LMFDB - Embedded newform 320.6.c.k.129.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(129,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("320.129");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 165x^{10} + 9528x^{8} + 254984x^{6} + 3245664x^{4} + 15975501x^{2} + 588289 $ x^12 + 165x^10 + 9528x^8 + 254984x^6 + 3245664x^4 + 15975501*x^2 + 588289
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{44}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.4
Root			$-5.17887i$ of defining polynomial
Character	$\chi$	$=$	320.129
Dual form			320.6.c.k.129.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.0114i q^{3} +(46.9000 + 30.4203i) q^{5} -91.0286i q^{7} -118.434 q^{9} +O(q^{10})$	$q-19.0114i q^{3} +(46.9000 + 30.4203i) q^{5} -91.0286i q^{7} -118.434 q^{9} -290.263 q^{11} +112.108i q^{13} +(578.332 - 891.635i) q^{15} +939.348i q^{17} -2055.29 q^{19} -1730.58 q^{21} -1684.18i q^{23} +(1274.22 + 2853.42i) q^{25} -2368.18i q^{27} -3960.68 q^{29} -7627.48 q^{31} +5518.31i q^{33} +(2769.11 - 4269.24i) q^{35} -9579.76i q^{37} +2131.33 q^{39} -281.653 q^{41} -6515.03i q^{43} +(-5554.53 - 3602.78i) q^{45} -1395.46i q^{47} +8520.79 q^{49} +17858.3 q^{51} -18538.0i q^{53} +(-13613.3 - 8829.87i) q^{55} +39074.0i q^{57} -51956.2 q^{59} -51541.5 q^{61} +10780.8i q^{63} +(-3410.35 + 5257.85i) q^{65} +39074.2i q^{67} -32018.7 q^{69} +30435.9 q^{71} +49694.2i q^{73} +(54247.5 - 24224.6i) q^{75} +26422.2i q^{77} +98240.4 q^{79} -73801.8 q^{81} +94470.1i q^{83} +(-28575.2 + 44055.4i) q^{85} +75298.0i q^{87} -98225.5 q^{89} +10205.0 q^{91} +145009. i q^{93} +(-96393.1 - 62522.5i) q^{95} +59010.4i q^{97} +34376.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 60 q^{5} - 1676 q^{9}+O(q^{10}) $ 12 * q + 60 * q^5 - 1676 * q^9	$ 12 q + 60 q^{5} - 1676 q^{9} - 688 q^{21} + 1500 q^{25} + 21304 q^{29} - 109672 q^{41} + 75140 q^{45} - 32348 q^{49} - 64552 q^{61} - 160 q^{65} + 166352 q^{69} - 173764 q^{81} - 151360 q^{85} - 3720 q^{89}+O(q^{100}) $ 12 * q + 60 * q^5 - 1676 * q^9 - 688 * q^21 + 1500 * q^25 + 21304 * q^29 - 109672 * q^41 + 75140 * q^45 - 32348 * q^49 - 64552 * q^61 - 160 * q^65 + 166352 * q^69 - 173764 * q^81 - 151360 * q^85 - 3720 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	19.0114i		−	1.21958i	−0.792562	−	0.609791i	$-0.791253\pi$
$3$		−	19.0114i		−	1.21958i	0.792562	−	0.609791i	$-0.208747\pi$
$4$		0			0
$5$	46.9000	+	30.4203i	0.838972	+	0.544174i
$5$	46.9000	+	30.4203i	0.838972	+	0.544174i
$6$		0			0
$7$		−	91.0286i		−	0.702155i	−0.936346	−	0.351077i	$-0.885815\pi$
$7$		−	91.0286i		−	0.702155i	0.936346	−	0.351077i	$-0.114185\pi$
$8$		0			0
$9$	−118.434			−0.487381
$10$		0			0
$11$	−290.263			−0.723286			−0.361643	−	0.932317i	$-0.617784\pi$
$11$	−290.263			−0.723286			−0.361643	+	0.932317i	$0.617784\pi$
$12$		0			0
$13$			112.108i			0.183983i	0.995760	+	0.0919914i	$0.0293232\pi$
$13$			112.108i			0.183983i	−0.995760	+	0.0919914i	$0.970677\pi$
$14$		0			0
$15$	578.332	−	891.635i	0.663665	−	1.02320i
$16$		0			0
$17$			939.348i			0.788323i	0.919041	+	0.394161i	$0.128965\pi$
$17$			939.348i			0.788323i	−0.919041	+	0.394161i	$0.871035\pi$
$18$		0			0
$19$	−2055.29			−1.30614			−0.653070	−	0.757298i	$-0.726519\pi$
$19$	−2055.29			−1.30614			−0.653070	+	0.757298i	$0.726519\pi$
$20$		0			0
$21$	−1730.58			−0.856336
$22$		0			0
$23$		−	1684.18i		−	0.663850i	−0.943306	−	0.331925i	$-0.892302\pi$
$23$		−	1684.18i		−	0.663850i	0.943306	−	0.331925i	$-0.107698\pi$
$24$		0			0
$25$	1274.22	+	2853.42i	0.407749	+	0.913094i
$26$		0			0
$27$		−	2368.18i		−	0.625181i
$28$		0			0
$29$	−3960.68			−0.874529			−0.437265	−	0.899333i	$-0.644053\pi$
$29$	−3960.68			−0.874529			−0.437265	+	0.899333i	$0.644053\pi$
$30$		0			0
$31$	−7627.48			−1.42553			−0.712765	−	0.701402i	$-0.752558\pi$
$31$	−7627.48			−1.42553			−0.712765	+	0.701402i	$0.752558\pi$
$32$		0			0
$33$			5518.31i			0.882106i
$34$		0			0
$35$	2769.11	−	4269.24i	0.382094	−	0.589089i
$36$		0			0
$37$		−	9579.76i		−	1.15040i	−0.818012	−	0.575202i	$-0.804924\pi$
$37$		−	9579.76i		−	1.15040i	0.818012	−	0.575202i	$-0.195076\pi$
$38$		0			0
$39$	2131.33			0.224382
$40$		0			0
$41$	−281.653			−0.0261671			−0.0130835	−	0.999914i	$-0.504165\pi$
$41$	−281.653			−0.0261671			−0.0130835	+	0.999914i	$0.504165\pi$
$42$		0			0
$43$		−	6515.03i		−	0.537335i	−0.963233	−	0.268668i	$-0.913417\pi$
$43$		−	6515.03i		−	0.537335i	0.963233	−	0.268668i	$-0.0865833\pi$
$44$		0			0
$45$	−5554.53	−	3602.78i	−0.408899	−	0.265220i
$46$		0			0
$47$		−	1395.46i		−	0.0921450i	−0.998938	−	0.0460725i	$-0.985329\pi$
$47$		−	1395.46i		−	0.0921450i	0.998938	−	0.0460725i	$-0.0146705\pi$
$48$		0			0
$49$	8520.79			0.506979
$50$		0			0
$51$	17858.3			0.961425
$52$		0			0
$53$		−	18538.0i		−	0.906510i	−0.891381	−	0.453255i	$-0.850263\pi$
$53$		−	18538.0i		−	0.906510i	0.891381	−	0.453255i	$-0.149737\pi$
$54$		0			0
$55$	−13613.3	−	8829.87i	−0.606817	−	0.393593i
$56$		0			0
$57$			39074.0i			1.59294i
$58$		0			0
$59$	−51956.2			−1.94315			−0.971577	−	0.236723i	$-0.923926\pi$
$59$	−51956.2			−1.94315			−0.971577	+	0.236723i	$0.923926\pi$
$60$		0			0
$61$	−51541.5			−1.77351			−0.886753	−	0.462243i	$-0.847045\pi$
$61$	−51541.5			−1.77351			−0.886753	+	0.462243i	$0.847045\pi$
$62$		0			0
$63$			10780.8i			0.342217i
$64$		0			0
$65$	−3410.35	+	5257.85i	−0.100119	+	0.154356i
$66$		0			0
$67$			39074.2i			1.06342i	0.846928	+	0.531708i	$0.178449\pi$
$67$			39074.2i			1.06342i	−0.846928	+	0.531708i	$0.821551\pi$
$68$		0			0
$69$	−32018.7			−0.809620
$70$		0			0
$71$	30435.9			0.716540			0.358270	−	0.933618i	$-0.383367\pi$
$71$	30435.9			0.716540			0.358270	+	0.933618i	$0.383367\pi$
$72$		0			0
$73$			49694.2i			1.09144i	0.837968	+	0.545719i	$0.183743\pi$
$73$			49694.2i			1.09144i	−0.837968	+	0.545719i	$0.816257\pi$
$74$		0			0
$75$	54247.5	−	24224.6i	1.11359	−	0.497284i
$76$		0			0
$77$			26422.2i			0.507859i
$78$		0			0
$79$	98240.4			1.77102			0.885508	−	0.464624i	$-0.153811\pi$
$79$	98240.4			1.77102			0.885508	+	0.464624i	$0.153811\pi$
$80$		0			0
$81$	−73801.8			−1.24984
$82$		0			0
$83$			94470.1i			1.50522i	0.658468	+	0.752608i	$0.271205\pi$
$83$			94470.1i			1.50522i	−0.658468	+	0.752608i	$0.728795\pi$
$84$		0			0
$85$	−28575.2	+	44055.4i	−0.428985	+	0.661381i
$86$		0			0
$87$			75298.0i			1.06656i
$88$		0			0
$89$	−98225.5			−1.31447			−0.657233	−	0.753688i	$-0.728273\pi$
$89$	−98225.5			−1.31447			−0.657233	+	0.753688i	$0.728273\pi$
$90$		0			0
$91$	10205.0			0.129184
$92$		0			0
$93$			145009.i			1.73855i
$94$		0			0
$95$	−96393.1	−	62522.5i	−1.09581	−	0.710767i
$96$		0			0
$97$			59010.4i			0.636795i	0.947957	+	0.318397i	$0.103145\pi$
$97$			59010.4i			0.636795i	−0.947957	+	0.318397i	$0.896855\pi$
$98$		0			0
$99$	34376.9			0.352516
$100$		0			0
$101$	8117.20			0.0791777			0.0395888	−	0.999216i	$-0.487395\pi$
$101$	8117.20			0.0791777			0.0395888	+	0.999216i	$0.487395\pi$
$102$		0			0
$103$			151537.i			1.40743i	0.710484	+	0.703713i	$0.248476\pi$
$103$			151537.i			1.40743i	−0.710484	+	0.703713i	$0.751524\pi$
$104$		0			0
$105$	−81164.3	−	52644.8i	−0.718442	−	0.465996i
$106$		0			0
$107$		−	95224.2i		−	0.804059i	−0.915627	−	0.402030i	$-0.868305\pi$
$107$		−	95224.2i		−	0.804059i	0.915627	−	0.402030i	$-0.131695\pi$
$108$		0			0
$109$	−65800.3			−0.530471			−0.265236	−	0.964184i	$-0.585450\pi$
$109$	−65800.3			−0.530471			−0.265236	+	0.964184i	$0.585450\pi$
$110$		0			0
$111$	−182125.			−1.40301
$112$		0			0
$113$		−	144361.i		−	1.06354i	−0.846888	−	0.531771i	$-0.821527\pi$
$113$		−	144361.i		−	1.06354i	0.846888	−	0.531771i	$-0.178473\pi$
$114$		0			0
$115$	51233.3	−	78988.2i	0.361250	−	0.556952i
$116$		0			0
$117$		−	13277.3i		−	0.0896697i
$118$		0			0
$119$	85507.6			0.553525
$120$		0			0
$121$	−76798.4			−0.476858
$122$		0			0
$123$			5354.63i			0.0319129i
$124$		0			0
$125$	−27041.0	+	172587.i	−0.154792	+	0.987947i
$126$		0			0
$127$		−	304065.i		−	1.67285i	−0.548083	−	0.836424i	$-0.684642\pi$
$127$		−	304065.i		−	1.67285i	0.548083	−	0.836424i	$-0.315358\pi$
$128$		0			0
$129$	−123860.			−0.655324
$130$		0			0
$131$	−79255.7			−0.403508			−0.201754	−	0.979436i	$-0.564664\pi$
$131$	−79255.7			−0.403508			−0.201754	+	0.979436i	$0.564664\pi$
$132$		0			0
$133$			187090.i			0.917112i
$134$		0			0
$135$	72040.7	−	111068.i	0.340207	−	0.524510i
$136$		0			0
$137$		−	224.012i		−	0.00101970i	−1.00000		0.000509848i	$-0.999838\pi$
$137$		−	224.012i		−	0.00101970i	1.00000		0.000509848i	$-0.000162290\pi$
$138$		0			0
$139$	−233668.			−1.02580			−0.512899	−	0.858449i	$-0.671428\pi$
$139$	−233668.			−1.02580			−0.512899	+	0.858449i	$0.671428\pi$
$140$		0			0
$141$	−26529.6			−0.112378
$142$		0			0
$143$		−	32540.7i		−	0.133072i
$144$		0			0
$145$	−185756.	−	120485.i	−0.733706	−	0.475896i
$146$		0			0
$147$		−	161992.i		−	0.618302i
$148$		0			0
$149$	248890.			0.918420			0.459210	−	0.888328i	$-0.348133\pi$
$149$	248890.			0.918420			0.459210	+	0.888328i	$0.348133\pi$
$150$		0			0
$151$	177518.			0.633577			0.316788	−	0.948496i	$-0.397395\pi$
$151$	177518.			0.633577			0.316788	+	0.948496i	$0.397395\pi$
$152$		0			0
$153$		−	111250.i		−	0.384214i
$154$		0			0
$155$	−357728.	−	232030.i	−1.19598	−	0.775737i
$156$		0			0
$157$		−	251590.i		−	0.814600i	−0.913294	−	0.407300i	$-0.866470\pi$
$157$		−	251590.i		−	0.814600i	0.913294	−	0.407300i	$-0.133530\pi$
$158$		0			0
$159$	−352433.			−1.10556
$160$		0			0
$161$	−153309.			−0.466125
$162$		0			0
$163$			551605.i			1.62615i	0.582162	+	0.813073i	$0.302207\pi$
$163$			551605.i			1.62615i	−0.582162	+	0.813073i	$0.697793\pi$
$164$		0			0
$165$	−167868.	+	258808.i	−0.480019	+	0.740063i
$166$		0			0
$167$		−	183161.i		−	0.508208i	−0.967177	−	0.254104i	$-0.918219\pi$
$167$		−	183161.i		−	0.508208i	0.967177	−	0.254104i	$-0.0817806\pi$
$168$		0			0
$169$	358725.			0.966150
$170$		0			0
$171$	243416.			0.636587
$172$		0			0
$173$		−	30633.5i		−	0.0778181i	−0.999243	−	0.0389091i	$-0.987612\pi$
$173$		−	30633.5i		−	0.0778181i	0.999243	−	0.0389091i	$-0.0123883\pi$
$174$		0			0
$175$	259743.	−	115990.i	0.641133	−	0.286303i
$176$		0			0
$177$			987760.i			2.36984i
$178$		0			0
$179$	548057.			1.27848			0.639239	−	0.769008i	$-0.279250\pi$
$179$	548057.			1.27848			0.639239	+	0.769008i	$0.279250\pi$
$180$		0			0
$181$	−60289.7			−0.136788			−0.0683938	−	0.997658i	$-0.521787\pi$
$181$	−60289.7			−0.136788			−0.0683938	+	0.997658i	$0.521787\pi$
$182$		0			0
$183$			979877.i			2.16294i
$184$		0			0
$185$	291419.	−	449290.i	0.626020	−	0.965156i
$186$		0			0
$187$		−	272658.i		−	0.570183i
$188$		0			0
$189$	−215572.			−0.438974
$190$		0			0
$191$	200579.			0.397833			0.198917	−	0.980016i	$-0.436258\pi$
$191$	200579.			0.397833			0.198917	+	0.980016i	$0.436258\pi$
$192$		0			0
$193$			290271.i			0.560931i	0.959864	+	0.280466i	$0.0904889\pi$
$193$			290271.i			0.560931i	−0.959864	+	0.280466i	$0.909511\pi$
$194$		0			0
$195$	99959.1	+	64835.5i	0.188250	+	0.122103i
$196$		0			0
$197$		−	1.03747e6i		−	1.90462i	−0.305135	−	0.952309i	$-0.598702\pi$
$197$		−	1.03747e6i		−	1.90462i	0.305135	−	0.952309i	$-0.401298\pi$
$198$		0			0
$199$	−603425.			−1.08017			−0.540083	−	0.841612i	$-0.681607\pi$
$199$	−603425.			−1.08017			−0.540083	+	0.841612i	$0.681607\pi$
$200$		0			0
$201$	742855.			1.29692
$202$		0			0
$203$			360535.i			0.614055i
$204$		0			0
$205$	−13209.5	−	8567.97i	−0.0219535	−	0.0142395i
$206$		0			0
$207$			199464.i			0.323548i
$208$		0			0
$209$	596575.			0.944712
$210$		0			0
$211$	−391555.			−0.605462			−0.302731	−	0.953076i	$-0.597898\pi$
$211$	−391555.			−0.605462			−0.302731	+	0.953076i	$0.597898\pi$
$212$		0			0
$213$		−	578629.i		−	0.873879i
$214$		0			0
$215$	198189.	−	305555.i	0.292404	−	0.450809i
$216$		0			0
$217$			694319.i			1.00094i
$218$		0			0
$219$	944757.			1.33110
$220$		0			0
$221$	−105308.			−0.145038
$222$		0			0
$223$			294000.i			0.395900i	0.980212	+	0.197950i	$0.0634284\pi$
$223$			294000.i			0.395900i	−0.980212	+	0.197950i	$0.936572\pi$
$224$		0			0
$225$	−150910.	−	337941.i	−0.198729	−	0.445025i
$226$		0			0
$227$		−	297246.i		−	0.382870i	−0.981505	−	0.191435i	$-0.938686\pi$
$227$		−	297246.i		−	0.382870i	0.981505	−	0.191435i	$-0.0613141\pi$
$228$		0			0
$229$	1.25335e6			1.57937			0.789685	−	0.613513i	$-0.210244\pi$
$229$	1.25335e6			1.57937			0.789685	+	0.613513i	$0.210244\pi$
$230$		0			0
$231$	502324.			0.619375
$232$		0			0
$233$		−	1.33740e6i		−	1.61388i	−0.590631	−	0.806942i	$-0.701121\pi$
$233$		−	1.33740e6i		−	1.61388i	0.590631	−	0.806942i	$-0.298879\pi$
$234$		0			0
$235$	42450.2	−	65446.9i	0.0501429	−	0.0773071i
$236$		0			0
$237$		−	1.86769e6i		−	2.15990i
$238$		0			0
$239$	−188428.			−0.213378			−0.106689	−	0.994292i	$-0.534025\pi$
$239$	−188428.			−0.213378			−0.106689	+	0.994292i	$0.534025\pi$
$240$		0			0
$241$	17282.8			0.0191678			0.00958390	−	0.999954i	$-0.496949\pi$
$241$	17282.8			0.0191678			0.00958390	+	0.999954i	$0.496949\pi$
$242$		0			0
$243$			827608.i			0.899103i
$244$		0			0
$245$	399625.	+	259205.i	0.425341	+	0.275885i
$246$		0			0
$247$		−	230414.i		−	0.240307i
$248$		0			0
$249$	1.79601e6			1.83574
$250$		0			0
$251$	−1.00912e6			−1.01102			−0.505511	−	0.862820i	$-0.668696\pi$
$251$	−1.00912e6			−1.01102			−0.505511	+	0.862820i	$0.668696\pi$
$252$		0			0
$253$			488856.i			0.480153i
$254$		0			0
$255$	837555.	+	543255.i	0.806609	+	0.523182i
$256$		0			0
$257$			261708.i			0.247164i	0.992334	+	0.123582i	$0.0394382\pi$
$257$			261708.i			0.247164i	−0.992334	+	0.123582i	$0.960562\pi$
$258$		0			0
$259$	−872032.			−0.807761
$260$		0			0
$261$	469077.			0.426229
$262$		0			0
$263$		−	981241.i		−	0.874755i	−0.899278	−	0.437377i	$-0.855907\pi$
$263$		−	981241.i		−	0.874755i	0.899278	−	0.437377i	$-0.144093\pi$
$264$		0			0
$265$	563930.	−	869431.i	0.493299	−	0.760537i
$266$		0			0
$267$			1.86740e6i			1.60310i
$268$		0			0
$269$	1.05920e6			0.892481			0.446241	−	0.894913i	$-0.352763\pi$
$269$	1.05920e6			0.892481			0.446241	+	0.894913i	$0.352763\pi$
$270$		0			0
$271$	−200978.			−0.166236			−0.0831180	−	0.996540i	$-0.526488\pi$
$271$	−200978.			−0.166236			−0.0831180	+	0.996540i	$0.526488\pi$
$272$		0			0
$273$		−	194012.i		−	0.157551i
$274$		0			0
$275$	−369858.	−	828242.i	−0.294919	−	0.660428i
$276$		0			0
$277$		−	2.08682e6i		−	1.63413i	−0.576548	−	0.817063i	$-0.695601\pi$
$277$		−	2.08682e6i		−	1.63413i	0.576548	−	0.817063i	$-0.304399\pi$
$278$		0			0
$279$	903349.			0.694777
$280$		0			0
$281$	−459978.			−0.347513			−0.173757	−	0.984789i	$-0.555591\pi$
$281$	−459978.			−0.347513			−0.173757	+	0.984789i	$0.555591\pi$
$282$		0			0
$283$		−	2.00444e6i		−	1.48774i	−0.668323	−	0.743871i	$-0.732987\pi$
$283$		−	2.00444e6i		−	1.48774i	0.668323	−	0.743871i	$-0.267013\pi$
$284$		0			0
$285$	−1.18864e6	+	1.83257e6i	−0.866839	+	1.33644i
$286$		0			0
$287$			25638.5i			0.0183733i
$288$		0			0
$289$	537482.			0.378547
$290$		0			0
$291$	1.12187e6			0.776623
$292$		0			0
$293$			1.60434e6i			1.09176i	0.837863	+	0.545880i	$0.183805\pi$
$293$			1.60434e6i			1.09176i	−0.837863	+	0.545880i	$0.816195\pi$
$294$		0			0
$295$	−2.43674e6	−	1.58052e6i	−1.63025	−	1.05741i
$296$		0			0
$297$			687396.i			0.452185i
$298$		0			0
$299$	188810.			0.122137
$300$		0			0
$301$	−593054.			−0.377293
$302$		0			0
$303$		−	154319.i		−	0.0965637i
$304$		0			0
$305$	−2.41730e6	−	1.56791e6i	−1.48792	−	0.965096i
$306$		0			0
$307$		−	2.20127e6i		−	1.33299i	−0.745510	−	0.666495i	$-0.767794\pi$
$307$		−	2.20127e6i		−	1.33299i	0.745510	−	0.666495i	$-0.232206\pi$
$308$		0			0
$309$	2.88093e6			1.71647
$310$		0			0
$311$	−2.31585e6			−1.35772			−0.678859	−	0.734269i	$-0.737525\pi$
$311$	−2.31585e6			−1.35772			−0.678859	+	0.734269i	$0.737525\pi$
$312$		0			0
$313$		−	1.42049e6i		−	0.819556i	−0.912185	−	0.409778i	$-0.865606\pi$
$313$		−	1.42049e6i		−	0.819556i	0.912185	−	0.409778i	$-0.134394\pi$
$314$		0			0
$315$	−327956.	+	505621.i	−0.186226	+	0.287111i
$316$		0			0
$317$			3.27102e6i			1.82825i	0.405433	+	0.914125i	$0.367121\pi$
$317$			3.27102e6i			1.82825i	−0.405433	+	0.914125i	$0.632879\pi$
$318$		0			0
$319$	1.14964e6			0.632535
$320$		0			0
$321$	−1.81035e6			−0.980616
$322$		0			0
$323$		−	1.93063e6i		−	1.02966i
$324$		0			0
$325$	−319890.	+	142849.i	−0.167994	+	0.0750188i
$326$		0			0
$327$			1.25096e6i			0.646953i
$328$		0			0
$329$	−127027.			−0.0647001
$330$		0			0
$331$	2.40502e6			1.20656			0.603280	−	0.797529i	$-0.293860\pi$
$331$	2.40502e6			1.20656			0.603280	+	0.797529i	$0.293860\pi$
$332$		0			0
$333$			1.13456e6i			0.560685i
$334$		0			0
$335$	−1.18865e6	+	1.83258e6i	−0.578683	+	0.892176i
$336$		0			0
$337$			1.36568e6i			0.655049i	0.944843	+	0.327525i	$0.106214\pi$
$337$			1.36568e6i			0.655049i	−0.944843	+	0.327525i	$0.893786\pi$
$338$		0			0
$339$	−2.74451e6			−1.29708
$340$		0			0
$341$	2.21397e6			1.03107
$342$		0			0
$343$		−	2.30555e6i		−	1.05813i
$344$		0			0
$345$	−1.50168e6	−	974017.i	−0.679248	−	0.440574i
$346$		0			0
$347$			1.95608e6i			0.872092i	0.899924	+	0.436046i	$0.143622\pi$
$347$			1.95608e6i			0.872092i	−0.899924	+	0.436046i	$0.856378\pi$
$348$		0			0
$349$	−900886.			−0.395919			−0.197959	−	0.980210i	$-0.563431\pi$
$349$	−900886.			−0.395919			−0.197959	+	0.980210i	$0.563431\pi$
$350$		0			0
$351$	265492.			0.115023
$352$		0			0
$353$		−	1.10129e6i		−	0.470398i	−0.971947	−	0.235199i	$-0.924426\pi$
$353$		−	1.10129e6i		−	0.470398i	0.971947	−	0.235199i	$-0.0755743\pi$
$354$		0			0
$355$	1.42744e6	+	925868.i	0.601157	+	0.389922i
$356$		0			0
$357$		−	1.62562e6i		−	0.675069i
$358$		0			0
$359$	919542.			0.376561			0.188281	−	0.982115i	$-0.439709\pi$
$359$	919542.			0.376561			0.188281	+	0.982115i	$0.439709\pi$
$360$		0			0
$361$	1.74812e6			0.705999
$362$		0			0
$363$			1.46005e6i			0.581567i
$364$		0			0
$365$	−1.51171e6	+	2.33066e6i	−0.593932	+	0.915686i
$366$		0			0
$367$			551955.i			0.213913i	0.994264	+	0.106957i	$0.0341106\pi$
$367$			551955.i			0.213913i	−0.994264	+	0.106957i	$0.965889\pi$
$368$		0			0
$369$	33357.2			0.0127533
$370$		0			0
$371$	−1.68749e6			−0.636510
$372$		0			0
$373$		−	2.39700e6i		−	0.892064i	−0.895017	−	0.446032i	$-0.852837\pi$
$373$		−	2.39700e6i		−	0.892064i	0.895017	−	0.446032i	$-0.147163\pi$
$374$		0			0
$375$	3.28113e6	+	514088.i	1.20488	+	0.188781i
$376$		0			0
$377$		−	444022.i		−	0.160898i
$378$		0			0
$379$	−2.89394e6			−1.03488			−0.517442	−	0.855719i	$-0.673116\pi$
$379$	−2.89394e6			−1.03488			−0.517442	+	0.855719i	$0.673116\pi$
$380$		0			0
$381$	−5.78070e6			−2.04018
$382$		0			0
$383$		−	1.31692e6i		−	0.458734i	−0.973340	−	0.229367i	$-0.926334\pi$
$383$		−	1.31692e6i		−	0.458734i	0.973340	−	0.229367i	$-0.0736656\pi$
$384$		0			0
$385$	−803771.	+	1.23920e6i	−0.276363	+	0.426079i
$386$		0			0
$387$			771598.i			0.261887i
$388$		0			0
$389$	2.08035e6			0.697047			0.348524	−	0.937300i	$-0.386683\pi$
$389$	2.08035e6			0.697047			0.348524	+	0.937300i	$0.386683\pi$
$390$		0			0
$391$	1.58203e6			0.523328
$392$		0			0
$393$			1.50676e6i			0.492112i
$394$		0			0
$395$	4.60747e6	+	2.98850e6i	1.48583	+	0.963741i
$396$		0			0
$397$		−	2.30011e6i		−	0.732440i	−0.930528	−	0.366220i	$-0.880652\pi$
$397$		−	2.30011e6i		−	0.732440i	0.930528	−	0.366220i	$-0.119348\pi$
$398$		0			0
$399$	3.55685e6			1.11849
$400$		0			0
$401$	−6.11490e6			−1.89902			−0.949508	−	0.313743i	$-0.898417\pi$
$401$	−6.11490e6			−1.89902			−0.949508	+	0.313743i	$0.898417\pi$
$402$		0			0
$403$		−	855099.i		−	0.262273i
$404$		0			0
$405$	−3.46131e6	−	2.24507e6i	−1.04858	−	0.680131i
$406$		0			0
$407$			2.78065e6i			0.832070i
$408$		0			0
$409$	6.46166e6			1.91001			0.955005	−	0.296590i	$-0.0958494\pi$
$409$	6.46166e6			1.91001			0.955005	+	0.296590i	$0.0958494\pi$
$410$		0			0
$411$	−4258.79			−0.00124360
$412$		0			0
$413$			4.72950e6i			1.36440i
$414$		0			0
$415$	−2.87380e6	+	4.43064e6i	−0.819100	+	1.26284i
$416$		0			0
$417$			4.44235e6i			1.25104i
$418$		0			0
$419$	3.06588e6			0.853139			0.426569	−	0.904455i	$-0.359722\pi$
$419$	3.06588e6			0.853139			0.426569	+	0.904455i	$0.359722\pi$
$420$		0			0
$421$	−856178.			−0.235428			−0.117714	−	0.993048i	$-0.537557\pi$
$421$	−856178.			−0.235428			−0.117714	+	0.993048i	$0.537557\pi$
$422$		0			0
$423$			165269.i			0.0449097i
$424$		0			0
$425$	−2.68035e6	+	1.19693e6i	−0.719813	+	0.321438i
$426$		0			0
$427$			4.69176e6i			1.24528i
$428$		0			0
$429$	−618645.			−0.162292
$430$		0			0
$431$	1.86381e6			0.483290			0.241645	−	0.970365i	$-0.422313\pi$
$431$	1.86381e6			0.483290			0.241645	+	0.970365i	$0.422313\pi$
$432$		0			0
$433$			5.96051e6i			1.52779i	0.645340	+	0.763895i	$0.276716\pi$
$433$			5.96051e6i			1.52779i	−0.645340	+	0.763895i	$0.723284\pi$
$434$		0			0
$435$	−2.29059e6	+	3.53148e6i	−0.580395	+	0.894815i
$436$		0			0
$437$			3.46149e6i			0.867080i
$438$		0			0
$439$	−3.11969e6			−0.772591			−0.386295	−	0.922375i	$-0.626245\pi$
$439$	−3.11969e6			−0.772591			−0.386295	+	0.922375i	$0.626245\pi$
$440$		0			0
$441$	−1.00915e6			−0.247092
$442$		0			0
$443$		−	8.04956e6i		−	1.94878i	−0.224863	−	0.974390i	$-0.572193\pi$
$443$		−	8.04956e6i		−	1.94878i	0.224863	−	0.974390i	$-0.427807\pi$
$444$		0			0
$445$	−4.60677e6	−	2.98804e6i	−1.10280	−	0.715298i
$446$		0			0
$447$		−	4.73174e6i		−	1.12009i
$448$		0			0
$449$	−2.68451e6			−0.628420			−0.314210	−	0.949354i	$-0.601740\pi$
$449$	−2.68451e6			−0.628420			−0.314210	+	0.949354i	$0.601740\pi$
$450$		0			0
$451$	81753.5			0.0189263
$452$		0			0
$453$		−	3.37486e6i		−	0.772699i
$454$		0			0
$455$	478615.	+	310439.i	0.108382	+	0.0702988i
$456$		0			0
$457$		−	981624.i		−	0.219864i	−0.993939	−	0.109932i	$-0.964937\pi$
$457$		−	981624.i		−	0.219864i	0.993939	−	0.109932i	$-0.0350634\pi$
$458$		0			0
$459$	2.22455e6			0.492845
$460$		0			0
$461$	−7.92524e6			−1.73684			−0.868420	−	0.495829i	$-0.834864\pi$
$461$	−7.92524e6			−1.73684			−0.868420	+	0.495829i	$0.834864\pi$
$462$		0			0
$463$		−	1.67531e6i		−	0.363198i	−0.983373	−	0.181599i	$-0.941873\pi$
$463$		−	1.67531e6i		−	0.363198i	0.983373	−	0.181599i	$-0.0581273\pi$
$464$		0			0
$465$	−4.41121e6	+	6.80092e6i	−0.946075	+	1.45860i
$466$		0			0
$467$		−	5.69218e6i		−	1.20778i	−0.797069	−	0.603888i	$-0.793617\pi$
$467$		−	5.69218e6i		−	1.20778i	0.797069	−	0.603888i	$-0.206383\pi$
$468$		0			0
$469$	3.55687e6			0.746682
$470$		0			0
$471$	−4.78308e6			−0.993472
$472$		0			0
$473$			1.89107e6i			0.388647i
$474$		0			0
$475$	−2.61889e6	−	5.86461e6i	−0.532577	−	1.19263i
$476$		0			0
$477$			2.19552e6i			0.441816i
$478$		0			0
$479$	−8.24150e6			−1.64122			−0.820612	−	0.571486i	$-0.806367\pi$
$479$	−8.24150e6			−1.64122			−0.820612	+	0.571486i	$0.806367\pi$
$480$		0			0
$481$	1.07396e6			0.211654
$482$		0			0
$483$			2.91462e6i			0.568478i
$484$		0			0
$485$	−1.79511e6	+	2.76759e6i	−0.346527	+	0.534253i
$486$		0			0
$487$			2.78291e6i			0.531713i	0.964013	+	0.265857i	$0.0856547\pi$
$487$			2.78291e6i			0.531713i	−0.964013	+	0.265857i	$0.914345\pi$
$488$		0			0
$489$	1.04868e7			1.98322
$490$		0			0
$491$	−3.71062e6			−0.694613			−0.347307	−	0.937752i	$-0.612904\pi$
$491$	−3.71062e6			−0.694613			−0.347307	+	0.937752i	$0.612904\pi$
$492$		0			0
$493$		−	3.72045e6i		−	0.689412i
$494$		0			0
$495$	1.61227e6	+	1.04575e6i	0.295751	+	0.191830i
$496$		0			0
$497$		−	2.77054e6i		−	0.503122i
$498$		0			0
$499$	3.70740e6			0.666528			0.333264	−	0.942833i	$-0.391850\pi$
$499$	3.70740e6			0.666528			0.333264	+	0.942833i	$0.391850\pi$
$500$		0			0
$501$	−3.48215e6			−0.619802
$502$		0			0
$503$		−	4.92831e6i		−	0.868516i	−0.900789	−	0.434258i	$-0.857011\pi$
$503$		−	4.92831e6i		−	0.868516i	0.900789	−	0.434258i	$-0.142989\pi$
$504$		0			0
$505$	380696.	+	246927.i	0.0664279	+	0.0430864i
$506$		0			0
$507$		−	6.81986e6i		−	1.17830i
$508$		0			0
$509$	4.72439e6			0.808261			0.404130	−	0.914701i	$-0.367574\pi$
$509$	4.72439e6			0.808261			0.404130	+	0.914701i	$0.367574\pi$
$510$		0			0
$511$	4.52360e6			0.766358
$512$		0			0
$513$			4.86731e6i			0.816574i
$514$		0			0
$515$	−4.60980e6	+	7.10709e6i	−0.765885	+	1.18079i
$516$		0			0
$517$			405050.i			0.0666472i
$518$		0			0
$519$	−582385.			−0.0949056
$520$		0			0
$521$	−4.72698e6			−0.762939			−0.381469	−	0.924382i	$-0.624582\pi$
$521$	−4.72698e6			−0.762939			−0.381469	+	0.924382i	$0.624582\pi$
$522$		0			0
$523$			1.43476e6i			0.229363i	0.993402	+	0.114682i	$0.0365848\pi$
$523$			1.43476e6i			0.229363i	−0.993402	+	0.114682i	$0.963415\pi$
$524$		0			0
$525$	−2.20514e6	−	4.93808e6i	−0.349170	−	0.781915i
$526$		0			0
$527$		−	7.16485e6i		−	1.12378i
$528$		0			0
$529$	3.59987e6			0.559303
$530$		0			0
$531$	6.15336e6			0.947056
$532$		0			0
$533$		−	31575.5i		−	0.00481429i
$534$		0			0
$535$	2.89675e6	−	4.46601e6i	0.437548	−	0.674584i
$536$		0			0
$537$		−	1.04193e7i		−	1.55921i
$538$		0			0
$539$	−2.47327e6			−0.366690
$540$		0			0
$541$	5.09101e6			0.747844			0.373922	−	0.927460i	$-0.378013\pi$
$541$	5.09101e6			0.747844			0.373922	+	0.927460i	$0.378013\pi$
$542$		0			0
$543$			1.14619e6i			0.166824i
$544$		0			0
$545$	−3.08603e6	−	2.00166e6i	−0.445051	−	0.288669i
$546$		0			0
$547$			1.96976e6i			0.281478i	0.990047	+	0.140739i	$0.0449478\pi$
$547$			1.96976e6i			0.281478i	−0.990047	+	0.140739i	$0.955052\pi$
$548$		0			0
$549$	6.10425e6			0.864373
$550$		0			0
$551$	8.14035e6			1.14226
$552$		0			0
$553$		−	8.94269e6i		−	1.24353i
$554$		0			0
$555$	−8.54164e6	−	5.54028e6i	−1.17709	−	0.763482i
$556$		0			0
$557$			1.43166e7i			1.95525i	0.210351	+	0.977626i	$0.432539\pi$
$557$			1.43166e7i			1.95525i	−0.210351	+	0.977626i	$0.567461\pi$
$558$		0			0
$559$	730385.			0.0988604
$560$		0			0
$561$	−5.18361e6			−0.695385
$562$		0			0
$563$			1.94499e6i			0.258610i	0.991605	+	0.129305i	$0.0412746\pi$
$563$			1.94499e6i			0.258610i	−0.991605	+	0.129305i	$0.958725\pi$
$564$		0			0
$565$	4.39151e6	−	6.77055e6i	0.578752	−	0.892283i
$566$		0			0
$567$			6.71808e6i			0.877582i
$568$		0			0
$569$	1.63638e6			0.211886			0.105943	−	0.994372i	$-0.466214\pi$
$569$	1.63638e6			0.211886			0.105943	+	0.994372i	$0.466214\pi$
$570$		0			0
$571$	−1.07950e7			−1.38558			−0.692789	−	0.721140i	$-0.743618\pi$
$571$	−1.07950e7			−1.38558			−0.692789	+	0.721140i	$0.743618\pi$
$572$		0			0
$573$		−	3.81328e6i		−	0.485190i
$574$		0			0
$575$	4.80568e6	−	2.14601e6i	0.606157	−	0.270684i
$576$		0			0
$577$		−	1.11502e7i		−	1.39426i	−0.716946	−	0.697129i	$-0.754460\pi$
$577$		−	1.11502e7i		−	1.39426i	0.716946	−	0.697129i	$-0.245540\pi$
$578$		0			0
$579$	5.51845e6			0.684102
$580$		0			0
$581$	8.59948e6			1.05690
$582$		0			0
$583$			5.38089e6i			0.655666i
$584$		0			0
$585$	403899.	−	622706.i	0.0487959	−	0.0752304i
$586$		0			0
$587$			1.50708e7i			1.80526i	0.430415	+	0.902631i	$0.358367\pi$
$587$			1.50708e7i			1.80526i	−0.430415	+	0.902631i	$0.641633\pi$
$588$		0			0
$589$	1.56767e7			1.86194
$590$		0			0
$591$	−1.97237e7			−2.32284
$592$		0			0
$593$		−	6.22941e6i		−	0.727462i	−0.931504	−	0.363731i	$-0.881503\pi$
$593$		−	6.22941e6i		−	0.727462i	0.931504	−	0.363731i	$-0.118497\pi$
$594$		0			0
$595$	4.01030e6	+	2.60116e6i	0.464392	+	0.301214i
$596$		0			0
$597$			1.14720e7i			1.31735i
$598$		0			0
$599$	2.19176e6			0.249589			0.124795	−	0.992183i	$-0.460173\pi$
$599$	2.19176e6			0.249589			0.124795	+	0.992183i	$0.460173\pi$
$600$		0			0
$601$	4.25014e6			0.479973			0.239987	−	0.970776i	$-0.422857\pi$
$601$	4.25014e6			0.479973			0.239987	+	0.970776i	$0.422857\pi$
$602$		0			0
$603$		−	4.62769e6i		−	0.518288i
$604$		0			0
$605$	−3.60184e6	−	2.33623e6i	−0.400071	−	0.259494i
$606$		0			0
$607$		−	3.17522e6i		−	0.349786i	−0.984587	−	0.174893i	$-0.944042\pi$
$607$		−	3.17522e6i		−	0.349786i	0.984587	−	0.174893i	$-0.0559579\pi$
$608$		0			0
$609$	6.85428e6			0.748891
$610$		0			0
$611$	156442.			0.0169531
$612$		0			0
$613$		−	3.00150e6i		−	0.322617i	−0.986904	−	0.161308i	$-0.948429\pi$
$613$		−	3.00150e6i		−	0.322617i	0.986904	−	0.161308i	$-0.0515714\pi$
$614$		0			0
$615$	−162889.	+	251132.i	−0.0173662	+	0.0267741i
$616$		0			0
$617$		−	1.47318e7i		−	1.55791i	−0.627079	−	0.778955i	$-0.715750\pi$
$617$		−	1.47318e7i		−	1.55791i	0.627079	−	0.778955i	$-0.284250\pi$
$618$		0			0
$619$	1.62754e7			1.70728			0.853639	−	0.520865i	$-0.174391\pi$
$619$	1.62754e7			1.70728			0.853639	+	0.520865i	$0.174391\pi$
$620$		0			0
$621$	−3.98846e6			−0.415026
$622$		0			0
$623$			8.94133e6i			0.922958i
$624$		0			0
$625$	−6.51837e6	+	7.27174e6i	−0.667481	+	0.744627i
$626$		0			0
$627$		−	1.13417e7i		−	1.15215i
$628$		0			0
$629$	8.99872e6			0.906889
$630$		0			0
$631$	7.53223e6			0.753095			0.376548	−	0.926397i	$-0.377111\pi$
$631$	7.53223e6			0.753095			0.376548	+	0.926397i	$0.377111\pi$
$632$		0			0
$633$			7.44402e6i			0.738411i
$634$		0			0
$635$	9.24972e6	−	1.42606e7i	0.910321	−	1.40347i
$636$		0			0
$637$			955246.i			0.0932753i
$638$		0			0
$639$	−3.60463e6			−0.349228
$640$		0			0
$641$	8.72066e6			0.838309			0.419155	−	0.907915i	$-0.362327\pi$
$641$	8.72066e6			0.838309			0.419155	+	0.907915i	$0.362327\pi$
$642$		0			0
$643$			1.43889e7i			1.37246i	0.727385	+	0.686230i	$0.240735\pi$
$643$			1.43889e7i			1.37246i	−0.727385	+	0.686230i	$0.759265\pi$
$644$		0			0
$645$	−5.80903e6	−	3.76785e6i	−0.549799	−	0.356611i
$646$		0			0
$647$			1.84992e7i			1.73737i	0.495362	+	0.868687i	$0.335035\pi$
$647$			1.84992e7i			1.73737i	−0.495362	+	0.868687i	$0.664965\pi$
$648$		0			0
$649$	1.50810e7			1.40546
$650$		0			0
$651$	1.32000e7			1.22073
$652$		0			0
$653$			4.95694e6i			0.454916i	0.973788	+	0.227458i	$0.0730414\pi$
$653$			4.95694e6i			0.454916i	−0.973788	+	0.227458i	$0.926959\pi$
$654$		0			0
$655$	−3.71709e6	−	2.41098e6i	−0.338532	−	0.219579i
$656$		0			0
$657$		−	5.88547e6i		−	0.531946i
$658$		0			0
$659$	−1.07824e7			−0.967170			−0.483585	−	0.875297i	$-0.660666\pi$
$659$	−1.07824e7			−0.967170			−0.483585	+	0.875297i	$0.660666\pi$
$660$		0			0
$661$	7.19928e6			0.640893			0.320446	−	0.947267i	$-0.396167\pi$
$661$	7.19928e6			0.640893			0.320446	+	0.947267i	$0.396167\pi$
$662$		0			0
$663$			2.00206e6i			0.176886i
$664$		0			0
$665$	−5.69134e6	+	8.77454e6i	−0.499069	+	0.769432i
$666$		0			0
$667$			6.67051e6i			0.580556i
$668$		0			0
$669$	5.58936e6			0.482833
$670$		0			0
$671$	1.49606e7			1.28275
$672$		0			0
$673$		−	9.15273e6i		−	0.778956i	−0.921036	−	0.389478i	$-0.872655\pi$
$673$		−	9.15273e6i		−	0.778956i	0.921036	−	0.389478i	$-0.127345\pi$
$674$		0			0
$675$	6.75742e6	−	3.01758e6i	0.570849	−	0.254917i
$676$		0			0
$677$		−	3.84650e6i		−	0.322548i	−0.986910	−	0.161274i	$-0.948440\pi$
$677$		−	3.84650e6i		−	0.322548i	0.986910	−	0.161274i	$-0.0515603\pi$
$678$		0			0
$679$	5.37164e6			0.447128
$680$		0			0
$681$	−5.65106e6			−0.466941
$682$		0			0
$683$			1.47653e7i			1.21113i	0.795797	+	0.605564i	$0.207052\pi$
$683$			1.47653e7i			1.21113i	−0.795797	+	0.605564i	$0.792948\pi$
$684$		0			0
$685$	6814.51	−	10506.2i	0.000554892	−	0.000855496i
$686$		0			0
$687$		−	2.38279e7i		−	1.92617i
$688$		0			0
$689$	2.07825e6			0.166782
$690$		0			0
$691$	−8.15013e6			−0.649336			−0.324668	−	0.945828i	$-0.605253\pi$
$691$	−8.15013e6			−0.649336			−0.324668	+	0.945828i	$0.605253\pi$
$692$		0			0
$693$		−	3.12928e6i		−	0.247521i
$694$		0			0
$695$	−1.09590e7	−	7.10823e6i	−0.860615	−	0.558212i
$696$		0			0
$697$		−	264571.i		−	0.0206281i
$698$		0			0
$699$	−2.54259e7			−1.96826
$700$		0			0
$701$	650995.			0.0500360			0.0250180	−	0.999687i	$-0.492036\pi$
$701$	650995.			0.0500360			0.0250180	+	0.999687i	$0.492036\pi$
$702$		0			0
$703$			1.96892e7i			1.50259i
$704$		0			0
$705$	−1.24424e6	−	807037.i	−0.0942824	−	0.0611534i
$706$		0			0
$707$		−	738897.i		−	0.0555950i
$708$		0			0
$709$	4.57134e6			0.341529			0.170764	−	0.985312i	$-0.445376\pi$
$709$	4.57134e6			0.341529			0.170764	+	0.985312i	$0.445376\pi$
$710$		0			0
$711$	−1.16350e7			−0.863159
$712$		0			0
$713$			1.28461e7i			0.946338i
$714$		0			0
$715$	989897.	−	1.52616e6i	0.0724144	−	0.111644i
$716$		0			0
$717$			3.58228e6i			0.260232i
$718$		0			0
$719$	5.79026e6			0.417711			0.208855	−	0.977947i	$-0.433026\pi$
$719$	5.79026e6			0.417711			0.208855	+	0.977947i	$0.433026\pi$
$720$		0			0
$721$	1.37942e7			0.988232
$722$		0			0
$723$		−	328571.i		−	0.0233767i
$724$		0			0
$725$	−5.04676e6	−	1.13015e7i	−0.356589	−	0.798528i
$726$		0			0
$727$			5.07311e6i			0.355991i	0.984031	+	0.177995i	$0.0569612\pi$
$727$			5.07311e6i			0.355991i	−0.984031	+	0.177995i	$0.943039\pi$
$728$		0			0
$729$	−2.19985e6			−0.153311
$730$		0			0
$731$	6.11988e6			0.423594
$732$		0			0
$733$		−	2.05426e7i		−	1.41220i	−0.708114	−	0.706098i	$-0.750454\pi$
$733$		−	2.05426e7i		−	1.41220i	0.708114	−	0.706098i	$-0.249546\pi$
$734$		0			0
$735$	4.92784e6	−	7.59743e6i	0.336464	−	0.518738i
$736$		0			0
$737$		−	1.13418e7i		−	0.769153i
$738$		0			0
$739$	−1.61223e7			−1.08596			−0.542982	−	0.839744i	$-0.682705\pi$
$739$	−1.61223e7			−1.08596			−0.542982	+	0.839744i	$0.682705\pi$
$740$		0			0
$741$	−4.38050e6			−0.293074
$742$		0			0
$743$		−	1.20905e7i		−	0.803473i	−0.915755	−	0.401736i	$-0.868407\pi$
$743$		−	1.20905e7i		−	0.803473i	0.915755	−	0.401736i	$-0.131593\pi$
$744$		0			0
$745$	1.16729e7	+	7.57129e6i	0.770529	+	0.499780i
$746$		0			0
$747$		−	1.11884e7i		−	0.733614i
$748$		0			0
$749$	−8.66813e6			−0.564574
$750$		0			0
$751$	1.13239e7			0.732650			0.366325	−	0.930487i	$-0.380616\pi$
$751$	1.13239e7			0.732650			0.366325	+	0.930487i	$0.380616\pi$
$752$		0			0
$753$			1.91849e7i			1.23302i
$754$		0			0
$755$	8.32558e6	+	5.40013e6i	0.531554	+	0.344776i
$756$		0			0
$757$		−	6.74048e6i		−	0.427514i	−0.976887	−	0.213757i	$-0.931430\pi$
$757$		−	6.74048e6i		−	0.427514i	0.976887	−	0.213757i	$-0.0685702\pi$
$758$		0			0
$759$	9.29384e6			0.585586
$760$		0			0
$761$	−1.49525e7			−0.935950			−0.467975	−	0.883742i	$-0.655016\pi$
$761$	−1.49525e7			−0.935950			−0.467975	+	0.883742i	$0.655016\pi$
$762$		0			0
$763$			5.98971e6i			0.372473i
$764$		0			0
$765$	3.38426e6	−	5.21764e6i	0.209079	−	0.322345i
$766$		0			0
$767$		−	5.82469e6i		−	0.357507i
$768$		0			0
$769$	2.56776e7			1.56581			0.782905	−	0.622142i	$-0.213737\pi$
$769$	2.56776e7			1.56581			0.782905	+	0.622142i	$0.213737\pi$
$770$		0			0
$771$	4.97545e6			0.301437
$772$		0			0
$773$		−	1.23034e7i		−	0.740588i	−0.928915	−	0.370294i	$-0.879257\pi$
$773$		−	1.23034e7i		−	0.740588i	0.928915	−	0.370294i	$-0.120743\pi$
$774$		0			0
$775$	−9.71905e6	−	2.17644e7i	−0.581259	−	1.30164i
$776$		0			0
$777$			1.65786e7i			0.985131i
$778$		0			0
$779$	578880.			0.0341779
$780$		0			0
$781$	−8.83441e6			−0.518263
$782$		0			0
$783$			9.37961e6i			0.546739i
$784$		0			0
$785$	7.65344e6	−	1.17996e7i	0.443284	−	0.683427i
$786$		0			0
$787$		−	2.78485e7i		−	1.60275i	−0.598163	−	0.801375i	$-0.704102\pi$
$787$		−	2.78485e7i		−	1.60275i	0.598163	−	0.801375i	$-0.295898\pi$
$788$		0			0
$789$	−1.86548e7			−1.06684
$790$		0			0
$791$	−1.31410e7			−0.746772
$792$		0			0
$793$		−	5.77821e6i		−	0.326295i
$794$		0			0
$795$	−1.65291e7	−	1.07211e7i	−0.927537	−	0.601619i
$796$		0			0
$797$		−	9.80382e6i		−	0.546701i	−0.961915	−	0.273350i	$-0.911868\pi$
$797$		−	9.80382e6i		−	0.546701i	0.961915	−	0.273350i	$-0.0881318\pi$
$798$		0			0
$799$	1.31082e6			0.0726401
$800$		0			0
$801$	1.16332e7			0.640645
$802$		0			0
$803$		−	1.44244e7i		−	0.789421i
$804$		0			0
$805$	−7.19019e6	−	4.66370e6i	−0.391066	−	0.253653i
$806$		0			0
$807$		−	2.01370e7i		−	1.08845i
$808$		0			0
$809$	−1.08607e7			−0.583428			−0.291714	−	0.956506i	$-0.594225\pi$
$809$	−1.08607e7			−0.583428			−0.291714	+	0.956506i	$0.594225\pi$
$810$		0			0
$811$	4.52355e6			0.241506			0.120753	−	0.992683i	$-0.461469\pi$
$811$	4.52355e6			0.241506			0.120753	+	0.992683i	$0.461469\pi$
$812$		0			0
$813$			3.82087e6i			0.202738i
$814$		0			0
$815$	−1.67800e7	+	2.58703e7i	−0.884906	+	1.36429i
$816$		0			0
$817$			1.33903e7i			0.701835i
$818$		0			0
$819$	−1.20862e6			−0.0629620
$820$		0			0
$821$	−2.49411e7			−1.29139			−0.645695	−	0.763596i	$-0.723432\pi$
$821$	−2.49411e7			−1.29139			−0.645695	+	0.763596i	$0.723432\pi$
$822$		0			0
$823$		−	1.44805e7i		−	0.745217i	−0.927989	−	0.372608i	$-0.878463\pi$
$823$		−	1.44805e7i		−	0.745217i	0.927989	−	0.372608i	$-0.121537\pi$
$824$		0			0
$825$	−1.57460e7	+	7.03152e6i	−0.805446	+	0.359678i
$826$		0			0
$827$			1.74301e7i			0.886207i	0.896470	+	0.443104i	$0.146123\pi$
$827$			1.74301e7i			0.886207i	−0.896470	+	0.443104i	$0.853877\pi$
$828$		0			0
$829$	1.17905e7			0.595860			0.297930	−	0.954588i	$-0.403704\pi$
$829$	1.17905e7			0.595860			0.297930	+	0.954588i	$0.403704\pi$
$830$		0			0
$831$	−3.96734e7			−1.99295
$832$		0			0
$833$			8.00398e6i			0.399663i
$834$		0			0
$835$	5.57181e6	−	8.59025e6i	0.276554	−	0.426373i
$836$		0			0
$837$			1.80633e7i			0.891215i
$838$		0			0
$839$	2.76099e6			0.135413			0.0677064	−	0.997705i	$-0.478432\pi$
$839$	2.76099e6			0.135413			0.0677064	+	0.997705i	$0.478432\pi$
$840$		0			0
$841$	−4.82419e6			−0.235198
$842$		0			0
$843$			8.74483e6i			0.423821i
$844$		0			0
$845$	1.68242e7	+	1.09125e7i	0.810573	+	0.525754i
$846$		0			0
$847$			6.99086e6i			0.334828i
$848$		0			0
$849$	−3.81073e7			−1.81442
$850$		0			0
$851$	−1.61341e7			−0.763695
$852$		0			0
$853$			3.24902e7i			1.52890i	0.644681	+	0.764452i	$0.276990\pi$
$853$			3.24902e7i			1.52890i	−0.644681	+	0.764452i	$0.723010\pi$
$854$		0			0
$855$	1.14162e7	+	7.40476e6i	0.534079	+	0.346414i
$856$		0			0
$857$			3.47075e7i			1.61425i	0.590378	+	0.807127i	$0.298979\pi$
$857$			3.47075e7i			1.61425i	−0.590378	+	0.807127i	$0.701021\pi$
$858$		0			0
$859$	2.01722e7			0.932762			0.466381	−	0.884584i	$-0.345558\pi$
$859$	2.01722e7			0.932762			0.466381	+	0.884584i	$0.345558\pi$
$860$		0			0
$861$	487424.			0.0224078
$862$		0			0
$863$			1.44410e7i			0.660041i	0.943974	+	0.330021i	$0.107056\pi$
$863$			1.44410e7i			0.660041i	−0.943974	+	0.330021i	$0.892944\pi$
$864$		0			0
$865$	931877.	−	1.43671e6i	0.0423466	−	0.0652872i
$866$		0			0
$867$		−	1.02183e7i		−	0.461669i
$868$		0			0
$869$	−2.85156e7			−1.28095
$870$		0			0
$871$	−4.38052e6			−0.195650
$872$		0			0
$873$		−	6.98881e6i		−	0.310362i
$874$		0			0
$875$	1.57104e7	+	2.46151e6i	0.693692	+	0.108688i
$876$		0			0
$877$		−	5.07628e6i		−	0.222868i	−0.993772	−	0.111434i	$-0.964456\pi$
$877$		−	5.07628e6i		−	0.222868i	0.993772	−	0.111434i	$-0.0355443\pi$
$878$		0			0
$879$	3.05008e7			1.33149
$880$		0			0
$881$	−2.41020e7			−1.04620			−0.523098	−	0.852273i	$-0.675224\pi$
$881$	−2.41020e7			−1.04620			−0.523098	+	0.852273i	$0.675224\pi$
$882$		0			0
$883$			247047.i			0.0106629i	0.999986	+	0.00533147i	$0.00169707\pi$
$883$			247047.i			0.0106629i	−0.999986	+	0.00533147i	$0.998303\pi$
$884$		0			0
$885$	−3.00479e7	+	4.63259e7i	−1.28960	+	1.98823i
$886$		0			0
$887$			1.63993e7i			0.699869i	0.936774	+	0.349935i	$0.113796\pi$
$887$			1.63993e7i			0.699869i	−0.936774	+	0.349935i	$0.886204\pi$
$888$		0			0
$889$	−2.76786e7			−1.17460
$890$		0			0
$891$	2.14219e7			0.903992
$892$		0			0
$893$			2.86807e6i			0.120354i
$894$		0			0
$895$	2.57039e7	+	1.66720e7i	1.07261	+	0.695714i
$896$		0			0
$897$		−	3.58954e6i		−	0.148956i
$898$		0			0
$899$	3.02100e7			1.24667
$900$		0			0
$901$	1.74136e7			0.714623
$902$		0			0
$903$			1.12748e7i			0.460139i
$904$		0			0
$905$	−2.82759e6	−	1.83403e6i	−0.114761	−	0.0744363i
$906$		0			0
$907$			2.99253e7i			1.20787i	0.797034	+	0.603934i	$0.206401\pi$
$907$			2.99253e7i			1.20787i	−0.797034	+	0.603934i	$0.793599\pi$
$908$		0			0
$909$	−961349.			−0.0385897
$910$		0			0
$911$	3.13449e7			1.25133			0.625663	−	0.780094i	$-0.284829\pi$
$911$	3.13449e7			1.25133			0.625663	+	0.780094i	$0.284829\pi$
$912$		0			0
$913$		−	2.74212e7i		−	1.08870i
$914$		0			0
$915$	−2.98081e7	+	4.59562e7i	−1.17701	+	1.81464i
$916$		0			0
$917$			7.21454e6i			0.283325i
$918$		0			0
$919$	−4.84736e7			−1.89329			−0.946643	−	0.322284i	$-0.895549\pi$
$919$	−4.84736e7			−1.89329			−0.946643	+	0.322284i	$0.895549\pi$
$920$		0			0
$921$	−4.18492e7			−1.62569
$922$		0			0
$923$			3.41210e6i			0.131831i
$924$		0			0
$925$	2.73351e7	−	1.22067e7i	1.05043	−	0.469076i
$926$		0			0
$927$		−	1.79471e7i		−	0.685953i
$928$		0			0
$929$	2.59358e7			0.985963			0.492981	−	0.870040i	$-0.335907\pi$
$929$	2.59358e7			0.985963			0.492981	+	0.870040i	$0.335907\pi$
$930$		0			0
$931$	−1.75127e7			−0.662184
$932$		0			0
$933$			4.40276e7i			1.65585i
$934$		0			0
$935$	8.29432e6	−	1.27876e7i	0.310279	−	0.478368i
$936$		0			0
$937$		−	1.52182e7i		−	0.566259i	−0.959082	−	0.283129i	$-0.908627\pi$
$937$		−	1.52182e7i		−	0.566259i	0.959082	−	0.283129i	$-0.0913726\pi$
$938$		0			0
$939$	−2.70056e7			−0.999516
$940$		0			0
$941$	−1.96215e7			−0.722366			−0.361183	−	0.932495i	$-0.617627\pi$
$941$	−1.96215e7			−0.722366			−0.361183	+	0.932495i	$0.617627\pi$
$942$		0			0
$943$			474356.i			0.0173710i
$944$		0			0
$945$	−1.01103e7	−	6.55777e6i	−0.368287	−	0.238878i
$946$		0			0
$947$		−	1.29697e6i		−	0.0469955i	−0.999724	−	0.0234977i	$-0.992520\pi$
$947$		−	1.29697e6i		−	0.0469955i	0.999724	−	0.0234977i	$-0.00748025\pi$
$948$		0			0
$949$	−5.57111e6			−0.200806
$950$		0			0
$951$	6.21867e7			2.22970
$952$		0			0
$953$			2.10228e7i			0.749823i	0.927061	+	0.374911i	$0.122327\pi$
$953$			2.10228e7i			0.749823i	−0.927061	+	0.374911i	$0.877673\pi$
$954$		0			0
$955$	9.40713e6	+	6.10165e6i	0.333771	+	0.216491i
$956$		0			0
$957$		−	2.18562e7i		−	0.771428i
$958$		0			0
$959$	−20391.5			−0.000715984
$960$		0			0
$961$	2.95492e7			1.03214
$962$		0			0
$963$			1.12777e7i			0.391883i
$964$		0			0
$965$	−8.83011e6	+	1.36137e7i	−0.305244	+	0.470606i
$966$		0			0
$967$			7.16230e6i			0.246312i	0.992387	+	0.123156i	$0.0393016\pi$
$967$			7.16230e6i			0.246312i	−0.992387	+	0.123156i	$0.960698\pi$
$968$		0			0
$969$	−3.67041e7			−1.25575
$970$		0			0
$971$	−3.00177e7			−1.02172			−0.510858	−	0.859665i	$-0.670672\pi$
$971$	−3.00177e7			−1.02172			−0.510858	+	0.859665i	$0.670672\pi$
$972$		0			0
$973$			2.12704e7i			0.720268i
$974$		0			0
$975$	2.71577e6	+	6.08156e6i	0.0914916	+	0.204882i
$976$		0			0
$977$			3.38644e7i			1.13503i	0.823364	+	0.567514i	$0.192095\pi$
$977$			3.38644e7i			1.13503i	−0.823364	+	0.567514i	$0.807905\pi$
$978$		0			0
$979$	2.85112e7			0.950734
$980$		0			0
$981$	7.79297e6			0.258542
$982$		0			0
$983$		−	3.47482e7i		−	1.14696i	−0.819219	−	0.573480i	$-0.805593\pi$
$983$		−	3.47482e7i		−	1.14696i	0.819219	−	0.573480i	$-0.194407\pi$
$984$		0			0
$985$	3.15600e7	−	4.86571e7i	1.03644	−	1.59792i
$986$		0			0
$987$			2.41495e6i			0.0789071i
$988$		0			0
$989$	−1.09725e7			−0.356710
$990$		0			0
$991$	5.51244e6			0.178303			0.0891517	−	0.996018i	$-0.471584\pi$
$991$	5.51244e6			0.178303			0.0891517	+	0.996018i	$0.471584\pi$
$992$		0			0
$993$		−	4.57228e7i		−	1.47150i
$994$		0			0
$995$	−2.83006e7	−	1.83563e7i	−0.906230	−	0.587799i
$996$		0			0
$997$		−	1.42212e7i		−	0.453105i	−0.973999	−	0.226553i	$-0.927254\pi$
$997$		−	1.42212e7i		−	0.453105i	0.973999	−	0.226553i	$-0.0727456\pi$
$998$		0			0
$999$	−2.26866e7			−0.719210

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.c.k.129.4		12
4.3	odd	2	inner	320.6.c.k.129.10		12
5.4	even	2	inner	320.6.c.k.129.9		12
8.3	odd	2		160.6.c.c.129.3	✓	12
8.5	even	2		160.6.c.c.129.9	yes	12
20.19	odd	2	inner	320.6.c.k.129.3		12
40.3	even	4		800.6.a.ba.1.5		6
40.13	odd	4		800.6.a.ba.1.2		6
40.19	odd	2		160.6.c.c.129.10	yes	12
40.27	even	4		800.6.a.z.1.2		6
40.29	even	2		160.6.c.c.129.4	yes	12
40.37	odd	4		800.6.a.z.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.c.c.129.3	✓	12	8.3	odd	2
160.6.c.c.129.4	yes	12	40.29	even	2
160.6.c.c.129.9	yes	12	8.5	even	2
160.6.c.c.129.10	yes	12	40.19	odd	2
320.6.c.k.129.3		12	20.19	odd	2	inner
320.6.c.k.129.4		12	1.1	even	1	trivial
320.6.c.k.129.9		12	5.4	even	2	inner
320.6.c.k.129.10		12	4.3	odd	2	inner
800.6.a.z.1.2		6	40.27	even	4
800.6.a.z.1.5		6	40.37	odd	4
800.6.a.ba.1.2		6	40.13	odd	4
800.6.a.ba.1.5		6	40.3	even	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-19.0114i q^{3} +(46.9000 + 30.4203i) q^{5} -91.0286i q^{7} -118.434 q^{9} +O(q^{10})\)	\(q-19.0114i q^{3} +(46.9000 + 30.4203i) q^{5} -91.0286i q^{7} -118.434 q^{9} -290.263 q^{11} +112.108i q^{13} +(578.332 - 891.635i) q^{15} +939.348i q^{17} -2055.29 q^{19} -1730.58 q^{21} -1684.18i q^{23} +(1274.22 + 2853.42i) q^{25} -2368.18i q^{27} -3960.68 q^{29} -7627.48 q^{31} +5518.31i q^{33} +(2769.11 - 4269.24i) q^{35} -9579.76i q^{37} +2131.33 q^{39} -281.653 q^{41} -6515.03i q^{43} +(-5554.53 - 3602.78i) q^{45} -1395.46i q^{47} +8520.79 q^{49} +17858.3 q^{51} -18538.0i q^{53} +(-13613.3 - 8829.87i) q^{55} +39074.0i q^{57} -51956.2 q^{59} -51541.5 q^{61} +10780.8i q^{63} +(-3410.35 + 5257.85i) q^{65} +39074.2i q^{67} -32018.7 q^{69} +30435.9 q^{71} +49694.2i q^{73} +(54247.5 - 24224.6i) q^{75} +26422.2i q^{77} +98240.4 q^{79} -73801.8 q^{81} +94470.1i q^{83} +(-28575.2 + 44055.4i) q^{85} +75298.0i q^{87} -98225.5 q^{89} +10205.0 q^{91} +145009. i q^{93} +(-96393.1 - 62522.5i) q^{95} +59010.4i q^{97} +34376.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 60 q^{5} - 1676 q^{9}+O(q^{10}) \) 12 * q + 60 * q^5 - 1676 * q^9	\( 12 q + 60 q^{5} - 1676 q^{9} - 688 q^{21} + 1500 q^{25} + 21304 q^{29} - 109672 q^{41} + 75140 q^{45} - 32348 q^{49} - 64552 q^{61} - 160 q^{65} + 166352 q^{69} - 173764 q^{81} - 151360 q^{85} - 3720 q^{89}+O(q^{100}) \) 12 * q + 60 * q^5 - 1676 * q^9 - 688 * q^21 + 1500 * q^25 + 21304 * q^29 - 109672 * q^41 + 75140 * q^45 - 32348 * q^49 - 64552 * q^61 - 160 * q^65 + 166352 * q^69 - 173764 * q^81 - 151360 * q^85 - 3720 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more