LMFDB - Embedded newform 384.4.d.a.193.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(193,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$2$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.4.d.a.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +8.00000i q^{5} -12.0000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +8.00000i q^{5} -12.0000 q^{7} -9.00000 q^{9} -12.0000i q^{11} +20.0000i q^{13} +24.0000 q^{15} +62.0000 q^{17} -108.000i q^{19} +36.0000i q^{21} +72.0000 q^{23} +61.0000 q^{25} +27.0000i q^{27} -128.000i q^{29} +204.000 q^{31} -36.0000 q^{33} -96.0000i q^{35} +228.000i q^{37} +60.0000 q^{39} -22.0000 q^{41} -204.000i q^{43} -72.0000i q^{45} +600.000 q^{47} -199.000 q^{49} -186.000i q^{51} -256.000i q^{53} +96.0000 q^{55} -324.000 q^{57} -828.000i q^{59} -84.0000i q^{61} +108.000 q^{63} -160.000 q^{65} -348.000i q^{67} -216.000i q^{69} -456.000 q^{71} +822.000 q^{73} -183.000i q^{75} +144.000i q^{77} +1356.00 q^{79} +81.0000 q^{81} -108.000i q^{83} +496.000i q^{85} -384.000 q^{87} -938.000 q^{89} -240.000i q^{91} -612.000i q^{93} +864.000 q^{95} +1278.00 q^{97} +108.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 24 q^{7} - 18 q^{9}+O(q^{10}) $ 2 * q - 24 * q^7 - 18 * q^9	$ 2 q - 24 q^{7} - 18 q^{9} + 48 q^{15} + 124 q^{17} + 144 q^{23} + 122 q^{25} + 408 q^{31} - 72 q^{33} + 120 q^{39} - 44 q^{41} + 1200 q^{47} - 398 q^{49} + 192 q^{55} - 648 q^{57} + 216 q^{63} - 320 q^{65} - 912 q^{71} + 1644 q^{73} + 2712 q^{79} + 162 q^{81} - 768 q^{87} - 1876 q^{89} + 1728 q^{95} + 2556 q^{97}+O(q^{100}) $ 2 * q - 24 * q^7 - 18 * q^9 + 48 * q^15 + 124 * q^17 + 144 * q^23 + 122 * q^25 + 408 * q^31 - 72 * q^33 + 120 * q^39 - 44 * q^41 + 1200 * q^47 - 398 * q^49 + 192 * q^55 - 648 * q^57 + 216 * q^63 - 320 * q^65 - 912 * q^71 + 1644 * q^73 + 2712 * q^79 + 162 * q^81 - 768 * q^87 - 1876 * q^89 + 1728 * q^95 + 2556 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	8.00000i	0.715542i	0.933809	+	0.357771i	$0.116463\pi$
$5$	8.00000i	0.715542i	−0.933809	+	0.357771i	$0.883537\pi$
$6$	0	0
$7$	−12.0000	−0.647939	−0.323970	−	0.946068i	$-0.605018\pi$
$7$	−12.0000	−0.647939	−0.323970	+	0.946068i	$0.605018\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 12.0000i	− 0.328921i	−0.986384	−	0.164461i	$-0.947412\pi$
$11$	− 12.0000i	− 0.328921i	0.986384	−	0.164461i	$-0.0525884\pi$
$12$	0	0
$13$	20.0000i	0.426692i	0.976977	+	0.213346i	$0.0684362\pi$
$13$	20.0000i	0.426692i	−0.976977	+	0.213346i	$0.931564\pi$
$14$	0	0
$15$	24.0000	0.413118
$16$	0	0
$17$	62.0000	0.884542	0.442271	−	0.896882i	$-0.354173\pi$
$17$	62.0000	0.884542	0.442271	+	0.896882i	$0.354173\pi$
$18$	0	0
$19$	− 108.000i	− 1.30405i	−0.758199	−	0.652024i	$-0.773920\pi$
$19$	− 108.000i	− 1.30405i	0.758199	−	0.652024i	$-0.226080\pi$
$20$	0	0
$21$	36.0000i	0.374088i
$22$	0	0
$23$	72.0000	0.652741	0.326370	−	0.945242i	$-0.394174\pi$
$23$	72.0000	0.652741	0.326370	+	0.945242i	$0.394174\pi$
$24$	0	0
$25$	61.0000	0.488000
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	− 128.000i	− 0.819621i	−0.912171	−	0.409810i	$-0.865595\pi$
$29$	− 128.000i	− 0.819621i	0.912171	−	0.409810i	$-0.134405\pi$
$30$	0	0
$31$	204.000	1.18192	0.590959	−	0.806701i	$-0.298749\pi$
$31$	204.000	1.18192	0.590959	+	0.806701i	$0.298749\pi$
$32$	0	0
$33$	−36.0000	−0.189903
$34$	0	0
$35$	− 96.0000i	− 0.463627i
$36$	0	0
$37$	228.000i	1.01305i	0.862224	+	0.506527i	$0.169071\pi$
$37$	228.000i	1.01305i	−0.862224	+	0.506527i	$0.830929\pi$
$38$	0	0
$39$	60.0000	0.246351
$40$	0	0
$41$	−22.0000	−0.0838006	−0.0419003	−	0.999122i	$-0.513341\pi$
$41$	−22.0000	−0.0838006	−0.0419003	+	0.999122i	$0.513341\pi$
$42$	0	0
$43$	− 204.000i	− 0.723482i	−0.932279	−	0.361741i	$-0.882183\pi$
$43$	− 204.000i	− 0.723482i	0.932279	−	0.361741i	$-0.117817\pi$
$44$	0	0
$45$	− 72.0000i	− 0.238514i
$46$	0	0
$47$	600.000	1.86211	0.931053	−	0.364884i	$-0.118891\pi$
$47$	600.000	1.86211	0.931053	+	0.364884i	$0.118891\pi$
$48$	0	0
$49$	−199.000	−0.580175
$50$	0	0
$51$	− 186.000i	− 0.510690i
$52$	0	0
$53$	− 256.000i	− 0.663477i	−0.943371	−	0.331739i	$-0.892365\pi$
$53$	− 256.000i	− 0.663477i	0.943371	−	0.331739i	$-0.107635\pi$
$54$	0	0
$55$	96.0000	0.235357
$56$	0	0
$57$	−324.000	−0.752892
$58$	0	0
$59$	− 828.000i	− 1.82706i	−0.406774	−	0.913529i	$-0.633346\pi$
$59$	− 828.000i	− 1.82706i	0.406774	−	0.913529i	$-0.366654\pi$
$60$	0	0
$61$	− 84.0000i	− 0.176313i	−0.996107	−	0.0881565i	$-0.971902\pi$
$61$	− 84.0000i	− 0.176313i	0.996107	−	0.0881565i	$-0.0280976\pi$
$62$	0	0
$63$	108.000	0.215980
$64$	0	0
$65$	−160.000	−0.305316
$66$	0	0
$67$	− 348.000i	− 0.634552i	−0.948333	−	0.317276i	$-0.897232\pi$
$67$	− 348.000i	− 0.634552i	0.948333	−	0.317276i	$-0.102768\pi$
$68$	0	0
$69$	− 216.000i	− 0.376860i
$70$	0	0
$71$	−456.000	−0.762215	−0.381107	−	0.924531i	$-0.624457\pi$
$71$	−456.000	−0.762215	−0.381107	+	0.924531i	$0.624457\pi$
$72$	0	0
$73$	822.000	1.31792	0.658958	−	0.752180i	$-0.270998\pi$
$73$	822.000	1.31792	0.658958	+	0.752180i	$0.270998\pi$
$74$	0	0
$75$	− 183.000i	− 0.281747i
$76$	0	0
$77$	144.000i	0.213121i
$78$	0	0
$79$	1356.00	1.93116	0.965582	−	0.260100i	$-0.0837554\pi$
$79$	1356.00	1.93116	0.965582	+	0.260100i	$0.0837554\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 108.000i	− 0.142826i	−0.997447	−	0.0714129i	$-0.977249\pi$
$83$	− 108.000i	− 0.142826i	0.997447	−	0.0714129i	$-0.0227508\pi$
$84$	0	0
$85$	496.000i	0.632927i
$86$	0	0
$87$	−384.000	−0.473208
$88$	0	0
$89$	−938.000	−1.11717	−0.558583	−	0.829449i	$-0.688655\pi$
$89$	−938.000	−1.11717	−0.558583	+	0.829449i	$0.688655\pi$
$90$	0	0
$91$	− 240.000i	− 0.276471i
$92$	0	0
$93$	− 612.000i	− 0.682381i
$94$	0	0
$95$	864.000	0.933100
$96$	0	0
$97$	1278.00	1.33774	0.668872	−	0.743377i	$-0.266777\pi$
$97$	1278.00	1.33774	0.668872	+	0.743377i	$0.266777\pi$
$98$	0	0
$99$	108.000i	0.109640i
$100$	0	0
$101$	− 608.000i	− 0.598993i	−0.954097	−	0.299496i	$-0.903181\pi$
$101$	− 608.000i	− 0.598993i	0.954097	−	0.299496i	$-0.0968186\pi$
$102$	0	0
$103$	−948.000	−0.906886	−0.453443	−	0.891285i	$-0.649804\pi$
$103$	−948.000	−0.906886	−0.453443	+	0.891285i	$0.649804\pi$
$104$	0	0
$105$	−288.000	−0.267675
$106$	0	0
$107$	1044.00i	0.943246i	0.881800	+	0.471623i	$0.156332\pi$
$107$	1044.00i	0.943246i	−0.881800	+	0.471623i	$0.843668\pi$
$108$	0	0
$109$	1780.00i	1.56416i	0.623180	+	0.782078i	$0.285840\pi$
$109$	1780.00i	1.56416i	−0.623180	+	0.782078i	$0.714160\pi$
$110$	0	0
$111$	684.000	0.584887
$112$	0	0
$113$	−622.000	−0.517813	−0.258906	−	0.965902i	$-0.583362\pi$
$113$	−622.000	−0.517813	−0.258906	+	0.965902i	$0.583362\pi$
$114$	0	0
$115$	576.000i	0.467063i
$116$	0	0
$117$	− 180.000i	− 0.142231i
$118$	0	0
$119$	−744.000	−0.573129
$120$	0	0
$121$	1187.00	0.891811
$122$	0	0
$123$	66.0000i	0.0483823i
$124$	0	0
$125$	1488.00i	1.06473i
$126$	0	0
$127$	−204.000	−0.142536	−0.0712680	−	0.997457i	$-0.522705\pi$
$127$	−204.000	−0.142536	−0.0712680	+	0.997457i	$0.522705\pi$
$128$	0	0
$129$	−612.000	−0.417702
$130$	0	0
$131$	− 348.000i	− 0.232098i	−0.993243	−	0.116049i	$-0.962977\pi$
$131$	− 348.000i	− 0.232098i	0.993243	−	0.116049i	$-0.0370230\pi$
$132$	0	0
$133$	1296.00i	0.844943i
$134$	0	0
$135$	−216.000	−0.137706
$136$	0	0
$137$	106.000	0.0661036	0.0330518	−	0.999454i	$-0.489477\pi$
$137$	106.000	0.0661036	0.0330518	+	0.999454i	$0.489477\pi$
$138$	0	0
$139$	1188.00i	0.724927i	0.931998	+	0.362463i	$0.118064\pi$
$139$	1188.00i	0.724927i	−0.931998	+	0.362463i	$0.881936\pi$
$140$	0	0
$141$	− 1800.00i	− 1.07509i
$142$	0	0
$143$	240.000	0.140348
$144$	0	0
$145$	1024.00	0.586473
$146$	0	0
$147$	597.000i	0.334964i
$148$	0	0
$149$	− 2648.00i	− 1.45592i	−0.685618	−	0.727962i	$-0.740468\pi$
$149$	− 2648.00i	− 1.45592i	0.685618	−	0.727962i	$-0.259532\pi$
$150$	0	0
$151$	−3420.00	−1.84315	−0.921575	−	0.388200i	$-0.873097\pi$
$151$	−3420.00	−1.84315	−0.921575	+	0.388200i	$0.873097\pi$
$152$	0	0
$153$	−558.000	−0.294847
$154$	0	0
$155$	1632.00i	0.845712i
$156$	0	0
$157$	60.0000i	0.0305001i	0.999884	+	0.0152501i	$0.00485444\pi$
$157$	60.0000i	0.0305001i	−0.999884	+	0.0152501i	$0.995146\pi$
$158$	0	0
$159$	−768.000	−0.383059
$160$	0	0
$161$	−864.000	−0.422936
$162$	0	0
$163$	228.000i	0.109560i	0.998498	+	0.0547802i	$0.0174458\pi$
$163$	228.000i	0.109560i	−0.998498	+	0.0547802i	$0.982554\pi$
$164$	0	0
$165$	− 288.000i	− 0.135883i
$166$	0	0
$167$	−4128.00	−1.91278	−0.956390	−	0.292093i	$-0.905648\pi$
$167$	−4128.00	−1.91278	−0.956390	+	0.292093i	$0.905648\pi$
$168$	0	0
$169$	1797.00	0.817934
$170$	0	0
$171$	972.000i	0.434682i
$172$	0	0
$173$	− 1352.00i	− 0.594166i	−0.954852	−	0.297083i	$-0.903986\pi$
$173$	− 1352.00i	− 0.594166i	0.954852	−	0.297083i	$-0.0960137\pi$
$174$	0	0
$175$	−732.000	−0.316194
$176$	0	0
$177$	−2484.00	−1.05485
$178$	0	0
$179$	1716.00i	0.716536i	0.933619	+	0.358268i	$0.116633\pi$
$179$	1716.00i	0.716536i	−0.933619	+	0.358268i	$0.883367\pi$
$180$	0	0
$181$	3692.00i	1.51616i	0.652164	+	0.758078i	$0.273861\pi$
$181$	3692.00i	1.51616i	−0.652164	+	0.758078i	$0.726139\pi$
$182$	0	0
$183$	−252.000	−0.101794
$184$	0	0
$185$	−1824.00	−0.724882
$186$	0	0
$187$	− 744.000i	− 0.290945i
$188$	0	0
$189$	− 324.000i	− 0.124696i
$190$	0	0
$191$	−48.0000	−0.0181841	−0.00909204	−	0.999959i	$-0.502894\pi$
$191$	−48.0000	−0.0181841	−0.00909204	+	0.999959i	$0.502894\pi$
$192$	0	0
$193$	2414.00	0.900329	0.450165	−	0.892946i	$-0.351365\pi$
$193$	2414.00	0.900329	0.450165	+	0.892946i	$0.351365\pi$
$194$	0	0
$195$	480.000i	0.176274i
$196$	0	0
$197$	− 4304.00i	− 1.55659i	−0.627902	−	0.778293i	$-0.716086\pi$
$197$	− 4304.00i	− 1.55659i	0.627902	−	0.778293i	$-0.283914\pi$
$198$	0	0
$199$	204.000	0.0726692	0.0363346	−	0.999340i	$-0.488432\pi$
$199$	204.000	0.0726692	0.0363346	+	0.999340i	$0.488432\pi$
$200$	0	0
$201$	−1044.00	−0.366359
$202$	0	0
$203$	1536.00i	0.531064i
$204$	0	0
$205$	− 176.000i	− 0.0599628i
$206$	0	0
$207$	−648.000	−0.217580
$208$	0	0
$209$	−1296.00	−0.428929
$210$	0	0
$211$	4020.00i	1.31160i	0.754933	+	0.655801i	$0.227669\pi$
$211$	4020.00i	1.31160i	−0.754933	+	0.655801i	$0.772331\pi$
$212$	0	0
$213$	1368.00i	0.440065i
$214$	0	0
$215$	1632.00	0.517681
$216$	0	0
$217$	−2448.00	−0.765811
$218$	0	0
$219$	− 2466.00i	− 0.760899i
$220$	0	0
$221$	1240.00i	0.377427i
$222$	0	0
$223$	516.000	0.154950	0.0774751	−	0.996994i	$-0.475314\pi$
$223$	516.000	0.154950	0.0774751	+	0.996994i	$0.475314\pi$
$224$	0	0
$225$	−549.000	−0.162667
$226$	0	0
$227$	1428.00i	0.417532i	0.977966	+	0.208766i	$0.0669446\pi$
$227$	1428.00i	0.417532i	−0.977966	+	0.208766i	$0.933055\pi$
$228$	0	0
$229$	6028.00i	1.73948i	0.493508	+	0.869741i	$0.335714\pi$
$229$	6028.00i	1.73948i	−0.493508	+	0.869741i	$0.664286\pi$
$230$	0	0
$231$	432.000	0.123046
$232$	0	0
$233$	2630.00	0.739472	0.369736	−	0.929137i	$-0.379448\pi$
$233$	2630.00	0.739472	0.369736	+	0.929137i	$0.379448\pi$
$234$	0	0
$235$	4800.00i	1.33241i
$236$	0	0
$237$	− 4068.00i	− 1.11496i
$238$	0	0
$239$	−4416.00	−1.19518	−0.597588	−	0.801803i	$-0.703874\pi$
$239$	−4416.00	−1.19518	−0.597588	+	0.801803i	$0.703874\pi$
$240$	0	0
$241$	4830.00	1.29099	0.645493	−	0.763766i	$-0.276652\pi$
$241$	4830.00	1.29099	0.645493	+	0.763766i	$0.276652\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 1592.00i	− 0.415139i
$246$	0	0
$247$	2160.00	0.556427
$248$	0	0
$249$	−324.000	−0.0824605
$250$	0	0
$251$	− 5532.00i	− 1.39114i	−0.718457	−	0.695571i	$-0.755151\pi$
$251$	− 5532.00i	− 1.39114i	0.718457	−	0.695571i	$-0.244849\pi$
$252$	0	0
$253$	− 864.000i	− 0.214700i
$254$	0	0
$255$	1488.00	0.365420
$256$	0	0
$257$	−254.000	−0.0616501	−0.0308251	−	0.999525i	$-0.509813\pi$
$257$	−254.000	−0.0616501	−0.0308251	+	0.999525i	$0.509813\pi$
$258$	0	0
$259$	− 2736.00i	− 0.656397i
$260$	0	0
$261$	1152.00i	0.273207i
$262$	0	0
$263$	4272.00	1.00161	0.500804	−	0.865561i	$-0.333038\pi$
$263$	4272.00	1.00161	0.500804	+	0.865561i	$0.333038\pi$
$264$	0	0
$265$	2048.00	0.474746
$266$	0	0
$267$	2814.00i	0.644996i
$268$	0	0
$269$	4544.00i	1.02994i	0.857210	+	0.514968i	$0.172196\pi$
$269$	4544.00i	1.02994i	−0.857210	+	0.514968i	$0.827804\pi$
$270$	0	0
$271$	−2076.00	−0.465343	−0.232672	−	0.972555i	$-0.574747\pi$
$271$	−2076.00	−0.465343	−0.232672	+	0.972555i	$0.574747\pi$
$272$	0	0
$273$	−720.000	−0.159620
$274$	0	0
$275$	− 732.000i	− 0.160514i
$276$	0	0
$277$	− 484.000i	− 0.104985i	−0.998621	−	0.0524923i	$-0.983283\pi$
$277$	− 484.000i	− 0.104985i	0.998621	−	0.0524923i	$-0.0167165\pi$
$278$	0	0
$279$	−1836.00	−0.393973
$280$	0	0
$281$	406.000	0.0861919	0.0430960	−	0.999071i	$-0.486278\pi$
$281$	406.000	0.0861919	0.0430960	+	0.999071i	$0.486278\pi$
$282$	0	0
$283$	− 8172.00i	− 1.71652i	−0.513216	−	0.858260i	$-0.671546\pi$
$283$	− 8172.00i	− 1.71652i	0.513216	−	0.858260i	$-0.328454\pi$
$284$	0	0
$285$	− 2592.00i	− 0.538726i
$286$	0	0
$287$	264.000	0.0542977
$288$	0	0
$289$	−1069.00	−0.217586
$290$	0	0
$291$	− 3834.00i	− 0.772347i
$292$	0	0
$293$	− 4960.00i	− 0.988963i	−0.869188	−	0.494482i	$-0.835358\pi$
$293$	− 4960.00i	− 0.988963i	0.869188	−	0.494482i	$-0.164642\pi$
$294$	0	0
$295$	6624.00	1.30734
$296$	0	0
$297$	324.000	0.0633010
$298$	0	0
$299$	1440.00i	0.278520i
$300$	0	0
$301$	2448.00i	0.468772i
$302$	0	0
$303$	−1824.00	−0.345829
$304$	0	0
$305$	672.000	0.126159
$306$	0	0
$307$	− 6684.00i	− 1.24259i	−0.783576	−	0.621296i	$-0.786606\pi$
$307$	− 6684.00i	− 1.24259i	0.783576	−	0.621296i	$-0.213394\pi$
$308$	0	0
$309$	2844.00i	0.523591i
$310$	0	0
$311$	4992.00	0.910194	0.455097	−	0.890442i	$-0.349605\pi$
$311$	4992.00	0.910194	0.455097	+	0.890442i	$0.349605\pi$
$312$	0	0
$313$	5402.00	0.975524	0.487762	−	0.872977i	$-0.337813\pi$
$313$	5402.00	0.975524	0.487762	+	0.872977i	$0.337813\pi$
$314$	0	0
$315$	864.000i	0.154542i
$316$	0	0
$317$	7856.00i	1.39191i	0.718083	+	0.695957i	$0.245020\pi$
$317$	7856.00i	1.39191i	−0.718083	+	0.695957i	$0.754980\pi$
$318$	0	0
$319$	−1536.00	−0.269591
$320$	0	0
$321$	3132.00	0.544583
$322$	0	0
$323$	− 6696.00i	− 1.15348i
$324$	0	0
$325$	1220.00i	0.208226i
$326$	0	0
$327$	5340.00	0.903066
$328$	0	0
$329$	−7200.00	−1.20653
$330$	0	0
$331$	3732.00i	0.619726i	0.950781	+	0.309863i	$0.100283\pi$
$331$	3732.00i	0.619726i	−0.950781	+	0.309863i	$0.899717\pi$
$332$	0	0
$333$	− 2052.00i	− 0.337684i
$334$	0	0
$335$	2784.00	0.454048
$336$	0	0
$337$	−5598.00	−0.904874	−0.452437	−	0.891796i	$-0.649445\pi$
$337$	−5598.00	−0.904874	−0.452437	+	0.891796i	$0.649445\pi$
$338$	0	0
$339$	1866.00i	0.298959i
$340$	0	0
$341$	− 2448.00i	− 0.388758i
$342$	0	0
$343$	6504.00	1.02386
$344$	0	0
$345$	1728.00	0.269659
$346$	0	0
$347$	− 8220.00i	− 1.27168i	−0.771821	−	0.635840i	$-0.780654\pi$
$347$	− 8220.00i	− 1.27168i	0.771821	−	0.635840i	$-0.219346\pi$
$348$	0	0
$349$	− 11844.0i	− 1.81660i	−0.418315	−	0.908302i	$-0.637379\pi$
$349$	− 11844.0i	− 1.81660i	0.418315	−	0.908302i	$-0.362621\pi$
$350$	0	0
$351$	−540.000	−0.0821170
$352$	0	0
$353$	−7006.00	−1.05635	−0.528175	−	0.849135i	$-0.677124\pi$
$353$	−7006.00	−1.05635	−0.528175	+	0.849135i	$0.677124\pi$
$354$	0	0
$355$	− 3648.00i	− 0.545396i
$356$	0	0
$357$	2232.00i	0.330896i
$358$	0	0
$359$	7512.00	1.10437	0.552184	−	0.833722i	$-0.313795\pi$
$359$	7512.00	1.10437	0.552184	+	0.833722i	$0.313795\pi$
$360$	0	0
$361$	−4805.00	−0.700539
$362$	0	0
$363$	− 3561.00i	− 0.514887i
$364$	0	0
$365$	6576.00i	0.943023i
$366$	0	0
$367$	−5076.00	−0.721976	−0.360988	−	0.932571i	$-0.617560\pi$
$367$	−5076.00	−0.721976	−0.360988	+	0.932571i	$0.617560\pi$
$368$	0	0
$369$	198.000	0.0279335
$370$	0	0
$371$	3072.00i	0.429893i
$372$	0	0
$373$	− 4860.00i	− 0.674641i	−0.941390	−	0.337321i	$-0.890479\pi$
$373$	− 4860.00i	− 0.674641i	0.941390	−	0.337321i	$-0.109521\pi$
$374$	0	0
$375$	4464.00	0.614720
$376$	0	0
$377$	2560.00	0.349726
$378$	0	0
$379$	− 5964.00i	− 0.808312i	−0.914690	−	0.404156i	$-0.867565\pi$
$379$	− 5964.00i	− 0.808312i	0.914690	−	0.404156i	$-0.132435\pi$
$380$	0	0
$381$	612.000i	0.0822932i
$382$	0	0
$383$	432.000	0.0576349	0.0288175	−	0.999585i	$-0.490826\pi$
$383$	432.000	0.0576349	0.0288175	+	0.999585i	$0.490826\pi$
$384$	0	0
$385$	−1152.00	−0.152497
$386$	0	0
$387$	1836.00i	0.241161i
$388$	0	0
$389$	− 8888.00i	− 1.15846i	−0.815165	−	0.579228i	$-0.803354\pi$
$389$	− 8888.00i	− 1.15846i	0.815165	−	0.579228i	$-0.196646\pi$
$390$	0	0
$391$	4464.00	0.577376
$392$	0	0
$393$	−1044.00	−0.134002
$394$	0	0
$395$	10848.0i	1.38183i
$396$	0	0
$397$	− 2676.00i	− 0.338299i	−0.985590	−	0.169149i	$-0.945898\pi$
$397$	− 2676.00i	− 0.338299i	0.985590	−	0.169149i	$-0.0541020\pi$
$398$	0	0
$399$	3888.00	0.487828
$400$	0	0
$401$	13790.0	1.71731	0.858653	−	0.512557i	$-0.171302\pi$
$401$	13790.0	1.71731	0.858653	+	0.512557i	$0.171302\pi$
$402$	0	0
$403$	4080.00i	0.504316i
$404$	0	0
$405$	648.000i	0.0795046i
$406$	0	0
$407$	2736.00	0.333215
$408$	0	0
$409$	−1974.00	−0.238650	−0.119325	−	0.992855i	$-0.538073\pi$
$409$	−1974.00	−0.238650	−0.119325	+	0.992855i	$0.538073\pi$
$410$	0	0
$411$	− 318.000i	− 0.0381649i
$412$	0	0
$413$	9936.00i	1.18382i
$414$	0	0
$415$	864.000	0.102198
$416$	0	0
$417$	3564.00	0.418537
$418$	0	0
$419$	− 4764.00i	− 0.555457i	−0.960660	−	0.277729i	$-0.910418\pi$
$419$	− 4764.00i	− 0.555457i	0.960660	−	0.277729i	$-0.0895816\pi$
$420$	0	0
$421$	92.0000i	0.0106504i	0.999986	+	0.00532518i	$0.00169507\pi$
$421$	92.0000i	0.0106504i	−0.999986	+	0.00532518i	$0.998305\pi$
$422$	0	0
$423$	−5400.00	−0.620702
$424$	0	0
$425$	3782.00	0.431656
$426$	0	0
$427$	1008.00i	0.114240i
$428$	0	0
$429$	− 720.000i	− 0.0810301i
$430$	0	0
$431$	−10488.0	−1.17213	−0.586066	−	0.810263i	$-0.699324\pi$
$431$	−10488.0	−1.17213	−0.586066	+	0.810263i	$0.699324\pi$
$432$	0	0
$433$	−13138.0	−1.45813	−0.729067	−	0.684442i	$-0.760046\pi$
$433$	−13138.0	−1.45813	−0.729067	+	0.684442i	$0.760046\pi$
$434$	0	0
$435$	− 3072.00i	− 0.338600i
$436$	0	0
$437$	− 7776.00i	− 0.851205i
$438$	0	0
$439$	3612.00	0.392691	0.196346	−	0.980535i	$-0.437093\pi$
$439$	3612.00	0.392691	0.196346	+	0.980535i	$0.437093\pi$
$440$	0	0
$441$	1791.00	0.193392
$442$	0	0
$443$	− 12972.0i	− 1.39124i	−0.718411	−	0.695619i	$-0.755130\pi$
$443$	− 12972.0i	− 1.39124i	0.718411	−	0.695619i	$-0.244870\pi$
$444$	0	0
$445$	− 7504.00i	− 0.799379i
$446$	0	0
$447$	−7944.00	−0.840578
$448$	0	0
$449$	5998.00	0.630430	0.315215	−	0.949020i	$-0.397923\pi$
$449$	5998.00	0.630430	0.315215	+	0.949020i	$0.397923\pi$
$450$	0	0
$451$	264.000i	0.0275638i
$452$	0	0
$453$	10260.0i	1.06414i
$454$	0	0
$455$	1920.00	0.197826
$456$	0	0
$457$	−8934.00	−0.914475	−0.457237	−	0.889345i	$-0.651161\pi$
$457$	−8934.00	−0.914475	−0.457237	+	0.889345i	$0.651161\pi$
$458$	0	0
$459$	1674.00i	0.170230i
$460$	0	0
$461$	7448.00i	0.752468i	0.926525	+	0.376234i	$0.122781\pi$
$461$	7448.00i	0.752468i	−0.926525	+	0.376234i	$0.877219\pi$
$462$	0	0
$463$	4356.00	0.437236	0.218618	−	0.975810i	$-0.429845\pi$
$463$	4356.00	0.437236	0.218618	+	0.975810i	$0.429845\pi$
$464$	0	0
$465$	4896.00	0.488272
$466$	0	0
$467$	8580.00i	0.850182i	0.905151	+	0.425091i	$0.139758\pi$
$467$	8580.00i	0.850182i	−0.905151	+	0.425091i	$0.860242\pi$
$468$	0	0
$469$	4176.00i	0.411151i
$470$	0	0
$471$	180.000	0.0176093
$472$	0	0
$473$	−2448.00	−0.237969
$474$	0	0
$475$	− 6588.00i	− 0.636375i
$476$	0	0
$477$	2304.00i	0.221159i
$478$	0	0
$479$	−12648.0	−1.20648	−0.603238	−	0.797561i	$-0.706123\pi$
$479$	−12648.0	−1.20648	−0.603238	+	0.797561i	$0.706123\pi$
$480$	0	0
$481$	−4560.00	−0.432262
$482$	0	0
$483$	2592.00i	0.244182i
$484$	0	0
$485$	10224.0i	0.957212i
$486$	0	0
$487$	−15036.0	−1.39907	−0.699534	−	0.714599i	$-0.746609\pi$
$487$	−15036.0	−1.39907	−0.699534	+	0.714599i	$0.746609\pi$
$488$	0	0
$489$	684.000	0.0632547
$490$	0	0
$491$	15684.0i	1.44157i	0.693161	+	0.720783i	$0.256218\pi$
$491$	15684.0i	1.44157i	−0.693161	+	0.720783i	$0.743782\pi$
$492$	0	0
$493$	− 7936.00i	− 0.724989i
$494$	0	0
$495$	−864.000	−0.0784523
$496$	0	0
$497$	5472.00	0.493869
$498$	0	0
$499$	16308.0i	1.46302i	0.681831	+	0.731509i	$0.261184\pi$
$499$	16308.0i	1.46302i	−0.681831	+	0.731509i	$0.738816\pi$
$500$	0	0
$501$	12384.0i	1.10434i
$502$	0	0
$503$	−10344.0	−0.916931	−0.458465	−	0.888712i	$-0.651601\pi$
$503$	−10344.0	−0.916931	−0.458465	+	0.888712i	$0.651601\pi$
$504$	0	0
$505$	4864.00	0.428604
$506$	0	0
$507$	− 5391.00i	− 0.472234i
$508$	0	0
$509$	− 5648.00i	− 0.491833i	−0.969291	−	0.245917i	$-0.920911\pi$
$509$	− 5648.00i	− 0.491833i	0.969291	−	0.245917i	$-0.0790890\pi$
$510$	0	0
$511$	−9864.00	−0.853929
$512$	0	0
$513$	2916.00	0.250964
$514$	0	0
$515$	− 7584.00i	− 0.648915i
$516$	0	0
$517$	− 7200.00i	− 0.612487i
$518$	0	0
$519$	−4056.00	−0.343042
$520$	0	0
$521$	19498.0	1.63958	0.819792	−	0.572662i	$-0.194089\pi$
$521$	19498.0	1.63958	0.819792	+	0.572662i	$0.194089\pi$
$522$	0	0
$523$	22596.0i	1.88920i	0.328217	+	0.944602i	$0.393552\pi$
$523$	22596.0i	1.88920i	−0.328217	+	0.944602i	$0.606448\pi$
$524$	0	0
$525$	2196.00i	0.182555i
$526$	0	0
$527$	12648.0	1.04546
$528$	0	0
$529$	−6983.00	−0.573929
$530$	0	0
$531$	7452.00i	0.609019i
$532$	0	0
$533$	− 440.000i	− 0.0357571i
$534$	0	0
$535$	−8352.00	−0.674932
$536$	0	0
$537$	5148.00	0.413692
$538$	0	0
$539$	2388.00i	0.190832i
$540$	0	0
$541$	− 812.000i	− 0.0645298i	−0.999479	−	0.0322649i	$-0.989728\pi$
$541$	− 812.000i	− 0.0645298i	0.999479	−	0.0322649i	$-0.0102720\pi$
$542$	0	0
$543$	11076.0	0.875353
$544$	0	0
$545$	−14240.0	−1.11922
$546$	0	0
$547$	6132.00i	0.479315i	0.970857	+	0.239658i	$0.0770352\pi$
$547$	6132.00i	0.479315i	−0.970857	+	0.239658i	$0.922965\pi$
$548$	0	0
$549$	756.000i	0.0587710i
$550$	0	0
$551$	−13824.0	−1.06882
$552$	0	0
$553$	−16272.0	−1.25128
$554$	0	0
$555$	5472.00i	0.418511i
$556$	0	0
$557$	2792.00i	0.212389i	0.994345	+	0.106195i	$0.0338667\pi$
$557$	2792.00i	0.212389i	−0.994345	+	0.106195i	$0.966133\pi$
$558$	0	0
$559$	4080.00	0.308704
$560$	0	0
$561$	−2232.00	−0.167977
$562$	0	0
$563$	6468.00i	0.484181i	0.970254	+	0.242090i	$0.0778330\pi$
$563$	6468.00i	0.484181i	−0.970254	+	0.242090i	$0.922167\pi$
$564$	0	0
$565$	− 4976.00i	− 0.370517i
$566$	0	0
$567$	−972.000	−0.0719932
$568$	0	0
$569$	10522.0	0.775229	0.387614	−	0.921822i	$-0.373299\pi$
$569$	10522.0	0.775229	0.387614	+	0.921822i	$0.373299\pi$
$570$	0	0
$571$	− 1068.00i	− 0.0782739i	−0.999234	−	0.0391370i	$-0.987539\pi$
$571$	− 1068.00i	− 0.0782739i	0.999234	−	0.0391370i	$-0.0124609\pi$
$572$	0	0
$573$	144.000i	0.0104986i
$574$	0	0
$575$	4392.00	0.318537
$576$	0	0
$577$	3602.00	0.259884	0.129942	−	0.991522i	$-0.458521\pi$
$577$	3602.00	0.259884	0.129942	+	0.991522i	$0.458521\pi$
$578$	0	0
$579$	− 7242.00i	− 0.519805i
$580$	0	0
$581$	1296.00i	0.0925424i
$582$	0	0
$583$	−3072.00	−0.218232
$584$	0	0
$585$	1440.00	0.101772
$586$	0	0
$587$	17940.0i	1.26144i	0.776012	+	0.630718i	$0.217240\pi$
$587$	17940.0i	1.26144i	−0.776012	+	0.630718i	$0.782760\pi$
$588$	0	0
$589$	− 22032.0i	− 1.54128i
$590$	0	0
$591$	−12912.0	−0.898695
$592$	0	0
$593$	18034.0	1.24885	0.624425	−	0.781085i	$-0.285334\pi$
$593$	18034.0	1.24885	0.624425	+	0.781085i	$0.285334\pi$
$594$	0	0
$595$	− 5952.00i	− 0.410098i
$596$	0	0
$597$	− 612.000i	− 0.0419556i
$598$	0	0
$599$	−12264.0	−0.836550	−0.418275	−	0.908320i	$-0.637365\pi$
$599$	−12264.0	−0.836550	−0.418275	+	0.908320i	$0.637365\pi$
$600$	0	0
$601$	−12634.0	−0.857490	−0.428745	−	0.903426i	$-0.641044\pi$
$601$	−12634.0	−0.857490	−0.428745	+	0.903426i	$0.641044\pi$
$602$	0	0
$603$	3132.00i	0.211517i
$604$	0	0
$605$	9496.00i	0.638128i
$606$	0	0
$607$	−2796.00	−0.186962	−0.0934812	−	0.995621i	$-0.529799\pi$
$607$	−2796.00	−0.186962	−0.0934812	+	0.995621i	$0.529799\pi$
$608$	0	0
$609$	4608.00	0.306610
$610$	0	0
$611$	12000.0i	0.794547i
$612$	0	0
$613$	− 13788.0i	− 0.908470i	−0.890882	−	0.454235i	$-0.849913\pi$
$613$	− 13788.0i	− 0.908470i	0.890882	−	0.454235i	$-0.150087\pi$
$614$	0	0
$615$	−528.000	−0.0346195
$616$	0	0
$617$	−18074.0	−1.17931	−0.589653	−	0.807657i	$-0.700736\pi$
$617$	−18074.0	−1.17931	−0.589653	+	0.807657i	$0.700736\pi$
$618$	0	0
$619$	5940.00i	0.385701i	0.981228	+	0.192850i	$0.0617732\pi$
$619$	5940.00i	0.385701i	−0.981228	+	0.192850i	$0.938227\pi$
$620$	0	0
$621$	1944.00i	0.125620i
$622$	0	0
$623$	11256.0	0.723856
$624$	0	0
$625$	−4279.00	−0.273856
$626$	0	0
$627$	3888.00i	0.247642i
$628$	0	0
$629$	14136.0i	0.896088i
$630$	0	0
$631$	−8700.00	−0.548877	−0.274439	−	0.961605i	$-0.588492\pi$
$631$	−8700.00	−0.548877	−0.274439	+	0.961605i	$0.588492\pi$
$632$	0	0
$633$	12060.0	0.757254
$634$	0	0
$635$	− 1632.00i	− 0.101990i
$636$	0	0
$637$	− 3980.00i	− 0.247556i
$638$	0	0
$639$	4104.00	0.254072
$640$	0	0
$641$	−16306.0	−1.00476	−0.502378	−	0.864648i	$-0.667541\pi$
$641$	−16306.0	−1.00476	−0.502378	+	0.864648i	$0.667541\pi$
$642$	0	0
$643$	− 22668.0i	− 1.39026i	−0.718883	−	0.695131i	$-0.755346\pi$
$643$	− 22668.0i	− 1.39026i	0.718883	−	0.695131i	$-0.244654\pi$
$644$	0	0
$645$	− 4896.00i	− 0.298883i
$646$	0	0
$647$	−11928.0	−0.724788	−0.362394	−	0.932025i	$-0.618041\pi$
$647$	−11928.0	−0.724788	−0.362394	+	0.932025i	$0.618041\pi$
$648$	0	0
$649$	−9936.00	−0.600959
$650$	0	0
$651$	7344.00i	0.442141i
$652$	0	0
$653$	− 2552.00i	− 0.152936i	−0.997072	−	0.0764682i	$-0.975636\pi$
$653$	− 2552.00i	− 0.152936i	0.997072	−	0.0764682i	$-0.0243644\pi$
$654$	0	0
$655$	2784.00	0.166076
$656$	0	0
$657$	−7398.00	−0.439305
$658$	0	0
$659$	2196.00i	0.129809i	0.997891	+	0.0649044i	$0.0206742\pi$
$659$	2196.00i	0.129809i	−0.997891	+	0.0649044i	$0.979326\pi$
$660$	0	0
$661$	4260.00i	0.250673i	0.992114	+	0.125336i	$0.0400010\pi$
$661$	4260.00i	0.250673i	−0.992114	+	0.125336i	$0.959999\pi$
$662$	0	0
$663$	3720.00	0.217908
$664$	0	0
$665$	−10368.0	−0.604592
$666$	0	0
$667$	− 9216.00i	− 0.535000i
$668$	0	0
$669$	− 1548.00i	− 0.0894606i
$670$	0	0
$671$	−1008.00	−0.0579932
$672$	0	0
$673$	−2018.00	−0.115584	−0.0577921	−	0.998329i	$-0.518406\pi$
$673$	−2018.00	−0.115584	−0.0577921	+	0.998329i	$0.518406\pi$
$674$	0	0
$675$	1647.00i	0.0939156i
$676$	0	0
$677$	− 9256.00i	− 0.525461i	−0.964869	−	0.262730i	$-0.915377\pi$
$677$	− 9256.00i	− 0.525461i	0.964869	−	0.262730i	$-0.0846230\pi$
$678$	0	0
$679$	−15336.0	−0.866777
$680$	0	0
$681$	4284.00	0.241062
$682$	0	0
$683$	− 29244.0i	− 1.63835i	−0.573546	−	0.819173i	$-0.694433\pi$
$683$	− 29244.0i	− 1.63835i	0.573546	−	0.819173i	$-0.305567\pi$
$684$	0	0
$685$	848.000i	0.0472999i
$686$	0	0
$687$	18084.0	1.00429
$688$	0	0
$689$	5120.00	0.283101
$690$	0	0
$691$	3684.00i	0.202816i	0.994845	+	0.101408i	$0.0323348\pi$
$691$	3684.00i	0.202816i	−0.994845	+	0.101408i	$0.967665\pi$
$692$	0	0
$693$	− 1296.00i	− 0.0710404i
$694$	0	0
$695$	−9504.00	−0.518715
$696$	0	0
$697$	−1364.00	−0.0741251
$698$	0	0
$699$	− 7890.00i	− 0.426934i
$700$	0	0
$701$	13456.0i	0.725002i	0.931983	+	0.362501i	$0.118077\pi$
$701$	13456.0i	0.725002i	−0.931983	+	0.362501i	$0.881923\pi$
$702$	0	0
$703$	24624.0	1.32107
$704$	0	0
$705$	14400.0	0.769270
$706$	0	0
$707$	7296.00i	0.388111i
$708$	0	0
$709$	6460.00i	0.342187i	0.985255	+	0.171093i	$0.0547300\pi$
$709$	6460.00i	0.342187i	−0.985255	+	0.171093i	$0.945270\pi$
$710$	0	0
$711$	−12204.0	−0.643721
$712$	0	0
$713$	14688.0	0.771487
$714$	0	0
$715$	1920.00i	0.100425i
$716$	0	0
$717$	13248.0i	0.690036i
$718$	0	0
$719$	17160.0	0.890070	0.445035	−	0.895513i	$-0.353191\pi$
$719$	17160.0	0.890070	0.445035	+	0.895513i	$0.353191\pi$
$720$	0	0
$721$	11376.0	0.587607
$722$	0	0
$723$	− 14490.0i	− 0.745351i
$724$	0	0
$725$	− 7808.00i	− 0.399975i
$726$	0	0
$727$	−11820.0	−0.602998	−0.301499	−	0.953466i	$-0.597487\pi$
$727$	−11820.0	−0.602998	−0.301499	+	0.953466i	$0.597487\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 12648.0i	− 0.639950i
$732$	0	0
$733$	23924.0i	1.20553i	0.797919	+	0.602765i	$0.205934\pi$
$733$	23924.0i	1.20553i	−0.797919	+	0.602765i	$0.794066\pi$
$734$	0	0
$735$	−4776.00	−0.239681
$736$	0	0
$737$	−4176.00	−0.208718
$738$	0	0
$739$	11796.0i	0.587176i	0.955932	+	0.293588i	$0.0948493\pi$
$739$	11796.0i	0.587176i	−0.955932	+	0.293588i	$0.905151\pi$
$740$	0	0
$741$	− 6480.00i	− 0.321253i
$742$	0	0
$743$	27024.0	1.33434	0.667170	−	0.744906i	$-0.267506\pi$
$743$	27024.0	1.33434	0.667170	+	0.744906i	$0.267506\pi$
$744$	0	0
$745$	21184.0	1.04177
$746$	0	0
$747$	972.000i	0.0476086i
$748$	0	0
$749$	− 12528.0i	− 0.611166i
$750$	0	0
$751$	17340.0	0.842537	0.421269	−	0.906936i	$-0.361585\pi$
$751$	17340.0	0.842537	0.421269	+	0.906936i	$0.361585\pi$
$752$	0	0
$753$	−16596.0	−0.803176
$754$	0	0
$755$	− 27360.0i	− 1.31885i
$756$	0	0
$757$	− 27236.0i	− 1.30767i	−0.756635	−	0.653837i	$-0.773158\pi$
$757$	− 27236.0i	− 1.30767i	0.756635	−	0.653837i	$-0.226842\pi$
$758$	0	0
$759$	−2592.00	−0.123957
$760$	0	0
$761$	−14758.0	−0.702992	−0.351496	−	0.936189i	$-0.614327\pi$
$761$	−14758.0	−0.702992	−0.351496	+	0.936189i	$0.614327\pi$
$762$	0	0
$763$	− 21360.0i	− 1.01348i
$764$	0	0
$765$	− 4464.00i	− 0.210976i
$766$	0	0
$767$	16560.0	0.779592
$768$	0	0
$769$	25774.0	1.20863	0.604314	−	0.796747i	$-0.293447\pi$
$769$	25774.0	1.20863	0.604314	+	0.796747i	$0.293447\pi$
$770$	0	0
$771$	762.000i	0.0355937i
$772$	0	0
$773$	37424.0i	1.74133i	0.491877	+	0.870665i	$0.336311\pi$
$773$	37424.0i	1.74133i	−0.491877	+	0.870665i	$0.663689\pi$
$774$	0	0
$775$	12444.0	0.576776
$776$	0	0
$777$	−8208.00	−0.378971
$778$	0	0
$779$	2376.00i	0.109280i
$780$	0	0
$781$	5472.00i	0.250709i
$782$	0	0
$783$	3456.00	0.157736
$784$	0	0
$785$	−480.000	−0.0218241
$786$	0	0
$787$	12804.0i	0.579941i	0.957036	+	0.289970i	$0.0936454\pi$
$787$	12804.0i	0.579941i	−0.957036	+	0.289970i	$0.906355\pi$
$788$	0	0
$789$	− 12816.0i	− 0.578278i
$790$	0	0
$791$	7464.00	0.335511
$792$	0	0
$793$	1680.00	0.0752315
$794$	0	0
$795$	− 6144.00i	− 0.274095i
$796$	0	0
$797$	− 32024.0i	− 1.42327i	−0.702548	−	0.711636i	$-0.747954\pi$
$797$	− 32024.0i	− 1.42327i	0.702548	−	0.711636i	$-0.252046\pi$
$798$	0	0
$799$	37200.0	1.64711
$800$	0	0
$801$	8442.00	0.372389
$802$	0	0
$803$	− 9864.00i	− 0.433491i
$804$	0	0
$805$	− 6912.00i	− 0.302629i
$806$	0	0
$807$	13632.0	0.594633
$808$	0	0
$809$	38090.0	1.65534	0.827672	−	0.561212i	$-0.189665\pi$
$809$	38090.0	1.65534	0.827672	+	0.561212i	$0.189665\pi$
$810$	0	0
$811$	− 10428.0i	− 0.451512i	−0.974184	−	0.225756i	$-0.927515\pi$
$811$	− 10428.0i	− 0.451512i	0.974184	−	0.225756i	$-0.0724853\pi$
$812$	0	0
$813$	6228.00i	0.268666i
$814$	0	0
$815$	−1824.00	−0.0783950
$816$	0	0
$817$	−22032.0	−0.943454
$818$	0	0
$819$	2160.00i	0.0921569i
$820$	0	0
$821$	7984.00i	0.339395i	0.985496	+	0.169698i	$0.0542791\pi$
$821$	7984.00i	0.339395i	−0.985496	+	0.169698i	$0.945721\pi$
$822$	0	0
$823$	28788.0	1.21930	0.609652	−	0.792669i	$-0.291309\pi$
$823$	28788.0	1.21930	0.609652	+	0.792669i	$0.291309\pi$
$824$	0	0
$825$	−2196.00	−0.0926726
$826$	0	0
$827$	468.000i	0.0196783i	0.999952	+	0.00983915i	$0.00313195\pi$
$827$	468.000i	0.0196783i	−0.999952	+	0.00983915i	$0.996868\pi$
$828$	0	0
$829$	28852.0i	1.20877i	0.796692	+	0.604386i	$0.206581\pi$
$829$	28852.0i	1.20877i	−0.796692	+	0.604386i	$0.793419\pi$
$830$	0	0
$831$	−1452.00	−0.0606129
$832$	0	0
$833$	−12338.0	−0.513189
$834$	0	0
$835$	− 33024.0i	− 1.36867i
$836$	0	0
$837$	5508.00i	0.227460i
$838$	0	0
$839$	−1944.00	−0.0799932	−0.0399966	−	0.999200i	$-0.512735\pi$
$839$	−1944.00	−0.0799932	−0.0399966	+	0.999200i	$0.512735\pi$
$840$	0	0
$841$	8005.00	0.328222
$842$	0	0
$843$	− 1218.00i	− 0.0497629i
$844$	0	0
$845$	14376.0i	0.585266i
$846$	0	0
$847$	−14244.0	−0.577839
$848$	0	0
$849$	−24516.0	−0.991033
$850$	0	0
$851$	16416.0i	0.661261i
$852$	0	0
$853$	37044.0i	1.48694i	0.668768	+	0.743472i	$0.266822\pi$
$853$	37044.0i	1.48694i	−0.668768	+	0.743472i	$0.733178\pi$
$854$	0	0
$855$	−7776.00	−0.311033
$856$	0	0
$857$	−15046.0	−0.599722	−0.299861	−	0.953983i	$-0.596940\pi$
$857$	−15046.0	−0.599722	−0.299861	+	0.953983i	$0.596940\pi$
$858$	0	0
$859$	12180.0i	0.483791i	0.970302	+	0.241895i	$0.0777691\pi$
$859$	12180.0i	0.483791i	−0.970302	+	0.241895i	$0.922231\pi$
$860$	0	0
$861$	− 792.000i	− 0.0313488i
$862$	0	0
$863$	28752.0	1.13410	0.567051	−	0.823683i	$-0.308084\pi$
$863$	28752.0	1.13410	0.567051	+	0.823683i	$0.308084\pi$
$864$	0	0
$865$	10816.0	0.425150
$866$	0	0
$867$	3207.00i	0.125623i
$868$	0	0
$869$	− 16272.0i	− 0.635201i
$870$	0	0
$871$	6960.00	0.270758
$872$	0	0
$873$	−11502.0	−0.445915
$874$	0	0
$875$	− 17856.0i	− 0.689878i
$876$	0	0
$877$	31884.0i	1.22765i	0.789443	+	0.613823i	$0.210369\pi$
$877$	31884.0i	1.22765i	−0.789443	+	0.613823i	$0.789631\pi$
$878$	0	0
$879$	−14880.0	−0.570978
$880$	0	0
$881$	30802.0	1.17792	0.588959	−	0.808163i	$-0.299538\pi$
$881$	30802.0	1.17792	0.588959	+	0.808163i	$0.299538\pi$
$882$	0	0
$883$	− 32460.0i	− 1.23711i	−0.785742	−	0.618554i	$-0.787719\pi$
$883$	− 32460.0i	− 1.23711i	0.785742	−	0.618554i	$-0.212281\pi$
$884$	0	0
$885$	− 19872.0i	− 0.754791i
$886$	0	0
$887$	−14832.0	−0.561454	−0.280727	−	0.959788i	$-0.590576\pi$
$887$	−14832.0	−0.561454	−0.280727	+	0.959788i	$0.590576\pi$
$888$	0	0
$889$	2448.00	0.0923547
$890$	0	0
$891$	− 972.000i	− 0.0365468i
$892$	0	0
$893$	− 64800.0i	− 2.42827i
$894$	0	0
$895$	−13728.0	−0.512711
$896$	0	0
$897$	4320.00	0.160803
$898$	0	0
$899$	− 26112.0i	− 0.968725i
$900$	0	0
$901$	− 15872.0i	− 0.586873i
$902$	0	0
$903$	7344.00	0.270646
$904$	0	0
$905$	−29536.0	−1.08487
$906$	0	0
$907$	6900.00i	0.252603i	0.991992	+	0.126301i	$0.0403106\pi$
$907$	6900.00i	0.252603i	−0.991992	+	0.126301i	$0.959689\pi$
$908$	0	0
$909$	5472.00i	0.199664i
$910$	0	0
$911$	−32832.0	−1.19404	−0.597021	−	0.802225i	$-0.703649\pi$
$911$	−32832.0	−1.19404	−0.597021	+	0.802225i	$0.703649\pi$
$912$	0	0
$913$	−1296.00	−0.0469785
$914$	0	0
$915$	− 2016.00i	− 0.0728381i
$916$	0	0
$917$	4176.00i	0.150386i
$918$	0	0
$919$	8340.00	0.299359	0.149680	−	0.988735i	$-0.452176\pi$
$919$	8340.00	0.299359	0.149680	+	0.988735i	$0.452176\pi$
$920$	0	0
$921$	−20052.0	−0.717411
$922$	0	0
$923$	− 9120.00i	− 0.325231i
$924$	0	0
$925$	13908.0i	0.494370i
$926$	0	0
$927$	8532.00	0.302295
$928$	0	0
$929$	−39826.0	−1.40651	−0.703255	−	0.710937i	$-0.748271\pi$
$929$	−39826.0	−1.40651	−0.703255	+	0.710937i	$0.748271\pi$
$930$	0	0
$931$	21492.0i	0.756576i
$932$	0	0
$933$	− 14976.0i	− 0.525501i
$934$	0	0
$935$	5952.00	0.208183
$936$	0	0
$937$	28550.0	0.995398	0.497699	−	0.867350i	$-0.334178\pi$
$937$	28550.0	0.995398	0.497699	+	0.867350i	$0.334178\pi$
$938$	0	0
$939$	− 16206.0i	− 0.563219i
$940$	0	0
$941$	− 50632.0i	− 1.75404i	−0.480450	−	0.877022i	$-0.659527\pi$
$941$	− 50632.0i	− 1.75404i	0.480450	−	0.877022i	$-0.340473\pi$
$942$	0	0
$943$	−1584.00	−0.0547000
$944$	0	0
$945$	2592.00	0.0892251
$946$	0	0
$947$	− 18204.0i	− 0.624657i	−0.949974	−	0.312329i	$-0.898891\pi$
$947$	− 18204.0i	− 0.624657i	0.949974	−	0.312329i	$-0.101109\pi$
$948$	0	0
$949$	16440.0i	0.562345i
$950$	0	0
$951$	23568.0	0.803622
$952$	0	0
$953$	−4934.00	−0.167710	−0.0838552	−	0.996478i	$-0.526723\pi$
$953$	−4934.00	−0.167710	−0.0838552	+	0.996478i	$0.526723\pi$
$954$	0	0
$955$	− 384.000i	− 0.0130115i
$956$	0	0
$957$	4608.00i	0.155648i
$958$	0	0
$959$	−1272.00	−0.0428311
$960$	0	0
$961$	11825.0	0.396932
$962$	0	0
$963$	− 9396.00i	− 0.314415i
$964$	0	0
$965$	19312.0i	0.644223i
$966$	0	0
$967$	−13284.0	−0.441763	−0.220881	−	0.975301i	$-0.570893\pi$
$967$	−13284.0	−0.441763	−0.220881	+	0.975301i	$0.570893\pi$
$968$	0	0
$969$	−20088.0	−0.665964
$970$	0	0
$971$	50820.0i	1.67960i	0.542896	+	0.839800i	$0.317328\pi$
$971$	50820.0i	1.67960i	−0.542896	+	0.839800i	$0.682672\pi$
$972$	0	0
$973$	− 14256.0i	− 0.469709i
$974$	0	0
$975$	3660.00	0.120219
$976$	0	0
$977$	11038.0	0.361450	0.180725	−	0.983534i	$-0.442156\pi$
$977$	11038.0	0.361450	0.180725	+	0.983534i	$0.442156\pi$
$978$	0	0
$979$	11256.0i	0.367460i
$980$	0	0
$981$	− 16020.0i	− 0.521386i
$982$	0	0
$983$	44112.0	1.43129	0.715643	−	0.698466i	$-0.246134\pi$
$983$	44112.0	1.43129	0.715643	+	0.698466i	$0.246134\pi$
$984$	0	0
$985$	34432.0	1.11380
$986$	0	0
$987$	21600.0i	0.696591i
$988$	0	0
$989$	− 14688.0i	− 0.472246i
$990$	0	0
$991$	−56196.0	−1.80134	−0.900668	−	0.434507i	$-0.856923\pi$
$991$	−56196.0	−1.80134	−0.900668	+	0.434507i	$0.856923\pi$
$992$	0	0
$993$	11196.0	0.357799
$994$	0	0
$995$	1632.00i	0.0519979i
$996$	0	0
$997$	45588.0i	1.44813i	0.689731	+	0.724065i	$0.257729\pi$
$997$	45588.0i	1.44813i	−0.689731	+	0.724065i	$0.742271\pi$
$998$	0	0
$999$	−6156.00	−0.194962

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.d.a.193.1	✓	2
3.2	odd	2		1152.4.d.b.577.1		2
4.3	odd	2		384.4.d.b.193.2	yes	2
8.3	odd	2		384.4.d.b.193.1	yes	2
8.5	even	2	inner	384.4.d.a.193.2	yes	2
12.11	even	2		1152.4.d.g.577.1		2
16.3	odd	4		768.4.a.b.1.1		1
16.5	even	4		768.4.a.a.1.1		1
16.11	odd	4		768.4.a.c.1.1		1
16.13	even	4		768.4.a.d.1.1		1
24.5	odd	2		1152.4.d.b.577.2		2
24.11	even	2		1152.4.d.g.577.2		2
48.5	odd	4		2304.4.a.l.1.1		1
48.11	even	4		2304.4.a.k.1.1		1
48.29	odd	4		2304.4.a.f.1.1		1
48.35	even	4		2304.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.a.193.1	✓	2	1.1	even	1	trivial
384.4.d.a.193.2	yes	2	8.5	even	2	inner
384.4.d.b.193.1	yes	2	8.3	odd	2
384.4.d.b.193.2	yes	2	4.3	odd	2
768.4.a.a.1.1		1	16.5	even	4
768.4.a.b.1.1		1	16.3	odd	4
768.4.a.c.1.1		1	16.11	odd	4
768.4.a.d.1.1		1	16.13	even	4
1152.4.d.b.577.1		2	3.2	odd	2
1152.4.d.b.577.2		2	24.5	odd	2
1152.4.d.g.577.1		2	12.11	even	2
1152.4.d.g.577.2		2	24.11	even	2
2304.4.a.e.1.1		1	48.35	even	4
2304.4.a.f.1.1		1	48.29	odd	4
2304.4.a.k.1.1		1	48.11	even	4
2304.4.a.l.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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +8.00000i q^{5} -12.0000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +8.00000i q^{5} -12.0000 q^{7} -9.00000 q^{9} -12.0000i q^{11} +20.0000i q^{13} +24.0000 q^{15} +62.0000 q^{17} -108.000i q^{19} +36.0000i q^{21} +72.0000 q^{23} +61.0000 q^{25} +27.0000i q^{27} -128.000i q^{29} +204.000 q^{31} -36.0000 q^{33} -96.0000i q^{35} +228.000i q^{37} +60.0000 q^{39} -22.0000 q^{41} -204.000i q^{43} -72.0000i q^{45} +600.000 q^{47} -199.000 q^{49} -186.000i q^{51} -256.000i q^{53} +96.0000 q^{55} -324.000 q^{57} -828.000i q^{59} -84.0000i q^{61} +108.000 q^{63} -160.000 q^{65} -348.000i q^{67} -216.000i q^{69} -456.000 q^{71} +822.000 q^{73} -183.000i q^{75} +144.000i q^{77} +1356.00 q^{79} +81.0000 q^{81} -108.000i q^{83} +496.000i q^{85} -384.000 q^{87} -938.000 q^{89} -240.000i q^{91} -612.000i q^{93} +864.000 q^{95} +1278.00 q^{97} +108.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 24 q^{7} - 18 q^{9}+O(q^{10}) \) 2 * q - 24 * q^7 - 18 * q^9	\( 2 q - 24 q^{7} - 18 q^{9} + 48 q^{15} + 124 q^{17} + 144 q^{23} + 122 q^{25} + 408 q^{31} - 72 q^{33} + 120 q^{39} - 44 q^{41} + 1200 q^{47} - 398 q^{49} + 192 q^{55} - 648 q^{57} + 216 q^{63} - 320 q^{65} - 912 q^{71} + 1644 q^{73} + 2712 q^{79} + 162 q^{81} - 768 q^{87} - 1876 q^{89} + 1728 q^{95} + 2556 q^{97}+O(q^{100}) \) 2 * q - 24 * q^7 - 18 * q^9 + 48 * q^15 + 124 * q^17 + 144 * q^23 + 122 * q^25 + 408 * q^31 - 72 * q^33 + 120 * q^39 - 44 * q^41 + 1200 * q^47 - 398 * q^49 + 192 * q^55 - 648 * q^57 + 216 * q^63 - 320 * q^65 - 912 * q^71 + 1644 * q^73 + 2712 * q^79 + 162 * q^81 - 768 * q^87 - 1876 * q^89 + 1728 * q^95 + 2556 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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