LMFDB - Embedded newform 384.4.d.c.193.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,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, 4, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
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:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.4
Root			$2.30278i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.4.d.c.193.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +18.4222i q^{5} -22.4222 q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +18.4222i q^{5} -22.4222 q^{7} -9.00000 q^{9} +53.6888i q^{11} +7.15559i q^{13} -55.2666 q^{15} +39.6888 q^{17} -125.689i q^{19} -67.2666i q^{21} -99.1556 q^{23} -214.378 q^{25} -27.0000i q^{27} +205.800i q^{29} +147.489 q^{31} -161.066 q^{33} -413.066i q^{35} -125.689i q^{37} -21.4668 q^{39} +506.444 q^{41} -413.689i q^{43} -165.800i q^{45} -313.911 q^{47} +159.755 q^{49} +119.066i q^{51} +44.3331i q^{53} -989.066 q^{55} +377.066 q^{57} -324.000i q^{59} +324.000i q^{61} +201.800 q^{63} -131.822 q^{65} +464.266i q^{67} -297.467i q^{69} -1052.84 q^{71} -1022.27 q^{73} -643.133i q^{75} -1203.82i q^{77} -602.910 q^{79} +81.0000 q^{81} +15.8217i q^{83} +731.156i q^{85} -617.400 q^{87} -381.378 q^{89} -160.444i q^{91} +442.466i q^{93} +2315.47 q^{95} +659.154 q^{97} -483.199i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{7} - 36 q^{9}+O(q^{10}) $ 4 * q - 32 * q^7 - 36 * q^9	$ 4 q - 32 q^{7} - 36 q^{9} - 48 q^{15} - 72 q^{17} - 512 q^{23} - 396 q^{25} - 160 q^{31} + 48 q^{33} - 432 q^{39} + 872 q^{41} - 448 q^{47} - 284 q^{49} - 3264 q^{55} + 816 q^{57} + 288 q^{63} + 1088 q^{65} - 4096 q^{71} - 1320 q^{73} + 992 q^{79} + 324 q^{81} - 912 q^{87} - 1064 q^{89} + 4416 q^{95} - 2440 q^{97}+O(q^{100}) $ 4 * q - 32 * q^7 - 36 * q^9 - 48 * q^15 - 72 * q^17 - 512 * q^23 - 396 * q^25 - 160 * q^31 + 48 * q^33 - 432 * q^39 + 872 * q^41 - 448 * q^47 - 284 * q^49 - 3264 * q^55 + 816 * q^57 + 288 * q^63 + 1088 * q^65 - 4096 * q^71 - 1320 * q^73 + 992 * q^79 + 324 * q^81 - 912 * q^87 - 1064 * q^89 + 4416 * q^95 - 2440 * 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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	18.4222i	1.64773i	0.566785	+	0.823866i	$0.308187\pi$
$5$	18.4222i	1.64773i	−0.566785	+	0.823866i	$0.691813\pi$
$6$	0	0
$7$	−22.4222	−1.21069	−0.605343	−	0.795965i	$-0.706964\pi$
$7$	−22.4222	−1.21069	−0.605343	+	0.795965i	$0.706964\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	53.6888i	1.47162i	0.677190	+	0.735809i	$0.263198\pi$
$11$	53.6888i	1.47162i	−0.677190	+	0.735809i	$0.736802\pi$
$12$	0	0
$13$	7.15559i	0.152662i	0.997083	+	0.0763309i	$0.0243205\pi$
$13$	7.15559i	0.152662i	−0.997083	+	0.0763309i	$0.975679\pi$
$14$	0	0
$15$	−55.2666	−0.951319
$16$	0	0
$17$	39.6888	0.566233	0.283116	−	0.959086i	$-0.408632\pi$
$17$	39.6888	0.566233	0.283116	+	0.959086i	$0.408632\pi$
$18$	0	0
$19$	− 125.689i	− 1.51763i	−0.651306	−	0.758816i	$-0.725778\pi$
$19$	− 125.689i	− 1.51763i	0.651306	−	0.758816i	$-0.274222\pi$
$20$	0	0
$21$	− 67.2666i	− 0.698989i
$22$	0	0
$23$	−99.1556	−0.898929	−0.449465	−	0.893298i	$-0.648385\pi$
$23$	−99.1556	−0.898929	−0.449465	+	0.893298i	$0.648385\pi$
$24$	0	0
$25$	−214.378	−1.71502
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	205.800i	1.31780i	0.752232	+	0.658898i	$0.228977\pi$
$29$	205.800i	1.31780i	−0.752232	+	0.658898i	$0.771023\pi$
$30$	0	0
$31$	147.489	0.854508	0.427254	−	0.904132i	$-0.359481\pi$
$31$	147.489	0.854508	0.427254	+	0.904132i	$0.359481\pi$
$32$	0	0
$33$	−161.066	−0.849639
$34$	0	0
$35$	− 413.066i	− 1.99489i
$36$	0	0
$37$	− 125.689i	− 0.558463i	−0.960224	−	0.279231i	$-0.909920\pi$
$37$	− 125.689i	− 0.558463i	0.960224	−	0.279231i	$-0.0900796\pi$
$38$	0	0
$39$	−21.4668	−0.0881393
$40$	0	0
$41$	506.444	1.92910	0.964552	−	0.263892i	$-0.0850063\pi$
$41$	506.444	1.92910	0.964552	+	0.263892i	$0.0850063\pi$
$42$	0	0
$43$	− 413.689i	− 1.46714i	−0.679615	−	0.733569i	$-0.737853\pi$
$43$	− 413.689i	− 1.46714i	0.679615	−	0.733569i	$-0.262147\pi$
$44$	0	0
$45$	− 165.800i	− 0.549244i
$46$	0	0
$47$	−313.911	−0.974226	−0.487113	−	0.873339i	$-0.661950\pi$
$47$	−313.911	−0.974226	−0.487113	+	0.873339i	$0.661950\pi$
$48$	0	0
$49$	159.755	0.465759
$50$	0	0
$51$	119.066i	0.326914i
$52$	0	0
$53$	44.3331i	0.114898i	0.998348	+	0.0574492i	$0.0182967\pi$
$53$	44.3331i	0.114898i	−0.998348	+	0.0574492i	$0.981703\pi$
$54$	0	0
$55$	−989.066	−2.42483
$56$	0	0
$57$	377.066	0.876205
$58$	0	0
$59$	− 324.000i	− 0.714936i	−0.933925	−	0.357468i	$-0.883640\pi$
$59$	− 324.000i	− 0.714936i	0.933925	−	0.357468i	$-0.116360\pi$
$60$	0	0
$61$	324.000i	0.680065i	0.940414	+	0.340032i	$0.110438\pi$
$61$	324.000i	0.680065i	−0.940414	+	0.340032i	$0.889562\pi$
$62$	0	0
$63$	201.800	0.403562
$64$	0	0
$65$	−131.822	−0.251546
$66$	0	0
$67$	464.266i	0.846554i	0.906000	+	0.423277i	$0.139120\pi$
$67$	464.266i	0.846554i	−0.906000	+	0.423277i	$0.860880\pi$
$68$	0	0
$69$	− 297.467i	− 0.518997i
$70$	0	0
$71$	−1052.84	−1.75985	−0.879927	−	0.475109i	$-0.842409\pi$
$71$	−1052.84	−1.75985	−0.879927	+	0.475109i	$0.842409\pi$
$72$	0	0
$73$	−1022.27	−1.63900	−0.819501	−	0.573078i	$-0.805749\pi$
$73$	−1022.27	−1.63900	−0.819501	+	0.573078i	$0.805749\pi$
$74$	0	0
$75$	− 643.133i	− 0.990168i
$76$	0	0
$77$	− 1203.82i	− 1.78167i
$78$	0	0
$79$	−602.910	−0.858642	−0.429321	−	0.903152i	$-0.641247\pi$
$79$	−602.910	−0.858642	−0.429321	+	0.903152i	$0.641247\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	15.8217i	0.0209236i	0.999945	+	0.0104618i	$0.00333016\pi$
$83$	15.8217i	0.0209236i	−0.999945	+	0.0104618i	$0.996670\pi$
$84$	0	0
$85$	731.156i	0.933000i
$86$	0	0
$87$	−617.400	−0.760830
$88$	0	0
$89$	−381.378	−0.454224	−0.227112	−	0.973869i	$-0.572928\pi$
$89$	−381.378	−0.454224	−0.227112	+	0.973869i	$0.572928\pi$
$90$	0	0
$91$	− 160.444i	− 0.184825i
$92$	0	0
$93$	442.466i	0.493350i
$94$	0	0
$95$	2315.47	2.50065
$96$	0	0
$97$	659.154	0.689969	0.344984	−	0.938608i	$-0.387884\pi$
$97$	659.154	0.689969	0.344984	+	0.938608i	$0.387884\pi$
$98$	0	0
$99$	− 483.199i	− 0.490539i
$100$	0	0
$101$	− 498.510i	− 0.491125i	−0.969381	−	0.245562i	$-0.921027\pi$
$101$	− 498.510i	− 0.491125i	0.969381	−	0.245562i	$-0.0789726\pi$
$102$	0	0
$103$	196.821	0.188285	0.0941425	−	0.995559i	$-0.469989\pi$
$103$	196.821	0.188285	0.0941425	+	0.995559i	$0.469989\pi$
$104$	0	0
$105$	1239.20	1.15175
$106$	0	0
$107$	− 359.378i	− 0.324695i	−0.986734	−	0.162347i	$-0.948093\pi$
$107$	− 359.378i	− 0.324695i	0.986734	−	0.162347i	$-0.0519065\pi$
$108$	0	0
$109$	1969.73i	1.73088i	0.501011	+	0.865441i	$0.332962\pi$
$109$	1969.73i	1.73088i	−0.501011	+	0.865441i	$0.667038\pi$
$110$	0	0
$111$	377.066	0.322429
$112$	0	0
$113$	−693.643	−0.577456	−0.288728	−	0.957411i	$-0.593232\pi$
$113$	−693.643	−0.577456	−0.288728	+	0.957411i	$0.593232\pi$
$114$	0	0
$115$	− 1826.66i	− 1.48119i
$116$	0	0
$117$	− 64.4003i	− 0.0508873i
$118$	0	0
$119$	−889.911	−0.685529
$120$	0	0
$121$	−1551.49	−1.16566
$122$	0	0
$123$	1519.33i	1.11377i
$124$	0	0
$125$	− 1646.53i	− 1.17816i
$126$	0	0
$127$	2656.78	1.85631	0.928153	−	0.372199i	$-0.121396\pi$
$127$	2656.78	1.85631	0.928153	+	0.372199i	$0.121396\pi$
$128$	0	0
$129$	1241.07	0.847053
$130$	0	0
$131$	615.734i	0.410664i	0.978692	+	0.205332i	$0.0658274\pi$
$131$	615.734i	0.410664i	−0.978692	+	0.205332i	$0.934173\pi$
$132$	0	0
$133$	2818.22i	1.83737i
$134$	0	0
$135$	497.400	0.317106
$136$	0	0
$137$	613.290	0.382459	0.191230	−	0.981545i	$-0.438753\pi$
$137$	613.290	0.382459	0.191230	+	0.981545i	$0.438753\pi$
$138$	0	0
$139$	1899.29i	1.15896i	0.814987	+	0.579480i	$0.196744\pi$
$139$	1899.29i	1.15896i	−0.814987	+	0.579480i	$0.803256\pi$
$140$	0	0
$141$	− 941.733i	− 0.562469i
$142$	0	0
$143$	−384.175	−0.224660
$144$	0	0
$145$	−3791.29	−2.17137
$146$	0	0
$147$	479.266i	0.268906i
$148$	0	0
$149$	976.377i	0.536832i	0.963303	+	0.268416i	$0.0865001\pi$
$149$	976.377i	0.536832i	−0.963303	+	0.268416i	$0.913500\pi$
$150$	0	0
$151$	−683.132	−0.368162	−0.184081	−	0.982911i	$-0.558931\pi$
$151$	−683.132	−0.368162	−0.184081	+	0.982911i	$0.558931\pi$
$152$	0	0
$153$	−357.199	−0.188744
$154$	0	0
$155$	2717.07i	1.40800i
$156$	0	0
$157$	− 511.109i	− 0.259815i	−0.991526	−	0.129907i	$-0.958532\pi$
$157$	− 511.109i	− 0.259815i	0.991526	−	0.129907i	$-0.0414680\pi$
$158$	0	0
$159$	−132.999	−0.0663366
$160$	0	0
$161$	2223.29	1.08832
$162$	0	0
$163$	2425.95i	1.16574i	0.812566	+	0.582869i	$0.198070\pi$
$163$	2425.95i	1.16574i	−0.812566	+	0.582869i	$0.801930\pi$
$164$	0	0
$165$	− 2967.20i	− 1.39998i
$166$	0	0
$167$	−337.332	−0.156309	−0.0781544	−	0.996941i	$-0.524903\pi$
$167$	−337.332	−0.156309	−0.0781544	+	0.996941i	$0.524903\pi$
$168$	0	0
$169$	2145.80	0.976694
$170$	0	0
$171$	1131.20i	0.505877i
$172$	0	0
$173$	− 2648.29i	− 1.16385i	−0.813243	−	0.581924i	$-0.802300\pi$
$173$	− 2648.29i	− 1.16385i	0.813243	−	0.581924i	$-0.197700\pi$
$174$	0	0
$175$	4806.82	2.07635
$176$	0	0
$177$	972.000	0.412768
$178$	0	0
$179$	2907.29i	1.21397i	0.794713	+	0.606986i	$0.207621\pi$
$179$	2907.29i	1.21397i	−0.794713	+	0.606986i	$0.792379\pi$
$180$	0	0
$181$	− 3682.80i	− 1.51238i	−0.654354	−	0.756188i	$-0.727060\pi$
$181$	− 3682.80i	− 1.51238i	0.654354	−	0.756188i	$-0.272940\pi$
$182$	0	0
$183$	−972.000	−0.392636
$184$	0	0
$185$	2315.47	0.920197
$186$	0	0
$187$	2130.85i	0.833277i
$188$	0	0
$189$	605.400i	0.232996i
$190$	0	0
$191$	1279.56	0.484740	0.242370	−	0.970184i	$-0.422075\pi$
$191$	1279.56	0.484740	0.242370	+	0.970184i	$0.422075\pi$
$192$	0	0
$193$	−4836.84	−1.80396	−0.901978	−	0.431783i	$-0.857885\pi$
$193$	−4836.84	−1.80396	−0.901978	+	0.431783i	$0.857885\pi$
$194$	0	0
$195$	− 395.465i	− 0.145230i
$196$	0	0
$197$	2869.31i	1.03771i	0.854861	+	0.518857i	$0.173642\pi$
$197$	2869.31i	1.03771i	−0.854861	+	0.518857i	$0.826358\pi$
$198$	0	0
$199$	−652.242	−0.232343	−0.116171	−	0.993229i	$-0.537062\pi$
$199$	−652.242	−0.232343	−0.116171	+	0.993229i	$0.537062\pi$
$200$	0	0
$201$	−1392.80	−0.488758
$202$	0	0
$203$	− 4614.49i	− 1.59544i
$204$	0	0
$205$	9329.82i	3.17865i
$206$	0	0
$207$	892.400	0.299643
$208$	0	0
$209$	6748.08	2.23337
$210$	0	0
$211$	537.511i	0.175373i	0.996148	+	0.0876866i	$0.0279474\pi$
$211$	537.511i	0.175373i	−0.996148	+	0.0876866i	$0.972053\pi$
$212$	0	0
$213$	− 3158.53i	− 1.01605i
$214$	0	0
$215$	7621.06	2.41745
$216$	0	0
$217$	−3307.02	−1.03454
$218$	0	0
$219$	− 3066.80i	− 0.946278i
$220$	0	0
$221$	283.997i	0.0864421i
$222$	0	0
$223$	−4041.98	−1.21377	−0.606885	−	0.794790i	$-0.707581\pi$
$223$	−4041.98	−1.21377	−0.606885	+	0.794790i	$0.707581\pi$
$224$	0	0
$225$	1929.40	0.571674
$226$	0	0
$227$	3070.22i	0.897699i	0.893607	+	0.448850i	$0.148166\pi$
$227$	3070.22i	0.897699i	−0.893607	+	0.448850i	$0.851834\pi$
$228$	0	0
$229$	205.110i	0.0591881i	0.999562	+	0.0295940i	$0.00942145\pi$
$229$	205.110i	0.0591881i	−0.999562	+	0.0295940i	$0.990579\pi$
$230$	0	0
$231$	3611.47	1.02864
$232$	0	0
$233$	−13.3776	−0.00376137	−0.00188068	−	0.999998i	$-0.500599\pi$
$233$	−13.3776	−0.00376137	−0.00188068	+	0.999998i	$0.500599\pi$
$234$	0	0
$235$	− 5782.93i	− 1.60526i
$236$	0	0
$237$	− 1808.73i	− 0.495737i
$238$	0	0
$239$	4327.99	1.17136	0.585679	−	0.810543i	$-0.300828\pi$
$239$	4327.99	1.17136	0.585679	+	0.810543i	$0.300828\pi$
$240$	0	0
$241$	−1508.31	−0.403150	−0.201575	−	0.979473i	$-0.564606\pi$
$241$	−1508.31	−0.403150	−0.201575	+	0.979473i	$0.564606\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	2943.04i	0.767446i
$246$	0	0
$247$	899.378	0.231684
$248$	0	0
$249$	−47.4652	−0.0120803
$250$	0	0
$251$	4871.47i	1.22504i	0.790456	+	0.612518i	$0.209843\pi$
$251$	4871.47i	1.22504i	−0.790456	+	0.612518i	$0.790157\pi$
$252$	0	0
$253$	− 5323.55i	− 1.32288i
$254$	0	0
$255$	−2193.47	−0.538668
$256$	0	0
$257$	−1665.55	−0.404258	−0.202129	−	0.979359i	$-0.564786\pi$
$257$	−1665.55	−0.404258	−0.202129	+	0.979359i	$0.564786\pi$
$258$	0	0
$259$	2818.22i	0.676122i
$260$	0	0
$261$	− 1852.20i	− 0.439265i
$262$	0	0
$263$	−7167.64	−1.68052	−0.840258	−	0.542188i	$-0.817596\pi$
$263$	−7167.64	−1.68052	−0.840258	+	0.542188i	$0.817596\pi$
$264$	0	0
$265$	−816.713	−0.189322
$266$	0	0
$267$	− 1144.13i	− 0.262246i
$268$	0	0
$269$	− 5453.84i	− 1.23616i	−0.786116	−	0.618079i	$-0.787911\pi$
$269$	− 5453.84i	− 1.23616i	0.786116	−	0.618079i	$-0.212089\pi$
$270$	0	0
$271$	−5416.20	−1.21406	−0.607031	−	0.794678i	$-0.707640\pi$
$271$	−5416.20	−1.21406	−0.607031	+	0.794678i	$0.707640\pi$
$272$	0	0
$273$	481.332	0.106709
$274$	0	0
$275$	− 11509.7i	− 2.52385i
$276$	0	0
$277$	2648.75i	0.574542i	0.957849	+	0.287271i	$0.0927479\pi$
$277$	2648.75i	0.574542i	−0.957849	+	0.287271i	$0.907252\pi$
$278$	0	0
$279$	−1327.40	−0.284836
$280$	0	0
$281$	−6664.57	−1.41486	−0.707429	−	0.706784i	$-0.750145\pi$
$281$	−6664.57	−1.41486	−0.707429	+	0.706784i	$0.750145\pi$
$282$	0	0
$283$	5630.84i	1.18275i	0.806396	+	0.591376i	$0.201415\pi$
$283$	5630.84i	1.18275i	−0.806396	+	0.591376i	$0.798585\pi$
$284$	0	0
$285$	6946.40i	1.44375i
$286$	0	0
$287$	−11355.6	−2.33554
$288$	0	0
$289$	−3337.80	−0.679381
$290$	0	0
$291$	1977.46i	0.398354i
$292$	0	0
$293$	− 908.374i	− 0.181119i	−0.995891	−	0.0905593i	$-0.971135\pi$
$293$	− 908.374i	− 0.181119i	0.995891	−	0.0905593i	$-0.0288655\pi$
$294$	0	0
$295$	5968.79	1.17802
$296$	0	0
$297$	1449.60	0.283213
$298$	0	0
$299$	− 709.517i	− 0.137232i
$300$	0	0
$301$	9275.82i	1.77624i
$302$	0	0
$303$	1495.53	0.283551
$304$	0	0
$305$	−5968.79	−1.12056
$306$	0	0
$307$	− 414.671i	− 0.0770896i	−0.999257	−	0.0385448i	$-0.987728\pi$
$307$	− 414.671i	− 0.0770896i	0.999257	−	0.0385448i	$-0.0122722\pi$
$308$	0	0
$309$	590.463i	0.108706i
$310$	0	0
$311$	−1615.91	−0.294629	−0.147315	−	0.989090i	$-0.547063\pi$
$311$	−1615.91	−0.294629	−0.147315	+	0.989090i	$0.547063\pi$
$312$	0	0
$313$	8479.33	1.53125	0.765623	−	0.643289i	$-0.222431\pi$
$313$	8479.33	1.53125	0.765623	+	0.643289i	$0.222431\pi$
$314$	0	0
$315$	3717.60i	0.664962i
$316$	0	0
$317$	− 6774.73i	− 1.20034i	−0.799873	−	0.600169i	$-0.795100\pi$
$317$	− 6774.73i	− 1.20034i	0.799873	−	0.600169i	$-0.204900\pi$
$318$	0	0
$319$	−11049.2	−1.93929
$320$	0	0
$321$	1078.13	0.187463
$322$	0	0
$323$	− 4988.44i	− 0.859332i
$324$	0	0
$325$	− 1534.00i	− 0.261818i
$326$	0	0
$327$	−5909.20	−0.999325
$328$	0	0
$329$	7038.57	1.17948
$330$	0	0
$331$	− 9292.36i	− 1.54306i	−0.636191	−	0.771532i	$-0.719491\pi$
$331$	− 9292.36i	− 1.54306i	0.636191	−	0.771532i	$-0.280509\pi$
$332$	0	0
$333$	1131.20i	0.186154i
$334$	0	0
$335$	−8552.80	−1.39489
$336$	0	0
$337$	6563.78	1.06098	0.530492	−	0.847690i	$-0.322007\pi$
$337$	6563.78	1.06098	0.530492	+	0.847690i	$0.322007\pi$
$338$	0	0
$339$	− 2080.93i	− 0.333394i
$340$	0	0
$341$	7918.49i	1.25751i
$342$	0	0
$343$	4108.75	0.646798
$344$	0	0
$345$	5479.99	0.855168
$346$	0	0
$347$	3870.93i	0.598855i	0.954119	+	0.299427i	$0.0967956\pi$
$347$	3870.93i	0.598855i	−0.954119	+	0.299427i	$0.903204\pi$
$348$	0	0
$349$	3474.57i	0.532922i	0.963846	+	0.266461i	$0.0858543\pi$
$349$	3474.57i	0.532922i	−0.963846	+	0.266461i	$0.914146\pi$
$350$	0	0
$351$	193.201	0.0293798
$352$	0	0
$353$	−4308.58	−0.649639	−0.324820	−	0.945776i	$-0.605304\pi$
$353$	−4308.58	−0.649639	−0.324820	+	0.945776i	$0.605304\pi$
$354$	0	0
$355$	− 19395.7i	− 2.89977i
$356$	0	0
$357$	− 2669.73i	− 0.395791i
$358$	0	0
$359$	8161.19	1.19981	0.599904	−	0.800072i	$-0.295205\pi$
$359$	8161.19	1.19981	0.599904	+	0.800072i	$0.295205\pi$
$360$	0	0
$361$	−8938.68	−1.30320
$362$	0	0
$363$	− 4654.47i	− 0.672992i
$364$	0	0
$365$	− 18832.4i	− 2.70064i
$366$	0	0
$367$	4427.66	0.629760	0.314880	−	0.949132i	$-0.398036\pi$
$367$	4427.66	0.629760	0.314880	+	0.949132i	$0.398036\pi$
$368$	0	0
$369$	−4558.00	−0.643035
$370$	0	0
$371$	− 994.045i	− 0.139106i
$372$	0	0
$373$	− 11278.5i	− 1.56562i	−0.622258	−	0.782812i	$-0.713785\pi$
$373$	− 11278.5i	− 1.56562i	0.622258	−	0.782812i	$-0.286215\pi$
$374$	0	0
$375$	4939.60	0.680213
$376$	0	0
$377$	−1472.62	−0.201177
$378$	0	0
$379$	− 709.683i	− 0.0961846i	−0.998843	−	0.0480923i	$-0.984686\pi$
$379$	− 709.683i	− 0.0961846i	0.998843	−	0.0480923i	$-0.0153142\pi$
$380$	0	0
$381$	7970.33i	1.07174i
$382$	0	0
$383$	−1233.69	−0.164592	−0.0822962	−	0.996608i	$-0.526225\pi$
$383$	−1233.69	−0.164592	−0.0822962	+	0.996608i	$0.526225\pi$
$384$	0	0
$385$	22177.1	2.93571
$386$	0	0
$387$	3723.20i	0.489046i
$388$	0	0
$389$	6830.06i	0.890226i	0.895474	+	0.445113i	$0.146836\pi$
$389$	6830.06i	0.890226i	−0.895474	+	0.445113i	$0.853164\pi$
$390$	0	0
$391$	−3935.37	−0.509003
$392$	0	0
$393$	−1847.20	−0.237097
$394$	0	0
$395$	− 11106.9i	− 1.41481i
$396$	0	0
$397$	− 11289.7i	− 1.42724i	−0.700535	−	0.713618i	$-0.747055\pi$
$397$	− 11289.7i	− 1.42724i	0.700535	−	0.713618i	$-0.252945\pi$
$398$	0	0
$399$	−8454.66	−1.06081
$400$	0	0
$401$	−3055.59	−0.380521	−0.190261	−	0.981734i	$-0.560933\pi$
$401$	−3055.59	−0.380521	−0.190261	+	0.981734i	$0.560933\pi$
$402$	0	0
$403$	1055.37i	0.130451i
$404$	0	0
$405$	1492.20i	0.183081i
$406$	0	0
$407$	6748.08	0.821843
$408$	0	0
$409$	−4089.01	−0.494349	−0.247175	−	0.968971i	$-0.579502\pi$
$409$	−4089.01	−0.494349	−0.247175	+	0.968971i	$0.579502\pi$
$410$	0	0
$411$	1839.87i	0.220813i
$412$	0	0
$413$	7264.79i	0.865562i
$414$	0	0
$415$	−291.471	−0.0344765
$416$	0	0
$417$	−5697.86	−0.669126
$418$	0	0
$419$	15397.0i	1.79520i	0.440806	+	0.897602i	$0.354693\pi$
$419$	15397.0i	1.79520i	−0.440806	+	0.897602i	$0.645307\pi$
$420$	0	0
$421$	− 1034.45i	− 0.119753i	−0.998206	−	0.0598766i	$-0.980929\pi$
$421$	− 1034.45i	− 0.119753i	0.998206	−	0.0598766i	$-0.0190707\pi$
$422$	0	0
$423$	2825.20	0.324742
$424$	0	0
$425$	−8508.40	−0.971101
$426$	0	0
$427$	− 7264.79i	− 0.823344i
$428$	0	0
$429$	− 1152.53i	− 0.129707i
$430$	0	0
$431$	−4943.86	−0.552523	−0.276261	−	0.961083i	$-0.589096\pi$
$431$	−4943.86	−0.552523	−0.276261	+	0.961083i	$0.589096\pi$
$432$	0	0
$433$	337.202	0.0374247	0.0187124	−	0.999825i	$-0.494043\pi$
$433$	337.202	0.0374247	0.0187124	+	0.999825i	$0.494043\pi$
$434$	0	0
$435$	− 11373.9i	− 1.25364i
$436$	0	0
$437$	12462.7i	1.36424i
$438$	0	0
$439$	−4493.93	−0.488573	−0.244286	−	0.969703i	$-0.578554\pi$
$439$	−4493.93	−0.488573	−0.244286	+	0.969703i	$0.578554\pi$
$440$	0	0
$441$	−1437.80	−0.155253
$442$	0	0
$443$	− 4292.26i	− 0.460341i	−0.973150	−	0.230171i	$-0.926072\pi$
$443$	− 4292.26i	− 0.460341i	0.973150	−	0.230171i	$-0.0739285\pi$
$444$	0	0
$445$	− 7025.82i	− 0.748440i
$446$	0	0
$447$	−2929.13	−0.309940
$448$	0	0
$449$	−4167.96	−0.438081	−0.219040	−	0.975716i	$-0.570293\pi$
$449$	−4167.96	−0.438081	−0.219040	+	0.975716i	$0.570293\pi$
$450$	0	0
$451$	27190.4i	2.83890i
$452$	0	0
$453$	− 2049.40i	− 0.212559i
$454$	0	0
$455$	2955.73	0.304543
$456$	0	0
$457$	−301.643	−0.0308759	−0.0154380	−	0.999881i	$-0.504914\pi$
$457$	−301.643	−0.0308759	−0.0154380	+	0.999881i	$0.504914\pi$
$458$	0	0
$459$	− 1071.60i	− 0.108971i
$460$	0	0
$461$	9611.88i	0.971085i	0.874213	+	0.485542i	$0.161378\pi$
$461$	9611.88i	0.971085i	−0.874213	+	0.485542i	$0.838622\pi$
$462$	0	0
$463$	−13251.0	−1.33008	−0.665041	−	0.746807i	$-0.731586\pi$
$463$	−13251.0	−1.33008	−0.665041	+	0.746807i	$0.731586\pi$
$464$	0	0
$465$	−8151.20	−0.812909
$466$	0	0
$467$	4432.00i	0.439161i	0.975594	+	0.219581i	$0.0704689\pi$
$467$	4432.00i	0.439161i	−0.975594	+	0.219581i	$0.929531\pi$
$468$	0	0
$469$	− 10409.9i	− 1.02491i
$470$	0	0
$471$	1533.33	0.150004
$472$	0	0
$473$	22210.5	2.15907
$474$	0	0
$475$	26944.9i	2.60277i
$476$	0	0
$477$	− 398.998i	− 0.0382995i
$478$	0	0
$479$	9076.49	0.865794	0.432897	−	0.901443i	$-0.357491\pi$
$479$	9076.49	0.865794	0.432897	+	0.901443i	$0.357491\pi$
$480$	0	0
$481$	899.378	0.0852559
$482$	0	0
$483$	6669.86i	0.628342i
$484$	0	0
$485$	12143.1i	1.13688i
$486$	0	0
$487$	3343.89	0.311142	0.155571	−	0.987825i	$-0.450278\pi$
$487$	3343.89	0.311142	0.155571	+	0.987825i	$0.450278\pi$
$488$	0	0
$489$	−7277.86	−0.673040
$490$	0	0
$491$	− 2423.73i	− 0.222773i	−0.993777	−	0.111386i	$-0.964471\pi$
$491$	− 2423.73i	− 0.222773i	0.993777	−	0.111386i	$-0.0355291\pi$
$492$	0	0
$493$	8167.95i	0.746179i
$494$	0	0
$495$	8901.60	0.808277
$496$	0	0
$497$	23607.1	2.13063
$498$	0	0
$499$	− 811.819i	− 0.0728296i	−0.999337	−	0.0364148i	$-0.988406\pi$
$499$	− 811.819i	− 0.0728296i	0.999337	−	0.0364148i	$-0.0115938\pi$
$500$	0	0
$501$	− 1012.00i	− 0.0902449i
$502$	0	0
$503$	−18192.4	−1.61264	−0.806320	−	0.591479i	$-0.798544\pi$
$503$	−18192.4	−1.61264	−0.806320	+	0.591479i	$0.798544\pi$
$504$	0	0
$505$	9183.65	0.809242
$506$	0	0
$507$	6437.39i	0.563895i
$508$	0	0
$509$	5645.44i	0.491610i	0.969319	+	0.245805i	$0.0790523\pi$
$509$	5645.44i	0.491610i	−0.969319	+	0.245805i	$0.920948\pi$
$510$	0	0
$511$	22921.5	1.98432
$512$	0	0
$513$	−3393.60	−0.292068
$514$	0	0
$515$	3625.88i	0.310243i
$516$	0	0
$517$	− 16853.5i	− 1.43369i
$518$	0	0
$519$	7944.87	0.671948
$520$	0	0
$521$	−12338.7	−1.03756	−0.518780	−	0.854908i	$-0.673614\pi$
$521$	−12338.7	−1.03756	−0.518780	+	0.854908i	$0.673614\pi$
$522$	0	0
$523$	− 10609.8i	− 0.887062i	−0.896259	−	0.443531i	$-0.853726\pi$
$523$	− 10609.8i	− 0.887062i	0.896259	−	0.443531i	$-0.146274\pi$
$524$	0	0
$525$	14420.5i	1.19878i
$526$	0	0
$527$	5853.65	0.483850
$528$	0	0
$529$	−2335.17	−0.191926
$530$	0	0
$531$	2916.00i	0.238312i
$532$	0	0
$533$	3623.91i	0.294501i
$534$	0	0
$535$	6620.53	0.535010
$536$	0	0
$537$	−8721.86	−0.700887
$538$	0	0
$539$	8577.07i	0.685419i
$540$	0	0
$541$	− 4035.42i	− 0.320696i	−0.987061	−	0.160348i	$-0.948738\pi$
$541$	− 4035.42i	− 0.320696i	0.987061	−	0.160348i	$-0.0512616\pi$
$542$	0	0
$543$	11048.4	0.873171
$544$	0	0
$545$	−36286.8	−2.85203
$546$	0	0
$547$	− 7407.45i	− 0.579012i	−0.957176	−	0.289506i	$-0.906509\pi$
$547$	− 7407.45i	− 0.579012i	0.957176	−	0.289506i	$-0.0934911\pi$
$548$	0	0
$549$	− 2916.00i	− 0.226688i
$550$	0	0
$551$	25866.7	1.99993
$552$	0	0
$553$	13518.6	1.03954
$554$	0	0
$555$	6946.40i	0.531276i
$556$	0	0
$557$	9500.77i	0.722730i	0.932424	+	0.361365i	$0.117689\pi$
$557$	9500.77i	0.722730i	−0.932424	+	0.361365i	$0.882311\pi$
$558$	0	0
$559$	2960.19	0.223976
$560$	0	0
$561$	−6392.54	−0.481093
$562$	0	0
$563$	− 2700.26i	− 0.202136i	−0.994880	−	0.101068i	$-0.967774\pi$
$563$	− 2700.26i	− 0.202136i	0.994880	−	0.101068i	$-0.0322259\pi$
$564$	0	0
$565$	− 12778.4i	− 0.951492i
$566$	0	0
$567$	−1816.20	−0.134521
$568$	0	0
$569$	−15904.9	−1.17183	−0.585913	−	0.810374i	$-0.699264\pi$
$569$	−15904.9	−1.17183	−0.585913	+	0.810374i	$0.699264\pi$
$570$	0	0
$571$	18234.0i	1.33638i	0.743992	+	0.668188i	$0.232930\pi$
$571$	18234.0i	1.33638i	−0.743992	+	0.668188i	$0.767070\pi$
$572$	0	0
$573$	3838.67i	0.279865i
$574$	0	0
$575$	21256.7	1.54168
$576$	0	0
$577$	4869.57	0.351339	0.175670	−	0.984449i	$-0.443791\pi$
$577$	4869.57	0.351339	0.175670	+	0.984449i	$0.443791\pi$
$578$	0	0
$579$	− 14510.5i	− 1.04151i
$580$	0	0
$581$	− 354.758i	− 0.0253319i
$582$	0	0
$583$	−2380.19	−0.169086
$584$	0	0
$585$	1186.40	0.0838486
$586$	0	0
$587$	− 1616.99i	− 0.113697i	−0.998383	−	0.0568485i	$-0.981895\pi$
$587$	− 1616.99i	− 0.113697i	0.998383	−	0.0568485i	$-0.0181052\pi$
$588$	0	0
$589$	− 18537.7i	− 1.29683i
$590$	0	0
$591$	−8607.93	−0.599125
$592$	0	0
$593$	8117.01	0.562101	0.281050	−	0.959693i	$-0.409317\pi$
$593$	8117.01	0.562101	0.281050	+	0.959693i	$0.409317\pi$
$594$	0	0
$595$	− 16394.1i	− 1.12957i
$596$	0	0
$597$	− 1956.73i	− 0.134143i
$598$	0	0
$599$	−9536.40	−0.650495	−0.325248	−	0.945629i	$-0.605448\pi$
$599$	−9536.40	−0.650495	−0.325248	+	0.945629i	$0.605448\pi$
$600$	0	0
$601$	16247.8	1.10276	0.551381	−	0.834253i	$-0.314101\pi$
$601$	16247.8	1.10276	0.551381	+	0.834253i	$0.314101\pi$
$602$	0	0
$603$	− 4178.39i	− 0.282185i
$604$	0	0
$605$	− 28581.9i	− 1.92069i
$606$	0	0
$607$	27725.7	1.85396	0.926980	−	0.375111i	$-0.122395\pi$
$607$	27725.7	1.85396	0.926980	+	0.375111i	$0.122395\pi$
$608$	0	0
$609$	13843.5	0.921125
$610$	0	0
$611$	− 2246.22i	− 0.148727i
$612$	0	0
$613$	927.190i	0.0610911i	0.999533	+	0.0305456i	$0.00972447\pi$
$613$	927.190i	0.0610911i	−0.999533	+	0.0305456i	$0.990276\pi$
$614$	0	0
$615$	−27989.5	−1.83519
$616$	0	0
$617$	18727.8	1.22196	0.610982	−	0.791644i	$-0.290775\pi$
$617$	18727.8	1.22196	0.610982	+	0.791644i	$0.290775\pi$
$618$	0	0
$619$	3210.22i	0.208449i	0.994554	+	0.104224i	$0.0332360\pi$
$619$	3210.22i	0.208449i	−0.994554	+	0.104224i	$0.966764\pi$
$620$	0	0
$621$	2677.20i	0.172999i
$622$	0	0
$623$	8551.33	0.549922
$624$	0	0
$625$	3535.57	0.226276
$626$	0	0
$627$	20244.3i	1.28944i
$628$	0	0
$629$	− 4988.44i	− 0.316220i
$630$	0	0
$631$	11911.2	0.751468	0.375734	−	0.926728i	$-0.377391\pi$
$631$	11911.2	0.751468	0.375734	+	0.926728i	$0.377391\pi$
$632$	0	0
$633$	−1612.53	−0.101252
$634$	0	0
$635$	48943.7i	3.05869i
$636$	0	0
$637$	1143.14i	0.0711036i
$638$	0	0
$639$	9475.60	0.586618
$640$	0	0
$641$	17232.6	1.06185	0.530924	−	0.847419i	$-0.321845\pi$
$641$	17232.6	1.06185	0.530924	+	0.847419i	$0.321845\pi$
$642$	0	0
$643$	12754.6i	0.782262i	0.920335	+	0.391131i	$0.127916\pi$
$643$	12754.6i	0.782262i	−0.920335	+	0.391131i	$0.872084\pi$
$644$	0	0
$645$	22863.2i	1.39572i
$646$	0	0
$647$	−9441.73	−0.573714	−0.286857	−	0.957973i	$-0.592610\pi$
$647$	−9441.73	−0.573714	−0.286857	+	0.957973i	$0.592610\pi$
$648$	0	0
$649$	17395.2	1.05211
$650$	0	0
$651$	− 9921.06i	− 0.597292i
$652$	0	0
$653$	5198.99i	0.311565i	0.987791	+	0.155783i	$0.0497900\pi$
$653$	5198.99i	0.311565i	−0.987791	+	0.155783i	$0.950210\pi$
$654$	0	0
$655$	−11343.2	−0.676664
$656$	0	0
$657$	9200.39	0.546334
$658$	0	0
$659$	− 6508.01i	− 0.384698i	−0.981327	−	0.192349i	$-0.938389\pi$
$659$	− 6508.01i	− 0.384698i	0.981327	−	0.192349i	$-0.0616105\pi$
$660$	0	0
$661$	25280.2i	1.48757i	0.668419	+	0.743785i	$0.266972\pi$
$661$	25280.2i	1.48757i	−0.668419	+	0.743785i	$0.733028\pi$
$662$	0	0
$663$	−851.991	−0.0499074
$664$	0	0
$665$	−51917.8	−3.02750
$666$	0	0
$667$	− 20406.2i	− 1.18460i
$668$	0	0
$669$	− 12125.9i	− 0.700770i
$670$	0	0
$671$	−17395.2	−1.00079
$672$	0	0
$673$	−5525.89	−0.316504	−0.158252	−	0.987399i	$-0.550586\pi$
$673$	−5525.89	−0.316504	−0.158252	+	0.987399i	$0.550586\pi$
$674$	0	0
$675$	5788.20i	0.330056i
$676$	0	0
$677$	− 6293.21i	− 0.357264i	−0.983916	−	0.178632i	$-0.942833\pi$
$677$	− 6293.21i	− 0.357264i	0.983916	−	0.178632i	$-0.0571671\pi$
$678$	0	0
$679$	−14779.7	−0.835335
$680$	0	0
$681$	−9210.66	−0.518287
$682$	0	0
$683$	− 5675.91i	− 0.317984i	−0.987280	−	0.158992i	$-0.949176\pi$
$683$	− 5675.91i	− 0.317984i	0.987280	−	0.158992i	$-0.0508243\pi$
$684$	0	0
$685$	11298.2i	0.630190i
$686$	0	0
$687$	−615.331	−0.0341722
$688$	0	0
$689$	−317.229	−0.0175406
$690$	0	0
$691$	3617.79i	0.199171i	0.995029	+	0.0995854i	$0.0317517\pi$
$691$	3617.79i	0.199171i	−0.995029	+	0.0995854i	$0.968248\pi$
$692$	0	0
$693$	10834.4i	0.593888i
$694$	0	0
$695$	−34989.1	−1.90966
$696$	0	0
$697$	20100.2	1.09232
$698$	0	0
$699$	− 40.1329i	− 0.00217163i
$700$	0	0
$701$	7938.43i	0.427718i	0.976865	+	0.213859i	$0.0686033\pi$
$701$	7938.43i	0.427718i	−0.976865	+	0.213859i	$0.931397\pi$
$702$	0	0
$703$	−15797.7	−0.847540
$704$	0	0
$705$	17348.8	0.926799
$706$	0	0
$707$	11177.7i	0.594597i
$708$	0	0
$709$	− 25691.6i	− 1.36088i	−0.732802	−	0.680442i	$-0.761788\pi$
$709$	− 25691.6i	− 1.36088i	0.732802	−	0.680442i	$-0.238212\pi$
$710$	0	0
$711$	5426.19	0.286214
$712$	0	0
$713$	−14624.3	−0.768142
$714$	0	0
$715$	− 7077.35i	− 0.370179i
$716$	0	0
$717$	12984.0i	0.676284i
$718$	0	0
$719$	36803.3	1.90895	0.954473	−	0.298298i	$-0.0964190\pi$
$719$	36803.3	1.90895	0.954473	+	0.298298i	$0.0964190\pi$
$720$	0	0
$721$	−4413.16	−0.227954
$722$	0	0
$723$	− 4524.94i	− 0.232759i
$724$	0	0
$725$	− 44118.9i	− 2.26005i
$726$	0	0
$727$	−28333.2	−1.44542	−0.722709	−	0.691152i	$-0.757103\pi$
$727$	−28333.2	−1.44542	−0.722709	+	0.691152i	$0.757103\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 16418.8i	− 0.830742i
$732$	0	0
$733$	10767.9i	0.542592i	0.962496	+	0.271296i	$0.0874522\pi$
$733$	10767.9i	0.542592i	−0.962496	+	0.271296i	$0.912548\pi$
$734$	0	0
$735$	−8829.13	−0.443085
$736$	0	0
$737$	−24925.9	−1.24580
$738$	0	0
$739$	17301.4i	0.861221i	0.902538	+	0.430610i	$0.141702\pi$
$739$	17301.4i	0.861221i	−0.902538	+	0.430610i	$0.858298\pi$
$740$	0	0
$741$	2698.13i	0.133763i
$742$	0	0
$743$	−24110.8	−1.19050	−0.595248	−	0.803542i	$-0.702946\pi$
$743$	−24110.8	−1.19050	−0.595248	+	0.803542i	$0.702946\pi$
$744$	0	0
$745$	−17987.0	−0.884555
$746$	0	0
$747$	− 142.396i	− 0.00697455i
$748$	0	0
$749$	8058.04i	0.393103i
$750$	0	0
$751$	30052.8	1.46024	0.730121	−	0.683318i	$-0.239464\pi$
$751$	30052.8	1.46024	0.730121	+	0.683318i	$0.239464\pi$
$752$	0	0
$753$	−14614.4	−0.707275
$754$	0	0
$755$	− 12584.8i	− 0.606633i
$756$	0	0
$757$	25599.4i	1.22910i	0.788879	+	0.614549i	$0.210662\pi$
$757$	25599.4i	1.22910i	−0.788879	+	0.614549i	$0.789338\pi$
$758$	0	0
$759$	15970.6	0.763765
$760$	0	0
$761$	28904.5	1.37685	0.688427	−	0.725306i	$-0.258302\pi$
$761$	28904.5	1.37685	0.688427	+	0.725306i	$0.258302\pi$
$762$	0	0
$763$	− 44165.7i	− 2.09555i
$764$	0	0
$765$	− 6580.40i	− 0.311000i
$766$	0	0
$767$	2318.41	0.109143
$768$	0	0
$769$	−8756.13	−0.410603	−0.205302	−	0.978699i	$-0.565818\pi$
$769$	−8756.13	−0.410603	−0.205302	+	0.978699i	$0.565818\pi$
$770$	0	0
$771$	− 4996.66i	− 0.233399i
$772$	0	0
$773$	28289.8i	1.31632i	0.752879	+	0.658159i	$0.228665\pi$
$773$	28289.8i	1.31632i	−0.752879	+	0.658159i	$0.771335\pi$
$774$	0	0
$775$	−31618.3	−1.46550
$776$	0	0
$777$	−8454.66	−0.390359
$778$	0	0
$779$	− 63654.4i	− 2.92767i
$780$	0	0
$781$	− 56526.0i	− 2.58983i
$782$	0	0
$783$	5556.60	0.253610
$784$	0	0
$785$	9415.75	0.428105
$786$	0	0
$787$	− 4859.74i	− 0.220116i	−0.993925	−	0.110058i	$-0.964896\pi$
$787$	− 4859.74i	− 0.220116i	0.993925	−	0.110058i	$-0.0351036\pi$
$788$	0	0
$789$	− 21502.9i	− 0.970246i
$790$	0	0
$791$	15553.0	0.699117
$792$	0	0
$793$	−2318.41	−0.103820
$794$	0	0
$795$	− 2450.14i	− 0.109305i
$796$	0	0
$797$	− 17361.7i	− 0.771623i	−0.922578	−	0.385811i	$-0.873922\pi$
$797$	− 17361.7i	− 0.771623i	0.922578	−	0.385811i	$-0.126078\pi$
$798$	0	0
$799$	−12458.8	−0.551638
$800$	0	0
$801$	3432.40	0.151408
$802$	0	0
$803$	− 54884.2i	− 2.41198i
$804$	0	0
$805$	40957.8i	1.79326i
$806$	0	0
$807$	16361.5	0.713696
$808$	0	0
$809$	24475.3	1.06367	0.531833	−	0.846849i	$-0.321503\pi$
$809$	24475.3	1.06367	0.531833	+	0.846849i	$0.321503\pi$
$810$	0	0
$811$	19875.4i	0.860566i	0.902694	+	0.430283i	$0.141586\pi$
$811$	19875.4i	0.860566i	−0.902694	+	0.430283i	$0.858414\pi$
$812$	0	0
$813$	− 16248.6i	− 0.700939i
$814$	0	0
$815$	−44691.4	−1.92083
$816$	0	0
$817$	−51996.1	−2.22658
$818$	0	0
$819$	1444.00i	0.0616085i
$820$	0	0
$821$	21682.1i	0.921693i	0.887480	+	0.460846i	$0.152454\pi$
$821$	21682.1i	0.921693i	−0.887480	+	0.460846i	$0.847546\pi$
$822$	0	0
$823$	−6698.17	−0.283698	−0.141849	−	0.989888i	$-0.545305\pi$
$823$	−6698.17	−0.283698	−0.141849	+	0.989888i	$0.545305\pi$
$824$	0	0
$825$	34529.0	1.45715
$826$	0	0
$827$	− 3390.85i	− 0.142577i	−0.997456	−	0.0712886i	$-0.977289\pi$
$827$	− 3390.85i	− 0.142577i	0.997456	−	0.0712886i	$-0.0227111\pi$
$828$	0	0
$829$	40093.4i	1.67974i	0.542789	+	0.839869i	$0.317368\pi$
$829$	40093.4i	1.67974i	−0.542789	+	0.839869i	$0.682632\pi$
$830$	0	0
$831$	−7946.25	−0.331712
$832$	0	0
$833$	6340.50	0.263728
$834$	0	0
$835$	− 6214.40i	− 0.257555i
$836$	0	0
$837$	− 3982.19i	− 0.164450i
$838$	0	0
$839$	6172.12	0.253975	0.126988	−	0.991904i	$-0.459469\pi$
$839$	6172.12	0.253975	0.126988	+	0.991904i	$0.459469\pi$
$840$	0	0
$841$	−17964.6	−0.736585
$842$	0	0
$843$	− 19993.7i	− 0.816869i
$844$	0	0
$845$	39530.3i	1.60933i
$846$	0	0
$847$	34787.8	1.41124
$848$	0	0
$849$	−16892.5	−0.682862
$850$	0	0
$851$	12462.7i	0.502018i
$852$	0	0
$853$	− 276.632i	− 0.0111040i	−0.999985	−	0.00555198i	$-0.998233\pi$
$853$	− 276.632i	− 0.0111040i	0.999985	−	0.00555198i	$-0.00176726\pi$
$854$	0	0
$855$	−20839.2	−0.833550
$856$	0	0
$857$	−3704.41	−0.147655	−0.0738274	−	0.997271i	$-0.523521\pi$
$857$	−3704.41	−0.147655	−0.0738274	+	0.997271i	$0.523521\pi$
$858$	0	0
$859$	26915.5i	1.06909i	0.845141	+	0.534544i	$0.179516\pi$
$859$	26915.5i	1.06909i	−0.845141	+	0.534544i	$0.820484\pi$
$860$	0	0
$861$	− 34066.8i	− 1.34842i
$862$	0	0
$863$	−23623.9	−0.931828	−0.465914	−	0.884830i	$-0.654274\pi$
$863$	−23623.9	−0.931828	−0.465914	+	0.884830i	$0.654274\pi$
$864$	0	0
$865$	48787.3	1.91771
$866$	0	0
$867$	− 10013.4i	− 0.392241i
$868$	0	0
$869$	− 32369.5i	− 1.26359i
$870$	0	0
$871$	−3322.10	−0.129236
$872$	0	0
$873$	−5932.39	−0.229990
$874$	0	0
$875$	36918.9i	1.42638i
$876$	0	0
$877$	− 14094.0i	− 0.542667i	−0.962485	−	0.271333i	$-0.912535\pi$
$877$	− 14094.0i	− 0.542667i	0.962485	−	0.271333i	$-0.0874646\pi$
$878$	0	0
$879$	2725.12	0.104569
$880$	0	0
$881$	18967.3	0.725341	0.362671	−	0.931917i	$-0.381865\pi$
$881$	18967.3	0.725341	0.362671	+	0.931917i	$0.381865\pi$
$882$	0	0
$883$	− 32886.0i	− 1.25334i	−0.779283	−	0.626672i	$-0.784417\pi$
$883$	− 32886.0i	− 1.25334i	0.779283	−	0.626672i	$-0.215583\pi$
$884$	0	0
$885$	17906.4i	0.680132i
$886$	0	0
$887$	27945.6	1.05786	0.528929	−	0.848666i	$-0.322594\pi$
$887$	27945.6	1.05786	0.528929	+	0.848666i	$0.322594\pi$
$888$	0	0
$889$	−59570.8	−2.24740
$890$	0	0
$891$	4348.79i	0.163513i
$892$	0	0
$893$	39455.1i	1.47852i
$894$	0	0
$895$	−53558.6	−2.00030
$896$	0	0
$897$	2128.55	0.0792310
$898$	0	0
$899$	30353.1i	1.12607i
$900$	0	0
$901$	1759.53i	0.0650592i
$902$	0	0
$903$	−27827.4	−1.02551
$904$	0	0
$905$	67845.2	2.49199
$906$	0	0
$907$	− 5544.89i	− 0.202993i	−0.994836	−	0.101497i	$-0.967637\pi$
$907$	− 5544.89i	− 0.202993i	0.994836	−	0.101497i	$-0.0323631\pi$
$908$	0	0
$909$	4486.59i	0.163708i
$910$	0	0
$911$	19638.9	0.714233	0.357116	−	0.934060i	$-0.383760\pi$
$911$	19638.9	0.714233	0.357116	+	0.934060i	$0.383760\pi$
$912$	0	0
$913$	−849.451	−0.0307916
$914$	0	0
$915$	− 17906.4i	− 0.646958i
$916$	0	0
$917$	− 13806.1i	− 0.497184i
$918$	0	0
$919$	−22128.7	−0.794297	−0.397149	−	0.917754i	$-0.630000\pi$
$919$	−22128.7	−0.794297	−0.397149	+	0.917754i	$0.630000\pi$
$920$	0	0
$921$	1244.01	0.0445077
$922$	0	0
$923$	− 7533.72i	− 0.268663i
$924$	0	0
$925$	26944.9i	0.957775i
$926$	0	0
$927$	−1771.39	−0.0627616
$928$	0	0
$929$	17599.3	0.621544	0.310772	−	0.950484i	$-0.399412\pi$
$929$	17599.3	0.621544	0.310772	+	0.950484i	$0.399412\pi$
$930$	0	0
$931$	− 20079.5i	− 0.706850i
$932$	0	0
$933$	− 4847.72i	− 0.170104i
$934$	0	0
$935$	−39254.9	−1.37302
$936$	0	0
$937$	−441.299	−0.0153859	−0.00769297	−	0.999970i	$-0.502449\pi$
$937$	−441.299	−0.0153859	−0.00769297	+	0.999970i	$0.502449\pi$
$938$	0	0
$939$	25438.0i	0.884065i
$940$	0	0
$941$	− 5408.01i	− 0.187350i	−0.995603	−	0.0936749i	$-0.970139\pi$
$941$	− 5408.01i	− 0.187350i	0.995603	−	0.0936749i	$-0.0298614\pi$
$942$	0	0
$943$	−50216.8	−1.73413
$944$	0	0
$945$	−11152.8	−0.383916
$946$	0	0
$947$	34237.1i	1.17482i	0.809289	+	0.587410i	$0.199852\pi$
$947$	34237.1i	1.17482i	−0.809289	+	0.587410i	$0.800148\pi$
$948$	0	0
$949$	− 7314.92i	− 0.250213i
$950$	0	0
$951$	20324.2	0.693015
$952$	0	0
$953$	−21410.7	−0.727766	−0.363883	−	0.931445i	$-0.618549\pi$
$953$	−21410.7	−0.727766	−0.363883	+	0.931445i	$0.618549\pi$
$954$	0	0
$955$	23572.2i	0.798722i
$956$	0	0
$957$	− 33147.5i	− 1.11965i
$958$	0	0
$959$	−13751.3	−0.463038
$960$	0	0
$961$	−8038.09	−0.269816
$962$	0	0
$963$	3234.40i	0.108232i
$964$	0	0
$965$	− 89105.3i	− 2.97243i
$966$	0	0
$967$	53874.6	1.79161	0.895806	−	0.444445i	$-0.146599\pi$
$967$	53874.6	1.79161	0.895806	+	0.444445i	$0.146599\pi$
$968$	0	0
$969$	14965.3	0.496136
$970$	0	0
$971$	42901.5i	1.41789i	0.705262	+	0.708947i	$0.250829\pi$
$971$	42901.5i	1.41789i	−0.705262	+	0.708947i	$0.749171\pi$
$972$	0	0
$973$	− 42586.2i	− 1.40314i
$974$	0	0
$975$	4602.00	0.151161
$976$	0	0
$977$	58636.5	1.92011	0.960055	−	0.279812i	$-0.0902721\pi$
$977$	58636.5	1.92011	0.960055	+	0.279812i	$0.0902721\pi$
$978$	0	0
$979$	− 20475.7i	− 0.668444i
$980$	0	0
$981$	− 17727.6i	− 0.576961i
$982$	0	0
$983$	25296.7	0.820793	0.410396	−	0.911907i	$-0.365390\pi$
$983$	25296.7	0.820793	0.410396	+	0.911907i	$0.365390\pi$
$984$	0	0
$985$	−52859.0	−1.70988
$986$	0	0
$987$	21115.7i	0.680973i
$988$	0	0
$989$	41019.6i	1.31885i
$990$	0	0
$991$	10605.2	0.339944	0.169972	−	0.985449i	$-0.445632\pi$
$991$	10605.2	0.339944	0.169972	+	0.985449i	$0.445632\pi$
$992$	0	0
$993$	27877.1	0.890888
$994$	0	0
$995$	− 12015.7i	− 0.382839i
$996$	0	0
$997$	− 5770.19i	− 0.183294i	−0.995792	−	0.0916469i	$-0.970787\pi$
$997$	− 5770.19i	− 0.183294i	0.995792	−	0.0916469i	$-0.0292131\pi$
$998$	0	0
$999$	−3393.60	−0.107476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.d.c.193.4	yes	4
3.2	odd	2		1152.4.d.i.577.1		4
4.3	odd	2		384.4.d.e.193.2	yes	4
8.3	odd	2		384.4.d.e.193.3	yes	4
8.5	even	2	inner	384.4.d.c.193.1	✓	4
12.11	even	2		1152.4.d.o.577.1		4
16.3	odd	4		768.4.a.p.1.2		2
16.5	even	4		768.4.a.k.1.1		2
16.11	odd	4		768.4.a.e.1.1		2
16.13	even	4		768.4.a.j.1.2		2
24.5	odd	2		1152.4.d.i.577.4		4
24.11	even	2		1152.4.d.o.577.4		4
48.5	odd	4		2304.4.a.bq.1.2		2
48.11	even	4		2304.4.a.bp.1.2		2
48.29	odd	4		2304.4.a.t.1.1		2
48.35	even	4		2304.4.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.c.193.1	✓	4	8.5	even	2	inner
384.4.d.c.193.4	yes	4	1.1	even	1	trivial
384.4.d.e.193.2	yes	4	4.3	odd	2
384.4.d.e.193.3	yes	4	8.3	odd	2
768.4.a.e.1.1		2	16.11	odd	4
768.4.a.j.1.2		2	16.13	even	4
768.4.a.k.1.1		2	16.5	even	4
768.4.a.p.1.2		2	16.3	odd	4
1152.4.d.i.577.1		4	3.2	odd	2
1152.4.d.i.577.4		4	24.5	odd	2
1152.4.d.o.577.1		4	12.11	even	2
1152.4.d.o.577.4		4	24.11	even	2
2304.4.a.s.1.1		2	48.35	even	4
2304.4.a.t.1.1		2	48.29	odd	4
2304.4.a.bp.1.2		2	48.11	even	4
2304.4.a.bq.1.2		2	48.5	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +18.4222i q^{5} -22.4222 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +18.4222i q^{5} -22.4222 q^{7} -9.00000 q^{9} +53.6888i q^{11} +7.15559i q^{13} -55.2666 q^{15} +39.6888 q^{17} -125.689i q^{19} -67.2666i q^{21} -99.1556 q^{23} -214.378 q^{25} -27.0000i q^{27} +205.800i q^{29} +147.489 q^{31} -161.066 q^{33} -413.066i q^{35} -125.689i q^{37} -21.4668 q^{39} +506.444 q^{41} -413.689i q^{43} -165.800i q^{45} -313.911 q^{47} +159.755 q^{49} +119.066i q^{51} +44.3331i q^{53} -989.066 q^{55} +377.066 q^{57} -324.000i q^{59} +324.000i q^{61} +201.800 q^{63} -131.822 q^{65} +464.266i q^{67} -297.467i q^{69} -1052.84 q^{71} -1022.27 q^{73} -643.133i q^{75} -1203.82i q^{77} -602.910 q^{79} +81.0000 q^{81} +15.8217i q^{83} +731.156i q^{85} -617.400 q^{87} -381.378 q^{89} -160.444i q^{91} +442.466i q^{93} +2315.47 q^{95} +659.154 q^{97} -483.199i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{7} - 36 q^{9}+O(q^{10}) \) 4 * q - 32 * q^7 - 36 * q^9	\( 4 q - 32 q^{7} - 36 q^{9} - 48 q^{15} - 72 q^{17} - 512 q^{23} - 396 q^{25} - 160 q^{31} + 48 q^{33} - 432 q^{39} + 872 q^{41} - 448 q^{47} - 284 q^{49} - 3264 q^{55} + 816 q^{57} + 288 q^{63} + 1088 q^{65} - 4096 q^{71} - 1320 q^{73} + 992 q^{79} + 324 q^{81} - 912 q^{87} - 1064 q^{89} + 4416 q^{95} - 2440 q^{97}+O(q^{100}) \) 4 * q - 32 * q^7 - 36 * q^9 - 48 * q^15 - 72 * q^17 - 512 * q^23 - 396 * q^25 - 160 * q^31 + 48 * q^33 - 432 * q^39 + 872 * q^41 - 448 * q^47 - 284 * q^49 - 3264 * q^55 + 816 * q^57 + 288 * q^63 + 1088 * q^65 - 4096 * q^71 - 1320 * q^73 + 992 * q^79 + 324 * q^81 - 912 * q^87 - 1064 * q^89 + 4416 * q^95 - 2440 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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