LMFDB - Embedded newform 684.6.k.f.505.3

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			505.3
Root			$-30.6056 - 54.7425i$ of defining polynomial
Character	$\chi$	$=$	684.505
Dual form			684.6.k.f.577.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-31.6056 + 54.7425i) q^{5} -80.2775 q^{7} +O(q^{10})$	$q+(-31.6056 + 54.7425i) q^{5} -80.2775 q^{7} +475.169 q^{11} +(-337.473 - 584.520i) q^{13} +(-866.499 + 1500.82i) q^{17} +(-574.366 - 1464.99i) q^{19} +(2424.72 + 4199.73i) q^{23} +(-435.326 - 754.007i) q^{25} +(-2394.08 - 4146.67i) q^{29} -127.218 q^{31} +(2537.22 - 4394.59i) q^{35} -13949.4 q^{37} +(-7883.24 + 13654.2i) q^{41} +(964.112 - 1669.89i) q^{43} +(-8099.51 - 14028.8i) q^{47} -10362.5 q^{49} +(-8925.98 - 15460.3i) q^{53} +(-15018.0 + 26011.9i) q^{55} +(-8424.75 + 14592.1i) q^{59} +(11641.7 + 20164.0i) q^{61} +42664.1 q^{65} +(13618.4 + 23587.8i) q^{67} +(37449.3 - 64864.1i) q^{71} +(34900.9 - 60450.1i) q^{73} -38145.4 q^{77} +(11408.5 - 19760.1i) q^{79} +58008.8 q^{83} +(-54772.4 - 94868.6i) q^{85} +(-8203.84 - 14209.5i) q^{89} +(27091.5 + 46923.8i) q^{91} +(98350.5 + 14859.7i) q^{95} +(65710.8 - 113814. i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−31.6056	+	54.7425i	−0.565378	+	0.979263i	0.431637	+	0.902048i	$0.357936\pi$
$5$	−31.6056	+	54.7425i	−0.565378	+	0.979263i	−0.997014	+	0.0772156i	$0.975397\pi$
$6$		0			0
$7$	−80.2775			−0.619225			−0.309613	−	0.950863i	$-0.600199\pi$
$7$	−80.2775			−0.619225			−0.309613	+	0.950863i	$0.600199\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	475.169			1.18404			0.592020	−	0.805923i	$-0.298331\pi$
$11$	475.169			1.18404			0.592020	+	0.805923i	$0.298331\pi$
$12$		0			0
$13$	−337.473	−	584.520i	−0.553835	−	0.959270i	−0.997993	−	0.0633218i	$-0.979831\pi$
$13$	−337.473	−	584.520i	−0.553835	−	0.959270i	0.444158	−	0.895948i	$-0.353503\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−866.499	+	1500.82i	−0.727186	+	1.25952i	0.230881	+	0.972982i	$0.425839\pi$
$17$	−866.499	+	1500.82i	−0.727186	+	1.25952i	−0.958068	+	0.286542i	$0.907494\pi$
$18$		0			0
$19$	−574.366	−	1464.99i	−0.365010	−	0.931004i
$19$	−574.366	−	1464.99i	−0.365010	−	0.931004i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2424.72	+	4199.73i	0.955743	+	1.65540i	0.732658	+	0.680597i	$0.238279\pi$
$23$	2424.72	+	4199.73i	0.955743	+	1.65540i	0.223085	+	0.974799i	$0.428387\pi$
$24$		0			0
$25$	−435.326	−	754.007i	−0.139304	−	0.241282i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2394.08	−	4146.67i	−0.528620	−	0.915597i	−0.999443	−	0.0333693i	$-0.989376\pi$
$29$	−2394.08	−	4146.67i	−0.528620	−	0.915597i	0.470823	−	0.882228i	$-0.343957\pi$
$30$		0			0
$31$	−127.218			−0.0237764			−0.0118882	−	0.999929i	$-0.503784\pi$
$31$	−127.218			−0.0237764			−0.0118882	+	0.999929i	$0.503784\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2537.22	−	4394.59i	0.350096	−	0.606384i
$36$		0			0
$37$	−13949.4			−1.67514			−0.837570	−	0.546331i	$-0.816024\pi$
$37$	−13949.4			−1.67514			−0.837570	+	0.546331i	$0.816024\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7883.24	+	13654.2i	−0.732394	+	1.26854i	0.223463	+	0.974712i	$0.428264\pi$
$41$	−7883.24	+	13654.2i	−0.732394	+	1.26854i	−0.955857	+	0.293831i	$0.905070\pi$
$42$		0			0
$43$	964.112	−	1669.89i	0.0795163	−	0.137726i	−0.823525	−	0.567280i	$-0.807996\pi$
$43$	964.112	−	1669.89i	0.0795163	−	0.137726i	0.903041	+	0.429554i	$0.141329\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8099.51	−	14028.8i	−0.534828	−	0.926349i	−0.999172	−	0.0406941i	$-0.987043\pi$
$47$	−8099.51	−	14028.8i	−0.534828	−	0.926349i	0.464344	−	0.885655i	$-0.346290\pi$
$48$		0			0
$49$	−10362.5			−0.616560
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−8925.98	−	15460.3i	−0.436482	−	0.756009i	0.560933	−	0.827861i	$-0.310442\pi$
$53$	−8925.98	−	15460.3i	−0.436482	−	0.756009i	−0.997415	+	0.0718520i	$0.977109\pi$
$54$		0			0
$55$	−15018.0	+	26011.9i	−0.669430	+	1.15949i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8424.75	+	14592.1i	−0.315084	+	0.545742i	−0.979455	−	0.201661i	$-0.935366\pi$
$59$	−8424.75	+	14592.1i	−0.315084	+	0.545742i	0.664371	+	0.747403i	$0.268699\pi$
$60$		0			0
$61$	11641.7	+	20164.0i	0.400583	+	0.693830i	0.993796	−	0.111215i	$-0.0354743\pi$
$61$	11641.7	+	20164.0i	0.400583	+	0.693830i	−0.593214	+	0.805045i	$0.702141\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	42664.1			1.25250
$66$		0			0
$67$	13618.4	+	23587.8i	0.370630	+	0.641950i	0.989663	−	0.143415i	$-0.0458085\pi$
$67$	13618.4	+	23587.8i	0.370630	+	0.641950i	−0.619033	+	0.785365i	$0.712475\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	37449.3	−	64864.1i	0.881654	−	1.52707i	0.0321524	−	0.999483i	$-0.489764\pi$
$71$	37449.3	−	64864.1i	0.881654	−	1.52707i	0.849501	−	0.527586i	$-0.176903\pi$
$72$		0			0
$73$	34900.9	−	60450.1i	0.766530	−	1.32767i	−0.172903	−	0.984939i	$-0.555315\pi$
$73$	34900.9	−	60450.1i	0.766530	−	1.32767i	0.939434	−	0.342731i	$-0.111352\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−38145.4			−0.733188
$78$		0			0
$79$	11408.5	−	19760.1i	0.205665	−	0.356222i	−0.744680	−	0.667422i	$-0.767398\pi$
$79$	11408.5	−	19760.1i	0.205665	−	0.356222i	0.950344	+	0.311200i	$0.100731\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	58008.8			0.924270			0.462135	−	0.886810i	$-0.347084\pi$
$83$	58008.8			0.924270			0.462135	+	0.886810i	$0.347084\pi$
$84$		0			0
$85$	−54772.4	−	94868.6i	−0.822270	−	1.42421i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8203.84	−	14209.5i	−0.109785	−	0.190153i	0.805898	−	0.592054i	$-0.201683\pi$
$89$	−8203.84	−	14209.5i	−0.109785	−	0.190153i	−0.915683	+	0.401901i	$0.868349\pi$
$90$		0			0
$91$	27091.5	+	46923.8i	0.342948	+	0.594004i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	98350.5	+	14859.7i	1.11807	+	0.168928i
$96$		0			0
$97$	65710.8	−	113814.i	0.709100	−	1.22820i	−0.256092	−	0.966653i	$-0.582435\pi$
$97$	65710.8	−	113814.i	0.709100	−	1.22820i	0.965191	−	0.261544i	$-0.0842318\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	46068.0	+	79792.2i	0.449362	+	0.778318i	0.998345	−	0.0575159i	$-0.0183180\pi$
$101$	46068.0	+	79792.2i	0.449362	+	0.778318i	−0.548983	+	0.835834i	$0.684985\pi$
$102$		0			0
$103$	149801.			1.39130			0.695651	−	0.718380i	$-0.255116\pi$
$103$	149801.			1.39130			0.695651	+	0.718380i	$0.255116\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−62370.7			−0.526649			−0.263324	−	0.964707i	$-0.584819\pi$
$107$	−62370.7			−0.526649			−0.263324	+	0.964707i	$0.584819\pi$
$108$		0			0
$109$	−27370.0	+	47406.1i	−0.220652	+	0.382180i	−0.955006	−	0.296586i	$-0.904152\pi$
$109$	−27370.0	+	47406.1i	−0.220652	+	0.382180i	0.734354	+	0.678767i	$0.237485\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	84395.3			0.621759			0.310879	−	0.950449i	$-0.399376\pi$
$113$	84395.3			0.621759			0.310879	+	0.950449i	$0.399376\pi$
$114$		0			0
$115$	−306538.			−2.16142
$116$		0			0
$117$		0			0
$118$		0			0
$119$	69560.4	−	120482.i	0.450292	−	0.779929i
$120$		0			0
$121$	64734.8			0.401952
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−142500.			−0.815717
$126$		0			0
$127$	59200.7	+	102539.i	0.325700	+	0.564128i	0.981654	−	0.190672i	$-0.0610668\pi$
$127$	59200.7	+	102539.i	0.325700	+	0.564128i	−0.655954	+	0.754801i	$0.727734\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	37228.3	−	64481.4i	0.189538	−	0.328289i	−0.755559	−	0.655081i	$-0.772634\pi$
$131$	37228.3	−	64481.4i	0.189538	−	0.328289i	0.945096	+	0.326792i	$0.105968\pi$
$132$		0			0
$133$	46108.6	+	117606.i	0.226023	+	0.576501i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	70227.1	+	121637.i	0.319671	+	0.553686i	0.980419	−	0.196921i	$-0.0630943\pi$
$137$	70227.1	+	121637.i	0.319671	+	0.553686i	−0.660748	+	0.750607i	$0.729761\pi$
$138$		0			0
$139$	−112322.	−	194548.i	−0.493093	−	0.854061i	0.506876	−	0.862019i	$-0.330800\pi$
$139$	−112322.	−	194548.i	−0.493093	−	0.854061i	−0.999968	+	0.00795774i	$0.997467\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−160357.	−	277746.i	−0.655763	−	1.13581i
$144$		0			0
$145$	302665.			1.19548
$146$		0			0
$147$		0			0
$148$		0			0
$149$	146994.	−	254601.i	0.542417	−	0.939494i	−0.456347	−	0.889802i	$-0.650843\pi$
$149$	146994.	−	254601.i	0.542417	−	0.939494i	0.998765	−	0.0496923i	$-0.0158241\pi$
$150$		0			0
$151$	366703.			1.30880			0.654398	−	0.756150i	$-0.272922\pi$
$151$	366703.			1.30880			0.654398	+	0.756150i	$0.272922\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4020.81	−	6964.26i	0.0134426	−	0.0232833i
$156$		0			0
$157$	14338.2	−	24834.5i	0.0464243	−	0.0804093i	−0.841880	−	0.539666i	$-0.818551\pi$
$157$	14338.2	−	24834.5i	0.0464243	−	0.0804093i	0.888304	+	0.459256i	$0.151884\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−194650.	−	337144.i	−0.591820	−	1.02506i
$162$		0			0
$163$	12767.3			0.0376383			0.0188192	−	0.999823i	$-0.494009\pi$
$163$	12767.3			0.0376383			0.0188192	+	0.999823i	$0.494009\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	121547.	+	210526.i	0.337252	+	0.584138i	0.983915	−	0.178638i	$-0.0571692\pi$
$167$	121547.	+	210526.i	0.337252	+	0.584138i	−0.646663	+	0.762776i	$0.723836\pi$
$168$		0			0
$169$	−42129.2	+	72969.8i	−0.113466	+	0.196529i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	213672.	−	370091.i	0.542792	−	0.940143i	−0.455951	−	0.890005i	$-0.650701\pi$
$173$	213672.	−	370091.i	0.542792	−	0.940143i	0.998742	−	0.0501376i	$-0.0159660\pi$
$174$		0			0
$175$	34946.9	+	60529.7i	0.0862607	+	0.149408i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−41174.1			−0.0960488			−0.0480244	−	0.998846i	$-0.515293\pi$
$179$	−41174.1			−0.0960488			−0.0480244	+	0.998846i	$0.515293\pi$
$180$		0			0
$181$	−241411.	−	418136.i	−0.547723	−	0.948684i	−0.998430	−	0.0560122i	$-0.982161\pi$
$181$	−241411.	−	418136.i	−0.547723	−	0.948684i	0.450707	−	0.892672i	$-0.351172\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	440879.	−	763624.i	0.947087	−	1.64040i
$186$		0			0
$187$	−411734.	+	713144.i	−0.861018	+	1.49133i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−218201.			−0.432786			−0.216393	−	0.976306i	$-0.569429\pi$
$191$	−218201.			−0.432786			−0.216393	+	0.976306i	$0.569429\pi$
$192$		0			0
$193$	357570.	−	619329.i	0.690983	−	1.19682i	−0.280533	−	0.959844i	$-0.590511\pi$
$193$	357570.	−	619329.i	0.690983	−	1.19682i	0.971516	−	0.236973i	$-0.0761554\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−17740.5			−0.0325686			−0.0162843	−	0.999867i	$-0.505184\pi$
$197$	−17740.5			−0.0325686			−0.0162843	+	0.999867i	$0.505184\pi$
$198$		0			0
$199$	−290101.	−	502469.i	−0.519297	−	0.899449i	−0.999748	−	0.0224275i	$-0.992861\pi$
$199$	−290101.	−	502469.i	−0.519297	−	0.899449i	0.480451	−	0.877021i	$-0.340473\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	192191.	+	332884.i	0.327335	+	0.566961i
$204$		0			0
$205$	−498309.	−	863096.i	−0.828159	−	1.43441i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−272921.	−	696119.i	−0.432186	−	1.10235i
$210$		0			0
$211$	418159.	−	724272.i	0.646599	−	1.11994i	−0.337330	−	0.941386i	$-0.609524\pi$
$211$	418159.	−	724272.i	0.646599	−	1.11994i	0.983930	−	0.178556i	$-0.0571427\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	60942.6	+	105556.i	0.0899135	+	0.155735i
$216$		0			0
$217$	10212.8			0.0147229
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.16968e6			1.61096
$222$		0			0
$223$	349535.	−	605413.i	0.470684	−	0.815248i	−0.528754	−	0.848775i	$-0.677341\pi$
$223$	349535.	−	605413.i	0.470684	−	0.815248i	0.999438	+	0.0335271i	$0.0106740\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−813403.			−1.04771			−0.523855	−	0.851808i	$-0.675507\pi$
$227$	−813403.			−1.04771			−0.523855	+	0.851808i	$0.675507\pi$
$228$		0			0
$229$	−1.18902e6			−1.49830			−0.749151	−	0.662399i	$-0.769538\pi$
$229$	−1.18902e6			−1.49830			−0.749151	+	0.662399i	$0.769538\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	106736.	−	184872.i	0.128801	−	0.223091i	−0.794411	−	0.607381i	$-0.792220\pi$
$233$	106736.	−	184872.i	0.128801	−	0.223091i	0.923212	+	0.384290i	$0.125554\pi$
$234$		0			0
$235$	1.02396e6			1.20952
$236$		0			0
$237$		0			0
$238$		0			0
$239$	965775.			1.09366			0.546829	−	0.837245i	$-0.315835\pi$
$239$	965775.			1.09366			0.546829	+	0.837245i	$0.315835\pi$
$240$		0			0
$241$	391235.	+	677639.i	0.433906	+	0.751547i	0.997206	−	0.0747059i	$-0.0238018\pi$
$241$	391235.	+	677639.i	0.433906	+	0.751547i	−0.563300	+	0.826252i	$0.690468\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	327514.	−	567270.i	0.348590	−	0.603775i
$246$		0			0
$247$	−662485.	+	830123.i	−0.690929	+	0.865765i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	448780.	+	777310.i	0.449624	+	0.778771i	0.998361	−	0.0572236i	$-0.0182248\pi$
$251$	448780.	+	777310.i	0.449624	+	0.778771i	−0.548738	+	0.835995i	$0.684891\pi$
$252$		0			0
$253$	1.15215e6	+	1.99558e6i	1.13164	+	1.96006i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	552328.	+	956660.i	0.521632	+	0.903493i	0.999683	+	0.0251612i	$0.00800989\pi$
$257$	552328.	+	956660.i	0.521632	+	0.903493i	−0.478051	+	0.878332i	$0.658657\pi$
$258$		0			0
$259$	1.11982e6			1.03729
$260$		0			0
$261$		0			0
$262$		0			0
$263$	306009.	−	530023.i	0.272800	−	0.472504i	−0.696778	−	0.717287i	$-0.745384\pi$
$263$	306009.	−	530023.i	0.272800	−	0.472504i	0.969578	+	0.244783i	$0.0787169\pi$
$264$		0			0
$265$	1.12844e6			0.987109
$266$		0			0
$267$		0			0
$268$		0			0
$269$	248820.	−	430969.i	0.209655	−	0.363133i	−0.741951	−	0.670454i	$-0.766099\pi$
$269$	248820.	−	430969.i	0.209655	−	0.363133i	0.951606	+	0.307321i	$0.0994326\pi$
$270$		0			0
$271$	724980.	−	1.25570e6i	0.599657	−	1.03864i	−0.393214	−	0.919447i	$-0.628637\pi$
$271$	724980.	−	1.25570e6i	0.599657	−	1.03864i	0.992871	−	0.119190i	$-0.0380297\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−206853.	−	358281.i	−0.164942	−	0.285688i
$276$		0			0
$277$	−87028.2			−0.0681492			−0.0340746	−	0.999419i	$-0.510848\pi$
$277$	−87028.2			−0.0681492			−0.0340746	+	0.999419i	$0.510848\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−949563.	−	1.64469e6i	−0.717394	−	1.24256i	−0.962029	−	0.272948i	$-0.912001\pi$
$281$	−949563.	−	1.64469e6i	−0.717394	−	1.24256i	0.244635	−	0.969615i	$-0.421332\pi$
$282$		0			0
$283$	−727473.	+	1.26002e6i	−0.539946	+	0.935214i	0.458960	+	0.888457i	$0.348222\pi$
$283$	−727473.	+	1.26002e6i	−0.539946	+	0.935214i	−0.998906	+	0.0467574i	$0.985111\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	632846.	−	1.09612e6i	0.453517	−	0.785514i
$288$		0			0
$289$	−791713.	−	1.37129e6i	−0.557600	−	0.965792i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−795288.			−0.541197			−0.270598	−	0.962692i	$-0.587222\pi$
$293$	−795288.			−0.541197			−0.270598	+	0.962692i	$0.587222\pi$
$294$		0			0
$295$	−532538.	−	922383.i	−0.356283	−	0.617101i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.63655e6	−	2.83459e6i	1.05865	−	1.83363i
$300$		0			0
$301$	−77396.5	+	134055.i	−0.0492385	+	0.0852836i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.47177e6			−0.905923
$306$		0			0
$307$	964887.	−	1.67123e6i	0.584293	−	1.01202i	−0.410670	−	0.911784i	$-0.634705\pi$
$307$	964887.	−	1.67123e6i	0.584293	−	1.01202i	0.994963	−	0.100241i	$-0.0319613\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.42463e6			−0.835220			−0.417610	−	0.908626i	$-0.637132\pi$
$311$	−1.42463e6			−0.835220			−0.417610	+	0.908626i	$0.637132\pi$
$312$		0			0
$313$	1.12004e6	+	1.93997e6i	0.646209	+	1.11927i	0.984021	+	0.178053i	$0.0569800\pi$
$313$	1.12004e6	+	1.93997e6i	0.646209	+	1.11927i	−0.337812	+	0.941214i	$0.609687\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	278493.	+	482363.i	0.155656	+	0.269604i	0.933298	−	0.359104i	$-0.116918\pi$
$317$	278493.	+	482363.i	0.155656	+	0.269604i	−0.777642	+	0.628708i	$0.783584\pi$
$318$		0			0
$319$	−1.13759e6	−	1.97037e6i	−0.625908	−	1.08410i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.69638e6	+	407395.i	1.43805	+	0.217275i
$324$		0			0
$325$	−293821.	+	508913.i	−0.154303	+	0.267261i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	650208.	+	1.12619e6i	0.331179	+	0.573619i
$330$		0			0
$331$	−1.85079e6			−0.928511			−0.464256	−	0.885701i	$-0.653678\pi$
$331$	−1.85079e6			−0.928511			−0.464256	+	0.885701i	$0.653678\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.72168e6			−0.838184
$336$		0			0
$337$	1.16626e6	−	2.02003e6i	0.559399	−	0.968907i	−0.438148	−	0.898903i	$-0.644365\pi$
$337$	1.16626e6	−	2.02003e6i	0.559399	−	0.968907i	0.997547	−	0.0700041i	$-0.0223013\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−60450.3			−0.0281522
$342$		0			0
$343$	2.18110e6			1.00101
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.33391e6	+	2.31041e6i	−0.594708	+	1.03006i	0.398880	+	0.917003i	$0.369399\pi$
$347$	−1.33391e6	+	2.31041e6i	−0.594708	+	1.03006i	−0.993588	+	0.113062i	$0.963934\pi$
$348$		0			0
$349$	−2.33528e6			−1.02630			−0.513152	−	0.858298i	$-0.671522\pi$
$349$	−2.33528e6			−1.02630			−0.513152	+	0.858298i	$0.671522\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.57168e6			1.09845			0.549225	−	0.835675i	$-0.314923\pi$
$353$	2.57168e6			1.09845			0.549225	+	0.835675i	$0.314923\pi$
$354$		0			0
$355$	2.36722e6	+	4.10014e6i	0.996935	+	1.72674i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.06014e6	−	3.56826e6i	0.843645	−	1.46124i	−0.0431473	−	0.999069i	$-0.513738\pi$
$359$	2.06014e6	−	3.56826e6i	0.843645	−	1.46124i	0.886793	−	0.462168i	$-0.152928\pi$
$360$		0			0
$361$	−1.81631e6	+	1.68288e6i	−0.733536	+	0.679651i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.20613e6	+	3.82112e6i	0.866759	+	1.50127i
$366$		0			0
$367$	−1.76851e6	−	3.06315e6i	−0.685398	−	1.18714i	−0.973312	−	0.229487i	$-0.926295\pi$
$367$	−1.76851e6	−	3.06315e6i	−0.685398	−	1.18714i	0.287914	−	0.957656i	$-0.407038\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	716555.	+	1.24111e6i	0.270281	+	0.468140i
$372$		0			0
$373$	−3.14142e6			−1.16910			−0.584552	−	0.811356i	$-0.698730\pi$
$373$	−3.14142e6			−1.16910			−0.584552	+	0.811356i	$0.698730\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.61587e6	+	2.79878e6i	−0.585537	+	1.01418i
$378$		0			0
$379$	1.24166e6			0.444021			0.222011	−	0.975044i	$-0.428738\pi$
$379$	1.24166e6			0.444021			0.222011	+	0.975044i	$0.428738\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.41220e6	+	2.44600e6i	−0.491924	+	0.852038i	−0.999957	−	0.00929990i	$-0.997040\pi$
$383$	−1.41220e6	+	2.44600e6i	−0.491924	+	0.852038i	0.508032	+	0.861338i	$0.330373\pi$
$384$		0			0
$385$	1.20561e6	−	2.08817e6i	0.414528	−	0.717984i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.26664e6	−	3.92593e6i	−0.759465	−	1.31543i	−0.943124	−	0.332442i	$-0.892127\pi$
$389$	−2.26664e6	−	3.92593e6i	−0.759465	−	1.31543i	0.183658	−	0.982990i	$-0.441206\pi$
$390$		0			0
$391$	−8.40405e6			−2.78001
$392$		0			0
$393$		0			0
$394$		0			0
$395$	721143.	+	1.24906e6i	0.232557	+	0.402800i
$396$		0			0
$397$	968813.	−	1.67803e6i	0.308506	−	0.534348i	−0.669530	−	0.742785i	$-0.733504\pi$
$397$	968813.	−	1.67803e6i	0.308506	−	0.534348i	0.978036	+	0.208437i	$0.0668377\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.21857e6	+	3.84268e6i	−0.688990	+	1.19336i	0.283176	+	0.959068i	$0.408612\pi$
$401$	−2.21857e6	+	3.84268e6i	−0.688990	+	1.19336i	−0.972165	+	0.234297i	$0.924721\pi$
$402$		0			0
$403$	42932.8	+	74361.7i	0.0131682	+	0.0228080i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.62832e6			−1.98343
$408$		0			0
$409$	3.20721e6	+	5.55506e6i	0.948025	+	1.64203i	0.749579	+	0.661915i	$0.230256\pi$
$409$	3.20721e6	+	5.55506e6i	0.948025	+	1.64203i	0.198446	+	0.980112i	$0.436411\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	676317.	−	1.17142e6i	0.195108	−	0.337937i
$414$		0			0
$415$	−1.83340e6	+	3.17555e6i	−0.522562	+	0.905103i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.16492e6			−0.602429			−0.301215	−	0.953556i	$-0.597392\pi$
$419$	−2.16492e6			−0.602429			−0.301215	+	0.953556i	$0.597392\pi$
$420$		0			0
$421$	3.39931e6	−	5.88777e6i	0.934727	−	1.61900i	0.159608	−	0.987180i	$-0.448977\pi$
$421$	3.39931e6	−	5.88777e6i	0.934727	−	1.61900i	0.775119	−	0.631815i	$-0.217690\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.50884e6			0.405201
$426$		0			0
$427$	−934567.	−	1.61872e6i	−0.248051	−	0.429637i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−101699.	−	176148.i	−0.0263708	−	0.0456756i	0.852539	−	0.522664i	$-0.175062\pi$
$431$	−101699.	−	176148.i	−0.0263708	−	0.0456756i	−0.878910	+	0.476988i	$0.841728\pi$
$432$		0			0
$433$	487850.	+	844981.i	0.125045	+	0.216584i	0.921751	−	0.387783i	$-0.126759\pi$
$433$	487850.	+	844981.i	0.125045	+	0.216584i	−0.796705	+	0.604368i	$0.793426\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.75990e6	−	5.96437e6i	1.19232	−	1.49404i
$438$		0			0
$439$	−1.96789e6	+	3.40848e6i	−0.487348	+	0.844111i	−0.999894	−	0.0145483i	$-0.995369\pi$
$439$	−1.96789e6	+	3.40848e6i	−0.487348	+	0.844111i	0.512546	+	0.858660i	$0.328702\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.39091e6	−	2.40913e6i	−0.336737	−	0.583246i	0.647080	−	0.762422i	$-0.275990\pi$
$443$	−1.39091e6	−	2.40913e6i	−0.336737	−	0.583246i	−0.983817	+	0.179176i	$0.942657\pi$
$444$		0			0
$445$	1.03715e6			0.248279
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.91764e6			−1.61936			−0.809678	−	0.586875i	$-0.800358\pi$
$449$	−6.91764e6			−1.61936			−0.809678	+	0.586875i	$0.800358\pi$
$450$		0			0
$451$	−3.74587e6	+	6.48804e6i	−0.867184	+	1.50201i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.42497e6			−0.775582
$456$		0			0
$457$	6.44928e6			1.44451			0.722256	−	0.691626i	$-0.243105\pi$
$457$	6.44928e6			1.44451			0.722256	+	0.691626i	$0.243105\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−610341.	+	1.05714e6i	−0.133758	+	0.231676i	−0.925122	−	0.379669i	$-0.876038\pi$
$461$	−610341.	+	1.05714e6i	−0.133758	+	0.231676i	0.791364	+	0.611345i	$0.209371\pi$
$462$		0			0
$463$	847245.			0.183678			0.0918388	−	0.995774i	$-0.470726\pi$
$463$	847245.			0.183678			0.0918388	+	0.995774i	$0.470726\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	76212.7			0.0161709			0.00808547	−	0.999967i	$-0.497426\pi$
$467$	76212.7			0.0161709			0.00808547	+	0.999967i	$0.497426\pi$
$468$		0			0
$469$	−1.09325e6	−	1.89357e6i	−0.229503	−	0.397512i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	458116.	−	793481.i	0.0941505	−	0.163074i
$474$		0			0
$475$	−854578.	+	1.07082e6i	−0.173787	+	0.217763i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.33045e6	+	2.30441e6i	0.264947	+	0.458902i	0.967550	−	0.252680i	$-0.0813121\pi$
$479$	1.33045e6	+	2.30441e6i	0.264947	+	0.458902i	−0.702602	+	0.711583i	$0.747979\pi$
$480$		0			0
$481$	4.70754e6	+	8.15370e6i	0.927750	+	1.60691i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.15366e6	+	7.19434e6i	0.801819	+	1.38879i
$486$		0			0
$487$	5.81426e6			1.11089			0.555446	−	0.831552i	$-0.312547\pi$
$487$	5.81426e6			1.11089			0.555446	+	0.831552i	$0.312547\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.27968e6	+	3.94853e6i	−0.426747	+	0.739148i	−0.996582	−	0.0826113i	$-0.973674\pi$
$491$	−2.27968e6	+	3.94853e6i	−0.426747	+	0.739148i	0.569834	+	0.821760i	$0.307007\pi$
$492$		0			0
$493$	8.29787e6			1.53762
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.00634e6	+	5.20713e6i	−0.545942	+	0.945600i
$498$		0			0
$499$	3.38613e6	−	5.86495e6i	0.608769	−	1.05442i	−0.382675	−	0.923883i	$-0.624997\pi$
$499$	3.38613e6	−	5.86495e6i	0.608769	−	1.05442i	0.991444	−	0.130536i	$-0.0416697\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.62169e6	−	4.54089e6i	−0.462020	−	0.800242i	0.537042	−	0.843556i	$-0.319542\pi$
$503$	−2.62169e6	−	4.54089e6i	−0.462020	−	0.800242i	−0.999062	+	0.0433140i	$0.986208\pi$
$504$		0			0
$505$	−5.82403e6			−1.01624
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.68578e6	+	8.11601e6i	0.801655	+	1.38851i	0.918526	+	0.395360i	$0.129380\pi$
$509$	4.68578e6	+	8.11601e6i	0.801655	+	1.38851i	−0.116872	+	0.993147i	$0.537287\pi$
$510$		0			0
$511$	−2.80175e6	+	4.85278e6i	−0.474655	+	0.822126i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.73455e6	+	8.20047e6i	−0.786611	+	1.36245i
$516$		0			0
$517$	−3.84864e6	−	6.66604e6i	−0.633258	−	1.09683i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−244496.			−0.0394619			−0.0197309	−	0.999805i	$-0.506281\pi$
$521$	−244496.			−0.0394619			−0.0197309	+	0.999805i	$0.506281\pi$
$522$		0			0
$523$	−2.04939e6	−	3.54964e6i	−0.327619	−	0.567453i	0.654420	−	0.756131i	$-0.272913\pi$
$523$	−2.04939e6	−	3.54964e6i	−0.327619	−	0.567453i	−0.982039	+	0.188678i	$0.939580\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	110235.	−	190932.i	0.0172899	−	0.0299469i
$528$		0			0
$529$	−8.54032e6	+	1.47923e7i	−1.32689	+	2.29824i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.06415e7			1.62250
$534$		0			0
$535$	1.97126e6	−	3.41433e6i	0.297756	−	0.515728i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.92395e6			−0.730032
$540$		0			0
$541$	227501.	+	394043.i	0.0334187	+	0.0578829i	0.882251	−	0.470779i	$-0.156027\pi$
$541$	227501.	+	394043.i	0.0334187	+	0.0578829i	−0.848832	+	0.528662i	$0.822694\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.73009e6	−	2.99660e6i	−0.249503	−	0.432153i
$546$		0			0
$547$	−4.19066e6	−	7.25843e6i	−0.598844	−	1.03723i	−0.992992	−	0.118182i	$-0.962293\pi$
$547$	−4.19066e6	−	7.25843e6i	−0.598844	−	1.03723i	0.394148	−	0.919047i	$-0.371040\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.69976e6	+	5.88901e6i	−0.659473	+	0.826349i
$552$		0			0
$553$	−915844.	+	1.58629e6i	−0.127353	+	0.220582i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.14702e6	+	3.71874e6i	0.293222	+	0.507876i	0.974570	−	0.224084i	$-0.0719390\pi$
$557$	2.14702e6	+	3.71874e6i	0.293222	+	0.507876i	−0.681347	+	0.731960i	$0.738606\pi$
$558$		0			0
$559$	−1.30145e6			−0.176156
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.43360e6			0.722465			0.361232	−	0.932476i	$-0.382356\pi$
$563$	5.43360e6			0.722465			0.361232	+	0.932476i	$0.382356\pi$
$564$		0			0
$565$	−2.66736e6	+	4.62001e6i	−0.351529	+	0.608866i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.53732e6			−0.587514			−0.293757	−	0.955880i	$-0.594906\pi$
$569$	−4.53732e6			−0.587514			−0.293757	+	0.955880i	$0.594906\pi$
$570$		0			0
$571$	−1.75132e6			−0.224789			−0.112395	−	0.993664i	$-0.535852\pi$
$571$	−1.75132e6			−0.224789			−0.112395	+	0.993664i	$0.535852\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.11108e6	−	3.65650e6i	0.266278	−	0.461208i
$576$		0			0
$577$	2.69353e6			0.336808			0.168404	−	0.985718i	$-0.446139\pi$
$577$	2.69353e6			0.336808			0.168404	+	0.985718i	$0.446139\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.65680e6			−0.572331
$582$		0			0
$583$	−4.24135e6	−	7.34624e6i	−0.516812	−	0.895145i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.95802e6	+	3.39138e6i	−0.234542	+	0.406239i	−0.959140	−	0.282934i	$-0.908692\pi$
$587$	−1.95802e6	+	3.39138e6i	−0.234542	+	0.406239i	0.724597	+	0.689172i	$0.242026\pi$
$588$		0			0
$589$	73069.9	+	186374.i	0.00867862	+	0.0221359i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.66866e6	+	9.81841e6i	0.661978	+	1.14658i	0.980095	+	0.198528i	$0.0636162\pi$
$593$	5.66866e6	+	9.81841e6i	0.661978	+	1.14658i	−0.318117	+	0.948051i	$0.603050\pi$
$594$		0			0
$595$	4.39699e6	+	7.61581e6i	0.509170	+	0.881909i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.86543e6	−	6.69512e6i	−0.440180	−	0.762414i	0.557522	−	0.830162i	$-0.311752\pi$
$599$	−3.86543e6	−	6.69512e6i	−0.440180	−	0.762414i	−0.997702	+	0.0677475i	$0.978419\pi$
$600$		0			0
$601$	−6.87953e6			−0.776913			−0.388457	−	0.921467i	$-0.626992\pi$
$601$	−6.87953e6			−0.776913			−0.388457	+	0.921467i	$0.626992\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.04598e6	+	3.54374e6i	−0.227255	+	0.393617i
$606$		0			0
$607$	1.19598e7			1.31750			0.658750	−	0.752362i	$-0.271085\pi$
$607$	1.19598e7			1.31750			0.658750	+	0.752362i	$0.271085\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.46673e6	+	9.46865e6i	−0.592413	+	1.02609i
$612$		0			0
$613$	1.49181e6	−	2.58389e6i	0.160347	−	0.277730i	−0.774646	−	0.632395i	$-0.782072\pi$
$613$	1.49181e6	−	2.58389e6i	0.160347	−	0.277730i	0.934993	+	0.354666i	$0.115405\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.62649e6	+	4.54922e6i	0.277756	+	0.481087i	0.970827	−	0.239782i	$-0.0770761\pi$
$617$	2.62649e6	+	4.54922e6i	0.277756	+	0.481087i	−0.693071	+	0.720869i	$0.743743\pi$
$618$		0			0
$619$	−1.09695e7			−1.15069			−0.575346	−	0.817910i	$-0.695133\pi$
$619$	−1.09695e7			−1.15069			−0.575346	+	0.817910i	$0.695133\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	658583.	+	1.14070e6i	0.0679815	+	0.117747i
$624$		0			0
$625$	5.86419e6	−	1.01571e7i	0.600493	−	1.04008i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.20871e7	−	2.09355e7i	1.21814	−	2.10988i
$630$		0			0
$631$	−3.73436e6	−	6.46811e6i	−0.373373	−	0.646701i	0.616709	−	0.787191i	$-0.288466\pi$
$631$	−3.73436e6	−	6.46811e6i	−0.373373	−	0.646701i	−0.990082	+	0.140490i	$0.955132\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.48429e6			−0.736573
$636$		0			0
$637$	3.49707e6	+	6.05710e6i	0.341473	+	0.591448i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.26419e6	−	1.43140e7i	0.794430	−	1.37599i	−0.128771	−	0.991674i	$-0.541103\pi$
$641$	8.26419e6	−	1.43140e7i	0.794430	−	1.37599i	0.923201	−	0.384318i	$-0.125563\pi$
$642$		0			0
$643$	−6.29383e6	+	1.09012e7i	−0.600326	+	1.03980i	0.392445	+	0.919775i	$0.371629\pi$
$643$	−6.29383e6	+	1.09012e7i	−0.600326	+	1.03980i	−0.992771	+	0.120020i	$0.961704\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.21912e7			−1.14494			−0.572472	−	0.819924i	$-0.694015\pi$
$647$	−1.21912e7			−1.14494			−0.572472	+	0.819924i	$0.694015\pi$
$648$		0			0
$649$	−4.00318e6	+	6.93371e6i	−0.373073	+	0.646181i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.52249e6			0.690365			0.345182	−	0.938536i	$-0.387817\pi$
$653$	7.52249e6			0.690365			0.345182	+	0.938536i	$0.387817\pi$
$654$		0			0
$655$	2.35325e6	+	4.07594e6i	0.214321	+	0.371214i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.02028e6	−	1.38915e7i	−0.719409	−	1.24605i	−0.961234	−	0.275733i	$-0.911079\pi$
$659$	−8.02028e6	−	1.38915e7i	−0.719409	−	1.24605i	0.241825	−	0.970320i	$-0.422254\pi$
$660$		0			0
$661$	6.26368e6	+	1.08490e7i	0.557604	+	0.965798i	0.997696	+	0.0678454i	$0.0216125\pi$
$661$	6.26368e6	+	1.08490e7i	0.557604	+	0.965798i	−0.440092	+	0.897953i	$0.645054\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−7.89533e6	−	1.19290e6i	−0.692335	−	0.104605i
$666$		0			0
$667$	1.16099e7	−	2.01090e7i	1.01045	−	1.75015i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.53178e6	+	9.58133e6i	0.474306	+	0.821523i
$672$		0			0
$673$	−7.50054e6			−0.638344			−0.319172	−	0.947697i	$-0.603405\pi$
$673$	−7.50054e6			−0.638344			−0.319172	+	0.947697i	$0.603405\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.19677e7			−1.00355			−0.501777	−	0.864997i	$-0.667320\pi$
$677$	−1.19677e7			−1.00355			−0.501777	+	0.864997i	$0.667320\pi$
$678$		0			0
$679$	−5.27510e6	+	9.13673e6i	−0.439092	+	0.760530i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.11577e6			−0.665699			−0.332850	−	0.942980i	$-0.608010\pi$
$683$	−8.11577e6			−0.665699			−0.332850	+	0.942980i	$0.608010\pi$
$684$		0			0
$685$	−8.87827e6			−0.722939
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.02455e6	+	1.04348e7i	−0.483478	+	0.837408i
$690$		0			0
$691$	−1.23034e7			−0.980235			−0.490117	−	0.871657i	$-0.663046\pi$
$691$	−1.23034e7			−0.980235			−0.490117	+	0.871657i	$0.663046\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.42000e7			1.11513
$696$		0			0
$697$	−1.36616e7	−	2.36626e7i	−1.06517	−	1.84494i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	7.04732e6	−	1.22063e7i	0.541663	−	0.938188i	−0.457146	−	0.889392i	$-0.651128\pi$
$701$	7.04732e6	−	1.22063e7i	0.541663	−	0.938188i	0.998809	−	0.0487959i	$-0.0155384\pi$
$702$		0			0
$703$	8.01205e6	+	2.04358e7i	0.611442	+	1.55956i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.69823e6	−	6.40552e6i	−0.278256	−	0.481954i
$708$		0			0
$709$	−2.13108e6	−	3.69113e6i	−0.159215	−	0.275768i	0.775371	−	0.631506i	$-0.217563\pi$
$709$	−2.13108e6	−	3.69113e6i	−0.159215	−	0.275768i	−0.934586	+	0.355738i	$0.884230\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−308469.	−	534283.i	−0.0227241	−	0.0393593i
$714$		0			0
$715$	2.02727e7			1.48302
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.85763e6	+	1.18778e7i	−0.494711	+	0.856865i	−0.999981	−	0.00609644i	$-0.998059\pi$
$719$	−6.85763e6	+	1.18778e7i	−0.494711	+	0.856865i	0.505270	+	0.862961i	$0.331393\pi$
$720$		0			0
$721$	−1.20256e7			−0.861529
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.08441e6	+	3.61031e6i	−0.147278	+	0.255093i
$726$		0			0
$727$	−1.05919e7	+	1.83457e7i	−0.743256	+	1.28736i	0.207749	+	0.978182i	$0.433386\pi$
$727$	−1.05919e7	+	1.83457e7i	−0.743256	+	1.28736i	−0.951005	+	0.309175i	$0.899947\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.67080e6	+	2.89392e6i	0.115646	+	0.200305i
$732$		0			0
$733$	6.01308e6			0.413368			0.206684	−	0.978408i	$-0.433733\pi$
$733$	6.01308e6			0.413368			0.206684	+	0.978408i	$0.433733\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.47107e6	+	1.12082e7i	0.438841	+	0.760095i
$738$		0			0
$739$	−8.41424e6	+	1.45739e7i	−0.566766	+	0.981667i	0.430117	+	0.902773i	$0.358472\pi$
$739$	−8.41424e6	+	1.45739e7i	−0.566766	+	0.981667i	−0.996883	+	0.0788942i	$0.974861\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.01935e6	−	3.49762e6i	0.134196	−	0.232434i	−0.791094	−	0.611695i	$-0.790488\pi$
$743$	2.01935e6	−	3.49762e6i	0.134196	−	0.232434i	0.925290	+	0.379260i	$0.123821\pi$
$744$		0			0
$745$	9.29165e6	+	1.60936e7i	0.613341	+	1.06234i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.00696e6			0.326114
$750$		0			0
$751$	−351347.	−	608550.i	−0.0227319	−	0.0393728i	0.854436	−	0.519557i	$-0.173903\pi$
$751$	−351347.	−	608550.i	−0.0227319	−	0.0393728i	−0.877168	+	0.480184i	$0.840570\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.15899e7	+	2.00742e7i	−0.739964	+	1.28166i
$756$		0			0
$757$	1.19882e7	−	2.07642e7i	0.760354	−	1.31697i	−0.182314	−	0.983240i	$-0.558359\pi$
$757$	1.19882e7	−	2.07642e7i	0.760354	−	1.31697i	0.942668	−	0.333731i	$-0.108308\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.29063e7			0.807869			0.403934	−	0.914788i	$-0.367642\pi$
$761$	1.29063e7			0.807869			0.403934	+	0.914788i	$0.367642\pi$
$762$		0			0
$763$	2.19719e6	−	3.80565e6i	0.136633	−	0.236656i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.13725e7			0.698019
$768$		0			0
$769$	−1.05367e7	−	1.82501e7i	−0.642524	−	1.11288i	−0.984867	−	0.173310i	$-0.944554\pi$
$769$	−1.05367e7	−	1.82501e7i	−0.642524	−	1.11288i	0.342343	−	0.939575i	$-0.388779\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.35234e6	−	7.53848e6i	−0.261984	−	0.453769i	0.704785	−	0.709421i	$-0.251043\pi$
$773$	−4.35234e6	−	7.53848e6i	−0.261984	−	0.453769i	−0.966769	+	0.255652i	$0.917710\pi$
$774$		0			0
$775$	55381.5	+	95923.6i	0.00331215	+	0.00573682i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.45311e7	+	3.70640e6i	1.44835	+	0.218831i
$780$		0			0
$781$	1.77948e7	−	3.08214e7i	1.04391	−	1.80811i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	906334.	+	1.56982e6i	0.0524946	+	0.0909232i
$786$		0			0
$787$	−2.69329e6			−0.155005			−0.0775026	−	0.996992i	$-0.524695\pi$
$787$	−2.69329e6			−0.155005			−0.0775026	+	0.996992i	$0.524695\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.77504e6			−0.385009
$792$		0			0
$793$	7.85752e6	−	1.36096e7i	0.443713	−	0.768534i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.67101e6			0.260474			0.130237	−	0.991483i	$-0.458426\pi$
$797$	4.67101e6			0.260474			0.130237	+	0.991483i	$0.458426\pi$
$798$		0			0
$799$	2.80729e7			1.55568
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.65838e7	−	2.87240e7i	0.907603	−	1.57201i
$804$		0			0
$805$	2.46081e7			1.33841
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.82792e7			0.981941			0.490970	−	0.871176i	$-0.336642\pi$
$809$	1.82792e7			0.981941			0.490970	+	0.871176i	$0.336642\pi$
$810$		0			0
$811$	−9.38560e6	−	1.62563e7i	−0.501083	−	0.867902i	−0.999999	−	0.00125119i	$-0.999602\pi$
$811$	−9.38560e6	−	1.62563e7i	−0.501083	−	0.867902i	0.498916	−	0.866650i	$-0.333732\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−403518.	+	698914.i	−0.0212799	+	0.0368578i
$816$		0			0
$817$	−3.00013e6	−	453289.i	−0.157248	−	0.0237585i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.00143e6	−	8.66273e6i	−0.258962	−	0.448536i	0.707002	−	0.707212i	$-0.250047\pi$
$821$	−5.00143e6	−	8.66273e6i	−0.258962	−	0.448536i	−0.965964	+	0.258676i	$0.916714\pi$
$822$		0			0
$823$	−5.31371e6	−	9.20361e6i	−0.273463	−	0.473651i	0.696283	−	0.717767i	$-0.254836\pi$
$823$	−5.31371e6	−	9.20361e6i	−0.273463	−	0.473651i	−0.969746	+	0.244116i	$0.921502\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.74700e6	+	4.75795e6i	0.139668	+	0.241911i	0.927371	−	0.374144i	$-0.122063\pi$
$827$	2.74700e6	+	4.75795e6i	0.139668	+	0.241911i	−0.787703	+	0.616055i	$0.788730\pi$
$828$		0			0
$829$	6.66282e6			0.336722			0.168361	−	0.985725i	$-0.446153\pi$
$829$	6.66282e6			0.336722			0.168361	+	0.985725i	$0.446153\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	8.97912e6	−	1.55523e7i	0.448354	−	0.776572i
$834$		0			0
$835$	−1.53663e7			−0.762699
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3.62638e6	−	6.28107e6i	0.177856	−	0.308055i	−0.763290	−	0.646056i	$-0.776417\pi$
$839$	3.62638e6	−	6.28107e6i	0.177856	−	0.308055i	0.941146	+	0.338001i	$0.109751\pi$
$840$		0			0
$841$	−1.20767e6	+	2.09174e6i	−0.0588786	+	0.101981i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.66303e6	−	4.61251e6i	−0.128302	−	0.222226i
$846$		0			0
$847$	−5.19674e6			−0.248899
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.38233e7	−	5.85837e7i	−1.60100	−	2.77302i
$852$		0			0
$853$	1.39644e7	−	2.41871e7i	0.657130	−	1.13818i	−0.324225	−	0.945980i	$-0.605104\pi$
$853$	1.39644e7	−	2.41871e7i	0.657130	−	1.13818i	0.981355	−	0.192202i	$-0.0615630\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.35275e6	−	2.34304e6i	0.0629169	−	0.108975i	−0.832851	−	0.553497i	$-0.813293\pi$
$857$	1.35275e6	−	2.34304e6i	0.0629169	−	0.108975i	0.895768	+	0.444522i	$0.146626\pi$
$858$		0			0
$859$	1.45030e7	+	2.51199e7i	0.670616	+	1.16154i	0.977730	+	0.209868i	$0.0673034\pi$
$859$	1.45030e7	+	2.51199e7i	0.670616	+	1.16154i	−0.307114	+	0.951673i	$0.599363\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.44738e6			0.386096			0.193048	−	0.981189i	$-0.438163\pi$
$863$	8.44738e6			0.386096			0.193048	+	0.981189i	$0.438163\pi$
$864$		0			0
$865$	1.35065e7	+	2.33939e7i	0.613765	+	1.06307i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	5.42096e6	−	9.38937e6i	0.243515	−	0.421781i
$870$		0			0
$871$	9.19171e6	−	1.59205e7i	0.410536	−	0.711068i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.14395e7			0.505113
$876$		0			0
$877$	−1.25597e7	+	2.17540e7i	−0.551416	+	0.955080i	0.446757	+	0.894655i	$0.352579\pi$
$877$	−1.25597e7	+	2.17540e7i	−0.551416	+	0.955080i	−0.998173	+	0.0604245i	$0.980755\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.95758e7			0.849728			0.424864	−	0.905257i	$-0.360322\pi$
$881$	1.95758e7			0.849728			0.424864	+	0.905257i	$0.360322\pi$
$882$		0			0
$883$	−1.40024e6	−	2.42528e6i	−0.0604366	−	0.104679i	0.834224	−	0.551426i	$-0.185916\pi$
$883$	−1.40024e6	−	2.42528e6i	−0.0604366	−	0.104679i	−0.894661	+	0.446746i	$0.852583\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−2.11985e7	−	3.67169e7i	−0.904682	−	1.56696i	−0.821343	−	0.570435i	$-0.806775\pi$
$887$	−2.11985e7	−	3.67169e7i	−0.904682	−	1.56696i	−0.0833393	−	0.996521i	$-0.526559\pi$
$888$		0			0
$889$	−4.75248e6	−	8.23154e6i	−0.201681	−	0.349322i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.59000e7	+	1.99234e7i	−0.667217	+	0.836053i
$894$		0			0
$895$	1.30133e6	−	2.25397e6i	0.0543039	−	0.0940571i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	304571.	+	527533.i	0.0125687	+	0.0217696i
$900$		0			0
$901$	3.09374e7			1.26962
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.05198e7			1.23868
$906$		0			0
$907$	1.00865e7	−	1.74703e7i	0.407118	−	0.705149i	−0.587447	−	0.809262i	$-0.699867\pi$
$907$	1.00865e7	−	1.74703e7i	0.407118	−	0.705149i	0.994565	+	0.104113i	$0.0332004\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.89915e7			1.55659			0.778294	−	0.627900i	$-0.216085\pi$
$911$	3.89915e7			1.55659			0.778294	+	0.627900i	$0.216085\pi$
$912$		0			0
$913$	2.75640e7			1.09437
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.98860e6	+	5.17640e6i	−0.117366	+	0.203285i
$918$		0			0
$919$	9.96447e6			0.389193			0.194597	−	0.980883i	$-0.437660\pi$
$919$	9.96447e6			0.389193			0.194597	+	0.980883i	$0.437660\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.05525e7			−1.95316
$924$		0			0
$925$	6.07253e6	+	1.05179e7i	0.233354	+	0.404181i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.00913e6	+	1.04081e7i	−0.228440	+	0.395670i	−0.957346	−	0.288944i	$-0.906696\pi$
$929$	−6.00913e6	+	1.04081e7i	−0.228440	+	0.395670i	0.728906	+	0.684614i	$0.240029\pi$
$930$		0			0
$931$	5.95188e6	+	1.51810e7i	0.225050	+	0.574020i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.60262e7	−	4.50786e7i	−0.973601	−	1.68633i
$936$		0			0
$937$	1.90689e7	+	3.30284e7i	0.709541	+	1.22896i	0.965027	+	0.262149i	$0.0844311\pi$
$937$	1.90689e7	+	3.30284e7i	0.709541	+	1.22896i	−0.255486	+	0.966813i	$0.582236\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.78331e6	−	1.52131e7i	−0.323358	−	0.560073i	0.657820	−	0.753175i	$-0.271479\pi$
$941$	−8.78331e6	−	1.52131e7i	−0.323358	−	0.560073i	−0.981179	+	0.193102i	$0.938145\pi$
$942$		0			0
$943$	−7.64584e7			−2.79992
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.41786e7	+	4.18786e7i	−0.876106	+	1.51746i	−0.0205267	+	0.999789i	$0.506534\pi$
$947$	−2.41786e7	+	4.18786e7i	−0.876106	+	1.51746i	−0.855580	+	0.517671i	$0.826799\pi$
$948$		0			0
$949$	−4.71124e7			−1.69812
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.89643e7	−	3.28471e7i	0.676402	−	1.17156i	−0.299656	−	0.954047i	$-0.596872\pi$
$953$	1.89643e7	−	3.28471e7i	0.676402	−	1.17156i	0.976057	−	0.217514i	$-0.0697949\pi$
$954$		0			0
$955$	6.89637e6	−	1.19449e7i	0.244688	−	0.423812i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5.63765e6	−	9.76470e6i	−0.197948	−	0.342856i
$960$		0			0
$961$	−2.86130e7			−0.999435
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.26024e7	+	3.91485e7i	0.781333	+	1.35331i
$966$		0			0
$967$	−8.31734e6	+	1.44061e7i	−0.286034	+	0.495426i	−0.972859	−	0.231397i	$-0.925670\pi$
$967$	−8.31734e6	+	1.44061e7i	−0.286034	+	0.495426i	0.686825	+	0.726823i	$0.259004\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.41242e7	−	2.44638e7i	0.480746	−	0.832677i	−0.519010	−	0.854768i	$-0.673699\pi$
$971$	1.41242e7	−	2.44638e7i	0.480746	−	0.832677i	0.999756	+	0.0220914i	$0.00703248\pi$
$972$		0			0
$973$	9.01694e6	+	1.56178e7i	0.305335	+	0.528856i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.01881e7			1.01181			0.505906	−	0.862589i	$-0.331158\pi$
$977$	3.01881e7			1.01181			0.505906	+	0.862589i	$0.331158\pi$
$978$		0			0
$979$	−3.89821e6	−	6.75190e6i	−0.129990	−	0.225149i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2321.02	−	4020.12i	7.66117e−5	−	0.000132695i	−0.865987	−	0.500066i	$-0.833309\pi$
$983$	2321.02	−	4020.12i	7.66117e−5	−	0.000132695i	0.866064	+	0.499934i	$0.166642\pi$
$984$		0			0
$985$	560698.	−	971158.i	0.0184136	−	0.0318933i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	9.35079e6			0.303989
$990$		0			0
$991$	1.56708e7	−	2.71426e7i	0.506882	−	0.877945i	−0.493087	−	0.869980i	$-0.664131\pi$
$991$	1.56708e7	−	2.71426e7i	0.506882	−	0.877945i	0.999968	−	0.00796469i	$-0.00253527\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3.66752e7			1.17440
$996$		0			0
$997$	−1.11856e7	−	1.93740e7i	−0.356387	−	0.617280i	0.630968	−	0.775809i	$-0.282658\pi$
$997$	−1.11856e7	−	1.93740e7i	−0.356387	−	0.617280i	−0.987354	+	0.158529i	$0.949325\pi$
$998$		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.505.3		18
3.2	odd	2		76.6.e.a.49.4	yes	18
12.11	even	2		304.6.i.d.49.6		18
19.7	even	3	inner	684.6.k.f.577.3		18
57.26	odd	6		76.6.e.a.45.4	✓	18
228.83	even	6		304.6.i.d.273.6		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.e.a.45.4	✓	18	57.26	odd	6
76.6.e.a.49.4	yes	18	3.2	odd	2
304.6.i.d.49.6		18	12.11	even	2
304.6.i.d.273.6		18	228.83	even	6
684.6.k.f.505.3		18	1.1	even	1	trivial
684.6.k.f.577.3		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)

\(f(q)\)	\(=\)	\(q+(-31.6056 + 54.7425i) q^{5} -80.2775 q^{7} +O(q^{10})\)	\(q+(-31.6056 + 54.7425i) q^{5} -80.2775 q^{7} +475.169 q^{11} +(-337.473 - 584.520i) q^{13} +(-866.499 + 1500.82i) q^{17} +(-574.366 - 1464.99i) q^{19} +(2424.72 + 4199.73i) q^{23} +(-435.326 - 754.007i) q^{25} +(-2394.08 - 4146.67i) q^{29} -127.218 q^{31} +(2537.22 - 4394.59i) q^{35} -13949.4 q^{37} +(-7883.24 + 13654.2i) q^{41} +(964.112 - 1669.89i) q^{43} +(-8099.51 - 14028.8i) q^{47} -10362.5 q^{49} +(-8925.98 - 15460.3i) q^{53} +(-15018.0 + 26011.9i) q^{55} +(-8424.75 + 14592.1i) q^{59} +(11641.7 + 20164.0i) q^{61} +42664.1 q^{65} +(13618.4 + 23587.8i) q^{67} +(37449.3 - 64864.1i) q^{71} +(34900.9 - 60450.1i) q^{73} -38145.4 q^{77} +(11408.5 - 19760.1i) q^{79} +58008.8 q^{83} +(-54772.4 - 94868.6i) q^{85} +(-8203.84 - 14209.5i) q^{89} +(27091.5 + 46923.8i) q^{91} +(98350.5 + 14859.7i) q^{95} +(65710.8 - 113814. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 11 q^{5} + 336 q^{7}+O(q^{10}) \) 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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