LMFDB - Embedded newform 684.5.h.f.37.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,5,Mod(37,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("684.37");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	684.h (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:	$70.7050547493$
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} - 2 x^{13} + 2574 x^{12} - 7948 x^{11} + 5136095 x^{10} - 4313010 x^{9} + 3526383758 x^{8} + \cdots + 13\!\cdots\!00 $ x^14 - 2x^13 + 2574x^12 - 7948x^11 + 5136095x^10 - 4313010x^9 + 3526383758x^8 + 42678039416x^7 + 1723219853217x^6 + 14955695567410x^5 + 301377432088480x^4 + 2343588994038264x^3 + 36239668281491904x^2 + 197066822628281760*x + 1359248846310158400
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{13} $
Twist minimal:	no (minimal twist has level 228)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.8
Root			$-4.56664 - 7.90965i$ of defining polynomial
Character	$\chi$	$=$	684.37
Dual form			684.5.h.f.37.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.13328 q^{5} -13.7213 q^{7} +O(q^{10})$	$q+8.13328 q^{5} -13.7213 q^{7} -214.704 q^{11} +116.434i q^{13} +396.330 q^{17} +(-24.2826 + 360.182i) q^{19} +541.165 q^{23} -558.850 q^{25} -249.019i q^{29} -416.965i q^{31} -111.599 q^{35} -2606.03i q^{37} -1044.37i q^{41} -1732.69 q^{43} +3608.58 q^{47} -2212.73 q^{49} -4226.68i q^{53} -1746.25 q^{55} +6182.61i q^{59} +3146.71 q^{61} +946.992i q^{65} +5599.68i q^{67} -7566.11i q^{71} -6233.92 q^{73} +2946.01 q^{77} +2164.36i q^{79} +2817.26 q^{83} +3223.46 q^{85} -12998.9i q^{89} -1597.62i q^{91} +(-197.498 + 2929.46i) q^{95} -11137.1i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 18 q^{5} - 86 q^{7}+O(q^{10}) $ 14 * q - 18 * q^5 - 86 * q^7	$ 14 q - 18 q^{5} - 86 q^{7} - 258 q^{11} - 498 q^{17} + 170 q^{19} + 588 q^{23} + 1560 q^{25} - 534 q^{35} + 1882 q^{43} + 222 q^{47} + 4104 q^{49} + 2702 q^{55} - 2462 q^{61} - 5774 q^{73} + 4578 q^{77} - 17988 q^{83} + 2342 q^{85} + 18270 q^{95}+O(q^{100}) $ 14 * q - 18 * q^5 - 86 * q^7 - 258 * q^11 - 498 * q^17 + 170 * q^19 + 588 * q^23 + 1560 * q^25 - 534 * q^35 + 1882 * q^43 + 222 * q^47 + 4104 * q^49 + 2702 * q^55 - 2462 * q^61 - 5774 * q^73 + 4578 * q^77 - 17988 * q^83 + 2342 * q^85 + 18270 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	8.13328			0.325331			0.162666	−	0.986681i	$-0.447991\pi$
$5$	8.13328			0.325331			0.162666	+	0.986681i	$0.447991\pi$
$6$		0			0
$7$	−13.7213			−0.280026			−0.140013	−	0.990150i	$-0.544714\pi$
$7$	−13.7213			−0.280026			−0.140013	+	0.990150i	$0.544714\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−214.704			−1.77441			−0.887207	−	0.461372i	$-0.847357\pi$
$11$	−214.704			−1.77441			−0.887207	+	0.461372i	$0.847357\pi$
$12$		0			0
$13$			116.434i			0.688960i	0.938794	+	0.344480i	$0.111945\pi$
$13$			116.434i			0.688960i	−0.938794	+	0.344480i	$0.888055\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	396.330			1.37138			0.685691	−	0.727893i	$-0.259500\pi$
$17$	396.330			1.37138			0.685691	+	0.727893i	$0.259500\pi$
$18$		0			0
$19$	−24.2826	+	360.182i	−0.0672649	+	0.997735i
$19$	−24.2826	+	360.182i	−0.0672649	+	0.997735i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	541.165			1.02300			0.511498	−	0.859285i	$-0.329091\pi$
$23$	541.165			1.02300			0.511498	+	0.859285i	$0.329091\pi$
$24$		0			0
$25$	−558.850			−0.894160
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	249.019i		−	0.296099i	−0.988980	−	0.148049i	$-0.952701\pi$
$29$		−	249.019i		−	0.296099i	0.988980	−	0.148049i	$-0.0472994\pi$
$30$		0			0
$31$		−	416.965i		−	0.433886i	−0.976184	−	0.216943i	$-0.930391\pi$
$31$		−	416.965i		−	0.433886i	0.976184	−	0.216943i	$-0.0696086\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−111.599			−0.0911010
$36$		0			0
$37$		−	2606.03i		−	1.90360i	−0.306720	−	0.951800i	$-0.599231\pi$
$37$		−	2606.03i		−	1.90360i	0.306720	−	0.951800i	$-0.400769\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1044.37i		−	0.621278i	−0.950528	−	0.310639i	$-0.899457\pi$
$41$		−	1044.37i		−	0.621278i	0.950528	−	0.310639i	$-0.100543\pi$
$42$		0			0
$43$	−1732.69			−0.937097			−0.468548	−	0.883438i	$-0.655223\pi$
$43$	−1732.69			−0.937097			−0.468548	+	0.883438i	$0.655223\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3608.58			1.63358			0.816791	−	0.576934i	$-0.195751\pi$
$47$	3608.58			1.63358			0.816791	+	0.576934i	$0.195751\pi$
$48$		0			0
$49$	−2212.73			−0.921586
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4226.68i		−	1.50469i	−0.658769	−	0.752346i	$-0.728922\pi$
$53$		−	4226.68i		−	1.50469i	0.658769	−	0.752346i	$-0.271078\pi$
$54$		0			0
$55$	−1746.25			−0.577272
$56$		0			0
$57$		0			0
$58$		0			0
$59$			6182.61i			1.77610i	0.459746	+	0.888050i	$0.347940\pi$
$59$			6182.61i			1.77610i	−0.459746	+	0.888050i	$0.652060\pi$
$60$		0			0
$61$	3146.71			0.845664			0.422832	−	0.906208i	$-0.361036\pi$
$61$	3146.71			0.845664			0.422832	+	0.906208i	$0.361036\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			946.992i			0.224140i
$66$		0			0
$67$			5599.68i			1.24742i	0.781655	+	0.623711i	$0.214376\pi$
$67$			5599.68i			1.24742i	−0.781655	+	0.623711i	$0.785624\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7566.11i		−	1.50091i	−0.660919	−	0.750457i	$-0.729833\pi$
$71$		−	7566.11i		−	1.50091i	0.660919	−	0.750457i	$-0.270167\pi$
$72$		0			0
$73$	−6233.92			−1.16981			−0.584905	−	0.811101i	$-0.698868\pi$
$73$	−6233.92			−1.16981			−0.584905	+	0.811101i	$0.698868\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2946.01			0.496881
$78$		0			0
$79$			2164.36i			0.346797i	0.984852	+	0.173398i	$0.0554748\pi$
$79$			2164.36i			0.346797i	−0.984852	+	0.173398i	$0.944525\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2817.26			0.408951			0.204475	−	0.978872i	$-0.434451\pi$
$83$	2817.26			0.408951			0.204475	+	0.978872i	$0.434451\pi$
$84$		0			0
$85$	3223.46			0.446153
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	12998.9i		−	1.64106i	−0.571600	−	0.820532i	$-0.693677\pi$
$89$		−	12998.9i		−	1.64106i	0.571600	−	0.820532i	$-0.306323\pi$
$90$		0			0
$91$		−	1597.62i		−	0.192926i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−197.498	+	2929.46i	−0.0218834	+	0.324594i
$96$		0			0
$97$		−	11137.1i		−	1.18367i	−0.806061	−	0.591833i	$-0.798405\pi$
$97$		−	11137.1i		−	1.18367i	0.806061	−	0.591833i	$-0.201595\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13509.1			−1.32430			−0.662148	−	0.749373i	$-0.730355\pi$
$101$	−13509.1			−1.32430			−0.662148	+	0.749373i	$0.730355\pi$
$102$		0			0
$103$		−	17783.0i		−	1.67622i	−0.545500	−	0.838111i	$-0.683660\pi$
$103$		−	17783.0i		−	1.67622i	0.545500	−	0.838111i	$-0.316340\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	14343.4i		−	1.25281i	−0.779499	−	0.626403i	$-0.784526\pi$
$107$		−	14343.4i		−	1.25281i	0.779499	−	0.626403i	$-0.215474\pi$
$108$		0			0
$109$			9397.44i			0.790964i	0.918474	+	0.395482i	$0.129422\pi$
$109$			9397.44i			0.790964i	−0.918474	+	0.395482i	$0.870578\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	3571.79i		−	0.279724i	−0.990171	−	0.139862i	$-0.955334\pi$
$113$		−	3571.79i		−	0.279724i	0.990171	−	0.139862i	$-0.0446659\pi$
$114$		0			0
$115$	4401.44			0.332812
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5438.14			−0.384022
$120$		0			0
$121$	31456.8			2.14854
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9628.58			−0.616229
$126$		0			0
$127$		−	18513.2i		−	1.14782i	−0.818918	−	0.573911i	$-0.805426\pi$
$127$		−	18513.2i		−	1.14782i	0.818918	−	0.573911i	$-0.194574\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1295.76			−0.0755063			−0.0377532	−	0.999287i	$-0.512020\pi$
$131$	−1295.76			−0.0755063			−0.0377532	+	0.999287i	$0.512020\pi$
$132$		0			0
$133$	333.188	−	4942.15i	0.0188359	−	0.279391i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1252.14			−0.0667130			−0.0333565	−	0.999444i	$-0.510620\pi$
$137$	−1252.14			−0.0667130			−0.0333565	+	0.999444i	$0.510620\pi$
$138$		0			0
$139$	−30565.5			−1.58198			−0.790992	−	0.611826i	$-0.790435\pi$
$139$	−30565.5			−1.58198			−0.790992	+	0.611826i	$0.790435\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	24998.9i		−	1.22250i
$144$		0			0
$145$		−	2025.34i		−	0.0963301i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−25079.6			−1.12966			−0.564831	−	0.825206i	$-0.691059\pi$
$149$	−25079.6			−1.12966			−0.564831	+	0.825206i	$0.691059\pi$
$150$		0			0
$151$			13110.5i			0.574997i	0.957781	+	0.287499i	$0.0928237\pi$
$151$			13110.5i			0.574997i	−0.957781	+	0.287499i	$0.907176\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	3391.29i		−	0.141157i
$156$		0			0
$157$	19102.1			0.774963			0.387481	−	0.921878i	$-0.373345\pi$
$157$	19102.1			0.774963			0.387481	+	0.921878i	$0.373345\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7425.46			−0.286465
$162$		0			0
$163$	36864.3			1.38749			0.693746	−	0.720220i	$-0.255959\pi$
$163$	36864.3			1.38749			0.693746	+	0.720220i	$0.255959\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	9595.83i		−	0.344072i	−0.985091	−	0.172036i	$-0.944965\pi$
$167$		−	9595.83i		−	0.344072i	0.985091	−	0.172036i	$-0.0550346\pi$
$168$		0			0
$169$	15004.1			0.525334
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	37043.4i		−	1.23771i	−0.785506	−	0.618854i	$-0.787597\pi$
$173$		−	37043.4i		−	1.23771i	0.785506	−	0.618854i	$-0.212403\pi$
$174$		0			0
$175$	7668.12			0.250388
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	19305.2i		−	0.602516i	−0.953543	−	0.301258i	$-0.902593\pi$
$179$		−	19305.2i		−	0.602516i	0.953543	−	0.301258i	$-0.0974065\pi$
$180$		0			0
$181$		−	40760.2i		−	1.24417i	−0.782950	−	0.622085i	$-0.786286\pi$
$181$		−	40760.2i		−	1.24417i	0.782950	−	0.622085i	$-0.213714\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	21195.6i		−	0.619300i
$186$		0			0
$187$	−85093.5			−2.43340
$188$		0			0
$189$		0			0
$190$		0			0
$191$	31160.5			0.854158			0.427079	−	0.904214i	$-0.359543\pi$
$191$	31160.5			0.854158			0.427079	+	0.904214i	$0.359543\pi$
$192$		0			0
$193$		−	45206.1i		−	1.21362i	−0.794847	−	0.606810i	$-0.792449\pi$
$193$		−	45206.1i		−	1.21362i	0.794847	−	0.606810i	$-0.207551\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	19964.6			0.514433			0.257217	−	0.966354i	$-0.417195\pi$
$197$	19964.6			0.514433			0.257217	+	0.966354i	$0.417195\pi$
$198$		0			0
$199$	15591.4			0.393713			0.196856	−	0.980432i	$-0.436927\pi$
$199$	15591.4			0.393713			0.196856	+	0.980432i	$0.436927\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			3416.85i			0.0829152i
$204$		0			0
$205$		−	8494.14i		−	0.202121i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5213.58	−	77332.6i	0.119356	−	1.77039i
$210$		0			0
$211$			74756.1i			1.67912i	0.543267	+	0.839560i	$0.317187\pi$
$211$			74756.1i			1.67912i	−0.543267	+	0.839560i	$0.682813\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−14092.5			−0.304867
$216$		0			0
$217$			5721.28i			0.121499i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			46146.3i			0.944827i
$222$		0			0
$223$		−	74052.1i		−	1.48911i	−0.667559	−	0.744557i	$-0.732661\pi$
$223$		−	74052.1i		−	1.48911i	0.667559	−	0.744557i	$-0.267339\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			9831.00i			0.190786i	0.995440	+	0.0953929i	$0.0304107\pi$
$227$			9831.00i			0.190786i	−0.995440	+	0.0953929i	$0.969589\pi$
$228$		0			0
$229$	−44215.9			−0.843155			−0.421578	−	0.906792i	$-0.638524\pi$
$229$	−44215.9			−0.843155			−0.421578	+	0.906792i	$0.638524\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−50311.4			−0.926732			−0.463366	−	0.886167i	$-0.653359\pi$
$233$	−50311.4			−0.926732			−0.463366	+	0.886167i	$0.653359\pi$
$234$		0			0
$235$	29349.6			0.531455
$236$		0			0
$237$		0			0
$238$		0			0
$239$	62652.5			1.09684			0.548419	−	0.836204i	$-0.315230\pi$
$239$	62652.5			1.09684			0.548419	+	0.836204i	$0.315230\pi$
$240$		0			0
$241$			68905.7i			1.18637i	0.805065	+	0.593186i	$0.202130\pi$
$241$			68905.7i			1.18637i	−0.805065	+	0.593186i	$0.797870\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−17996.7			−0.299821
$246$		0			0
$247$	−41937.6	−	2827.33i	−0.687400	−	0.0463429i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	107091.			1.69983			0.849917	−	0.526917i	$-0.176652\pi$
$251$	107091.			1.69983			0.849917	+	0.526917i	$0.176652\pi$
$252$		0			0
$253$	−116190.			−1.81522
$254$		0			0
$255$		0			0
$256$		0			0
$257$			49327.7i			0.746835i	0.927663	+	0.373418i	$0.121814\pi$
$257$			49327.7i			0.746835i	−0.927663	+	0.373418i	$0.878186\pi$
$258$		0			0
$259$			35758.0i			0.533057i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−20318.6			−0.293753			−0.146877	−	0.989155i	$-0.546922\pi$
$263$	−20318.6			−0.293753			−0.146877	+	0.989155i	$0.546922\pi$
$264$		0			0
$265$		−	34376.8i		−	0.489523i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			105240.i			1.45437i	0.686442	+	0.727184i	$0.259171\pi$
$269$			105240.i			1.45437i	−0.686442	+	0.727184i	$0.740829\pi$
$270$		0			0
$271$	−90904.2			−1.23779			−0.618893	−	0.785475i	$-0.712419\pi$
$271$	−90904.2			−1.23779			−0.618893	+	0.785475i	$0.712419\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	119987.			1.58661
$276$		0			0
$277$	145046.			1.89036			0.945181	−	0.326547i	$-0.105885\pi$
$277$	145046.			1.89036			0.945181	+	0.326547i	$0.105885\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	67150.2i		−	0.850422i	−0.905094	−	0.425211i	$-0.860200\pi$
$281$		−	67150.2i		−	0.850422i	0.905094	−	0.425211i	$-0.139800\pi$
$282$		0			0
$283$	−58821.9			−0.734457			−0.367229	−	0.930131i	$-0.619693\pi$
$283$	−58821.9			−0.734457			−0.367229	+	0.930131i	$0.619693\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			14330.0i			0.173974i
$288$		0			0
$289$	73556.1			0.880690
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	111403.i		−	1.29766i	−0.760935	−	0.648828i	$-0.775259\pi$
$293$		−	111403.i		−	1.29766i	0.760935	−	0.648828i	$-0.224741\pi$
$294$		0			0
$295$			50284.9i			0.577821i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			63010.1i			0.704803i
$300$		0			0
$301$	23774.7			0.262411
$302$		0			0
$303$		0			0
$304$		0			0
$305$	25593.1			0.275121
$306$		0			0
$307$		−	43727.9i		−	0.463961i	−0.972720	−	0.231981i	$-0.925479\pi$
$307$		−	43727.9i		−	0.463961i	0.972720	−	0.231981i	$-0.0745206\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−115389.			−1.19301			−0.596507	−	0.802608i	$-0.703445\pi$
$311$	−115389.			−1.19301			−0.596507	+	0.802608i	$0.703445\pi$
$312$		0			0
$313$	15878.3			0.162075			0.0810376	−	0.996711i	$-0.474177\pi$
$313$	15878.3			0.162075			0.0810376	+	0.996711i	$0.474177\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	103989.i		−	1.03483i	−0.855735	−	0.517414i	$-0.826895\pi$
$317$		−	103989.i		−	1.03483i	0.855735	−	0.517414i	$-0.173105\pi$
$318$		0			0
$319$			53465.4i			0.525401i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−9623.93	+	142751.i	−0.0922460	+	1.36828i
$324$		0			0
$325$		−	65069.2i		−	0.616040i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−49514.3			−0.457445
$330$		0			0
$331$			540.007i			0.00492883i	0.999997	+	0.00246441i	$0.000784448\pi$
$331$			540.007i			0.00492883i	−0.999997	+	0.00246441i	$0.999216\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			45543.7i			0.405825i
$336$		0			0
$337$			148740.i			1.30969i	0.755765	+	0.654843i	$0.227265\pi$
$337$			148740.i			1.30969i	−0.755765	+	0.654843i	$0.772735\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			89524.0i			0.769893i
$342$		0			0
$343$	63306.1			0.538093
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−98774.9			−0.820328			−0.410164	−	0.912012i	$-0.634529\pi$
$347$	−98774.9			−0.820328			−0.410164	+	0.912012i	$0.634529\pi$
$348$		0			0
$349$	−17880.5			−0.146801			−0.0734005	−	0.997303i	$-0.523385\pi$
$349$	−17880.5			−0.146801			−0.0734005	+	0.997303i	$0.523385\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	65400.7			0.524848			0.262424	−	0.964953i	$-0.415478\pi$
$353$	65400.7			0.524848			0.262424	+	0.964953i	$0.415478\pi$
$354$		0			0
$355$		−	61537.3i		−	0.488294i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−126291.			−0.979906			−0.489953	−	0.871749i	$-0.662986\pi$
$359$	−126291.			−0.979906			−0.489953	+	0.871749i	$0.662986\pi$
$360$		0			0
$361$	−129142.	−	17492.4i	−0.990951	−	0.134225i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−50702.2			−0.380576
$366$		0			0
$367$	65887.2			0.489180			0.244590	−	0.969627i	$-0.421347\pi$
$367$	65887.2			0.489180			0.244590	+	0.969627i	$0.421347\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			57995.3i			0.421352i
$372$		0			0
$373$		−	135411.i		−	0.973279i	−0.873603	−	0.486640i	$-0.838223\pi$
$373$		−	135411.i		−	0.973279i	0.873603	−	0.486640i	$-0.161777\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	28994.3			0.204000
$378$		0			0
$379$			104824.i			0.729762i	0.931054	+	0.364881i	$0.118890\pi$
$379$			104824.i			0.729762i	−0.931054	+	0.364881i	$0.881110\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	15810.6i		−	0.107783i	−0.998547	−	0.0538917i	$-0.982837\pi$
$383$		−	15810.6i		−	0.107783i	0.998547	−	0.0538917i	$-0.0171626\pi$
$384$		0			0
$385$	23960.7			0.161651
$386$		0			0
$387$		0			0
$388$		0			0
$389$	17585.1			0.116210			0.0581052	−	0.998310i	$-0.481494\pi$
$389$	17585.1			0.116210			0.0581052	+	0.998310i	$0.481494\pi$
$390$		0			0
$391$	214480.			1.40292
$392$		0			0
$393$		0			0
$394$		0			0
$395$			17603.3i			0.112824i
$396$		0			0
$397$	242133.			1.53629			0.768143	−	0.640278i	$-0.221181\pi$
$397$	242133.			1.53629			0.768143	+	0.640278i	$0.221181\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			87965.1i			0.547043i	0.961866	+	0.273522i	$0.0881885\pi$
$401$			87965.1i			0.547043i	−0.961866	+	0.273522i	$0.911811\pi$
$402$		0			0
$403$	48549.0			0.298930
$404$		0			0
$405$		0			0
$406$		0			0
$407$			559525.i			3.37777i
$408$		0			0
$409$			101645.i			0.607629i	0.952731	+	0.303815i	$0.0982603\pi$
$409$			101645.i			0.607629i	−0.952731	+	0.303815i	$0.901740\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	84833.1i		−	0.497353i
$414$		0			0
$415$	22913.6			0.133044
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16719.9			0.0952369			0.0476185	−	0.998866i	$-0.484837\pi$
$419$	16719.9			0.0952369			0.0476185	+	0.998866i	$0.484837\pi$
$420$		0			0
$421$			168414.i			0.950201i	0.879932	+	0.475100i	$0.157588\pi$
$421$			168414.i			0.950201i	−0.879932	+	0.475100i	$0.842412\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−221489.			−1.22623
$426$		0			0
$427$	−43176.9			−0.236807
$428$		0			0
$429$		0			0
$430$		0			0
$431$			146952.i			0.791079i	0.918449	+	0.395539i	$0.129442\pi$
$431$			146952.i			0.791079i	−0.918449	+	0.395539i	$0.870558\pi$
$432$		0			0
$433$			139872.i			0.746030i	0.927825	+	0.373015i	$0.121676\pi$
$433$			139872.i			0.746030i	−0.927825	+	0.373015i	$0.878324\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−13140.9	+	194918.i	−0.0688118	+	1.02068i
$438$		0			0
$439$			306182.i			1.58873i	0.607440	+	0.794366i	$0.292197\pi$
$439$			306182.i			1.58873i	−0.607440	+	0.794366i	$0.707803\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	17455.2			0.0889443			0.0444722	−	0.999011i	$-0.485839\pi$
$443$	17455.2			0.0889443			0.0444722	+	0.999011i	$0.485839\pi$
$444$		0			0
$445$		−	105723.i		−	0.533889i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			59146.0i			0.293381i	0.989182	+	0.146691i	$0.0468622\pi$
$449$			59146.0i			0.293381i	−0.989182	+	0.146691i	$0.953138\pi$
$450$		0			0
$451$			224230.i			1.10240i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	12993.9i		−	0.0627650i
$456$		0			0
$457$	−355441.			−1.70190			−0.850952	−	0.525244i	$-0.823974\pi$
$457$	−355441.			−1.70190			−0.850952	+	0.525244i	$0.823974\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	51628.7			0.242935			0.121467	−	0.992595i	$-0.461240\pi$
$461$	51628.7			0.242935			0.121467	+	0.992595i	$0.461240\pi$
$462$		0			0
$463$	82045.7			0.382731			0.191366	−	0.981519i	$-0.438708\pi$
$463$	82045.7			0.382731			0.191366	+	0.981519i	$0.438708\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−100428.			−0.460490			−0.230245	−	0.973133i	$-0.573953\pi$
$467$	−100428.			−0.460490			−0.230245	+	0.973133i	$0.573953\pi$
$468$		0			0
$469$		−	76834.6i		−	0.349310i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	372016.			1.66280
$474$		0			0
$475$	13570.4	−	201288.i	0.0601456	−	0.892135i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−36946.0			−0.161026			−0.0805130	−	0.996754i	$-0.525656\pi$
$479$	−36946.0			−0.161026			−0.0805130	+	0.996754i	$0.525656\pi$
$480$		0			0
$481$	303431.			1.31150
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	90581.2i		−	0.385083i
$486$		0			0
$487$		−	75737.6i		−	0.319340i	−0.987170	−	0.159670i	$-0.948957\pi$
$487$		−	75737.6i		−	0.319340i	0.987170	−	0.159670i	$-0.0510431\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−82125.1			−0.340653			−0.170327	−	0.985388i	$-0.554482\pi$
$491$	−82125.1			−0.340653			−0.170327	+	0.985388i	$0.554482\pi$
$492$		0			0
$493$		−	98693.5i		−	0.406064i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			103817.i			0.420294i
$498$		0			0
$499$	−77351.6			−0.310648			−0.155324	−	0.987864i	$-0.549642\pi$
$499$	−77351.6			−0.310648			−0.155324	+	0.987864i	$0.549642\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−92824.8			−0.366883			−0.183442	−	0.983031i	$-0.558724\pi$
$503$	−92824.8			−0.366883			−0.183442	+	0.983031i	$0.558724\pi$
$504$		0			0
$505$	−109874.			−0.430835
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	204620.i		−	0.789792i	−0.918726	−	0.394896i	$-0.870781\pi$
$509$		−	204620.i		−	0.789792i	0.918726	−	0.394896i	$-0.129219\pi$
$510$		0			0
$511$	85537.2			0.327577
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	144634.i		−	0.545327i
$516$		0			0
$517$	−774777.			−2.89865
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	180623.i		−	0.665424i	−0.943029	−	0.332712i	$-0.892036\pi$
$521$		−	180623.i		−	0.665424i	0.943029	−	0.332712i	$-0.107964\pi$
$522$		0			0
$523$		−	217248.i		−	0.794240i	−0.917767	−	0.397120i	$-0.870010\pi$
$523$		−	217248.i		−	0.794240i	0.917767	−	0.397120i	$-0.129990\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	165255.i		−	0.595024i
$528$		0			0
$529$	13018.3			0.0465204
$530$		0			0
$531$		0			0
$532$		0			0
$533$	121600.			0.428036
$534$		0			0
$535$		−	116659.i		−	0.407577i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	475081.			1.63527
$540$		0			0
$541$	−64299.5			−0.219691			−0.109846	−	0.993949i	$-0.535036\pi$
$541$	−64299.5			−0.219691			−0.109846	+	0.993949i	$0.535036\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			76432.0i			0.257325i
$546$		0			0
$547$			436288.i			1.45814i	0.684440	+	0.729069i	$0.260047\pi$
$547$			436288.i			1.45814i	−0.684440	+	0.729069i	$0.739953\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	89692.2	+	6046.84i	0.295428	+	0.0199171i
$552$		0			0
$553$		−	29697.7i		−	0.0971119i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	530935.			1.71132			0.855659	−	0.517539i	$-0.173152\pi$
$557$	530935.			1.71132			0.855659	+	0.517539i	$0.173152\pi$
$558$		0			0
$559$		−	201745.i		−	0.645622i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			499386.i			1.57551i	0.615991	+	0.787753i	$0.288756\pi$
$563$			499386.i			1.57551i	−0.615991	+	0.787753i	$0.711244\pi$
$564$		0			0
$565$		−	29050.4i		−	0.0910029i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	292016.i		−	0.901949i	−0.892537	−	0.450975i	$-0.851076\pi$
$569$		−	292016.i		−	0.901949i	0.892537	−	0.450975i	$-0.148924\pi$
$570$		0			0
$571$	361137.			1.10764			0.553822	−	0.832635i	$-0.313169\pi$
$571$	361137.			1.10764			0.553822	+	0.832635i	$0.313169\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−302430.			−0.914722
$576$		0			0
$577$	−328994.			−0.988180			−0.494090	−	0.869411i	$-0.664499\pi$
$577$	−328994.			−0.988180			−0.494090	+	0.869411i	$0.664499\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−38656.3			−0.114517
$582$		0			0
$583$			907485.i			2.66994i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	87176.3			0.253001			0.126501	−	0.991967i	$-0.459625\pi$
$587$	87176.3			0.253001			0.126501	+	0.991967i	$0.459625\pi$
$588$		0			0
$589$	150183.	+	10125.0i	0.432904	+	0.0291853i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−480588.			−1.36667			−0.683335	−	0.730105i	$-0.739471\pi$
$593$	−480588.			−1.36667			−0.683335	+	0.730105i	$0.739471\pi$
$594$		0			0
$595$	−44229.9			−0.124934
$596$		0			0
$597$		0			0
$598$		0			0
$599$			389059.i			1.08433i	0.840272	+	0.542166i	$0.182395\pi$
$599$			389059.i			1.08433i	−0.840272	+	0.542166i	$0.817605\pi$
$600$		0			0
$601$		−	462636.i		−	1.28083i	−0.768030	−	0.640413i	$-0.778763\pi$
$601$		−	462636.i		−	1.28083i	0.768030	−	0.640413i	$-0.221237\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	255847.			0.698988
$606$		0			0
$607$			36769.4i			0.0997952i	0.998754	+	0.0498976i	$0.0158895\pi$
$607$			36769.4i			0.0997952i	−0.998754	+	0.0498976i	$0.984111\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			420162.i			1.12547i
$612$		0			0
$613$	−173851.			−0.462655			−0.231327	−	0.972876i	$-0.574307\pi$
$613$	−173851.			−0.462655			−0.231327	+	0.972876i	$0.574307\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	339821.			0.892648			0.446324	−	0.894871i	$-0.352733\pi$
$617$	339821.			0.892648			0.446324	+	0.894871i	$0.352733\pi$
$618$		0			0
$619$	−718519.			−1.87524			−0.937620	−	0.347663i	$-0.886975\pi$
$619$	−718519.			−1.87524			−0.937620	+	0.347663i	$0.886975\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			178361.i			0.459540i
$624$		0			0
$625$	270969.			0.693681
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.03285e6i		−	2.61056i
$630$		0			0
$631$	311326.			0.781910			0.390955	−	0.920410i	$-0.372145\pi$
$631$	311326.			0.781910			0.390955	+	0.920410i	$0.372145\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	150573.i		−	0.373422i
$636$		0			0
$637$		−	257637.i		−	0.634936i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	514998.i		−	1.25340i	−0.779261	−	0.626699i	$-0.784405\pi$
$641$		−	514998.i		−	1.25340i	0.779261	−	0.626699i	$-0.215595\pi$
$642$		0			0
$643$	−142162.			−0.343845			−0.171922	−	0.985110i	$-0.554998\pi$
$643$	−142162.			−0.343845			−0.171922	+	0.985110i	$0.554998\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	169979.			0.406057			0.203028	−	0.979173i	$-0.434922\pi$
$647$	169979.			0.406057			0.203028	+	0.979173i	$0.434922\pi$
$648$		0			0
$649$		−	1.32743e6i		−	3.15154i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−292912.			−0.686927			−0.343464	−	0.939166i	$-0.611600\pi$
$653$	−292912.			−0.686927			−0.343464	+	0.939166i	$0.611600\pi$
$654$		0			0
$655$	−10538.8			−0.0245646
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	385655.i		−	0.888031i	−0.896019	−	0.444015i	$-0.853554\pi$
$659$		−	385655.i		−	0.888031i	0.896019	−	0.444015i	$-0.146446\pi$
$660$		0			0
$661$			127679.i			0.292223i	0.989268	+	0.146112i	$0.0466759\pi$
$661$			127679.i			0.292223i	−0.989268	+	0.146112i	$0.953324\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2709.91	−	40195.9i	0.00612791	−	0.0908947i
$666$		0			0
$667$		−	134760.i		−	0.302908i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−675612.			−1.50056
$672$		0			0
$673$			317504.i			0.701003i	0.936562	+	0.350501i	$0.113989\pi$
$673$			317504.i			0.701003i	−0.936562	+	0.350501i	$0.886011\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	52934.6i		−	0.115495i	−0.998331	−	0.0577474i	$-0.981608\pi$
$677$		−	52934.6i		−	0.115495i	0.998331	−	0.0577474i	$-0.0183918\pi$
$678$		0			0
$679$			152815.i			0.331456i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	178002.i		−	0.381578i	−0.981631	−	0.190789i	$-0.938895\pi$
$683$		−	178002.i		−	0.381578i	0.981631	−	0.190789i	$-0.0611047\pi$
$684$		0			0
$685$	−10184.0			−0.0217038
$686$		0			0
$687$		0			0
$688$		0			0
$689$	492130.			1.03667
$690$		0			0
$691$	−93370.8			−0.195549			−0.0977743	−	0.995209i	$-0.531172\pi$
$691$	−93370.8			−0.195549			−0.0977743	+	0.995209i	$0.531172\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−248598.			−0.514669
$696$		0			0
$697$		−	413914.i		−	0.852010i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−311636.			−0.634179			−0.317089	−	0.948396i	$-0.602705\pi$
$701$	−311636.			−0.634179			−0.317089	+	0.948396i	$0.602705\pi$
$702$		0			0
$703$	938645.	+	63281.3i	1.89929	+	0.128046i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	185362.			0.370837
$708$		0			0
$709$	546619.			1.08741			0.543704	−	0.839277i	$-0.317021\pi$
$709$	546619.			1.08741			0.543704	+	0.839277i	$0.317021\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	225647.i		−	0.443864i
$714$		0			0
$715$		−	203323.i		−	0.397717i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−215257.			−0.416390			−0.208195	−	0.978087i	$-0.566759\pi$
$719$	−215257.			−0.416390			−0.208195	+	0.978087i	$0.566759\pi$
$720$		0			0
$721$			244006.i			0.469385i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			139164.i			0.264759i
$726$		0			0
$727$	607019.			1.14851			0.574253	−	0.818678i	$-0.305292\pi$
$727$	607019.			1.14851			0.574253	+	0.818678i	$0.305292\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−686717.			−1.28512
$732$		0			0
$733$	−180665.			−0.336253			−0.168126	−	0.985765i	$-0.553772\pi$
$733$	−180665.			−0.336253			−0.168126	+	0.985765i	$0.553772\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1.20227e6i		−	2.21344i
$738$		0			0
$739$	−697195.			−1.27663			−0.638315	−	0.769775i	$-0.720368\pi$
$739$	−697195.			−1.27663			−0.638315	+	0.769775i	$0.720368\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	597293.i		−	1.08196i	−0.841037	−	0.540978i	$-0.818054\pi$
$743$		−	597293.i		−	1.08196i	0.841037	−	0.540978i	$-0.181946\pi$
$744$		0			0
$745$	−203980.			−0.367514
$746$		0			0
$747$		0			0
$748$		0			0
$749$			196809.i			0.350818i
$750$		0			0
$751$		−	491817.i		−	0.872014i	−0.899943	−	0.436007i	$-0.856392\pi$
$751$		−	491817.i		−	0.872014i	0.899943	−	0.436007i	$-0.143608\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			106631.i			0.187065i
$756$		0			0
$757$	531456.			0.927417			0.463709	−	0.885988i	$-0.346518\pi$
$757$	531456.			0.927417			0.463709	+	0.885988i	$0.346518\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−725369.			−1.25253			−0.626267	−	0.779609i	$-0.715418\pi$
$761$	−725369.			−1.25253			−0.626267	+	0.779609i	$0.715418\pi$
$762$		0			0
$763$		−	128945.i		−	0.221490i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−719867.			−1.22366
$768$		0			0
$769$	−222021.			−0.375440			−0.187720	−	0.982223i	$-0.560110\pi$
$769$	−222021.			−0.375440			−0.187720	+	0.982223i	$0.560110\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			25512.9i			0.0426974i	0.999772	+	0.0213487i	$0.00679601\pi$
$773$			25512.9i			0.0426974i	−0.999772	+	0.0213487i	$0.993204\pi$
$774$		0			0
$775$			233021.i			0.387964i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	376163.	+	25360.0i	0.619871	+	0.0417902i
$780$		0			0
$781$			1.62447e6i			2.66324i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	155362.			0.252120
$786$		0			0
$787$		−	259729.i		−	0.419344i	−0.977772	−	0.209672i	$-0.932760\pi$
$787$		−	259729.i		−	0.419344i	0.977772	−	0.209672i	$-0.0672397\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			49009.5i			0.0783298i
$792$		0			0
$793$			366385.i			0.582628i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	376571.i		−	0.592830i	−0.955059	−	0.296415i	$-0.904209\pi$
$797$		−	376571.i		−	0.592830i	0.955059	−	0.296415i	$-0.0957912\pi$
$798$		0			0
$799$	1.43019e6			2.24026
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.33845e6			2.07573
$804$		0			0
$805$	−60393.3			−0.0931960
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−60963.5			−0.0931478			−0.0465739	−	0.998915i	$-0.514830\pi$
$809$	−60963.5			−0.0931478			−0.0465739	+	0.998915i	$0.514830\pi$
$810$		0			0
$811$		−	1.00210e6i		−	1.52359i	−0.647818	−	0.761795i	$-0.724318\pi$
$811$		−	1.00210e6i		−	1.52359i	0.647818	−	0.761795i	$-0.275682\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	299827.			0.451394
$816$		0			0
$817$	42074.3	−	624085.i	0.0630338	−	0.934974i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−918552.			−1.36275			−0.681377	−	0.731933i	$-0.738619\pi$
$821$	−918552.			−1.36275			−0.681377	+	0.731933i	$0.738619\pi$
$822$		0			0
$823$	−297161.			−0.438725			−0.219363	−	0.975643i	$-0.570398\pi$
$823$	−297161.			−0.438725			−0.219363	+	0.975643i	$0.570398\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			276553.i			0.404359i	0.979348	+	0.202180i	$0.0648025\pi$
$827$			276553.i			0.404359i	−0.979348	+	0.202180i	$0.935197\pi$
$828$		0			0
$829$		−	515453.i		−	0.750033i	−0.927018	−	0.375016i	$-0.877637\pi$
$829$		−	515453.i		−	0.750033i	0.927018	−	0.375016i	$-0.122363\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−876969.			−1.26385
$834$		0			0
$835$		−	78045.6i		−	0.111937i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			749151.i			1.06425i	0.846664	+	0.532127i	$0.178607\pi$
$839$			749151.i			1.06425i	−0.846664	+	0.532127i	$0.821393\pi$
$840$		0			0
$841$	645271.			0.912326
$842$		0			0
$843$		0			0
$844$		0			0
$845$	122032.			0.170908
$846$		0			0
$847$	−431627.			−0.601647
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1.41029e6i		−	1.94737i
$852$		0			0
$853$	298633.			0.410431			0.205215	−	0.978717i	$-0.434211\pi$
$853$	298633.			0.410431			0.205215	+	0.978717i	$0.434211\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	476959.i		−	0.649410i	−0.945815	−	0.324705i	$-0.894735\pi$
$857$		−	476959.i		−	0.649410i	0.945815	−	0.324705i	$-0.105265\pi$
$858$		0			0
$859$	−250855.			−0.339966			−0.169983	−	0.985447i	$-0.554371\pi$
$859$	−250855.			−0.339966			−0.169983	+	0.985447i	$0.554371\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1.08428e6i			1.45586i	0.685653	+	0.727928i	$0.259517\pi$
$863$			1.08428e6i			1.45586i	−0.685653	+	0.727928i	$0.740483\pi$
$864$		0			0
$865$		−	301284.i		−	0.402665i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	464696.i		−	0.615360i
$870$		0			0
$871$	−651994.			−0.859424
$872$		0			0
$873$		0			0
$874$		0			0
$875$	132116.			0.172560
$876$		0			0
$877$			96641.8i			0.125651i	0.998025	+	0.0628255i	$0.0200112\pi$
$877$			96641.8i			0.125651i	−0.998025	+	0.0628255i	$0.979989\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	726842.			0.936457			0.468229	−	0.883607i	$-0.344892\pi$
$881$	726842.			0.936457			0.468229	+	0.883607i	$0.344892\pi$
$882$		0			0
$883$	−998833.			−1.28107			−0.640533	−	0.767931i	$-0.721286\pi$
$883$	−998833.			−1.28107			−0.640533	+	0.767931i	$0.721286\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1.36198e6i		−	1.73111i	−0.500813	−	0.865556i	$-0.666966\pi$
$887$		−	1.36198e6i		−	1.73111i	0.500813	−	0.865556i	$-0.333034\pi$
$888$		0			0
$889$			254024.i			0.321419i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−87625.9	+	1.29975e6i	−0.109883	+	1.62988i
$894$		0			0
$895$		−	157015.i		−	0.196017i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−103832.			−0.128473
$900$		0			0
$901$		−	1.67516e6i		−	2.06351i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	331514.i		−	0.404767i
$906$		0			0
$907$			536799.i			0.652525i	0.945279	+	0.326263i	$0.105789\pi$
$907$			536799.i			0.652525i	−0.945279	+	0.326263i	$0.894211\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			183647.i			0.221283i	0.993860	+	0.110641i	$0.0352905\pi$
$911$			183647.i			0.221283i	−0.993860	+	0.110641i	$0.964709\pi$
$912$		0			0
$913$	−604877.			−0.725647
$914$		0			0
$915$		0			0
$916$		0			0
$917$	17779.5			0.0211437
$918$		0			0
$919$	−606257.			−0.717837			−0.358919	−	0.933369i	$-0.616854\pi$
$919$	−606257.			−0.717837			−0.358919	+	0.933369i	$0.616854\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	880954.			1.03407
$924$		0			0
$925$			1.45638e6i			1.70212i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	379543.			0.439774			0.219887	−	0.975525i	$-0.429431\pi$
$929$	379543.			0.439774			0.219887	+	0.975525i	$0.429431\pi$
$930$		0			0
$931$	53730.9	−	796985.i	0.0619904	−	0.919498i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−692089.			−0.791661
$936$		0			0
$937$	−188522.			−0.214725			−0.107362	−	0.994220i	$-0.534241\pi$
$937$	−188522.			−0.214725			−0.107362	+	0.994220i	$0.534241\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	634679.i		−	0.716762i	−0.933576	−	0.358381i	$-0.883329\pi$
$941$		−	634679.i		−	0.716762i	0.933576	−	0.358381i	$-0.116671\pi$
$942$		0			0
$943$		−	565175.i		−	0.635565i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.25543e6			1.39989			0.699944	−	0.714198i	$-0.253208\pi$
$947$	1.25543e6			1.39989			0.699944	+	0.714198i	$0.253208\pi$
$948$		0			0
$949$		−	725842.i		−	0.805953i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			6883.08i			0.00757874i	0.999993	+	0.00378937i	$0.00120620\pi$
$953$			6883.08i			0.00757874i	−0.999993	+	0.00378937i	$0.998794\pi$
$954$		0			0
$955$	253437.			0.277884
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17180.9			0.0186813
$960$		0			0
$961$	749661.			0.811743
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	367674.i		−	0.394828i
$966$		0			0
$967$	−1.23457e6			−1.32027			−0.660136	−	0.751146i	$-0.729501\pi$
$967$	−1.23457e6			−1.32027			−0.660136	+	0.751146i	$0.729501\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			959171.i			1.01732i	0.860967	+	0.508660i	$0.169859\pi$
$971$			959171.i			1.01732i	−0.860967	+	0.508660i	$0.830141\pi$
$972$		0			0
$973$	419397.			0.442996
$974$		0			0
$975$		0			0
$976$		0			0
$977$			937337.i			0.981989i	0.871163	+	0.490994i	$0.163366\pi$
$977$			937337.i			0.981989i	−0.871163	+	0.490994i	$0.836634\pi$
$978$		0			0
$979$			2.79091e6i			2.91193i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1.19526e6i		−	1.23696i	−0.785800	−	0.618481i	$-0.787748\pi$
$983$		−	1.19526e6i		−	1.23696i	0.785800	−	0.618481i	$-0.212252\pi$
$984$		0			0
$985$	162378.			0.167361
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−937672.			−0.958646
$990$		0			0
$991$		−	1.46820e6i		−	1.49499i	−0.664266	−	0.747496i	$-0.731256\pi$
$991$		−	1.46820e6i		−	1.49499i	0.664266	−	0.747496i	$-0.268744\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	126809.			0.128087
$996$		0			0
$997$	−104406.			−0.105035			−0.0525175	−	0.998620i	$-0.516725\pi$
$997$	−104406.			−0.105035			−0.0525175	+	0.998620i	$0.516725\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.5.h.f.37.8		14
3.2	odd	2		228.5.h.a.37.11	yes	14
12.11	even	2		912.5.o.c.721.4		14
19.18	odd	2	inner	684.5.h.f.37.7		14
57.56	even	2		228.5.h.a.37.4	✓	14
228.227	odd	2		912.5.o.c.721.11		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.5.h.a.37.4	✓	14	57.56	even	2
228.5.h.a.37.11	yes	14	3.2	odd	2
684.5.h.f.37.7		14	19.18	odd	2	inner
684.5.h.f.37.8		14	1.1	even	1	trivial
912.5.o.c.721.4		14	12.11	even	2
912.5.o.c.721.11		14	228.227	odd	2

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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.13328 q^{5} -13.7213 q^{7} +O(q^{10})\)	\(q+8.13328 q^{5} -13.7213 q^{7} -214.704 q^{11} +116.434i q^{13} +396.330 q^{17} +(-24.2826 + 360.182i) q^{19} +541.165 q^{23} -558.850 q^{25} -249.019i q^{29} -416.965i q^{31} -111.599 q^{35} -2606.03i q^{37} -1044.37i q^{41} -1732.69 q^{43} +3608.58 q^{47} -2212.73 q^{49} -4226.68i q^{53} -1746.25 q^{55} +6182.61i q^{59} +3146.71 q^{61} +946.992i q^{65} +5599.68i q^{67} -7566.11i q^{71} -6233.92 q^{73} +2946.01 q^{77} +2164.36i q^{79} +2817.26 q^{83} +3223.46 q^{85} -12998.9i q^{89} -1597.62i q^{91} +(-197.498 + 2929.46i) q^{95} -11137.1i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 18 q^{5} - 86 q^{7}+O(q^{10}) \) 14 * q - 18 * q^5 - 86 * q^7	\( 14 q - 18 q^{5} - 86 q^{7} - 258 q^{11} - 498 q^{17} + 170 q^{19} + 588 q^{23} + 1560 q^{25} - 534 q^{35} + 1882 q^{43} + 222 q^{47} + 4104 q^{49} + 2702 q^{55} - 2462 q^{61} - 5774 q^{73} + 4578 q^{77} - 17988 q^{83} + 2342 q^{85} + 18270 q^{95}+O(q^{100}) \) 14 * q - 18 * q^5 - 86 * q^7 - 258 * q^11 - 498 * q^17 + 170 * q^19 + 588 * q^23 + 1560 * q^25 - 534 * q^35 + 1882 * q^43 + 222 * q^47 + 4104 * q^49 + 2702 * q^55 - 2462 * q^61 - 5774 * q^73 + 4578 * q^77 - 17988 * q^83 + 2342 * q^85 + 18270 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more