LMFDB - Embedded newform 196.6.e.e.165.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [196,6,Mod(165,196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(196, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("196.165");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 196 = 2^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	196.e (of order $3$, degree $2$, not 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:	$31.4352286833$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			165.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	196.165
Dual form			196.6.e.e.177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{3} +(-48.0000 - 83.1384i) q^{5} +(119.500 + 206.980i) q^{9} +O(q^{10})$	\(q+(-1.00000 + 1.73205i) q^{3} +(-48.0000 - 83.1384i) q^{5} +(119.500 + 206.980i) q^{9} +(360.000 - 623.538i) q^{11} -572.000 q^{13} +192.000 q^{15} +(627.000 - 1086.00i) q^{17} +(-47.0000 - 81.4064i) q^{19} +(-48.0000 - 83.1384i) q^{23} +(-3045.50 + 5274.96i) q^{25} -964.000 q^{27} -4374.00 q^{29} +(-3122.00 + 5407.46i) q^{31} +(720.000 + 1247.08i) q^{33} +(5399.00 + 9351.34i) q^{37} +(572.000 - 990.733i) q^{39} -12006.0 q^{41} -9160.00 q^{43} +(11472.0 - 19870.1i) q^{45} +(-12918.0 - 22374.6i) q^{47} +(1254.00 + 2171.99i) q^{51} +(-507.000 + 878.150i) q^{53} -69120.0 q^{55} +188.000 q^{57} +(621.000 - 1075.60i) q^{59} +(3796.00 + 6574.86i) q^{61} +(27456.0 + 47555.2i) q^{65} +(-20566.0 + 35621.4i) q^{67} +192.000 q^{69} -37632.0 q^{71} +(-6719.00 + 11637.6i) q^{73} +(-6091.00 - 10549.9i) q^{75} +(-3124.00 - 5410.93i) q^{79} +(-28074.5 + 48626.5i) q^{81} +25254.0 q^{83} -120384. q^{85} +(4374.00 - 7575.99i) q^{87} +(-22563.0 - 39080.3i) q^{89} +(-6244.00 - 10814.9i) q^{93} +(-4512.00 + 7815.01i) q^{95} -107222. q^{97} +172080. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 96 q^{5} + 239 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 96 * q^5 + 239 * q^9	$ 2 q - 2 q^{3} - 96 q^{5} + 239 q^{9} + 720 q^{11} - 1144 q^{13} + 384 q^{15} + 1254 q^{17} - 94 q^{19} - 96 q^{23} - 6091 q^{25} - 1928 q^{27} - 8748 q^{29} - 6244 q^{31} + 1440 q^{33} + 10798 q^{37} + 1144 q^{39} - 24012 q^{41} - 18320 q^{43} + 22944 q^{45} - 25836 q^{47} + 2508 q^{51} - 1014 q^{53} - 138240 q^{55} + 376 q^{57} + 1242 q^{59} + 7592 q^{61} + 54912 q^{65} - 41132 q^{67} + 384 q^{69} - 75264 q^{71} - 13438 q^{73} - 12182 q^{75} - 6248 q^{79} - 56149 q^{81} + 50508 q^{83} - 240768 q^{85} + 8748 q^{87} - 45126 q^{89} - 12488 q^{93} - 9024 q^{95} - 214444 q^{97} + 344160 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 96 * q^5 + 239 * q^9 + 720 * q^11 - 1144 * q^13 + 384 * q^15 + 1254 * q^17 - 94 * q^19 - 96 * q^23 - 6091 * q^25 - 1928 * q^27 - 8748 * q^29 - 6244 * q^31 + 1440 * q^33 + 10798 * q^37 + 1144 * q^39 - 24012 * q^41 - 18320 * q^43 + 22944 * q^45 - 25836 * q^47 + 2508 * q^51 - 1014 * q^53 - 138240 * q^55 + 376 * q^57 + 1242 * q^59 + 7592 * q^61 + 54912 * q^65 - 41132 * q^67 + 384 * q^69 - 75264 * q^71 - 13438 * q^73 - 12182 * q^75 - 6248 * q^79 - 56149 * q^81 + 50508 * q^83 - 240768 * q^85 + 8748 * q^87 - 45126 * q^89 - 12488 * q^93 - 9024 * q^95 - 214444 * q^97 + 344160 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/196\mathbb{Z}\right)^\times$.

$n$	$99$	$101$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	−1.00000	+	1.73205i	−0.0641500	+	0.111111i	−0.896317	−	0.443415i	$-0.853767\pi$
$3$	−1.00000	+	1.73205i	−0.0641500	+	0.111111i	0.832167	+	0.554526i	$0.187100\pi$
$4$		0			0
$5$	−48.0000	−	83.1384i	−0.858650	−	1.48723i	−0.873217	−	0.487332i	$-0.837970\pi$
$5$	−48.0000	−	83.1384i	−0.858650	−	1.48723i	0.0145668	−	0.999894i	$-0.495363\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	119.500	+	206.980i	0.491770	+	0.851770i
$10$		0			0
$11$	360.000	−	623.538i	0.897059	−	1.55375i	0.0658225	−	0.997831i	$-0.479033\pi$
$11$	360.000	−	623.538i	0.897059	−	1.55375i	0.831236	−	0.555920i	$-0.187634\pi$
$12$		0			0
$13$	−572.000			−0.938723			−0.469362	−	0.883006i	$-0.655516\pi$
$13$	−572.000			−0.938723			−0.469362	+	0.883006i	$0.655516\pi$
$14$		0			0
$15$	192.000			0.220330
$16$		0			0
$17$	627.000	−	1086.00i	0.526193	−	0.911393i	−0.473341	−	0.880879i	$-0.656952\pi$
$17$	627.000	−	1086.00i	0.526193	−	0.911393i	0.999534	−	0.0305142i	$-0.00971447\pi$
$18$		0			0
$19$	−47.0000	−	81.4064i	−0.0298685	−	0.0517338i	0.850705	−	0.525644i	$-0.176176\pi$
$19$	−47.0000	−	81.4064i	−0.0298685	−	0.0517338i	−0.880573	+	0.473910i	$0.842842\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−48.0000	−	83.1384i	−0.0189200	−	0.0327704i	0.856410	−	0.516296i	$-0.172689\pi$
$23$	−48.0000	−	83.1384i	−0.0189200	−	0.0327704i	−0.875330	+	0.483525i	$0.839356\pi$
$24$		0			0
$25$	−3045.50	+	5274.96i	−0.974560	+	1.68799i
$26$		0			0
$27$	−964.000			−0.254488
$28$		0			0
$29$	−4374.00			−0.965792			−0.482896	−	0.875678i	$-0.660415\pi$
$29$	−4374.00			−0.965792			−0.482896	+	0.875678i	$0.660415\pi$
$30$		0			0
$31$	−3122.00	+	5407.46i	−0.583484	+	1.01062i	0.411579	+	0.911374i	$0.364977\pi$
$31$	−3122.00	+	5407.46i	−0.583484	+	1.01062i	−0.995063	+	0.0992492i	$0.968356\pi$
$32$		0			0
$33$	720.000	+	1247.08i	0.115093	+	0.199346i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5399.00	+	9351.34i	0.648349	+	1.12297i	0.983517	+	0.180815i	$0.0578737\pi$
$37$	5399.00	+	9351.34i	0.648349	+	1.12297i	−0.335168	+	0.942158i	$0.608793\pi$
$38$		0			0
$39$	572.000	−	990.733i	0.0602191	−	0.104303i
$40$		0			0
$41$	−12006.0			−1.11542			−0.557710	−	0.830036i	$-0.688320\pi$
$41$	−12006.0			−1.11542			−0.557710	+	0.830036i	$0.688320\pi$
$42$		0			0
$43$	−9160.00			−0.755482			−0.377741	−	0.925911i	$-0.623299\pi$
$43$	−9160.00			−0.755482			−0.377741	+	0.925911i	$0.623299\pi$
$44$		0			0
$45$	11472.0	−	19870.1i	0.844516	−	1.46274i
$46$		0			0
$47$	−12918.0	−	22374.6i	−0.853003	−	1.47744i	−0.878485	−	0.477770i	$-0.841445\pi$
$47$	−12918.0	−	22374.6i	−0.853003	−	1.47744i	0.0254821	−	0.999675i	$-0.491888\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	1254.00	+	2171.99i	0.0675106	+	0.116932i
$52$		0			0
$53$	−507.000	+	878.150i	−0.0247924	+	0.0429417i	−0.878155	−	0.478375i	$-0.841226\pi$
$53$	−507.000	+	878.150i	−0.0247924	+	0.0429417i	0.853363	+	0.521317i	$0.174559\pi$
$54$		0			0
$55$	−69120.0			−3.08104
$56$		0			0
$57$	188.000			0.00766427
$58$		0			0
$59$	621.000	−	1075.60i	0.0232253	−	0.0402274i	−0.854179	−	0.519979i	$-0.825940\pi$
$59$	621.000	−	1075.60i	0.0232253	−	0.0402274i	0.877404	+	0.479751i	$0.159273\pi$
$60$		0			0
$61$	3796.00	+	6574.86i	0.130618	+	0.226236i	0.923915	−	0.382598i	$-0.124971\pi$
$61$	3796.00	+	6574.86i	0.130618	+	0.226236i	−0.793297	+	0.608835i	$0.791637\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	27456.0	+	47555.2i	0.806035	+	1.39609i
$66$		0			0
$67$	−20566.0	+	35621.4i	−0.559710	+	0.969446i	0.437811	+	0.899067i	$0.355754\pi$
$67$	−20566.0	+	35621.4i	−0.559710	+	0.969446i	−0.997520	+	0.0703784i	$0.977579\pi$
$68$		0			0
$69$	192.000			0.00485488
$70$		0			0
$71$	−37632.0			−0.885955			−0.442977	−	0.896533i	$-0.646078\pi$
$71$	−37632.0			−0.885955			−0.442977	+	0.896533i	$0.646078\pi$
$72$		0			0
$73$	−6719.00	+	11637.6i	−0.147570	+	0.255598i	−0.930329	−	0.366727i	$-0.880478\pi$
$73$	−6719.00	+	11637.6i	−0.147570	+	0.255598i	0.782759	+	0.622325i	$0.213812\pi$
$74$		0			0
$75$	−6091.00	−	10549.9i	−0.125036	−	0.216569i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−3124.00	−	5410.93i	−0.0563175	−	0.0975448i	0.836492	−	0.547979i	$-0.184603\pi$
$79$	−3124.00	−	5410.93i	−0.0563175	−	0.0975448i	−0.892810	+	0.450434i	$0.851269\pi$
$80$		0			0
$81$	−28074.5	+	48626.5i	−0.475444	+	0.823493i
$82$		0			0
$83$	25254.0			0.402379			0.201189	−	0.979552i	$-0.435519\pi$
$83$	25254.0			0.402379			0.201189	+	0.979552i	$0.435519\pi$
$84$		0			0
$85$	−120384.			−1.80726
$86$		0			0
$87$	4374.00	−	7575.99i	0.0619556	−	0.107310i
$88$		0			0
$89$	−22563.0	−	39080.3i	−0.301941	−	0.522977i	0.674635	−	0.738152i	$-0.264301\pi$
$89$	−22563.0	−	39080.3i	−0.301941	−	0.522977i	−0.976576	+	0.215175i	$0.930968\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−6244.00	−	10814.9i	−0.0748610	−	0.129663i
$94$		0			0
$95$	−4512.00	+	7815.01i	−0.0512932	+	0.0888425i
$96$		0			0
$97$	−107222.			−1.15706			−0.578528	−	0.815662i	$-0.696373\pi$
$97$	−107222.			−1.15706			−0.578528	+	0.815662i	$0.696373\pi$
$98$		0			0
$99$	172080.			1.76458
$100$		0			0
$101$	23568.0	−	40821.0i	0.229890	−	0.398180i	−0.727886	−	0.685699i	$-0.759497\pi$
$101$	23568.0	−	40821.0i	0.229890	−	0.398180i	0.957775	+	0.287518i	$0.0928302\pi$
$102$		0			0
$103$	61102.0	+	105832.i	0.567495	+	0.982931i	0.996813	+	0.0797772i	$0.0254209\pi$
$103$	61102.0	+	105832.i	0.567495	+	0.982931i	−0.429317	+	0.903154i	$0.641246\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	64818.0	+	112268.i	0.547314	+	0.947975i	0.998457	+	0.0555236i	$0.0176828\pi$
$107$	64818.0	+	112268.i	0.547314	+	0.947975i	−0.451144	+	0.892451i	$0.648984\pi$
$108$		0			0
$109$	110279.	−	191009.i	0.889051	−	1.53988i	0.0480515	−	0.998845i	$-0.484699\pi$
$109$	110279.	−	191009.i	0.889051	−	1.53988i	0.840999	−	0.541036i	$-0.181968\pi$
$110$		0			0
$111$	−21596.0			−0.166366
$112$		0			0
$113$	170694.			1.25754			0.628770	−	0.777591i	$-0.283559\pi$
$113$	170694.			1.25754			0.628770	+	0.777591i	$0.283559\pi$
$114$		0			0
$115$	−4608.00	+	7981.29i	−0.0324914	+	0.0562767i
$116$		0			0
$117$	−68354.0	−	118393.i	−0.461636	−	0.799576i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−178674.	−	309473.i	−1.10943	−	1.92159i
$122$		0			0
$123$	12006.0	−	20795.0i	0.0715543	−	0.123936i
$124$		0			0
$125$	284736.			1.62992
$126$		0			0
$127$	−249808.			−1.37435			−0.687175	−	0.726492i	$-0.741149\pi$
$127$	−249808.			−1.37435			−0.687175	+	0.726492i	$0.741149\pi$
$128$		0			0
$129$	9160.00	−	15865.6i	0.0484642	−	0.0839425i
$130$		0			0
$131$	6105.00	+	10574.2i	0.0310819	+	0.0538354i	0.881148	−	0.472841i	$-0.156771\pi$
$131$	6105.00	+	10574.2i	0.0310819	+	0.0538354i	−0.850066	+	0.526676i	$0.823438\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	46272.0	+	80145.5i	0.218516	+	0.378481i
$136$		0			0
$137$	6951.00	−	12039.5i	0.0316407	−	0.0548033i	−0.849771	−	0.527151i	$-0.823260\pi$
$137$	6951.00	−	12039.5i	0.0316407	−	0.0548033i	0.881412	+	0.472348i	$0.156593\pi$
$138$		0			0
$139$	431794.			1.89557			0.947785	−	0.318911i	$-0.103317\pi$
$139$	431794.			1.89557			0.947785	+	0.318911i	$0.103317\pi$
$140$		0			0
$141$	51672.0			0.218881
$142$		0			0
$143$	−205920.	+	356664.i	−0.842090	+	1.45854i
$144$		0			0
$145$	209952.	+	363648.i	0.829278	+	1.43635i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−163407.	−	283029.i	−0.602983	−	1.04440i	−0.992367	−	0.123321i	$-0.960645\pi$
$149$	−163407.	−	283029.i	−0.602983	−	1.04440i	0.389384	−	0.921076i	$-0.372688\pi$
$150$		0			0
$151$	−86740.0	+	150238.i	−0.309583	+	0.536213i	−0.978271	−	0.207330i	$-0.933523\pi$
$151$	−86740.0	+	150238.i	−0.309583	+	0.536213i	0.668688	+	0.743543i	$0.266856\pi$
$152$		0			0
$153$	299706.			1.03506
$154$		0			0
$155$	599424.			2.00403
$156$		0			0
$157$	−27266.0	+	47226.1i	−0.0882820	+	0.152909i	−0.906785	−	0.421593i	$-0.861471\pi$
$157$	−27266.0	+	47226.1i	−0.0882820	+	0.152909i	0.818503	+	0.574502i	$0.194804\pi$
$158$		0			0
$159$	−1014.00	−	1756.30i	−0.00318086	−	0.00550942i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−52480.0	−	90898.0i	−0.154712	−	0.267970i	0.778242	−	0.627965i	$-0.216112\pi$
$163$	−52480.0	−	90898.0i	−0.154712	−	0.267970i	−0.932954	+	0.359995i	$0.882778\pi$
$164$		0			0
$165$	69120.0	−	119719.i	0.197649	−	0.342338i
$166$		0			0
$167$	−160788.			−0.446131			−0.223066	−	0.974803i	$-0.571606\pi$
$167$	−160788.			−0.446131			−0.223066	+	0.974803i	$0.571606\pi$
$168$		0			0
$169$	−44109.0			−0.118798
$170$		0			0
$171$	11233.0	−	19456.1i	0.0293769	−	0.0508822i
$172$		0			0
$173$	180282.	+	312258.i	0.457970	+	0.793227i	0.998854	−	0.0478701i	$-0.0152433\pi$
$173$	180282.	+	312258.i	0.457970	+	0.793227i	−0.540883	+	0.841098i	$0.681910\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	1242.00	+	2151.21i	0.00297981	+	0.00516118i
$178$		0			0
$179$	156366.	−	270834.i	0.364762	−	0.631787i	−0.623976	−	0.781444i	$-0.714484\pi$
$179$	156366.	−	270834.i	0.364762	−	0.631787i	0.988738	+	0.149657i	$0.0478170\pi$
$180$		0			0
$181$	123820.			0.280928			0.140464	−	0.990086i	$-0.455141\pi$
$181$	123820.			0.280928			0.140464	+	0.990086i	$0.455141\pi$
$182$		0			0
$183$	−15184.0			−0.0335165
$184$		0			0
$185$	518304.	−	897729.i	1.11341	−	1.92848i
$186$		0			0
$187$	−451440.	−	781917.i	−0.944052	−	1.63515i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−161724.	−	280114.i	−0.320768	−	0.555586i	0.659879	−	0.751372i	$-0.270608\pi$
$191$	−161724.	−	280114.i	−0.320768	−	0.555586i	−0.980647	+	0.195786i	$0.937274\pi$
$192$		0			0
$193$	309977.	−	536896.i	0.599013	−	1.03752i	−0.393954	−	0.919130i	$-0.628893\pi$
$193$	309977.	−	536896.i	0.599013	−	1.03752i	0.992967	−	0.118391i	$-0.0377735\pi$
$194$		0			0
$195$	−109824.			−0.206829
$196$		0			0
$197$	−499362.			−0.916748			−0.458374	−	0.888759i	$-0.651568\pi$
$197$	−499362.			−0.916748			−0.458374	+	0.888759i	$0.651568\pi$
$198$		0			0
$199$	−392966.	+	680637.i	−0.703432	+	1.21838i	0.263822	+	0.964571i	$0.415017\pi$
$199$	−392966.	+	680637.i	−0.703432	+	1.21838i	−0.967254	+	0.253809i	$0.918317\pi$
$200$		0			0
$201$	−41132.0	−	71242.7i	−0.0718108	−	0.124380i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	576288.	+	998160.i	0.957756	+	1.65888i
$206$		0			0
$207$	11472.0	−	19870.1i	0.0186086	−	0.0322310i
$208$		0			0
$209$	−67680.0			−0.107175
$210$		0			0
$211$	1.06276e6			1.64335			0.821676	−	0.569955i	$-0.193039\pi$
$211$	1.06276e6			1.64335			0.821676	+	0.569955i	$0.193039\pi$
$212$		0			0
$213$	37632.0	−	65180.5i	0.0568340	−	0.0984394i
$214$		0			0
$215$	439680.	+	761548.i	0.648695	+	1.12357i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−13438.0	−	23275.3i	−0.0189332	−	0.0327933i
$220$		0			0
$221$	−358644.	+	621190.i	−0.493950	+	0.855546i
$222$		0			0
$223$	−707720.			−0.953014			−0.476507	−	0.879171i	$-0.658097\pi$
$223$	−707720.			−0.953014			−0.476507	+	0.879171i	$0.658097\pi$
$224$		0			0
$225$	−1.45575e6			−1.91704
$226$		0			0
$227$	−522183.	+	904447.i	−0.672602	+	1.16498i	0.304562	+	0.952493i	$0.401490\pi$
$227$	−522183.	+	904447.i	−0.672602	+	1.16498i	−0.977164	+	0.212488i	$0.931843\pi$
$228$		0			0
$229$	−269858.	−	467408.i	−0.340053	−	0.588989i	0.644389	−	0.764698i	$-0.277112\pi$
$229$	−269858.	−	467408.i	−0.340053	−	0.588989i	−0.984442	+	0.175709i	$0.943778\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−88557.0	−	153385.i	−0.106864	−	0.185095i	0.807634	−	0.589684i	$-0.200748\pi$
$233$	−88557.0	−	153385.i	−0.106864	−	0.185095i	−0.914498	+	0.404590i	$0.867414\pi$
$234$		0			0
$235$	−1.24013e6	+	2.14796e6i	−1.46486	+	2.53722i
$236$		0			0
$237$	12496.0			0.0144511
$238$		0			0
$239$	−655464.			−0.742257			−0.371128	−	0.928582i	$-0.621029\pi$
$239$	−655464.			−0.742257			−0.371128	+	0.928582i	$0.621029\pi$
$240$		0			0
$241$	693547.	−	1.20126e6i	0.769189	−	1.33228i	−0.168814	−	0.985648i	$-0.553994\pi$
$241$	693547.	−	1.20126e6i	0.769189	−	1.33228i	0.938003	−	0.346627i	$-0.112673\pi$
$242$		0			0
$243$	−173275.	−	300121.i	−0.188244	−	0.326047i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	26884.0	+	46564.5i	0.0280383	+	0.0485637i
$248$		0			0
$249$	−25254.0	+	43741.2i	−0.0258126	+	0.0447087i
$250$		0			0
$251$	−1.88811e6			−1.89166			−0.945830	−	0.324663i	$-0.894749\pi$
$251$	−1.88811e6			−1.89166			−0.945830	+	0.324663i	$0.894749\pi$
$252$		0			0
$253$	−69120.0			−0.0678895
$254$		0			0
$255$	120384.	−	208511.i	0.115936	−	0.200807i
$256$		0			0
$257$	173097.	+	299813.i	0.163477	+	0.283151i	0.936113	−	0.351698i	$-0.114396\pi$
$257$	173097.	+	299813.i	0.163477	+	0.283151i	−0.772636	+	0.634849i	$0.781062\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−522693.	−	905331.i	−0.474947	−	0.822633i
$262$		0			0
$263$	464544.	−	804614.i	0.414131	−	0.717296i	−0.581206	−	0.813756i	$-0.697419\pi$
$263$	464544.	−	804614.i	0.414131	−	0.717296i	0.995337	+	0.0964609i	$0.0307523\pi$
$264$		0			0
$265$	97344.0			0.0851519
$266$		0			0
$267$	90252.0			0.0774780
$268$		0			0
$269$	191034.	−	330881.i	0.160964	−	0.278799i	−0.774250	−	0.632879i	$-0.781873\pi$
$269$	191034.	−	330881.i	0.160964	−	0.278799i	0.935215	+	0.354081i	$0.115206\pi$
$270$		0			0
$271$	−790280.	−	1.36881e6i	−0.653669	−	1.13219i	−0.982226	−	0.187703i	$-0.939896\pi$
$271$	−790280.	−	1.36881e6i	−0.653669	−	1.13219i	0.328557	−	0.944484i	$-0.393438\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.19276e6	+	3.79797e6i	1.74847	+	3.02845i
$276$		0			0
$277$	684557.	−	1.18569e6i	0.536056	−	0.928476i	−0.463055	−	0.886329i	$-0.653247\pi$
$277$	684557.	−	1.18569e6i	0.536056	−	0.928476i	0.999111	−	0.0421469i	$-0.0134198\pi$
$278$		0			0
$279$	−1.49232e6			−1.14776
$280$		0			0
$281$	−394854.			−0.298312			−0.149156	−	0.988814i	$-0.547656\pi$
$281$	−394854.			−0.298312			−0.149156	+	0.988814i	$0.547656\pi$
$282$		0			0
$283$	336517.	−	582865.i	0.249770	−	0.432615i	−0.713692	−	0.700460i	$-0.752978\pi$
$283$	336517.	−	582865.i	0.249770	−	0.432615i	0.963462	+	0.267845i	$0.0863115\pi$
$284$		0			0
$285$	−9024.00	−	15630.0i	−0.00658092	−	0.0113985i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−76329.5	−	132207.i	−0.0537586	−	0.0931126i
$290$		0			0
$291$	107222.	−	185714.i	0.0742252	−	0.128562i
$292$		0			0
$293$	−1.83468e6			−1.24851			−0.624254	−	0.781222i	$-0.714597\pi$
$293$	−1.83468e6			−1.24851			−0.624254	+	0.781222i	$0.714597\pi$
$294$		0			0
$295$	−119232.			−0.0797697
$296$		0			0
$297$	−347040.	+	601091.i	−0.228291	+	0.395411i
$298$		0			0
$299$	27456.0	+	47555.2i	0.0177607	+	0.0307624i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	47136.0	+	81641.9i	0.0294948	+	0.0510866i
$304$		0			0
$305$	364416.	−	631187.i	0.224310	−	0.388516i
$306$		0			0
$307$	1.51056e6			0.914727			0.457363	−	0.889280i	$-0.348794\pi$
$307$	1.51056e6			0.914727			0.457363	+	0.889280i	$0.348794\pi$
$308$		0			0
$309$	−244408.			−0.145619
$310$		0			0
$311$	−937644.	+	1.62405e6i	−0.549714	+	0.952133i	0.448580	+	0.893743i	$0.351930\pi$
$311$	−937644.	+	1.62405e6i	−0.549714	+	0.952133i	−0.998294	+	0.0583901i	$0.981403\pi$
$312$		0			0
$313$	−755381.	−	1.30836e6i	−0.435818	−	0.754859i	0.561544	−	0.827447i	$-0.310208\pi$
$313$	−755381.	−	1.30836e6i	−0.435818	−	0.754859i	−0.997362	+	0.0725879i	$0.976874\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.01355e6	−	1.75551e6i	−0.566495	−	0.981197i	−0.996909	−	0.0785659i	$-0.974966\pi$
$317$	−1.01355e6	−	1.75551e6i	−0.566495	−	0.981197i	0.430414	−	0.902631i	$-0.358367\pi$
$318$		0			0
$319$	−1.57464e6	+	2.72736e6i	−0.866372	+	1.50060i
$320$		0			0
$321$	−259272.			−0.140441
$322$		0			0
$323$	−117876.			−0.0628665
$324$		0			0
$325$	1.74203e6	−	3.01728e6i	0.914842	−	1.58455i
$326$		0			0
$327$	220558.	+	382018.i	0.114065	+	0.197567i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−770044.	−	1.33376e6i	−0.386319	−	0.669123i	0.605633	−	0.795744i	$-0.292920\pi$
$331$	−770044.	−	1.33376e6i	−0.386319	−	0.669123i	−0.991951	+	0.126621i	$0.959587\pi$
$332$		0			0
$333$	−1.29036e6	+	2.23497e6i	−0.637677	+	1.10449i
$334$		0			0
$335$	3.94867e6			1.92238
$336$		0			0
$337$	1.01166e6			0.485245			0.242622	−	0.970121i	$-0.421992\pi$
$337$	1.01166e6			0.485245			0.242622	+	0.970121i	$0.421992\pi$
$338$		0			0
$339$	−170694.	+	295651.i	−0.0806713	+	0.139727i
$340$		0			0
$341$	2.24784e6	+	3.89337e6i	1.04684	+	1.81318i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−9216.00	−	15962.6i	−0.00416864	−	0.00722030i
$346$		0			0
$347$	1.07874e6	−	1.86843e6i	0.480942	−	0.833017i	−0.518818	−	0.854884i	$-0.673628\pi$
$347$	1.07874e6	−	1.86843e6i	0.480942	−	0.833017i	0.999761	+	0.0218677i	$0.00696127\pi$
$348$		0			0
$349$	1.15798e6			0.508906			0.254453	−	0.967085i	$-0.418105\pi$
$349$	1.15798e6			0.508906			0.254453	+	0.967085i	$0.418105\pi$
$350$		0			0
$351$	551408.			0.238894
$352$		0			0
$353$	−1.58783e6	+	2.75020e6i	−0.678215	+	1.17470i	0.297303	+	0.954783i	$0.403913\pi$
$353$	−1.58783e6	+	2.75020e6i	−0.678215	+	1.17470i	−0.975518	+	0.219920i	$0.929420\pi$
$354$		0			0
$355$	1.80634e6	+	3.12867e6i	0.760725	+	1.31761i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	37308.0	+	64619.4i	0.0152780	+	0.0264622i	0.873563	−	0.486711i	$-0.161803\pi$
$359$	37308.0	+	64619.4i	0.0152780	+	0.0264622i	−0.858285	+	0.513173i	$0.828470\pi$
$360$		0			0
$361$	1.23363e6	−	2.13671e6i	0.498216	−	0.862935i
$362$		0			0
$363$	714698.			0.284679
$364$		0			0
$365$	1.29005e6			0.506843
$366$		0			0
$367$	−899036.	+	1.55718e6i	−0.348427	+	0.603493i	−0.985970	−	0.166921i	$-0.946617\pi$
$367$	−899036.	+	1.55718e6i	−0.348427	+	0.603493i	0.637543	+	0.770415i	$0.279951\pi$
$368$		0			0
$369$	−1.43472e6	−	2.48500e6i	−0.548530	−	0.950082i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.10230e6	−	1.90924e6i	−0.410231	−	0.710542i	0.584683	−	0.811262i	$-0.301219\pi$
$373$	−1.10230e6	−	1.90924e6i	−0.410231	−	0.710542i	−0.994915	+	0.100720i	$0.967885\pi$
$374$		0			0
$375$	−284736.	+	493177.i	−0.104560	+	0.181103i
$376$		0			0
$377$	2.50193e6			0.906612
$378$		0			0
$379$	−177568.			−0.0634990			−0.0317495	−	0.999496i	$-0.510108\pi$
$379$	−177568.			−0.0634990			−0.0317495	+	0.999496i	$0.510108\pi$
$380$		0			0
$381$	249808.	−	432680.i	0.0881645	−	0.152705i
$382$		0			0
$383$	−1.43734e6	−	2.48955e6i	−0.500683	−	0.867209i	−1.00000		0.000789287i	$-0.999749\pi$
$383$	−1.43734e6	−	2.48955e6i	−0.500683	−	0.867209i	0.499316	−	0.866420i	$-0.333585\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.09462e6	−	1.89594e6i	−0.371523	−	0.643497i
$388$		0			0
$389$	2.39982e6	−	4.15662e6i	0.804091	−	1.39273i	−0.112812	−	0.993616i	$-0.535986\pi$
$389$	2.39982e6	−	4.15662e6i	0.804091	−	1.39273i	0.916903	−	0.399110i	$-0.130681\pi$
$390$		0			0
$391$	−120384.			−0.0398223
$392$		0			0
$393$	−24420.0			−0.00797562
$394$		0			0
$395$	−299904.	+	519449.i	−0.0967140	+	0.167514i
$396$		0			0
$397$	−1.40821e6	−	2.43910e6i	−0.448428	−	0.776699i	0.549856	−	0.835259i	$-0.314682\pi$
$397$	−1.40821e6	−	2.43910e6i	−0.448428	−	0.776699i	−0.998284	+	0.0585598i	$0.981349\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.41898e6	+	2.45775e6i	0.440673	+	0.763269i	0.997740	−	0.0671994i	$-0.0214064\pi$
$401$	1.41898e6	+	2.45775e6i	0.440673	+	0.763269i	−0.557066	+	0.830468i	$0.688073\pi$
$402$		0			0
$403$	1.78578e6	−	3.09307e6i	0.547730	−	0.948696i
$404$		0			0
$405$	5.39030e6			1.63296
$406$		0			0
$407$	7.77456e6			2.32643
$408$		0			0
$409$	77143.0	−	133616.i	0.0228028	−	0.0394956i	−0.854399	−	0.519618i	$-0.826074\pi$
$409$	77143.0	−	133616.i	0.0228028	−	0.0394956i	0.877202	+	0.480122i	$0.159408\pi$
$410$		0			0
$411$	13902.0	+	24079.0i	0.00405950	+	0.00703126i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.21219e6	−	2.09958e6i	−0.345502	−	0.598428i
$416$		0			0
$417$	−431794.	+	747889.i	−0.121601	+	0.210619i
$418$		0			0
$419$	−3.72865e6			−1.03757			−0.518783	−	0.854906i	$-0.673615\pi$
$419$	−3.72865e6			−1.03757			−0.518783	+	0.854906i	$0.673615\pi$
$420$		0			0
$421$	−2.32623e6			−0.639658			−0.319829	−	0.947475i	$-0.603626\pi$
$421$	−2.32623e6			−0.639658			−0.319829	+	0.947475i	$0.603626\pi$
$422$		0			0
$423$	3.08740e6	−	5.34754e6i	0.838962	−	1.45312i
$424$		0			0
$425$	3.81906e6	+	6.61480e6i	1.02561	+	1.77642i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−411840.	−	713328.i	−0.108040	−	0.187131i
$430$		0			0
$431$	1.30741e6	−	2.26450e6i	0.339015	−	0.587192i	−0.645232	−	0.763986i	$-0.723239\pi$
$431$	1.30741e6	−	2.26450e6i	0.339015	−	0.587192i	0.984248	+	0.176795i	$0.0565728\pi$
$432$		0			0
$433$	1.19226e6			0.305598			0.152799	−	0.988257i	$-0.451171\pi$
$433$	1.19226e6			0.305598			0.152799	+	0.988257i	$0.451171\pi$
$434$		0			0
$435$	−839808.			−0.212793
$436$		0			0
$437$	−4512.00	+	7815.01i	−0.00113023	+	0.00195761i
$438$		0			0
$439$	528964.	+	916193.i	0.130998	+	0.226895i	0.924062	−	0.382244i	$-0.124849\pi$
$439$	528964.	+	916193.i	0.130998	+	0.226895i	−0.793064	+	0.609139i	$0.791515\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.06378e6	−	3.57457e6i	−0.499636	−	0.865395i	0.500364	−	0.865815i	$-0.333200\pi$
$443$	−2.06378e6	−	3.57457e6i	−0.499636	−	0.865395i	−1.00000		0.000420339i	$0.999866\pi$
$444$		0			0
$445$	−2.16605e6	+	3.75171e6i	−0.518523	+	0.898108i
$446$		0			0
$447$	653628.			0.154725
$448$		0			0
$449$	3.75823e6			0.879766			0.439883	−	0.898055i	$-0.355020\pi$
$449$	3.75823e6			0.879766			0.439883	+	0.898055i	$0.355020\pi$
$450$		0			0
$451$	−4.32216e6	+	7.48620e6i	−1.00060	+	1.73309i
$452$		0			0
$453$	−173480.	−	300476.i	−0.0397195	−	0.0687962i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	225557.	+	390676.i	0.0505203	+	0.0875037i	0.890180	−	0.455610i	$-0.150579\pi$
$457$	225557.	+	390676.i	0.0505203	+	0.0875037i	−0.839659	+	0.543113i	$0.817245\pi$
$458$		0			0
$459$	−604428.	+	1.04690e6i	−0.133910	+	0.231939i
$460$		0			0
$461$	1.95186e6			0.427756			0.213878	−	0.976860i	$-0.431390\pi$
$461$	1.95186e6			0.427756			0.213878	+	0.976860i	$0.431390\pi$
$462$		0			0
$463$	−7.20218e6			−1.56139			−0.780695	−	0.624913i	$-0.785135\pi$
$463$	−7.20218e6			−1.56139			−0.780695	+	0.624913i	$0.785135\pi$
$464$		0			0
$465$	−599424.	+	1.03823e6i	−0.128559	+	0.222670i
$466$		0			0
$467$	3.58900e6	+	6.21634e6i	0.761521	+	1.31899i	0.942066	+	0.335426i	$0.108880\pi$
$467$	3.58900e6	+	6.21634e6i	0.761521	+	1.31899i	−0.180545	+	0.983567i	$0.557786\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−54532.0	−	94452.2i	−0.0113266	−	0.0196182i
$472$		0			0
$473$	−3.29760e6	+	5.71161e6i	−0.677712	+	1.17383i
$474$		0			0
$475$	572554.			0.116435
$476$		0			0
$477$	−242346.			−0.0487686
$478$		0			0
$479$	3.08816e6	−	5.34885e6i	0.614980	−	1.06518i	−0.375408	−	0.926859i	$-0.622498\pi$
$479$	3.08816e6	−	5.34885e6i	0.614980	−	1.06518i	0.990388	−	0.138316i	$-0.0441691\pi$
$480$		0			0
$481$	−3.08823e6	−	5.34897e6i	−0.608621	−	1.05416i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.14666e6	+	8.91427e6i	0.993507	+	1.72080i
$486$		0			0
$487$	−3.79665e6	+	6.57599e6i	−0.725401	+	1.25643i	0.233408	+	0.972379i	$0.425012\pi$
$487$	−3.79665e6	+	6.57599e6i	−0.725401	+	1.25643i	−0.958809	+	0.284052i	$0.908321\pi$
$488$		0			0
$489$	209920.			0.0396992
$490$		0			0
$491$	−1.51878e6			−0.284309			−0.142155	−	0.989844i	$-0.545403\pi$
$491$	−1.51878e6			−0.284309			−0.142155	+	0.989844i	$0.545403\pi$
$492$		0			0
$493$	−2.74250e6	+	4.75015e6i	−0.508193	+	0.880217i
$494$		0			0
$495$	−8.25984e6	−	1.43065e7i	−1.51516	−	2.62433i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−737878.	−	1.27804e6i	−0.132658	−	0.229770i	0.792042	−	0.610466i	$-0.209018\pi$
$499$	−737878.	−	1.27804e6i	−0.132658	−	0.229770i	−0.924700	+	0.380696i	$0.875685\pi$
$500$		0			0
$501$	160788.	−	278493.i	0.0286193	−	0.0495701i
$502$		0			0
$503$	1.31309e6			0.231406			0.115703	−	0.993284i	$-0.463088\pi$
$503$	1.31309e6			0.231406			0.115703	+	0.993284i	$0.463088\pi$
$504$		0			0
$505$	−4.52506e6			−0.789579
$506$		0			0
$507$	44109.0	−	76399.0i	0.00762092	−	0.0131998i
$508$		0			0
$509$	2.20466e6	+	3.81858e6i	0.377178	+	0.653292i	0.990651	−	0.136424i	$-0.0435610\pi$
$509$	2.20466e6	+	3.81858e6i	0.377178	+	0.653292i	−0.613472	+	0.789716i	$0.710228\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	45308.0	+	78475.8i	0.00760119	+	0.0131656i
$514$		0			0
$515$	5.86579e6	−	1.01598e7i	0.974560	−	1.68799i
$516$		0			0
$517$	−1.86019e7			−3.06078
$518$		0			0
$519$	−721128.			−0.117515
$520$		0			0
$521$	1.48815e6	−	2.57755e6i	0.240188	−	0.416018i	−0.720580	−	0.693372i	$-0.756124\pi$
$521$	1.48815e6	−	2.57755e6i	0.240188	−	0.416018i	0.960768	+	0.277354i	$0.0894575\pi$
$522$		0			0
$523$	3.17314e6	+	5.49603e6i	0.507265	+	0.878608i	0.999965	+	0.00840891i	$0.00267667\pi$
$523$	3.17314e6	+	5.49603e6i	0.507265	+	0.878608i	−0.492700	+	0.870199i	$0.663990\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.91499e6	+	6.78096e6i	0.614050	+	1.06357i
$528$		0			0
$529$	3.21356e6	−	5.56606e6i	0.499284	−	0.864785i
$530$		0			0
$531$	296838.			0.0456860
$532$		0			0
$533$	6.86743e6			1.04707
$534$		0			0
$535$	6.22253e6	−	1.07777e7i	0.939902	−	1.62796i
$536$		0			0
$537$	312732.	+	541668.i	0.0467990	+	0.0810583i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	683333.	+	1.18357e6i	0.100378	+	0.173860i	0.911840	−	0.410545i	$-0.134661\pi$
$541$	683333.	+	1.18357e6i	0.100378	+	0.173860i	−0.811462	+	0.584405i	$0.801328\pi$
$542$		0			0
$543$	−123820.	+	214463.i	−0.0180215	+	0.0312142i
$544$		0			0
$545$	−2.11736e7			−3.05353
$546$		0			0
$547$	−9.55818e6			−1.36586			−0.682931	−	0.730483i	$-0.739295\pi$
$547$	−9.55818e6			−1.36586			−0.682931	+	0.730483i	$0.739295\pi$
$548$		0			0
$549$	−907244.	+	1.57139e6i	−0.128467	+	0.222512i
$550$		0			0
$551$	205578.	+	356072.i	0.0288468	+	0.0499641i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	1.03661e6	+	1.79546e6i	0.142851	+	0.247425i
$556$		0			0
$557$	−3.47144e6	+	6.01270e6i	−0.474101	+	0.821167i	−0.999560	−	0.0296515i	$-0.990560\pi$
$557$	−3.47144e6	+	6.01270e6i	−0.474101	+	0.821167i	0.525459	+	0.850819i	$0.323894\pi$
$558$		0			0
$559$	5.23952e6			0.709189
$560$		0			0
$561$	1.80576e6			0.242244
$562$		0			0
$563$	2.62331e6	−	4.54371e6i	0.348802	−	0.604143i	−0.637235	−	0.770670i	$-0.719922\pi$
$563$	2.62331e6	−	4.54371e6i	0.348802	−	0.604143i	0.986037	+	0.166527i	$0.0532553\pi$
$564$		0			0
$565$	−8.19331e6	−	1.41912e7i	−1.07979	−	1.87025i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.73276e6	−	3.00122e6i	−0.224366	−	0.388613i	0.731763	−	0.681559i	$-0.238698\pi$
$569$	−1.73276e6	−	3.00122e6i	−0.224366	−	0.388613i	−0.956129	+	0.292946i	$0.905364\pi$
$570$		0			0
$571$	−2.45034e6	+	4.24412e6i	−0.314512	+	0.544750i	−0.979334	−	0.202252i	$-0.935174\pi$
$571$	−2.45034e6	+	4.24412e6i	−0.314512	+	0.544750i	0.664822	+	0.747002i	$0.268507\pi$
$572$		0			0
$573$	646896.			0.0823091
$574$		0			0
$575$	584736.			0.0737548
$576$		0			0
$577$	1.14173e6	−	1.97753e6i	0.142766	−	0.247277i	−0.785772	−	0.618517i	$-0.787734\pi$
$577$	1.14173e6	−	1.97753e6i	0.142766	−	0.247277i	0.928537	+	0.371240i	$0.121067\pi$
$578$		0			0
$579$	619954.	+	1.07379e6i	0.0768534	+	0.133114i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	365040.	+	632268.i	0.0444804	+	0.0770424i
$584$		0			0
$585$	−6.56198e6	+	1.13657e7i	−0.792767	+	1.37311i
$586$		0			0
$587$	−1.03157e7			−1.23568			−0.617838	−	0.786305i	$-0.711991\pi$
$587$	−1.03157e7			−1.23568			−0.617838	+	0.786305i	$0.711991\pi$
$588$		0			0
$589$	586936.			0.0697112
$590$		0			0
$591$	499362.	−	864920.i	0.0588094	−	0.101861i
$592$		0			0
$593$	1.76419e6	+	3.05566e6i	0.206020	+	0.356836i	0.950457	−	0.310856i	$-0.100616\pi$
$593$	1.76419e6	+	3.05566e6i	0.206020	+	0.356836i	−0.744438	+	0.667692i	$0.767282\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−785932.	−	1.36127e6i	−0.0902504	−	0.156318i
$598$		0			0
$599$	−1.46130e6	+	2.53105e6i	−0.166407	+	0.288226i	−0.937154	−	0.348916i	$-0.886550\pi$
$599$	−1.46130e6	+	2.53105e6i	−0.166407	+	0.288226i	0.770747	+	0.637141i	$0.219883\pi$
$600$		0			0
$601$	1.17567e7			1.32770			0.663849	−	0.747866i	$-0.268922\pi$
$601$	1.17567e7			1.32770			0.663849	+	0.747866i	$0.268922\pi$
$602$		0			0
$603$	−9.83055e6			−1.10099
$604$		0			0
$605$	−1.71528e7	+	2.97094e7i	−1.90522	+	3.29994i
$606$		0			0
$607$	−2.35746e6	−	4.08323e6i	−0.259700	−	0.449814i	0.706462	−	0.707751i	$-0.250290\pi$
$607$	−2.35746e6	−	4.08323e6i	−0.259700	−	0.449814i	−0.966161	+	0.257938i	$0.916957\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.38910e6	+	1.27983e7i	0.800734	+	1.38691i
$612$		0			0
$613$	−106921.	+	185193.i	−0.0114924	+	0.0199055i	−0.871714	−	0.490014i	$-0.836992\pi$
$613$	−106921.	+	185193.i	−0.0114924	+	0.0199055i	0.860222	+	0.509920i	$0.170325\pi$
$614$		0			0
$615$	−2.30515e6			−0.245760
$616$		0			0
$617$	−336666.			−0.0356030			−0.0178015	−	0.999842i	$-0.505667\pi$
$617$	−336666.			−0.0356030			−0.0178015	+	0.999842i	$0.505667\pi$
$618$		0			0
$619$	7.13276e6	−	1.23543e7i	0.748223	−	1.29596i	−0.200450	−	0.979704i	$-0.564240\pi$
$619$	7.13276e6	−	1.23543e7i	0.748223	−	1.29596i	0.948674	−	0.316257i	$-0.102426\pi$
$620$		0			0
$621$	46272.0	+	80145.5i	0.00481492	+	0.00833969i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−4.15014e6	−	7.18825e6i	−0.424974	−	0.736077i
$626$		0			0
$627$	67680.0	−	117225.i	0.00687530	−	0.0119084i
$628$		0			0
$629$	1.35407e7			1.36463
$630$		0			0
$631$	−6.59637e6			−0.659525			−0.329763	−	0.944064i	$-0.606969\pi$
$631$	−6.59637e6			−0.659525			−0.329763	+	0.944064i	$0.606969\pi$
$632$		0			0
$633$	−1.06276e6	+	1.84076e6i	−0.105421	+	0.182595i
$634$		0			0
$635$	1.19908e7	+	2.07686e7i	1.18008	+	2.04397i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−4.49702e6	−	7.78907e6i	−0.435685	−	0.754629i
$640$		0			0
$641$	5.12449e6	−	8.87588e6i	0.492613	−	0.853230i	−0.507351	−	0.861739i	$-0.669375\pi$
$641$	5.12449e6	−	8.87588e6i	0.492613	−	0.853230i	0.999964	+	0.00850936i	$0.00270865\pi$
$642$		0			0
$643$	4.16543e6			0.397312			0.198656	−	0.980069i	$-0.436342\pi$
$643$	4.16543e6			0.397312			0.198656	+	0.980069i	$0.436342\pi$
$644$		0			0
$645$	−1.75872e6			−0.166455
$646$		0			0
$647$	−1.67525e6	+	2.90163e6i	−0.157333	+	0.272509i	−0.933906	−	0.357518i	$-0.883623\pi$
$647$	−1.67525e6	+	2.90163e6i	−0.157333	+	0.272509i	0.776573	+	0.630027i	$0.216956\pi$
$648$		0			0
$649$	−447120.	−	774435.i	−0.0416689	−	0.0721727i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.52704e6	+	7.84106e6i	0.415462	+	0.719601i	0.995477	−	0.0950046i	$-0.0302866\pi$
$653$	4.52704e6	+	7.84106e6i	0.415462	+	0.719601i	−0.580015	+	0.814606i	$0.696953\pi$
$654$		0			0
$655$	586080.	−	1.01512e6i	0.0533769	−	0.0924516i
$656$		0			0
$657$	−3.21168e6			−0.290281
$658$		0			0
$659$	6.45382e6			0.578899			0.289450	−	0.957193i	$-0.406528\pi$
$659$	6.45382e6			0.578899			0.289450	+	0.957193i	$0.406528\pi$
$660$		0			0
$661$	7.15836e6	−	1.23987e7i	0.637250	−	1.10375i	−0.348783	−	0.937203i	$-0.613405\pi$
$661$	7.15836e6	−	1.23987e7i	0.637250	−	1.10375i	0.986034	−	0.166547i	$-0.0532616\pi$
$662$		0			0
$663$	−717288.	−	1.24238e6i	−0.0633738	−	0.109767i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	209952.	+	363648.i	0.0182728	+	0.0316494i
$668$		0			0
$669$	707720.	−	1.22581e6i	0.0611359	−	0.105890i
$670$		0			0
$671$	5.46624e6			0.468686
$672$		0			0
$673$	2.27250e7			1.93404			0.967020	−	0.254701i	$-0.0819771\pi$
$673$	2.27250e7			1.93404			0.967020	+	0.254701i	$0.0819771\pi$
$674$		0			0
$675$	2.93586e6	−	5.08506e6i	0.248014	−	0.429573i
$676$		0			0
$677$	−4.26625e6	−	7.38935e6i	−0.357746	−	0.619633i	0.629838	−	0.776726i	$-0.283121\pi$
$677$	−4.26625e6	−	7.38935e6i	−0.357746	−	0.619633i	−0.987584	+	0.157093i	$0.949788\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−1.04437e6	−	1.80889e6i	−0.0862949	−	0.149467i
$682$		0			0
$683$	1.12460e7	−	1.94787e7i	0.922461	−	1.59775i	0.126867	−	0.991920i	$-0.459508\pi$
$683$	1.12460e7	−	1.94787e7i	0.922461	−	1.59775i	0.795594	−	0.605830i	$-0.207159\pi$
$684$		0			0
$685$	−1.33459e6			−0.108673
$686$		0			0
$687$	1.07943e6			0.0872576
$688$		0			0
$689$	290004.	−	502302.i	0.0232732	−	0.0403103i
$690$		0			0
$691$	−6.34471e6	−	1.09894e7i	−0.505495	−	0.875543i	−0.999980	−	0.00635656i	$-0.997977\pi$
$691$	−6.34471e6	−	1.09894e7i	−0.505495	−	0.875543i	0.494485	−	0.869186i	$-0.335357\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.07261e7	−	3.58987e7i	−1.62763	−	2.81914i
$696$		0			0
$697$	−7.52776e6	+	1.30385e7i	−0.586927	+	1.01659i
$698$		0			0
$699$	354228.			0.0274214
$700$		0			0
$701$	−5.13939e6			−0.395018			−0.197509	−	0.980301i	$-0.563285\pi$
$701$	−5.13939e6			−0.395018			−0.197509	+	0.980301i	$0.563285\pi$
$702$		0			0
$703$	507506.	−	879026.i	0.0387305	−	0.0670832i
$704$		0			0
$705$	−2.48026e6	−	4.29593e6i	−0.187942	−	0.325525i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	5.80325e6	+	1.00515e7i	0.433566	+	0.750959i	0.997177	−	0.0750815i	$-0.0239217\pi$
$709$	5.80325e6	+	1.00515e7i	0.433566	+	0.750959i	−0.563611	+	0.826040i	$0.690588\pi$
$710$		0			0
$711$	746636.	−	1.29321e6i	0.0553905	−	0.0959391i
$712$		0			0
$713$	599424.			0.0441581
$714$		0			0
$715$	3.95366e7			2.89224
$716$		0			0
$717$	655464.	−	1.13530e6i	0.0476158	−	0.0824730i
$718$		0			0
$719$	7.54988e6	+	1.30768e7i	0.544650	+	0.943362i	0.998629	+	0.0523493i	$0.0166709\pi$
$719$	7.54988e6	+	1.30768e7i	0.544650	+	0.943362i	−0.453979	+	0.891013i	$0.649996\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	1.38709e6	+	2.40252e6i	0.0986870	+	0.170931i
$724$		0			0
$725$	1.33210e7	−	2.30727e7i	0.941223	−	1.63025i
$726$		0			0
$727$	2.32536e7			1.63175			0.815874	−	0.578229i	$-0.196256\pi$
$727$	2.32536e7			1.63175			0.815874	+	0.578229i	$0.196256\pi$
$728$		0			0
$729$	−1.29511e7			−0.902585
$730$		0			0
$731$	−5.74332e6	+	9.94772e6i	−0.397530	+	0.688542i
$732$		0			0
$733$	1.18907e7	+	2.05953e7i	0.817423	+	1.41582i	0.907575	+	0.419890i	$0.137931\pi$
$733$	1.18907e7	+	2.05953e7i	0.817423	+	1.41582i	−0.0901518	+	0.995928i	$0.528735\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.48075e7	+	2.56474e7i	1.00418	+	1.73930i
$738$		0			0
$739$	1.25696e6	−	2.17712e6i	0.0846663	−	0.146646i	−0.820583	−	0.571528i	$-0.806351\pi$
$739$	1.25696e6	−	2.17712e6i	0.0846663	−	0.146646i	0.905249	+	0.424882i	$0.139684\pi$
$740$		0			0
$741$	−107536.			−0.00719463
$742$		0			0
$743$	−2.22646e7			−1.47959			−0.739797	−	0.672830i	$-0.765078\pi$
$743$	−2.22646e7			−1.47959			−0.739797	+	0.672830i	$0.765078\pi$
$744$		0			0
$745$	−1.56871e7	+	2.71708e7i	−1.03550	+	1.79354i
$746$		0			0
$747$	3.01785e6	+	5.22707e6i	0.197878	+	0.342734i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.15054e7	−	1.99279e7i	−0.744392	−	1.28932i	−0.950479	−	0.310790i	$-0.899406\pi$
$751$	−1.15054e7	−	1.99279e7i	−0.744392	−	1.28932i	0.206087	−	0.978534i	$-0.433927\pi$
$752$		0			0
$753$	1.88811e6	−	3.27030e6i	0.121350	−	0.210184i
$754$		0			0
$755$	1.66541e7			1.06329
$756$		0			0
$757$	1.59335e7			1.01058			0.505290	−	0.862950i	$-0.331386\pi$
$757$	1.59335e7			1.01058			0.505290	+	0.862950i	$0.331386\pi$
$758$		0			0
$759$	69120.0	−	119719.i	0.00435511	−	0.00754327i
$760$		0			0
$761$	8.43143e6	+	1.46037e7i	0.527764	+	0.914114i	0.999476	+	0.0323613i	$0.0103027\pi$
$761$	8.43143e6	+	1.46037e7i	0.527764	+	0.914114i	−0.471712	+	0.881752i	$0.656364\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1.43859e7	−	2.49171e7i	−0.888757	−	1.53937i
$766$		0			0
$767$	−355212.	+	615245.i	−0.0218021	+	0.0377624i
$768$		0			0
$769$	2.75402e7			1.67939			0.839694	−	0.543060i	$-0.182735\pi$
$769$	2.75402e7			1.67939			0.839694	+	0.543060i	$0.182735\pi$
$770$		0			0
$771$	−692388.			−0.0419482
$772$		0			0
$773$	1.13410e7	−	1.96432e7i	0.682658	−	1.18240i	−0.291508	−	0.956568i	$-0.594157\pi$
$773$	1.13410e7	−	1.96432e7i	0.682658	−	1.18240i	0.974167	−	0.225831i	$-0.0725096\pi$
$774$		0			0
$775$	−1.90161e7	−	3.29369e7i	−1.13728	−	1.96983i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	564282.	+	977365.i	0.0333160	+	0.0577050i
$780$		0			0
$781$	−1.35475e7	+	2.34650e7i	−0.794753	+	1.37655i
$782$		0			0
$783$	4.21654e6			0.245783
$784$		0			0
$785$	5.23507e6

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:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	196.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{3} +(-48.0000 - 83.1384i) q^{5} +(119.500 + 206.980i) q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{3} +(-48.0000 - 83.1384i) q^{5} +(119.500 + 206.980i) q^{9} +(360.000 - 623.538i) q^{11} -572.000 q^{13} +192.000 q^{15} +(627.000 - 1086.00i) q^{17} +(-47.0000 - 81.4064i) q^{19} +(-48.0000 - 83.1384i) q^{23} +(-3045.50 + 5274.96i) q^{25} -964.000 q^{27} -4374.00 q^{29} +(-3122.00 + 5407.46i) q^{31} +(720.000 + 1247.08i) q^{33} +(5399.00 + 9351.34i) q^{37} +(572.000 - 990.733i) q^{39} -12006.0 q^{41} -9160.00 q^{43} +(11472.0 - 19870.1i) q^{45} +(-12918.0 - 22374.6i) q^{47} +(1254.00 + 2171.99i) q^{51} +(-507.000 + 878.150i) q^{53} -69120.0 q^{55} +188.000 q^{57} +(621.000 - 1075.60i) q^{59} +(3796.00 + 6574.86i) q^{61} +(27456.0 + 47555.2i) q^{65} +(-20566.0 + 35621.4i) q^{67} +192.000 q^{69} -37632.0 q^{71} +(-6719.00 + 11637.6i) q^{73} +(-6091.00 - 10549.9i) q^{75} +(-3124.00 - 5410.93i) q^{79} +(-28074.5 + 48626.5i) q^{81} +25254.0 q^{83} -120384. q^{85} +(4374.00 - 7575.99i) q^{87} +(-22563.0 - 39080.3i) q^{89} +(-6244.00 - 10814.9i) q^{93} +(-4512.00 + 7815.01i) q^{95} -107222. q^{97} +172080. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 96 q^{5} + 239 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 96 * q^5 + 239 * q^9	\( 2 q - 2 q^{3} - 96 q^{5} + 239 q^{9} + 720 q^{11} - 1144 q^{13} + 384 q^{15} + 1254 q^{17} - 94 q^{19} - 96 q^{23} - 6091 q^{25} - 1928 q^{27} - 8748 q^{29} - 6244 q^{31} + 1440 q^{33} + 10798 q^{37} + 1144 q^{39} - 24012 q^{41} - 18320 q^{43} + 22944 q^{45} - 25836 q^{47} + 2508 q^{51} - 1014 q^{53} - 138240 q^{55} + 376 q^{57} + 1242 q^{59} + 7592 q^{61} + 54912 q^{65} - 41132 q^{67} + 384 q^{69} - 75264 q^{71} - 13438 q^{73} - 12182 q^{75} - 6248 q^{79} - 56149 q^{81} + 50508 q^{83} - 240768 q^{85} + 8748 q^{87} - 45126 q^{89} - 12488 q^{93} - 9024 q^{95} - 214444 q^{97} + 344160 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 96 * q^5 + 239 * q^9 + 720 * q^11 - 1144 * q^13 + 384 * q^15 + 1254 * q^17 - 94 * q^19 - 96 * q^23 - 6091 * q^25 - 1928 * q^27 - 8748 * q^29 - 6244 * q^31 + 1440 * q^33 + 10798 * q^37 + 1144 * q^39 - 24012 * q^41 - 18320 * q^43 + 22944 * q^45 - 25836 * q^47 + 2508 * q^51 - 1014 * q^53 - 138240 * q^55 + 376 * q^57 + 1242 * q^59 + 7592 * q^61 + 54912 * q^65 - 41132 * q^67 + 384 * q^69 - 75264 * q^71 - 13438 * q^73 - 12182 * q^75 - 6248 * q^79 - 56149 * q^81 + 50508 * q^83 - 240768 * q^85 + 8748 * q^87 - 45126 * q^89 - 12488 * q^93 - 9024 * q^95 - 214444 * q^97 + 344160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more