LMFDB - Embedded newform 768.4.d.f.385.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(385,768)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, 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("768.385");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.d (of order $2$, degree $1$, not 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:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			385.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	768.385
Dual form			768.4.d.f.385.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +4.00000i q^{5} -10.0000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +4.00000i q^{5} -10.0000 q^{7} -9.00000 q^{9} +4.00000i q^{11} -26.0000i q^{13} +12.0000 q^{15} +14.0000 q^{17} +8.00000i q^{19} +30.0000i q^{21} -148.000 q^{23} +109.000 q^{25} +27.0000i q^{27} -72.0000i q^{29} +18.0000 q^{31} +12.0000 q^{33} -40.0000i q^{35} +262.000i q^{37} -78.0000 q^{39} +378.000 q^{41} +432.000i q^{43} -36.0000i q^{45} +148.000 q^{47} -243.000 q^{49} -42.0000i q^{51} +360.000i q^{53} -16.0000 q^{55} +24.0000 q^{57} +428.000i q^{59} +442.000i q^{61} +90.0000 q^{63} +104.000 q^{65} -692.000i q^{67} +444.000i q^{69} -540.000 q^{71} +1018.00 q^{73} -327.000i q^{75} -40.0000i q^{77} +386.000 q^{79} +81.0000 q^{81} +108.000i q^{83} +56.0000i q^{85} -216.000 q^{87} +382.000 q^{89} +260.000i q^{91} -54.0000i q^{93} -32.0000 q^{95} +298.000 q^{97} -36.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20 q^{7} - 18 q^{9}+O(q^{10}) $ 2 * q - 20 * q^7 - 18 * q^9	$ 2 q - 20 q^{7} - 18 q^{9} + 24 q^{15} + 28 q^{17} - 296 q^{23} + 218 q^{25} + 36 q^{31} + 24 q^{33} - 156 q^{39} + 756 q^{41} + 296 q^{47} - 486 q^{49} - 32 q^{55} + 48 q^{57} + 180 q^{63} + 208 q^{65} - 1080 q^{71} + 2036 q^{73} + 772 q^{79} + 162 q^{81} - 432 q^{87} + 764 q^{89} - 64 q^{95} + 596 q^{97}+O(q^{100}) $ 2 * q - 20 * q^7 - 18 * q^9 + 24 * q^15 + 28 * q^17 - 296 * q^23 + 218 * q^25 + 36 * q^31 + 24 * q^33 - 156 * q^39 + 756 * q^41 + 296 * q^47 - 486 * q^49 - 32 * q^55 + 48 * q^57 + 180 * q^63 + 208 * q^65 - 1080 * q^71 + 2036 * q^73 + 772 * q^79 + 162 * q^81 - 432 * q^87 + 764 * q^89 - 64 * q^95 + 596 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	4.00000i	0.357771i	0.983870	+	0.178885i	$0.0572491\pi$
$5$	4.00000i	0.357771i	−0.983870	+	0.178885i	$0.942751\pi$
$6$	0	0
$7$	−10.0000	−0.539949	−0.269975	−	0.962867i	$-0.587015\pi$
$7$	−10.0000	−0.539949	−0.269975	+	0.962867i	$0.587015\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	4.00000i	0.109640i	0.998496	+	0.0548202i	$0.0174586\pi$
$11$	4.00000i	0.109640i	−0.998496	+	0.0548202i	$0.982541\pi$
$12$	0	0
$13$	− 26.0000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 26.0000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	12.0000	0.206559
$16$	0	0
$17$	14.0000	0.199735	0.0998676	−	0.995001i	$-0.468158\pi$
$17$	14.0000	0.199735	0.0998676	+	0.995001i	$0.468158\pi$
$18$	0	0
$19$	8.00000i	0.0965961i	0.998833	+	0.0482980i	$0.0153797\pi$
$19$	8.00000i	0.0965961i	−0.998833	+	0.0482980i	$0.984620\pi$
$20$	0	0
$21$	30.0000i	0.311740i
$22$	0	0
$23$	−148.000	−1.34174	−0.670872	−	0.741573i	$-0.734080\pi$
$23$	−148.000	−1.34174	−0.670872	+	0.741573i	$0.734080\pi$
$24$	0	0
$25$	109.000	0.872000
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	− 72.0000i	− 0.461037i	−0.973068	−	0.230518i	$-0.925958\pi$
$29$	− 72.0000i	− 0.461037i	0.973068	−	0.230518i	$-0.0740422\pi$
$30$	0	0
$31$	18.0000	0.104287	0.0521435	−	0.998640i	$-0.483395\pi$
$31$	18.0000	0.104287	0.0521435	+	0.998640i	$0.483395\pi$
$32$	0	0
$33$	12.0000	0.0633010
$34$	0	0
$35$	− 40.0000i	− 0.193178i
$36$	0	0
$37$	262.000i	1.16412i	0.813145	+	0.582061i	$0.197754\pi$
$37$	262.000i	1.16412i	−0.813145	+	0.582061i	$0.802246\pi$
$38$	0	0
$39$	−78.0000	−0.320256
$40$	0	0
$41$	378.000	1.43985	0.719923	−	0.694054i	$-0.244177\pi$
$41$	378.000	1.43985	0.719923	+	0.694054i	$0.244177\pi$
$42$	0	0
$43$	432.000i	1.53208i	0.642794	+	0.766039i	$0.277775\pi$
$43$	432.000i	1.53208i	−0.642794	+	0.766039i	$0.722225\pi$
$44$	0	0
$45$	− 36.0000i	− 0.119257i
$46$	0	0
$47$	148.000	0.459320	0.229660	−	0.973271i	$-0.426239\pi$
$47$	148.000	0.459320	0.229660	+	0.973271i	$0.426239\pi$
$48$	0	0
$49$	−243.000	−0.708455
$50$	0	0
$51$	− 42.0000i	− 0.115317i
$52$	0	0
$53$	360.000i	0.933015i	0.884517	+	0.466508i	$0.154488\pi$
$53$	360.000i	0.933015i	−0.884517	+	0.466508i	$0.845512\pi$
$54$	0	0
$55$	−16.0000	−0.0392262
$56$	0	0
$57$	24.0000	0.0557698
$58$	0	0
$59$	428.000i	0.944421i	0.881486	+	0.472211i	$0.156544\pi$
$59$	428.000i	0.944421i	−0.881486	+	0.472211i	$0.843456\pi$
$60$	0	0
$61$	442.000i	0.927743i	0.885903	+	0.463871i	$0.153540\pi$
$61$	442.000i	0.927743i	−0.885903	+	0.463871i	$0.846460\pi$
$62$	0	0
$63$	90.0000	0.179983
$64$	0	0
$65$	104.000	0.198456
$66$	0	0
$67$	− 692.000i	− 1.26181i	−0.775860	−	0.630905i	$-0.782684\pi$
$67$	− 692.000i	− 1.26181i	0.775860	−	0.630905i	$-0.217316\pi$
$68$	0	0
$69$	444.000i	0.774657i
$70$	0	0
$71$	−540.000	−0.902623	−0.451311	−	0.892367i	$-0.649044\pi$
$71$	−540.000	−0.902623	−0.451311	+	0.892367i	$0.649044\pi$
$72$	0	0
$73$	1018.00	1.63216	0.816081	−	0.577937i	$-0.196142\pi$
$73$	1018.00	1.63216	0.816081	+	0.577937i	$0.196142\pi$
$74$	0	0
$75$	− 327.000i	− 0.503449i
$76$	0	0
$77$	− 40.0000i	− 0.0592003i
$78$	0	0
$79$	386.000	0.549726	0.274863	−	0.961483i	$-0.411367\pi$
$79$	386.000	0.549726	0.274863	+	0.961483i	$0.411367\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	108.000i	0.142826i	0.997447	+	0.0714129i	$0.0227508\pi$
$83$	108.000i	0.142826i	−0.997447	+	0.0714129i	$0.977249\pi$
$84$	0	0
$85$	56.0000i	0.0714594i
$86$	0	0
$87$	−216.000	−0.266180
$88$	0	0
$89$	382.000	0.454965	0.227483	−	0.973782i	$-0.426950\pi$
$89$	382.000	0.454965	0.227483	+	0.973782i	$0.426950\pi$
$90$	0	0
$91$	260.000i	0.299510i
$92$	0	0
$93$	− 54.0000i	− 0.0602101i
$94$	0	0
$95$	−32.0000	−0.0345593
$96$	0	0
$97$	298.000	0.311931	0.155966	−	0.987762i	$-0.450151\pi$
$97$	298.000	0.311931	0.155966	+	0.987762i	$0.450151\pi$
$98$	0	0
$99$	− 36.0000i	− 0.0365468i
$100$	0	0
$101$	1800.00i	1.77333i	0.462409	+	0.886667i	$0.346985\pi$
$101$	1800.00i	1.77333i	−0.462409	+	0.886667i	$0.653015\pi$
$102$	0	0
$103$	818.000	0.782524	0.391262	−	0.920279i	$-0.372039\pi$
$103$	818.000	0.782524	0.391262	+	0.920279i	$0.372039\pi$
$104$	0	0
$105$	−120.000	−0.111531
$106$	0	0
$107$	− 716.000i	− 0.646900i	−0.946245	−	0.323450i	$-0.895157\pi$
$107$	− 716.000i	− 0.646900i	0.946245	−	0.323450i	$-0.104843\pi$
$108$	0	0
$109$	1134.00i	0.996491i	0.867036	+	0.498245i	$0.166022\pi$
$109$	1134.00i	0.996491i	−0.867036	+	0.498245i	$0.833978\pi$
$110$	0	0
$111$	786.000	0.672106
$112$	0	0
$113$	1858.00	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$113$	1858.00	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$114$	0	0
$115$	− 592.000i	− 0.480037i
$116$	0	0
$117$	234.000i	0.184900i
$118$	0	0
$119$	−140.000	−0.107847
$120$	0	0
$121$	1315.00	0.987979
$122$	0	0
$123$	− 1134.00i	− 0.831295i
$124$	0	0
$125$	936.000i	0.669747i
$126$	0	0
$127$	−1402.00	−0.979586	−0.489793	−	0.871839i	$-0.662928\pi$
$127$	−1402.00	−0.979586	−0.489793	+	0.871839i	$0.662928\pi$
$128$	0	0
$129$	1296.00	0.884546
$130$	0	0
$131$	1188.00i	0.792336i	0.918178	+	0.396168i	$0.129660\pi$
$131$	1188.00i	0.792336i	−0.918178	+	0.396168i	$0.870340\pi$
$132$	0	0
$133$	− 80.0000i	− 0.0521570i
$134$	0	0
$135$	−108.000	−0.0688530
$136$	0	0
$137$	−734.000	−0.457736	−0.228868	−	0.973457i	$-0.573502\pi$
$137$	−734.000	−0.457736	−0.228868	+	0.973457i	$0.573502\pi$
$138$	0	0
$139$	− 404.000i	− 0.246524i	−0.992374	−	0.123262i	$-0.960664\pi$
$139$	− 404.000i	− 0.246524i	0.992374	−	0.123262i	$-0.0393355\pi$
$140$	0	0
$141$	− 444.000i	− 0.265188i
$142$	0	0
$143$	104.000	0.0608176
$144$	0	0
$145$	288.000	0.164946
$146$	0	0
$147$	729.000i	0.409027i
$148$	0	0
$149$	− 684.000i	− 0.376077i	−0.982162	−	0.188038i	$-0.939787\pi$
$149$	− 684.000i	− 0.376077i	0.982162	−	0.188038i	$-0.0602130\pi$
$150$	0	0
$151$	3502.00	1.88734	0.943671	−	0.330885i	$-0.107347\pi$
$151$	3502.00	1.88734	0.943671	+	0.330885i	$0.107347\pi$
$152$	0	0
$153$	−126.000	−0.0665784
$154$	0	0
$155$	72.0000i	0.0373108i
$156$	0	0
$157$	3258.00i	1.65616i	0.560612	+	0.828079i	$0.310566\pi$
$157$	3258.00i	1.65616i	−0.560612	+	0.828079i	$0.689434\pi$
$158$	0	0
$159$	1080.00	0.538677
$160$	0	0
$161$	1480.00	0.724474
$162$	0	0
$163$	− 856.000i	− 0.411332i	−0.978622	−	0.205666i	$-0.934064\pi$
$163$	− 856.000i	− 0.411332i	0.978622	−	0.205666i	$-0.0659360\pi$
$164$	0	0
$165$	48.0000i	0.0226472i
$166$	0	0
$167$	−1552.00	−0.719146	−0.359573	−	0.933117i	$-0.617078\pi$
$167$	−1552.00	−0.719146	−0.359573	+	0.933117i	$0.617078\pi$
$168$	0	0
$169$	1521.00	0.692308
$170$	0	0
$171$	− 72.0000i	− 0.0321987i
$172$	0	0
$173$	1548.00i	0.680302i	0.940371	+	0.340151i	$0.110478\pi$
$173$	1548.00i	0.680302i	−0.940371	+	0.340151i	$0.889522\pi$
$174$	0	0
$175$	−1090.00	−0.470836
$176$	0	0
$177$	1284.00	0.545262
$178$	0	0
$179$	3420.00i	1.42806i	0.700115	+	0.714030i	$0.253132\pi$
$179$	3420.00i	1.42806i	−0.700115	+	0.714030i	$0.746868\pi$
$180$	0	0
$181$	− 3502.00i	− 1.43813i	−0.694943	−	0.719065i	$-0.744570\pi$
$181$	− 3502.00i	− 1.43813i	0.694943	−	0.719065i	$-0.255430\pi$
$182$	0	0
$183$	1326.00	0.535632
$184$	0	0
$185$	−1048.00	−0.416489
$186$	0	0
$187$	56.0000i	0.0218991i
$188$	0	0
$189$	− 270.000i	− 0.103913i
$190$	0	0
$191$	−4792.00	−1.81538	−0.907688	−	0.419645i	$-0.862155\pi$
$191$	−4792.00	−1.81538	−0.907688	+	0.419645i	$0.862155\pi$
$192$	0	0
$193$	−98.0000	−0.0365502	−0.0182751	−	0.999833i	$-0.505817\pi$
$193$	−98.0000	−0.0365502	−0.0182751	+	0.999833i	$0.505817\pi$
$194$	0	0
$195$	− 312.000i	− 0.114578i
$196$	0	0
$197$	1184.00i	0.428206i	0.976811	+	0.214103i	$0.0686828\pi$
$197$	1184.00i	0.428206i	−0.976811	+	0.214103i	$0.931317\pi$
$198$	0	0
$199$	1466.00	0.522221	0.261111	−	0.965309i	$-0.415911\pi$
$199$	1466.00	0.522221	0.261111	+	0.965309i	$0.415911\pi$
$200$	0	0
$201$	−2076.00	−0.728506
$202$	0	0
$203$	720.000i	0.248936i
$204$	0	0
$205$	1512.00i	0.515135i
$206$	0	0
$207$	1332.00	0.447248
$208$	0	0
$209$	−32.0000	−0.0105908
$210$	0	0
$211$	− 2692.00i	− 0.878317i	−0.898410	−	0.439159i	$-0.855277\pi$
$211$	− 2692.00i	− 0.878317i	0.898410	−	0.439159i	$-0.144723\pi$
$212$	0	0
$213$	1620.00i	0.521129i
$214$	0	0
$215$	−1728.00	−0.548133
$216$	0	0
$217$	−180.000	−0.0563097
$218$	0	0
$219$	− 3054.00i	− 0.942330i
$220$	0	0
$221$	− 364.000i	− 0.110793i
$222$	0	0
$223$	−4554.00	−1.36753	−0.683763	−	0.729704i	$-0.739658\pi$
$223$	−4554.00	−1.36753	−0.683763	+	0.729704i	$0.739658\pi$
$224$	0	0
$225$	−981.000	−0.290667
$226$	0	0
$227$	5580.00i	1.63153i	0.578383	+	0.815766i	$0.303684\pi$
$227$	5580.00i	1.63153i	−0.578383	+	0.815766i	$0.696316\pi$
$228$	0	0
$229$	− 902.000i	− 0.260288i	−0.991495	−	0.130144i	$-0.958456\pi$
$229$	− 902.000i	− 0.260288i	0.991495	−	0.130144i	$-0.0415439\pi$
$230$	0	0
$231$	−120.000	−0.0341793
$232$	0	0
$233$	−1962.00	−0.551652	−0.275826	−	0.961208i	$-0.588951\pi$
$233$	−1962.00	−0.551652	−0.275826	+	0.961208i	$0.588951\pi$
$234$	0	0
$235$	592.000i	0.164331i
$236$	0	0
$237$	− 1158.00i	− 0.317385i
$238$	0	0
$239$	2016.00	0.545624	0.272812	−	0.962067i	$-0.412046\pi$
$239$	2016.00	0.545624	0.272812	+	0.962067i	$0.412046\pi$
$240$	0	0
$241$	−1826.00	−0.488062	−0.244031	−	0.969767i	$-0.578470\pi$
$241$	−1826.00	−0.488062	−0.244031	+	0.969767i	$0.578470\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 972.000i	− 0.253464i
$246$	0	0
$247$	208.000	0.0535819
$248$	0	0
$249$	324.000	0.0824605
$250$	0	0
$251$	3060.00i	0.769504i	0.923020	+	0.384752i	$0.125713\pi$
$251$	3060.00i	0.769504i	−0.923020	+	0.384752i	$0.874287\pi$
$252$	0	0
$253$	− 592.000i	− 0.147110i
$254$	0	0
$255$	168.000	0.0412571
$256$	0	0
$257$	−558.000	−0.135436	−0.0677181	−	0.997704i	$-0.521572\pi$
$257$	−558.000	−0.135436	−0.0677181	+	0.997704i	$0.521572\pi$
$258$	0	0
$259$	− 2620.00i	− 0.628567i
$260$	0	0
$261$	648.000i	0.153679i
$262$	0	0
$263$	−5544.00	−1.29984	−0.649920	−	0.760003i	$-0.725197\pi$
$263$	−5544.00	−1.29984	−0.649920	+	0.760003i	$0.725197\pi$
$264$	0	0
$265$	−1440.00	−0.333806
$266$	0	0
$267$	− 1146.00i	− 0.262674i
$268$	0	0
$269$	− 4856.00i	− 1.10065i	−0.834950	−	0.550326i	$-0.814503\pi$
$269$	− 4856.00i	− 1.10065i	0.834950	−	0.550326i	$-0.185497\pi$
$270$	0	0
$271$	7030.00	1.57580	0.787901	−	0.615803i	$-0.211168\pi$
$271$	7030.00	1.57580	0.787901	+	0.615803i	$0.211168\pi$
$272$	0	0
$273$	780.000	0.172922
$274$	0	0
$275$	436.000i	0.0956065i
$276$	0	0
$277$	738.000i	0.160080i	0.996792	+	0.0800399i	$0.0255048\pi$
$277$	738.000i	0.160080i	−0.996792	+	0.0800399i	$0.974495\pi$
$278$	0	0
$279$	−162.000	−0.0347623
$280$	0	0
$281$	1854.00	0.393596	0.196798	−	0.980444i	$-0.436946\pi$
$281$	1854.00	0.393596	0.196798	+	0.980444i	$0.436946\pi$
$282$	0	0
$283$	− 1404.00i	− 0.294909i	−0.989069	−	0.147454i	$-0.952892\pi$
$283$	− 1404.00i	− 0.294909i	0.989069	−	0.147454i	$-0.0471079\pi$
$284$	0	0
$285$	96.0000i	0.0199528i
$286$	0	0
$287$	−3780.00	−0.777444
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	− 894.000i	− 0.180093i
$292$	0	0
$293$	− 2056.00i	− 0.409941i	−0.978768	−	0.204971i	$-0.934290\pi$
$293$	− 2056.00i	− 0.409941i	0.978768	−	0.204971i	$-0.0657099\pi$
$294$	0	0
$295$	−1712.00	−0.337886
$296$	0	0
$297$	−108.000	−0.0211003
$298$	0	0
$299$	3848.00i	0.744266i
$300$	0	0
$301$	− 4320.00i	− 0.827245i
$302$	0	0
$303$	5400.00	1.02383
$304$	0	0
$305$	−1768.00	−0.331919
$306$	0	0
$307$	− 2340.00i	− 0.435019i	−0.976058	−	0.217510i	$-0.930207\pi$
$307$	− 2340.00i	− 0.435019i	0.976058	−	0.217510i	$-0.0697933\pi$
$308$	0	0
$309$	− 2454.00i	− 0.451790i
$310$	0	0
$311$	−1616.00	−0.294646	−0.147323	−	0.989088i	$-0.547066\pi$
$311$	−1616.00	−0.294646	−0.147323	+	0.989088i	$0.547066\pi$
$312$	0	0
$313$	−3286.00	−0.593405	−0.296702	−	0.954970i	$-0.595887\pi$
$313$	−3286.00	−0.593405	−0.296702	+	0.954970i	$0.595887\pi$
$314$	0	0
$315$	360.000i	0.0643927i
$316$	0	0
$317$	− 9936.00i	− 1.76045i	−0.474560	−	0.880223i	$-0.657393\pi$
$317$	− 9936.00i	− 1.76045i	0.474560	−	0.880223i	$-0.342607\pi$
$318$	0	0
$319$	288.000	0.0505483
$320$	0	0
$321$	−2148.00	−0.373488
$322$	0	0
$323$	112.000i	0.0192936i
$324$	0	0
$325$	− 2834.00i	− 0.483699i
$326$	0	0
$327$	3402.00	0.575324
$328$	0	0
$329$	−1480.00	−0.248009
$330$	0	0
$331$	2116.00i	0.351377i	0.984446	+	0.175689i	$0.0562152\pi$
$331$	2116.00i	0.351377i	−0.984446	+	0.175689i	$0.943785\pi$
$332$	0	0
$333$	− 2358.00i	− 0.388041i
$334$	0	0
$335$	2768.00	0.451439
$336$	0	0
$337$	−7874.00	−1.27277	−0.636386	−	0.771371i	$-0.719571\pi$
$337$	−7874.00	−1.27277	−0.636386	+	0.771371i	$0.719571\pi$
$338$	0	0
$339$	− 5574.00i	− 0.893033i
$340$	0	0
$341$	72.0000i	0.0114341i
$342$	0	0
$343$	5860.00	0.922479
$344$	0	0
$345$	−1776.00	−0.277150
$346$	0	0
$347$	10476.0i	1.62069i	0.585950	+	0.810347i	$0.300722\pi$
$347$	10476.0i	1.62069i	−0.585950	+	0.810347i	$0.699278\pi$
$348$	0	0
$349$	− 7030.00i	− 1.07824i	−0.842228	−	0.539122i	$-0.818756\pi$
$349$	− 7030.00i	− 1.07824i	0.842228	−	0.539122i	$-0.181244\pi$
$350$	0	0
$351$	702.000	0.106752
$352$	0	0
$353$	−5310.00	−0.800631	−0.400316	−	0.916377i	$-0.631099\pi$
$353$	−5310.00	−0.800631	−0.400316	+	0.916377i	$0.631099\pi$
$354$	0	0
$355$	− 2160.00i	− 0.322932i
$356$	0	0
$357$	420.000i	0.0622654i
$358$	0	0
$359$	−7628.00	−1.12142	−0.560711	−	0.828012i	$-0.689472\pi$
$359$	−7628.00	−1.12142	−0.560711	+	0.828012i	$0.689472\pi$
$360$	0	0
$361$	6795.00	0.990669
$362$	0	0
$363$	− 3945.00i	− 0.570410i
$364$	0	0
$365$	4072.00i	0.583940i
$366$	0	0
$367$	13778.0	1.95969	0.979844	−	0.199763i	$-0.0640171\pi$
$367$	13778.0	1.95969	0.979844	+	0.199763i	$0.0640171\pi$
$368$	0	0
$369$	−3402.00	−0.479949
$370$	0	0
$371$	− 3600.00i	− 0.503781i
$372$	0	0
$373$	2566.00i	0.356200i	0.984012	+	0.178100i	$0.0569950\pi$
$373$	2566.00i	0.356200i	−0.984012	+	0.178100i	$0.943005\pi$
$374$	0	0
$375$	2808.00	0.386679
$376$	0	0
$377$	−1872.00	−0.255737
$378$	0	0
$379$	− 7120.00i	− 0.964986i	−0.875900	−	0.482493i	$-0.839731\pi$
$379$	− 7120.00i	− 0.964986i	0.875900	−	0.482493i	$-0.160269\pi$
$380$	0	0
$381$	4206.00i	0.565564i
$382$	0	0
$383$	−14040.0	−1.87313	−0.936567	−	0.350488i	$-0.886016\pi$
$383$	−14040.0	−1.87313	−0.936567	+	0.350488i	$0.886016\pi$
$384$	0	0
$385$	160.000	0.0211801
$386$	0	0
$387$	− 3888.00i	− 0.510693i
$388$	0	0
$389$	− 4972.00i	− 0.648047i	−0.946049	−	0.324024i	$-0.894964\pi$
$389$	− 4972.00i	− 0.648047i	0.946049	−	0.324024i	$-0.105036\pi$
$390$	0	0
$391$	−2072.00	−0.267994
$392$	0	0
$393$	3564.00	0.457456
$394$	0	0
$395$	1544.00i	0.196676i
$396$	0	0
$397$	− 8766.00i	− 1.10819i	−0.832452	−	0.554097i	$-0.813064\pi$
$397$	− 8766.00i	− 1.10819i	0.832452	−	0.554097i	$-0.186936\pi$
$398$	0	0
$399$	−240.000	−0.0301129
$400$	0	0
$401$	−12042.0	−1.49962	−0.749811	−	0.661652i	$-0.769856\pi$
$401$	−12042.0	−1.49962	−0.749811	+	0.661652i	$0.769856\pi$
$402$	0	0
$403$	− 468.000i	− 0.0578480i
$404$	0	0
$405$	324.000i	0.0397523i
$406$	0	0
$407$	−1048.00	−0.127635
$408$	0	0
$409$	−8002.00	−0.967417	−0.483708	−	0.875229i	$-0.660710\pi$
$409$	−8002.00	−0.967417	−0.483708	+	0.875229i	$0.660710\pi$
$410$	0	0
$411$	2202.00i	0.264274i
$412$	0	0
$413$	− 4280.00i	− 0.509940i
$414$	0	0
$415$	−432.000	−0.0510989
$416$	0	0
$417$	−1212.00	−0.142331
$418$	0	0
$419$	364.000i	0.0424405i	0.999775	+	0.0212202i	$0.00675512\pi$
$419$	364.000i	0.0424405i	−0.999775	+	0.0212202i	$0.993245\pi$
$420$	0	0
$421$	− 1502.00i	− 0.173879i	−0.996214	−	0.0869394i	$-0.972291\pi$
$421$	− 1502.00i	− 0.173879i	0.996214	−	0.0869394i	$-0.0277087\pi$
$422$	0	0
$423$	−1332.00	−0.153107
$424$	0	0
$425$	1526.00	0.174169
$426$	0	0
$427$	− 4420.00i	− 0.500934i
$428$	0	0
$429$	− 312.000i	− 0.0351131i
$430$	0	0
$431$	3532.00	0.394734	0.197367	−	0.980330i	$-0.436761\pi$
$431$	3532.00	0.394734	0.197367	+	0.980330i	$0.436761\pi$
$432$	0	0
$433$	8946.00	0.992881	0.496440	−	0.868071i	$-0.334640\pi$
$433$	8946.00	0.992881	0.496440	+	0.868071i	$0.334640\pi$
$434$	0	0
$435$	− 864.000i	− 0.0952313i
$436$	0	0
$437$	− 1184.00i	− 0.129607i
$438$	0	0
$439$	1602.00	0.174167	0.0870835	−	0.996201i	$-0.472245\pi$
$439$	1602.00	0.174167	0.0870835	+	0.996201i	$0.472245\pi$
$440$	0	0
$441$	2187.00	0.236152
$442$	0	0
$443$	− 4356.00i	− 0.467178i	−0.972335	−	0.233589i	$-0.924953\pi$
$443$	− 4356.00i	− 0.467178i	0.972335	−	0.233589i	$-0.0750470\pi$
$444$	0	0
$445$	1528.00i	0.162773i
$446$	0	0
$447$	−2052.00	−0.217128
$448$	0	0
$449$	2110.00	0.221775	0.110888	−	0.993833i	$-0.464631\pi$
$449$	2110.00	0.221775	0.110888	+	0.993833i	$0.464631\pi$
$450$	0	0
$451$	1512.00i	0.157865i
$452$	0	0
$453$	− 10506.0i	− 1.08966i
$454$	0	0
$455$	−1040.00	−0.107156
$456$	0	0
$457$	−13698.0	−1.40211	−0.701056	−	0.713106i	$-0.747288\pi$
$457$	−13698.0	−1.40211	−0.701056	+	0.713106i	$0.747288\pi$
$458$	0	0
$459$	378.000i	0.0384391i
$460$	0	0
$461$	12028.0i	1.21518i	0.794249	+	0.607592i	$0.207864\pi$
$461$	12028.0i	1.21518i	−0.794249	+	0.607592i	$0.792136\pi$
$462$	0	0
$463$	10286.0	1.03246	0.516232	−	0.856449i	$-0.327334\pi$
$463$	10286.0	1.03246	0.516232	+	0.856449i	$0.327334\pi$
$464$	0	0
$465$	216.000	0.0215414
$466$	0	0
$467$	11516.0i	1.14111i	0.821260	+	0.570553i	$0.193271\pi$
$467$	11516.0i	1.14111i	−0.821260	+	0.570553i	$0.806729\pi$
$468$	0	0
$469$	6920.00i	0.681313i
$470$	0	0
$471$	9774.00	0.956183
$472$	0	0
$473$	−1728.00	−0.167978
$474$	0	0
$475$	872.000i	0.0842318i
$476$	0	0
$477$	− 3240.00i	− 0.311005i
$478$	0	0
$479$	−13932.0	−1.32895	−0.664477	−	0.747308i	$-0.731346\pi$
$479$	−13932.0	−1.32895	−0.664477	+	0.747308i	$0.731346\pi$
$480$	0	0
$481$	6812.00	0.645739
$482$	0	0
$483$	− 4440.00i	− 0.418275i
$484$	0	0
$485$	1192.00i	0.111600i
$486$	0	0
$487$	−10682.0	−0.993938	−0.496969	−	0.867768i	$-0.665554\pi$
$487$	−10682.0	−0.993938	−0.496969	+	0.867768i	$0.665554\pi$
$488$	0	0
$489$	−2568.00	−0.237483
$490$	0	0
$491$	11660.0i	1.07171i	0.844311	+	0.535854i	$0.180010\pi$
$491$	11660.0i	1.07171i	−0.844311	+	0.535854i	$0.819990\pi$
$492$	0	0
$493$	− 1008.00i	− 0.0920853i
$494$	0	0
$495$	144.000	0.0130754
$496$	0	0
$497$	5400.00	0.487370
$498$	0	0
$499$	12404.0i	1.11278i	0.830920	+	0.556392i	$0.187815\pi$
$499$	12404.0i	1.11278i	−0.830920	+	0.556392i	$0.812185\pi$
$500$	0	0
$501$	4656.00i	0.415199i
$502$	0	0
$503$	2308.00	0.204590	0.102295	−	0.994754i	$-0.467381\pi$
$503$	2308.00	0.204590	0.102295	+	0.994754i	$0.467381\pi$
$504$	0	0
$505$	−7200.00	−0.634447
$506$	0	0
$507$	− 4563.00i	− 0.399704i
$508$	0	0
$509$	1984.00i	0.172769i	0.996262	+	0.0863843i	$0.0275313\pi$
$509$	1984.00i	0.172769i	−0.996262	+	0.0863843i	$0.972469\pi$
$510$	0	0
$511$	−10180.0	−0.881285
$512$	0	0
$513$	−216.000	−0.0185899
$514$	0	0
$515$	3272.00i	0.279964i
$516$	0	0
$517$	592.000i	0.0503600i
$518$	0	0
$519$	4644.00	0.392773
$520$	0	0
$521$	−3982.00	−0.334846	−0.167423	−	0.985885i	$-0.553545\pi$
$521$	−3982.00	−0.334846	−0.167423	+	0.985885i	$0.553545\pi$
$522$	0	0
$523$	− 18368.0i	− 1.53571i	−0.640623	−	0.767855i	$-0.721324\pi$
$523$	− 18368.0i	− 1.53571i	0.640623	−	0.767855i	$-0.278676\pi$
$524$	0	0
$525$	3270.00i	0.271837i
$526$	0	0
$527$	252.000	0.0208298
$528$	0	0
$529$	9737.00	0.800279
$530$	0	0
$531$	− 3852.00i	− 0.314807i
$532$	0	0
$533$	− 9828.00i	− 0.798683i
$534$	0	0
$535$	2864.00	0.231442
$536$	0	0
$537$	10260.0	0.824491
$538$	0	0
$539$	− 972.000i	− 0.0776753i
$540$	0	0
$541$	2870.00i	0.228079i	0.993476	+	0.114040i	$0.0363791\pi$
$541$	2870.00i	0.228079i	−0.993476	+	0.114040i	$0.963621\pi$
$542$	0	0
$543$	−10506.0	−0.830305
$544$	0	0
$545$	−4536.00	−0.356515
$546$	0	0
$547$	14176.0i	1.10808i	0.832489	+	0.554042i	$0.186915\pi$
$547$	14176.0i	1.10808i	−0.832489	+	0.554042i	$0.813085\pi$
$548$	0	0
$549$	− 3978.00i	− 0.309248i
$550$	0	0
$551$	576.000	0.0445343
$552$	0	0
$553$	−3860.00	−0.296824
$554$	0	0
$555$	3144.00i	0.240460i
$556$	0	0
$557$	12884.0i	0.980094i	0.871696	+	0.490047i	$0.163020\pi$
$557$	12884.0i	0.980094i	−0.871696	+	0.490047i	$0.836980\pi$
$558$	0	0
$559$	11232.0	0.849844
$560$	0	0
$561$	168.000	0.0126434
$562$	0	0
$563$	15588.0i	1.16688i	0.812155	+	0.583442i	$0.198295\pi$
$563$	15588.0i	1.16688i	−0.812155	+	0.583442i	$0.801705\pi$
$564$	0	0
$565$	7432.00i	0.553392i
$566$	0	0
$567$	−810.000	−0.0599944
$568$	0	0
$569$	4730.00	0.348492	0.174246	−	0.984702i	$-0.444251\pi$
$569$	4730.00	0.348492	0.174246	+	0.984702i	$0.444251\pi$
$570$	0	0
$571$	− 4724.00i	− 0.346223i	−0.984902	−	0.173111i	$-0.944618\pi$
$571$	− 4724.00i	− 0.346223i	0.984902	−	0.173111i	$-0.0553821\pi$
$572$	0	0
$573$	14376.0i	1.04811i
$574$	0	0
$575$	−16132.0	−1.17000
$576$	0	0
$577$	16838.0	1.21486	0.607431	−	0.794373i	$-0.292200\pi$
$577$	16838.0	1.21486	0.607431	+	0.794373i	$0.292200\pi$
$578$	0	0
$579$	294.000i	0.0211023i
$580$	0	0
$581$	− 1080.00i	− 0.0771187i
$582$	0	0
$583$	−1440.00	−0.102296
$584$	0	0
$585$	−936.000	−0.0661519
$586$	0	0
$587$	9932.00i	0.698360i	0.937056	+	0.349180i	$0.113540\pi$
$587$	9932.00i	0.698360i	−0.937056	+	0.349180i	$0.886460\pi$
$588$	0	0
$589$	144.000i	0.0100737i
$590$	0	0
$591$	3552.00	0.247225
$592$	0	0
$593$	15714.0	1.08819	0.544095	−	0.839024i	$-0.316873\pi$
$593$	15714.0	1.08819	0.544095	+	0.839024i	$0.316873\pi$
$594$	0	0
$595$	− 560.000i	− 0.0385845i
$596$	0	0
$597$	− 4398.00i	− 0.301504i
$598$	0	0
$599$	20340.0	1.38743	0.693714	−	0.720250i	$-0.255973\pi$
$599$	20340.0	1.38743	0.693714	+	0.720250i	$0.255973\pi$
$600$	0	0
$601$	16434.0	1.11540	0.557701	−	0.830042i	$-0.311684\pi$
$601$	16434.0	1.11540	0.557701	+	0.830042i	$0.311684\pi$
$602$	0	0
$603$	6228.00i	0.420603i
$604$	0	0
$605$	5260.00i	0.353470i
$606$	0	0
$607$	−8234.00	−0.550589	−0.275295	−	0.961360i	$-0.588775\pi$
$607$	−8234.00	−0.550589	−0.275295	+	0.961360i	$0.588775\pi$
$608$	0	0
$609$	2160.00	0.143724
$610$	0	0
$611$	− 3848.00i	− 0.254785i
$612$	0	0
$613$	− 6506.00i	− 0.428670i	−0.976760	−	0.214335i	$-0.931242\pi$
$613$	− 6506.00i	− 0.428670i	0.976760	−	0.214335i	$-0.0687585\pi$
$614$	0	0
$615$	4536.00	0.297413
$616$	0	0
$617$	20542.0	1.34034	0.670170	−	0.742208i	$-0.266221\pi$
$617$	20542.0	1.34034	0.670170	+	0.742208i	$0.266221\pi$
$618$	0	0
$619$	− 11116.0i	− 0.721793i	−0.932606	−	0.360896i	$-0.882471\pi$
$619$	− 11116.0i	− 0.721793i	0.932606	−	0.360896i	$-0.117529\pi$
$620$	0	0
$621$	− 3996.00i	− 0.258219i
$622$	0	0
$623$	−3820.00	−0.245658
$624$	0	0
$625$	9881.00	0.632384
$626$	0	0
$627$	96.0000i	0.00611463i
$628$	0	0
$629$	3668.00i	0.232516i
$630$	0	0
$631$	3574.00	0.225481	0.112741	−	0.993624i	$-0.464037\pi$
$631$	3574.00	0.225481	0.112741	+	0.993624i	$0.464037\pi$
$632$	0	0
$633$	−8076.00	−0.507097
$634$	0	0
$635$	− 5608.00i	− 0.350467i
$636$	0	0
$637$	6318.00i	0.392980i
$638$	0	0
$639$	4860.00	0.300874
$640$	0	0
$641$	7110.00	0.438109	0.219055	−	0.975713i	$-0.429703\pi$
$641$	7110.00	0.438109	0.219055	+	0.975713i	$0.429703\pi$
$642$	0	0
$643$	− 1728.00i	− 0.105981i	−0.998595	−	0.0529904i	$-0.983125\pi$
$643$	− 1728.00i	− 0.105981i	0.998595	−	0.0529904i	$-0.0168753\pi$
$644$	0	0
$645$	5184.00i	0.316465i
$646$	0	0
$647$	−14724.0	−0.894683	−0.447342	−	0.894363i	$-0.647629\pi$
$647$	−14724.0	−0.894683	−0.447342	+	0.894363i	$0.647629\pi$
$648$	0	0
$649$	−1712.00	−0.103547
$650$	0	0
$651$	540.000i	0.0325104i
$652$	0	0
$653$	23076.0i	1.38290i	0.722424	+	0.691450i	$0.243028\pi$
$653$	23076.0i	1.38290i	−0.722424	+	0.691450i	$0.756972\pi$
$654$	0	0
$655$	−4752.00	−0.283475
$656$	0	0
$657$	−9162.00	−0.544054
$658$	0	0
$659$	− 27252.0i	− 1.61091i	−0.592660	−	0.805453i	$-0.701922\pi$
$659$	− 27252.0i	− 1.61091i	0.592660	−	0.805453i	$-0.298078\pi$
$660$	0	0
$661$	18910.0i	1.11273i	0.830938	+	0.556364i	$0.187804\pi$
$661$	18910.0i	1.11273i	−0.830938	+	0.556364i	$0.812196\pi$
$662$	0	0
$663$	−1092.00	−0.0639665
$664$	0	0
$665$	320.000	0.0186603
$666$	0	0
$667$	10656.0i	0.618594i
$668$	0	0
$669$	13662.0i	0.789542i
$670$	0	0
$671$	−1768.00	−0.101718
$672$	0	0
$673$	−350.000	−0.0200468	−0.0100234	−	0.999950i	$-0.503191\pi$
$673$	−350.000	−0.0200468	−0.0100234	+	0.999950i	$0.503191\pi$
$674$	0	0
$675$	2943.00i	0.167816i
$676$	0	0
$677$	13468.0i	0.764575i	0.924043	+	0.382288i	$0.124864\pi$
$677$	13468.0i	0.764575i	−0.924043	+	0.382288i	$0.875136\pi$
$678$	0	0
$679$	−2980.00	−0.168427
$680$	0	0
$681$	16740.0	0.941965
$682$	0	0
$683$	− 12852.0i	− 0.720012i	−0.932950	−	0.360006i	$-0.882775\pi$
$683$	− 12852.0i	− 0.720012i	0.932950	−	0.360006i	$-0.117225\pi$
$684$	0	0
$685$	− 2936.00i	− 0.163765i
$686$	0	0
$687$	−2706.00	−0.150277
$688$	0	0
$689$	9360.00	0.517544
$690$	0	0
$691$	− 1080.00i	− 0.0594575i	−0.999558	−	0.0297288i	$-0.990536\pi$
$691$	− 1080.00i	− 0.0594575i	0.999558	−	0.0297288i	$-0.00946435\pi$
$692$	0	0
$693$	360.000i	0.0197334i
$694$	0	0
$695$	1616.00	0.0881991
$696$	0	0
$697$	5292.00	0.287588
$698$	0	0
$699$	5886.00i	0.318496i
$700$	0	0
$701$	− 22720.0i	− 1.22414i	−0.790803	−	0.612070i	$-0.790337\pi$
$701$	− 22720.0i	− 1.22414i	0.790803	−	0.612070i	$-0.209663\pi$
$702$	0	0
$703$	−2096.00	−0.112450
$704$	0	0
$705$	1776.00	0.0948766
$706$	0	0
$707$	− 18000.0i	− 0.957510i
$708$	0	0
$709$	22346.0i	1.18367i	0.806059	+	0.591835i	$0.201596\pi$
$709$	22346.0i	1.18367i	−0.806059	+	0.591835i	$0.798404\pi$
$710$	0	0
$711$	−3474.00	−0.183242
$712$	0	0
$713$	−2664.00	−0.139926
$714$	0	0
$715$	416.000i	0.0217588i
$716$	0	0
$717$	− 6048.00i	− 0.315016i
$718$	0	0
$719$	35244.0	1.82807	0.914033	−	0.405640i	$-0.132951\pi$
$719$	35244.0	1.82807	0.914033	+	0.405640i	$0.132951\pi$
$720$	0	0
$721$	−8180.00	−0.422523
$722$	0	0
$723$	5478.00i	0.281783i
$724$	0	0
$725$	− 7848.00i	− 0.402024i
$726$	0	0
$727$	−25074.0	−1.27915	−0.639576	−	0.768728i	$-0.720890\pi$
$727$	−25074.0	−1.27915	−0.639576	+	0.768728i	$0.720890\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	6048.00i	0.306010i
$732$	0	0
$733$	23742.0i	1.19636i	0.801362	+	0.598179i	$0.204109\pi$
$733$	23742.0i	1.19636i	−0.801362	+	0.598179i	$0.795891\pi$
$734$	0	0
$735$	−2916.00	−0.146338
$736$	0	0
$737$	2768.00	0.138345
$738$	0	0
$739$	− 4932.00i	− 0.245503i	−0.992437	−	0.122751i	$-0.960828\pi$
$739$	− 4932.00i	− 0.245503i	0.992437	−	0.122751i	$-0.0391718\pi$
$740$	0	0
$741$	− 624.000i	− 0.0309355i
$742$	0	0
$743$	26248.0	1.29602	0.648012	−	0.761630i	$-0.275601\pi$
$743$	26248.0	1.29602	0.648012	+	0.761630i	$0.275601\pi$
$744$	0	0
$745$	2736.00	0.134549
$746$	0	0
$747$	− 972.000i	− 0.0476086i
$748$	0	0
$749$	7160.00i	0.349293i
$750$	0	0
$751$	26146.0	1.27041	0.635207	−	0.772342i	$-0.280915\pi$
$751$	26146.0	1.27041	0.635207	+	0.772342i	$0.280915\pi$
$752$	0	0
$753$	9180.00	0.444273
$754$	0	0
$755$	14008.0i	0.675236i
$756$	0	0
$757$	4202.00i	0.201749i	0.994899	+	0.100875i	$0.0321641\pi$
$757$	4202.00i	0.201749i	−0.994899	+	0.100875i	$0.967836\pi$
$758$	0	0
$759$	−1776.00	−0.0849337
$760$	0	0
$761$	31018.0	1.47753	0.738766	−	0.673962i	$-0.235409\pi$
$761$	31018.0	1.47753	0.738766	+	0.673962i	$0.235409\pi$
$762$	0	0
$763$	− 11340.0i	− 0.538054i
$764$	0	0
$765$	− 504.000i	− 0.0238198i
$766$	0	0
$767$	11128.0	0.523871
$768$	0	0
$769$	−7570.00	−0.354982	−0.177491	−	0.984122i	$-0.556798\pi$
$769$	−7570.00	−0.354982	−0.177491	+	0.984122i	$0.556798\pi$
$770$	0	0
$771$	1674.00i	0.0781941i
$772$	0	0
$773$	31392.0i	1.46066i	0.683093	+	0.730331i	$0.260634\pi$
$773$	31392.0i	1.46066i	−0.683093	+	0.730331i	$0.739366\pi$
$774$	0	0
$775$	1962.00	0.0909382
$776$	0	0
$777$	−7860.00	−0.362903
$778$	0	0
$779$	3024.00i	0.139083i
$780$	0	0
$781$	− 2160.00i	− 0.0989640i
$782$	0	0
$783$	1944.00	0.0887266
$784$	0	0
$785$	−13032.0	−0.592525
$786$	0	0
$787$	− 14608.0i	− 0.661651i	−0.943692	−	0.330825i	$-0.892673\pi$
$787$	− 14608.0i	− 0.661651i	0.943692	−	0.330825i	$-0.107327\pi$
$788$	0	0
$789$	16632.0i	0.750462i
$790$	0	0
$791$	−18580.0	−0.835182
$792$	0	0
$793$	11492.0	0.514619
$794$	0	0
$795$	4320.00i	0.192723i
$796$	0	0
$797$	− 30204.0i	− 1.34238i	−0.741283	−	0.671192i	$-0.765782\pi$
$797$	− 30204.0i	− 1.34238i	0.741283	−	0.671192i	$-0.234218\pi$
$798$	0	0
$799$	2072.00	0.0917423
$800$	0	0
$801$	−3438.00	−0.151655
$802$	0	0
$803$	4072.00i	0.178951i
$804$	0	0
$805$	5920.00i	0.259196i
$806$	0	0
$807$	−14568.0	−0.635462
$808$	0	0
$809$	306.000	0.0132984	0.00664919	−	0.999978i	$-0.497883\pi$
$809$	306.000	0.0132984	0.00664919	+	0.999978i	$0.497883\pi$
$810$	0	0
$811$	34920.0i	1.51197i	0.654589	+	0.755985i	$0.272842\pi$
$811$	34920.0i	1.51197i	−0.654589	+	0.755985i	$0.727158\pi$
$812$	0	0
$813$	− 21090.0i	− 0.909789i
$814$	0	0
$815$	3424.00	0.147163
$816$	0	0
$817$	−3456.00	−0.147993
$818$	0	0
$819$	− 2340.00i	− 0.0998367i
$820$	0	0
$821$	14720.0i	0.625739i	0.949796	+	0.312869i	$0.101290\pi$
$821$	14720.0i	0.625739i	−0.949796	+	0.312869i	$0.898710\pi$
$822$	0	0
$823$	14030.0	0.594235	0.297117	−	0.954841i	$-0.403975\pi$
$823$	14030.0	0.594235	0.297117	+	0.954841i	$0.403975\pi$
$824$	0	0
$825$	1308.00	0.0551984
$826$	0	0
$827$	− 24764.0i	− 1.04127i	−0.853780	−	0.520634i	$-0.825696\pi$
$827$	− 24764.0i	− 1.04127i	0.853780	−	0.520634i	$-0.174304\pi$
$828$	0	0
$829$	− 1810.00i	− 0.0758310i	−0.999281	−	0.0379155i	$-0.987928\pi$
$829$	− 1810.00i	− 0.0758310i	0.999281	−	0.0379155i	$-0.0120718\pi$
$830$	0	0
$831$	2214.00	0.0924222
$832$	0	0
$833$	−3402.00	−0.141503
$834$	0	0
$835$	− 6208.00i	− 0.257289i
$836$	0	0
$837$	486.000i	0.0200700i
$838$	0	0
$839$	−4500.00	−0.185170	−0.0925848	−	0.995705i	$-0.529513\pi$
$839$	−4500.00	−0.185170	−0.0925848	+	0.995705i	$0.529513\pi$
$840$	0	0
$841$	19205.0	0.787445
$842$	0	0
$843$	− 5562.00i	− 0.227243i
$844$	0	0
$845$	6084.00i	0.247688i
$846$	0	0
$847$	−13150.0	−0.533458
$848$	0	0
$849$	−4212.00	−0.170266
$850$	0	0
$851$	− 38776.0i	− 1.56196i
$852$	0	0
$853$	− 29458.0i	− 1.18244i	−0.806510	−	0.591221i	$-0.798646\pi$
$853$	− 29458.0i	− 1.18244i	0.806510	−	0.591221i	$-0.201354\pi$
$854$	0	0
$855$	288.000	0.0115198
$856$	0	0
$857$	−15030.0	−0.599084	−0.299542	−	0.954083i	$-0.596834\pi$
$857$	−15030.0	−0.599084	−0.299542	+	0.954083i	$0.596834\pi$
$858$	0	0
$859$	27368.0i	1.08706i	0.839390	+	0.543530i	$0.182912\pi$
$859$	27368.0i	1.08706i	−0.839390	+	0.543530i	$0.817088\pi$
$860$	0	0
$861$	11340.0i	0.448857i
$862$	0	0
$863$	39352.0	1.55221	0.776105	−	0.630603i	$-0.217193\pi$
$863$	39352.0	1.55221	0.776105	+	0.630603i	$0.217193\pi$
$864$	0	0
$865$	−6192.00	−0.243392
$866$	0	0
$867$	14151.0i	0.554317i
$868$	0	0
$869$	1544.00i	0.0602723i
$870$	0	0
$871$	−17992.0	−0.699926
$872$	0	0
$873$	−2682.00	−0.103977
$874$	0	0
$875$	− 9360.00i	− 0.361629i
$876$	0	0
$877$	− 32246.0i	− 1.24159i	−0.783975	−	0.620793i	$-0.786811\pi$
$877$	− 32246.0i	− 1.24159i	0.783975	−	0.620793i	$-0.213189\pi$
$878$	0	0
$879$	−6168.00	−0.236680
$880$	0	0
$881$	1714.00	0.0655461	0.0327731	−	0.999463i	$-0.489566\pi$
$881$	1714.00	0.0655461	0.0327731	+	0.999463i	$0.489566\pi$
$882$	0	0
$883$	1288.00i	0.0490879i	0.999699	+	0.0245440i	$0.00781337\pi$
$883$	1288.00i	0.0490879i	−0.999699	+	0.0245440i	$0.992187\pi$
$884$	0	0
$885$	5136.00i	0.195079i
$886$	0	0
$887$	12888.0	0.487865	0.243933	−	0.969792i	$-0.421562\pi$
$887$	12888.0	0.487865	0.243933	+	0.969792i	$0.421562\pi$
$888$	0	0
$889$	14020.0	0.528927
$890$	0	0
$891$	324.000i	0.0121823i
$892$	0	0
$893$	1184.00i	0.0443685i
$894$	0	0
$895$	−13680.0	−0.510918
$896$	0	0
$897$	11544.0	0.429702
$898$	0	0
$899$	− 1296.00i	− 0.0480801i
$900$	0	0
$901$	5040.00i	0.186356i
$902$	0	0
$903$	−12960.0	−0.477610
$904$	0	0
$905$	14008.0	0.514521
$906$	0	0
$907$	44920.0i	1.64448i	0.569140	+	0.822240i	$0.307276\pi$
$907$	44920.0i	1.64448i	−0.569140	+	0.822240i	$0.692724\pi$
$908$	0	0
$909$	− 16200.0i	− 0.591111i
$910$	0	0
$911$	12384.0	0.450384	0.225192	−	0.974314i	$-0.427699\pi$
$911$	12384.0	0.450384	0.225192	+	0.974314i	$0.427699\pi$
$912$	0	0
$913$	−432.000	−0.0156595
$914$	0	0
$915$	5304.00i	0.191634i
$916$	0	0
$917$	− 11880.0i	− 0.427821i
$918$	0	0
$919$	8190.00	0.293975	0.146988	−	0.989138i	$-0.453042\pi$
$919$	8190.00	0.293975	0.146988	+	0.989138i	$0.453042\pi$
$920$	0	0
$921$	−7020.00	−0.251158
$922$	0	0
$923$	14040.0i	0.500685i
$924$	0	0
$925$	28558.0i	1.01511i
$926$	0	0
$927$	−7362.00	−0.260841
$928$	0	0
$929$	−4698.00	−0.165916	−0.0829582	−	0.996553i	$-0.526437\pi$
$929$	−4698.00	−0.165916	−0.0829582	+	0.996553i	$0.526437\pi$
$930$	0	0
$931$	− 1944.00i	− 0.0684340i
$932$	0	0
$933$	4848.00i	0.170114i
$934$	0	0
$935$	−224.000	−0.00783485
$936$	0	0
$937$	10646.0	0.371174	0.185587	−	0.982628i	$-0.440581\pi$
$937$	10646.0	0.371174	0.185587	+	0.982628i	$0.440581\pi$
$938$	0	0
$939$	9858.00i	0.342602i
$940$	0	0
$941$	25852.0i	0.895591i	0.894136	+	0.447795i	$0.147791\pi$
$941$	25852.0i	0.895591i	−0.894136	+	0.447795i	$0.852209\pi$
$942$	0	0
$943$	−55944.0	−1.93191
$944$	0	0
$945$	1080.00	0.0371771
$946$	0	0
$947$	− 3172.00i	− 0.108845i	−0.998518	−	0.0544225i	$-0.982668\pi$
$947$	− 3172.00i	− 0.108845i	0.998518	−	0.0544225i	$-0.0173318\pi$
$948$	0	0
$949$	− 26468.0i	− 0.905361i
$950$	0	0
$951$	−29808.0	−1.01639
$952$	0	0
$953$	−53150.0	−1.80661	−0.903304	−	0.429001i	$-0.858866\pi$
$953$	−53150.0	−1.80661	−0.903304	+	0.429001i	$0.858866\pi$
$954$	0	0
$955$	− 19168.0i	− 0.649489i
$956$	0	0
$957$	− 864.000i	− 0.0291841i
$958$	0	0
$959$	7340.00	0.247154
$960$	0	0
$961$	−29467.0	−0.989124
$962$	0	0
$963$	6444.00i	0.215633i
$964$	0	0
$965$	− 392.000i	− 0.0130766i
$966$	0	0
$967$	9802.00	0.325968	0.162984	−	0.986629i	$-0.447888\pi$
$967$	9802.00	0.325968	0.162984	+	0.986629i	$0.447888\pi$
$968$	0	0
$969$	336.000	0.0111392
$970$	0	0
$971$	− 46660.0i	− 1.54211i	−0.636767	−	0.771056i	$-0.719729\pi$
$971$	− 46660.0i	− 1.54211i	0.636767	−	0.771056i	$-0.280271\pi$
$972$	0	0
$973$	4040.00i	0.133110i
$974$	0	0
$975$	−8502.00	−0.279264
$976$	0	0
$977$	−41994.0	−1.37514	−0.687568	−	0.726120i	$-0.741322\pi$
$977$	−41994.0	−1.37514	−0.687568	+	0.726120i	$0.741322\pi$
$978$	0	0
$979$	1528.00i	0.0498826i
$980$	0	0
$981$	− 10206.0i	− 0.332164i
$982$	0	0
$983$	11848.0	0.384428	0.192214	−	0.981353i	$-0.438433\pi$
$983$	11848.0	0.384428	0.192214	+	0.981353i	$0.438433\pi$
$984$	0	0
$985$	−4736.00	−0.153200
$986$	0	0
$987$	4440.00i	0.143188i
$988$	0	0
$989$	− 63936.0i	− 2.05566i
$990$	0	0
$991$	−7390.00	−0.236883	−0.118442	−	0.992961i	$-0.537790\pi$
$991$	−7390.00	−0.236883	−0.118442	+	0.992961i	$0.537790\pi$
$992$	0	0
$993$	6348.00	0.202868
$994$	0	0
$995$	5864.00i	0.186835i
$996$	0	0
$997$	− 47682.0i	− 1.51465i	−0.653039	−	0.757324i	$-0.726506\pi$
$997$	− 47682.0i	− 1.51465i	0.653039	−	0.757324i	$-0.273494\pi$
$998$	0	0
$999$	−7074.00	−0.224035

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.d.f.385.1		2
4.3	odd	2		768.4.d.k.385.2		2
8.3	odd	2		768.4.d.k.385.1		2
8.5	even	2	inner	768.4.d.f.385.2		2
16.3	odd	4		384.4.a.c.1.1	yes	1
16.5	even	4		384.4.a.b.1.1	✓	1
16.11	odd	4		384.4.a.f.1.1	yes	1
16.13	even	4		384.4.a.g.1.1	yes	1
48.5	odd	4		1152.4.a.h.1.1		1
48.11	even	4		1152.4.a.g.1.1		1
48.29	odd	4		1152.4.a.f.1.1		1
48.35	even	4		1152.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.a.b.1.1	✓	1	16.5	even	4
384.4.a.c.1.1	yes	1	16.3	odd	4
384.4.a.f.1.1	yes	1	16.11	odd	4
384.4.a.g.1.1	yes	1	16.13	even	4
768.4.d.f.385.1		2	1.1	even	1	trivial
768.4.d.f.385.2		2	8.5	even	2	inner
768.4.d.k.385.1		2	8.3	odd	2
768.4.d.k.385.2		2	4.3	odd	2
1152.4.a.e.1.1		1	48.35	even	4
1152.4.a.f.1.1		1	48.29	odd	4
1152.4.a.g.1.1		1	48.11	even	4
1152.4.a.h.1.1		1	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +4.00000i q^{5} -10.0000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +4.00000i q^{5} -10.0000 q^{7} -9.00000 q^{9} +4.00000i q^{11} -26.0000i q^{13} +12.0000 q^{15} +14.0000 q^{17} +8.00000i q^{19} +30.0000i q^{21} -148.000 q^{23} +109.000 q^{25} +27.0000i q^{27} -72.0000i q^{29} +18.0000 q^{31} +12.0000 q^{33} -40.0000i q^{35} +262.000i q^{37} -78.0000 q^{39} +378.000 q^{41} +432.000i q^{43} -36.0000i q^{45} +148.000 q^{47} -243.000 q^{49} -42.0000i q^{51} +360.000i q^{53} -16.0000 q^{55} +24.0000 q^{57} +428.000i q^{59} +442.000i q^{61} +90.0000 q^{63} +104.000 q^{65} -692.000i q^{67} +444.000i q^{69} -540.000 q^{71} +1018.00 q^{73} -327.000i q^{75} -40.0000i q^{77} +386.000 q^{79} +81.0000 q^{81} +108.000i q^{83} +56.0000i q^{85} -216.000 q^{87} +382.000 q^{89} +260.000i q^{91} -54.0000i q^{93} -32.0000 q^{95} +298.000 q^{97} -36.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20 q^{7} - 18 q^{9}+O(q^{10}) \) 2 * q - 20 * q^7 - 18 * q^9	\( 2 q - 20 q^{7} - 18 q^{9} + 24 q^{15} + 28 q^{17} - 296 q^{23} + 218 q^{25} + 36 q^{31} + 24 q^{33} - 156 q^{39} + 756 q^{41} + 296 q^{47} - 486 q^{49} - 32 q^{55} + 48 q^{57} + 180 q^{63} + 208 q^{65} - 1080 q^{71} + 2036 q^{73} + 772 q^{79} + 162 q^{81} - 432 q^{87} + 764 q^{89} - 64 q^{95} + 596 q^{97}+O(q^{100}) \) 2 * q - 20 * q^7 - 18 * q^9 + 24 * q^15 + 28 * q^17 - 296 * q^23 + 218 * q^25 + 36 * q^31 + 24 * q^33 - 156 * q^39 + 756 * q^41 + 296 * q^47 - 486 * q^49 - 32 * q^55 + 48 * q^57 + 180 * q^63 + 208 * q^65 - 1080 * q^71 + 2036 * q^73 + 772 * q^79 + 162 * q^81 - 432 * q^87 + 764 * q^89 - 64 * q^95 + 596 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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