LMFDB - Embedded newform 196.6.e.h.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:	yes
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.h.177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(8.00000 - 13.8564i) q^{3} +(-8.00000 - 13.8564i) q^{5} +(-6.50000 - 11.2583i) q^{9} +O(q^{10})$	\(q+(8.00000 - 13.8564i) q^{3} +(-8.00000 - 13.8564i) q^{5} +(-6.50000 - 11.2583i) q^{9} +(38.0000 - 65.8179i) q^{11} +880.000 q^{13} -256.000 q^{15} +(528.000 - 914.523i) q^{17} +(-968.000 - 1676.63i) q^{19} +(-468.000 - 810.600i) q^{23} +(1434.50 - 2484.63i) q^{25} +3680.00 q^{27} -3982.00 q^{29} +(-784.000 + 1357.93i) q^{31} +(-608.000 - 1053.09i) q^{33} +(-2469.00 - 4276.43i) q^{37} +(7040.00 - 12193.6i) q^{39} -15840.0 q^{41} -16412.0 q^{43} +(-104.000 + 180.133i) q^{45} +(10384.0 + 17985.6i) q^{47} +(-8448.00 - 14632.4i) q^{51} +(18701.0 - 32391.1i) q^{53} -1216.00 q^{55} -30976.0 q^{57} +(-10568.0 + 18304.3i) q^{59} +(1496.00 + 2591.15i) q^{61} +(-7040.00 - 12193.6i) q^{65} +(22918.0 - 39695.1i) q^{67} -14976.0 q^{69} -49840.0 q^{71} +(28160.0 - 48774.6i) q^{73} +(-22952.0 - 39754.0i) q^{75} +(-20372.0 - 35285.3i) q^{79} +(31019.5 - 53727.4i) q^{81} +112464. q^{83} -16896.0 q^{85} +(-31856.0 + 55176.2i) q^{87} +(-32128.0 - 55647.3i) q^{89} +(12544.0 + 21726.8i) q^{93} +(-15488.0 + 26826.0i) q^{95} -2272.00 q^{97} -988.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{3} - 16 q^{5} - 13 q^{9}+O(q^{10}) $ 2 * q + 16 * q^3 - 16 * q^5 - 13 * q^9	$ 2 q + 16 q^{3} - 16 q^{5} - 13 q^{9} + 76 q^{11} + 1760 q^{13} - 512 q^{15} + 1056 q^{17} - 1936 q^{19} - 936 q^{23} + 2869 q^{25} + 7360 q^{27} - 7964 q^{29} - 1568 q^{31} - 1216 q^{33} - 4938 q^{37} + 14080 q^{39} - 31680 q^{41} - 32824 q^{43} - 208 q^{45} + 20768 q^{47} - 16896 q^{51} + 37402 q^{53} - 2432 q^{55} - 61952 q^{57} - 21136 q^{59} + 2992 q^{61} - 14080 q^{65} + 45836 q^{67} - 29952 q^{69} - 99680 q^{71} + 56320 q^{73} - 45904 q^{75} - 40744 q^{79} + 62039 q^{81} + 224928 q^{83} - 33792 q^{85} - 63712 q^{87} - 64256 q^{89} + 25088 q^{93} - 30976 q^{95} - 4544 q^{97} - 1976 q^{99}+O(q^{100}) $ 2 * q + 16 * q^3 - 16 * q^5 - 13 * q^9 + 76 * q^11 + 1760 * q^13 - 512 * q^15 + 1056 * q^17 - 1936 * q^19 - 936 * q^23 + 2869 * q^25 + 7360 * q^27 - 7964 * q^29 - 1568 * q^31 - 1216 * q^33 - 4938 * q^37 + 14080 * q^39 - 31680 * q^41 - 32824 * q^43 - 208 * q^45 + 20768 * q^47 - 16896 * q^51 + 37402 * q^53 - 2432 * q^55 - 61952 * q^57 - 21136 * q^59 + 2992 * q^61 - 14080 * q^65 + 45836 * q^67 - 29952 * q^69 - 99680 * q^71 + 56320 * q^73 - 45904 * q^75 - 40744 * q^79 + 62039 * q^81 + 224928 * q^83 - 33792 * q^85 - 63712 * q^87 - 64256 * q^89 + 25088 * q^93 - 30976 * q^95 - 4544 * q^97 - 1976 * 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
$3$	8.00000	−	13.8564i	0.513200	−	0.888889i	−0.486683	−	0.873579i	$-0.661793\pi$
$3$	8.00000	−	13.8564i	0.513200	−	0.888889i	0.999883	−	0.0153100i	$-0.00487351\pi$
$4$		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
$9$	−6.50000	−	11.2583i	−0.0267490	−	0.0463306i
$10$		0			0
$11$	38.0000	−	65.8179i	0.0946895	−	0.164007i	−0.814789	−	0.579757i	$-0.803148\pi$
$11$	38.0000	−	65.8179i	0.0946895	−	0.164007i	0.909479	+	0.415750i	$0.136481\pi$
$12$		0			0
$13$	880.000			1.44419			0.722095	−	0.691794i	$-0.243179\pi$
$13$	880.000			1.44419			0.722095	+	0.691794i	$0.243179\pi$
$14$		0			0
$15$	−256.000			−0.293773
$16$		0			0
$17$	528.000	−	914.523i	0.443110	−	0.767489i	−0.554808	−	0.831978i	$-0.687208\pi$
$17$	528.000	−	914.523i	0.443110	−	0.767489i	0.997918	+	0.0644890i	$0.0205418\pi$
$18$		0			0
$19$	−968.000	−	1676.63i	−0.615165	−	1.06550i	−0.990356	−	0.138549i	$-0.955756\pi$
$19$	−968.000	−	1676.63i	−0.615165	−	1.06550i	0.375191	−	0.926948i	$-0.377577\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−468.000	−	810.600i	−0.184470	−	0.319512i	0.758928	−	0.651175i	$-0.225724\pi$
$23$	−468.000	−	810.600i	−0.184470	−	0.319512i	−0.943398	+	0.331663i	$0.892390\pi$
$24$		0			0
$25$	1434.50	−	2484.63i	0.459040	−	0.795081i
$26$		0			0
$27$	3680.00			0.971490
$28$		0			0
$29$	−3982.00			−0.879238			−0.439619	−	0.898184i	$-0.644886\pi$
$29$	−3982.00			−0.879238			−0.439619	+	0.898184i	$0.644886\pi$
$30$		0			0
$31$	−784.000	+	1357.93i	−0.146525	+	0.253789i	−0.929941	−	0.367709i	$-0.880142\pi$
$31$	−784.000	+	1357.93i	−0.146525	+	0.253789i	0.783416	+	0.621498i	$0.213476\pi$
$32$		0			0
$33$	−608.000	−	1053.09i	−0.0971894	−	0.168337i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2469.00	−	4276.43i	−0.296495	−	0.513544i	0.678837	−	0.734289i	$-0.262484\pi$
$37$	−2469.00	−	4276.43i	−0.296495	−	0.513544i	−0.975331	+	0.220745i	$0.929151\pi$
$38$		0			0
$39$	7040.00	−	12193.6i	0.741159	−	1.28372i
$40$		0			0
$41$	−15840.0			−1.47162			−0.735810	−	0.677188i	$-0.763198\pi$
$41$	−15840.0			−1.47162			−0.735810	+	0.677188i	$0.763198\pi$
$42$		0			0
$43$	−16412.0			−1.35360			−0.676800	−	0.736167i	$-0.736634\pi$
$43$	−16412.0			−1.35360			−0.676800	+	0.736167i	$0.736634\pi$
$44$		0			0
$45$	−104.000	+	180.133i	−0.00765600	+	0.0132606i
$46$		0			0
$47$	10384.0	+	17985.6i	0.685678	+	1.18763i	0.973223	+	0.229862i	$0.0738274\pi$
$47$	10384.0	+	17985.6i	0.685678	+	1.18763i	−0.287546	+	0.957767i	$0.592839\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−8448.00	−	14632.4i	−0.454808	−	0.787751i
$52$		0			0
$53$	18701.0	−	32391.1i	0.914482	−	1.58393i	0.106824	−	0.994278i	$-0.465932\pi$
$53$	18701.0	−	32391.1i	0.914482	−	1.58393i	0.807658	−	0.589651i	$-0.200735\pi$
$54$		0			0
$55$	−1216.00			−0.0542034
$56$		0			0
$57$	−30976.0			−1.26281
$58$		0			0
$59$	−10568.0	+	18304.3i	−0.395242	+	0.684579i	−0.993132	−	0.116999i	$-0.962673\pi$
$59$	−10568.0	+	18304.3i	−0.395242	+	0.684579i	0.597890	+	0.801578i	$0.296006\pi$
$60$		0			0
$61$	1496.00	+	2591.15i	0.0514763	+	0.0891595i	0.890615	−	0.454757i	$-0.150274\pi$
$61$	1496.00	+	2591.15i	0.0514763	+	0.0891595i	−0.839139	+	0.543917i	$0.816941\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7040.00	−	12193.6i	−0.206676	−	0.357973i
$66$		0			0
$67$	22918.0	−	39695.1i	0.623720	−	1.08031i	−0.365067	−	0.930981i	$-0.618954\pi$
$67$	22918.0	−	39695.1i	0.623720	−	1.08031i	0.988787	−	0.149334i	$-0.0477128\pi$
$68$		0			0
$69$	−14976.0			−0.378681
$70$		0			0
$71$	−49840.0			−1.17336			−0.586681	−	0.809818i	$-0.699566\pi$
$71$	−49840.0			−1.17336			−0.586681	+	0.809818i	$0.699566\pi$
$72$		0			0
$73$	28160.0	−	48774.6i	0.618480	−	1.07124i	−0.371283	−	0.928520i	$-0.621082\pi$
$73$	28160.0	−	48774.6i	0.618480	−	1.07124i	0.989763	−	0.142719i	$-0.0455845\pi$
$74$		0			0
$75$	−22952.0	−	39754.0i	−0.471159	−	0.816071i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−20372.0	−	35285.3i	−0.367253	−	0.636102i	0.621882	−	0.783111i	$-0.286369\pi$
$79$	−20372.0	−	35285.3i	−0.367253	−	0.636102i	−0.989135	+	0.147010i	$0.953035\pi$
$80$		0			0
$81$	31019.5	−	53727.4i	0.525318	−	0.909877i
$82$		0			0
$83$	112464.			1.79192			0.895959	−	0.444136i	$-0.146489\pi$
$83$	112464.			1.79192			0.895959	+	0.444136i	$0.146489\pi$
$84$		0			0
$85$	−16896.0			−0.253651
$86$		0			0
$87$	−31856.0	+	55176.2i	−0.451225	+	0.781545i
$88$		0			0
$89$	−32128.0	−	55647.3i	−0.429941	−	0.744679i	0.566927	−	0.823768i	$-0.308132\pi$
$89$	−32128.0	−	55647.3i	−0.429941	−	0.744679i	−0.996868	+	0.0790889i	$0.974799\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	12544.0	+	21726.8i	0.150393	+	0.260489i
$94$		0			0
$95$	−15488.0	+	26826.0i	−0.176070	+	0.304963i
$96$		0			0
$97$	−2272.00			−0.0245177			−0.0122588	−	0.999925i	$-0.503902\pi$
$97$	−2272.00			−0.0245177			−0.0122588	+	0.999925i	$0.503902\pi$
$98$		0			0
$99$	−988.000			−0.0101314
$100$		0			0
$101$	55000.0	−	95262.8i	0.536487	−	0.929223i	−0.462603	−	0.886566i	$-0.653084\pi$
$101$	55000.0	−	95262.8i	0.536487	−	0.929223i	0.999090	−	0.0426572i	$-0.0135823\pi$
$102$		0			0
$103$	42064.0	+	72857.0i	0.390677	+	0.676672i	0.992539	−	0.121928i	$-0.0389077\pi$
$103$	42064.0	+	72857.0i	0.390677	+	0.676672i	−0.601862	+	0.798600i	$0.705574\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6974.00	−	12079.3i	−0.0588874	−	0.101996i	0.835079	−	0.550130i	$-0.185422\pi$
$107$	−6974.00	−	12079.3i	−0.0588874	−	0.101996i	−0.893966	+	0.448134i	$0.852089\pi$
$108$		0			0
$109$	−11297.0	+	19567.0i	−0.0910745	+	0.157746i	−0.907964	−	0.419049i	$-0.862363\pi$
$109$	−11297.0	+	19567.0i	−0.0910745	+	0.157746i	0.816889	+	0.576795i	$0.195697\pi$
$110$		0			0
$111$	−79008.0			−0.608644
$112$		0			0
$113$	94786.0			0.698310			0.349155	−	0.937065i	$-0.386469\pi$
$113$	94786.0			0.698310			0.349155	+	0.937065i	$0.386469\pi$
$114$		0			0
$115$	−7488.00	+	12969.6i	−0.0527985	+	0.0914496i
$116$		0			0
$117$	−5720.00	−	9907.33i	−0.0386306	−	0.0669101i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	77637.5	+	134472.i	0.482068	+	0.834966i
$122$		0			0
$123$	−126720.	+	219485.i	−0.755235	+	1.30811i
$124$		0			0
$125$	−95904.0			−0.548987
$126$		0			0
$127$	140624.			0.773660			0.386830	−	0.922151i	$-0.373570\pi$
$127$	140624.			0.773660			0.386830	+	0.922151i	$0.373570\pi$
$128$		0			0
$129$	−131296.	+	227411.i	−0.694668	+	1.20320i
$130$		0			0
$131$	121176.	+	209883.i	0.616934	+	1.06856i	0.990042	+	0.140772i	$0.0449586\pi$
$131$	121176.	+	209883.i	0.616934	+	1.06856i	−0.373108	+	0.927788i	$0.621708\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−29440.0	−	50991.6i	−0.139028	−	0.240804i
$136$		0			0
$137$	−86443.0	+	149724.i	−0.393485	+	0.681536i	−0.992907	−	0.118897i	$-0.962064\pi$
$137$	−86443.0	+	149724.i	−0.393485	+	0.681536i	0.599421	+	0.800434i	$0.295397\pi$
$138$		0			0
$139$	−167376.			−0.734778			−0.367389	−	0.930067i	$-0.619748\pi$
$139$	−167376.			−0.734778			−0.367389	+	0.930067i	$0.619748\pi$
$140$		0			0
$141$	332288.			1.40756
$142$		0			0
$143$	33440.0	−	57919.8i	0.136750	−	0.236857i
$144$		0			0
$145$	31856.0	+	55176.2i	0.125826	+	0.217937i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	202301.	+	350396.i	0.746504	+	1.29298i	0.949489	+	0.313802i	$0.101603\pi$
$149$	202301.	+	350396.i	0.746504	+	1.29298i	−0.202984	+	0.979182i	$0.565064\pi$
$150$		0			0
$151$	−222596.	+	385548.i	−0.794465	+	1.37605i	0.128713	+	0.991682i	$0.458916\pi$
$151$	−222596.	+	385548.i	−0.794465	+	1.37605i	−0.923178	+	0.384373i	$0.874418\pi$
$152$		0			0
$153$	−13728.0			−0.0474110
$154$		0			0
$155$	25088.0			0.0838758
$156$		0			0
$157$	86248.0	−	149386.i	0.279254	−	0.483683i	−0.691945	−	0.721950i	$-0.743246\pi$
$157$	86248.0	−	149386.i	0.279254	−	0.483683i	0.971200	+	0.238267i	$0.0765794\pi$
$158$		0			0
$159$	−299216.	−	518257.i	−0.938625	−	1.62575i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−83742.0	−	145045.i	−0.246873	−	0.427597i	0.715783	−	0.698323i	$-0.246070\pi$
$163$	−83742.0	−	145045.i	−0.246873	−	0.427597i	−0.962657	+	0.270725i	$0.912737\pi$
$164$		0			0
$165$	−9728.00	+	16849.4i	−0.0278172	+	0.0481808i
$166$		0			0
$167$	206624.			0.573310			0.286655	−	0.958034i	$-0.407457\pi$
$167$	206624.			0.573310			0.286655	+	0.958034i	$0.407457\pi$
$168$		0			0
$169$	403107.			1.08568
$170$		0			0
$171$	−12584.0	+	21796.1i	−0.0329100	+	0.0570019i
$172$		0			0
$173$	169928.	+	294324.i	0.431668	+	0.747671i	0.997017	−	0.0771810i	$-0.0245919\pi$
$173$	169928.	+	294324.i	0.431668	+	0.747671i	−0.565349	+	0.824852i	$0.691259\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	169088.	+	292869.i	0.405676	+	0.702652i
$178$		0			0
$179$	−259770.	+	449935.i	−0.605977	+	1.04958i	0.385919	+	0.922533i	$0.373884\pi$
$179$	−259770.	+	449935.i	−0.605977	+	1.04958i	−0.991896	+	0.127051i	$0.959449\pi$
$180$		0			0
$181$	830000.			1.88314			0.941568	−	0.336823i	$-0.109352\pi$
$181$	830000.			1.88314			0.941568	+	0.336823i	$0.109352\pi$
$182$		0			0
$183$	47872.0			0.105671
$184$		0			0
$185$	−39504.0	+	68422.9i	−0.0848617	+	0.146985i
$186$		0			0
$187$	−40128.0	−	69503.7i	−0.0839158	−	0.145346i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−150804.	−	261200.i	−0.299109	−	0.518072i	0.676823	−	0.736145i	$-0.263356\pi$
$191$	−150804.	−	261200.i	−0.299109	−	0.518072i	−0.975932	+	0.218074i	$0.930023\pi$
$192$		0			0
$193$	242231.	−	419556.i	0.468098	−	0.810769i	−0.531238	−	0.847223i	$-0.678273\pi$
$193$	242231.	−	419556.i	0.468098	−	0.810769i	0.999335	+	0.0364539i	$0.0116062\pi$
$194$		0			0
$195$	−225280.			−0.424264
$196$		0			0
$197$	−183018.			−0.335991			−0.167996	−	0.985788i	$-0.553729\pi$
$197$	−183018.			−0.335991			−0.167996	+	0.985788i	$0.553729\pi$
$198$		0			0
$199$	452144.	−	783136.i	0.809364	−	1.40186i	−0.103940	−	0.994584i	$-0.533145\pi$
$199$	452144.	−	783136.i	0.809364	−	1.40186i	0.913305	−	0.407277i	$-0.133522\pi$
$200$		0			0
$201$	−366688.	−	635122.i	−0.640187	−	1.10884i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	126720.	+	219485.i	0.210601	+	0.364772i
$206$		0			0
$207$	−6084.00	+	10537.8i	−0.00986878	+	0.0170932i
$208$		0			0
$209$	−147136.			−0.232999
$210$		0			0
$211$	−494428.			−0.764534			−0.382267	−	0.924052i	$-0.624857\pi$
$211$	−494428.			−0.764534			−0.382267	+	0.924052i	$0.624857\pi$
$212$		0			0
$213$	−398720.	+	690603.i	−0.602170	+	1.04299i
$214$		0			0
$215$	131296.	+	227411.i	0.193711	+	0.335518i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−450560.	−	780393.i	−0.634808	−	1.09952i
$220$		0			0
$221$	464640.	−	804780.i	0.639935	−	1.10840i
$222$		0			0
$223$	−389824.			−0.524936			−0.262468	−	0.964941i	$-0.584536\pi$
$223$	−389824.			−0.524936			−0.262468	+	0.964941i	$0.584536\pi$
$224$		0			0
$225$	−37297.0			−0.0491154
$226$		0			0
$227$	39512.0	−	68436.8i	0.0508937	−	0.0881505i	−0.839456	−	0.543427i	$-0.817126\pi$
$227$	39512.0	−	68436.8i	0.0508937	−	0.0881505i	0.890350	+	0.455277i	$0.150460\pi$
$228$		0			0
$229$	629896.	+	1.09101e6i	0.793743	+	1.37480i	0.923634	+	0.383276i	$0.125204\pi$
$229$	629896.	+	1.09101e6i	0.793743	+	1.37480i	−0.129891	+	0.991528i	$0.541463\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	386309.	+	669107.i	0.466171	+	0.807431i	0.999254	−	0.0386317i	$-0.0122999\pi$
$233$	386309.	+	669107.i	0.466171	+	0.807431i	−0.533083	+	0.846063i	$0.678967\pi$
$234$		0			0
$235$	166144.	−	287770.i	0.196252	−	0.339919i
$236$		0			0
$237$	−651904.			−0.753898
$238$		0			0
$239$	−1.42507e6			−1.61377			−0.806886	−	0.590707i	$-0.798849\pi$
$239$	−1.42507e6			−1.61377			−0.806886	+	0.590707i	$0.798849\pi$
$240$		0			0
$241$	−574288.	+	994696.i	−0.636923	+	1.10318i	0.349181	+	0.937055i	$0.386460\pi$
$241$	−574288.	+	994696.i	−0.636923	+	1.10318i	−0.986104	+	0.166128i	$0.946873\pi$
$242$		0			0
$243$	−49192.0	−	85203.0i	−0.0534415	−	0.0925634i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−851840.	−	1.47543e6i	−0.888415	−	1.53878i
$248$		0			0
$249$	899712.	−	1.55835e6i	0.919613	−	1.59282i
$250$		0			0
$251$	−278096.			−0.278619			−0.139309	−	0.990249i	$-0.544488\pi$
$251$	−278096.			−0.278619			−0.139309	+	0.990249i	$0.544488\pi$
$252$		0			0
$253$	−71136.0			−0.0698696
$254$		0			0
$255$	−135168.	+	234118.i	−0.130174	+	0.225468i
$256$		0			0
$257$	−178816.	−	309718.i	−0.168878	−	0.292506i	0.769148	−	0.639071i	$-0.220681\pi$
$257$	−178816.	−	309718.i	−0.168878	−	0.292506i	−0.938026	+	0.346566i	$0.887348\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	25883.0	+	44830.7i	0.0235187	+	0.0407356i
$262$		0			0
$263$	42416.0	−	73466.7i	0.0378129	−	0.0654939i	−0.846500	−	0.532389i	$-0.821294\pi$
$263$	42416.0	−	73466.7i	0.0378129	−	0.0654939i	0.884313	+	0.466895i	$0.154628\pi$
$264$		0			0
$265$	−598432.			−0.523480
$266$		0			0
$267$	−1.02810e6			−0.882583
$268$		0			0
$269$	1.09886e6	−	1.90327e6i	0.925891	−	1.60369i	0.135770	−	0.990740i	$-0.456649\pi$
$269$	1.09886e6	−	1.90327e6i	0.925891	−	1.60369i	0.790121	−	0.612951i	$-0.210018\pi$
$270$		0			0
$271$	237952.	+	412145.i	0.196819	+	0.340900i	0.947495	−	0.319770i	$-0.103606\pi$
$271$	237952.	+	412145.i	0.196819	+	0.340900i	−0.750677	+	0.660670i	$0.770272\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−109022.	−	188832.i	−0.0869325	−	0.150572i
$276$		0			0
$277$	797005.	−	1.38045e6i	0.624111	−	1.08099i	−0.364601	−	0.931164i	$-0.618795\pi$
$277$	797005.	−	1.38045e6i	0.624111	−	1.08099i	0.988712	−	0.149828i	$-0.0478719\pi$
$278$		0			0
$279$	20384.0			0.0156776
$280$		0			0
$281$	−1.31558e6			−0.993919			−0.496959	−	0.867774i	$-0.665550\pi$
$281$	−1.31558e6			−0.993919			−0.496959	+	0.867774i	$0.665550\pi$
$282$		0			0
$283$	−504328.	+	873522.i	−0.374323	+	0.648347i	−0.990226	−	0.139476i	$-0.955458\pi$
$283$	−504328.	+	873522.i	−0.374323	+	0.648347i	0.615902	+	0.787823i	$0.288792\pi$
$284$		0			0
$285$	247808.	+	429216.i	0.180719	+	0.313014i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	152360.	+	263896.i	0.107307	+	0.185861i
$290$		0			0
$291$	−18176.0	+	31481.8i	−0.0125825	+	0.0217935i
$292$		0			0
$293$	2.05762e6			1.40022			0.700108	−	0.714037i	$-0.253135\pi$
$293$	2.05762e6			1.40022			0.700108	+	0.714037i	$0.253135\pi$
$294$		0			0
$295$	338176.			0.226250
$296$		0			0
$297$	139840.	−	242210.i	0.0919899	−	0.159331i
$298$		0			0
$299$	−411840.	−	713328.i	−0.266410	−	0.461436i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−880000.	−	1.52420e6i	−0.550651	−	0.953755i
$304$		0			0
$305$	23936.0	−	41458.4i	0.0147334	−	0.0255189i
$306$		0			0
$307$	2.07099e6			1.25410			0.627050	−	0.778979i	$-0.284262\pi$
$307$	2.07099e6			1.25410			0.627050	+	0.778979i	$0.284262\pi$
$308$		0			0
$309$	1.34605e6			0.801981
$310$		0			0
$311$	−476608.	+	825509.i	−0.279422	+	0.483973i	−0.971241	−	0.238098i	$-0.923476\pi$
$311$	−476608.	+	825509.i	−0.279422	+	0.483973i	0.691819	+	0.722071i	$0.256809\pi$
$312$		0			0
$313$	−1.42920e6	−	2.47545e6i	−0.824579	−	1.42821i	−0.902241	−	0.431233i	$-0.858079\pi$
$313$	−1.42920e6	−	2.47545e6i	−0.824579	−	1.42821i	0.0776620	−	0.996980i	$-0.475255\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	993801.	+	1.72131e6i	0.555458	+	0.962082i	0.997868	+	0.0652685i	$0.0207904\pi$
$317$	993801.	+	1.72131e6i	0.555458	+	0.962082i	−0.442410	+	0.896813i	$0.645876\pi$
$318$		0			0
$319$	−151316.	+	262087.i	−0.0832546	+	0.144201i
$320$		0			0
$321$	−223168.			−0.120884
$322$		0			0
$323$	−2.04442e6			−1.09034
$324$		0			0
$325$	1.26236e6	−	2.18647e6i	0.662941	−	1.14825i
$326$		0			0
$327$	180752.	+	313072.i	0.0934789	+	0.161910i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−197290.	−	341716.i	−0.0989772	−	0.171434i	0.812284	−	0.583262i	$-0.198224\pi$
$331$	−197290.	−	341716.i	−0.0989772	−	0.171434i	−0.911262	+	0.411828i	$0.864890\pi$
$332$		0			0
$333$	−32097.0	+	55593.6i	−0.0158619	+	0.0274735i
$334$		0			0
$335$	−733376.			−0.357038
$336$		0			0
$337$	2.19606e6			1.05334			0.526672	−	0.850069i	$-0.323440\pi$
$337$	2.19606e6			1.05334			0.526672	+	0.850069i	$0.323440\pi$
$338$		0			0
$339$	758288.	−	1.31339e6i	0.358373	−	0.620720i
$340$		0			0
$341$	59584.0	+	103203.i	0.0277488	+	0.0480623i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	119808.	+	207514.i	0.0541924	+	0.0938639i
$346$		0			0
$347$	−2.01359e6	+	3.48765e6i	−0.897735	+	1.55492i	−0.0673525	+	0.997729i	$0.521455\pi$
$347$	−2.01359e6	+	3.48765e6i	−0.897735	+	1.55492i	−0.830383	+	0.557194i	$0.811878\pi$
$348$		0			0
$349$	−1.96469e6			−0.863436			−0.431718	−	0.902009i	$-0.642092\pi$
$349$	−1.96469e6			−0.863436			−0.431718	+	0.902009i	$0.642092\pi$
$350$		0			0
$351$	3.23840e6			1.40302
$352$		0			0
$353$	−1.67261e6	+	2.89704e6i	−0.714426	+	1.23742i	0.248754	+	0.968567i	$0.419979\pi$
$353$	−1.67261e6	+	2.89704e6i	−0.714426	+	1.23742i	−0.963180	+	0.268856i	$0.913354\pi$
$354$		0			0
$355$	398720.	+	690603.i	0.167918	+	0.290842i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.14756e6	+	1.98764e6i	0.469938	+	0.813957i	0.999409	−	0.0343712i	$-0.0109428\pi$
$359$	1.14756e6	+	1.98764e6i	0.469938	+	0.813957i	−0.529471	+	0.848328i	$0.677610\pi$
$360$		0			0
$361$	−635998.	+	1.10158e6i	−0.256855	+	0.444886i
$362$		0			0
$363$	2.48440e6			0.989589
$364$		0			0
$365$	−901120.			−0.354038
$366$		0			0
$367$	−1.46934e6	+	2.54498e6i	−0.569454	+	0.986323i	0.427166	+	0.904173i	$0.359512\pi$
$367$	−1.46934e6	+	2.54498e6i	−0.569454	+	0.986323i	−0.996620	+	0.0821495i	$0.973821\pi$
$368$		0			0
$369$	102960.	+	178332.i	0.0393643	+	0.0681810i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	207229.	+	358931.i	0.0771220	+	0.133579i	0.902007	−	0.431721i	$-0.142094\pi$
$373$	207229.	+	358931.i	0.0771220	+	0.133579i	−0.824885	+	0.565300i	$0.808760\pi$
$374$		0			0
$375$	−767232.	+	1.32888e6i	−0.281740	+	0.487988i
$376$		0			0
$377$	−3.50416e6			−1.26979
$378$		0			0
$379$	2.57111e6			0.919438			0.459719	−	0.888065i	$-0.347950\pi$
$379$	2.57111e6			0.919438			0.459719	+	0.888065i	$0.347950\pi$
$380$		0			0
$381$	1.12499e6	−	1.94854e6i	0.397042	−	0.687698i
$382$		0			0
$383$	1.28366e6	+	2.22337e6i	0.447151	+	0.774489i	0.998199	−	0.0599849i	$-0.0191052\pi$
$383$	1.28366e6	+	2.22337e6i	0.447151	+	0.774489i	−0.551048	+	0.834474i	$0.685772\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	106678.	+	184772.i	0.0362074	+	0.0627131i
$388$		0			0
$389$	−1.68904e6	+	2.92551e6i	−0.565936	+	0.980230i	0.431026	+	0.902339i	$0.358152\pi$
$389$	−1.68904e6	+	2.92551e6i	−0.565936	+	0.980230i	−0.996962	+	0.0778902i	$0.975182\pi$
$390$		0			0
$391$	−988416.			−0.326962
$392$		0			0
$393$	3.87763e6			1.26644
$394$		0			0
$395$	−325952.	+	564565.i	−0.105114	+	0.182063i
$396$		0			0
$397$	1.09886e6	+	1.90327e6i	0.349917	+	0.606073i	0.986234	−	0.165354i	$-0.0528766\pi$
$397$	1.09886e6	+	1.90327e6i	0.349917	+	0.606073i	−0.636318	+	0.771427i	$0.719543\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.73835e6	+	3.01092e6i	0.539855	+	0.935056i	0.998911	+	0.0466490i	$0.0148542\pi$
$401$	1.73835e6	+	3.01092e6i	0.539855	+	0.935056i	−0.459056	+	0.888407i	$0.651812\pi$
$402$		0			0
$403$	−689920.	+	1.19498e6i	−0.211610	+	0.366519i
$404$		0			0
$405$	−992624.			−0.300710
$406$		0			0
$407$	−375288.			−0.112300
$408$		0			0
$409$	842160.	−	1.45866e6i	0.248935	−	0.431168i	−0.714295	−	0.699844i	$-0.753253\pi$
$409$	842160.	−	1.45866e6i	0.248935	−	0.431168i	0.963231	+	0.268676i	$0.0865860\pi$
$410$		0			0
$411$	1.38309e6	+	2.39558e6i	0.403873	+	0.699529i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−899712.	−	1.55835e6i	−0.256439	−	0.444165i
$416$		0			0
$417$	−1.33901e6	+	2.31923e6i	−0.377088	+	0.653136i
$418$		0			0
$419$	−4.40475e6			−1.22571			−0.612853	−	0.790197i	$-0.709978\pi$
$419$	−4.40475e6			−1.22571			−0.612853	+	0.790197i	$0.709978\pi$
$420$		0			0
$421$	3.06601e6			0.843078			0.421539	−	0.906810i	$-0.361490\pi$
$421$	3.06601e6			0.843078			0.421539	+	0.906810i	$0.361490\pi$
$422$		0			0
$423$	134992.	−	233813.i	0.0366823	−	0.0635357i
$424$		0			0
$425$	−1.51483e6	−	2.62377e6i	−0.406810	−	0.704616i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−535040.	−	926716.i	−0.140360	−	0.243110i
$430$		0			0
$431$	−660176.	+	1.14346e6i	−0.171185	+	0.296502i	−0.938835	−	0.344369i	$-0.888093\pi$
$431$	−660176.	+	1.14346e6i	−0.171185	+	0.296502i	0.767649	+	0.640870i	$0.221426\pi$
$432$		0			0
$433$	2.91510e6			0.747196			0.373598	−	0.927591i	$-0.378124\pi$
$433$	2.91510e6			0.747196			0.373598	+	0.927591i	$0.378124\pi$
$434$		0			0
$435$	1.01939e6			0.258296
$436$		0			0
$437$	−906048.	+	1.56932e6i	−0.226959	+	0.393105i
$438$		0			0
$439$	−3.53126e6	−	6.11633e6i	−0.874518	−	1.51471i	−0.857275	−	0.514859i	$-0.827844\pi$
$439$	−3.53126e6	−	6.11633e6i	−0.874518	−	1.51471i	−0.0172432	−	0.999851i	$-0.505489\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.86906e6	+	6.70141e6i	0.936690	+	1.62239i	0.771592	+	0.636118i	$0.219461\pi$
$443$	3.86906e6	+	6.70141e6i	0.936690	+	1.62239i	0.165098	+	0.986277i	$0.447206\pi$
$444$		0			0
$445$	−514048.	+	890357.i	−0.123056	+	0.213140i
$446$		0			0
$447$	6.47363e6			1.53242
$448$		0			0
$449$	−5.62457e6			−1.31666			−0.658330	−	0.752729i	$-0.728737\pi$
$449$	−5.62457e6			−1.31666			−0.658330	+	0.752729i	$0.728737\pi$
$450$		0			0
$451$	−601920.	+	1.04256e6i	−0.139347	+	0.241356i
$452$		0			0
$453$	3.56154e6	+	6.16876e6i	0.815440	+	1.41238i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2.95758e6	−	5.12268e6i	−0.662439	−	1.14738i	−0.979973	−	0.199132i	$-0.936188\pi$
$457$	−2.95758e6	−	5.12268e6i	−0.662439	−	1.14738i	0.317533	−	0.948247i	$-0.397145\pi$
$458$		0			0
$459$	1.94304e6	−	3.36544e6i	0.430477	−	0.745608i
$460$		0			0
$461$	7.21195e6			1.58052			0.790261	−	0.612770i	$-0.209945\pi$
$461$	7.21195e6			1.58052			0.790261	+	0.612770i	$0.209945\pi$
$462$		0			0
$463$	5.22092e6			1.13186			0.565932	−	0.824452i	$-0.308516\pi$
$463$	5.22092e6			1.13186			0.565932	+	0.824452i	$0.308516\pi$
$464$		0			0
$465$	200704.	−	347630.i	0.0430451	−	0.0745563i
$466$		0			0
$467$	−2.90769e6	−	5.03626e6i	−0.616958	−	1.06860i	−0.990038	−	0.140803i	$-0.955031\pi$
$467$	−2.90769e6	−	5.03626e6i	−0.616958	−	1.06860i	0.373080	−	0.927799i	$-0.378302\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1.37997e6	−	2.39017e6i	−0.286627	−	0.496452i
$472$		0			0
$473$	−623656.	+	1.08020e6i	−0.128172	+	0.222000i
$474$		0			0
$475$	−5.55438e6			−1.12954
$476$		0			0
$477$	−486226.			−0.0978458
$478$		0			0
$479$	3.76939e6	−	6.52878e6i	0.750641	−	1.30015i	−0.196871	−	0.980429i	$-0.563078\pi$
$479$	3.76939e6	−	6.52878e6i	0.750641	−	1.30015i	0.947512	−	0.319720i	$-0.103589\pi$
$480$		0			0
$481$	−2.17272e6	−	3.76326e6i	−0.428194	−	0.741655i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	18176.0	+	31481.8i	0.00350868	+	0.00607721i
$486$		0			0
$487$	3.29830e6	−	5.71282e6i	0.630185	−	1.09151i	−0.357329	−	0.933979i	$-0.616313\pi$
$487$	3.29830e6	−	5.71282e6i	0.630185	−	1.09151i	0.987514	−	0.157533i	$-0.0503541\pi$
$488$		0			0
$489$	−2.67974e6			−0.506782
$490$		0			0
$491$	−9.96666e6			−1.86572			−0.932859	−	0.360242i	$-0.882694\pi$
$491$	−9.96666e6			−1.86572			−0.932859	+	0.360242i	$0.882694\pi$
$492$		0			0
$493$	−2.10250e6	+	3.64163e6i	−0.389599	+	0.674805i
$494$		0			0
$495$	7904.00	+	13690.1i	0.00144989	+	0.00251128i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.47135e6	+	2.54845e6i	0.264524	+	0.458169i	0.967439	−	0.253105i	$-0.0814519\pi$
$499$	1.47135e6	+	2.54845e6i	0.264524	+	0.458169i	−0.702915	+	0.711274i	$0.748119\pi$
$500$		0			0
$501$	1.65299e6	−	2.86307e6i	0.294223	−	0.509609i
$502$		0			0
$503$	4.35142e6			0.766852			0.383426	−	0.923572i	$-0.374744\pi$
$503$	4.35142e6			0.766852			0.383426	+	0.923572i	$0.374744\pi$
$504$		0			0
$505$	−1.76000e6			−0.307103
$506$		0			0
$507$	3.22486e6	−	5.58561e6i	0.557173	−	0.965053i
$508$		0			0
$509$	−2.59247e6	−	4.49029e6i	−0.443527	−	0.768211i	0.554422	−	0.832236i	$-0.312939\pi$
$509$	−2.59247e6	−	4.49029e6i	−0.443527	−	0.768211i	−0.997948	+	0.0640254i	$0.979606\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−3.56224e6	−	6.16998e6i	−0.597626	−	1.03512i
$514$		0			0
$515$	673024.	−	1.16571e6i	0.111818	−	0.193675i
$516$		0			0
$517$	1.57837e6			0.259706
$518$		0			0
$519$	5.43770e6			0.886128
$520$		0			0
$521$	−3.70840e6	+	6.42314e6i	−0.598539	+	1.03670i	0.394498	+	0.918897i	$0.370918\pi$
$521$	−3.70840e6	+	6.42314e6i	−0.598539	+	1.03670i	−0.993037	+	0.117803i	$0.962415\pi$
$522$		0			0
$523$	3.23673e6	+	5.60618e6i	0.517430	+	0.896216i	0.999795	+	0.0202453i	$0.00644471\pi$
$523$	3.23673e6	+	5.60618e6i	0.517430	+	0.896216i	−0.482365	+	0.875971i	$0.660222\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	827904.	+	1.43397e6i	0.129853	+	0.224913i
$528$		0			0
$529$	2.78012e6	−	4.81532e6i	0.431941	−	0.748145i
$530$		0			0
$531$	274768.			0.0422892
$532$		0			0
$533$	−1.39392e7			−2.12530
$534$		0			0
$535$	−111584.	+	193269.i	−0.0168546	+	0.0291930i
$536$		0			0
$537$	4.15632e6	+	7.19896e6i	0.621975	+	1.07729i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−4.55494e6	−	7.88938e6i	−0.669097	−	1.15891i	−0.978157	−	0.207867i	$-0.933348\pi$
$541$	−4.55494e6	−	7.88938e6i	−0.669097	−	1.15891i	0.309060	−	0.951043i	$-0.399986\pi$
$542$		0			0
$543$	6.64000e6	−	1.15008e7i	0.966426	−	1.67390i
$544$		0			0
$545$	361504.			0.0521341
$546$		0			0
$547$	−5.63776e6			−0.805635			−0.402818	−	0.915280i	$-0.631969\pi$
$547$	−5.63776e6			−0.805635			−0.402818	+	0.915280i	$0.631969\pi$
$548$		0			0
$549$	19448.0	−	33684.9i	0.00275387	−	0.00476985i
$550$		0			0
$551$	3.85458e6	+	6.67632e6i	0.540876	+	0.936825i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	632064.	+	1.09477e6i	0.0871021	+	0.150865i
$556$		0			0
$557$	−1.85536e6	+	3.21358e6i	−0.253390	+	0.438885i	−0.964457	−	0.264240i	$-0.914879\pi$
$557$	−1.85536e6	+	3.21358e6i	−0.253390	+	0.438885i	0.711067	+	0.703124i	$0.248212\pi$
$558$		0			0
$559$	−1.44426e7			−1.95486
$560$		0			0
$561$	−1.28410e6			−0.172262
$562$		0			0
$563$	−2.63058e6	+	4.55631e6i	−0.349769	+	0.605818i	−0.986208	−	0.165509i	$-0.947073\pi$
$563$	−2.63058e6	+	4.55631e6i	−0.349769	+	0.605818i	0.636439	+	0.771327i	$0.280407\pi$
$564$		0			0
$565$	−758288.	−	1.31339e6i	−0.0999340	−	0.173091i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.40464e6	+	2.43292e6i	0.181880	+	0.315026i	0.942521	−	0.334147i	$-0.108448\pi$
$569$	1.40464e6	+	2.43292e6i	0.181880	+	0.315026i	−0.760640	+	0.649173i	$0.775115\pi$
$570$		0			0
$571$	4.09853e6	−	7.09887e6i	0.526064	−	0.911169i	−0.473475	−	0.880807i	$-0.657001\pi$
$571$	4.09853e6	−	7.09887e6i	0.526064	−	0.911169i	0.999539	−	0.0303619i	$-0.00966596\pi$
$572$		0			0
$573$	−4.82573e6			−0.614011
$574$		0			0
$575$	−2.68538e6			−0.338717
$576$		0			0
$577$	1.84861e6	−	3.20188e6i	0.231156	−	0.400374i	−0.726993	−	0.686645i	$-0.759083\pi$
$577$	1.84861e6	−	3.20188e6i	0.231156	−	0.400374i	0.958149	+	0.286271i	$0.0924159\pi$
$578$		0			0
$579$	−3.87570e6	−	6.71290e6i	−0.480456	−	0.832174i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.42128e6	−	2.46172e6i	−0.173184	−	0.299963i
$584$		0			0
$585$	−91520.0	+	158517.i	−0.0110567	+	0.0191508i
$586$		0			0
$587$	−8.25282e6			−0.988569			−0.494284	−	0.869300i	$-0.664570\pi$
$587$	−8.25282e6			−0.988569			−0.494284	+	0.869300i	$0.664570\pi$
$588$		0			0
$589$	3.03565e6			0.360548
$590$		0			0
$591$	−1.46414e6	+	2.53597e6i	−0.172431	+	0.298659i
$592$		0			0
$593$	3.15638e6	+	5.46702e6i	0.368598	+	0.638431i	0.989347	−	0.145579i	$-0.0465045\pi$
$593$	3.15638e6	+	5.46702e6i	0.368598	+	0.638431i	−0.620749	+	0.784010i	$0.713171\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.23430e6	−	1.25302e7i	−0.830732	−	1.43887i
$598$		0			0
$599$	3.46488e6	−	6.00135e6i	0.394567	−	0.683411i	−0.598479	−	0.801139i	$-0.704228\pi$
$599$	3.46488e6	−	6.00135e6i	0.394567	−	0.683411i	0.993046	+	0.117728i	$0.0375611\pi$
$600$		0			0
$601$	1.10092e7			1.24328			0.621638	−	0.783305i	$-0.286467\pi$
$601$	1.10092e7			1.24328			0.621638	+	0.783305i	$0.286467\pi$
$602$		0			0
$603$	−595868.			−0.0667355
$604$		0			0
$605$	1.24220e6	−	2.15155e6i	0.137976	−	0.238981i
$606$		0			0
$607$	166848.	+	288989.i	0.0183802	+	0.0318354i	0.875069	−	0.483998i	$-0.160816\pi$
$607$	166848.	+	288989.i	0.0183802	+	0.0318354i	−0.856689	+	0.515833i	$0.827482\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.13792e6	+	1.58273e7i	0.990249	+	1.71516i
$612$		0			0
$613$	−879637.	+	1.52358e6i	−0.0945480	+	0.163762i	−0.909420	−	0.415879i	$-0.863474\pi$
$613$	−879637.	+	1.52358e6i	−0.0945480	+	0.163762i	0.814872	+	0.579641i	$0.196807\pi$
$614$		0			0
$615$	4.05504e6			0.432322
$616$		0			0
$617$	−1.46142e7			−1.54548			−0.772738	−	0.634725i	$-0.781113\pi$
$617$	−1.46142e7			−1.54548			−0.772738	+	0.634725i	$0.781113\pi$
$618$		0			0
$619$	1.15772e6	−	2.00523e6i	0.121444	−	0.210348i	−0.798893	−	0.601473i	$-0.794581\pi$
$619$	1.15772e6	−	2.00523e6i	0.121444	−	0.210348i	0.920337	+	0.391125i	$0.127914\pi$
$620$		0			0
$621$	−1.72224e6	−	2.98301e6i	−0.179211	−	0.310403i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−3.71558e6	−	6.43557e6i	−0.380475	−	0.659003i
$626$		0			0
$627$	−1.17709e6	+	2.03878e6i	−0.119575	+	0.207110i
$628$		0			0
$629$	−5.21453e6			−0.525519
$630$		0			0
$631$	1.47152e7			1.47127			0.735636	−	0.677377i	$-0.236883\pi$
$631$	1.47152e7			1.47127			0.735636	+	0.677377i	$0.236883\pi$
$632$		0			0
$633$	−3.95542e6	+	6.85100e6i	−0.392359	+	0.679586i
$634$		0			0
$635$	−1.12499e6	−	1.94854e6i	−0.110717	−	0.191768i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	323960.	+	561115.i	0.0313862	+	0.0543626i
$640$		0			0
$641$	−5.37775e6	+	9.31454e6i	−0.516958	+	0.895398i	0.482848	+	0.875704i	$0.339602\pi$
$641$	−5.37775e6	+	9.31454e6i	−0.516958	+	0.895398i	−0.999806	+	0.0196939i	$0.993731\pi$
$642$		0			0
$643$	1.81097e7			1.72736			0.863681	−	0.504039i	$-0.168153\pi$
$643$	1.81097e7			1.72736			0.863681	+	0.504039i	$0.168153\pi$
$644$		0			0
$645$	4.20147e6			0.397651
$646$		0			0
$647$	−1.93838e6	+	3.35738e6i	−0.182045	+	0.315311i	−0.942577	−	0.333989i	$-0.891605\pi$
$647$	−1.93838e6	+	3.35738e6i	−0.182045	+	0.315311i	0.760532	+	0.649301i	$0.224938\pi$
$648$		0			0
$649$	803168.	+	1.39113e6i	0.0748505	+	0.129645i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.91278e6	+	5.04509e6i	0.267316	+	0.463005i	0.968168	−	0.250302i	$-0.0805299\pi$
$653$	2.91278e6	+	5.04509e6i	0.267316	+	0.463005i	−0.700852	+	0.713307i	$0.747197\pi$
$654$		0			0
$655$	1.93882e6	−	3.35813e6i	0.176577	−	0.305840i
$656$		0			0
$657$	−732160.			−0.0661748
$658$		0			0
$659$	−6.46989e6			−0.580341			−0.290171	−	0.956975i	$-0.593712\pi$
$659$	−6.46989e6			−0.580341			−0.290171	+	0.956975i	$0.593712\pi$
$660$		0			0
$661$	2.09001e6	−	3.62000e6i	0.186056	−	0.322259i	−0.757876	−	0.652399i	$-0.773763\pi$
$661$	2.09001e6	−	3.62000e6i	0.186056	−	0.322259i	0.943932	+	0.330140i	$0.107096\pi$
$662$		0			0
$663$	−7.43424e6	−	1.28765e7i	−0.656830	−	1.13766i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.86358e6	+	3.22781e6i	0.162193	+	0.280927i
$668$		0			0
$669$	−3.11859e6	+	5.40156e6i	−0.269397	+	0.466610i
$670$		0			0
$671$	227392.			0.0194970
$672$		0			0
$673$	1.27159e7			1.08221			0.541104	−	0.840956i	$-0.318007\pi$
$673$	1.27159e7			1.08221			0.541104	+	0.840956i	$0.318007\pi$
$674$		0			0
$675$	5.27896e6	−	9.14343e6i	0.445953	−	0.772413i
$676$		0			0
$677$	−5.93551e6	−	1.02806e7i	−0.497722	−	0.862079i	0.502275	−	0.864708i	$-0.332497\pi$
$677$	−5.93551e6	−	1.02806e7i	−0.497722	−	0.862079i	−0.999997	+	0.00262884i	$0.999163\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−632192.	−	1.09499e6i	−0.0522374	−	0.0904778i
$682$		0			0
$683$	−6.99889e6	+	1.21224e7i	−0.574087	+	0.994348i	0.422053	+	0.906571i	$0.361310\pi$
$683$	−6.99889e6	+	1.21224e7i	−0.574087	+	0.994348i	−0.996140	+	0.0877767i	$0.972024\pi$
$684$		0			0
$685$	2.76618e6			0.225244
$686$		0			0
$687$	2.01567e7			1.62940
$688$		0			0
$689$	1.64569e7	−	2.85042e7i	1.32069	−	2.28749i
$690$		0			0
$691$	4.90018e6	+	8.48737e6i	0.390407	+	0.676204i	0.992503	−	0.122219i	$-0.0390011\pi$
$691$	4.90018e6	+	8.48737e6i	0.390407	+	0.676204i	−0.602096	+	0.798423i	$0.705668\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.33901e6	+	2.31923e6i	0.105153	+	0.182130i
$696$		0			0
$697$	−8.36352e6	+	1.44860e7i	−0.652089	+	1.12945i
$698$		0			0
$699$	1.23619e7			0.956956
$700$		0			0
$701$	−1.80481e6			−0.138719			−0.0693597	−	0.997592i	$-0.522096\pi$
$701$	−1.80481e6			−0.138719			−0.0693597	+	0.997592i	$0.522096\pi$
$702$		0			0
$703$	−4.77998e6	+	8.27918e6i	−0.364786	+	0.631828i
$704$		0			0
$705$	−2.65830e6	−	4.60432e6i	−0.201434	−	0.348893i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	3.23702e6	+	5.60668e6i	0.241841	+	0.418881i	0.961239	−	0.275718i	$-0.0889155\pi$
$709$	3.23702e6	+	5.60668e6i	0.241841	+	0.418881i	−0.719398	+	0.694598i	$0.755582\pi$
$710$		0			0
$711$	−264836.	+	458709.i	−0.0196473	+	0.0340301i
$712$		0			0
$713$	1.46765e6			0.108118
$714$		0			0
$715$	−1.07008e6			−0.0782801
$716$		0			0
$717$	−1.14006e7	+	1.97464e7i	−0.828188	+	1.43446i
$718$		0			0
$719$	1.17032e7	+	2.02706e7i	0.844273	+	1.46232i	0.886251	+	0.463206i	$0.153301\pi$
$719$	1.17032e7	+	2.02706e7i	0.844273	+	1.46232i	−0.0419777	+	0.999119i	$0.513366\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	9.18861e6	+	1.59151e7i	0.653738	+	1.13231i
$724$		0			0
$725$	−5.71218e6	+	9.89378e6i	−0.403605	+	0.699065i
$726$		0			0
$727$	−491936.			−0.0345201			−0.0172601	−	0.999851i	$-0.505494\pi$
$727$	−491936.			−0.0345201			−0.0172601	+	0.999851i	$0.505494\pi$
$728$		0			0
$729$	1.35013e7			0.940931
$730$		0			0
$731$	−8.66554e6	+	1.50091e7i	−0.599794	+	1.03887i
$732$		0			0
$733$	−3.93210e6	−	6.81060e6i	−0.270312	−	0.468194i	0.698630	−	0.715483i	$-0.253793\pi$
$733$	−3.93210e6	−	6.81060e6i	−0.270312	−	0.468194i	−0.968942	+	0.247289i	$0.920460\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.74177e6	−	3.01683e6i	−0.118119	−	0.204589i
$738$		0			0
$739$	−3.46273e6	+	5.99763e6i	−0.233243	+	0.403988i	−0.958761	−	0.284215i	$-0.908267\pi$
$739$	−3.46273e6	+	5.99763e6i	−0.233243	+	0.403988i	0.725518	+	0.688203i	$0.241600\pi$
$740$		0			0
$741$	−2.72589e7			−1.82374
$742$		0			0
$743$	1.59306e7			1.05867			0.529333	−	0.848414i	$-0.322442\pi$
$743$	1.59306e7			1.05867			0.529333	+	0.848414i	$0.322442\pi$
$744$		0			0
$745$	3.23682e6	−	5.60633e6i	0.213662	−	0.370073i
$746$		0			0
$747$	−731016.	−	1.26616e6i	−0.0479320	−	0.0830206i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−5.00762e6	−	8.67346e6i	−0.323990	−	0.561167i	0.657317	−	0.753614i	$-0.271691\pi$
$751$	−5.00762e6	−	8.67346e6i	−0.323990	−	0.561167i	−0.981307	+	0.192446i	$0.938358\pi$
$752$		0			0
$753$	−2.22477e6	+	3.85341e6i	−0.142987	+	0.247661i
$754$		0			0
$755$	7.12307e6			0.454779
$756$		0			0
$757$	3.79619e6			0.240773			0.120387	−	0.992727i	$-0.461587\pi$
$757$	3.79619e6			0.240773			0.120387	+	0.992727i	$0.461587\pi$
$758$		0			0
$759$	−569088.	+	985689.i	−0.0358571	+	0.0621063i
$760$		0			0
$761$	5.17422e6	+	8.96202e6i	0.323880	+	0.560976i	0.981285	−	0.192561i	$-0.0616794\pi$
$761$	5.17422e6	+	8.96202e6i	0.323880	+	0.560976i	−0.657405	+	0.753537i	$0.728346\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	109824.	+	190221.i	0.00678490	+	0.0117518i
$766$		0			0
$767$	−9.29984e6	+	1.61078e7i	−0.570804	+	0.988662i
$768$		0			0
$769$	−1.51131e7			−0.921591			−0.460796	−	0.887506i	$-0.652436\pi$
$769$	−1.51131e7			−0.921591			−0.460796	+	0.887506i	$0.652436\pi$
$770$		0			0
$771$	−5.72211e6			−0.346673
$772$		0			0
$773$	2.83087e6	−	4.90321e6i	0.170401	−	0.295143i	−0.768159	−	0.640259i	$-0.778827\pi$
$773$	2.83087e6	−	4.90321e6i	0.170401	−	0.295143i	0.938560	+	0.345116i	$0.112160\pi$
$774$		0			0
$775$	2.24930e6	+	3.89589e6i	0.134522	+	0.232998i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.53331e7	+	2.65577e7i	0.905288	+	1.56801i
$780$		0			0
$781$	−1.89392e6	+	3.28037e6i	−0.111105	+	0.192440i
$782$		0			0
$783$	−1.46538e7			−0.854171
$784$		0			0
$785$	−2.75994e6			−0.159855
$786$		0

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)

\(f(q)\)	\(=\)	\(q+(8.00000 - 13.8564i) q^{3} +(-8.00000 - 13.8564i) q^{5} +(-6.50000 - 11.2583i) q^{9} +O(q^{10})\)	\(q+(8.00000 - 13.8564i) q^{3} +(-8.00000 - 13.8564i) q^{5} +(-6.50000 - 11.2583i) q^{9} +(38.0000 - 65.8179i) q^{11} +880.000 q^{13} -256.000 q^{15} +(528.000 - 914.523i) q^{17} +(-968.000 - 1676.63i) q^{19} +(-468.000 - 810.600i) q^{23} +(1434.50 - 2484.63i) q^{25} +3680.00 q^{27} -3982.00 q^{29} +(-784.000 + 1357.93i) q^{31} +(-608.000 - 1053.09i) q^{33} +(-2469.00 - 4276.43i) q^{37} +(7040.00 - 12193.6i) q^{39} -15840.0 q^{41} -16412.0 q^{43} +(-104.000 + 180.133i) q^{45} +(10384.0 + 17985.6i) q^{47} +(-8448.00 - 14632.4i) q^{51} +(18701.0 - 32391.1i) q^{53} -1216.00 q^{55} -30976.0 q^{57} +(-10568.0 + 18304.3i) q^{59} +(1496.00 + 2591.15i) q^{61} +(-7040.00 - 12193.6i) q^{65} +(22918.0 - 39695.1i) q^{67} -14976.0 q^{69} -49840.0 q^{71} +(28160.0 - 48774.6i) q^{73} +(-22952.0 - 39754.0i) q^{75} +(-20372.0 - 35285.3i) q^{79} +(31019.5 - 53727.4i) q^{81} +112464. q^{83} -16896.0 q^{85} +(-31856.0 + 55176.2i) q^{87} +(-32128.0 - 55647.3i) q^{89} +(12544.0 + 21726.8i) q^{93} +(-15488.0 + 26826.0i) q^{95} -2272.00 q^{97} -988.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{3} - 16 q^{5} - 13 q^{9}+O(q^{10}) \) 2 * q + 16 * q^3 - 16 * q^5 - 13 * q^9	\( 2 q + 16 q^{3} - 16 q^{5} - 13 q^{9} + 76 q^{11} + 1760 q^{13} - 512 q^{15} + 1056 q^{17} - 1936 q^{19} - 936 q^{23} + 2869 q^{25} + 7360 q^{27} - 7964 q^{29} - 1568 q^{31} - 1216 q^{33} - 4938 q^{37} + 14080 q^{39} - 31680 q^{41} - 32824 q^{43} - 208 q^{45} + 20768 q^{47} - 16896 q^{51} + 37402 q^{53} - 2432 q^{55} - 61952 q^{57} - 21136 q^{59} + 2992 q^{61} - 14080 q^{65} + 45836 q^{67} - 29952 q^{69} - 99680 q^{71} + 56320 q^{73} - 45904 q^{75} - 40744 q^{79} + 62039 q^{81} + 224928 q^{83} - 33792 q^{85} - 63712 q^{87} - 64256 q^{89} + 25088 q^{93} - 30976 q^{95} - 4544 q^{97} - 1976 q^{99}+O(q^{100}) \) 2 * q + 16 * q^3 - 16 * q^5 - 13 * q^9 + 76 * q^11 + 1760 * q^13 - 512 * q^15 + 1056 * q^17 - 1936 * q^19 - 936 * q^23 + 2869 * q^25 + 7360 * q^27 - 7964 * q^29 - 1568 * q^31 - 1216 * q^33 - 4938 * q^37 + 14080 * q^39 - 31680 * q^41 - 32824 * q^43 - 208 * q^45 + 20768 * q^47 - 16896 * q^51 + 37402 * q^53 - 2432 * q^55 - 61952 * q^57 - 21136 * q^59 + 2992 * q^61 - 14080 * q^65 + 45836 * q^67 - 29952 * q^69 - 99680 * q^71 + 56320 * q^73 - 45904 * q^75 - 40744 * q^79 + 62039 * q^81 + 224928 * q^83 - 33792 * q^85 - 63712 * q^87 - 64256 * q^89 + 25088 * q^93 - 30976 * q^95 - 4544 * q^97 - 1976 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more