LMFDB - Embedded newform 684.6.k.f.505.9

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			505.9
Root			$48.4871 + 82.2500i$ of defining polynomial
Character	$\chi$	$=$	684.505
Dual form			684.6.k.f.577.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(47.4871 - 82.2500i) q^{5} +189.860 q^{7} +O(q^{10})$	$q+(47.4871 - 82.2500i) q^{5} +189.860 q^{7} +530.005 q^{11} +(-353.895 - 612.964i) q^{13} +(764.589 - 1324.31i) q^{17} +(654.031 - 1431.20i) q^{19} +(-497.425 - 861.565i) q^{23} +(-2947.54 - 5105.29i) q^{25} +(1290.66 + 2235.49i) q^{29} -2790.81 q^{31} +(9015.88 - 15616.0i) q^{35} +7238.14 q^{37} +(-2181.63 + 3778.69i) q^{41} +(-3121.70 + 5406.94i) q^{43} +(-12320.9 - 21340.4i) q^{47} +19239.7 q^{49} +(15497.6 + 26842.6i) q^{53} +(25168.4 - 43592.9i) q^{55} +(18177.7 - 31484.8i) q^{59} +(2886.93 + 5000.30i) q^{61} -67221.7 q^{65} +(30388.4 + 52634.2i) q^{67} +(-14829.0 + 25684.5i) q^{71} +(-39002.3 + 67553.9i) q^{73} +100627. q^{77} +(-33427.5 + 57898.2i) q^{79} +21919.5 q^{83} +(-72616.2 - 125775. i) q^{85} +(45374.8 + 78591.5i) q^{89} +(-67190.4 - 116377. i) q^{91} +(-86658.4 - 121758. i) q^{95} +(-30757.3 + 53273.1i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	47.4871	−	82.2500i	0.849474	−	1.47133i	−0.0322040	−	0.999481i	$-0.510253\pi$
$5$	47.4871	−	82.2500i	0.849474	−	1.47133i	0.881678	−	0.471851i	$-0.156414\pi$
$6$		0			0
$7$	189.860			1.46449			0.732247	−	0.681039i	$-0.238472\pi$
$7$	189.860			1.46449			0.732247	+	0.681039i	$0.238472\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	530.005			1.32068			0.660341	−	0.750966i	$-0.270412\pi$
$11$	530.005			1.32068			0.660341	+	0.750966i	$0.270412\pi$
$12$		0			0
$13$	−353.895	−	612.964i	−0.580786	−	1.00595i	−0.995386	−	0.0959473i	$-0.969412\pi$
$13$	−353.895	−	612.964i	−0.580786	−	1.00595i	0.414600	−	0.910004i	$-0.363921\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	764.589	−	1324.31i	0.641661	−	1.11139i	−0.343401	−	0.939189i	$-0.611579\pi$
$17$	764.589	−	1324.31i	0.641661	−	1.11139i	0.985062	−	0.172201i	$-0.0550878\pi$
$18$		0			0
$19$	654.031	−	1431.20i	0.415637	−	0.909530i
$19$	654.031	−	1431.20i	0.415637	−	0.909530i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−497.425	−	861.565i	−0.196068	−	0.339600i	0.751182	−	0.660095i	$-0.229484\pi$
$23$	−497.425	−	861.565i	−0.196068	−	0.339600i	−0.947250	+	0.320495i	$0.896151\pi$
$24$		0			0
$25$	−2947.54	−	5105.29i	−0.943213	−	1.63369i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1290.66	+	2235.49i	0.284982	+	0.493603i	0.972605	−	0.232465i	$-0.0746791\pi$
$29$	1290.66	+	2235.49i	0.284982	+	0.493603i	−0.687623	+	0.726068i	$0.741346\pi$
$30$		0			0
$31$	−2790.81			−0.521585			−0.260793	−	0.965395i	$-0.583984\pi$
$31$	−2790.81			−0.521585			−0.260793	+	0.965395i	$0.583984\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9015.88	−	15616.0i	1.24405	−	2.15476i
$36$		0			0
$37$	7238.14			0.869206			0.434603	−	0.900622i	$-0.356889\pi$
$37$	7238.14			0.869206			0.434603	+	0.900622i	$0.356889\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2181.63	+	3778.69i	−0.202685	+	0.351060i	−0.949393	−	0.314092i	$-0.898300\pi$
$41$	−2181.63	+	3778.69i	−0.202685	+	0.351060i	0.746708	+	0.665152i	$0.231633\pi$
$42$		0			0
$43$	−3121.70	+	5406.94i	−0.257466	+	0.445944i	−0.965562	−	0.260172i	$-0.916221\pi$
$43$	−3121.70	+	5406.94i	−0.257466	+	0.445944i	0.708096	+	0.706116i	$0.249554\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−12320.9	−	21340.4i	−0.813576	−	1.40915i	−0.910346	−	0.413848i	$-0.864185\pi$
$47$	−12320.9	−	21340.4i	−0.813576	−	1.40915i	0.0967705	−	0.995307i	$-0.469149\pi$
$48$		0			0
$49$	19239.7			1.14474
$50$		0			0
$51$		0			0
$52$		0			0
$53$	15497.6	+	26842.6i	0.757833	+	1.31261i	0.943953	+	0.330079i	$0.107075\pi$
$53$	15497.6	+	26842.6i	0.757833	+	1.31261i	−0.186120	+	0.982527i	$0.559591\pi$
$54$		0			0
$55$	25168.4	−	43592.9i	1.12188	−	1.94316i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	18177.7	−	31484.8i	0.679845	−	1.17753i	−0.295182	−	0.955441i	$-0.595380\pi$
$59$	18177.7	−	31484.8i	0.679845	−	1.17753i	0.975027	−	0.222085i	$-0.0712863\pi$
$60$		0			0
$61$	2886.93	+	5000.30i	0.0993370	+	0.172057i	0.911410	−	0.411499i	$-0.134994\pi$
$61$	2886.93	+	5000.30i	0.0993370	+	0.172057i	−0.812073	+	0.583555i	$0.801661\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−67221.7			−1.97345
$66$		0			0
$67$	30388.4	+	52634.2i	0.827029	+	1.43246i	0.900359	+	0.435148i	$0.143304\pi$
$67$	30388.4	+	52634.2i	0.827029	+	1.43246i	−0.0733301	+	0.997308i	$0.523363\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−14829.0	+	25684.5i	−0.349112	+	0.604680i	−0.986092	−	0.166201i	$-0.946850\pi$
$71$	−14829.0	+	25684.5i	−0.349112	+	0.604680i	0.636980	+	0.770880i	$0.280183\pi$
$72$		0			0
$73$	−39002.3	+	67553.9i	−0.856609	+	1.48369i	0.0185353	+	0.999828i	$0.494100\pi$
$73$	−39002.3	+	67553.9i	−0.856609	+	1.48369i	−0.875144	+	0.483862i	$0.839234\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	100627.			1.93413
$78$		0			0
$79$	−33427.5	+	57898.2i	−0.602611	+	1.04375i	0.389814	+	0.920894i	$0.372539\pi$
$79$	−33427.5	+	57898.2i	−0.602611	+	1.04375i	−0.992424	+	0.122858i	$0.960794\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	21919.5			0.349250			0.174625	−	0.984635i	$-0.444129\pi$
$83$	21919.5			0.349250			0.174625	+	0.984635i	$0.444129\pi$
$84$		0			0
$85$	−72616.2	−	125775.i	−1.09015	−	1.88819i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	45374.8	+	78591.5i	0.607211	+	1.05172i	0.991698	+	0.128590i	$0.0410452\pi$
$89$	45374.8	+	78591.5i	0.607211	+	1.05172i	−0.384487	+	0.923131i	$0.625622\pi$
$90$		0			0
$91$	−67190.4	−	116377.i	−0.850558	−	1.47321i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−86658.4	−	121758.i	−0.985149	−	1.38416i
$96$		0			0
$97$	−30757.3	+	53273.1i	−0.331908	+	0.574882i	−0.982886	−	0.184215i	$-0.941026\pi$
$97$	−30757.3	+	53273.1i	−0.331908	+	0.574882i	0.650978	+	0.759097i	$0.274359\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5882.91	−	10189.5i	−0.0573837	−	0.0993915i	0.835906	−	0.548872i	$-0.184943\pi$
$101$	−5882.91	−	10189.5i	−0.0573837	−	0.0993915i	−0.893290	+	0.449480i	$0.851609\pi$
$102$		0			0
$103$	−57682.4			−0.535735			−0.267868	−	0.963456i	$-0.586319\pi$
$103$	−57682.4			−0.535735			−0.267868	+	0.963456i	$0.586319\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	79892.9			0.674604			0.337302	−	0.941397i	$-0.390486\pi$
$107$	79892.9			0.674604			0.337302	+	0.941397i	$0.390486\pi$
$108$		0			0
$109$	−98323.4	+	170301.i	−0.792667	+	1.37294i	0.131643	+	0.991297i	$0.457975\pi$
$109$	−98323.4	+	170301.i	−0.792667	+	1.37294i	−0.924310	+	0.381642i	$0.875359\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−205714.			−1.51554			−0.757770	−	0.652521i	$-0.773711\pi$
$113$	−205714.			−1.51554			−0.757770	+	0.652521i	$0.773711\pi$
$114$		0			0
$115$	−94484.9			−0.666220
$116$		0			0
$117$		0			0
$118$		0			0
$119$	145165.	−	251433.i	0.939709	−	1.62762i
$120$		0			0
$121$	119854.			0.744199
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−263086.			−1.50599
$126$		0			0
$127$	−143.620	−	248.756i	−0.000790141	−	0.00136856i	0.865630	−	0.500684i	$-0.166918\pi$
$127$	−143.620	−	248.756i	−0.000790141	−	0.00136856i	−0.866420	+	0.499316i	$0.833585\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	21086.6	−	36523.1i	0.107357	−	0.185947i	−0.807342	−	0.590084i	$-0.799095\pi$
$131$	21086.6	−	36523.1i	0.107357	−	0.185947i	0.914699	+	0.404137i	$0.132428\pi$
$132$		0			0
$133$	124174.	−	271728.i	0.608698	−	1.33200i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	52318.5	+	90618.2i	0.238152	+	0.412491i	0.960184	−	0.279369i	$-0.0901251\pi$
$137$	52318.5	+	90618.2i	0.238152	+	0.412491i	−0.722032	+	0.691859i	$0.756792\pi$
$138$		0			0
$139$	64404.7	+	111552.i	0.282736	+	0.489713i	0.972058	−	0.234743i	$-0.0754247\pi$
$139$	64404.7	+	111552.i	0.282736	+	0.489713i	−0.689322	+	0.724455i	$0.742091\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−187566.	−	324874.i	−0.767033	−	1.32854i
$144$		0			0
$145$	245159.			0.968339
$146$		0			0
$147$		0			0
$148$		0			0
$149$	136506.	−	236435.i	0.503716	−	0.872462i	−0.496275	−	0.868165i	$-0.665299\pi$
$149$	136506.	−	236435.i	0.503716	−	0.872462i	0.999991	−	0.00429614i	$-0.00136751\pi$
$150$		0			0
$151$	68761.9			0.245418			0.122709	−	0.992443i	$-0.460842\pi$
$151$	68761.9			0.245418			0.122709	+	0.992443i	$0.460842\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−132527.	+	229544.i	−0.443073	+	0.767425i
$156$		0			0
$157$	275998.	−	478042.i	0.893628	−	1.54781i	0.0581337	−	0.998309i	$-0.481485\pi$
$157$	275998.	−	478042.i	0.893628	−	1.54781i	0.835494	−	0.549500i	$-0.185182\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−94440.9	−	163576.i	−0.287141	−	0.497343i
$162$		0			0
$163$	−153395.			−0.452211			−0.226106	−	0.974103i	$-0.572599\pi$
$163$	−153395.			−0.452211			−0.226106	+	0.974103i	$0.572599\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−146626.	−	253963.i	−0.406836	−	0.704660i	0.587697	−	0.809081i	$-0.300035\pi$
$167$	−146626.	−	253963.i	−0.406836	−	0.704660i	−0.994533	+	0.104421i	$0.966701\pi$
$168$		0			0
$169$	−64836.9	+	112301.i	−0.174625	+	0.302459i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	114249.	−	197885.i	0.290227	−	0.502688i	−0.683636	−	0.729823i	$-0.739602\pi$
$173$	114249.	−	197885.i	0.290227	−	0.502688i	0.973863	+	0.227135i	$0.0729358\pi$
$174$		0			0
$175$	−559619.	−	969289.i	−1.38133	−	2.39253i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−247143.			−0.576521			−0.288261	−	0.957552i	$-0.593077\pi$
$179$	−247143.			−0.576521			−0.288261	+	0.957552i	$0.593077\pi$
$180$		0			0
$181$	353864.	+	612910.i	0.802859	+	1.39059i	0.917727	+	0.397213i	$0.130022\pi$
$181$	353864.	+	612910.i	0.802859	+	1.39059i	−0.114867	+	0.993381i	$0.536644\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	343718.	−	595337.i	0.738368	−	1.27889i
$186$		0			0
$187$	405236.	−	701889.i	0.847430	−	1.46779i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10502.1			−0.0208301			−0.0104151	−	0.999946i	$-0.503315\pi$
$191$	−10502.1			−0.0208301			−0.0104151	+	0.999946i	$0.503315\pi$
$192$		0			0
$193$	496794.	−	860472.i	0.960025	−	1.66281i	0.237601	−	0.971363i	$-0.423639\pi$
$193$	496794.	−	860472.i	0.960025	−	1.66281i	0.722424	−	0.691450i	$-0.243028\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	185300.			0.340182			0.170091	−	0.985428i	$-0.445594\pi$
$197$	185300.			0.340182			0.170091	+	0.985428i	$0.445594\pi$
$198$		0			0
$199$	43149.9	+	74737.8i	0.0772408	+	0.133785i	0.902059	−	0.431614i	$-0.142056\pi$
$199$	43149.9	+	74737.8i	0.0772408	+	0.133785i	−0.824818	+	0.565399i	$0.808722\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	245045.	+	424430.i	0.417354	+	0.722879i
$204$		0			0
$205$	207198.	+	358878.i	0.344351	+	0.596433i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	346640.	−	758544.i	0.548924	−	1.20120i
$210$		0			0
$211$	151234.	−	261945.i	0.233853	−	0.405045i	−0.725086	−	0.688659i	$-0.758200\pi$
$211$	151234.	−	261945.i	0.233853	−	0.405045i	0.958939	+	0.283613i	$0.0915332\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	296481.	+	513519.i	0.437422	+	0.757636i
$216$		0			0
$217$	−529861.			−0.763858
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.08234e6			−1.49067
$222$		0			0
$223$	−339486.	+	588007.i	−0.457151	+	0.791809i	−0.998809	−	0.0487901i	$-0.984463\pi$
$223$	−339486.	+	588007.i	−0.457151	+	0.791809i	0.541658	+	0.840599i	$0.317797\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−231581.			−0.298289			−0.149145	−	0.988815i	$-0.547652\pi$
$227$	−231581.			−0.298289			−0.149145	+	0.988815i	$0.547652\pi$
$228$		0			0
$229$	33377.9			0.0420601			0.0210300	−	0.999779i	$-0.493305\pi$
$229$	33377.9			0.0420601			0.0210300	+	0.999779i	$0.493305\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−542190.	+	939100.i	−0.654277	+	1.13324i	0.327798	+	0.944748i	$0.393694\pi$
$233$	−542190.	+	939100.i	−0.654277	+	1.13324i	−0.982075	+	0.188493i	$0.939640\pi$
$234$		0			0
$235$	−2.34033e6			−2.76445
$236$		0			0
$237$		0			0
$238$		0			0
$239$	284041.			0.321652			0.160826	−	0.986983i	$-0.448584\pi$
$239$	284041.			0.321652			0.160826	+	0.986983i	$0.448584\pi$
$240$		0			0
$241$	306804.	+	531400.i	0.340266	+	0.589357i	0.984482	−	0.175486i	$-0.0561497\pi$
$241$	306804.	+	531400.i	0.340266	+	0.589357i	−0.644216	+	0.764843i	$0.722816\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	913637.	−	1.58246e6i	0.972430	−	1.68430i
$246$		0			0
$247$	−1.10873e6	+	105598.i	−1.15634	+	0.110132i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−369089.	−	639282.i	−0.369783	−	0.640484i	0.619748	−	0.784801i	$-0.287235\pi$
$251$	−369089.	−	639282.i	−0.369783	−	0.640484i	−0.989531	+	0.144317i	$0.953901\pi$
$252$		0			0
$253$	−263637.	−	456633.i	−0.258944	−	0.448504i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−686359.	−	1.18881e6i	−0.648214	−	1.12274i	−0.983549	−	0.180641i	$-0.942183\pi$
$257$	−686359.	−	1.18881e6i	−0.648214	−	1.12274i	0.335335	−	0.942099i	$-0.391151\pi$
$258$		0			0
$259$	1.37423e6			1.27295
$260$		0			0
$261$		0			0
$262$		0			0
$263$	146329.	−	253449.i	0.130449	−	0.225944i	−0.793401	−	0.608699i	$-0.791692\pi$
$263$	146329.	−	253449.i	0.130449	−	0.225944i	0.923850	+	0.382756i	$0.125025\pi$
$264$		0			0
$265$	2.94373e6			2.57504
$266$		0			0
$267$		0			0
$268$		0			0
$269$	977718.	−	1.69346e6i	0.823821	−	1.42690i	−0.0789962	−	0.996875i	$-0.525172\pi$
$269$	977718.	−	1.69346e6i	0.823821	−	1.42690i	0.902817	−	0.430025i	$-0.141495\pi$
$270$		0			0
$271$	−713502.	+	1.23582e6i	−0.590163	+	1.02219i	0.404047	+	0.914738i	$0.367603\pi$
$271$	−713502.	+	1.23582e6i	−0.590163	+	1.02219i	−0.994210	+	0.107454i	$0.965730\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.56221e6	−	2.70583e6i	−1.24568	−	2.15759i
$276$		0			0
$277$	−1.20887e6			−0.946629			−0.473315	−	0.880893i	$-0.656943\pi$
$277$	−1.20887e6			−0.946629			−0.473315	+	0.880893i	$0.656943\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	996432.	+	1.72587e6i	0.752804	+	1.30389i	0.946459	+	0.322825i	$0.104633\pi$
$281$	996432.	+	1.72587e6i	0.752804	+	1.30389i	−0.193655	+	0.981070i	$0.562034\pi$
$282$		0			0
$283$	33435.3	−	57911.6i	0.0248164	−	0.0429832i	−0.853350	−	0.521338i	$-0.825433\pi$
$283$	33435.3	−	57911.6i	0.0248164	−	0.0429832i	0.878167	+	0.478354i	$0.158767\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−414203.	+	717421.i	−0.296831	+	0.514126i
$288$		0			0
$289$	−459264.	−	795469.i	−0.323458	−	0.560246i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.98637e6			−1.35173			−0.675866	−	0.737025i	$-0.736230\pi$
$293$	−1.98637e6			−1.35173			−0.675866	+	0.737025i	$0.736230\pi$
$294$		0			0
$295$	−1.72641e6	−	2.99024e6i	−1.15502	−	2.00056i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−352072.	+	609807.i	−0.227748	+	0.394470i
$300$		0			0
$301$	−592685.	+	1.02656e6i	−0.377058	+	0.653083i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	548366.			0.337537
$306$		0			0
$307$	932152.	−	1.61454e6i	0.564470	−	0.977691i	−0.432629	−	0.901572i	$-0.642414\pi$
$307$	932152.	−	1.61454e6i	0.564470	−	0.977691i	0.997099	−	0.0761187i	$-0.0242528\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.73570e6			−1.01759			−0.508796	−	0.860887i	$-0.669909\pi$
$311$	−1.73570e6			−1.01759			−0.508796	+	0.860887i	$0.669909\pi$
$312$		0			0
$313$	26566.2	+	46014.0i	0.0153274	+	0.0265478i	0.873587	−	0.486667i	$-0.161788\pi$
$313$	26566.2	+	46014.0i	0.0153274	+	0.0265478i	−0.858260	+	0.513215i	$0.828454\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	690554.	+	1.19607e6i	0.385966	+	0.668514i	0.991903	−	0.127000i	$-0.0405347\pi$
$317$	690554.	+	1.19607e6i	0.385966	+	0.668514i	−0.605936	+	0.795513i	$0.707201\pi$
$318$		0			0
$319$	684056.	+	1.18482e6i	0.376370	+	0.651892i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.39529e6	−	1.96042e6i	−0.744144	−	1.04555i
$324$		0			0
$325$	−2.08624e6	+	3.61347e6i	−1.09561	+	1.89765i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.33924e6	−	4.05169e6i	−1.19148	−	2.06370i
$330$		0			0
$331$	−79218.6			−0.0397427			−0.0198713	−	0.999803i	$-0.506326\pi$
$331$	−79218.6			−0.0397427			−0.0198713	+	0.999803i	$0.506326\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.77222e6			2.81016
$336$		0			0
$337$	1.06653e6	−	1.84729e6i	0.511563	−	0.886053i	−0.488347	−	0.872649i	$-0.662400\pi$
$337$	1.06653e6	−	1.84729e6i	0.511563	−	0.886053i	0.999910	−	0.0134034i	$-0.00426657\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.47914e6			−0.688848
$342$		0			0
$343$	461871.			0.211976
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−988867.	+	1.71277e6i	−0.440874	+	0.763616i	−0.997755	−	0.0669767i	$-0.978665\pi$
$347$	−988867.	+	1.71277e6i	−0.440874	+	0.763616i	0.556881	+	0.830592i	$0.311998\pi$
$348$		0			0
$349$	−2.01158e6			−0.884043			−0.442022	−	0.897004i	$-0.645739\pi$
$349$	−2.01158e6			−0.884043			−0.442022	+	0.897004i	$0.645739\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.89132e6			1.23498			0.617490	−	0.786579i	$-0.288150\pi$
$353$	2.89132e6			1.23498			0.617490	+	0.786579i	$0.288150\pi$
$354$		0			0
$355$	1.40837e6	+	2.43936e6i	0.593123	+	1.02732i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	371558.	−	643557.i	0.152156	−	0.263543i	−0.779864	−	0.625950i	$-0.784712\pi$
$359$	371558.	−	643557.i	0.152156	−	0.263543i	0.932020	+	0.362407i	$0.118045\pi$
$360$		0			0
$361$	−1.62059e6	−	1.87210e6i	−0.654491	−	0.756070i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.70420e6	+	6.41587e6i	1.45533	+	2.52071i
$366$		0			0
$367$	2.21010e6	+	3.82801e6i	0.856540	+	1.48357i	0.875209	+	0.483745i	$0.160724\pi$
$367$	2.21010e6	+	3.82801e6i	0.856540	+	1.48357i	−0.0186695	+	0.999826i	$0.505943\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.94236e6	+	5.09632e6i	1.10984	+	1.92230i
$372$		0			0
$373$	−2.07890e6			−0.773680			−0.386840	−	0.922147i	$-0.626433\pi$
$373$	−2.07890e6			−0.773680			−0.386840	+	0.922147i	$0.626433\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	913517.	−	1.58226e6i	0.331027	−	0.573355i
$378$		0			0
$379$	3.87208e6			1.38467			0.692335	−	0.721576i	$-0.256582\pi$
$379$	3.87208e6			1.38467			0.692335	+	0.721576i	$0.256582\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	72504.1	−	125581.i	0.0252560	−	0.0437448i	−0.853121	−	0.521713i	$-0.825293\pi$
$383$	72504.1	−	125581.i	0.0252560	−	0.0437448i	0.878377	+	0.477968i	$0.158627\pi$
$384$		0			0
$385$	4.77846e6	−	8.27653e6i	1.64299	−	2.84575i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−670618.	−	1.16154e6i	−0.224699	−	0.389190i	0.731530	−	0.681809i	$-0.238807\pi$
$389$	−670618.	−	1.16154e6i	−0.224699	−	0.389190i	−0.956229	+	0.292619i	$0.905473\pi$
$390$		0			0
$391$	−1.52130e6			−0.503238
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.17475e6	+	5.49883e6i	1.02380	+	1.77328i
$396$		0			0
$397$	−635788.	+	1.10122e6i	−0.202458	+	0.350668i	−0.949320	−	0.314311i	$-0.898226\pi$
$397$	−635788.	+	1.10122e6i	−0.202458	+	0.350668i	0.746862	+	0.664980i	$0.231560\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−722427.	+	1.25128e6i	−0.224354	+	0.388592i	−0.956125	−	0.292958i	$-0.905360\pi$
$401$	−722427.	+	1.25128e6i	−0.224354	+	0.388592i	0.731772	+	0.681550i	$0.238694\pi$
$402$		0			0
$403$	987652.	+	1.71066e6i	0.302929	+	0.524689i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.83625e6			1.14794
$408$		0			0
$409$	306511.	+	530893.i	0.0906020	+	0.156927i	0.907765	−	0.419480i	$-0.137788\pi$
$409$	306511.	+	530893.i	0.0906020	+	0.156927i	−0.817163	+	0.576407i	$0.804454\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.45122e6	−	5.97769e6i	0.995629	−	1.72448i
$414$		0			0
$415$	1.04089e6	−	1.80288e6i	0.296679	−	0.513862i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.30914e6			−1.75564			−0.877820	−	0.478991i	$-0.841003\pi$
$419$	−6.30914e6			−1.75564			−0.877820	+	0.478991i	$0.841003\pi$
$420$		0			0
$421$	−1.28732e6	+	2.22970e6i	−0.353981	+	0.613113i	−0.986943	−	0.161070i	$-0.948506\pi$
$421$	−1.28732e6	+	2.22970e6i	−0.353981	+	0.613113i	0.632962	+	0.774183i	$0.281839\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.01463e6			−2.42089
$426$		0			0
$427$	548111.	+	949356.i	0.145478	+	0.251976i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	921074.	+	1.59535e6i	0.238837	+	0.413678i	0.960381	−	0.278691i	$-0.0899005\pi$
$431$	921074.	+	1.59535e6i	0.238837	+	0.413678i	−0.721544	+	0.692369i	$0.756567\pi$
$432$		0			0
$433$	−2.48716e6	−	4.30788e6i	−0.637505	−	1.10419i	−0.985979	−	0.166872i	$-0.946633\pi$
$433$	−2.48716e6	−	4.30788e6i	−0.637505	−	1.10419i	0.348474	−	0.937318i	$-0.386700\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.55841e6	+	148425.i	−0.390370	+	0.0371796i
$438$		0			0
$439$	1.97430e6	−	3.41959e6i	0.488936	−	0.846863i	−0.510983	−	0.859591i	$-0.670718\pi$
$439$	1.97430e6	−	3.41959e6i	0.488936	−	0.846863i	0.999919	+	0.0127283i	$0.00405165\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.42436e6	−	5.93117e6i	−0.829031	−	1.43592i	−0.898799	−	0.438360i	$-0.855560\pi$
$443$	−3.42436e6	−	5.93117e6i	−0.829031	−	1.43592i	0.0697689	−	0.997563i	$-0.477774\pi$
$444$		0			0
$445$	8.61886e6			2.06324
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.05530e6			0.715216			0.357608	−	0.933872i	$-0.383592\pi$
$449$	3.05530e6			0.715216			0.357608	+	0.933872i	$0.383592\pi$
$450$		0			0
$451$	−1.15627e6	+	2.00272e6i	−0.267682	+	0.463639i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.27627e7			−2.89011
$456$		0			0
$457$	−5.86875e6			−1.31448			−0.657242	−	0.753680i	$-0.728277\pi$
$457$	−5.86875e6			−1.31448			−0.657242	+	0.753680i	$0.728277\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−970738.	+	1.68137e6i	−0.212740	+	0.368477i	−0.952571	−	0.304316i	$-0.901572\pi$
$461$	−970738.	+	1.68137e6i	−0.212740	+	0.368477i	0.739831	+	0.672793i	$0.234905\pi$
$462$		0			0
$463$	2.37394e6			0.514656			0.257328	−	0.966324i	$-0.417158\pi$
$463$	2.37394e6			0.514656			0.257328	+	0.966324i	$0.417158\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.72608e6			0.578425			0.289212	−	0.957265i	$-0.406607\pi$
$467$	2.72608e6			0.578425			0.289212	+	0.957265i	$0.406607\pi$
$468$		0			0
$469$	5.76953e6	+	9.99312e6i	1.21118	+	2.09782i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.65452e6	+	2.86570e6i	−0.340031	+	0.588950i
$474$		0			0
$475$	−9.23449e6	+	879510.i	−1.87793	+	0.178857i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.64917e6	+	4.58849e6i	0.527558	+	0.913757i	0.999484	+	0.0321192i	$0.0102256\pi$
$479$	2.64917e6	+	4.58849e6i	0.527558	+	0.913757i	−0.471926	+	0.881638i	$0.656441\pi$
$480$		0			0
$481$	−2.56154e6	−	4.43672e6i	−0.504823	−	0.874378i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.92114e6	+	5.05957e6i	0.563895	+	0.976695i
$486$		0			0
$487$	5.45937e6			1.04309			0.521543	−	0.853225i	$-0.325356\pi$
$487$	5.45937e6			1.04309			0.521543	+	0.853225i	$0.325356\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.95495e6	−	6.85017e6i	0.740350	−	1.28232i	−0.211986	−	0.977273i	$-0.567993\pi$
$491$	3.95495e6	−	6.85017e6i	0.740350	−	1.28232i	0.952336	−	0.305051i	$-0.0986735\pi$
$492$		0			0
$493$	3.94730e6			0.731447
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.81542e6	+	4.87645e6i	−0.511272	+	0.885550i
$498$		0			0
$499$	−4.75343e6	+	8.23318e6i	−0.854586	+	1.48019i	0.0224433	+	0.999748i	$0.492855\pi$
$499$	−4.75343e6	+	8.23318e6i	−0.854586	+	1.48019i	−0.877029	+	0.480438i	$0.840478\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.72038e6	+	6.44389e6i	0.655643	+	1.13561i	0.981732	+	0.190268i	$0.0609356\pi$
$503$	3.72038e6	+	6.44389e6i	0.655643	+	1.13561i	−0.326090	+	0.945339i	$0.605731\pi$
$504$		0			0
$505$	−1.11745e6			−0.194984
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−513257.	−	888987.i	−0.0878092	−	0.152090i	0.818776	−	0.574114i	$-0.194653\pi$
$509$	−513257.	−	888987.i	−0.0878092	−	0.152090i	−0.906585	+	0.422024i	$0.861320\pi$
$510$		0			0
$511$	−7.40496e6	+	1.28258e7i	−1.25450	+	2.17286i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.73917e6	+	4.74438e6i	−0.455093	+	0.788245i
$516$		0			0
$517$	−6.53014e6	−	1.13105e7i	−1.07447	−	1.86104i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5.83600e6			−0.941936			−0.470968	−	0.882150i	$-0.656095\pi$
$521$	−5.83600e6			−0.941936			−0.470968	+	0.882150i	$0.656095\pi$
$522$		0			0
$523$	1.68243e6	+	2.91406e6i	0.268958	+	0.465848i	0.968593	−	0.248652i	$-0.0799875\pi$
$523$	1.68243e6	+	2.91406e6i	0.268958	+	0.465848i	−0.699635	+	0.714500i	$0.746654\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.13382e6	+	3.69588e6i	−0.334681	+	0.579684i
$528$		0			0
$529$	2.72331e6	−	4.71691e6i	0.423114	−	0.732856i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.08827e6			0.470866
$534$		0			0
$535$	3.79388e6	−	6.57119e6i	0.573058	−	0.992566i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.01971e7			1.51184
$540$		0			0
$541$	4.79104e6	+	8.29833e6i	0.703780	+	1.21898i	0.967130	+	0.254282i	$0.0818393\pi$
$541$	4.79104e6	+	8.29833e6i	0.703780	+	1.21898i	−0.263350	+	0.964700i	$0.584827\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.33818e6	+	1.61742e7i	1.34670	+	2.33255i
$546$		0			0
$547$	−2.53036e6	−	4.38271e6i	−0.361588	−	0.626289i	0.626634	−	0.779314i	$-0.284432\pi$
$547$	−2.53036e6	−	4.38271e6i	−0.361588	−	0.626289i	−0.988222	+	0.153024i	$0.951099\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4.04357e6	−	385117.i	0.567396	−	0.0540398i
$552$		0			0
$553$	−6.34654e6	+	1.09925e7i	−0.882520	+	1.52857i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.89885e6	+	6.75300e6i	0.532474	+	0.922271i	0.999281	+	0.0379124i	$0.0120708\pi$
$557$	3.89885e6	+	6.75300e6i	0.532474	+	0.922271i	−0.466807	+	0.884359i	$0.654596\pi$
$558$		0			0
$559$	4.41902e6			0.598131
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.30439e6			−0.572323			−0.286161	−	0.958181i	$-0.592379\pi$
$563$	−4.30439e6			−0.572323			−0.286161	+	0.958181i	$0.592379\pi$
$564$		0			0
$565$	−9.76875e6	+	1.69200e7i	−1.28741	+	2.22986i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9.79289e6			1.26803			0.634016	−	0.773320i	$-0.281405\pi$
$569$	9.79289e6			1.26803			0.634016	+	0.773320i	$0.281405\pi$
$570$		0			0
$571$	−6.14671e6			−0.788955			−0.394477	−	0.918906i	$-0.629074\pi$
$571$	−6.14671e6			−0.788955			−0.394477	+	0.918906i	$0.629074\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.93236e6	+	5.07899e6i	−0.369869	+	0.640631i
$576$		0			0
$577$	−704051.			−0.0880369			−0.0440185	−	0.999031i	$-0.514016\pi$
$577$	−704051.			−0.0880369			−0.0440185	+	0.999031i	$0.514016\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.16163e6			0.511474
$582$		0			0
$583$	8.21378e6	+	1.42267e7i	1.00086	+	1.73353i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.80763e6	+	8.32707e6i	−0.575885	+	0.997463i	0.420059	+	0.907497i	$0.362009\pi$
$587$	−4.80763e6	+	8.32707e6i	−0.575885	+	0.997463i	−0.995945	+	0.0899662i	$0.971324\pi$
$588$		0			0
$589$	−1.82527e6	+	3.99421e6i	−0.216790	+	0.474398i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5.57887e6	−	9.66289e6i	−0.651493	−	1.12842i	−0.982761	−	0.184882i	$-0.940810\pi$
$593$	−5.57887e6	−	9.66289e6i	−0.651493	−	1.12842i	0.331268	−	0.943537i	$-0.392524\pi$
$594$		0			0
$595$	−1.37869e7	−	2.38796e7i	−1.59652	−	2.76525i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.00898e6	−	6.94376e6i	−0.456528	−	0.790729i	0.542247	−	0.840219i	$-0.317574\pi$
$599$	−4.00898e6	−	6.94376e6i	−0.456528	−	0.790729i	−0.998775	+	0.0494902i	$0.984240\pi$
$600$		0			0
$601$	1.34115e7			1.51458			0.757290	−	0.653079i	$-0.226523\pi$
$601$	1.34115e7			1.51458			0.757290	+	0.653079i	$0.226523\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.69151e6	−	9.85799e6i	0.632178	−	1.09496i
$606$		0			0
$607$	3.14504e6			0.346461			0.173231	−	0.984881i	$-0.444579\pi$
$607$	3.14504e6			0.346461			0.173231	+	0.984881i	$0.444579\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.72061e6	+	1.51045e7i	−0.945027	+	1.63683i
$612$		0			0
$613$	1.86161e6	−	3.22440e6i	0.200095	−	0.346575i	−0.748464	−	0.663176i	$-0.769208\pi$
$613$	1.86161e6	−	3.22440e6i	0.200095	−	0.346575i	0.948559	+	0.316601i	$0.102542\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	650699.	+	1.12704e6i	0.0688125	+	0.119187i	0.898379	−	0.439221i	$-0.144746\pi$
$617$	650699.	+	1.12704e6i	0.0688125	+	0.119187i	−0.829566	+	0.558408i	$0.811412\pi$
$618$		0			0
$619$	1.44161e7			1.51224			0.756118	−	0.654435i	$-0.227093\pi$
$619$	1.44161e7			1.51224			0.756118	+	0.654435i	$0.227093\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.61485e6	+	1.49214e7i	0.889257	+	1.54024i
$624$		0			0
$625$	−3.28211e6	+	5.68478e6i	−0.336088	+	0.582122i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.53420e6	−	9.58552e6i	0.557736	−	0.966026i
$630$		0			0
$631$	1.56517e6	+	2.71095e6i	0.156490	+	0.271049i	0.933601	−	0.358315i	$-0.116649\pi$
$631$	1.56517e6	+	2.71095e6i	0.156490	+	0.271049i	−0.777110	+	0.629364i	$0.783315\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−27280.3			−0.00268482
$636$		0			0
$637$	−6.80883e6	−	1.17932e7i	−0.664851	−	1.15156i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.12325e6	−	3.67757e6i	0.204106	−	0.353522i	−0.745742	−	0.666235i	$-0.767905\pi$
$641$	2.12325e6	−	3.67757e6i	0.204106	−	0.353522i	0.949848	+	0.312713i	$0.101238\pi$
$642$		0			0
$643$	5.08032e6	−	8.79938e6i	0.484578	−	0.839314i	−0.515265	−	0.857031i	$-0.672306\pi$
$643$	5.08032e6	−	8.79938e6i	0.484578	−	0.839314i	0.999843	+	0.0177168i	$0.00563972\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.33300e6			−0.313022			−0.156511	−	0.987676i	$-0.550025\pi$
$647$	−3.33300e6			−0.313022			−0.156511	+	0.987676i	$0.550025\pi$
$648$		0			0
$649$	9.63429e6	−	1.66871e7i	0.897859	−	1.55514i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.56244e6			0.326938			0.163469	−	0.986549i	$-0.447732\pi$
$653$	3.56244e6			0.326938			0.163469	+	0.986549i	$0.447732\pi$
$654$		0			0
$655$	−2.00269e6	−	3.46875e6i	−0.182394	−	0.315915i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.65732e6	−	1.67270e7i	−0.866249	−	1.50039i	−0.865801	−	0.500388i	$-0.833191\pi$
$659$	−9.65732e6	−	1.67270e7i	−0.866249	−	1.50039i	−0.000448081	−	1.00000i	$-0.500143\pi$
$660$		0			0
$661$	5.55793e6	+	9.62662e6i	0.494777	+	0.856979i	0.999982	−	0.00602036i	$-0.00191635\pi$
$661$	5.55793e6	+	9.62662e6i	0.494777	+	0.856979i	−0.505205	+	0.863000i	$0.668583\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.64529e7	−	2.31169e7i	−1.44274	−	2.02710i
$666$		0			0
$667$	1.28401e6	−	2.22398e6i	0.111752	−	0.193560i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.53008e6	+	2.65018e6i	0.131193	+	0.227232i
$672$		0			0
$673$	4.82842e6			0.410929			0.205465	−	0.978665i	$-0.434129\pi$
$673$	4.82842e6			0.410929			0.205465	+	0.978665i	$0.434129\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.20527e6			0.268777			0.134389	−	0.990929i	$-0.457093\pi$
$677$	3.20527e6			0.268777			0.134389	+	0.990929i	$0.457093\pi$
$678$		0			0
$679$	−5.83956e6	+	1.01144e7i	−0.486078	+	0.841912i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.29678e7			−1.88395			−0.941973	−	0.335688i	$-0.891031\pi$
$683$	−2.29678e7			−1.88395			−0.941973	+	0.335688i	$0.891031\pi$
$684$		0			0
$685$	9.93780e6			0.809215
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.09690e7	−	1.89989e7i	0.880278	−	1.52469i
$690$		0			0
$691$	7.10876e6			0.566368			0.283184	−	0.959066i	$-0.408609\pi$
$691$	7.10876e6			0.566368			0.283184	+	0.959066i	$0.408609\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.22336e7			0.960707
$696$		0			0
$697$	3.33610e6	+	5.77829e6i	0.260110	+	0.450523i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7.73351e6	+	1.33948e7i	−0.594403	+	1.02954i	0.399227	+	0.916852i	$0.369278\pi$
$701$	−7.73351e6	+	1.33948e7i	−0.594403	+	1.02954i	−0.993631	+	0.112685i	$0.964055\pi$
$702$		0			0
$703$	4.73397e6	−	1.03592e7i	0.361274	−	0.790569i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.11693e6	−	1.93458e6i	−0.0840381	−	0.145558i
$708$		0			0
$709$	6.30537e6	+	1.09212e7i	0.471081	+	0.815936i	0.999453	−	0.0330772i	$-0.0105307\pi$
$709$	6.30537e6	+	1.09212e7i	0.471081	+	0.815936i	−0.528372	+	0.849013i	$0.677197\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.38822e6	+	2.40446e6i	0.102266	+	0.177131i
$714$		0			0
$715$	−3.56278e7			−2.60630
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.02749e7	−	1.77967e7i	0.741234	−	1.28386i	−0.210699	−	0.977551i	$-0.567574\pi$
$719$	1.02749e7	−	1.77967e7i	0.741234	−	1.28386i	0.951934	−	0.306304i	$-0.0990926\pi$
$720$		0			0
$721$	−1.09516e7			−0.784581
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7.60855e6	−	1.31784e7i	0.537597	−	0.931146i
$726$		0			0
$727$	−8.28557e6	+	1.43510e7i	−0.581415	+	1.00704i	0.413897	+	0.910324i	$0.364167\pi$
$727$	−8.28557e6	+	1.43510e7i	−0.581415	+	1.00704i	−0.995312	+	0.0967169i	$0.969166\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.77363e6	+	8.26818e6i	0.330412	+	0.572290i
$732$		0			0
$733$	2.52568e7			1.73628			0.868139	−	0.496321i	$-0.165316\pi$
$733$	2.52568e7			1.73628			0.868139	+	0.496321i	$0.165316\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.61060e7	+	2.78964e7i	1.09224	+	1.89182i
$738$		0			0
$739$	6.54248e6	−	1.13319e7i	0.440688	−	0.763294i	−0.557053	−	0.830477i	$-0.688068\pi$
$739$	6.54248e6	−	1.13319e7i	0.440688	−	0.763294i	0.997741	+	0.0671833i	$0.0214012\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.18603e6	+	2.05426e6i	−0.0788177	+	0.136516i	−0.902740	−	0.430186i	$-0.858448\pi$
$743$	−1.18603e6	+	2.05426e6i	−0.0788177	+	0.136516i	0.823922	+	0.566703i	$0.191781\pi$
$744$		0			0
$745$	−1.29645e7	−	2.24552e7i	−0.855787	−	1.48227i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.51684e7			0.987953
$750$		0			0
$751$	−1.34919e7	−	2.33686e7i	−0.872916	−	1.51193i	−0.858966	−	0.512032i	$-0.828893\pi$
$751$	−1.34919e7	−	2.33686e7i	−0.872916	−	1.51193i	−0.0139493	−	0.999903i	$-0.504440\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.26530e6	−	5.65567e6i	0.208476	−	0.361091i
$756$		0			0
$757$	1.46311e7	−	2.53419e7i	0.927980	−	1.60731i	0.141284	−	0.989969i	$-0.454877\pi$
$757$	1.46311e7	−	2.53419e7i	0.927980	−	1.60731i	0.786697	−	0.617340i	$-0.211790\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.62075e7			1.64045			0.820226	−	0.572040i	$-0.193848\pi$
$761$	2.62075e7			1.64045			0.820226	+	0.572040i	$0.193848\pi$
$762$		0			0
$763$	−1.86677e7	+	3.23333e7i	−1.16086	+	2.01066i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.57321e7			−1.57938
$768$		0			0
$769$	2.54995e6	+	4.41664e6i	0.155495	+	0.269325i	0.933239	−	0.359256i	$-0.116969\pi$
$769$	2.54995e6	+	4.41664e6i	0.155495	+	0.269325i	−0.777744	+	0.628581i	$0.783636\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8.67472e6	+	1.50250e7i	0.522164	+	0.904414i	0.999668	+	0.0257842i	$0.00820828\pi$
$773$	8.67472e6	+	1.50250e7i	0.522164	+	0.904414i	−0.477504	+	0.878630i	$0.658458\pi$
$774$		0			0
$775$	8.22601e6	+	1.42479e7i	0.491966	+	0.852110i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.98122e6	+	5.59373e6i	0.235057	+	0.330262i
$780$		0			0
$781$	−7.85942e6	+	1.36129e7i	−0.461066	+	0.798589i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.62126e7	−	4.54016e7i	−1.51823	−	2.62965i
$786$		0			0
$787$	3.89085e6			0.223928			0.111964	−	0.993712i	$-0.464286\pi$
$787$	3.89085e6			0.223928			0.111964	+	0.993712i	$0.464286\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.90568e7			−2.21950
$792$		0			0
$793$	2.04334e6	−	3.53917e6i	0.115387	−	0.199856i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.46516e6			0.137467			0.0687336	−	0.997635i	$-0.478104\pi$
$797$	2.46516e6			0.137467			0.0687336	+	0.997635i	$0.478104\pi$
$798$		0			0
$799$	−3.76817e7			−2.08816
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.06714e7	+	3.58039e7i	−1.13131	+	1.95948i
$804$		0			0
$805$	−1.79389e7			−0.975676
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.17605e7			−0.631762			−0.315881	−	0.948799i	$-0.602300\pi$
$809$	−1.17605e7			−0.631762			−0.315881	+	0.948799i	$0.602300\pi$
$810$		0			0
$811$	4.70469e6	+	8.14875e6i	0.251176	+	0.435050i	0.963850	−	0.266446i	$-0.0858493\pi$
$811$	4.70469e6	+	8.14875e6i	0.251176	+	0.435050i	−0.712674	+	0.701496i	$0.752516\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.28427e6	+	1.26167e7i	−0.384142	+	0.665353i
$816$		0			0
$817$	5.69674e6	+	8.00409e6i	0.298587	+	0.419524i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.22161e6	−	5.57999e6i	−0.166807	−	0.288918i	0.770488	−	0.637454i	$-0.220012\pi$
$821$	−3.22161e6	−	5.57999e6i	−0.166807	−	0.288918i	−0.937296	+	0.348536i	$0.886679\pi$
$822$		0			0
$823$	5.61870e6	+	9.73187e6i	0.289158	+	0.500837i	0.973609	−	0.228222i	$-0.0732912\pi$
$823$	5.61870e6	+	9.73187e6i	0.289158	+	0.500837i	−0.684451	+	0.729059i	$0.739958\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−5.80309e6	−	1.00513e7i	−0.295050	−	0.511042i	0.679946	−	0.733262i	$-0.262003\pi$
$827$	−5.80309e6	−	1.00513e7i	−0.295050	−	0.511042i	−0.974997	+	0.222220i	$0.928670\pi$
$828$		0			0
$829$	−2.34279e7			−1.18399			−0.591993	−	0.805943i	$-0.701659\pi$
$829$	−2.34279e7			−1.18399			−0.591993	+	0.805943i	$0.701659\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.47105e7	−	2.54793e7i	0.734537	−	1.27226i
$834$		0			0
$835$	−2.78513e7			−1.38239
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.33046e7	−	2.30442e7i	0.652523	−	1.13020i	−0.329986	−	0.943986i	$-0.607044\pi$
$839$	1.33046e7	−	2.30442e7i	0.652523	−	1.13020i	0.982509	−	0.186217i	$-0.0596227\pi$
$840$		0			0
$841$	6.92396e6	−	1.19927e7i	0.337571	−	0.584690i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.15783e6	+	1.06657e7i	0.296678	+	0.513862i
$846$		0			0
$847$	2.27554e7			1.08988
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.60043e6	−	6.23613e6i	−0.170424	−	0.295183i
$852$		0			0
$853$	1.75001e7	−	3.03111e7i	0.823510	−	1.42636i	−0.0795426	−	0.996831i	$-0.525346\pi$
$853$	1.75001e7	−	3.03111e7i	0.823510	−	1.42636i	0.903053	−	0.429530i	$-0.141321\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.77062e6	−	8.26296e6i	0.221882	−	0.384312i	−0.733497	−	0.679692i	$-0.762113\pi$
$857$	4.77062e6	−	8.26296e6i	0.221882	−	0.384312i	0.955380	+	0.295381i	$0.0954466\pi$
$858$		0			0
$859$	2.79816e6	+	4.84656e6i	0.129387	+	0.224105i	0.923439	−	0.383745i	$-0.125366\pi$
$859$	2.79816e6	+	4.84656e6i	0.129387	+	0.224105i	−0.794052	+	0.607849i	$0.792032\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−6.93765e6			−0.317092			−0.158546	−	0.987352i	$-0.550681\pi$
$863$	−6.93765e6			−0.317092			−0.158546	+	0.987352i	$0.550681\pi$
$864$		0			0
$865$	−1.08507e7	−	1.87940e7i	−0.493081	−	0.854041i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.77168e7	+	3.06863e7i	−0.795856	+	1.37846i
$870$		0			0
$871$	2.15086e7	−	3.72540e7i	0.960653	−	1.66390i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.99494e7			−2.20552
$876$		0			0
$877$	−5.50749e6	+	9.53926e6i	−0.241799	+	0.418809i	−0.961227	−	0.275759i	$-0.911071\pi$
$877$	−5.50749e6	+	9.53926e6i	−0.241799	+	0.418809i	0.719428	+	0.694567i	$0.244404\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	3.01700e7			1.30959			0.654796	−	0.755806i	$-0.272754\pi$
$881$	3.01700e7			1.30959			0.654796	+	0.755806i	$0.272754\pi$
$882$		0			0
$883$	1.52557e7	+	2.64237e7i	0.658464	+	1.14049i	0.981013	+	0.193940i	$0.0621267\pi$
$883$	1.52557e7	+	2.64237e7i	0.658464	+	1.14049i	−0.322550	+	0.946552i	$0.604540\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.89963e6	−	6.75436e6i	−0.166423	−	0.288254i	0.770736	−	0.637154i	$-0.219889\pi$
$887$	−3.89963e6	−	6.75436e6i	−0.166423	−	0.288254i	−0.937160	+	0.348900i	$0.886555\pi$
$888$		0			0
$889$	−27267.6	−	47228.8i	−0.00115716	−	0.00200425i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3.86007e7	+	3.67641e6i	−1.61982	+	0.154275i
$894$		0			0
$895$	−1.17361e7	+	2.03275e7i	−0.489740	+	0.848254i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.60198e6	−	6.23882e6i	−0.148642	−	0.257456i
$900$		0			0
$901$	4.73971e7			1.94509
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.72158e7			2.72803
$906$		0			0
$907$	−3.48982e6	+	6.04454e6i	−0.140859	+	0.243975i	−0.927820	−	0.373027i	$-0.878320\pi$
$907$	−3.48982e6	+	6.04454e6i	−0.140859	+	0.243975i	0.786961	+	0.617002i	$0.211653\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.39353e7			0.955528			0.477764	−	0.878488i	$-0.341447\pi$
$911$	2.39353e7			0.955528			0.477764	+	0.878488i	$0.341447\pi$
$912$		0			0
$913$	1.16175e7			0.461247
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.00350e6	−	6.93427e6i	0.157223	−	0.272319i
$918$		0			0
$919$	−9.03520e6			−0.352898			−0.176449	−	0.984310i	$-0.556461\pi$
$919$	−9.03520e6			−0.352898			−0.176449	+	0.984310i	$0.556461\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.09916e7			0.811037
$924$		0			0
$925$	−2.13347e7	−	3.69528e7i	−0.819846	−	1.42002i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.55430e6	−	2.69212e6i	0.0590874	−	0.102342i	−0.834969	−	0.550298i	$-0.814514\pi$
$929$	1.55430e6	−	2.69212e6i	0.0590874	−	0.102342i	0.894056	+	0.447955i	$0.147848\pi$
$930$		0			0
$931$	1.25834e7	−	2.75359e7i	0.475798	−	1.04118i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.84869e7	−	6.66613e7i	−1.43974	−	2.49370i
$936$		0			0
$937$	−1.85807e7	−	3.21827e7i	−0.691374	−	1.19750i	−0.971388	−	0.237499i	$-0.923672\pi$
$937$	−1.85807e7	−	3.21827e7i	−0.691374	−	1.19750i	0.280013	−	0.959996i	$-0.409661\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.14651e6	−	1.41102e7i	−0.299915	−	0.519467i	0.676202	−	0.736717i	$-0.263625\pi$
$941$	−8.14651e6	−	1.41102e7i	−0.299915	−	0.519467i	−0.976116	+	0.217249i	$0.930292\pi$
$942$		0			0
$943$	4.34078e6			0.158960
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.90876e7	+	3.30607e7i	−0.691634	+	1.19795i	0.279668	+	0.960097i	$0.409776\pi$
$947$	−1.90876e7	+	3.30607e7i	−0.691634	+	1.19795i	−0.971302	+	0.237849i	$0.923558\pi$
$948$		0			0
$949$	5.52108e7			1.99003
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.93217e6	−	6.81072e6i	0.140249	−	0.242919i	−0.787341	−	0.616517i	$-0.788543\pi$
$953$	3.93217e6	−	6.81072e6i	0.140249	−	0.242919i	0.927590	+	0.373599i	$0.121876\pi$
$954$		0			0
$955$	−498713.	+	863795.i	−0.0176946	+	0.0306480i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	9.93317e6	+	1.72047e7i	0.348772	+	0.604090i
$960$		0			0
$961$	−2.08406e7			−0.727949
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.71825e7	−	8.17225e7i	−1.63103	−	2.82503i
$966$		0			0
$967$	1.96743e6	−	3.40769e6i	0.0676601	−	0.117191i	−0.830211	−	0.557450i	$-0.811780\pi$
$967$	1.96743e6	−	3.40769e6i	0.0676601	−	0.117191i	0.897871	+	0.440259i	$0.145113\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.28377e6	−	3.95561e6i	0.0777329	−	0.134637i	−0.824539	−	0.565806i	$-0.808565\pi$
$971$	2.28377e6	−	3.95561e6i	0.0777329	−	0.134637i	0.902271	+	0.431168i	$0.141899\pi$
$972$		0			0
$973$	1.22279e7	+	2.11793e7i	0.414065	+	0.717181i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.77743e7			0.595739			0.297870	−	0.954607i	$-0.403724\pi$
$977$	1.77743e7			0.595739			0.297870	+	0.954607i	$0.403724\pi$
$978$		0			0
$979$	2.40489e7	+	4.16539e7i	0.801933	+	1.38899i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−9.38981e6	+	1.62636e7i	−0.309937	+	0.536826i	−0.978348	−	0.206966i	$-0.933641\pi$
$983$	−9.38981e6	+	1.62636e7i	−0.309937	+	0.536826i	0.668412	+	0.743792i	$0.266975\pi$
$984$		0			0
$985$	8.79937e6	−	1.52410e7i	0.288975	−	0.500520i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.21124e6			0.201924
$990$		0			0
$991$	−2.05047e7	+	3.55152e7i	−0.663239	+	1.14876i	0.316521	+	0.948586i	$0.397485\pi$
$991$	−2.05047e7	+	3.55152e7i	−0.663239	+	1.14876i	−0.979760	+	0.200178i	$0.935848\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.19624e6			0.262456
$996$		0			0
$997$	−1.23888e7	−	2.14579e7i	−0.394720	−	0.683676i	0.598345	−	0.801239i	$-0.295825\pi$
$997$	−1.23888e7	−	2.14579e7i	−0.394720	−	0.683676i	−0.993065	+	0.117563i	$0.962492\pi$
$998$		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.505.9		18
3.2	odd	2		76.6.e.a.49.3	yes	18
12.11	even	2		304.6.i.d.49.7		18
19.7	even	3	inner	684.6.k.f.577.9		18
57.26	odd	6		76.6.e.a.45.3	✓	18
228.83	even	6		304.6.i.d.273.7		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.e.a.45.3	✓	18	57.26	odd	6
76.6.e.a.49.3	yes	18	3.2	odd	2
304.6.i.d.49.7		18	12.11	even	2
304.6.i.d.273.7		18	228.83	even	6
684.6.k.f.505.9		18	1.1	even	1	trivial
684.6.k.f.577.9		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)

\(f(q)\)	\(=\)	\(q+(47.4871 - 82.2500i) q^{5} +189.860 q^{7} +O(q^{10})\)	\(q+(47.4871 - 82.2500i) q^{5} +189.860 q^{7} +530.005 q^{11} +(-353.895 - 612.964i) q^{13} +(764.589 - 1324.31i) q^{17} +(654.031 - 1431.20i) q^{19} +(-497.425 - 861.565i) q^{23} +(-2947.54 - 5105.29i) q^{25} +(1290.66 + 2235.49i) q^{29} -2790.81 q^{31} +(9015.88 - 15616.0i) q^{35} +7238.14 q^{37} +(-2181.63 + 3778.69i) q^{41} +(-3121.70 + 5406.94i) q^{43} +(-12320.9 - 21340.4i) q^{47} +19239.7 q^{49} +(15497.6 + 26842.6i) q^{53} +(25168.4 - 43592.9i) q^{55} +(18177.7 - 31484.8i) q^{59} +(2886.93 + 5000.30i) q^{61} -67221.7 q^{65} +(30388.4 + 52634.2i) q^{67} +(-14829.0 + 25684.5i) q^{71} +(-39002.3 + 67553.9i) q^{73} +100627. q^{77} +(-33427.5 + 57898.2i) q^{79} +21919.5 q^{83} +(-72616.2 - 125775. i) q^{85} +(45374.8 + 78591.5i) q^{89} +(-67190.4 - 116377. i) q^{91} +(-86658.4 - 121758. i) q^{95} +(-30757.3 + 53273.1i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 11 q^{5} + 336 q^{7}+O(q^{10}) \) 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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