LMFDB - Embedded newform 384.6.c.b.383.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,6,Mod(383,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("384.383");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	384.c (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:	$61.5873868082$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 306 x^{18} + 37827 x^{16} + 2442168 x^{14} + 88368509 x^{12} + 1774000974 x^{10} + \cdots + 2870280625 $ x^20 + 306x^18 + 37827x^16 + 2442168x^14 + 88368509x^12 + 1774000974x^10 + 18093172325x^8 + 74958811500x^6 + 79355888475x^4 + 28497132750*x^2 + 2870280625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{88}\cdot 3^{14}\cdot 41^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.9
Root			$-5.94193i$ of defining polynomial
Character	$\chi$	$=$	384.383
Dual form			384.6.c.b.383.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.25525 - 15.4245i) q^{3} +86.6842i q^{5} -110.468i q^{7} +(-232.828 + 69.5720i) q^{9} +O(q^{10})$	\(q+(-2.25525 - 15.4245i) q^{3} +86.6842i q^{5} -110.468i q^{7} +(-232.828 + 69.5720i) q^{9} +212.115 q^{11} -539.316 q^{13} +(1337.06 - 195.494i) q^{15} -24.1619i q^{17} +1930.54i q^{19} +(-1703.91 + 249.133i) q^{21} +275.110 q^{23} -4389.15 q^{25} +(1598.19 + 3434.34i) q^{27} -7337.17i q^{29} -7478.16i q^{31} +(-478.373 - 3271.76i) q^{33} +9575.83 q^{35} +7681.01 q^{37} +(1216.29 + 8318.65i) q^{39} +19394.9i q^{41} -4494.69i q^{43} +(-6030.79 - 20182.5i) q^{45} -3629.25 q^{47} +4603.83 q^{49} +(-372.684 + 54.4911i) q^{51} +3676.39i q^{53} +18387.1i q^{55} +(29777.5 - 4353.85i) q^{57} +23949.3 q^{59} +43340.3 q^{61} +(7685.47 + 25720.0i) q^{63} -46750.1i q^{65} -65528.4i q^{67} +(-620.442 - 4243.43i) q^{69} -30068.2 q^{71} -43260.2 q^{73} +(9898.63 + 67700.3i) q^{75} -23432.0i q^{77} -90539.6i q^{79} +(49368.5 - 32396.6i) q^{81} +109867. q^{83} +2094.46 q^{85} +(-113172. + 16547.1i) q^{87} -138798. i q^{89} +59577.1i q^{91} +(-115347. + 16865.1i) q^{93} -167347. q^{95} -4237.49 q^{97} +(-49386.3 + 14757.3i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{3}+O(q^{10}) $ 20 * q - 2 * q^3	$ 20 q - 2 q^{3} - 948 q^{11} + 852 q^{15} + 1640 q^{21} + 328 q^{23} - 12500 q^{25} - 2030 q^{27} + 2836 q^{33} + 7184 q^{35} + 15056 q^{37} - 12980 q^{39} + 11800 q^{45} + 36640 q^{47} - 33388 q^{49} - 1936 q^{51} + 15404 q^{57} - 62908 q^{59} + 73264 q^{61} + 23608 q^{63} - 84024 q^{69} + 34888 q^{71} + 52568 q^{73} - 115698 q^{75} + 55444 q^{81} + 225172 q^{83} - 30112 q^{85} - 225700 q^{87} - 148016 q^{93} + 418616 q^{95} + 7600 q^{97} - 378260 q^{99}+O(q^{100}) $ 20 * q - 2 * q^3 - 948 * q^11 + 852 * q^15 + 1640 * q^21 + 328 * q^23 - 12500 * q^25 - 2030 * q^27 + 2836 * q^33 + 7184 * q^35 + 15056 * q^37 - 12980 * q^39 + 11800 * q^45 + 36640 * q^47 - 33388 * q^49 - 1936 * q^51 + 15404 * q^57 - 62908 * q^59 + 73264 * q^61 + 23608 * q^63 - 84024 * q^69 + 34888 * q^71 + 52568 * q^73 - 115698 * q^75 + 55444 * q^81 + 225172 * q^83 - 30112 * q^85 - 225700 * q^87 - 148016 * q^93 + 418616 * q^95 + 7600 * q^97 - 378260 * q^99

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
$3$	−2.25525	−	15.4245i	−0.144674	−	0.989479i
$3$	−2.25525	−	15.4245i	−0.144674	−	0.989479i
$4$		0			0
$5$			86.6842i			1.55065i	0.631560	+	0.775327i	$0.282415\pi$
$5$			86.6842i			1.55065i	−0.631560	+	0.775327i	$0.717585\pi$
$6$		0			0
$7$		−	110.468i		−	0.852101i	−0.904699	−	0.426051i	$-0.859905\pi$
$7$		−	110.468i		−	0.852101i	0.904699	−	0.426051i	$-0.140095\pi$
$8$		0			0
$9$	−232.828	+	69.5720i	−0.958139	+	0.286304i
$10$		0			0
$11$	212.115			0.528555			0.264278	−	0.964447i	$-0.414866\pi$
$11$	212.115			0.528555			0.264278	+	0.964447i	$0.414866\pi$
$12$		0			0
$13$	−539.316			−0.885084			−0.442542	−	0.896748i	$-0.645923\pi$
$13$	−539.316			−0.885084			−0.442542	+	0.896748i	$0.645923\pi$
$14$		0			0
$15$	1337.06	−	195.494i	1.53434	−	0.224340i
$16$		0			0
$17$		−	24.1619i		−	0.0202773i	−0.999949	−	0.0101386i	$-0.996773\pi$
$17$		−	24.1619i		−	0.0202773i	0.999949	−	0.0101386i	$-0.00322728\pi$
$18$		0			0
$19$			1930.54i			1.22686i	0.789749	+	0.613430i	$0.210211\pi$
$19$			1930.54i			1.22686i	−0.789749	+	0.613430i	$0.789789\pi$
$20$		0			0
$21$	−1703.91	+	249.133i	−0.843137	+	0.123277i
$22$		0			0
$23$	275.110			0.108439			0.0542197	−	0.998529i	$-0.482733\pi$
$23$	275.110			0.108439			0.0542197	+	0.998529i	$0.482733\pi$
$24$		0			0
$25$	−4389.15			−1.40453
$26$		0			0
$27$	1598.19	+	3434.34i	0.421910	+	0.906638i
$28$		0			0
$29$		−	7337.17i		−	1.62007i	−0.586382	−	0.810034i	$-0.699448\pi$
$29$		−	7337.17i		−	1.62007i	0.586382	−	0.810034i	$-0.300552\pi$
$30$		0			0
$31$		−	7478.16i		−	1.39762i	−0.715305	−	0.698812i	$-0.753712\pi$
$31$		−	7478.16i		−	1.39762i	0.715305	−	0.698812i	$-0.246288\pi$
$32$		0			0
$33$	−478.373	−	3271.76i	−0.0764683	−	0.522995i
$34$		0			0
$35$	9575.83			1.32131
$36$		0			0
$37$	7681.01			0.922389			0.461195	−	0.887299i	$-0.347421\pi$
$37$	7681.01			0.922389			0.461195	+	0.887299i	$0.347421\pi$
$38$		0			0
$39$	1216.29	+	8318.65i	0.128049	+	0.875773i
$40$		0			0
$41$			19394.9i			1.80188i	0.433939	+	0.900942i	$0.357123\pi$
$41$			19394.9i			1.80188i	−0.433939	+	0.900942i	$0.642877\pi$
$42$		0			0
$43$		−	4494.69i		−	0.370705i	−0.982672	−	0.185352i	$-0.940657\pi$
$43$		−	4494.69i		−	0.370705i	0.982672	−	0.185352i	$-0.0593427\pi$
$44$		0			0
$45$	−6030.79	−	20182.5i	−0.443959	−	1.48574i
$46$		0			0
$47$	−3629.25			−0.239647			−0.119824	−	0.992795i	$-0.538233\pi$
$47$	−3629.25			−0.239647			−0.119824	+	0.992795i	$0.538233\pi$
$48$		0			0
$49$	4603.83			0.273923
$50$		0			0
$51$	−372.684	+	54.4911i	−0.0200639	+	0.00293360i
$52$		0			0
$53$			3676.39i			0.179776i	0.995952	+	0.0898880i	$0.0286509\pi$
$53$			3676.39i			0.179776i	−0.995952	+	0.0898880i	$0.971349\pi$
$54$		0			0
$55$			18387.1i			0.819606i
$56$		0			0
$57$	29777.5	−	4353.85i	1.21395	−	0.177495i
$58$		0			0
$59$	23949.3			0.895702			0.447851	−	0.894108i	$-0.352189\pi$
$59$	23949.3			0.895702			0.447851	+	0.894108i	$0.352189\pi$
$60$		0			0
$61$	43340.3			1.49131			0.745653	−	0.666334i	$-0.232137\pi$
$61$	43340.3			1.49131			0.745653	+	0.666334i	$0.232137\pi$
$62$		0			0
$63$	7685.47	+	25720.0i	0.243960	+	0.816431i
$64$		0			0
$65$		−	46750.1i		−	1.37246i
$66$		0			0
$67$		−	65528.4i		−	1.78337i	−0.452652	−	0.891687i	$-0.649522\pi$
$67$		−	65528.4i		−	1.78337i	0.452652	−	0.891687i	$-0.350478\pi$
$68$		0			0
$69$	−620.442	−	4243.43i	−0.0156884	−	0.107299i
$70$		0			0
$71$	−30068.2			−0.707882			−0.353941	−	0.935268i	$-0.615159\pi$
$71$	−30068.2			−0.707882			−0.353941	+	0.935268i	$0.615159\pi$
$72$		0			0
$73$	−43260.2			−0.950127			−0.475063	−	0.879952i	$-0.657575\pi$
$73$	−43260.2			−0.950127			−0.475063	+	0.879952i	$0.657575\pi$
$74$		0			0
$75$	9898.63	+	67700.3i	0.203199	+	1.38975i
$76$		0			0
$77$		−	23432.0i		−	0.450383i
$78$		0			0
$79$		−	90539.6i		−	1.63219i	−0.577917	−	0.816096i	$-0.696134\pi$
$79$		−	90539.6i		−	1.63219i	0.577917	−	0.816096i	$-0.303866\pi$
$80$		0			0
$81$	49368.5	−	32396.6i	0.836060	−	0.548639i
$82$		0			0
$83$	109867.			1.75053			0.875266	−	0.483641i	$-0.160686\pi$
$83$	109867.			1.75053			0.875266	+	0.483641i	$0.160686\pi$
$84$		0			0
$85$	2094.46			0.0314430
$86$		0			0
$87$	−113172.	+	16547.1i	−1.60302	+	0.234382i
$88$		0			0
$89$		−	138798.i		−	1.85741i	−0.370822	−	0.928704i	$-0.620924\pi$
$89$		−	138798.i		−	1.85741i	0.370822	−	0.928704i	$-0.379076\pi$
$90$		0			0
$91$			59577.1i			0.754181i
$92$		0			0
$93$	−115347.	+	16865.1i	−1.38292	+	0.202200i
$94$		0			0
$95$	−167347.			−1.90243
$96$		0			0
$97$	−4237.49			−0.0457277			−0.0228638	−	0.999739i	$-0.507278\pi$
$97$	−4237.49			−0.0457277			−0.0228638	+	0.999739i	$0.507278\pi$
$98$		0			0
$99$	−49386.3	+	14757.3i	−0.506429	+	0.151328i
$100$		0			0
$101$		−	162305.i		−	1.58317i	−0.611057	−	0.791587i	$-0.709255\pi$
$101$		−	162305.i		−	1.58317i	0.611057	−	0.791587i	$-0.290745\pi$
$102$		0			0
$103$		−	91064.8i		−	0.845780i	−0.906181	−	0.422890i	$-0.861016\pi$
$103$		−	91064.8i		−	0.845780i	0.906181	−	0.422890i	$-0.138984\pi$
$104$		0			0
$105$	−21595.9	−	147702.i	−0.191160	−	1.30741i
$106$		0			0
$107$	−156580.			−1.32214			−0.661068	−	0.750326i	$-0.729897\pi$
$107$	−156580.			−1.32214			−0.661068	+	0.750326i	$0.729897\pi$
$108$		0			0
$109$	−113832.			−0.917692			−0.458846	−	0.888516i	$-0.651737\pi$
$109$	−113832.			−0.917692			−0.458846	+	0.888516i	$0.651737\pi$
$110$		0			0
$111$	−17322.6	−	118475.i	−0.133446	−	0.912685i
$112$		0			0
$113$			9212.03i			0.0678671i	0.999424	+	0.0339336i	$0.0108035\pi$
$113$			9212.03i			0.0678671i	−0.999424	+	0.0339336i	$0.989197\pi$
$114$		0			0
$115$			23847.7i			0.168152i
$116$		0			0
$117$	125568.	−	37521.2i	0.848033	−	0.253403i
$118$		0			0
$119$	−2669.12			−0.0172783
$120$		0			0
$121$	−116058.			−0.720629
$122$		0			0
$123$	299155.	−	43740.2i	1.78293	−	0.260686i
$124$		0			0
$125$		−	109582.i		−	0.627284i
$126$		0			0
$127$			82080.7i			0.451577i	0.974176	+	0.225788i	$0.0724958\pi$
$127$			82080.7i			0.451577i	−0.974176	+	0.225788i	$0.927504\pi$
$128$		0			0
$129$	−69328.1	+	10136.6i	−0.366805	+	0.0536315i
$130$		0			0
$131$	192564.			0.980387			0.490193	−	0.871614i	$-0.336926\pi$
$131$	192564.			0.980387			0.490193	+	0.871614i	$0.336926\pi$
$132$		0			0
$133$	213263.			1.04541
$134$		0			0
$135$	−297703.	+	138538.i	−1.40588	+	0.654237i
$136$		0			0
$137$		−	262306.i		−	1.19401i	−0.802238	−	0.597004i	$-0.796358\pi$
$137$		−	262306.i		−	1.19401i	0.802238	−	0.597004i	$-0.203642\pi$
$138$		0			0
$139$		−	33737.6i		−	0.148108i	−0.997254	−	0.0740539i	$-0.976406\pi$
$139$		−	33737.6i		−	0.148108i	0.997254	−	0.0740539i	$-0.0235937\pi$
$140$		0			0
$141$	8184.86	+	55979.2i	0.0346708	+	0.237126i
$142$		0			0
$143$	−114397.			−0.467816
$144$		0			0
$145$	636017.			2.51217
$146$		0			0
$147$	−10382.8	−	71011.5i	−0.0396296	−	0.271041i
$148$		0			0
$149$		−	322123.i		−	1.18866i	−0.804223	−	0.594328i	$-0.797418\pi$
$149$		−	322123.i		−	1.18866i	0.804223	−	0.594328i	$-0.202582\pi$
$150$		0			0
$151$			190296.i			0.679183i	0.940573	+	0.339592i	$0.110289\pi$
$151$			190296.i			0.679183i	−0.940573	+	0.339592i	$0.889711\pi$
$152$		0			0
$153$	1680.99	+	5625.56i	0.00580547	+	0.0194284i
$154$		0			0
$155$	648238.			2.16723
$156$		0			0
$157$	−6464.74			−0.0209316			−0.0104658	−	0.999945i	$-0.503331\pi$
$157$	−6464.74			−0.0209316			−0.0104658	+	0.999945i	$0.503331\pi$
$158$		0			0
$159$	56706.3	−	8291.16i	0.177885	−	0.0260089i
$160$		0			0
$161$		−	30390.9i		−	0.0924014i
$162$		0			0
$163$			333250.i			0.982428i	0.871039	+	0.491214i	$0.163447\pi$
$163$			333250.i			0.982428i	−0.871039	+	0.491214i	$0.836553\pi$
$164$		0			0
$165$	283610.	−	41467.4i	0.810984	−	0.118576i
$166$		0			0
$167$	430857.			1.19548			0.597740	−	0.801690i	$-0.296066\pi$
$167$	430857.			1.19548			0.597740	+	0.801690i	$0.296066\pi$
$168$		0			0
$169$	−80431.7			−0.216626
$170$		0			0
$171$	−134311.	−	449483.i	−0.351255	−	1.17550i
$172$		0			0
$173$		−	720397.i		−	1.83002i	−0.403428	−	0.915011i	$-0.632181\pi$
$173$		−	720397.i		−	1.83002i	0.403428	−	0.915011i	$-0.367819\pi$
$174$		0			0
$175$			484861.i			1.19680i
$176$		0			0
$177$	−54011.7	−	369406.i	−0.129585	−	0.886279i
$178$		0			0
$179$	822982.			1.91981			0.959904	−	0.280329i	$-0.0904437\pi$
$179$	822982.			1.91981			0.959904	+	0.280329i	$0.0904437\pi$
$180$		0			0
$181$	86567.9			0.196409			0.0982043	−	0.995166i	$-0.468690\pi$
$181$	86567.9			0.196409			0.0982043	+	0.995166i	$0.468690\pi$
$182$		0			0
$183$	−97743.1	−	668500.i	−0.215754	−	1.47562i
$184$		0			0
$185$			665822.i			1.43031i
$186$		0			0
$187$		−	5125.11i		−	0.0107176i
$188$		0			0
$189$	379384.	−	176549.i	0.772547	−	0.359510i
$190$		0			0
$191$	−93397.9			−0.185248			−0.0926240	−	0.995701i	$-0.529525\pi$
$191$	−93397.9			−0.185248			−0.0926240	+	0.995701i	$0.529525\pi$
$192$		0			0
$193$	63345.4			0.122411			0.0612057	−	0.998125i	$-0.480505\pi$
$193$	63345.4			0.122411			0.0612057	+	0.998125i	$0.480505\pi$
$194$		0			0
$195$	−721096.	+	105433.i	−1.35802	+	0.198560i
$196$		0			0
$197$			603734.i			1.10836i	0.832397	+	0.554179i	$0.186968\pi$
$197$			603734.i			1.10836i	−0.832397	+	0.554179i	$0.813032\pi$
$198$		0			0
$199$		−	324247.i		−	0.580420i	−0.956963	−	0.290210i	$-0.906275\pi$
$199$		−	324247.i		−	0.580420i	0.956963	−	0.290210i	$-0.0937252\pi$
$200$		0			0
$201$	−1.01074e6	+	147783.i	−1.76461	+	0.258008i
$202$		0			0
$203$	−810522.			−1.38046
$204$		0			0
$205$	−1.68123e6			−2.79410
$206$		0			0
$207$	−64053.3	+	19140.0i	−0.103900	+	0.0310467i
$208$		0			0
$209$			409497.i			0.648463i
$210$		0			0
$211$			660510.i			1.02135i	0.859775	+	0.510673i	$0.170604\pi$
$211$			660510.i			1.02135i	−0.859775	+	0.510673i	$0.829396\pi$
$212$		0			0
$213$	67811.2	+	463785.i	0.102412	+	0.700435i
$214$		0			0
$215$	389618.			0.574835
$216$		0			0
$217$	−826097.			−1.19092
$218$		0			0
$219$	97562.5	+	667265.i	0.137459	+	0.940131i
$220$		0			0
$221$			13030.9i			0.0179471i
$222$		0			0
$223$			162536.i			0.218870i	0.993994	+	0.109435i	$0.0349042\pi$
$223$			162536.i			0.218870i	−0.993994	+	0.109435i	$0.965096\pi$
$224$		0			0
$225$	1.02192e6	−	305362.i	1.34573	−	0.402123i
$226$		0			0
$227$	−667969.			−0.860383			−0.430192	−	0.902738i	$-0.641554\pi$
$227$	−667969.			−0.860383			−0.430192	+	0.902738i	$0.641554\pi$
$228$		0			0
$229$	945355.			1.19126			0.595630	−	0.803259i	$-0.296903\pi$
$229$	945355.			1.19126			0.595630	+	0.803259i	$0.296903\pi$
$230$		0			0
$231$	−361425.	+	52844.9i	−0.445644	+	0.0651588i
$232$		0			0
$233$			666944.i			0.804821i	0.915459	+	0.402411i	$0.131828\pi$
$233$			666944.i			0.804821i	−0.915459	+	0.402411i	$0.868172\pi$
$234$		0			0
$235$		−	314599.i		−	0.371610i
$236$		0			0
$237$	−1.39652e6	+	204189.i	−1.61502	+	0.236136i
$238$		0			0
$239$	409633.			0.463874			0.231937	−	0.972731i	$-0.425494\pi$
$239$	409633.			0.463874			0.231937	+	0.972731i	$0.425494\pi$
$240$		0			0
$241$	427196.			0.473789			0.236894	−	0.971535i	$-0.423870\pi$
$241$	427196.			0.473789			0.236894	+	0.971535i	$0.423870\pi$
$242$		0			0
$243$	−611038.	−	688420.i	−0.663823	−	0.747890i
$244$		0			0
$245$			399079.i			0.424760i
$246$		0			0
$247$		−	1.04117e6i		−	1.08587i
$248$		0			0
$249$	−247776.	−	1.69463e6i	−0.253257	−	1.73212i
$250$		0			0
$251$	881998.			0.883656			0.441828	−	0.897100i	$-0.354330\pi$
$251$	881998.			0.883656			0.441828	+	0.897100i	$0.354330\pi$
$252$		0			0
$253$	58355.1			0.0573162
$254$		0			0
$255$	−4723.52	−	32305.9i	−0.00454899	−	0.0311122i
$256$		0			0
$257$			599689.i			0.566361i	0.959067	+	0.283181i	$0.0913896\pi$
$257$			599689.i			0.566361i	−0.959067	+	0.283181i	$0.908610\pi$
$258$		0			0
$259$		−	848506.i		−	0.785969i
$260$		0			0
$261$	510461.	+	1.70830e6i	0.463833	+	1.55225i
$262$		0			0
$263$	456966.			0.407375			0.203687	−	0.979036i	$-0.434707\pi$
$263$	456966.			0.407375			0.203687	+	0.979036i	$0.434707\pi$
$264$		0			0
$265$	−318685.			−0.278770
$266$		0			0
$267$	−2.14088e6	+	313023.i	−1.83787	+	0.268719i
$268$		0			0
$269$			1.10405e6i			0.930267i	0.885241	+	0.465134i	$0.153994\pi$
$269$			1.10405e6i			0.930267i	−0.885241	+	0.465134i	$0.846006\pi$
$270$		0			0
$271$		−	33195.9i		−	0.0274575i	−0.999906	−	0.0137287i	$-0.995630\pi$
$271$		−	33195.9i		−	0.0274575i	0.999906	−	0.0137287i	$-0.00437013\pi$
$272$		0			0
$273$	918944.	−	134361.i	0.746247	−	0.109111i
$274$		0			0
$275$	−931006.			−0.742371
$276$		0			0
$277$	1.31847e6			1.03246			0.516229	−	0.856451i	$-0.327336\pi$
$277$	1.31847e6			1.03246			0.516229	+	0.856451i	$0.327336\pi$
$278$		0			0
$279$	520270.	+	1.74112e6i	0.400146	+	1.33912i
$280$		0			0
$281$			401388.i			0.303248i	0.988438	+	0.151624i	$0.0484504\pi$
$281$			401388.i			0.303248i	−0.988438	+	0.151624i	$0.951550\pi$
$282$		0			0
$283$		−	2.13872e6i		−	1.58741i	−0.608305	−	0.793703i	$-0.708150\pi$
$283$		−	2.13872e6i		−	1.58741i	0.608305	−	0.793703i	$-0.291850\pi$
$284$		0			0
$285$	377410.	+	2.58124e6i	0.275233	+	1.88242i
$286$		0			0
$287$	2.14251e6			1.53539
$288$		0			0
$289$	1.41927e6			0.999589
$290$		0			0
$291$	9556.58	+	65360.9i	0.00661561	+	0.0452466i
$292$		0			0
$293$		−	1.22189e6i		−	0.831499i	−0.909479	−	0.415750i	$-0.863519\pi$
$293$		−	1.22189e6i		−	0.831499i	0.909479	−	0.415750i	$-0.136481\pi$
$294$		0			0
$295$			2.07603e6i			1.38892i
$296$		0			0
$297$	339001.	+	728476.i	0.223003	+	0.479208i
$298$		0			0
$299$	−148371.			−0.0959781
$300$		0			0
$301$	−496519.			−0.315878
$302$		0			0
$303$	−2.50347e6	+	366038.i	−1.56652	+	0.229044i
$304$		0			0
$305$			3.75692e6i			2.31250i
$306$		0			0
$307$			58956.8i			0.0357016i	0.999841	+	0.0178508i	$0.00568240\pi$
$307$			58956.8i			0.0357016i	−0.999841	+	0.0178508i	$0.994318\pi$
$308$		0			0
$309$	−1.40463e6	+	205374.i	−0.836882	+	0.122363i
$310$		0			0
$311$	1.02518e6			0.601034			0.300517	−	0.953776i	$-0.402841\pi$
$311$	1.02518e6			0.601034			0.300517	+	0.953776i	$0.402841\pi$
$312$		0			0
$313$	−1.63335e6			−0.942365			−0.471183	−	0.882036i	$-0.656173\pi$
$313$	−1.63335e6			−0.942365			−0.471183	+	0.882036i	$0.656173\pi$
$314$		0			0
$315$	−2.22952e6	+	666209.i	−1.26600	+	0.378298i
$316$		0			0
$317$			596082.i			0.333164i	0.986028	+	0.166582i	$0.0532731\pi$
$317$			596082.i			0.333164i	−0.986028	+	0.166582i	$0.946727\pi$
$318$		0			0
$319$		−	1.55633e6i		−	0.856296i
$320$		0			0
$321$	353126.	+	2.41516e6i	0.191279	+	1.30823i
$322$		0			0
$323$	46645.5			0.0248773
$324$		0			0
$325$	2.36714e6			1.24313
$326$		0			0
$327$	256719.	+	1.75579e6i	0.132766	+	0.908038i
$328$		0			0
$329$			400916.i			0.204204i
$330$		0			0
$331$			1.40214e6i			0.703428i	0.936107	+	0.351714i	$0.114401\pi$
$331$			1.40214e6i			0.703428i	−0.936107	+	0.351714i	$0.885599\pi$
$332$		0			0
$333$	−1.78835e6	+	534383.i	−0.883777	+	0.264084i
$334$		0			0
$335$	5.68028e6			2.76540
$336$		0			0
$337$	−2.84572e6			−1.36495			−0.682477	−	0.730907i	$-0.739097\pi$
$337$	−2.84572e6			−1.36495			−0.682477	+	0.730907i	$0.739097\pi$
$338$		0			0
$339$	142091.	−	20775.4i	0.0671531	−	0.00981862i
$340$		0			0
$341$		−	1.58623e6i		−	0.738722i
$342$		0			0
$343$		−	2.36521e6i		−	1.08551i
$344$		0			0
$345$	367838.	−	53782.5i	0.166383	−	0.0243273i
$346$		0			0
$347$	121815.			0.0543098			0.0271549	−	0.999631i	$-0.491355\pi$
$347$	121815.			0.0543098			0.0271549	+	0.999631i	$0.491355\pi$
$348$		0			0
$349$	−324699.			−0.142698			−0.0713488	−	0.997451i	$-0.522730\pi$
$349$	−324699.			−0.142698			−0.0713488	+	0.997451i	$0.522730\pi$
$350$		0			0
$351$	−861931.	−	1.85219e6i	−0.373426	−	0.802451i
$352$		0			0
$353$			244816.i			0.104569i	0.998632	+	0.0522846i	$0.0166503\pi$
$353$			244816.i			0.104569i	−0.998632	+	0.0522846i	$0.983350\pi$
$354$		0			0
$355$		−	2.60643e6i		−	1.09768i
$356$		0			0
$357$	6019.52	+	41169.7i	0.00249972	+	0.0170965i
$358$		0			0
$359$	793556.			0.324969			0.162484	−	0.986711i	$-0.448049\pi$
$359$	793556.			0.324969			0.162484	+	0.986711i	$0.448049\pi$
$360$		0			0
$361$	−1.25088e6			−0.505183
$362$		0			0
$363$	261740.	+	1.79013e6i	0.104257	+	0.713048i
$364$		0			0
$365$		−	3.74998e6i		−	1.47332i
$366$		0			0
$367$		−	785525.i		−	0.304435i	−0.988347	−	0.152218i	$-0.951359\pi$
$367$		−	785525.i		−	0.304435i	0.988347	−	0.152218i	$-0.0486415\pi$
$368$		0			0
$369$	−1.34934e6	−	4.51566e6i	−0.515887	−	1.72646i
$370$		0			0
$371$	406123.			0.153187
$372$		0			0
$373$	−1.97945e6			−0.736671			−0.368335	−	0.929693i	$-0.620072\pi$
$373$	−1.97945e6			−0.736671			−0.368335	+	0.929693i	$0.620072\pi$
$374$		0			0
$375$	−1.69024e6	+	247135.i	−0.620684	+	0.0907518i
$376$		0			0
$377$			3.95705e6i			1.43390i
$378$		0			0
$379$			3.50429e6i			1.25315i	0.779362	+	0.626574i	$0.215544\pi$
$379$			3.50429e6i			1.25315i	−0.779362	+	0.626574i	$0.784456\pi$
$380$		0			0
$381$	1.26605e6	−	185112.i	0.446826	−	0.0653315i
$382$		0			0
$383$	1.16489e6			0.405779			0.202889	−	0.979202i	$-0.434967\pi$
$383$	1.16489e6			0.405779			0.202889	+	0.979202i	$0.434967\pi$
$384$		0			0
$385$	2.03118e6			0.698388
$386$		0			0
$387$	312704.	+	1.04649e6i	0.106134	+	0.355187i
$388$		0			0
$389$		−	65857.4i		−	0.0220664i	−0.999939	−	0.0110332i	$-0.996488\pi$
$389$		−	65857.4i		−	0.0220664i	0.999939	−	0.0110332i	$-0.00351204\pi$
$390$		0			0
$391$		−	6647.19i		−	0.00219885i
$392$		0			0
$393$	−434280.	−	2.97020e6i	−0.141837	−	0.970073i
$394$		0			0
$395$	7.84836e6			2.53096
$396$		0			0
$397$	−4.01996e6			−1.28011			−0.640053	−	0.768331i	$-0.721087\pi$
$397$	−4.01996e6			−1.28011			−0.640053	+	0.768331i	$0.721087\pi$
$398$		0			0
$399$	−480960.	−	3.28946e6i	−0.151244	−	1.03441i
$400$		0			0
$401$			1.82280e6i			0.566081i	0.959108	+	0.283040i	$0.0913430\pi$
$401$			1.82280e6i			0.566081i	−0.959108	+	0.283040i	$0.908657\pi$
$402$		0			0
$403$			4.03309e6i			1.23702i
$404$		0			0
$405$	2.80827e6	+	4.27947e6i	0.850749	+	1.29644i
$406$		0			0
$407$	1.62926e6			0.487534
$408$		0			0
$409$	−1.23169e6			−0.364077			−0.182039	−	0.983291i	$-0.558270\pi$
$409$	−1.23169e6			−0.364077			−0.182039	+	0.983291i	$0.558270\pi$
$410$		0			0
$411$	−4.04593e6	+	591566.i	−1.18145	+	0.172742i
$412$		0			0
$413$		−	2.64564e6i		−	0.763229i
$414$		0			0
$415$			9.52369e6i			2.71447i
$416$		0			0
$417$	−520385.	+	76086.8i	−0.146550	+	0.0214274i
$418$		0			0
$419$	3.26757e6			0.909264			0.454632	−	0.890679i	$-0.349771\pi$
$419$	3.26757e6			0.909264			0.454632	+	0.890679i	$0.349771\pi$
$420$		0			0
$421$	−2.58190e6			−0.709961			−0.354981	−	0.934874i	$-0.615513\pi$
$421$	−2.58190e6			−0.709961			−0.354981	+	0.934874i	$0.615513\pi$
$422$		0			0
$423$	844990.	−	252494.i	0.229615	−	0.0686120i
$424$		0			0
$425$			106050.i			0.0284800i
$426$		0			0
$427$		−	4.78771e6i		−	1.27074i
$428$		0			0
$429$	257994.	+	1.76451e6i	0.0676809	+	0.462894i
$430$		0			0
$431$	−3.97814e6			−1.03154			−0.515771	−	0.856727i	$-0.672494\pi$
$431$	−3.97814e6			−1.03154			−0.515771	+	0.856727i	$0.672494\pi$
$432$		0			0
$433$	5.30560e6			1.35992			0.679962	−	0.733247i	$-0.261996\pi$
$433$	5.30560e6			1.35992			0.679962	+	0.733247i	$0.261996\pi$
$434$		0			0
$435$	−1.43438e6	−	9.81021e6i	−0.363446	−	2.48574i
$436$		0			0
$437$			531111.i			0.133040i
$438$		0			0
$439$		−	2.79565e6i		−	0.692344i	−0.938171	−	0.346172i	$-0.887481\pi$
$439$		−	2.79565e6i		−	0.692344i	0.938171	−	0.346172i	$-0.112519\pi$
$440$		0			0
$441$	−1.07190e6	+	320297.i	−0.262456	+	0.0784254i
$442$		0			0
$443$	−3.79176e6			−0.917976			−0.458988	−	0.888442i	$-0.651788\pi$
$443$	−3.79176e6			−0.917976			−0.458988	+	0.888442i	$0.651788\pi$
$444$		0			0
$445$	1.20316e7			2.88020
$446$		0			0
$447$	−4.96858e6	+	726468.i	−1.17615	+	0.171968i
$448$		0			0
$449$		−	6.90181e6i		−	1.61565i	−0.589422	−	0.807825i	$-0.700645\pi$
$449$		−	6.90181e6i		−	1.61565i	0.589422	−	0.807825i	$-0.299355\pi$
$450$		0			0
$451$			4.11395e6i			0.952396i
$452$		0			0
$453$	2.93521e6	−	429164.i	0.672038	−	0.0982603i
$454$		0			0
$455$	−5.16439e6			−1.16947
$456$		0			0
$457$	5.98001e6			1.33940			0.669702	−	0.742630i	$-0.266422\pi$
$457$	5.98001e6			1.33940			0.669702	+	0.742630i	$0.266422\pi$
$458$		0			0
$459$	82980.2	−	38615.4i	0.0183841	−	0.00855518i
$460$		0			0
$461$		−	5.66582e6i		−	1.24168i	−0.783936	−	0.620841i	$-0.786791\pi$
$461$		−	5.66582e6i		−	1.24168i	0.783936	−	0.620841i	$-0.213209\pi$
$462$		0			0
$463$		−	1.78722e6i		−	0.387458i	−0.981055	−	0.193729i	$-0.937942\pi$
$463$		−	1.78722e6i		−	0.387458i	0.981055	−	0.193729i	$-0.0620583\pi$
$464$		0			0
$465$	−1.46194e6	−	9.99872e6i	−0.313543	−	2.14443i
$466$		0			0
$467$	−654822.			−0.138941			−0.0694706	−	0.997584i	$-0.522131\pi$
$467$	−654822.			−0.138941			−0.0694706	+	0.997584i	$0.522131\pi$
$468$		0			0
$469$	−7.23879e6			−1.51962
$470$		0			0
$471$	14579.6	+	99715.0i	0.00302826	+	0.0207114i
$472$		0			0
$473$		−	953392.i		−	0.195938i
$474$		0			0
$475$		−	8.47343e6i		−	1.72316i
$476$		0			0
$477$	−255773.	−	855965.i	−0.0514706	−	0.172250i
$478$		0			0
$479$	−4.61770e6			−0.919575			−0.459788	−	0.888029i	$-0.652074\pi$
$479$	−4.61770e6			−0.919575			−0.459788	+	0.888029i	$0.652074\pi$
$480$		0			0
$481$	−4.14249e6			−0.816392
$482$		0			0
$483$	−468763.	+	68539.0i	−0.0914293	+	0.0133681i
$484$		0			0
$485$		−	367323.i		−	0.0709078i
$486$		0			0
$487$			2.36141e6i			0.451178i	0.974223	+	0.225589i	$0.0724307\pi$
$487$			2.36141e6i			0.451178i	−0.974223	+	0.225589i	$0.927569\pi$
$488$		0			0
$489$	5.14020e6	−	751561.i	0.972093	−	0.142132i
$490$		0			0
$491$	8.12401e6			1.52078			0.760390	−	0.649466i	$-0.225008\pi$
$491$	8.12401e6			1.52078			0.760390	+	0.649466i	$0.225008\pi$
$492$		0			0
$493$	−177280.			−0.0328505
$494$		0			0
$495$	−1.27922e6	−	4.28102e6i	−0.234657	−	0.785297i
$496$		0			0
$497$			3.32157e6i			0.603187i
$498$		0			0
$499$			6.53711e6i			1.17526i	0.809129	+	0.587631i	$0.199939\pi$
$499$			6.53711e6i			1.17526i	−0.809129	+	0.587631i	$0.800061\pi$
$500$		0			0
$501$	−971689.	−	6.64573e6i	−0.172955	−	1.18290i
$502$		0			0
$503$	−7.65689e6			−1.34937			−0.674687	−	0.738104i	$-0.735721\pi$
$503$	−7.65689e6			−1.34937			−0.674687	+	0.738104i	$0.735721\pi$
$504$		0			0
$505$	1.40693e7			2.45495
$506$		0			0
$507$	181393.	+	1.24061e6i	0.0313402	+	0.214347i
$508$		0			0
$509$			2.52236e6i			0.431532i	0.976445	+	0.215766i	$0.0692248\pi$
$509$			2.52236e6i			0.431532i	−0.976445	+	0.215766i	$0.930775\pi$
$510$		0			0
$511$			4.77887e6i			0.809604i
$512$		0			0
$513$	−6.63013e6	+	3.08538e6i	−1.11232	+	0.517624i
$514$		0			0
$515$	7.89388e6			1.31151
$516$		0			0
$517$	−769820.			−0.126667
$518$		0			0
$519$	−1.11117e7	+	1.62467e6i	−1.81077	+	0.264757i
$520$		0			0
$521$		−	4.05284e6i		−	0.654131i	−0.945002	−	0.327065i	$-0.893940\pi$
$521$		−	4.05284e6i		−	0.654131i	0.945002	−	0.327065i	$-0.106060\pi$
$522$		0			0
$523$			4.38894e6i			0.701625i	0.936446	+	0.350813i	$0.114095\pi$
$523$			4.38894e6i			0.701625i	−0.936446	+	0.350813i	$0.885905\pi$
$524$		0			0
$525$	7.47871e6	−	1.09348e6i	1.18421	−	0.173146i
$526$		0			0
$527$	−180687.			−0.0283400
$528$		0			0
$529$	−6.36066e6			−0.988241
$530$		0			0
$531$	−5.57607e6	+	1.66620e6i	−0.858207	+	0.256443i
$532$		0			0
$533$		−	1.04600e7i		−	1.59482i
$534$		0			0
$535$		−	1.35730e7i		−	2.05018i
$536$		0			0
$537$	−1.85603e6	−	1.26940e7i	−0.277747	−	1.89961i
$538$		0			0
$539$	976543.			0.144784
$540$		0			0
$541$	−5.53968e6			−0.813751			−0.406875	−	0.913484i	$-0.633382\pi$
$541$	−5.53968e6			−0.813751			−0.406875	+	0.913484i	$0.633382\pi$
$542$		0			0
$543$	−195232.	−	1.33526e6i	−0.0284153	−	0.194342i
$544$		0			0
$545$		−	9.86741e6i		−	1.42302i
$546$		0			0
$547$			1.15951e7i			1.65694i	0.560034	+	0.828470i	$0.310788\pi$
$547$			1.15951e7i			1.65694i	−0.560034	+	0.828470i	$0.689212\pi$
$548$		0			0
$549$	−1.00908e7	+	3.01527e6i	−1.42888	+	0.426968i
$550$		0			0
$551$	1.41647e7			1.98760
$552$		0			0
$553$	−1.00017e7			−1.39079
$554$		0			0
$555$	1.02699e7	−	1.50159e6i	1.41526	−	0.206929i
$556$		0			0
$557$			7.96541e6i			1.08785i	0.839133	+	0.543927i	$0.183063\pi$
$557$			7.96541e6i			1.08785i	−0.839133	+	0.543927i	$0.816937\pi$
$558$		0			0
$559$			2.42405e6i			0.328105i
$560$		0			0
$561$	−79052.1	+	11558.4i	−0.0106049	+	0.00155057i
$562$		0			0
$563$	5.14209e6			0.683705			0.341853	−	0.939754i	$-0.388946\pi$
$563$	5.14209e6			0.683705			0.341853	+	0.939754i	$0.388946\pi$
$564$		0			0
$565$	−798538.			−0.105238
$566$		0			0
$567$	−3.57878e6	−	5.45364e6i	−0.467496	−	0.712408i
$568$		0			0
$569$		−	3.04215e6i		−	0.393913i	−0.980412	−	0.196956i	$-0.936894\pi$
$569$		−	3.04215e6i		−	0.393913i	0.980412	−	0.196956i	$-0.0631057\pi$
$570$		0			0
$571$		−	9.13371e6i		−	1.17235i	−0.810185	−	0.586175i	$-0.800633\pi$
$571$		−	9.13371e6i		−	1.17235i	0.810185	−	0.586175i	$-0.199367\pi$
$572$		0			0
$573$	210635.	+	1.44061e6i	0.0268006	+	0.183299i
$574$		0			0
$575$	−1.20750e6			−0.152306
$576$		0			0
$577$	−9.31021e6			−1.16418			−0.582090	−	0.813125i	$-0.697765\pi$
$577$	−9.31021e6			−1.16418			−0.582090	+	0.813125i	$0.697765\pi$
$578$		0			0
$579$	−142860.	−	977069.i	−0.0177098	−	0.121124i
$580$		0			0
$581$		−	1.21367e7i		−	1.49163i
$582$		0			0
$583$			779818.i			0.0950215i
$584$		0			0
$585$	3.25250e6	+	1.08847e7i	0.392941	+	1.31501i
$586$		0			0
$587$	−5.65177e6			−0.677001			−0.338500	−	0.940966i	$-0.609920\pi$
$587$	−5.65177e6			−0.677001			−0.338500	+	0.940966i	$0.609920\pi$
$588$		0			0
$589$	1.44369e7			1.71469
$590$		0			0
$591$	9.31227e6	−	1.36157e6i	1.09670	−	0.160351i
$592$		0			0
$593$			1.25646e7i			1.46728i	0.679541	+	0.733638i	$0.262179\pi$
$593$			1.25646e7i			1.46728i	−0.679541	+	0.733638i	$0.737821\pi$
$594$		0			0
$595$		−	231370.i		−	0.0267926i
$596$		0			0
$597$	−5.00133e6	+	731256.i	−0.574314	+	0.0839719i
$598$		0			0
$599$	1.66264e7			1.89335			0.946676	−	0.322186i	$-0.104418\pi$
$599$	1.66264e7			1.89335			0.946676	+	0.322186i	$0.104418\pi$
$600$		0			0
$601$	4.65512e6			0.525708			0.262854	−	0.964836i	$-0.415336\pi$
$601$	4.65512e6			0.525708			0.262854	+	0.964836i	$0.415336\pi$
$602$		0			0
$603$	4.55894e6	+	1.52568e7i	0.510588	+	1.70872i
$604$		0			0
$605$		−	1.00604e7i		−	1.11745i
$606$		0			0
$607$		−	4.60458e6i		−	0.507245i	−0.967303	−	0.253623i	$-0.918378\pi$
$607$		−	4.60458e6i		−	0.507245i	0.967303	−	0.253623i	$-0.0816222\pi$
$608$		0			0
$609$	1.82793e6	+	1.25019e7i	0.199717	+	1.36594i
$610$		0			0
$611$	1.95731e6			0.212108
$612$		0			0
$613$	−5.06680e6			−0.544606			−0.272303	−	0.962212i	$-0.587785\pi$
$613$	−5.06680e6			−0.544606			−0.272303	+	0.962212i	$0.587785\pi$
$614$		0			0
$615$	3.79159e6	+	2.59320e7i	0.404234	+	2.76470i
$616$		0			0
$617$		−	5.69843e6i		−	0.602618i	−0.953527	−	0.301309i	$-0.902576\pi$
$617$		−	5.69843e6i		−	0.602618i	0.953527	−	0.301309i	$-0.0974235\pi$
$618$		0			0
$619$		−	4.94045e6i		−	0.518251i	−0.965844	−	0.259125i	$-0.916566\pi$
$619$		−	4.94045e6i		−	0.518251i	0.965844	−	0.259125i	$-0.0834343\pi$
$620$		0			0
$621$	439680.	+	944822.i	0.0457517	+	0.0983153i
$622$		0			0
$623$	−1.53327e7			−1.58270
$624$		0			0
$625$	−4.21707e6			−0.431828
$626$		0			0
$627$	6.31627e6	−	923518.i	0.641641	−	0.0938159i
$628$		0			0
$629$		−	185588.i		−	0.0187035i
$630$		0			0
$631$		−	1.04200e7i		−	1.04182i	−0.853612	−	0.520910i	$-0.825593\pi$
$631$		−	1.04200e7i		−	1.04182i	0.853612	−	0.520910i	$-0.174407\pi$
$632$		0			0
$633$	1.01880e7	−	1.48961e6i	1.01060	−	0.147762i
$634$		0			0
$635$	−7.11510e6			−0.700240
$636$		0			0
$637$	−2.48292e6			−0.242445
$638$		0			0
$639$	7.00070e6	−	2.09190e6i	0.678249	−	0.202670i
$640$		0			0
$641$		−	2.75177e6i		−	0.264525i	−0.991215	−	0.132263i	$-0.957776\pi$
$641$		−	2.75177e6i		−	0.264525i	0.991215	−	0.132263i	$-0.0422242\pi$
$642$		0			0
$643$		−	1.96874e7i		−	1.87785i	−0.344123	−	0.938925i	$-0.611824\pi$
$643$		−	1.96874e7i		−	1.87785i	0.344123	−	0.938925i	$-0.388176\pi$
$644$		0			0
$645$	−878686.	−	6.00965e6i	−0.0831638	−	0.568787i
$646$		0			0
$647$	−2.00089e7			−1.87915			−0.939575	−	0.342343i	$-0.888780\pi$
$647$	−2.00089e7			−1.87915			−0.939575	+	0.342343i	$0.888780\pi$
$648$		0			0
$649$	5.08002e6			0.473428
$650$		0			0
$651$	1.86305e6	+	1.27421e7i	0.172295	+	1.17839i
$652$		0			0
$653$			3.44288e6i			0.315965i	0.987442	+	0.157982i	$0.0504989\pi$
$653$			3.44288e6i			0.315965i	−0.987442	+	0.157982i	$0.949501\pi$
$654$		0			0
$655$			1.66923e7i			1.52024i
$656$		0			0
$657$	1.00722e7	−	3.00970e6i	0.910353	−	0.272025i
$658$		0			0
$659$	2.86205e6			0.256722			0.128361	−	0.991727i	$-0.459028\pi$
$659$	2.86205e6			0.256722			0.128361	+	0.991727i	$0.459028\pi$
$660$		0			0
$661$	1.51348e7			1.34733			0.673663	−	0.739039i	$-0.264720\pi$
$661$	1.51348e7			1.34733			0.673663	+	0.739039i	$0.264720\pi$
$662$		0			0
$663$	200995.	−	29387.9i	0.0177583	−	0.00259648i
$664$		0			0
$665$			1.84865e7i			1.62107i
$666$		0			0
$667$		−	2.01853e6i		−	0.175679i
$668$		0			0
$669$	2.50703e6	−	366559.i	0.216568	−	0.0316649i
$670$		0			0
$671$	9.19314e6			0.788238
$672$		0			0
$673$	−6.06636e6			−0.516286			−0.258143	−	0.966107i	$-0.583111\pi$
$673$	−6.06636e6			−0.516286			−0.258143	+	0.966107i	$0.583111\pi$
$674$		0			0
$675$	−7.01472e6	−	1.50738e7i	−0.592585	−	1.27340i
$676$		0			0
$677$			1.23983e7i			1.03966i	0.854270	+	0.519829i	$0.174005\pi$
$677$			1.23983e7i			1.03966i	−0.854270	+	0.519829i	$0.825995\pi$
$678$		0			0
$679$			468106.i			0.0389646i
$680$		0			0
$681$	1.50644e6	+	1.03031e7i	0.124475	+	0.851331i
$682$		0			0
$683$	−1.41080e7			−1.15722			−0.578608	−	0.815605i	$-0.696404\pi$
$683$	−1.41080e7			−1.15722			−0.578608	+	0.815605i	$0.696404\pi$
$684$		0			0
$685$	2.27378e7			1.85149
$686$		0			0
$687$	−2.13201e6	−	1.45816e7i	−0.172345	−	1.17873i
$688$		0			0
$689$		−	1.98273e6i		−	0.159117i
$690$		0			0
$691$			5.34373e6i			0.425745i	0.977080	+	0.212872i	$0.0682818\pi$
$691$			5.34373e6i			0.425745i	−0.977080	+	0.212872i	$0.931718\pi$
$692$		0			0
$693$	1.63021e6	+	5.45561e6i	0.128947	+	0.431529i
$694$		0			0
$695$	2.92452e6			0.229664
$696$		0			0
$697$	468617.			0.0365373
$698$		0			0
$699$	1.02872e7	−	1.50412e6i	0.796354	−	0.116437i
$700$		0			0
$701$			4.64930e6i			0.357349i	0.983908	+	0.178674i	$0.0571809\pi$
$701$			4.64930e6i			0.357349i	−0.983908	+	0.178674i	$0.942819\pi$
$702$		0			0
$703$			1.48285e7i			1.13164i
$704$		0			0
$705$	−4.85251e6	+	709498.i	−0.367700	+	0.0537624i
$706$		0			0
$707$	−1.79295e7			−1.34902
$708$		0			0
$709$	−7.37828e6			−0.551239			−0.275619	−	0.961267i	$-0.588883\pi$
$709$	−7.37828e6			−0.551239			−0.275619	+	0.961267i	$0.588883\pi$
$710$		0			0
$711$	6.29902e6	+	2.10801e7i	0.467303	+	1.56387i
$712$		0			0
$713$		−	2.05732e6i		−	0.151558i
$714$		0			0
$715$		−	9.91642e6i		−	0.725421i
$716$		0			0
$717$	−923823.	−	6.31836e6i	−0.0671106	−	0.458994i
$718$		0			0
$719$	2.11646e7			1.52682			0.763410	−	0.645914i	$-0.223523\pi$
$719$	2.11646e7			1.52682			0.763410	+	0.645914i	$0.223523\pi$
$720$		0			0
$721$	−1.00597e7			−0.720691
$722$		0			0
$723$	−963434.	−	6.58927e6i	−0.0685451	−	0.468804i
$724$		0			0
$725$			3.22039e7i			2.27543i
$726$		0			0
$727$		−	1.97307e7i		−	1.38454i	−0.721636	−	0.692272i	$-0.756610\pi$
$727$		−	1.97307e7i		−	1.38454i	0.721636	−	0.692272i	$-0.243390\pi$
$728$		0			0
$729$	−9.24046e6	+	1.09775e7i	−0.643984	+	0.765039i
$730$		0			0
$731$	−108600.			−0.00751688
$732$		0			0
$733$	−1.30762e7			−0.898924			−0.449462	−	0.893299i	$-0.648384\pi$
$733$	−1.30762e7			−0.898924			−0.449462	+	0.893299i	$0.648384\pi$
$734$		0			0
$735$	6.15558e6	−	900023.i	0.420291	−	0.0614519i
$736$		0			0
$737$		−	1.38996e7i		−	0.942612i
$738$		0			0
$739$			9.42776e6i			0.635035i	0.948252	+	0.317517i	$0.102849\pi$
$739$			9.42776e6i			0.635035i	−0.948252	+	0.317517i	$0.897151\pi$
$740$		0			0
$741$	−1.60595e7	+	2.34810e6i	−1.07445	+	0.157098i
$742$		0			0
$743$	5.00117e6			0.332353			0.166177	−	0.986096i	$-0.446858\pi$
$743$	5.00117e6			0.332353			0.166177	+	0.986096i	$0.446858\pi$
$744$		0			0
$745$	2.79230e7			1.84320
$746$		0			0
$747$	−2.55800e7	+	7.64363e6i	−1.67725	+	0.501185i
$748$		0			0
$749$			1.72971e7i			1.12659i
$750$		0			0
$751$		−	2.94559e7i		−	1.90578i	−0.303314	−	0.952891i	$-0.598093\pi$
$751$		−	2.94559e7i		−	1.90578i	0.303314	−	0.952891i	$-0.401907\pi$
$752$		0			0
$753$	−1.98912e6	−	1.36043e7i	−0.127842	−	0.874360i
$754$		0			0
$755$	−1.64956e7			−1.05318
$756$		0			0
$757$	−2.64703e7			−1.67888			−0.839440	−	0.543452i	$-0.817117\pi$
$757$	−2.64703e7			−1.67888			−0.839440	+	0.543452i	$0.817117\pi$
$758$		0			0
$759$	−131605.	−	900096.i	−0.00829218	−	0.0567132i
$760$		0			0
$761$			1.16223e7i			0.727494i	0.931498	+	0.363747i	$0.118503\pi$
$761$			1.16223e7i			0.727494i	−0.931498	+	0.363747i	$0.881497\pi$
$762$		0			0
$763$			1.25748e7i			0.781967i
$764$		0			0
$765$	−487648.	+	145715.i	−0.0301268	+	0.00900227i
$766$		0			0
$767$	−1.29163e7			−0.792772
$768$		0			0
$769$	−4.74960e6			−0.289628			−0.144814	−	0.989459i	$-0.546258\pi$
$769$	−4.74960e6			−0.289628			−0.144814	+	0.989459i	$0.546258\pi$
$770$		0			0
$771$	9.24988e6	−	1.35245e6i	0.560403	−	0.0819379i
$772$		0			0
$773$			2.39153e7i			1.43955i	0.694208	+	0.719774i	$0.255755\pi$
$773$			2.39153e7i			1.43955i	−0.694208	+	0.719774i	$0.744245\pi$
$774$		0			0
$775$			3.28228e7i			1.96300i
$776$		0			0
$777$	−1.30877e7	+	1.91359e6i	−0.777700	+	0.113709i
$778$		0			0
$779$	−3.74425e7			−2.21066
$780$		0			0
$781$	−6.37792e6			−0.374155
$782$		0			0
$783$	2.51983e7	−	1.17262e7i	1.46882	−	0.683524i
$784$		0			0
$785$		−	560391.i		−	0.0324576i
$786$		0			0
$787$			773120.i			0.0444949i	0.999752	+	0.0222475i	$0.00708217\pi$
$787$			773120.i			0.0444949i	−0.999752	+	0.0222475i	$0.992918\pi$
$788$		0			0
$789$	−1.03057e6	−	7.04844e6i	−0.0589366	−	0.403089i
$790$		0			0
$791$	1.01763e6			0.0578297
$792$		0			0
$793$	−2.33741e7			−1.31993
$794$		0			0
$795$	718713.	+	4.91554e6i	0.0403309	+	0.275837i
$796$		0			0
$797$			2.84342e7i			1.58560i	0.609480	+	0.792802i	$0.291378\pi$
$797$			2.84342e7i			1.58560i	−0.609480	+	0.792802i	$0.708622\pi$
$798$		0			0
$799$			87689.7i			0.00485939i
$800$		0			0
$801$	9.65643e6	+	3.23160e7i	0.531784	+	1.77965i
$802$		0			0
$803$	−9.17616e6			−0.502194
$804$		0			0
$805$	2.63441e6			0.143283
$806$		0			0
$807$	1.70294e7	−	2.48990e6i	0.920480	−	0.134586i
$808$		0			0
$809$		−	1.04767e7i		−	0.562800i	−0.959591	−	0.281400i	$-0.909201\pi$
$809$		−	1.04767e7i		−	0.562800i	0.959591	−	0.281400i	$-0.0907988\pi$
$810$		0			0
$811$		−	1.11593e7i		−	0.595776i	−0.954601	−	0.297888i	$-0.903718\pi$
$811$		−	1.11593e7i		−	0.595776i	0.954601	−	0.297888i	$-0.0962822\pi$
$812$		0			0
$813$	−512028.	+	74864.9i	−0.0271686	+	0.00397239i
$814$		0			0
$815$	−2.88875e7			−1.52341
$816$		0			0
$817$	8.67717e6			0.454803
$818$		0			0
$819$	−4.14490e6	−	1.38712e7i	−0.215925	−	0.722611i
$820$		0			0
$821$			1.20649e7i			0.624690i	0.949969	+	0.312345i	$0.101114\pi$
$821$			1.20649e7i			0.624690i	−0.949969	+	0.312345i	$0.898886\pi$
$822$		0			0
$823$		−	2.97522e7i		−	1.53115i	−0.643344	−	0.765577i	$-0.722454\pi$
$823$		−	2.97522e7i		−	1.53115i	0.643344	−	0.765577i	$-0.277546\pi$
$824$		0			0
$825$	2.09965e6	+	1.43603e7i	0.107402	+	0.734561i
$826$		0			0
$827$	2.54258e7			1.29274			0.646369	−	0.763025i	$-0.276287\pi$
$827$	2.54258e7			1.29274			0.646369	+	0.763025i	$0.276287\pi$
$828$		0			0
$829$	−2.74678e7			−1.38815			−0.694077	−	0.719901i	$-0.744187\pi$
$829$	−2.74678e7			−1.38815			−0.694077	+	0.719901i	$0.744187\pi$
$830$		0			0
$831$	−2.97349e6	−	2.03367e7i	−0.149370	−	1.02159i
$832$		0			0
$833$		−	111237.i		−	0.00555441i
$834$		0			0
$835$			3.73485e7i			1.85377i
$836$		0			0
$837$	2.56825e7	−	1.19515e7i	1.26714	−	0.589672i
$838$		0			0
$839$	−2.10051e7			−1.03019			−0.515097	−	0.857132i	$-0.672244\pi$
$839$	−2.10051e7			−1.03019			−0.515097	+	0.857132i	$0.672244\pi$
$840$		0			0
$841$	−3.33229e7			−1.62462
$842$		0			0
$843$	6.19119e6	−	905230.i	0.300058	−	0.0438722i
$844$		0			0
$845$		−	6.97216e6i		−	0.335912i
$846$		0			0
$847$			1.28207e7i			0.614049i
$848$		0			0
$849$	−3.29886e7	+	4.82335e6i	−1.57071	+	0.229657i
$850$		0			0
$851$	2.11313e6			0.100023
$852$		0			0
$853$	2.25309e7			1.06024			0.530121	−	0.847922i	$-0.322146\pi$
$853$	2.25309e7			1.06024			0.530121	+	0.847922i	$0.322146\pi$
$854$		0			0
$855$	3.89631e7	−	1.16427e7i	1.82280	−	0.544675i
$856$		0			0
$857$		−	2.37134e7i		−	1.10292i	−0.834203	−	0.551458i	$-0.814072\pi$
$857$		−	2.37134e7i		−	1.10292i	0.834203	−	0.551458i	$-0.185928\pi$
$858$		0			0
$859$		−	1.08107e7i		−	0.499887i	−0.968260	−	0.249944i	$-0.919588\pi$
$859$		−	1.08107e7i		−	0.499887i	0.968260	−	0.249944i	$-0.0804121\pi$
$860$		0			0
$861$	−4.83189e6	−	3.30471e7i	−0.222131	−	1.51924i
$862$		0			0
$863$	1.86848e7			0.854006			0.427003	−	0.904250i	$-0.359569\pi$
$863$	1.86848e7			0.854006			0.427003	+	0.904250i	$0.359569\pi$
$864$		0			0
$865$	6.24470e7			2.83773
$866$		0			0
$867$	−3.20081e6	−	2.18915e7i	−0.144615	−	0.989072i
$868$		0			0
$869$		−	1.92048e7i		−	0.862703i
$870$		0			0
$871$			3.53405e7i			1.57844i
$872$		0			0
$873$	986604.	−	294810.i	0.0438134	−	0.0130920i
$874$		0			0
$875$	−1.21053e7			−0.534509
$876$		0			0
$877$	1.16971e7			0.513544			0.256772	−	0.966472i	$-0.417341\pi$
$877$	1.16971e7			0.513544			0.256772	+	0.966472i	$0.417341\pi$
$878$		0			0
$879$	−1.88469e7	+	2.75566e6i	−0.822751	+	0.120296i
$880$		0			0
$881$			2.81207e7i			1.22064i	0.792156	+	0.610318i	$0.208958\pi$
$881$			2.81207e7i			1.22064i	−0.792156	+	0.610318i	$0.791042\pi$
$882$		0			0
$883$			1.71174e7i			0.738816i	0.929267	+	0.369408i	$0.120439\pi$
$883$			1.71174e7i			0.738816i	−0.929267	+	0.369408i	$0.879561\pi$
$884$		0			0
$885$	3.20216e7	−	4.68196e6i	1.37431	−	0.200942i
$886$		0			0
$887$	−2.58331e7			−1.10247			−0.551237	−	0.834349i	$-0.685844\pi$
$887$	−2.58331e7			−1.10247			−0.551237	+	0.834349i	$0.685844\pi$
$888$		0			0
$889$	9.06729e6			0.384789
$890$		0			0
$891$	1.04718e7	−	6.87181e6i	0.441904	−	0.289986i
$892$		0			0
$893$		−	7.00641e6i		−	0.294013i
$894$		0			0
$895$			7.13395e7i			2.97696i
$896$		0			0
$897$	334614.	+	2.28855e6i	0.0138856	+	0.0949683i
$898$		0			0
$899$	−5.48685e7			−2.26425
$900$		0			0
$901$	88828.6			0.00364536
$902$		0			0
$903$	1.11977e6	+	7.65854e6i	0.0456994	+	0.312555i
$904$		0			0
$905$			7.50407e6i			0.304562i
$906$		0			0
$907$		−	2.84483e7i		−	1.14825i	−0.818766	−	0.574127i	$-0.805341\pi$
$907$		−	2.84483e7i		−	1.14825i	0.818766	−	0.574127i	$-0.194659\pi$
$908$		0			0
$909$	1.12919e7	+	3.77891e7i	0.453269	+	1.51690i
$910$		0			0
$911$	4.51021e7			1.80053			0.900266	−	0.435340i	$-0.143372\pi$
$911$	4.51021e7			1.80053			0.900266	+	0.435340i	$0.143372\pi$
$912$		0			0
$913$	2.33044e7			0.925253
$914$		0			0
$915$	5.79484e7	−	8.47278e6i	2.28817	−	0.334559i
$916$		0			0
$917$		−	2.12722e7i		−	0.835389i
$918$		0			0
$919$		−	4.39345e7i		−	1.71600i	−0.513651	−	0.857999i	$-0.671707\pi$
$919$		−	4.39345e7i		−	1.71600i	0.513651	−	0.857999i	$-0.328293\pi$
$920$		0			0
$921$	909377.	−	132962.i	0.0353260	−	0.00516511i
$922$		0			0
$923$	1.62162e7			0.626535
$924$		0			0
$925$	−3.37131e7			−1.29552
$926$		0			0
$927$	6.33556e6	+	2.12024e7i	0.242151	+	0.810375i
$928$		0			0
$929$			2.25804e7i			0.858405i	0.903208	+	0.429203i	$0.141205\pi$
$929$			2.25804e7i			0.858405i	−0.903208	+	0.429203i	$0.858795\pi$
$930$		0			0
$931$			8.88787e6i			0.336065i
$932$		0			0
$933$	−2.31204e6	−	1.58128e7i	−0.0869542	−	0.594711i
$934$		0			0
$935$	444266.			0.0166194
$936$		0			0
$937$	8.13233e6			0.302598			0.151299	−	0.988488i	$-0.451654\pi$
$937$	8.13233e6			0.302598			0.151299	+	0.988488i	$0.451654\pi$
$938$		0			0
$939$	3.68362e6	+	2.51936e7i	0.136336	+	0.932451i
$940$		0			0
$941$			1.83529e7i			0.675665i	0.941206	+	0.337833i	$0.109694\pi$
$941$			1.83529e7i			0.675665i	−0.941206	+	0.337833i	$0.890306\pi$
$942$		0			0
$943$			5.33573e6i			0.195395i
$944$		0			0
$945$	1.53040e7	+	3.28866e7i	0.557476	+	1.19795i
$946$		0			0
$947$	3.42017e6			0.123929			0.0619644	−	0.998078i	$-0.480263\pi$
$947$	3.42017e6			0.123929			0.0619644	+	0.998078i	$0.480263\pi$
$948$		0			0
$949$	2.33309e7			0.840942
$950$		0			0
$951$	9.19425e6	−	1.34431e6i	0.329659	−	0.0482003i
$952$		0			0
$953$		−	4.57241e7i		−	1.63084i	−0.578867	−	0.815422i	$-0.696505\pi$
$953$		−	4.57241e7i		−	1.63084i	0.578867	−	0.815422i	$-0.303495\pi$
$954$		0			0
$955$		−	8.09612e6i		−	0.287256i
$956$		0			0
$957$	−2.40055e7	+	3.50990e6i	−0.847287	+	0.123884i
$958$		0			0
$959$	−2.89765e7			−1.01742
$960$		0			0
$961$	−2.72937e7			−0.953354
$962$		0			0
$963$	3.64561e7	−	1.08936e7i	1.26679	−	0.378533i
$964$		0			0
$965$			5.49105e6i			0.189818i
$966$		0			0
$967$		−	6.01434e6i		−	0.206834i	−0.994638	−	0.103417i	$-0.967022\pi$
$967$		−	6.01434e6i		−	0.206834i	0.994638	−	0.103417i	$-0.0329776\pi$
$968$		0			0
$969$	−105197.	−	719482.i	−0.00359911	−	0.0246156i
$970$		0			0
$971$	4.11774e7			1.40156			0.700779	−	0.713378i	$-0.252836\pi$
$971$	4.11774e7			1.40156			0.700779	+	0.713378i	$0.252836\pi$
$972$		0			0
$973$	−3.72693e6			−0.126203
$974$		0			0
$975$	−5.33848e6	−	3.65118e7i	−0.179848	−	1.23005i
$976$		0			0
$977$		−	2.82069e7i		−	0.945409i	−0.881221	−	0.472704i	$-0.843278\pi$
$977$		−	2.82069e7i		−	0.945409i	0.881221	−	0.472704i	$-0.156722\pi$
$978$		0			0
$979$		−	2.94411e7i		−	0.981743i
$980$		0			0
$981$	2.65032e7	−	7.91950e6i	0.879277	−	0.262739i
$982$		0			0
$983$	6.06646e6			0.200240			0.100120	−	0.994975i	$-0.468077\pi$
$983$	6.06646e6			0.200240			0.100120	+	0.994975i	$0.468077\pi$
$984$		0			0
$985$	−5.23342e7			−1.71868
$986$		0			0
$987$	6.18391e6	−	904165.i	0.202055	−	0.0295430i
$988$		0			0
$989$		−	1.23653e6i		−	0.0401990i
$990$		0			0
$991$			2.99468e7i			0.968649i	0.874888	+	0.484324i	$0.160935\pi$
$991$			2.99468e7i			0.968649i	−0.874888	+	0.484324i	$0.839065\pi$
$992$		0			0
$993$	2.16272e7	−	3.16216e6i	0.696028	−	0.101768i
$994$		0			0
$995$	2.81071e7			0.900031
$996$		0			0
$997$	3.82255e7			1.21791			0.608954	−	0.793205i	$-0.291589\pi$
$997$	3.82255e7			1.21791			0.608954	+	0.793205i	$0.291589\pi$
$998$		0			0
$999$	1.22757e7	+	2.63792e7i	0.389165	+	0.836273i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.6.c.b.383.9	yes	20
3.2	odd	2		384.6.c.c.383.11	yes	20
4.3	odd	2		384.6.c.c.383.12	yes	20
8.3	odd	2		384.6.c.a.383.9	✓	20
8.5	even	2		384.6.c.d.383.12	yes	20
12.11	even	2	inner	384.6.c.b.383.10	yes	20
24.5	odd	2		384.6.c.a.383.10	yes	20
24.11	even	2		384.6.c.d.383.11	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.c.a.383.9	✓	20	8.3	odd	2
384.6.c.a.383.10	yes	20	24.5	odd	2
384.6.c.b.383.9	yes	20	1.1	even	1	trivial
384.6.c.b.383.10	yes	20	12.11	even	2	inner
384.6.c.c.383.11	yes	20	3.2	odd	2
384.6.c.c.383.12	yes	20	4.3	odd	2
384.6.c.d.383.11	yes	20	24.11	even	2
384.6.c.d.383.12	yes	20	8.5	even	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	384.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.25525 - 15.4245i) q^{3} +86.6842i q^{5} -110.468i q^{7} +(-232.828 + 69.5720i) q^{9} +O(q^{10})\)	\(q+(-2.25525 - 15.4245i) q^{3} +86.6842i q^{5} -110.468i q^{7} +(-232.828 + 69.5720i) q^{9} +212.115 q^{11} -539.316 q^{13} +(1337.06 - 195.494i) q^{15} -24.1619i q^{17} +1930.54i q^{19} +(-1703.91 + 249.133i) q^{21} +275.110 q^{23} -4389.15 q^{25} +(1598.19 + 3434.34i) q^{27} -7337.17i q^{29} -7478.16i q^{31} +(-478.373 - 3271.76i) q^{33} +9575.83 q^{35} +7681.01 q^{37} +(1216.29 + 8318.65i) q^{39} +19394.9i q^{41} -4494.69i q^{43} +(-6030.79 - 20182.5i) q^{45} -3629.25 q^{47} +4603.83 q^{49} +(-372.684 + 54.4911i) q^{51} +3676.39i q^{53} +18387.1i q^{55} +(29777.5 - 4353.85i) q^{57} +23949.3 q^{59} +43340.3 q^{61} +(7685.47 + 25720.0i) q^{63} -46750.1i q^{65} -65528.4i q^{67} +(-620.442 - 4243.43i) q^{69} -30068.2 q^{71} -43260.2 q^{73} +(9898.63 + 67700.3i) q^{75} -23432.0i q^{77} -90539.6i q^{79} +(49368.5 - 32396.6i) q^{81} +109867. q^{83} +2094.46 q^{85} +(-113172. + 16547.1i) q^{87} -138798. i q^{89} +59577.1i q^{91} +(-115347. + 16865.1i) q^{93} -167347. q^{95} -4237.49 q^{97} +(-49386.3 + 14757.3i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3}+O(q^{10}) \) 20 * q - 2 * q^3	\( 20 q - 2 q^{3} - 948 q^{11} + 852 q^{15} + 1640 q^{21} + 328 q^{23} - 12500 q^{25} - 2030 q^{27} + 2836 q^{33} + 7184 q^{35} + 15056 q^{37} - 12980 q^{39} + 11800 q^{45} + 36640 q^{47} - 33388 q^{49} - 1936 q^{51} + 15404 q^{57} - 62908 q^{59} + 73264 q^{61} + 23608 q^{63} - 84024 q^{69} + 34888 q^{71} + 52568 q^{73} - 115698 q^{75} + 55444 q^{81} + 225172 q^{83} - 30112 q^{85} - 225700 q^{87} - 148016 q^{93} + 418616 q^{95} + 7600 q^{97} - 378260 q^{99}+O(q^{100}) \) 20 * q - 2 * q^3 - 948 * q^11 + 852 * q^15 + 1640 * q^21 + 328 * q^23 - 12500 * q^25 - 2030 * q^27 + 2836 * q^33 + 7184 * q^35 + 15056 * q^37 - 12980 * q^39 + 11800 * q^45 + 36640 * q^47 - 33388 * q^49 - 1936 * q^51 + 15404 * q^57 - 62908 * q^59 + 73264 * q^61 + 23608 * q^63 - 84024 * q^69 + 34888 * q^71 + 52568 * q^73 - 115698 * q^75 + 55444 * q^81 + 225172 * q^83 - 30112 * q^85 - 225700 * q^87 - 148016 * q^93 + 418616 * q^95 + 7600 * q^97 - 378260 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more