LMFDB - Embedded newform 384.8.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,Mod(1,384)]

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, 0, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	384.a (trivial)

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:	yes
Analytic conductor:	$119.955849786$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 620x^{2} - 700x + 83625 $ x^4 - 620x^2 - 700x + 83625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-17.7724$ of defining polynomial
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0000 q^{3} -282.264 q^{5} -1362.23 q^{7} +729.000 q^{9} +O(q^{10})$	$q+27.0000 q^{3} -282.264 q^{5} -1362.23 q^{7} +729.000 q^{9} -3814.16 q^{11} -14857.6 q^{13} -7621.12 q^{15} -4637.82 q^{17} +3218.11 q^{19} -36780.1 q^{21} -39347.2 q^{23} +1547.80 q^{25} +19683.0 q^{27} -154692. q^{29} -71211.7 q^{31} -102982. q^{33} +384507. q^{35} +261898. q^{37} -401156. q^{39} +325681. q^{41} -531827. q^{43} -205770. q^{45} -895746. q^{47} +1.03211e6 q^{49} -125221. q^{51} -1.02021e6 q^{53} +1.07660e6 q^{55} +86889.1 q^{57} -1.89747e6 q^{59} +3.00735e6 q^{61} -993062. q^{63} +4.19377e6 q^{65} +2.51954e6 q^{67} -1.06237e6 q^{69} +4.15228e6 q^{71} -4.35349e6 q^{73} +41790.5 q^{75} +5.19575e6 q^{77} +3.62436e6 q^{79} +531441. q^{81} +5.94931e6 q^{83} +1.30909e6 q^{85} -4.17669e6 q^{87} +1.21553e6 q^{89} +2.02394e7 q^{91} -1.92272e6 q^{93} -908357. q^{95} -1.04112e7 q^{97} -2.78053e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{3} - 336 q^{5} + 680 q^{7} + 2916 q^{9}+O(q^{10}) $ 4 * q + 108 * q^3 - 336 * q^5 + 680 * q^7 + 2916 * q^9	$ 4 q + 108 q^{3} - 336 q^{5} + 680 q^{7} + 2916 q^{9} + 3856 q^{11} - 10680 q^{13} - 9072 q^{15} + 26232 q^{17} - 15456 q^{19} + 18360 q^{21} - 11312 q^{23} + 159052 q^{25} + 78732 q^{27} + 1856 q^{29} + 71752 q^{31} + 104112 q^{33} - 179040 q^{35} + 180088 q^{37} - 288360 q^{39} + 11224 q^{41} + 66688 q^{43} - 244944 q^{45} - 1334448 q^{47} + 2401140 q^{49} + 708264 q^{51} + 864576 q^{53} - 3304896 q^{55} - 417312 q^{57} + 1878448 q^{59} + 1901176 q^{61} + 495720 q^{63} + 4366944 q^{65} + 5505488 q^{67} - 305424 q^{69} - 967696 q^{71} + 3244760 q^{73} + 4294404 q^{75} + 8979488 q^{77} + 6471816 q^{79} + 2125764 q^{81} + 17019600 q^{83} - 12122592 q^{85} + 50112 q^{87} + 13559816 q^{89} + 6692304 q^{91} + 1937304 q^{93} + 22523904 q^{95} + 2180520 q^{97} + 2811024 q^{99}+O(q^{100}) $ 4 * q + 108 * q^3 - 336 * q^5 + 680 * q^7 + 2916 * q^9 + 3856 * q^11 - 10680 * q^13 - 9072 * q^15 + 26232 * q^17 - 15456 * q^19 + 18360 * q^21 - 11312 * q^23 + 159052 * q^25 + 78732 * q^27 + 1856 * q^29 + 71752 * q^31 + 104112 * q^33 - 179040 * q^35 + 180088 * q^37 - 288360 * q^39 + 11224 * q^41 + 66688 * q^43 - 244944 * q^45 - 1334448 * q^47 + 2401140 * q^49 + 708264 * q^51 + 864576 * q^53 - 3304896 * q^55 - 417312 * q^57 + 1878448 * q^59 + 1901176 * q^61 + 495720 * q^63 + 4366944 * q^65 + 5505488 * q^67 - 305424 * q^69 - 967696 * q^71 + 3244760 * q^73 + 4294404 * q^75 + 8979488 * q^77 + 6471816 * q^79 + 2125764 * q^81 + 17019600 * q^83 - 12122592 * q^85 + 50112 * q^87 + 13559816 * q^89 + 6692304 * q^91 + 1937304 * q^93 + 22523904 * q^95 + 2180520 * q^97 + 2811024 * q^99

Display 100 coefficients 10 coefficients

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
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	0	0
$5$	−282.264	−1.00986	−0.504929	−	0.863161i	$-0.668481\pi$
$5$	−282.264	−1.00986	−0.504929	+	0.863161i	$0.668481\pi$
$6$	0	0
$7$	−1362.23	−1.50109	−0.750543	−	0.660821i	$-0.770208\pi$
$7$	−1362.23	−1.50109	−0.750543	+	0.660821i	$0.770208\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−3814.16	−0.864022	−0.432011	−	0.901868i	$-0.642196\pi$
$11$	−3814.16	−0.864022	−0.432011	+	0.901868i	$0.642196\pi$
$12$	0	0
$13$	−14857.6	−1.87563	−0.937817	−	0.347130i	$-0.887156\pi$
$13$	−14857.6	−1.87563	−0.937817	+	0.347130i	$0.887156\pi$
$14$	0	0
$15$	−7621.12	−0.583041
$16$	0	0
$17$	−4637.82	−0.228951	−0.114476	−	0.993426i	$-0.536519\pi$
$17$	−4637.82	−0.228951	−0.114476	+	0.993426i	$0.536519\pi$
$18$	0	0
$19$	3218.11	0.107638	0.0538188	−	0.998551i	$-0.482861\pi$
$19$	3218.11	0.107638	0.0538188	+	0.998551i	$0.482861\pi$
$20$	0	0
$21$	−36780.1	−0.866653
$22$	0	0
$23$	−39347.2	−0.674320	−0.337160	−	0.941447i	$-0.609466\pi$
$23$	−39347.2	−0.674320	−0.337160	+	0.941447i	$0.609466\pi$
$24$	0	0
$25$	1547.80	0.0198118
$26$	0	0
$27$	19683.0	0.192450
$28$	0	0
$29$	−154692.	−1.17781	−0.588906	−	0.808202i	$-0.700441\pi$
$29$	−154692.	−1.17781	−0.588906	+	0.808202i	$0.700441\pi$
$30$	0	0
$31$	−71211.7	−0.429324	−0.214662	−	0.976688i	$-0.568865\pi$
$31$	−71211.7	−0.429324	−0.214662	+	0.976688i	$0.568865\pi$
$32$	0	0
$33$	−102982.	−0.498843
$34$	0	0
$35$	384507.	1.51588
$36$	0	0
$37$	261898.	0.850013	0.425006	−	0.905190i	$-0.360272\pi$
$37$	261898.	0.850013	0.425006	+	0.905190i	$0.360272\pi$
$38$	0	0
$39$	−401156.	−1.08290
$40$	0	0
$41$	325681.	0.737987	0.368994	−	0.929432i	$-0.379702\pi$
$41$	325681.	0.737987	0.368994	+	0.929432i	$0.379702\pi$
$42$	0	0
$43$	−531827.	−1.02007	−0.510036	−	0.860153i	$-0.670368\pi$
$43$	−531827.	−1.02007	−0.510036	+	0.860153i	$0.670368\pi$
$44$	0	0
$45$	−205770.	−0.336619
$46$	0	0
$47$	−895746.	−1.25847	−0.629234	−	0.777216i	$-0.716631\pi$
$47$	−895746.	−1.25847	−0.629234	+	0.777216i	$0.716631\pi$
$48$	0	0
$49$	1.03211e6	1.25326
$50$	0	0
$51$	−125221.	−0.132185
$52$	0	0
$53$	−1.02021e6	−0.941287	−0.470643	−	0.882323i	$-0.655978\pi$
$53$	−1.02021e6	−0.941287	−0.470643	+	0.882323i	$0.655978\pi$
$54$	0	0
$55$	1.07660e6	0.872539
$56$	0	0
$57$	86889.1	0.0621446
$58$	0	0
$59$	−1.89747e6	−1.20280	−0.601400	−	0.798948i	$-0.705390\pi$
$59$	−1.89747e6	−1.20280	−0.601400	+	0.798948i	$0.705390\pi$
$60$	0	0
$61$	3.00735e6	1.69640	0.848202	−	0.529672i	$-0.177685\pi$
$61$	3.00735e6	1.69640	0.848202	+	0.529672i	$0.177685\pi$
$62$	0	0
$63$	−993062.	−0.500362
$64$	0	0
$65$	4.19377e6	1.89412
$66$	0	0
$67$	2.51954e6	1.02343	0.511717	−	0.859154i	$-0.329009\pi$
$67$	2.51954e6	1.02343	0.511717	+	0.859154i	$0.329009\pi$
$68$	0	0
$69$	−1.06237e6	−0.389319
$70$	0	0
$71$	4.15228e6	1.37684	0.688418	−	0.725314i	$-0.258305\pi$
$71$	4.15228e6	1.37684	0.688418	+	0.725314i	$0.258305\pi$
$72$	0	0
$73$	−4.35349e6	−1.30981	−0.654904	−	0.755712i	$-0.727291\pi$
$73$	−4.35349e6	−1.30981	−0.654904	+	0.755712i	$0.727291\pi$
$74$	0	0
$75$	41790.5	0.0114384
$76$	0	0
$77$	5.19575e6	1.29697
$78$	0	0
$79$	3.62436e6	0.827059	0.413530	−	0.910491i	$-0.364296\pi$
$79$	3.62436e6	0.827059	0.413530	+	0.910491i	$0.364296\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	5.94931e6	1.14207	0.571036	−	0.820925i	$-0.306542\pi$
$83$	5.94931e6	1.14207	0.571036	+	0.820925i	$0.306542\pi$
$84$	0	0
$85$	1.30909e6	0.231208
$86$	0	0
$87$	−4.17669e6	−0.680010
$88$	0	0
$89$	1.21553e6	0.182769	0.0913843	−	0.995816i	$-0.470871\pi$
$89$	1.21553e6	0.182769	0.0913843	+	0.995816i	$0.470871\pi$
$90$	0	0
$91$	2.02394e7	2.81549
$92$	0	0
$93$	−1.92272e6	−0.247870
$94$	0	0
$95$	−908357.	−0.108699
$96$	0	0
$97$	−1.04112e7	−1.15824	−0.579119	−	0.815243i	$-0.696603\pi$
$97$	−1.04112e7	−1.15824	−0.579119	+	0.815243i	$0.696603\pi$
$98$	0	0
$99$	−2.78053e6	−0.288007
$100$	0	0
$101$	−7.52279e6	−0.726531	−0.363266	−	0.931686i	$-0.618338\pi$
$101$	−7.52279e6	−0.726531	−0.363266	+	0.931686i	$0.618338\pi$
$102$	0	0
$103$	−1.26662e7	−1.14214	−0.571068	−	0.820903i	$-0.693471\pi$
$103$	−1.26662e7	−1.14214	−0.571068	+	0.820903i	$0.693471\pi$
$104$	0	0
$105$	1.03817e7	0.875196
$106$	0	0
$107$	−2.03306e7	−1.60438	−0.802188	−	0.597072i	$-0.796331\pi$
$107$	−2.03306e7	−1.60438	−0.802188	+	0.597072i	$0.796331\pi$
$108$	0	0
$109$	−2.31105e7	−1.70930	−0.854648	−	0.519208i	$-0.826227\pi$
$109$	−2.31105e7	−1.70930	−0.854648	+	0.519208i	$0.826227\pi$
$110$	0	0
$111$	7.07124e6	0.490755
$112$	0	0
$113$	1.55087e7	1.01111	0.505557	−	0.862793i	$-0.331287\pi$
$113$	1.55087e7	1.01111	0.505557	+	0.862793i	$0.331287\pi$
$114$	0	0
$115$	1.11063e7	0.680967
$116$	0	0
$117$	−1.08312e7	−0.625211
$118$	0	0
$119$	6.31776e6	0.343676
$120$	0	0
$121$	−4.93933e6	−0.253466
$122$	0	0
$123$	8.79339e6	0.426077
$124$	0	0
$125$	2.16150e7	0.989850
$126$	0	0
$127$	4.52189e7	1.95888	0.979439	−	0.201742i	$-0.0646603\pi$
$127$	4.52189e7	1.95888	0.979439	+	0.201742i	$0.0646603\pi$
$128$	0	0
$129$	−1.43593e7	−0.588939
$130$	0	0
$131$	−1.44228e7	−0.560532	−0.280266	−	0.959922i	$-0.590423\pi$
$131$	−1.44228e7	−0.560532	−0.280266	+	0.959922i	$0.590423\pi$
$132$	0	0
$133$	−4.38380e6	−0.161573
$134$	0	0
$135$	−5.55580e6	−0.194347
$136$	0	0
$137$	2.63066e7	0.874063	0.437032	−	0.899446i	$-0.356030\pi$
$137$	2.63066e7	0.874063	0.437032	+	0.899446i	$0.356030\pi$
$138$	0	0
$139$	−1.79784e7	−0.567806	−0.283903	−	0.958853i	$-0.591629\pi$
$139$	−1.79784e7	−0.567806	−0.283903	+	0.958853i	$0.591629\pi$
$140$	0	0
$141$	−2.41851e7	−0.726577
$142$	0	0
$143$	5.66695e7	1.62059
$144$	0	0
$145$	4.36640e7	1.18942
$146$	0	0
$147$	2.78671e7	0.723571
$148$	0	0
$149$	−5.16114e7	−1.27819	−0.639093	−	0.769129i	$-0.720690\pi$
$149$	−5.16114e7	−1.27819	−0.639093	+	0.769129i	$0.720690\pi$
$150$	0	0
$151$	6.67253e6	0.157714	0.0788572	−	0.996886i	$-0.474873\pi$
$151$	6.67253e6	0.157714	0.0788572	+	0.996886i	$0.474873\pi$
$152$	0	0
$153$	−3.38097e6	−0.0763171
$154$	0	0
$155$	2.01005e7	0.433556
$156$	0	0
$157$	−5.35184e7	−1.10371	−0.551854	−	0.833941i	$-0.686079\pi$
$157$	−5.35184e7	−1.10371	−0.551854	+	0.833941i	$0.686079\pi$
$158$	0	0
$159$	−2.75455e7	−0.543452
$160$	0	0
$161$	5.35997e7	1.01221
$162$	0	0
$163$	−3.49634e7	−0.632350	−0.316175	−	0.948701i	$-0.602399\pi$
$163$	−3.49634e7	−0.632350	−0.316175	+	0.948701i	$0.602399\pi$
$164$	0	0
$165$	2.90682e7	0.503761
$166$	0	0
$167$	3.78924e7	0.629570	0.314785	−	0.949163i	$-0.398068\pi$
$167$	3.78924e7	0.629570	0.314785	+	0.949163i	$0.398068\pi$
$168$	0	0
$169$	1.58001e8	2.51800
$170$	0	0
$171$	2.34600e6	0.0358792
$172$	0	0
$173$	−2.40284e7	−0.352828	−0.176414	−	0.984316i	$-0.556450\pi$
$173$	−2.40284e7	−0.352828	−0.176414	+	0.984316i	$0.556450\pi$
$174$	0	0
$175$	−2.10845e6	−0.0297392
$176$	0	0
$177$	−5.12318e7	−0.694437
$178$	0	0
$179$	1.29176e8	1.68344	0.841720	−	0.539915i	$-0.181544\pi$
$179$	1.29176e8	1.68344	0.841720	+	0.539915i	$0.181544\pi$
$180$	0	0
$181$	−7.22849e7	−0.906092	−0.453046	−	0.891487i	$-0.649663\pi$
$181$	−7.22849e7	−0.906092	−0.453046	+	0.891487i	$0.649663\pi$
$182$	0	0
$183$	8.11984e7	0.979420
$184$	0	0
$185$	−7.39242e7	−0.858392
$186$	0	0
$187$	1.76894e7	0.197819
$188$	0	0
$189$	−2.68127e7	−0.288884
$190$	0	0
$191$	5.31417e7	0.551846	0.275923	−	0.961180i	$-0.411016\pi$
$191$	5.31417e7	0.551846	0.275923	+	0.961180i	$0.411016\pi$
$192$	0	0
$193$	−1.37647e8	−1.37821	−0.689105	−	0.724662i	$-0.741996\pi$
$193$	−1.37647e8	−1.37821	−0.689105	+	0.724662i	$0.741996\pi$
$194$	0	0
$195$	1.13232e8	1.09357
$196$	0	0
$197$	−6.55467e7	−0.610828	−0.305414	−	0.952220i	$-0.598795\pi$
$197$	−6.55467e7	−0.610828	−0.305414	+	0.952220i	$0.598795\pi$
$198$	0	0
$199$	3.65315e7	0.328611	0.164305	−	0.986410i	$-0.447462\pi$
$199$	3.65315e7	0.328611	0.164305	+	0.986410i	$0.447462\pi$
$200$	0	0
$201$	6.80277e7	0.590880
$202$	0	0
$203$	2.10726e8	1.76800
$204$	0	0
$205$	−9.19279e7	−0.745262
$206$	0	0
$207$	−2.86841e7	−0.224773
$208$	0	0
$209$	−1.22744e7	−0.0930012
$210$	0	0
$211$	1.37911e8	1.01067	0.505335	−	0.862923i	$-0.331369\pi$
$211$	1.37911e8	1.01067	0.505335	+	0.862923i	$0.331369\pi$
$212$	0	0
$213$	1.12112e8	0.794917
$214$	0	0
$215$	1.50116e8	1.03013
$216$	0	0
$217$	9.70064e7	0.644453
$218$	0	0
$219$	−1.17544e8	−0.756218
$220$	0	0
$221$	6.89071e7	0.429429
$222$	0	0
$223$	6.89464e7	0.416337	0.208168	−	0.978093i	$-0.433250\pi$
$223$	6.89464e7	0.416337	0.208168	+	0.978093i	$0.433250\pi$
$224$	0	0
$225$	1.12834e6	0.00660394
$226$	0	0
$227$	1.41524e8	0.803044	0.401522	−	0.915849i	$-0.368481\pi$
$227$	1.41524e8	0.803044	0.401522	+	0.915849i	$0.368481\pi$
$228$	0	0
$229$	−1.38090e8	−0.759868	−0.379934	−	0.925014i	$-0.624053\pi$
$229$	−1.38090e8	−0.759868	−0.379934	+	0.925014i	$0.624053\pi$
$230$	0	0
$231$	1.40285e8	0.748807
$232$	0	0
$233$	3.71716e8	1.92515	0.962577	−	0.271007i	$-0.0873566\pi$
$233$	3.71716e8	1.92515	0.962577	+	0.271007i	$0.0873566\pi$
$234$	0	0
$235$	2.52837e8	1.27087
$236$	0	0
$237$	9.78577e7	0.477503
$238$	0	0
$239$	−4.40837e7	−0.208875	−0.104437	−	0.994531i	$-0.533304\pi$
$239$	−4.40837e7	−0.208875	−0.104437	+	0.994531i	$0.533304\pi$
$240$	0	0
$241$	1.62773e8	0.749069	0.374535	−	0.927213i	$-0.377802\pi$
$241$	1.62773e8	0.749069	0.374535	+	0.927213i	$0.377802\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	0	0
$245$	−2.91328e8	−1.26561
$246$	0	0
$247$	−4.78136e7	−0.201889
$248$	0	0
$249$	1.60631e8	0.659376
$250$	0	0
$251$	−1.77876e8	−0.710001	−0.355001	−	0.934866i	$-0.615519\pi$
$251$	−1.77876e8	−0.710001	−0.355001	+	0.934866i	$0.615519\pi$
$252$	0	0
$253$	1.50077e8	0.582628
$254$	0	0
$255$	3.53454e7	0.133488
$256$	0	0
$257$	3.50129e8	1.28665	0.643327	−	0.765592i	$-0.277554\pi$
$257$	3.50129e8	1.28665	0.643327	+	0.765592i	$0.277554\pi$
$258$	0	0
$259$	−3.56764e8	−1.27594
$260$	0	0
$261$	−1.12771e8	−0.392604
$262$	0	0
$263$	−3.08649e8	−1.04621	−0.523106	−	0.852268i	$-0.675227\pi$
$263$	−3.08649e8	−1.04621	−0.523106	+	0.852268i	$0.675227\pi$
$264$	0	0
$265$	2.87967e8	0.950565
$266$	0	0
$267$	3.28194e7	0.105521
$268$	0	0
$269$	2.10933e8	0.660712	0.330356	−	0.943856i	$-0.392831\pi$
$269$	2.10933e8	0.660712	0.330356	+	0.943856i	$0.392831\pi$
$270$	0	0
$271$	5.48682e8	1.67467	0.837334	−	0.546692i	$-0.184113\pi$
$271$	5.48682e8	1.67467	0.837334	+	0.546692i	$0.184113\pi$
$272$	0	0
$273$	5.46465e8	1.62552
$274$	0	0
$275$	−5.90355e6	−0.0171178
$276$	0	0
$277$	−5.01494e8	−1.41771	−0.708854	−	0.705355i	$-0.750788\pi$
$277$	−5.01494e8	−1.41771	−0.708854	+	0.705355i	$0.750788\pi$
$278$	0	0
$279$	−5.19133e7	−0.143108
$280$	0	0
$281$	−5.11695e8	−1.37575	−0.687874	−	0.725830i	$-0.741456\pi$
$281$	−5.11695e8	−1.37575	−0.687874	+	0.725830i	$0.741456\pi$
$282$	0	0
$283$	2.43080e8	0.637523	0.318761	−	0.947835i	$-0.396733\pi$
$283$	2.43080e8	0.637523	0.318761	+	0.947835i	$0.396733\pi$
$284$	0	0
$285$	−2.45256e7	−0.0627571
$286$	0	0
$287$	−4.43651e8	−1.10778
$288$	0	0
$289$	−3.88829e8	−0.947581
$290$	0	0
$291$	−2.81101e8	−0.668709
$292$	0	0
$293$	−4.62895e8	−1.07509	−0.537547	−	0.843234i	$-0.680649\pi$
$293$	−4.62895e8	−1.07509	−0.537547	+	0.843234i	$0.680649\pi$
$294$	0	0
$295$	5.35588e8	1.21466
$296$	0	0
$297$	−7.50742e7	−0.166281
$298$	0	0
$299$	5.84606e8	1.26478
$300$	0	0
$301$	7.24468e8	1.53122
$302$	0	0
$303$	−2.03115e8	−0.419463
$304$	0	0
$305$	−8.48866e8	−1.71313
$306$	0	0
$307$	−3.82064e8	−0.753619	−0.376810	−	0.926291i	$-0.622979\pi$
$307$	−3.82064e8	−0.753619	−0.376810	+	0.926291i	$0.622979\pi$
$308$	0	0
$309$	−3.41989e8	−0.659412
$310$	0	0
$311$	7.31257e7	0.137851	0.0689253	−	0.997622i	$-0.478043\pi$
$311$	7.31257e7	0.137851	0.0689253	+	0.997622i	$0.478043\pi$
$312$	0	0
$313$	−3.79042e8	−0.698687	−0.349343	−	0.936995i	$-0.613595\pi$
$313$	−3.79042e8	−0.698687	−0.349343	+	0.936995i	$0.613595\pi$
$314$	0	0
$315$	2.80305e8	0.505294
$316$	0	0
$317$	−6.03233e8	−1.06360	−0.531799	−	0.846871i	$-0.678484\pi$
$317$	−6.03233e8	−1.06360	−0.531799	+	0.846871i	$0.678484\pi$
$318$	0	0
$319$	5.90022e8	1.01766
$320$	0	0
$321$	−5.48925e8	−0.926287
$322$	0	0
$323$	−1.49250e7	−0.0246437
$324$	0	0
$325$	−2.29966e7	−0.0371597
$326$	0	0
$327$	−6.23984e8	−0.986862
$328$	0	0
$329$	1.22021e9	1.88907
$330$	0	0
$331$	3.21985e8	0.488021	0.244010	−	0.969773i	$-0.421537\pi$
$331$	3.21985e8	0.488021	0.244010	+	0.969773i	$0.421537\pi$
$332$	0	0
$333$	1.90923e8	0.283338
$334$	0	0
$335$	−7.11176e8	−1.03352
$336$	0	0
$337$	−6.22663e8	−0.886233	−0.443117	−	0.896464i	$-0.646127\pi$
$337$	−6.22663e8	−0.886233	−0.443117	+	0.896464i	$0.646127\pi$
$338$	0	0
$339$	4.18734e8	0.583767
$340$	0	0
$341$	2.71613e8	0.370946
$342$	0	0
$343$	−2.84121e8	−0.380167
$344$	0	0
$345$	2.99870e8	0.393157
$346$	0	0
$347$	−1.23835e9	−1.59108	−0.795538	−	0.605903i	$-0.792812\pi$
$347$	−1.23835e9	−1.59108	−0.795538	+	0.605903i	$0.792812\pi$
$348$	0	0
$349$	7.64841e7	0.0963124	0.0481562	−	0.998840i	$-0.484665\pi$
$349$	7.64841e7	0.0963124	0.0481562	+	0.998840i	$0.484665\pi$
$350$	0	0
$351$	−2.92443e8	−0.360966
$352$	0	0
$353$	−9.94452e8	−1.20330	−0.601648	−	0.798762i	$-0.705489\pi$
$353$	−9.94452e8	−1.20330	−0.601648	+	0.798762i	$0.705489\pi$
$354$	0	0
$355$	−1.17204e9	−1.39041
$356$	0	0
$357$	1.70579e8	0.198421
$358$	0	0
$359$	1.17743e9	1.34308	0.671542	−	0.740966i	$-0.265632\pi$
$359$	1.17743e9	1.34308	0.671542	+	0.740966i	$0.265632\pi$
$360$	0	0
$361$	−8.83515e8	−0.988414
$362$	0	0
$363$	−1.33362e8	−0.146338
$364$	0	0
$365$	1.22883e9	1.32272
$366$	0	0
$367$	−1.89983e8	−0.200624	−0.100312	−	0.994956i	$-0.531984\pi$
$367$	−1.89983e8	−0.200624	−0.100312	+	0.994956i	$0.531984\pi$
$368$	0	0
$369$	2.37421e8	0.245996
$370$	0	0
$371$	1.38975e9	1.41295
$372$	0	0
$373$	−5.55522e8	−0.554269	−0.277134	−	0.960831i	$-0.589385\pi$
$373$	−5.55522e8	−0.554269	−0.277134	+	0.960831i	$0.589385\pi$
$374$	0	0
$375$	5.83604e8	0.571490
$376$	0	0
$377$	2.29836e9	2.20914
$378$	0	0
$379$	−1.30691e9	−1.23313	−0.616565	−	0.787304i	$-0.711476\pi$
$379$	−1.30691e9	−1.23313	−0.616565	+	0.787304i	$0.711476\pi$
$380$	0	0
$381$	1.22091e9	1.13096
$382$	0	0
$383$	1.42228e8	0.129357	0.0646784	−	0.997906i	$-0.479398\pi$
$383$	1.42228e8	0.129357	0.0646784	+	0.997906i	$0.479398\pi$
$384$	0	0
$385$	−1.46657e9	−1.30976
$386$	0	0
$387$	−3.87702e8	−0.340024
$388$	0	0
$389$	1.79969e9	1.55015	0.775077	−	0.631866i	$-0.217711\pi$
$389$	1.79969e9	1.55015	0.775077	+	0.631866i	$0.217711\pi$
$390$	0	0
$391$	1.82485e8	0.154386
$392$	0	0
$393$	−3.89416e8	−0.323623
$394$	0	0
$395$	−1.02303e9	−0.835212
$396$	0	0
$397$	−2.73058e8	−0.219022	−0.109511	−	0.993986i	$-0.534929\pi$
$397$	−2.73058e8	−0.219022	−0.109511	+	0.993986i	$0.534929\pi$
$398$	0	0
$399$	−1.18362e8	−0.0932844
$400$	0	0
$401$	−2.13136e9	−1.65064	−0.825320	−	0.564666i	$-0.809005\pi$
$401$	−2.13136e9	−1.65064	−0.825320	+	0.564666i	$0.809005\pi$
$402$	0	0
$403$	1.05804e9	0.805255
$404$	0	0
$405$	−1.50007e8	−0.112206
$406$	0	0
$407$	−9.98920e8	−0.734430
$408$	0	0
$409$	−8.20609e7	−0.0593068	−0.0296534	−	0.999560i	$-0.509440\pi$
$409$	−8.20609e7	−0.0593068	−0.0296534	+	0.999560i	$0.509440\pi$
$410$	0	0
$411$	7.10278e8	0.504641
$412$	0	0
$413$	2.58479e9	1.80551
$414$	0	0
$415$	−1.67928e9	−1.15333
$416$	0	0
$417$	−4.85418e8	−0.327823
$418$	0	0
$419$	−1.97020e9	−1.30847	−0.654233	−	0.756293i	$-0.727008\pi$
$419$	−1.97020e9	−1.30847	−0.654233	+	0.756293i	$0.727008\pi$
$420$	0	0
$421$	−2.07781e9	−1.35712	−0.678559	−	0.734546i	$-0.737395\pi$
$421$	−2.07781e9	−1.35712	−0.678559	+	0.734546i	$0.737395\pi$
$422$	0	0
$423$	−6.52999e8	−0.419490
$424$	0	0
$425$	−7.17841e6	−0.00453594
$426$	0	0
$427$	−4.09669e9	−2.54645
$428$	0	0
$429$	1.53008e9	0.935648
$430$	0	0
$431$	1.88880e9	1.13636	0.568181	−	0.822904i	$-0.307648\pi$
$431$	1.88880e9	1.13636	0.568181	+	0.822904i	$0.307648\pi$
$432$	0	0
$433$	−1.10906e9	−0.656517	−0.328258	−	0.944588i	$-0.606462\pi$
$433$	−1.10906e9	−0.656517	−0.328258	+	0.944588i	$0.606462\pi$
$434$	0	0
$435$	1.17893e9	0.686713
$436$	0	0
$437$	−1.26624e8	−0.0725822
$438$	0	0
$439$	−1.94201e9	−1.09553	−0.547766	−	0.836632i	$-0.684521\pi$
$439$	−1.94201e9	−1.09553	−0.547766	+	0.836632i	$0.684521\pi$
$440$	0	0
$441$	7.52411e8	0.417754
$442$	0	0
$443$	1.10899e9	0.606057	0.303028	−	0.952982i	$-0.402002\pi$
$443$	1.10899e9	0.606057	0.303028	+	0.952982i	$0.402002\pi$
$444$	0	0
$445$	−3.43101e8	−0.184570
$446$	0	0
$447$	−1.39351e9	−0.737961
$448$	0	0
$449$	2.19688e9	1.14537	0.572683	−	0.819777i	$-0.305902\pi$
$449$	2.19688e9	1.14537	0.572683	+	0.819777i	$0.305902\pi$
$450$	0	0
$451$	−1.24220e9	−0.637637
$452$	0	0
$453$	1.80158e8	0.0910564
$454$	0	0
$455$	−5.71286e9	−2.84324
$456$	0	0
$457$	2.84116e8	0.139248	0.0696240	−	0.997573i	$-0.477820\pi$
$457$	2.84116e8	0.139248	0.0696240	+	0.997573i	$0.477820\pi$
$458$	0	0
$459$	−9.12863e7	−0.0440617
$460$	0	0
$461$	1.29061e9	0.613541	0.306770	−	0.951784i	$-0.400752\pi$
$461$	1.29061e9	0.613541	0.306770	+	0.951784i	$0.400752\pi$
$462$	0	0
$463$	−4.12483e9	−1.93140	−0.965701	−	0.259658i	$-0.916390\pi$
$463$	−4.12483e9	−1.93140	−0.965701	+	0.259658i	$0.916390\pi$
$464$	0	0
$465$	5.42713e8	0.250314
$466$	0	0
$467$	−2.27629e9	−1.03424	−0.517118	−	0.855914i	$-0.672995\pi$
$467$	−2.27629e9	−1.03424	−0.517118	+	0.855914i	$0.672995\pi$
$468$	0	0
$469$	−3.43219e9	−1.53626
$470$	0	0
$471$	−1.44500e9	−0.637226
$472$	0	0
$473$	2.02848e9	0.881365
$474$	0	0
$475$	4.98099e6	0.00213249
$476$	0	0
$477$	−7.43730e8	−0.313762
$478$	0	0
$479$	−6.83485e8	−0.284155	−0.142077	−	0.989856i	$-0.545378\pi$
$479$	−6.83485e8	−0.284155	−0.142077	+	0.989856i	$0.545378\pi$
$480$	0	0
$481$	−3.89118e9	−1.59431
$482$	0	0
$483$	1.44719e9	0.584402
$484$	0	0
$485$	2.93869e9	1.16966
$486$	0	0
$487$	3.70483e9	1.45351	0.726753	−	0.686899i	$-0.241028\pi$
$487$	3.70483e9	1.45351	0.726753	+	0.686899i	$0.241028\pi$
$488$	0	0
$489$	−9.44013e8	−0.365087
$490$	0	0
$491$	2.89597e9	1.10410	0.552050	−	0.833811i	$-0.313846\pi$
$491$	2.89597e9	1.10410	0.552050	+	0.833811i	$0.313846\pi$
$492$	0	0
$493$	7.17436e8	0.269661
$494$	0	0
$495$	7.84841e8	0.290846
$496$	0	0
$497$	−5.65634e9	−2.06675
$498$	0	0
$499$	−1.72733e9	−0.622334	−0.311167	−	0.950355i	$-0.600720\pi$
$499$	−1.72733e9	−0.622334	−0.311167	+	0.950355i	$0.600720\pi$
$500$	0	0
$501$	1.02309e9	0.363483
$502$	0	0
$503$	−2.71038e9	−0.949604	−0.474802	−	0.880093i	$-0.657480\pi$
$503$	−2.71038e9	−0.949604	−0.474802	+	0.880093i	$0.657480\pi$
$504$	0	0
$505$	2.12341e9	0.733693
$506$	0	0
$507$	4.26603e9	1.45377
$508$	0	0
$509$	2.23564e9	0.751432	0.375716	−	0.926735i	$-0.377397\pi$
$509$	2.23564e9	0.751432	0.375716	+	0.926735i	$0.377397\pi$
$510$	0	0
$511$	5.93043e9	1.96613
$512$	0	0
$513$	6.33421e7	0.0207149
$514$	0	0
$515$	3.57522e9	1.15339
$516$	0	0
$517$	3.41652e9	1.08734
$518$	0	0
$519$	−6.48767e8	−0.203706
$520$	0	0
$521$	−3.88054e9	−1.20215	−0.601076	−	0.799192i	$-0.705261\pi$
$521$	−3.88054e9	−1.20215	−0.601076	+	0.799192i	$0.705261\pi$
$522$	0	0
$523$	−2.91868e9	−0.892135	−0.446067	−	0.894999i	$-0.647176\pi$
$523$	−2.91868e9	−0.892135	−0.446067	+	0.894999i	$0.647176\pi$
$524$	0	0
$525$	−5.69281e7	−0.0171700
$526$	0	0
$527$	3.30267e8	0.0982943
$528$	0	0
$529$	−1.85662e9	−0.545292
$530$	0	0
$531$	−1.38326e9	−0.400933
$532$	0	0
$533$	−4.83885e9	−1.38419
$534$	0	0
$535$	5.73858e9	1.62019
$536$	0	0
$537$	3.48776e9	0.971934
$538$	0	0
$539$	−3.93665e9	−1.08285
$540$	0	0
$541$	3.32189e9	0.901976	0.450988	−	0.892530i	$-0.351072\pi$
$541$	3.32189e9	0.901976	0.450988	+	0.892530i	$0.351072\pi$
$542$	0	0
$543$	−1.95169e9	−0.523133
$544$	0	0
$545$	6.52326e9	1.72614
$546$	0	0
$547$	−2.18673e9	−0.571267	−0.285634	−	0.958339i	$-0.592204\pi$
$547$	−2.18673e9	−0.571267	−0.285634	+	0.958339i	$0.592204\pi$
$548$	0	0
$549$	2.19236e9	0.565468
$550$	0	0
$551$	−4.97817e8	−0.126777
$552$	0	0
$553$	−4.93720e9	−1.24149
$554$	0	0
$555$	−1.99595e9	−0.495593
$556$	0	0
$557$	4.20230e9	1.03037	0.515186	−	0.857078i	$-0.327723\pi$
$557$	4.20230e9	1.03037	0.515186	+	0.857078i	$0.327723\pi$
$558$	0	0
$559$	7.90170e9	1.91328
$560$	0	0
$561$	4.77614e8	0.114211
$562$	0	0
$563$	−9.23236e8	−0.218039	−0.109019	−	0.994040i	$-0.534771\pi$
$563$	−9.23236e8	−0.218039	−0.109019	+	0.994040i	$0.534771\pi$
$564$	0	0
$565$	−4.37754e9	−1.02108
$566$	0	0
$567$	−7.23942e8	−0.166787
$568$	0	0
$569$	−4.79957e8	−0.109222	−0.0546109	−	0.998508i	$-0.517392\pi$
$569$	−4.79957e8	−0.109222	−0.0546109	+	0.998508i	$0.517392\pi$
$570$	0	0
$571$	−3.38377e9	−0.760632	−0.380316	−	0.924857i	$-0.624185\pi$
$571$	−3.38377e9	−0.760632	−0.380316	+	0.924857i	$0.624185\pi$
$572$	0	0
$573$	1.43482e9	0.318609
$574$	0	0
$575$	−6.09015e7	−0.0133595
$576$	0	0
$577$	−3.30613e9	−0.716481	−0.358241	−	0.933629i	$-0.616623\pi$
$577$	−3.30613e9	−0.716481	−0.358241	+	0.933629i	$0.616623\pi$
$578$	0	0
$579$	−3.71646e9	−0.795710
$580$	0	0
$581$	−8.10431e9	−1.71435
$582$	0	0
$583$	3.89123e9	0.813293
$584$	0	0
$585$	3.05726e9	0.631374
$586$	0	0
$587$	7.73199e9	1.57782	0.788910	−	0.614508i	$-0.210646\pi$
$587$	7.73199e9	1.57782	0.788910	+	0.614508i	$0.210646\pi$
$588$	0	0
$589$	−2.29167e8	−0.0462114
$590$	0	0
$591$	−1.76976e9	−0.352662
$592$	0	0
$593$	7.78755e9	1.53359	0.766795	−	0.641891i	$-0.221850\pi$
$593$	7.78755e9	1.53359	0.766795	+	0.641891i	$0.221850\pi$
$594$	0	0
$595$	−1.78327e9	−0.347063
$596$	0	0
$597$	9.86351e8	0.189724
$598$	0	0
$599$	2.28551e8	0.0434500	0.0217250	−	0.999764i	$-0.493084\pi$
$599$	2.28551e8	0.0434500	0.0217250	+	0.999764i	$0.493084\pi$
$600$	0	0
$601$	−6.38900e9	−1.20053	−0.600264	−	0.799802i	$-0.704938\pi$
$601$	−6.38900e9	−1.20053	−0.600264	+	0.799802i	$0.704938\pi$
$602$	0	0
$603$	1.83675e9	0.341145
$604$	0	0
$605$	1.39419e9	0.255964
$606$	0	0
$607$	−5.84357e9	−1.06052	−0.530259	−	0.847836i	$-0.677905\pi$
$607$	−5.84357e9	−1.06052	−0.530259	+	0.847836i	$0.677905\pi$
$608$	0	0
$609$	5.68960e9	1.02075
$610$	0	0
$611$	1.33087e10	2.36043
$612$	0	0
$613$	−6.34845e9	−1.11316	−0.556578	−	0.830796i	$-0.687886\pi$
$613$	−6.34845e9	−1.11316	−0.556578	+	0.830796i	$0.687886\pi$
$614$	0	0
$615$	−2.48205e9	−0.430277
$616$	0	0
$617$	3.22881e9	0.553406	0.276703	−	0.960955i	$-0.410758\pi$
$617$	3.22881e9	0.553406	0.276703	+	0.960955i	$0.410758\pi$
$618$	0	0
$619$	3.22164e9	0.545959	0.272980	−	0.962020i	$-0.411991\pi$
$619$	3.22164e9	0.545959	0.272980	+	0.962020i	$0.411991\pi$
$620$	0	0
$621$	−7.74471e8	−0.129773
$622$	0	0
$623$	−1.65583e9	−0.274351
$624$	0	0
$625$	−6.22204e9	−1.01942
$626$	0	0
$627$	−3.31409e8	−0.0536943
$628$	0	0
$629$	−1.21463e9	−0.194611
$630$	0	0
$631$	4.72340e9	0.748431	0.374216	−	0.927342i	$-0.377912\pi$
$631$	4.72340e9	0.748431	0.374216	+	0.927342i	$0.377912\pi$
$632$	0	0
$633$	3.72359e9	0.583510
$634$	0	0
$635$	−1.27637e10	−1.97819
$636$	0	0
$637$	−1.53348e10	−2.35066
$638$	0	0
$639$	3.02701e9	0.458945
$640$	0	0
$641$	−3.27899e9	−0.491742	−0.245871	−	0.969303i	$-0.579074\pi$
$641$	−3.27899e9	−0.491742	−0.245871	+	0.969303i	$0.579074\pi$
$642$	0	0
$643$	7.67620e9	1.13870	0.569348	−	0.822097i	$-0.307196\pi$
$643$	7.67620e9	1.13870	0.569348	+	0.822097i	$0.307196\pi$
$644$	0	0
$645$	4.05312e9	0.594744
$646$	0	0
$647$	1.62477e9	0.235845	0.117923	−	0.993023i	$-0.462377\pi$
$647$	1.62477e9	0.235845	0.117923	+	0.993023i	$0.462377\pi$
$648$	0	0
$649$	7.23727e9	1.03925
$650$	0	0
$651$	2.61917e9	0.372075
$652$	0	0
$653$	−1.03058e10	−1.44838	−0.724192	−	0.689598i	$-0.757787\pi$
$653$	−1.03058e10	−1.44838	−0.724192	+	0.689598i	$0.757787\pi$
$654$	0	0
$655$	4.07104e9	0.566058
$656$	0	0
$657$	−3.17369e9	−0.436602
$658$	0	0
$659$	2.92364e9	0.397947	0.198973	−	0.980005i	$-0.436239\pi$
$659$	2.92364e9	0.397947	0.198973	+	0.980005i	$0.436239\pi$
$660$	0	0
$661$	−5.43749e9	−0.732307	−0.366154	−	0.930554i	$-0.619325\pi$
$661$	−5.43749e9	−0.732307	−0.366154	+	0.930554i	$0.619325\pi$
$662$	0	0
$663$	1.86049e9	0.247931
$664$	0	0
$665$	1.23739e9	0.163166
$666$	0	0
$667$	6.08671e9	0.794222
$668$	0	0
$669$	1.86155e9	0.240372
$670$	0	0
$671$	−1.14705e10	−1.46573
$672$	0	0
$673$	−3.90783e9	−0.494177	−0.247089	−	0.968993i	$-0.579474\pi$
$673$	−3.90783e9	−0.494177	−0.247089	+	0.968993i	$0.579474\pi$
$674$	0	0
$675$	3.04653e7	0.00381278
$676$	0	0
$677$	7.18167e9	0.889540	0.444770	−	0.895645i	$-0.353285\pi$
$677$	7.18167e9	0.889540	0.444770	+	0.895645i	$0.353285\pi$
$678$	0	0
$679$	1.41823e10	1.73862
$680$	0	0
$681$	3.82115e9	0.463638
$682$	0	0
$683$	1.25450e10	1.50660	0.753302	−	0.657674i	$-0.228460\pi$
$683$	1.25450e10	1.50660	0.753302	+	0.657674i	$0.228460\pi$
$684$	0	0
$685$	−7.42540e9	−0.882679
$686$	0	0
$687$	−3.72843e9	−0.438710
$688$	0	0
$689$	1.51578e10	1.76551
$690$	0	0
$691$	−1.50101e10	−1.73065	−0.865327	−	0.501208i	$-0.832889\pi$
$691$	−1.50101e10	−1.73065	−0.865327	+	0.501208i	$0.832889\pi$
$692$	0	0
$693$	3.78770e9	0.432324
$694$	0	0
$695$	5.07466e9	0.573403
$696$	0	0
$697$	−1.51045e9	−0.168963
$698$	0	0
$699$	1.00363e10	1.11149
$700$	0	0
$701$	−1.24579e10	−1.36594	−0.682968	−	0.730448i	$-0.739311\pi$
$701$	−1.24579e10	−1.36594	−0.682968	+	0.730448i	$0.739311\pi$
$702$	0	0
$703$	8.42816e8	0.0914933
$704$	0	0
$705$	6.82659e9	0.733739
$706$	0	0
$707$	1.02477e10	1.09059
$708$	0	0
$709$	6.91477e9	0.728645	0.364323	−	0.931273i	$-0.381301\pi$
$709$	6.91477e9	0.728645	0.364323	+	0.931273i	$0.381301\pi$
$710$	0	0
$711$	2.64216e9	0.275686
$712$	0	0
$713$	2.80198e9	0.289502
$714$	0	0
$715$	−1.59957e10	−1.63656
$716$	0	0
$717$	−1.19026e9	−0.120594
$718$	0	0
$719$	9.16082e9	0.919143	0.459572	−	0.888141i	$-0.348003\pi$
$719$	9.16082e9	0.919143	0.459572	+	0.888141i	$0.348003\pi$
$720$	0	0
$721$	1.72543e10	1.71444
$722$	0	0
$723$	4.39486e9	0.432475
$724$	0	0
$725$	−2.39432e8	−0.0233346
$726$	0	0
$727$	−1.28357e9	−0.123894	−0.0619469	−	0.998079i	$-0.519731\pi$
$727$	−1.28357e9	−0.123894	−0.0619469	+	0.998079i	$0.519731\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	2.46652e9	0.233547
$732$	0	0
$733$	−1.81802e10	−1.70505	−0.852523	−	0.522690i	$-0.824928\pi$
$733$	−1.81802e10	−1.70505	−0.852523	+	0.522690i	$0.824928\pi$
$734$	0	0
$735$	−7.86587e9	−0.730703
$736$	0	0
$737$	−9.60995e9	−0.884270
$738$	0	0
$739$	−1.74466e10	−1.59021	−0.795107	−	0.606469i	$-0.792586\pi$
$739$	−1.74466e10	−1.59021	−0.795107	+	0.606469i	$0.792586\pi$
$740$	0	0
$741$	−1.29097e9	−0.116560
$742$	0	0
$743$	−8.52480e9	−0.762471	−0.381236	−	0.924478i	$-0.624501\pi$
$743$	−8.52480e9	−0.762471	−0.381236	+	0.924478i	$0.624501\pi$
$744$	0	0
$745$	1.45680e10	1.29079
$746$	0	0
$747$	4.33705e9	0.380691
$748$	0	0
$749$	2.76948e10	2.40831
$750$	0	0
$751$	1.26184e10	1.08708	0.543542	−	0.839382i	$-0.317083\pi$
$751$	1.26184e10	1.08708	0.543542	+	0.839382i	$0.317083\pi$
$752$	0	0
$753$	−4.80265e9	−0.409920
$754$	0	0
$755$	−1.88341e9	−0.159269
$756$	0	0
$757$	1.38035e10	1.15652	0.578259	−	0.815854i	$-0.303732\pi$
$757$	1.38035e10	1.15652	0.578259	+	0.815854i	$0.303732\pi$
$758$	0	0
$759$	4.05207e9	0.336380
$760$	0	0
$761$	1.26732e10	1.04241	0.521205	−	0.853432i	$-0.325483\pi$
$761$	1.26732e10	1.04241	0.521205	+	0.853432i	$0.325483\pi$
$762$	0	0
$763$	3.14817e10	2.56580
$764$	0	0
$765$	9.54326e8	0.0770693
$766$	0	0
$767$	2.81920e10	2.25601
$768$	0	0
$769$	1.76471e10	1.39936	0.699681	−	0.714455i	$-0.253325\pi$
$769$	1.76471e10	1.39936	0.699681	+	0.714455i	$0.253325\pi$
$770$	0	0
$771$	9.45347e9	0.742850
$772$	0	0
$773$	2.20323e10	1.71566	0.857829	−	0.513935i	$-0.171813\pi$
$773$	2.20323e10	1.71566	0.857829	+	0.513935i	$0.171813\pi$
$774$	0	0
$775$	−1.10221e8	−0.00850569
$776$	0	0
$777$	−9.63261e9	−0.736666
$778$	0	0
$779$	1.04808e9	0.0794351
$780$	0	0
$781$	−1.58375e10	−1.18962
$782$	0	0
$783$	−3.04481e9	−0.226670
$784$	0	0
$785$	1.51063e10	1.11459
$786$	0	0
$787$	3.57574e9	0.261489	0.130745	−	0.991416i	$-0.458263\pi$
$787$	3.57574e9	0.261489	0.130745	+	0.991416i	$0.458263\pi$
$788$	0	0
$789$	−8.33351e9	−0.604030
$790$	0	0
$791$	−2.11263e10	−1.51777
$792$	0	0
$793$	−4.46821e10	−3.18183
$794$	0	0
$795$	7.77511e9	0.548809
$796$	0	0
$797$	−2.23299e9	−0.156237	−0.0781183	−	0.996944i	$-0.524891\pi$
$797$	−2.23299e9	−0.156237	−0.0781183	+	0.996944i	$0.524891\pi$
$798$	0	0
$799$	4.15431e9	0.288128
$800$	0	0
$801$	8.86123e8	0.0609228
$802$	0	0
$803$	1.66049e10	1.13170
$804$	0	0
$805$	−1.51293e10	−1.02219
$806$	0	0
$807$	5.69520e9	0.381462
$808$	0	0
$809$	−1.89092e10	−1.25560	−0.627802	−	0.778373i	$-0.716045\pi$
$809$	−1.89092e10	−1.25560	−0.627802	+	0.778373i	$0.716045\pi$
$810$	0	0
$811$	−9.97440e9	−0.656620	−0.328310	−	0.944570i	$-0.606479\pi$
$811$	−9.97440e9	−0.656620	−0.328310	+	0.944570i	$0.606479\pi$
$812$	0	0
$813$	1.48144e10	0.966870
$814$	0	0
$815$	9.86891e9	0.638583
$816$	0	0
$817$	−1.71148e9	−0.109798
$818$	0	0
$819$	1.47546e10	0.938496
$820$	0	0
$821$	2.26783e10	1.43024	0.715120	−	0.699002i	$-0.246372\pi$
$821$	2.26783e10	1.43024	0.715120	+	0.699002i	$0.246372\pi$
$822$	0	0
$823$	3.64946e9	0.228207	0.114103	−	0.993469i	$-0.463600\pi$
$823$	3.64946e9	0.228207	0.114103	+	0.993469i	$0.463600\pi$
$824$	0	0
$825$	−1.59396e8	−0.00988299
$826$	0	0
$827$	1.51192e9	0.0929521	0.0464761	−	0.998919i	$-0.485201\pi$
$827$	1.51192e9	0.0929521	0.0464761	+	0.998919i	$0.485201\pi$
$828$	0	0
$829$	1.26210e10	0.769398	0.384699	−	0.923042i	$-0.374305\pi$
$829$	1.26210e10	0.769398	0.384699	+	0.923042i	$0.374305\pi$
$830$	0	0
$831$	−1.35403e10	−0.818514
$832$	0	0
$833$	−4.78676e9	−0.286936
$834$	0	0
$835$	−1.06956e10	−0.635776
$836$	0	0
$837$	−1.40166e9	−0.0826235
$838$	0	0
$839$	1.67517e10	0.979244	0.489622	−	0.871935i	$-0.337135\pi$
$839$	1.67517e10	0.979244	0.489622	+	0.871935i	$0.337135\pi$
$840$	0	0
$841$	6.67984e9	0.387240
$842$	0	0
$843$	−1.38158e10	−0.794289
$844$	0	0
$845$	−4.45979e10	−2.54282
$846$	0	0
$847$	6.72848e9	0.380474
$848$	0	0
$849$	6.56315e9	0.368074
$850$	0	0
$851$	−1.03049e10	−0.573181
$852$	0	0
$853$	−3.42742e10	−1.89080	−0.945401	−	0.325909i	$-0.894330\pi$
$853$	−3.42742e10	−1.89080	−0.945401	+	0.325909i	$0.894330\pi$
$854$	0	0
$855$	−6.62192e8	−0.0362329
$856$	0	0
$857$	6.94990e9	0.377177	0.188589	−	0.982056i	$-0.439609\pi$
$857$	6.94990e9	0.377177	0.188589	+	0.982056i	$0.439609\pi$
$858$	0	0
$859$	−2.55440e9	−0.137503	−0.0687516	−	0.997634i	$-0.521902\pi$
$859$	−2.55440e9	−0.137503	−0.0687516	+	0.997634i	$0.521902\pi$
$860$	0	0
$861$	−1.19786e10	−0.639579
$862$	0	0
$863$	−6.10627e9	−0.323399	−0.161699	−	0.986840i	$-0.551697\pi$
$863$	−6.10627e9	−0.323399	−0.161699	+	0.986840i	$0.551697\pi$
$864$	0	0
$865$	6.78235e9	0.356306
$866$	0	0
$867$	−1.04984e10	−0.547086
$868$	0	0
$869$	−1.38239e10	−0.714598
$870$	0	0
$871$	−3.74345e10	−1.91959
$872$	0	0
$873$	−7.58973e9	−0.386079
$874$	0	0
$875$	−2.94444e10	−1.48585
$876$	0	0
$877$	−1.66173e9	−0.0831882	−0.0415941	−	0.999135i	$-0.513244\pi$
$877$	−1.66173e9	−0.0831882	−0.0415941	+	0.999135i	$0.513244\pi$
$878$	0	0
$879$	−1.24982e10	−0.620705
$880$	0	0
$881$	1.86590e10	0.919332	0.459666	−	0.888092i	$-0.347969\pi$
$881$	1.86590e10	0.919332	0.459666	+	0.888092i	$0.347969\pi$
$882$	0	0
$883$	8.45215e8	0.0413147	0.0206573	−	0.999787i	$-0.493424\pi$
$883$	8.45215e8	0.0413147	0.0206573	+	0.999787i	$0.493424\pi$
$884$	0	0
$885$	1.44609e10	0.701282
$886$	0	0
$887$	−3.73395e10	−1.79654	−0.898269	−	0.439447i	$-0.855175\pi$
$887$	−3.73395e10	−1.79654	−0.898269	+	0.439447i	$0.855175\pi$
$888$	0	0
$889$	−6.15984e10	−2.94044
$890$	0	0
$891$	−2.02700e9	−0.0960025
$892$	0	0
$893$	−2.88261e9	−0.135458
$894$	0	0
$895$	−3.64618e10	−1.70003
$896$	0	0
$897$	1.57844e10	0.730220
$898$	0	0
$899$	1.10159e10	0.505663
$900$	0	0
$901$	4.73153e9	0.215509
$902$	0	0
$903$	1.95606e10	0.884048
$904$	0	0
$905$	2.04034e10	0.915024
$906$	0	0
$907$	1.22684e10	0.545960	0.272980	−	0.962020i	$-0.411991\pi$
$907$	1.22684e10	0.545960	0.272980	+	0.962020i	$0.411991\pi$
$908$	0	0
$909$	−5.48412e9	−0.242177
$910$	0	0
$911$	3.21931e10	1.41074	0.705372	−	0.708837i	$-0.250780\pi$
$911$	3.21931e10	1.41074	0.705372	+	0.708837i	$0.250780\pi$
$912$	0	0
$913$	−2.26917e10	−0.986776
$914$	0	0
$915$	−2.29194e10	−0.989074
$916$	0	0
$917$	1.96471e10	0.841407
$918$	0	0
$919$	3.10394e10	1.31919	0.659597	−	0.751620i	$-0.270727\pi$
$919$	3.10394e10	1.31919	0.659597	+	0.751620i	$0.270727\pi$
$920$	0	0
$921$	−1.03157e10	−0.435102
$922$	0	0
$923$	−6.16931e10	−2.58244
$924$	0	0
$925$	4.05364e8	0.0168403
$926$	0	0
$927$	−9.23369e9	−0.380712
$928$	0	0
$929$	7.41429e9	0.303399	0.151700	−	0.988427i	$-0.451525\pi$
$929$	7.41429e9	0.303399	0.151700	+	0.988427i	$0.451525\pi$
$930$	0	0
$931$	3.32146e9	0.134898
$932$	0	0
$933$	1.97439e9	0.0795880
$934$	0	0
$935$	−4.99308e9	−0.199769
$936$	0	0
$937$	6.86618e9	0.272663	0.136332	−	0.990663i	$-0.456469\pi$
$937$	6.86618e9	0.272663	0.136332	+	0.990663i	$0.456469\pi$
$938$	0	0
$939$	−1.02341e10	−0.403387
$940$	0	0
$941$	−7.55613e9	−0.295621	−0.147811	−	0.989016i	$-0.547223\pi$
$941$	−7.55613e9	−0.295621	−0.147811	+	0.989016i	$0.547223\pi$
$942$	0	0
$943$	−1.28146e10	−0.497640
$944$	0	0
$945$	7.56825e9	0.291732
$946$	0	0
$947$	2.18917e10	0.837636	0.418818	−	0.908070i	$-0.362444\pi$
$947$	2.18917e10	0.837636	0.418818	+	0.908070i	$0.362444\pi$
$948$	0	0
$949$	6.46826e10	2.45672
$950$	0	0
$951$	−1.62873e10	−0.614069
$952$	0	0
$953$	−1.91574e10	−0.716986	−0.358493	−	0.933532i	$-0.616709\pi$
$953$	−1.91574e10	−0.716986	−0.358493	+	0.933532i	$0.616709\pi$
$954$	0	0
$955$	−1.50000e10	−0.557286
$956$	0	0
$957$	1.59306e10	0.587543
$958$	0	0
$959$	−3.58355e10	−1.31204
$960$	0	0
$961$	−2.24415e10	−0.815681
$962$	0	0
$963$	−1.48210e10	−0.534792
$964$	0	0
$965$	3.88527e10	1.39180
$966$	0	0
$967$	4.49432e10	1.59835	0.799174	−	0.601099i	$-0.205270\pi$
$967$	4.49432e10	1.59835	0.799174	+	0.601099i	$0.205270\pi$
$968$	0	0
$969$	−4.02976e8	−0.0142281
$970$	0	0
$971$	−1.58327e10	−0.554994	−0.277497	−	0.960726i	$-0.589505\pi$
$971$	−1.58327e10	−0.554994	−0.277497	+	0.960726i	$0.589505\pi$
$972$	0	0
$973$	2.44907e10	0.852327
$974$	0	0
$975$	−6.20909e8	−0.0214542
$976$	0	0
$977$	−2.34667e10	−0.805048	−0.402524	−	0.915409i	$-0.631867\pi$
$977$	−2.34667e10	−0.805048	−0.402524	+	0.915409i	$0.631867\pi$
$978$	0	0
$979$	−4.63624e9	−0.157916
$980$	0	0
$981$	−1.68476e10	−0.569765
$982$	0	0
$983$	1.54097e10	0.517438	0.258719	−	0.965953i	$-0.416700\pi$
$983$	1.54097e10	0.517438	0.258719	+	0.965953i	$0.416700\pi$
$984$	0	0
$985$	1.85015e10	0.616849
$986$	0	0
$987$	3.29456e10	1.09066
$988$	0	0
$989$	2.09259e10	0.687856
$990$	0	0
$991$	2.37648e10	0.775669	0.387835	−	0.921729i	$-0.373223\pi$
$991$	2.37648e10	0.775669	0.387835	+	0.921729i	$0.373223\pi$
$992$	0	0
$993$	8.69361e9	0.281759
$994$	0	0
$995$	−1.03115e10	−0.331850
$996$	0	0
$997$	3.48653e10	1.11419	0.557096	−	0.830448i	$-0.311916\pi$
$997$	3.48653e10	1.11419	0.557096	+	0.830448i	$0.311916\pi$
$998$	0	0
$999$	5.15493e9	0.163585

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.a.q.1.2	yes	4
4.3	odd	2		384.8.a.m.1.2	✓	4
8.3	odd	2		384.8.a.t.1.3	yes	4
8.5	even	2		384.8.a.p.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.a.m.1.2	✓	4	4.3	odd	2
384.8.a.p.1.3	yes	4	8.5	even	2
384.8.a.q.1.2	yes	4	1.1	even	1	trivial
384.8.a.t.1.3	yes	4	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q+27.0000 q^{3} -282.264 q^{5} -1362.23 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q+27.0000 q^{3} -282.264 q^{5} -1362.23 q^{7} +729.000 q^{9} -3814.16 q^{11} -14857.6 q^{13} -7621.12 q^{15} -4637.82 q^{17} +3218.11 q^{19} -36780.1 q^{21} -39347.2 q^{23} +1547.80 q^{25} +19683.0 q^{27} -154692. q^{29} -71211.7 q^{31} -102982. q^{33} +384507. q^{35} +261898. q^{37} -401156. q^{39} +325681. q^{41} -531827. q^{43} -205770. q^{45} -895746. q^{47} +1.03211e6 q^{49} -125221. q^{51} -1.02021e6 q^{53} +1.07660e6 q^{55} +86889.1 q^{57} -1.89747e6 q^{59} +3.00735e6 q^{61} -993062. q^{63} +4.19377e6 q^{65} +2.51954e6 q^{67} -1.06237e6 q^{69} +4.15228e6 q^{71} -4.35349e6 q^{73} +41790.5 q^{75} +5.19575e6 q^{77} +3.62436e6 q^{79} +531441. q^{81} +5.94931e6 q^{83} +1.30909e6 q^{85} -4.17669e6 q^{87} +1.21553e6 q^{89} +2.02394e7 q^{91} -1.92272e6 q^{93} -908357. q^{95} -1.04112e7 q^{97} -2.78053e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{3} - 336 q^{5} + 680 q^{7} + 2916 q^{9}+O(q^{10}) \) 4 * q + 108 * q^3 - 336 * q^5 + 680 * q^7 + 2916 * q^9	\( 4 q + 108 q^{3} - 336 q^{5} + 680 q^{7} + 2916 q^{9} + 3856 q^{11} - 10680 q^{13} - 9072 q^{15} + 26232 q^{17} - 15456 q^{19} + 18360 q^{21} - 11312 q^{23} + 159052 q^{25} + 78732 q^{27} + 1856 q^{29} + 71752 q^{31} + 104112 q^{33} - 179040 q^{35} + 180088 q^{37} - 288360 q^{39} + 11224 q^{41} + 66688 q^{43} - 244944 q^{45} - 1334448 q^{47} + 2401140 q^{49} + 708264 q^{51} + 864576 q^{53} - 3304896 q^{55} - 417312 q^{57} + 1878448 q^{59} + 1901176 q^{61} + 495720 q^{63} + 4366944 q^{65} + 5505488 q^{67} - 305424 q^{69} - 967696 q^{71} + 3244760 q^{73} + 4294404 q^{75} + 8979488 q^{77} + 6471816 q^{79} + 2125764 q^{81} + 17019600 q^{83} - 12122592 q^{85} + 50112 q^{87} + 13559816 q^{89} + 6692304 q^{91} + 1937304 q^{93} + 22523904 q^{95} + 2180520 q^{97} + 2811024 q^{99}+O(q^{100}) \) 4 * q + 108 * q^3 - 336 * q^5 + 680 * q^7 + 2916 * q^9 + 3856 * q^11 - 10680 * q^13 - 9072 * q^15 + 26232 * q^17 - 15456 * q^19 + 18360 * q^21 - 11312 * q^23 + 159052 * q^25 + 78732 * q^27 + 1856 * q^29 + 71752 * q^31 + 104112 * q^33 - 179040 * q^35 + 180088 * q^37 - 288360 * q^39 + 11224 * q^41 + 66688 * q^43 - 244944 * q^45 - 1334448 * q^47 + 2401140 * q^49 + 708264 * q^51 + 864576 * q^53 - 3304896 * q^55 - 417312 * q^57 + 1878448 * q^59 + 1901176 * q^61 + 495720 * q^63 + 4366944 * q^65 + 5505488 * q^67 - 305424 * q^69 - 967696 * q^71 + 3244760 * q^73 + 4294404 * q^75 + 8979488 * q^77 + 6471816 * q^79 + 2125764 * q^81 + 17019600 * q^83 - 12122592 * q^85 + 50112 * q^87 + 13559816 * q^89 + 6692304 * q^91 + 1937304 * q^93 + 22523904 * q^95 + 2180520 * q^97 + 2811024 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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