LMFDB - Embedded newform 100.6.l.a.47.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,6,Mod(3,100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 7]))

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

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

chi := DirichletCharacter("100.3");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 100 = 2^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	100.l (of order $20$, degree $8$, 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:	$16.0383819813$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

Embedding invariants

Embedding label			47.1
Root			$0.587785 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	100.47
Dual form			100.6.l.a.83.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.04029 - 2.56816i) q^{2} +(18.8091 - 25.8885i) q^{4} +(-54.8418 - 10.8339i) q^{5} +(28.3177 - 178.791i) q^{8} +(231.107 - 75.0911i) q^{9} +O(q^{10})$	\(q+(5.04029 - 2.56816i) q^{2} +(18.8091 - 25.8885i) q^{4} +(-54.8418 - 10.8339i) q^{5} +(28.3177 - 178.791i) q^{8} +(231.107 - 75.0911i) q^{9} +(-304.242 + 86.2367i) q^{10} +(-1084.77 - 552.720i) q^{13} +(-316.433 - 973.882i) q^{16} +(-1716.39 - 271.849i) q^{17} +(972.000 - 972.000i) q^{18} +(-1312.00 + 1216.00i) q^{20} +(2890.26 + 1188.30i) q^{25} -6887.05 q^{26} +(3204.63 - 4410.79i) q^{29} +(-4096.00 - 4096.00i) q^{32} +(-9349.24 + 3037.75i) q^{34} +(2402.92 - 7395.42i) q^{36} +(2624.63 - 5151.12i) q^{37} +(-3489.99 + 9498.42i) q^{40} +(2567.27 + 7901.23i) q^{41} +(-13487.8 + 1614.36i) q^{45} +16807.0i q^{49} +(17619.5 - 1433.26i) q^{50} +(-34712.8 + 17687.0i) q^{52} +(40386.1 - 6396.54i) q^{53} +(4824.66 - 30461.7i) q^{58} +(17179.0 - 52871.5i) q^{61} +(-31164.2 - 10125.9i) q^{64} +(53502.9 + 42064.5i) q^{65} +(-39321.5 + 39321.5i) q^{68} +(-6881.19 - 43446.1i) q^{72} +(2725.61 + 5349.31i) q^{73} -32703.6i q^{74} +(6802.89 + 56837.7i) q^{80} +(47771.6 - 34708.1i) q^{81} +(33231.4 + 33231.4i) q^{82} +(91184.5 + 33503.8i) q^{85} +(-4893.82 - 1590.10i) q^{89} +(-63836.8 + 42775.8i) q^{90} +(24263.7 + 153195. i) q^{97} +(43163.0 + 84712.2i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 76 q^{5} + 256 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 - 76 * q^5 + 256 * q^8	$ 8 q - 8 q^{2} - 76 q^{5} + 256 q^{8} + 24 q^{10} - 1438 q^{13} + 2048 q^{16} + 1434 q^{17} + 7776 q^{18} - 10496 q^{20} + 474 q^{25} - 11504 q^{26} - 32768 q^{32} - 61000 q^{34} - 15552 q^{36} - 32926 q^{37} + 20224 q^{40} - 9904 q^{41} - 19926 q^{45} + 26824 q^{50} - 46016 q^{52} + 153682 q^{53} + 68512 q^{58} + 109896 q^{61} + 163278 q^{65} - 45888 q^{68} - 62208 q^{72} + 68662 q^{73} + 77824 q^{80} + 118098 q^{81} - 39616 q^{82} + 608828 q^{85} - 255250 q^{89} - 153576 q^{90} - 68666 q^{97} + 134456 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 - 76 * q^5 + 256 * q^8 + 24 * q^10 - 1438 * q^13 + 2048 * q^16 + 1434 * q^17 + 7776 * q^18 - 10496 * q^20 + 474 * q^25 - 11504 * q^26 - 32768 * q^32 - 61000 * q^34 - 15552 * q^36 - 32926 * q^37 + 20224 * q^40 - 9904 * q^41 - 19926 * q^45 + 26824 * q^50 - 46016 * q^52 + 153682 * q^53 + 68512 * q^58 + 109896 * q^61 + 163278 * q^65 - 45888 * q^68 - 62208 * q^72 + 68662 * q^73 + 77824 * q^80 + 118098 * q^81 - 39616 * q^82 + 608828 * q^85 - 255250 * q^89 - 153576 * q^90 - 68666 * q^97 + 134456 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$.

$n$	$51$	$77$
$\chi(n)$	$-1$	$e\left(\frac{17}{20}\right)$

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 $

$2$	5.04029	−	2.56816i	0.891007	−	0.453990i
$2$	5.04029	−	2.56816i	0.891007	−	0.453990i
$3$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$3$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$4$	18.8091	−	25.8885i	0.587785	−	0.809017i
$5$	−54.8418	−	10.8339i	−0.981041	−	0.193802i
$5$	−54.8418	−	10.8339i	−0.981041	−	0.193802i
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	28.3177	−	178.791i	0.156434	−	0.987688i
$9$	231.107	−	75.0911i	0.951057	−	0.309017i
$10$	−304.242	+	86.2367i	−0.962098	+	0.272704i
$11$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$11$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$12$		0			0
$13$	−1084.77	−	552.720i	−1.78025	−	0.907082i	−0.910320	−	0.413904i	$-0.864165\pi$
$13$	−1084.77	−	552.720i	−1.78025	−	0.907082i	−0.869929	−	0.493178i	$-0.835835\pi$
$14$		0			0
$15$		0			0
$16$	−316.433	−	973.882i	−0.309017	−	0.951057i
$17$	−1716.39	−	271.849i	−1.44043	−	0.228142i	−0.613166	−	0.789954i	$-0.710104\pi$
$17$	−1716.39	−	271.849i	−1.44043	−	0.228142i	−0.827265	+	0.561812i	$0.810104\pi$
$18$	972.000	−	972.000i	0.707107	−	0.707107i
$19$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$19$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$20$	−1312.00	+	1216.00i	−0.733430	+	0.679765i
$21$		0			0
$22$		0			0
$23$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$23$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$24$		0			0
$25$	2890.26	+	1188.30i	0.924882	+	0.380255i
$26$	−6887.05			−1.99802
$27$		0			0
$28$		0			0
$29$	3204.63	−	4410.79i	0.707591	−	0.973916i	−0.292254	−	0.956341i	$-0.594405\pi$
$29$	3204.63	−	4410.79i	0.707591	−	0.973916i	0.999846	−	0.0175754i	$-0.00559471\pi$
$30$		0			0
$31$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$31$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$32$	−4096.00	−	4096.00i	−0.707107	−	0.707107i
$33$		0			0
$34$	−9349.24	+	3037.75i	−1.38701	+	0.450666i
$35$		0			0
$36$	2402.92	−	7395.42i	0.309017	−	0.951057i
$37$	2624.63	−	5151.12i	0.315184	−	0.618583i	−0.678011	−	0.735052i	$-0.737158\pi$
$37$	2624.63	−	5151.12i	0.315184	−	0.618583i	0.993194	+	0.116469i	$0.0371577\pi$
$38$		0			0
$39$		0			0
$40$	−3489.99	+	9498.42i	−0.344884	+	0.938645i
$41$	2567.27	+	7901.23i	0.238513	+	0.734066i	0.996636	+	0.0819552i	$0.0261164\pi$
$41$	2567.27	+	7901.23i	0.238513	+	0.734066i	−0.758123	+	0.652111i	$0.773884\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	−13487.8	+	1614.36i	−0.992913	+	0.118842i
$46$		0			0
$47$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$47$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$48$		0			0
$49$			16807.0i			1.00000i
$50$	17619.5	−	1433.26i	0.996708	−	0.0810777i
$51$		0			0
$52$	−34712.8	+	17687.0i	−1.78025	+	0.907082i
$53$	40386.1	−	6396.54i	1.97489	−	0.312792i	0.983969	−	0.178339i	$-0.0570723\pi$
$53$	40386.1	−	6396.54i	1.97489	−	0.312792i	0.990920	−	0.134453i	$-0.0429277\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	4824.66	−	30461.7i	0.188320	−	1.18901i
$59$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$59$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$60$		0			0
$61$	17179.0	−	52871.5i	0.591117	−	1.81927i	0.0179385	−	0.999839i	$-0.494290\pi$
$61$	17179.0	−	52871.5i	0.591117	−	1.81927i	0.573178	−	0.819431i	$-0.305710\pi$
$62$		0			0
$63$		0			0
$64$	−31164.2	−	10125.9i	−0.951057	−	0.309017i
$65$	53502.9	+	42064.5i	1.57070	+	1.23490i
$66$		0			0
$67$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$67$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$68$	−39321.5	+	39321.5i	−1.03123	+	1.03123i
$69$		0			0
$70$		0			0
$71$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$71$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$72$	−6881.19	−	43446.1i	−0.156434	−	0.987688i
$73$	2725.61	+	5349.31i	0.0598628	+	0.117487i	0.918999	−	0.394259i	$-0.128999\pi$
$73$	2725.61	+	5349.31i	0.0598628	+	0.117487i	−0.859137	+	0.511746i	$0.828999\pi$
$74$		−	32703.6i		−	0.694251i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$79$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$80$	6802.89	+	56837.7i	0.118842	+	0.992913i
$81$	47771.6	−	34708.1i	0.809017	−	0.587785i
$82$	33231.4	+	33231.4i	0.545775	+	0.545775i
$83$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$83$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$84$		0			0
$85$	91184.5	+	33503.8i	1.36891	+	0.502975i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4893.82	−	1590.10i	−0.0654898	−	0.0212789i	0.276089	−	0.961132i	$-0.410962\pi$
$89$	−4893.82	−	1590.10i	−0.0654898	−	0.0212789i	−0.341579	+	0.939853i	$0.610962\pi$
$90$	−63836.8	+	42775.8i	−0.830739	+	0.556662i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	24263.7	+	153195.i	0.261835	+	1.65316i	0.671560	+	0.740950i	$0.265625\pi$
$97$	24263.7	+	153195.i	0.261835	+	1.65316i	−0.409725	+	0.912209i	$0.634375\pi$
$98$	43163.0	+	84712.2i	0.453990	+	0.891007i
$99$		0			0
$100$	85126.5	−	52473.7i	0.851265	−	0.524737i
$101$	201570.			1.96617			0.983086	−	0.183144i	$-0.0586276\pi$
$101$	201570.			1.96617			0.983086	+	0.183144i	$0.0586276\pi$
$102$		0			0
$103$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$103$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$104$	−129539.	+	178296.i	−1.17441	+	1.61643i
$105$		0			0
$106$	187131.	−	135958.i	1.61763	−	1.17528i
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$	−173943.	+	56517.5i	−1.40230	+	0.455635i	−0.909934	−	0.414753i	$-0.863868\pi$
$109$	−173943.	+	56517.5i	−1.40230	+	0.455635i	−0.492366	+	0.870388i	$0.663868\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−192998.	−	98337.5i	−1.42186	−	0.724474i	−0.437267	−	0.899332i	$-0.644053\pi$
$113$	−192998.	−	98337.5i	−1.42186	−	0.724474i	−0.984594	+	0.174858i	$0.944053\pi$
$114$		0			0
$115$		0			0
$116$	−53912.7	−	165926.i	−0.372003	−	1.14491i
$117$	−292203.	−	46280.4i	−1.97342	−	0.312559i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−130293.	−	94663.4i	−0.809017	−	0.587785i
$122$	−49195.2	−	310606.i	−0.299242	−	1.88934i
$123$		0			0
$124$		0			0
$125$	−145633.	−	96481.0i	−0.833652	−	0.552290i
$126$		0			0
$127$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$127$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$128$	−183082.	+	28997.3i	−0.987688	+	0.156434i
$129$		0			0
$130$	377698.	+	74613.3i	1.96014	+	0.387220i
$131$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$131$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−97208.0	+	299176.i	−0.450666	+	1.38701i
$137$	199445.	−	391432.i	0.907865	−	1.78179i	0.439696	−	0.898147i	$-0.355086\pi$
$137$	199445.	−	391432.i	0.907865	−	1.78179i	0.468169	−	0.883639i	$-0.344914\pi$
$138$		0			0
$139$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$139$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−146260.	−	201309.i	−0.587785	−	0.809017i
$145$	−223534.	+	207177.i	−0.882923	+	0.818319i
$146$	27475.8	+	19962.3i	0.106676	+	0.0775048i
$147$		0			0
$148$	−83988.1	−	164836.i	−0.315184	−	0.618583i
$149$			278825.i			1.02888i	0.857526	+	0.514441i	$0.172001\pi$
$149$			278825.i			1.02888i	−0.857526	+	0.514441i	$0.827999\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−417082.	+	66059.2i	−1.44043	+	0.228142i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−392402.	−	392402.i	−1.27052	−	1.27052i	−0.945815	−	0.324705i	$-0.894735\pi$
$157$	−392402.	−	392402.i	−1.27052	−	1.27052i	−0.324705	−	0.945815i	$-0.605265\pi$
$158$		0			0
$159$		0			0
$160$	180257.	+	269008.i	0.556662	+	0.830739i
$161$		0			0
$162$	151647.	−	297624.i	0.453990	−	0.891007i
$163$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$163$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$164$	252839.	+	82152.5i	0.734066	+	0.238513i
$165$		0			0
$166$		0			0
$167$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$167$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$168$		0			0
$169$	652994.	+	898769.i	1.75870	+	2.42065i
$170$	545640.	−	65307.6i	1.44805	−	0.173317i
$171$		0			0
$172$		0			0
$173$	−337407.	−	662199.i	−0.857115	−	1.68218i	−0.722596	−	0.691271i	$-0.757051\pi$
$173$	−337407.	−	662199.i	−0.857115	−	1.68218i	−0.134519	−	0.990911i	$-0.542949\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−28749.9	+	4553.54i	−0.0680122	+	0.0107721i
$179$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$179$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$180$	−211901.	+	379545.i	−0.487475	+	0.873137i
$181$	477785.	−	347131.i	1.08402	−	0.787585i	0.105638	−	0.994405i	$-0.466311\pi$
$181$	477785.	−	347131.i	1.08402	−	0.787585i	0.978379	+	0.206820i	$0.0663115\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−199746.	+	254062.i	−0.429090	+	0.545771i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$191$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$192$		0			0
$193$	−246910.	+	246910.i	−0.477139	+	0.477139i	−0.904215	−	0.427077i	$-0.859543\pi$
$193$	−246910.	+	246910.i	−0.477139	+	0.477139i	0.427077	+	0.904215i	$0.359543\pi$
$194$	515725.	+	709834.i	0.983815	+	1.35411i
$195$		0			0
$196$	435109.	+	316125.i	0.809017	+	0.587785i
$197$	148434.	+	937174.i	0.272500	+	1.72050i	0.621532	+	0.783389i	$0.286511\pi$
$197$	148434.	+	937174.i	0.272500	+	1.72050i	−0.349031	+	0.937111i	$0.613489\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	294302.	−	483101.i	0.520257	−	0.854010i
$201$		0			0
$202$	1.01597e6	−	517663.i	1.75187	−	0.892623i
$203$		0			0
$204$		0			0
$205$	−55192.8	−	461132.i	−0.0917271	−	0.766373i
$206$		0			0
$207$		0			0
$208$	−195025.	+	1.23134e6i	−0.312559	+	1.97342i
$209$		0			0
$210$		0			0
$211$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$211$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$212$	594031.	−	1.16585e6i	0.907757	−	1.78157i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−731578.	+	731578.i	−1.04260	+	1.04260i
$219$		0			0
$220$		0			0
$221$	1.71163e6	+	1.24357e6i	2.35738	+	1.71274i
$222$		0			0
$223$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$223$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$224$		0			0
$225$	757188.	+	57591.0i	0.997120	+	0.0758400i
$226$	−1.22531e6			−1.59579
$227$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$227$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$228$		0			0
$229$	−32122.1	+	44212.3i	−0.0404776	+	0.0557127i	−0.828776	−	0.559580i	$-0.810962\pi$
$229$	−32122.1	+	44212.3i	−0.0404776	+	0.0557127i	0.788298	+	0.615293i	$0.210962\pi$
$230$		0			0
$231$		0			0
$232$	−697861.	−	697861.i	−0.851234	−	0.851234i
$233$	222325.	−	1.40371e6i	0.268286	−	1.69389i	−0.373999	−	0.927429i	$-0.622014\pi$
$233$	222325.	−	1.40371e6i	0.268286	−	1.69389i	0.642286	−	0.766465i	$-0.277986\pi$
$234$	−1.59164e6	+	517156.i	−1.90023	+	0.617422i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$239$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$240$		0			0
$241$	−306295.	−	942680.i	−0.339702	−	1.04549i	−0.964360	−	0.264595i	$-0.914762\pi$
$241$	−306295.	−	942680.i	−0.339702	−	1.04549i	0.624658	−	0.780898i	$-0.285238\pi$
$242$	−899826.	−	142518.i	−0.987688	−	0.156434i
$243$		0			0
$244$	−1.04564e6	−	1.43921e6i	−1.12437	−	1.54756i
$245$	182085.	−	921727.i	0.193802	−	0.981041i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−981812.	−	112284.i	−0.993524	−	0.113623i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−848316.	+	616338.i	−0.809017	+	0.587785i
$257$	677064.	+	677064.i	0.639436	+	0.639436i	0.950416	−	0.310980i	$-0.100657\pi$
$257$	677064.	+	677064.i	0.639436	+	0.639436i	−0.310980	+	0.950416i	$0.600657\pi$
$258$		0			0
$259$		0			0
$260$	2.09533e6	−	593916.i	1.92229	−	0.544869i
$261$	409400.	−	1.26000e6i	0.372003	−	1.14491i
$262$		0			0
$263$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$263$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$264$		0			0
$265$	−2.28415e6	−	86739.9i	−1.99807	−	0.0758759i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.00626e6	+	1.38500e6i	0.847870	+	1.16699i	0.984328	+	0.176347i	$0.0564280\pi$
$269$	1.00626e6	+	1.38500e6i	0.847870	+	1.16699i	−0.136458	+	0.990646i	$0.543572\pi$
$270$		0			0
$271$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$271$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$272$	278373.	+	1.75758e6i	0.228142	+	1.44043i
$273$		0			0
$274$		−	2.48514e6i		−	1.99974i
$275$		0			0
$276$		0			0
$277$	−2.23628e6	+	1.13944e6i	−1.75117	+	0.892264i	−0.791478	+	0.611197i	$0.790688\pi$
$277$	−2.23628e6	+	1.13944e6i	−1.75117	+	0.892264i	−0.959688	+	0.281067i	$0.909312\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.35447e6	−	984077.i	1.02330	−	0.743470i	0.0563421	−	0.998412i	$-0.482056\pi$
$281$	1.35447e6	−	984077.i	1.02330	−	0.743470i	0.966956	+	0.254942i	$0.0820562\pi$
$282$		0			0
$283$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$283$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.25419e6	−	639040.i	−0.891007	−	0.453990i
$289$	1.52171e6	+	494434.i	1.07174	+	0.348228i
$290$	−594610.	+	1.61830e6i	−0.415181	+	1.12997i
$291$		0			0
$292$	189752.	+	30053.8i	0.130236	+	0.0206273i
$293$	1.02179e6	−	1.02179e6i	0.695335	−	0.695335i	−0.268065	−	0.963401i	$-0.586384\pi$
$293$	1.02179e6	−	1.02179e6i	0.695335	−	0.695335i	0.963401	+	0.268065i	$0.0863844\pi$
$294$		0			0
$295$		0			0
$296$	−846650.	−	615127.i	−0.561661	−	0.408071i
$297$		0			0
$298$	716066.	+	1.40536e6i	0.467103	+	0.916741i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.51493e6	+	2.71346e6i	−0.932487	+	1.67022i
$306$	−1.93256e6	+	1.40409e6i	−1.17986	+	0.857218i
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$311$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$312$		0			0
$313$	613302.	+	312493.i	0.353845	+	0.180293i	0.621873	−	0.783118i	$-0.286372\pi$
$313$	613302.	+	312493.i	0.353845	+	0.180293i	−0.268028	+	0.963411i	$0.586372\pi$
$314$	−2.98557e6	−	970070.i	−1.70885	−	0.555238i
$315$		0			0
$316$		0			0
$317$	1.98725e6	+	314750.i	1.11072	+	0.175921i	0.684724	−	0.728803i	$-0.259923\pi$
$317$	1.98725e6	+	314750.i	1.11072	+	0.175921i	0.425998	+	0.904724i	$0.359923\pi$
$318$		0			0
$319$		0			0
$320$	1.59940e6	+	892950.i	0.873137	+	0.487475i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	1.88957e6i		−	1.00000i
$325$	−2.47848e6	−	2.88653e6i	−1.30160	−	1.51589i
$326$		0			0
$327$		0			0
$328$	1.48537e6	−	235259.i	0.762340	−	0.120743i
$329$		0			0
$330$		0			0
$331$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$331$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$332$		0			0
$333$	219766.	−	1.38755e6i	0.108605	−	0.685704i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.85308e6	+	3.63687e6i	−0.888829	+	1.74443i	−0.262378	+	0.964965i	$0.584507\pi$
$337$	−1.85308e6	+	3.63687e6i	−0.888829	+	1.74443i	−0.626451	+	0.779461i	$0.715493\pi$
$338$	5.59946e6	+	2.85307e6i	2.66597	+	1.35838i
$339$		0			0
$340$	2.58247e6	−	1.73046e6i	1.21154	−	0.811828i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−3.40126e6	−	2.47116e6i	−1.52739	−	1.10971i
$347$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$347$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$348$		0			0
$349$		−	4.52898e6i		−	1.99038i	−0.0979399	−	0.995192i	$-0.531225\pi$
$349$		−	4.52898e6i		−	1.99038i	0.0979399	−	0.995192i	$-0.468775\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.56787e6	−	723479.i	1.95109	−	0.309022i	0.951087	−	0.308924i	$-0.0999688\pi$
$353$	4.56787e6	−	723479.i	1.95109	−	0.309022i	1.00000		9.80566e-5i	$-3.12124e-5\pi$
$354$		0			0
$355$		0			0
$356$	−133214.	+	96785.6i	−0.0557089	+	0.0404749i
$357$		0			0
$358$		0			0
$359$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$359$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$360$	−93312.0	+	2.45722e6i	−0.0379473	+	0.999280i
$361$	−765157.	+	2.35491e6i	−0.309017	+	0.951057i
$362$	1.51669e6	−	2.97667e6i	0.608310	−	1.19388i
$363$		0			0
$364$		0			0
$365$	−91523.8	−	322895.i	−0.0359585	−	0.126861i
$366$		0			0
$367$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$367$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$368$		0			0
$369$	1.18663e6	+	1.63325e6i	0.453678	+	0.624434i
$370$	−354307.	+	1.79353e6i	−0.134547	+	0.681089i
$371$		0			0
$372$		0			0
$373$	−2.13906e6	−	4.19815e6i	−0.796071	−	1.56238i	−0.826565	−	0.562841i	$-0.809708\pi$
$373$	−2.13906e6	−	4.19815e6i	−0.796071	−	1.56238i	0.0304944	−	0.999535i	$-0.490292\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.91423e6	+	3.01345e6i	−2.14311	+	1.09197i
$378$		0			0
$379$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$379$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$383$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$384$		0			0
$385$		0			0
$386$	−610394.	+	1.87860e6i	−0.208517	+	0.641750i
$387$		0			0
$388$	4.42237e6	+	2.25331e6i	1.49134	+	0.759874i
$389$	2.51773e6	+	818059.i	0.843597	+	0.274101i	0.698762	−	0.715355i	$-0.253735\pi$
$389$	2.51773e6	+	818059.i	0.843597	+	0.274101i	0.144835	+	0.989456i	$0.453735\pi$
$390$		0			0
$391$		0			0
$392$	3.00494e6	+	475935.i	0.987688	+	0.156434i
$393$		0			0
$394$	3.15496e6	+	4.34243e6i	1.02389	+	1.40926i
$395$		0			0
$396$		0			0
$397$	665391.	+	4.20112e6i	0.211885	+	1.33779i	0.832652	+	0.553796i	$0.186821\pi$
$397$	665391.	+	4.20112e6i	0.211885	+	1.33779i	−0.620767	+	0.783995i	$0.713179\pi$
$398$		0			0
$399$		0			0
$400$	242688.	−	3.19078e6i	0.0758400	−	0.997120i
$401$	2.37994e6			0.739103			0.369552	−	0.929210i	$-0.379511\pi$
$401$	2.37994e6			0.739103			0.369552	+	0.929210i	$0.379511\pi$
$402$		0			0
$403$		0			0
$404$	3.79135e6	−	5.21834e6i	1.15569	−	1.59067i
$405$	−2.99591e6	+	1.38591e6i	−0.907592	+	0.419852i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−3.15367e6	+	1.02469e6i	−0.932197	+	0.302889i	−0.735461	−	0.677567i	$-0.763034\pi$
$409$	−3.15367e6	+	1.02469e6i	−0.932197	+	0.302889i	−0.196736	+	0.980456i	$0.563034\pi$
$410$	−1.46245e6	−	2.18250e6i	−0.429656	−	0.641200i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	2.17929e6	+	6.70717e6i	0.617422	+	1.90023i
$417$		0			0
$418$		0			0
$419$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$419$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$420$		0			0
$421$	−5.62338e6	−	4.08563e6i	−1.54630	−	1.12345i	−0.946228	−	0.323499i	$-0.895141\pi$
$421$	−5.62338e6	−	4.08563e6i	−1.54630	−	1.12345i	−0.600067	−	0.799950i	$-0.704859\pi$
$422$		0			0
$423$		0			0
$424$		−	7.40180e6i		−	1.99951i
$425$	−4.63775e6	−	2.82529e6i	−1.24548	−	0.758735i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$		0			0
$433$	−108251.	+	683471.i	−0.0277468	+	0.175186i	−0.997670	−	0.0682249i	$-0.978266\pi$
$433$	−108251.	+	683471.i	−0.0277468	+	0.175186i	0.969923	+	0.243411i	$0.0782665\pi$
$434$		0			0
$435$		0			0
$436$	−1.80856e6	+	5.56618e6i	−0.455635	+	1.40230i
$437$		0			0
$438$		0			0
$439$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$439$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$440$		0			0
$441$	1.26206e6	+	3.88421e6i	0.309017	+	0.951057i
$442$	1.18208e7	+	1.87224e6i	2.87801	+	0.455832i
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	251159.	+	140223.i	0.0601242	+	0.0335675i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			1.71061e6i			0.400438i	0.979751	+	0.200219i	$0.0641654\pi$
$449$			1.71061e6i			0.400438i	−0.979751	+	0.200219i	$0.935835\pi$
$450$	3.96435e6	−	1.65430e6i	0.922871	−	0.385109i
$451$		0			0
$452$	−6.17594e6	+	3.14680e6i	−1.42186	+	0.724474i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	979183.	+	979183.i	0.219318	+	0.219318i	0.808211	−	0.588893i	$-0.200436\pi$
$457$	979183.	+	979183.i	0.219318	+	0.219318i	−0.588893	+	0.808211i	$0.700436\pi$
$458$	−48360.7	+	305337.i	−0.0107728	+	0.0680168i
$459$		0			0
$460$		0			0
$461$	2.42254e6	−	7.45580e6i	0.530907	−	1.63396i	−0.221424	−	0.975178i	$-0.571071\pi$
$461$	2.42254e6	−	7.45580e6i	0.530907	−	1.63396i	0.752331	−	0.658785i	$-0.228929\pi$
$462$		0			0
$463$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$463$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$464$	−5.30964e6	−	1.72521e6i	−1.14491	−	0.372003i
$465$		0			0
$466$	−2.48435e6	−	7.64606e6i	−0.529967	−	1.63107i
$467$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$467$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$468$	−6.69421e6	+	6.69421e6i	−1.41281	+	1.41281i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	8.85319e6	−	4.51092e6i	1.78157	−	0.907757i
$478$		0			0
$479$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$479$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$480$		0			0
$481$	−5.69426e6	+	4.13712e6i	−1.12221	+	0.815334i
$482$	−3.96477e6	−	3.96477e6i	−0.777321	−	0.777321i
$483$		0			0
$484$	−4.90140e6	+	1.59256e6i	−0.951057	+	0.309017i
$485$	329026.	−	8.66436e6i	0.0635150	−	1.67256i
$486$		0			0
$487$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$487$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$488$	−8.96646e6	−	4.56864e6i	−1.70440	−	0.868436i
$489$		0			0
$490$	−1.44938e6	−	5.11340e6i	−0.272704	−	0.962098i
$491$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$491$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$492$		0			0
$493$	−6.69944e6	+	6.69944e6i	−1.24143	+	1.24143i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−5.23698e6	+	1.95550e6i	−0.936820	+	0.349811i
$501$		0			0
$502$		0			0
$503$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$503$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$504$		0			0
$505$	−1.10544e7	−	2.18378e6i	−1.92889	−	0.381048i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.00057e6	−	2.59954e6i	1.36876	−	0.444736i	0.469800	−	0.882773i	$-0.344326\pi$
$509$	8.00057e6	−	2.59954e6i	1.36876	−	0.444736i	0.898957	+	0.438037i	$0.144326\pi$
$510$		0			0
$511$		0			0
$512$	−2.69291e6	+	5.28513e6i	−0.453990	+	0.891007i
$513$		0			0
$514$	5.15141e6	+	1.67379e6i	0.860039	+	0.279444i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	9.03581e6	−	8.37465e6i	1.46541	−	1.35818i
$521$	9.53137e6	+	6.92494e6i	1.53837	+	1.11769i	0.951347	+	0.308121i	$0.0997002\pi$
$521$	9.53137e6	+	6.92494e6i	1.53837	+	1.11769i	0.587023	+	0.809570i	$0.300300\pi$
$522$	−1.17239e6	−	7.40219e6i	−0.188320	−	1.18901i
$523$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$523$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.78319e6	+	5.20711e6i	−0.587785	+	0.809017i
$530$	−1.17355e7	+	5.42886e6i	−1.81474	+	0.839497i
$531$		0			0
$532$		0			0
$533$	1.58227e6	−	9.99003e6i	0.241247	−	1.52317i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	8.62873e6	+	4.39656e6i	1.28526	+	0.654873i
$539$		0			0
$540$		0			0
$541$	2.39413e6	+	7.36838e6i	0.351686	+	1.08238i	0.957906	+	0.287081i	$0.0926849\pi$
$541$	2.39413e6	+	7.36838e6i	0.351686	+	1.08238i	−0.606221	+	0.795297i	$0.707315\pi$
$542$		0			0
$543$		0			0
$544$	5.91682e6	+	8.14381e6i	0.857218	+	1.17986i
$545$	1.01517e7	−	1.21505e6i	1.46402	−	0.175228i
$546$		0			0
$547$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$547$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$548$	−6.38223e6	−	1.25258e7i	−0.907865	−	1.78179i
$549$		−	1.35090e7i		−	1.91289i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−8.34526e6	+	1.14863e7i	−1.15522	+	1.59003i
$555$		0			0
$556$		0			0
$557$	−9.06359e6	−	9.06359e6i	−1.23783	−	1.23783i	−0.960884	−	0.276950i	$-0.910676\pi$
$557$	−9.06359e6	−	9.06359e6i	−1.23783	−	1.23783i	−0.276950	−	0.960884i	$-0.589324\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	4.29964e6	−	8.43852e6i	0.574237	−	1.12700i
$563$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$563$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$564$		0			0
$565$	9.51900e6	+	7.48392e6i	1.25450	+	0.986298i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9.06537e6	+	1.24774e7i	1.17383	+	1.61564i	0.630938	+	0.775834i	$0.282670\pi$
$569$	9.06537e6	+	1.24774e7i	1.17383	+	1.61564i	0.542891	+	0.839803i	$0.317330\pi$
$570$		0			0
$571$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$571$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−7.96262e6			−1.00000
$577$	7.77640e6	−	3.96227e6i	0.972386	−	0.495456i	0.105748	−	0.994393i	$-0.466276\pi$
$577$	7.77640e6	−	3.96227e6i	0.972386	−	0.495456i	0.866638	+	0.498937i	$0.166276\pi$
$578$	8.93966e6	−	1.41590e6i	1.11302	−	0.176284i
$579$		0			0
$580$	1.15905e6	+	9.68379e6i	0.143065	+	1.19530i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	1.03359e6	−	335834.i	0.125405	−	0.0407467i
$585$	1.55235e7	+	5.70379e6i	1.87543	+	0.689086i
$586$	2.52601e6	−	7.77427e6i	0.303873	−	0.935224i
$587$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$587$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−5.84711e6	−	926091.i	−0.685704	−	0.108605i
$593$	−1.12979e7	+	1.12979e7i	−1.31935	+	1.31935i	−0.405062	+	0.914289i	$0.632750\pi$
$593$	−1.12979e7	+	1.12979e7i	−1.31935	+	1.31935i	−0.914289	+	0.405062i	$0.867250\pi$
$594$		0			0
$595$		0			0
$596$	7.21837e6	+	5.24445e6i	0.832383	+	0.604762i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−1.49469e6			−0.168797			−0.0843986	−	0.996432i	$-0.526897\pi$
$601$	−1.49469e6			−0.168797			−0.0843986	+	0.996432i	$0.526897\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.11994e6	+	6.60309e6i	0.679765	+	0.733430i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$	−667109.	+	1.75672e7i	−0.0725892	+	1.91152i
$611$		0			0
$612$	−6.13476e6	+	1.20402e7i	−0.662093	+	1.29943i
$613$	1.63364e7	+	8.32380e6i	1.75592	+	0.894685i	0.955367	+	0.295423i	$0.0954605\pi$
$613$	1.63364e7	+	8.32380e6i	1.75592	+	0.894685i	0.800552	+	0.599263i	$0.204540\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.72552e6	+	590065.i	0.393980	+	0.0624003i	0.350282	−	0.936644i	$-0.386086\pi$
$617$	3.72552e6	+	590065.i	0.393980	+	0.0624003i	0.0436985	+	0.999045i	$0.486086\pi$
$618$		0			0
$619$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$619$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	6.94152e6	+	6.86896e6i	0.710812	+	0.703382i
$626$	3.89375e6			0.397130
$627$		0			0
$628$	−1.75394e7	+	2.77797e6i	−1.77467	+	0.281079i
$629$	−5.90520e6	+	8.12781e6i	−0.595125	+	0.819119i
$630$		0			0
$631$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$631$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$632$		0			0
$633$		0			0
$634$	1.08247e7	−	3.51715e6i	1.06953	−	0.347510i
$635$		0			0
$636$		0			0
$637$	9.28956e6	−	1.82318e7i	0.907082	−	1.78025i
$638$		0			0
$639$		0			0
$640$	1.03547e7	+	393216.i	0.999280	+	0.0379473i
$641$	−4.58181e6	−	1.41014e7i	−0.440445	−	1.35555i	−0.887402	−	0.460996i	$-0.847492\pi$
$641$	−4.58181e6	−	1.41014e7i	−0.440445	−	1.35555i	0.446957	−	0.894555i	$-0.352508\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$647$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$648$	−4.85271e6	−	9.52398e6i	−0.453990	−	0.891007i
$649$		0			0
$650$	−1.99053e7	−	8.18386e6i	−1.84793	−	0.759757i
$651$		0			0
$652$		0			0
$653$	39019.5	−	6180.08i	0.00358096	−	0.000567168i	−0.154644	−	0.987970i	$-0.549423\pi$
$653$	39019.5	−	6180.08i	0.00358096	−	0.000567168i	0.158225	+	0.987403i	$0.449423\pi$
$654$		0			0
$655$		0			0
$656$	6.88250e6	−	5.00043e6i	0.624434	−	0.453678i
$657$	1.03159e6	+	1.03159e6i	0.0932384	+	0.0932384i
$658$		0			0
$659$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$659$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$660$		0			0
$661$	6.41305e6	−	1.97373e7i	0.570901	−	1.75705i	−0.0788303	−	0.996888i	$-0.525119\pi$
$661$	6.41305e6	−	1.97373e7i	0.570901	−	1.75705i	0.649731	−	0.760164i	$-0.274881\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−2.45575e6	−	7.55803e6i	−0.214535	−	0.660272i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−3.78230e6	−	7.42318e6i	−0.321898	−	0.631760i	0.672185	−	0.740383i	$-0.265356\pi$
$673$	−3.78230e6	−	7.42318e6i	−0.321898	−	0.631760i	−0.994083	+	0.108623i	$0.965356\pi$
$674$			2.30899e7i			1.95781i
$675$		0			0
$676$	3.55501e7			2.99208
$677$	−1.76207e7	+	8.97820e6i	−1.47758	+	0.752865i	−0.992574	−	0.121644i	$-0.961183\pi$
$677$	−1.76207e7	+	8.97820e6i	−1.47758	+	0.752865i	−0.485008	+	0.874510i	$0.661183\pi$
$678$		0			0
$679$		0			0
$680$	8.57229e6	−	1.53542e7i	0.710926	−	1.27337i
$681$		0			0
$682$		0			0
$683$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$683$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$684$		0			0
$685$	−1.51786e7	+	1.93061e7i	−1.23597	+	1.57206i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.73453e7	−	1.53834e7i	−3.79952	−	1.23454i
$690$		0			0
$691$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$691$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$692$	−2.34897e7	−	3.72040e6i	−1.86471	−	0.295342i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.25848e6	−	1.42595e7i	−0.176090	−	1.11179i
$698$	−1.16311e7	−	2.28274e7i	−0.903616	−	1.77345i
$699$		0			0
$700$		0			0
$701$	−2.46868e7			−1.89745			−0.948724	−	0.316107i	$-0.897624\pi$
$701$	−2.46868e7			−1.89745			−0.948724	+	0.316107i	$0.897624\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	2.11654e7	−	1.53775e7i	1.59814	−	1.16112i
$707$		0			0
$708$		0			0
$709$	−2.25964e7	+	7.34202e6i	−1.68820	+	0.548529i	−0.986474	−	0.163918i	$-0.947587\pi$
$709$	−2.25964e7	+	7.34202e6i	−1.68820	+	0.548529i	−0.701726	+	0.712447i	$0.747587\pi$
$710$		0			0
$711$		0			0
$712$	−422877.	+	829942.i	−0.0312618	+	0.0613547i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$719$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$720$	5.84020e6	+	1.26247e7i	0.419852	+	0.907592i
$721$		0			0
$722$	2.19117e6	+	1.38345e7i	0.156434	+	0.987688i
$723$		0			0
$724$		−	1.88984e7i		−	1.33992i
$725$	1.45035e7	−	8.94026e6i	1.02477	−	0.631692i
$726$		0			0
$727$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$727$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$728$		0			0
$729$	8.43408e6	−	1.16085e7i	0.587785	−	0.809017i
$730$	−1.29055e6	−	1.39244e6i	−0.0896331	−	0.0967094i
$731$		0			0
$732$		0			0
$733$	−4.16355e6	+	2.62876e7i	−0.286223	+	1.80714i	0.255750	+	0.966743i	$0.417678\pi$
$733$	−4.16355e6	+	2.62876e7i	−0.286223	+	1.80714i	−0.541973	+	0.840396i	$0.682322\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	1.01754e7	+	5.18462e6i	0.687717	+	0.350409i
$739$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$739$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$740$	2.82025e6	+	9.94982e6i	0.189325	+	0.667938i
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	3.02075e6	−	1.52913e7i	0.199399	−	1.00938i
$746$	−2.15630e7	−	1.56664e7i	−1.41861	−	1.03068i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$	−2.20704e7	+	3.03773e7i	−1.41378	+	1.94590i
$755$		0			0
$756$		0			0
$757$	−5.57057e6	−	5.57057e6i	−0.353313	−	0.353313i	0.508028	−	0.861341i	$-0.330375\pi$
$757$	−5.57057e6	−	5.57057e6i	−0.353313	−	0.353313i	−0.861341	+	0.508028i	$0.830375\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.47457e6	+	2.91597e7i	−0.593059	+	1.82525i	−0.0288986	+	0.999582i	$0.509200\pi$
$761$	−9.47457e6	+	2.91597e7i	−0.593059	+	1.82525i	−0.564160	+	0.825665i	$0.690800\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	2.35892e7	+	895792.i	1.45734	+	0.0553419i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.51488e7	−	2.08505e7i	−0.923766	−	1.27146i	−0.962242	−	0.272196i	$-0.912250\pi$
$769$	−1.51488e7	−	2.08505e7i	−0.923766	−	1.27146i	0.0384755	−	0.999260i	$-0.487750\pi$
$770$		0			0
$771$		0			0
$772$	1.74798e6	+	1.10363e7i	0.105558	+	0.666468i
$773$	−4.52871e6	−	8.88809e6i	−0.272600	−	0.535007i	0.713603	−	0.700551i	$-0.247062\pi$
$773$	−4.52871e6	−	8.88809e6i	−0.272600	−	0.535007i	−0.986203	+	0.165543i	$0.947062\pi$
$774$		0			0
$775$		0			0
$776$	2.80769e7			1.67377
$777$		0			0
$778$	1.47910e7	−	2.34266e6i	0.876089	−	0.138759i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.63680e7	−	5.31830e6i	0.951057	−	0.309017i
$785$	1.72688e7	+	2.57712e7i	1.00020	+	1.49266i
$786$		0			0
$787$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$787$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$788$	2.70540e7	+	1.37847e7i	1.55209	+	0.790827i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−4.78584e7	+	4.78584e7i	−2.70256	+	2.70256i
$794$	1.41429e7	+	1.94660e7i	0.796136	+	1.09579i
$795$		0			0
$796$		0			0
$797$	−4.66990e6	−	2.94846e7i	−0.260412	−	1.64418i	−0.677653	−	0.735382i	$-0.737003\pi$
$797$	−4.66990e6	−	2.94846e7i	−0.260412	−	1.64418i	0.417241	−	0.908796i	$-0.362997\pi$
$798$		0			0
$799$		0			0
$800$	−6.97122e6	−	1.67058e7i	−0.385109	−	0.922871i
$801$	−1.25040e6			−0.0688600
$802$	1.19956e7	−	6.11206e6i	0.658546	−	0.335546i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	5.70798e6	−	3.60388e7i	0.307577	−	1.94197i
$809$	3.29726e7	−	1.07134e7i	1.77126	−	0.575516i	0.772992	−	0.634415i	$-0.218759\pi$
$809$	3.29726e7	−	1.07134e7i	1.77126	−	0.575516i	0.998264	+	0.0588994i	$0.0187591\pi$
$810$	−1.15410e7	+	1.46793e7i	−0.618062	+	0.786129i
$811$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$811$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.32639e7	+	1.32639e7i	−0.693085	+	0.693085i
$819$		0			0
$820$	−1.29762e7	−	7.24462e6i	−0.673925	−	0.376254i
$821$	1.73641e7	+	1.26157e7i	0.899069	+	0.653212i	0.938227	−	0.346021i	$-0.112467\pi$
$821$	1.73641e7	+	1.26157e7i	0.899069	+	0.653212i	−0.0391575	+	0.999233i	$0.512467\pi$
$822$		0			0
$823$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$823$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$827$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$828$		0			0
$829$	−9.60357e6	+	1.32182e7i	−0.485340	+	0.668014i	−0.979520	−	0.201346i	$-0.935468\pi$
$829$	−9.60357e6	+	1.32182e7i	−0.485340	+	0.668014i	0.494180	+	0.869360i	$0.335468\pi$
$830$		0			0
$831$		0			0
$832$	2.82094e7	+	2.82094e7i	1.41281	+	1.41281i
$833$	4.56896e6	−	2.88473e7i	0.228142	−	1.44043i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$839$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$840$		0			0
$841$	−2.84715e6	−	8.76263e6i	−0.138810	−	0.427213i
$842$	−3.88360e7	−	6.15102e6i	−1.88779	−	0.298997i
$843$		0			0
$844$		0			0
$845$	−2.60743e7	−	5.63646e7i	−1.25623	−	2.71559i
$846$		0			0
$847$		0			0
$848$	−1.90090e7	−	3.73073e7i	−0.907757	−	1.78157i
$849$		0			0
$850$	−3.06314e7	−	2.32980e6i	−1.45419	−	0.110604i
$851$		0			0
$852$		0			0
$853$	4.19683e7	−	6.64713e6i	1.97492	−	0.312796i	0.984087	−	0.177690i	$-0.0568623\pi$
$853$	4.19683e7	−	6.64713e6i	1.97492	−	0.312796i	0.990831	−	0.135107i	$-0.0431377\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.02544e7	+	3.02544e7i	1.40714	+	1.40714i	0.774210	+	0.632929i	$0.218148\pi$
$857$	3.02544e7	+	3.02544e7i	1.40714	+	1.40714i	0.632929	+	0.774210i	$0.281852\pi$
$858$		0			0
$859$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$859$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$863$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$864$		0			0
$865$	1.13299e7	+	3.99716e7i	0.514854	+	1.81640i
$866$	1.20964e6	+	3.72290e6i	0.0548104	+	0.168689i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	5.17915e6	+	3.26999e7i	0.230657	+	1.45631i
$873$	1.71111e7	+	3.35824e7i	0.759874	+	1.49134i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3.08398e7	−	1.57136e7i	1.35398	−	0.689887i	0.381826	−	0.924234i	$-0.375295\pi$
$877$	3.08398e7	−	1.57136e7i	1.35398	−	0.689887i	0.972153	+	0.234347i	$0.0752953\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.08385e7	+	7.87465e6i	−0.470468	+	0.341815i	−0.797624	−	0.603155i	$-0.793910\pi$
$881$	−1.08385e7	+	7.87465e6i	−0.470468	+	0.341815i	0.327155	+	0.944971i	$0.393910\pi$
$882$	1.63364e7	+	1.63364e7i	0.707107	+	0.707107i
$883$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$883$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$884$	6.43887e7	−	2.09211e7i	2.77127	−	0.900440i
$885$		0			0
$886$		0			0
$887$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$887$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$888$		0			0
$889$		0			0
$890$	1.62603e6	+	61748.1i	0.0688104	+	0.00261305i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	4.39312e6	+	8.62198e6i	0.181795	+	0.356793i
$899$		0			0
$900$	1.57330e7	−	1.85193e7i	0.647448	−	0.762109i
$901$	−7.10571e7			−2.91605
$902$		0			0
$903$		0			0
$904$	−2.30471e7	+	3.17216e7i	−0.937983	+	1.29102i
$905$	−2.99634e7	+	1.38611e7i	−1.21610	+	0.562568i
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$	4.65841e7	−	1.51361e7i	1.86994	−	0.607581i
$910$		0			0
$911$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$911$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$912$		0			0
$913$		0			0
$914$	7.45007e6	+	2.42067e6i	0.294981	+	0.0958453i
$915$		0			0
$916$	540403.	+	1.66319e6i	0.0212803	+	0.0654942i
$917$		0			0
$918$		0			0
$919$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$919$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$920$		0			0
$921$		0			0
$922$	−6.93738e6	−	4.38009e7i	−0.268762	−	1.69690i
$923$		0			0
$924$		0			0
$925$	1.37069e7	−	1.17692e7i	0.526727	−	0.452265i
$926$		0			0
$927$		0			0
$928$	−3.11928e7	+	4.94045e6i	−1.18901	+	0.188320i
$929$	−3.79296e6	+	5.22056e6i	−0.144191	+	0.198462i	−0.875004	−	0.484116i	$-0.839141\pi$
$929$	−3.79296e6	+	5.22056e6i	−0.144191	+	0.198462i	0.730813	+	0.682578i	$0.239141\pi$
$930$		0			0
$931$		0			0
$932$	−3.21582e7	−	3.21582e7i	−1.21269	−	1.21269i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−1.65490e7	+	5.09326e7i	−0.617422	+	1.90023i
$937$	1.90074e7	−	3.73042e7i	0.707252	−	1.38806i	−0.205135	−	0.978734i	$-0.565763\pi$
$937$	1.90074e7	−	3.73042e7i	0.707252	−	1.38806i	0.912388	−	0.409327i	$-0.134237\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.63656e7	+	5.03683e7i	0.602503	+	1.85431i	0.513121	+	0.858316i	$0.328489\pi$
$941$	1.63656e7	+	5.03683e7i	0.602503	+	1.85431i	0.0893816	+	0.995997i	$0.471511\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$947$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$948$		0			0
$949$		−	7.30929e6i		−	0.263457i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	5.49069e7	−	8.69640e6i	1.95837	−	0.310175i	0.958701	−	0.284414i	$-0.0917991\pi$
$953$	5.49069e7	−	8.69640e6i	1.95837	−	0.310175i	0.999668	−	0.0257609i	$-0.00820087\pi$
$954$	3.30379e7	−	4.54728e7i	1.17528	−	1.61763i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	8.84689e6	−	2.72279e7i	0.309017	−	0.951057i
$962$	−1.80759e7	+	3.54760e7i	−0.629743	+	1.23594i
$963$		0			0
$964$	−3.01657e7	−	9.80145e6i	−1.04549	−	0.339702i
$965$	1.62160e7	−	1.08660e7i	0.560563	−	0.375622i
$966$		0			0
$967$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$967$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$968$	−2.06145e7	+	2.06145e7i	−0.707107	+	0.707107i
$969$		0			0
$970$	−2.05930e7	−	4.45159e7i	−0.702735	−	1.51910i
$971$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$971$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−5.69266e7			−1.91289
$977$	3.70920e7	−	1.88993e7i	1.24321	−	0.633445i	0.296343	−	0.955081i	$-0.404233\pi$
$977$	3.70920e7	−	1.88993e7i	1.24321	−	0.633445i	0.946863	+	0.321636i	$0.104233\pi$
$978$		0			0
$979$		0			0
$980$	−2.04373e7	−	2.20508e7i	−0.679765	−	0.733430i
$981$	−3.59555e7	+	2.61232e7i	−1.19287	+	0.866669i
$982$		0			0
$983$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$983$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$984$		0			0
$985$	2.01283e6	−	5.30045e7i	0.0661022	−	1.74069i
$986$	−1.65619e7	+	5.09724e7i	−0.542524	+	1.66972i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$991$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9.67732e6	+	6.11002e7i	0.308331	+	1.94673i	0.321775	+	0.946816i	$0.395720\pi$
$997$	9.67732e6	+	6.11002e7i	0.308331	+	1.94673i	−0.0134441	+	0.999910i	$0.504280\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.6.l.a.47.1	✓	8
4.3	odd	2	CM	100.6.l.a.47.1	✓	8
25.8	odd	20	inner	100.6.l.a.83.1	yes	8
100.83	even	20	inner	100.6.l.a.83.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.6.l.a.47.1	✓	8	1.1	even	1	trivial
100.6.l.a.47.1	✓	8	4.3	odd	2	CM
100.6.l.a.83.1	yes	8	25.8	odd	20	inner
100.6.l.a.83.1	yes	8	100.83	even	20	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:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	100.l (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(5.04029 - 2.56816i) q^{2} +(18.8091 - 25.8885i) q^{4} +(-54.8418 - 10.8339i) q^{5} +(28.3177 - 178.791i) q^{8} +(231.107 - 75.0911i) q^{9} +O(q^{10})\)	\(q+(5.04029 - 2.56816i) q^{2} +(18.8091 - 25.8885i) q^{4} +(-54.8418 - 10.8339i) q^{5} +(28.3177 - 178.791i) q^{8} +(231.107 - 75.0911i) q^{9} +(-304.242 + 86.2367i) q^{10} +(-1084.77 - 552.720i) q^{13} +(-316.433 - 973.882i) q^{16} +(-1716.39 - 271.849i) q^{17} +(972.000 - 972.000i) q^{18} +(-1312.00 + 1216.00i) q^{20} +(2890.26 + 1188.30i) q^{25} -6887.05 q^{26} +(3204.63 - 4410.79i) q^{29} +(-4096.00 - 4096.00i) q^{32} +(-9349.24 + 3037.75i) q^{34} +(2402.92 - 7395.42i) q^{36} +(2624.63 - 5151.12i) q^{37} +(-3489.99 + 9498.42i) q^{40} +(2567.27 + 7901.23i) q^{41} +(-13487.8 + 1614.36i) q^{45} +16807.0i q^{49} +(17619.5 - 1433.26i) q^{50} +(-34712.8 + 17687.0i) q^{52} +(40386.1 - 6396.54i) q^{53} +(4824.66 - 30461.7i) q^{58} +(17179.0 - 52871.5i) q^{61} +(-31164.2 - 10125.9i) q^{64} +(53502.9 + 42064.5i) q^{65} +(-39321.5 + 39321.5i) q^{68} +(-6881.19 - 43446.1i) q^{72} +(2725.61 + 5349.31i) q^{73} -32703.6i q^{74} +(6802.89 + 56837.7i) q^{80} +(47771.6 - 34708.1i) q^{81} +(33231.4 + 33231.4i) q^{82} +(91184.5 + 33503.8i) q^{85} +(-4893.82 - 1590.10i) q^{89} +(-63836.8 + 42775.8i) q^{90} +(24263.7 + 153195. i) q^{97} +(43163.0 + 84712.2i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 76 q^{5} + 256 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 - 76 * q^5 + 256 * q^8	\( 8 q - 8 q^{2} - 76 q^{5} + 256 q^{8} + 24 q^{10} - 1438 q^{13} + 2048 q^{16} + 1434 q^{17} + 7776 q^{18} - 10496 q^{20} + 474 q^{25} - 11504 q^{26} - 32768 q^{32} - 61000 q^{34} - 15552 q^{36} - 32926 q^{37} + 20224 q^{40} - 9904 q^{41} - 19926 q^{45} + 26824 q^{50} - 46016 q^{52} + 153682 q^{53} + 68512 q^{58} + 109896 q^{61} + 163278 q^{65} - 45888 q^{68} - 62208 q^{72} + 68662 q^{73} + 77824 q^{80} + 118098 q^{81} - 39616 q^{82} + 608828 q^{85} - 255250 q^{89} - 153576 q^{90} - 68666 q^{97} + 134456 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 - 76 * q^5 + 256 * q^8 + 24 * q^10 - 1438 * q^13 + 2048 * q^16 + 1434 * q^17 + 7776 * q^18 - 10496 * q^20 + 474 * q^25 - 11504 * q^26 - 32768 * q^32 - 61000 * q^34 - 15552 * q^36 - 32926 * q^37 + 20224 * q^40 - 9904 * q^41 - 19926 * q^45 + 26824 * q^50 - 46016 * q^52 + 153682 * q^53 + 68512 * q^58 + 109896 * q^61 + 163278 * q^65 - 45888 * q^68 - 62208 * q^72 + 68662 * q^73 + 77824 * q^80 + 118098 * q^81 - 39616 * q^82 + 608828 * q^85 - 255250 * q^89 - 153576 * q^90 - 68666 * q^97 + 134456 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more