LMFDB - Embedded newform 684.6.k.f.505.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,6,Mod(505,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("684.505");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	684.k (of order $3$, degree $2$, 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:	$109.702532752$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 7 x^{17} + 17616 x^{16} - 456301 x^{15} + 216301789 x^{14} - 6076762674 x^{13} + \cdots + 55\!\cdots\!41 $ x^18 - 7x^17 + 17616x^16 - 456301x^15 + 216301789x^14 - 6076762674x^13 + 1365504992251x^12 - 54625271848675x^11 + 6304911261795867x^10 - 242577082836320134x^9 + 18704695568091019759x^8 - 704026968823840581411x^7 + 38243816152807979345695x^6 - 1065755622605542033556002x^5 + 37300806349205873742869889x^4 - 655224799021744228503297133x^3 + 19168823714556940699533734824x^2 - 236484931518202417328772433599*x + 5519876026771720332419776049541
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{7}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			505.8
Root			$26.2470 + 43.7291i$ of defining polynomial
Character	$\chi$	$=$	684.505
Dual form			684.6.k.f.577.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(25.2470 - 43.7291i) q^{5} -187.942 q^{7} +O(q^{10})$	$q+(25.2470 - 43.7291i) q^{5} -187.942 q^{7} -411.164 q^{11} +(286.794 + 496.742i) q^{13} +(-559.243 + 968.637i) q^{17} +(-1048.98 - 1172.92i) q^{19} +(72.0483 + 124.791i) q^{23} +(287.675 + 498.268i) q^{25} +(2616.14 + 4531.29i) q^{29} -6724.09 q^{31} +(-4744.99 + 8218.56i) q^{35} +12337.9 q^{37} +(5964.03 - 10330.0i) q^{41} +(-2611.95 + 4524.03i) q^{43} +(10457.1 + 18112.3i) q^{47} +18515.4 q^{49} +(-4550.48 - 7881.66i) q^{53} +(-10380.7 + 17979.9i) q^{55} +(12030.3 - 20837.0i) q^{59} +(-21555.9 - 37335.8i) q^{61} +28962.8 q^{65} +(-8051.46 - 13945.5i) q^{67} +(39902.4 - 69113.1i) q^{71} +(-17210.0 + 29808.6i) q^{73} +77275.3 q^{77} +(16379.2 - 28369.7i) q^{79} -49610.5 q^{83} +(28238.4 + 48910.4i) q^{85} +(39238.1 + 67962.3i) q^{89} +(-53900.8 - 93359.0i) q^{91} +(-77774.3 + 16258.6i) q^{95} +(62657.4 - 108526. i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 11 q^{5} + 336 q^{7}+O(q^{10}) $ 18 * q - 11 * q^5 + 336 * q^7	$ 18 q - 11 q^{5} + 336 q^{7} + 320 q^{11} + 227 q^{13} - 179 q^{17} - 868 q^{19} + 3425 q^{23} - 7054 q^{25} + 7349 q^{29} - 9960 q^{31} - 15888 q^{35} + 26444 q^{37} + 7311 q^{41} - 8283 q^{43} - 37603 q^{47} + 124738 q^{49} + 20337 q^{53} + 716 q^{55} + 74455 q^{59} - 7569 q^{61} - 188998 q^{65} - 26177 q^{67} + 53463 q^{71} - 14103 q^{73} + 31960 q^{77} + 31825 q^{79} - 82600 q^{83} - 50787 q^{85} + 155197 q^{89} - 2800 q^{91} - 49315 q^{95} + 111241 q^{97}+O(q^{100}) $ 18 * q - 11 * q^5 + 336 * q^7 + 320 * q^11 + 227 * q^13 - 179 * q^17 - 868 * q^19 + 3425 * q^23 - 7054 * q^25 + 7349 * q^29 - 9960 * q^31 - 15888 * q^35 + 26444 * q^37 + 7311 * q^41 - 8283 * q^43 - 37603 * q^47 + 124738 * q^49 + 20337 * q^53 + 716 * q^55 + 74455 * q^59 - 7569 * q^61 - 188998 * q^65 - 26177 * q^67 + 53463 * q^71 - 14103 * q^73 + 31960 * q^77 + 31825 * q^79 - 82600 * q^83 - 50787 * q^85 + 155197 * q^89 - 2800 * q^91 - 49315 * q^95 + 111241 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$5$	25.2470	−	43.7291i	0.451633	−	0.782251i	−0.546855	−	0.837227i	$-0.684175\pi$
$5$	25.2470	−	43.7291i	0.451633	−	0.782251i	0.998488	+	0.0549767i	$0.0175085\pi$
$6$		0			0
$7$	−187.942			−1.44971			−0.724853	−	0.688904i	$-0.758092\pi$
$7$	−187.942			−1.44971			−0.724853	+	0.688904i	$0.758092\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−411.164			−1.02455			−0.512276	−	0.858821i	$-0.671197\pi$
$11$	−411.164			−1.02455			−0.512276	+	0.858821i	$0.671197\pi$
$12$		0			0
$13$	286.794	+	496.742i	0.470665	+	0.815216i	0.999437	−	0.0335480i	$-0.0106807\pi$
$13$	286.794	+	496.742i	0.470665	+	0.815216i	−0.528772	+	0.848764i	$0.677347\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−559.243	+	968.637i	−0.469330	+	0.812903i	−0.999385	−	0.0350601i	$-0.988838\pi$
$17$	−559.243	+	968.637i	−0.469330	+	0.812903i	0.530056	+	0.847963i	$0.322171\pi$
$18$		0			0
$19$	−1048.98	−	1172.92i	−0.666630	−	0.745388i
$19$	−1048.98	−	1172.92i	−0.666630	−	0.745388i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	72.0483	+	124.791i	0.0283991	+	0.0491886i	0.879876	−	0.475204i	$-0.157626\pi$
$23$	72.0483	+	124.791i	0.0283991	+	0.0491886i	−0.851477	+	0.524393i	$0.824292\pi$
$24$		0			0
$25$	287.675	+	498.268i	0.0920560	+	0.159446i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2616.14	+	4531.29i	0.577652	+	1.00052i	0.995748	+	0.0921194i	$0.0293641\pi$
$29$	2616.14	+	4531.29i	0.577652	+	1.00052i	−0.418096	+	0.908403i	$0.637303\pi$
$30$		0			0
$31$	−6724.09			−1.25669			−0.628347	−	0.777934i	$-0.716268\pi$
$31$	−6724.09			−1.25669			−0.628347	+	0.777934i	$0.716268\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4744.99	+	8218.56i	−0.654734	+	1.13403i
$36$		0			0
$37$	12337.9			1.48162			0.740811	−	0.671713i	$-0.234441\pi$
$37$	12337.9			1.48162			0.740811	+	0.671713i	$0.234441\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5964.03	−	10330.0i	0.554090	−	0.959712i	−0.443884	−	0.896084i	$-0.646400\pi$
$41$	5964.03	−	10330.0i	0.554090	−	0.959712i	0.997974	−	0.0636278i	$-0.0202670\pi$
$42$		0			0
$43$	−2611.95	+	4524.03i	−0.215424	+	0.373125i	−0.953404	−	0.301698i	$-0.902447\pi$
$43$	−2611.95	+	4524.03i	−0.215424	+	0.373125i	0.737980	+	0.674823i	$0.235780\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	10457.1	+	18112.3i	0.690507	+	1.19599i	0.971672	+	0.236333i	$0.0759458\pi$
$47$	10457.1	+	18112.3i	0.690507	+	1.19599i	−0.281165	+	0.959659i	$0.590721\pi$
$48$		0			0
$49$	18515.4			1.10165
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4550.48	−	7881.66i	−0.222519	−	0.385414i	0.733053	−	0.680171i	$-0.238095\pi$
$53$	−4550.48	−	7881.66i	−0.222519	−	0.385414i	−0.955572	+	0.294757i	$0.904761\pi$
$54$		0			0
$55$	−10380.7	+	17979.9i	−0.462721	+	0.801456i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12030.3	−	20837.0i	0.449930	−	0.779301i	−0.548451	−	0.836183i	$-0.684782\pi$
$59$	12030.3	−	20837.0i	0.449930	−	0.779301i	0.998381	+	0.0568813i	$0.0181157\pi$
$60$		0			0
$61$	−21555.9	−	37335.8i	−0.741721	−	1.28470i	−0.951711	−	0.306995i	$-0.900676\pi$
$61$	−21555.9	−	37335.8i	−0.741721	−	1.28470i	0.209990	−	0.977704i	$-0.432657\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	28962.8			0.850271
$66$		0			0
$67$	−8051.46	−	13945.5i	−0.219123	−	0.379532i	0.735417	−	0.677615i	$-0.236986\pi$
$67$	−8051.46	−	13945.5i	−0.219123	−	0.379532i	−0.954540	+	0.298083i	$0.903653\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	39902.4	−	69113.1i	0.939407	−	1.62710i	0.172826	−	0.984952i	$-0.444710\pi$
$71$	39902.4	−	69113.1i	0.939407	−	1.62710i	0.766581	−	0.642148i	$-0.221956\pi$
$72$		0			0
$73$	−17210.0	+	29808.6i	−0.377985	+	0.654688i	−0.990769	−	0.135562i	$-0.956716\pi$
$73$	−17210.0	+	29808.6i	−0.377985	+	0.654688i	0.612784	+	0.790250i	$0.290049\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	77275.3			1.48530
$78$		0			0
$79$	16379.2	−	28369.7i	0.295274	−	0.511430i	−0.679774	−	0.733421i	$-0.737922\pi$
$79$	16379.2	−	28369.7i	0.295274	−	0.511430i	0.975049	+	0.221991i	$0.0712556\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−49610.5			−0.790457			−0.395229	−	0.918583i	$-0.629335\pi$
$83$	−49610.5			−0.790457			−0.395229	+	0.918583i	$0.629335\pi$
$84$		0			0
$85$	28238.4	+	48910.4i	0.423929	+	0.734267i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	39238.1	+	67962.3i	0.525089	+	0.909480i	0.999573	+	0.0292164i	$0.00930119\pi$
$89$	39238.1	+	67962.3i	0.525089	+	0.909480i	−0.474484	+	0.880264i	$0.657365\pi$
$90$		0			0
$91$	−53900.8	−	93359.0i	−0.682326	−	1.18182i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−77774.3	+	16258.6i	−0.884153	+	0.184830i
$96$		0			0
$97$	62657.4	−	108526.i	0.676149	−	1.17113i	−0.299982	−	0.953945i	$-0.596981\pi$
$97$	62657.4	−	108526.i	0.676149	−	1.17113i	0.976131	−	0.217180i	$-0.0696860\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	64376.2	+	111503.i	0.627945	+	1.08763i	0.987963	+	0.154688i	$0.0494371\pi$
$101$	64376.2	+	111503.i	0.627945	+	1.08763i	−0.360018	+	0.932945i	$0.617230\pi$
$102$		0			0
$103$	−185375.			−1.72170			−0.860851	−	0.508857i	$-0.830068\pi$
$103$	−185375.			−1.72170			−0.860851	+	0.508857i	$0.830068\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	51720.8			0.436723			0.218361	−	0.975868i	$-0.429929\pi$
$107$	51720.8			0.436723			0.218361	+	0.975868i	$0.429929\pi$
$108$		0			0
$109$	−10666.1	+	18474.1i	−0.0859879	+	0.148935i	−0.905812	−	0.423680i	$-0.860738\pi$
$109$	−10666.1	+	18474.1i	−0.0859879	+	0.148935i	0.819824	+	0.572616i	$0.194071\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−181009.			−1.33353			−0.666767	−	0.745266i	$-0.732322\pi$
$113$	−181009.			−1.33353			−0.666767	+	0.745266i	$0.732322\pi$
$114$		0			0
$115$	7276.02			0.0513038
$116$		0			0
$117$		0			0
$118$		0			0
$119$	105105.	−	182048.i	0.680390	−	1.17847i
$120$		0			0
$121$	8005.17			0.0497058
$122$		0			0
$123$		0			0
$124$		0			0
$125$	186846.			1.06957
$126$		0			0
$127$	909.283	+	1574.92i	0.00500253	+	0.00866464i	0.868516	−	0.495661i	$-0.165074\pi$
$127$	909.283	+	1574.92i	0.00500253	+	0.00866464i	−0.863513	+	0.504326i	$0.831741\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	39930.7	−	69162.1i	0.203296	−	0.352119i	−0.746292	−	0.665618i	$-0.768168\pi$
$131$	39930.7	−	69162.1i	0.203296	−	0.352119i	0.949589	+	0.313499i	$0.101501\pi$
$132$		0			0
$133$	197149.	+	220441.i	0.966418	+	1.08059i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−99388.3	−	172146.i	−0.452412	−	0.783600i	0.546123	−	0.837705i	$-0.316103\pi$
$137$	−99388.3	−	172146.i	−0.452412	−	0.783600i	−0.998535	+	0.0541044i	$0.982770\pi$
$138$		0			0
$139$	−188358.	−	326246.i	−0.826890	−	1.43222i	−0.900466	−	0.434926i	$-0.856775\pi$
$139$	−188358.	−	326246.i	−0.826890	−	1.43222i	0.0735761	−	0.997290i	$-0.476559\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−117920.	−	204243.i	−0.482221	−	0.835231i
$144$		0			0
$145$	264199.			1.04355
$146$		0			0
$147$		0			0
$148$		0			0
$149$	115828.	−	200620.i	0.427413	−	0.740300i	−0.569230	−	0.822179i	$-0.692758\pi$
$149$	115828.	−	200620.i	0.427413	−	0.740300i	0.996642	+	0.0818781i	$0.0260918\pi$
$150$		0			0
$151$	465504.			1.66143			0.830714	−	0.556700i	$-0.187933\pi$
$151$	465504.			1.66143			0.830714	+	0.556700i	$0.187933\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−169763.	+	294039.i	−0.567564	+	0.983049i
$156$		0			0
$157$	−256177.	+	443712.i	−0.829451	+	1.43665i	0.0690177	+	0.997615i	$0.478013\pi$
$157$	−256177.	+	443712.i	−0.829451	+	1.43665i	−0.898469	+	0.439037i	$0.855320\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−13540.9	−	23453.6i	−0.0411703	−	0.0713090i
$162$		0			0
$163$	551349.			1.62539			0.812694	−	0.582690i	$-0.198000\pi$
$163$	551349.			1.62539			0.812694	+	0.582690i	$0.198000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−24265.6	−	42029.2i	−0.0673286	−	0.116617i	0.830396	−	0.557174i	$-0.188114\pi$
$167$	−24265.6	−	42029.2i	−0.0673286	−	0.116617i	−0.897725	+	0.440557i	$0.854781\pi$
$168$		0			0
$169$	21144.6	−	36623.6i	0.0569487	−	0.0986380i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	84590.8	−	146516.i	0.214886	−	0.372193i	−0.738351	−	0.674416i	$-0.764395\pi$
$173$	84590.8	−	146516.i	0.214886	−	0.372193i	0.953237	+	0.302223i	$0.0977287\pi$
$174$		0			0
$175$	−54066.3	−	93645.7i	−0.133454	−	0.231149i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	560804.			1.30821			0.654106	−	0.756403i	$-0.273045\pi$
$179$	560804.			1.30821			0.654106	+	0.756403i	$0.273045\pi$
$180$		0			0
$181$	186731.	+	323428.i	0.423663	+	0.733806i	0.996295	−	0.0860068i	$-0.0274107\pi$
$181$	186731.	+	323428.i	0.423663	+	0.733806i	−0.572631	+	0.819813i	$0.694077\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	311496.	−	539526.i	0.669149	−	1.15900i
$186$		0			0
$187$	229941.	−	398269.i	0.480852	−	0.832861i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	603335.			1.19667			0.598336	−	0.801245i	$-0.295829\pi$
$191$	603335.			1.19667			0.598336	+	0.801245i	$0.295829\pi$
$192$		0			0
$193$	52.8895	−	91.6073i	0.000102206	−	0.000177026i	−0.865974	−	0.500089i	$-0.833301\pi$
$193$	52.8895	−	91.6073i	0.000102206	−	0.000177026i	0.866077	+	0.499911i	$0.166634\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	704077.			1.29257			0.646285	−	0.763096i	$-0.276322\pi$
$197$	704077.			1.29257			0.646285	+	0.763096i	$0.276322\pi$
$198$		0			0
$199$	81423.6	+	141030.i	0.145753	+	0.252452i	0.929654	−	0.368435i	$-0.120106\pi$
$199$	81423.6	+	141030.i	0.145753	+	0.252452i	−0.783901	+	0.620886i	$0.786773\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−491684.	−	851622.i	−0.837425	−	1.45046i
$204$		0			0
$205$	−301148.	−	521604.i	−0.500490	−	0.866875i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	431305.	+	482261.i	0.682997	+	0.763689i
$210$		0			0
$211$	349140.	−	604728.i	0.539875	−	0.935091i	−0.459035	−	0.888418i	$-0.651805\pi$
$211$	349140.	−	604728.i	0.539875	−	0.935091i	0.998910	−	0.0466728i	$-0.0148618\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	131888.	+	228436.i	0.194585	+	0.337030i
$216$		0			0
$217$	1.26374e6			1.82184
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−641550.			−0.883588
$222$		0			0
$223$	−93703.3	+	162299.i	−0.126181	+	0.218551i	−0.922194	−	0.386728i	$-0.873605\pi$
$223$	−93703.3	+	162299.i	−0.126181	+	0.218551i	0.796013	+	0.605279i	$0.206939\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−606601.			−0.781338			−0.390669	−	0.920531i	$-0.627756\pi$
$227$	−606601.			−0.781338			−0.390669	+	0.920531i	$0.627756\pi$
$228$		0			0
$229$	−215431.			−0.271469			−0.135734	−	0.990745i	$-0.543339\pi$
$229$	−215431.			−0.271469			−0.135734	+	0.990745i	$0.543339\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	511411.	−	885789.i	0.617135	−	1.06891i	−0.372871	−	0.927883i	$-0.621627\pi$
$233$	511411.	−	885789.i	0.617135	−	1.06891i	0.990006	−	0.141026i	$-0.0450400\pi$
$234$		0			0
$235$	1.05605e6			1.24742
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.31113e6			1.48474			0.742371	−	0.669989i	$-0.233701\pi$
$239$	1.31113e6			1.48474			0.742371	+	0.669989i	$0.233701\pi$
$240$		0			0
$241$	4784.45	+	8286.91i	0.00530627	+	0.00919073i	0.868666	−	0.495398i	$-0.164978\pi$
$241$	4784.45	+	8286.91i	0.00530627	+	0.00919073i	−0.863360	+	0.504588i	$0.831644\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	467458.	−	809662.i	0.497540	−	0.861764i
$246$		0			0
$247$	281794.	−	857460.i	0.293893	−	0.894276i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	228222.	+	395293.i	0.228651	+	0.396036i	0.957409	−	0.288736i	$-0.0932351\pi$
$251$	228222.	+	395293.i	0.228651	+	0.396036i	−0.728757	+	0.684772i	$0.759902\pi$
$252$		0			0
$253$	−29623.7	−	51309.7i	−0.0290963	−	0.0503963i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−259837.	−	450050.i	−0.245396	−	0.425038i	0.716847	−	0.697231i	$-0.245585\pi$
$257$	−259837.	−	450050.i	−0.245396	−	0.425038i	−0.962243	+	0.272192i	$0.912251\pi$
$258$		0			0
$259$	−2.31882e6			−2.14792
$260$		0			0
$261$		0			0
$262$		0			0
$263$	159146.	−	275650.i	0.141876	−	0.245736i	−0.786327	−	0.617810i	$-0.788020\pi$
$263$	159146.	−	275650.i	0.141876	−	0.245736i	0.928203	+	0.372074i	$0.121353\pi$
$264$		0			0
$265$	−459544.			−0.401987
$266$		0			0
$267$		0			0
$268$		0			0
$269$	71668.3	−	124133.i	0.0603874	−	0.104594i	−0.834251	−	0.551385i	$-0.814100\pi$
$269$	71668.3	−	124133.i	0.0603874	−	0.104594i	0.894639	+	0.446790i	$0.147433\pi$
$270$		0			0
$271$	304619.	−	527616.i	0.251961	−	0.436410i	−0.712104	−	0.702074i	$-0.752258\pi$
$271$	304619.	−	527616.i	0.251961	−	0.436410i	0.964066	+	0.265664i	$0.0855911\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−118282.	−	204870.i	−0.0943161	−	0.163360i
$276$		0			0
$277$	−1.94146e6			−1.52030			−0.760151	−	0.649746i	$-0.774875\pi$
$277$	−1.94146e6			−1.52030			−0.760151	+	0.649746i	$0.774875\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	286688.	+	496557.i	0.216592	+	0.375149i	0.953764	−	0.300557i	$-0.0971724\pi$
$281$	286688.	+	496557.i	0.216592	+	0.375149i	−0.737172	+	0.675706i	$0.763839\pi$
$282$		0			0
$283$	−1.04413e6	+	1.80849e6i	−0.774977	+	1.34230i	0.159831	+	0.987144i	$0.448905\pi$
$283$	−1.04413e6	+	1.80849e6i	−0.774977	+	1.34230i	−0.934808	+	0.355155i	$0.884428\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.12090e6	+	1.94145e6i	−0.803268	+	1.39130i
$288$		0			0
$289$	84423.8	+	146226.i	0.0594593	+	0.102987i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	598199.			0.407077			0.203538	−	0.979067i	$-0.434756\pi$
$293$	598199.			0.407077			0.203538	+	0.979067i	$0.434756\pi$
$294$		0			0
$295$	−607456.	−	1.05215e6i	−0.406406	−	0.703916i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−41326.0	+	71578.8i	−0.0267329	+	0.0463027i
$300$		0			0
$301$	490896.	−	850257.i	0.312301	−	0.540921i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.17689e6			−1.33994
$306$		0			0
$307$	71331.2	−	123549.i	0.0431950	−	0.0748159i	−0.843620	−	0.536941i	$-0.819580\pi$
$307$	71331.2	−	123549.i	0.0431950	−	0.0748159i	0.886815	+	0.462125i	$0.152913\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.33734e6			1.95659			0.978294	−	0.207219i	$-0.0664414\pi$
$311$	3.33734e6			1.95659			0.978294	+	0.207219i	$0.0664414\pi$
$312$		0			0
$313$	1.12350e6	+	1.94596e6i	0.648205	+	1.12272i	0.983551	+	0.180628i	$0.0578131\pi$
$313$	1.12350e6	+	1.94596e6i	0.648205	+	1.12272i	−0.335347	+	0.942095i	$0.608854\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	29587.8	+	51247.6i	0.0165373	+	0.0286434i	0.874176	−	0.485610i	$-0.161402\pi$
$317$	29587.8	+	51247.6i	0.0165373	+	0.0286434i	−0.857638	+	0.514253i	$0.828069\pi$
$318$		0			0
$319$	−1.07566e6	−	1.86310e6i	−0.591834	−	1.02509i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.72277e6	−	360141.i	0.918798	−	0.192073i
$324$		0			0
$325$	−165007.	+	285801.i	−0.0866551	+	0.150091i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.96534e6	−	3.40407e6i	−1.00103	−	1.73384i
$330$		0			0
$331$	2.35196e6			1.17994			0.589970	−	0.807425i	$-0.299140\pi$
$331$	2.35196e6			1.17994			0.589970	+	0.807425i	$0.299140\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−813102.			−0.395852
$336$		0			0
$337$	−82853.0	+	143506.i	−0.0397405	+	0.0688326i	−0.885212	−	0.465189i	$-0.845986\pi$
$337$	−82853.0	+	143506.i	−0.0397405	+	0.0688326i	0.845471	+	0.534021i	$0.179320\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.76471e6			1.28755
$342$		0			0
$343$	−321077.			−0.147358
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−407820.	+	706365.i	−0.181821	+	0.314924i	−0.942501	−	0.334204i	$-0.891533\pi$
$347$	−407820.	+	706365.i	−0.181821	+	0.314924i	0.760679	+	0.649128i	$0.224866\pi$
$348$		0			0
$349$	1.07255e6			0.471359			0.235680	−	0.971831i	$-0.424268\pi$
$349$	1.07255e6			0.471359			0.235680	+	0.971831i	$0.424268\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.70760e6			1.58364			0.791819	−	0.610756i	$-0.209134\pi$
$353$	3.70760e6			1.58364			0.791819	+	0.610756i	$0.209134\pi$
$354$		0			0
$355$	−2.01484e6	−	3.48980e6i	−0.848533	−	1.46970i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	440125.	−	762318.i	0.180235	−	0.312176i	−0.761725	−	0.647900i	$-0.775647\pi$
$359$	440125.	−	762318.i	0.180235	−	0.312176i	0.941961	+	0.335724i	$0.108981\pi$
$360$		0			0
$361$	−275361.	+	2.46074e6i	−0.111208	+	0.993797i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	869003.	+	1.50516e6i	0.341420	+	0.591357i
$366$		0			0
$367$	1.32405e6	+	2.29332e6i	0.513143	+	0.888789i	0.999884	+	0.0152431i	$0.00485222\pi$
$367$	1.32405e6	+	2.29332e6i	0.513143	+	0.888789i	−0.486741	+	0.873546i	$0.661814\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	855228.	+	1.48130e6i	0.322587	+	0.558737i
$372$		0			0
$373$	1.42105e6			0.528856			0.264428	−	0.964405i	$-0.414817\pi$
$373$	1.42105e6			0.528856			0.264428	+	0.964405i	$0.414817\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.50059e6	+	2.59909e6i	−0.543761	+	0.941822i
$378$		0			0
$379$	4.14092e6			1.48081			0.740405	−	0.672161i	$-0.234634\pi$
$379$	4.14092e6			1.48081			0.740405	+	0.672161i	$0.234634\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−932604.	+	1.61532e6i	−0.324863	+	0.562679i	−0.981485	−	0.191541i	$-0.938652\pi$
$383$	−932604.	+	1.61532e6i	−0.324863	+	0.562679i	0.656622	+	0.754220i	$0.271985\pi$
$384$		0			0
$385$	1.95097e6	−	3.37918e6i	0.670809	−	1.16188i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	225886.	+	391246.i	0.0756859	+	0.131092i	0.901384	−	0.433020i	$-0.142552\pi$
$389$	225886.	+	391246.i	0.0756859	+	0.131092i	−0.825698	+	0.564112i	$0.809219\pi$
$390$		0			0
$391$	−161170.			−0.0533141
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−827054.	−	1.43250e6i	−0.266711	−	0.461957i
$396$		0			0
$397$	387696.	−	671509.i	0.123457	−	0.213833i	−0.797672	−	0.603092i	$-0.793935\pi$
$397$	387696.	−	671509.i	0.123457	−	0.213833i	0.921129	+	0.389258i	$0.127269\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.11762e6	−	5.39987e6i	0.968193	−	1.67696i	0.267412	−	0.963582i	$-0.413832\pi$
$401$	3.11762e6	−	5.39987e6i	0.968193	−	1.67696i	0.700781	−	0.713376i	$-0.252835\pi$
$402$		0			0
$403$	−1.92843e6	−	3.34014e6i	−0.591482	−	1.02448i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.07291e6			−1.51800
$408$		0			0
$409$	−2.43708e6	−	4.22114e6i	−0.720379	−	1.24773i	−0.960848	−	0.277077i	$-0.910634\pi$
$409$	−2.43708e6	−	4.22114e6i	−0.720379	−	1.24773i	0.240468	−	0.970657i	$-0.422699\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.26100e6	+	3.91616e6i	−0.652266	+	1.12976i
$414$		0			0
$415$	−1.25252e6	+	2.16942e6i	−0.356996	+	0.618336i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.11548e6			−0.866943			−0.433471	−	0.901167i	$-0.642711\pi$
$419$	−3.11548e6			−0.866943			−0.433471	+	0.901167i	$0.642711\pi$
$420$		0			0
$421$	−2.17883e6	+	3.77384e6i	−0.599125	+	1.03771i	0.393826	+	0.919185i	$0.371151\pi$
$421$	−2.17883e6	+	3.77384e6i	−0.599125	+	1.03771i	−0.992951	+	0.118529i	$0.962182\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−643520.			−0.172818
$426$		0			0
$427$	4.05126e6	+	7.01699e6i	1.07528	+	1.86244i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.39109e6	−	5.87353e6i	−0.879317	−	1.52302i	−0.852091	−	0.523393i	$-0.824666\pi$
$431$	−3.39109e6	−	5.87353e6i	−0.879317	−	1.52302i	−0.0272262	−	0.999629i	$-0.508667\pi$
$432$		0			0
$433$	−16420.8	−	28441.7i	−0.00420896	−	0.00729014i	0.863913	−	0.503641i	$-0.168006\pi$
$433$	−16420.8	−	28441.7i	−0.00420896	−	0.00729014i	−0.868122	+	0.496351i	$0.834673\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	70792.0	−	215411.i	0.0177329	−	0.0539589i
$438$		0			0
$439$	−2.63969e6	+	4.57208e6i	−0.653720	+	1.13228i	0.328493	+	0.944506i	$0.393459\pi$
$439$	−2.63969e6	+	4.57208e6i	−0.653720	+	1.13228i	−0.982213	+	0.187770i	$0.939874\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.86131e6	+	6.68798e6i	0.934813	+	1.61914i	0.774967	+	0.632002i	$0.217767\pi$
$443$	3.86131e6	+	6.68798e6i	0.934813	+	1.61914i	0.159846	+	0.987142i	$0.448900\pi$
$444$		0			0
$445$	3.96258e6			0.948589
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.68110e6			−1.56398			−0.781992	−	0.623288i	$-0.785796\pi$
$449$	−6.68110e6			−1.56398			−0.781992	+	0.623288i	$0.785796\pi$
$450$		0			0
$451$	−2.45220e6	+	4.24733e6i	−0.567694	+	0.983275i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.44334e6			−1.23264
$456$		0			0
$457$	−1.15454e6			−0.258593			−0.129297	−	0.991606i	$-0.541272\pi$
$457$	−1.15454e6			−0.258593			−0.129297	+	0.991606i	$0.541272\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.99643e6	+	6.92202e6i	−0.875830	+	1.51698i	−0.0199544	+	0.999801i	$0.506352\pi$
$461$	−3.99643e6	+	6.92202e6i	−0.875830	+	1.51698i	−0.855876	+	0.517181i	$0.826981\pi$
$462$		0			0
$463$	−4.73362e6			−1.02622			−0.513111	−	0.858322i	$-0.671507\pi$
$463$	−4.73362e6			−1.02622			−0.513111	+	0.858322i	$0.671507\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.37127e6			0.715322			0.357661	−	0.933851i	$-0.383574\pi$
$467$	3.37127e6			0.715322			0.357661	+	0.933851i	$0.383574\pi$
$468$		0			0
$469$	1.51321e6	+	2.62096e6i	0.317664	+	0.550210i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.07394e6	−	1.86012e6i	0.220713	−	0.382285i
$474$		0			0
$475$	282659.	−	860093.i	0.0574816	−	0.174909i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.45446e6	−	5.98330e6i	−0.687925	−	1.19152i	−0.972508	−	0.232870i	$-0.925188\pi$
$479$	−3.45446e6	−	5.98330e6i	−0.687925	−	1.19152i	0.284583	−	0.958652i	$-0.408145\pi$
$480$		0			0
$481$	3.53844e6	+	6.12876e6i	0.697348	+	1.20784i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.16382e6	−	5.47990e6i	−0.610742	−	1.05784i
$486$		0			0
$487$	−3.84736e6			−0.735089			−0.367545	−	0.930006i	$-0.619801\pi$
$487$	−3.84736e6			−0.735089			−0.367545	+	0.930006i	$0.619801\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.95587e6	+	8.58382e6i	−0.927719	+	1.60686i	−0.140590	+	0.990068i	$0.544900\pi$
$491$	−4.95587e6	+	8.58382e6i	−0.927719	+	1.60686i	−0.787129	+	0.616788i	$0.788433\pi$
$492$		0			0
$493$	−5.85223e6			−1.08444
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.49937e6	+	1.29893e7i	−1.36186	+	2.35882i
$498$		0			0
$499$	−3.50894e6	+	6.07766e6i	−0.630847	+	1.09266i	0.356532	+	0.934283i	$0.383959\pi$
$499$	−3.50894e6	+	6.07766e6i	−0.630847	+	1.09266i	−0.987379	+	0.158376i	$0.949374\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.11612e6	−	1.93318e6i	−0.196694	−	0.340684i	0.750760	−	0.660575i	$-0.229687\pi$
$503$	−1.11612e6	−	1.93318e6i	−0.196694	−	0.340684i	−0.947455	+	0.319890i	$0.896354\pi$
$504$		0			0
$505$	6.50123e6			1.13440
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−787950.	−	1.36477e6i	−0.134804	−	0.233488i	0.790718	−	0.612180i	$-0.209707\pi$
$509$	−787950.	−	1.36477e6i	−0.134804	−	0.233488i	−0.925523	+	0.378692i	$0.876374\pi$
$510$		0			0
$511$	3.23449e6	−	5.60231e6i	0.547966	−	0.949106i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.68017e6	+	8.10629e6i	−0.777577	+	1.34680i
$516$		0			0
$517$	−4.29960e6	−	7.44713e6i	−0.707460	−	1.22536i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4.35615e6			0.703086			0.351543	−	0.936172i	$-0.385657\pi$
$521$	4.35615e6			0.703086			0.351543	+	0.936172i	$0.385657\pi$
$522$		0			0
$523$	−2.64694e6	−	4.58464e6i	−0.423146	−	0.732911i	0.573099	−	0.819486i	$-0.305741\pi$
$523$	−2.64694e6	−	4.58464e6i	−0.423146	−	0.732911i	−0.996245	+	0.0865752i	$0.972408\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.76040e6	−	6.51320e6i	0.589803	−	1.02157i
$528$		0			0
$529$	3.20779e6	−	5.55605e6i	0.498387	−	0.863232i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.84180e6			1.04316
$534$		0			0
$535$	1.30580e6	−	2.26171e6i	0.197238	−	0.341627i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7.61286e6			−1.12869
$540$		0			0
$541$	−5.45983e6	−	9.45670e6i	−0.802021	−	1.38914i	−0.918284	−	0.395923i	$-0.870425\pi$
$541$	−5.45983e6	−	9.45670e6i	−0.802021	−	1.38914i	0.116263	−	0.993218i	$-0.462909\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	538572.	+	932835.i	0.0776699	+	0.134528i
$546$		0			0
$547$	−1.49890e6	−	2.59617e6i	−0.214192	−	0.370992i	0.738830	−	0.673892i	$-0.235379\pi$
$547$	−1.49890e6	−	2.59617e6i	−0.214192	−	0.370992i	−0.953022	+	0.302900i	$0.902045\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.57053e6	−	7.82176e6i	0.360697	−	1.09755i
$552$		0			0
$553$	−3.07835e6	+	5.33186e6i	−0.428061	+	0.741423i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−197093.	−	341375.i	−0.0269174	−	0.0466223i	0.852253	−	0.523130i	$-0.175236\pi$
$557$	−197093.	−	341375.i	−0.0269174	−	0.0466223i	−0.879170	+	0.476508i	$0.841902\pi$
$558$		0			0
$559$	−2.99637e6			−0.405569
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.10461e7			−1.46872			−0.734361	−	0.678760i	$-0.762518\pi$
$563$	−1.10461e7			−1.46872			−0.734361	+	0.678760i	$0.762518\pi$
$564$		0			0
$565$	−4.56994e6	+	7.91537e6i	−0.602267	+	1.04316i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	7.21256e6			0.933918			0.466959	−	0.884279i	$-0.345349\pi$
$569$	7.21256e6			0.933918			0.466959	+	0.884279i	$0.345349\pi$
$570$		0			0
$571$	3.09113e6			0.396760			0.198380	−	0.980125i	$-0.436432\pi$
$571$	3.09113e6			0.396760			0.198380	+	0.980125i	$0.436432\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−41453.0	+	71798.6i	−0.00522861	+	0.00905621i
$576$		0			0
$577$	−2.16381e6			−0.270570			−0.135285	−	0.990807i	$-0.543195\pi$
$577$	−2.16381e6			−0.270570			−0.135285	+	0.990807i	$0.543195\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	9.32392e6			1.14593
$582$		0			0
$583$	1.87099e6	+	3.24066e6i	0.227982	+	0.394877i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	632835.	−	1.09610e6i	0.0758045	−	0.131297i	−0.825631	−	0.564210i	$-0.809181\pi$
$587$	632835.	−	1.09610e6i	0.0758045	−	0.131297i	0.901436	+	0.432913i	$0.142514\pi$
$588$		0			0
$589$	7.05347e6	+	7.88679e6i	0.837750	+	0.936725i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.38325e6	+	1.45202e7i	0.978984	+	1.69565i	0.666106	+	0.745857i	$0.267960\pi$
$593$	8.38325e6	+	1.45202e7i	0.978984	+	1.69565i	0.312878	+	0.949793i	$0.398707\pi$
$594$		0			0
$595$	−5.30720e6	−	9.19234e6i	−0.614573	−	1.06447i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.31238e6	−	4.00516e6i	−0.263325	−	0.456092i	0.703798	−	0.710400i	$-0.251486\pi$
$599$	−2.31238e6	−	4.00516e6i	−0.263325	−	0.456092i	−0.967123	+	0.254308i	$0.918152\pi$
$600$		0			0
$601$	−568699.			−0.0642238			−0.0321119	−	0.999484i	$-0.510223\pi$
$601$	−568699.			−0.0642238			−0.0321119	+	0.999484i	$0.510223\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	202107.	−	350059.i	0.0224488	−	0.0388824i
$606$		0			0
$607$	9.98344e6			1.09979			0.549893	−	0.835235i	$-0.314668\pi$
$607$	9.98344e6			1.09979			0.549893	+	0.835235i	$0.314668\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.99809e6	+	1.03890e7i	−0.649995	+	1.12582i
$612$		0			0
$613$	−2.93664e6	+	5.08641e6i	−0.315645	+	0.546714i	−0.979574	−	0.201082i	$-0.935554\pi$
$613$	−2.93664e6	+	5.08641e6i	−0.315645	+	0.546714i	0.663929	+	0.747796i	$0.268888\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.31577e6	+	7.47513e6i	0.456400	+	0.790508i	0.998767	−	0.0496336i	$-0.0158053\pi$
$617$	4.31577e6	+	7.47513e6i	0.456400	+	0.790508i	−0.542368	+	0.840141i	$0.682472\pi$
$618$		0			0
$619$	−6.29725e6			−0.660578			−0.330289	−	0.943880i	$-0.607146\pi$
$619$	−6.29725e6			−0.660578			−0.330289	+	0.943880i	$0.607146\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.37450e6	−	1.27730e7i	−0.761224	−	1.31848i
$624$		0			0
$625$	3.81831e6	−	6.61351e6i	0.390995	−	0.677224i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.89989e6	+	1.19510e7i	−0.695369	+	1.20441i
$630$		0			0
$631$	2.14448e6	+	3.71434e6i	0.214411	+	0.371371i	0.953090	−	0.302686i	$-0.0978833\pi$
$631$	2.14448e6	+	3.71434e6i	0.214411	+	0.371371i	−0.738679	+	0.674057i	$0.764550\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	91826.8			0.00903723
$636$		0			0
$637$	5.31010e6	+	9.19737e6i	0.518507	+	0.898080i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5.61057e6	−	9.71779e6i	0.539339	−	0.934163i	−0.459600	−	0.888126i	$-0.652007\pi$
$641$	5.61057e6	−	9.71779e6i	0.539339	−	0.934163i	0.998940	−	0.0460372i	$-0.0146593\pi$
$642$		0			0
$643$	−2.20567e6	+	3.82032e6i	−0.210384	+	0.364395i	−0.951835	−	0.306612i	$-0.900805\pi$
$643$	−2.20567e6	+	3.82032e6i	−0.210384	+	0.364395i	0.741451	+	0.671007i	$0.234138\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.20279e6			−0.676457			−0.338229	−	0.941064i	$-0.609828\pi$
$647$	−7.20279e6			−0.676457			−0.338229	+	0.941064i	$0.609828\pi$
$648$		0			0
$649$	−4.94641e6	+	8.56744e6i	−0.460976	+	0.798434i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.23537e6			0.388695			0.194347	−	0.980933i	$-0.437741\pi$
$653$	4.23537e6			0.388695			0.194347	+	0.980933i	$0.437741\pi$
$654$		0			0
$655$	−2.01627e6	−	3.49227e6i	−0.183630	−	0.318057i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.24491e6	+	5.62036e6i	0.291065	+	0.504139i	0.974062	−	0.226282i	$-0.0726572\pi$
$659$	3.24491e6	+	5.62036e6i	0.291065	+	0.504139i	−0.682997	+	0.730421i	$0.739324\pi$
$660$		0			0
$661$	−5.60734e6	−	9.71219e6i	−0.499175	−	0.864597i	0.500824	−	0.865549i	$-0.333030\pi$
$661$	−5.60734e6	−	9.71219e6i	−0.499175	−	0.864597i	−1.00000		0.000952002i	$0.999697\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.46171e7	−	3.05568e6i	1.28176	−	0.267950i
$666$		0			0
$667$	−376977.	+	652943.i	−0.0328095	+	0.0568278i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8.86300e6	+	1.53512e7i	0.759932	+	1.31624i
$672$		0			0
$673$	301881.			0.0256920			0.0128460	−	0.999917i	$-0.495911\pi$
$673$	301881.			0.0256920			0.0128460	+	0.999917i	$0.495911\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	470872.			0.0394849			0.0197424	−	0.999805i	$-0.493715\pi$
$677$	470872.			0.0394849			0.0197424	+	0.999805i	$0.493715\pi$
$678$		0			0
$679$	−1.17760e7	+	2.03966e7i	−0.980218	+	1.69779i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.01992e7			0.836589			0.418295	−	0.908311i	$-0.362628\pi$
$683$	1.01992e7			0.836589			0.418295	+	0.908311i	$0.362628\pi$
$684$		0			0
$685$	−1.00370e7			−0.817296
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.61010e6	−	4.52083e6i	0.209464	−	0.362802i
$690$		0			0
$691$	4.81669e6			0.383754			0.191877	−	0.981419i	$-0.438542\pi$
$691$	4.81669e6			0.383754			0.191877	+	0.981419i	$0.438542\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.90220e7			−1.49380
$696$		0			0
$697$	6.67068e6	+	1.15540e7i	0.520102	+	0.900843i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.09052e6	+	3.62088e6i	−0.160679	+	0.278304i	−0.935112	−	0.354351i	$-0.884702\pi$
$701$	−2.09052e6	+	3.62088e6i	−0.160679	+	0.278304i	0.774434	+	0.632655i	$0.218035\pi$
$702$		0			0
$703$	−1.29423e7	−	1.44713e7i	−0.987694	−	1.10438i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.20990e7	−	2.09561e7i	−0.910336	−	1.57675i
$708$		0			0
$709$	−4.41436e6	−	7.64589e6i	−0.329801	−	0.571232i	0.652671	−	0.757641i	$-0.273648\pi$
$709$	−4.41436e6	−	7.64589e6i	−0.329801	−	0.571232i	−0.982472	+	0.186409i	$0.940315\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−484459.	−	839107.i	−0.0356889	−	0.0618150i
$714$		0			0
$715$	−1.19085e7			−0.871146
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.62278e6	−	4.54279e6i	0.189208	−	0.327718i	−0.755778	−	0.654828i	$-0.772741\pi$
$719$	2.62278e6	−	4.54279e6i	0.189208	−	0.327718i	0.944986	+	0.327109i	$0.106075\pi$
$720$		0			0
$721$	3.48398e7			2.49596
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.50520e6	+	2.60708e6i	−0.106353	+	0.184208i
$726$		0			0
$727$	9.18888e6	−	1.59156e7i	0.644802	−	1.11683i	−0.339545	−	0.940590i	$-0.610273\pi$
$727$	9.18888e6	−	1.59156e7i	0.644802	−	1.11683i	0.984347	−	0.176241i	$-0.0563937\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.92142e6	−	5.06006e6i	−0.202209	−	0.350237i
$732$		0			0
$733$	2.42837e7			1.66938			0.834689	−	0.550721i	$-0.185647\pi$
$733$	2.42837e7			1.66938			0.834689	+	0.550721i	$0.185647\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.31048e6	+	5.73391e6i	0.224503	+	0.388850i
$738$		0			0
$739$	−1.29261e6	+	2.23886e6i	−0.0870674	+	0.150805i	−0.906270	−	0.422699i	$-0.861083\pi$
$739$	−1.29261e6	+	2.23886e6i	−0.0870674	+	0.150805i	0.819203	+	0.573504i	$0.194416\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9.08526e6	+	1.57361e7i	−0.603761	+	1.04575i	0.388485	+	0.921455i	$0.372999\pi$
$743$	−9.08526e6	+	1.57361e7i	−0.603761	+	1.04575i	−0.992246	+	0.124290i	$0.960335\pi$
$744$		0			0
$745$	−5.84862e6	−	1.01301e7i	−0.386067	−	0.668688i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−9.72054e6			−0.633120
$750$		0			0
$751$	−7.21562e6	−	1.24978e7i	−0.466846	−	0.808601i	0.532437	−	0.846470i	$-0.321276\pi$
$751$	−7.21562e6	−	1.24978e7i	−0.466846	−	0.808601i	−0.999283	+	0.0378689i	$0.987943\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.17526e7	−	2.03561e7i	0.750355	−	1.29965i
$756$		0			0
$757$	−1.51265e6	+	2.61999e6i	−0.0959399	+	0.166173i	−0.910001	−	0.414607i	$-0.863919\pi$
$757$	−1.51265e6	+	2.61999e6i	−0.0959399	+	0.166173i	0.814061	+	0.580780i	$0.197252\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.91756e7			−1.20029			−0.600147	−	0.799890i	$-0.704891\pi$
$761$	−1.91756e7			−1.20029			−0.600147	+	0.799890i	$0.704891\pi$
$762$		0			0
$763$	2.00460e6	−	3.47208e6i	0.124657	−	0.215913i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.38008e7			0.847065
$768$		0			0
$769$	−1.08868e7	−	1.88565e7i	−0.663872	−	1.14986i	−0.979590	−	0.201007i	$-0.935579\pi$
$769$	−1.08868e7	−	1.88565e7i	−0.663872	−	1.14986i	0.315718	−	0.948853i	$-0.397755\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.04601e6	−	1.81174e6i	−0.0629631	−	0.109055i	0.832826	−	0.553536i	$-0.186722\pi$
$773$	−1.04601e6	−	1.81174e6i	−0.0629631	−	0.109055i	−0.895789	+	0.444480i	$0.853388\pi$
$774$		0			0
$775$	−1.93435e6	−	3.35040e6i	−0.115686	−	0.200374i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.83724e7	+	3.84072e6i	−1.08473	+	0.226761i
$780$		0			0
$781$	−1.64065e7	+	2.84168e7i	−0.962471	+	1.66705i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.29354e7	+	2.24048e7i	0.749215	+	1.29768i
$786$		0			0
$787$	3.41901e7			1.96772			0.983860	−	0.178942i	$-0.0572673\pi$
$787$	3.41901e7			1.96772			0.983860	+	0.178942i	$0.0572673\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.40193e7			1.93323
$792$		0			0
$793$	1.23642e7	−	2.14154e7i	0.698205	−	1.20933i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.39978e7			0.780573			0.390287	−	0.920693i	$-0.372376\pi$
$797$	1.39978e7			0.780573			0.390287	+	0.920693i	$0.372376\pi$
$798$		0			0
$799$	−2.33923e7			−1.29630
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.07614e6	−	1.22562e7i	0.387265	−	0.670762i
$804$		0			0
$805$	−1.36747e6			−0.0743754
$806$		0			0
$807$		0			0
$808$		0			0
$809$	7.41687e6			0.398428			0.199214	−	0.979956i	$-0.436161\pi$
$809$	7.41687e6			0.398428			0.199214	+	0.979956i	$0.436161\pi$
$810$		0			0
$811$	1.16311e7	+	2.01456e7i	0.620965	+	1.07554i	0.989306	+	0.145852i	$0.0465924\pi$
$811$	1.16311e7	+	2.01456e7i	0.620965	+	1.07554i	−0.368341	+	0.929691i	$0.620074\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.39199e7	−	2.41100e7i	0.734079	−	1.27146i
$816$		0			0
$817$	8.04619e6	−	1.68204e6i	0.421731	−	0.0881620i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4.30361e6	−	7.45407e6i	−0.222831	−	0.385954i	0.732836	−	0.680406i	$-0.238196\pi$
$821$	−4.30361e6	−	7.45407e6i	−0.222831	−	0.385954i	−0.955666	+	0.294452i	$0.904863\pi$
$822$		0			0
$823$	7.89709e6	+	1.36782e7i	0.406413	+	0.703928i	0.994485	−	0.104881i	$-0.0334460\pi$
$823$	7.89709e6	+	1.36782e7i	0.406413	+	0.703928i	−0.588072	+	0.808809i	$0.700113\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.47559e6	−	1.12161e7i	−0.329242	−	0.570264i	0.653119	−	0.757255i	$-0.273460\pi$
$827$	−6.47559e6	−	1.12161e7i	−0.329242	−	0.570264i	−0.982362	+	0.186990i	$0.940127\pi$
$828$		0			0
$829$	−1.28769e7			−0.650765			−0.325383	−	0.945582i	$-0.605493\pi$
$829$	−1.28769e7			−0.650765			−0.325383	+	0.945582i	$0.605493\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.03546e7	+	1.79347e7i	−0.517036	+	0.895532i
$834$		0			0
$835$	−2.45054e6			−0.121631
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.39980e7	−	2.42452e7i	0.686530	−	1.18911i	−0.286423	−	0.958103i	$-0.592466\pi$
$839$	1.39980e7	−	2.42452e7i	0.686530	−	1.18911i	0.972953	−	0.231002i	$-0.0742003\pi$
$840$		0			0
$841$	−3.43281e6	+	5.94580e6i	−0.167363	+	0.289881i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.06768e6	−	1.84927e6i	−0.0514398	−	0.0890963i
$846$		0			0
$847$	−1.50451e6			−0.0720588
$848$		0			0
$849$		0			0
$850$		0			0
$851$	888925.	+	1.53966e6i	0.0420767	+	0.0728789i
$852$		0			0
$853$	7.33528e6	−	1.27051e7i	0.345179	−	0.597867i	−0.640208	−	0.768202i	$-0.721152\pi$
$853$	7.33528e6	−	1.27051e7i	0.345179	−	0.597867i	0.985386	+	0.170335i	$0.0544850\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.31078e7	−	2.27033e7i	0.609645	−	1.05594i	−0.381653	−	0.924306i	$-0.624645\pi$
$857$	1.31078e7	−	2.27033e7i	0.609645	−	1.05594i	0.991299	−	0.131631i	$-0.0420215\pi$
$858$		0			0
$859$	1.60178e7	+	2.77437e7i	0.740664	+	1.28287i	0.952193	+	0.305496i	$0.0988222\pi$
$859$	1.60178e7	+	2.77437e7i	0.740664	+	1.28287i	−0.211530	+	0.977372i	$0.567844\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.79175e6			0.127600			0.0637998	−	0.997963i	$-0.479678\pi$
$863$	2.79175e6			0.127600			0.0637998	+	0.997963i	$0.479678\pi$
$864$		0			0
$865$	−4.27133e6	−	7.39817e6i	−0.194099	−	0.336189i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.73456e6	+	1.16646e7i	−0.302524	+	0.523986i
$870$		0			0
$871$	4.61823e6	−	7.99900e6i	0.206267	−	0.357265i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.51162e7			−1.55056
$876$		0			0
$877$	1.02849e7	−	1.78139e7i	0.451543	−	0.782096i	−0.546939	−	0.837173i	$-0.684207\pi$
$877$	1.02849e7	−	1.78139e7i	0.451543	−	0.782096i	0.998482	+	0.0550767i	$0.0175403\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	884571.			0.0383966			0.0191983	−	0.999816i	$-0.493889\pi$
$881$	884571.			0.0383966			0.0191983	+	0.999816i	$0.493889\pi$
$882$		0			0
$883$	2.71324e6	+	4.69946e6i	0.117108	+	0.202837i	0.918620	−	0.395141i	$-0.129304\pi$
$883$	2.71324e6	+	4.69946e6i	0.117108	+	0.202837i	−0.801513	+	0.597978i	$0.795971\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7.62216e6	−	1.32020e7i	−0.325289	−	0.563417i	0.656282	−	0.754516i	$-0.272128\pi$
$887$	−7.62216e6	−	1.32020e7i	−0.325289	−	0.563417i	−0.981571	+	0.191099i	$0.938795\pi$
$888$		0			0
$889$	−170893.	−	295995.i	−0.00725220	−	0.0125612i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.02748e7	−	3.12648e7i	0.431166	−	1.31198i
$894$		0			0
$895$	1.41586e7	−	2.45235e7i	0.590831	−	1.02335i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.75912e7	−	3.04688e7i	−0.725931	−	1.25735i
$900$		0			0
$901$	1.01793e7			0.417739
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.88576e7			0.765360
$906$		0			0
$907$	3.62809e6	−	6.28404e6i	0.146440	−	0.253642i	−0.783469	−	0.621431i	$-0.786552\pi$
$907$	3.62809e6	−	6.28404e6i	0.146440	−	0.253642i	0.929909	+	0.367789i	$0.119885\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.21717e6			−0.128433			−0.0642166	−	0.997936i	$-0.520455\pi$
$911$	−3.21717e6			−0.128433			−0.0642166	+	0.997936i	$0.520455\pi$
$912$		0			0
$913$	2.03981e7			0.809864
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7.50468e6	+	1.29985e7i	−0.294720	+	0.510469i
$918$		0			0
$919$	−5.78840e6			−0.226084			−0.113042	−	0.993590i	$-0.536059\pi$
$919$	−5.78840e6			−0.226084			−0.113042	+	0.993590i	$0.536059\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.57752e7			1.76858
$924$		0			0
$925$	3.54931e6	+	6.14758e6i	0.136392	+	0.236238i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.26109e7	−	2.18428e7i	0.479411	−	0.830364i	−0.520310	−	0.853977i	$-0.674184\pi$
$929$	1.26109e7	−	2.18428e7i	0.479411	−	0.830364i	0.999721	+	0.0236130i	$0.00751694\pi$
$930$		0			0
$931$	−1.94223e7	−	2.17170e7i	−0.734391	−	0.821155i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.16106e7	−	2.01102e7i	−0.434337	−	0.752294i
$936$		0			0
$937$	1.25018e7	+	2.16537e7i	0.465182	+	0.805719i	0.999210	−	0.0397478i	$-0.0126555\pi$
$937$	1.25018e7	+	2.16537e7i	0.465182	+	0.805719i	−0.534027	+	0.845467i	$0.679322\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.25003e7	+	2.16511e7i	0.460198	+	0.797087i	0.998970	−	0.0453646i	$-0.0144450\pi$
$941$	1.25003e7	+	2.16511e7i	0.460198	+	0.797087i	−0.538772	+	0.842452i	$0.681112\pi$
$942$		0			0
$943$	1.71879e6			0.0629425
$944$		0			0
$945$		0			0
$946$		0			0
$947$	155598.	−	269504.i	0.00563806	−	0.00976540i	−0.863193	−	0.504875i	$-0.831539\pi$
$947$	155598.	−	269504.i	0.00563806	−	0.00976540i	0.868831	+	0.495109i	$0.164872\pi$
$948$		0			0
$949$	−1.97429e7			−0.711617
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.37486e7	−	4.11338e7i	0.847045	−	1.46712i	−0.0367888	−	0.999323i	$-0.511713\pi$
$953$	2.37486e7	−	4.11338e7i	0.847045	−	1.46712i	0.883834	−	0.467802i	$-0.154954\pi$
$954$		0			0
$955$	1.52324e7	−	2.63833e7i	0.540456	−	0.936098i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.86793e7	+	3.23535e7i	0.655864	+	1.13599i
$960$		0			0
$961$	1.65842e7			0.579278
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2670.60	−	4625.62i	−9.23190e−5	−	0.000159901i
$966$		0			0
$967$	−1.00225e7	+	1.73594e7i	−0.344674	+	0.596992i	−0.985294	−	0.170865i	$-0.945344\pi$
$967$	−1.00225e7	+	1.73594e7i	−0.344674	+	0.596992i	0.640621	+	0.767857i	$0.278677\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.33282e7	+	2.30850e7i	−0.453651	+	0.785747i	−0.998610	−	0.0527164i	$-0.983212\pi$
$971$	−1.33282e7	+	2.30850e7i	−0.453651	+	0.785747i	0.544959	+	0.838463i	$0.316545\pi$
$972$		0			0
$973$	3.54005e7	+	6.13155e7i	1.19875	+	2.07629i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.83585e7			0.950487			0.475243	−	0.879854i	$-0.342360\pi$
$977$	2.83585e7			0.950487			0.475243	+	0.879854i	$0.342360\pi$
$978$		0			0
$979$	−1.61333e7	−	2.79437e7i	−0.537980	−	0.931809i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−8.09422e6	+	1.40196e7i	−0.267172	+	0.462756i	−0.968130	−	0.250446i	$-0.919423\pi$
$983$	−8.09422e6	+	1.40196e7i	−0.267172	+	0.462756i	0.700958	+	0.713202i	$0.252756\pi$
$984$		0			0
$985$	1.77758e7	−	3.07887e7i	0.583767	−	1.01111i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−752745.			−0.0244713
$990$		0			0
$991$	−1.72685e7	+	2.99099e7i	−0.558561	+	0.967456i	0.439056	+	0.898460i	$0.355313\pi$
$991$	−1.72685e7	+	2.99099e7i	−0.558561	+	0.967456i	−0.997617	+	0.0689965i	$0.978020\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.22282e6			0.263307
$996$		0			0
$997$	1.12806e7	+	1.95385e7i	0.359412	+	0.622521i	0.987863	−	0.155329i	$-0.0496438\pi$
$997$	1.12806e7	+	1.95385e7i	0.359412	+	0.622521i	−0.628450	+	0.777850i	$0.716310\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.6.k.f.505.8		18
3.2	odd	2		76.6.e.a.49.2	yes	18
12.11	even	2		304.6.i.d.49.8		18
19.7	even	3	inner	684.6.k.f.577.8		18
57.26	odd	6		76.6.e.a.45.2	✓	18
228.83	even	6		304.6.i.d.273.8		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.e.a.45.2	✓	18	57.26	odd	6
76.6.e.a.49.2	yes	18	3.2	odd	2
304.6.i.d.49.8		18	12.11	even	2
304.6.i.d.273.8		18	228.83	even	6
684.6.k.f.505.8		18	1.1	even	1	trivial
684.6.k.f.577.8		18	19.7	even	3	inner

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	684.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(25.2470 - 43.7291i) q^{5} -187.942 q^{7} +O(q^{10})\)	\(q+(25.2470 - 43.7291i) q^{5} -187.942 q^{7} -411.164 q^{11} +(286.794 + 496.742i) q^{13} +(-559.243 + 968.637i) q^{17} +(-1048.98 - 1172.92i) q^{19} +(72.0483 + 124.791i) q^{23} +(287.675 + 498.268i) q^{25} +(2616.14 + 4531.29i) q^{29} -6724.09 q^{31} +(-4744.99 + 8218.56i) q^{35} +12337.9 q^{37} +(5964.03 - 10330.0i) q^{41} +(-2611.95 + 4524.03i) q^{43} +(10457.1 + 18112.3i) q^{47} +18515.4 q^{49} +(-4550.48 - 7881.66i) q^{53} +(-10380.7 + 17979.9i) q^{55} +(12030.3 - 20837.0i) q^{59} +(-21555.9 - 37335.8i) q^{61} +28962.8 q^{65} +(-8051.46 - 13945.5i) q^{67} +(39902.4 - 69113.1i) q^{71} +(-17210.0 + 29808.6i) q^{73} +77275.3 q^{77} +(16379.2 - 28369.7i) q^{79} -49610.5 q^{83} +(28238.4 + 48910.4i) q^{85} +(39238.1 + 67962.3i) q^{89} +(-53900.8 - 93359.0i) q^{91} +(-77774.3 + 16258.6i) q^{95} +(62657.4 - 108526. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 11 q^{5} + 336 q^{7}+O(q^{10}) \) 18 * q - 11 * q^5 + 336 * q^7	\( 18 q - 11 q^{5} + 336 q^{7} + 320 q^{11} + 227 q^{13} - 179 q^{17} - 868 q^{19} + 3425 q^{23} - 7054 q^{25} + 7349 q^{29} - 9960 q^{31} - 15888 q^{35} + 26444 q^{37} + 7311 q^{41} - 8283 q^{43} - 37603 q^{47} + 124738 q^{49} + 20337 q^{53} + 716 q^{55} + 74455 q^{59} - 7569 q^{61} - 188998 q^{65} - 26177 q^{67} + 53463 q^{71} - 14103 q^{73} + 31960 q^{77} + 31825 q^{79} - 82600 q^{83} - 50787 q^{85} + 155197 q^{89} - 2800 q^{91} - 49315 q^{95} + 111241 q^{97}+O(q^{100}) \) 18 * q - 11 * q^5 + 336 * q^7 + 320 * q^11 + 227 * q^13 - 179 * q^17 - 868 * q^19 + 3425 * q^23 - 7054 * q^25 + 7349 * q^29 - 9960 * q^31 - 15888 * q^35 + 26444 * q^37 + 7311 * q^41 - 8283 * q^43 - 37603 * q^47 + 124738 * q^49 + 20337 * q^53 + 716 * q^55 + 74455 * q^59 - 7569 * q^61 - 188998 * q^65 - 26177 * q^67 + 53463 * q^71 - 14103 * q^73 + 31960 * q^77 + 31825 * q^79 - 82600 * q^83 - 50787 * q^85 + 155197 * q^89 - 2800 * q^91 - 49315 * q^95 + 111241 * q^97

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