LMFDB - Embedded newform 124.5.c.a.61.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [124,5,Mod(61,124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(124, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("124.61");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 124 = 2^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	124.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:	$12.8178754224$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 478x^{8} + 69668x^{6} + 4198200x^{4} + 101304000x^{2} + 622080000 $ x^10 + 478x^8 + 69668x^6 + 4198200x^4 + 101304000x^2 + 622080000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			61.5
Root			$-3.01087i$ of defining polynomial
Character	$\chi$	$=$	124.61
Dual form			124.5.c.a.61.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.01087i q^{3} +16.1544 q^{5} -66.9407 q^{7} +71.9347 q^{9} +O(q^{10})$	$q-3.01087i q^{3} +16.1544 q^{5} -66.9407 q^{7} +71.9347 q^{9} -221.831i q^{11} +65.3555i q^{13} -48.6389i q^{15} -59.5422i q^{17} -328.553 q^{19} +201.550i q^{21} -623.356i q^{23} -364.034 q^{25} -460.466i q^{27} -1273.53i q^{29} +(328.048 - 903.275i) q^{31} -667.902 q^{33} -1081.39 q^{35} -477.045i q^{37} +196.777 q^{39} +329.250 q^{41} +3327.48i q^{43} +1162.06 q^{45} +742.126 q^{47} +2080.06 q^{49} -179.274 q^{51} -1744.76i q^{53} -3583.55i q^{55} +989.229i q^{57} -4726.44 q^{59} +4981.21i q^{61} -4815.36 q^{63} +1055.78i q^{65} +6596.77 q^{67} -1876.84 q^{69} +8230.46 q^{71} +7160.56i q^{73} +1096.06i q^{75} +14849.5i q^{77} -6391.96i q^{79} +4440.31 q^{81} +9782.20i q^{83} -961.871i q^{85} -3834.42 q^{87} -2278.45i q^{89} -4374.95i q^{91} +(-2719.64 - 987.710i) q^{93} -5307.58 q^{95} -3409.25 q^{97} -15957.3i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9}+O(q^{10}) $ 10 * q - 6 * q^5 - 2 * q^7 - 146 * q^9	$ 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9} - 310 q^{19} - 756 q^{25} + 734 q^{31} - 372 q^{33} - 666 q^{35} - 1924 q^{39} - 210 q^{41} + 650 q^{45} - 1992 q^{47} + 188 q^{49} + 1352 q^{51} + 5610 q^{59} + 3478 q^{63} - 5420 q^{67} + 10160 q^{69} - 1734 q^{71} + 11598 q^{81} - 13244 q^{87} + 3088 q^{93} + 4302 q^{95} + 2942 q^{97}+O(q^{100}) $ 10 * q - 6 * q^5 - 2 * q^7 - 146 * q^9 - 310 * q^19 - 756 * q^25 + 734 * q^31 - 372 * q^33 - 666 * q^35 - 1924 * q^39 - 210 * q^41 + 650 * q^45 - 1992 * q^47 + 188 * q^49 + 1352 * q^51 + 5610 * q^59 + 3478 * q^63 - 5420 * q^67 + 10160 * q^69 - 1734 * q^71 + 11598 * q^81 - 13244 * q^87 + 3088 * q^93 + 4302 * q^95 + 2942 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/124\mathbb{Z}\right)^\times$.

$n$	$63$	$65$
$\chi(n)$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	3.01087i		−	0.334541i	−0.985911	−	0.167270i	$-0.946505\pi$
$3$		−	3.01087i		−	0.334541i	0.985911	−	0.167270i	$-0.0534953\pi$
$4$		0			0
$5$	16.1544			0.646177			0.323089	−	0.946369i	$-0.395279\pi$
$5$	16.1544			0.646177			0.323089	+	0.946369i	$0.395279\pi$
$6$		0			0
$7$	−66.9407			−1.36614			−0.683069	−	0.730354i	$-0.739355\pi$
$7$	−66.9407			−1.36614			−0.683069	+	0.730354i	$0.739355\pi$
$8$		0			0
$9$	71.9347			0.888082
$10$		0			0
$11$		−	221.831i		−	1.83331i	−0.399679	−	0.916655i	$-0.630878\pi$
$11$		−	221.831i		−	1.83331i	0.399679	−	0.916655i	$-0.369122\pi$
$12$		0			0
$13$			65.3555i			0.386719i	0.981128	+	0.193360i	$0.0619384\pi$
$13$			65.3555i			0.386719i	−0.981128	+	0.193360i	$0.938062\pi$
$14$		0			0
$15$		−	48.6389i		−	0.216173i
$16$		0			0
$17$		−	59.5422i		−	0.206028i	−0.994680	−	0.103014i	$-0.967151\pi$
$17$		−	59.5422i		−	0.206028i	0.994680	−	0.103014i	$-0.0328487\pi$
$18$		0			0
$19$	−328.553			−0.910118			−0.455059	−	0.890461i	$-0.650382\pi$
$19$	−328.553			−0.910118			−0.455059	+	0.890461i	$0.650382\pi$
$20$		0			0
$21$			201.550i			0.457029i
$22$		0			0
$23$		−	623.356i		−	1.17837i	−0.807999	−	0.589183i	$-0.799450\pi$
$23$		−	623.356i		−	1.17837i	0.807999	−	0.589183i	$-0.200550\pi$
$24$		0			0
$25$	−364.034			−0.582455
$26$		0			0
$27$		−	460.466i		−	0.631641i
$28$		0			0
$29$		−	1273.53i		−	1.51430i	−0.653240	−	0.757151i	$-0.726591\pi$
$29$		−	1273.53i		−	1.51430i	0.653240	−	0.757151i	$-0.273409\pi$
$30$		0			0
$31$	328.048	−	903.275i	0.341361	−	0.939932i
$31$	328.048	−	903.275i	0.341361	−	0.939932i
$32$		0			0
$33$	−667.902			−0.613317
$34$		0			0
$35$	−1081.39			−0.882767
$36$		0			0
$37$		−	477.045i		−	0.348462i	−0.984705	−	0.174231i	$-0.944256\pi$
$37$		−	477.045i		−	0.348462i	0.984705	−	0.174231i	$-0.0557440\pi$
$38$		0			0
$39$	196.777			0.129373
$40$		0			0
$41$	329.250			0.195866			0.0979328	−	0.995193i	$-0.468777\pi$
$41$	329.250			0.195866			0.0979328	+	0.995193i	$0.468777\pi$
$42$		0			0
$43$			3327.48i			1.79961i	0.436293	+	0.899804i	$0.356291\pi$
$43$			3327.48i			1.79961i	−0.436293	+	0.899804i	$0.643709\pi$
$44$		0			0
$45$	1162.06			0.573859
$46$		0			0
$47$	742.126			0.335956			0.167978	−	0.985791i	$-0.446276\pi$
$47$	742.126			0.335956			0.167978	+	0.985791i	$0.446276\pi$
$48$		0			0
$49$	2080.06			0.866332
$50$		0			0
$51$	−179.274			−0.0689249
$52$		0			0
$53$		−	1744.76i		−	0.621131i	−0.950552	−	0.310566i	$-0.899482\pi$
$53$		−	1744.76i		−	0.621131i	0.950552	−	0.310566i	$-0.100518\pi$
$54$		0			0
$55$		−	3583.55i		−	1.18464i
$56$		0			0
$57$			989.229i			0.304472i
$58$		0			0
$59$	−4726.44			−1.35778			−0.678891	−	0.734239i	$-0.737539\pi$
$59$	−4726.44			−1.35778			−0.678891	+	0.734239i	$0.737539\pi$
$60$		0			0
$61$			4981.21i			1.33867i	0.742959	+	0.669337i	$0.233422\pi$
$61$			4981.21i			1.33867i	−0.742959	+	0.669337i	$0.766578\pi$
$62$		0			0
$63$	−4815.36			−1.21324
$64$		0			0
$65$			1055.78i			0.249889i
$66$		0			0
$67$	6596.77			1.46954			0.734770	−	0.678316i	$-0.237290\pi$
$67$	6596.77			1.46954			0.734770	+	0.678316i	$0.237290\pi$
$68$		0			0
$69$	−1876.84			−0.394212
$70$		0			0
$71$	8230.46			1.63270			0.816351	−	0.577555i	$-0.195993\pi$
$71$	8230.46			1.63270			0.816351	+	0.577555i	$0.195993\pi$
$72$		0			0
$73$			7160.56i			1.34370i	0.740688	+	0.671849i	$0.234500\pi$
$73$			7160.56i			1.34370i	−0.740688	+	0.671849i	$0.765500\pi$
$74$		0			0
$75$			1096.06i			0.194855i
$76$		0			0
$77$			14849.5i			2.50455i
$78$		0			0
$79$		−	6391.96i		−	1.02419i	−0.858929	−	0.512094i	$-0.828870\pi$
$79$		−	6391.96i		−	1.02419i	0.858929	−	0.512094i	$-0.171130\pi$
$80$		0			0
$81$	4440.31			0.676773
$82$		0			0
$83$			9782.20i			1.41997i	0.704215	+	0.709987i	$0.251299\pi$
$83$			9782.20i			1.41997i	−0.704215	+	0.709987i	$0.748701\pi$
$84$		0			0
$85$		−	961.871i		−	0.133131i
$86$		0			0
$87$	−3834.42			−0.506596
$88$		0			0
$89$		−	2278.45i		−	0.287647i	−0.989603	−	0.143824i	$-0.954060\pi$
$89$		−	2278.45i		−	0.287647i	0.989603	−	0.143824i	$-0.0459398\pi$
$90$		0			0
$91$		−	4374.95i		−	0.528311i
$92$		0			0
$93$	−2719.64	−	987.710i	−0.314446	−	0.114199i
$94$		0			0
$95$	−5307.58			−0.588098
$96$		0			0
$97$	−3409.25			−0.362339			−0.181170	−	0.983452i	$-0.557988\pi$
$97$	−3409.25			−0.362339			−0.181170	+	0.983452i	$0.557988\pi$
$98$		0			0
$99$		−	15957.3i		−	1.62813i
$100$		0			0
$101$	15736.6			1.54265			0.771327	−	0.636439i	$-0.219593\pi$
$101$	15736.6			1.54265			0.771327	+	0.636439i	$0.219593\pi$
$102$		0			0
$103$	−9560.46			−0.901165			−0.450583	−	0.892735i	$-0.648784\pi$
$103$	−9560.46			−0.901165			−0.450583	+	0.892735i	$0.648784\pi$
$104$		0			0
$105$			3255.92i			0.295322i
$106$		0			0
$107$	11127.3			0.971899			0.485949	−	0.873987i	$-0.338474\pi$
$107$	11127.3			0.971899			0.485949	+	0.873987i	$0.338474\pi$
$108$		0			0
$109$	114.812			0.00966352			0.00483176	−	0.999988i	$-0.498462\pi$
$109$	114.812			0.00966352			0.00483176	+	0.999988i	$0.498462\pi$
$110$		0			0
$111$	−1436.32			−0.116575
$112$		0			0
$113$	15726.7			1.23163			0.615815	−	0.787891i	$-0.288827\pi$
$113$	15726.7			1.23163			0.615815	+	0.787891i	$0.288827\pi$
$114$		0			0
$115$		−	10070.0i		−	0.761434i
$116$		0			0
$117$			4701.33i			0.343438i
$118$		0			0
$119$			3985.80i			0.281463i
$120$		0			0
$121$	−34567.8			−2.36103
$122$		0			0
$123$		−	991.328i		−	0.0655250i
$124$		0			0
$125$	−15977.3			−1.02255
$126$		0			0
$127$		−	15947.7i		−	0.988759i	−0.869246	−	0.494380i	$-0.835395\pi$
$127$		−	15947.7i		−	0.988759i	0.869246	−	0.494380i	$-0.164605\pi$
$128$		0			0
$129$	10018.6			0.602043
$130$		0			0
$131$	−26656.7			−1.55333			−0.776666	−	0.629913i	$-0.783091\pi$
$131$	−26656.7			−1.55333			−0.776666	+	0.629913i	$0.783091\pi$
$132$		0			0
$133$	21993.6			1.24335
$134$		0			0
$135$		−	7438.57i		−	0.408152i
$136$		0			0
$137$		−	1658.69i		−	0.0883741i	−0.999023	−	0.0441870i	$-0.985930\pi$
$137$		−	1658.69i		−	0.0883741i	0.999023	−	0.0441870i	$-0.0140697\pi$
$138$		0			0
$139$		−	24655.9i		−	1.27612i	−0.769987	−	0.638059i	$-0.779737\pi$
$139$		−	24655.9i		−	1.27612i	0.769987	−	0.638059i	$-0.220263\pi$
$140$		0			0
$141$		−	2234.44i		−	0.112391i
$142$		0			0
$143$	14497.9			0.708976
$144$		0			0
$145$		−	20573.1i		−	0.978507i
$146$		0			0
$147$		−	6262.79i		−	0.289823i
$148$		0			0
$149$	−21164.6			−0.953317			−0.476658	−	0.879089i	$-0.658152\pi$
$149$	−21164.6			−0.953317			−0.476658	+	0.879089i	$0.658152\pi$
$150$		0			0
$151$			15194.7i			0.666406i	0.942855	+	0.333203i	$0.108129\pi$
$151$			15194.7i			0.666406i	−0.942855	+	0.333203i	$0.891871\pi$
$152$		0			0
$153$		−	4283.15i		−	0.182970i
$154$		0			0
$155$	5299.44	−	14591.9i	0.220580	−	0.607363i
$156$		0			0
$157$	−695.584			−0.0282196			−0.0141098	−	0.999900i	$-0.504491\pi$
$157$	−695.584			−0.0282196			−0.0141098	+	0.999900i	$0.504491\pi$
$158$		0			0
$159$	−5253.23			−0.207794
$160$		0			0
$161$			41727.9i			1.60981i
$162$		0			0
$163$	−14204.3			−0.534619			−0.267309	−	0.963611i	$-0.586135\pi$
$163$	−14204.3			−0.534619			−0.267309	+	0.963611i	$0.586135\pi$
$164$		0			0
$165$	−10789.6			−0.396312
$166$		0			0
$167$		−	14959.2i		−	0.536384i	−0.963365	−	0.268192i	$-0.913574\pi$
$167$		−	14959.2i		−	0.536384i	0.963365	−	0.268192i	$-0.0864262\pi$
$168$		0			0
$169$	24289.7			0.850448
$170$		0			0
$171$	−23634.3			−0.808260
$172$		0			0
$173$	18495.1			0.617965			0.308983	−	0.951068i	$-0.400011\pi$
$173$	18495.1			0.617965			0.308983	+	0.951068i	$0.400011\pi$
$174$		0			0
$175$	24368.7			0.795713
$176$		0			0
$177$			14230.7i			0.454234i
$178$		0			0
$179$			16289.5i			0.508394i	0.967152	+	0.254197i	$0.0818112\pi$
$179$			16289.5i			0.508394i	−0.967152	+	0.254197i	$0.918189\pi$
$180$		0			0
$181$		−	23603.4i		−	0.720471i	−0.932861	−	0.360236i	$-0.882696\pi$
$181$		−	23603.4i		−	0.720471i	0.932861	−	0.360236i	$-0.117304\pi$
$182$		0			0
$183$	14997.8			0.447841
$184$		0			0
$185$		−	7706.39i		−	0.225168i
$186$		0			0
$187$	−13208.3			−0.377714
$188$		0			0
$189$			30823.9i			0.862908i
$190$		0			0
$191$	9319.69			0.255467			0.127733	−	0.991809i	$-0.459230\pi$
$191$	9319.69			0.255467			0.127733	+	0.991809i	$0.459230\pi$
$192$		0			0
$193$	41176.9			1.10545			0.552725	−	0.833364i	$-0.313588\pi$
$193$	41176.9			1.10545			0.552725	+	0.833364i	$0.313588\pi$
$194$		0			0
$195$	3178.82			0.0835981
$196$		0			0
$197$		−	30012.5i		−	0.773339i	−0.922218	−	0.386669i	$-0.873625\pi$
$197$		−	30012.5i		−	0.773339i	0.922218	−	0.386669i	$-0.126375\pi$
$198$		0			0
$199$			57165.7i			1.44354i	0.692132	+	0.721771i	$0.256672\pi$
$199$			57165.7i			1.44354i	−0.692132	+	0.721771i	$0.743328\pi$
$200$		0			0
$201$		−	19862.0i		−	0.491621i
$202$		0			0
$203$			85250.9i			2.06874i
$204$		0			0
$205$	5318.85			0.126564
$206$		0			0
$207$		−	44840.9i		−	1.04649i
$208$		0			0
$209$			72883.0i			1.66853i
$210$		0			0
$211$	−56296.3			−1.26449			−0.632244	−	0.774769i	$-0.717866\pi$
$211$	−56296.3			−1.26449			−0.632244	+	0.774769i	$0.717866\pi$
$212$		0			0
$213$		−	24780.8i		−	0.546206i
$214$		0			0
$215$			53753.5i			1.16287i
$216$		0			0
$217$	−21959.8	+	60465.9i	−0.466347	+	1.28408i
$218$		0			0
$219$	21559.5			0.449522
$220$		0			0
$221$	3891.41			0.0796751
$222$		0			0
$223$		−	5569.33i		−	0.111994i	−0.998431	−	0.0559968i	$-0.982166\pi$
$223$		−	5569.33i		−	0.111994i	0.998431	−	0.0559968i	$-0.0178337\pi$
$224$		0			0
$225$	−26186.7			−0.517268
$226$		0			0
$227$	47715.7			0.925996			0.462998	−	0.886359i	$-0.346774\pi$
$227$	47715.7			0.925996			0.462998	+	0.886359i	$0.346774\pi$
$228$		0			0
$229$		−	68963.9i		−	1.31508i	−0.753422	−	0.657538i	$-0.771598\pi$
$229$		−	68963.9i		−	1.31508i	0.753422	−	0.657538i	$-0.228402\pi$
$230$		0			0
$231$	44709.9			0.837876
$232$		0			0
$233$	7500.46			0.138158			0.0690790	−	0.997611i	$-0.477994\pi$
$233$	7500.46			0.138158			0.0690790	+	0.997611i	$0.477994\pi$
$234$		0			0
$235$	11988.6			0.217087
$236$		0			0
$237$	−19245.3			−0.342633
$238$		0			0
$239$			52994.8i			0.927763i	0.885897	+	0.463882i	$0.153544\pi$
$239$			52994.8i			0.927763i	−0.885897	+	0.463882i	$0.846456\pi$
$240$		0			0
$241$			56610.9i			0.974689i	0.873210	+	0.487345i	$0.162034\pi$
$241$			56610.9i			0.974689i	−0.873210	+	0.487345i	$0.837966\pi$
$242$		0			0
$243$		−	50666.9i		−	0.858049i
$244$		0			0
$245$	33602.2			0.559804
$246$		0			0
$247$		−	21472.7i		−	0.351960i
$248$		0			0
$249$	29452.9			0.475039
$250$		0			0
$251$		−	30642.8i		−	0.486387i	−0.969978	−	0.243193i	$-0.921805\pi$
$251$		−	30642.8i		−	0.486387i	0.969978	−	0.243193i	$-0.0781949\pi$
$252$		0			0
$253$	−138279.			−2.16031
$254$		0			0
$255$	−2896.07			−0.0445377
$256$		0			0
$257$	−3213.28			−0.0486500			−0.0243250	−	0.999704i	$-0.507744\pi$
$257$	−3213.28			−0.0486500			−0.0243250	+	0.999704i	$0.507744\pi$
$258$		0			0
$259$			31933.7i			0.476047i
$260$		0			0
$261$		−	91610.8i		−	1.34482i
$262$		0			0
$263$			69055.0i			0.998352i	0.866501	+	0.499176i	$0.166364\pi$
$263$			69055.0i			0.998352i	−0.866501	+	0.499176i	$0.833636\pi$
$264$		0			0
$265$		−	28185.6i		−	0.401361i
$266$		0			0
$267$	−6860.12			−0.0962297
$268$		0			0
$269$		−	49330.7i		−	0.681731i	−0.940112	−	0.340866i	$-0.889280\pi$
$269$		−	49330.7i		−	0.681731i	0.940112	−	0.340866i	$-0.110720\pi$
$270$		0			0
$271$		−	56069.0i		−	0.763457i	−0.924275	−	0.381728i	$-0.875329\pi$
$271$		−	56069.0i		−	0.763457i	0.924275	−	0.381728i	$-0.124671\pi$
$272$		0			0
$273$	−13172.4			−0.176742
$274$		0			0
$275$			80753.9i			1.06782i
$276$		0			0
$277$		−	43505.6i		−	0.567003i	−0.958972	−	0.283502i	$-0.908504\pi$
$277$		−	43505.6i		−	0.567003i	0.958972	−	0.283502i	$-0.0914961\pi$
$278$		0			0
$279$	23598.1	−	64976.8i	0.303157	−	0.834737i
$280$		0			0
$281$	76538.0			0.969314			0.484657	−	0.874704i	$-0.338944\pi$
$281$	76538.0			0.969314			0.484657	+	0.874704i	$0.338944\pi$
$282$		0			0
$283$	32018.4			0.399785			0.199893	−	0.979818i	$-0.435941\pi$
$283$	32018.4			0.399785			0.199893	+	0.979818i	$0.435941\pi$
$284$		0			0
$285$			15980.4i			0.196743i
$286$		0			0
$287$	−22040.2			−0.267579
$288$		0			0
$289$	79975.7			0.957552
$290$		0			0
$291$			10264.8i			0.121217i
$292$		0			0
$293$	−83976.9			−0.978194			−0.489097	−	0.872230i	$-0.662674\pi$
$293$	−83976.9			−0.978194			−0.489097	+	0.872230i	$0.662674\pi$
$294$		0			0
$295$	−76353.0			−0.877368
$296$		0			0
$297$	−102145.			−1.15799
$298$		0			0
$299$	40739.8			0.455697
$300$		0			0
$301$		−	222744.i		−	2.45851i
$302$		0			0
$303$		−	47380.9i		−	0.516081i
$304$		0			0
$305$			80468.6i			0.865021i
$306$		0			0
$307$	180118.			1.91109			0.955544	−	0.294849i	$-0.0952694\pi$
$307$	180118.			1.91109			0.955544	+	0.294849i	$0.0952694\pi$
$308$		0			0
$309$			28785.3i			0.301477i
$310$		0			0
$311$	−14764.7			−0.152653			−0.0763265	−	0.997083i	$-0.524319\pi$
$311$	−14764.7			−0.152653			−0.0763265	+	0.997083i	$0.524319\pi$
$312$		0			0
$313$		−	117215.i		−	1.19645i	−0.801328	−	0.598225i	$-0.795873\pi$
$313$		−	117215.i		−	1.19645i	0.801328	−	0.598225i	$-0.204127\pi$
$314$		0			0
$315$	−77789.4			−0.783970
$316$		0			0
$317$	39990.7			0.397961			0.198980	−	0.980003i	$-0.436237\pi$
$317$	39990.7			0.397961			0.198980	+	0.980003i	$0.436237\pi$
$318$		0			0
$319$	−282507.			−2.77618
$320$		0			0
$321$		−	33502.7i		−	0.325140i
$322$		0			0
$323$			19562.8i			0.187510i
$324$		0			0
$325$		−	23791.6i		−	0.225246i
$326$		0			0
$327$		−	345.684i		−	0.00323284i
$328$		0			0
$329$	−49678.5			−0.458962
$330$		0			0
$331$			118136.i			1.07827i	0.842219	+	0.539135i	$0.181249\pi$
$331$			118136.i			1.07827i	−0.842219	+	0.539135i	$0.818751\pi$
$332$		0			0
$333$		−	34316.0i		−	0.309463i
$334$		0			0
$335$	106567.			0.949584
$336$		0			0
$337$		−	92291.6i		−	0.812648i	−0.913729	−	0.406324i	$-0.866811\pi$
$337$		−	92291.6i		−	0.812648i	0.913729	−	0.406324i	$-0.133189\pi$
$338$		0			0
$339$		−	47351.0i		−	0.412030i
$340$		0			0
$341$	−200374.	−	72771.1i	−1.72319	−	0.625821i
$342$		0			0
$343$	21483.8			0.182609
$344$		0			0
$345$	−30319.3			−0.254731
$346$		0			0
$347$		−	127823.i		−	1.06157i	−0.847506	−	0.530786i	$-0.821897\pi$
$347$		−	127823.i		−	1.06157i	0.847506	−	0.530786i	$-0.178103\pi$
$348$		0			0
$349$	132077.			1.08437			0.542183	−	0.840260i	$-0.317598\pi$
$349$	132077.			1.08437			0.542183	+	0.840260i	$0.317598\pi$
$350$		0			0
$351$	30094.0			0.244268
$352$		0			0
$353$			155311.i			1.24639i	0.782068	+	0.623193i	$0.214165\pi$
$353$			155311.i			1.24639i	−0.782068	+	0.623193i	$0.785835\pi$
$354$		0			0
$355$	132958.			1.05502
$356$		0			0
$357$	12000.7			0.0941609
$358$		0			0
$359$	97187.0			0.754083			0.377042	−	0.926196i	$-0.376941\pi$
$359$	97187.0			0.754083			0.377042	+	0.926196i	$0.376941\pi$
$360$		0			0
$361$	−22374.1			−0.171685
$362$		0			0
$363$			104079.i			0.789860i
$364$		0			0
$365$			115675.i			0.868267i
$366$		0			0
$367$		−	12139.6i		−	0.0901302i	−0.998984	−	0.0450651i	$-0.985650\pi$
$367$		−	12139.6i		−	0.0901302i	0.998984	−	0.0450651i	$-0.0143495\pi$
$368$		0			0
$369$	23684.5			0.173945
$370$		0			0
$371$			116795.i			0.848550i
$372$		0			0
$373$	−60713.5			−0.436383			−0.218191	−	0.975906i	$-0.570016\pi$
$373$	−60713.5			−0.436383			−0.218191	+	0.975906i	$0.570016\pi$
$374$		0			0
$375$			48105.5i			0.342084i
$376$		0			0
$377$	83232.1			0.585609
$378$		0			0
$379$	−17514.0			−0.121929			−0.0609645	−	0.998140i	$-0.519418\pi$
$379$	−17514.0			−0.121929			−0.0609645	+	0.998140i	$0.519418\pi$
$380$		0			0
$381$	−48016.4			−0.330780
$382$		0			0
$383$			167299.i			1.14050i	0.821471	+	0.570250i	$0.193154\pi$
$383$			167299.i			1.14050i	−0.821471	+	0.570250i	$0.806846\pi$
$384$		0			0
$385$			239885.i			1.61839i
$386$		0			0
$387$			239361.i			1.59820i
$388$		0			0
$389$		−	117046.i		−	0.773492i	−0.922186	−	0.386746i	$-0.873599\pi$
$389$		−	117046.i		−	0.773492i	0.922186	−	0.386746i	$-0.126401\pi$
$390$		0			0
$391$	−37116.0			−0.242777
$392$		0			0
$393$			80259.9i			0.519653i
$394$		0			0
$395$		−	103259.i		−	0.661808i
$396$		0			0
$397$	138674.			0.879858			0.439929	−	0.898033i	$-0.355004\pi$
$397$	138674.			0.879858			0.439929	+	0.898033i	$0.355004\pi$
$398$		0			0
$399$		−	66219.7i		−	0.415950i
$400$		0			0
$401$		−	243942.i		−	1.51704i	−0.651650	−	0.758520i	$-0.725923\pi$
$401$		−	243942.i		−	1.51704i	0.651650	−	0.758520i	$-0.274077\pi$
$402$		0			0
$403$	59034.0	+	21439.8i	0.363490	+	0.132011i
$404$		0			0
$405$	71730.6			0.437315
$406$		0			0
$407$	−105823.			−0.638839
$408$		0			0
$409$		−	11473.9i		−	0.0685907i	−0.999412	−	0.0342954i	$-0.989081\pi$
$409$		−	11473.9i		−	0.0685907i	0.999412	−	0.0342954i	$-0.0109187\pi$
$410$		0			0
$411$	−4994.10			−0.0295647
$412$		0			0
$413$	316391.			1.85492
$414$		0			0
$415$			158026.i			0.917555i
$416$		0			0
$417$	−74235.6			−0.426914
$418$		0			0
$419$	−64305.6			−0.366286			−0.183143	−	0.983086i	$-0.558627\pi$
$419$	−64305.6			−0.366286			−0.183143	+	0.983086i	$0.558627\pi$
$420$		0			0
$421$	−178277.			−1.00585			−0.502923	−	0.864331i	$-0.667742\pi$
$421$	−178277.			−1.00585			−0.502923	+	0.864331i	$0.667742\pi$
$422$		0			0
$423$	53384.6			0.298356
$424$		0			0
$425$			21675.4i			0.120002i
$426$		0			0
$427$		−	333446.i		−	1.82881i
$428$		0			0
$429$		−	43651.1i		−	0.237181i
$430$		0			0
$431$	153293.			0.825214			0.412607	−	0.910909i	$-0.364618\pi$
$431$	153293.			0.825214			0.412607	+	0.910909i	$0.364618\pi$
$432$		0			0
$433$			234185.i			1.24906i	0.781000	+	0.624530i	$0.214710\pi$
$433$			234185.i			1.24906i	−0.781000	+	0.624530i	$0.785290\pi$
$434$		0			0
$435$	−61942.9			−0.327351
$436$		0			0
$437$			204805.i			1.07245i
$438$		0			0
$439$	15890.9			0.0824553			0.0412276	−	0.999150i	$-0.486873\pi$
$439$	15890.9			0.0824553			0.0412276	+	0.999150i	$0.486873\pi$
$440$		0			0
$441$	149629.			0.769374
$442$		0			0
$443$	−262700.			−1.33860			−0.669302	−	0.742990i	$-0.733407\pi$
$443$	−262700.			−1.33860			−0.669302	+	0.742990i	$0.733407\pi$
$444$		0			0
$445$		−	36807.1i		−	0.185871i
$446$		0			0
$447$			63723.8i			0.318923i
$448$		0			0
$449$			79750.8i			0.395587i	0.980244	+	0.197794i	$0.0633776\pi$
$449$			79750.8i			0.395587i	−0.980244	+	0.197794i	$0.936622\pi$
$450$		0			0
$451$		−	73037.7i		−	0.359082i
$452$		0			0
$453$	45749.3			0.222940
$454$		0			0
$455$		−	70674.8i		−	0.341383i
$456$		0			0
$457$			377226.i			1.80621i	0.429416	+	0.903107i	$0.358719\pi$
$457$			377226.i			1.80621i	−0.429416	+	0.903107i	$0.641281\pi$
$458$		0			0
$459$	−27417.2			−0.130136
$460$		0			0
$461$			119252.i			0.561130i	0.959835	+	0.280565i	$0.0905218\pi$
$461$			119252.i			0.561130i	−0.959835	+	0.280565i	$0.909478\pi$
$462$		0			0
$463$			286927.i			1.33847i	0.743050	+	0.669236i	$0.233379\pi$
$463$			286927.i			1.33847i	−0.743050	+	0.669236i	$0.766621\pi$
$464$		0			0
$465$	−43934.3	−	15955.9i	−0.203188	−	0.0737930i
$466$		0			0
$467$	−366554.			−1.68075			−0.840377	−	0.542002i	$-0.817666\pi$
$467$	−366554.			−1.68075			−0.840377	+	0.542002i	$0.817666\pi$
$468$		0			0
$469$	−441593.			−2.00759
$470$		0			0
$471$			2094.31i			0.00944059i
$472$		0			0
$473$	738136.			3.29924
$474$		0			0
$475$	119604.			0.530103
$476$		0			0
$477$		−	125509.i		−	0.551616i
$478$		0			0
$479$	191449.			0.834414			0.417207	−	0.908812i	$-0.363009\pi$
$479$	191449.			0.834414			0.417207	+	0.908812i	$0.363009\pi$
$480$		0			0
$481$	31177.5			0.134757
$482$		0			0
$483$	125637.			0.538547
$484$		0			0
$485$	−55074.5			−0.234136
$486$		0			0
$487$		−	263460.i		−	1.11085i	−0.831566	−	0.555426i	$-0.812555\pi$
$487$		−	263460.i		−	1.11085i	0.831566	−	0.555426i	$-0.187445\pi$
$488$		0			0
$489$			42767.2i			0.178852i
$490$		0			0
$491$		−	172003.i		−	0.713464i	−0.934207	−	0.356732i	$-0.883891\pi$
$491$		−	172003.i		−	0.713464i	0.934207	−	0.356732i	$-0.116109\pi$
$492$		0			0
$493$	−75828.6			−0.311989
$494$		0			0
$495$		−	257781.i		−	1.05206i
$496$		0			0
$497$	−550953.			−2.23050
$498$		0			0
$499$		−	264134.i		−	1.06077i	−0.847756	−	0.530387i	$-0.822047\pi$
$499$		−	264134.i		−	1.06077i	0.847756	−	0.530387i	$-0.177953\pi$
$500$		0			0
$501$	−45040.2			−0.179442
$502$		0			0
$503$	72423.1			0.286247			0.143124	−	0.989705i	$-0.454285\pi$
$503$	72423.1			0.286247			0.143124	+	0.989705i	$0.454285\pi$
$504$		0			0
$505$	254216.			0.996829
$506$		0			0
$507$		−	73132.9i		−	0.284510i
$508$		0			0
$509$		−	378639.i		−	1.46147i	−0.682662	−	0.730734i	$-0.739178\pi$
$509$		−	378639.i		−	1.46147i	0.682662	−	0.730734i	$-0.260822\pi$
$510$		0			0
$511$		−	479333.i		−	1.83568i
$512$		0			0
$513$			151287.i			0.574868i
$514$		0			0
$515$	−154444.			−0.582313
$516$		0			0
$517$		−	164626.i		−	0.615911i
$518$		0			0
$519$		−	55686.3i		−	0.206735i
$520$		0			0
$521$	−306109.			−1.12772			−0.563859	−	0.825871i	$-0.690684\pi$
$521$	−306109.			−1.12772			−0.563859	+	0.825871i	$0.690684\pi$
$522$		0			0
$523$			87718.5i			0.320692i	0.987061	+	0.160346i	$0.0512610\pi$
$523$			87718.5i			0.320692i	−0.987061	+	0.160346i	$0.948739\pi$
$524$		0			0
$525$		−	73371.0i		−	0.266199i
$526$		0			0
$527$	−53783.0	−	19532.7i	−0.193653	−	0.0703301i
$528$		0			0
$529$	−108732.			−0.388548
$530$		0			0
$531$	−339995.			−1.20582
$532$		0			0
$533$			21518.3i			0.0757450i
$534$		0			0
$535$	179755.			0.628019
$536$		0			0
$537$	49045.4			0.170079
$538$		0			0
$539$		−	461421.i		−	1.58825i
$540$		0			0
$541$	114508.			0.391240			0.195620	−	0.980680i	$-0.437328\pi$
$541$	114508.			0.391240			0.195620	+	0.980680i	$0.437328\pi$
$542$		0			0
$543$	−71066.6			−0.241027
$544$		0			0
$545$	1854.73			0.00624435
$546$		0			0
$547$	415415.			1.38838			0.694188	−	0.719794i	$-0.255764\pi$
$547$	415415.			1.38838			0.694188	+	0.719794i	$0.255764\pi$
$548$		0			0
$549$			358321.i			1.18885i
$550$		0			0
$551$			418421.i			1.37819i
$552$		0			0
$553$			427883.i			1.39918i
$554$		0			0
$555$	−23202.9			−0.0753280
$556$		0			0
$557$		−	522689.i		−	1.68474i	−0.538900	−	0.842370i	$-0.681160\pi$
$557$		−	522689.i		−	1.68474i	0.538900	−	0.842370i	$-0.318840\pi$
$558$		0			0
$559$	−217469.			−0.695943
$560$		0			0
$561$			39768.4i			0.126361i
$562$		0			0
$563$	383486.			1.20985			0.604926	−	0.796282i	$-0.293203\pi$
$563$	383486.			1.20985			0.604926	+	0.796282i	$0.293203\pi$
$564$		0			0
$565$	254056.			0.795851
$566$		0			0
$567$	−297237.			−0.924565
$568$		0			0
$569$		−	73389.1i		−	0.226677i	−0.993556	−	0.113338i	$-0.963846\pi$
$569$		−	73389.1i		−	0.226677i	0.993556	−	0.113338i	$-0.0361544\pi$
$570$		0			0
$571$		−	60865.1i		−	0.186679i	−0.995634	−	0.0933397i	$-0.970246\pi$
$571$		−	60865.1i		−	0.186679i	0.995634	−	0.0933397i	$-0.0297543\pi$
$572$		0			0
$573$		−	28060.3i		−	0.0854641i
$574$		0			0
$575$			226923.i			0.686345i
$576$		0			0
$577$	561989.			1.68802			0.844008	−	0.536331i	$-0.180190\pi$
$577$	561989.			1.68802			0.844008	+	0.536331i	$0.180190\pi$
$578$		0			0
$579$		−	123978.i		−	0.369818i
$580$		0			0
$581$		−	654827.i		−	1.93988i
$582$		0			0
$583$	−387040.			−1.13873
$584$		0			0
$585$			75947.3i			0.221922i
$586$		0			0
$587$		−	18401.2i		−	0.0534036i	−0.999643	−	0.0267018i	$-0.991500\pi$
$587$		−	18401.2i		−	0.0534036i	0.999643	−	0.0267018i	$-0.00850045\pi$
$588$		0			0
$589$	−107781.	+	296773.i	−0.310679	+	0.855449i
$590$		0			0
$591$	−90363.7			−0.258713
$592$		0			0
$593$	−376357.			−1.07026			−0.535132	−	0.844769i	$-0.679738\pi$
$593$	−376357.			−1.07026			−0.535132	+	0.844769i	$0.679738\pi$
$594$		0			0
$595$			64388.3i			0.181875i
$596$		0			0
$597$	172118.			0.482924
$598$		0			0
$599$	−287857.			−0.802275			−0.401137	−	0.916018i	$-0.631385\pi$
$599$	−287857.			−0.802275			−0.401137	+	0.916018i	$0.631385\pi$
$600$		0			0
$601$		−	89490.0i		−	0.247757i	−0.992297	−	0.123878i	$-0.960467\pi$
$601$		−	89490.0i		−	0.247757i	0.992297	−	0.123878i	$-0.0395333\pi$
$602$		0			0
$603$	474536.			1.30507
$604$		0			0
$605$	−558423.			−1.52564
$606$		0			0
$607$	−391296.			−1.06201			−0.531004	−	0.847369i	$-0.678185\pi$
$607$	−391296.			−1.06201			−0.531004	+	0.847369i	$0.678185\pi$
$608$		0			0
$609$	256679.			0.692079
$610$		0			0
$611$			48502.1i			0.129921i
$612$		0			0
$613$			189074.i			0.503165i	0.967836	+	0.251583i	$0.0809510\pi$
$613$			189074.i			0.503165i	−0.967836	+	0.251583i	$0.919049\pi$
$614$		0			0
$615$		−	16014.3i		−	0.0423408i
$616$		0			0
$617$	110621.			0.290581			0.145291	−	0.989389i	$-0.453588\pi$
$617$	110621.			0.290581			0.145291	+	0.989389i	$0.453588\pi$
$618$		0			0
$619$			235880.i			0.615616i	0.951448	+	0.307808i	$0.0995955\pi$
$619$			235880.i			0.615616i	−0.951448	+	0.307808i	$0.900405\pi$
$620$		0			0
$621$	−287034.			−0.744304
$622$		0			0
$623$			152521.i			0.392966i
$624$		0			0
$625$	−30582.7			−0.0782917
$626$		0			0
$627$	219441.			0.558191
$628$		0			0
$629$	−28404.3			−0.0717931
$630$		0			0
$631$		−	725427.i		−	1.82194i	−0.412468	−	0.910972i	$-0.635333\pi$
$631$		−	725427.i		−	1.82194i	0.412468	−	0.910972i	$-0.364667\pi$
$632$		0			0
$633$			169501.i			0.423023i
$634$		0			0
$635$		−	257626.i		−	0.638914i
$636$		0			0
$637$			135944.i			0.335027i
$638$		0			0
$639$	592055.			1.44997
$640$		0			0
$641$			42234.9i			0.102791i	0.998678	+	0.0513955i	$0.0163669\pi$
$641$			42234.9i			0.102791i	−0.998678	+	0.0513955i	$0.983633\pi$
$642$		0			0
$643$			765165.i			1.85069i	0.379130	+	0.925343i	$0.376223\pi$
$643$			765165.i			1.85069i	−0.379130	+	0.925343i	$0.623777\pi$
$644$		0			0
$645$	161845.			0.389026
$646$		0			0
$647$		−	137859.i		−	0.329327i	−0.986350	−	0.164663i	$-0.947346\pi$
$647$		−	137859.i		−	0.329327i	0.986350	−	0.164663i	$-0.0526538\pi$
$648$		0			0
$649$			1.04847e6i			2.48924i
$650$		0			0
$651$	182055.	+	66118.0i	0.429576	+	0.156012i
$652$		0			0
$653$	−101837.			−0.238825			−0.119412	−	0.992845i	$-0.538101\pi$
$653$	−101837.			−0.238825			−0.119412	+	0.992845i	$0.538101\pi$
$654$		0			0
$655$	−430624.			−1.00373
$656$		0			0
$657$			515093.i			1.19331i
$658$		0			0
$659$	479706.			1.10460			0.552300	−	0.833646i	$-0.313750\pi$
$659$	479706.			1.10460			0.552300	+	0.833646i	$0.313750\pi$
$660$		0			0
$661$	600294.			1.37392			0.686959	−	0.726696i	$-0.258945\pi$
$661$	600294.			1.37392			0.686959	+	0.726696i	$0.258945\pi$
$662$		0			0
$663$		−	11716.5i		−	0.0266546i
$664$		0			0
$665$	355294.			0.803423
$666$		0			0
$667$	−793861.			−1.78440
$668$		0			0
$669$	−16768.5			−0.0374664
$670$		0			0
$671$	1.10498e6			2.45420
$672$		0			0
$673$			467721.i			1.03266i	0.856390	+	0.516329i	$0.172702\pi$
$673$			467721.i			1.03266i	−0.856390	+	0.516329i	$0.827298\pi$
$674$		0			0
$675$			167625.i			0.367902i
$676$		0			0
$677$		−	518012.i		−	1.13022i	−0.825016	−	0.565109i	$-0.808834\pi$
$677$		−	518012.i		−	1.13022i	0.825016	−	0.565109i	$-0.191166\pi$
$678$		0			0
$679$	228218.			0.495005
$680$		0			0
$681$		−	143666.i		−	0.309784i
$682$		0			0
$683$	−341401.			−0.731852			−0.365926	−	0.930644i	$-0.619248\pi$
$683$	−341401.			−0.731852			−0.365926	+	0.930644i	$0.619248\pi$
$684$		0			0
$685$		−	26795.2i		−	0.0571053i
$686$		0			0
$687$	−207641.			−0.439946
$688$		0			0
$689$	114030.			0.240203
$690$		0			0
$691$	−561520.			−1.17600			−0.588002	−	0.808859i	$-0.700085\pi$
$691$	−561520.			−1.17600			−0.588002	+	0.808859i	$0.700085\pi$
$692$		0			0
$693$			1.06819e6i			2.22425i
$694$		0			0
$695$		−	398302.i		−	0.824599i
$696$		0			0
$697$		−	19604.3i		−	0.0403539i
$698$		0			0
$699$		−	22582.9i		−	0.0462195i
$700$		0			0
$701$	−48521.5			−0.0987412			−0.0493706	−	0.998781i	$-0.515722\pi$
$701$	−48521.5			−0.0987412			−0.0493706	+	0.998781i	$0.515722\pi$
$702$		0			0
$703$			156734.i			0.317142i
$704$		0			0
$705$		−	36096.2i		−	0.0726245i
$706$		0			0
$707$	−1.05342e6			−2.10748
$708$		0			0
$709$		−	329285.i		−	0.655059i	−0.944841	−	0.327529i	$-0.893784\pi$
$709$		−	329285.i		−	0.655059i	0.944841	−	0.327529i	$-0.106216\pi$
$710$		0			0
$711$		−	459804.i		−	0.909564i
$712$		0			0
$713$	−563062.	−	204491.i	−1.10758	−	0.402249i
$714$		0			0
$715$	234205.			0.458124
$716$		0			0
$717$	159560.			0.310375
$718$		0			0
$719$			829911.i			1.60537i	0.596407	+	0.802683i	$0.296595\pi$
$719$			829911.i			1.60537i	−0.596407	+	0.802683i	$0.703405\pi$
$720$		0			0
$721$	639984.			1.23112
$722$		0			0
$723$	170448.			0.326073
$724$		0			0
$725$			463608.i			0.882012i
$726$		0			0
$727$	1.01055e6			1.91201			0.956004	−	0.293353i	$-0.0947712\pi$
$727$	1.01055e6			1.91201			0.956004	+	0.293353i	$0.0947712\pi$
$728$		0			0
$729$	207113.			0.389720
$730$		0			0
$731$	198125.			0.370771
$732$		0			0
$733$	540023.			1.00509			0.502544	−	0.864552i	$-0.332398\pi$
$733$	540023.			1.00509			0.502544	+	0.864552i	$0.332398\pi$
$734$		0			0
$735$		−	101172.i		−	0.187277i
$736$		0			0
$737$		−	1.46336e6i		−	2.69412i
$738$		0			0
$739$			69857.3i			0.127915i	0.997953	+	0.0639577i	$0.0203723\pi$
$739$			69857.3i			0.127915i	−0.997953	+	0.0639577i	$0.979628\pi$
$740$		0			0
$741$	−64651.6			−0.117745
$742$		0			0
$743$			64118.2i			0.116146i	0.998312	+	0.0580729i	$0.0184956\pi$
$743$			64118.2i			0.116146i	−0.998312	+	0.0580729i	$0.981504\pi$
$744$		0			0
$745$	−341902.			−0.616012
$746$		0			0
$747$			703679.i			1.26105i
$748$		0			0
$749$	−744868.			−1.32775
$750$		0			0
$751$	217979.			0.386488			0.193244	−	0.981151i	$-0.438099\pi$
$751$	217979.			0.386488			0.193244	+	0.981151i	$0.438099\pi$
$752$		0			0
$753$	−92261.5			−0.162716
$754$		0			0
$755$			245462.i			0.430616i
$756$		0			0
$757$			270275.i			0.471643i	0.971796	+	0.235822i	$0.0757781\pi$
$757$			270275.i			0.471643i	−0.971796	+	0.235822i	$0.924222\pi$
$758$		0			0
$759$			416341.i			0.722712i
$760$		0			0
$761$		−	218980.i		−	0.378124i	−0.981965	−	0.189062i	$-0.939455\pi$
$761$		−	218980.i		−	0.378124i	0.981965	−	0.189062i	$-0.0605448\pi$
$762$		0			0
$763$	−7685.62			−0.0132017
$764$		0			0
$765$		−	69191.9i		−	0.118231i
$766$		0			0
$767$		−	308899.i		−	0.525080i
$768$		0			0
$769$	−686317.			−1.16057			−0.580286	−	0.814413i	$-0.697059\pi$
$769$	−686317.			−1.16057			−0.580286	+	0.814413i	$0.697059\pi$
$770$		0			0
$771$			9674.76i			0.0162754i
$772$		0			0
$773$		−	750379.i		−	1.25580i	−0.778293	−	0.627901i	$-0.783914\pi$
$773$		−	750379.i		−	1.25580i	0.778293	−	0.627901i	$-0.216086\pi$
$774$		0			0
$775$	−119421.	+	328823.i	−0.198828	+	0.547468i
$776$		0			0
$777$	96148.2			0.159257
$778$		0			0
$779$	−108176.			−0.178261
$780$		0			0
$781$		−	1.82577e6i		−	2.99325i
$782$		0			0
$783$	−586416.			−0.956494
$784$		0			0
$785$	−11236.8			−0.0182348
$786$		0			0
$787$			727065.i			1.17388i	0.809630	+	0.586940i	$0.199668\pi$
$787$			727065.i			1.17388i	−0.809630	+	0.586940i	$0.800332\pi$
$788$		0			0
$789$	207916.			0.333990
$790$		0			0
$791$	−1.05276e6			−1.68258
$792$		0			0
$793$	−325549.			−0.517691
$794$		0			0
$795$	−84863.0			−0.134272
$796$		0			0
$797$		−	101857.i		−	0.160352i	−0.996781	−	0.0801758i	$-0.974452\pi$
$797$		−	101857.i		−	0.160352i	0.996781	−	0.0801758i	$-0.0255482\pi$
$798$		0			0
$799$		−	44187.8i		−	0.0692164i
$800$		0			0
$801$		−	163900.i		−	0.255454i
$802$		0			0
$803$	1.58843e6			2.46341
$804$		0			0
$805$			674091.i			1.04022i
$806$		0			0
$807$	−148528.			−0.228067
$808$		0			0
$809$		−	994694.i		−	1.51982i	−0.650028	−	0.759911i	$-0.725243\pi$
$809$		−	994694.i		−	1.51982i	0.650028	−	0.759911i	$-0.274757\pi$
$810$		0			0
$811$	−968962.			−1.47321			−0.736606	−	0.676322i	$-0.763573\pi$
$811$	−968962.			−1.47321			−0.736606	+	0.676322i	$0.763573\pi$
$812$		0			0
$813$	−168816.			−0.255407
$814$		0			0
$815$	−229462.			−0.345458
$816$		0			0
$817$		−	1.09325e6i		−	1.63786i
$818$		0			0
$819$		−	314710.i		−	0.469184i
$820$		0			0
$821$		−	25477.6i		−	0.0377983i	−0.999821	−	0.0188991i	$-0.993984\pi$
$821$		−	25477.6i		−	0.0377983i	0.999821	−	0.0188991i	$-0.00601614\pi$
$822$		0			0
$823$		−	1.23265e6i		−	1.81988i	−0.414745	−	0.909938i	$-0.636129\pi$
$823$		−	1.23265e6i		−	1.81988i	0.414745	−	0.909938i	$-0.363871\pi$
$824$		0			0
$825$	243139.			0.357229
$826$		0			0
$827$			923364.i			1.35009i	0.737778	+	0.675044i	$0.235875\pi$
$827$			923364.i			1.35009i	−0.737778	+	0.675044i	$0.764125\pi$
$828$		0			0
$829$		−	309988.i		−	0.451061i	−0.974236	−	0.225531i	$-0.927588\pi$
$829$		−	309988.i		−	0.451061i	0.974236	−	0.225531i	$-0.0724116\pi$
$830$		0			0
$831$	−130990.			−0.189686
$832$		0			0
$833$		−	123851.i		−	0.178489i
$834$		0			0
$835$		−	241658.i		−	0.346599i
$836$		0			0
$837$	−415927.	−	151055.i	−0.593699	−	0.215618i
$838$		0			0
$839$	1.26783e6			1.80109			0.900547	−	0.434759i	$-0.143166\pi$
$839$	1.26783e6			1.80109			0.900547	+	0.434759i	$0.143166\pi$
$840$		0			0
$841$	−914592.			−1.29311
$842$		0			0
$843$		−	230446.i		−	0.324275i
$844$		0			0
$845$	392386.			0.549541
$846$		0			0
$847$	2.31399e6			3.22549
$848$		0			0
$849$		−	96403.2i		−	0.133745i
$850$		0			0
$851$	−297369.			−0.410616
$852$		0			0
$853$	240725.			0.330844			0.165422	−	0.986223i	$-0.447101\pi$
$853$	240725.			0.330844			0.165422	+	0.986223i	$0.447101\pi$
$854$		0			0
$855$	−381799.			−0.522279
$856$		0			0
$857$	−209842.			−0.285714			−0.142857	−	0.989743i	$-0.545629\pi$
$857$	−209842.			−0.285714			−0.142857	+	0.989743i	$0.545629\pi$
$858$		0			0
$859$		−	733682.i		−	0.994310i	−0.867662	−	0.497155i	$-0.834378\pi$
$859$		−	733682.i		−	0.994310i	0.867662	−	0.497155i	$-0.165622\pi$
$860$		0			0
$861$			66360.2i			0.0895162i
$862$		0			0
$863$			1.46308e6i			1.96447i	0.187648	+	0.982236i	$0.439914\pi$
$863$			1.46308e6i			1.96447i	−0.187648	+	0.982236i	$0.560086\pi$
$864$		0			0
$865$	298778.			0.399315
$866$		0			0
$867$		−	240796.i		−	0.320340i
$868$		0			0
$869$	−1.41793e6			−1.87766
$870$		0			0
$871$			431135.i			0.568299i
$872$		0			0
$873$	−245243.			−0.321787
$874$		0			0
$875$	1.06953e6			1.39694
$876$		0			0
$877$	−1.34294e6			−1.74605			−0.873024	−	0.487678i	$-0.837844\pi$
$877$	−1.34294e6			−1.74605			−0.873024	+	0.487678i	$0.837844\pi$
$878$		0			0
$879$			252843.i			0.327246i
$880$		0			0
$881$			259075.i			0.333790i	0.985975	+	0.166895i	$0.0533741\pi$
$881$			259075.i			0.333790i	−0.985975	+	0.166895i	$0.946626\pi$
$882$		0			0
$883$			41148.7i			0.0527758i	0.999652	+	0.0263879i	$0.00840051\pi$
$883$			41148.7i			0.0527758i	−0.999652	+	0.0263879i	$0.991599\pi$
$884$		0			0
$885$			229889.i			0.293515i
$886$		0			0
$887$	−519280.			−0.660016			−0.330008	−	0.943978i	$-0.607051\pi$
$887$	−519280.			−0.660016			−0.330008	+	0.943978i	$0.607051\pi$
$888$		0			0
$889$			1.06755e6i			1.35078i
$890$		0			0
$891$		−	984996.i		−	1.24073i
$892$		0			0
$893$	−243828.			−0.305760
$894$		0			0
$895$			263147.i			0.328513i
$896$		0			0
$897$		−	122662.i		−	0.152449i
$898$		0			0
$899$	−1.15035e6	−	417779.i	−1.42334	−	0.516924i
$900$		0			0
$901$	−103887.			−0.127971
$902$		0			0
$903$	−670652.			−0.822473
$904$		0			0
$905$		−	381299.i		−	0.465552i
$906$		0			0
$907$	463052.			0.562879			0.281439	−	0.959579i	$-0.409188\pi$
$907$	463052.			0.562879			0.281439	+	0.959579i	$0.409188\pi$
$908$		0			0
$909$	1.13201e6			1.37000
$910$		0			0
$911$		−	499296.i		−	0.601618i	−0.953684	−	0.300809i	$-0.902743\pi$
$911$		−	499296.i		−	0.601618i	0.953684	−	0.300809i	$-0.0972567\pi$
$912$		0			0
$913$	2.16999e6			2.60325
$914$		0			0
$915$	242280.			0.289385
$916$		0			0
$917$	1.78442e6			2.12207
$918$		0			0
$919$	263056.			0.311471			0.155736	−	0.987799i	$-0.450225\pi$
$919$	263056.			0.311471			0.155736	+	0.987799i	$0.450225\pi$
$920$		0			0
$921$		−	542312.i		−	0.639337i
$922$		0			0
$923$			537906.i			0.631397i
$924$		0			0
$925$			173661.i			0.202963i
$926$		0			0
$927$	−687729.			−0.800309
$928$		0			0
$929$		−	696545.i		−	0.807082i	−0.914962	−	0.403541i	$-0.867779\pi$
$929$		−	696545.i		−	0.807082i	0.914962	−	0.403541i	$-0.132221\pi$
$930$		0			0
$931$	−683410.			−0.788464
$932$		0			0
$933$			44454.7i			0.0510686i
$934$		0			0
$935$	−213372.			−0.244070
$936$		0			0
$937$	−684691.			−0.779858			−0.389929	−	0.920845i	$-0.627500\pi$
$937$	−684691.			−0.779858			−0.389929	+	0.920845i	$0.627500\pi$
$938$		0			0
$939$	−352919.			−0.400261
$940$		0			0
$941$			1.09654e6i			1.23835i	0.785252	+	0.619176i	$0.212533\pi$
$941$			1.09654e6i			1.23835i	−0.785252	+	0.619176i	$0.787467\pi$
$942$		0			0
$943$		−	205240.i		−	0.230801i
$944$		0			0
$945$			497943.i			0.557592i
$946$		0			0
$947$		−	439438.i		−	0.490002i	−0.969523	−	0.245001i	$-0.921212\pi$
$947$		−	439438.i		−	0.490002i	0.969523	−	0.245001i	$-0.0787883\pi$
$948$		0			0
$949$	−467982.			−0.519633
$950$		0			0
$951$		−	120407.i		−	0.133134i
$952$		0			0
$953$			701926.i			0.772868i	0.922317	+	0.386434i	$0.126293\pi$
$953$			701926.i			0.772868i	−0.922317	+	0.386434i	$0.873707\pi$
$954$		0			0
$955$	150554.			0.165077
$956$		0			0
$957$			850592.i			0.928747i
$958$		0			0
$959$			111034.i			0.120731i
$960$		0			0
$961$	−708290.	−	592636.i	−0.766945	−	0.641713i
$962$		0			0
$963$	800437.			0.863126
$964$		0			0
$965$	665190.			0.714317
$966$		0			0
$967$			554085.i			0.592548i	0.955103	+	0.296274i	$0.0957442\pi$
$967$			554085.i			0.592548i	−0.955103	+	0.296274i	$0.904256\pi$
$968$		0			0
$969$	58900.9			0.0627298
$970$		0			0
$971$	902421.			0.957129			0.478565	−	0.878052i	$-0.341157\pi$
$971$	902421.			0.957129			0.478565	+	0.878052i	$0.341157\pi$
$972$		0			0
$973$			1.65048e6i			1.74335i
$974$		0			0
$975$	−71633.5			−0.0753541
$976$		0			0
$977$	−1.65718e6			−1.73612			−0.868060	−	0.496459i	$-0.834633\pi$
$977$	−1.65718e6			−1.73612			−0.868060	+	0.496459i	$0.834633\pi$
$978$		0			0
$979$	−505431.			−0.527347
$980$		0			0
$981$	8258.98			0.00858200
$982$		0			0
$983$			919314.i			0.951386i	0.879611	+	0.475693i	$0.157803\pi$
$983$			919314.i			0.951386i	−0.879611	+	0.475693i	$0.842197\pi$
$984$		0			0
$985$		−	484835.i		−	0.499714i
$986$		0			0
$987$			149575.i			0.153541i
$988$		0			0
$989$	2.07420e6			2.12060
$990$		0			0
$991$			857735.i			0.873385i	0.899611	+	0.436692i	$0.143850\pi$
$991$			857735.i			0.873385i	−0.899611	+	0.436692i	$0.856150\pi$
$992$		0			0
$993$	355693.			0.360725
$994$		0			0
$995$			923479.i			0.932784i
$996$		0			0
$997$	−1.04357e6			−1.04986			−0.524928	−	0.851147i	$-0.675908\pi$
$997$	−1.04357e6			−1.04986			−0.524928	+	0.851147i	$0.675908\pi$
$998$		0			0
$999$	−219663.			−0.220103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.5.c.a.61.5	✓	10
3.2	odd	2		1116.5.h.b.433.4		10
4.3	odd	2		496.5.e.c.433.6		10
31.30	odd	2	inner	124.5.c.a.61.6	yes	10
93.92	even	2		1116.5.h.b.433.3		10
124.123	even	2		496.5.e.c.433.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.5.c.a.61.5	✓	10	1.1	even	1	trivial
124.5.c.a.61.6	yes	10	31.30	odd	2	inner
496.5.e.c.433.5		10	124.123	even	2
496.5.e.c.433.6		10	4.3	odd	2
1116.5.h.b.433.3		10	93.92	even	2
1116.5.h.b.433.4		10	3.2	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:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	124.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.01087i q^{3} +16.1544 q^{5} -66.9407 q^{7} +71.9347 q^{9} +O(q^{10})\)	\(q-3.01087i q^{3} +16.1544 q^{5} -66.9407 q^{7} +71.9347 q^{9} -221.831i q^{11} +65.3555i q^{13} -48.6389i q^{15} -59.5422i q^{17} -328.553 q^{19} +201.550i q^{21} -623.356i q^{23} -364.034 q^{25} -460.466i q^{27} -1273.53i q^{29} +(328.048 - 903.275i) q^{31} -667.902 q^{33} -1081.39 q^{35} -477.045i q^{37} +196.777 q^{39} +329.250 q^{41} +3327.48i q^{43} +1162.06 q^{45} +742.126 q^{47} +2080.06 q^{49} -179.274 q^{51} -1744.76i q^{53} -3583.55i q^{55} +989.229i q^{57} -4726.44 q^{59} +4981.21i q^{61} -4815.36 q^{63} +1055.78i q^{65} +6596.77 q^{67} -1876.84 q^{69} +8230.46 q^{71} +7160.56i q^{73} +1096.06i q^{75} +14849.5i q^{77} -6391.96i q^{79} +4440.31 q^{81} +9782.20i q^{83} -961.871i q^{85} -3834.42 q^{87} -2278.45i q^{89} -4374.95i q^{91} +(-2719.64 - 987.710i) q^{93} -5307.58 q^{95} -3409.25 q^{97} -15957.3i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9}+O(q^{10}) \) 10 * q - 6 * q^5 - 2 * q^7 - 146 * q^9	\( 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9} - 310 q^{19} - 756 q^{25} + 734 q^{31} - 372 q^{33} - 666 q^{35} - 1924 q^{39} - 210 q^{41} + 650 q^{45} - 1992 q^{47} + 188 q^{49} + 1352 q^{51} + 5610 q^{59} + 3478 q^{63} - 5420 q^{67} + 10160 q^{69} - 1734 q^{71} + 11598 q^{81} - 13244 q^{87} + 3088 q^{93} + 4302 q^{95} + 2942 q^{97}+O(q^{100}) \) 10 * q - 6 * q^5 - 2 * q^7 - 146 * q^9 - 310 * q^19 - 756 * q^25 + 734 * q^31 - 372 * q^33 - 666 * q^35 - 1924 * q^39 - 210 * q^41 + 650 * q^45 - 1992 * q^47 + 188 * q^49 + 1352 * q^51 + 5610 * q^59 + 3478 * q^63 - 5420 * q^67 + 10160 * q^69 - 1734 * q^71 + 11598 * q^81 - 13244 * q^87 + 3088 * q^93 + 4302 * q^95 + 2942 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more