LMFDB - Embedded newform 220.6.b.b.89.14

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			89.14
Root			$26.2077i$ of defining polynomial
Character	$\chi$	$=$	220.89
Dual form			220.6.b.b.89.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+26.2077i q^{3} +(12.7997 + 54.4166i) q^{5} -64.0890i q^{7} -443.842 q^{9} +121.000 q^{11} -528.657i q^{13} +(-1426.13 + 335.450i) q^{15} -1541.39i q^{17} -3107.81 q^{19} +1679.62 q^{21} -1789.78i q^{23} +(-2797.34 + 1393.03i) q^{25} -5263.61i q^{27} -4892.43 q^{29} +1172.26 q^{31} +3171.13i q^{33} +(3487.50 - 820.319i) q^{35} +12632.9i q^{37} +13854.9 q^{39} +1520.48 q^{41} -11538.1i q^{43} +(-5681.05 - 24152.4i) q^{45} -19780.6i q^{47} +12699.6 q^{49} +40396.2 q^{51} +16273.8i q^{53} +(1548.76 + 6584.41i) q^{55} -81448.4i q^{57} -39459.2 q^{59} +18830.5 q^{61} +28445.4i q^{63} +(28767.7 - 6766.65i) q^{65} +40415.1i q^{67} +46906.1 q^{69} -65200.4 q^{71} -46369.3i q^{73} +(-36508.1 - 73311.7i) q^{75} -7754.77i q^{77} +20145.5 q^{79} +30093.4 q^{81} +85967.1i q^{83} +(83877.2 - 19729.3i) q^{85} -128219. i q^{87} -12067.3 q^{89} -33881.1 q^{91} +30722.2i q^{93} +(-39779.0 - 169116. i) q^{95} -137100. i q^{97} -53704.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			26.2077i			1.68122i	0.541638	+	0.840612i	$0.317804\pi$
$3$			26.2077i			1.68122i	−0.541638	+	0.840612i	$0.682196\pi$
$4$		0			0
$5$	12.7997	+	54.4166i	0.228968	+	0.973434i
$5$	12.7997	+	54.4166i	0.228968	+	0.973434i
$6$		0			0
$7$		−	64.0890i		−	0.494354i	−0.968970	−	0.247177i	$-0.920497\pi$
$7$		−	64.0890i		−	0.494354i	0.968970	−	0.247177i	$-0.0795029\pi$
$8$		0			0
$9$	−443.842			−1.82651
$10$		0			0
$11$	121.000			0.301511
$11$	121.000			0.301511
$12$		0			0
$13$		−	528.657i		−	0.867592i	−0.901011	−	0.433796i	$-0.857174\pi$
$13$		−	528.657i		−	0.867592i	0.901011	−	0.433796i	$-0.142826\pi$
$14$		0			0
$15$	−1426.13	+	335.450i	−1.63656	+	0.384946i
$16$		0			0
$17$		−	1541.39i		−	1.29357i	−0.762672	−	0.646785i	$-0.776113\pi$
$17$		−	1541.39i		−	1.29357i	0.762672	−	0.646785i	$-0.223887\pi$
$18$		0			0
$19$	−3107.81			−1.97501			−0.987507	−	0.157576i	$-0.949632\pi$
$19$	−3107.81			−1.97501			−0.987507	+	0.157576i	$0.949632\pi$
$20$		0			0
$21$	1679.62			0.831120
$22$		0			0
$23$		−	1789.78i		−	0.705474i	−0.935722	−	0.352737i	$-0.885251\pi$
$23$		−	1789.78i		−	0.705474i	0.935722	−	0.352737i	$-0.114749\pi$
$24$		0			0
$25$	−2797.34	+	1393.03i	−0.895147	+	0.445770i
$26$		0			0
$27$		−	5263.61i		−	1.38955i
$28$		0			0
$29$	−4892.43			−1.08026			−0.540131	−	0.841581i	$-0.681625\pi$
$29$	−4892.43			−1.08026			−0.540131	+	0.841581i	$0.681625\pi$
$30$		0			0
$31$	1172.26			0.219089			0.109544	−	0.993982i	$-0.465061\pi$
$31$	1172.26			0.219089			0.109544	+	0.993982i	$0.465061\pi$
$32$		0			0
$33$			3171.13i			0.506908i
$34$		0			0
$35$	3487.50	−	820.319i	0.481221	−	0.113191i
$36$		0			0
$37$			12632.9i			1.51705i	0.651644	+	0.758525i	$0.274080\pi$
$37$			12632.9i			1.51705i	−0.651644	+	0.758525i	$0.725920\pi$
$38$		0			0
$39$	13854.9			1.45862
$40$		0			0
$41$	1520.48			0.141261			0.0706305	−	0.997503i	$-0.477499\pi$
$41$	1520.48			0.141261			0.0706305	+	0.997503i	$0.477499\pi$
$42$		0			0
$43$		−	11538.1i		−	0.951620i	−0.879548	−	0.475810i	$-0.842155\pi$
$43$		−	11538.1i		−	0.951620i	0.879548	−	0.475810i	$-0.157845\pi$
$44$		0			0
$45$	−5681.05	−	24152.4i	−0.418213	−	1.77799i
$46$		0			0
$47$		−	19780.6i		−	1.30615i	−0.757292	−	0.653076i	$-0.773478\pi$
$47$		−	19780.6i		−	1.30615i	0.757292	−	0.653076i	$-0.226522\pi$
$48$		0			0
$49$	12699.6			0.755614
$50$		0			0
$51$	40396.2			2.17478
$52$		0			0
$53$			16273.8i			0.795790i	0.917431	+	0.397895i	$0.130259\pi$
$53$			16273.8i			0.795790i	−0.917431	+	0.397895i	$0.869741\pi$
$54$		0			0
$55$	1548.76	+	6584.41i	0.0690364	+	0.293501i
$56$		0			0
$57$		−	81448.4i		−	3.32044i
$58$		0			0
$59$	−39459.2			−1.47577			−0.737884	−	0.674927i	$-0.764175\pi$
$59$	−39459.2			−1.47577			−0.737884	+	0.674927i	$0.764175\pi$
$60$		0			0
$61$	18830.5			0.647943			0.323972	−	0.946067i	$-0.394982\pi$
$61$	18830.5			0.647943			0.323972	+	0.946067i	$0.394982\pi$
$62$		0			0
$63$			28445.4i			0.902944i
$64$		0			0
$65$	28767.7	−	6766.65i	0.844543	−	0.198651i
$66$		0			0
$67$			40415.1i			1.09991i	0.835194	+	0.549955i	$0.185355\pi$
$67$			40415.1i			1.09991i	−0.835194	+	0.549955i	$0.814645\pi$
$68$		0			0
$69$	46906.1			1.18606
$70$		0			0
$71$	−65200.4			−1.53499			−0.767493	−	0.641057i	$-0.778496\pi$
$71$	−65200.4			−1.53499			−0.767493	+	0.641057i	$0.778496\pi$
$72$		0			0
$73$		−	46369.3i		−	1.01841i	−0.860645	−	0.509205i	$-0.829939\pi$
$73$		−	46369.3i		−	1.01841i	0.860645	−	0.509205i	$-0.170061\pi$
$74$		0			0
$75$	−36508.1	−	73311.7i	−0.749439	−	1.50494i
$76$		0			0
$77$		−	7754.77i		−	0.149053i
$78$		0			0
$79$	20145.5			0.363170			0.181585	−	0.983375i	$-0.441877\pi$
$79$	20145.5			0.363170			0.181585	+	0.983375i	$0.441877\pi$
$80$		0			0
$81$	30093.4			0.509634
$82$		0			0
$83$			85967.1i			1.36974i	0.728667	+	0.684869i	$0.240140\pi$
$83$			85967.1i			1.36974i	−0.728667	+	0.684869i	$0.759860\pi$
$84$		0			0
$85$	83877.2	−	19729.3i	1.25920	−	0.296186i
$86$		0			0
$87$		−	128219.i		−	1.81616i
$88$		0			0
$89$	−12067.3			−0.161486			−0.0807430	−	0.996735i	$-0.525729\pi$
$89$	−12067.3			−0.161486			−0.0807430	+	0.996735i	$0.525729\pi$
$90$		0			0
$91$	−33881.1			−0.428898
$92$		0			0
$93$			30722.2i			0.368337i
$94$		0			0
$95$	−39779.0	−	169116.i	−0.452215	−	1.92255i
$96$		0			0
$97$		−	137100.i		−	1.47947i	−0.672896	−	0.739737i	$-0.734950\pi$
$97$		−	137100.i		−	1.47947i	0.672896	−	0.739737i	$-0.265050\pi$
$98$		0			0
$99$	−53704.9			−0.550714
$100$		0			0
$101$	−47877.0			−0.467008			−0.233504	−	0.972356i	$-0.575019\pi$
$101$	−47877.0			−0.467008			−0.233504	+	0.972356i	$0.575019\pi$
$102$		0			0
$103$			83825.2i			0.778541i	0.921123	+	0.389271i	$0.127273\pi$
$103$			83825.2i			0.778541i	−0.921123	+	0.389271i	$0.872727\pi$
$104$		0			0
$105$	21498.7	+	91399.4i	0.190300	+	0.809040i
$106$		0			0
$107$		−	25557.2i		−	0.215801i	−0.994162	−	0.107901i	$-0.965587\pi$
$107$		−	25557.2i		−	0.215801i	0.994162	−	0.107901i	$-0.0344128\pi$
$108$		0			0
$109$	−54744.9			−0.441344			−0.220672	−	0.975348i	$-0.570825\pi$
$109$	−54744.9			−0.441344			−0.220672	+	0.975348i	$0.570825\pi$
$110$		0			0
$111$	−331080.			−2.55050
$112$		0			0
$113$			72063.0i			0.530904i	0.964124	+	0.265452i	$0.0855212\pi$
$113$			72063.0i			0.530904i	−0.964124	+	0.265452i	$0.914479\pi$
$114$		0			0
$115$	97394.0	−	22908.7i	0.686732	−	0.161531i
$116$		0			0
$117$			234640.i			1.58467i
$118$		0			0
$119$	−98786.0			−0.639482
$120$		0			0
$121$	14641.0			0.0909091
$122$		0			0
$123$			39848.4i			0.237491i
$124$		0			0
$125$	−111609.	−	134391.i	−0.638888	−	0.769300i
$126$		0			0
$127$			113193.i			0.622745i	0.950288	+	0.311372i	$0.100789\pi$
$127$			113193.i			0.622745i	−0.950288	+	0.311372i	$0.899211\pi$
$128$		0			0
$129$	302387.			1.59989
$130$		0			0
$131$	142156.			0.723746			0.361873	−	0.932227i	$-0.382137\pi$
$131$	142156.			0.723746			0.361873	+	0.932227i	$0.382137\pi$
$132$		0			0
$133$			199176.i			0.976356i
$134$		0			0
$135$	286428.	−	67372.6i	1.35264	−	0.318163i
$136$		0			0
$137$		−	206954.i		−	0.942045i	−0.882121	−	0.471022i	$-0.843885\pi$
$137$		−	206954.i		−	0.942045i	0.882121	−	0.471022i	$-0.156115\pi$
$138$		0			0
$139$	−432970.			−1.90073			−0.950367	−	0.311132i	$-0.899292\pi$
$139$	−432970.			−1.90073			−0.950367	+	0.311132i	$0.899292\pi$
$140$		0			0
$141$	518402.			2.19593
$142$		0			0
$143$		−	63967.5i		−	0.261589i
$144$		0			0
$145$	−62621.6	−	266229.i	−0.247345	−	1.05156i
$146$		0			0
$147$			332827.i			1.27036i
$148$		0			0
$149$	171827.			0.634052			0.317026	−	0.948417i	$-0.397316\pi$
$149$	171827.			0.634052			0.317026	+	0.948417i	$0.397316\pi$
$150$		0			0
$151$	295056.			1.05308			0.526542	−	0.850149i	$-0.323488\pi$
$151$	295056.			1.05308			0.526542	+	0.850149i	$0.323488\pi$
$152$		0			0
$153$			684134.i			2.36272i
$154$		0			0
$155$	15004.6	+	63790.5i	0.0501643	+	0.213268i
$156$		0			0
$157$			133587.i			0.432529i	0.976335	+	0.216264i	$0.0693873\pi$
$157$			133587.i			0.432529i	−0.976335	+	0.216264i	$0.930613\pi$
$158$		0			0
$159$	−426498.			−1.33790
$160$		0			0
$161$	−114705.			−0.348754
$162$		0			0
$163$			326639.i			0.962939i	0.876463	+	0.481470i	$0.159897\pi$
$163$			326639.i			0.962939i	−0.876463	+	0.481470i	$0.840103\pi$
$164$		0			0
$165$	−172562.	+	40589.5i	−0.493441	+	0.116066i
$166$		0			0
$167$			614485.i			1.70498i	0.522741	+	0.852491i	$0.324909\pi$
$167$			614485.i			1.70498i	−0.522741	+	0.852491i	$0.675091\pi$
$168$		0			0
$169$	91814.9			0.247284
$170$		0			0
$171$	1.37938e6			3.60739
$172$		0			0
$173$			500953.i			1.27257i	0.771454	+	0.636285i	$0.219530\pi$
$173$			500953.i			1.27257i	−0.771454	+	0.636285i	$0.780470\pi$
$174$		0			0
$175$	89278.0	+	179278.i	0.220368	+	0.442520i
$176$		0			0
$177$		−	1.03413e6i		−	2.48110i
$178$		0			0
$179$	−305612.			−0.712915			−0.356457	−	0.934312i	$-0.616016\pi$
$179$	−305612.			−0.712915			−0.356457	+	0.934312i	$0.616016\pi$
$180$		0			0
$181$	−52085.1			−0.118173			−0.0590864	−	0.998253i	$-0.518819\pi$
$181$	−52085.1			−0.118173			−0.0590864	+	0.998253i	$0.518819\pi$
$182$		0			0
$183$			493503.i			1.08934i
$184$		0			0
$185$	−687442.	+	161698.i	−1.47675	+	0.347356i
$186$		0			0
$187$		−	186508.i		−	0.390026i
$188$		0			0
$189$	−337340.			−0.686930
$190$		0			0
$191$	−894767.			−1.77471			−0.887354	−	0.461089i	$-0.847459\pi$
$191$	−894767.			−1.77471			−0.887354	+	0.461089i	$0.847459\pi$
$192$		0			0
$193$		−	456312.i		−	0.881797i	−0.897557	−	0.440899i	$-0.854660\pi$
$193$		−	456312.i		−	0.881797i	0.897557	−	0.440899i	$-0.145340\pi$
$194$		0			0
$195$	177338.	+	753935.i	0.333976	+	1.41987i
$196$		0			0
$197$		−	423709.i		−	0.777861i	−0.921267	−	0.388931i	$-0.872845\pi$
$197$		−	423709.i		−	0.777861i	0.921267	−	0.388931i	$-0.127155\pi$
$198$		0			0
$199$	−1.09199e6			−1.95472			−0.977360	−	0.211581i	$-0.932139\pi$
$199$	−1.09199e6			−1.95472			−0.977360	+	0.211581i	$0.932139\pi$
$200$		0			0
$201$	−1.05919e6			−1.84919
$202$		0			0
$203$			313551.i			0.534032i
$204$		0			0
$205$	19461.7	+	82739.6i	0.0323442	+	0.137508i
$206$		0			0
$207$			794382.i			1.28856i
$208$		0			0
$209$	−376045.			−0.595489
$210$		0			0
$211$	−741511.			−1.14660			−0.573299	−	0.819346i	$-0.694337\pi$
$211$	−741511.			−1.14660			−0.573299	+	0.819346i	$0.694337\pi$
$212$		0			0
$213$		−	1.70875e6i		−	2.58065i
$214$		0			0
$215$	627865.	−	147684.i	0.926340	−	0.217890i
$216$		0			0
$217$		−	75129.0i		−	0.108307i
$218$		0			0
$219$	1.21523e6			1.71218
$220$		0			0
$221$	−814866.			−1.12229
$222$		0			0
$223$		−	1.01755e6i		−	1.37023i	−0.728436	−	0.685113i	$-0.759753\pi$
$223$		−	1.01755e6i		−	1.37023i	0.728436	−	0.685113i	$-0.240247\pi$
$224$		0			0
$225$	1.24158e6	−	618287.i	1.63500	−	0.814205i
$226$		0			0
$227$			51060.6i			0.0657690i	0.999459	+	0.0328845i	$0.0104694\pi$
$227$			51060.6i			0.0657690i	−0.999459	+	0.0328845i	$0.989531\pi$
$228$		0			0
$229$	−405426.			−0.510885			−0.255443	−	0.966824i	$-0.582221\pi$
$229$	−405426.			−0.510885			−0.255443	+	0.966824i	$0.582221\pi$
$230$		0			0
$231$	203234.			0.250592
$232$		0			0
$233$		−	175866.i		−	0.212223i	−0.994354	−	0.106111i	$-0.966160\pi$
$233$		−	175866.i		−	0.212223i	0.994354	−	0.106111i	$-0.0338400\pi$
$234$		0			0
$235$	1.07639e6	−	253185.i	1.27145	−	0.299067i
$236$		0			0
$237$			527967.i			0.610570i
$238$		0			0
$239$	−310598.			−0.351726			−0.175863	−	0.984415i	$-0.556272\pi$
$239$	−310598.			−0.351726			−0.175863	+	0.984415i	$0.556272\pi$
$240$		0			0
$241$	−62501.7			−0.0693185			−0.0346592	−	0.999399i	$-0.511035\pi$
$241$	−62501.7			−0.0693185			−0.0346592	+	0.999399i	$0.511035\pi$
$242$		0			0
$243$		−	490381.i		−	0.532743i
$244$		0			0
$245$	162551.	+	691069.i	0.173011	+	0.735540i
$246$		0			0
$247$			1.64296e6i			1.71351i
$248$		0			0
$249$	−2.25300e6			−2.30283
$250$		0			0
$251$	1.65573e6			1.65885			0.829423	−	0.558621i	$-0.188670\pi$
$251$	1.65573e6			1.65885			0.829423	+	0.558621i	$0.188670\pi$
$252$		0			0
$253$		−	216564.i		−	0.212708i
$254$		0			0
$255$	517059.	+	2.19823e6i	0.497955	+	2.11700i
$256$		0			0
$257$			11379.3i			0.0107469i	0.999986	+	0.00537344i	$0.00171043\pi$
$257$			11379.3i			0.0107469i	−0.999986	+	0.00537344i	$0.998290\pi$
$258$		0			0
$259$	809632.			0.749960
$260$		0			0
$261$	2.17147e6			1.97311
$262$		0			0
$263$		−	1.42857e6i		−	1.27354i	−0.771053	−	0.636771i	$-0.780270\pi$
$263$		−	1.42857e6i		−	1.27354i	0.771053	−	0.636771i	$-0.219730\pi$
$264$		0			0
$265$	−885564.	+	208299.i	−0.774649	+	0.182210i
$266$		0			0
$267$		−	316256.i		−	0.271494i
$268$		0			0
$269$	1.52305e6			1.28331			0.641657	−	0.766991i	$-0.278247\pi$
$269$	1.52305e6			1.28331			0.641657	+	0.766991i	$0.278247\pi$
$270$		0			0
$271$	−629994.			−0.521090			−0.260545	−	0.965462i	$-0.583902\pi$
$271$	−629994.			−0.521090			−0.260545	+	0.965462i	$0.583902\pi$
$272$		0			0
$273$		−	887944.i		−	0.721073i
$274$		0			0
$275$	−338478.	+	168557.i	−0.269897	+	0.134405i
$276$		0			0
$277$		−	2.27661e6i		−	1.78274i	−0.453275	−	0.891371i	$-0.649744\pi$
$277$		−	2.27661e6i		−	1.78274i	0.453275	−	0.891371i	$-0.350256\pi$
$278$		0			0
$279$	−520299.			−0.400168
$280$		0			0
$281$	496611.			0.375189			0.187595	−	0.982247i	$-0.439931\pi$
$281$	496611.			0.375189			0.187595	+	0.982247i	$0.439931\pi$
$282$		0			0
$283$		−	1.50166e6i		−	1.11456i	−0.830323	−	0.557282i	$-0.811844\pi$
$283$		−	1.50166e6i		−	1.11456i	0.830323	−	0.557282i	$-0.188156\pi$
$284$		0			0
$285$	4.43215e6	−	1.04251e6i	3.23223	−	0.760274i
$286$		0			0
$287$		−	97446.3i		−	0.0698330i
$288$		0			0
$289$	−956023.			−0.673323
$290$		0			0
$291$	3.59307e6			2.48733
$292$		0			0
$293$		−	1.42593e6i		−	0.970355i	−0.874416	−	0.485177i	$-0.838755\pi$
$293$		−	1.42593e6i		−	0.970355i	0.874416	−	0.485177i	$-0.161245\pi$
$294$		0			0
$295$	−505066.	−	2.14724e6i	−0.337904	−	1.43656i
$296$		0			0
$297$		−	636897.i		−	0.418965i
$298$		0			0
$299$	−946182.			−0.612064
$300$		0			0
$301$	−739466.			−0.470438
$302$		0			0
$303$		−	1.25475e6i		−	0.785144i
$304$		0			0
$305$	241025.	+	1.02469e6i	0.148358	+	0.630730i
$306$		0			0
$307$			1.86573e6i			1.12981i	0.825157	+	0.564903i	$0.191086\pi$
$307$			1.86573e6i			1.12981i	−0.825157	+	0.564903i	$0.808914\pi$
$308$		0			0
$309$	−2.19686e6			−1.30890
$310$		0			0
$311$	61240.1			0.0359034			0.0179517	−	0.999839i	$-0.494285\pi$
$311$	61240.1			0.0359034			0.0179517	+	0.999839i	$0.494285\pi$
$312$		0			0
$313$		−	152008.i		−	0.0877011i	−0.999038	−	0.0438506i	$-0.986037\pi$
$313$		−	152008.i		−	0.0877011i	0.999038	−	0.0438506i	$-0.0139625\pi$
$314$		0			0
$315$	−1.54790e6	+	364092.i	−0.878956	+	0.206745i
$316$		0			0
$317$		−	1.21712e6i		−	0.680277i	−0.940375	−	0.340138i	$-0.889526\pi$
$317$		−	1.21712e6i		−	0.680277i	0.940375	−	0.340138i	$-0.110474\pi$
$318$		0			0
$319$	−591984.			−0.325711
$320$		0			0
$321$	669795.			0.362810
$322$		0			0
$323$			4.79034e6i			2.55482i
$324$		0			0
$325$	736436.	+	1.47883e6i	0.386747	+	0.776623i
$326$		0			0
$327$		−	1.43474e6i		−	0.741999i
$328$		0			0
$329$	−1.26772e6			−0.645702
$330$		0			0
$331$	−2.90401e6			−1.45690			−0.728448	−	0.685101i	$-0.759758\pi$
$331$	−2.90401e6			−1.45690			−0.728448	+	0.685101i	$0.759758\pi$
$332$		0			0
$333$		−	5.60703e6i		−	2.77091i
$334$		0			0
$335$	−2.19926e6	+	517302.i	−1.07069	+	0.251844i
$336$		0			0
$337$		−	577357.i		−	0.276930i	−0.990367	−	0.138465i	$-0.955783\pi$
$337$		−	577357.i		−	0.276930i	0.990367	−	0.138465i	$-0.0442168\pi$
$338$		0			0
$339$	−1.88860e6			−0.892568
$340$		0			0
$341$	141844.			0.0660577
$342$		0			0
$343$		−	1.89105e6i		−	0.867895i
$344$		0			0
$345$	600384.	+	2.55247e6i	0.271570	+	1.15455i
$346$		0			0
$347$			2.99345e6i			1.33459i	0.744793	+	0.667296i	$0.232548\pi$
$347$			2.99345e6i			1.33459i	−0.744793	+	0.667296i	$0.767452\pi$
$348$		0			0
$349$	−2.05019e6			−0.901012			−0.450506	−	0.892773i	$-0.648756\pi$
$349$	−2.05019e6			−0.901012			−0.450506	+	0.892773i	$0.648756\pi$
$350$		0			0
$351$	−2.78264e6			−1.20556
$352$		0			0
$353$			566987.i			0.242179i	0.992642	+	0.121089i	$0.0386388\pi$
$353$			566987.i			0.242179i	−0.992642	+	0.121089i	$0.961361\pi$
$354$		0			0
$355$	−834545.	−	3.54799e6i	−0.351463	−	1.49421i
$356$		0			0
$357$		−	2.58895e6i		−	1.07511i
$358$		0			0
$359$	−1.82570e6			−0.747643			−0.373821	−	0.927501i	$-0.621953\pi$
$359$	−1.82570e6			−0.747643			−0.373821	+	0.927501i	$0.621953\pi$
$360$		0			0
$361$	7.18237e6			2.90068
$362$		0			0
$363$			383707.i			0.152838i
$364$		0			0
$365$	2.52326e6	−	593512.i	0.991356	−	0.233183i
$366$		0			0
$367$			1.33009e6i			0.515485i	0.966214	+	0.257742i	$0.0829785\pi$
$367$			1.33009e6i			0.515485i	−0.966214	+	0.257742i	$0.917021\pi$
$368$		0			0
$369$	−674856.			−0.258015
$370$		0			0
$371$	1.04297e6			0.393402
$372$		0			0
$373$			475043.i			0.176791i	0.996085	+	0.0883956i	$0.0281740\pi$
$373$			475043.i			0.176791i	−0.996085	+	0.0883956i	$0.971826\pi$
$374$		0			0
$375$	3.52208e6	−	2.92502e6i	1.29336	−	1.07411i
$376$		0			0
$377$			2.58642e6i			0.937227i
$378$		0			0
$379$	−1.05251e6			−0.376382			−0.188191	−	0.982132i	$-0.560262\pi$
$379$	−1.05251e6			−0.376382			−0.188191	+	0.982132i	$0.560262\pi$
$380$		0			0
$381$	−2.96652e6			−1.04697
$382$		0			0
$383$		−	1.05438e6i		−	0.367283i	−0.982993	−	0.183641i	$-0.941211\pi$
$383$		−	1.05438e6i		−	0.367283i	0.982993	−	0.183641i	$-0.0587885\pi$
$384$		0			0
$385$	421988.	−	99258.6i	0.145094	−	0.0341284i
$386$		0			0
$387$			5.12110e6i			1.73815i
$388$		0			0
$389$	−4.41847e6			−1.48046			−0.740231	−	0.672352i	$-0.765284\pi$
$389$	−4.41847e6			−1.48046			−0.740231	+	0.672352i	$0.765284\pi$
$390$		0			0
$391$	−2.75875e6			−0.912580
$392$		0			0
$393$			3.72557e6i			1.21678i
$394$		0			0
$395$	257856.	+	1.09625e6i	0.0831543	+	0.353522i
$396$		0			0
$397$			1.02930e6i			0.327766i	0.986480	+	0.163883i	$0.0524020\pi$
$397$			1.02930e6i			0.327766i	−0.986480	+	0.163883i	$0.947598\pi$
$398$		0			0
$399$	−5.21995e6			−1.64147
$400$		0			0
$401$	4.82524e6			1.49850			0.749252	−	0.662285i	$-0.230413\pi$
$401$	4.82524e6			1.49850			0.749252	+	0.662285i	$0.230413\pi$
$402$		0			0
$403$		−	619724.i		−	0.190080i
$404$		0			0
$405$	385186.	+	1.63758e6i	0.116690	+	0.496095i
$406$		0			0
$407$			1.52859e6i			0.457408i
$408$		0			0
$409$	5.24511e6			1.55041			0.775205	−	0.631710i	$-0.217647\pi$
$409$	5.24511e6			1.55041			0.775205	+	0.631710i	$0.217647\pi$
$410$		0			0
$411$	5.42377e6			1.58379
$412$		0			0
$413$			2.52890e6i			0.729552i
$414$		0			0
$415$	−4.67804e6	+	1.10035e6i	−1.33335	+	0.313626i
$416$		0			0
$417$		−	1.13472e7i		−	3.19556i
$418$		0			0
$419$	588049.			0.163636			0.0818180	−	0.996647i	$-0.473927\pi$
$419$	588049.			0.163636			0.0818180	+	0.996647i	$0.473927\pi$
$420$		0			0
$421$	4.83665e6			1.32996			0.664982	−	0.746860i	$-0.268439\pi$
$421$	4.83665e6			1.32996			0.664982	+	0.746860i	$0.268439\pi$
$422$		0			0
$423$			8.77945e6i			2.38570i
$424$		0			0
$425$	2.14720e6	+	4.31178e6i	0.576635	+	1.15794i
$426$		0			0
$427$		−	1.20683e6i		−	0.320314i
$428$		0			0
$429$	1.67644e6			0.439789
$430$		0			0
$431$	−3.40144e6			−0.882003			−0.441002	−	0.897506i	$-0.645377\pi$
$431$	−3.40144e6			−0.882003			−0.441002	+	0.897506i	$0.645377\pi$
$432$		0			0
$433$			5.81707e6i			1.49102i	0.666493	+	0.745512i	$0.267795\pi$
$433$			5.81707e6i			1.49102i	−0.666493	+	0.745512i	$0.732205\pi$
$434$		0			0
$435$	6.97725e6	−	1.64117e6i	1.76791	−	0.415843i
$436$		0			0
$437$			5.56230e6i			1.39332i
$438$		0			0
$439$	7.33653e6			1.81689			0.908446	−	0.418001i	$-0.137269\pi$
$439$	7.33653e6			1.81689			0.908446	+	0.418001i	$0.137269\pi$
$440$		0			0
$441$	−5.63662e6			−1.38014
$442$		0			0
$443$		−	1.29709e6i		−	0.314022i	−0.987597	−	0.157011i	$-0.949814\pi$
$443$		−	1.29709e6i		−	0.314022i	0.987597	−	0.157011i	$-0.0501858\pi$
$444$		0			0
$445$	−154458.	−	656661.i	−0.0369751	−	0.157196i
$446$		0			0
$447$			4.50317e6i			1.06598i
$448$		0			0
$449$	−4.47446e6			−1.04743			−0.523715	−	0.851894i	$-0.675454\pi$
$449$	−4.47446e6			−1.04743			−0.523715	+	0.851894i	$0.675454\pi$
$450$		0			0
$451$	183979.			0.0425918
$452$		0			0
$453$			7.73275e6i			1.77047i
$454$		0			0
$455$	−433667.	−	1.84369e6i	−0.0982038	−	0.417504i
$456$		0			0
$457$		−	5.49383e6i		−	1.23051i	−0.788329	−	0.615254i	$-0.789054\pi$
$457$		−	5.49383e6i		−	1.23051i	0.788329	−	0.615254i	$-0.210946\pi$
$458$		0			0
$459$	−8.11327e6			−1.79748
$460$		0			0
$461$	−4.40705e6			−0.965819			−0.482909	−	0.875670i	$-0.660420\pi$
$461$	−4.40705e6			−0.965819			−0.482909	+	0.875670i	$0.660420\pi$
$462$		0			0
$463$			5.34156e6i			1.15802i	0.815321	+	0.579009i	$0.196560\pi$
$463$			5.34156e6i			1.15802i	−0.815321	+	0.579009i	$0.803440\pi$
$464$		0			0
$465$	−1.67180e6	+	393235.i	−0.358552	+	0.0843374i
$466$		0			0
$467$			4.24423e6i			0.900548i	0.892890	+	0.450274i	$0.148674\pi$
$467$			4.24423e6i			0.900548i	−0.892890	+	0.450274i	$0.851326\pi$
$468$		0			0
$469$	2.59017e6			0.543745
$470$		0			0
$471$	−3.50100e6			−0.727177
$472$		0			0
$473$		−	1.39611e6i		−	0.286924i
$474$		0			0
$475$	8.69358e6	−	4.32928e6i	1.76793	−	0.880402i
$476$		0			0
$477$		−	7.22299e6i		−	1.45352i
$478$		0			0
$479$	−127031.			−0.0252970			−0.0126485	−	0.999920i	$-0.504026\pi$
$479$	−127031.			−0.0252970			−0.0126485	+	0.999920i	$0.504026\pi$
$480$		0			0
$481$	6.67849e6			1.31618
$482$		0			0
$483$		−	3.00616e6i		−	0.586333i
$484$		0			0
$485$	7.46050e6	−	1.75483e6i	1.44017	−	0.338752i
$486$		0			0
$487$		−	5.88474e6i		−	1.12436i	−0.827015	−	0.562179i	$-0.809963\pi$
$487$		−	5.88474e6i		−	1.12436i	0.827015	−	0.562179i	$-0.190037\pi$
$488$		0			0
$489$	−8.56044e6			−1.61892
$490$		0			0
$491$	670926.			0.125595			0.0627973	−	0.998026i	$-0.479998\pi$
$491$	670926.			0.125595			0.0627973	+	0.998026i	$0.479998\pi$
$492$		0			0
$493$			7.54113e6i			1.39740i
$494$		0			0
$495$	−687407.	−	2.92244e6i	−0.126096	−	0.536084i
$496$		0			0
$497$			4.17863e6i			0.758827i
$498$		0			0
$499$	4.23099e6			0.760661			0.380330	−	0.924851i	$-0.375810\pi$
$499$	4.23099e6			0.760661			0.380330	+	0.924851i	$0.375810\pi$
$500$		0			0
$501$	−1.61042e7			−2.86646
$502$		0			0
$503$		−	4.42693e6i		−	0.780157i	−0.920782	−	0.390079i	$-0.872448\pi$
$503$		−	4.42693e6i		−	0.780157i	0.920782	−	0.390079i	$-0.127552\pi$
$504$		0			0
$505$	−612812.	−	2.60531e6i	−0.106930	−	0.454601i
$506$		0			0
$507$			2.40626e6i			0.415740i
$508$		0			0
$509$	7.57091e6			1.29525			0.647625	−	0.761959i	$-0.275762\pi$
$509$	7.57091e6			1.29525			0.647625	+	0.761959i	$0.275762\pi$
$510$		0			0
$511$	−2.97176e6			−0.503456
$512$		0			0
$513$			1.63583e7i			2.74438i
$514$		0			0
$515$	−4.56149e6	+	1.07294e6i	−0.757859	+	0.178261i
$516$		0			0
$517$		−	2.39345e6i		−	0.393820i
$518$		0			0
$519$	−1.31288e7			−2.13948
$520$		0			0
$521$	−4.67098e6			−0.753900			−0.376950	−	0.926234i	$-0.623027\pi$
$521$	−4.67098e6			−0.753900			−0.376950	+	0.926234i	$0.623027\pi$
$522$		0			0
$523$		−	3.45852e6i		−	0.552887i	−0.961030	−	0.276444i	$-0.910844\pi$
$523$		−	3.45852e6i		−	0.552887i	0.961030	−	0.276444i	$-0.0891559\pi$
$524$		0			0
$525$	−4.69847e6	+	2.33977e6i	−0.743975	+	0.370488i
$526$		0			0
$527$		−	1.80691e6i		−	0.283407i
$528$		0			0
$529$	3.23302e6			0.502306
$530$		0			0
$531$	1.75137e7			2.69551
$532$		0			0
$533$		−	803815.i		−	0.122557i
$534$		0			0
$535$	1.39074e6	−	327124.i	0.210068	−	0.0494115i
$536$		0			0
$537$		−	8.00938e6i		−	1.19857i
$538$		0			0
$539$	1.53665e6			0.227826
$540$		0			0
$541$	−9.09362e6			−1.33581			−0.667904	−	0.744248i	$-0.732808\pi$
$541$	−9.09362e6			−1.33581			−0.667904	+	0.744248i	$0.732808\pi$
$542$		0			0
$543$		−	1.36503e6i		−	0.198675i
$544$		0			0
$545$	−700718.	−	2.97903e6i	−0.101054	−	0.429620i
$546$		0			0
$547$			5.08979e6i			0.727331i	0.931530	+	0.363665i	$0.118475\pi$
$547$			5.08979e6i			0.727331i	−0.931530	+	0.363665i	$0.881525\pi$
$548$		0			0
$549$	−8.35777e6			−1.18348
$550$		0			0
$551$	1.52047e7			2.13353
$552$		0			0
$553$		−	1.29110e6i		−	0.179535i
$554$		0			0
$555$	−4.23772e6	−	1.80163e7i	−0.583983	−	2.48274i
$556$		0			0
$557$			9.80352e6i			1.33889i	0.742863	+	0.669443i	$0.233467\pi$
$557$			9.80352e6i			1.33889i	−0.742863	+	0.669443i	$0.766533\pi$
$558$		0			0
$559$	−6.09970e6			−0.825618
$560$		0			0
$561$	4.88794e6			0.655721
$562$		0			0
$563$			1.28530e7i			1.70897i	0.519478	+	0.854484i	$0.326126\pi$
$563$			1.28530e7i			1.70897i	−0.519478	+	0.854484i	$0.673874\pi$
$564$		0			0
$565$	−3.92142e6	+	922384.i	−0.516800	+	0.121560i
$566$		0			0
$567$		−	1.92865e6i		−	0.251940i
$568$		0			0
$569$	−1.43558e6			−0.185886			−0.0929429	−	0.995671i	$-0.529627\pi$
$569$	−1.43558e6			−0.185886			−0.0929429	+	0.995671i	$0.529627\pi$
$570$		0			0
$571$	−4.63588e6			−0.595034			−0.297517	−	0.954716i	$-0.596159\pi$
$571$	−4.63588e6			−0.595034			−0.297517	+	0.954716i	$0.596159\pi$
$572$		0			0
$573$		−	2.34498e7i		−	2.98368i
$574$		0			0
$575$	2.49323e6	+	5.00663e6i	0.314479	+	0.631503i
$576$		0			0
$577$			591099.i			0.0739130i	0.999317	+	0.0369565i	$0.0117663\pi$
$577$			591099.i			0.0739130i	−0.999317	+	0.0369565i	$0.988234\pi$
$578$		0			0
$579$	1.19589e7			1.48250
$580$		0			0
$581$	5.50955e6			0.677135
$582$		0			0
$583$			1.96913e6i			0.239940i
$584$		0			0
$585$	−1.27683e7	+	3.00332e6i	−1.54257	+	0.362838i
$586$		0			0
$587$			2.87938e6i			0.344908i	0.985018	+	0.172454i	$0.0551696\pi$
$587$			2.87938e6i			0.344908i	−0.985018	+	0.172454i	$0.944830\pi$
$588$		0			0
$589$	−3.64316e6			−0.432703
$590$		0			0
$591$	1.11044e7			1.30776
$592$		0			0
$593$			1.87004e6i			0.218381i	0.994021	+	0.109190i	$0.0348258\pi$
$593$			1.87004e6i			0.218381i	−0.994021	+	0.109190i	$0.965174\pi$
$594$		0			0
$595$	−1.26443e6	−	5.37560e6i	−0.146421	−	0.622493i
$596$		0			0
$597$		−	2.86184e7i		−	3.28632i
$598$		0			0
$599$	415288.			0.0472915			0.0236457	−	0.999720i	$-0.492473\pi$
$599$	415288.			0.0472915			0.0236457	+	0.999720i	$0.492473\pi$
$600$		0			0
$601$	1.31366e7			1.48353			0.741764	−	0.670661i	$-0.233990\pi$
$601$	1.31366e7			1.48353			0.741764	+	0.670661i	$0.233990\pi$
$602$		0			0
$603$		−	1.79380e7i		−	2.00900i
$604$		0			0
$605$	187400.	+	796714.i	0.0208153	+	0.0884940i
$606$		0			0
$607$		−	18756.3i		−	0.00206622i	−0.999999	−	0.00103311i	$-0.999671\pi$
$607$		−	18756.3i		−	0.00206622i	0.999999	−	0.00103311i	$-0.000328849\pi$
$608$		0			0
$609$	−8.21744e6			−0.897828
$610$		0			0
$611$	−1.04571e7			−1.13321
$612$		0			0
$613$		−	5.55708e6i		−	0.597304i	−0.954362	−	0.298652i	$-0.903463\pi$
$613$		−	5.55708e6i		−	0.597304i	0.954362	−	0.298652i	$-0.0965370\pi$
$614$		0			0
$615$	−2.16841e6	+	510047.i	−0.231182	+	0.0543779i
$616$		0			0
$617$			1.68269e7i			1.77947i	0.456478	+	0.889735i	$0.349111\pi$
$617$			1.68269e7i			1.77947i	−0.456478	+	0.889735i	$0.650889\pi$
$618$		0			0
$619$	6.25620e6			0.656273			0.328136	−	0.944630i	$-0.393579\pi$
$619$	6.25620e6			0.656273			0.328136	+	0.944630i	$0.393579\pi$
$620$		0			0
$621$	−9.42073e6			−0.980292
$622$		0			0
$623$			773381.i			0.0798313i
$624$		0			0
$625$	5.88455e6	−	7.79356e6i	0.602578	−	0.798060i
$626$		0			0
$627$		−	9.85526e6i		−	1.00115i
$628$		0			0
$629$	1.94723e7			1.96241
$630$		0			0
$631$	1.28950e7			1.28928			0.644641	−	0.764486i	$-0.277007\pi$
$631$	1.28950e7			1.28928			0.644641	+	0.764486i	$0.277007\pi$
$632$		0			0
$633$		−	1.94333e7i		−	1.92769i
$634$		0			0
$635$	−6.15958e6	+	1.44884e6i	−0.606201	+	0.142589i
$636$		0			0
$637$		−	6.71373e6i		−	0.655565i
$638$		0			0
$639$	2.89387e7			2.80367
$640$		0			0
$641$	−1.14483e7			−1.10051			−0.550257	−	0.834995i	$-0.685470\pi$
$641$	−1.14483e7			−1.10051			−0.550257	+	0.834995i	$0.685470\pi$
$642$		0			0
$643$		−	9.95257e6i		−	0.949309i	−0.880172	−	0.474655i	$-0.842573\pi$
$643$		−	9.95257e6i		−	0.949309i	0.880172	−	0.474655i	$-0.157427\pi$
$644$		0			0
$645$	3.87046e6	+	1.64549e7i	0.366323	+	1.55738i
$646$		0			0
$647$		−	4.71410e6i		−	0.442729i	−0.975191	−	0.221365i	$-0.928949\pi$
$647$		−	4.71410e6i		−	0.442729i	0.975191	−	0.221365i	$-0.0710511\pi$
$648$		0			0
$649$	−4.77456e6			−0.444961
$650$		0			0
$651$	1.96896e6			0.182089
$652$		0			0
$653$			1.66636e7i			1.52928i	0.644459	+	0.764639i	$0.277083\pi$
$653$			1.66636e7i			1.52928i	−0.644459	+	0.764639i	$0.722917\pi$
$654$		0			0
$655$	1.81955e6	+	7.73564e6i	0.165715	+	0.704519i
$656$		0			0
$657$			2.05806e7i			1.86014i
$658$		0			0
$659$	−7.73348e6			−0.693684			−0.346842	−	0.937924i	$-0.612746\pi$
$659$	−7.73348e6			−0.693684			−0.346842	+	0.937924i	$0.612746\pi$
$660$		0			0
$661$	8.58080e6			0.763878			0.381939	−	0.924188i	$-0.375256\pi$
$661$	8.58080e6			0.763878			0.381939	+	0.924188i	$0.375256\pi$
$662$		0			0
$663$		−	2.13557e7i		−	1.88682i
$664$		0			0
$665$	−1.08385e7	+	2.54939e6i	−0.950418	+	0.223554i
$666$		0			0
$667$			8.75639e6i			0.762097i
$668$		0			0
$669$	2.66675e7			2.30366
$670$		0			0
$671$	2.27849e6			0.195362
$672$		0			0
$673$			1.06321e7i			0.904859i	0.891800	+	0.452429i	$0.149443\pi$
$673$			1.06321e7i			0.904859i	−0.891800	+	0.452429i	$0.850557\pi$
$674$		0			0
$675$	7.33238e6	+	1.47241e7i	0.619420	+	1.24385i
$676$		0			0
$677$			1.42422e7i			1.19427i	0.802139	+	0.597137i	$0.203695\pi$
$677$			1.42422e7i			1.19427i	−0.802139	+	0.597137i	$0.796305\pi$
$678$		0			0
$679$	−8.78658e6			−0.731384
$680$		0			0
$681$	−1.33818e6			−0.110572
$682$		0			0
$683$			1.10334e7i			0.905020i	0.891759	+	0.452510i	$0.149471\pi$
$683$			1.10334e7i			0.905020i	−0.891759	+	0.452510i	$0.850529\pi$
$684$		0			0
$685$	1.12617e7	−	2.64894e6i	0.917018	−	0.215698i
$686$		0			0
$687$		−	1.06253e7i		−	0.858912i
$688$		0			0
$689$	8.60324e6			0.690421
$690$		0			0
$691$	−3.50968e6			−0.279623			−0.139811	−	0.990178i	$-0.544650\pi$
$691$	−3.50968e6			−0.279623			−0.139811	+	0.990178i	$0.544650\pi$
$692$		0			0
$693$			3.44189e6i			0.272248i
$694$		0			0
$695$	−5.54189e6	−	2.35608e7i	−0.435207	−	1.85024i
$696$		0			0
$697$		−	2.34366e6i		−	0.182731i
$698$		0			0
$699$	4.60904e6			0.356794
$700$		0			0
$701$	−2.06987e7			−1.59092			−0.795458	−	0.606009i	$-0.792770\pi$
$701$	−2.06987e7			−1.59092			−0.795458	+	0.606009i	$0.792770\pi$
$702$		0			0
$703$		−	3.92607e7i		−	2.99620i
$704$		0			0
$705$	6.63539e6	+	2.82097e7i	0.502798	+	2.13760i
$706$		0			0
$707$			3.06839e6i			0.230867i
$708$		0			0
$709$	−1.39218e7			−1.04011			−0.520057	−	0.854132i	$-0.674089\pi$
$709$	−1.39218e7			−1.04011			−0.520057	+	0.854132i	$0.674089\pi$
$710$		0			0
$711$	−8.94142e6			−0.663335
$712$		0			0
$713$		−	2.09809e6i		−	0.154561i
$714$		0			0
$715$	3.48089e6	−	818764.i	0.254639	−	0.0598954i
$716$		0			0
$717$		−	8.14006e6i		−	0.591330i
$718$		0			0
$719$	3.96939e6			0.286353			0.143176	−	0.989697i	$-0.454268\pi$
$719$	3.96939e6			0.286353			0.143176	+	0.989697i	$0.454268\pi$
$720$		0			0
$721$	5.37227e6			0.384875
$722$		0			0
$723$		−	1.63802e6i		−	0.116540i
$724$		0			0
$725$	1.36858e7	−	6.81531e6i	0.966994	−	0.481549i
$726$		0			0
$727$		−	1.28442e7i		−	0.901300i	−0.892701	−	0.450650i	$-0.851192\pi$
$727$		−	1.28442e7i		−	0.901300i	0.892701	−	0.450650i	$-0.148808\pi$
$728$		0			0
$729$	2.01644e7			1.40529
$730$		0			0
$731$	−1.77847e7			−1.23099
$732$		0			0
$733$		−	1.51085e7i		−	1.03863i	−0.854582	−	0.519316i	$-0.826187\pi$
$733$		−	1.51085e7i		−	1.03863i	0.854582	−	0.519316i	$-0.173813\pi$
$734$		0			0
$735$	−1.81113e7	+	4.26009e6i	−1.23661	+	0.290871i
$736$		0			0
$737$			4.89023e6i			0.331635i
$738$		0			0
$739$	−2.27094e7			−1.52966			−0.764829	−	0.644234i	$-0.777176\pi$
$739$	−2.27094e7			−1.52966			−0.764829	+	0.644234i	$0.777176\pi$
$740$		0			0
$741$	−4.30583e7			−2.88079
$742$		0			0
$743$		−	1.54143e6i		−	0.102436i	−0.998688	−	0.0512179i	$-0.983690\pi$
$743$		−	1.54143e6i		−	0.102436i	0.998688	−	0.0512179i	$-0.0163103\pi$
$744$		0			0
$745$	2.19933e6	+	9.35022e6i	0.145177	+	0.617207i
$746$		0			0
$747$		−	3.81559e7i		−	2.50184i
$748$		0			0
$749$	−1.63793e6			−0.106682
$750$		0			0
$751$	−7.76182e6			−0.502185			−0.251093	−	0.967963i	$-0.580790\pi$
$751$	−7.76182e6			−0.502185			−0.251093	+	0.967963i	$0.580790\pi$
$752$		0			0
$753$			4.33929e7i			2.78889i
$754$		0			0
$755$	3.77663e6	+	1.60560e7i	0.241122	+	1.02511i
$756$		0			0
$757$		−	69075.5i		−	0.00438111i	−0.999998	−	0.00219055i	$-0.999303\pi$
$757$		−	69075.5i		−	0.00438111i	0.999998	−	0.00219055i	$-0.000697276\pi$
$758$		0			0
$759$	5.67564e6			0.357610
$760$		0			0
$761$	−9.54346e6			−0.597371			−0.298686	−	0.954352i	$-0.596548\pi$
$761$	−9.54346e6			−0.597371			−0.298686	+	0.954352i	$0.596548\pi$
$762$		0			0
$763$			3.50855e6i			0.218180i
$764$		0			0
$765$	−3.72282e7	+	8.75670e6i	−2.29995	+	0.540987i
$766$		0			0
$767$			2.08604e7i			1.28036i
$768$		0			0
$769$	1.65278e7			1.00786			0.503929	−	0.863745i	$-0.331887\pi$
$769$	1.65278e7			1.00786			0.503929	+	0.863745i	$0.331887\pi$
$770$		0			0
$771$	−298225.			−0.0180679
$772$		0			0
$773$		−	2.71610e7i		−	1.63492i	−0.575985	−	0.817460i	$-0.695381\pi$
$773$		−	2.71610e7i		−	1.63492i	0.575985	−	0.817460i	$-0.304619\pi$
$774$		0			0
$775$	−3.27921e6	+	1.63300e6i	−0.196117	+	0.0976632i
$776$		0			0
$777$			2.12186e7i			1.26085i
$778$		0			0
$779$	−4.72537e6			−0.278992
$780$		0			0
$781$	−7.88925e6			−0.462816
$782$		0			0
$783$			2.57518e7i			1.50108i
$784$		0			0
$785$	−7.26935e6	+	1.70987e6i	−0.421038	+	0.0990352i
$786$		0			0
$787$			2.98471e7i			1.71777i	0.512167	+	0.858886i	$0.328843\pi$
$787$			2.98471e7i			1.71777i	−0.512167	+	0.858886i	$0.671157\pi$
$788$		0			0
$789$	3.74396e7			2.14111
$790$		0			0
$791$	4.61844e6			0.262455
$792$		0			0
$793$		−	9.95487e6i		−	0.562150i
$794$		0			0
$795$	−5.45904e6	−	2.32086e7i	−0.306336	−	1.30236i
$796$		0			0
$797$			9.31780e6i			0.519598i	0.965663	+	0.259799i	$0.0836564\pi$
$797$			9.31780e6i			0.519598i	−0.965663	+	0.259799i	$0.916344\pi$
$798$		0			0
$799$	−3.04895e7			−1.68960
$800$		0			0
$801$	5.35598e6			0.294956
$802$		0			0
$803$		−	5.61068e6i		−	0.307062i
$804$		0			0
$805$	−1.46819e6	−	6.24188e6i	−0.0798535	−	0.339489i
$806$		0			0
$807$			3.99156e7i			2.15754i
$808$		0			0
$809$	−3.50340e7			−1.88200			−0.940998	−	0.338411i	$-0.890111\pi$
$809$	−3.50340e7			−1.88200			−0.940998	+	0.338411i	$0.890111\pi$
$810$		0			0
$811$	−3.42814e7			−1.83023			−0.915117	−	0.403190i	$-0.867902\pi$
$811$	−3.42814e7			−1.83023			−0.915117	+	0.403190i	$0.867902\pi$
$812$		0			0
$813$		−	1.65107e7i		−	0.876069i
$814$		0			0
$815$	−1.77746e7	+	4.18088e6i	−0.937358	+	0.220482i
$816$		0			0
$817$			3.58582e7i			1.87946i
$818$		0			0
$819$	1.50379e7			0.783387
$820$		0			0
$821$	2.38494e7			1.23487			0.617433	−	0.786623i	$-0.288173\pi$
$821$	2.38494e7			1.23487			0.617433	+	0.786623i	$0.288173\pi$
$822$		0			0
$823$		−	5.89850e6i		−	0.303558i	−0.988414	−	0.151779i	$-0.951500\pi$
$823$		−	5.89850e6i		−	0.303558i	0.988414	−	0.151779i	$-0.0485002\pi$
$824$		0			0
$825$	−4.41748e6	−	8.87071e6i	−0.225964	−	0.453757i
$826$		0			0
$827$		−	1.39931e7i		−	0.711460i	−0.934589	−	0.355730i	$-0.884232\pi$
$827$		−	1.39931e7i		−	0.711460i	0.934589	−	0.355730i	$-0.115768\pi$
$828$		0			0
$829$	1.94439e7			0.982644			0.491322	−	0.870978i	$-0.336514\pi$
$829$	1.94439e7			0.982644			0.491322	+	0.870978i	$0.336514\pi$
$830$		0			0
$831$	5.96646e7			2.99719
$832$		0			0
$833$		−	1.95750e7i		−	0.977440i
$834$		0			0
$835$	−3.34382e7	+	7.86522e6i	−1.65969	+	0.390386i
$836$		0			0
$837$		−	6.17033e6i		−	0.304435i
$838$		0			0
$839$	1.46211e7			0.717092			0.358546	−	0.933512i	$-0.383273\pi$
$839$	1.46211e7			0.717092			0.358546	+	0.933512i	$0.383273\pi$
$840$		0			0
$841$	3.42470e6			0.166968
$842$		0			0
$843$			1.30150e7i			0.630777i
$844$		0			0
$845$	1.17520e6	+	4.99626e6i	0.0566202	+	0.240715i
$846$		0			0
$847$		−	938327.i		−	0.0449413i
$848$		0			0
$849$	3.93550e7			1.87383
$850$		0			0
$851$	2.26102e7			1.07024
$852$		0			0
$853$		−	1.92349e7i		−	0.905144i	−0.891728	−	0.452572i	$-0.850507\pi$
$853$		−	1.92349e7i		−	0.905144i	0.891728	−	0.452572i	$-0.149493\pi$
$854$		0			0
$855$	1.76556e7	+	7.50610e7i	0.825976	+	3.51155i
$856$		0			0
$857$			3.91181e6i			0.181939i	0.995854	+	0.0909694i	$0.0289966\pi$
$857$			3.91181e6i			0.181939i	−0.995854	+	0.0909694i	$0.971003\pi$
$858$		0			0
$859$	1.32707e7			0.613638			0.306819	−	0.951768i	$-0.400735\pi$
$859$	1.32707e7			0.613638			0.306819	+	0.951768i	$0.400735\pi$
$860$		0			0
$861$	2.55384e6			0.117405
$862$		0			0
$863$			2.05251e7i			0.938121i	0.883166	+	0.469061i	$0.155407\pi$
$863$			2.05251e7i			0.938121i	−0.883166	+	0.469061i	$0.844593\pi$
$864$		0			0
$865$	−2.72602e7	+	6.41205e6i	−1.23876	+	0.291378i
$866$		0			0
$867$		−	2.50551e7i		−	1.13201i
$868$		0			0
$869$	2.43760e6			0.109500
$870$		0			0
$871$	2.13657e7			0.954273
$872$		0			0
$873$			6.08507e7i			2.70228i
$874$		0			0
$875$	−8.61299e6	+	7.15291e6i	−0.380307	+	0.315837i
$876$		0			0
$877$		−	2.33070e7i		−	1.02326i	−0.859205	−	0.511632i	$-0.829041\pi$
$877$		−	2.33070e7i		−	1.02326i	0.859205	−	0.511632i	$-0.170959\pi$
$878$		0			0
$879$	3.73704e7			1.63138
$880$		0			0
$881$	−9.31106e6			−0.404165			−0.202083	−	0.979368i	$-0.564771\pi$
$881$	−9.31106e6			−0.404165			−0.202083	+	0.979368i	$0.564771\pi$
$882$		0			0
$883$		−	3.85172e7i		−	1.66247i	−0.555925	−	0.831233i	$-0.687636\pi$
$883$		−	3.85172e7i		−	1.66247i	0.555925	−	0.831233i	$-0.312364\pi$
$884$		0			0
$885$	5.62741e7	−	1.32366e7i	2.41518	−	0.568091i
$886$		0			0
$887$			2.82081e7i			1.20383i	0.798560	+	0.601915i	$0.205595\pi$
$887$			2.82081e7i			1.20383i	−0.798560	+	0.601915i	$0.794405\pi$
$888$		0			0
$889$	7.25442e6			0.307856
$890$		0			0
$891$	3.64130e6			0.153660
$892$		0			0
$893$			6.14742e7i			2.57967i
$894$		0			0
$895$	−3.91174e6	−	1.66304e7i	−0.163235	−	0.693976i
$896$		0			0
$897$		−	2.47972e7i		−	1.02902i
$898$		0			0
$899$	−5.73520e6			−0.236673
$900$		0			0
$901$	2.50842e7			1.02941
$902$		0			0
$903$		−	1.93797e7i		−	0.790911i
$904$		0			0
$905$	−666674.	−	2.83430e6i	−0.0270578	−	0.115033i
$906$		0			0
$907$		−	1.41463e7i		−	0.570985i	−0.958381	−	0.285492i	$-0.907843\pi$
$907$		−	1.41463e7i		−	0.570985i	0.958381	−	0.285492i	$-0.0921571\pi$
$908$		0			0
$909$	2.12499e7			0.852995
$910$		0			0
$911$	−3.41700e7			−1.36411			−0.682055	−	0.731301i	$-0.738913\pi$
$911$	−3.41700e7			−1.36411			−0.682055	+	0.731301i	$0.738913\pi$
$912$		0			0
$913$			1.04020e7i			0.412991i
$914$		0			0
$915$	−2.68548e7	+	6.31669e6i	−1.06040	+	0.249423i
$916$		0			0
$917$		−	9.11062e6i		−	0.357787i
$918$		0			0
$919$	9.77047e6			0.381616			0.190808	−	0.981627i	$-0.438889\pi$
$919$	9.77047e6			0.381616			0.190808	+	0.981627i	$0.438889\pi$
$920$		0			0
$921$	−4.88966e7			−1.89946
$922$		0			0
$923$			3.44687e7i			1.33174i
$924$		0			0
$925$	−1.75981e7	−	3.53386e7i	−0.676256	−	1.35798i
$926$		0			0
$927$		−	3.72052e7i		−	1.42202i
$928$		0			0
$929$	3.67368e7			1.39657			0.698285	−	0.715820i	$-0.253947\pi$
$929$	3.67368e7			1.39657			0.698285	+	0.715820i	$0.253947\pi$
$930$		0			0
$931$	−3.94679e7			−1.49235
$932$		0			0
$933$			1.60496e6i			0.0603616i
$934$		0			0
$935$	1.01491e7	−	2.38725e6i	0.379665	−	0.0893034i
$936$		0			0
$937$			3.55132e7i			1.32142i	0.750641	+	0.660711i	$0.229745\pi$
$937$			3.55132e7i			1.32142i	−0.750641	+	0.660711i	$0.770255\pi$
$938$		0			0
$939$	3.98377e6			0.147445
$940$		0			0
$941$	8.61941e6			0.317325			0.158662	−	0.987333i	$-0.449282\pi$
$941$	8.61941e6			0.317325			0.158662	+	0.987333i	$0.449282\pi$
$942$		0			0
$943$		−	2.72134e6i		−	0.0996560i
$944$		0			0
$945$	−4.31784e6	−	1.83569e7i	−0.157285	−	0.668681i
$946$		0			0
$947$			5.88715e6i			0.213319i	0.994296	+	0.106660i	$0.0340155\pi$
$947$			5.88715e6i			0.213319i	−0.994296	+	0.106660i	$0.965984\pi$
$948$		0			0
$949$	−2.45134e7			−0.883565
$950$		0			0
$951$	3.18979e7			1.14370
$952$		0			0
$953$		−	8.65335e6i		−	0.308640i	−0.988021	−	0.154320i	$-0.950681\pi$
$953$		−	8.65335e6i		−	0.308640i	0.988021	−	0.154320i	$-0.0493187\pi$
$954$		0			0
$955$	−1.14527e7	−	4.86902e7i	−0.406351	−	1.72756i
$956$		0			0
$957$		−	1.55145e7i		−	0.547594i
$958$		0			0
$959$	−1.32634e7			−0.465704
$960$		0			0
$961$	−2.72550e7			−0.952000
$962$		0			0
$963$			1.13434e7i			0.394163i
$964$		0			0
$965$	2.48310e7	−	5.84066e6i	0.858371	−	0.201903i
$966$		0			0
$967$			3.86242e7i			1.32829i	0.747603	+	0.664146i	$0.231205\pi$
$967$			3.86242e7i			1.32829i	−0.747603	+	0.664146i	$0.768795\pi$
$968$		0			0
$969$	−1.25544e8			−4.29522
$970$		0			0
$971$	−1.98124e7			−0.674357			−0.337179	−	0.941441i	$-0.609473\pi$
$971$	−1.98124e7			−0.674357			−0.337179	+	0.941441i	$0.609473\pi$
$972$		0			0
$973$			2.77486e7i			0.939636i
$974$		0			0
$975$	−3.87567e7	+	1.93003e7i	−1.30568	+	0.650207i
$976$		0			0
$977$			4.08355e7i			1.36868i	0.729163	+	0.684340i	$0.239910\pi$
$977$			4.08355e7i			1.36868i	−0.729163	+	0.684340i	$0.760090\pi$
$978$		0			0
$979$	−1.46014e6			−0.0486899
$980$		0			0
$981$	2.42981e7			0.806121
$982$		0			0
$983$		−	1.60569e7i		−	0.530003i	−0.964248	−	0.265002i	$-0.914628\pi$
$983$		−	1.60569e7i		−	0.530003i	0.964248	−	0.265002i	$-0.0853725\pi$
$984$		0			0
$985$	2.30568e7	−	5.42335e6i	0.757197	−	0.178105i
$986$		0			0
$987$		−	3.32239e7i		−	1.08557i
$988$		0			0
$989$	−2.06507e7			−0.671343
$990$		0			0
$991$	3.67047e6			0.118724			0.0593619	−	0.998237i	$-0.481093\pi$
$991$	3.67047e6			0.118724			0.0593619	+	0.998237i	$0.481093\pi$
$992$		0			0
$993$		−	7.61075e7i		−	2.44937i
$994$		0			0
$995$	−1.39771e7	−	5.94222e7i	−0.447568	−	1.90279i
$996$		0			0
$997$			1.14427e7i			0.364579i	0.983245	+	0.182289i	$0.0583507\pi$
$997$			1.14427e7i			0.364579i	−0.983245	+	0.182289i	$0.941649\pi$
$998$		0			0
$999$	6.64949e7			2.10802

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	220.6.b.b.89.14	yes	14
5.2	odd	4		1100.6.a.m.1.14		14
5.3	odd	4		1100.6.a.m.1.1		14
5.4	even	2	inner	220.6.b.b.89.1	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.6.b.b.89.1	✓	14	5.4	even	2	inner
220.6.b.b.89.14	yes	14	1.1	even	1	trivial
1100.6.a.m.1.1		14	5.3	odd	4
1100.6.a.m.1.14		14	5.2	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)

\(f(q)\)	\(=\)	\(q+26.2077i q^{3} +(12.7997 + 54.4166i) q^{5} -64.0890i q^{7} -443.842 q^{9} +121.000 q^{11} -528.657i q^{13} +(-1426.13 + 335.450i) q^{15} -1541.39i q^{17} -3107.81 q^{19} +1679.62 q^{21} -1789.78i q^{23} +(-2797.34 + 1393.03i) q^{25} -5263.61i q^{27} -4892.43 q^{29} +1172.26 q^{31} +3171.13i q^{33} +(3487.50 - 820.319i) q^{35} +12632.9i q^{37} +13854.9 q^{39} +1520.48 q^{41} -11538.1i q^{43} +(-5681.05 - 24152.4i) q^{45} -19780.6i q^{47} +12699.6 q^{49} +40396.2 q^{51} +16273.8i q^{53} +(1548.76 + 6584.41i) q^{55} -81448.4i q^{57} -39459.2 q^{59} +18830.5 q^{61} +28445.4i q^{63} +(28767.7 - 6766.65i) q^{65} +40415.1i q^{67} +46906.1 q^{69} -65200.4 q^{71} -46369.3i q^{73} +(-36508.1 - 73311.7i) q^{75} -7754.77i q^{77} +20145.5 q^{79} +30093.4 q^{81} +85967.1i q^{83} +(83877.2 - 19729.3i) q^{85} -128219. i q^{87} -12067.3 q^{89} -33881.1 q^{91} +30722.2i q^{93} +(-39779.0 - 169116. i) q^{95} -137100. i q^{97} -53704.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 44 q^{5} - 1142 q^{9} + 1694 q^{11} + 326 q^{15} - 4540 q^{19} + 3824 q^{21} - 3816 q^{25} - 9972 q^{29} + 19076 q^{31} + 5136 q^{35} - 13616 q^{39} + 15052 q^{41} - 8374 q^{45} - 55346 q^{49} - 13380 q^{51}+ \cdots - 138182 q^{99}+O(q^{100}) \) 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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