LMFDB - Embedded newform 384.6.c.c.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			$-6.14819i$ of defining polynomial
Character	$\chi$	$=$	384.383
Dual form			384.6.c.c.383.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.67629 - 15.4981i) q^{3} +6.76651i q^{5} -132.568i q^{7} +(-237.380 - 51.9584i) q^{9} +O(q^{10})$	\(q+(1.67629 - 15.4981i) q^{3} +6.76651i q^{5} -132.568i q^{7} +(-237.380 - 51.9584i) q^{9} -687.674 q^{11} +609.929 q^{13} +(104.868 + 11.3426i) q^{15} -550.567i q^{17} -232.742i q^{19} +(-2054.54 - 222.221i) q^{21} -4487.15 q^{23} +3079.21 q^{25} +(-1203.17 + 3591.84i) q^{27} -2046.54i q^{29} +6671.05i q^{31} +(-1152.74 + 10657.6i) q^{33} +897.020 q^{35} +2608.75 q^{37} +(1022.42 - 9452.73i) q^{39} +12755.2i q^{41} -9781.38i q^{43} +(351.577 - 1606.23i) q^{45} -16955.6 q^{47} -767.166 q^{49} +(-8532.72 - 922.907i) q^{51} -22440.7i q^{53} -4653.15i q^{55} +(-3607.05 - 390.142i) q^{57} +16109.1 q^{59} -8678.41 q^{61} +(-6888.00 + 31468.9i) q^{63} +4127.09i q^{65} +53373.9i q^{67} +(-7521.74 + 69542.1i) q^{69} -45069.3 q^{71} +7818.42 q^{73} +(5161.64 - 47721.9i) q^{75} +91163.3i q^{77} -6497.77i q^{79} +(53649.7 + 24667.8i) q^{81} +101487. q^{83} +3725.42 q^{85} +(-31717.4 - 3430.59i) q^{87} +59204.9i q^{89} -80856.9i q^{91} +(103388. + 11182.6i) q^{93} +1574.85 q^{95} -146508. q^{97} +(163240. + 35730.4i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.67629	−	15.4981i	0.107534	−	0.994201i
$3$	1.67629	−	15.4981i	0.107534	−	0.994201i
$4$		0			0
$5$			6.76651i			0.121043i	0.998167	+	0.0605215i	$0.0192764\pi$
$5$			6.76651i			0.121043i	−0.998167	+	0.0605215i	$0.980724\pi$
$6$		0			0
$7$		−	132.568i		−	1.02257i	−0.859412	−	0.511284i	$-0.829170\pi$
$7$		−	132.568i		−	1.02257i	0.859412	−	0.511284i	$-0.170830\pi$
$8$		0			0
$9$	−237.380	−	51.9584i	−0.976873	−	0.213820i
$10$		0			0
$11$	−687.674			−1.71357			−0.856783	−	0.515677i	$-0.827541\pi$
$11$	−687.674			−1.71357			−0.856783	+	0.515677i	$0.827541\pi$
$12$		0			0
$13$	609.929			1.00097			0.500485	−	0.865745i	$-0.333155\pi$
$13$	609.929			1.00097			0.500485	+	0.865745i	$0.333155\pi$
$14$		0			0
$15$	104.868	+	11.3426i	0.120341	+	0.0130162i
$16$		0			0
$17$		−	550.567i		−	0.462049i	−0.972948	−	0.231024i	$-0.925792\pi$
$17$		−	550.567i		−	0.462049i	0.972948	−	0.231024i	$-0.0742077\pi$
$18$		0			0
$19$		−	232.742i		−	0.147908i	−0.997262	−	0.0739538i	$-0.976438\pi$
$19$		−	232.742i		−	0.147908i	0.997262	−	0.0739538i	$-0.0235617\pi$
$20$		0			0
$21$	−2054.54	−	222.221i	−1.01664	−	0.109961i
$22$		0			0
$23$	−4487.15			−1.76869			−0.884343	−	0.466838i	$-0.845393\pi$
$23$	−4487.15			−1.76869			−0.884343	+	0.466838i	$0.845393\pi$
$24$		0			0
$25$	3079.21			0.985349
$26$		0			0
$27$	−1203.17	+	3591.84i	−0.317627	+	0.948216i
$28$		0			0
$29$		−	2046.54i		−	0.451882i	−0.974141	−	0.225941i	$-0.927454\pi$
$29$		−	2046.54i		−	0.451882i	0.974141	−	0.225941i	$-0.0725457\pi$
$30$		0			0
$31$			6671.05i			1.24678i	0.781911	+	0.623391i	$0.214245\pi$
$31$			6671.05i			1.24678i	−0.781911	+	0.623391i	$0.785755\pi$
$32$		0			0
$33$	−1152.74	+	10657.6i	−0.184266	+	1.70363i
$34$		0			0
$35$	897.020			0.123775
$36$		0			0
$37$	2608.75			0.313277			0.156639	−	0.987656i	$-0.449934\pi$
$37$	2608.75			0.313277			0.156639	+	0.987656i	$0.449934\pi$
$38$		0			0
$39$	1022.42	−	9452.73i	0.107638	−	0.995166i
$40$		0			0
$41$			12755.2i			1.18502i	0.805563	+	0.592511i	$0.201863\pi$
$41$			12755.2i			1.18502i	−0.805563	+	0.592511i	$0.798137\pi$
$42$		0			0
$43$		−	9781.38i		−	0.806732i	−0.915039	−	0.403366i	$-0.867840\pi$
$43$		−	9781.38i		−	0.806732i	0.915039	−	0.403366i	$-0.132160\pi$
$44$		0			0
$45$	351.577	−	1606.23i	0.0258815	−	0.118244i
$46$		0			0
$47$	−16955.6			−1.11961			−0.559807	−	0.828623i	$-0.689125\pi$
$47$	−16955.6			−1.11961			−0.559807	+	0.828623i	$0.689125\pi$
$48$		0			0
$49$	−767.166			−0.0456456
$50$		0			0
$51$	−8532.72	−	922.907i	−0.459370	−	0.0496858i
$52$		0			0
$53$		−	22440.7i		−	1.09735i	−0.836034	−	0.548677i	$-0.815132\pi$
$53$		−	22440.7i		−	1.09735i	0.836034	−	0.548677i	$-0.184868\pi$
$54$		0			0
$55$		−	4653.15i		−	0.207415i
$56$		0			0
$57$	−3607.05	−	390.142i	−0.147050	−	0.0159051i
$58$		0			0
$59$	16109.1			0.602477			0.301239	−	0.953549i	$-0.402600\pi$
$59$	16109.1			0.602477			0.301239	+	0.953549i	$0.402600\pi$
$60$		0			0
$61$	−8678.41			−0.298618			−0.149309	−	0.988791i	$-0.547705\pi$
$61$	−8678.41			−0.298618			−0.149309	+	0.988791i	$0.547705\pi$
$62$		0			0
$63$	−6888.00	+	31468.9i	−0.218646	+	0.998919i
$64$		0			0
$65$			4127.09i			0.121160i
$66$		0			0
$67$			53373.9i			1.45259i	0.687385	+	0.726293i	$0.258759\pi$
$67$			53373.9i			1.45259i	−0.687385	+	0.726293i	$0.741241\pi$
$68$		0			0
$69$	−7521.74	+	69542.1i	−0.190193	+	1.75843i
$70$		0			0
$71$	−45069.3			−1.06105			−0.530524	−	0.847670i	$-0.678005\pi$
$71$	−45069.3			−1.06105			−0.530524	+	0.847670i	$0.678005\pi$
$72$		0			0
$73$	7818.42			0.171716			0.0858582	−	0.996307i	$-0.472637\pi$
$73$	7818.42			0.171716			0.0858582	+	0.996307i	$0.472637\pi$
$74$		0			0
$75$	5161.64	−	47721.9i	0.105958	−	0.979635i
$76$		0			0
$77$			91163.3i			1.75224i
$78$		0			0
$79$		−	6497.77i		−	0.117138i	−0.998283	−	0.0585688i	$-0.981346\pi$
$79$		−	6497.77i		−	0.117138i	0.998283	−	0.0585688i	$-0.0186537\pi$
$80$		0			0
$81$	53649.7	+	24667.8i	0.908562	+	0.417751i
$82$		0			0
$83$	101487.			1.61701			0.808506	−	0.588488i	$-0.200277\pi$
$83$	101487.			1.61701			0.808506	+	0.588488i	$0.200277\pi$
$84$		0			0
$85$	3725.42			0.0559278
$86$		0			0
$87$	−31717.4	−	3430.59i	−0.449262	−	0.0485926i
$88$		0			0
$89$			59204.9i			0.792288i	0.918188	+	0.396144i	$0.129652\pi$
$89$			59204.9i			0.792288i	−0.918188	+	0.396144i	$0.870348\pi$
$90$		0			0
$91$		−	80856.9i		−	1.02356i
$92$		0			0
$93$	103388.	+	11182.6i	1.23955	+	0.134071i
$94$		0			0
$95$	1574.85			0.0179032
$96$		0			0
$97$	−146508.			−1.58100			−0.790500	−	0.612462i	$-0.790179\pi$
$97$	−146508.			−1.58100			−0.790500	+	0.612462i	$0.790179\pi$
$98$		0			0
$99$	163240.	+	35730.4i	1.67394	+	0.366396i
$100$		0			0
$101$			168923.i			1.64773i	0.566789	+	0.823863i	$0.308186\pi$
$101$			168923.i			1.64773i	−0.566789	+	0.823863i	$0.691814\pi$
$102$		0			0
$103$		−	65641.0i		−	0.609652i	−0.952408	−	0.304826i	$-0.901402\pi$
$103$		−	65641.0i		−	0.609652i	0.952408	−	0.304826i	$-0.0985983\pi$
$104$		0			0
$105$	1503.66	−	13902.1i	0.0133100	−	0.123057i
$106$		0			0
$107$	115602.			0.976128			0.488064	−	0.872808i	$-0.337703\pi$
$107$	115602.			0.976128			0.488064	+	0.872808i	$0.337703\pi$
$108$		0			0
$109$	45690.7			0.368351			0.184175	−	0.982893i	$-0.441039\pi$
$109$	45690.7			0.368351			0.184175	+	0.982893i	$0.441039\pi$
$110$		0			0
$111$	4373.02	−	40430.6i	0.0336879	−	0.311461i
$112$		0			0
$113$			141844.i			1.04500i	0.852641	+	0.522498i	$0.175000\pi$
$113$			141844.i			1.04500i	−0.852641	+	0.522498i	$0.825000\pi$
$114$		0			0
$115$		−	30362.3i		−	0.214087i
$116$		0			0
$117$	−144785.	−	31690.9i	−0.977821	−	0.214028i
$118$		0			0
$119$	−72987.3			−0.472476
$120$		0			0
$121$	311845.			1.93631
$122$		0			0
$123$	197680.	+	21381.3i	1.17815	+	0.127430i
$124$		0			0
$125$			41980.9i			0.240312i
$126$		0			0
$127$			228668.i			1.25804i	0.777387	+	0.629022i	$0.216545\pi$
$127$			228668.i			1.25804i	−0.777387	+	0.629022i	$0.783455\pi$
$128$		0			0
$129$	−151593.	−	16396.4i	−0.802054	−	0.0867509i
$130$		0			0
$131$	−94798.2			−0.482638			−0.241319	−	0.970446i	$-0.577580\pi$
$131$	−94798.2			−0.482638			−0.241319	+	0.970446i	$0.577580\pi$
$132$		0			0
$133$	−30854.0			−0.151246
$134$		0			0
$135$	−24304.2	−	8141.27i	−0.114775	−	0.0384466i
$136$		0			0
$137$			229049.i			1.04262i	0.853367	+	0.521311i	$0.174557\pi$
$137$			229049.i			1.04262i	−0.853367	+	0.521311i	$0.825443\pi$
$138$		0			0
$139$			156067.i			0.685133i	0.939494	+	0.342566i	$0.111296\pi$
$139$			156067.i			0.685133i	−0.939494	+	0.342566i	$0.888704\pi$
$140$		0			0
$141$	−28422.4	+	262779.i	−0.120396	+	1.11312i
$142$		0			0
$143$	−419433.			−1.71523
$144$		0			0
$145$	13847.9			0.0546972
$146$		0			0
$147$	−1285.99	+	11889.6i	−0.00490845	+	0.0453810i
$148$		0			0
$149$			299333.i			1.10456i	0.833659	+	0.552279i	$0.186242\pi$
$149$			299333.i			1.10456i	−0.833659	+	0.552279i	$0.813758\pi$
$150$		0			0
$151$			70935.1i			0.253174i	0.991956	+	0.126587i	$0.0404023\pi$
$151$			70935.1i			0.253174i	−0.991956	+	0.126587i	$0.959598\pi$
$152$		0			0
$153$	−28606.6	+	130694.i	−0.0987955	+	0.451363i
$154$		0			0
$155$	−45139.7			−0.150914
$156$		0			0
$157$	387310.			1.25403			0.627017	−	0.779006i	$-0.284276\pi$
$157$	387310.			1.25403			0.627017	+	0.779006i	$0.284276\pi$
$158$		0			0
$159$	−347788.	−	37617.0i	−1.09099	−	0.118003i
$160$		0			0
$161$			594850.i			1.80860i
$162$		0			0
$163$		−	520550.i		−	1.53459i	−0.641291	−	0.767297i	$-0.721601\pi$
$163$		−	520550.i		−	1.53459i	0.641291	−	0.767297i	$-0.278399\pi$
$164$		0			0
$165$	−72114.9	−	7800.01i	−0.206212	−	0.0223041i
$166$		0			0
$167$	−474775.			−1.31734			−0.658668	−	0.752434i	$-0.728880\pi$
$167$	−474775.			−1.31734			−0.658668	+	0.752434i	$0.728880\pi$
$168$		0			0
$169$	720.801			0.00194133
$170$		0			0
$171$	−12092.9	+	55248.3i	−0.0316257	+	0.144487i
$172$		0			0
$173$		−	428218.i		−	1.08780i	−0.839149	−	0.543901i	$-0.816947\pi$
$173$		−	428218.i		−	1.08780i	0.839149	−	0.543901i	$-0.183053\pi$
$174$		0			0
$175$		−	408204.i		−	1.00759i
$176$		0			0
$177$	27003.4	−	249660.i	0.0647867	−	0.598984i
$178$		0			0
$179$	68071.9			0.158794			0.0793972	−	0.996843i	$-0.474700\pi$
$179$	68071.9			0.158794			0.0793972	+	0.996843i	$0.474700\pi$
$180$		0			0
$181$	−607181.			−1.37760			−0.688798	−	0.724953i	$-0.741861\pi$
$181$	−607181.			−1.37760			−0.688798	+	0.724953i	$0.741861\pi$
$182$		0			0
$183$	−14547.5	+	134499.i	−0.0321115	+	0.296886i
$184$		0			0
$185$			17652.2i			0.0379200i
$186$		0			0
$187$			378611.i			0.791751i
$188$		0			0
$189$	476161.	+	159502.i	0.969615	+	0.324796i
$190$		0			0
$191$	188197.			0.373275			0.186637	−	0.982429i	$-0.440241\pi$
$191$	188197.			0.373275			0.186637	+	0.982429i	$0.440241\pi$
$192$		0			0
$193$	−597386.			−1.15441			−0.577207	−	0.816598i	$-0.695857\pi$
$193$	−597386.			−1.15441			−0.577207	+	0.816598i	$0.695857\pi$
$194$		0			0
$195$	63961.9	+	6918.18i	0.120458	+	0.0130288i
$196$		0			0
$197$		−	983345.i		−	1.80526i	−0.430416	−	0.902631i	$-0.641633\pi$
$197$		−	983345.i		−	1.80526i	0.430416	−	0.902631i	$-0.358367\pi$
$198$		0			0
$199$			698819.i			1.25093i	0.780254	+	0.625463i	$0.215090\pi$
$199$			698819.i			1.25093i	−0.780254	+	0.625463i	$0.784910\pi$
$200$		0			0
$201$	827193.	+	89469.9i	1.44416	+	0.156202i
$202$		0			0
$203$	−271305.			−0.462080
$204$		0			0
$205$	−86307.9			−0.143439
$206$		0			0
$207$	1.06516e6	+	233145.i	1.72778	+	0.378181i
$208$		0			0
$209$			160051.i			0.253450i
$210$		0			0
$211$			558698.i			0.863915i	0.901894	+	0.431957i	$0.142177\pi$
$211$			558698.i			0.863915i	−0.901894	+	0.431957i	$0.857823\pi$
$212$		0			0
$213$	−75549.0	+	698487.i	−0.114098	+	1.05489i
$214$		0			0
$215$	66185.8			0.0976492
$216$		0			0
$217$	884366.			1.27492
$218$		0			0
$219$	13105.9	−	121170.i	0.0184653	−	0.170721i
$220$		0			0
$221$		−	335807.i		−	0.462497i
$222$		0			0
$223$		−	1.31975e6i		−	1.77717i	−0.458710	−	0.888586i	$-0.651688\pi$
$223$		−	1.31975e6i		−	1.77717i	0.458710	−	0.888586i	$-0.348312\pi$
$224$		0			0
$225$	−730944.	−	159991.i	−0.962560	−	0.210688i
$226$		0			0
$227$	−473989.			−0.610526			−0.305263	−	0.952268i	$-0.598744\pi$
$227$	−473989.			−0.610526			−0.305263	+	0.952268i	$0.598744\pi$
$228$		0			0
$229$	−891437.			−1.12332			−0.561658	−	0.827369i	$-0.689836\pi$
$229$	−891437.			−1.12332			−0.561658	+	0.827369i	$0.689836\pi$
$230$		0			0
$231$	1.41285e6	+	152816.i	1.74208	+	0.188425i
$232$		0			0
$233$		−	726371.i		−	0.876534i	−0.898845	−	0.438267i	$-0.855592\pi$
$233$		−	726371.i		−	0.876534i	0.898845	−	0.438267i	$-0.144408\pi$
$234$		0			0
$235$		−	114730.i		−	0.135521i
$236$		0			0
$237$	−100703.	−	10892.1i	−0.116458	−	0.0125962i
$238$		0			0
$239$	−870021.			−0.985224			−0.492612	−	0.870249i	$-0.663958\pi$
$239$	−870021.			−0.985224			−0.492612	+	0.870249i	$0.663958\pi$
$240$		0			0
$241$	−1.14608e6			−1.27108			−0.635538	−	0.772070i	$-0.719221\pi$
$241$	−1.14608e6			−1.27108			−0.635538	+	0.772070i	$0.719221\pi$
$242$		0			0
$243$	472235.	−	790116.i	0.513030	−	0.858371i
$244$		0			0
$245$		−	5191.04i		−	0.00552508i
$246$		0			0
$247$		−	141956.i		−	0.148051i
$248$		0			0
$249$	170120.	−	1.57284e6i	0.173883	−	1.60763i
$250$		0			0
$251$	−138933.			−0.139194			−0.0695969	−	0.997575i	$-0.522171\pi$
$251$	−138933.			−0.139194			−0.0695969	+	0.997575i	$0.522171\pi$
$252$		0			0
$253$	3.08569e6			3.03076
$254$		0			0
$255$	6244.86	−	57736.7i	0.00601412	−	0.0556035i
$256$		0			0
$257$		−	1.75878e6i		−	1.66103i	−0.556994	−	0.830517i	$-0.688045\pi$
$257$		−	1.75878e6i		−	1.66103i	0.556994	−	0.830517i	$-0.311955\pi$
$258$		0			0
$259$		−	345836.i		−	0.320347i
$260$		0			0
$261$	−106335.	+	485808.i	−0.0966217	+	0.441432i
$262$		0			0
$263$	636768.			0.567665			0.283832	−	0.958874i	$-0.408394\pi$
$263$	636768.			0.567665			0.283832	+	0.958874i	$0.408394\pi$
$264$		0			0
$265$	151845.			0.132827
$266$		0			0
$267$	917562.	+	99244.4i	0.787694	+	0.0851977i
$268$		0			0
$269$		−	330293.i		−	0.278304i	−0.990271	−	0.139152i	$-0.955562\pi$
$269$		−	330293.i		−	0.278304i	0.990271	−	0.139152i	$-0.0444376\pi$
$270$		0			0
$271$			1.31762e6i			1.08985i	0.838485	+	0.544925i	$0.183442\pi$
$271$			1.31762e6i			1.08985i	−0.838485	+	0.544925i	$0.816558\pi$
$272$		0			0
$273$	−1.25313e6	−	135539.i	−1.01763	−	0.110067i
$274$		0			0
$275$	−2.11750e6			−1.68846
$276$		0			0
$277$	−1.99956e6			−1.56579			−0.782897	−	0.622152i	$-0.786259\pi$
$277$	−1.99956e6			−1.56579			−0.782897	+	0.622152i	$0.786259\pi$
$278$		0			0
$279$	346617.	−	1.58358e6i	0.266587	−	1.21795i
$280$		0			0
$281$			755636.i			0.570882i	0.958396	+	0.285441i	$0.0921401\pi$
$281$			755636.i			0.570882i	−0.958396	+	0.285441i	$0.907860\pi$
$282$		0			0
$283$		−	500952.i		−	0.371817i	−0.982567	−	0.185909i	$-0.940477\pi$
$283$		−	500952.i		−	0.371817i	0.982567	−	0.185909i	$-0.0595229\pi$
$284$		0			0
$285$	2639.90	−	24407.1i	0.00192520	−	0.0177994i
$286$		0			0
$287$	1.69092e6			1.21177
$288$		0			0
$289$	1.11673e6			0.786511
$290$		0			0
$291$	−245589.	+	2.27059e6i	−0.170011	+	1.57183i
$292$		0			0
$293$		−	1.43780e6i		−	0.978431i	−0.872163	−	0.489215i	$-0.837283\pi$
$293$		−	1.43780e6i		−	0.978431i	0.872163	−	0.489215i	$-0.162717\pi$
$294$		0			0
$295$			109002.i			0.0729257i
$296$		0			0
$297$	827390.	−	2.47001e6i	0.544276	−	1.62483i
$298$		0			0
$299$	−2.73684e6			−1.77040
$300$		0			0
$301$	−1.29669e6			−0.824938
$302$		0			0
$303$	2.61798e6	+	283163.i	1.63817	+	0.177186i
$304$		0			0
$305$		−	58722.6i		−	0.0361456i
$306$		0			0
$307$		−	730785.i		−	0.442531i	−0.975214	−	0.221266i	$-0.928981\pi$
$307$		−	730785.i		−	0.442531i	0.975214	−	0.221266i	$-0.0710188\pi$
$308$		0			0
$309$	−1.01731e6	−	110033.i	−0.606117	−	0.0655582i
$310$		0			0
$311$	648099.			0.379962			0.189981	−	0.981788i	$-0.439157\pi$
$311$	648099.			0.379962			0.189981	+	0.981788i	$0.439157\pi$
$312$		0			0
$313$	131249.			0.0757243			0.0378622	−	0.999283i	$-0.487945\pi$
$313$	131249.			0.0757243			0.0378622	+	0.999283i	$0.487945\pi$
$314$		0			0
$315$	−212935.	−	46607.7i	−0.120912	−	0.0264656i
$316$		0			0
$317$		−	3.23573e6i		−	1.80852i	−0.426978	−	0.904262i	$-0.640422\pi$
$317$		−	3.23573e6i		−	1.80852i	0.426978	−	0.904262i	$-0.359578\pi$
$318$		0			0
$319$			1.40735e6i			0.774330i
$320$		0			0
$321$	193782.	−	1.79161e6i	0.104967	−	0.970468i
$322$		0			0
$323$	−128140.			−0.0683406
$324$		0			0
$325$	1.87810e6			0.986305
$326$		0			0
$327$	76590.6	−	708117.i	0.0396101	−	0.366215i
$328$		0			0
$329$			2.24776e6i			1.14488i
$330$		0			0
$331$			1.75600e6i			0.880956i	0.897763	+	0.440478i	$0.145191\pi$
$331$			1.75600e6i			0.880956i	−0.897763	+	0.440478i	$0.854809\pi$
$332$		0			0
$333$	−619266.	−	135547.i	−0.306032	−	0.0669851i
$334$		0			0
$335$	−361155.			−0.175825
$336$		0			0
$337$	−2.56913e6			−1.23229			−0.616143	−	0.787634i	$-0.711306\pi$
$337$	−2.56913e6			−1.23229			−0.616143	+	0.787634i	$0.711306\pi$
$338$		0			0
$339$	2.19830e6	+	237771.i	1.03894	+	0.112372i
$340$		0			0
$341$		−	4.58751e6i		−	2.13644i
$342$		0			0
$343$		−	2.12636e6i		−	0.975892i
$344$		0			0
$345$	−470557.	−	50895.9i	−0.212846	−	0.0230216i
$346$		0			0
$347$	−3.29984e6			−1.47119			−0.735596	−	0.677420i	$-0.763098\pi$
$347$	−3.29984e6			−1.47119			−0.735596	+	0.677420i	$0.763098\pi$
$348$		0			0
$349$	1.40218e6			0.616227			0.308113	−	0.951350i	$-0.400302\pi$
$349$	1.40218e6			0.616227			0.308113	+	0.951350i	$0.400302\pi$
$350$		0			0
$351$	−733849.	+	2.19077e6i	−0.317936	+	0.949136i
$352$		0			0
$353$			4.52203e6i			1.93151i	0.259456	+	0.965755i	$0.416457\pi$
$353$			4.52203e6i			1.93151i	−0.259456	+	0.965755i	$0.583543\pi$
$354$		0			0
$355$		−	304962.i		−	0.128432i
$356$		0			0
$357$	−122348.	+	1.13116e6i	−0.0508072	+	0.469737i
$358$		0			0
$359$	−3.40082e6			−1.39267			−0.696335	−	0.717717i	$-0.745187\pi$
$359$	−3.40082e6			−1.39267			−0.696335	+	0.717717i	$0.745187\pi$
$360$		0			0
$361$	2.42193e6			0.978123
$362$		0			0
$363$	522741.	−	4.83299e6i	0.208219	−	1.92508i
$364$		0			0
$365$			52903.4i			0.0207851i
$366$		0			0
$367$		−	2.90470e6i		−	1.12573i	−0.826547	−	0.562867i	$-0.809698\pi$
$367$		−	2.90470e6i		−	1.12573i	0.826547	−	0.562867i	$-0.190302\pi$
$368$		0			0
$369$	662737.	−	3.02782e6i	0.253382	−	1.15762i
$370$		0			0
$371$	−2.97491e6			−1.12212
$372$		0			0
$373$	−1.58294e6			−0.589104			−0.294552	−	0.955635i	$-0.595170\pi$
$373$	−1.58294e6			−0.589104			−0.294552	+	0.955635i	$0.595170\pi$
$374$		0			0
$375$	650622.	+	70371.9i	0.238919	+	0.0258417i
$376$		0			0
$377$		−	1.24824e6i		−	0.452321i
$378$		0			0
$379$			112527.i			0.0402401i	0.999798	+	0.0201201i	$0.00640485\pi$
$379$			112527.i			0.0402401i	−0.999798	+	0.0201201i	$0.993595\pi$
$380$		0			0
$381$	3.54391e6	+	383313.i	1.25075	+	0.135282i
$382$		0			0
$383$	−1.84859e6			−0.643937			−0.321969	−	0.946750i	$-0.604345\pi$
$383$	−1.84859e6			−0.643937			−0.321969	+	0.946750i	$0.604345\pi$
$384$		0			0
$385$	−616857.			−0.212096
$386$		0			0
$387$	−508225.	+	2.32191e6i	−0.172496	+	0.788074i
$388$		0			0
$389$			3.58651e6i			1.20171i	0.799360	+	0.600853i	$0.205172\pi$
$389$			3.58651e6i			1.20171i	−0.799360	+	0.600853i	$0.794828\pi$
$390$		0			0
$391$			2.47047e6i			0.817219i
$392$		0			0
$393$	−158909.	+	1.46919e6i	−0.0518999	+	0.479840i
$394$		0			0
$395$	43967.2			0.0141787
$396$		0			0
$397$	−3.21573e6			−1.02401			−0.512004	−	0.858983i	$-0.671097\pi$
$397$	−3.21573e6			−1.02401			−0.512004	+	0.858983i	$0.671097\pi$
$398$		0			0
$399$	−51720.2	+	478178.i	−0.0162640	+	0.150369i
$400$		0			0
$401$			2.52639e6i			0.784583i	0.919841	+	0.392291i	$0.128318\pi$
$401$			2.52639e6i			0.784583i	−0.919841	+	0.392291i	$0.871682\pi$
$402$		0			0
$403$			4.06887e6i			1.24799i
$404$		0			0
$405$	−166915.	+	363021.i	−0.0505658	+	0.109975i
$406$		0			0
$407$	−1.79397e6			−0.536821
$408$		0			0
$409$	−1.17254e6			−0.346594			−0.173297	−	0.984870i	$-0.555442\pi$
$409$	−1.17254e6			−0.346594			−0.173297	+	0.984870i	$0.555442\pi$
$410$		0			0
$411$	3.54982e6	+	383952.i	1.03658	+	0.112117i
$412$		0			0
$413$		−	2.13554e6i		−	0.616074i
$414$		0			0
$415$			686709.i			0.195728i
$416$		0			0
$417$	2.41874e6	+	261613.i	0.681160	+	0.0736749i
$418$		0			0
$419$	−1.27821e6			−0.355685			−0.177842	−	0.984059i	$-0.556912\pi$
$419$	−1.27821e6			−0.355685			−0.177842	+	0.984059i	$0.556912\pi$
$420$		0			0
$421$	−1.02494e6			−0.281834			−0.140917	−	0.990021i	$-0.545005\pi$
$421$	−1.02494e6			−0.281834			−0.140917	+	0.990021i	$0.545005\pi$
$422$		0			0
$423$	4.02492e6	+	880985.i	1.09372	+	0.239396i
$424$		0			0
$425$		−	1.69531e6i		−	0.455279i
$426$		0			0
$427$			1.15048e6i			0.305357i
$428$		0			0
$429$	−703089.	+	6.50039e6i	−0.184445	+	1.70528i
$430$		0			0
$431$	−6.06484e6			−1.57263			−0.786315	−	0.617826i	$-0.788014\pi$
$431$	−6.06484e6			−1.57263			−0.786315	+	0.617826i	$0.788014\pi$
$432$		0			0
$433$	924491.			0.236964			0.118482	−	0.992956i	$-0.462197\pi$
$433$	924491.			0.236964			0.118482	+	0.992956i	$0.462197\pi$
$434$		0			0
$435$	23213.1	−	214616.i	0.00588179	−	0.0543800i
$436$		0			0
$437$			1.04435e6i			0.261602i
$438$		0			0
$439$			2.18052e6i			0.540007i	0.962859	+	0.270004i	$0.0870249\pi$
$439$			2.18052e6i			0.540007i	−0.962859	+	0.270004i	$0.912975\pi$
$440$		0			0
$441$	182110.	+	39860.7i	0.0445900	+	0.00975997i
$442$		0			0
$443$	2.19111e6			0.530463			0.265231	−	0.964185i	$-0.414552\pi$
$443$	2.19111e6			0.530463			0.265231	+	0.964185i	$0.414552\pi$
$444$		0			0
$445$	−400611.			−0.0959009
$446$		0			0
$447$	4.63908e6	+	501767.i	1.09815	+	0.118777i
$448$		0			0
$449$		−	2.31214e6i		−	0.541252i	−0.962685	−	0.270626i	$-0.912769\pi$
$449$		−	2.31214e6i		−	0.541252i	0.962685	−	0.270626i	$-0.0872306\pi$
$450$		0			0
$451$		−	8.77139e6i		−	2.03061i
$452$		0			0
$453$	1.09936e6	+	118907.i	0.251706	+	0.0272247i
$454$		0			0
$455$	547119.			0.123895
$456$		0			0
$457$	−2.20376e6			−0.493599			−0.246799	−	0.969067i	$-0.579379\pi$
$457$	−2.20376e6			−0.493599			−0.246799	+	0.969067i	$0.579379\pi$
$458$		0			0
$459$	1.97755e6	+	662426.i	0.438122	+	0.146759i
$460$		0			0
$461$		−	1.43150e6i		−	0.313718i	−0.987621	−	0.156859i	$-0.949863\pi$
$461$		−	1.43150e6i		−	0.313718i	0.987621	−	0.156859i	$-0.0501369\pi$
$462$		0			0
$463$		−	7.31236e6i		−	1.58528i	−0.609692	−	0.792639i	$-0.708707\pi$
$463$		−	7.31236e6i		−	1.58528i	0.609692	−	0.792639i	$-0.291293\pi$
$464$		0			0
$465$	−75667.1	+	699579.i	−0.0162284	+	0.150039i
$466$		0			0
$467$	8.26234e6			1.75312			0.876559	−	0.481295i	$-0.159833\pi$
$467$	8.26234e6			1.75312			0.876559	+	0.481295i	$0.159833\pi$
$468$		0			0
$469$	7.07565e6			1.48537
$470$		0			0
$471$	649241.	−	6.00255e6i	0.134851	−	1.24676i
$472$		0			0
$473$			6.72640e6i			1.38239i
$474$		0			0
$475$		−	716662.i		−	0.145741i
$476$		0			0
$477$	−1.16598e6	+	5.32698e6i	−0.234637	+	1.07198i
$478$		0			0
$479$	−1.73045e6			−0.344604			−0.172302	−	0.985044i	$-0.555121\pi$
$479$	−1.73045e6			−0.344604			−0.172302	+	0.985044i	$0.555121\pi$
$480$		0			0
$481$	1.59116e6			0.313581
$482$		0			0
$483$	9.21903e6	+	997139.i	1.79811	+	0.194486i
$484$		0			0
$485$		−	991348.i		−	0.191369i
$486$		0			0
$487$			7.85313e6i			1.50045i	0.661185	+	0.750223i	$0.270054\pi$
$487$			7.85313e6i			1.50045i	−0.661185	+	0.750223i	$0.729946\pi$
$488$		0			0
$489$	−8.06752e6	−	872591.i	−1.52570	−	0.165021i
$490$		0			0
$491$	6.99056e6			1.30860			0.654302	−	0.756233i	$-0.272963\pi$
$491$	6.99056e6			1.30860			0.654302	+	0.756233i	$0.272963\pi$
$492$		0			0
$493$	−1.12676e6			−0.208792
$494$		0			0
$495$	−241770.	+	1.10457e6i	−0.0443496	+	0.202618i
$496$		0			0
$497$			5.97473e6i			1.08499i
$498$		0			0
$499$		−	5.17073e6i		−	0.929609i	−0.885413	−	0.464805i	$-0.846125\pi$
$499$		−	5.17073e6i		−	0.929609i	0.885413	−	0.464805i	$-0.153875\pi$
$500$		0			0
$501$	−795858.	+	7.35809e6i	−0.141658	+	1.30970i
$502$		0			0
$503$	−7.39136e6			−1.30258			−0.651290	−	0.758829i	$-0.725772\pi$
$503$	−7.39136e6			−1.30258			−0.651290	+	0.758829i	$0.725772\pi$
$504$		0			0
$505$	−1.14302e6			−0.199446
$506$		0			0
$507$	1208.27	−	11171.0i	0.000208758	−	0.00193007i
$508$		0			0
$509$			8.05541e6i			1.37814i	0.724695	+	0.689070i	$0.241981\pi$
$509$			8.05541e6i			1.37814i	−0.724695	+	0.689070i	$0.758019\pi$
$510$		0			0
$511$		−	1.03647e6i		−	0.175592i
$512$		0			0
$513$	835971.	+	280028.i	0.140248	+	0.0469795i
$514$		0			0
$515$	444160.			0.0737941
$516$		0			0
$517$	1.16599e7			1.91853
$518$		0			0
$519$	−6.63655e6	−	717816.i	−1.08149	−	0.116975i
$520$		0			0
$521$			6.37512e6i			1.02895i	0.857506	+	0.514475i	$0.172013\pi$
$521$			6.37512e6i			1.02895i	−0.857506	+	0.514475i	$0.827987\pi$
$522$		0			0
$523$			9.99175e6i			1.59730i	0.601794	+	0.798651i	$0.294453\pi$
$523$			9.99175e6i			1.59730i	−0.601794	+	0.798651i	$0.705547\pi$
$524$		0			0
$525$	−6.32637e6	−	684267.i	−1.00174	−	0.108350i
$526$		0			0
$527$	3.67286e6			0.576074
$528$		0			0
$529$	1.36981e7			2.12825
$530$		0			0
$531$	−3.82398e6	−	837002.i	−0.588544	−	0.128822i
$532$		0			0
$533$			7.77975e6i			1.18617i
$534$		0			0
$535$			782223.i			0.118153i
$536$		0			0
$537$	114108.	−	1.05498e6i	0.0170758	−	0.157874i
$538$		0			0
$539$	527560.			0.0782169
$540$		0			0
$541$	−1.10975e7			−1.63017			−0.815087	−	0.579339i	$-0.803311\pi$
$541$	−1.10975e7			−1.63017			−0.815087	+	0.579339i	$0.803311\pi$
$542$		0			0
$543$	−1.01781e6	+	9.41014e6i	−0.148138	+	1.36961i
$544$		0			0
$545$			309166.i			0.0445862i
$546$		0			0
$547$		−	7.87837e6i		−	1.12582i	−0.826519	−	0.562909i	$-0.809682\pi$
$547$		−	7.87837e6i		−	1.12582i	0.826519	−	0.562909i	$-0.190318\pi$
$548$		0			0
$549$	2.06008e6	+	450916.i	0.291712	+	0.0638506i
$550$		0			0
$551$	−476316.			−0.0668369
$552$		0			0
$553$	−861393.			−0.119781
$554$		0			0
$555$	273574.	+	29590.0i	0.0377001	+	0.00407768i
$556$		0			0
$557$			2.96402e6i			0.404802i	0.979303	+	0.202401i	$0.0648744\pi$
$557$			2.96402e6i			0.404802i	−0.979303	+	0.202401i	$0.935126\pi$
$558$		0			0
$559$		−	5.96595e6i		−	0.807514i
$560$		0			0
$561$	5.86773e6	+	634660.i	0.787160	+	0.0851400i
$562$		0			0
$563$	−1.10467e6			−0.146879			−0.0734395	−	0.997300i	$-0.523398\pi$
$563$	−1.10467e6			−0.146879			−0.0734395	+	0.997300i	$0.523398\pi$
$564$		0			0
$565$	−959787.			−0.126489
$566$		0			0
$567$	3.27015e6	−	7.11221e6i	0.427179	−	0.929066i
$568$		0			0
$569$			706266.i			0.0914509i	0.998954	+	0.0457254i	$0.0145599\pi$
$569$			706266.i			0.0914509i	−0.998954	+	0.0457254i	$0.985440\pi$
$570$		0			0
$571$			8.81546e6i			1.13150i	0.824577	+	0.565750i	$0.191413\pi$
$571$			8.81546e6i			1.13150i	−0.824577	+	0.565750i	$0.808587\pi$
$572$		0			0
$573$	315471.	−	2.91668e6i	0.0401396	−	0.371110i
$574$		0			0
$575$	−1.38169e7			−1.74277
$576$		0			0
$577$	−6.30662e6			−0.788601			−0.394300	−	0.918982i	$-0.629013\pi$
$577$	−6.30662e6			−0.788601			−0.394300	+	0.918982i	$0.629013\pi$
$578$		0			0
$579$	−1.00139e6	+	9.25832e6i	−0.124138	+	1.14772i
$580$		0			0
$581$		−	1.34538e7i		−	1.65350i
$582$		0			0
$583$			1.54319e7i			1.88039i
$584$		0			0
$585$	214437.	−	979690.i	0.0259066	−	0.118358i
$586$		0			0
$587$	−4.68327e6			−0.560988			−0.280494	−	0.959856i	$-0.590498\pi$
$587$	−4.68327e6			−0.560988			−0.280494	+	0.959856i	$0.590498\pi$
$588$		0			0
$589$	1.55263e6			0.184409
$590$		0			0
$591$	−1.52399e7	−	1.64837e6i	−1.79479	−	0.194127i
$592$		0			0
$593$		−	5.23289e6i		−	0.611089i	−0.952178	−	0.305545i	$-0.901161\pi$
$593$		−	5.23289e6i		−	0.611089i	0.952178	−	0.305545i	$-0.0988385\pi$
$594$		0			0
$595$		−	493869.i		−	0.0571899i
$596$		0			0
$597$	1.08303e7	+	1.17142e6i	1.24367	+	0.134517i
$598$		0			0
$599$	−370351.			−0.0421741			−0.0210871	−	0.999778i	$-0.506713\pi$
$599$	−370351.			−0.0421741			−0.0210871	+	0.999778i	$0.506713\pi$
$600$		0			0
$601$	−1.55229e7			−1.75302			−0.876511	−	0.481382i	$-0.840135\pi$
$601$	−1.55229e7			−1.75302			−0.876511	+	0.481382i	$0.840135\pi$
$602$		0			0
$603$	2.77322e6	−	1.26699e7i	0.310593	−	1.41899i
$604$		0			0
$605$			2.11010e6i			0.234377i
$606$		0			0
$607$		−	4.07624e6i		−	0.449043i	−0.974469	−	0.224522i	$-0.927918\pi$
$607$		−	4.07624e6i		−	0.449043i	0.974469	−	0.224522i	$-0.0720820\pi$
$608$		0			0
$609$	−454784.	+	4.20470e6i	−0.0496892	+	0.459401i
$610$		0			0
$611$	−1.03417e7			−1.12070
$612$		0			0
$613$	−1.13469e7			−1.21962			−0.609811	−	0.792547i	$-0.708755\pi$
$613$	−1.13469e7			−1.21962			−0.609811	+	0.792547i	$0.708755\pi$
$614$		0			0
$615$	−144677.	+	1.33761e6i	−0.0154245	+	0.142607i
$616$		0			0
$617$			7.85036e6i			0.830188i	0.909779	+	0.415094i	$0.136251\pi$
$617$			7.85036e6i			0.830188i	−0.909779	+	0.415094i	$0.863749\pi$
$618$		0			0
$619$		−	5.05257e6i		−	0.530012i	−0.964247	−	0.265006i	$-0.914626\pi$
$619$		−	5.05257e6i		−	0.530012i	0.964247	−	0.265006i	$-0.0853740\pi$
$620$		0			0
$621$	5.39880e6	−	1.61171e7i	0.561783	−	1.67710i
$622$		0			0
$623$	7.84866e6			0.810168
$624$		0			0
$625$	9.33848e6			0.956260
$626$		0			0
$627$	2.48048e6	+	268291.i	0.251980	+	0.0272544i
$628$		0			0
$629$		−	1.43629e6i		−	0.144749i
$630$		0			0
$631$		−	1.88223e7i		−	1.88191i	−0.338530	−	0.940956i	$-0.609930\pi$
$631$		−	1.88223e7i		−	1.88191i	0.338530	−	0.940956i	$-0.390070\pi$
$632$		0			0
$633$	8.65874e6	+	936537.i	0.858905	+	0.0929000i
$634$		0			0
$635$	−1.54728e6			−0.152277
$636$		0			0
$637$	−467917.			−0.0456899
$638$		0			0
$639$	1.06986e7	+	2.34173e6i	1.03651	+	0.226874i
$640$		0			0
$641$			1.21548e7i			1.16843i	0.811597	+	0.584217i	$0.198598\pi$
$641$			1.21548e7i			1.16843i	−0.811597	+	0.584217i	$0.801402\pi$
$642$		0			0
$643$			6.75899e6i			0.644695i	0.946621	+	0.322348i	$0.104472\pi$
$643$			6.75899e6i			0.644695i	−0.946621	+	0.322348i	$0.895528\pi$
$644$		0			0
$645$	110946.	−	1.02575e6i	0.0105006	−	0.0970830i
$646$		0			0
$647$	7.10161e6			0.666954			0.333477	−	0.942758i	$-0.391778\pi$
$647$	7.10161e6			0.666954			0.333477	+	0.942758i	$0.391778\pi$
$648$		0			0
$649$	−1.10778e7			−1.03239
$650$		0			0
$651$	1.48245e6	−	1.37060e7i	0.137097	−	1.26753i
$652$		0			0
$653$			3.87673e6i			0.355781i	0.984050	+	0.177891i	$0.0569273\pi$
$653$			3.87673e6i			0.355781i	−0.984050	+	0.177891i	$0.943073\pi$
$654$		0			0
$655$		−	641453.i		−	0.0584200i
$656$		0			0
$657$	−1.85594e6	−	406232.i	−0.167745	−	0.0367165i
$658$		0			0
$659$	−3.63662e6			−0.326201			−0.163100	−	0.986609i	$-0.552149\pi$
$659$	−3.63662e6			−0.326201			−0.163100	+	0.986609i	$0.552149\pi$
$660$		0			0
$661$	8.07769e6			0.719091			0.359545	−	0.933128i	$-0.382932\pi$
$661$	8.07769e6			0.719091			0.359545	+	0.933128i	$0.382932\pi$
$662$		0			0
$663$	−5.20436e6	−	562908.i	−0.459815	−	0.0497341i
$664$		0			0
$665$		−	208774.i		−	0.0183072i
$666$		0			0
$667$			9.18313e6i			0.799238i
$668$		0			0
$669$	−2.04536e7	−	2.21228e6i	−1.76687	−	0.191106i
$670$		0			0
$671$	5.96792e6			0.511702
$672$		0			0
$673$	1.48594e7			1.26463			0.632317	−	0.774710i	$-0.282104\pi$
$673$	1.48594e7			1.26463			0.632317	+	0.774710i	$0.282104\pi$
$674$		0			0
$675$	−3.70482e6	+	1.10600e7i	−0.312974	+	0.934323i
$676$		0			0
$677$			8.98129e6i			0.753125i	0.926391	+	0.376562i	$0.122894\pi$
$677$			8.98129e6i			0.753125i	−0.926391	+	0.376562i	$0.877106\pi$
$678$		0			0
$679$			1.94222e7i			1.61668i
$680$		0			0
$681$	−794542.	+	7.34592e6i	−0.0656521	+	0.606986i
$682$		0			0
$683$	−2.64006e6			−0.216552			−0.108276	−	0.994121i	$-0.534533\pi$
$683$	−2.64006e6			−0.216552			−0.108276	+	0.994121i	$0.534533\pi$
$684$		0			0
$685$	−1.54986e6			−0.126202
$686$		0			0
$687$	−1.49430e6	+	1.38156e7i	−0.120794	+	1.11680i
$688$		0			0
$689$		−	1.36872e7i		−	1.09842i
$690$		0			0
$691$		−	3.95325e6i		−	0.314963i	−0.987522	−	0.157482i	$-0.949663\pi$
$691$		−	3.95325e6i		−	0.314963i	0.987522	−	0.157482i	$-0.0503375\pi$
$692$		0			0
$693$	4.73670e6	−	2.16404e7i	0.374664	−	1.71171i
$694$		0			0
$695$	−1.05603e6			−0.0829305
$696$		0			0
$697$	7.02257e6			0.547538
$698$		0			0
$699$	−1.12573e7	−	1.21761e6i	−0.871451	−	0.0942570i
$700$		0			0
$701$		−	6.18289e6i		−	0.475222i	−0.971360	−	0.237611i	$-0.923636\pi$
$701$		−	6.18289e6i		−	0.475222i	0.971360	−	0.237611i	$-0.0763643\pi$
$702$		0			0
$703$		−	607166.i		−	0.0463361i
$704$		0			0
$705$	−1.77809e6	−	192320.i	−0.134736	−	0.0145731i
$706$		0			0
$707$	2.23937e7			1.68491
$708$		0			0
$709$	1.01154e7			0.755734			0.377867	−	0.925860i	$-0.376658\pi$
$709$	1.01154e7			0.755734			0.377867	+	0.925860i	$0.376658\pi$
$710$		0			0
$711$	−337613.	+	1.54244e6i	−0.0250464	+	0.114429i
$712$		0			0
$713$		−	2.99340e7i		−	2.20516i
$714$		0			0
$715$		−	2.83809e6i		−	0.207616i
$716$		0			0
$717$	−1.45840e6	+	1.34836e7i	−0.105945	+	0.979511i
$718$		0			0
$719$	2.51532e7			1.81456			0.907279	−	0.420530i	$-0.138156\pi$
$719$	2.51532e7			1.81456			0.907279	+	0.420530i	$0.138156\pi$
$720$		0			0
$721$	−8.70187e6			−0.623411
$722$		0			0
$723$	−1.92115e6	+	1.77620e7i	−0.136683	+	1.26370i
$724$		0			0
$725$		−	6.30174e6i		−	0.445262i
$726$		0			0
$727$		−	1.70218e7i		−	1.19445i	−0.802072	−	0.597227i	$-0.796269\pi$
$727$		−	1.70218e7i		−	1.19445i	0.802072	−	0.597227i	$-0.203731\pi$
$728$		0			0
$729$	−1.14537e7	−	8.64319e6i	−0.798226	−	0.602359i
$730$		0			0
$731$	−5.38531e6			−0.372749
$732$		0			0
$733$	−1.61715e7			−1.11171			−0.555853	−	0.831281i	$-0.687608\pi$
$733$	−1.61715e7			−1.11171			−0.555853	+	0.831281i	$0.687608\pi$
$734$		0			0
$735$	−80451.0	−	8701.66i	−0.00549305	−	0.000594133i
$736$		0			0
$737$		−	3.67039e7i		−	2.48910i
$738$		0			0
$739$		−	3.81667e6i		−	0.257083i	−0.991704	−	0.128542i	$-0.958970\pi$
$739$		−	3.81667e6i		−	0.257083i	0.991704	−	0.128542i	$-0.0410296\pi$
$740$		0			0
$741$	−2.20005e6	−	237959.i	−0.147193	−	0.0159205i
$742$		0			0
$743$	1.59769e7			1.06175			0.530873	−	0.847451i	$-0.321864\pi$
$743$	1.59769e7			1.06175			0.530873	+	0.847451i	$0.321864\pi$
$744$		0			0
$745$	−2.02544e6			−0.133699
$746$		0			0
$747$	−2.40909e7	−	5.27307e6i	−1.57961	−	0.345750i
$748$		0			0
$749$		−	1.53251e7i		−	0.998157i
$750$		0			0
$751$		−	2.48347e6i		−	0.160679i	−0.996768	−	0.0803395i	$-0.974400\pi$
$751$		−	2.48347e6i		−	0.160679i	0.996768	−	0.0803395i	$-0.0256005\pi$
$752$		0			0
$753$	−232891.	+	2.15319e6i	−0.0149680	+	0.138387i
$754$		0			0
$755$	−479983.			−0.0306449
$756$		0			0
$757$	2.17289e6			0.137815			0.0689077	−	0.997623i	$-0.478049\pi$
$757$	2.17289e6			0.137815			0.0689077	+	0.997623i	$0.478049\pi$
$758$		0			0
$759$	5.17250e6	−	4.78223e7i	0.325909	−	3.01319i
$760$		0			0
$761$		−	1.95287e7i		−	1.22240i	−0.791478	−	0.611198i	$-0.790688\pi$
$761$		−	1.95287e7i		−	1.22240i	0.791478	−	0.611198i	$-0.209312\pi$
$762$		0			0
$763$		−	6.05710e6i		−	0.376664i
$764$		0			0
$765$	−884340.	−	193567.i	−0.0546343	−	0.0119585i
$766$		0			0
$767$	9.82540e6			0.603062
$768$		0			0
$769$	−1.16573e6			−0.0710857			−0.0355428	−	0.999368i	$-0.511316\pi$
$769$	−1.16573e6			−0.0710857			−0.0355428	+	0.999368i	$0.511316\pi$
$770$		0			0
$771$	−2.72577e7	−	2.94822e6i	−1.65140	−	0.178617i
$772$		0			0
$773$			1.92482e7i			1.15862i	0.815107	+	0.579311i	$0.196678\pi$
$773$			1.92482e7i			1.15862i	−0.815107	+	0.579311i	$0.803322\pi$
$774$		0			0
$775$			2.05416e7i			1.22851i
$776$		0			0
$777$	−5.35979e6	−	579720.i	−0.318490	−	0.0344481i
$778$		0			0
$779$	2.96866e6			0.175274
$780$		0			0
$781$	3.09930e7			1.81818
$782$		0			0
$783$	7.35084e6	+	2.46234e6i	0.428482	+	0.143530i
$784$		0			0
$785$			2.62073e6i			0.151792i
$786$		0			0
$787$		−	2.41154e7i		−	1.38790i	−0.720024	−	0.693949i	$-0.755869\pi$
$787$		−	2.41154e7i		−	1.38790i	0.720024	−	0.693949i	$-0.244131\pi$
$788$		0			0
$789$	1.06741e6	−	9.86867e6i	0.0610431	−	0.564373i
$790$		0			0
$791$	1.88039e7			1.06858
$792$		0			0
$793$	−5.29322e6			−0.298908
$794$		0			0
$795$	254536.	−	2.35331e6i	0.0142834	−	0.132057i
$796$		0			0
$797$			2.46187e6i			0.137284i	0.997641	+	0.0686420i	$0.0218666\pi$
$797$			2.46187e6i			0.137284i	−0.997641	+	0.0686420i	$0.978133\pi$
$798$		0			0
$799$			9.33519e6i			0.517316i
$800$		0			0
$801$	3.07619e6	−	1.40541e7i	0.169407	−	0.773965i
$802$		0			0
$803$	−5.37653e6			−0.294248
$804$		0			0
$805$	−4.02506e6			−0.218919
$806$		0			0
$807$	−5.11891e6	−	553666.i	−0.276690	−	0.0299270i
$808$		0			0
$809$		−	9.92214e6i		−	0.533008i	−0.963834	−	0.266504i	$-0.914131\pi$
$809$		−	9.92214e6i		−	0.533008i	0.963834	−	0.266504i	$-0.0858686\pi$
$810$		0			0
$811$			2.34389e7i			1.25137i	0.780076	+	0.625684i	$0.215180\pi$
$811$			2.34389e7i			1.25137i	−0.780076	+	0.625684i	$0.784820\pi$
$812$		0			0
$813$	2.04206e7	+	2.20871e6i	1.08353	+	0.117196i
$814$		0			0
$815$	3.52231e6			0.185752
$816$		0			0
$817$	−2.27654e6			−0.119322
$818$		0			0
$819$	−4.20119e6	+	1.91938e7i	−0.218858	+	0.999888i
$820$		0			0
$821$		−	1.50269e7i		−	0.778059i	−0.921225	−	0.389029i	$-0.872810\pi$
$821$		−	1.50269e7i		−	0.778059i	0.921225	−	0.389029i	$-0.127190\pi$
$822$		0			0
$823$		−	3.54090e6i		−	0.182228i	−0.995840	−	0.0911138i	$-0.970957\pi$
$823$		−	3.54090e6i		−	0.182228i	0.995840	−	0.0911138i	$-0.0290427\pi$
$824$		0			0
$825$	−3.54953e6	+	3.28171e7i	−0.181567	+	1.67867i
$826$		0			0
$827$	3.15599e7			1.60462			0.802308	−	0.596910i	$-0.203605\pi$
$827$	3.15599e7			1.60462			0.802308	+	0.596910i	$0.203605\pi$
$828$		0			0
$829$	−1.34618e7			−0.680326			−0.340163	−	0.940367i	$-0.610482\pi$
$829$	−1.34618e7			−0.680326			−0.340163	+	0.940367i	$0.610482\pi$
$830$		0			0
$831$	−3.35183e6	+	3.09893e7i	−0.168376	+	1.55671i
$832$		0			0
$833$			422376.i			0.0210905i
$834$		0			0
$835$		−	3.21257e6i		−	0.159454i
$836$		0			0
$837$	−2.39613e7	−	8.02642e6i	−1.18222	−	0.396012i
$838$		0			0
$839$	1.65806e7			0.813199			0.406599	−	0.913607i	$-0.366715\pi$
$839$	1.65806e7			0.813199			0.406599	+	0.913607i	$0.366715\pi$
$840$		0			0
$841$	1.63228e7			0.795802
$842$		0			0
$843$	1.17109e7	+	1.26666e6i	0.567572	+	0.0613891i
$844$		0			0
$845$			4877.30i			0.000234984i
$846$		0			0
$847$		−	4.13405e7i		−	1.98001i
$848$		0			0
$849$	−7.76378e6	−	839738.i	−0.369661	−	0.0399829i
$850$		0			0
$851$	−1.17059e7			−0.554089
$852$		0			0
$853$	−2.35123e7			−1.10643			−0.553214	−	0.833039i	$-0.686599\pi$
$853$	−2.35123e7			−1.10643			−0.553214	+	0.833039i	$0.686599\pi$
$854$		0			0
$855$	−373838.	−	81826.7i	−0.0174891	−	0.00382807i
$856$		0			0
$857$		−	2.12776e7i		−	0.989627i	−0.868999	−	0.494813i	$-0.835236\pi$
$857$		−	2.12776e7i		−	0.989627i	0.868999	−	0.494813i	$-0.164764\pi$
$858$		0			0
$859$			9.20510e6i			0.425643i	0.977091	+	0.212821i	$0.0682653\pi$
$859$			9.20510e6i			0.425643i	−0.977091	+	0.212821i	$0.931735\pi$
$860$		0			0
$861$	2.83447e6	−	2.62060e7i	0.130306	−	1.20474i
$862$		0			0
$863$	7.62622e6			0.348564			0.174282	−	0.984696i	$-0.444240\pi$
$863$	7.62622e6			0.348564			0.174282	+	0.984696i	$0.444240\pi$
$864$		0			0
$865$	2.89754e6			0.131671
$866$		0			0
$867$	1.87196e6	−	1.73072e7i	0.0845765	−	0.781950i
$868$		0			0
$869$			4.46835e6i			0.200723i
$870$		0			0
$871$			3.25543e7i			1.45400i
$872$		0			0
$873$	3.47781e7	+	7.61232e6i	1.54444	+	0.338050i
$874$		0			0
$875$	5.56530e6			0.245736
$876$		0			0
$877$	3.68055e6			0.161590			0.0807948	−	0.996731i	$-0.474254\pi$
$877$	3.68055e6			0.161590			0.0807948	+	0.996731i	$0.474254\pi$
$878$		0			0
$879$	−2.22832e7	−	2.41017e6i	−0.972757	−	0.105214i
$880$		0			0
$881$			1.12805e7i			0.489653i	0.969567	+	0.244826i	$0.0787310\pi$
$881$			1.12805e7i			0.489653i	−0.969567	+	0.244826i	$0.921269\pi$
$882$		0			0
$883$			9.26250e6i			0.399785i	0.979818	+	0.199893i	$0.0640593\pi$
$883$			9.26250e6i			0.399785i	−0.979818	+	0.199893i	$0.935941\pi$
$884$		0			0
$885$	1.68932e6	+	182719.i	0.0725028	+	0.00784197i
$886$		0			0
$887$	−1.85163e6			−0.0790216			−0.0395108	−	0.999219i	$-0.512580\pi$
$887$	−1.85163e6			−0.0790216			−0.0395108	+	0.999219i	$0.512580\pi$
$888$		0			0
$889$	3.03140e7			1.28644
$890$		0			0
$891$	−3.68935e7	−	1.69634e7i	−1.55688	−	0.715844i
$892$		0			0
$893$			3.94628e6i			0.165599i
$894$		0			0
$895$			460609.i			0.0192209i
$896$		0			0
$897$	−4.58773e6	+	4.24158e7i	−0.190378	+	1.76014i
$898$		0			0
$899$	1.36526e7			0.563398
$900$		0			0
$901$	−1.23551e7			−0.507031
$902$		0			0
$903$	−2.17363e6	+	2.00963e7i	−0.0887087	+	0.820155i
$904$		0			0
$905$		−	4.10850e6i		−	0.166748i
$906$		0			0
$907$			1.81822e7i			0.733886i	0.930243	+	0.366943i	$0.119596\pi$
$907$			1.81822e7i			0.733886i	−0.930243	+	0.366943i	$0.880404\pi$
$908$		0			0
$909$	8.77696e6	−	4.00989e7i	0.352318	−	1.60962i
$910$		0			0
$911$	−2.97211e7			−1.18650			−0.593251	−	0.805018i	$-0.702156\pi$
$911$	−2.97211e7			−1.18650			−0.593251	+	0.805018i	$0.702156\pi$
$912$		0			0
$913$	−6.97896e7			−2.77086
$914$		0			0
$915$	−910086.	−	98435.8i	−0.0359360	−	0.00388687i
$916$		0			0
$917$			1.25672e7i			0.493531i
$918$		0			0
$919$		−	1.67014e7i		−	0.652325i	−0.945314	−	0.326163i	$-0.894244\pi$
$919$		−	1.67014e7i		−	0.652325i	0.945314	−	0.326163i	$-0.105756\pi$
$920$		0			0
$921$	−1.13258e7	−	1.22501e6i	−0.439965	−	0.0475871i
$922$		0			0
$923$	−2.74891e7			−1.06208
$924$		0			0
$925$	8.03291e6			0.308687
$926$		0			0
$927$	−3.41060e6	+	1.55819e7i	−0.130356	+	0.595553i
$928$		0			0
$929$		−	7.30629e6i		−	0.277752i	−0.990310	−	0.138876i	$-0.955651\pi$
$929$		−	7.30629e6i		−	0.277752i	0.990310	−	0.138876i	$-0.0443490\pi$
$930$		0			0
$931$			178552.i			0.00675134i
$932$		0			0
$933$	1.08640e6	−	1.00443e7i	0.0408588	−	0.377759i
$934$		0			0
$935$	−2.56187e6			−0.0958359
$936$		0			0
$937$	1.58175e7			0.588558			0.294279	−	0.955720i	$-0.404921\pi$
$937$	1.58175e7			0.588558			0.294279	+	0.955720i	$0.404921\pi$
$938$		0			0
$939$	220011.	−	2.03411e6i	0.00814292	−	0.0752852i
$940$		0			0
$941$			2.17746e7i			0.801633i	0.916158	+	0.400816i	$0.131273\pi$
$941$			2.17746e7i			0.801633i	−0.916158	+	0.400816i	$0.868727\pi$
$942$		0			0
$943$		−	5.72343e7i		−	2.09593i
$944$		0			0
$945$	−1.07927e6	+	3.22195e6i	−0.0393142	+	0.117365i
$946$		0			0
$947$	1.81364e7			0.657169			0.328584	−	0.944475i	$-0.393428\pi$
$947$	1.81364e7			0.657169			0.328584	+	0.944475i	$0.393428\pi$
$948$		0			0
$949$	4.76869e6			0.171883
$950$		0			0
$951$	−5.01476e7	−	5.42401e6i	−1.79804	−	0.194477i
$952$		0			0
$953$			2.50157e7i			0.892239i	0.894973	+	0.446119i	$0.147194\pi$
$953$			2.50157e7i			0.892239i	−0.894973	+	0.446119i	$0.852806\pi$
$954$		0			0
$955$			1.27343e6i			0.0451823i
$956$		0			0
$957$	2.18112e7	+	2.35913e6i	0.769840	+	0.0832667i
$958$		0			0
$959$	3.03645e7			1.06615
$960$		0			0
$961$	−1.58738e7			−0.554463
$962$		0			0
$963$	−2.74417e7	−	6.00650e6i	−0.953553	−	0.208716i
$964$		0			0
$965$		−	4.04221e6i		−	0.139734i
$966$		0			0
$967$			5.10696e7i			1.75629i	0.478395	+	0.878145i	$0.341219\pi$
$967$			5.10696e7i			1.75629i	−0.478395	+	0.878145i	$0.658781\pi$
$968$		0			0
$969$	−214799.	+	1.98592e6i	−0.00734892	+	0.0679443i
$970$		0			0
$971$	−3.26999e7			−1.11301			−0.556505	−	0.830844i	$-0.687858\pi$
$971$	−3.26999e7			−1.11301			−0.556505	+	0.830844i	$0.687858\pi$
$972$		0			0
$973$	2.06895e7			0.700595
$974$		0			0
$975$	3.14824e6	−	2.91070e7i	0.106061	−	0.980585i
$976$		0			0
$977$		−	3.15607e7i		−	1.05782i	−0.848679	−	0.528909i	$-0.822601\pi$
$977$		−	3.15607e7i		−	1.05782i	0.848679	−	0.528909i	$-0.177399\pi$
$978$		0			0
$979$		−	4.07137e7i		−	1.35764i
$980$		0			0
$981$	−1.08461e7	−	2.37401e6i	−0.359832	−	0.0787609i
$982$		0			0
$983$	−1.82318e7			−0.601791			−0.300896	−	0.953657i	$-0.597286\pi$
$983$	−1.82318e7			−0.601791			−0.300896	+	0.953657i	$0.597286\pi$
$984$		0			0
$985$	6.65381e6			0.218514
$986$		0			0
$987$	3.48360e7	+	3.76789e6i	1.13824	+	0.123113i
$988$		0			0
$989$			4.38905e7i			1.42685i
$990$		0			0
$991$			1.46491e7i			0.473836i	0.971530	+	0.236918i	$0.0761372\pi$
$991$			1.46491e7i			0.473836i	−0.971530	+	0.236918i	$0.923863\pi$
$992$		0			0
$993$	2.72146e7	+	2.94356e6i	0.875848	+	0.0947325i
$994$		0			0
$995$	−4.72856e6			−0.151416
$996$		0			0
$997$	−2.14521e7			−0.683490			−0.341745	−	0.939793i	$-0.611018\pi$
$997$	−2.14521e7			−0.683490			−0.341745	+	0.939793i	$0.611018\pi$
$998$		0			0
$999$	−3.13878e6	+	9.37021e6i	−0.0995054	+	0.297054i

Twists

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

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

\(f(q)\)	\(=\)	\(q+(1.67629 - 15.4981i) q^{3} +6.76651i q^{5} -132.568i q^{7} +(-237.380 - 51.9584i) q^{9} +O(q^{10})\)	\(q+(1.67629 - 15.4981i) q^{3} +6.76651i q^{5} -132.568i q^{7} +(-237.380 - 51.9584i) q^{9} -687.674 q^{11} +609.929 q^{13} +(104.868 + 11.3426i) q^{15} -550.567i q^{17} -232.742i q^{19} +(-2054.54 - 222.221i) q^{21} -4487.15 q^{23} +3079.21 q^{25} +(-1203.17 + 3591.84i) q^{27} -2046.54i q^{29} +6671.05i q^{31} +(-1152.74 + 10657.6i) q^{33} +897.020 q^{35} +2608.75 q^{37} +(1022.42 - 9452.73i) q^{39} +12755.2i q^{41} -9781.38i q^{43} +(351.577 - 1606.23i) q^{45} -16955.6 q^{47} -767.166 q^{49} +(-8532.72 - 922.907i) q^{51} -22440.7i q^{53} -4653.15i q^{55} +(-3607.05 - 390.142i) q^{57} +16109.1 q^{59} -8678.41 q^{61} +(-6888.00 + 31468.9i) q^{63} +4127.09i q^{65} +53373.9i q^{67} +(-7521.74 + 69542.1i) q^{69} -45069.3 q^{71} +7818.42 q^{73} +(5161.64 - 47721.9i) q^{75} +91163.3i q^{77} -6497.77i q^{79} +(53649.7 + 24667.8i) q^{81} +101487. q^{83} +3725.42 q^{85} +(-31717.4 - 3430.59i) q^{87} +59204.9i q^{89} -80856.9i q^{91} +(103388. + 11182.6i) q^{93} +1574.85 q^{95} -146508. q^{97} +(163240. + 35730.4i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3}+O(q^{10}) \) 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