LMFDB - Embedded newform 196.6.e.a.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.a.177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-13.0000 + 22.5167i) q^{3} +(-8.00000 - 13.8564i) q^{5} +(-216.500 - 374.989i) q^{9} +O(q^{10})$	\(q+(-13.0000 + 22.5167i) q^{3} +(-8.00000 - 13.8564i) q^{5} +(-216.500 - 374.989i) q^{9} +(-4.00000 + 6.92820i) q^{11} +684.000 q^{13} +416.000 q^{15} +(1109.00 - 1920.84i) q^{17} +(1349.00 + 2336.54i) q^{19} +(-1672.00 - 2895.99i) q^{23} +(1434.50 - 2484.63i) q^{25} +4940.00 q^{27} -3254.00 q^{29} +(-2394.00 + 4146.53i) q^{31} +(-104.000 - 180.133i) q^{33} +(5735.00 + 9933.31i) q^{37} +(-8892.00 + 15401.4i) q^{39} +13350.0 q^{41} -928.000 q^{43} +(-3464.00 + 5999.82i) q^{45} +(-606.000 - 1049.62i) q^{47} +(28834.0 + 49942.0i) q^{51} +(-6555.00 + 11353.6i) q^{53} +128.000 q^{55} -70148.0 q^{57} +(-17351.0 + 30052.8i) q^{59} +(516.000 + 893.738i) q^{61} +(-5472.00 - 9477.78i) q^{65} +(-5054.00 + 8753.78i) q^{67} +86944.0 q^{69} +62720.0 q^{71} +(9463.00 - 16390.4i) q^{73} +(37297.0 + 64600.3i) q^{75} +(-5700.00 - 9872.69i) q^{79} +(-11610.5 + 20110.0i) q^{81} +88958.0 q^{83} -35488.0 q^{85} +(42302.0 - 73269.2i) q^{87} +(-9861.00 - 17079.8i) q^{89} +(-62244.0 - 107810. i) q^{93} +(21584.0 - 37384.6i) q^{95} +17062.0 q^{97} +3464.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 26 q^{3} - 16 q^{5} - 433 q^{9}+O(q^{10}) $ 2 * q - 26 * q^3 - 16 * q^5 - 433 * q^9	$ 2 q - 26 q^{3} - 16 q^{5} - 433 q^{9} - 8 q^{11} + 1368 q^{13} + 832 q^{15} + 2218 q^{17} + 2698 q^{19} - 3344 q^{23} + 2869 q^{25} + 9880 q^{27} - 6508 q^{29} - 4788 q^{31} - 208 q^{33} + 11470 q^{37} - 17784 q^{39} + 26700 q^{41} - 1856 q^{43} - 6928 q^{45} - 1212 q^{47} + 57668 q^{51} - 13110 q^{53} + 256 q^{55} - 140296 q^{57} - 34702 q^{59} + 1032 q^{61} - 10944 q^{65} - 10108 q^{67} + 173888 q^{69} + 125440 q^{71} + 18926 q^{73} + 74594 q^{75} - 11400 q^{79} - 23221 q^{81} + 177916 q^{83} - 70976 q^{85} + 84604 q^{87} - 19722 q^{89} - 124488 q^{93} + 43168 q^{95} + 34124 q^{97} + 6928 q^{99}+O(q^{100}) $ 2 * q - 26 * q^3 - 16 * q^5 - 433 * q^9 - 8 * q^11 + 1368 * q^13 + 832 * q^15 + 2218 * q^17 + 2698 * q^19 - 3344 * q^23 + 2869 * q^25 + 9880 * q^27 - 6508 * q^29 - 4788 * q^31 - 208 * q^33 + 11470 * q^37 - 17784 * q^39 + 26700 * q^41 - 1856 * q^43 - 6928 * q^45 - 1212 * q^47 + 57668 * q^51 - 13110 * q^53 + 256 * q^55 - 140296 * q^57 - 34702 * q^59 + 1032 * q^61 - 10944 * q^65 - 10108 * q^67 + 173888 * q^69 + 125440 * q^71 + 18926 * q^73 + 74594 * q^75 - 11400 * q^79 - 23221 * q^81 + 177916 * q^83 - 70976 * q^85 + 84604 * q^87 - 19722 * q^89 - 124488 * q^93 + 43168 * q^95 + 34124 * q^97 + 6928 * 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
$3$	−13.0000	+	22.5167i	−0.833950	+	1.44444i	0.0609318	+	0.998142i	$0.480593\pi$
$3$	−13.0000	+	22.5167i	−0.833950	+	1.44444i	−0.894882	+	0.446303i	$0.852741\pi$
$4$		0			0
$5$	−8.00000	−	13.8564i	−0.143108	−	0.247871i	0.785557	−	0.618789i	$-0.212376\pi$
$5$	−8.00000	−	13.8564i	−0.143108	−	0.247871i	−0.928666	+	0.370918i	$0.879043\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−216.500	−	374.989i	−0.890947	−	1.54316i
$10$		0			0
$11$	−4.00000	+	6.92820i	−0.00996732	+	0.0172639i	−0.870966	−	0.491343i	$-0.836506\pi$
$11$	−4.00000	+	6.92820i	−0.00996732	+	0.0172639i	0.860999	+	0.508607i	$0.169839\pi$
$12$		0			0
$13$	684.000			1.12253			0.561265	−	0.827636i	$-0.310315\pi$
$13$	684.000			1.12253			0.561265	+	0.827636i	$0.310315\pi$
$14$		0			0
$15$	416.000			0.477381
$16$		0			0
$17$	1109.00	−	1920.84i	0.930699	−	1.61202i	0.148570	−	0.988902i	$-0.452533\pi$
$17$	1109.00	−	1920.84i	0.930699	−	1.61202i	0.782129	−	0.623116i	$-0.214134\pi$
$18$		0			0
$19$	1349.00	+	2336.54i	0.857290	+	1.48487i	0.874504	+	0.485019i	$0.161187\pi$
$19$	1349.00	+	2336.54i	0.857290	+	1.48487i	−0.0172134	+	0.999852i	$0.505479\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1672.00	−	2895.99i	−0.659047	−	1.14150i	−0.980863	−	0.194701i	$-0.937626\pi$
$23$	−1672.00	−	2895.99i	−0.659047	−	1.14150i	0.321815	−	0.946803i	$-0.395707\pi$
$24$		0			0
$25$	1434.50	−	2484.63i	0.459040	−	0.795081i
$26$		0			0
$27$	4940.00			1.30412
$28$		0			0
$29$	−3254.00			−0.718493			−0.359247	−	0.933243i	$-0.616966\pi$
$29$	−3254.00			−0.718493			−0.359247	+	0.933243i	$0.616966\pi$
$30$		0			0
$31$	−2394.00	+	4146.53i	−0.447425	+	0.774962i	−0.998218	−	0.0596797i	$-0.980992\pi$
$31$	−2394.00	+	4146.53i	−0.447425	+	0.774962i	0.550793	+	0.834642i	$0.314325\pi$
$32$		0			0
$33$	−104.000	−	180.133i	−0.0166245	−	0.0287945i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5735.00	+	9933.31i	0.688698	+	1.19286i	0.972259	+	0.233906i	$0.0751509\pi$
$37$	5735.00	+	9933.31i	0.688698	+	1.19286i	−0.283561	+	0.958954i	$0.591516\pi$
$38$		0			0
$39$	−8892.00	+	15401.4i	−0.936134	+	1.62143i
$40$		0			0
$41$	13350.0			1.24029			0.620143	−	0.784489i	$-0.287075\pi$
$41$	13350.0			1.24029			0.620143	+	0.784489i	$0.287075\pi$
$42$		0			0
$43$	−928.000			−0.0765380			−0.0382690	−	0.999267i	$-0.512184\pi$
$43$	−928.000			−0.0765380			−0.0382690	+	0.999267i	$0.512184\pi$
$44$		0			0
$45$	−3464.00	+	5999.82i	−0.255004	+	0.441679i
$46$		0			0
$47$	−606.000	−	1049.62i	−0.0400155	−	0.0693088i	0.845324	−	0.534254i	$-0.179407\pi$
$47$	−606.000	−	1049.62i	−0.0400155	−	0.0693088i	−0.885340	+	0.464945i	$0.846074\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	28834.0	+	49942.0i	1.55231	+	2.68869i
$52$		0			0
$53$	−6555.00	+	11353.6i	−0.320541	+	0.555193i	−0.980600	−	0.196021i	$-0.937198\pi$
$53$	−6555.00	+	11353.6i	−0.320541	+	0.555193i	0.660059	+	0.751214i	$0.270531\pi$
$54$		0			0
$55$	128.000			0.00570563
$56$		0			0
$57$	−70148.0			−2.85975
$58$		0			0
$59$	−17351.0	+	30052.8i	−0.648925	+	1.12397i	0.334455	+	0.942412i	$0.391448\pi$
$59$	−17351.0	+	30052.8i	−0.648925	+	1.12397i	−0.983380	+	0.181559i	$0.941886\pi$
$60$		0			0
$61$	516.000	+	893.738i	0.0177552	+	0.0307529i	0.874766	−	0.484545i	$-0.161015\pi$
$61$	516.000	+	893.738i	0.0177552	+	0.0307529i	−0.857011	+	0.515298i	$0.827681\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5472.00	−	9477.78i	−0.160643	−	0.278242i
$66$		0			0
$67$	−5054.00	+	8753.78i	−0.137546	+	0.238237i	−0.926567	−	0.376129i	$-0.877255\pi$
$67$	−5054.00	+	8753.78i	−0.137546	+	0.238237i	0.789021	+	0.614366i	$0.210588\pi$
$68$		0			0
$69$	86944.0			2.19845
$70$		0			0
$71$	62720.0			1.47659			0.738295	−	0.674477i	$-0.235631\pi$
$71$	62720.0			1.47659			0.738295	+	0.674477i	$0.235631\pi$
$72$		0			0
$73$	9463.00	−	16390.4i	0.207836	−	0.359983i	−0.743196	−	0.669073i	$-0.766691\pi$
$73$	9463.00	−	16390.4i	0.207836	−	0.359983i	0.951033	+	0.309090i	$0.100024\pi$
$74$		0			0
$75$	37297.0	+	64600.3i	0.765633	+	1.32612i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5700.00	−	9872.69i	−0.102756	−	0.177979i	0.810063	−	0.586343i	$-0.199433\pi$
$79$	−5700.00	−	9872.69i	−0.102756	−	0.177979i	−0.912819	+	0.408364i	$0.866099\pi$
$80$		0			0
$81$	−11610.5	+	20110.0i	−0.196625	+	0.340564i
$82$		0			0
$83$	88958.0			1.41739			0.708696	−	0.705514i	$-0.249284\pi$
$83$	88958.0			1.41739			0.708696	+	0.705514i	$0.249284\pi$
$84$		0			0
$85$	−35488.0			−0.532763
$86$		0			0
$87$	42302.0	−	73269.2i	0.599188	−	1.03782i
$88$		0			0
$89$	−9861.00	−	17079.8i	−0.131961	−	0.228563i	0.792471	−	0.609909i	$-0.208794\pi$
$89$	−9861.00	−	17079.8i	−0.131961	−	0.228563i	−0.924432	+	0.381346i	$0.875461\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−62244.0	−	107810.i	−0.746260	−	1.29256i
$94$		0			0
$95$	21584.0	−	37384.6i	0.245371	−	0.424995i
$96$		0			0
$97$	17062.0			0.184120			0.0920599	−	0.995753i	$-0.470655\pi$
$97$	17062.0			0.184120			0.0920599	+	0.995753i	$0.470655\pi$
$98$		0			0
$99$	3464.00			0.0355214
$100$		0			0
$101$	−22952.0	+	39754.0i	−0.223881	+	0.387773i	−0.955983	−	0.293422i	$-0.905206\pi$
$101$	−22952.0	+	39754.0i	−0.223881	+	0.387773i	0.732102	+	0.681195i	$0.238539\pi$
$102$		0			0
$103$	68006.0	+	117790.i	0.631618	+	1.09399i	0.987221	+	0.159357i	$0.0509420\pi$
$103$	68006.0	+	117790.i	0.631618	+	1.09399i	−0.355604	+	0.934637i	$0.615725\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	34578.0	+	59890.9i	0.291971	+	0.505709i	0.974276	−	0.225359i	$-0.0723554\pi$
$107$	34578.0	+	59890.9i	0.291971	+	0.505709i	−0.682304	+	0.731068i	$0.739022\pi$
$108$		0			0
$109$	73207.0	−	126798.i	0.590183	−	1.02223i	−0.404025	−	0.914748i	$-0.632389\pi$
$109$	73207.0	−	126798.i	0.590183	−	1.02223i	0.994207	−	0.107478i	$-0.0342776\pi$
$110$		0			0
$111$	−298220.			−2.29736
$112$		0			0
$113$	−80186.0			−0.590748			−0.295374	−	0.955382i	$-0.595444\pi$
$113$	−80186.0			−0.590748			−0.295374	+	0.955382i	$0.595444\pi$
$114$		0			0
$115$	−26752.0	+	46335.8i	−0.188630	+	0.326717i
$116$		0			0
$117$	−148086.	−	256492.i	−1.00011	−	1.73225i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	80493.5	+	139419.i	0.499801	+	0.865681i
$122$		0			0
$123$	−173550.	+	300597.i	−1.03434	+	1.79152i
$124$		0			0
$125$	−95904.0			−0.548987
$126$		0			0
$127$	274800.			1.51185			0.755923	−	0.654661i	$-0.227189\pi$
$127$	274800.			1.51185			0.755923	+	0.654661i	$0.227189\pi$
$128$		0			0
$129$	12064.0	−	20895.5i	0.0638289	−	0.110555i
$130$		0			0
$131$	−90371.0	−	156527.i	−0.460099	−	0.796914i	0.538867	−	0.842391i	$-0.318853\pi$
$131$	−90371.0	−	156527.i	−0.460099	−	0.796914i	−0.998965	+	0.0454769i	$0.985519\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−39520.0	−	68450.6i	−0.186630	−	0.323253i
$136$		0			0
$137$	104839.	−	181586.i	0.477223	−	0.826575i	−0.522436	−	0.852678i	$-0.674977\pi$
$137$	104839.	−	181586.i	0.477223	−	0.826575i	0.999659	+	0.0261038i	$0.00831003\pi$
$138$		0			0
$139$	17242.0			0.0756921			0.0378461	−	0.999284i	$-0.487950\pi$
$139$	17242.0			0.0756921			0.0378461	+	0.999284i	$0.487950\pi$
$140$		0			0
$141$	31512.0			0.133484
$142$		0			0
$143$	−2736.00	+	4738.89i	−0.0111886	+	0.0193792i
$144$		0			0
$145$	26032.0	+	45088.7i	0.102822	+	0.178094i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−29679.0	−	51405.5i	−0.109518	−	0.189690i	0.806057	−	0.591837i	$-0.201597\pi$
$149$	−29679.0	−	51405.5i	−0.109518	−	0.189690i	−0.915575	+	0.402147i	$0.868264\pi$
$150$		0			0
$151$	168172.	−	291282.i	0.600221	−	1.03961i	−0.392566	−	0.919724i	$-0.628413\pi$
$151$	168172.	−	291282.i	0.600221	−	1.03961i	0.992787	−	0.119890i	$-0.0382541\pi$
$152$		0			0
$153$	−960394.			−3.31681
$154$		0			0
$155$	76608.0			0.256121
$156$		0			0
$157$	−232294.	+	402345.i	−0.752123	+	1.30272i	0.194669	+	0.980869i	$0.437637\pi$
$157$	−232294.	+	402345.i	−0.752123	+	1.30272i	−0.946792	+	0.321846i	$0.895697\pi$
$158$		0			0
$159$	−170430.	−	295193.i	−0.534630	−	0.926006i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−157396.	−	272618.i	−0.464007	−	0.803684i	0.535149	−	0.844758i	$-0.320255\pi$
$163$	−157396.	−	272618.i	−0.464007	−	0.803684i	−0.999156	+	0.0410737i	$0.986922\pi$
$164$		0			0
$165$	−1664.00	+	2882.13i	−0.00475821	+	0.00824146i
$166$		0			0
$167$	285724.			0.792785			0.396393	−	0.918081i	$-0.370262\pi$
$167$	285724.			0.792785			0.396393	+	0.918081i	$0.370262\pi$
$168$		0			0
$169$	96563.0			0.260072
$170$		0			0
$171$	584117.	−	1.01172e6i	1.52760	−	2.64588i
$172$		0			0
$173$	354574.	+	614140.i	0.900724	+	1.56010i	0.826557	+	0.562854i	$0.190296\pi$
$173$	354574.	+	614140.i	0.900724	+	1.56010i	0.0741671	+	0.997246i	$0.476370\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−451126.	−	781373.i	−1.08234	−	1.87467i
$178$		0			0
$179$	308574.	−	534466.i	0.719825	−	1.24677i	−0.241244	−	0.970464i	$-0.577555\pi$
$179$	308574.	−	534466.i	0.719825	−	1.24677i	0.961069	−	0.276309i	$-0.0891112\pi$
$180$		0			0
$181$	237828.			0.539593			0.269797	−	0.962917i	$-0.413044\pi$
$181$	237828.			0.539593			0.269797	+	0.962917i	$0.413044\pi$
$182$		0			0
$183$	−26832.0			−0.0592278
$184$		0			0
$185$	91760.0	−	158933.i	0.197117	−	0.341417i
$186$		0			0
$187$	8872.00	+	15366.8i	0.0185531	+	0.0321350i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	66756.0	+	115625.i	0.132406	+	0.229334i	0.924603	−	0.380931i	$-0.124396\pi$
$191$	66756.0	+	115625.i	0.132406	+	0.229334i	−0.792198	+	0.610265i	$0.791063\pi$
$192$		0			0
$193$	−135223.	+	234213.i	−0.261311	+	0.452604i	−0.966590	−	0.256326i	$-0.917488\pi$
$193$	−135223.	+	234213.i	−0.261311	+	0.452604i	0.705280	+	0.708929i	$0.250821\pi$
$194$		0			0
$195$	284544.			0.535874
$196$		0			0
$197$	875102.			1.60655			0.803273	−	0.595611i	$-0.203090\pi$
$197$	875102.			1.60655			0.803273	+	0.595611i	$0.203090\pi$
$198$		0			0
$199$	173810.	−	301048.i	0.311130	−	0.538893i	−0.667477	−	0.744630i	$-0.732626\pi$
$199$	173810.	−	301048.i	0.311130	−	0.538893i	0.978607	+	0.205737i	$0.0659592\pi$
$200$		0			0
$201$	−131404.	−	227598.i	−0.229413	−	0.397355i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−106800.	−	184983.i	−0.177495	−	0.307431i
$206$		0			0
$207$	−723976.	+	1.25396e6i	−1.17435	+	2.03404i
$208$		0			0
$209$	−21584.0			−0.0341795
$210$		0			0
$211$	−425380.			−0.657765			−0.328883	−	0.944371i	$-0.606672\pi$
$211$	−425380.			−0.657765			−0.328883	+	0.944371i	$0.606672\pi$
$212$		0			0
$213$	−815360.	+	1.41224e6i	−1.23140	+	2.13285i
$214$		0			0
$215$	7424.00	+	12858.7i	0.0109532	+	0.0189715i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	246038.	+	426150.i	0.346651	+	0.600416i
$220$		0			0
$221$	758556.	−	1.31386e6i	1.04474	−	1.80954i
$222$		0			0
$223$	481592.			0.648511			0.324255	−	0.945970i	$-0.394886\pi$
$223$	481592.			0.648511			0.324255	+	0.945970i	$0.394886\pi$
$224$		0			0
$225$	−1.24228e6			−1.63592
$226$		0			0
$227$	3021.00	−	5232.53i	0.00389122	−	0.00673980i	−0.864073	−	0.503366i	$-0.832095\pi$
$227$	3021.00	−	5232.53i	0.00389122	−	0.00673980i	0.867964	+	0.496626i	$0.165428\pi$
$228$		0			0
$229$	−902.000	−	1562.31i	−0.00113663	−	0.00196870i	0.865457	−	0.500984i	$-0.167028\pi$
$229$	−902.000	−	1562.31i	−0.00113663	−	0.00196870i	−0.866593	+	0.499015i	$0.833695\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	805763.	+	1.39562e6i	0.972339	+	1.68414i	0.688452	+	0.725282i	$0.258291\pi$
$233$	805763.	+	1.39562e6i	0.972339	+	1.68414i	0.283887	+	0.958858i	$0.408376\pi$
$234$		0			0
$235$	−9696.00	+	16794.0i	−0.0114531	+	0.0198373i
$236$		0			0
$237$	296400.			0.342774
$238$		0			0
$239$	−987096.			−1.11780			−0.558901	−	0.829235i	$-0.688777\pi$
$239$	−987096.			−1.11780			−0.558901	+	0.829235i	$0.688777\pi$
$240$		0			0
$241$	−446755.	+	773802.i	−0.495481	+	0.858198i	−0.999986	−	0.00521050i	$-0.998341\pi$
$241$	−446755.	+	773802.i	−0.495481	+	0.858198i	0.504506	+	0.863408i	$0.331675\pi$
$242$		0			0
$243$	298337.	+	516735.i	0.324109	+	0.561374i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	922716.	+	1.59819e6i	0.962334	+	1.66681i
$248$		0			0
$249$	−1.15645e6	+	2.00304e6i	−1.18203	+	2.04734i
$250$		0			0
$251$	365946.			0.366634			0.183317	−	0.983054i	$-0.441317\pi$
$251$	365946.			0.366634			0.183317	+	0.983054i	$0.441317\pi$
$252$		0			0
$253$	26752.0			0.0262757
$254$		0			0
$255$	461344.	−	799071.i	0.444298	−	0.769547i
$256$		0			0
$257$	−702297.	−	1.21641e6i	−0.663266	−	1.14881i	−0.979752	−	0.200214i	$-0.935836\pi$
$257$	−702297.	−	1.21641e6i	−0.663266	−	1.14881i	0.316486	−	0.948597i	$-0.397497\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	704491.	+	1.22021e6i	0.640139	+	1.10875i
$262$		0			0
$263$	−549840.	+	952351.i	−0.490170	+	0.849000i	−0.999936	−	0.0113135i	$-0.996399\pi$
$263$	−549840.	+	952351.i	−0.490170	+	0.849000i	0.509766	+	0.860313i	$0.329732\pi$
$264$		0			0
$265$	209760.			0.183488
$266$		0			0
$267$	512772.			0.440196
$268$		0			0
$269$	−407474.	+	705766.i	−0.343336	+	0.594675i	−0.985050	−	0.172269i	$-0.944890\pi$
$269$	−407474.	+	705766.i	−0.343336	+	0.594675i	0.641714	+	0.766944i	$0.278224\pi$
$270$		0			0
$271$	849528.	+	1.47143e6i	0.702675	+	1.21707i	0.967524	+	0.252779i	$0.0813445\pi$
$271$	849528.	+	1.47143e6i	0.702675	+	1.21707i	−0.264849	+	0.964290i	$0.585322\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	11476.0	+	19877.0i	0.00915079	+	0.0158496i
$276$		0			0
$277$	682541.	−	1.18220e6i	0.534477	−	0.925742i	−0.464711	−	0.885462i	$-0.653842\pi$
$277$	682541.	−	1.18220e6i	0.534477	−	0.925742i	0.999188	−	0.0402796i	$-0.0128249\pi$
$278$		0			0
$279$	2.07320e6			1.59453
$280$		0			0
$281$	−715846.			−0.540821			−0.270411	−	0.962745i	$-0.587159\pi$
$281$	−715846.			−0.540821			−0.270411	+	0.962745i	$0.587159\pi$
$282$		0			0
$283$	−108863.	+	188556.i	−0.0808005	+	0.139951i	−0.903594	−	0.428390i	$-0.859081\pi$
$283$	−108863.	+	188556.i	−0.0808005	+	0.139951i	0.822793	+	0.568340i	$0.192414\pi$
$284$		0			0
$285$	561184.	+	971999.i	0.409254	+	0.708849i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.74983e6	−	3.03080e6i	−1.23240	−	2.13458i
$290$		0			0
$291$	−221806.	+	384179.i	−0.153547	+	0.265951i
$292$		0			0
$293$	1.50708e6			1.02557			0.512787	−	0.858516i	$-0.328613\pi$
$293$	1.50708e6			1.02557			0.512787	+	0.858516i	$0.328613\pi$
$294$		0			0
$295$	555232.			0.371466
$296$		0			0
$297$	−19760.0	+	34225.3i	−0.0129986	+	0.0225142i
$298$		0			0
$299$	−1.14365e6	−	1.98086e6i	−0.739800	−	1.28137i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−596752.	−	1.03360e6i	−0.373411	−	0.646767i
$304$		0			0
$305$	8256.00	−	14299.8i	0.00508183	−	0.00880199i
$306$		0			0
$307$	12502.0			0.00757066			0.00378533	−	0.999993i	$-0.498795\pi$
$307$	12502.0			0.00757066			0.00378533	+	0.999993i	$0.498795\pi$
$308$		0			0
$309$	−3.53631e6			−2.10695
$310$		0			0
$311$	323716.	−	560693.i	0.189786	−	0.328718i	−0.755393	−	0.655272i	$-0.772554\pi$
$311$	323716.	−	560693.i	0.189786	−	0.328718i	0.945179	+	0.326554i	$0.105887\pi$
$312$		0			0
$313$	467989.	+	810581.i	0.270007	+	0.467666i	0.968863	−	0.247596i	$-0.0796407\pi$
$313$	467989.	+	810581.i	0.270007	+	0.467666i	−0.698856	+	0.715262i	$0.746307\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−352971.	−	611364.i	−0.197284	−	0.341705i	0.750363	−	0.661026i	$-0.229879\pi$
$317$	−352971.	−	611364.i	−0.197284	−	0.341705i	−0.947647	+	0.319321i	$0.896545\pi$
$318$		0			0
$319$	13016.0	−	22544.4i	0.00716145	−	0.0124040i
$320$		0			0
$321$	−1.79806e6			−0.973959
$322$		0			0
$323$	5.98416e6			3.19152
$324$		0			0
$325$	981198.	−	1.69948e6i	0.515286	−	0.892501i
$326$		0			0
$327$	1.90338e6	+	3.29675e6i	0.984366	+	1.70497i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	571520.	+	989902.i	0.286722	+	0.496618i	0.973025	−	0.230698i	$-0.0741010\pi$
$331$	571520.	+	989902.i	0.286722	+	0.496618i	−0.686303	+	0.727316i	$0.740768\pi$
$332$		0			0
$333$	2.48326e6	−	4.30112e6i	1.22719	−	2.12555i
$334$		0			0
$335$	161728.			0.0787360
$336$		0			0
$337$	−2.36402e6			−1.13390			−0.566952	−	0.823751i	$-0.691877\pi$
$337$	−2.36402e6			−1.13390			−0.566952	+	0.823751i	$0.691877\pi$
$338$		0			0
$339$	1.04242e6	−	1.80552e6i	0.492655	−	0.853303i
$340$		0			0
$341$	−19152.0	−	33172.2i	−0.00891925	−	0.0154486i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−695552.	−	1.20473e6i	−0.314617	−	0.544932i
$346$		0			0
$347$	−363120.	+	628942.i	−0.161892	+	0.280406i	−0.935547	−	0.353201i	$-0.885093\pi$
$347$	−363120.	+	628942.i	−0.161892	+	0.280406i	0.773655	+	0.633607i	$0.218426\pi$
$348$		0			0
$349$	136180.			0.0598480			0.0299240	−	0.999552i	$-0.490473\pi$
$349$	136180.			0.0598480			0.0299240	+	0.999552i	$0.490473\pi$
$350$		0			0
$351$	3.37896e6			1.46391
$352$		0			0
$353$	584535.	−	1.01244e6i	0.249674	−	0.432448i	−0.713761	−	0.700389i	$-0.753010\pi$
$353$	584535.	−	1.01244e6i	0.249674	−	0.432448i	0.963435	+	0.267941i	$0.0863431\pi$
$354$		0			0
$355$	−501760.	−	869074.i	−0.211312	−	0.366004i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2140.00	+	3706.59i	0.000876350	+	0.00151788i	0.866463	−	0.499241i	$-0.166388\pi$
$359$	2140.00	+	3706.59i	0.000876350	+	0.00151788i	−0.865587	+	0.500759i	$0.833054\pi$
$360$		0			0
$361$	−2.40155e6	+	4.15961e6i	−0.969894	+	1.67990i
$362$		0			0
$363$	−4.18566e6			−1.66724
$364$		0			0
$365$	−302816.			−0.118973
$366$		0			0
$367$	−1.22398e6	+	2.12000e6i	−0.474361	+	0.821618i	−0.999569	−	0.0293563i	$-0.990654\pi$
$367$	−1.22398e6	+	2.12000e6i	−0.474361	+	0.821618i	0.525208	+	0.850974i	$0.323988\pi$
$368$		0			0
$369$	−2.89028e6	−	5.00610e6i	−1.10503	−	1.91396i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	452257.	+	783332.i	0.168311	+	0.291524i	0.937826	−	0.347105i	$-0.112835\pi$
$373$	452257.	+	783332.i	0.168311	+	0.291524i	−0.769515	+	0.638629i	$0.779502\pi$
$374$		0			0
$375$	1.24675e6	−	2.15944e6i	0.457828	−	0.792981i
$376$		0			0
$377$	−2.22574e6			−0.806530
$378$		0			0
$379$	−4.23034e6			−1.51279			−0.756393	−	0.654117i	$-0.773040\pi$
$379$	−4.23034e6			−1.51279			−0.756393	+	0.654117i	$0.773040\pi$
$380$		0			0
$381$	−3.57240e6	+	6.18758e6i	−1.26080	+	2.18378i
$382$		0			0
$383$	−2.27700e6	−	3.94388e6i	−0.793169	−	1.37381i	−0.923995	−	0.382404i	$-0.875096\pi$
$383$	−2.27700e6	−	3.94388e6i	−0.793169	−	1.37381i	0.130826	−	0.991405i	$-0.458237\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	200912.	+	347990.i	0.0681912	+	0.118111i
$388$		0			0
$389$	1.99270e6	−	3.45146e6i	0.667680	−	1.15646i	−0.310871	−	0.950452i	$-0.600621\pi$
$389$	1.99270e6	−	3.45146e6i	0.667680	−	1.15646i	0.978551	−	0.206004i	$-0.0660460\pi$
$390$		0			0
$391$	−7.41699e6			−2.45350
$392$		0			0
$393$	4.69929e6			1.53480
$394$		0			0
$395$	−91200.0	+	157963.i	−0.0294105	+	0.0509404i
$396$		0			0
$397$	−276210.	−	478410.i	−0.0879555	−	0.152343i	0.818691	−	0.574234i	$-0.194700\pi$
$397$	−276210.	−	478410.i	−0.0879555	−	0.152343i	−0.906647	+	0.421890i	$0.861367\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−19095.0	−	33073.5i	−0.00593006	−	0.0102712i	0.863045	−	0.505127i	$-0.168554\pi$
$401$	−19095.0	−	33073.5i	−0.00593006	−	0.0102712i	−0.868975	+	0.494856i	$0.835221\pi$
$402$		0			0
$403$	−1.63750e6	+	2.83623e6i	−0.502247	+	0.869918i
$404$		0			0
$405$	371536.			0.112555
$406$		0			0
$407$	−91760.0			−0.0274579
$408$		0			0
$409$	1.96238e6	−	3.39894e6i	0.580062	−	1.00470i	−0.415410	−	0.909634i	$-0.636362\pi$
$409$	1.96238e6	−	3.39894e6i	0.580062	−	1.00470i	0.995471	−	0.0950619i	$-0.0303049\pi$
$410$		0			0
$411$	2.72581e6	+	4.72125e6i	0.795961	+	1.37864i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−711664.	−	1.23264e6i	−0.202841	−	0.351330i
$416$		0			0
$417$	−224146.	+	388232.i	−0.0631235	+	0.109333i
$418$		0			0
$419$	598386.			0.166512			0.0832562	−	0.996528i	$-0.473468\pi$
$419$	598386.			0.166512			0.0832562	+	0.996528i	$0.473468\pi$
$420$		0			0
$421$	4.61597e6			1.26928			0.634641	−	0.772807i	$-0.281148\pi$
$421$	4.61597e6			1.26928			0.634641	+	0.772807i	$0.281148\pi$
$422$		0			0
$423$	−262398.	+	454487.i	−0.0713033	+	0.123501i
$424$		0			0
$425$	−3.18172e6	−	5.51090e6i	−0.854456	−	1.47996i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−71136.0	−	123211.i	−0.0186615	−	0.0323226i
$430$		0			0
$431$	30780.0	−	53312.5i	0.00798133	−	0.0138241i	−0.862007	−	0.506896i	$-0.830793\pi$
$431$	30780.0	−	53312.5i	0.00798133	−	0.0138241i	0.869988	+	0.493072i	$0.164126\pi$
$432$		0			0
$433$	−3.79727e6			−0.973310			−0.486655	−	0.873594i	$-0.661783\pi$
$433$	−3.79727e6			−0.973310			−0.486655	+	0.873594i	$0.661783\pi$
$434$		0			0
$435$	−1.35366e6			−0.342995
$436$		0			0
$437$	4.51106e6	−	7.81338e6i	1.12999	−	1.95720i
$438$		0			0
$439$	1.14426e6	+	1.98192e6i	0.283376	+	0.490822i	0.972214	−	0.234093i	$-0.0752122\pi$
$439$	1.14426e6	+	1.98192e6i	0.283376	+	0.490822i	−0.688838	+	0.724915i	$0.741879\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.37988e6	+	4.12207e6i	0.576163	+	0.997944i	0.995914	+	0.0903044i	$0.0287840\pi$
$443$	2.37988e6	+	4.12207e6i	0.576163	+	0.997944i	−0.419751	+	0.907639i	$0.637883\pi$
$444$		0			0
$445$	−157776.	+	273276.i	−0.0377695	+	0.0654186i
$446$		0			0
$447$	1.54331e6			0.365329
$448$		0			0
$449$	−4.36715e6			−1.02231			−0.511155	−	0.859489i	$-0.670782\pi$
$449$	−4.36715e6			−1.02231			−0.511155	+	0.859489i	$0.670782\pi$
$450$		0			0
$451$	−53400.0	+	92491.5i	−0.0123623	+	0.0214122i
$452$		0			0
$453$	4.37247e6	+	7.57334e6i	1.00111	+	1.73397i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2.72497e6	−	4.71979e6i	−0.610339	−	1.05714i	−0.991183	−	0.132499i	$-0.957700\pi$
$457$	−2.72497e6	−	4.71979e6i	−0.610339	−	1.05714i	0.380844	−	0.924639i	$-0.375634\pi$
$458$		0			0
$459$	5.47846e6	−	9.48897e6i	1.21374	−	2.10226i
$460$		0			0
$461$	1.66966e6			0.365911			0.182956	−	0.983121i	$-0.441434\pi$
$461$	1.66966e6			0.365911			0.182956	+	0.983121i	$0.441434\pi$
$462$		0			0
$463$	70768.0			0.0153421			0.00767104	−	0.999971i	$-0.497558\pi$
$463$	70768.0			0.0153421			0.00767104	+	0.999971i	$0.497558\pi$
$464$		0			0
$465$	−995904.	+	1.72496e6i	−0.213592	+	0.369952i
$466$		0			0
$467$	−2.83041e6	−	4.90242e6i	−0.600562	−	1.04020i	−0.992736	−	0.120313i	$-0.961610\pi$
$467$	−2.83041e6	−	4.90242e6i	−0.600562	−	1.04020i	0.392174	−	0.919891i	$-0.371723\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−6.03964e6	−	1.04610e7i	−1.25447	−	2.17280i
$472$		0			0
$473$	3712.00	−	6429.37i	0.000762878	−	0.00132134i
$474$		0			0
$475$	7.74056e6			1.57412
$476$		0			0
$477$	5.67663e6			1.14234
$478$		0			0
$479$	724742.	−	1.25529e6i	0.144326	−	0.249980i	−0.784795	−	0.619755i	$-0.787232\pi$
$479$	724742.	−	1.25529e6i	0.144326	−	0.249980i	0.929121	+	0.369775i	$0.120565\pi$
$480$		0			0
$481$	3.92274e6	+	6.79438e6i	0.773084	+	1.33902i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−136496.	−	236418.i	−0.0263491	−	0.0456380i
$486$		0			0
$487$	−2.03752e6	+	3.52909e6i	−0.389296	+	0.674280i	−0.992355	−	0.123417i	$-0.960615\pi$
$487$	−2.03752e6	+	3.52909e6i	−0.389296	+	0.674280i	0.603059	+	0.797696i	$0.293948\pi$
$488$		0			0
$489$	8.18459e6			1.54784
$490$		0			0
$491$	986100.			0.184594			0.0922969	−	0.995732i	$-0.470579\pi$
$491$	986100.			0.184594			0.0922969	+	0.995732i	$0.470579\pi$
$492$		0			0
$493$	−3.60869e6	+	6.25043e6i	−0.668701	+	1.15822i
$494$		0			0
$495$	−27712.0	−	47998.6i	−0.00508341	−	0.00880472i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.99171e6	−	5.18179e6i	−0.537859	−	0.931598i	−0.999019	−	0.0442817i	$-0.985900\pi$
$499$	−2.99171e6	−	5.18179e6i	−0.537859	−	0.931598i	0.461160	−	0.887317i	$-0.347433\pi$
$500$		0			0
$501$	−3.71441e6	+	6.43355e6i	−0.661144	+	1.14513i
$502$		0			0
$503$	−3.49373e6			−0.615700			−0.307850	−	0.951435i	$-0.599609\pi$
$503$	−3.49373e6			−0.615700			−0.307850	+	0.951435i	$0.599609\pi$
$504$		0			0
$505$	734464.			0.128157
$506$		0			0
$507$	−1.25532e6	+	2.17428e6i	−0.216887	+	0.375660i
$508$		0			0
$509$	−1.07855e6	−	1.86811e6i	−0.184522	−	0.319601i	0.758894	−	0.651215i	$-0.225740\pi$
$509$	−1.07855e6	−	1.86811e6i	−0.184522	−	0.319601i	−0.943415	+	0.331614i	$0.892407\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	6.66406e6	+	1.15425e7i	1.11801	+	1.93645i
$514$		0			0
$515$	1.08810e6	−	1.88464e6i	0.180779	−	0.313119i
$516$		0			0
$517$	9696.00			0.00159539
$518$		0			0
$519$	−1.84378e7			−3.00464
$520$		0			0
$521$	3.32908e6	−	5.76614e6i	0.537317	−	0.930660i	−0.461730	−	0.887020i	$-0.652771\pi$
$521$	3.32908e6	−	5.76614e6i	0.537317	−	0.930660i	0.999047	−	0.0436400i	$-0.0138955\pi$
$522$		0			0
$523$	−2.97611e6	−	5.15478e6i	−0.475768	−	0.824055i	0.523846	−	0.851813i	$-0.324497\pi$
$523$	−2.97611e6	−	5.15478e6i	−0.475768	−	0.824055i	−0.999615	+	0.0277579i	$0.991163\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5.30989e6	+	9.19700e6i	0.832835	+	1.44251i
$528$		0			0
$529$	−2.37300e6	+	4.11015e6i	−0.368687	+	0.638585i
$530$		0			0
$531$	1.50260e7			2.31263
$532$		0			0
$533$	9.13140e6			1.39226
$534$		0			0
$535$	553248.	−	958254.i	0.0835671	−	0.144742i
$536$		0			0
$537$	8.02292e6	+	1.38961e7i	1.20060	+	2.07949i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	3.19840e6	+	5.53980e6i	0.469830	+	0.813769i	0.999405	−	0.0344943i	$-0.0109821\pi$
$541$	3.19840e6	+	5.53980e6i	0.469830	+	0.813769i	−0.529575	+	0.848263i	$0.677649\pi$
$542$		0			0
$543$	−3.09176e6	+	5.35509e6i	−0.449994	+	0.779413i
$544$		0			0
$545$	−2.34262e6			−0.337840
$546$		0			0
$547$	−5.51851e6			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$547$	−5.51851e6			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$548$		0			0
$549$	223428.	−	386989.i	0.0316378	−	0.0547983i
$550$		0			0
$551$	−4.38965e6	−	7.60309e6i	−0.615957	−	1.06687i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	2.38576e6	+	4.13226e6i	0.328772	+	0.569449i
$556$		0			0
$557$	−1.01080e6	+	1.75075e6i	−0.138046	+	0.239103i	−0.926757	−	0.375661i	$-0.877416\pi$
$557$	−1.01080e6	+	1.75075e6i	−0.138046	+	0.239103i	0.788711	+	0.614765i	$0.210749\pi$
$558$		0			0
$559$	−634752.			−0.0859161
$560$		0			0
$561$	−461344.			−0.0618896
$562$		0			0
$563$	−4.07339e6	+	7.05532e6i	−0.541608	+	0.938092i	0.457204	+	0.889362i	$0.348851\pi$
$563$	−4.07339e6	+	7.05532e6i	−0.541608	+	0.938092i	−0.998812	+	0.0487304i	$0.984482\pi$
$564$		0			0
$565$	641488.	+	1.11109e6i	0.0845410	+	0.146429i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.99121e6	+	1.03771e7i	0.775772	+	1.34368i	0.934360	+	0.356332i	$0.115973\pi$
$569$	5.99121e6	+	1.03771e7i	0.775772	+	1.34368i	−0.158588	+	0.987345i	$0.550694\pi$
$570$		0			0
$571$	−697908.	+	1.20881e6i	−0.0895793	+	0.155156i	−0.907333	−	0.420412i	$-0.861886\pi$
$571$	−697908.	+	1.20881e6i	−0.0895793	+	0.155156i	0.817754	+	0.575568i	$0.195219\pi$
$572$		0			0
$573$	−3.47131e6			−0.441679
$574$		0			0
$575$	−9.59394e6			−1.21012
$576$		0			0
$577$	−983921.	+	1.70420e6i	−0.123033	+	0.213099i	−0.920962	−	0.389652i	$-0.872595\pi$
$577$	−983921.	+	1.70420e6i	−0.123033	+	0.213099i	0.797930	+	0.602751i	$0.205929\pi$
$578$		0			0
$579$	−3.51580e6	−	6.08954e6i	−0.435840	−	0.754898i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−52440.0	−	90828.7i	−0.00638986	−	0.0110676i
$584$		0			0
$585$	−2.36938e6	+	4.10388e6i	−0.286249	+	0.495798i
$586$		0			0
$587$	−3.18897e6			−0.381993			−0.190997	−	0.981591i	$-0.561172\pi$
$587$	−3.18897e6			−0.381993			−0.190997	+	0.981591i	$0.561172\pi$
$588$		0			0
$589$	−1.29180e7			−1.53429
$590$		0			0
$591$	−1.13763e7	+	1.97044e7i	−1.33978	+	2.32057i
$592$		0			0
$593$	−837501.	−	1.45059e6i	−0.0978022	−	0.169398i	0.812972	−	0.582302i	$-0.197848\pi$
$593$	−837501.	−	1.45059e6i	−0.0978022	−	0.169398i	−0.910775	+	0.412904i	$0.864515\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	4.51906e6	+	7.82724e6i	0.518934	+	0.898820i
$598$		0			0
$599$	5.03173e6	−	8.71522e6i	0.572994	−	0.992456i	−0.423262	−	0.906007i	$-0.639115\pi$
$599$	5.03173e6	−	8.71522e6i	0.572994	−	0.992456i	0.996256	−	0.0864482i	$-0.0275517\pi$
$600$		0			0
$601$	1.72798e6			0.195143			0.0975713	−	0.995229i	$-0.468893\pi$
$601$	1.72798e6			0.195143			0.0975713	+	0.995229i	$0.468893\pi$
$602$		0			0
$603$	4.37676e6			0.490185
$604$		0			0
$605$	1.28790e6	−	2.23070e6i	0.143051	−	0.247772i
$606$		0			0
$607$	8.47613e6	+	1.46811e7i	0.933740	+	1.61728i	0.776866	+	0.629666i	$0.216808\pi$
$607$	8.47613e6	+	1.46811e7i	0.933740	+	1.61728i	0.156873	+	0.987619i	$0.449859\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−414504.	−	717942.i	−0.0449185	−	0.0778012i
$612$		0			0
$613$	−5.09708e6	+	8.82840e6i	−0.547861	+	0.948923i	0.450560	+	0.892746i	$0.351224\pi$
$613$	−5.09708e6	+	8.82840e6i	−0.547861	+	0.948923i	−0.998421	+	0.0561768i	$0.982109\pi$
$614$		0			0
$615$	5.55360e6			0.592089
$616$		0			0
$617$	1.57452e7			1.66508			0.832540	−	0.553965i	$-0.186886\pi$
$617$	1.57452e7			1.66508			0.832540	+	0.553965i	$0.186886\pi$
$618$		0			0
$619$	166345.	−	288118.i	0.0174495	−	0.0302234i	−0.857169	−	0.515036i	$-0.827779\pi$
$619$	166345.	−	288118.i	0.0174495	−	0.0302234i	0.874618	+	0.484812i	$0.161112\pi$
$620$		0			0
$621$	−8.25968e6	−	1.43062e7i	−0.859477	−	1.48866i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−3.71558e6	−	6.43557e6i	−0.380475	−	0.659003i
$626$		0			0
$627$	280592.	−	486000.i	0.0285040	−	0.0493704i
$628$		0			0
$629$	2.54405e7			2.56388
$630$		0			0
$631$	3.59720e6			0.359659			0.179830	−	0.983698i	$-0.442445\pi$
$631$	3.59720e6			0.359659			0.179830	+	0.983698i	$0.442445\pi$
$632$		0			0
$633$	5.52994e6	−	9.57814e6i	0.548544	−	0.950105i
$634$		0			0
$635$	−2.19840e6	−	3.80774e6i	−0.216358	−	0.374743i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−1.35789e7	−	2.35193e7i	−1.31556	−	2.27862i
$640$		0			0
$641$	7.31946e6	−	1.26777e7i	0.703614	−	1.21869i	−0.263576	−	0.964639i	$-0.584902\pi$
$641$	7.31946e6	−	1.26777e7i	0.703614	−	1.21869i	0.967190	−	0.254056i	$-0.0817647\pi$
$642$		0			0
$643$	−1.38386e7			−1.31997			−0.659987	−	0.751277i	$-0.729438\pi$
$643$	−1.38386e7			−1.31997			−0.659987	+	0.751277i	$0.729438\pi$
$644$		0			0
$645$	−386048.			−0.0365378
$646$		0			0
$647$	7.01791e6	−	1.21554e7i	0.659093	−	1.14158i	−0.321758	−	0.946822i	$-0.604274\pi$
$647$	7.01791e6	−	1.21554e7i	0.659093	−	1.14158i	0.980851	−	0.194761i	$-0.0623931\pi$
$648$		0			0
$649$	−138808.	−	240423.i	−0.0129361	−	0.0224060i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.05314e6	−	1.39484e7i	−0.739064	−	1.28010i	−0.952917	−	0.303231i	$-0.901935\pi$
$653$	−8.05314e6	−	1.39484e7i	−0.739064	−	1.28010i	0.213853	−	0.976866i	$-0.431399\pi$
$654$		0			0
$655$	−1.44594e6	+	2.50443e6i	−0.131688	+	0.228090i
$656$		0			0
$657$	−8.19496e6			−0.740685
$658$		0			0
$659$	4.80075e6			0.430622			0.215311	−	0.976546i	$-0.430924\pi$
$659$	4.80075e6			0.430622			0.215311	+	0.976546i	$0.430924\pi$
$660$		0			0
$661$	8.82824e6	−	1.52910e7i	0.785905	−	1.36123i	−0.142551	−	0.989787i	$-0.545531\pi$
$661$	8.82824e6	−	1.52910e7i	0.785905	−	1.36123i	0.928457	−	0.371441i	$-0.121136\pi$
$662$		0			0
$663$	1.97225e7	+	3.41603e7i	1.74252	+	3.01813i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	5.44069e6	+	9.42355e6i	0.473521	+	0.820162i
$668$		0			0
$669$	−6.26070e6	+	1.08438e7i	−0.540826	+	0.936738i
$670$		0			0
$671$	−8256.00			−0.000707886
$672$		0			0
$673$	6.59225e6			0.561043			0.280521	−	0.959848i	$-0.409493\pi$
$673$	6.59225e6			0.561043			0.280521	+	0.959848i	$0.409493\pi$
$674$		0			0
$675$	7.08643e6	−	1.22741e7i	0.598643	−	1.03688i
$676$		0			0
$677$	−4.88589e6	−	8.46261e6i	−0.409706	−	0.709631i	0.585151	−	0.810924i	$-0.301035\pi$
$677$	−4.88589e6	−	8.46261e6i	−0.409706	−	0.709631i	−0.994857	+	0.101293i	$0.967702\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	78546.0	+	136046.i	0.00649017	+	0.0112413i
$682$		0			0
$683$	9.43316e6	−	1.63387e7i	0.773758	−	1.34019i	−0.161731	−	0.986835i	$-0.551708\pi$
$683$	9.43316e6	−	1.63387e7i	0.773758	−	1.34019i	0.935490	−	0.353354i	$-0.114959\pi$
$684$		0			0
$685$	−3.35485e6			−0.273178
$686$		0			0
$687$	46904.0			0.00379156
$688$		0			0
$689$	−4.48362e6	+	7.76586e6i	−0.359816	+	0.623220i
$690$		0			0
$691$	−4.33509e6	−	7.50860e6i	−0.345385	−	0.598224i	0.640039	−	0.768343i	$-0.278918\pi$
$691$	−4.33509e6	−	7.50860e6i	−0.345385	−	0.598224i	−0.985424	+	0.170119i	$0.945585\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−137936.	−	238912.i	−0.0108322	−	0.0187619i
$696$		0			0
$697$	1.48052e7	−	2.56433e7i	1.15433	−	1.99936i
$698$		0			0
$699$	−4.18997e7			−3.24353
$700$		0			0
$701$	7.93482e6			0.609877			0.304938	−	0.952372i	$-0.401364\pi$
$701$	7.93482e6			0.609877			0.304938	+	0.952372i	$0.401364\pi$
$702$		0			0
$703$	−1.54730e7	+	2.68001e7i	−1.18083	+	2.04526i
$704$		0			0
$705$	−252096.	−	436643.i	−0.0191026	−	0.0330867i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.31300e7	−	2.27419e7i	−0.980956	−	1.69907i	−0.658685	−	0.752419i	$-0.728887\pi$
$709$	−1.31300e7	−	2.27419e7i	−0.980956	−	1.69907i	−0.322272	−	0.946647i	$-0.604446\pi$
$710$		0			0
$711$	−2.46810e6	+	4.27487e6i	−0.183100	+	0.317139i
$712$		0			0
$713$	1.60111e7			1.17950
$714$		0			0
$715$	87552.0			0.00640473
$716$		0			0
$717$	1.28322e7	−	2.22261e7i	0.932191	−	1.61460i
$718$		0			0
$719$	−1.10381e7	−	1.91186e7i	−0.796295	−	1.37922i	−0.922014	−	0.387157i	$-0.873457\pi$
$719$	−1.10381e7	−	1.91186e7i	−0.796295	−	1.37922i	0.125719	−	0.992066i	$-0.459876\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−1.16156e7	−	2.01189e7i	−0.826413	−	1.43139i
$724$		0			0
$725$	−4.66786e6	+	8.08498e6i	−0.329817	+	0.571260i
$726$		0			0
$727$	8.49245e6			0.595933			0.297966	−	0.954576i	$-0.403692\pi$
$727$	8.49245e6			0.595933			0.297966	+	0.954576i	$0.403692\pi$
$728$		0			0
$729$	−2.11562e7			−1.47441
$730$		0			0
$731$	−1.02915e6	+	1.78254e6i	−0.0712338	+	0.123381i
$732$		0			0
$733$	9.53564e6	+	1.65162e7i	0.655526	+	1.13540i	0.981762	+	0.190116i	$0.0608865\pi$
$733$	9.53564e6	+	1.65162e7i	0.655526	+	1.13540i	−0.326235	+	0.945289i	$0.605780\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−40432.0	−	70030.3i	−0.00274193	−	0.00474916i
$738$		0			0
$739$	7.34161e6	−	1.27160e7i	0.494516	−	0.856527i	−0.505464	−	0.862848i	$-0.668679\pi$
$739$	7.34161e6	−	1.27160e7i	0.494516	−	0.856527i	0.999980	+	0.00632086i	$0.00201201\pi$
$740$		0			0
$741$	−4.79812e7			−3.21015
$742$		0			0
$743$	1.64265e7			1.09162			0.545812	−	0.837908i	$-0.316221\pi$
$743$	1.64265e7			1.09162			0.545812	+	0.837908i	$0.316221\pi$
$744$		0			0
$745$	−474864.	+	822489.i	−0.0313457	+	0.0542924i
$746$		0			0
$747$	−1.92594e7	−	3.33583e7i	−1.26282	−	2.18727i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.22178e7	+	2.11619e7i	0.790486	+	1.36916i	0.925666	+	0.378342i	$0.123506\pi$
$751$	1.22178e7	+	2.11619e7i	0.790486	+	1.36916i	−0.135180	+	0.990821i	$0.543161\pi$
$752$		0			0
$753$	−4.75730e6	+	8.23988e6i	−0.305755	+	0.529582i
$754$		0			0
$755$	−5.38150e6			−0.343587
$756$		0			0
$757$	−295566.			−0.0187463			−0.00937313	−	0.999956i	$-0.502984\pi$
$757$	−295566.			−0.0187463			−0.00937313	+	0.999956i	$0.502984\pi$
$758$		0			0
$759$	−347776.	+	602366.i	−0.0219127	+	0.0379538i
$760$		0			0
$761$	236921.	+	410359.i	0.0148300	+	0.0256864i	0.873345	−	0.487102i	$-0.161946\pi$
$761$	236921.	+	410359.i	0.0148300	+	0.0256864i	−0.858515	+	0.512788i	$0.828613\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	7.68315e6	+	1.33076e7i	0.474664	+	0.822141i
$766$		0			0
$767$	−1.18681e7	+	2.05561e7i	−0.728437	+	1.26169i
$768$		0			0
$769$	2.33241e7			1.42229			0.711145	−	0.703045i	$-0.248177\pi$
$769$	2.33241e7			1.42229			0.711145	+	0.703045i	$0.248177\pi$
$770$		0			0
$771$	3.65194e7			2.21253
$772$		0			0
$773$	−7.77914e6	+	1.34739e7i	−0.468255	+	0.811042i	−0.999342	−	0.0362754i	$-0.988451\pi$
$773$	−7.77914e6	+	1.34739e7i	−0.468255	+	0.811042i	0.531086	+	0.847318i	$0.321784\pi$
$774$		0			0
$775$	6.86839e6	+	1.18964e7i	0.410772	+	0.711477i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.80092e7	+	3.11928e7i	1.06328	+	1.84166i
$780$		0			0
$781$	−250880.	+	434537.i	−0.0147176	+	0.0254917i
$782$		0			0
$783$	−1.60748e7			−0.937001
$784$		0			0
$785$

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)

\(f(q)\)	\(=\)	\(q+(-13.0000 + 22.5167i) q^{3} +(-8.00000 - 13.8564i) q^{5} +(-216.500 - 374.989i) q^{9} +O(q^{10})\)	\(q+(-13.0000 + 22.5167i) q^{3} +(-8.00000 - 13.8564i) q^{5} +(-216.500 - 374.989i) q^{9} +(-4.00000 + 6.92820i) q^{11} +684.000 q^{13} +416.000 q^{15} +(1109.00 - 1920.84i) q^{17} +(1349.00 + 2336.54i) q^{19} +(-1672.00 - 2895.99i) q^{23} +(1434.50 - 2484.63i) q^{25} +4940.00 q^{27} -3254.00 q^{29} +(-2394.00 + 4146.53i) q^{31} +(-104.000 - 180.133i) q^{33} +(5735.00 + 9933.31i) q^{37} +(-8892.00 + 15401.4i) q^{39} +13350.0 q^{41} -928.000 q^{43} +(-3464.00 + 5999.82i) q^{45} +(-606.000 - 1049.62i) q^{47} +(28834.0 + 49942.0i) q^{51} +(-6555.00 + 11353.6i) q^{53} +128.000 q^{55} -70148.0 q^{57} +(-17351.0 + 30052.8i) q^{59} +(516.000 + 893.738i) q^{61} +(-5472.00 - 9477.78i) q^{65} +(-5054.00 + 8753.78i) q^{67} +86944.0 q^{69} +62720.0 q^{71} +(9463.00 - 16390.4i) q^{73} +(37297.0 + 64600.3i) q^{75} +(-5700.00 - 9872.69i) q^{79} +(-11610.5 + 20110.0i) q^{81} +88958.0 q^{83} -35488.0 q^{85} +(42302.0 - 73269.2i) q^{87} +(-9861.00 - 17079.8i) q^{89} +(-62244.0 - 107810. i) q^{93} +(21584.0 - 37384.6i) q^{95} +17062.0 q^{97} +3464.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 26 q^{3} - 16 q^{5} - 433 q^{9}+O(q^{10}) \) 2 * q - 26 * q^3 - 16 * q^5 - 433 * q^9	\( 2 q - 26 q^{3} - 16 q^{5} - 433 q^{9} - 8 q^{11} + 1368 q^{13} + 832 q^{15} + 2218 q^{17} + 2698 q^{19} - 3344 q^{23} + 2869 q^{25} + 9880 q^{27} - 6508 q^{29} - 4788 q^{31} - 208 q^{33} + 11470 q^{37} - 17784 q^{39} + 26700 q^{41} - 1856 q^{43} - 6928 q^{45} - 1212 q^{47} + 57668 q^{51} - 13110 q^{53} + 256 q^{55} - 140296 q^{57} - 34702 q^{59} + 1032 q^{61} - 10944 q^{65} - 10108 q^{67} + 173888 q^{69} + 125440 q^{71} + 18926 q^{73} + 74594 q^{75} - 11400 q^{79} - 23221 q^{81} + 177916 q^{83} - 70976 q^{85} + 84604 q^{87} - 19722 q^{89} - 124488 q^{93} + 43168 q^{95} + 34124 q^{97} + 6928 q^{99}+O(q^{100}) \) 2 * q - 26 * q^3 - 16 * q^5 - 433 * q^9 - 8 * q^11 + 1368 * q^13 + 832 * q^15 + 2218 * q^17 + 2698 * q^19 - 3344 * q^23 + 2869 * q^25 + 9880 * q^27 - 6508 * q^29 - 4788 * q^31 - 208 * q^33 + 11470 * q^37 - 17784 * q^39 + 26700 * q^41 - 1856 * q^43 - 6928 * q^45 - 1212 * q^47 + 57668 * q^51 - 13110 * q^53 + 256 * q^55 - 140296 * q^57 - 34702 * q^59 + 1032 * q^61 - 10944 * q^65 - 10108 * q^67 + 173888 * q^69 + 125440 * q^71 + 18926 * q^73 + 74594 * q^75 - 11400 * q^79 - 23221 * q^81 + 177916 * q^83 - 70976 * q^85 + 84604 * q^87 - 19722 * q^89 - 124488 * q^93 + 43168 * q^95 + 34124 * q^97 + 6928 * q^99

Introduction

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