LMFDB - Embedded newform 196.6.e.f.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.f.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	−0.832167	−	0.554526i	$-0.812900\pi$
$3$	1.00000	−	1.73205i	0.0641500	−	0.111111i	0.896317	+	0.443415i	$0.146233\pi$
$4$		0			0
$5$	48.0000	+	83.1384i	0.858650	+	1.48723i	0.873217	+	0.487332i	$0.162030\pi$
$5$	48.0000	+	83.1384i	0.858650	+	1.48723i	−0.0145668	+	0.999894i	$0.504637\pi$
$6$		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	$-0.344484\pi$
$13$	572.000			0.938723			0.469362	+	0.883006i	$0.344484\pi$
$14$		0			0
$15$	192.000			0.220330
$16$		0			0
$17$	−627.000	+	1086.00i	−0.526193	+	0.911393i	0.473341	+	0.880879i	$0.343048\pi$
$17$	−627.000	+	1086.00i	−0.526193	+	0.911393i	−0.999534	+	0.0305142i	$0.990286\pi$
$18$		0			0
$19$	47.0000	+	81.4064i	0.0298685	+	0.0517338i	0.880573	−	0.473910i	$-0.157158\pi$
$19$	47.0000	+	81.4064i	0.0298685	+	0.0517338i	−0.850705	+	0.525644i	$0.823824\pi$
$20$		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	$-0.635023\pi$
$31$	3122.00	−	5407.46i	0.583484	−	1.01062i	0.995063	−	0.0992492i	$-0.0316441\pi$
$32$		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	$-0.311680\pi$
$41$	12006.0			1.11542			0.557710	+	0.830036i	$0.311680\pi$
$42$		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	$0.158555\pi$
$47$	12918.0	+	22374.6i	0.853003	+	1.47744i	−0.0254821	+	0.999675i	$0.508112\pi$
$48$		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	−0.877404	−	0.479751i	$-0.840727\pi$
$59$	−621.000	+	1075.60i	−0.0232253	+	0.0402274i	0.854179	+	0.519979i	$0.174060\pi$
$60$		0			0
$61$	−3796.00	−	6574.86i	−0.130618	−	0.226236i	0.793297	−	0.608835i	$-0.208363\pi$
$61$	−3796.00	−	6574.86i	−0.130618	−	0.226236i	−0.923915	+	0.382598i	$0.875029\pi$
$62$		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	−0.782759	−	0.622325i	$-0.786188\pi$
$73$	6719.00	−	11637.6i	0.147570	−	0.255598i	0.930329	+	0.366727i	$0.119522\pi$
$74$		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	$-0.564481\pi$
$83$	−25254.0			−0.402379			−0.201189	+	0.979552i	$0.564481\pi$
$84$		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	0.976576	−	0.215175i	$-0.0690322\pi$
$89$	22563.0	+	39080.3i	0.301941	+	0.522977i	−0.674635	+	0.738152i	$0.735699\pi$
$90$		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	$-0.303627\pi$
$97$	107222.			1.15706			0.578528	+	0.815662i	$0.303627\pi$
$98$		0			0
$99$	172080.			1.76458
$100$		0			0
$101$	−23568.0	+	40821.0i	−0.229890	+	0.398180i	−0.957775	−	0.287518i	$-0.907170\pi$
$101$	−23568.0	+	40821.0i	−0.229890	+	0.398180i	0.727886	+	0.685699i	$0.240503\pi$
$102$		0			0
$103$	−61102.0	−	105832.i	−0.567495	−	0.982931i	−0.996813	−	0.0797772i	$-0.974579\pi$
$103$	−61102.0	−	105832.i	−0.567495	−	0.982931i	0.429317	−	0.903154i	$-0.358754\pi$
$104$		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	0.850066	−	0.526676i	$-0.176562\pi$
$131$	−6105.00	−	10574.2i	−0.0310819	−	0.0538354i	−0.881148	+	0.472841i	$0.843229\pi$
$132$		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	$-0.896683\pi$
$139$	−431794.			−1.89557			−0.947785	+	0.318911i	$0.896683\pi$
$140$		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	−0.818503	−	0.574502i	$-0.805196\pi$
$157$	27266.0	−	47226.1i	0.0882820	−	0.152909i	0.906785	+	0.421593i	$0.138529\pi$
$158$		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	$-0.428394\pi$
$167$	160788.			0.446131			0.223066	+	0.974803i	$0.428394\pi$
$168$		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	0.540883	−	0.841098i	$-0.318090\pi$
$173$	−180282.	−	312258.i	−0.457970	−	0.793227i	−0.998854	+	0.0478701i	$0.984757\pi$
$174$		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	$-0.544859\pi$
$181$	−123820.			−0.280928			−0.140464	+	0.990086i	$0.544859\pi$
$182$		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	$-0.584983\pi$
$199$	392966.	−	680637.i	0.703432	−	1.21838i	0.967254	−	0.253809i	$-0.0816835\pi$
$200$		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	$-0.341903\pi$
$223$	707720.			0.953014			0.476507	+	0.879171i	$0.341903\pi$
$224$		0			0
$225$	−1.45575e6			−1.91704
$226$		0			0
$227$	522183.	−	904447.i	0.672602	−	1.16498i	−0.304562	−	0.952493i	$-0.598510\pi$
$227$	522183.	−	904447.i	0.672602	−	1.16498i	0.977164	−	0.212488i	$-0.0681567\pi$
$228$		0			0
$229$	269858.	+	467408.i	0.340053	+	0.588989i	0.984442	−	0.175709i	$-0.0562217\pi$
$229$	269858.	+	467408.i	0.340053	+	0.588989i	−0.644389	+	0.764698i	$0.722888\pi$
$230$		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	$0.446006\pi$
$241$	−693547.	+	1.20126e6i	−0.769189	+	1.33228i	−0.938003	+	0.346627i	$0.887327\pi$
$242$		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	$-0.105251\pi$
$251$	1.88811e6			1.89166			0.945830	+	0.324663i	$0.105251\pi$
$252$		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	0.772636	−	0.634849i	$-0.218938\pi$
$257$	−173097.	−	299813.i	−0.163477	−	0.283151i	−0.936113	+	0.351698i	$0.885604\pi$
$258$		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	−0.935215	−	0.354081i	$-0.884794\pi$
$269$	−191034.	+	330881.i	−0.160964	+	0.278799i	0.774250	+	0.632879i	$0.218127\pi$
$270$		0			0
$271$	790280.	+	1.36881e6i	0.653669	+	1.13219i	0.982226	+	0.187703i	$0.0601044\pi$
$271$	790280.	+	1.36881e6i	0.653669	+	1.13219i	−0.328557	+	0.944484i	$0.606562\pi$
$272$		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	−0.963462	−	0.267845i	$-0.913688\pi$
$283$	−336517.	+	582865.i	−0.249770	+	0.432615i	0.713692	+	0.700460i	$0.247022\pi$
$284$		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	$-0.285403\pi$
$293$	1.83468e6			1.24851			0.624254	+	0.781222i	$0.285403\pi$
$294$		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	$-0.651206\pi$
$307$	−1.51056e6			−0.914727			−0.457363	+	0.889280i	$0.651206\pi$
$308$		0			0
$309$	−244408.			−0.145619
$310$		0			0
$311$	937644.	−	1.62405e6i	0.549714	−	0.952133i	−0.448580	−	0.893743i	$-0.648070\pi$
$311$	937644.	−	1.62405e6i	0.549714	−	0.952133i	0.998294	−	0.0583901i	$-0.0185967\pi$
$312$		0			0
$313$	755381.	+	1.30836e6i	0.435818	+	0.754859i	0.997362	−	0.0725879i	$-0.0231258\pi$
$313$	755381.	+	1.30836e6i	0.435818	+	0.754859i	−0.561544	+	0.827447i	$0.689792\pi$
$314$		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	$-0.581895\pi$
$349$	−1.15798e6			−0.508906			−0.254453	+	0.967085i	$0.581895\pi$
$350$		0			0
$351$	551408.			0.238894
$352$		0			0
$353$	1.58783e6	−	2.75020e6i	0.678215	−	1.17470i	−0.297303	−	0.954783i	$-0.596087\pi$
$353$	1.58783e6	−	2.75020e6i	0.678215	−	1.17470i	0.975518	−	0.219920i	$-0.0705796\pi$
$354$		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	−0.637543	−	0.770415i	$-0.720049\pi$
$367$	899036.	−	1.55718e6i	0.348427	−	0.603493i	0.985970	+	0.166921i	$0.0533826\pi$
$368$		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	$0.000251238\pi$
$383$	1.43734e6	+	2.48955e6i	0.500683	+	0.867209i	−0.499316	+	0.866420i	$0.666415\pi$
$384$		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	0.998284	−	0.0585598i	$-0.0186508\pi$
$397$	1.40821e6	+	2.43910e6i	0.448428	+	0.776699i	−0.549856	+	0.835259i	$0.685318\pi$
$398$		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	−0.877202	−	0.480122i	$-0.840592\pi$
$409$	−77143.0	+	133616.i	−0.0228028	+	0.0394956i	0.854399	+	0.519618i	$0.173926\pi$
$410$		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	$-0.326385\pi$
$419$	3.72865e6			1.03757			0.518783	+	0.854906i	$0.326385\pi$
$420$		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	$-0.548829\pi$
$433$	−1.19226e6			−0.305598			−0.152799	+	0.988257i	$0.548829\pi$
$434$		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	0.793064	−	0.609139i	$-0.208485\pi$
$439$	−528964.	−	916193.i	−0.130998	−	0.226895i	−0.924062	+	0.382244i	$0.875151\pi$
$440$		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	$-0.568610\pi$
$461$	−1.95186e6			−0.427756			−0.213878	+	0.976860i	$0.568610\pi$
$462$		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	$-0.891120\pi$
$467$	−3.58900e6	−	6.21634e6i	−0.761521	−	1.31899i	0.180545	−	0.983567i	$-0.442214\pi$
$468$		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	$0.377502\pi$
$479$	−3.08816e6	+	5.34885e6i	−0.614980	+	1.06518i	−0.990388	+	0.138316i	$0.955831\pi$
$480$		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	$-0.536912\pi$
$503$	−1.31309e6			−0.231406			−0.115703	+	0.993284i	$0.536912\pi$
$504$		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	0.613472	−	0.789716i	$-0.289772\pi$
$509$	−2.20466e6	−	3.81858e6i	−0.377178	−	0.653292i	−0.990651	+	0.136424i	$0.956439\pi$
$510$		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	−0.960768	−	0.277354i	$-0.910542\pi$
$521$	−1.48815e6	+	2.57755e6i	−0.240188	+	0.416018i	0.720580	+	0.693372i	$0.243876\pi$
$522$		0			0
$523$	−3.17314e6	−	5.49603e6i	−0.507265	−	0.878608i	−0.999965	−	0.00840891i	$-0.997323\pi$
$523$	−3.17314e6	−	5.49603e6i	−0.507265	−	0.878608i	0.492700	−	0.870199i	$-0.336010\pi$
$524$		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	−0.986037	−	0.166527i	$-0.946745\pi$
$563$	−2.62331e6	+	4.54371e6i	−0.348802	+	0.604143i	0.637235	+	0.770670i	$0.280078\pi$
$564$		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	−0.928537	−	0.371240i	$-0.878933\pi$
$577$	−1.14173e6	+	1.97753e6i	−0.142766	+	0.247277i	0.785772	+	0.618517i	$0.212266\pi$
$578$		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	$-0.288009\pi$
$587$	1.03157e7			1.23568			0.617838	+	0.786305i	$0.288009\pi$
$588$		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	0.744438	−	0.667692i	$-0.232718\pi$
$593$	−1.76419e6	−	3.05566e6i	−0.206020	−	0.356836i	−0.950457	+	0.310856i	$0.899384\pi$
$594$		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	$-0.731078\pi$
$601$	−1.17567e7			−1.32770			−0.663849	+	0.747866i	$0.731078\pi$
$602$		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	0.966161	−	0.257938i	$-0.0830430\pi$
$607$	2.35746e6	+	4.08323e6i	0.259700	+	0.449814i	−0.706462	+	0.707751i	$0.749710\pi$
$608$		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	$0.435760\pi$
$619$	−7.13276e6	+	1.23543e7i	−0.748223	+	1.29596i	−0.948674	+	0.316257i	$0.897574\pi$
$620$		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	$-0.563658\pi$
$643$	−4.16543e6			−0.397312			−0.198656	+	0.980069i	$0.563658\pi$
$644$		0			0
$645$	−1.75872e6			−0.166455
$646$		0			0
$647$	1.67525e6	−	2.90163e6i	0.157333	−	0.272509i	−0.776573	−	0.630027i	$-0.783044\pi$
$647$	1.67525e6	−	2.90163e6i	0.157333	−	0.272509i	0.933906	+	0.357518i	$0.116377\pi$
$648$		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	$0.386595\pi$
$661$	−7.15836e6	+	1.23987e7i	−0.637250	+	1.10375i	−0.986034	+	0.166547i	$0.946738\pi$
$662$		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	0.987584	−	0.157093i	$-0.0502122\pi$
$677$	4.26625e6	+	7.38935e6i	0.357746	+	0.619633i	−0.629838	+	0.776726i	$0.716879\pi$
$678$		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	$0.00202337\pi$
$691$	6.34471e6	+	1.09894e7i	0.505495	+	0.875543i	−0.494485	+	0.869186i	$0.664643\pi$
$692$		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	$-0.983329\pi$
$719$	−7.54988e6	−	1.30768e7i	−0.544650	−	0.943362i	0.453979	−	0.891013i	$-0.350004\pi$
$720$		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	$-0.803744\pi$
$727$	−2.32536e7			−1.63175			−0.815874	+	0.578229i	$0.803744\pi$
$728$		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	$-0.862069\pi$
$733$	−1.18907e7	−	2.05953e7i	−0.817423	−	1.41582i	0.0901518	−	0.995928i	$-0.471265\pi$
$734$		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	$-0.989697\pi$
$761$	−8.43143e6	−	1.46037e7i	−0.527764	−	0.914114i	0.471712	−	0.881752i	$-0.343636\pi$
$762$		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	$-0.817265\pi$
$769$	−2.75402e7			−1.67939			−0.839694	+	0.543060i	$0.817265\pi$
$770$		0			0
$771$	−692388.			−0.0419482
$772$		0			0
$773$	−1.13410e7	+	1.96432e7i	−0.682658	+	1.18240i	0.291508	+	0.956568i	$0.405843\pi$
$773$	−1.13410e7	+	1.96432e7i	−0.682658	+	1.18240i	−0.974167	+	0.225831i	$0.927490\pi$
$774$		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			0.303213

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

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