LMFDB - Embedded newform 378.5.b.a.323.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,5,Mod(323,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("378.323");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	378.b (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:	$39.0738460457$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.5443747577856.29
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324 $ x^8 + 24x^6 + 180x^4 + 488*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.4
Root			$-2.39656i$ of defining polynomial
Character	$\chi$	$=$	378.323
Dual form			378.5.b.a.323.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} +14.9735i q^{5} -18.5203 q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} +14.9735i q^{5} -18.5203 q^{7} +22.6274i q^{8} +42.3515 q^{10} -207.672i q^{11} -62.8805 q^{13} +52.3832i q^{14} +64.0000 q^{16} +187.297i q^{17} +462.879 q^{19} -119.788i q^{20} -587.385 q^{22} -455.668i q^{23} +400.793 q^{25} +177.853i q^{26} +148.162 q^{28} +1181.31i q^{29} -1516.38 q^{31} -181.019i q^{32} +529.755 q^{34} -277.314i q^{35} -2092.12 q^{37} -1309.22i q^{38} -338.812 q^{40} +2995.00i q^{41} -3082.43 q^{43} +1661.38i q^{44} -1288.82 q^{46} +1652.15i q^{47} +343.000 q^{49} -1133.62i q^{50} +503.044 q^{52} +2863.40i q^{53} +3109.58 q^{55} -419.066i q^{56} +3341.26 q^{58} +3120.03i q^{59} +3220.68 q^{61} +4288.97i q^{62} -512.000 q^{64} -941.542i q^{65} -5231.48 q^{67} -1498.37i q^{68} -784.361 q^{70} -2620.11i q^{71} -6236.52 q^{73} +5917.41i q^{74} -3703.04 q^{76} +3846.14i q^{77} +3408.06 q^{79} +958.306i q^{80} +8471.13 q^{82} -6087.85i q^{83} -2804.49 q^{85} +8718.43i q^{86} +4699.08 q^{88} +13322.1i q^{89} +1164.56 q^{91} +3645.34i q^{92} +4672.99 q^{94} +6930.94i q^{95} -6073.09 q^{97} -970.151i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4}+O(q^{10}) $ 8 * q - 64 * q^4	$ 8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100}) $ 8 * q - 64 * q^4 - 224 * q^10 - 376 * q^13 + 512 * q^16 + 1120 * q^19 - 1120 * q^22 + 792 * q^25 - 880 * q^31 - 1792 * q^34 + 1576 * q^37 + 1792 * q^40 - 5768 * q^43 - 160 * q^46 + 2744 * q^49 + 3008 * q^52 + 488 * q^55 + 7552 * q^58 - 2560 * q^61 - 4096 * q^64 + 23784 * q^67 + 1568 * q^70 - 13176 * q^73 - 8960 * q^76 - 9592 * q^79 - 3360 * q^82 - 39880 * q^85 + 8960 * q^88 + 5096 * q^91 + 4608 * q^94 - 14016 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	14.9735i	0.598941i	0.954106	+	0.299471i	$0.0968100\pi$
$5$	14.9735i	0.598941i	−0.954106	+	0.299471i	$0.903190\pi$
$6$	0	0
$7$	−18.5203	−0.377964
$7$	−18.5203	−0.377964
$8$	22.6274i	0.353553i
$9$	0	0
$10$	42.3515	0.423515
$11$	− 207.672i	− 1.71630i	−0.513401	−	0.858149i	$-0.671615\pi$
$11$	− 207.672i	− 1.71630i	0.513401	−	0.858149i	$-0.328385\pi$
$12$	0	0
$13$	−62.8805	−0.372074	−0.186037	−	0.982543i	$-0.559564\pi$
$13$	−62.8805	−0.372074	−0.186037	+	0.982543i	$0.559564\pi$
$14$	52.3832i	0.267261i
$15$	0	0
$16$	64.0000	0.250000
$17$	187.297i	0.648085i	0.946042	+	0.324043i	$0.105042\pi$
$17$	187.297i	0.648085i	−0.946042	+	0.324043i	$0.894958\pi$
$18$	0	0
$19$	462.879	1.28221	0.641107	−	0.767451i	$-0.278475\pi$
$19$	462.879	1.28221	0.641107	+	0.767451i	$0.278475\pi$
$20$	− 119.788i	− 0.299471i
$21$	0	0
$22$	−587.385	−1.21361
$23$	− 455.668i	− 0.861376i	−0.902501	−	0.430688i	$-0.858271\pi$
$23$	− 455.668i	− 0.861376i	0.902501	−	0.430688i	$-0.141729\pi$
$24$	0	0
$25$	400.793	0.641270
$26$	177.853i	0.263096i
$27$	0	0
$28$	148.162	0.188982
$29$	1181.31i	1.40465i	0.711855	+	0.702326i	$0.247855\pi$
$29$	1181.31i	1.40465i	−0.711855	+	0.702326i	$0.752145\pi$
$30$	0	0
$31$	−1516.38	−1.57792	−0.788960	−	0.614445i	$-0.789380\pi$
$31$	−1516.38	−1.57792	−0.788960	+	0.614445i	$0.789380\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	529.755	0.458265
$35$	− 277.314i	− 0.226378i
$36$	0	0
$37$	−2092.12	−1.52821	−0.764105	−	0.645092i	$-0.776819\pi$
$37$	−2092.12	−1.52821	−0.764105	+	0.645092i	$0.776819\pi$
$38$	− 1309.22i	− 0.906663i
$39$	0	0
$40$	−338.812	−0.211758
$41$	2995.00i	1.78168i	0.454321	+	0.890838i	$0.349882\pi$
$41$	2995.00i	1.78168i	−0.454321	+	0.890838i	$0.650118\pi$
$42$	0	0
$43$	−3082.43	−1.66708	−0.833540	−	0.552459i	$-0.813690\pi$
$43$	−3082.43	−1.66708	−0.833540	+	0.552459i	$0.813690\pi$
$44$	1661.38i	0.858149i
$45$	0	0
$46$	−1288.82	−0.609085
$47$	1652.15i	0.747918i	0.927445	+	0.373959i	$0.122000\pi$
$47$	1652.15i	0.747918i	−0.927445	+	0.373959i	$0.878000\pi$
$48$	0	0
$49$	343.000	0.142857
$50$	− 1133.62i	− 0.453446i
$51$	0	0
$52$	503.044	0.186037
$53$	2863.40i	1.01936i	0.860363	+	0.509682i	$0.170237\pi$
$53$	2863.40i	1.01936i	−0.860363	+	0.509682i	$0.829763\pi$
$54$	0	0
$55$	3109.58	1.02796
$56$	− 419.066i	− 0.133631i
$57$	0	0
$58$	3341.26	0.993239
$59$	3120.03i	0.896302i	0.893958	+	0.448151i	$0.147917\pi$
$59$	3120.03i	0.896302i	−0.893958	+	0.448151i	$0.852083\pi$
$60$	0	0
$61$	3220.68	0.865540	0.432770	−	0.901504i	$-0.357536\pi$
$61$	3220.68	0.865540	0.432770	+	0.901504i	$0.357536\pi$
$62$	4288.97i	1.11576i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 941.542i	− 0.222850i
$66$	0	0
$67$	−5231.48	−1.16540	−0.582700	−	0.812688i	$-0.698004\pi$
$67$	−5231.48	−1.16540	−0.582700	+	0.812688i	$0.698004\pi$
$68$	− 1498.37i	− 0.324043i
$69$	0	0
$70$	−784.361	−0.160074
$71$	− 2620.11i	− 0.519759i	−0.965641	−	0.259880i	$-0.916317\pi$
$71$	− 2620.11i	− 0.519759i	0.965641	−	0.259880i	$-0.0836828\pi$
$72$	0	0
$73$	−6236.52	−1.17030	−0.585149	−	0.810926i	$-0.698964\pi$
$73$	−6236.52	−1.17030	−0.585149	+	0.810926i	$0.698964\pi$
$74$	5917.41i	1.08061i
$75$	0	0
$76$	−3703.04	−0.641107
$77$	3846.14i	0.648700i
$78$	0	0
$79$	3408.06	0.546076	0.273038	−	0.962003i	$-0.411971\pi$
$79$	3408.06	0.546076	0.273038	+	0.962003i	$0.411971\pi$
$80$	958.306i	0.149735i
$81$	0	0
$82$	8471.13	1.25984
$83$	− 6087.85i	− 0.883706i	−0.897087	−	0.441853i	$-0.854321\pi$
$83$	− 6087.85i	− 0.883706i	0.897087	−	0.441853i	$-0.145679\pi$
$84$	0	0
$85$	−2804.49	−0.388165
$86$	8718.43i	1.17880i
$87$	0	0
$88$	4699.08	0.606803
$89$	13322.1i	1.68187i	0.541134	+	0.840936i	$0.317995\pi$
$89$	13322.1i	1.68187i	−0.541134	+	0.840936i	$0.682005\pi$
$90$	0	0
$91$	1164.56	0.140631
$92$	3645.34i	0.430688i
$93$	0	0
$94$	4672.99	0.528858
$95$	6930.94i	0.767971i
$96$	0	0
$97$	−6073.09	−0.645455	−0.322728	−	0.946492i	$-0.604600\pi$
$97$	−6073.09	−0.645455	−0.322728	+	0.946492i	$0.604600\pi$
$98$	− 970.151i	− 0.101015i
$99$	0	0
$100$	−3206.35	−0.320635
$101$	5450.61i	0.534321i	0.963652	+	0.267160i	$0.0860854\pi$
$101$	5450.61i	0.534321i	−0.963652	+	0.267160i	$0.913915\pi$
$102$	0	0
$103$	4836.88	0.455922	0.227961	−	0.973670i	$-0.426794\pi$
$103$	4836.88	0.455922	0.227961	+	0.973670i	$0.426794\pi$
$104$	− 1422.82i	− 0.131548i
$105$	0	0
$106$	8098.91	0.720800
$107$	7429.22i	0.648897i	0.945903	+	0.324448i	$0.105179\pi$
$107$	7429.22i	0.648897i	−0.945903	+	0.324448i	$0.894821\pi$
$108$	0	0
$109$	15978.9	1.34492	0.672458	−	0.740136i	$-0.265239\pi$
$109$	15978.9	1.34492	0.672458	+	0.740136i	$0.265239\pi$
$110$	− 8795.23i	− 0.726878i
$111$	0	0
$112$	−1185.30	−0.0944911
$113$	− 8283.62i	− 0.648729i	−0.945932	−	0.324364i	$-0.894850\pi$
$113$	− 8283.62i	− 0.648729i	0.945932	−	0.324364i	$-0.105150\pi$
$114$	0	0
$115$	6822.96	0.515914
$116$	− 9450.50i	− 0.702326i
$117$	0	0
$118$	8824.77	0.633781
$119$	− 3468.78i	− 0.244953i
$120$	0	0
$121$	−28486.7	−1.94568
$122$	− 9109.45i	− 0.612029i
$123$	0	0
$124$	12131.0	0.788960
$125$	15359.7i	0.983024i
$126$	0	0
$127$	25189.7	1.56176	0.780882	−	0.624679i	$-0.214770\pi$
$127$	25189.7	1.56176	0.780882	+	0.624679i	$0.214770\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	−2663.08	−0.157579
$131$	10668.0i	0.621644i	0.950468	+	0.310822i	$0.100604\pi$
$131$	10668.0i	0.621644i	−0.950468	+	0.310822i	$0.899396\pi$
$132$	0	0
$133$	−8572.65	−0.484632
$134$	14796.9i	0.824062i
$135$	0	0
$136$	−4238.04	−0.229133
$137$	− 9682.43i	− 0.515873i	−0.966162	−	0.257937i	$-0.916957\pi$
$137$	− 9682.43i	− 0.515873i	0.966162	−	0.257937i	$-0.0830426\pi$
$138$	0	0
$139$	−17916.1	−0.927285	−0.463643	−	0.886022i	$-0.653458\pi$
$139$	−17916.1	−0.927285	−0.463643	+	0.886022i	$0.653458\pi$
$140$	2218.51i	0.113189i
$141$	0	0
$142$	−7410.78	−0.367525
$143$	13058.5i	0.638589i
$144$	0	0
$145$	−17688.4	−0.841304
$146$	17639.5i	0.827526i
$147$	0	0
$148$	16737.0	0.764105
$149$	− 17588.5i	− 0.792240i	−0.918199	−	0.396120i	$-0.870356\pi$
$149$	− 17588.5i	− 0.792240i	0.918199	−	0.396120i	$-0.129644\pi$
$150$	0	0
$151$	−35597.0	−1.56120	−0.780602	−	0.625028i	$-0.785088\pi$
$151$	−35597.0	−1.56120	−0.780602	+	0.625028i	$0.785088\pi$
$152$	10473.8i	0.453331i
$153$	0	0
$154$	10878.5	0.458700
$155$	− 22705.6i	− 0.945081i
$156$	0	0
$157$	32966.9	1.33745	0.668726	−	0.743509i	$-0.266840\pi$
$157$	32966.9	1.33745	0.668726	+	0.743509i	$0.266840\pi$
$158$	− 9639.46i	− 0.386134i
$159$	0	0
$160$	2710.50	0.105879
$161$	8439.09i	0.325570i
$162$	0	0
$163$	−20460.9	−0.770102	−0.385051	−	0.922895i	$-0.625816\pi$
$163$	−20460.9	−0.770102	−0.385051	+	0.922895i	$0.625816\pi$
$164$	− 23960.0i	− 0.890838i
$165$	0	0
$166$	−17219.1	−0.624875
$167$	− 19104.1i	− 0.685006i	−0.939517	−	0.342503i	$-0.888725\pi$
$167$	− 19104.1i	− 0.685006i	0.939517	−	0.342503i	$-0.111275\pi$
$168$	0	0
$169$	−24607.0	−0.861561
$170$	7932.30i	0.274474i
$171$	0	0
$172$	24659.5	0.833540
$173$	27932.8i	0.933302i	0.884442	+	0.466651i	$0.154540\pi$
$173$	27932.8i	0.933302i	−0.884442	+	0.466651i	$0.845460\pi$
$174$	0	0
$175$	−7422.80	−0.242377
$176$	− 13291.0i	− 0.429074i
$177$	0	0
$178$	37680.6	1.18926
$179$	26824.0i	0.837179i	0.908176	+	0.418589i	$0.137475\pi$
$179$	26824.0i	0.837179i	−0.908176	+	0.418589i	$0.862525\pi$
$180$	0	0
$181$	−8214.15	−0.250729	−0.125365	−	0.992111i	$-0.540010\pi$
$181$	−8214.15	−0.250729	−0.125365	+	0.992111i	$0.540010\pi$
$182$	− 3293.88i	− 0.0994409i
$183$	0	0
$184$	10310.6	0.304543
$185$	− 31326.4i	− 0.915308i
$186$	0	0
$187$	38896.3	1.11231
$188$	− 13217.2i	− 0.373959i
$189$	0	0
$190$	19603.7	0.543038
$191$	19528.1i	0.535296i	0.963517	+	0.267648i	$0.0862464\pi$
$191$	19528.1i	0.535296i	−0.963517	+	0.267648i	$0.913754\pi$
$192$	0	0
$193$	1684.59	0.0452252	0.0226126	−	0.999744i	$-0.492802\pi$
$193$	1684.59	0.0452252	0.0226126	+	0.999744i	$0.492802\pi$
$194$	17177.3i	0.456406i
$195$	0	0
$196$	−2744.00	−0.0714286
$197$	− 25114.6i	− 0.647134i	−0.946205	−	0.323567i	$-0.895118\pi$
$197$	− 25114.6i	− 0.647134i	0.946205	−	0.323567i	$-0.104882\pi$
$198$	0	0
$199$	−48183.6	−1.21673	−0.608363	−	0.793659i	$-0.708173\pi$
$199$	−48183.6	−1.21673	−0.608363	+	0.793659i	$0.708173\pi$
$200$	9068.92i	0.226723i
$201$	0	0
$202$	15416.6	0.377822
$203$	− 21878.2i	− 0.530909i
$204$	0	0
$205$	−44845.7	−1.06712
$206$	− 13680.8i	− 0.322386i
$207$	0	0
$208$	−4024.35	−0.0930184
$209$	− 96127.1i	− 2.20066i
$210$	0	0
$211$	−37374.5	−0.839480	−0.419740	−	0.907644i	$-0.637879\pi$
$211$	−37374.5	−0.839480	−0.419740	+	0.907644i	$0.637879\pi$
$212$	− 22907.2i	− 0.509682i
$213$	0	0
$214$	21013.0	0.458839
$215$	− 46154.9i	− 0.998483i
$216$	0	0
$217$	28083.8	0.596397
$218$	− 45195.3i	− 0.950999i
$219$	0	0
$220$	−24876.7	−0.513981
$221$	− 11777.3i	− 0.241135i
$222$	0	0
$223$	−14929.9	−0.300225	−0.150113	−	0.988669i	$-0.547964\pi$
$223$	−14929.9	−0.300225	−0.150113	+	0.988669i	$0.547964\pi$
$224$	3352.53i	0.0668153i
$225$	0	0
$226$	−23429.6	−0.458720
$227$	− 81222.3i	− 1.57625i	−0.615518	−	0.788123i	$-0.711053\pi$
$227$	− 81222.3i	− 1.57625i	0.615518	−	0.788123i	$-0.288947\pi$
$228$	0	0
$229$	−26288.7	−0.501300	−0.250650	−	0.968078i	$-0.580644\pi$
$229$	−26288.7	−0.501300	−0.250650	+	0.968078i	$0.580644\pi$
$230$	− 19298.2i	− 0.364806i
$231$	0	0
$232$	−26730.1	−0.496620
$233$	77377.7i	1.42529i	0.701523	+	0.712647i	$0.252504\pi$
$233$	77377.7i	1.42529i	−0.701523	+	0.712647i	$0.747496\pi$
$234$	0	0
$235$	−24738.5	−0.447959
$236$	− 24960.2i	− 0.448151i
$237$	0	0
$238$	−9811.20	−0.173208
$239$	− 214.251i	− 0.00375083i	−0.999998	−	0.00187541i	$-0.999403\pi$
$239$	− 214.251i	− 0.00375083i	0.999998	−	0.00187541i	$-0.000596963\pi$
$240$	0	0
$241$	3165.30	0.0544980	0.0272490	−	0.999629i	$-0.491325\pi$
$241$	3165.30	0.0544980	0.0272490	+	0.999629i	$0.491325\pi$
$242$	80572.4i	1.37580i
$243$	0	0
$244$	−25765.4	−0.432770
$245$	5135.92i	0.0855630i
$246$	0	0
$247$	−29106.1	−0.477078
$248$	− 34311.8i	− 0.557879i
$249$	0	0
$250$	43443.9	0.695103
$251$	31889.6i	0.506177i	0.967443	+	0.253088i	$0.0814464\pi$
$251$	31889.6i	0.506177i	−0.967443	+	0.253088i	$0.918554\pi$
$252$	0	0
$253$	−94629.5	−1.47838
$254$	− 71247.2i	− 1.10433i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 89826.0i	− 1.35999i	−0.733217	−	0.679995i	$-0.761982\pi$
$257$	− 89826.0i	− 1.35999i	0.733217	−	0.679995i	$-0.238018\pi$
$258$	0	0
$259$	38746.6	0.577609
$260$	7532.34i	0.111425i
$261$	0	0
$262$	30173.8	0.439569
$263$	− 23175.4i	− 0.335055i	−0.985867	−	0.167528i	$-0.946422\pi$
$263$	− 23175.4i	− 0.335055i	0.985867	−	0.167528i	$-0.0535783\pi$
$264$	0	0
$265$	−42875.1	−0.610539
$266$	24247.1i	0.342686i
$267$	0	0
$268$	41851.8	0.582700
$269$	56937.5i	0.786853i	0.919356	+	0.393427i	$0.128710\pi$
$269$	56937.5i	0.786853i	−0.919356	+	0.393427i	$0.871290\pi$
$270$	0	0
$271$	123059.	1.67561	0.837807	−	0.545967i	$-0.183838\pi$
$271$	123059.	1.67561	0.837807	+	0.545967i	$0.183838\pi$
$272$	11987.0i	0.162021i
$273$	0	0
$274$	−27386.0	−0.364778
$275$	− 83233.6i	− 1.10061i
$276$	0	0
$277$	−10073.2	−0.131283	−0.0656415	−	0.997843i	$-0.520909\pi$
$277$	−10073.2	−0.131283	−0.0656415	+	0.997843i	$0.520909\pi$
$278$	50674.3i	0.655690i
$279$	0	0
$280$	6274.89	0.0800369
$281$	− 123794.i	− 1.56778i	−0.620898	−	0.783892i	$-0.713232\pi$
$281$	− 123794.i	− 1.56778i	0.620898	−	0.783892i	$-0.286768\pi$
$282$	0	0
$283$	112317.	1.40240	0.701201	−	0.712964i	$-0.252648\pi$
$283$	112317.	1.40240	0.701201	+	0.712964i	$0.252648\pi$
$284$	20960.8i	0.259880i
$285$	0	0
$286$	36935.0	0.451551
$287$	− 55468.1i	− 0.673410i
$288$	0	0
$289$	48441.0	0.579986
$290$	50030.4i	0.594892i
$291$	0	0
$292$	49892.2	0.585149
$293$	64511.2i	0.751449i	0.926731	+	0.375725i	$0.122606\pi$
$293$	64511.2i	0.751449i	−0.926731	+	0.375725i	$0.877394\pi$
$294$	0	0
$295$	−46717.8	−0.536832
$296$	− 47339.3i	− 0.540304i
$297$	0	0
$298$	−49747.9	−0.560198
$299$	28652.6i	0.320496i
$300$	0	0
$301$	57087.4	0.630097
$302$	100684.i	1.10394i
$303$	0	0
$304$	29624.3	0.320554
$305$	48224.9i	0.518408i
$306$	0	0
$307$	11483.3	0.121840	0.0609199	−	0.998143i	$-0.480597\pi$
$307$	11483.3	0.121840	0.0609199	+	0.998143i	$0.480597\pi$
$308$	− 30769.1i	− 0.324350i
$309$	0	0
$310$	−64221.0	−0.668273
$311$	− 11256.9i	− 0.116385i	−0.998305	−	0.0581926i	$-0.981466\pi$
$311$	− 11256.9i	− 0.116385i	0.998305	−	0.0581926i	$-0.0185337\pi$
$312$	0	0
$313$	−112068.	−1.14392	−0.571959	−	0.820282i	$-0.693816\pi$
$313$	−112068.	−1.14392	−0.571959	+	0.820282i	$0.693816\pi$
$314$	− 93244.4i	− 0.945722i
$315$	0	0
$316$	−27264.5	−0.273038
$317$	47792.1i	0.475595i	0.971315	+	0.237798i	$0.0764255\pi$
$317$	47792.1i	0.475595i	−0.971315	+	0.237798i	$0.923574\pi$
$318$	0	0
$319$	245326.	2.41080
$320$	− 7666.45i	− 0.0748676i
$321$	0	0
$322$	23869.4	0.230213
$323$	86695.8i	0.830984i
$324$	0	0
$325$	−25202.1	−0.238600
$326$	57872.0i	0.544545i
$327$	0	0
$328$	−67769.0	−0.629918
$329$	− 30598.3i	− 0.282686i
$330$	0	0
$331$	−108343.	−0.988885	−0.494443	−	0.869210i	$-0.664628\pi$
$331$	−108343.	−0.988885	−0.494443	+	0.869210i	$0.664628\pi$
$332$	48702.8i	0.441853i
$333$	0	0
$334$	−54034.7	−0.484373
$335$	− 78333.7i	− 0.698006i
$336$	0	0
$337$	163814.	1.44241	0.721207	−	0.692719i	$-0.243588\pi$
$337$	163814.	1.44241	0.721207	+	0.692719i	$0.243588\pi$
$338$	69599.2i	0.609216i
$339$	0	0
$340$	22435.9	0.194082
$341$	314910.i	2.70818i
$342$	0	0
$343$	−6352.45	−0.0539949
$344$	− 69747.5i	− 0.589402i
$345$	0	0
$346$	79005.8	0.659944
$347$	199534.i	1.65713i	0.559892	+	0.828566i	$0.310843\pi$
$347$	199534.i	1.65713i	−0.559892	+	0.828566i	$0.689157\pi$
$348$	0	0
$349$	28305.5	0.232391	0.116196	−	0.993226i	$-0.462930\pi$
$349$	28305.5	0.232391	0.116196	+	0.993226i	$0.462930\pi$
$350$	20994.8i	0.171386i
$351$	0	0
$352$	−37592.6	−0.303401
$353$	− 134579.i	− 1.08001i	−0.841661	−	0.540007i	$-0.818422\pi$
$353$	− 134579.i	− 1.08001i	0.841661	−	0.540007i	$-0.181578\pi$
$354$	0	0
$355$	39232.2	0.311305
$356$	− 106577.i	− 0.840936i
$357$	0	0
$358$	75869.8	0.591975
$359$	125745.i	0.975666i	0.872937	+	0.487833i	$0.162213\pi$
$359$	125745.i	0.975666i	−0.872937	+	0.487833i	$0.837787\pi$
$360$	0	0
$361$	83936.4	0.644074
$362$	23233.1i	0.177292i
$363$	0	0
$364$	−9316.50	−0.0703153
$365$	− 93382.7i	− 0.700940i
$366$	0	0
$367$	−22221.8	−0.164986	−0.0824931	−	0.996592i	$-0.526288\pi$
$367$	−22221.8	−0.164986	−0.0824931	+	0.996592i	$0.526288\pi$
$368$	− 29162.8i	− 0.215344i
$369$	0	0
$370$	−88604.5	−0.647221
$371$	− 53030.8i	− 0.385284i
$372$	0	0
$373$	34876.0	0.250674	0.125337	−	0.992114i	$-0.459999\pi$
$373$	34876.0	0.250674	0.125337	+	0.992114i	$0.459999\pi$
$374$	− 110015.i	− 0.786520i
$375$	0	0
$376$	−37383.9	−0.264429
$377$	− 74281.5i	− 0.522634i
$378$	0	0
$379$	−102131.	−0.711016	−0.355508	−	0.934673i	$-0.615692\pi$
$379$	−102131.	−0.711016	−0.355508	+	0.934673i	$0.615692\pi$
$380$	− 55447.5i	− 0.383986i
$381$	0	0
$382$	55233.9	0.378511
$383$	12013.5i	0.0818981i	0.999161	+	0.0409490i	$0.0130381\pi$
$383$	12013.5i	0.0818981i	−0.999161	+	0.0409490i	$0.986962\pi$
$384$	0	0
$385$	−57590.3	−0.388533
$386$	− 4764.75i	− 0.0319791i
$387$	0	0
$388$	48584.7	0.322728
$389$	102479.i	0.677227i	0.940926	+	0.338614i	$0.109958\pi$
$389$	102479.i	0.677227i	−0.940926	+	0.338614i	$0.890042\pi$
$390$	0	0
$391$	85345.1	0.558245
$392$	7761.20i	0.0505076i
$393$	0	0
$394$	−71034.9	−0.457593
$395$	51030.7i	0.327068i
$396$	0	0
$397$	102844.	0.652528	0.326264	−	0.945279i	$-0.394210\pi$
$397$	102844.	0.652528	0.326264	+	0.945279i	$0.394210\pi$
$398$	136284.i	0.860355i
$399$	0	0
$400$	25650.8	0.160317
$401$	252453.i	1.56997i	0.619514	+	0.784986i	$0.287330\pi$
$401$	252453.i	1.56997i	−0.619514	+	0.784986i	$0.712670\pi$
$402$	0	0
$403$	95350.7	0.587102
$404$	− 43604.9i	− 0.267160i
$405$	0	0
$406$	−61881.0	−0.375409
$407$	434475.i	2.62286i
$408$	0	0
$409$	−42140.0	−0.251911	−0.125956	−	0.992036i	$-0.540200\pi$
$409$	−42140.0	−0.251911	−0.125956	+	0.992036i	$0.540200\pi$
$410$	126843.i	0.754567i
$411$	0	0
$412$	−38695.0	−0.227961
$413$	− 57783.7i	− 0.338770i
$414$	0	0
$415$	91156.7	0.529288
$416$	11382.6i	0.0657740i
$417$	0	0
$418$	−271889.	−1.55610
$419$	− 219896.i	− 1.25254i	−0.779608	−	0.626268i	$-0.784582\pi$
$419$	− 219896.i	− 1.25254i	0.779608	−	0.626268i	$-0.215418\pi$
$420$	0	0
$421$	−148584.	−0.838316	−0.419158	−	0.907913i	$-0.637675\pi$
$421$	−148584.	−0.838316	−0.419158	+	0.907913i	$0.637675\pi$
$422$	105711.i	0.593602i
$423$	0	0
$424$	−64791.2	−0.360400
$425$	75067.3i	0.415597i
$426$	0	0
$427$	−59647.7	−0.327143
$428$	− 59433.7i	− 0.324448i
$429$	0	0
$430$	−130546.	−0.706034
$431$	340490.i	1.83295i	0.400097	+	0.916473i	$0.368976\pi$
$431$	340490.i	1.83295i	−0.400097	+	0.916473i	$0.631024\pi$
$432$	0	0
$433$	−341800.	−1.82304	−0.911519	−	0.411257i	$-0.865090\pi$
$433$	−341800.	−1.82304	−0.911519	+	0.411257i	$0.865090\pi$
$434$	− 79432.9i	− 0.421717i
$435$	0	0
$436$	−127831.	−0.672458
$437$	− 210919.i	− 1.10447i
$438$	0	0
$439$	34198.2	0.177449	0.0887246	−	0.996056i	$-0.471721\pi$
$439$	34198.2	0.177449	0.0887246	+	0.996056i	$0.471721\pi$
$440$	70361.8i	0.363439i
$441$	0	0
$442$	−33311.2	−0.170509
$443$	− 127446.i	− 0.649409i	−0.945815	−	0.324705i	$-0.894735\pi$
$443$	− 127446.i	− 0.649409i	0.945815	−	0.324705i	$-0.105265\pi$
$444$	0	0
$445$	−199479.	−1.00734
$446$	42228.1i	0.212291i
$447$	0	0
$448$	9482.37	0.0472456
$449$	− 250976.i	− 1.24491i	−0.782655	−	0.622456i	$-0.786135\pi$
$449$	− 250976.i	− 1.24491i	0.782655	−	0.622456i	$-0.213865\pi$
$450$	0	0
$451$	621977.	3.05789
$452$	66268.9i	0.324364i
$453$	0	0
$454$	−229731.	−1.11457
$455$	17437.6i	0.0842295i
$456$	0	0
$457$	−311362.	−1.49085	−0.745425	−	0.666590i	$-0.767753\pi$
$457$	−311362.	−1.49085	−0.745425	+	0.666590i	$0.767753\pi$
$458$	74355.6i	0.354473i
$459$	0	0
$460$	−54583.7	−0.257957
$461$	− 299717.i	− 1.41029i	−0.709061	−	0.705147i	$-0.750881\pi$
$461$	− 299717.i	− 1.41029i	0.709061	−	0.705147i	$-0.249119\pi$
$462$	0	0
$463$	350647.	1.63572	0.817859	−	0.575418i	$-0.195161\pi$
$463$	350647.	1.63572	0.817859	+	0.575418i	$0.195161\pi$
$464$	75604.0i	0.351163i
$465$	0	0
$466$	218857.	1.00783
$467$	− 187935.i	− 0.861734i	−0.902416	−	0.430867i	$-0.858208\pi$
$467$	− 187935.i	− 0.861734i	0.902416	−	0.430867i	$-0.141792\pi$
$468$	0	0
$469$	96888.3	0.440479
$470$	69971.1i	0.316755i
$471$	0	0
$472$	−70598.1	−0.316891
$473$	640135.i	2.86121i
$474$	0	0
$475$	185519.	0.822245
$476$	27750.3i	0.122477i
$477$	0	0
$478$	−605.993	−0.00265223
$479$	440865.i	1.92148i	0.277459	+	0.960738i	$0.410508\pi$
$479$	440865.i	1.92148i	−0.277459	+	0.960738i	$0.589492\pi$
$480$	0	0
$481$	131553.	0.568607
$482$	− 8952.81i	− 0.0385359i
$483$	0	0
$484$	227893.	0.972839
$485$	− 90935.6i	− 0.386590i
$486$	0	0
$487$	−309791.	−1.30620	−0.653101	−	0.757270i	$-0.726532\pi$
$487$	−309791.	−1.30620	−0.653101	+	0.757270i	$0.726532\pi$
$488$	72875.6i	0.306015i
$489$	0	0
$490$	14526.6	0.0605022
$491$	− 43693.9i	− 0.181241i	−0.995885	−	0.0906207i	$-0.971115\pi$
$491$	− 43693.9i	− 0.181241i	0.995885	−	0.0906207i	$-0.0288851\pi$
$492$	0	0
$493$	−221256.	−0.910335
$494$	82324.4i	0.337345i
$495$	0	0
$496$	−97048.4	−0.394480
$497$	48525.0i	0.196450i
$498$	0	0
$499$	−18463.9	−0.0741521	−0.0370760	−	0.999312i	$-0.511804\pi$
$499$	−18463.9	−0.0741521	−0.0370760	+	0.999312i	$0.511804\pi$
$500$	− 122878.i	− 0.491512i
$501$	0	0
$502$	90197.5	0.357921
$503$	− 186650.i	− 0.737722i	−0.929484	−	0.368861i	$-0.879748\pi$
$503$	− 186650.i	− 0.737722i	0.929484	−	0.368861i	$-0.120252\pi$
$504$	0	0
$505$	−81614.8	−0.320027
$506$	267653.i	1.04537i
$507$	0	0
$508$	−201518.	−0.780882
$509$	158194.i	0.610595i	0.952257	+	0.305298i	$0.0987559\pi$
$509$	158194.i	0.610595i	−0.952257	+	0.305298i	$0.901244\pi$
$510$	0	0
$511$	115502.	0.442331
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	−254066.	−0.961658
$515$	72425.1i	0.273070i
$516$	0	0
$517$	343106.	1.28365
$518$	− 109592.i	− 0.408431i
$519$	0	0
$520$	21304.7	0.0787895
$521$	− 352048.i	− 1.29696i	−0.761232	−	0.648479i	$-0.775405\pi$
$521$	− 352048.i	− 1.29696i	0.761232	−	0.648479i	$-0.224595\pi$
$522$	0	0
$523$	182994.	0.669013	0.334506	−	0.942393i	$-0.391430\pi$
$523$	182994.	0.669013	0.334506	+	0.942393i	$0.391430\pi$
$524$	− 85344.3i	− 0.310822i
$525$	0	0
$526$	−65550.0	−0.236920
$527$	− 284013.i	− 1.02263i
$528$	0	0
$529$	72207.6	0.258031
$530$	121269.i	0.431717i
$531$	0	0
$532$	68581.2	0.242316
$533$	− 188327.i	− 0.662915i
$534$	0	0
$535$	−111242.	−0.388651
$536$	− 118375.i	− 0.412031i
$537$	0	0
$538$	161044.	0.556389
$539$	− 71231.5i	− 0.245185i
$540$	0	0
$541$	216902.	0.741087	0.370543	−	0.928815i	$-0.379172\pi$
$541$	216902.	0.741087	0.370543	+	0.928815i	$0.379172\pi$
$542$	− 348063.i	− 1.18484i
$543$	0	0
$544$	33904.3	0.114566
$545$	239261.i	0.805525i
$546$	0	0
$547$	−307754.	−1.02856	−0.514279	−	0.857623i	$-0.671940\pi$
$547$	−307754.	−1.02856	−0.514279	+	0.857623i	$0.671940\pi$
$548$	77459.4i	0.257937i
$549$	0	0
$550$	−235420.	−0.778248
$551$	546805.i	1.80107i
$552$	0	0
$553$	−63118.2	−0.206397
$554$	28491.4i	0.0928311i
$555$	0	0
$556$	143329.	0.463643
$557$	− 74238.2i	− 0.239286i	−0.992817	−	0.119643i	$-0.961825\pi$
$557$	− 74238.2i	− 0.239286i	0.992817	−	0.119643i	$-0.0381750\pi$
$558$	0	0
$559$	193825.	0.620277
$560$	− 17748.1i	− 0.0565946i
$561$	0	0
$562$	−350142.	−1.10859
$563$	− 279024.i	− 0.880286i	−0.897928	−	0.440143i	$-0.854928\pi$
$563$	− 279024.i	− 0.880286i	0.897928	−	0.440143i	$-0.145072\pi$
$564$	0	0
$565$	124035.	0.388550
$566$	− 317680.i	− 0.991648i
$567$	0	0
$568$	59286.2	0.183763
$569$	154079.i	0.475902i	0.971277	+	0.237951i	$0.0764758\pi$
$569$	154079.i	0.475902i	−0.971277	+	0.237951i	$0.923524\pi$
$570$	0	0
$571$	202500.	0.621087	0.310543	−	0.950559i	$-0.399489\pi$
$571$	202500.	0.621087	0.310543	+	0.950559i	$0.399489\pi$
$572$	− 104468.i	− 0.319295i
$573$	0	0
$574$	−156888.	−0.476173
$575$	− 182629.i	− 0.552374i
$576$	0	0
$577$	−404034.	−1.21358	−0.606788	−	0.794864i	$-0.707542\pi$
$577$	−404034.	−1.21358	−0.606788	+	0.794864i	$0.707542\pi$
$578$	− 137012.i	− 0.410112i
$579$	0	0
$580$	141507.	0.420652
$581$	112749.i	0.334010i
$582$	0	0
$583$	594647.	1.74953
$584$	− 141116.i	− 0.413763i
$585$	0	0
$586$	182465.	0.531355
$587$	− 204467.i	− 0.593399i	−0.954971	−	0.296700i	$-0.904114\pi$
$587$	− 204467.i	− 0.593399i	0.954971	−	0.296700i	$-0.0958860\pi$
$588$	0	0
$589$	−701901.	−2.02323
$590$	132138.i	0.379598i
$591$	0	0
$592$	−133896.	−0.382053
$593$	− 226013.i	− 0.642723i	−0.946957	−	0.321361i	$-0.895860\pi$
$593$	− 226013.i	− 0.642723i	0.946957	−	0.321361i	$-0.104140\pi$
$594$	0	0
$595$	51939.9	0.146713
$596$	140708.i	0.396120i
$597$	0	0
$598$	81041.8	0.226625
$599$	45246.9i	0.126106i	0.998010	+	0.0630530i	$0.0200837\pi$
$599$	45246.9i	0.126106i	−0.998010	+	0.0630530i	$0.979916\pi$
$600$	0	0
$601$	24718.9	0.0684352	0.0342176	−	0.999414i	$-0.489106\pi$
$601$	24718.9	0.0684352	0.0342176	+	0.999414i	$0.489106\pi$
$602$	− 161468.i	− 0.445546i
$603$	0	0
$604$	284776.	0.780602
$605$	− 426546.i	− 1.16535i
$606$	0	0
$607$	34864.1	0.0946241	0.0473120	−	0.998880i	$-0.484934\pi$
$607$	34864.1	0.0946241	0.0473120	+	0.998880i	$0.484934\pi$
$608$	− 83790.1i	− 0.226666i
$609$	0	0
$610$	136401.	0.366570
$611$	− 103888.i	− 0.278281i
$612$	0	0
$613$	−175378.	−0.466717	−0.233359	−	0.972391i	$-0.574972\pi$
$613$	−175378.	−0.466717	−0.233359	+	0.972391i	$0.574972\pi$
$614$	− 32479.6i	− 0.0861537i
$615$	0	0
$616$	−87028.2	−0.229350
$617$	− 192394.i	− 0.505383i	−0.967547	−	0.252691i	$-0.918684\pi$
$617$	− 192394.i	− 0.505383i	0.967547	−	0.252691i	$-0.0813158\pi$
$618$	0	0
$619$	−235328.	−0.614174	−0.307087	−	0.951681i	$-0.599354\pi$
$619$	−235328.	−0.614174	−0.307087	+	0.951681i	$0.599354\pi$
$620$	181645.i	0.472540i
$621$	0	0
$622$	−31839.3	−0.0822967
$623$	− 246729.i	− 0.635688i
$624$	0	0
$625$	20506.3	0.0524961
$626$	316977.i	0.808872i
$627$	0	0
$628$	−263735.	−0.668726
$629$	− 391847.i	− 0.990411i
$630$	0	0
$631$	534962.	1.34358	0.671791	−	0.740741i	$-0.265525\pi$
$631$	534962.	1.34358	0.671791	+	0.740741i	$0.265525\pi$
$632$	77115.7i	0.193067i
$633$	0	0
$634$	135176.	0.336297
$635$	377179.i	0.935405i
$636$	0	0
$637$	−21568.0	−0.0531534
$638$	− 693886.i	− 1.70469i
$639$	0	0
$640$	−21684.0	−0.0529394
$641$	715669.i	1.74179i	0.491469	+	0.870895i	$0.336460\pi$
$641$	715669.i	1.74179i	−0.491469	+	0.870895i	$0.663540\pi$
$642$	0	0
$643$	401264.	0.970528	0.485264	−	0.874368i	$-0.338723\pi$
$643$	401264.	0.970528	0.485264	+	0.874368i	$0.338723\pi$
$644$	− 67512.7i	− 0.162785i
$645$	0	0
$646$	245213.	0.587595
$647$	− 761674.i	− 1.81953i	−0.415119	−	0.909767i	$-0.636260\pi$
$647$	− 761674.i	− 1.81953i	0.415119	−	0.909767i	$-0.363740\pi$
$648$	0	0
$649$	647942.	1.53832
$650$	71282.2i	0.168715i
$651$	0	0
$652$	163687.	0.385051
$653$	− 225493.i	− 0.528819i	−0.964411	−	0.264409i	$-0.914823\pi$
$653$	− 225493.i	− 0.528819i	0.964411	−	0.264409i	$-0.0851770\pi$
$654$	0	0
$655$	−159738.	−0.372328
$656$	191680.i	0.445419i
$657$	0	0
$658$	−86545.0	−0.199890
$659$	− 844001.i	− 1.94344i	−0.236127	−	0.971722i	$-0.575878\pi$
$659$	− 844001.i	− 1.94344i	0.236127	−	0.971722i	$-0.424122\pi$
$660$	0	0
$661$	−233508.	−0.534441	−0.267220	−	0.963635i	$-0.586105\pi$
$661$	−233508.	−0.534441	−0.267220	+	0.963635i	$0.586105\pi$
$662$	306441.i	0.699247i
$663$	0	0
$664$	137752.	0.312437
$665$	− 128363.i	− 0.290266i
$666$	0	0
$667$	538287.	1.20993
$668$	152833.i	0.342503i
$669$	0	0
$670$	−221561.	−0.493564
$671$	− 668844.i	− 1.48552i
$672$	0	0
$673$	−380168.	−0.839355	−0.419677	−	0.907673i	$-0.637857\pi$
$673$	−380168.	−0.839355	−0.419677	+	0.907673i	$0.637857\pi$
$674$	− 463335.i	− 1.01994i
$675$	0	0
$676$	196856.	0.430781
$677$	390571.i	0.852163i	0.904685	+	0.426082i	$0.140106\pi$
$677$	390571.i	0.852163i	−0.904685	+	0.426082i	$0.859894\pi$
$678$	0	0
$679$	112475.	0.243959
$680$	− 63458.4i	− 0.137237i
$681$	0	0
$682$	890699.	1.91497
$683$	483186.i	1.03579i	0.855443	+	0.517896i	$0.173285\pi$
$683$	483186.i	1.03579i	−0.855443	+	0.517896i	$0.826715\pi$
$684$	0	0
$685$	144980.	0.308978
$686$	17967.4i	0.0381802i
$687$	0	0
$688$	−197276.	−0.416770
$689$	− 180052.i	− 0.379279i
$690$	0	0
$691$	−145791.	−0.305334	−0.152667	−	0.988278i	$-0.548786\pi$
$691$	−145791.	−0.305334	−0.152667	+	0.988278i	$0.548786\pi$
$692$	− 223462.i	− 0.466651i
$693$	0	0
$694$	564366.	1.17177
$695$	− 268267.i	− 0.555389i
$696$	0	0
$697$	−560953.	−1.15468
$698$	− 80060.0i	− 0.164325i
$699$	0	0
$700$	59382.4	0.121189
$701$	− 427557.i	− 0.870078i	−0.900412	−	0.435039i	$-0.856735\pi$
$701$	− 427557.i	− 0.870078i	0.900412	−	0.435039i	$-0.143265\pi$
$702$	0	0
$703$	−968400.	−1.95949
$704$	106328.i	0.214537i
$705$	0	0
$706$	−380648.	−0.763685
$707$	− 100947.i	− 0.201954i
$708$	0	0
$709$	−568967.	−1.13186	−0.565932	−	0.824452i	$-0.691484\pi$
$709$	−568967.	−1.13186	−0.565932	+	0.824452i	$0.691484\pi$
$710$	− 110965.i	− 0.220126i
$711$	0	0
$712$	−301445.	−0.594632
$713$	690966.i	1.35918i
$714$	0	0
$715$	−195532.	−0.382477
$716$	− 214592.i	− 0.418589i
$717$	0	0
$718$	355660.	0.689900
$719$	− 460546.i	− 0.890873i	−0.895314	−	0.445436i	$-0.853049\pi$
$719$	− 460546.i	− 0.890873i	0.895314	−	0.445436i	$-0.146951\pi$
$720$	0	0
$721$	−89580.2	−0.172322
$722$	− 237408.i	− 0.455429i
$723$	0	0
$724$	65713.2	0.125365
$725$	473462.i	0.900761i
$726$	0	0
$727$	551066.	1.04264	0.521321	−	0.853361i	$-0.325439\pi$
$727$	551066.	1.04264	0.521321	+	0.853361i	$0.325439\pi$
$728$	26351.0i	0.0497204i
$729$	0	0
$730$	−264126.	−0.495639
$731$	− 577329.i	− 1.08041i
$732$	0	0
$733$	−280002.	−0.521138	−0.260569	−	0.965455i	$-0.583910\pi$
$733$	−280002.	−0.521138	−0.260569	+	0.965455i	$0.583910\pi$
$734$	62852.8i	0.116663i
$735$	0	0
$736$	−82484.7	−0.152271
$737$	1.08643e6i	2.00017i
$738$	0	0
$739$	8667.21	0.0158705	0.00793525	−	0.999969i	$-0.497474\pi$
$739$	8667.21	0.0158705	0.00793525	+	0.999969i	$0.497474\pi$
$740$	250611.i	0.457654i
$741$	0	0
$742$	−149994.	−0.272437
$743$	370104.i	0.670419i	0.942144	+	0.335210i	$0.108807\pi$
$743$	370104.i	0.670419i	−0.942144	+	0.335210i	$0.891193\pi$
$744$	0	0
$745$	263362.	0.474505
$746$	− 98644.2i	− 0.177253i
$747$	0	0
$748$	−311170.	−0.556154
$749$	− 137591.i	− 0.245260i
$750$	0	0
$751$	508370.	0.901363	0.450681	−	0.892685i	$-0.351181\pi$
$751$	508370.	0.901363	0.450681	+	0.892685i	$0.351181\pi$
$752$	105738.i	0.186980i
$753$	0	0
$754$	−210100.	−0.369558
$755$	− 533013.i	− 0.935070i
$756$	0	0
$757$	832192.	1.45222	0.726109	−	0.687580i	$-0.241327\pi$
$757$	832192.	1.45222	0.726109	+	0.687580i	$0.241327\pi$
$758$	288870.i	0.502765i
$759$	0	0
$760$	−156829.	−0.271519
$761$	− 599649.i	− 1.03545i	−0.855548	−	0.517724i	$-0.826780\pi$
$761$	− 599649.i	− 1.03545i	0.855548	−	0.517724i	$-0.173220\pi$
$762$	0	0
$763$	−295934.	−0.508330
$764$	− 156225.i	− 0.267648i
$765$	0	0
$766$	33979.4	0.0579107
$767$	− 196189.i	− 0.333490i
$768$	0	0
$769$	122085.	0.206447	0.103224	−	0.994658i	$-0.467084\pi$
$769$	122085.	0.206447	0.103224	+	0.994658i	$0.467084\pi$
$770$	162890.i	0.274734i
$771$	0	0
$772$	−13476.8	−0.0226126
$773$	655751.i	1.09744i	0.836007	+	0.548719i	$0.184884\pi$
$773$	655751.i	1.09744i	−0.836007	+	0.548719i	$0.815116\pi$
$774$	0	0
$775$	−607755.	−1.01187
$776$	− 137418.i	− 0.228203i
$777$	0	0
$778$	289854.	0.478872
$779$	1.38632e6i	2.28449i
$780$	0	0
$781$	−544123.	−0.892061
$782$	− 241392.i	− 0.394739i
$783$	0	0
$784$	21952.0	0.0357143
$785$	493630.i	0.801055i
$786$	0	0
$787$	208274.	0.336268	0.168134	−	0.985764i	$-0.446226\pi$
$787$	208274.	0.336268	0.168134	+	0.985764i	$0.446226\pi$
$788$	200917.i	0.323567i
$789$	0	0
$790$	144337.	0.231272
$791$	153415.i	0.245196i
$792$	0	0
$793$	−202518.	−0.322045
$794$	− 290888.i	− 0.461407i
$795$	0	0
$796$	385468.	0.608363
$797$	− 464263.i	− 0.730883i	−0.930834	−	0.365442i	$-0.880918\pi$
$797$	− 464263.i	− 0.730883i	0.930834	−	0.365442i	$-0.119082\pi$
$798$	0	0
$799$	−309442.	−0.484715
$800$	− 72551.4i	− 0.113362i
$801$	0	0
$802$	714045.	1.11014
$803$	1.29515e6i	2.00858i
$804$	0	0
$805$	−126363.	−0.194997
$806$	− 269693.i	− 0.415144i
$807$	0	0
$808$	−123333.	−0.188911
$809$	− 653300.i	− 0.998195i	−0.866546	−	0.499097i	$-0.833665\pi$
$809$	− 653300.i	− 0.998195i	0.866546	−	0.499097i	$-0.166335\pi$
$810$	0	0
$811$	−1.15300e6	−1.75302	−0.876510	−	0.481384i	$-0.840134\pi$
$811$	−1.15300e6	−1.75302	−0.876510	+	0.481384i	$0.840134\pi$
$812$	175026.i	0.265454i
$813$	0	0
$814$	1.22888e6	1.85464
$815$	− 306371.i	− 0.461246i
$816$	0	0
$817$	−1.42679e6	−2.13755
$818$	119190.i	0.178128i
$819$	0	0
$820$	358765.	0.533559
$821$	− 440108.i	− 0.652939i	−0.945208	−	0.326470i	$-0.894141\pi$
$821$	− 440108.i	− 0.652939i	0.945208	−	0.326470i	$-0.105859\pi$
$822$	0	0
$823$	628536.	0.927963	0.463981	−	0.885845i	$-0.346420\pi$
$823$	628536.	0.927963	0.463981	+	0.885845i	$0.346420\pi$
$824$	109446.i	0.161193i
$825$	0	0
$826$	−163437.	−0.239547
$827$	92273.2i	0.134916i	0.997722	+	0.0674582i	$0.0214889\pi$
$827$	92273.2i	0.134916i	−0.997722	+	0.0674582i	$0.978511\pi$
$828$	0	0
$829$	−745793.	−1.08520	−0.542599	−	0.839992i	$-0.682560\pi$
$829$	−745793.	−1.08520	−0.542599	+	0.839992i	$0.682560\pi$
$830$	− 257830.i	− 0.374263i
$831$	0	0
$832$	32194.8	0.0465092
$833$	64242.7i	0.0925836i
$834$	0	0
$835$	286056.	0.410278
$836$	769017.i	1.10033i
$837$	0	0
$838$	−621961.	−0.885677
$839$	957819.i	1.36069i	0.732891	+	0.680346i	$0.238170\pi$
$839$	957819.i	1.36069i	−0.732891	+	0.680346i	$0.761830\pi$
$840$	0	0
$841$	−688219.	−0.973049
$842$	420259.i	0.592779i
$843$	0	0
$844$	298996.	0.419740
$845$	− 368454.i	− 0.516024i
$846$	0	0
$847$	527580.	0.735397
$848$	183257.i	0.254841i
$849$	0	0
$850$	212322.	0.293872
$851$	953312.i	1.31636i
$852$	0	0
$853$	−943000.	−1.29603	−0.648013	−	0.761629i	$-0.724400\pi$
$853$	−943000.	−1.29603	−0.648013	+	0.761629i	$0.724400\pi$
$854$	168709.i	0.231325i
$855$	0	0
$856$	−168104.	−0.229420
$857$	− 601580.i	− 0.819090i	−0.912290	−	0.409545i	$-0.865687\pi$
$857$	− 601580.i	− 0.819090i	0.912290	−	0.409545i	$-0.134313\pi$
$858$	0	0
$859$	456174.	0.618221	0.309111	−	0.951026i	$-0.399969\pi$
$859$	456174.	0.618221	0.309111	+	0.951026i	$0.399969\pi$
$860$	369239.i	0.499241i
$861$	0	0
$862$	963051.	1.29609
$863$	− 387585.i	− 0.520410i	−0.965553	−	0.260205i	$-0.916210\pi$
$863$	− 387585.i	− 0.520410i	0.965553	−	0.260205i	$-0.0837901\pi$
$864$	0	0
$865$	−418252.	−0.558993
$866$	966756.i	1.28908i
$867$	0	0
$868$	−224670.	−0.298199
$869$	− 707759.i	− 0.937230i
$870$	0	0
$871$	328958.	0.433614
$872$	361562.i	0.475499i
$873$	0	0
$874$	−596570.	−0.780978
$875$	− 284467.i	− 0.371548i
$876$	0	0
$877$	382540.	0.497368	0.248684	−	0.968585i	$-0.420002\pi$
$877$	382540.	0.497368	0.248684	+	0.968585i	$0.420002\pi$
$878$	− 96727.1i	− 0.125476i
$879$	0	0
$880$	199013.	0.256990
$881$	− 166044.i	− 0.213930i	−0.994263	−	0.106965i	$-0.965887\pi$
$881$	− 166044.i	− 0.213930i	0.994263	−	0.106965i	$-0.0341132\pi$
$882$	0	0
$883$	−882718.	−1.13214	−0.566071	−	0.824357i	$-0.691537\pi$
$883$	−882718.	−1.13214	−0.566071	+	0.824357i	$0.691537\pi$
$884$	94218.4i	0.120568i
$885$	0	0
$886$	−360472.	−0.459202
$887$	233588.i	0.296896i	0.988920	+	0.148448i	$0.0474277\pi$
$887$	233588.i	0.296896i	−0.988920	+	0.148448i	$0.952572\pi$
$888$	0	0
$889$	−466520.	−0.590291
$890$	564212.i	0.712299i
$891$	0	0
$892$	119439.	0.150113
$893$	764747.i	0.958992i
$894$	0	0
$895$	−401651.	−0.501421
$896$	− 26820.2i	− 0.0334077i
$897$	0	0
$898$	−709866.	−0.880286
$899$	− 1.79132e6i	− 2.21643i
$900$	0	0
$901$	−536304.	−0.660635
$902$	− 1.75922e6i	− 2.16225i
$903$	0	0
$904$	187437.	0.229360
$905$	− 122995.i	− 0.150172i
$906$	0	0
$907$	−192790.	−0.234353	−0.117176	−	0.993111i	$-0.537384\pi$
$907$	−192790.	−0.234353	−0.117176	+	0.993111i	$0.537384\pi$
$908$	649779.i	0.788123i
$909$	0	0
$910$	49321.0	0.0595592
$911$	1.15573e6i	1.39258i	0.717759	+	0.696292i	$0.245168\pi$
$911$	1.15573e6i	1.39258i	−0.717759	+	0.696292i	$0.754832\pi$
$912$	0	0
$913$	−1.26428e6	−1.51670
$914$	880666.i	1.05419i
$915$	0	0
$916$	210309.	0.250650
$917$	− 197575.i	− 0.234959i
$918$	0	0
$919$	1.17909e6	1.39610	0.698049	−	0.716050i	$-0.254052\pi$
$919$	1.17909e6	1.39610	0.698049	+	0.716050i	$0.254052\pi$
$920$	154386.i	0.182403i
$921$	0	0
$922$	−847728.	−0.997229
$923$	164753.i	0.193389i
$924$	0	0
$925$	−838508.	−0.979995
$926$	− 991780.i	− 1.15663i
$927$	0	0
$928$	213840.	0.248310
$929$	630809.i	0.730915i	0.930828	+	0.365457i	$0.119087\pi$
$929$	630809.i	0.730915i	−0.930828	+	0.365457i	$0.880913\pi$
$930$	0	0
$931$	158768.	0.183174
$932$	− 619022.i	− 0.712647i
$933$	0	0
$934$	−531559.	−0.609338
$935$	582414.i	0.666206i
$936$	0	0
$937$	936497.	1.06666	0.533331	−	0.845906i	$-0.320940\pi$
$937$	936497.	1.06666	0.533331	+	0.845906i	$0.320940\pi$
$938$	− 274042.i	− 0.311466i
$939$	0	0
$940$	197908.	0.223979
$941$	1.58769e6i	1.79303i	0.443016	+	0.896514i	$0.353908\pi$
$941$	1.58769e6i	1.79303i	−0.443016	+	0.896514i	$0.646092\pi$
$942$	0	0
$943$	1.36472e6	1.53469
$944$	199682.i	0.224075i
$945$	0	0
$946$	1.81057e6	2.02318
$947$	75297.7i	0.0839618i	0.999118	+	0.0419809i	$0.0133669\pi$
$947$	75297.7i	0.0839618i	−0.999118	+	0.0419809i	$0.986633\pi$
$948$	0	0
$949$	392155.	0.435437
$950$	− 524727.i	− 0.581415i
$951$	0	0
$952$	78489.6	0.0866040
$953$	− 20401.2i	− 0.0224632i	−0.999937	−	0.0112316i	$-0.996425\pi$
$953$	− 20401.2i	− 0.0224632i	0.999937	−	0.0112316i	$-0.00357520\pi$
$954$	0	0
$955$	−292405.	−0.320611
$956$	1714.01i	0.00187541i
$957$	0	0
$958$	1.24696e6	1.35869
$959$	179321.i	0.194982i
$960$	0	0
$961$	1.37589e6	1.48983
$962$	− 372089.i	− 0.402066i
$963$	0	0
$964$	−25322.4	−0.0272490
$965$	25224.3i	0.0270873i
$966$	0	0
$967$	−619443.	−0.662443	−0.331222	−	0.943553i	$-0.607461\pi$
$967$	−619443.	−0.662443	−0.331222	+	0.943553i	$0.607461\pi$
$968$	− 644580.i	− 0.687901i
$969$	0	0
$970$	−257205.	−0.273360
$971$	1.61783e6i	1.71591i	0.513725	+	0.857955i	$0.328265\pi$
$971$	1.61783e6i	1.71591i	−0.513725	+	0.857955i	$0.671735\pi$
$972$	0	0
$973$	331810.	0.350481
$974$	876221.i	0.923625i
$975$	0	0
$976$	206123.	0.216385
$977$	− 320209.i	− 0.335463i	−0.985833	−	0.167732i	$-0.946356\pi$
$977$	− 320209.i	− 0.335463i	0.985833	−	0.167732i	$-0.0536442\pi$
$978$	0	0
$979$	2.76663e6	2.88659
$980$	− 41087.4i	− 0.0427815i
$981$	0	0
$982$	−123585.	−0.128157
$983$	947603.i	0.980662i	0.871536	+	0.490331i	$0.163124\pi$
$983$	947603.i	0.980662i	−0.871536	+	0.490331i	$0.836876\pi$
$984$	0	0
$985$	376055.	0.387595
$986$	625806.i	0.643704i
$987$	0	0
$988$	232849.	0.238539
$989$	1.40457e6i	1.43598i
$990$	0	0
$991$	−309271.	−0.314914	−0.157457	−	0.987526i	$-0.550330\pi$
$991$	−309271.	−0.314914	−0.157457	+	0.987526i	$0.550330\pi$
$992$	274494.i	0.278939i
$993$	0	0
$994$	137250.	0.138911
$995$	− 721478.i	− 0.728747i
$996$	0	0
$997$	149332.	0.150232	0.0751159	−	0.997175i	$-0.476067\pi$
$997$	149332.	0.150232	0.0751159	+	0.997175i	$0.476067\pi$
$998$	52223.9i	0.0524334i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.5.b.a.323.4	✓	8
3.2	odd	2	inner	378.5.b.a.323.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.5.b.a.323.4	✓	8	1.1	even	1	trivial
378.5.b.a.323.5	yes	8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	378.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} +14.9735i q^{5} -18.5203 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} +14.9735i q^{5} -18.5203 q^{7} +22.6274i q^{8} +42.3515 q^{10} -207.672i q^{11} -62.8805 q^{13} +52.3832i q^{14} +64.0000 q^{16} +187.297i q^{17} +462.879 q^{19} -119.788i q^{20} -587.385 q^{22} -455.668i q^{23} +400.793 q^{25} +177.853i q^{26} +148.162 q^{28} +1181.31i q^{29} -1516.38 q^{31} -181.019i q^{32} +529.755 q^{34} -277.314i q^{35} -2092.12 q^{37} -1309.22i q^{38} -338.812 q^{40} +2995.00i q^{41} -3082.43 q^{43} +1661.38i q^{44} -1288.82 q^{46} +1652.15i q^{47} +343.000 q^{49} -1133.62i q^{50} +503.044 q^{52} +2863.40i q^{53} +3109.58 q^{55} -419.066i q^{56} +3341.26 q^{58} +3120.03i q^{59} +3220.68 q^{61} +4288.97i q^{62} -512.000 q^{64} -941.542i q^{65} -5231.48 q^{67} -1498.37i q^{68} -784.361 q^{70} -2620.11i q^{71} -6236.52 q^{73} +5917.41i q^{74} -3703.04 q^{76} +3846.14i q^{77} +3408.06 q^{79} +958.306i q^{80} +8471.13 q^{82} -6087.85i q^{83} -2804.49 q^{85} +8718.43i q^{86} +4699.08 q^{88} +13322.1i q^{89} +1164.56 q^{91} +3645.34i q^{92} +4672.99 q^{94} +6930.94i q^{95} -6073.09 q^{97} -970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4}+O(q^{10}) \) 8 * q - 64 * q^4	\( 8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100}) \) 8 * q - 64 * q^4 - 224 * q^10 - 376 * q^13 + 512 * q^16 + 1120 * q^19 - 1120 * q^22 + 792 * q^25 - 880 * q^31 - 1792 * q^34 + 1576 * q^37 + 1792 * q^40 - 5768 * q^43 - 160 * q^46 + 2744 * q^49 + 3008 * q^52 + 488 * q^55 + 7552 * q^58 - 2560 * q^61 - 4096 * q^64 + 23784 * q^67 + 1568 * q^70 - 13176 * q^73 - 8960 * q^76 - 9592 * q^79 - 3360 * q^82 - 39880 * q^85 + 8960 * q^88 + 5096 * q^91 + 4608 * q^94 - 14016 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

Properties

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