LMFDB - Embedded newform 384.5.b.c.319.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,5,Mod(319,384)]

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, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("384.319");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	384.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$39.6940658242$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.5
Root			$0.228425 - 1.39564i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.5.b.c.319.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{3} -39.7490i q^{5} +46.0431i q^{7} +27.0000 q^{9} +O(q^{10})$	$q+5.19615 q^{3} -39.7490i q^{5} +46.0431i q^{7} +27.0000 q^{9} -181.283 q^{11} +183.498i q^{13} -206.542i q^{15} +427.992 q^{17} -668.558 q^{19} +239.247i q^{21} +882.463i q^{23} -954.984 q^{25} +140.296 q^{27} +807.247i q^{29} -391.276i q^{31} -941.976 q^{33} +1830.17 q^{35} -466.980i q^{37} +953.484i q^{39} +2159.93 q^{41} +509.182 q^{43} -1073.22i q^{45} +2056.83i q^{47} +281.031 q^{49} +2223.91 q^{51} +753.725i q^{53} +7205.84i q^{55} -3473.93 q^{57} -1309.43 q^{59} +801.913i q^{61} +1243.16i q^{63} +7293.87 q^{65} +505.813 q^{67} +4585.41i q^{69} -2170.94i q^{71} -2297.97 q^{73} -4962.24 q^{75} -8346.85i q^{77} +10705.3i q^{79} +729.000 q^{81} -2977.81 q^{83} -17012.3i q^{85} +4194.58i q^{87} -6493.92 q^{89} -8448.82 q^{91} -2033.13i q^{93} +26574.5i q^{95} -8189.92 q^{97} -4894.65 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 216 q^{9}+O(q^{10}) $ 8 * q + 216 * q^9	$ 8 q + 216 q^{9} + 1392 q^{17} - 3576 q^{25} - 1440 q^{33} - 1008 q^{41} + 10376 q^{49} - 9504 q^{57} + 23808 q^{65} - 10256 q^{73} + 5832 q^{81} - 31632 q^{89} - 45200 q^{97}+O(q^{100}) $ 8 * q + 216 * q^9 + 1392 * q^17 - 3576 * q^25 - 1440 * q^33 - 1008 * q^41 + 10376 * q^49 - 9504 * q^57 + 23808 * q^65 - 10256 * q^73 + 5832 * q^81 - 31632 * q^89 - 45200 * 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
$3$	5.19615	0.577350
$3$	5.19615	0.577350
$4$	0	0
$5$	− 39.7490i	− 1.58996i	−0.606635	−	0.794980i	$-0.707481\pi$
$5$	− 39.7490i	− 1.58996i	0.606635	−	0.794980i	$-0.292519\pi$
$6$	0	0
$7$	46.0431i	0.939655i	0.882758	+	0.469828i	$0.155684\pi$
$7$	46.0431i	0.939655i	−0.882758	+	0.469828i	$0.844316\pi$
$8$	0	0
$9$	27.0000	0.333333
$10$	0	0
$11$	−181.283	−1.49821	−0.749105	−	0.662451i	$-0.769516\pi$
$11$	−181.283	−1.49821	−0.749105	+	0.662451i	$0.769516\pi$
$12$	0	0
$13$	183.498i	1.08579i	0.839801	+	0.542894i	$0.182671\pi$
$13$	183.498i	1.08579i	−0.839801	+	0.542894i	$0.817329\pi$
$14$	0	0
$15$	− 206.542i	− 0.917964i
$16$	0	0
$17$	427.992	1.48094	0.740471	−	0.672089i	$-0.234603\pi$
$17$	427.992	1.48094	0.740471	+	0.672089i	$0.234603\pi$
$18$	0	0
$19$	−668.558	−1.85196	−0.925981	−	0.377571i	$-0.876759\pi$
$19$	−668.558	−1.85196	−0.925981	+	0.377571i	$0.876759\pi$
$20$	0	0
$21$	239.247i	0.542510i
$22$	0	0
$23$	882.463i	1.66817i	0.551635	+	0.834086i	$0.314004\pi$
$23$	882.463i	1.66817i	−0.551635	+	0.834086i	$0.685996\pi$
$24$	0	0
$25$	−954.984	−1.52797
$26$	0	0
$27$	140.296	0.192450
$28$	0	0
$29$	807.247i	0.959866i	0.877305	+	0.479933i	$0.159339\pi$
$29$	807.247i	0.959866i	−0.877305	+	0.479933i	$0.840661\pi$
$30$	0	0
$31$	− 391.276i	− 0.407155i	−0.979059	−	0.203577i	$-0.934743\pi$
$31$	− 391.276i	− 0.407155i	0.979059	−	0.203577i	$-0.0652569\pi$
$32$	0	0
$33$	−941.976	−0.864992
$34$	0	0
$35$	1830.17	1.49402
$36$	0	0
$37$	− 466.980i	− 0.341111i	−0.985348	−	0.170555i	$-0.945444\pi$
$37$	− 466.980i	− 0.341111i	0.985348	−	0.170555i	$-0.0545561\pi$
$38$	0	0
$39$	953.484i	0.626880i
$40$	0	0
$41$	2159.93	1.28491	0.642454	−	0.766325i	$-0.277917\pi$
$41$	2159.93	1.28491	0.642454	+	0.766325i	$0.277917\pi$
$42$	0	0
$43$	509.182	0.275382	0.137691	−	0.990475i	$-0.456032\pi$
$43$	509.182	0.275382	0.137691	+	0.990475i	$0.456032\pi$
$44$	0	0
$45$	− 1073.22i	− 0.529987i
$46$	0	0
$47$	2056.83i	0.931116i	0.885017	+	0.465558i	$0.154146\pi$
$47$	2056.83i	0.931116i	−0.885017	+	0.465558i	$0.845854\pi$
$48$	0	0
$49$	281.031	0.117048
$50$	0	0
$51$	2223.91	0.855022
$52$	0	0
$53$	753.725i	0.268325i	0.990959	+	0.134163i	$0.0428344\pi$
$53$	753.725i	0.268325i	−0.990959	+	0.134163i	$0.957166\pi$
$54$	0	0
$55$	7205.84i	2.38210i
$56$	0	0
$57$	−3473.93	−1.06923
$58$	0	0
$59$	−1309.43	−0.376165	−0.188083	−	0.982153i	$-0.560227\pi$
$59$	−1309.43	−0.376165	−0.188083	+	0.982153i	$0.560227\pi$
$60$	0	0
$61$	801.913i	0.215510i	0.994177	+	0.107755i	$0.0343662\pi$
$61$	801.913i	0.215510i	−0.994177	+	0.107755i	$0.965634\pi$
$62$	0	0
$63$	1243.16i	0.313218i
$64$	0	0
$65$	7293.87	1.72636
$66$	0	0
$67$	505.813	0.112678	0.0563392	−	0.998412i	$-0.482057\pi$
$67$	505.813	0.112678	0.0563392	+	0.998412i	$0.482057\pi$
$68$	0	0
$69$	4585.41i	0.963119i
$70$	0	0
$71$	− 2170.94i	− 0.430658i	−0.976542	−	0.215329i	$-0.930918\pi$
$71$	− 2170.94i	− 0.430658i	0.976542	−	0.215329i	$-0.0690823\pi$
$72$	0	0
$73$	−2297.97	−0.431219	−0.215610	−	0.976480i	$-0.569174\pi$
$73$	−2297.97	−0.431219	−0.215610	+	0.976480i	$0.569174\pi$
$74$	0	0
$75$	−4962.24	−0.882177
$76$	0	0
$77$	− 8346.85i	− 1.40780i
$78$	0	0
$79$	10705.3i	1.71532i	0.514219	+	0.857659i	$0.328082\pi$
$79$	10705.3i	1.71532i	−0.514219	+	0.857659i	$0.671918\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	−2977.81	−0.432256	−0.216128	−	0.976365i	$-0.569343\pi$
$83$	−2977.81	−0.432256	−0.216128	+	0.976365i	$0.569343\pi$
$84$	0	0
$85$	− 17012.3i	− 2.35464i
$86$	0	0
$87$	4194.58i	0.554179i
$88$	0	0
$89$	−6493.92	−0.819836	−0.409918	−	0.912122i	$-0.634443\pi$
$89$	−6493.92	−0.819836	−0.409918	+	0.912122i	$0.634443\pi$
$90$	0	0
$91$	−8448.82	−1.02027
$92$	0	0
$93$	− 2033.13i	− 0.235071i
$94$	0	0
$95$	26574.5i	2.94455i
$96$	0	0
$97$	−8189.92	−0.870435	−0.435217	−	0.900325i	$-0.643328\pi$
$97$	−8189.92	−0.870435	−0.435217	+	0.900325i	$0.643328\pi$
$98$	0	0
$99$	−4894.65	−0.499403
$100$	0	0
$101$	15576.5i	1.52696i	0.645833	+	0.763479i	$0.276510\pi$
$101$	15576.5i	1.52696i	−0.645833	+	0.763479i	$0.723490\pi$
$102$	0	0
$103$	− 6628.58i	− 0.624807i	−0.949950	−	0.312403i	$-0.898866\pi$
$103$	− 6628.58i	− 0.624807i	0.949950	−	0.312403i	$-0.101134\pi$
$104$	0	0
$105$	9509.83	0.862570
$106$	0	0
$107$	−11796.4	−1.03035	−0.515173	−	0.857086i	$-0.672272\pi$
$107$	−11796.4	−1.03035	−0.515173	+	0.857086i	$0.672272\pi$
$108$	0	0
$109$	13286.1i	1.11826i	0.829080	+	0.559130i	$0.188865\pi$
$109$	13286.1i	1.11826i	−0.829080	+	0.559130i	$0.811135\pi$
$110$	0	0
$111$	− 2426.50i	− 0.196940i
$112$	0	0
$113$	−11713.6	−0.917350	−0.458675	−	0.888604i	$-0.651676\pi$
$113$	−11713.6	−0.917350	−0.458675	+	0.888604i	$0.651676\pi$
$114$	0	0
$115$	35077.0	2.65233
$116$	0	0
$117$	4954.45i	0.361929i
$118$	0	0
$119$	19706.1i	1.39157i
$120$	0	0
$121$	18222.7	1.24463
$122$	0	0
$123$	11223.3	0.741842
$124$	0	0
$125$	13116.5i	0.839459i
$126$	0	0
$127$	5640.55i	0.349715i	0.984594	+	0.174858i	$0.0559465\pi$
$127$	5640.55i	0.349715i	−0.984594	+	0.174858i	$0.944054\pi$
$128$	0	0
$129$	2645.79	0.158992
$130$	0	0
$131$	31922.5	1.86018	0.930089	−	0.367335i	$-0.119730\pi$
$131$	31922.5	1.86018	0.930089	+	0.367335i	$0.119730\pi$
$132$	0	0
$133$	− 30782.5i	− 1.74021i
$134$	0	0
$135$	− 5576.63i	− 0.305988i
$136$	0	0
$137$	−10515.6	−0.560263	−0.280132	−	0.959962i	$-0.590378\pi$
$137$	−10515.6	−0.560263	−0.280132	+	0.959962i	$0.590378\pi$
$138$	0	0
$139$	−14985.9	−0.775626	−0.387813	−	0.921738i	$-0.626769\pi$
$139$	−14985.9	−0.775626	−0.387813	+	0.921738i	$0.626769\pi$
$140$	0	0
$141$	10687.6i	0.537580i
$142$	0	0
$143$	− 33265.2i	− 1.62674i
$144$	0	0
$145$	32087.3	1.52615
$146$	0	0
$147$	1460.28	0.0675775
$148$	0	0
$149$	11795.4i	0.531298i	0.964070	+	0.265649i	$0.0855863\pi$
$149$	11795.4i	0.531298i	−0.964070	+	0.265649i	$0.914414\pi$
$150$	0	0
$151$	33792.9i	1.48208i	0.671460	+	0.741040i	$0.265667\pi$
$151$	33792.9i	1.48208i	−0.671460	+	0.741040i	$0.734333\pi$
$152$	0	0
$153$	11555.8	0.493647
$154$	0	0
$155$	−15552.8	−0.647360
$156$	0	0
$157$	− 28788.9i	− 1.16796i	−0.811770	−	0.583978i	$-0.801496\pi$
$157$	− 28788.9i	− 1.16796i	0.811770	−	0.583978i	$-0.198504\pi$
$158$	0	0
$159$	3916.47i	0.154918i
$160$	0	0
$161$	−40631.3	−1.56751
$162$	0	0
$163$	−13409.0	−0.504684	−0.252342	−	0.967638i	$-0.581201\pi$
$163$	−13409.0	−0.504684	−0.252342	+	0.967638i	$0.581201\pi$
$164$	0	0
$165$	37442.6i	1.37530i
$166$	0	0
$167$	− 13206.7i	− 0.473546i	−0.971565	−	0.236773i	$-0.923910\pi$
$167$	− 13206.7i	− 0.473546i	0.971565	−	0.236773i	$-0.0760898\pi$
$168$	0	0
$169$	−5110.53	−0.178934
$170$	0	0
$171$	−18051.1	−0.617320
$172$	0	0
$173$	− 364.956i	− 0.0121940i	−0.999981	−	0.00609702i	$-0.998059\pi$
$173$	− 364.956i	− 0.0121940i	0.999981	−	0.00609702i	$-0.00194076\pi$
$174$	0	0
$175$	− 43970.5i	− 1.43577i
$176$	0	0
$177$	−6804.00	−0.217179
$178$	0	0
$179$	14372.0	0.448549	0.224275	−	0.974526i	$-0.427999\pi$
$179$	14372.0	0.448549	0.224275	+	0.974526i	$0.427999\pi$
$180$	0	0
$181$	4050.54i	0.123639i	0.998087	+	0.0618195i	$0.0196903\pi$
$181$	4050.54i	0.123639i	−0.998087	+	0.0618195i	$0.980310\pi$
$182$	0	0
$183$	4166.86i	0.124425i
$184$	0	0
$185$	−18562.0	−0.542352
$186$	0	0
$187$	−77587.9	−2.21876
$188$	0	0
$189$	6459.67i	0.180837i
$190$	0	0
$191$	− 49880.9i	− 1.36731i	−0.729805	−	0.683656i	$-0.760389\pi$
$191$	− 49880.9i	− 1.36731i	0.729805	−	0.683656i	$-0.239611\pi$
$192$	0	0
$193$	−48425.7	−1.30005	−0.650026	−	0.759912i	$-0.725242\pi$
$193$	−48425.7	−1.30005	−0.650026	+	0.759912i	$0.725242\pi$
$194$	0	0
$195$	37900.0	0.996714
$196$	0	0
$197$	5556.74i	0.143182i	0.997434	+	0.0715908i	$0.0228076\pi$
$197$	5556.74i	0.143182i	−0.997434	+	0.0715908i	$0.977192\pi$
$198$	0	0
$199$	− 60594.7i	− 1.53013i	−0.643952	−	0.765066i	$-0.722706\pi$
$199$	− 60594.7i	− 1.53013i	0.643952	−	0.765066i	$-0.277294\pi$
$200$	0	0
$201$	2628.28	0.0650549
$202$	0	0
$203$	−37168.2	−0.901943
$204$	0	0
$205$	− 85855.1i	− 2.04295i
$206$	0	0
$207$	23826.5i	0.556057i
$208$	0	0
$209$	121198.	2.77463
$210$	0	0
$211$	27539.9	0.618583	0.309292	−	0.950967i	$-0.399908\pi$
$211$	27539.9	0.618583	0.309292	+	0.950967i	$0.399908\pi$
$212$	0	0
$213$	− 11280.6i	− 0.248640i
$214$	0	0
$215$	− 20239.5i	− 0.437847i
$216$	0	0
$217$	18015.6	0.382585
$218$	0	0
$219$	−11940.6	−0.248965
$220$	0	0
$221$	78535.7i	1.60799i
$222$	0	0
$223$	− 3021.35i	− 0.0607564i	−0.999538	−	0.0303782i	$-0.990329\pi$
$223$	− 3021.35i	− 0.0607564i	0.999538	−	0.0303782i	$-0.00967116\pi$
$224$	0	0
$225$	−25784.6	−0.509325
$226$	0	0
$227$	−7077.28	−0.137346	−0.0686728	−	0.997639i	$-0.521876\pi$
$227$	−7077.28	−0.137346	−0.0686728	+	0.997639i	$0.521876\pi$
$228$	0	0
$229$	− 102761.i	− 1.95956i	−0.200088	−	0.979778i	$-0.564123\pi$
$229$	− 102761.i	− 1.95956i	0.200088	−	0.979778i	$-0.435877\pi$
$230$	0	0
$231$	− 43371.5i	− 0.812795i
$232$	0	0
$233$	22976.6	0.423227	0.211613	−	0.977353i	$-0.432128\pi$
$233$	22976.6	0.423227	0.211613	+	0.977353i	$0.432128\pi$
$234$	0	0
$235$	81757.1	1.48044
$236$	0	0
$237$	55626.4i	0.990339i
$238$	0	0
$239$	− 35566.7i	− 0.622656i	−0.950303	−	0.311328i	$-0.899226\pi$
$239$	− 35566.7i	− 0.622656i	0.950303	−	0.311328i	$-0.100774\pi$
$240$	0	0
$241$	−92683.4	−1.59576	−0.797881	−	0.602816i	$-0.794045\pi$
$241$	−92683.4	−1.59576	−0.797881	+	0.602816i	$0.794045\pi$
$242$	0	0
$243$	3788.00	0.0641500
$244$	0	0
$245$	− 11170.7i	− 0.186101i
$246$	0	0
$247$	− 122679.i	− 2.01084i
$248$	0	0
$249$	−15473.2	−0.249563
$250$	0	0
$251$	27590.5	0.437937	0.218968	−	0.975732i	$-0.429731\pi$
$251$	27590.5	0.437937	0.218968	+	0.975732i	$0.429731\pi$
$252$	0	0
$253$	− 159976.i	− 2.49927i
$254$	0	0
$255$	− 88398.3i	− 1.35945i
$256$	0	0
$257$	92304.6	1.39752	0.698758	−	0.715358i	$-0.253736\pi$
$257$	92304.6	1.39752	0.698758	+	0.715358i	$0.253736\pi$
$258$	0	0
$259$	21501.2	0.320526
$260$	0	0
$261$	21795.7i	0.319955i
$262$	0	0
$263$	− 13884.4i	− 0.200732i	−0.994951	−	0.100366i	$-0.967999\pi$
$263$	− 13884.4i	− 0.200732i	0.994951	−	0.100366i	$-0.0320014\pi$
$264$	0	0
$265$	29959.8	0.426626
$266$	0	0
$267$	−33743.4	−0.473333
$268$	0	0
$269$	52698.1i	0.728266i	0.931347	+	0.364133i	$0.118635\pi$
$269$	52698.1i	0.728266i	−0.931347	+	0.364133i	$0.881365\pi$
$270$	0	0
$271$	47028.8i	0.640361i	0.947356	+	0.320181i	$0.103744\pi$
$271$	47028.8i	0.640361i	−0.947356	+	0.320181i	$0.896256\pi$
$272$	0	0
$273$	−43901.4	−0.589051
$274$	0	0
$275$	173123.	2.28923
$276$	0	0
$277$	108393.i	1.41268i	0.707873	+	0.706340i	$0.249655\pi$
$277$	108393.i	1.41268i	−0.707873	+	0.706340i	$0.750345\pi$
$278$	0	0
$279$	− 10564.4i	− 0.135718i
$280$	0	0
$281$	−73090.7	−0.925656	−0.462828	−	0.886448i	$-0.653165\pi$
$281$	−73090.7	−0.925656	−0.462828	+	0.886448i	$0.653165\pi$
$282$	0	0
$283$	44763.1	0.558917	0.279458	−	0.960158i	$-0.409845\pi$
$283$	44763.1	0.558917	0.279458	+	0.960158i	$0.409845\pi$
$284$	0	0
$285$	138085.i	1.70003i
$286$	0	0
$287$	99449.9i	1.20737i
$288$	0	0
$289$	99656.3	1.19319
$290$	0	0
$291$	−42556.1	−0.502546
$292$	0	0
$293$	42030.2i	0.489583i	0.969576	+	0.244792i	$0.0787195\pi$
$293$	42030.2i	0.489583i	−0.969576	+	0.244792i	$0.921280\pi$
$294$	0	0
$295$	52048.6i	0.598088i
$296$	0	0
$297$	−25433.4	−0.288331
$298$	0	0
$299$	−161930.	−1.81128
$300$	0	0
$301$	23444.3i	0.258765i
$302$	0	0
$303$	80937.9i	0.881590i
$304$	0	0
$305$	31875.3	0.342653
$306$	0	0
$307$	8820.31	0.0935852	0.0467926	−	0.998905i	$-0.485100\pi$
$307$	8820.31	0.0935852	0.0467926	+	0.998905i	$0.485100\pi$
$308$	0	0
$309$	− 34443.1i	− 0.360732i
$310$	0	0
$311$	− 155059.i	− 1.60315i	−0.597893	−	0.801576i	$-0.703995\pi$
$311$	− 155059.i	− 1.60315i	0.597893	−	0.801576i	$-0.296005\pi$
$312$	0	0
$313$	−179666.	−1.83390	−0.916951	−	0.398999i	$-0.869358\pi$
$313$	−179666.	−1.83390	−0.916951	+	0.398999i	$0.869358\pi$
$314$	0	0
$315$	49414.6	0.498005
$316$	0	0
$317$	6188.77i	0.0615865i	0.999526	+	0.0307933i	$0.00980335\pi$
$317$	6188.77i	0.0615865i	−0.999526	+	0.0307933i	$0.990197\pi$
$318$	0	0
$319$	− 146341.i	− 1.43808i
$320$	0	0
$321$	−61296.0	−0.594870
$322$	0	0
$323$	−286138.	−2.74265
$324$	0	0
$325$	− 175238.i	− 1.65906i
$326$	0	0
$327$	69036.4i	0.645628i
$328$	0	0
$329$	−94703.1	−0.874928
$330$	0	0
$331$	−132159.	−1.20626	−0.603130	−	0.797643i	$-0.706080\pi$
$331$	−132159.	−1.20626	−0.603130	+	0.797643i	$0.706080\pi$
$332$	0	0
$333$	− 12608.5i	− 0.113704i
$334$	0	0
$335$	− 20105.6i	− 0.179154i
$336$	0	0
$337$	70040.0	0.616718	0.308359	−	0.951270i	$-0.400220\pi$
$337$	70040.0	0.616718	0.308359	+	0.951270i	$0.400220\pi$
$338$	0	0
$339$	−60865.8	−0.529632
$340$	0	0
$341$	70931.8i	0.610004i
$342$	0	0
$343$	123489.i	1.04964i
$344$	0	0
$345$	182266.	1.53132
$346$	0	0
$347$	157722.	1.30988	0.654942	−	0.755680i	$-0.272693\pi$
$347$	157722.	1.30988	0.654942	+	0.755680i	$0.272693\pi$
$348$	0	0
$349$	− 198747.i	− 1.63174i	−0.578238	−	0.815868i	$-0.696259\pi$
$349$	− 198747.i	− 1.63174i	0.578238	−	0.815868i	$-0.303741\pi$
$350$	0	0
$351$	25744.1i	0.208960i
$352$	0	0
$353$	13376.1	0.107345	0.0536724	−	0.998559i	$-0.482907\pi$
$353$	13376.1	0.107345	0.0536724	+	0.998559i	$0.482907\pi$
$354$	0	0
$355$	−86292.9	−0.684729
$356$	0	0
$357$	102396.i	0.803426i
$358$	0	0
$359$	190011.i	1.47432i	0.675721	+	0.737158i	$0.263832\pi$
$359$	190011.i	1.47432i	−0.675721	+	0.737158i	$0.736168\pi$
$360$	0	0
$361$	316649.	2.42976
$362$	0	0
$363$	94687.8	0.718590
$364$	0	0
$365$	91342.0i	0.685622i
$366$	0	0
$367$	94140.5i	0.698947i	0.936946	+	0.349474i	$0.113640\pi$
$367$	94140.5i	0.698947i	−0.936946	+	0.349474i	$0.886360\pi$
$368$	0	0
$369$	58318.1	0.428302
$370$	0	0
$371$	−34703.9	−0.252133
$372$	0	0
$373$	66152.4i	0.475475i	0.971329	+	0.237738i	$0.0764058\pi$
$373$	66152.4i	0.475475i	−0.971329	+	0.237738i	$0.923594\pi$
$374$	0	0
$375$	68155.6i	0.484662i
$376$	0	0
$377$	−148128.	−1.04221
$378$	0	0
$379$	64395.2	0.448306	0.224153	−	0.974554i	$-0.428038\pi$
$379$	64395.2	0.448306	0.224153	+	0.974554i	$0.428038\pi$
$380$	0	0
$381$	29309.2i	0.201908i
$382$	0	0
$383$	30714.5i	0.209385i	0.994505	+	0.104693i	$0.0333859\pi$
$383$	30714.5i	0.209385i	−0.994505	+	0.104693i	$0.966614\pi$
$384$	0	0
$385$	−331779.	−2.23835
$386$	0	0
$387$	13747.9	0.0917941
$388$	0	0
$389$	199122.i	1.31589i	0.753066	+	0.657945i	$0.228574\pi$
$389$	199122.i	1.31589i	−0.753066	+	0.657945i	$0.771426\pi$
$390$	0	0
$391$	377687.i	2.47046i
$392$	0	0
$393$	165874.	1.07397
$394$	0	0
$395$	425525.	2.72729
$396$	0	0
$397$	− 68565.8i	− 0.435037i	−0.976056	−	0.217519i	$-0.930204\pi$
$397$	− 68565.8i	− 0.435037i	0.976056	−	0.217519i	$-0.0697963\pi$
$398$	0	0
$399$	− 159951.i	− 1.00471i
$400$	0	0
$401$	−21797.1	−0.135553	−0.0677765	−	0.997701i	$-0.521590\pi$
$401$	−21797.1	−0.135553	−0.0677765	+	0.997701i	$0.521590\pi$
$402$	0	0
$403$	71798.3	0.442084
$404$	0	0
$405$	− 28977.0i	− 0.176662i
$406$	0	0
$407$	84655.8i	0.511055i
$408$	0	0
$409$	230211.	1.37620	0.688098	−	0.725618i	$-0.258446\pi$
$409$	230211.	1.37620	0.688098	+	0.725618i	$0.258446\pi$
$410$	0	0
$411$	−54640.6	−0.323468
$412$	0	0
$413$	− 60290.3i	− 0.353465i
$414$	0	0
$415$	118365.i	0.687270i
$416$	0	0
$417$	−77868.9	−0.447808
$418$	0	0
$419$	17360.5	0.0988859	0.0494430	−	0.998777i	$-0.484255\pi$
$419$	17360.5	0.0988859	0.0494430	+	0.998777i	$0.484255\pi$
$420$	0	0
$421$	− 186829.i	− 1.05410i	−0.849836	−	0.527048i	$-0.823299\pi$
$421$	− 186829.i	− 1.05410i	0.849836	−	0.527048i	$-0.176701\pi$
$422$	0	0
$423$	55534.5i	0.310372i
$424$	0	0
$425$	−408726.	−2.26284
$426$	0	0
$427$	−36922.6	−0.202505
$428$	0	0
$429$	− 172851.i	− 0.939197i
$430$	0	0
$431$	153753.i	0.827693i	0.910347	+	0.413846i	$0.135815\pi$
$431$	153753.i	0.827693i	−0.910347	+	0.413846i	$0.864185\pi$
$432$	0	0
$433$	9168.87	0.0489035	0.0244517	−	0.999701i	$-0.492216\pi$
$433$	9168.87	0.0489035	0.0244517	+	0.999701i	$0.492216\pi$
$434$	0	0
$435$	166730.	0.881122
$436$	0	0
$437$	− 589978.i	− 3.08939i
$438$	0	0
$439$	− 178760.i	− 0.927561i	−0.885950	−	0.463780i	$-0.846493\pi$
$439$	− 178760.i	− 0.927561i	0.885950	−	0.463780i	$-0.153507\pi$
$440$	0	0
$441$	7587.85	0.0390159
$442$	0	0
$443$	−17798.2	−0.0906920	−0.0453460	−	0.998971i	$-0.514439\pi$
$443$	−17798.2	−0.0906920	−0.0453460	+	0.998971i	$0.514439\pi$
$444$	0	0
$445$	258127.i	1.30351i
$446$	0	0
$447$	61290.5i	0.306745i
$448$	0	0
$449$	242915.	1.20493	0.602464	−	0.798146i	$-0.294186\pi$
$449$	242915.	1.20493	0.602464	+	0.798146i	$0.294186\pi$
$450$	0	0
$451$	−391559.	−1.92506
$452$	0	0
$453$	175593.i	0.855680i
$454$	0	0
$455$	335832.i	1.62218i
$456$	0	0
$457$	−190762.	−0.913396	−0.456698	−	0.889622i	$-0.650968\pi$
$457$	−190762.	−0.913396	−0.456698	+	0.889622i	$0.650968\pi$
$458$	0	0
$459$	60045.6	0.285007
$460$	0	0
$461$	− 155121.i	− 0.729909i	−0.931025	−	0.364955i	$-0.881084\pi$
$461$	− 155121.i	− 0.729909i	0.931025	−	0.364955i	$-0.118916\pi$
$462$	0	0
$463$	230925.i	1.07723i	0.842551	+	0.538616i	$0.181053\pi$
$463$	230925.i	1.07723i	−0.842551	+	0.538616i	$0.818947\pi$
$464$	0	0
$465$	−80814.9	−0.373754
$466$	0	0
$467$	−154112.	−0.706647	−0.353323	−	0.935501i	$-0.614948\pi$
$467$	−154112.	−0.706647	−0.353323	+	0.935501i	$0.614948\pi$
$468$	0	0
$469$	23289.2i	0.105879i
$470$	0	0
$471$	− 149592.i	− 0.674319i
$472$	0	0
$473$	−92306.3	−0.412581
$474$	0	0
$475$	638462.	2.82975
$476$	0	0
$477$	20350.6i	0.0894417i
$478$	0	0
$479$	405668.i	1.76807i	0.467420	+	0.884035i	$0.345184\pi$
$479$	405668.i	1.76807i	−0.467420	+	0.884035i	$0.654816\pi$
$480$	0	0
$481$	85690.0	0.370373
$482$	0	0
$483$	−211127.	−0.905000
$484$	0	0
$485$	325541.i	1.38396i
$486$	0	0
$487$	− 426114.i	− 1.79667i	−0.439314	−	0.898334i	$-0.644778\pi$
$487$	− 426114.i	− 1.79667i	0.439314	−	0.898334i	$-0.355222\pi$
$488$	0	0
$489$	−69675.0	−0.291379
$490$	0	0
$491$	−273125.	−1.13292	−0.566459	−	0.824090i	$-0.691687\pi$
$491$	−273125.	−1.13292	−0.566459	+	0.824090i	$0.691687\pi$
$492$	0	0
$493$	345495.i	1.42151i
$494$	0	0
$495$	194558.i	0.794032i
$496$	0	0
$497$	99957.1	0.404670
$498$	0	0
$499$	−103548.	−0.415855	−0.207928	−	0.978144i	$-0.566672\pi$
$499$	−103548.	−0.415855	−0.207928	+	0.978144i	$0.566672\pi$
$500$	0	0
$501$	− 68624.2i	− 0.273402i
$502$	0	0
$503$	− 217063.i	− 0.857925i	−0.903322	−	0.428963i	$-0.858879\pi$
$503$	− 217063.i	− 0.857925i	0.903322	−	0.428963i	$-0.141121\pi$
$504$	0	0
$505$	619151.	2.42780
$506$	0	0
$507$	−26555.1	−0.103307
$508$	0	0
$509$	115098.i	0.444256i	0.975018	+	0.222128i	$0.0713003\pi$
$509$	115098.i	0.444256i	−0.975018	+	0.222128i	$0.928700\pi$
$510$	0	0
$511$	− 105806.i	− 0.405198i
$512$	0	0
$513$	−93796.1	−0.356410
$514$	0	0
$515$	−263479.	−0.993418
$516$	0	0
$517$	− 372870.i	− 1.39501i
$518$	0	0
$519$	− 1896.37i	− 0.00704024i
$520$	0	0
$521$	288543.	1.06300	0.531502	−	0.847057i	$-0.321628\pi$
$521$	288543.	1.06300	0.531502	+	0.847057i	$0.321628\pi$
$522$	0	0
$523$	−176826.	−0.646461	−0.323231	−	0.946320i	$-0.604769\pi$
$523$	−176826.	−0.646461	−0.323231	+	0.946320i	$0.604769\pi$
$524$	0	0
$525$	− 228477.i	− 0.828942i
$526$	0	0
$527$	− 167463.i	− 0.602973i
$528$	0	0
$529$	−498900.	−1.78280
$530$	0	0
$531$	−35354.6	−0.125388
$532$	0	0
$533$	396343.i	1.39514i
$534$	0	0
$535$	468896.i	1.63821i
$536$	0	0
$537$	74678.9	0.258970
$538$	0	0
$539$	−50946.4	−0.175362
$540$	0	0
$541$	425220.i	1.45284i	0.687249	+	0.726422i	$0.258818\pi$
$541$	425220.i	1.45284i	−0.687249	+	0.726422i	$0.741182\pi$
$542$	0	0
$543$	21047.2i	0.0713830i
$544$	0	0
$545$	528108.	1.77799
$546$	0	0
$547$	−495505.	−1.65605	−0.828024	−	0.560692i	$-0.810535\pi$
$547$	−495505.	−1.65605	−0.828024	+	0.560692i	$0.810535\pi$
$548$	0	0
$549$	21651.7i	0.0718367i
$550$	0	0
$551$	− 539691.i	− 1.77763i
$552$	0	0
$553$	−492905.	−1.61181
$554$	0	0
$555$	−96451.0	−0.313127
$556$	0	0
$557$	− 257895.i	− 0.831251i	−0.909536	−	0.415626i	$-0.863563\pi$
$557$	− 257895.i	− 0.831251i	0.909536	−	0.415626i	$-0.136437\pi$
$558$	0	0
$559$	93433.9i	0.299007i
$560$	0	0
$561$	−403158.	−1.28100
$562$	0	0
$563$	417799.	1.31811	0.659054	−	0.752096i	$-0.270957\pi$
$563$	417799.	1.31811	0.659054	+	0.752096i	$0.270957\pi$
$564$	0	0
$565$	465606.i	1.45855i
$566$	0	0
$567$	33565.4i	0.104406i
$568$	0	0
$569$	−100935.	−0.311756	−0.155878	−	0.987776i	$-0.549821\pi$
$569$	−100935.	−0.311756	−0.155878	+	0.987776i	$0.549821\pi$
$570$	0	0
$571$	453320.	1.39038	0.695188	−	0.718828i	$-0.255321\pi$
$571$	453320.	1.39038	0.695188	+	0.718828i	$0.255321\pi$
$572$	0	0
$573$	− 259189.i	− 0.789418i
$574$	0	0
$575$	− 842738.i	− 2.54892i
$576$	0	0
$577$	12119.6	0.0364031	0.0182015	−	0.999834i	$-0.494206\pi$
$577$	12119.6	0.0364031	0.0182015	+	0.999834i	$0.494206\pi$
$578$	0	0
$579$	−251627.	−0.750586
$580$	0	0
$581$	− 137108.i	− 0.406172i
$582$	0	0
$583$	− 136638.i	− 0.402008i
$584$	0	0
$585$	196934.	0.575453
$586$	0	0
$587$	327468.	0.950370	0.475185	−	0.879886i	$-0.342381\pi$
$587$	327468.	0.950370	0.475185	+	0.879886i	$0.342381\pi$
$588$	0	0
$589$	261591.i	0.754035i
$590$	0	0
$591$	28873.7i	0.0826660i
$592$	0	0
$593$	610182.	1.73520	0.867601	−	0.497260i	$-0.165661\pi$
$593$	610182.	1.73520	0.867601	+	0.497260i	$0.165661\pi$
$594$	0	0
$595$	783298.	2.21255
$596$	0	0
$597$	− 314859.i	− 0.883422i
$598$	0	0
$599$	499941.i	1.39337i	0.717379	+	0.696683i	$0.245342\pi$
$599$	499941.i	1.39337i	−0.717379	+	0.696683i	$0.754658\pi$
$600$	0	0
$601$	−486965.	−1.34818	−0.674091	−	0.738649i	$-0.735464\pi$
$601$	−486965.	−1.34818	−0.674091	+	0.738649i	$0.735464\pi$
$602$	0	0
$603$	13657.0	0.0375595
$604$	0	0
$605$	− 724334.i	− 1.97892i
$606$	0	0
$607$	41693.4i	0.113159i	0.998398	+	0.0565796i	$0.0180195\pi$
$607$	41693.4i	0.113159i	−0.998398	+	0.0565796i	$0.981981\pi$
$608$	0	0
$609$	−193131.	−0.520737
$610$	0	0
$611$	−377425.	−1.01099
$612$	0	0
$613$	542042.i	1.44249i	0.692681	+	0.721244i	$0.256429\pi$
$613$	542042.i	1.44249i	−0.692681	+	0.721244i	$0.743571\pi$
$614$	0	0
$615$	− 446116.i	− 1.17950i
$616$	0	0
$617$	−146814.	−0.385653	−0.192827	−	0.981233i	$-0.561765\pi$
$617$	−146814.	−0.385653	−0.192827	+	0.981233i	$0.561765\pi$
$618$	0	0
$619$	72720.6	0.189791	0.0948956	−	0.995487i	$-0.469748\pi$
$619$	72720.6	0.189791	0.0948956	+	0.995487i	$0.469748\pi$
$620$	0	0
$621$	123806.i	0.321040i
$622$	0	0
$623$	− 299000.i	− 0.770363i
$624$	0	0
$625$	−75495.2	−0.193268
$626$	0	0
$627$	629766.	1.60193
$628$	0	0
$629$	− 199864.i	− 0.505165i
$630$	0	0
$631$	− 568570.i	− 1.42799i	−0.700151	−	0.713995i	$-0.746884\pi$
$631$	− 568570.i	− 1.42799i	0.700151	−	0.713995i	$-0.253116\pi$
$632$	0	0
$633$	143102.	0.357139
$634$	0	0
$635$	224206.	0.556033
$636$	0	0
$637$	51568.7i	0.127089i
$638$	0	0
$639$	− 58615.5i	− 0.143553i
$640$	0	0
$641$	292494.	0.711870	0.355935	−	0.934511i	$-0.384162\pi$
$641$	292494.	0.711870	0.355935	+	0.934511i	$0.384162\pi$
$642$	0	0
$643$	103161.	0.249514	0.124757	−	0.992187i	$-0.460185\pi$
$643$	103161.	0.249514	0.124757	+	0.992187i	$0.460185\pi$
$644$	0	0
$645$	− 105167.i	− 0.252791i
$646$	0	0
$647$	− 372569.i	− 0.890017i	−0.895526	−	0.445009i	$-0.853201\pi$
$647$	− 372569.i	− 0.890017i	0.895526	−	0.445009i	$-0.146799\pi$
$648$	0	0
$649$	237378.	0.563574
$650$	0	0
$651$	93611.6	0.220886
$652$	0	0
$653$	85619.0i	0.200791i	0.994948	+	0.100395i	$0.0320108\pi$
$653$	85619.0i	0.200791i	−0.994948	+	0.100395i	$0.967989\pi$
$654$	0	0
$655$	− 1.26889e6i	− 2.95761i
$656$	0	0
$657$	−62045.1	−0.143740
$658$	0	0
$659$	258812.	0.595954	0.297977	−	0.954573i	$-0.403688\pi$
$659$	258812.	0.595954	0.297977	+	0.954573i	$0.403688\pi$
$660$	0	0
$661$	327660.i	0.749928i	0.927039	+	0.374964i	$0.122345\pi$
$661$	327660.i	0.749928i	−0.927039	+	0.374964i	$0.877655\pi$
$662$	0	0
$663$	408084.i	0.928372i
$664$	0	0
$665$	−1.22357e6	−2.76686
$666$	0	0
$667$	−712366.	−1.60122
$668$	0	0
$669$	− 15699.4i	− 0.0350777i
$670$	0	0
$671$	− 145374.i	− 0.322880i
$672$	0	0
$673$	137121.	0.302742	0.151371	−	0.988477i	$-0.451631\pi$
$673$	137121.	0.302742	0.151371	+	0.988477i	$0.451631\pi$
$674$	0	0
$675$	−133981.	−0.294059
$676$	0	0
$677$	− 573451.i	− 1.25118i	−0.780153	−	0.625589i	$-0.784859\pi$
$677$	− 573451.i	− 1.25118i	0.780153	−	0.625589i	$-0.215141\pi$
$678$	0	0
$679$	− 377089.i	− 0.817909i
$680$	0	0
$681$	−36774.6	−0.0792965
$682$	0	0
$683$	−175872.	−0.377012	−0.188506	−	0.982072i	$-0.560364\pi$
$683$	−175872.	−0.377012	−0.188506	+	0.982072i	$0.560364\pi$
$684$	0	0
$685$	417984.i	0.890797i
$686$	0	0
$687$	− 533962.i	− 1.13135i
$688$	0	0
$689$	−138307.	−0.291344
$690$	0	0
$691$	−7216.66	−0.0151140	−0.00755702	−	0.999971i	$-0.502405\pi$
$691$	−7216.66	−0.0151140	−0.00755702	+	0.999971i	$0.502405\pi$
$692$	0	0
$693$	− 225365.i	− 0.469267i
$694$	0	0
$695$	595673.i	1.23321i
$696$	0	0
$697$	924433.	1.90287
$698$	0	0
$699$	119390.	0.244350
$700$	0	0
$701$	502543.i	1.02267i	0.859380	+	0.511337i	$0.170850\pi$
$701$	502543.i	1.02267i	−0.859380	+	0.511337i	$0.829150\pi$
$702$	0	0
$703$	312203.i	0.631723i
$704$	0	0
$705$	424823.	0.854731
$706$	0	0
$707$	−717191.	−1.43481
$708$	0	0
$709$	95591.1i	0.190163i	0.995470	+	0.0950813i	$0.0303111\pi$
$709$	95591.1i	0.190163i	−0.995470	+	0.0950813i	$0.969689\pi$
$710$	0	0
$711$	289043.i	0.571772i
$712$	0	0
$713$	345286.	0.679204
$714$	0	0
$715$	−1.32226e6	−2.58645
$716$	0	0
$717$	− 184810.i	− 0.359491i
$718$	0	0
$719$	475789.i	0.920357i	0.887826	+	0.460178i	$0.152215\pi$
$719$	475789.i	0.920357i	−0.887826	+	0.460178i	$0.847785\pi$
$720$	0	0
$721$	305200.	0.587103
$722$	0	0
$723$	−481597.	−0.921313
$724$	0	0
$725$	− 770908.i	− 1.46665i
$726$	0	0
$727$	193262.i	0.365661i	0.983144	+	0.182830i	$0.0585259\pi$
$727$	193262.i	0.365661i	−0.983144	+	0.182830i	$0.941474\pi$
$728$	0	0
$729$	19683.0	0.0370370
$730$	0	0
$731$	217926.	0.407825
$732$	0	0
$733$	− 650799.i	− 1.21126i	−0.795745	−	0.605632i	$-0.792920\pi$
$733$	− 650799.i	− 1.21126i	0.795745	−	0.605632i	$-0.207080\pi$
$734$	0	0
$735$	− 58044.8i	− 0.107446i
$736$	0	0
$737$	−91695.6	−0.168816
$738$	0	0
$739$	99472.6	0.182144	0.0910719	−	0.995844i	$-0.470971\pi$
$739$	99472.6	0.182144	0.0910719	+	0.995844i	$0.470971\pi$
$740$	0	0
$741$	− 637459.i	− 1.16096i
$742$	0	0
$743$	− 715681.i	− 1.29641i	−0.761466	−	0.648205i	$-0.775520\pi$
$743$	− 715681.i	− 1.29641i	0.761466	−	0.648205i	$-0.224480\pi$
$744$	0	0
$745$	468854.	0.844744
$746$	0	0
$747$	−80401.0	−0.144085
$748$	0	0
$749$	− 543144.i	− 0.968170i
$750$	0	0
$751$	− 603532.i	− 1.07009i	−0.844824	−	0.535045i	$-0.820295\pi$
$751$	− 603532.i	− 1.07009i	0.844824	−	0.535045i	$-0.179705\pi$
$752$	0	0
$753$	143364.	0.252843
$754$	0	0
$755$	1.34324e6	2.35645
$756$	0	0
$757$	784192.i	1.36846i	0.729268	+	0.684228i	$0.239861\pi$
$757$	784192.i	1.36846i	−0.729268	+	0.684228i	$0.760139\pi$
$758$	0	0
$759$	− 831259.i	− 1.44296i
$760$	0	0
$761$	−568076.	−0.980927	−0.490464	−	0.871462i	$-0.663173\pi$
$761$	−568076.	−0.980927	−0.490464	+	0.871462i	$0.663173\pi$
$762$	0	0
$763$	−611731.	−1.05078
$764$	0	0
$765$	− 459331.i	− 0.784880i
$766$	0	0
$767$	− 240278.i	− 0.408435i
$768$	0	0
$769$	−531757.	−0.899209	−0.449605	−	0.893228i	$-0.648435\pi$
$769$	−531757.	−0.899209	−0.449605	+	0.893228i	$0.648435\pi$
$770$	0	0
$771$	479629.	0.806856
$772$	0	0
$773$	− 260161.i	− 0.435394i	−0.976016	−	0.217697i	$-0.930145\pi$
$773$	− 260161.i	− 0.435394i	0.976016	−	0.217697i	$-0.0698545\pi$
$774$	0	0
$775$	373662.i	0.622122i
$776$	0	0
$777$	111724.	0.185056
$778$	0	0
$779$	−1.44404e6	−2.37960
$780$	0	0
$781$	393556.i	0.645216i
$782$	0	0
$783$	113254.i	0.184726i
$784$	0	0
$785$	−1.14433e6	−1.85700
$786$	0	0
$787$	724342.	1.16948	0.584742	−	0.811219i	$-0.301196\pi$
$787$	724342.	1.16948	0.584742	+	0.811219i	$0.301196\pi$
$788$	0	0
$789$	− 72145.7i	− 0.115893i
$790$	0	0
$791$	− 539332.i	− 0.861993i
$792$	0	0
$793$	−147150.	−0.233998
$794$	0	0
$795$	155676.	0.246313
$796$	0	0
$797$	1.04249e6i	1.64117i	0.571524	+	0.820586i	$0.306353\pi$
$797$	1.04249e6i	1.64117i	−0.571524	+	0.820586i	$0.693647\pi$
$798$	0	0
$799$	880309.i	1.37893i
$800$	0	0
$801$	−175336.	−0.273279
$802$	0	0
$803$	416584.	0.646057
$804$	0	0
$805$	1.61506e6i	2.49227i
$806$	0	0
$807$	273827.i	0.420465i
$808$	0	0
$809$	−647222.	−0.988909	−0.494455	−	0.869204i	$-0.664632\pi$
$809$	−647222.	−0.988909	−0.494455	+	0.869204i	$0.664632\pi$
$810$	0	0
$811$	1.24087e6	1.88662	0.943309	−	0.331916i	$-0.107695\pi$
$811$	1.24087e6	1.88662	0.943309	+	0.331916i	$0.107695\pi$
$812$	0	0
$813$	244369.i	0.369713i
$814$	0	0
$815$	532993.i	0.802428i
$816$	0	0
$817$	−340418.	−0.509997
$818$	0	0
$819$	−228118.	−0.340089
$820$	0	0
$821$	819276.i	1.21547i	0.794141	+	0.607734i	$0.207921\pi$
$821$	819276.i	1.21547i	−0.794141	+	0.607734i	$0.792079\pi$
$822$	0	0
$823$	515929.i	0.761710i	0.924635	+	0.380855i	$0.124370\pi$
$823$	515929.i	0.761710i	−0.924635	+	0.380855i	$0.875630\pi$
$824$	0	0
$825$	899573.	1.32169
$826$	0	0
$827$	−140674.	−0.205685	−0.102842	−	0.994698i	$-0.532794\pi$
$827$	−140674.	−0.205685	−0.102842	+	0.994698i	$0.532794\pi$
$828$	0	0
$829$	137443.i	0.199993i	0.994988	+	0.0999964i	$0.0318831\pi$
$829$	137443.i	0.199993i	−0.994988	+	0.0999964i	$0.968117\pi$
$830$	0	0
$831$	563229.i	0.815611i
$832$	0	0
$833$	120279.	0.173341
$834$	0	0
$835$	−524955.	−0.752920
$836$	0	0
$837$	− 54894.5i	− 0.0783570i
$838$	0	0
$839$	− 591422.i	− 0.840182i	−0.907482	−	0.420091i	$-0.861998\pi$
$839$	− 591422.i	− 0.840182i	0.907482	−	0.420091i	$-0.138002\pi$
$840$	0	0
$841$	55633.2	0.0786579
$842$	0	0
$843$	−379790.	−0.534428
$844$	0	0
$845$	203138.i	0.284498i
$846$	0	0
$847$	839029.i	1.16953i
$848$	0	0
$849$	232596.	0.322691
$850$	0	0
$851$	412093.	0.569031
$852$	0	0
$853$	− 169773.i	− 0.233331i	−0.993171	−	0.116665i	$-0.962780\pi$
$853$	− 169773.i	− 0.233331i	0.993171	−	0.116665i	$-0.0372205\pi$
$854$	0	0
$855$	717512.i	0.981515i
$856$	0	0
$857$	1.05083e6	1.43077	0.715384	−	0.698732i	$-0.246252\pi$
$857$	1.05083e6	1.43077	0.715384	+	0.698732i	$0.246252\pi$
$858$	0	0
$859$	1.05896e6	1.43514	0.717569	−	0.696487i	$-0.245255\pi$
$859$	1.05896e6	1.43514	0.717569	+	0.696487i	$0.245255\pi$
$860$	0	0
$861$	516757.i	0.697075i
$862$	0	0
$863$	1.22011e6i	1.63824i	0.573621	+	0.819121i	$0.305538\pi$
$863$	1.22011e6i	1.63824i	−0.573621	+	0.819121i	$0.694462\pi$
$864$	0	0
$865$	−14506.6	−0.0193881
$866$	0	0
$867$	517829.	0.688887
$868$	0	0
$869$	− 1.94069e6i	− 2.56991i
$870$	0	0
$871$	92815.8i	0.122345i
$872$	0	0
$873$	−221128.	−0.290145
$874$	0	0
$875$	−603927.	−0.788802
$876$	0	0
$877$	− 707060.i	− 0.919299i	−0.888100	−	0.459650i	$-0.847975\pi$
$877$	− 707060.i	− 0.919299i	0.888100	−	0.459650i	$-0.152025\pi$
$878$	0	0
$879$	218395.i	0.282661i
$880$	0	0
$881$	430965.	0.555252	0.277626	−	0.960689i	$-0.410452\pi$
$881$	430965.	0.555252	0.277626	+	0.960689i	$0.410452\pi$
$882$	0	0
$883$	1.08382e6	1.39007	0.695037	−	0.718974i	$-0.255388\pi$
$883$	1.08382e6	1.39007	0.695037	+	0.718974i	$0.255388\pi$
$884$	0	0
$885$	270452.i	0.345306i
$886$	0	0
$887$	− 620229.i	− 0.788324i	−0.919041	−	0.394162i	$-0.871035\pi$
$887$	− 620229.i	− 0.788324i	0.919041	−	0.394162i	$-0.128965\pi$
$888$	0	0
$889$	−259709.	−0.328612
$890$	0	0
$891$	−132156.	−0.166468
$892$	0	0
$893$	− 1.37511e6i	− 1.72439i
$894$	0	0
$895$	− 571272.i	− 0.713176i
$896$	0	0
$897$	−841414.	−1.04574
$898$	0	0
$899$	315856.	0.390814
$900$	0	0
$901$	322589.i	0.397374i
$902$	0	0
$903$	121820.i	0.149398i
$904$	0	0
$905$	161005.	0.196581
$906$	0	0
$907$	−155091.	−0.188526	−0.0942629	−	0.995547i	$-0.530049\pi$
$907$	−155091.	−0.188526	−0.0942629	+	0.995547i	$0.530049\pi$
$908$	0	0
$909$	420566.i	0.508986i
$910$	0	0
$911$	670134.i	0.807467i	0.914877	+	0.403734i	$0.132288\pi$
$911$	670134.i	0.807467i	−0.914877	+	0.403734i	$0.867712\pi$
$912$	0	0
$913$	539828.	0.647611
$914$	0	0
$915$	165629.	0.197831
$916$	0	0
$917$	1.46981e6i	1.74793i
$918$	0	0
$919$	− 631844.i	− 0.748134i	−0.927402	−	0.374067i	$-0.877963\pi$
$919$	− 631844.i	− 0.748134i	0.927402	−	0.374067i	$-0.122037\pi$
$920$	0	0
$921$	45831.7	0.0540314
$922$	0	0
$923$	398364.	0.467602
$924$	0	0
$925$	445959.i	0.521208i
$926$	0	0
$927$	− 178972.i	− 0.208269i
$928$	0	0
$929$	1.16204e6	1.34645	0.673225	−	0.739437i	$-0.264908\pi$
$929$	1.16204e6	1.34645	0.673225	+	0.739437i	$0.264908\pi$
$930$	0	0
$931$	−187886.	−0.216768
$932$	0	0
$933$	− 805708.i	− 0.925580i
$934$	0	0
$935$	3.08404e6i	3.52774i
$936$	0	0
$937$	−1.18305e6	−1.34748	−0.673741	−	0.738967i	$-0.735314\pi$
$937$	−1.18305e6	−1.34748	−0.673741	+	0.738967i	$0.735314\pi$
$938$	0	0
$939$	−933570.	−1.05880
$940$	0	0
$941$	− 159241.i	− 0.179836i	−0.995949	−	0.0899179i	$-0.971340\pi$
$941$	− 159241.i	− 0.179836i	0.995949	−	0.0899179i	$-0.0286605\pi$
$942$	0	0
$943$	1.90606e6i	2.14345i
$944$	0	0
$945$	256766.	0.287523
$946$	0	0
$947$	940316.	1.04851	0.524257	−	0.851560i	$-0.324343\pi$
$947$	940316.	1.04851	0.524257	+	0.851560i	$0.324343\pi$
$948$	0	0
$949$	− 421673.i	− 0.468213i
$950$	0	0
$951$	32157.8i	0.0355570i
$952$	0	0
$953$	577950.	0.636362	0.318181	−	0.948030i	$-0.396928\pi$
$953$	577950.	0.636362	0.318181	+	0.948030i	$0.396928\pi$
$954$	0	0
$955$	−1.98272e6	−2.17397
$956$	0	0
$957$	− 760408.i	− 0.830276i
$958$	0	0
$959$	− 484170.i	− 0.526454i
$960$	0	0
$961$	770424.	0.834225
$962$	0	0
$963$	−318504.	−0.343449
$964$	0	0
$965$	1.92487e6i	2.06703i
$966$	0	0
$967$	785047.i	0.839542i	0.907630	+	0.419771i	$0.137890\pi$
$967$	785047.i	0.839542i	−0.907630	+	0.419771i	$0.862110\pi$
$968$	0	0
$969$	−1.48681e6	−1.58347
$970$	0	0
$971$	−1.05236e6	−1.11616	−0.558082	−	0.829786i	$-0.688462\pi$
$971$	−1.05236e6	−1.11616	−0.558082	+	0.829786i	$0.688462\pi$
$972$	0	0
$973$	− 689996.i	− 0.728821i
$974$	0	0
$975$	− 910562.i	− 0.957856i
$976$	0	0
$977$	20197.7	0.0211598	0.0105799	−	0.999944i	$-0.496632\pi$
$977$	20197.7	0.0211598	0.0105799	+	0.999944i	$0.496632\pi$
$978$	0	0
$979$	1.17724e6	1.22829
$980$	0	0
$981$	358723.i	0.372754i
$982$	0	0
$983$	1.43076e6i	1.48068i	0.672234	+	0.740338i	$0.265335\pi$
$983$	1.43076e6i	1.48068i	−0.672234	+	0.740338i	$0.734665\pi$
$984$	0	0
$985$	220875.	0.227653
$986$	0	0
$987$	−492092.	−0.505140
$988$	0	0
$989$	449334.i	0.459385i
$990$	0	0
$991$	− 787866.i	− 0.802241i	−0.916025	−	0.401120i	$-0.868621\pi$
$991$	− 787866.i	− 0.802241i	0.916025	−	0.401120i	$-0.131379\pi$
$992$	0	0
$993$	−686718.	−0.696434
$994$	0	0
$995$	−2.40858e6	−2.43285
$996$	0	0
$997$	− 453709.i	− 0.456444i	−0.973609	−	0.228222i	$-0.926709\pi$
$997$	− 453709.i	− 0.456444i	0.973609	−	0.228222i	$-0.0732912\pi$
$998$	0	0
$999$	− 65515.5i	− 0.0656468i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.5.b.c.319.5	yes	8
3.2	odd	2		1152.5.b.k.703.8		8
4.3	odd	2	inner	384.5.b.c.319.1	✓	8
8.3	odd	2	inner	384.5.b.c.319.8	yes	8
8.5	even	2	inner	384.5.b.c.319.4	yes	8
12.11	even	2		1152.5.b.k.703.7		8
16.3	odd	4		768.5.g.c.511.1		4
16.5	even	4		768.5.g.g.511.2		4
16.11	odd	4		768.5.g.g.511.4		4
16.13	even	4		768.5.g.c.511.3		4
24.5	odd	2		1152.5.b.k.703.2		8
24.11	even	2		1152.5.b.k.703.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.b.c.319.1	✓	8	4.3	odd	2	inner
384.5.b.c.319.4	yes	8	8.5	even	2	inner
384.5.b.c.319.5	yes	8	1.1	even	1	trivial
384.5.b.c.319.8	yes	8	8.3	odd	2	inner
768.5.g.c.511.1		4	16.3	odd	4
768.5.g.c.511.3		4	16.13	even	4
768.5.g.g.511.2		4	16.5	even	4
768.5.g.g.511.4		4	16.11	odd	4
1152.5.b.k.703.1		8	24.11	even	2
1152.5.b.k.703.2		8	24.5	odd	2
1152.5.b.k.703.7		8	12.11	even	2
1152.5.b.k.703.8		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	384.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{3} -39.7490i q^{5} +46.0431i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q+5.19615 q^{3} -39.7490i q^{5} +46.0431i q^{7} +27.0000 q^{9} -181.283 q^{11} +183.498i q^{13} -206.542i q^{15} +427.992 q^{17} -668.558 q^{19} +239.247i q^{21} +882.463i q^{23} -954.984 q^{25} +140.296 q^{27} +807.247i q^{29} -391.276i q^{31} -941.976 q^{33} +1830.17 q^{35} -466.980i q^{37} +953.484i q^{39} +2159.93 q^{41} +509.182 q^{43} -1073.22i q^{45} +2056.83i q^{47} +281.031 q^{49} +2223.91 q^{51} +753.725i q^{53} +7205.84i q^{55} -3473.93 q^{57} -1309.43 q^{59} +801.913i q^{61} +1243.16i q^{63} +7293.87 q^{65} +505.813 q^{67} +4585.41i q^{69} -2170.94i q^{71} -2297.97 q^{73} -4962.24 q^{75} -8346.85i q^{77} +10705.3i q^{79} +729.000 q^{81} -2977.81 q^{83} -17012.3i q^{85} +4194.58i q^{87} -6493.92 q^{89} -8448.82 q^{91} -2033.13i q^{93} +26574.5i q^{95} -8189.92 q^{97} -4894.65 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 216 q^{9}+O(q^{10}) \) 8 * q + 216 * q^9	\( 8 q + 216 q^{9} + 1392 q^{17} - 3576 q^{25} - 1440 q^{33} - 1008 q^{41} + 10376 q^{49} - 9504 q^{57} + 23808 q^{65} - 10256 q^{73} + 5832 q^{81} - 31632 q^{89} - 45200 q^{97}+O(q^{100}) \) 8 * q + 216 * q^9 + 1392 * q^17 - 3576 * q^25 - 1440 * q^33 - 1008 * q^41 + 10376 * q^49 - 9504 * q^57 + 23808 * q^65 - 10256 * q^73 + 5832 * q^81 - 31632 * q^89 - 45200 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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