20365e6	1.56171	0.780854	+	0.624713i	$0.214784\pi$
$464$	0	0
$465$	−2.08386e6	−0.446926
$466$	0	0
$467$	5.37048e6i	1.13952i	0.821812	+	0.569758i	$0.192963\pi$
$467$	5.37048e6i	1.13952i	−0.821812	+	0.569758i	$0.807037\pi$
$468$	0	0
$469$	1.33888e6i	0.281066i
$470$	0	0
$471$	975996.	0.202720
$472$	0	0
$473$	−281083.	−0.0577673
$474$	0	0
$475$	− 1.18322e6i	− 0.240620i
$476$	0	0
$477$	− 283476.i	− 0.0570453i
$478$	0	0
$479$	1.58556e6	0.315751	0.157875	−	0.987459i	$-0.449536\pi$
$479$	1.58556e6	0.315751	0.157875	+	0.987459i	$0.449536\pi$
$480$	0	0
$481$	4.37143e6	0.861511
$482$	0	0
$483$	3.45433e6i	0.673746i
$484$	0	0
$485$	3.89030e6i	0.750981i
$486$	0	0
$487$	6.18324e6	1.18139	0.590696	−	0.806895i	$-0.298853\pi$
$487$	6.18324e6	1.18139	0.590696	+	0.806895i	$0.298853\pi$
$488$	0	0
$489$	5.54583e6	1.04880
$490$	0	0
$491$	− 5.07372e6i	− 0.949779i	−0.880045	−	0.474889i	$-0.842488\pi$
$491$	− 5.07372e6i	− 0.949779i	0.880045	−	0.474889i	$-0.157512\pi$
$492$	0	0
$493$	554073.i	0.102671i
$494$	0	0
$495$	−144046.	−0.0264234
$496$	0	0
$497$	1.06792e7	1.93931
$498$	0	0
$499$	1.46917e6i	0.264132i	0.991241	+	0.132066i	$0.0421610\pi$
$499$	1.46917e6i	0.264132i	−0.991241	+	0.132066i	$0.957839\pi$
$500$	0	0
$501$	1.12242e6i	0.199784i
$502$	0	0
$503$	7.90833e6	1.39369	0.696843	−	0.717223i	$-0.254587\pi$
$503$	7.90833e6	1.39369	0.696843	+	0.717223i	$0.254587\pi$
$504$	0	0
$505$	3.78516e6	0.660474
$506$	0	0
$507$	1.72320e6i	0.297725i
$508$	0	0
$509$	9.22730e6i	1.57863i	0.613989	+	0.789315i	$0.289564\pi$
$509$	9.22730e6i	1.57863i	−0.613989	+	0.789315i	$0.710436\pi$
$510$	0	0
$511$	−1.09266e7	−1.85112
$512$	0	0
$513$	−641091.	−0.107554
$514$	0	0
$515$	6.34776e6i	1.05464i
$516$	0	0
$517$	− 107411.i	− 0.0176735i
$518$	0	0
$519$	1.51672e6	0.247166
$520$	0	0
$521$	−2.61030e6	−0.421304	−0.210652	−	0.977561i	$-0.567559\pi$
$521$	−2.61030e6	−0.421304	−0.210652	+	0.977561i	$0.567559\pi$
$522$	0	0
$523$	− 9.17412e6i	− 1.46659i	−0.679908	−	0.733297i	$-0.737980\pi$
$523$	− 9.17412e6i	− 1.46659i	0.679908	−	0.733297i	$-0.262020\pi$
$524$	0	0
$525$	− 1.68878e6i	− 0.267409i
$526$	0	0
$527$	−554035.	−0.0868981
$528$	0	0
$529$	1.13768e6	0.176759
$530$	0	0
$531$	− 197519.i	− 0.0303999i
$532$	0	0
$533$	− 3.88898e6i	− 0.592949i
$534$	0	0
$535$	−5.17484e6	−0.781650
$536$	0	0
$537$	3.14683e6	0.470909
$538$	0	0
$539$	− 111414.i	− 0.0165184i
$540$	0	0
$541$	− 1.28020e7i	− 1.88055i	−0.340421	−	0.940273i	$-0.610570\pi$
$541$	− 1.28020e7i	− 1.88055i	0.340421	−	0.940273i	$-0.389430\pi$
$542$	0	0
$543$	−1.97238e6	−0.287072
$544$	0	0
$545$	1.00683e7	1.45199
$546$	0	0
$547$	7.29906e6i	1.04303i	0.853241	+	0.521517i	$0.174634\pi$
$547$	7.29906e6i	1.04303i	−0.853241	+	0.521517i	$0.825366\pi$
$548$	0	0
$549$	− 2.49704e6i	− 0.353586i
$550$	0	0
$551$	−4.82719e6	−0.677354
$552$	0	0
$553$	4.09455e6	0.569368
$554$	0	0
$555$	− 3.91374e6i	− 0.539336i
$556$	0	0
$557$	− 8.42637e6i	− 1.15081i	−0.817869	−	0.575404i	$-0.804845\pi$
$557$	− 8.42637e6i	− 1.15081i	0.817869	−	0.575404i	$-0.195155\pi$
$558$	0	0
$559$	−2.82747e6	−0.382709
$560$	0	0
$561$	−38297.5	−0.00513764
$562$	0	0
$563$	− 357556.i	− 0.0475415i	−0.999717	−	0.0237707i	$-0.992433\pi$
$563$	− 357556.i	− 0.0475415i	0.999717	−	0.0237707i	$-0.00756718\pi$
$564$	0	0
$565$	5.92372e6i	0.780681i
$566$	0	0
$567$	−915015.	−0.119528
$568$	0	0
$569$	−4.73291e6	−0.612840	−0.306420	−	0.951896i	$-0.599131\pi$
$569$	−4.73291e6	−0.612840	−0.306420	+	0.951896i	$0.599131\pi$
$570$	0	0
$571$	− 3.07290e6i	− 0.394420i	−0.980361	−	0.197210i	$-0.936812\pi$
$571$	− 3.07290e6i	− 0.394420i	0.980361	−	0.197210i	$-0.0631880\pi$
$572$	0	0
$573$	6.92930e6i	0.881664i
$574$	0	0
$575$	−3.70285e6	−0.467054
$576$	0	0
$577$	−3.13293e6	−0.391752	−0.195876	−	0.980629i	$-0.562755\pi$
$577$	−3.13293e6	−0.391752	−0.195876	+	0.980629i	$0.562755\pi$
$578$	0	0
$579$	891471.i	0.110512i
$580$	0	0
$581$	− 2.56401e6i	− 0.315122i
$582$	0	0
$583$	147535.	0.0179773
$584$	0	0
$585$	−1.44899e6	−0.175055
$586$	0	0
$587$	− 9.87159e6i	− 1.18247i	−0.806498	−	0.591237i	$-0.798640\pi$
$587$	− 9.87159e6i	− 1.18247i	0.806498	−	0.591237i	$-0.201360\pi$
$588$	0	0
$589$	− 4.82686e6i	− 0.573293i
$590$	0	0
$591$	6.04095e6	0.711437
$592$	0	0
$593$	8.54711e6	0.998120	0.499060	−	0.866568i	$-0.333679\pi$
$593$	8.54711e6	0.998120	0.499060	+	0.866568i	$0.333679\pi$
$594$	0	0
$595$	593849.i	0.0687675i
$596$	0	0
$597$	748875.i	0.0859950i
$598$	0	0
$599$	−1.33918e6	−0.152501	−0.0762506	−	0.997089i	$-0.524295\pi$
$599$	−1.33918e6	−0.152501	−0.0762506	+	0.997089i	$0.524295\pi$
$600$	0	0
$601$	7.76048e6	0.876400	0.438200	−	0.898877i	$-0.355616\pi$
$601$	7.76048e6	0.876400	0.438200	+	0.898877i	$0.355616\pi$
$602$	0	0
$603$	777620.i	0.0870912i
$604$	0	0
$605$	6.71889e6i	0.746292i
$606$	0	0
$607$	9.80613e6	1.08025	0.540127	−	0.841583i	$-0.318376\pi$
$607$	9.80613e6	1.08025	0.540127	+	0.841583i	$0.318376\pi$
$608$	0	0
$609$	−6.88975e6	−0.752767
$610$	0	0
$611$	− 1.08047e6i	− 0.117087i
$612$	0	0
$613$	− 4.69548e6i	− 0.504695i	−0.967637	−	0.252348i	$-0.918797\pi$
$613$	− 4.69548e6i	− 0.504695i	0.967637	−	0.252348i	$-0.0812026\pi$
$614$	0	0
$615$	−3.48180e6	−0.371207
$616$	0	0
$617$	−1.15220e7	−1.21847	−0.609235	−	0.792990i	$-0.708523\pi$
$617$	−1.15220e7	−1.21847	−0.609235	+	0.792990i	$0.708523\pi$
$618$	0	0
$619$	− 1.16086e7i	− 1.21773i	−0.793273	−	0.608866i	$-0.791625\pi$
$619$	− 1.16086e7i	− 1.21773i	0.793273	−	0.608866i	$-0.208375\pi$
$620$	0	0
$621$	2.00628e6i	0.208767i
$622$	0	0
$623$	−1.63952e7	−1.69238
$624$	0	0
$625$	−3.75076e6	−0.384077
$626$	0	0
$627$	− 333655.i	− 0.0338945i
$628$	0	0
$629$	− 1.04055e6i	− 0.104866i
$630$	0	0
$631$	1.68312e7	1.68283	0.841417	−	0.540386i	$-0.181722\pi$
$631$	1.68312e7	1.68283	0.841417	+	0.540386i	$0.181722\pi$
$632$	0	0
$633$	1.07944e7	1.07075
$634$	0	0
$635$	− 923850.i	− 0.0909216i
$636$	0	0
$637$	− 1.12074e6i	− 0.109435i
$638$	0	0
$639$	6.20248e6	0.600915
$640$	0	0
$641$	−3.39644e6	−0.326497	−0.163248	−	0.986585i	$-0.552197\pi$
$641$	−3.39644e6	−0.326497	−0.163248	+	0.986585i	$0.552197\pi$
$642$	0	0
$643$	1.07812e7i	1.02835i	0.857685	+	0.514176i	$0.171902\pi$
$643$	1.07812e7i	1.02835i	−0.857685	+	0.514176i	$0.828098\pi$
$644$	0	0
$645$	2.53144e6i	0.239589i
$646$	0	0
$647$	7.43042e6	0.697835	0.348917	−	0.937153i	$-0.386549\pi$
$647$	7.43042e6	0.697835	0.348917	+	0.937153i	$0.386549\pi$
$648$	0	0
$649$	102799.	0.00958021
$650$	0	0
$651$	− 6.88927e6i	− 0.637120i
$652$	0	0
$653$	1.65134e7i	1.51550i	0.652548	+	0.757748i	$0.273700\pi$
$653$	1.65134e7i	1.51550i	−0.652548	+	0.757748i	$0.726300\pi$
$654$	0	0
$655$	5.69436e6	0.518611
$656$	0	0
$657$	−6.34619e6	−0.573588
$658$	0	0
$659$	1.01361e6i	0.0909200i	0.998966	+	0.0454600i	$0.0144754\pi$
$659$	1.01361e6i	0.0909200i	−0.998966	+	0.0454600i	$0.985525\pi$
$660$	0	0
$661$	− 4.22743e6i	− 0.376333i	−0.982137	−	0.188167i	$-0.939745\pi$
$661$	− 4.22743e6i	− 0.376333i	0.982137	−	0.188167i	$-0.0602545\pi$
$662$	0	0
$663$	−385242.	−0.0340369
$664$	0	0
$665$	−5.17372e6	−0.453680
$666$	0	0
$667$	1.51066e7i	1.31477i
$668$	0	0
$669$	4.32712e6i	0.373795i
$670$	0	0
$671$	1.29959e6	0.111429
$672$	0	0
$673$	−1.18192e7	−1.00589	−0.502945	−	0.864318i	$-0.667750\pi$
$673$	−1.18192e7	−1.00589	−0.502945	+	0.864318i	$0.667750\pi$
$674$	0	0
$675$	− 980846.i	− 0.0828593i
$676$	0	0
$677$	3.45016e6i	0.289313i	0.989482	+	0.144657i	$0.0462077\pi$
$677$	3.45016e6i	0.289313i	−0.989482	+	0.144657i	$0.953792\pi$
$678$	0	0
$679$	−1.28614e7	−1.07057
$680$	0	0
$681$	−408333.	−0.0337402
$682$	0	0
$683$	− 1.60796e7i	− 1.31893i	−0.751735	−	0.659466i	$-0.770783\pi$
$683$	− 1.60796e7i	− 1.31893i	0.751735	−	0.659466i	$-0.229217\pi$
$684$	0	0
$685$	8.80612e6i	0.717065i
$686$	0	0
$687$	−1.09198e6	−0.0882717
$688$	0	0
$689$	1.48408e6	0.119100
$690$	0	0
$691$	1.34443e6i	0.107113i	0.998565	+	0.0535566i	$0.0170558\pi$
$691$	1.34443e6i	0.107113i	−0.998565	+	0.0535566i	$0.982944\pi$
$692$	0	0
$693$	− 476219.i	− 0.0376681i
$694$	0	0
$695$	1.67099e6	0.131223
$696$	0	0
$697$	−925705.	−0.0721756
$698$	0	0
$699$	− 4.12552e6i	− 0.319364i
$700$	0	0
$701$	2.08681e6i	0.160394i	0.996779	+	0.0801970i	$0.0255549\pi$
$701$	2.08681e6i	0.160394i	−0.996779	+	0.0801970i	$0.974445\pi$
$702$	0	0
$703$	9.06543e6	0.691831
$704$	0	0
$705$	−967344.	−0.0733007
$706$	0	0
$707$	1.25138e7i	0.941545i
$708$	0	0
$709$	3.40863e6i	0.254662i	0.991860	+	0.127331i	$0.0406411\pi$
$709$	3.40863e6i	0.254662i	−0.991860	+	0.127331i	$0.959359\pi$
$710$	0	0
$711$	2.37811e6	0.176424
$712$	0	0
$713$	−1.51055e7	−1.11279
$714$	0	0
$715$	− 754126.i	− 0.0551669i
$716$	0	0
$717$	− 1.39662e7i	− 1.01457i
$718$	0	0
$719$	−1.65800e6	−0.119608	−0.0598041	−	0.998210i	$-0.519048\pi$
$719$	−1.65800e6	−0.119608	−0.0598041	+	0.998210i	$0.519048\pi$
$720$	0	0
$721$	−2.09858e7	−1.50345
$722$	0	0
$723$	− 8.23552e6i	− 0.585929i
$724$	0	0
$725$	− 7.38543e6i	− 0.521832i
$726$	0	0
$727$	−2.05012e7	−1.43861	−0.719305	−	0.694694i	$-0.755540\pi$
$727$	−2.05012e7	−1.43861	−0.719305	+	0.694694i	$0.755540\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	673032.i	0.0465846i
$732$	0	0
$733$	− 3.05065e6i	− 0.209717i	−0.994487	−	0.104858i	$-0.966561\pi$
$733$	− 3.05065e6i	− 0.209717i	0.994487	−	0.104858i	$-0.0334389\pi$
$734$	0	0
$735$	−1.00339e6	−0.0685098
$736$	0	0
$737$	−404712.	−0.0274459
$738$	0	0
$739$	2.65769e7i	1.79016i	0.445902	+	0.895082i	$0.352883\pi$
$739$	2.65769e7i	1.79016i	−0.445902	+	0.895082i	$0.647117\pi$
$740$	0	0
$741$	− 3.35631e6i	− 0.224552i
$742$	0	0
$743$	−3.54700e6	−0.235716	−0.117858	−	0.993030i	$-0.537603\pi$
$743$	−3.54700e6	−0.235716	−0.117858	+	0.993030i	$0.537603\pi$
$744$	0	0
$745$	−9.06207e6	−0.598187
$746$	0	0
$747$	− 1.48918e6i	− 0.0976437i
$748$	0	0
$749$	− 1.71081e7i	− 1.11429i
$750$	0	0
$751$	−8.47107e6	−0.548073	−0.274037	−	0.961719i	$-0.588359\pi$
$751$	−8.47107e6	−0.548073	−0.274037	+	0.961719i	$0.588359\pi$
$752$	0	0
$753$	−1.56316e7	−1.00465
$754$	0	0
$755$	− 1.26153e7i	− 0.805433i
$756$	0	0
$757$	1.27633e7i	0.809511i	0.914425	+	0.404756i	$0.132643\pi$
$757$	1.27633e7i	0.809511i	−0.914425	+	0.404756i	$0.867357\pi$
$758$	0	0
$759$	−1.04417e6	−0.0657908
$760$	0	0
$761$	−2.30382e7	−1.44207	−0.721036	−	0.692898i	$-0.756334\pi$
$761$	−2.30382e7	−1.44207	−0.721036	+	0.692898i	$0.756334\pi$
$762$	0	0
$763$	3.32859e7i	2.06990i
$764$	0	0
$765$	344907.i	0.0213083i
$766$	0	0
$767$	1.03407e6	0.0634691
$768$	0	0
$769$	−2.35154e7	−1.43396	−0.716978	−	0.697095i	$-0.754475\pi$
$769$	−2.35154e7	−1.43396	−0.716978	+	0.697095i	$0.754475\pi$
$770$	0	0
$771$	− 4.82885e6i	− 0.292555i
$772$	0	0
$773$	− 1.06592e7i	− 0.641618i	−0.947144	−	0.320809i	$-0.896045\pi$
$773$	− 1.06592e7i	− 0.641618i	0.947144	−	0.320809i	$-0.103955\pi$
$774$	0	0
$775$	7.38492e6	0.441664
$776$	0	0
$777$	1.29389e7	0.768856
$778$	0	0
$779$	− 8.06492e6i	− 0.476164i
$780$	0	0
$781$	3.22808e6i	0.189372i
$782$	0	0
$783$	−4.00157e6	−0.233252
$784$	0	0
$785$	4.57466e6	0.264962
$786$	0	0
$787$	9.74081e6i	0.560607i	0.959911	+	0.280303i	$0.0904351\pi$
$787$	9.74081e6i	0.560607i	−0.959911	+	0.280303i	$0.909565\pi$
$788$	0	0
$789$	− 1.02280e6i	− 0.0584923i
$790$	0	0
$791$	−1.95839e7	−1.11291
$792$	0	0
$793$	1.30728e7	0.738220
$794$	0	0
$795$	− 1.32870e6i	− 0.0745605i
$796$	0	0
$797$	− 2.43301e7i	− 1.35675i	−0.734717	−	0.678374i	$-0.762685\pi$
$797$	− 2.43301e7i	− 1.35675i	0.734717	−	0.678374i	$-0.237315\pi$
$798$	0	0
$799$	−257188.	−0.0142522
$800$	0	0
$801$	−9.52235e6	−0.524400
$802$	0	0
$803$	− 3.30287e6i	− 0.180760i
$804$	0	0
$805$	1.61910e7i	0.880613i
$806$	0	0
$807$	1.09625e6	0.0592553
$808$	0	0
$809$	−2.35926e7	−1.26737	−0.633686	−	0.773591i	$-0.718459\pi$
$809$	−2.35926e7	−1.26737	−0.633686	+	0.773591i	$0.718459\pi$
$810$	0	0
$811$	− 4.23483e6i	− 0.226092i	−0.993590	−	0.113046i	$-0.963939\pi$
$811$	− 4.23483e6i	− 0.226092i	0.993590	−	0.113046i	$-0.0360607\pi$
$812$	0	0
$813$	5.64406e6i	0.299478i
$814$	0	0
$815$	2.59942e7	1.37083
$816$	0	0
$817$	−5.86359e6	−0.307332
$818$	0	0
$819$	− 4.79039e6i	− 0.249552i
$820$	0	0
$821$	2.35083e7i	1.21721i	0.793475	+	0.608603i	$0.208270\pi$
$821$	2.35083e7i	1.21721i	−0.793475	+	0.608603i	$0.791730\pi$
$822$	0	0
$823$	6.83692e6	0.351853	0.175926	−	0.984403i	$-0.443708\pi$
$823$	6.83692e6	0.351853	0.175926	+	0.984403i	$0.443708\pi$
$824$	0	0
$825$	510481.	0.0261123
$826$	0	0
$827$	2.10512e7i	1.07032i	0.844751	+	0.535159i	$0.179748\pi$
$827$	2.10512e7i	1.07032i	−0.844751	+	0.535159i	$0.820252\pi$
$828$	0	0
$829$	− 4.65552e6i	− 0.235278i	−0.993056	−	0.117639i	$-0.962467\pi$
$829$	− 4.65552e6i	− 0.235278i	0.993056	−	0.117639i	$-0.0375326\pi$
$830$	0	0
$831$	5.97581e6	0.300189
$832$	0	0
$833$	−266772.	−0.0133207
$834$	0	0
$835$	5.26098e6i	0.261126i
$836$	0	0
$837$	− 4.00129e6i	− 0.197418i
$838$	0	0
$839$	3.33033e7	1.63336	0.816681	−	0.577089i	$-0.195812\pi$
$839$	3.33033e7	1.63336	0.816681	+	0.577089i	$0.195812\pi$
$840$	0	0
$841$	−9.61929e6	−0.468978
$842$	0	0
$843$	− 1.83607e7i	− 0.889856i
$844$	0	0
$845$	8.07691e6i	0.389138i
$846$	0	0
$847$	−2.22128e7	−1.06388
$848$	0	0
$849$	1.45523e6	0.0692887
$850$	0	0
$851$	− 2.83700e7i	− 1.34288i
$852$	0	0
$853$	− 3.29374e7i	− 1.54995i	−0.631994	−	0.774973i	$-0.717763\pi$
$853$	− 3.29374e7i	− 1.54995i	0.631994	−	0.774973i	$-0.282237\pi$
$854$	0	0
$855$	−3.00490e6	−0.140577
$856$	0	0
$857$	−1.26881e7	−0.590123	−0.295062	−	0.955478i	$-0.595340\pi$
$857$	−1.26881e7	−0.590123	−0.295062	+	0.955478i	$0.595340\pi$
$858$	0	0
$859$	4.07691e7i	1.88516i	0.333983	+	0.942579i	$0.391607\pi$
$859$	4.07691e7i	1.88516i	−0.333983	+	0.942579i	$0.608393\pi$
$860$	0	0
$861$	− 1.15109e7i	− 0.529177i
$862$	0	0
$863$	9.70061e6	0.443376	0.221688	−	0.975118i	$-0.428843\pi$
$863$	9.70061e6	0.443376	0.221688	+	0.975118i	$0.428843\pi$
$864$	0	0
$865$	7.10914e6	0.323055
$866$	0	0
$867$	− 1.26870e7i	− 0.573207i
$868$	0	0
$869$	1.23769e6i	0.0555984i
$870$	0	0
$871$	−4.07108e6	−0.181830
$872$	0	0
$873$	−7.46991e6	−0.331726
$874$	0	0
$875$	− 2.63005e7i	− 1.16130i
$876$	0	0
$877$	2.65092e7i	1.16385i	0.813242	+	0.581926i	$0.197701\pi$
$877$	2.65092e7i	1.16385i	−0.813242	+	0.581926i	$0.802299\pi$
$878$	0	0
$879$	−1.78129e7	−0.777609
$880$	0	0
$881$	2.39141e7	1.03804	0.519020	−	0.854762i	$-0.326297\pi$
$881$	2.39141e7	1.03804	0.519020	+	0.854762i	$0.326297\pi$
$882$	0	0
$883$	− 2.94092e6i	− 0.126935i	−0.997984	−	0.0634675i	$-0.979784\pi$
$883$	− 2.94092e6i	− 0.126935i	0.997984	−	0.0634675i	$-0.0202159\pi$
$884$	0	0
$885$	− 925803.i	− 0.0397338i
$886$	0	0
$887$	−3.66901e6	−0.156581	−0.0782907	−	0.996931i	$-0.524946\pi$
$887$	−3.66901e6	−0.156581	−0.0782907	+	0.996931i	$0.524946\pi$
$888$	0	0
$889$	3.05427e6	0.129614
$890$	0	0
$891$	− 276588.i	− 0.0116718i
$892$	0	0
$893$	− 2.24067e6i	− 0.0940262i
$894$	0	0
$895$	1.47497e7	0.615497
$896$	0	0
$897$	−1.05035e7	−0.435865
$898$	0	0
$899$	− 3.01284e7i	− 1.24330i
$900$	0	0
$901$	− 353261.i	− 0.0144972i
$902$	0	0
$903$	−8.36898e6	−0.341549
$904$	0	0
$905$	−9.24487e6	−0.375214
$906$	0	0
$907$	2.08057e7i	0.839776i	0.907576	+	0.419888i	$0.137931\pi$
$907$	2.08057e7i	0.839776i	−0.907576	+	0.419888i	$0.862069\pi$
$908$	0	0
$909$	7.26802e6i	0.291747i
$910$	0	0
$911$	6.68661e6	0.266938	0.133469	−	0.991053i	$-0.457388\pi$
$911$	6.68661e6	0.266938	0.133469	+	0.991053i	$0.457388\pi$
$912$	0	0
$913$	775041.	0.0307714
$914$	0	0
$915$	− 1.17041e7i	− 0.462151i
$916$	0	0
$917$	1.88257e7i	0.739311i
$918$	0	0
$919$	4.47652e7	1.74844	0.874222	−	0.485526i	$-0.161372\pi$
$919$	4.47652e7	1.74844	0.874222	+	0.485526i	$0.161372\pi$
$920$	0	0
$921$	2.69437e7	1.04667
$922$	0	0
$923$	3.24719e7i	1.25459i
$924$	0	0
$925$	1.38698e7i	0.532985i
$926$	0	0
$927$	−1.21886e7	−0.465858
$928$	0	0
$929$	1.74528e7	0.663477	0.331738	−	0.943371i	$-0.392365\pi$
$929$	1.74528e7	0.663477	0.331738	+	0.943371i	$0.392365\pi$
$930$	0	0
$931$	− 2.32417e6i	− 0.0878807i
$932$	0	0
$933$	1.50655e7i	0.566604i
$934$	0	0
$935$	−179507.	−0.00671509
$936$	0	0
$937$	−5.05211e6	−0.187985	−0.0939925	−	0.995573i	$-0.529963\pi$
$937$	−5.05211e6	−0.187985	−0.0939925	+	0.995573i	$0.529963\pi$
$938$	0	0
$939$	− 8.96770e6i	− 0.331908i
$940$	0	0
$941$	1.55690e7i	0.573175i	0.958054	+	0.286588i	$0.0925209\pi$
$941$	1.55690e7i	0.573175i	−0.958054	+	0.286588i	$0.907479\pi$
$942$	0	0
$943$	−2.52390e7	−0.924256
$944$	0	0
$945$	−4.28883e6	−0.156228
$946$	0	0
$947$	− 3.00092e7i	− 1.08738i	−0.839287	−	0.543688i	$-0.817028\pi$
$947$	− 3.00092e7i	− 1.08738i	0.839287	−	0.543688i	$-0.182972\pi$
$948$	0	0
$949$	− 3.32243e7i	− 1.19754i
$950$	0	0
$951$	1.30730e7	0.468731
$952$	0	0
$953$	−5.68719e6	−0.202845	−0.101423	−	0.994843i	$-0.532339\pi$
$953$	−5.68719e6	−0.202845	−0.101423	+	0.994843i	$0.532339\pi$
$954$	0	0
$955$	3.24788e7i	1.15237i
$956$	0	0
$957$	− 2.08261e6i	− 0.0735071i
$958$	0	0
$959$	−2.91132e7	−1.02222
$960$	0	0
$961$	1.49711e6	0.0522933
$962$	0	0
$963$	− 9.93640e6i	− 0.345274i
$964$	0	0
$965$	4.17847e6i	0.144444i
$966$	0	0
$967$	−1.04709e7	−0.360096	−0.180048	−	0.983658i	$-0.557625\pi$
$967$	−1.04709e7	−0.360096	−0.180048	+	0.983658i	$0.557625\pi$
$968$	0	0
$969$	−798912.	−0.0273331
$970$	0	0
$971$	− 3.98022e7i	− 1.35475i	−0.735638	−	0.677375i	$-0.763117\pi$
$971$	− 3.98022e7i	− 1.35475i	0.735638	−	0.677375i	$-0.236883\pi$
$972$	0	0
$973$	5.52431e6i	0.187066i
$974$	0	0
$975$	5.13503e6	0.172994
$976$	0	0
$977$	3.52817e7	1.18253	0.591267	−	0.806476i	$-0.298628\pi$
$977$	3.52817e7	1.18253	0.591267	+	0.806476i	$0.298628\pi$
$978$	0	0
$979$	− 4.95591e6i	− 0.165259i
$980$	0	0
$981$	1.93324e7i	0.641378i
$982$	0	0
$983$	−4.51814e7	−1.49134	−0.745669	−	0.666317i	$-0.767870\pi$
$983$	−4.51814e7	−1.49134	−0.745669	+	0.666317i	$0.767870\pi$
$984$	0	0
$985$	2.83149e7	0.929876
$986$	0	0
$987$	− 3.19806e6i	− 0.104494i
$988$	0	0
$989$	1.83499e7i	0.596546i
$990$	0	0
$991$	−2.99927e7	−0.970134	−0.485067	−	0.874477i	$-0.661205\pi$
$991$	−2.99927e7	−0.970134	−0.485067	+	0.874477i	$0.661205\pi$
$992$	0	0
$993$	2.96178e7	0.953191
$994$	0	0
$995$	3.51010e6i	0.112399i
$996$	0	0
$997$	− 1.79150e7i	− 0.570795i	−0.958409	−	0.285397i	$-0.907874\pi$
$997$	− 1.79150e7i	− 0.570795i	0.958409	−	0.285397i	$-0.0921256\pi$
$998$	0	0
$999$	7.51491e6	0.238238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.6.d.i.193.7	yes	8
4.3	odd	2	inner	384.6.d.i.193.3	yes	8
8.3	odd	2	inner	384.6.d.i.193.6	yes	8
8.5	even	2	inner	384.6.d.i.193.2	✓	8
16.3	odd	4		768.6.a.y.1.3		4
16.5	even	4		768.6.a.y.1.2		4
16.11	odd	4		768.6.a.x.1.2		4
16.13	even	4		768.6.a.x.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.d.i.193.2	✓	8	8.5	even	2	inner
384.6.d.i.193.3	yes	8	4.3	odd	2	inner
384.6.d.i.193.6	yes	8	8.3	odd	2	inner
384.6.d.i.193.7	yes	8	1.1	even	1	trivial
768.6.a.x.1.2		4	16.11	odd	4
768.6.a.x.1.3		4	16.13	even	4
768.6.a.y.1.2		4	16.5	even	4
768.6.a.y.1.3		4	16.3	odd	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} +42.1845i q^{5} -139.463 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} +42.1845i q^{5} -139.463 q^{7} -81.0000 q^{9} -42.1564i q^{11} -424.060i q^{13} -379.661 q^{15} -100.940 q^{17} -879.411i q^{19} -1255.17i q^{21} -2752.09 q^{23} +1345.47 q^{25} -729.000i q^{27} -5489.12i q^{29} +5488.74 q^{31} +379.408 q^{33} -5883.17i q^{35} +10308.5i q^{37} +3816.54 q^{39} +9170.82 q^{41} -6667.63i q^{43} -3416.94i q^{45} +2547.92 q^{47} +2642.87 q^{49} -908.462i q^{51} +3499.70i q^{53} +1778.35 q^{55} +7914.70 q^{57} +2438.50i q^{59} +30827.7i q^{61} +11296.5 q^{63} +17888.8 q^{65} -9600.25i q^{67} -24768.8i q^{69} -76573.8 q^{71} +78348.0 q^{73} +12109.2i q^{75} +5879.25i q^{77} -29359.4 q^{79} +6561.00 q^{81} +18384.9i q^{83} -4258.11i q^{85} +49402.1 q^{87} +117560. q^{89} +59140.6i q^{91} +49398.7i q^{93} +37097.5 q^{95} +92221.1 q^{97} +3414.67i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 648 q^{9}+O(q^{10}) \) 8 * q - 648 * q^9	\( 8 q - 648 q^{9} + 4080 q^{17} - 5528 q^{25} - 18144 q^{33} + 58704 q^{41} - 44024 q^{49} + 38880 q^{57} - 203904 q^{65} + 248816 q^{73} + 52488 q^{81} - 46800 q^{89} - 262544 q^{97}+O(q^{100}) \) 8 * q - 648 * q^9 + 4080 * q^17 - 5528 * q^25 - 18144 * q^33 + 58704 * q^41 - 44024 * q^49 + 38880 * q^57 - 203904 * q^65 + 248816 * q^73 + 52488 * q^81 - 46800 * q^89 - 262544 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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