LMFDB - Embedded newform 1536.4.d.f.769.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,4,Mod(769,1536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1536, 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("1536.769");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1536.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:	$90.6269337688$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3317760000.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 17x^{4} + 256 $ x^8 + 17*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.2
Root			$1.72286 - 1.01575i$ of defining polynomial
Character	$\chi$	$=$	1536.769
Dual form			1536.4.d.f.769.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -2.26874i q^{5} -20.9692 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -2.26874i q^{5} -20.9692 q^{7} -9.00000 q^{9} -1.51472i q^{11} +3.20848i q^{13} -6.80622 q^{15} -10.2010 q^{17} +31.1127i q^{19} +62.9075i q^{21} +197.959 q^{23} +119.853 q^{25} +27.0000i q^{27} +152.973i q^{29} -188.722 q^{31} -4.54416 q^{33} +47.5736i q^{35} +274.868i q^{37} +9.62545 q^{39} +95.5736 q^{41} -263.563i q^{43} +20.4187i q^{45} +116.284 q^{47} +96.7056 q^{49} +30.6030i q^{51} -87.2464i q^{53} -3.43650 q^{55} +93.3381 q^{57} -323.029i q^{59} -409.435i q^{61} +188.722 q^{63} +7.27922 q^{65} -264.833i q^{67} -593.876i q^{69} -308.604 q^{71} +331.529 q^{73} -359.558i q^{75} +31.7624i q^{77} -1.58468 q^{79} +81.0000 q^{81} +25.8680i q^{83} +23.1435i q^{85} +458.919 q^{87} +842.548 q^{89} -67.2792i q^{91} +566.167i q^{93} +70.5867 q^{95} -519.088 q^{97} +13.6325i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 72 q^{9}+O(q^{10}) $ 8 * q - 72 * q^9	$ 8 q - 72 q^{9} - 240 q^{17} + 280 q^{25} - 240 q^{33} + 1104 q^{41} - 584 q^{49} - 960 q^{65} + 480 q^{73} + 648 q^{81} + 3120 q^{89} - 4560 q^{97}+O(q^{100}) $ 8 * q - 72 * q^9 - 240 * q^17 + 280 * q^25 - 240 * q^33 + 1104 * q^41 - 584 * q^49 - 960 * q^65 + 480 * q^73 + 648 * q^81 + 3120 * q^89 - 4560 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times$.

$n$	$511$	$517$	$1025$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	− 2.26874i	− 0.202922i	−0.994839	−	0.101461i	$-0.967648\pi$
$5$	− 2.26874i	− 0.202922i	0.994839	−	0.101461i	$-0.0323518\pi$
$6$	0	0
$7$	−20.9692	−1.13223	−0.566114	−	0.824327i	$-0.691554\pi$
$7$	−20.9692	−1.13223	−0.566114	+	0.824327i	$0.691554\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 1.51472i	− 0.0415186i	−0.999785	−	0.0207593i	$-0.993392\pi$
$11$	− 1.51472i	− 0.0415186i	0.999785	−	0.0207593i	$-0.00660837\pi$
$12$	0	0
$13$	3.20848i	0.0684518i	0.999414	+	0.0342259i	$0.0108966\pi$
$13$	3.20848i	0.0684518i	−0.999414	+	0.0342259i	$0.989103\pi$
$14$	0	0
$15$	−6.80622	−0.117157
$16$	0	0
$17$	−10.2010	−0.145536	−0.0727679	−	0.997349i	$-0.523183\pi$
$17$	−10.2010	−0.145536	−0.0727679	+	0.997349i	$0.523183\pi$
$18$	0	0
$19$	31.1127i	0.375671i	0.982201	+	0.187835i	$0.0601471\pi$
$19$	31.1127i	0.375671i	−0.982201	+	0.187835i	$0.939853\pi$
$20$	0	0
$21$	62.9075i	0.653692i
$22$	0	0
$23$	197.959	1.79466	0.897331	−	0.441358i	$-0.145503\pi$
$23$	197.959	1.79466	0.897331	+	0.441358i	$0.145503\pi$
$24$	0	0
$25$	119.853	0.958823
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	152.973i	0.979530i	0.871854	+	0.489765i	$0.162918\pi$
$29$	152.973i	0.979530i	−0.871854	+	0.489765i	$0.837082\pi$
$30$	0	0
$31$	−188.722	−1.09340	−0.546702	−	0.837327i	$-0.684117\pi$
$31$	−188.722	−1.09340	−0.546702	+	0.837327i	$0.684117\pi$
$32$	0	0
$33$	−4.54416	−0.0239708
$34$	0	0
$35$	47.5736i	0.229754i
$36$	0	0
$37$	274.868i	1.22130i	0.791902	+	0.610649i	$0.209091\pi$
$37$	274.868i	1.22130i	−0.791902	+	0.610649i	$0.790909\pi$
$38$	0	0
$39$	9.62545	0.0395207
$40$	0	0
$41$	95.5736	0.364051	0.182025	−	0.983294i	$-0.441735\pi$
$41$	95.5736	0.364051	0.182025	+	0.983294i	$0.441735\pi$
$42$	0	0
$43$	− 263.563i	− 0.934722i	−0.884067	−	0.467361i	$-0.845205\pi$
$43$	− 263.563i	− 0.934722i	0.884067	−	0.467361i	$-0.154795\pi$
$44$	0	0
$45$	20.4187i	0.0676408i
$46$	0	0
$47$	116.284	0.360888	0.180444	−	0.983585i	$-0.442246\pi$
$47$	116.284	0.360888	0.180444	+	0.983585i	$0.442246\pi$
$48$	0	0
$49$	96.7056	0.281941
$50$	0	0
$51$	30.6030i	0.0840251i
$52$	0	0
$53$	− 87.2464i	− 0.226117i	−0.993588	−	0.113059i	$-0.963935\pi$
$53$	− 87.2464i	− 0.226117i	0.993588	−	0.113059i	$-0.0360648\pi$
$54$	0	0
$55$	−3.43650	−0.00842506
$56$	0	0
$57$	93.3381	0.216894
$58$	0	0
$59$	− 323.029i	− 0.712794i	−0.934335	−	0.356397i	$-0.884005\pi$
$59$	− 323.029i	− 0.712794i	0.934335	−	0.356397i	$-0.115995\pi$
$60$	0	0
$61$	− 409.435i	− 0.859390i	−0.902974	−	0.429695i	$-0.858621\pi$
$61$	− 409.435i	− 0.859390i	0.902974	−	0.429695i	$-0.141379\pi$
$62$	0	0
$63$	188.722	0.377409
$64$	0	0
$65$	7.27922	0.0138904
$66$	0	0
$67$	− 264.833i	− 0.482902i	−0.970413	−	0.241451i	$-0.922377\pi$
$67$	− 264.833i	− 0.482902i	0.970413	−	0.241451i	$-0.0776234\pi$
$68$	0	0
$69$	− 593.876i	− 1.03615i
$70$	0	0
$71$	−308.604	−0.515839	−0.257920	−	0.966166i	$-0.583037\pi$
$71$	−308.604	−0.515839	−0.257920	+	0.966166i	$0.583037\pi$
$72$	0	0
$73$	331.529	0.531542	0.265771	−	0.964036i	$-0.414374\pi$
$73$	331.529	0.531542	0.265771	+	0.964036i	$0.414374\pi$
$74$	0	0
$75$	− 359.558i	− 0.553576i
$76$	0	0
$77$	31.7624i	0.0470086i
$78$	0	0
$79$	−1.58468	−0.00225684	−0.00112842	−	0.999999i	$-0.500359\pi$
$79$	−1.58468	−0.00225684	−0.00112842	+	0.999999i	$0.500359\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	25.8680i	0.0342094i	0.999854	+	0.0171047i	$0.00544486\pi$
$83$	25.8680i	0.0342094i	−0.999854	+	0.0171047i	$0.994555\pi$
$84$	0	0
$85$	23.1435i	0.0295325i
$86$	0	0
$87$	458.919	0.565532
$88$	0	0
$89$	842.548	1.00348	0.501741	−	0.865018i	$-0.332693\pi$
$89$	842.548	1.00348	0.501741	+	0.865018i	$0.332693\pi$
$90$	0	0
$91$	− 67.2792i	− 0.0775031i
$92$	0	0
$93$	566.167i	0.631278i
$94$	0	0
$95$	70.5867	0.0762320
$96$	0	0
$97$	−519.088	−0.543355	−0.271677	−	0.962388i	$-0.587578\pi$
$97$	−519.088	−0.543355	−0.271677	+	0.962388i	$0.587578\pi$
$98$	0	0
$99$	13.6325i	0.0138395i
$100$	0	0
$101$	431.561i	0.425167i	0.977143	+	0.212584i	$0.0681878\pi$
$101$	431.561i	0.425167i	−0.977143	+	0.212584i	$0.931812\pi$
$102$	0	0
$103$	−1317.62	−1.26048	−0.630238	−	0.776402i	$-0.717043\pi$
$103$	−1317.62	−1.26048	−0.630238	+	0.776402i	$0.717043\pi$
$104$	0	0
$105$	142.721	0.132649
$106$	0	0
$107$	− 1593.65i	− 1.43985i	−0.694054	−	0.719923i	$-0.744177\pi$
$107$	− 1593.65i	− 1.43985i	0.694054	−	0.719923i	$-0.255823\pi$
$108$	0	0
$109$	− 1163.92i	− 1.02279i	−0.859347	−	0.511393i	$-0.829130\pi$
$109$	− 1163.92i	− 1.02279i	0.859347	−	0.511393i	$-0.170870\pi$
$110$	0	0
$111$	824.603	0.705116
$112$	0	0
$113$	−286.325	−0.238364	−0.119182	−	0.992872i	$-0.538027\pi$
$113$	−286.325	−0.238364	−0.119182	+	0.992872i	$0.538027\pi$
$114$	0	0
$115$	− 449.117i	− 0.364177i
$116$	0	0
$117$	− 28.8764i	− 0.0228173i
$118$	0	0
$119$	213.907	0.164780
$120$	0	0
$121$	1328.71	0.998276
$122$	0	0
$123$	− 286.721i	− 0.210185i
$124$	0	0
$125$	− 555.508i	− 0.397489i
$126$	0	0
$127$	−219.117	−0.153098	−0.0765491	−	0.997066i	$-0.524390\pi$
$127$	−219.117	−0.153098	−0.0765491	+	0.997066i	$0.524390\pi$
$128$	0	0
$129$	−790.690	−0.539662
$130$	0	0
$131$	− 1463.70i	− 0.976217i	−0.872783	−	0.488108i	$-0.837687\pi$
$131$	− 1463.70i	− 0.976217i	0.872783	−	0.488108i	$-0.162313\pi$
$132$	0	0
$133$	− 652.407i	− 0.425345i
$134$	0	0
$135$	61.2560	0.0390524
$136$	0	0
$137$	957.868	0.597344	0.298672	−	0.954356i	$-0.403456\pi$
$137$	957.868	0.597344	0.298672	+	0.954356i	$0.403456\pi$
$138$	0	0
$139$	− 1631.21i	− 0.995379i	−0.867355	−	0.497689i	$-0.834182\pi$
$139$	− 1631.21i	− 0.995379i	0.867355	−	0.497689i	$-0.165818\pi$
$140$	0	0
$141$	− 348.852i	− 0.208359i
$142$	0	0
$143$	4.85995	0.00284202
$144$	0	0
$145$	347.056	0.198769
$146$	0	0
$147$	− 290.117i	− 0.162778i
$148$	0	0
$149$	− 2722.62i	− 1.49695i	−0.663162	−	0.748476i	$-0.730786\pi$
$149$	− 2722.62i	− 1.49695i	0.663162	−	0.748476i	$-0.269214\pi$
$150$	0	0
$151$	−1556.20	−0.838690	−0.419345	−	0.907827i	$-0.637740\pi$
$151$	−1556.20	−0.838690	−0.419345	+	0.907827i	$0.637740\pi$
$152$	0	0
$153$	91.8091	0.0485119
$154$	0	0
$155$	428.162i	0.221876i
$156$	0	0
$157$	1337.39i	0.679845i	0.940453	+	0.339923i	$0.110401\pi$
$157$	1337.39i	0.679845i	−0.940453	+	0.339923i	$0.889599\pi$
$158$	0	0
$159$	−261.739	−0.130549
$160$	0	0
$161$	−4151.03	−2.03197
$162$	0	0
$163$	− 2234.93i	− 1.07395i	−0.843599	−	0.536974i	$-0.819567\pi$
$163$	− 2234.93i	− 1.07395i	0.843599	−	0.536974i	$-0.180433\pi$
$164$	0	0
$165$	10.3095i	0.00486421i
$166$	0	0
$167$	2661.95	1.23346	0.616731	−	0.787174i	$-0.288457\pi$
$167$	2661.95	1.23346	0.616731	+	0.787174i	$0.288457\pi$
$168$	0	0
$169$	2186.71	0.995314
$170$	0	0
$171$	− 280.014i	− 0.125224i
$172$	0	0
$173$	3433.94i	1.50912i	0.656232	+	0.754559i	$0.272149\pi$
$173$	3433.94i	1.50912i	−0.656232	+	0.754559i	$0.727851\pi$
$174$	0	0
$175$	−2513.21	−1.08561
$176$	0	0
$177$	−969.088	−0.411532
$178$	0	0
$179$	− 861.263i	− 0.359630i	−0.983700	−	0.179815i	$-0.942450\pi$
$179$	− 861.263i	− 0.359630i	0.983700	−	0.179815i	$-0.0575500\pi$
$180$	0	0
$181$	4452.81i	1.82859i	0.405051	+	0.914294i	$0.367254\pi$
$181$	4452.81i	1.82859i	−0.405051	+	0.914294i	$0.632746\pi$
$182$	0	0
$183$	−1228.31	−0.496169
$184$	0	0
$185$	623.604	0.247828
$186$	0	0
$187$	15.4517i	0.00604245i
$188$	0	0
$189$	− 566.167i	− 0.217897i
$190$	0	0
$191$	20.8079	0.00788277	0.00394138	−	0.999992i	$-0.498745\pi$
$191$	20.8079	0.00788277	0.00394138	+	0.999992i	$0.498745\pi$
$192$	0	0
$193$	−3823.76	−1.42612	−0.713058	−	0.701105i	$-0.752690\pi$
$193$	−3823.76	−1.42612	−0.713058	+	0.701105i	$0.752690\pi$
$194$	0	0
$195$	− 21.8377i	− 0.00801963i
$196$	0	0
$197$	3929.22i	1.42104i	0.703675	+	0.710522i	$0.251541\pi$
$197$	3929.22i	1.42104i	−0.703675	+	0.710522i	$0.748459\pi$
$198$	0	0
$199$	4711.35	1.67829	0.839143	−	0.543910i	$-0.183057\pi$
$199$	4711.35	1.67829	0.839143	+	0.543910i	$0.183057\pi$
$200$	0	0
$201$	−794.498	−0.278804
$202$	0	0
$203$	− 3207.72i	− 1.10905i
$204$	0	0
$205$	− 216.832i	− 0.0738741i
$206$	0	0
$207$	−1781.63	−0.598221
$208$	0	0
$209$	47.1270	0.0155973
$210$	0	0
$211$	− 4873.25i	− 1.58999i	−0.606614	−	0.794996i	$-0.707473\pi$
$211$	− 4873.25i	− 1.58999i	0.606614	−	0.794996i	$-0.292527\pi$
$212$	0	0
$213$	925.812i	0.297820i
$214$	0	0
$215$	−597.957	−0.189676
$216$	0	0
$217$	3957.35	1.23798
$218$	0	0
$219$	− 994.587i	− 0.306886i
$220$	0	0
$221$	− 32.7298i	− 0.00996219i
$222$	0	0
$223$	2177.17	0.653786	0.326893	−	0.945061i	$-0.393998\pi$
$223$	2177.17	0.653786	0.326893	+	0.945061i	$0.393998\pi$
$224$	0	0
$225$	−1078.68	−0.319608
$226$	0	0
$227$	− 3934.74i	− 1.15048i	−0.817986	−	0.575238i	$-0.804909\pi$
$227$	− 3934.74i	− 1.15048i	0.817986	−	0.575238i	$-0.195091\pi$
$228$	0	0
$229$	− 4057.08i	− 1.17074i	−0.810767	−	0.585370i	$-0.800949\pi$
$229$	− 4057.08i	− 1.17074i	0.810767	−	0.585370i	$-0.199051\pi$
$230$	0	0
$231$	95.2871	0.0271404
$232$	0	0
$233$	−1508.56	−0.424160	−0.212080	−	0.977252i	$-0.568024\pi$
$233$	−1508.56	−0.424160	−0.212080	+	0.977252i	$0.568024\pi$
$234$	0	0
$235$	− 263.818i	− 0.0732323i
$236$	0	0
$237$	4.75404i	0.00130299i
$238$	0	0
$239$	−1135.85	−0.307414	−0.153707	−	0.988116i	$-0.549121\pi$
$239$	−1135.85	−0.307414	−0.153707	+	0.988116i	$0.549121\pi$
$240$	0	0
$241$	288.671	0.0771574	0.0385787	−	0.999256i	$-0.487717\pi$
$241$	288.671	0.0771574	0.0385787	+	0.999256i	$0.487717\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 219.400i	− 0.0572121i
$246$	0	0
$247$	−99.8246	−0.0257153
$248$	0	0
$249$	77.6039	0.0197508
$250$	0	0
$251$	− 4547.16i	− 1.14348i	−0.820434	−	0.571741i	$-0.806268\pi$
$251$	− 4547.16i	− 1.14348i	0.820434	−	0.571741i	$-0.193732\pi$
$252$	0	0
$253$	− 299.852i	− 0.0745119i
$254$	0	0
$255$	69.4304	0.0170506
$256$	0	0
$257$	−6856.90	−1.66429	−0.832144	−	0.554560i	$-0.812887\pi$
$257$	−6856.90	−1.66429	−0.832144	+	0.554560i	$0.812887\pi$
$258$	0	0
$259$	− 5763.75i	− 1.38279i
$260$	0	0
$261$	− 1376.76i	− 0.326510i
$262$	0	0
$263$	−5473.62	−1.28334	−0.641669	−	0.766981i	$-0.721758\pi$
$263$	−5473.62	−1.28334	−0.641669	+	0.766981i	$0.721758\pi$
$264$	0	0
$265$	−197.939	−0.0458842
$266$	0	0
$267$	− 2527.65i	− 0.579361i
$268$	0	0
$269$	− 5249.36i	− 1.18981i	−0.803795	−	0.594906i	$-0.797189\pi$
$269$	− 5249.36i	− 1.18981i	0.803795	−	0.594906i	$-0.202811\pi$
$270$	0	0
$271$	6829.69	1.53090	0.765450	−	0.643495i	$-0.222516\pi$
$271$	6829.69	1.53090	0.765450	+	0.643495i	$0.222516\pi$
$272$	0	0
$273$	−201.838	−0.0447464
$274$	0	0
$275$	− 181.543i	− 0.0398090i
$276$	0	0
$277$	− 6282.87i	− 1.36282i	−0.731902	−	0.681410i	$-0.761367\pi$
$277$	− 6282.87i	− 1.36282i	0.731902	−	0.681410i	$-0.238633\pi$
$278$	0	0
$279$	1698.50	0.364468
$280$	0	0
$281$	5470.24	1.16131	0.580654	−	0.814151i	$-0.302797\pi$
$281$	5470.24	1.16131	0.580654	+	0.814151i	$0.302797\pi$
$282$	0	0
$283$	2263.84i	0.475516i	0.971324	+	0.237758i	$0.0764126\pi$
$283$	2263.84i	0.475516i	−0.971324	+	0.237758i	$0.923587\pi$
$284$	0	0
$285$	− 211.760i	− 0.0440126i
$286$	0	0
$287$	−2004.10	−0.412189
$288$	0	0
$289$	−4808.94	−0.978819
$290$	0	0
$291$	1557.26i	0.313706i
$292$	0	0
$293$	2760.69i	0.550448i	0.961380	+	0.275224i	$0.0887520\pi$
$293$	2760.69i	0.550448i	−0.961380	+	0.275224i	$0.911248\pi$
$294$	0	0
$295$	−732.870	−0.144642
$296$	0	0
$297$	40.8974	0.00799026
$298$	0	0
$299$	635.147i	0.122848i
$300$	0	0
$301$	5526.70i	1.05832i
$302$	0	0
$303$	1294.68	0.245470
$304$	0	0
$305$	−928.903	−0.174390
$306$	0	0
$307$	− 6617.77i	− 1.23028i	−0.788418	−	0.615140i	$-0.789099\pi$
$307$	− 6617.77i	− 1.23028i	0.788418	−	0.615140i	$-0.210901\pi$
$308$	0	0
$309$	3952.86i	0.727736i
$310$	0	0
$311$	−896.109	−0.163388	−0.0816940	−	0.996657i	$-0.526033\pi$
$311$	−896.109	−0.163388	−0.0816940	+	0.996657i	$0.526033\pi$
$312$	0	0
$313$	6257.93	1.13009	0.565047	−	0.825059i	$-0.308858\pi$
$313$	6257.93	1.13009	0.565047	+	0.825059i	$0.308858\pi$
$314$	0	0
$315$	− 428.162i	− 0.0765848i
$316$	0	0
$317$	− 5996.04i	− 1.06237i	−0.847256	−	0.531185i	$-0.821747\pi$
$317$	− 5996.04i	− 1.06237i	0.847256	−	0.531185i	$-0.178253\pi$
$318$	0	0
$319$	231.711	0.0406688
$320$	0	0
$321$	−4780.94	−0.831295
$322$	0	0
$323$	− 317.381i	− 0.0546735i
$324$	0	0
$325$	384.546i	0.0656331i
$326$	0	0
$327$	−3491.77	−0.590506
$328$	0	0
$329$	−2438.38	−0.408608
$330$	0	0
$331$	585.068i	0.0971548i	0.998819	+	0.0485774i	$0.0154688\pi$
$331$	585.068i	0.0971548i	−0.998819	+	0.0485774i	$0.984531\pi$
$332$	0	0
$333$	− 2473.81i	− 0.407099i
$334$	0	0
$335$	−600.837	−0.0979917
$336$	0	0
$337$	10603.5	1.71398	0.856989	−	0.515335i	$-0.172333\pi$
$337$	10603.5	1.71398	0.856989	+	0.515335i	$0.172333\pi$
$338$	0	0
$339$	858.974i	0.137620i
$340$	0	0
$341$	285.861i	0.0453967i
$342$	0	0
$343$	5164.59	0.813007
$344$	0	0
$345$	−1347.35	−0.210258
$346$	0	0
$347$	− 6576.57i	− 1.01743i	−0.860935	−	0.508716i	$-0.830120\pi$
$347$	− 6576.57i	− 1.01743i	0.860935	−	0.508716i	$-0.169880\pi$
$348$	0	0
$349$	1752.56i	0.268804i	0.990927	+	0.134402i	$0.0429113\pi$
$349$	1752.56i	0.268804i	−0.990927	+	0.134402i	$0.957089\pi$
$350$	0	0
$351$	−86.6291	−0.0131736
$352$	0	0
$353$	−9605.58	−1.44831	−0.724155	−	0.689638i	$-0.757770\pi$
$353$	−9605.58	−1.44831	−0.724155	+	0.689638i	$0.757770\pi$
$354$	0	0
$355$	700.143i	0.104675i
$356$	0	0
$357$	− 641.720i	− 0.0951356i
$358$	0	0
$359$	689.753	0.101403	0.0507016	−	0.998714i	$-0.483854\pi$
$359$	689.753	0.101403	0.0507016	+	0.998714i	$0.483854\pi$
$360$	0	0
$361$	5891.00	0.858872
$362$	0	0
$363$	− 3986.12i	− 0.576355i
$364$	0	0
$365$	− 752.153i	− 0.107862i
$366$	0	0
$367$	−1361.33	−0.193626	−0.0968130	−	0.995303i	$-0.530865\pi$
$367$	−1361.33	−0.193626	−0.0968130	+	0.995303i	$0.530865\pi$
$368$	0	0
$369$	−860.162	−0.121350
$370$	0	0
$371$	1829.48i	0.256016i
$372$	0	0
$373$	− 2967.35i	− 0.411913i	−0.978561	−	0.205957i	$-0.933969\pi$
$373$	− 2967.35i	− 0.411913i	0.978561	−	0.205957i	$-0.0660305\pi$
$374$	0	0
$375$	−1666.52	−0.229490
$376$	0	0
$377$	−490.812	−0.0670506
$378$	0	0
$379$	− 537.530i	− 0.0728524i	−0.999336	−	0.0364262i	$-0.988403\pi$
$379$	− 537.530i	− 0.0728524i	0.999336	−	0.0364262i	$-0.0115974\pi$
$380$	0	0
$381$	657.350i	0.0883912i
$382$	0	0
$383$	4851.60	0.647272	0.323636	−	0.946182i	$-0.395095\pi$
$383$	4851.60	0.647272	0.323636	+	0.946182i	$0.395095\pi$
$384$	0	0
$385$	72.0606	0.00953909
$386$	0	0
$387$	2372.07i	0.311574i
$388$	0	0
$389$	− 11567.8i	− 1.50774i	−0.657026	−	0.753868i	$-0.728186\pi$
$389$	− 11567.8i	− 1.50774i	0.657026	−	0.753868i	$-0.271814\pi$
$390$	0	0
$391$	−2019.38	−0.261188
$392$	0	0
$393$	−4391.11	−0.563619
$394$	0	0
$395$	3.59523i	0 0.000457964i
$396$	0	0
$397$	− 2444.46i	− 0.309028i	−0.987991	−	0.154514i	$-0.950619\pi$
$397$	− 2444.46i	− 0.309028i	0.987991	−	0.154514i	$-0.0493811\pi$
$398$	0	0
$399$	−1957.22	−0.245573
$400$	0	0
$401$	9707.07	1.20885	0.604424	−	0.796663i	$-0.293403\pi$
$401$	9707.07	1.20885	0.604424	+	0.796663i	$0.293403\pi$
$402$	0	0
$403$	− 605.513i	− 0.0748455i
$404$	0	0
$405$	− 183.768i	− 0.0225469i
$406$	0	0
$407$	416.347	0.0507066
$408$	0	0
$409$	13592.7	1.64331	0.821655	−	0.569985i	$-0.193051\pi$
$409$	13592.7	1.64331	0.821655	+	0.569985i	$0.193051\pi$
$410$	0	0
$411$	− 2873.60i	− 0.344877i
$412$	0	0
$413$	6773.66i	0.807046i
$414$	0	0
$415$	58.6877	0.00694185
$416$	0	0
$417$	−4893.64	−0.574682
$418$	0	0
$419$	1914.14i	0.223178i	0.993754	+	0.111589i	$0.0355940\pi$
$419$	1914.14i	0.223178i	−0.993754	+	0.111589i	$0.964406\pi$
$420$	0	0
$421$	9614.08i	1.11297i	0.830857	+	0.556486i	$0.187851\pi$
$421$	9614.08i	1.11297i	−0.830857	+	0.556486i	$0.812149\pi$
$422$	0	0
$423$	−1046.56	−0.120296
$424$	0	0
$425$	−1222.62	−0.139543
$426$	0	0
$427$	8585.51i	0.973026i
$428$	0	0
$429$	− 14.5799i	− 0.00164084i
$430$	0	0
$431$	6515.13	0.728127	0.364064	−	0.931374i	$-0.381389\pi$
$431$	6515.13	0.728127	0.364064	+	0.931374i	$0.381389\pi$
$432$	0	0
$433$	14836.2	1.64661	0.823306	−	0.567598i	$-0.192127\pi$
$433$	14836.2	1.64661	0.823306	+	0.567598i	$0.192127\pi$
$434$	0	0
$435$	− 1041.17i	− 0.114759i
$436$	0	0
$437$	6159.03i	0.674202i
$438$	0	0
$439$	757.009	0.0823008	0.0411504	−	0.999153i	$-0.486898\pi$
$439$	757.009	0.0823008	0.0411504	+	0.999153i	$0.486898\pi$
$440$	0	0
$441$	−870.351	−0.0939802
$442$	0	0
$443$	− 8729.10i	− 0.936189i	−0.883678	−	0.468095i	$-0.844941\pi$
$443$	− 8729.10i	− 0.936189i	0.883678	−	0.468095i	$-0.155059\pi$
$444$	0	0
$445$	− 1911.52i	− 0.203629i
$446$	0	0
$447$	−8167.86	−0.864265
$448$	0	0
$449$	−6574.28	−0.691001	−0.345500	−	0.938419i	$-0.612291\pi$
$449$	−6574.28	−0.691001	−0.345500	+	0.938419i	$0.612291\pi$
$450$	0	0
$451$	− 144.767i	− 0.0151149i
$452$	0	0
$453$	4668.61i	0.484218i
$454$	0	0
$455$	−152.639	−0.0157271
$456$	0	0
$457$	31.0173	0.00317490	0.00158745	−	0.999999i	$-0.499495\pi$
$457$	31.0173	0.00317490	0.00158745	+	0.999999i	$0.499495\pi$
$458$	0	0
$459$	− 275.427i	− 0.0280084i
$460$	0	0
$461$	14702.5i	1.48539i	0.669631	+	0.742694i	$0.266452\pi$
$461$	14702.5i	1.48539i	−0.669631	+	0.742694i	$0.733548\pi$
$462$	0	0
$463$	−9162.02	−0.919644	−0.459822	−	0.888011i	$-0.652087\pi$
$463$	−9162.02	−0.919644	−0.459822	+	0.888011i	$0.652087\pi$
$464$	0	0
$465$	1284.49	0.128100
$466$	0	0
$467$	15539.7i	1.53981i	0.638161	+	0.769903i	$0.279695\pi$
$467$	15539.7i	1.53981i	−0.638161	+	0.769903i	$0.720305\pi$
$468$	0	0
$469$	5553.32i	0.546756i
$470$	0	0
$471$	4012.18	0.392509
$472$	0	0
$473$	−399.225	−0.0388084
$474$	0	0
$475$	3728.94i	0.360201i
$476$	0	0
$477$	785.217i	0.0753724i
$478$	0	0
$479$	−15249.7	−1.45464	−0.727322	−	0.686296i	$-0.759235\pi$
$479$	−15249.7	−1.45464	−0.727322	+	0.686296i	$0.759235\pi$
$480$	0	0
$481$	−881.909	−0.0836000
$482$	0	0
$483$	12453.1i	1.17316i
$484$	0	0
$485$	1177.68i	0.110259i
$486$	0	0
$487$	−2321.14	−0.215977	−0.107988	−	0.994152i	$-0.534441\pi$
$487$	−2321.14	−0.215977	−0.107988	+	0.994152i	$0.534441\pi$
$488$	0	0
$489$	−6704.80	−0.620044
$490$	0	0
$491$	2296.72i	0.211099i	0.994414	+	0.105550i	$0.0336602\pi$
$491$	2296.72i	0.211099i	−0.994414	+	0.105550i	$0.966340\pi$
$492$	0	0
$493$	− 1560.48i	− 0.142557i
$494$	0	0
$495$	30.9285	0.00280835
$496$	0	0
$497$	6471.17	0.584047
$498$	0	0
$499$	− 16770.7i	− 1.50453i	−0.658862	−	0.752264i	$-0.728962\pi$
$499$	− 16770.7i	− 1.50453i	0.658862	−	0.752264i	$-0.271038\pi$
$500$	0	0
$501$	− 7985.86i	− 0.712140i
$502$	0	0
$503$	13487.8	1.19561	0.597805	−	0.801641i	$-0.296040\pi$
$503$	13487.8	1.19561	0.597805	+	0.801641i	$0.296040\pi$
$504$	0	0
$505$	979.100	0.0862760
$506$	0	0
$507$	− 6560.12i	− 0.574645i
$508$	0	0
$509$	4223.36i	0.367774i	0.982947	+	0.183887i	$0.0588681\pi$
$509$	4223.36i	0.367774i	−0.982947	+	0.183887i	$0.941132\pi$
$510$	0	0
$511$	−6951.88	−0.601826
$512$	0	0
$513$	−840.043	−0.0722979
$514$	0	0
$515$	2989.34i	0.255779i
$516$	0	0
$517$	− 176.137i	− 0.0149836i
$518$	0	0
$519$	10301.8	0.871290
$520$	0	0
$521$	−13328.7	−1.12080	−0.560402	−	0.828221i	$-0.689353\pi$
$521$	−13328.7	−1.12080	−0.560402	+	0.828221i	$0.689353\pi$
$522$	0	0
$523$	11372.5i	0.950833i	0.879761	+	0.475416i	$0.157703\pi$
$523$	11372.5i	0.950833i	−0.879761	+	0.475416i	$0.842297\pi$
$524$	0	0
$525$	7539.64i	0.626775i
$526$	0	0
$527$	1925.16	0.159130
$528$	0	0
$529$	27020.6	2.22081
$530$	0	0
$531$	2907.26i	0.237598i
$532$	0	0
$533$	306.646i	0.0249199i
$534$	0	0
$535$	−3615.57	−0.292177
$536$	0	0
$537$	−2583.79	−0.207633
$538$	0	0
$539$	− 146.482i	− 0.0117058i
$540$	0	0
$541$	− 405.445i	− 0.0322208i	−0.999870	−	0.0161104i	$-0.994872\pi$
$541$	− 405.445i	− 0.0322208i	0.999870	−	0.0161104i	$-0.00512832\pi$
$542$	0	0
$543$	13358.4	1.05574
$544$	0	0
$545$	−2640.64	−0.207546
$546$	0	0
$547$	− 1506.93i	− 0.117791i	−0.998264	−	0.0588956i	$-0.981242\pi$
$547$	− 1506.93i	− 0.117791i	0.998264	−	0.0588956i	$-0.0187579\pi$
$548$	0	0
$549$	3684.92i	0.286463i
$550$	0	0
$551$	−4759.40	−0.367981
$552$	0	0
$553$	33.2294	0.00255526
$554$	0	0
$555$	− 1870.81i	− 0.143084i
$556$	0	0
$557$	− 25711.9i	− 1.95592i	−0.208791	−	0.977960i	$-0.566953\pi$
$557$	− 25711.9i	− 1.95592i	0.208791	−	0.977960i	$-0.433047\pi$
$558$	0	0
$559$	845.639	0.0639834
$560$	0	0
$561$	46.3550	0.00348861
$562$	0	0
$563$	15804.0i	1.18305i	0.806286	+	0.591525i	$0.201474\pi$
$563$	15804.0i	1.18305i	−0.806286	+	0.591525i	$0.798526\pi$
$564$	0	0
$565$	649.597i	0.0483694i
$566$	0	0
$567$	−1698.50	−0.125803
$568$	0	0
$569$	−12466.7	−0.918505	−0.459253	−	0.888306i	$-0.651883\pi$
$569$	−12466.7	−0.918505	−0.459253	+	0.888306i	$0.651883\pi$
$570$	0	0
$571$	15570.2i	1.14114i	0.821248	+	0.570572i	$0.193278\pi$
$571$	15570.2i	1.14114i	−0.821248	+	0.570572i	$0.806722\pi$
$572$	0	0
$573$	− 62.4238i	− 0.00455112i
$574$	0	0
$575$	23725.9	1.72076
$576$	0	0
$577$	6957.65	0.501994	0.250997	−	0.967988i	$-0.419242\pi$
$577$	6957.65	0.501994	0.250997	+	0.967988i	$0.419242\pi$
$578$	0	0
$579$	11471.3i	0.823368i
$580$	0	0
$581$	− 542.429i	− 0.0387328i
$582$	0	0
$583$	−132.154	−0.00938807
$584$	0	0
$585$	−65.5130	−0.00463013
$586$	0	0
$587$	295.840i	0.0208017i	0.999946	+	0.0104009i	$0.00331076\pi$
$587$	295.840i	0.0208017i	−0.999946	+	0.0104009i	$0.996689\pi$
$588$	0	0
$589$	− 5871.66i	− 0.410760i
$590$	0	0
$591$	11787.7	0.820440
$592$	0	0
$593$	14779.0	1.02344	0.511720	−	0.859152i	$-0.329009\pi$
$593$	14779.0	1.02344	0.511720	+	0.859152i	$0.329009\pi$
$594$	0	0
$595$	− 485.299i	− 0.0334375i
$596$	0	0
$597$	− 14134.1i	− 0.968959i
$598$	0	0
$599$	−21096.5	−1.43903	−0.719514	−	0.694478i	$-0.755635\pi$
$599$	−21096.5	−1.43903	−0.719514	+	0.694478i	$0.755635\pi$
$600$	0	0
$601$	9409.10	0.638611	0.319305	−	0.947652i	$-0.396550\pi$
$601$	9409.10	0.638611	0.319305	+	0.947652i	$0.396550\pi$
$602$	0	0
$603$	2383.49i	0.160967i
$604$	0	0
$605$	− 3014.49i	− 0.202573i
$606$	0	0
$607$	6091.78	0.407344	0.203672	−	0.979039i	$-0.434712\pi$
$607$	6091.78	0.407344	0.203672	+	0.979039i	$0.434712\pi$
$608$	0	0
$609$	−9623.15	−0.640311
$610$	0	0
$611$	373.095i	0.0247035i
$612$	0	0
$613$	− 7884.11i	− 0.519472i	−0.965680	−	0.259736i	$-0.916364\pi$
$613$	− 7884.11i	− 0.519472i	0.965680	−	0.259736i	$-0.0836355\pi$
$614$	0	0
$615$	−650.495	−0.0426512
$616$	0	0
$617$	−14573.9	−0.950928	−0.475464	−	0.879735i	$-0.657720\pi$
$617$	−14573.9	−0.950928	−0.475464	+	0.879735i	$0.657720\pi$
$618$	0	0
$619$	26984.9i	1.75220i	0.482128	+	0.876101i	$0.339864\pi$
$619$	26984.9i	1.75220i	−0.482128	+	0.876101i	$0.660136\pi$
$620$	0	0
$621$	5344.88i	0.345383i
$622$	0	0
$623$	−17667.5	−1.13617
$624$	0	0
$625$	13721.3	0.878163
$626$	0	0
$627$	− 141.381i	− 0.00900512i
$628$	0	0
$629$	− 2803.93i	− 0.177742i
$630$	0	0
$631$	−28881.6	−1.82212	−0.911061	−	0.412272i	$-0.864735\pi$
$631$	−28881.6	−1.82212	−0.911061	+	0.412272i	$0.864735\pi$
$632$	0	0
$633$	−14619.8	−0.917983
$634$	0	0
$635$	497.119i	0.0310670i
$636$	0	0
$637$	310.279i	0.0192993i
$638$	0	0
$639$	2777.44	0.171946
$640$	0	0
$641$	−16448.9	−1.01356	−0.506781	−	0.862075i	$-0.669165\pi$
$641$	−16448.9	−1.01356	−0.506781	+	0.862075i	$0.669165\pi$
$642$	0	0
$643$	8947.26i	0.548749i	0.961623	+	0.274375i	$0.0884708\pi$
$643$	8947.26i	0.548749i	−0.961623	+	0.274375i	$0.911529\pi$
$644$	0	0
$645$	1793.87i	0.109510i
$646$	0	0
$647$	18260.7	1.10959	0.554794	−	0.831988i	$-0.312797\pi$
$647$	18260.7	1.10959	0.554794	+	0.831988i	$0.312797\pi$
$648$	0	0
$649$	−489.299	−0.0295942
$650$	0	0
$651$	− 11872.1i	− 0.714750i
$652$	0	0
$653$	− 7847.64i	− 0.470294i	−0.971960	−	0.235147i	$-0.924443\pi$
$653$	− 7847.64i	− 0.470294i	0.971960	−	0.235147i	$-0.0755571\pi$
$654$	0	0
$655$	−3320.77	−0.198096
$656$	0	0
$657$	−2983.76	−0.177181
$658$	0	0
$659$	31707.2i	1.87426i	0.348978	+	0.937131i	$0.386529\pi$
$659$	31707.2i	1.87426i	−0.348978	+	0.937131i	$0.613471\pi$
$660$	0	0
$661$	14271.9i	0.839805i	0.907569	+	0.419903i	$0.137936\pi$
$661$	14271.9i	0.839805i	−0.907569	+	0.419903i	$0.862064\pi$
$662$	0	0
$663$	−98.1893	−0.00575167
$664$	0	0
$665$	−1480.14	−0.0863120
$666$	0	0
$667$	30282.3i	1.75793i
$668$	0	0
$669$	− 6531.52i	− 0.377463i
$670$	0	0
$671$	−620.179	−0.0356807
$672$	0	0
$673$	8792.89	0.503627	0.251813	−	0.967776i	$-0.418973\pi$
$673$	8792.89	0.503627	0.251813	+	0.967776i	$0.418973\pi$
$674$	0	0
$675$	3236.03i	0.184525i
$676$	0	0
$677$	− 24521.0i	− 1.39205i	−0.718017	−	0.696026i	$-0.754950\pi$
$677$	− 24521.0i	− 1.39205i	0.718017	−	0.696026i	$-0.245050\pi$
$678$	0	0
$679$	10884.8	0.615202
$680$	0	0
$681$	−11804.2	−0.664228
$682$	0	0
$683$	23406.0i	1.31128i	0.755072	+	0.655641i	$0.227602\pi$
$683$	23406.0i	1.31128i	−0.755072	+	0.655641i	$0.772398\pi$
$684$	0	0
$685$	− 2173.15i	− 0.121215i
$686$	0	0
$687$	−12171.2	−0.675927
$688$	0	0
$689$	279.929	0.0154781
$690$	0	0
$691$	− 13121.2i	− 0.722367i	−0.932495	−	0.361183i	$-0.882373\pi$
$691$	− 13121.2i	− 0.722367i	0.932495	−	0.361183i	$-0.117627\pi$
$692$	0	0
$693$	− 285.861i	− 0.0156695i
$694$	0	0
$695$	−3700.80	−0.201985
$696$	0	0
$697$	−974.947	−0.0529824
$698$	0	0
$699$	4525.69i	0.244889i
$700$	0	0
$701$	16603.5i	0.894585i	0.894388	+	0.447293i	$0.147612\pi$
$701$	16603.5i	0.894585i	−0.894388	+	0.447293i	$0.852388\pi$
$702$	0	0
$703$	−8551.88	−0.458805
$704$	0	0
$705$	−791.455	−0.0422807
$706$	0	0
$707$	− 9049.47i	− 0.481386i
$708$	0	0
$709$	16453.8i	0.871557i	0.900054	+	0.435778i	$0.143527\pi$
$709$	16453.8i	0.871557i	−0.900054	+	0.435778i	$0.856473\pi$
$710$	0	0
$711$	14.2621	0.000752281 0
$712$	0	0
$713$	−37359.2	−1.96229
$714$	0	0
$715$	− 11.0260i	0 0.000576710i
$716$	0	0
$717$	3407.55i	0.177486i
$718$	0	0
$719$	17050.6	0.884393	0.442197	−	0.896918i	$-0.354199\pi$
$719$	17050.6	0.884393	0.442197	+	0.896918i	$0.354199\pi$
$720$	0	0
$721$	27629.4	1.42715
$722$	0	0
$723$	− 866.013i	− 0.0445468i
$724$	0	0
$725$	18334.3i	0.939196i
$726$	0	0
$727$	20024.2	1.02153	0.510767	−	0.859719i	$-0.329361\pi$
$727$	20024.2	1.02153	0.510767	+	0.859719i	$0.329361\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	2688.61i	0.136036i
$732$	0	0
$733$	− 34181.9i	− 1.72243i	−0.508243	−	0.861213i	$-0.669705\pi$
$733$	− 34181.9i	− 1.72243i	0.508243	−	0.861213i	$-0.330295\pi$
$734$	0	0
$735$	−658.200	−0.0330314
$736$	0	0
$737$	−401.147	−0.0200494
$738$	0	0
$739$	− 951.634i	− 0.0473700i	−0.999719	−	0.0236850i	$-0.992460\pi$
$739$	− 951.634i	− 0.0473700i	0.999719	−	0.0236850i	$-0.00753987\pi$
$740$	0	0
$741$	299.474i	0.0148468i
$742$	0	0
$743$	16905.7	0.834737	0.417368	−	0.908737i	$-0.362953\pi$
$743$	16905.7	0.834737	0.417368	+	0.908737i	$0.362953\pi$
$744$	0	0
$745$	−6176.92	−0.303765
$746$	0	0
$747$	− 232.812i	− 0.0114031i
$748$	0	0
$749$	33417.4i	1.63023i
$750$	0	0
$751$	23154.0	1.12503	0.562517	−	0.826786i	$-0.309833\pi$
$751$	23154.0	1.12503	0.562517	+	0.826786i	$0.309833\pi$
$752$	0	0
$753$	−13641.5	−0.660190
$754$	0	0
$755$	3530.63i	0.170189i
$756$	0	0
$757$	− 29839.7i	− 1.43268i	−0.697749	−	0.716342i	$-0.745815\pi$
$757$	− 29839.7i	− 1.43268i	0.697749	−	0.716342i	$-0.254185\pi$
$758$	0	0
$759$	−899.555	−0.0430195
$760$	0	0
$761$	−15872.3	−0.756074	−0.378037	−	0.925791i	$-0.623401\pi$
$761$	−15872.3	−0.756074	−0.378037	+	0.925791i	$0.623401\pi$
$762$	0	0
$763$	24406.5i	1.15803i
$764$	0	0
$765$	− 208.291i	− 0.00984416i
$766$	0	0
$767$	1036.43	0.0487920
$768$	0	0
$769$	27878.8	1.30733	0.653664	−	0.756785i	$-0.273231\pi$
$769$	27878.8	1.30733	0.653664	+	0.756785i	$0.273231\pi$
$770$	0	0
$771$	20570.7i	0.960877i
$772$	0	0
$773$	9894.81i	0.460403i	0.973143	+	0.230201i	$0.0739385\pi$
$773$	9894.81i	0.460403i	−0.973143	+	0.230201i	$0.926061\pi$
$774$	0	0
$775$	−22618.9	−1.04838
$776$	0	0
$777$	−17291.2	−0.798352
$778$	0	0
$779$	2973.55i	0.136763i
$780$	0	0
$781$	467.448i	0.0214169i
$782$	0	0
$783$	−4130.27	−0.188511
$784$	0	0
$785$	3034.20	0.137956
$786$	0	0
$787$	− 5394.90i	− 0.244355i	−0.992508	−	0.122177i	$-0.961012\pi$
$787$	− 5394.90i	− 0.244355i	0.992508	−	0.122177i	$-0.0389877\pi$
$788$	0	0
$789$	16420.9i	0.740936i
$790$	0	0
$791$	6003.99	0.269883
$792$	0	0
$793$	1313.67	0.0588268
$794$	0	0
$795$	593.818i	0.0264913i
$796$	0	0
$797$	− 4227.39i	− 0.187882i	−0.995578	−	0.0939410i	$-0.970053\pi$
$797$	− 4227.39i	− 0.187882i	0.995578	−	0.0939410i	$-0.0299465\pi$
$798$	0	0
$799$	−1186.21	−0.0525222
$800$	0	0
$801$	−7582.94	−0.334494
$802$	0	0
$803$	− 502.173i	− 0.0220689i
$804$	0	0
$805$	9417.60i	0.412332i
$806$	0	0
$807$	−15748.1	−0.686938
$808$	0	0
$809$	3420.55	0.148653	0.0743264	−	0.997234i	$-0.476319\pi$
$809$	3420.55	0.148653	0.0743264	+	0.997234i	$0.476319\pi$
$810$	0	0
$811$	14917.4i	0.645894i	0.946417	+	0.322947i	$0.104674\pi$
$811$	14917.4i	0.645894i	−0.946417	+	0.322947i	$0.895326\pi$
$812$	0	0
$813$	− 20489.1i	− 0.883866i
$814$	0	0
$815$	−5070.49	−0.217928
$816$	0	0
$817$	8200.17	0.351148
$818$	0	0
$819$	605.513i	0.0258344i
$820$	0	0
$821$	− 448.581i	− 0.0190689i	−0.999955	−	0.00953447i	$-0.996965\pi$
$821$	− 448.581i	− 0.0190689i	0.999955	−	0.00953447i	$-0.00303496\pi$
$822$	0	0
$823$	20199.4	0.855536	0.427768	−	0.903889i	$-0.359300\pi$
$823$	20199.4	0.855536	0.427768	+	0.903889i	$0.359300\pi$
$824$	0	0
$825$	−544.630	−0.0229837
$826$	0	0
$827$	− 12961.0i	− 0.544979i	−0.962159	−	0.272490i	$-0.912153\pi$
$827$	− 12961.0i	− 0.544979i	0.962159	−	0.272490i	$-0.0878471\pi$
$828$	0	0
$829$	31314.2i	1.31193i	0.754792	+	0.655964i	$0.227738\pi$
$829$	31314.2i	1.31193i	−0.754792	+	0.655964i	$0.772262\pi$
$830$	0	0
$831$	−18848.6	−0.786824
$832$	0	0
$833$	−986.495	−0.0410324
$834$	0	0
$835$	− 6039.29i	− 0.250297i
$836$	0	0
$837$	− 5095.51i	− 0.210426i
$838$	0	0
$839$	−33718.0	−1.38746	−0.693728	−	0.720237i	$-0.744033\pi$
$839$	−33718.0	−1.38746	−0.693728	+	0.720237i	$0.744033\pi$
$840$	0	0
$841$	988.242	0.0405200
$842$	0	0
$843$	− 16410.7i	− 0.670481i
$844$	0	0
$845$	− 4961.07i	− 0.201972i
$846$	0	0
$847$	−27861.8	−1.13028
$848$	0	0
$849$	6791.51	0.274540
$850$	0	0
$851$	54412.5i	2.19182i
$852$	0	0
$853$	− 13279.6i	− 0.533040i	−0.963829	−	0.266520i	$-0.914126\pi$
$853$	− 13279.6i	− 0.533040i	0.963829	−	0.266520i	$-0.0858738\pi$
$854$	0	0
$855$	−635.280	−0.0254107
$856$	0	0
$857$	−20285.5	−0.808564	−0.404282	−	0.914634i	$-0.632478\pi$
$857$	−20285.5	−0.808564	−0.404282	+	0.914634i	$0.632478\pi$
$858$	0	0
$859$	42382.1i	1.68342i	0.539929	+	0.841711i	$0.318451\pi$
$859$	42382.1i	1.68342i	−0.539929	+	0.841711i	$0.681549\pi$
$860$	0	0
$861$	6012.29i	0.237977i
$862$	0	0
$863$	−19943.5	−0.786657	−0.393328	−	0.919398i	$-0.628676\pi$
$863$	−19943.5	−0.786657	−0.393328	+	0.919398i	$0.628676\pi$
$864$	0	0
$865$	7790.71	0.306234
$866$	0	0
$867$	14426.8i	0.565122i
$868$	0	0
$869$	2.40035i	0 9.37010e-5i
$870$	0	0
$871$	849.711	0.0330555
$872$	0	0
$873$	4671.79	0.181118
$874$	0	0
$875$	11648.5i	0.450048i
$876$	0	0
$877$	25607.7i	0.985988i	0.870033	+	0.492994i	$0.164097\pi$
$877$	25607.7i	0.985988i	−0.870033	+	0.492994i	$0.835903\pi$
$878$	0	0
$879$	8282.07	0.317801
$880$	0	0
$881$	−4881.96	−0.186694	−0.0933470	−	0.995634i	$-0.529757\pi$
$881$	−4881.96	−0.186694	−0.0933470	+	0.995634i	$0.529757\pi$
$882$	0	0
$883$	− 34096.9i	− 1.29949i	−0.760151	−	0.649747i	$-0.774875\pi$
$883$	− 34096.9i	− 1.29949i	0.760151	−	0.649747i	$-0.225125\pi$
$884$	0	0
$885$	2198.61i	0.0835090i
$886$	0	0
$887$	42658.9	1.61482	0.807410	−	0.589991i	$-0.200869\pi$
$887$	42658.9	1.61482	0.807410	+	0.589991i	$0.200869\pi$
$888$	0	0
$889$	4594.69	0.173342
$890$	0	0
$891$	− 122.692i	− 0.00461318i
$892$	0	0
$893$	3617.91i	0.135575i
$894$	0	0
$895$	−1953.98	−0.0729770
$896$	0	0
$897$	1905.44	0.0709262
$898$	0	0
$899$	− 28869.4i	− 1.07102i
$900$	0	0
$901$	890.001i	0.0329081i
$902$	0	0
$903$	16580.1	0.611021
$904$	0	0
$905$	10102.3	0.371062
$906$	0	0
$907$	− 14321.8i	− 0.524310i	−0.965026	−	0.262155i	$-0.915567\pi$
$907$	− 14321.8i	− 0.524310i	0.965026	−	0.262155i	$-0.0844331\pi$
$908$	0	0
$909$	− 3884.05i	− 0.141722i
$910$	0	0
$911$	−15159.6	−0.551329	−0.275664	−	0.961254i	$-0.588898\pi$
$911$	−15159.6	−0.551329	−0.275664	+	0.961254i	$0.588898\pi$
$912$	0	0
$913$	39.1827	0.00142033
$914$	0	0
$915$	2786.71i	0.100684i
$916$	0	0
$917$	30692.6i	1.10530i
$918$	0	0
$919$	21391.5	0.767833	0.383917	−	0.923368i	$-0.374575\pi$
$919$	21391.5	0.767833	0.383917	+	0.923368i	$0.374575\pi$
$920$	0	0
$921$	−19853.3	−0.710302
$922$	0	0
$923$	− 990.152i	− 0.0353101i
$924$	0	0
$925$	32943.7i	1.17101i
$926$	0	0
$927$	11858.6	0.420159
$928$	0	0
$929$	16845.7	0.594931	0.297465	−	0.954733i	$-0.403859\pi$
$929$	16845.7	0.594931	0.297465	+	0.954733i	$0.403859\pi$
$930$	0	0
$931$	3008.77i	0.105917i
$932$	0	0
$933$	2688.33i	0.0943321i
$934$	0	0
$935$	35.0558	0.00122615
$936$	0	0
$937$	25445.1	0.887147	0.443573	−	0.896238i	$-0.353711\pi$
$937$	25445.1	0.887147	0.443573	+	0.896238i	$0.353711\pi$
$938$	0	0
$939$	− 18773.8i	− 0.652460i
$940$	0	0
$941$	− 16942.8i	− 0.586948i	−0.955967	−	0.293474i	$-0.905189\pi$
$941$	− 16942.8i	− 0.586948i	0.955967	−	0.293474i	$-0.0948114\pi$
$942$	0	0
$943$	18919.6	0.653348
$944$	0	0
$945$	−1284.49	−0.0442163
$946$	0	0
$947$	21984.0i	0.754367i	0.926139	+	0.377184i	$0.123107\pi$
$947$	21984.0i	0.754367i	−0.926139	+	0.377184i	$0.876893\pi$
$948$	0	0
$949$	1063.71i	0.0363850i
$950$	0	0
$951$	−17988.1	−0.613360
$952$	0	0
$953$	−17078.9	−0.580523	−0.290262	−	0.956947i	$-0.593742\pi$
$953$	−17078.9	−0.580523	−0.290262	+	0.956947i	$0.593742\pi$
$954$	0	0
$955$	− 47.2078i	− 0.00159959i
$956$	0	0
$957$	− 695.133i	− 0.0234801i
$958$	0	0
$959$	−20085.7	−0.676330
$960$	0	0
$961$	5825.16	0.195534
$962$	0	0
$963$	14342.8i	0.479949i
$964$	0	0
$965$	8675.12i	0.289391i
$966$	0	0
$967$	21385.4	0.711178	0.355589	−	0.934642i	$-0.384280\pi$
$967$	21385.4	0.711178	0.355589	+	0.934642i	$0.384280\pi$
$968$	0	0
$969$	−952.143	−0.0315658
$970$	0	0
$971$	− 32629.9i	− 1.07842i	−0.842172	−	0.539209i	$-0.818723\pi$
$971$	− 32629.9i	− 1.07842i	0.842172	−	0.539209i	$-0.181277\pi$
$972$	0	0
$973$	34205.2i	1.12700i
$974$	0	0
$975$	1153.64	0.0378933
$976$	0	0
$977$	−13577.8	−0.444618	−0.222309	−	0.974976i	$-0.571359\pi$
$977$	−13577.8	−0.444618	−0.222309	+	0.974976i	$0.571359\pi$
$978$	0	0
$979$	− 1276.22i	− 0.0416632i
$980$	0	0
$981$	10475.3i	0.340929i
$982$	0	0
$983$	−50233.2	−1.62990	−0.814949	−	0.579533i	$-0.803235\pi$
$983$	−50233.2	−1.62990	−0.814949	+	0.579533i	$0.803235\pi$
$984$	0	0
$985$	8914.39	0.288361
$986$	0	0
$987$	7315.13i	0.235910i
$988$	0	0
$989$	− 52174.7i	− 1.67751i
$990$	0	0
$991$	38733.4	1.24158	0.620791	−	0.783976i	$-0.286812\pi$
$991$	38733.4	1.24158	0.620791	+	0.783976i	$0.286812\pi$
$992$	0	0
$993$	1755.20	0.0560924
$994$	0	0
$995$	− 10688.8i	− 0.340562i
$996$	0	0
$997$	− 42613.4i	− 1.35364i	−0.736148	−	0.676820i	$-0.763357\pi$
$997$	− 42613.4i	− 1.35364i	0.736148	−	0.676820i	$-0.236643\pi$
$998$	0	0
$999$	−7421.43	−0.235039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.4.d.f.769.2		8
4.3	odd	2	inner	1536.4.d.f.769.6		8
8.3	odd	2	inner	1536.4.d.f.769.3		8
8.5	even	2	inner	1536.4.d.f.769.7		8
16.3	odd	4		1536.4.a.i.1.2	✓	4
16.5	even	4		1536.4.a.i.1.3	yes	4
16.11	odd	4		1536.4.a.l.1.3	yes	4
16.13	even	4		1536.4.a.l.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.4.a.i.1.2	✓	4	16.3	odd	4
1536.4.a.i.1.3	yes	4	16.5	even	4
1536.4.a.l.1.2	yes	4	16.13	even	4
1536.4.a.l.1.3	yes	4	16.11	odd	4
1536.4.d.f.769.2		8	1.1	even	1	trivial
1536.4.d.f.769.3		8	8.3	odd	2	inner
1536.4.d.f.769.6		8	4.3	odd	2	inner
1536.4.d.f.769.7		8	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1536.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -2.26874i q^{5} -20.9692 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -2.26874i q^{5} -20.9692 q^{7} -9.00000 q^{9} -1.51472i q^{11} +3.20848i q^{13} -6.80622 q^{15} -10.2010 q^{17} +31.1127i q^{19} +62.9075i q^{21} +197.959 q^{23} +119.853 q^{25} +27.0000i q^{27} +152.973i q^{29} -188.722 q^{31} -4.54416 q^{33} +47.5736i q^{35} +274.868i q^{37} +9.62545 q^{39} +95.5736 q^{41} -263.563i q^{43} +20.4187i q^{45} +116.284 q^{47} +96.7056 q^{49} +30.6030i q^{51} -87.2464i q^{53} -3.43650 q^{55} +93.3381 q^{57} -323.029i q^{59} -409.435i q^{61} +188.722 q^{63} +7.27922 q^{65} -264.833i q^{67} -593.876i q^{69} -308.604 q^{71} +331.529 q^{73} -359.558i q^{75} +31.7624i q^{77} -1.58468 q^{79} +81.0000 q^{81} +25.8680i q^{83} +23.1435i q^{85} +458.919 q^{87} +842.548 q^{89} -67.2792i q^{91} +566.167i q^{93} +70.5867 q^{95} -519.088 q^{97} +13.6325i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{9}+O(q^{10}) \) 8 * q - 72 * q^9	\( 8 q - 72 q^{9} - 240 q^{17} + 280 q^{25} - 240 q^{33} + 1104 q^{41} - 584 q^{49} - 960 q^{65} + 480 q^{73} + 648 q^{81} + 3120 q^{89} - 4560 q^{97}+O(q^{100}) \) 8 * q - 72 * q^9 - 240 * q^17 + 280 * q^25 - 240 * q^33 + 1104 * q^41 - 584 * q^49 - 960 * q^65 + 480 * q^73 + 648 * q^81 + 3120 * q^89 - 4560 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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