LMFDB - Embedded newform 220.6.b.b.89.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [220,6,Mod(89,220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(220, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("220.89");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 220 = 2^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	220.b (of order $2$, degree $1$, 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:	$35.2844403589$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 2272 x^{12} + 1983198 x^{10} + 827062096 x^{8} + 165415157329 x^{6} + 13843733383152 x^{4} + \cdots + 27\!\cdots\!24 $ x^14 + 2272x^12 + 1983198x^10 + 827062096x^8 + 165415157329x^6 + 13843733383152x^4 + 402818574022128x^2 + 2766293773242624
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{2}\cdot 5^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.2
Root			$-24.5916i$ of defining polynomial
Character	$\chi$	$=$	220.89
Dual form			220.6.b.b.89.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-24.5916i q^{3} +(39.2421 + 39.8128i) q^{5} +68.6167i q^{7} -361.745 q^{9} +121.000 q^{11} -322.943i q^{13} +(979.058 - 965.025i) q^{15} +1078.94i q^{17} +2901.39 q^{19} +1687.39 q^{21} +1136.77i q^{23} +(-45.1117 + 3124.67i) q^{25} +2920.12i q^{27} -732.481 q^{29} +6971.80 q^{31} -2975.58i q^{33} +(-2731.82 + 2692.66i) q^{35} +3293.91i q^{37} -7941.68 q^{39} +15016.5 q^{41} -15377.2i q^{43} +(-14195.6 - 14402.1i) q^{45} -6002.58i q^{47} +12098.7 q^{49} +26532.8 q^{51} -35344.7i q^{53} +(4748.30 + 4817.34i) q^{55} -71349.6i q^{57} -20735.0 q^{59} -44901.6 q^{61} -24821.7i q^{63} +(12857.3 - 12673.0i) q^{65} +67912.8i q^{67} +27954.9 q^{69} +11219.8 q^{71} -51188.9i q^{73} +(76840.6 + 1109.37i) q^{75} +8302.62i q^{77} +49679.5 q^{79} -16093.8 q^{81} -1551.85i q^{83} +(-42955.5 + 42339.9i) q^{85} +18012.8i q^{87} +103366. q^{89} +22159.3 q^{91} -171447. i q^{93} +(113857. + 115512. i) q^{95} -541.081i q^{97} -43771.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 44 q^{5} - 1142 q^{9} + 1694 q^{11} + 326 q^{15} - 4540 q^{19} + 3824 q^{21} - 3816 q^{25} - 9972 q^{29} + 19076 q^{31} + 5136 q^{35} - 13616 q^{39} + 15052 q^{41} - 8374 q^{45} - 55346 q^{49} - 13380 q^{51}+ \cdots - 138182 q^{99}+O(q^{100}) $ 14 * q - 44 * q^5 - 1142 * q^9 + 1694 * q^11 + 326 * q^15 - 4540 * q^19 + 3824 * q^21 - 3816 * q^25 - 9972 * q^29 + 19076 * q^31 + 5136 * q^35 - 13616 * q^39 + 15052 * q^41 - 8374 * q^45 - 55346 * q^49 - 13380 * q^51 - 5324 * q^55 - 2108 * q^59 + 11660 * q^61 - 65664 * q^65 + 1620 * q^69 - 89284 * q^71 + 48362 * q^75 + 19432 * q^79 - 94714 * q^81 + 117404 * q^85 - 38836 * q^89 + 39632 * q^91 + 231336 * q^95 - 138182 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/220\mathbb{Z}\right)^\times$.

$n$	$101$	$111$	$177$
$\chi(n)$	$1$	$1$	$-1$

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
$3$		−	24.5916i		−	1.57755i	−0.614683	−	0.788775i	$-0.710716\pi$
$3$		−	24.5916i		−	1.57755i	0.614683	−	0.788775i	$-0.289284\pi$
$4$		0			0
$5$	39.2421	+	39.8128i	0.701984	+	0.712192i
$5$	39.2421	+	39.8128i	0.701984	+	0.712192i
$6$		0			0
$7$			68.6167i			0.529279i	0.964347	+	0.264640i	$0.0852530\pi$
$7$			68.6167i			0.529279i	−0.964347	+	0.264640i	$0.914747\pi$
$8$		0			0
$9$	−361.745			−1.48866
$10$		0			0
$11$	121.000			0.301511
$11$	121.000			0.301511
$12$		0			0
$13$		−	322.943i		−	0.529991i	−0.964250	−	0.264995i	$-0.914630\pi$
$13$		−	322.943i		−	0.529991i	0.964250	−	0.264995i	$-0.0853704\pi$
$14$		0			0
$15$	979.058	−	965.025i	1.12352	−	1.10741i
$16$		0			0
$17$			1078.94i			0.905471i	0.891645	+	0.452736i	$0.149552\pi$
$17$			1078.94i			0.905471i	−0.891645	+	0.452736i	$0.850448\pi$
$18$		0			0
$19$	2901.39			1.84383			0.921917	−	0.387387i	$-0.126622\pi$
$19$	2901.39			1.84383			0.921917	+	0.387387i	$0.126622\pi$
$20$		0			0
$21$	1687.39			0.834964
$22$		0			0
$23$			1136.77i			0.448076i	0.974580	+	0.224038i	$0.0719240\pi$
$23$			1136.77i			0.448076i	−0.974580	+	0.224038i	$0.928076\pi$
$24$		0			0
$25$	−45.1117	+	3124.67i	−0.0144357	+	0.999896i
$26$		0			0
$27$			2920.12i			0.770887i
$28$		0			0
$29$	−732.481			−0.161734			−0.0808670	−	0.996725i	$-0.525769\pi$
$29$	−732.481			−0.161734			−0.0808670	+	0.996725i	$0.525769\pi$
$30$		0			0
$31$	6971.80			1.30299			0.651495	−	0.758653i	$-0.274142\pi$
$31$	6971.80			1.30299			0.651495	+	0.758653i	$0.274142\pi$
$32$		0			0
$33$		−	2975.58i		−	0.475649i
$34$		0			0
$35$	−2731.82	+	2692.66i	−0.376948	+	0.371546i
$36$		0			0
$37$			3293.91i			0.395556i	0.980247	+	0.197778i	$0.0633725\pi$
$37$			3293.91i			0.395556i	−0.980247	+	0.197778i	$0.936627\pi$
$38$		0			0
$39$	−7941.68			−0.836086
$40$		0			0
$41$	15016.5			1.39511			0.697555	−	0.716531i	$-0.254271\pi$
$41$	15016.5			1.39511			0.697555	+	0.716531i	$0.254271\pi$
$42$		0			0
$43$		−	15377.2i		−	1.26825i	−0.773229	−	0.634127i	$-0.781360\pi$
$43$		−	15377.2i		−	1.26825i	0.773229	−	0.634127i	$-0.218640\pi$
$44$		0			0
$45$	−14195.6	−	14402.1i	−1.04502	−	1.06021i
$46$		0			0
$47$		−	6002.58i		−	0.396363i	−0.980165	−	0.198182i	$-0.936496\pi$
$47$		−	6002.58i		−	0.396363i	0.980165	−	0.198182i	$-0.0635036\pi$
$48$		0			0
$49$	12098.7			0.719864
$50$		0			0
$51$	26532.8			1.42842
$52$		0			0
$53$		−	35344.7i		−	1.72836i	−0.503183	−	0.864180i	$-0.667838\pi$
$53$		−	35344.7i		−	1.72836i	0.503183	−	0.864180i	$-0.332162\pi$
$54$		0			0
$55$	4748.30	+	4817.34i	0.211656	+	0.214734i
$56$		0			0
$57$		−	71349.6i		−	2.90874i
$58$		0			0
$59$	−20735.0			−0.775486			−0.387743	−	0.921768i	$-0.626745\pi$
$59$	−20735.0			−0.775486			−0.387743	+	0.921768i	$0.626745\pi$
$60$		0			0
$61$	−44901.6			−1.54503			−0.772516	−	0.634995i	$-0.781002\pi$
$61$	−44901.6			−1.54503			−0.772516	+	0.634995i	$0.781002\pi$
$62$		0			0
$63$		−	24821.7i		−	0.787917i
$64$		0			0
$65$	12857.3	−	12673.0i	0.377455	−	0.372045i
$66$		0			0
$67$			67912.8i			1.84827i	0.382071	+	0.924133i	$0.375211\pi$
$67$			67912.8i			1.84827i	−0.382071	+	0.924133i	$0.624789\pi$
$68$		0			0
$69$	27954.9			0.706862
$70$		0			0
$71$	11219.8			0.264142			0.132071	−	0.991240i	$-0.457837\pi$
$71$	11219.8			0.264142			0.132071	+	0.991240i	$0.457837\pi$
$72$		0			0
$73$		−	51188.9i		−	1.12426i	−0.827047	−	0.562132i	$-0.809981\pi$
$73$		−	51188.9i		−	1.12426i	0.827047	−	0.562132i	$-0.190019\pi$
$74$		0			0
$75$	76840.6	+	1109.37i	1.57738	+	0.0227731i
$76$		0			0
$77$			8302.62i			0.159584i
$78$		0			0
$79$	49679.5			0.895591			0.447795	−	0.894136i	$-0.352209\pi$
$79$	49679.5			0.895591			0.447795	+	0.894136i	$0.352209\pi$
$80$		0			0
$81$	−16093.8			−0.272549
$82$		0			0
$83$		−	1551.85i		−	0.0247260i	−0.999924	−	0.0123630i	$-0.996065\pi$
$83$		−	1551.85i		−	0.0247260i	0.999924	−	0.0123630i	$-0.00393537\pi$
$84$		0			0
$85$	−42955.5	+	42339.9i	−0.644870	+	0.635627i
$86$		0			0
$87$			18012.8i			0.255143i
$88$		0			0
$89$	103366.			1.38325			0.691627	−	0.722255i	$-0.256894\pi$
$89$	103366.			1.38325			0.691627	+	0.722255i	$0.256894\pi$
$90$		0			0
$91$	22159.3			0.280513
$92$		0			0
$93$		−	171447.i		−	2.05553i
$94$		0			0
$95$	113857.	+	115512.i	1.29434	+	1.31316i
$96$		0			0
$97$		−	541.081i		−	0.00583893i	−0.999996	−	0.00291946i	$-0.999071\pi$
$97$		−	541.081i		−	0.00583893i	0.999996	−	0.00291946i	$-0.000929295\pi$
$98$		0			0
$99$	−43771.1			−0.448848
$100$		0			0
$101$	6404.26			0.0624691			0.0312346	−	0.999512i	$-0.490056\pi$
$101$	6404.26			0.0624691			0.0312346	+	0.999512i	$0.490056\pi$
$102$		0			0
$103$			45740.8i			0.424826i	0.977180	+	0.212413i	$0.0681322\pi$
$103$			45740.8i			0.424826i	−0.977180	+	0.212413i	$0.931868\pi$
$104$		0			0
$105$	66216.8	+	67179.7i	0.586132	+	0.594655i
$106$		0			0
$107$			95191.4i			0.803782i	0.915687	+	0.401891i	$0.131647\pi$
$107$			95191.4i			0.803782i	−0.915687	+	0.401891i	$0.868353\pi$
$108$		0			0
$109$	55413.3			0.446733			0.223366	−	0.974735i	$-0.428295\pi$
$109$	55413.3			0.446733			0.223366	+	0.974735i	$0.428295\pi$
$110$		0			0
$111$	81002.5			0.624009
$112$		0			0
$113$		−	93724.8i		−	0.690492i	−0.938512	−	0.345246i	$-0.887796\pi$
$113$		−	93724.8i		−	0.690492i	0.938512	−	0.345246i	$-0.112204\pi$
$114$		0			0
$115$	−45257.8	+	44609.2i	−0.319117	+	0.314543i
$116$		0			0
$117$			116823.i			0.788976i
$118$		0			0
$119$	−74033.2			−0.479247
$120$		0			0
$121$	14641.0			0.0909091
$122$		0			0
$123$		−	369279.i		−	2.20085i
$124$		0			0
$125$	−126172.	+	120823.i	−0.722252	+	0.691630i
$126$		0			0
$127$		−	29470.4i		−	0.162135i	−0.996709	−	0.0810675i	$-0.974167\pi$
$127$		−	29470.4i		−	0.162135i	0.996709	−	0.0810675i	$-0.0258329\pi$
$128$		0			0
$129$	−378149.			−2.00073
$130$		0			0
$131$	−39243.8			−0.199799			−0.0998993	−	0.994998i	$-0.531852\pi$
$131$	−39243.8			−0.199799			−0.0998993	+	0.994998i	$0.531852\pi$
$132$		0			0
$133$			199084.i			0.975903i
$134$		0			0
$135$	−116258.	+	114592.i	−0.549020	+	0.541150i
$136$		0			0
$137$		−	337986.i		−	1.53850i	−0.638949	−	0.769249i	$-0.720631\pi$
$137$		−	337986.i		−	1.53850i	0.638949	−	0.769249i	$-0.279369\pi$
$138$		0			0
$139$	−18346.5			−0.0805407			−0.0402704	−	0.999189i	$-0.512822\pi$
$139$	−18346.5			−0.0805407			−0.0402704	+	0.999189i	$0.512822\pi$
$140$		0			0
$141$	−147613.			−0.625282
$142$		0			0
$143$		−	39076.2i		−	0.159798i
$144$		0			0
$145$	−28744.1	−	29162.1i	−0.113535	−	0.115186i
$146$		0			0
$147$		−	297527.i		−	1.13562i
$148$		0			0
$149$	−146424.			−0.540314			−0.270157	−	0.962816i	$-0.587076\pi$
$149$	−146424.			−0.540314			−0.270157	+	0.962816i	$0.587076\pi$
$150$		0			0
$151$	290139.			1.03553			0.517766	−	0.855522i	$-0.326764\pi$
$151$	290139.			1.03553			0.517766	+	0.855522i	$0.326764\pi$
$152$		0			0
$153$		−	390300.i		−	1.34794i
$154$		0			0
$155$	273588.	+	277567.i	0.914678	+	0.927979i
$156$		0			0
$157$			597096.i			1.93328i	0.256138	+	0.966640i	$0.417550\pi$
$157$			597096.i			1.93328i	−0.256138	+	0.966640i	$0.582450\pi$
$158$		0			0
$159$	−869180.			−2.72657
$160$		0			0
$161$	−78001.2			−0.237157
$162$		0			0
$163$			112140.i			0.330591i	0.986244	+	0.165296i	$0.0528578\pi$
$163$			112140.i			0.330591i	−0.986244	+	0.165296i	$0.947142\pi$
$164$		0			0
$165$	118466.	−	116768.i	0.338754	−	0.333898i
$166$		0			0
$167$			504282.i			1.39921i	0.714530	+	0.699604i	$0.246640\pi$
$167$			504282.i			1.39921i	−0.714530	+	0.699604i	$0.753360\pi$
$168$		0			0
$169$	267001.			0.719110
$170$		0			0
$171$	−1.04956e6			−2.74484
$172$		0			0
$173$			143639.i			0.364886i	0.983216	+	0.182443i	$0.0584004\pi$
$173$			143639.i			0.364886i	−0.983216	+	0.182443i	$0.941600\pi$
$174$		0			0
$175$	−214405.	−	3095.41i	−0.529224	−	0.00764053i
$176$		0			0
$177$			509906.i			1.22337i
$178$		0			0
$179$	−429976.			−1.00302			−0.501512	−	0.865151i	$-0.667223\pi$
$179$	−429976.			−1.00302			−0.501512	+	0.865151i	$0.667223\pi$
$180$		0			0
$181$	491559.			1.11527			0.557634	−	0.830087i	$-0.311709\pi$
$181$	491559.			1.11527			0.557634	+	0.830087i	$0.311709\pi$
$182$		0			0
$183$			1.10420e6i			2.43736i
$184$		0			0
$185$	−131140.	+	129260.i	−0.281712	+	0.277674i
$186$		0			0
$187$			130552.i			0.273010i
$188$		0			0
$189$	−200369.			−0.408014
$190$		0			0
$191$	514282.			1.02004			0.510021	−	0.860162i	$-0.329638\pi$
$191$	514282.			1.02004			0.510021	+	0.860162i	$0.329638\pi$
$192$		0			0
$193$		−	484037.i		−	0.935374i	−0.883894	−	0.467687i	$-0.845087\pi$
$193$		−	484037.i		−	0.935374i	0.883894	−	0.467687i	$-0.154913\pi$
$194$		0			0
$195$	−311648.	−	316180.i	−0.586919	−	0.595454i
$196$		0			0
$197$		−	44622.0i		−	0.0819188i	−0.999161	−	0.0409594i	$-0.986959\pi$
$197$		−	44622.0i		−	0.0819188i	0.999161	−	0.0409594i	$-0.0130414\pi$
$198$		0			0
$199$	550206.			0.984900			0.492450	−	0.870341i	$-0.336101\pi$
$199$	550206.			0.984900			0.492450	+	0.870341i	$0.336101\pi$
$200$		0			0
$201$	1.67008e6			2.91573
$202$		0			0
$203$		−	50260.4i		−	0.0856024i
$204$		0			0
$205$	589278.	+	597847.i	0.979345	+	0.993587i
$206$		0			0
$207$		−	411219.i		−	0.667034i
$208$		0			0
$209$	351068.			0.555937
$210$		0			0
$211$	−1.17040e6			−1.80979			−0.904894	−	0.425636i	$-0.860050\pi$
$211$	−1.17040e6			−1.80979			−0.904894	+	0.425636i	$0.860050\pi$
$212$		0			0
$213$		−	275911.i		−	0.416697i
$214$		0			0
$215$	612209.	−	603434.i	0.903241	−	0.890294i
$216$		0			0
$217$			478382.i			0.689645i
$218$		0			0
$219$	−1.25881e6			−1.77358
$220$		0			0
$221$	348436.			0.479891
$222$		0			0
$223$			957130.i			1.28887i	0.764660	+	0.644434i	$0.222907\pi$
$223$			957130.i			1.28887i	−0.764660	+	0.644434i	$0.777093\pi$
$224$		0			0
$225$	16318.9	−	1.13033e6i	0.0214899	−	1.48851i
$226$		0			0
$227$			411895.i			0.530544i	0.964174	+	0.265272i	$0.0854617\pi$
$227$			411895.i			0.530544i	−0.964174	+	0.265272i	$0.914538\pi$
$228$		0			0
$229$	304437.			0.383627			0.191813	−	0.981431i	$-0.438563\pi$
$229$	304437.			0.383627			0.191813	+	0.981431i	$0.438563\pi$
$230$		0			0
$231$	204174.			0.251751
$232$		0			0
$233$		−	15539.4i		−	0.0187519i	−0.999956	−	0.00937593i	$-0.997016\pi$
$233$		−	15539.4i		−	0.0187519i	0.999956	−	0.00937593i	$-0.00298450\pi$
$234$		0			0
$235$	238979.	−	235554.i	0.282287	−	0.278241i
$236$		0			0
$237$		−	1.22170e6i		−	1.41284i
$238$		0			0
$239$	−794336.			−0.899517			−0.449758	−	0.893150i	$-0.648490\pi$
$239$	−794336.			−0.899517			−0.449758	+	0.893150i	$0.648490\pi$
$240$		0			0
$241$	−1.42354e6			−1.57880			−0.789400	−	0.613879i	$-0.789608\pi$
$241$	−1.42354e6			−1.57880			−0.789400	+	0.613879i	$0.789608\pi$
$242$		0			0
$243$			1.10536e6i			1.20085i
$244$		0			0
$245$	474781.	+	481685.i	0.505333	+	0.512681i
$246$		0			0
$247$		−	936984.i		−	0.977215i
$248$		0			0
$249$	−38162.4			−0.0390065
$250$		0			0
$251$	45704.8			0.0457908			0.0228954	−	0.999738i	$-0.492712\pi$
$251$	45704.8			0.0457908			0.0228954	+	0.999738i	$0.492712\pi$
$252$		0			0
$253$			137549.i			0.135100i
$254$		0			0
$255$	1.04120e6	+	1.05634e6i	1.00273	+	1.01731i
$256$		0			0
$257$			258018.i			0.243679i	0.992550	+	0.121839i	$0.0388793\pi$
$257$			258018.i			0.243679i	−0.992550	+	0.121839i	$0.961121\pi$
$258$		0			0
$259$	−226018.			−0.209360
$260$		0			0
$261$	264971.			0.240767
$262$		0			0
$263$			523132.i			0.466360i	0.972434	+	0.233180i	$0.0749132\pi$
$263$			523132.i			0.466360i	−0.972434	+	0.233180i	$0.925087\pi$
$264$		0			0
$265$	1.40717e6	−	1.38700e6i	1.23092	−	1.21328i
$266$		0			0
$267$		−	2.54193e6i		−	2.18215i
$268$		0			0
$269$	−1.02600e6			−0.864503			−0.432251	−	0.901753i	$-0.642281\pi$
$269$	−1.02600e6			−0.864503			−0.432251	+	0.901753i	$0.642281\pi$
$270$		0			0
$271$	1.55354e6			1.28499			0.642494	−	0.766291i	$-0.277900\pi$
$271$	1.55354e6			1.28499			0.642494	+	0.766291i	$0.277900\pi$
$272$		0			0
$273$		−	544932.i		−	0.442523i
$274$		0			0
$275$	−5458.51	+	378086.i	−0.00435254	+	0.301480i
$276$		0			0
$277$		−	742412.i		−	0.581361i	−0.956820	−	0.290680i	$-0.906118\pi$
$277$		−	742412.i		−	0.581361i	0.956820	−	0.290680i	$-0.0938816\pi$
$278$		0			0
$279$	−2.52201e6			−1.93971
$280$		0			0
$281$	−1.47286e6			−1.11274			−0.556372	−	0.830933i	$-0.687807\pi$
$281$	−1.47286e6			−1.11274			−0.556372	+	0.830933i	$0.687807\pi$
$282$		0			0
$283$		−	1.26344e6i		−	0.937756i	−0.883263	−	0.468878i	$-0.844658\pi$
$283$		−	1.26344e6i		−	0.937756i	0.883263	−	0.468878i	$-0.155342\pi$
$284$		0			0
$285$	2.84063e6	−	2.79991e6i	2.07158	−	2.04189i
$286$		0			0
$287$			1.03038e6i			0.738403i
$288$		0			0
$289$	255748.			0.180122
$290$		0			0
$291$	−13306.0			−0.00921119
$292$		0			0
$293$			1.50653e6i			1.02520i	0.858627	+	0.512600i	$0.171318\pi$
$293$			1.50653e6i			1.02520i	−0.858627	+	0.512600i	$0.828682\pi$
$294$		0			0
$295$	−813685.	−	825517.i	−0.544379	−	0.552295i
$296$		0			0
$297$			353334.i			0.232431i
$298$		0			0
$299$	367112.			0.237476
$300$		0			0
$301$	1.05513e6			0.671260
$302$		0			0
$303$		−	157491.i		−	0.0985481i
$304$		0			0
$305$	−1.76204e6	−	1.78766e6i	−1.08459	−	1.10036i
$306$		0			0
$307$		−	2.16775e6i		−	1.31269i	−0.754460	−	0.656346i	$-0.772101\pi$
$307$		−	2.16775e6i		−	1.31269i	0.754460	−	0.656346i	$-0.227899\pi$
$308$		0			0
$309$	1.12484e6			0.670184
$310$		0			0
$311$	1.82214e6			1.06827			0.534135	−	0.845399i	$-0.320637\pi$
$311$	1.82214e6			1.06827			0.534135	+	0.845399i	$0.320637\pi$
$312$		0			0
$313$		−	475864.i		−	0.274550i	−0.990533	−	0.137275i	$-0.956166\pi$
$313$		−	475864.i		−	0.274550i	0.990533	−	0.137275i	$-0.0438344\pi$
$314$		0			0
$315$	988221.	−	974057.i	0.561148	−	0.553106i
$316$		0			0
$317$		−	763607.i		−	0.426797i	−0.976965	−	0.213399i	$-0.931547\pi$
$317$		−	763607.i		−	0.426797i	0.976965	−	0.213399i	$-0.0684533\pi$
$318$		0			0
$319$	−88630.2			−0.0487646
$320$		0			0
$321$	2.34091e6			1.26801
$322$		0			0
$323$			3.13042e6i			1.66954i
$324$		0			0
$325$	1.00909e6	+	14568.5i	0.529935	+	0.00765080i
$326$		0			0
$327$		−	1.36270e6i		−	0.704743i
$328$		0			0
$329$	411877.			0.209787
$330$		0			0
$331$	1.13499e6			0.569407			0.284703	−	0.958616i	$-0.408105\pi$
$331$	1.13499e6			0.569407			0.284703	+	0.958616i	$0.408105\pi$
$332$		0			0
$333$		−	1.19156e6i		−	0.588849i
$334$		0			0
$335$	−2.70379e6	+	2.66504e6i	−1.31632	+	1.29745i
$336$		0			0
$337$			1.08011e6i			0.518078i	0.965867	+	0.259039i	$0.0834058\pi$
$337$			1.08011e6i			0.518078i	−0.965867	+	0.259039i	$0.916594\pi$
$338$		0			0
$339$	−2.30484e6			−1.08928
$340$		0			0
$341$	843588.			0.392866
$342$		0			0
$343$			1.98342e6i			0.910288i
$344$		0			0
$345$	1.09701e6	+	1.11296e6i	0.496206	+	0.503422i
$346$		0			0
$347$		−	2.31228e6i		−	1.03090i	−0.856920	−	0.515449i	$-0.827625\pi$
$347$		−	2.31228e6i		−	1.03090i	0.856920	−	0.515449i	$-0.172375\pi$
$348$		0			0
$349$	−3.32947e6			−1.46323			−0.731614	−	0.681719i	$-0.761233\pi$
$349$	−3.32947e6			−1.46323			−0.731614	+	0.681719i	$0.761233\pi$
$350$		0			0
$351$	943032.			0.408563
$352$		0			0
$353$			3.35633e6i			1.43360i	0.697278	+	0.716801i	$0.254394\pi$
$353$			3.35633e6i			1.43360i	−0.697278	+	0.716801i	$0.745606\pi$
$354$		0			0
$355$	440287.	+	446690.i	0.185424	+	0.188120i
$356$		0			0
$357$			1.82059e6i			0.756035i
$358$		0			0
$359$	−600243.			−0.245805			−0.122903	−	0.992419i	$-0.539220\pi$
$359$	−600243.			−0.245805			−0.122903	+	0.992419i	$0.539220\pi$
$360$		0			0
$361$	5.94195e6			2.39972
$362$		0			0
$363$		−	360045.i		−	0.143414i
$364$		0			0
$365$	2.03797e6	−	2.00876e6i	0.800693	−	0.789216i
$366$		0			0
$367$			2.60152e6i			1.00824i	0.863635	+	0.504118i	$0.168182\pi$
$367$			2.60152e6i			1.00824i	−0.863635	+	0.504118i	$0.831818\pi$
$368$		0			0
$369$	−5.43213e6			−2.07685
$370$		0			0
$371$	2.42523e6			0.914784
$372$		0			0
$373$		−	4.00605e6i		−	1.49089i	−0.666570	−	0.745443i	$-0.732238\pi$
$373$		−	4.00605e6i		−	1.49089i	0.666570	−	0.745443i	$-0.267762\pi$
$374$		0			0
$375$	2.97122e6	+	3.10277e6i	1.09108	+	1.13939i
$376$		0			0
$377$			236550.i			0.0857175i
$378$		0			0
$379$	−1.27812e6			−0.457059			−0.228529	−	0.973537i	$-0.573392\pi$
$379$	−1.27812e6			−0.457059			−0.228529	+	0.973537i	$0.573392\pi$
$380$		0			0
$381$	−724723.			−0.255776
$382$		0			0
$383$			1.93797e6i			0.675073i	0.941312	+	0.337536i	$0.109594\pi$
$383$			1.93797e6i			0.675073i	−0.941312	+	0.337536i	$0.890406\pi$
$384$		0			0
$385$	−330550.	+	325812.i	−0.113654	+	0.112025i
$386$		0			0
$387$			5.56262e6i			1.88800i
$388$		0			0
$389$	1.01122e6			0.338823			0.169412	−	0.985545i	$-0.445813\pi$
$389$	1.01122e6			0.338823			0.169412	+	0.985545i	$0.445813\pi$
$390$		0			0
$391$	−1.22650e6			−0.405720
$392$		0			0
$393$			965065.i			0.315192i
$394$		0			0
$395$	1.94953e6	+	1.97788e6i	0.628691	+	0.637833i
$396$		0			0
$397$		−	2.31935e6i		−	0.738566i	−0.929317	−	0.369283i	$-0.879603\pi$
$397$		−	2.31935e6i		−	0.738566i	0.929317	−	0.369283i	$-0.120397\pi$
$398$		0			0
$399$	4.89578e6			1.53953
$400$		0			0
$401$	159220.			0.0494467			0.0247233	−	0.999694i	$-0.492130\pi$
$401$	159220.			0.0494467			0.0247233	+	0.999694i	$0.492130\pi$
$402$		0			0
$403$		−	2.25150e6i		−	0.690572i
$404$		0			0
$405$	−631554.	−	640737.i	−0.191325	−	0.194108i
$406$		0			0
$407$			398564.i			0.119265i
$408$		0			0
$409$	−5.97006e6			−1.76470			−0.882348	−	0.470597i	$-0.844039\pi$
$409$	−5.97006e6			−1.76470			−0.882348	+	0.470597i	$0.844039\pi$
$410$		0			0
$411$	−8.31159e6			−2.42706
$412$		0			0
$413$		−	1.42277e6i		−	0.410448i
$414$		0			0
$415$	61783.4	−	60897.8i	0.0176097	−	0.0173573i
$416$		0			0
$417$			451168.i			0.127057i
$418$		0			0
$419$	−5.36018e6			−1.49157			−0.745786	−	0.666186i	$-0.767926\pi$
$419$	−5.36018e6			−1.49157			−0.745786	+	0.666186i	$0.767926\pi$
$420$		0			0
$421$	−4.23113e6			−1.16346			−0.581729	−	0.813383i	$-0.697624\pi$
$421$	−4.23113e6			−1.16346			−0.581729	+	0.813383i	$0.697624\pi$
$422$		0			0
$423$			2.17140e6i			0.590050i
$424$		0			0
$425$	−3.37133e6	−	48672.7i	−0.905377	−	0.0130711i
$426$		0			0
$427$		−	3.08100e6i		−	0.817753i
$428$		0			0
$429$	−960944.			−0.252089
$430$		0			0
$431$	−7.37436e6			−1.91219			−0.956096	−	0.293055i	$-0.905328\pi$
$431$	−7.37436e6			−1.91219			−0.956096	+	0.293055i	$0.905328\pi$
$432$		0			0
$433$		−	4.04144e6i		−	1.03590i	−0.855412	−	0.517948i	$-0.826696\pi$
$433$		−	4.04144e6i		−	1.03590i	0.855412	−	0.517948i	$-0.173304\pi$
$434$		0			0
$435$	−717141.	+	706862.i	−0.181711	+	0.179107i
$436$		0			0
$437$			3.29820e6i			0.826178i
$438$		0			0
$439$	3.79231e6			0.939165			0.469583	−	0.882889i	$-0.344404\pi$
$439$	3.79231e6			0.939165			0.469583	+	0.882889i	$0.344404\pi$
$440$		0			0
$441$	−4.37666e6			−1.07163
$442$		0			0
$443$		−	5.95941e6i		−	1.44276i	−0.692540	−	0.721380i	$-0.743508\pi$
$443$		−	5.95941e6i		−	1.44276i	0.692540	−	0.721380i	$-0.256492\pi$
$444$		0			0
$445$	4.05630e6	+	4.11528e6i	0.971023	+	0.985143i
$446$		0			0
$447$			3.60079e6i			0.852372i
$448$		0			0
$449$	−5.90973e6			−1.38341			−0.691706	−	0.722179i	$-0.743141\pi$
$449$	−5.90973e6			−1.38341			−0.691706	+	0.722179i	$0.743141\pi$
$450$		0			0
$451$	1.81699e6			0.420641
$452$		0			0
$453$		−	7.13497e6i		−	1.63360i
$454$		0			0
$455$	869579.	+	882223.i	0.196916	+	0.199779i
$456$		0			0
$457$			3.15000e6i			0.705537i	0.935711	+	0.352769i	$0.114760\pi$
$457$			3.15000e6i			0.705537i	−0.935711	+	0.352769i	$0.885240\pi$
$458$		0			0
$459$	−3.15063e6			−0.698016
$460$		0			0
$461$	−8.43560e6			−1.84869			−0.924344	−	0.381561i	$-0.875387\pi$
$461$	−8.43560e6			−1.84869			−0.924344	+	0.381561i	$0.875387\pi$
$462$		0			0
$463$		−	4.81421e6i		−	1.04369i	−0.853040	−	0.521846i	$-0.825244\pi$
$463$		−	4.81421e6i		−	1.04369i	0.853040	−	0.521846i	$-0.174756\pi$
$464$		0			0
$465$	6.82580e6	−	6.72796e6i	1.46393	−	1.44295i
$466$		0			0
$467$		−	1.34298e6i		−	0.284956i	−0.989798	−	0.142478i	$-0.954493\pi$
$467$		−	1.34298e6i		−	0.284956i	0.989798	−	0.142478i	$-0.0455071\pi$
$468$		0			0
$469$	−4.65995e6			−0.978248
$470$		0			0
$471$	1.46835e7			3.04984
$472$		0			0
$473$		−	1.86064e6i		−	0.382393i
$474$		0			0
$475$	−130886.	+	9.06589e6i	−0.0266171	+	1.84364i
$476$		0			0
$477$			1.27857e7i			2.57294i
$478$		0			0
$479$	2.57179e6			0.512149			0.256074	−	0.966657i	$-0.417571\pi$
$479$	2.57179e6			0.512149			0.256074	+	0.966657i	$0.417571\pi$
$480$		0			0
$481$	1.06375e6			0.209641
$482$		0			0
$483$			1.91817e6i			0.374127i
$484$		0			0
$485$	21541.9	−	21233.2i	0.00415844	−	0.00409884i
$486$		0			0
$487$		−	5.80826e6i		−	1.10975i	−0.831935	−	0.554874i	$-0.812767\pi$
$487$		−	5.80826e6i		−	1.10975i	0.831935	−	0.554874i	$-0.187233\pi$
$488$		0			0
$489$	2.75770e6			0.521524
$490$		0			0
$491$	−7.13394e6			−1.33544			−0.667722	−	0.744411i	$-0.732730\pi$
$491$	−7.13394e6			−1.33544			−0.667722	+	0.744411i	$0.732730\pi$
$492$		0			0
$493$		−	790302.i		−	0.146445i
$494$		0			0
$495$	−1.71767e6	−	1.74265e6i	−0.315084	−	0.319666i
$496$		0			0
$497$			769863.i			0.139805i
$498$		0			0
$499$	−5.11035e6			−0.918755			−0.459377	−	0.888241i	$-0.651927\pi$
$499$	−5.11035e6			−0.918755			−0.459377	+	0.888241i	$0.651927\pi$
$500$		0			0
$501$	1.24011e7			2.20732
$502$		0			0
$503$			9.08121e6i			1.60038i	0.599745	+	0.800191i	$0.295268\pi$
$503$			9.08121e6i			1.60038i	−0.599745	+	0.800191i	$0.704732\pi$
$504$		0			0
$505$	251317.	+	254971.i	0.0438523	+	0.0444900i
$506$		0			0
$507$		−	6.56596e6i		−	1.13443i
$508$		0			0
$509$	−8.26612e6			−1.41419			−0.707094	−	0.707119i	$-0.749994\pi$
$509$	−8.26612e6			−1.41419			−0.707094	+	0.707119i	$0.749994\pi$
$510$		0			0
$511$	3.51241e6			0.595050
$512$		0			0
$513$			8.47239e6i			1.42139i
$514$		0			0
$515$	−1.82107e6	+	1.79497e6i	−0.302558	+	0.298221i
$516$		0			0
$517$		−	726312.i		−	0.119508i
$518$		0			0
$519$	3.53230e6			0.575625
$520$		0			0
$521$	2.07813e6			0.335412			0.167706	−	0.985837i	$-0.446364\pi$
$521$	2.07813e6			0.335412			0.167706	+	0.985837i	$0.446364\pi$
$522$		0			0
$523$			1.94810e6i			0.311427i	0.987802	+	0.155714i	$0.0497677\pi$
$523$			1.94810e6i			0.311427i	−0.987802	+	0.155714i	$0.950232\pi$
$524$		0			0
$525$	−76121.0	+	5.27255e6i	−0.0120533	+	0.834877i
$526$		0			0
$527$			7.52215e6i			1.17982i
$528$		0			0
$529$	5.14410e6			0.799228
$530$		0			0
$531$	7.50077e6			1.15444
$532$		0			0
$533$		−	4.84947e6i		−	0.739395i
$534$		0			0
$535$	−3.78983e6	+	3.73551e6i	−0.572448	+	0.564243i
$536$		0			0
$537$			1.05738e7i			1.58232i
$538$		0			0
$539$	1.46395e6			0.217047
$540$		0			0
$541$	−8.56903e6			−1.25875			−0.629374	−	0.777103i	$-0.716688\pi$
$541$	−8.56903e6			−1.25875			−0.629374	+	0.777103i	$0.716688\pi$
$542$		0			0
$543$		−	1.20882e7i		−	1.75939i
$544$		0			0
$545$	2.17454e6	+	2.20616e6i	0.313599	+	0.318160i
$546$		0			0
$547$		−	5.59536e6i		−	0.799577i	−0.916607	−	0.399788i	$-0.869084\pi$
$547$		−	5.59536e6i		−	0.799577i	0.916607	−	0.399788i	$-0.130916\pi$
$548$		0			0
$549$	1.62429e7			2.30003
$550$		0			0
$551$	−2.12521e6			−0.298211
$552$		0			0
$553$			3.40884e6i			0.474017i
$554$		0			0
$555$	3.17871e6	+	3.22493e6i	0.438045	+	0.444414i
$556$		0			0
$557$		−	774159.i		−	0.105728i	−0.998602	−	0.0528642i	$-0.983165\pi$
$557$		−	774159.i		−	0.105728i	0.998602	−	0.0528642i	$-0.0168351\pi$
$558$		0			0
$559$	−4.96597e6			−0.672163
$560$		0			0
$561$	3.21047e6			0.430686
$562$		0			0
$563$			7.62044e6i			1.01323i	0.862172	+	0.506616i	$0.169104\pi$
$563$			7.62044e6i			1.01323i	−0.862172	+	0.506616i	$0.830896\pi$
$564$		0			0
$565$	3.73144e6	−	3.67796e6i	0.491763	−	0.484714i
$566$		0			0
$567$		−	1.10430e6i		−	0.144255i
$568$		0			0
$569$	−592534.			−0.0767243			−0.0383622	−	0.999264i	$-0.512214\pi$
$569$	−592534.			−0.0767243			−0.0383622	+	0.999264i	$0.512214\pi$
$570$		0			0
$571$	1.30108e7			1.66999			0.834995	−	0.550257i	$-0.185470\pi$
$571$	1.30108e7			1.66999			0.834995	+	0.550257i	$0.185470\pi$
$572$		0			0
$573$		−	1.26470e7i		−	1.60917i
$574$		0			0
$575$	−3.55203e6	−	51281.5i	−0.448030	−	0.00646831i
$576$		0			0
$577$			6.70946e6i			0.838973i	0.907762	+	0.419486i	$0.137790\pi$
$577$			6.70946e6i			0.838973i	−0.907762	+	0.419486i	$0.862210\pi$
$578$		0			0
$579$	−1.19032e7			−1.47560
$580$		0			0
$581$	106483.			0.0130870
$582$		0			0
$583$		−	4.27670e6i		−	0.521120i
$584$		0			0
$585$	−4.65105e6	+	4.58438e6i	−0.561903	+	0.553849i
$586$		0			0
$587$		−	1.19616e7i		−	1.43283i	−0.697674	−	0.716415i	$-0.745782\pi$
$587$		−	1.19616e7i		−	1.43283i	0.697674	−	0.716415i	$-0.254218\pi$
$588$		0			0
$589$	2.02279e7			2.40250
$590$		0			0
$591$	−1.09732e6			−0.129231
$592$		0			0
$593$		−	1.36302e7i		−	1.59172i	−0.605481	−	0.795860i	$-0.707019\pi$
$593$		−	1.36302e7i		−	1.59172i	0.605481	−	0.795860i	$-0.292981\pi$
$594$		0			0
$595$	−2.90522e6	−	2.94747e6i	−0.336424	−	0.341316i
$596$		0			0
$597$		−	1.35304e7i		−	1.55373i
$598$		0			0
$599$	−3.87186e6			−0.440913			−0.220456	−	0.975397i	$-0.570755\pi$
$599$	−3.87186e6			−0.440913			−0.220456	+	0.975397i	$0.570755\pi$
$600$		0			0
$601$	−2.64964e6			−0.299227			−0.149613	−	0.988745i	$-0.547803\pi$
$601$	−2.64964e6			−0.299227			−0.149613	+	0.988745i	$0.547803\pi$
$602$		0			0
$603$		−	2.45671e7i		−	2.75144i
$604$		0			0
$605$	574544.	+	582899.i	0.0638168	+	0.0647448i
$606$		0			0
$607$		−	1.36476e7i		−	1.50343i	−0.659486	−	0.751717i	$-0.729226\pi$
$607$		−	1.36476e7i		−	1.50343i	0.659486	−	0.751717i	$-0.270774\pi$
$608$		0			0
$609$	−1.23598e6			−0.135042
$610$		0			0
$611$	−1.93849e6			−0.210069
$612$		0			0
$613$			1.60364e7i			1.72367i	0.507185	+	0.861837i	$0.330686\pi$
$613$			1.60364e7i			1.72367i	−0.507185	+	0.861837i	$0.669314\pi$
$614$		0			0
$615$	1.47020e7	−	1.44913e7i	1.56743	−	1.54497i
$616$		0			0
$617$		−	7.42285e6i		−	0.784979i	−0.919757	−	0.392489i	$-0.871614\pi$
$617$		−	7.42285e6i		−	0.784979i	0.919757	−	0.392489i	$-0.128386\pi$
$618$		0			0
$619$	1.37805e7			1.44557			0.722784	−	0.691074i	$-0.242862\pi$
$619$	1.37805e7			1.44557			0.722784	+	0.691074i	$0.242862\pi$
$620$		0			0
$621$	−3.31949e6			−0.345416
$622$		0			0
$623$			7.09262e6i			0.732128i
$624$		0			0
$625$	−9.76155e6	−	281919.i	−0.999583	−	0.0288685i
$626$		0			0
$627$		−	8.63331e6i		−	0.877018i
$628$		0			0
$629$	−3.55393e6			−0.358165
$630$		0			0
$631$	2.22868e6			0.222830			0.111415	−	0.993774i	$-0.464462\pi$
$631$	2.22868e6			0.222830			0.111415	+	0.993774i	$0.464462\pi$
$632$		0			0
$633$			2.87819e7i			2.85503i
$634$		0			0
$635$	1.17330e6	−	1.15648e6i	0.115471	−	0.113816i
$636$		0			0
$637$		−	3.90721e6i		−	0.381521i
$638$		0			0
$639$	−4.05869e6			−0.393218
$640$		0			0
$641$	1.75304e6			0.168519			0.0842593	−	0.996444i	$-0.473148\pi$
$641$	1.75304e6			0.168519			0.0842593	+	0.996444i	$0.473148\pi$
$642$		0			0
$643$		−	1.68507e7i		−	1.60727i	−0.595121	−	0.803636i	$-0.702896\pi$
$643$		−	1.68507e7i		−	1.60727i	0.595121	−	0.803636i	$-0.297104\pi$
$644$		0			0
$645$	−1.48394e7	−	1.50552e7i	−1.40448	−	1.42491i
$646$		0			0
$647$		−	1.86231e6i		−	0.174901i	−0.996169	−	0.0874505i	$-0.972128\pi$
$647$		−	1.86231e6i		−	0.174901i	0.996169	−	0.0874505i	$-0.0278720\pi$
$648$		0			0
$649$	−2.50893e6			−0.233818
$650$		0			0
$651$	1.17642e7			1.08795
$652$		0			0
$653$			1.64223e7i			1.50713i	0.657375	+	0.753563i	$0.271667\pi$
$653$			1.64223e7i			1.50713i	−0.657375	+	0.753563i	$0.728333\pi$
$654$		0			0
$655$	−1.54001e6	−	1.56240e6i	−0.140256	−	0.142295i
$656$		0			0
$657$			1.85173e7i			1.67365i
$658$		0			0
$659$	6.16058e6			0.552596			0.276298	−	0.961072i	$-0.410892\pi$
$659$	6.16058e6			0.552596			0.276298	+	0.961072i	$0.410892\pi$
$660$		0			0
$661$	−2.06032e7			−1.83413			−0.917065	−	0.398737i	$-0.869449\pi$
$661$	−2.06032e7			−1.83413			−0.917065	+	0.398737i	$0.869449\pi$
$662$		0			0
$663$		−	8.56859e6i		−	0.757052i
$664$		0			0
$665$	−7.92607e6	+	7.81247e6i	−0.695030	+	0.685069i
$666$		0			0
$667$		−	832660.i		−	0.0724692i
$668$		0			0
$669$	2.35373e7			2.03325
$670$		0			0
$671$	−5.43310e6			−0.465845
$672$		0			0
$673$		−	1.14267e7i		−	0.972488i	−0.873823	−	0.486244i	$-0.838367\pi$
$673$		−	1.14267e7i		−	0.972488i	0.873823	−	0.486244i	$-0.161633\pi$
$674$		0			0
$675$	−9.12441e6	−	131731.i	−0.770806	−	0.0111283i
$676$		0			0
$677$		−	679572.i		−	0.0569854i	−0.999594	−	0.0284927i	$-0.990929\pi$
$677$		−	679572.i		−	0.0569854i	0.999594	−	0.0284927i	$-0.00907074\pi$
$678$		0			0
$679$	37127.2			0.00309042
$680$		0			0
$681$	1.01291e7			0.836959
$682$		0			0
$683$			6.88710e6i			0.564917i	0.959280	+	0.282458i	$0.0911499\pi$
$683$			6.88710e6i			0.564917i	−0.959280	+	0.282458i	$0.908850\pi$
$684$		0			0
$685$	1.34561e7	−	1.32633e7i	1.09571	−	1.08000i
$686$		0			0
$687$		−	7.48659e6i		−	0.605190i
$688$		0			0
$689$	−1.14143e7			−0.916014
$690$		0			0
$691$	4.64486e6			0.370065			0.185032	−	0.982732i	$-0.440761\pi$
$691$	4.64486e6			0.370065			0.185032	+	0.982732i	$0.440761\pi$
$692$		0			0
$693$		−	3.00343e6i		−	0.237566i
$694$		0			0
$695$	−719954.	−	730424.i	−0.0565383	−	0.0573605i
$696$		0			0
$697$			1.62019e7i			1.26323i
$698$		0			0
$699$	−382138.			−0.0295820
$700$		0			0
$701$	−3.04696e6			−0.234192			−0.117096	−	0.993121i	$-0.537358\pi$
$701$	−3.04696e6			−0.234192			−0.117096	+	0.993121i	$0.537358\pi$
$702$		0			0
$703$			9.55692e6i			0.729340i
$704$		0			0
$705$	−5.79264e6	−	5.87687e6i	−0.438938	−	0.445321i
$706$		0			0
$707$			439439.i			0.0330636i
$708$		0			0
$709$	1.50817e7			1.12677			0.563385	−	0.826194i	$-0.309499\pi$
$709$	1.50817e7			1.12677			0.563385	+	0.826194i	$0.309499\pi$
$710$		0			0
$711$	−1.79713e7			−1.33323
$712$		0			0
$713$			7.92532e6i			0.583839i
$714$		0			0
$715$	1.55573e6	−	1.53343e6i	0.113807	−	0.112176i
$716$		0			0
$717$			1.95339e7i			1.41903i
$718$		0			0
$719$	−1.20848e7			−0.871803			−0.435901	−	0.899994i	$-0.643570\pi$
$719$	−1.20848e7			−0.871803			−0.435901	+	0.899994i	$0.643570\pi$
$720$		0			0
$721$	−3.13858e6			−0.224851
$722$		0			0
$723$			3.50071e7i			2.49063i
$724$		0			0
$725$	33043.4	−	2.28876e6i	0.00233475	−	0.161717i
$726$		0			0
$727$		−	2.52669e7i		−	1.77303i	−0.462697	−	0.886516i	$-0.653118\pi$
$727$		−	2.52669e7i		−	1.77303i	0.462697	−	0.886516i	$-0.346882\pi$
$728$		0			0
$729$	2.32717e7			1.62185
$730$		0			0
$731$	1.65911e7			1.14837
$732$		0			0
$733$			3.21976e6i			0.221342i	0.993857	+	0.110671i	$0.0352999\pi$
$733$			3.21976e6i			0.221342i	−0.993857	+	0.110671i	$0.964700\pi$
$734$		0			0
$735$	1.18454e7	−	1.16756e7i	0.808780	−	0.797188i
$736$		0			0
$737$			8.21744e6i			0.557273i
$738$		0			0
$739$	1.83282e7			1.23455			0.617274	−	0.786748i	$-0.288237\pi$
$739$	1.83282e7			1.23455			0.617274	+	0.786748i	$0.288237\pi$
$740$		0			0
$741$	−2.30419e7			−1.54160
$742$		0			0
$743$			2.56985e7i			1.70779i	0.520444	+	0.853896i	$0.325767\pi$
$743$			2.56985e7i			1.70779i	−0.520444	+	0.853896i	$0.674233\pi$
$744$		0			0
$745$	−5.74599e6	−	5.82954e6i	−0.379292	−	0.384808i
$746$		0			0
$747$			561373.i			0.0368086i
$748$		0			0
$749$	−6.53172e6			−0.425425
$750$		0			0
$751$	1.44321e7			0.933748			0.466874	−	0.884324i	$-0.345380\pi$
$751$	1.44321e7			0.933748			0.466874	+	0.884324i	$0.345380\pi$
$752$		0			0
$753$		−	1.12395e6i		−	0.0722372i
$754$		0			0
$755$	1.13857e7	+	1.15512e7i	0.726927	+	0.737498i
$756$		0			0
$757$		−	2.39198e7i		−	1.51712i	−0.651606	−	0.758558i	$-0.725904\pi$
$757$		−	2.39198e7i		−	1.51712i	0.651606	−	0.758558i	$-0.274096\pi$
$758$		0			0
$759$	3.38254e6			0.213127
$760$		0			0
$761$	−2.78047e7			−1.74043			−0.870215	−	0.492672i	$-0.836020\pi$
$761$	−2.78047e7			−1.74043			−0.870215	+	0.492672i	$0.836020\pi$
$762$		0			0
$763$			3.80228e6i			0.236446i
$764$		0			0
$765$	1.55389e7	−	1.53162e7i	0.959992	−	0.946232i
$766$		0			0
$767$			6.69623e6i			0.411000i
$768$		0			0
$769$	−1.25298e7			−0.764063			−0.382031	−	0.924149i	$-0.624775\pi$
$769$	−1.25298e7			−0.764063			−0.382031	+	0.924149i	$0.624775\pi$
$770$		0			0
$771$	6.34507e6			0.384415
$772$		0			0
$773$			2.17300e7i			1.30801i	0.756490	+	0.654006i	$0.226913\pi$
$773$			2.17300e7i			1.30801i	−0.756490	+	0.654006i	$0.773087\pi$
$774$		0			0
$775$	−314510.	+	2.17846e7i	−0.0188096	+	1.30285i
$776$		0			0
$777$			5.55812e6i			0.330275i
$778$		0			0
$779$	4.35686e7			2.57235
$780$		0			0
$781$	1.35759e6			0.0796419
$782$		0			0
$783$		−	2.13893e6i		−	0.124679i
$784$		0			0
$785$	−2.37720e7	+	2.34313e7i	−1.37687	+	1.35713i
$786$		0			0
$787$			7.47679e6i			0.430307i	0.976580	+	0.215153i	$0.0690252\pi$
$787$			7.47679e6i			0.430307i	−0.976580	+	0.215153i	$0.930975\pi$
$788$		0			0
$789$	1.28646e7			0.735706
$790$		0			0
$791$	6.43109e6			0.365463
$792$		0			0
$793$			1.45007e7i			0.818852i
$794$		0			0
$795$	−3.41085e7	−	3.46045e7i	−1.91401	−	1.94184i
$796$		0			0
$797$			8.58463e6i			0.478714i	0.970932	+	0.239357i	$0.0769366\pi$
$797$			8.58463e6i			0.478714i	−0.970932	+	0.239357i	$0.923063\pi$
$798$		0			0
$799$	6.47642e6			0.358895
$800$		0			0
$801$	−3.73920e7			−2.05920
$802$		0			0
$803$		−	6.19386e6i		−	0.338979i
$804$		0			0
$805$	−3.06093e6	−	3.10544e6i	−0.166481	−	0.168902i
$806$		0			0
$807$			2.52309e7i			1.36380i
$808$		0			0
$809$	−2.00739e7			−1.07835			−0.539176	−	0.842193i	$-0.681264\pi$
$809$	−2.00739e7			−1.07835			−0.539176	+	0.842193i	$0.681264\pi$
$810$		0			0
$811$	345528.			0.0184472			0.00922362	−	0.999957i	$-0.497064\pi$
$811$	345528.			0.0184472			0.00922362	+	0.999957i	$0.497064\pi$
$812$		0			0
$813$		−	3.82039e7i		−	2.02713i
$814$		0			0
$815$	−4.46460e6	+	4.40061e6i	−0.235445	+	0.232070i
$816$		0			0
$817$		−	4.46152e7i		−	2.33845i
$818$		0			0
$819$	−8.01601e6			−0.417589
$820$		0			0
$821$	−3.55582e7			−1.84112			−0.920560	−	0.390601i	$-0.872267\pi$
$821$	−3.55582e7			−1.84112			−0.920560	+	0.390601i	$0.872267\pi$
$822$		0			0
$823$			2.18881e7i			1.12644i	0.826307	+	0.563220i	$0.190438\pi$
$823$			2.18881e7i			1.12644i	−0.826307	+	0.563220i	$0.809562\pi$
$824$		0			0
$825$	9.29771e6	+	134233.i	0.475599	+	0.00686634i
$826$		0			0
$827$			3.08664e7i			1.56936i	0.619901	+	0.784680i	$0.287173\pi$
$827$			3.08664e7i			1.56936i	−0.619901	+	0.784680i	$0.712827\pi$
$828$		0			0
$829$	1.02344e7			0.517219			0.258610	−	0.965982i	$-0.416736\pi$
$829$	1.02344e7			0.517219			0.258610	+	0.965982i	$0.416736\pi$
$830$		0			0
$831$	−1.82571e7			−0.917125
$832$		0			0
$833$			1.30538e7i			0.651816i
$834$		0			0
$835$	−2.00769e7	+	1.97891e7i	−0.996506	+	0.982223i
$836$		0			0
$837$			2.03585e7i			1.00446i
$838$		0			0
$839$	1.42644e7			0.699596			0.349798	−	0.936825i	$-0.386250\pi$
$839$	1.42644e7			0.699596			0.349798	+	0.936825i	$0.386250\pi$
$840$		0			0
$841$	−1.99746e7			−0.973842
$842$		0			0
$843$			3.62199e7i			1.75541i
$844$		0			0
$845$	1.04777e7	+	1.06300e7i	0.504804	+	0.512145i
$846$		0			0
$847$			1.00462e6i			0.0481163i
$848$		0			0
$849$	−3.10700e7			−1.47936
$850$		0			0
$851$	−3.74441e6			−0.177239
$852$		0			0
$853$		−	6.85288e6i		−	0.322478i	−0.986915	−	0.161239i	$-0.948451\pi$
$853$		−	6.85288e6i		−	0.322478i	0.986915	−	0.161239i	$-0.0515491\pi$
$854$		0			0
$855$	−4.11870e7	−	4.17859e7i	−1.92684	−	1.95486i
$856$		0			0
$857$		−	3.51342e7i		−	1.63410i	−0.576567	−	0.817050i	$-0.695608\pi$
$857$		−	3.51342e7i		−	1.63410i	0.576567	−	0.817050i	$-0.304392\pi$
$858$		0			0
$859$	−8.48281e6			−0.392244			−0.196122	−	0.980579i	$-0.562835\pi$
$859$	−8.48281e6			−0.392244			−0.196122	+	0.980579i	$0.562835\pi$
$860$		0			0
$861$	2.53387e7			1.16487
$862$		0			0
$863$		−	3.84664e7i		−	1.75815i	−0.476687	−	0.879073i	$-0.658162\pi$
$863$		−	3.84664e7i		−	1.75815i	0.476687	−	0.879073i	$-0.341838\pi$
$864$		0			0
$865$	−5.71866e6	+	5.63669e6i	−0.259869	+	0.256144i
$866$		0			0
$867$		−	6.28924e6i		−	0.284152i
$868$		0			0
$869$	6.01122e6			0.270031
$870$		0			0
$871$	2.19320e7			0.979563
$872$		0			0
$873$			195733.i			0.00869218i
$874$		0			0
$875$	−8.29046e6	−	8.65752e6i	−0.366065	−	0.382273i
$876$		0			0
$877$		−	1.87813e7i		−	0.824567i	−0.911056	−	0.412283i	$-0.864731\pi$
$877$		−	1.87813e7i		−	0.824567i	0.911056	−	0.412283i	$-0.135269\pi$
$878$		0			0
$879$	3.70479e7			1.61730
$880$		0			0
$881$	4.01661e7			1.74349			0.871745	−	0.489960i	$-0.162989\pi$
$881$	4.01661e7			1.74349			0.871745	+	0.489960i	$0.162989\pi$
$882$		0			0
$883$			5.55332e6i			0.239690i	0.992793	+	0.119845i	$0.0382398\pi$
$883$			5.55332e6i			0.239690i	−0.992793	+	0.119845i	$0.961760\pi$
$884$		0			0
$885$	−2.03008e7	+	2.00098e7i	−0.871272	+	0.858784i
$886$		0			0
$887$			2.28130e7i			0.973584i	0.873518	+	0.486792i	$0.161833\pi$
$887$			2.28130e7i			0.973584i	−0.873518	+	0.486792i	$0.838167\pi$
$888$		0			0
$889$	2.02216e6			0.0858147
$890$		0			0
$891$	−1.94735e6			−0.0821767
$892$		0			0
$893$		−	1.74158e7i		−	0.730828i
$894$		0			0
$895$	−1.68732e7	−	1.71185e7i	−0.704107	−	0.714346i
$896$		0			0
$897$		−	9.02785e6i		−	0.374630i
$898$		0			0
$899$	−5.10671e6			−0.210738
$900$		0			0
$901$	3.81347e7			1.56498
$902$		0			0
$903$		−	2.59474e7i		−	1.05895i
$904$		0			0
$905$	1.92898e7	+	1.95703e7i	0.782901	+	0.794285i
$906$		0			0
$907$			3.90037e7i			1.57430i	0.616761	+	0.787150i	$0.288444\pi$
$907$			3.90037e7i			1.57430i	−0.616761	+	0.787150i	$0.711556\pi$
$908$		0			0
$909$	−2.31671e6			−0.0929953
$910$		0			0
$911$	−3.68587e7			−1.47145			−0.735723	−	0.677283i	$-0.763157\pi$
$911$	−3.68587e7			−1.47145			−0.735723	+	0.677283i	$0.763157\pi$
$912$		0			0
$913$		−	187774.i		−	0.00745517i
$914$		0			0
$915$	−4.39613e7	+	4.33312e7i	−1.73587	+	1.71099i
$916$		0			0
$917$		−	2.69278e6i		−	0.105749i
$918$		0			0
$919$	−1.03588e7			−0.404597			−0.202299	−	0.979324i	$-0.564841\pi$
$919$	−1.03588e7			−0.404597			−0.202299	+	0.979324i	$0.564841\pi$
$920$		0			0
$921$	−5.33083e7			−2.07084
$922$		0			0
$923$		−	3.62335e6i		−	0.139993i
$924$		0			0
$925$	−1.02924e7	−	148594.i	−0.395515	−	0.00571014i
$926$		0			0
$927$		−	1.65465e7i		−	0.632422i
$928$		0			0
$929$	−9.13177e6			−0.347149			−0.173574	−	0.984821i	$-0.555532\pi$
$929$	−9.13177e6			−0.347149			−0.173574	+	0.984821i	$0.555532\pi$
$930$		0			0
$931$	3.51032e7			1.32731
$932$		0			0
$933$		−	4.48093e7i		−	1.68525i
$934$		0			0
$935$	−5.19762e6	+	5.12312e6i	−0.194435	+	0.191649i
$936$		0			0
$937$		−	219135.i		−	0.00815384i	−0.999992	−	0.00407692i	$-0.998702\pi$
$937$		−	219135.i		−	0.00815384i	0.999992	−	0.00407692i	$-0.00129773\pi$
$938$		0			0
$939$	−1.17022e7			−0.433116
$940$		0			0
$941$	1.48284e7			0.545907			0.272954	−	0.962027i	$-0.411999\pi$
$941$	1.48284e7			0.545907			0.272954	+	0.962027i	$0.411999\pi$
$942$		0			0
$943$			1.70702e7i			0.625116i
$944$		0			0
$945$	−7.86289e6	−	7.97723e6i	−0.286420	−	0.290585i
$946$		0			0
$947$			5.18145e7i			1.87748i	0.344621	+	0.938742i	$0.388007\pi$
$947$			5.18145e7i			1.87748i	−0.344621	+	0.938742i	$0.611993\pi$
$948$		0			0
$949$	−1.65311e7			−0.595850
$950$		0			0
$951$	−1.87783e7			−0.673294
$952$		0			0
$953$			3.57043e7i			1.27347i	0.771083	+	0.636735i	$0.219715\pi$
$953$			3.57043e7i			1.27347i	−0.771083	+	0.636735i	$0.780285\pi$
$954$		0			0
$955$	2.01815e7	+	2.04750e7i	0.716054	+	0.726466i
$956$		0			0
$957$			2.17955e6i			0.0769286i
$958$		0			0
$959$	2.31915e7			0.814295
$960$		0			0
$961$	1.99769e7			0.697781
$962$		0			0
$963$		−	3.44350e7i		−	1.19656i
$964$		0			0
$965$	1.92708e7	−	1.89946e7i	0.666166	−	0.656618i
$966$		0			0
$967$		−	9.61627e6i		−	0.330705i	−0.986235	−	0.165352i	$-0.947124\pi$
$967$		−	9.61627e6i		−	0.330705i	0.986235	−	0.165352i	$-0.0528762\pi$
$968$		0			0
$969$	7.69819e7			2.63378
$970$		0			0
$971$	2.39248e7			0.814330			0.407165	−	0.913355i	$-0.366517\pi$
$971$	2.39248e7			0.814330			0.407165	+	0.913355i	$0.366517\pi$
$972$		0			0
$973$		−	1.25887e6i		−	0.0426285i
$974$		0			0
$975$	358262.	−	2.48152e7i	0.0120695	−	0.835999i
$976$		0			0
$977$		−	3.46267e6i		−	0.116058i	−0.998315	−	0.0580290i	$-0.981518\pi$
$977$		−	3.46267e6i		−	0.116058i	0.998315	−	0.0580290i	$-0.0184816\pi$
$978$		0			0
$979$	1.25073e7			0.417067
$980$		0			0
$981$	−2.00455e7			−0.665033
$982$		0			0
$983$		−	3.74695e7i		−	1.23678i	−0.785870	−	0.618392i	$-0.787784\pi$
$983$		−	3.74695e7i		−	1.23678i	0.785870	−	0.618392i	$-0.212216\pi$
$984$		0			0
$985$	1.77652e6	−	1.75106e6i	0.0583419	−	0.0575057i
$986$		0			0
$987$		−	1.01287e7i		−	0.330949i
$988$		0			0
$989$	1.74803e7			0.568275
$990$		0			0
$991$	2.73653e7			0.885150			0.442575	−	0.896732i	$-0.354065\pi$
$991$	2.73653e7			0.885150			0.442575	+	0.896732i	$0.354065\pi$
$992$		0			0
$993$		−	2.79112e7i		−	0.898267i
$994$		0			0
$995$	2.15912e7	+	2.19052e7i	0.691385	+	0.701438i
$996$		0			0
$997$		−	1.37185e7i		−	0.437087i	−0.975827	−	0.218543i	$-0.929869\pi$
$997$		−	1.37185e7i		−	0.437087i	0.975827	−	0.218543i	$-0.0701305\pi$
$998$		0			0
$999$	−9.61861e6			−0.304929

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	220.6.b.b.89.2	✓	14
5.2	odd	4		1100.6.a.m.1.2		14
5.3	odd	4		1100.6.a.m.1.13		14
5.4	even	2	inner	220.6.b.b.89.13	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.6.b.b.89.2	✓	14	1.1	even	1	trivial
220.6.b.b.89.13	yes	14	5.4	even	2	inner
1100.6.a.m.1.2		14	5.2	odd	4
1100.6.a.m.1.13		14	5.3	odd	4

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:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	220.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-24.5916i q^{3} +(39.2421 + 39.8128i) q^{5} +68.6167i q^{7} -361.745 q^{9} +121.000 q^{11} -322.943i q^{13} +(979.058 - 965.025i) q^{15} +1078.94i q^{17} +2901.39 q^{19} +1687.39 q^{21} +1136.77i q^{23} +(-45.1117 + 3124.67i) q^{25} +2920.12i q^{27} -732.481 q^{29} +6971.80 q^{31} -2975.58i q^{33} +(-2731.82 + 2692.66i) q^{35} +3293.91i q^{37} -7941.68 q^{39} +15016.5 q^{41} -15377.2i q^{43} +(-14195.6 - 14402.1i) q^{45} -6002.58i q^{47} +12098.7 q^{49} +26532.8 q^{51} -35344.7i q^{53} +(4748.30 + 4817.34i) q^{55} -71349.6i q^{57} -20735.0 q^{59} -44901.6 q^{61} -24821.7i q^{63} +(12857.3 - 12673.0i) q^{65} +67912.8i q^{67} +27954.9 q^{69} +11219.8 q^{71} -51188.9i q^{73} +(76840.6 + 1109.37i) q^{75} +8302.62i q^{77} +49679.5 q^{79} -16093.8 q^{81} -1551.85i q^{83} +(-42955.5 + 42339.9i) q^{85} +18012.8i q^{87} +103366. q^{89} +22159.3 q^{91} -171447. i q^{93} +(113857. + 115512. i) q^{95} -541.081i q^{97} -43771.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 44 q^{5} - 1142 q^{9} + 1694 q^{11} + 326 q^{15} - 4540 q^{19} + 3824 q^{21} - 3816 q^{25} - 9972 q^{29} + 19076 q^{31} + 5136 q^{35} - 13616 q^{39} + 15052 q^{41} - 8374 q^{45} - 55346 q^{49} - 13380 q^{51}+ \cdots - 138182 q^{99}+O(q^{100}) \) 14 * q - 44 * q^5 - 1142 * q^9 + 1694 * q^11 + 326 * q^15 - 4540 * q^19 + 3824 * q^21 - 3816 * q^25 - 9972 * q^29 + 19076 * q^31 + 5136 * q^35 - 13616 * q^39 + 15052 * q^41 - 8374 * q^45 - 55346 * q^49 - 13380 * q^51 - 5324 * q^55 - 2108 * q^59 + 11660 * q^61 - 65664 * q^65 + 1620 * q^69 - 89284 * q^71 + 48362 * q^75 + 19432 * q^79 - 94714 * q^81 + 117404 * q^85 - 38836 * q^89 + 39632 * q^91 + 231336 * q^95 - 138182 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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