LMFDB - Embedded newform 684.6.k.f.505.2

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			505.2
Root			$-34.6401 - 61.7304i$ of defining polynomial
Character	$\chi$	$=$	684.505
Dual form			684.6.k.f.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-35.6401 + 61.7304i) q^{5} +252.315 q^{7} +O(q^{10})$	$q+(-35.6401 + 61.7304i) q^{5} +252.315 q^{7} -88.0323 q^{11} +(307.862 + 533.232i) q^{13} +(-285.543 + 494.575i) q^{17} +(361.524 + 1531.47i) q^{19} +(1214.28 + 2103.20i) q^{23} +(-977.931 - 1693.83i) q^{25} +(-1142.82 - 1979.43i) q^{29} +3684.20 q^{31} +(-8992.54 + 15575.5i) q^{35} +3064.28 q^{37} +(1246.89 - 2159.67i) q^{41} +(-2450.11 + 4243.71i) q^{43} +(-8786.51 - 15218.7i) q^{47} +46856.0 q^{49} +(12713.9 + 22021.1i) q^{53} +(3137.48 - 5434.27i) q^{55} +(11756.9 - 20363.5i) q^{59} +(9886.00 + 17123.1i) q^{61} -43888.8 q^{65} +(-13694.3 - 23719.2i) q^{67} +(16750.0 - 29011.9i) q^{71} +(-8986.08 + 15564.3i) q^{73} -22211.9 q^{77} +(41242.9 - 71434.8i) q^{79} -40274.4 q^{83} +(-20353.5 - 35253.4i) q^{85} +(27641.5 + 47876.5i) q^{89} +(77678.2 + 134543. i) q^{91} +(-107423. - 32264.7i) q^{95} +(13348.8 - 23120.7i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−35.6401	+	61.7304i	−0.637549	+	1.10427i	0.348420	+	0.937339i	$0.386718\pi$
$5$	−35.6401	+	61.7304i	−0.637549	+	1.10427i	−0.985969	+	0.166929i	$0.946615\pi$
$6$		0			0
$7$	252.315			1.94625			0.973125	−	0.230279i	$-0.0739637\pi$
$7$	252.315			1.94625			0.973125	+	0.230279i	$0.0739637\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−88.0323			−0.219362			−0.109681	−	0.993967i	$-0.534983\pi$
$11$	−88.0323			−0.219362			−0.109681	+	0.993967i	$0.534983\pi$
$12$		0			0
$13$	307.862	+	533.232i	0.505239	+	0.875100i	0.999982	+	0.00606030i	$0.00192907\pi$
$13$	307.862	+	533.232i	0.505239	+	0.875100i	−0.494742	+	0.869040i	$0.664738\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−285.543	+	494.575i	−0.239634	+	0.415059i	−0.960609	−	0.277902i	$-0.910361\pi$
$17$	−285.543	+	494.575i	−0.239634	+	0.415059i	0.720975	+	0.692961i	$0.243694\pi$
$18$		0			0
$19$	361.524	+	1531.47i	0.229748	+	0.973250i
$19$	361.524	+	1531.47i	0.229748	+	0.973250i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1214.28	+	2103.20i	0.478630	+	0.829011i	0.999700	−	0.0245027i	$-0.00780023\pi$
$23$	1214.28	+	2103.20i	0.478630	+	0.829011i	−0.521070	+	0.853514i	$0.674467\pi$
$24$		0			0
$25$	−977.931	−	1693.83i	−0.312938	−	0.542024i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1142.82	−	1979.43i	−0.252339	−	0.437064i	0.711830	−	0.702351i	$-0.247866\pi$
$29$	−1142.82	−	1979.43i	−0.252339	−	0.437064i	−0.964169	+	0.265288i	$0.914533\pi$
$30$		0			0
$31$	3684.20			0.688555			0.344278	−	0.938868i	$-0.388124\pi$
$31$	3684.20			0.688555			0.344278	+	0.938868i	$0.388124\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8992.54	+	15575.5i	−1.24083	+	2.14918i
$36$		0			0
$37$	3064.28			0.367980			0.183990	−	0.982928i	$-0.441099\pi$
$37$	3064.28			0.367980			0.183990	+	0.982928i	$0.441099\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1246.89	−	2159.67i	0.115842	−	0.200645i	−0.802274	−	0.596956i	$-0.796377\pi$
$41$	1246.89	−	2159.67i	0.115842	−	0.200645i	0.918116	+	0.396311i	$0.129710\pi$
$42$		0			0
$43$	−2450.11	+	4243.71i	−0.202075	+	0.350005i	−0.949197	−	0.314682i	$-0.898102\pi$
$43$	−2450.11	+	4243.71i	−0.202075	+	0.350005i	0.747122	+	0.664687i	$0.231435\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8786.51	−	15218.7i	−0.580192	−	1.00492i	−0.995456	−	0.0952210i	$-0.969644\pi$
$47$	−8786.51	−	15218.7i	−0.580192	−	1.00492i	0.415264	−	0.909701i	$-0.363689\pi$
$48$		0			0
$49$	46856.0			2.78789
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12713.9	+	22021.1i	0.621712	+	1.07684i	0.989167	+	0.146795i	$0.0468958\pi$
$53$	12713.9	+	22021.1i	0.621712	+	1.07684i	−0.367455	+	0.930041i	$0.619771\pi$
$54$		0			0
$55$	3137.48	−	5434.27i	0.139854	−	0.242234i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	11756.9	−	20363.5i	0.439706	−	0.761592i	−0.557961	−	0.829867i	$-0.688416\pi$
$59$	11756.9	−	20363.5i	0.439706	−	0.761592i	0.997667	+	0.0682748i	$0.0217495\pi$
$60$		0			0
$61$	9886.00	+	17123.1i	0.340170	+	0.589192i	0.984464	−	0.175586i	$-0.0561820\pi$
$61$	9886.00	+	17123.1i	0.340170	+	0.589192i	−0.644294	+	0.764778i	$0.722849\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−43888.8			−1.28846
$66$		0			0
$67$	−13694.3	−	23719.2i	−0.372694	−	0.645525i	0.617285	−	0.786740i	$-0.288233\pi$
$67$	−13694.3	−	23719.2i	−0.372694	−	0.645525i	−0.989979	+	0.141214i	$0.954899\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	16750.0	−	29011.9i	0.394339	−	0.683014i	−0.598678	−	0.800990i	$-0.704307\pi$
$71$	16750.0	−	29011.9i	0.394339	−	0.683014i	0.993017	+	0.117975i	$0.0376404\pi$
$72$		0			0
$73$	−8986.08	+	15564.3i	−0.197362	+	0.341841i	−0.947672	−	0.319245i	$-0.896571\pi$
$73$	−8986.08	+	15564.3i	−0.197362	+	0.341841i	0.750310	+	0.661086i	$0.229904\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−22211.9			−0.426932
$78$		0			0
$79$	41242.9	−	71434.8i	0.743501	−	1.28778i	−0.207391	−	0.978258i	$-0.566497\pi$
$79$	41242.9	−	71434.8i	0.743501	−	1.28778i	0.950892	−	0.309523i	$-0.100169\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−40274.4			−0.641702			−0.320851	−	0.947130i	$-0.603969\pi$
$83$	−40274.4			−0.641702			−0.320851	+	0.947130i	$0.603969\pi$
$84$		0			0
$85$	−20353.5	−	35253.4i	−0.305557	−	0.529241i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	27641.5	+	47876.5i	0.369902	+	0.640689i	0.989550	−	0.144191i	$-0.0460579\pi$
$89$	27641.5	+	47876.5i	0.369902	+	0.640689i	−0.619648	+	0.784880i	$0.712725\pi$
$90$		0			0
$91$	77678.2	+	134543.i	0.983322	+	1.70316i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−107423.	−	32264.7i	−1.22120	−	0.366791i
$96$		0			0
$97$	13348.8	−	23120.7i	0.144049	−	0.249501i	−0.784968	−	0.619536i	$-0.787321\pi$
$97$	13348.8	−	23120.7i	0.144049	−	0.249501i	0.929018	+	0.370035i	$0.120654\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	60668.2	+	105080.i	0.591777	+	1.02499i	0.993993	+	0.109443i	$0.0349067\pi$
$101$	60668.2	+	105080.i	0.591777	+	1.02499i	−0.402216	+	0.915545i	$0.631760\pi$
$102$		0			0
$103$	−123258.			−1.14478			−0.572392	−	0.819980i	$-0.693985\pi$
$103$	−123258.			−1.14478			−0.572392	+	0.819980i	$0.693985\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−225615.			−1.90506			−0.952529	−	0.304449i	$-0.901528\pi$
$107$	−225615.			−1.90506			−0.952529	+	0.304449i	$0.901528\pi$
$108$		0			0
$109$	−25583.8	+	44312.5i	−0.206253	+	0.357240i	−0.950531	−	0.310629i	$-0.899460\pi$
$109$	−25583.8	+	44312.5i	−0.206253	+	0.357240i	0.744279	+	0.667869i	$0.232793\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	113870.			0.838909			0.419454	−	0.907776i	$-0.362221\pi$
$113$	113870.			0.838909			0.419454	+	0.907776i	$0.362221\pi$
$114$		0			0
$115$	−173108.			−1.22060
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−72046.9	+	124789.i	−0.466388	+	0.807808i
$120$		0			0
$121$	−153301.			−0.951881
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−83336.4			−0.477045
$126$		0			0
$127$	−23234.7	−	40243.6i	−0.127828	−	0.221405i	0.795007	−	0.606601i	$-0.207467\pi$
$127$	−23234.7	−	40243.6i	−0.127828	−	0.221405i	−0.922835	+	0.385196i	$0.874134\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−142237.	+	246361.i	−0.724157	+	1.25428i	0.235163	+	0.971956i	$0.424438\pi$
$131$	−142237.	+	246361.i	−0.724157	+	1.25428i	−0.959320	+	0.282321i	$0.908896\pi$
$132$		0			0
$133$	91217.9	+	386413.i	0.447148	+	1.89419i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	143955.	+	249337.i	0.655277	+	1.13497i	0.981824	+	0.189792i	$0.0607814\pi$
$137$	143955.	+	249337.i	0.655277	+	1.13497i	−0.326547	+	0.945181i	$0.605885\pi$
$138$		0			0
$139$	88821.1	+	153843.i	0.389923	+	0.675367i	0.992439	−	0.122740i	$-0.0391681\pi$
$139$	88821.1	+	153843.i	0.389923	+	0.675367i	−0.602515	+	0.798107i	$0.705835\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−27101.8	−	46941.6i	−0.110830	−	0.191963i
$144$		0			0
$145$	162921.			0.643514
$146$		0			0
$147$		0			0
$148$		0			0
$149$	81129.1	−	140520.i	0.299372	−	0.518527i	−0.676621	−	0.736332i	$-0.736556\pi$
$149$	81129.1	−	140520.i	0.299372	−	0.518527i	0.975992	+	0.217805i	$0.0698896\pi$
$150$		0			0
$151$	−449650.			−1.60484			−0.802420	−	0.596760i	$-0.796455\pi$
$151$	−449650.			−1.60484			−0.802420	+	0.596760i	$0.796455\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−131305.	+	227427.i	−0.438988	+	0.760349i
$156$		0			0
$157$	−27389.5	+	47440.1i	−0.0886820	+	0.153602i	−0.906954	−	0.421229i	$-0.861599\pi$
$157$	−27389.5	+	47440.1i	−0.0886820	+	0.153602i	0.818272	+	0.574831i	$0.194932\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	306382.	+	530669.i	0.931533	+	1.61346i
$162$		0			0
$163$	207755.			0.612467			0.306233	−	0.951956i	$-0.400931\pi$
$163$	207755.			0.612467			0.306233	+	0.951956i	$0.400931\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	136696.	+	236765.i	0.379284	+	0.656940i	0.990958	−	0.134170i	$-0.0428369\pi$
$167$	136696.	+	236765.i	0.379284	+	0.656940i	−0.611674	+	0.791110i	$0.709504\pi$
$168$		0			0
$169$	−3910.94	+	6773.94i	−0.0105333	+	0.0182442i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	262508.	−	454677.i	0.666848	−	1.15501i	−0.311933	−	0.950104i	$-0.600976\pi$
$173$	262508.	−	454677.i	0.666848	−	1.15501i	0.978781	−	0.204910i	$-0.0656902\pi$
$174$		0			0
$175$	−246747.	−	427378.i	−0.609055	−	1.05491i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	13590.9			0.0317041			0.0158520	−	0.999874i	$-0.494954\pi$
$179$	13590.9			0.0317041			0.0158520	+	0.999874i	$0.494954\pi$
$180$		0			0
$181$	−36970.5	−	64034.9i	−0.0838802	−	0.145285i	0.821033	−	0.570880i	$-0.193398\pi$
$181$	−36970.5	−	64034.9i	−0.0838802	−	0.145285i	−0.904913	+	0.425596i	$0.860065\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−109211.	+	189160.i	−0.234606	+	0.406349i
$186$		0			0
$187$	25137.0	−	43538.6i	0.0525666	−	0.0910480i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	522570.			1.03648			0.518240	−	0.855235i	$-0.326587\pi$
$191$	522570.			1.03648			0.518240	+	0.855235i	$0.326587\pi$
$192$		0			0
$193$	12411.0	−	21496.6i	0.0239836	−	0.0415409i	−0.853784	−	0.520627i	$-0.825698\pi$
$193$	12411.0	−	21496.6i	0.0239836	−	0.0415409i	0.877768	+	0.479086i	$0.159032\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	363318.			0.666993			0.333496	−	0.942751i	$-0.391772\pi$
$197$	363318.			0.666993			0.333496	+	0.942751i	$0.391772\pi$
$198$		0			0
$199$	40673.4	+	70448.5i	0.0728078	+	0.126107i	0.900131	−	0.435620i	$-0.143471\pi$
$199$	40673.4	+	70448.5i	0.0728078	+	0.126107i	−0.827323	+	0.561726i	$0.810137\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−288352.	−	499440.i	−0.491114	−	0.850635i
$204$		0			0
$205$	88878.3	+	153942.i	0.147710	+	0.255842i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−31825.8	−	134819.i	−0.0503980	−	0.213494i
$210$		0			0
$211$	−235569.	+	408017.i	−0.364260	+	0.630917i	−0.988657	−	0.150190i	$-0.952011\pi$
$211$	−235569.	+	408017.i	−0.364260	+	0.630917i	0.624397	+	0.781107i	$0.285345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−174644.	−	302492.i	−0.257666	−	0.446291i
$216$		0			0
$217$	929580.			1.34010
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−351631.			−0.484291
$222$		0			0
$223$	−650115.	+	1.12603e6i	−0.875443	+	1.51631i	−0.0191530	+	0.999817i	$0.506097\pi$
$223$	−650115.	+	1.12603e6i	−0.875443	+	1.51631i	−0.856290	+	0.516495i	$0.827236\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	73342.4			0.0944692			0.0472346	−	0.998884i	$-0.484959\pi$
$227$	73342.4			0.0944692			0.0472346	+	0.998884i	$0.484959\pi$
$228$		0			0
$229$	−1.25234e6			−1.57809			−0.789046	−	0.614335i	$-0.789424\pi$
$229$	−1.25234e6			−1.57809			−0.789046	+	0.614335i	$0.789424\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−544644.	+	943351.i	−0.657238	+	1.13837i	0.324089	+	0.946027i	$0.394942\pi$
$233$	−544644.	+	943351.i	−0.657238	+	1.13837i	−0.981328	+	0.192344i	$0.938391\pi$
$234$		0			0
$235$	1.25261e6			1.47960
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−928359.			−1.05129			−0.525644	−	0.850705i	$-0.676175\pi$
$239$	−928359.			−1.05129			−0.525644	+	0.850705i	$0.676175\pi$
$240$		0			0
$241$	−49744.9	−	86160.6i	−0.0551703	−	0.0955578i	0.837121	−	0.547017i	$-0.184237\pi$
$241$	−49744.9	−	86160.6i	−0.0551703	−	0.0955578i	−0.892292	+	0.451460i	$0.850904\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.66995e6	+	2.89244e6i	−1.77742	+	3.07857i
$246$		0			0
$247$	−705329.	+	664256.i	−0.735613	+	0.692777i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	190030.	+	329141.i	0.190387	+	0.329760i	0.945379	−	0.325975i	$-0.105692\pi$
$251$	190030.	+	329141.i	0.190387	+	0.329760i	−0.754992	+	0.655735i	$0.772359\pi$
$252$		0			0
$253$	−106896.	−	185149.i	−0.104993	−	0.181853i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	552645.	+	957209.i	0.521931	+	0.904012i	0.999675	+	0.0255121i	$0.00812165\pi$
$257$	552645.	+	957209.i	0.521931	+	0.904012i	−0.477743	+	0.878500i	$0.658545\pi$
$258$		0			0
$259$	773166.			0.716182
$260$		0			0
$261$		0			0
$262$		0			0
$263$	276003.	−	478051.i	0.246050	−	0.426172i	−0.716376	−	0.697714i	$-0.754200\pi$
$263$	276003.	−	478051.i	0.246050	−	0.426172i	0.962426	+	0.271543i	$0.0875338\pi$
$264$		0			0
$265$	−1.81250e6			−1.58549
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−859906.	+	1.48940e6i	−0.724553	+	1.25496i	0.234604	+	0.972091i	$0.424621\pi$
$269$	−859906.	+	1.48940e6i	−0.724553	+	1.25496i	−0.959158	+	0.282872i	$0.908713\pi$
$270$		0			0
$271$	451795.	−	782532.i	0.373696	−	0.647260i	−0.616435	−	0.787406i	$-0.711424\pi$
$271$	451795.	−	782532.i	0.373696	−	0.647260i	0.990131	+	0.140146i	$0.0447571\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	86089.5	+	149111.i	0.0686465	+	0.118899i
$276$		0			0
$277$	−1.04518e6			−0.818447			−0.409223	−	0.912434i	$-0.634200\pi$
$277$	−1.04518e6			−0.818447			−0.409223	+	0.912434i	$0.634200\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	213159.	+	369202.i	0.161041	+	0.278932i	0.935242	−	0.354008i	$-0.115181\pi$
$281$	213159.	+	369202.i	0.161041	+	0.278932i	−0.774201	+	0.632940i	$0.781848\pi$
$282$		0			0
$283$	−29882.7	+	51758.4i	−0.0221796	+	0.0384162i	−0.876902	−	0.480669i	$-0.840394\pi$
$283$	−29882.7	+	51758.4i	−0.0221796	+	0.0384162i	0.854723	+	0.519085i	$0.173727\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	314609.	−	544918.i	0.225458	−	0.390505i
$288$		0			0
$289$	546859.	+	947188.i	0.385151	+	0.667101i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.17326e6			−0.798410			−0.399205	−	0.916862i	$-0.630714\pi$
$293$	−1.17326e6			−0.798410			−0.399205	+	0.916862i	$0.630714\pi$
$294$		0			0
$295$	838032.	+	1.45151e6i	0.560668	+	0.971105i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−747661.	+	1.29499e6i	−0.483645	+	0.837698i
$300$		0			0
$301$	−618199.	+	1.07075e6i	−0.393289	+	0.681197i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.40935e6			−0.867501
$306$		0			0
$307$	−252040.	+	436545.i	−0.152624	+	0.264353i	−0.932191	−	0.361966i	$-0.882106\pi$
$307$	−252040.	+	436545.i	−0.152624	+	0.264353i	0.779567	+	0.626318i	$0.215439\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.39525e6			−1.40426			−0.702132	−	0.712046i	$-0.747769\pi$
$311$	−2.39525e6			−1.40426			−0.702132	+	0.712046i	$0.747769\pi$
$312$		0			0
$313$	−1.06455e6	−	1.84385e6i	−0.614193	−	1.06381i	−0.990525	−	0.137329i	$-0.956148\pi$
$313$	−1.06455e6	−	1.84385e6i	−0.614193	−	1.06381i	0.376332	−	0.926485i	$-0.377185\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	399285.	+	691581.i	0.223169	+	0.386541i	0.955769	−	0.294120i	$-0.0950264\pi$
$317$	399285.	+	691581.i	0.223169	+	0.386541i	−0.732599	+	0.680660i	$0.761693\pi$
$318$		0			0
$319$	100605.	+	174254.i	0.0553535	+	0.0958750i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−860657.	−	258500.i	−0.459012	−	0.137865i
$324$		0			0
$325$	602135.	−	1.04293e6i	0.316217	−	0.547704i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.21697e6	−	3.83991e6i	−1.12920	−	1.95583i
$330$		0			0
$331$	−224790.			−0.112773			−0.0563867	−	0.998409i	$-0.517958\pi$
$331$	−224790.			−0.112773			−0.0563867	+	0.998409i	$0.517958\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.95226e6			0.950444
$336$		0			0
$337$	1.02795e6	−	1.78046e6i	0.493056	−	0.853998i	−0.506912	−	0.861998i	$-0.669213\pi$
$337$	1.02795e6	−	1.78046e6i	0.493056	−	0.853998i	0.999968	+	0.00799964i	$0.00254639\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−324329.			−0.151043
$342$		0			0
$343$	7.58183e6			3.47967
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.48248e6	−	2.56773e6i	0.660946	−	1.14479i	−0.319422	−	0.947613i	$-0.603489\pi$
$347$	1.48248e6	−	2.56773e6i	0.660946	−	1.14479i	0.980368	−	0.197179i	$-0.0631780\pi$
$348$		0			0
$349$	2.25013e6			0.988883			0.494441	−	0.869211i	$-0.335373\pi$
$349$	2.25013e6			0.988883			0.494441	+	0.869211i	$0.335373\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.72711e6			0.737706			0.368853	−	0.929488i	$-0.379751\pi$
$353$	1.72711e6			0.737706			0.368853	+	0.929488i	$0.379751\pi$
$354$		0			0
$355$	1.19394e6	+	2.06797e6i	0.502820	+	0.870910i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	526967.	−	912734.i	0.215798	−	0.373773i	−0.737721	−	0.675106i	$-0.764098\pi$
$359$	526967.	−	912734.i	0.215798	−	0.373773i	0.953519	+	0.301332i	$0.0974314\pi$
$360$		0			0
$361$	−2.21470e6	+	1.10732e6i	−0.894431	+	0.447205i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−640529.	−	1.10943e6i	−0.251656	−	0.435881i
$366$		0			0
$367$	−1.90021e6	−	3.29126e6i	−0.736438	−	1.27555i	−0.954089	−	0.299522i	$-0.903173\pi$
$367$	−1.90021e6	−	3.29126e6i	−0.736438	−	1.27555i	0.217651	−	0.976027i	$-0.430160\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.20791e6	+	5.55627e6i	1.21001	+	2.09579i
$372$		0			0
$373$	−1.16264e6			−0.432685			−0.216342	−	0.976318i	$-0.569413\pi$
$373$	−1.16264e6			−0.432685			−0.216342	+	0.976318i	$0.569413\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	703663.	−	1.21878e6i	0.254983	−	0.441643i
$378$		0			0
$379$	−1.49509e6			−0.534648			−0.267324	−	0.963607i	$-0.586139\pi$
$379$	−1.49509e6			−0.534648			−0.267324	+	0.963607i	$0.586139\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.52086e6	−	4.36626e6i	0.878115	−	1.52094i	0.0247088	−	0.999695i	$-0.492134\pi$
$383$	2.52086e6	−	4.36626e6i	0.878115	−	1.52094i	0.853407	−	0.521246i	$-0.174533\pi$
$384$		0			0
$385$	791634.	−	1.37115e6i	0.272190	−	0.471448i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−612495.	−	1.06087e6i	−0.205224	−	0.355459i	0.744980	−	0.667087i	$-0.232459\pi$
$389$	−612495.	−	1.06087e6i	−0.205224	−	0.355459i	−0.950204	+	0.311628i	$0.899126\pi$
$390$		0			0
$391$	−1.38692e6			−0.458785
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.93980e6	+	5.09188e6i	0.948036	+	1.64205i
$396$		0			0
$397$	−639677.	+	1.10795e6i	−0.203697	+	0.352813i	−0.949717	−	0.313110i	$-0.898629\pi$
$397$	−639677.	+	1.10795e6i	−0.203697	+	0.352813i	0.746020	+	0.665924i	$0.231962\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	473386.	−	819928.i	0.147013	−	0.254633i	−0.783109	−	0.621884i	$-0.786368\pi$
$401$	473386.	−	819928.i	0.147013	−	0.254633i	0.930122	+	0.367251i	$0.119701\pi$
$402$		0			0
$403$	1.13422e6	+	1.96453e6i	0.347885	+	0.602555i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−269756.			−0.0807208
$408$		0			0
$409$	−1.78713e6	−	3.09540e6i	−0.528260	−	0.914974i	−0.999457	−	0.0329456i	$-0.989511\pi$
$409$	−1.78713e6	−	3.09540e6i	−0.528260	−	0.914974i	0.471197	−	0.882028i	$-0.343822\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.96644e6	−	5.13803e6i	0.855777	−	1.48225i
$414$		0			0
$415$	1.43538e6	−	2.48616e6i	0.409117	−	0.708611i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.64582e6			1.29279			0.646395	−	0.763003i	$-0.276276\pi$
$419$	4.64582e6			1.29279			0.646395	+	0.763003i	$0.276276\pi$
$420$		0			0
$421$	1.59986e6	−	2.77104e6i	0.439923	−	0.761970i	−0.557760	−	0.830002i	$-0.688339\pi$
$421$	1.59986e6	−	2.77104e6i	0.439923	−	0.761970i	0.997683	+	0.0680329i	$0.0216723\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.11697e6			0.299963
$426$		0			0
$427$	2.49439e6	+	4.32041e6i	0.662056	+	1.14671i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.46054e6	−	4.26178e6i	−0.638025	−	1.10509i	−0.985866	−	0.167537i	$-0.946418\pi$
$431$	−2.46054e6	−	4.26178e6i	−0.638025	−	1.10509i	0.347841	−	0.937553i	$-0.386915\pi$
$432$		0			0
$433$	1.48931e6	+	2.57956e6i	0.381738	+	0.661189i	0.991311	−	0.131541i	$-0.0419924\pi$
$433$	1.48931e6	+	2.57956e6i	0.381738	+	0.661189i	−0.609573	+	0.792730i	$0.708659\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.78199e6	+	2.61999e6i	−0.696871	+	0.656291i
$438$		0			0
$439$	1.08758e6	−	1.88374e6i	0.269339	−	0.466509i	−0.699352	−	0.714777i	$-0.746528\pi$
$439$	1.08758e6	−	1.88374e6i	0.269339	−	0.466509i	0.968691	+	0.248268i	$0.0798614\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	189407.	+	328062.i	0.0458549	+	0.0794231i	0.888042	−	0.459763i	$-0.152065\pi$
$443$	189407.	+	328062.i	0.0458549	+	0.0794231i	−0.842187	+	0.539186i	$0.818732\pi$
$444$		0			0
$445$	−3.94058e6			−0.943323
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5.64067e6			1.32043			0.660214	−	0.751077i	$-0.270466\pi$
$449$	5.64067e6			1.32043			0.660214	+	0.751077i	$0.270466\pi$
$450$		0			0
$451$	−109766.	+	190121.i	−0.0254114	+	0.0440138i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.10738e7			−2.50766
$456$		0			0
$457$	−317551.			−0.0711252			−0.0355626	−	0.999367i	$-0.511322\pi$
$457$	−317551.			−0.0711252			−0.0355626	+	0.999367i	$0.511322\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.63141e6	+	2.82569e6i	−0.357529	+	0.619259i	−0.987547	−	0.157322i	$-0.949714\pi$
$461$	−1.63141e6	+	2.82569e6i	−0.357529	+	0.619259i	0.630018	+	0.776580i	$0.283047\pi$
$462$		0			0
$463$	−7.65368e6			−1.65927			−0.829636	−	0.558305i	$-0.811452\pi$
$463$	−7.65368e6			−1.65927			−0.829636	+	0.558305i	$0.811452\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−885237.			−0.187831			−0.0939155	−	0.995580i	$-0.529938\pi$
$467$	−885237.			−0.187831			−0.0939155	+	0.995580i	$0.529938\pi$
$468$		0			0
$469$	−3.45528e6	−	5.98472e6i	−0.725356	−	1.25635i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	215689.	−	373583.i	0.0443276	−	0.0767777i
$474$		0			0
$475$	2.24050e6	−	2.11003e6i	0.455628	−	0.429096i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.58332e6	−	4.47445e6i	−0.514447	−	0.891047i	−0.999859	−	0.0167625i	$-0.994664\pi$
$479$	−2.58332e6	−	4.47445e6i	−0.514447	−	0.891047i	0.485413	−	0.874285i	$-0.338669\pi$
$480$		0			0
$481$	943375.	+	1.63397e6i	0.185918	+	0.322020i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	951502.	+	1.64805e6i	0.183677	+	0.318138i
$486$		0			0
$487$	−4.53740e6			−0.866931			−0.433465	−	0.901170i	$-0.642709\pi$
$487$	−4.53740e6			−0.866931			−0.433465	+	0.901170i	$0.642709\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.58323e6	−	7.93839e6i	0.857963	−	1.48603i	−0.0159066	−	0.999873i	$-0.505063\pi$
$491$	4.58323e6	−	7.93839e6i	0.857963	−	1.48603i	0.873869	−	0.486161i	$-0.161603\pi$
$492$		0			0
$493$	1.30530e6			0.241876
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.22628e6	−	7.32014e6i	0.767481	−	1.32932i
$498$		0			0
$499$	−4.86531e6	+	8.42697e6i	−0.874700	+	1.51503i	−0.0176190	+	0.999845i	$0.505609\pi$
$499$	−4.86531e6	+	8.42697e6i	−0.874700	+	1.51503i	−0.857081	+	0.515181i	$0.827725\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5.37084e6	+	9.30257e6i	0.946503	+	1.63939i	0.752713	+	0.658349i	$0.228745\pi$
$503$	5.37084e6	+	9.30257e6i	0.946503	+	1.63939i	0.193791	+	0.981043i	$0.437922\pi$
$504$		0			0
$505$	−8.64888e6			−1.50915
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.77946e6	+	4.81416e6i	0.475516	+	0.823618i	0.999607	−	0.0280444i	$-0.00892797\pi$
$509$	2.77946e6	+	4.81416e6i	0.475516	+	0.823618i	−0.524090	+	0.851663i	$0.675595\pi$
$510$		0			0
$511$	−2.26733e6	+	3.92712e6i	−0.384115	+	0.665307i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.39294e6	−	7.60880e6i	0.729856	−	1.26415i
$516$		0			0
$517$	773497.	+	1.33974e6i	0.127272	+	0.220441i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.44377e6			0.394427			0.197213	−	0.980361i	$-0.436811\pi$
$521$	2.44377e6			0.394427			0.197213	+	0.980361i	$0.436811\pi$
$522$		0			0
$523$	2.06842e6	+	3.58260e6i	0.330662	+	0.572723i	0.982642	−	0.185513i	$-0.0593948\pi$
$523$	2.06842e6	+	3.58260e6i	0.330662	+	0.572723i	−0.651980	+	0.758236i	$0.726061\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.05200e6	+	1.82211e6i	−0.165002	+	0.285791i
$528$		0			0
$529$	269211.	−	466288.i	0.0418268	−	0.0724461i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.53547e6			0.234112
$534$		0			0
$535$	8.04093e6	−	1.39273e7i	1.21457	−	2.10369i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.12485e6			−0.611555
$540$		0			0
$541$	332076.	+	575172.i	0.0487803	+	0.0844899i	0.889385	−	0.457160i	$-0.151133\pi$
$541$	332076.	+	575172.i	0.0487803	+	0.0844899i	−0.840604	+	0.541650i	$0.817800\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.82362e6	−	3.15860e6i	−0.262992	−	0.455516i
$546$		0			0
$547$	−2.27290e6	−	3.93678e6i	−0.324797	−	0.562565i	0.656674	−	0.754174i	$-0.271963\pi$
$547$	−2.27290e6	−	3.93678e6i	−0.324797	−	0.562565i	−0.981471	+	0.191609i	$0.938629\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.61828e6	−	2.46581e6i	0.367398	−	0.346004i
$552$		0			0
$553$	1.04062e7	−	1.80241e7i	1.44704	−	2.50634i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.10469e6	−	7.10954e6i	−0.560586	−	0.970964i	−0.997445	−	0.0714342i	$-0.977242\pi$
$557$	−4.10469e6	−	7.10954e6i	−0.560586	−	0.970964i	0.436859	−	0.899530i	$-0.356091\pi$
$558$		0			0
$559$	−3.01717e6			−0.408386
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.32458e7			1.76119			0.880597	−	0.473866i	$-0.157142\pi$
$563$	1.32458e7			1.76119			0.880597	+	0.473866i	$0.157142\pi$
$564$		0			0
$565$	−4.05835e6	+	7.02927e6i	−0.534846	+	0.926380i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.11112e7			−1.43873			−0.719365	−	0.694632i	$-0.755567\pi$
$569$	−1.11112e7			−1.43873			−0.719365	+	0.694632i	$0.755567\pi$
$570$		0			0
$571$	1.44176e7			1.85055			0.925277	−	0.379291i	$-0.123832\pi$
$571$	1.44176e7			1.85055			0.925277	+	0.379291i	$0.123832\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.37497e6	−	4.11356e6i	0.299563	−	0.518858i
$576$		0			0
$577$	−4.63470e6			−0.579538			−0.289769	−	0.957097i	$-0.593578\pi$
$577$	−4.63470e6			−0.579538			−0.289769	+	0.957097i	$0.593578\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.01618e7			−1.24891
$582$		0			0
$583$	−1.11923e6	−	1.93857e6i	−0.136380	−	0.236217i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	87378.3	−	151344.i	0.0104667	−	0.0181288i	−0.860745	−	0.509037i	$-0.830002\pi$
$587$	87378.3	−	151344.i	0.0104667	−	0.0181288i	0.871211	+	0.490908i	$0.163335\pi$
$588$		0			0
$589$	1.33193e6	+	5.64224e6i	0.158195	+	0.670137i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−63610.1	−	110176.i	−0.00742830	−	0.0128662i	0.862287	−	0.506419i	$-0.169031\pi$
$593$	−63610.1	−	110176.i	−0.00742830	−	0.0128662i	−0.869716	+	0.493553i	$0.835698\pi$
$594$		0			0
$595$	−5.13551e6	−	8.89497e6i	−0.594691	−	1.03003i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.00216e6	−	1.38601e7i	−0.911255	−	1.57834i	−0.812293	−	0.583249i	$-0.801781\pi$
$599$	−8.00216e6	−	1.38601e7i	−0.911255	−	1.57834i	−0.0989618	−	0.995091i	$-0.531552\pi$
$600$		0			0
$601$	1.52423e7			1.72134			0.860668	−	0.509166i	$-0.170046\pi$
$601$	1.52423e7			1.72134			0.860668	+	0.509166i	$0.170046\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.46367e6	−	9.46336e6i	0.606871	−	1.05113i
$606$		0			0
$607$	1.49576e7			1.64774			0.823870	−	0.566778i	$-0.191810\pi$
$607$	1.49576e7			1.64774			0.823870	+	0.566778i	$0.191810\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.41005e6	−	9.37049e6i	0.586271	−	1.01545i
$612$		0			0
$613$	889245.	−	1.54022e6i	0.0955807	−	0.165551i	−0.814270	−	0.580486i	$-0.802863\pi$
$613$	889245.	−	1.54022e6i	0.0955807	−	0.165551i	0.909851	+	0.414936i	$0.136196\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.97293e6	−	1.38095e7i	−0.843150	−	1.46038i	−0.887218	−	0.461350i	$-0.847365\pi$
$617$	−7.97293e6	−	1.38095e7i	−0.843150	−	1.46038i	0.0440683	−	0.999029i	$-0.485968\pi$
$618$		0			0
$619$	−1.53524e7			−1.61046			−0.805231	−	0.592962i	$-0.797959\pi$
$619$	−1.53524e7			−1.61046			−0.805231	+	0.592962i	$0.797959\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.97438e6	+	1.20800e7i	0.719922	+	1.24694i
$624$		0			0
$625$	6.02615e6	−	1.04376e7i	0.617078	−	1.06881i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−874985.	+	1.51552e6i	−0.0881807	+	0.152734i
$630$		0			0
$631$	1.84277e6	+	3.19177e6i	0.184246	+	0.319123i	0.943322	−	0.331879i	$-0.107682\pi$
$631$	1.84277e6	+	3.19177e6i	0.184246	+	0.319123i	−0.759076	+	0.651002i	$0.774349\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.31234e6			0.325988
$636$		0			0
$637$	1.44252e7	+	2.49851e7i	1.40855	+	2.43968i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.60486e6	+	4.51176e6i	−0.250403	+	0.433711i	−0.963637	−	0.267215i	$-0.913897\pi$
$641$	−2.60486e6	+	4.51176e6i	−0.250403	+	0.433711i	0.713234	+	0.700926i	$0.247230\pi$
$642$		0			0
$643$	3.62947e6	−	6.28642e6i	0.346191	−	0.599620i	−0.639379	−	0.768892i	$-0.720808\pi$
$643$	3.62947e6	−	6.28642e6i	0.346191	−	0.599620i	0.985569	+	0.169272i	$0.0541417\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.49406e6			0.797728			0.398864	−	0.917010i	$-0.369405\pi$
$647$	8.49406e6			0.797728			0.398864	+	0.917010i	$0.369405\pi$
$648$		0			0
$649$	−1.03499e6	+	1.79265e6i	−0.0964545	+	0.167064i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.16393e7			1.06818			0.534089	−	0.845428i	$-0.320655\pi$
$653$	1.16393e7			1.06818			0.534089	+	0.845428i	$0.320655\pi$
$654$		0			0
$655$	−1.01386e7	−	1.75606e7i	−0.923372	−	1.59933i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.86735e6	+	1.70907e7i	0.885089	+	1.53302i	0.845612	+	0.533798i	$0.179236\pi$
$659$	9.86735e6	+	1.70907e7i	0.885089	+	1.53302i	0.0394769	+	0.999220i	$0.487431\pi$
$660$		0			0
$661$	7.97821e6	+	1.38187e7i	0.710235	+	1.23016i	0.964769	+	0.263099i	$0.0847446\pi$
$661$	7.97821e6	+	1.38187e7i	0.710235	+	1.23016i	−0.254534	+	0.967064i	$0.581922\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.71045e7	−	8.14088e6i	−2.37677	−	0.713867i
$666$		0			0
$667$	2.77542e6	−	4.80717e6i	0.241554	−	0.418384i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−870288.	−	1.50738e6i	−0.0746202	−	0.129246i
$672$		0			0
$673$	1.18685e7			1.01008			0.505041	−	0.863096i	$-0.331477\pi$
$673$	1.18685e7			1.01008			0.505041	+	0.863096i	$0.331477\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.13643e7			0.952952			0.476476	−	0.879188i	$-0.341914\pi$
$677$	1.13643e7			0.952952			0.476476	+	0.879188i	$0.341914\pi$
$678$		0			0
$679$	3.36810e6	−	5.83372e6i	0.280356	−	0.485591i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.33134e7			−1.09204			−0.546018	−	0.837774i	$-0.683857\pi$
$683$	−1.33134e7			−1.09204			−0.546018	+	0.837774i	$0.683857\pi$
$684$		0			0
$685$	−2.05223e7			−1.67109
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.82824e6	+	1.35589e7i	−0.628226	+	1.08812i
$690$		0			0
$691$	1.77578e7			1.41480			0.707400	−	0.706813i	$-0.249868\pi$
$691$	1.77578e7			1.41480			0.707400	+	0.706813i	$0.249868\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.26624e7			−0.994381
$696$		0			0
$697$	712079.	+	1.23336e6i	0.0555196	+	0.0961628i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.58452e6	+	4.47652e6i	−0.198648	+	0.344069i	−0.948090	−	0.318001i	$-0.896988\pi$
$701$	−2.58452e6	+	4.47652e6i	−0.198648	+	0.344069i	0.749442	+	0.662070i	$0.230322\pi$
$702$		0			0
$703$	1.10781e6	+	4.69286e6i	0.0845429	+	0.358137i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.53075e7	+	2.65134e7i	1.15175	+	1.99488i
$708$		0			0
$709$	6.35081e6	+	1.09999e7i	0.474476	+	0.821816i	0.999573	−	0.0292265i	$-0.00930441\pi$
$709$	6.35081e6	+	1.09999e7i	0.474476	+	0.821816i	−0.525097	+	0.851042i	$0.675971\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.47366e6	+	7.74860e6i	0.329563	+	0.570820i
$714$		0			0
$715$	3.86364e6			0.282638
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.17643e7	+	2.03763e7i	−0.848678	+	1.46995i	0.0337101	+	0.999432i	$0.489268\pi$
$719$	−1.17643e7	+	2.03763e7i	−0.848678	+	1.46995i	−0.882388	+	0.470522i	$0.844066\pi$
$720$		0			0
$721$	−3.11000e7			−2.22804
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.23520e6	+	3.87149e6i	−0.157933	+	0.273548i
$726$		0			0
$727$	1.08765e7	−	1.88387e7i	0.763229	−	1.32195i	−0.177949	−	0.984040i	$-0.556946\pi$
$727$	1.08765e7	−	1.88387e7i	0.763229	−	1.32195i	0.941178	−	0.337911i	$-0.109720\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.39922e6	−	2.42352e6i	−0.0968485	−	0.167746i
$732$		0			0
$733$	−9.80139e6			−0.673795			−0.336897	−	0.941541i	$-0.609378\pi$
$733$	−9.80139e6			−0.673795			−0.336897	+	0.941541i	$0.609378\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.20554e6	+	2.08806e6i	0.0817548	+	0.141603i
$738$		0			0
$739$	−7.08883e6	+	1.22782e7i	−0.477489	+	0.827035i	−0.999667	−	0.0258011i	$-0.991786\pi$
$739$	−7.08883e6	+	1.22782e7i	−0.477489	+	0.827035i	0.522178	+	0.852837i	$0.325120\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.04168e6	−	5.26834e6i	0.202135	−	0.350108i	−0.747081	−	0.664733i	$-0.768545\pi$
$743$	3.04168e6	−	5.26834e6i	0.202135	−	0.350108i	0.949216	+	0.314625i	$0.101879\pi$
$744$		0			0
$745$	5.78289e6	+	1.00163e7i	0.381728	+	0.661173i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.69261e7			−3.70772
$750$		0			0
$751$	6.05235e6	+	1.04830e7i	0.391584	+	0.678243i	0.992659	−	0.120950i	$-0.0385941\pi$
$751$	6.05235e6	+	1.04830e7i	0.391584	+	0.678243i	−0.601075	+	0.799193i	$0.705261\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.60255e7	−	2.77571e7i	1.02316	−	1.77217i
$756$		0			0
$757$	−837126.	+	1.44994e6i	−0.0530947	+	0.0919627i	−0.891351	−	0.453313i	$-0.850242\pi$
$757$	−837126.	+	1.44994e6i	−0.0530947	+	0.0919627i	0.838257	+	0.545276i	$0.183575\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.71108e7			1.07105			0.535525	−	0.844520i	$-0.320114\pi$
$761$	1.71108e7			1.07105			0.535525	+	0.844520i	$0.320114\pi$
$762$		0			0
$763$	−6.45519e6	+	1.11807e7i	−0.401419	+	0.695278i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.44780e7			0.888626
$768$		0			0
$769$	−4.88795e6	−	8.46617e6i	−0.298065	−	0.516263i	0.677628	−	0.735404i	$-0.263008\pi$
$769$	−4.88795e6	−	8.46617e6i	−0.298065	−	0.516263i	−0.975693	+	0.219141i	$0.929674\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.30084e6	+	1.09134e7i	0.379271	+	0.656916i	0.990956	−	0.134185i	$-0.0428415\pi$
$773$	6.30084e6	+	1.09134e7i	0.379271	+	0.656916i	−0.611686	+	0.791101i	$0.709508\pi$
$774$		0			0
$775$	−3.60289e6	−	6.24039e6i	−0.215475	−	0.373214i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.75825e6	+	1.12880e6i	0.221892	+	0.0666457i
$780$		0			0
$781$	−1.47454e6	+	2.55398e6i	−0.0865027	+	0.149827i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.95233e6	−	3.38154e6i	−0.113078	−	0.195857i
$786$		0			0
$787$	−1.91406e7			−1.10159			−0.550795	−	0.834641i	$-0.685675\pi$
$787$	−1.91406e7			−1.10159			−0.550795	+	0.834641i	$0.685675\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.87312e7			1.63273
$792$		0			0
$793$	−6.08704e6	+	1.05431e7i	−0.343735	+	0.595366i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.28733e6			0.0717865			0.0358932	−	0.999356i	$-0.488572\pi$
$797$	1.28733e6			0.0717865			0.0358932	+	0.999356i	$0.488572\pi$
$798$		0			0
$799$	1.00357e7			0.556136
$800$		0			0
$801$		0			0
$802$		0			0
$803$	791066.	−	1.37017e6i	0.0432936	−	0.0749867i
$804$		0			0
$805$	−4.36779e7			−2.37559
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.73676e7			−1.47016			−0.735082	−	0.677978i	$-0.762856\pi$
$809$	−2.73676e7			−1.47016			−0.735082	+	0.677978i	$0.762856\pi$
$810$		0			0
$811$	−3.68597e6	−	6.38429e6i	−0.196789	−	0.340848i	0.750697	−	0.660647i	$-0.229718\pi$
$811$	−3.68597e6	−	6.38429e6i	−0.196789	−	0.340848i	−0.947485	+	0.319799i	$0.896385\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.40441e6	+	1.28248e7i	−0.390478	+	0.676327i
$816$		0			0
$817$	−7.38488e6	−	2.21806e6i	−0.387069	−	0.116257i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.13053e7	−	1.95813e7i	−0.585360	−	1.01387i	−0.994830	−	0.101550i	$-0.967620\pi$
$821$	−1.13053e7	−	1.95813e7i	−0.585360	−	1.01387i	0.409471	−	0.912323i	$-0.365713\pi$
$822$		0			0
$823$	1.42337e7	+	2.46536e7i	0.732520	+	1.26876i	0.955803	+	0.294008i	$0.0949893\pi$
$823$	1.42337e7	+	2.46536e7i	0.732520	+	1.26876i	−0.223283	+	0.974754i	$0.571677\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5.08685e6	+	8.81069e6i	0.258634	+	0.447967i	0.965876	−	0.259004i	$-0.0833944\pi$
$827$	5.08685e6	+	8.81069e6i	0.258634	+	0.447967i	−0.707242	+	0.706971i	$0.750061\pi$
$828$		0			0
$829$	4.96805e6			0.251073			0.125536	−	0.992089i	$-0.459935\pi$
$829$	4.96805e6			0.251073			0.125536	+	0.992089i	$0.459935\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.33794e7	+	2.31738e7i	−0.668074	+	1.15714i
$834$		0			0
$835$	−1.94874e7			−0.967250
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.19839e6	−	9.00387e6i	0.254955	−	0.441595i	−0.709928	−	0.704274i	$-0.751273\pi$
$839$	5.19839e6	−	9.00387e6i	0.254955	−	0.441595i	0.964883	+	0.262679i	$0.0846060\pi$
$840$		0			0
$841$	7.64348e6	−	1.32389e7i	0.372650	−	0.645449i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−278772.	−	482847.i	−0.0134310	−	0.0232631i
$846$		0			0
$847$	−3.86803e7			−1.85260
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.72090e6	+	6.44479e6i	0.176126	+	0.305060i
$852$		0			0
$853$	1.62456e6	−	2.81382e6i	0.0764475	−	0.132411i	−0.825267	−	0.564742i	$-0.808976\pi$
$853$	1.62456e6	−	2.81382e6i	0.0764475	−	0.132411i	0.901715	+	0.432331i	$0.142309\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.16911e7	−	2.02495e7i	0.543754	−	0.941810i	−0.454930	−	0.890527i	$-0.650336\pi$
$857$	1.16911e7	−	2.02495e7i	0.543754	−	0.941810i	0.998684	−	0.0512826i	$-0.0163309\pi$
$858$		0			0
$859$	469598.	+	813368.i	0.0217142	+	0.0376101i	0.876678	−	0.481077i	$-0.159754\pi$
$859$	469598.	+	813368.i	0.0217142	+	0.0376101i	−0.854964	+	0.518687i	$0.826421\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	3.88566e7			1.77598			0.887989	−	0.459864i	$-0.152102\pi$
$863$	3.88566e7			1.77598			0.887989	+	0.459864i	$0.152102\pi$
$864$		0			0
$865$	1.87116e7	+	3.24094e7i	0.850296	+	1.47276i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.63071e6	+	6.28857e6i	−0.163095	+	0.282490i
$870$		0			0
$871$	8.43189e6	−	1.46045e7i	0.376599	−	0.652289i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.10270e7			−0.928449
$876$		0			0
$877$	6.05008e6	−	1.04790e7i	0.265621	−	0.460069i	−0.702105	−	0.712073i	$-0.747756\pi$
$877$	6.05008e6	−	1.04790e7i	0.265621	−	0.460069i	0.967726	+	0.252004i	$0.0810897\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.25012e7			−0.542642			−0.271321	−	0.962489i	$-0.587461\pi$
$881$	−1.25012e7			−0.542642			−0.271321	+	0.962489i	$0.587461\pi$
$882$		0			0
$883$	−1.22509e7	−	2.12191e7i	−0.528767	−	0.915852i	−0.999437	−	0.0335424i	$-0.989321\pi$
$883$	−1.22509e7	−	2.12191e7i	−0.528767	−	0.915852i	0.470670	−	0.882309i	$-0.344012\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.48831e6	−	6.04194e6i	−0.148870	−	0.257850i	0.781940	−	0.623353i	$-0.214230\pi$
$887$	−3.48831e6	−	6.04194e6i	−0.148870	−	0.257850i	−0.930810	+	0.365503i	$0.880897\pi$
$888$		0			0
$889$	−5.86246e6	−	1.01541e7i	−0.248786	−	0.430910i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.01304e7	−	1.89582e7i	0.844742	−	0.795551i
$894$		0			0
$895$	−484380.	+	838971.i	−0.0202129	+	0.0350098i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.21039e6	−	7.29261e6i	−0.173749	−	0.300943i
$900$		0			0
$901$	−1.45215e7			−0.595934
$902$		0			0
$903$		0			0
$904$		0			0
$905$	5.27053e6			0.213911
$906$		0			0
$907$	1.41679e7	−	2.45394e7i	0.571855	−	0.990482i	−0.424520	−	0.905418i	$-0.639557\pi$
$907$	1.41679e7	−	2.45394e7i	0.571855	−	0.990482i	0.996376	−	0.0850637i	$-0.0271094\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	7.34867e6			0.293368			0.146684	−	0.989183i	$-0.453140\pi$
$911$	7.34867e6			0.293368			0.146684	+	0.989183i	$0.453140\pi$
$912$		0			0
$913$	3.54545e6			0.140765
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3.58885e7	+	6.21606e7i	−1.40939	+	2.44114i
$918$		0			0
$919$	−1.01362e7			−0.395899			−0.197949	−	0.980212i	$-0.563428\pi$
$919$	−1.01362e7			−0.395899			−0.197949	+	0.980212i	$0.563428\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.06267e7			0.796941
$924$		0			0
$925$	−2.99666e6	−	5.19036e6i	−0.115155	−	0.199454i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.13628e7	+	1.96809e7i	−0.431961	+	0.748179i	−0.997042	−	0.0768569i	$-0.975512\pi$
$929$	−1.13628e7	+	1.96809e7i	−0.431961	+	0.748179i	0.565081	+	0.825035i	$0.308845\pi$
$930$		0			0
$931$	1.69396e7	+	7.17586e7i	0.640513	+	2.71331i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.79177e6	+	3.10344e6i	0.0670275	+	0.116095i
$936$		0			0
$937$	−1.38676e6	−	2.40195e6i	−0.0516005	−	0.0893747i	0.839071	−	0.544021i	$-0.183099\pi$
$937$	−1.38676e6	−	2.40195e6i	−0.0516005	−	0.0893747i	−0.890672	+	0.454647i	$0.849766\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.59541e7	+	2.76333e7i	0.587351	+	1.01732i	0.994578	+	0.103994i	$0.0331623\pi$
$941$	1.59541e7	+	2.76333e7i	0.587351	+	1.01732i	−0.407227	+	0.913327i	$0.633504\pi$
$942$		0			0
$943$	6.05629e6			0.221782
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.89457e6	+	1.36738e7i	−0.286058	+	0.495466i	−0.972865	−	0.231373i	$-0.925678\pi$
$947$	−7.89457e6	+	1.36738e7i	−0.286058	+	0.495466i	0.686808	+	0.726839i	$0.259012\pi$
$948$		0			0
$949$	−1.10659e7			−0.398860
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−8.02282e6	+	1.38959e7i	−0.286151	+	0.495628i	−0.972888	−	0.231278i	$-0.925709\pi$
$953$	−8.02282e6	+	1.38959e7i	−0.286151	+	0.495628i	0.686737	+	0.726906i	$0.259042\pi$
$954$		0			0
$955$	−1.86244e7	+	3.22585e7i	−0.660808	+	1.14455i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.63220e7	+	6.29116e7i	1.27533	+	2.20894i
$960$		0			0
$961$	−1.50558e7			−0.525891
$962$		0			0
$963$		0			0
$964$		0			0
$965$	884662.	+	1.53228e6i	0.0305815	+	0.0529687i
$966$		0			0
$967$	3.12207e6	−	5.40759e6i	0.107368	−	0.185968i	−0.807335	−	0.590093i	$-0.799091\pi$
$967$	3.12207e6	−	5.40759e6i	0.107368	−	0.185968i	0.914703	+	0.404126i	$0.132424\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.44348e7	−	2.50019e7i	0.491319	−	0.850990i	−0.508631	−	0.860985i	$-0.669848\pi$
$971$	1.44348e7	−	2.50019e7i	0.491319	−	0.850990i	0.999950	+	0.00999502i	$0.00318157\pi$
$972$		0			0
$973$	2.24109e7	+	3.88169e7i	0.758888	+	1.31443i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.53409e6			−0.0514179			−0.0257089	−	0.999669i	$-0.508184\pi$
$977$	−1.53409e6			−0.0514179			−0.0257089	+	0.999669i	$0.508184\pi$
$978$		0			0
$979$	−2.43335e6	−	4.21468e6i	−0.0811423	−	0.140543i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.37733e7	+	2.38561e7i	−0.454627	+	0.787437i	−0.998667	−	0.0516223i	$-0.983561\pi$
$983$	−1.37733e7	+	2.38561e7i	−0.454627	+	0.787437i	0.544040	+	0.839060i	$0.316894\pi$
$984$		0			0
$985$	−1.29487e7	+	2.24278e7i	−0.425241	+	0.736538i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.19005e7			−0.386878
$990$		0			0
$991$	1.81799e7	−	3.14885e7i	0.588040	−	1.01851i	−0.406449	−	0.913673i	$-0.633233\pi$
$991$	1.81799e7	−	3.14885e7i	0.588040	−	1.01851i	0.994489	−	0.104841i	$-0.0334334\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.79842e6			−0.185674
$996$		0			0
$997$	1.15365e7	+	1.99819e7i	0.367568	+	0.636646i	0.989185	−	0.146675i	$-0.0468572\pi$
$997$	1.15365e7	+	1.99819e7i	0.367568	+	0.636646i	−0.621617	+	0.783322i	$0.713524\pi$
$998$		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.505.2		18
3.2	odd	2		76.6.e.a.49.1	yes	18
12.11	even	2		304.6.i.d.49.9		18
19.7	even	3	inner	684.6.k.f.577.2		18
57.26	odd	6		76.6.e.a.45.1	✓	18
228.83	even	6		304.6.i.d.273.9		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.e.a.45.1	✓	18	57.26	odd	6
76.6.e.a.49.1	yes	18	3.2	odd	2
304.6.i.d.49.9		18	12.11	even	2
304.6.i.d.273.9		18	228.83	even	6
684.6.k.f.505.2		18	1.1	even	1	trivial
684.6.k.f.577.2		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)

\(f(q)\)	\(=\)	\(q+(-35.6401 + 61.7304i) q^{5} +252.315 q^{7} +O(q^{10})\)	\(q+(-35.6401 + 61.7304i) q^{5} +252.315 q^{7} -88.0323 q^{11} +(307.862 + 533.232i) q^{13} +(-285.543 + 494.575i) q^{17} +(361.524 + 1531.47i) q^{19} +(1214.28 + 2103.20i) q^{23} +(-977.931 - 1693.83i) q^{25} +(-1142.82 - 1979.43i) q^{29} +3684.20 q^{31} +(-8992.54 + 15575.5i) q^{35} +3064.28 q^{37} +(1246.89 - 2159.67i) q^{41} +(-2450.11 + 4243.71i) q^{43} +(-8786.51 - 15218.7i) q^{47} +46856.0 q^{49} +(12713.9 + 22021.1i) q^{53} +(3137.48 - 5434.27i) q^{55} +(11756.9 - 20363.5i) q^{59} +(9886.00 + 17123.1i) q^{61} -43888.8 q^{65} +(-13694.3 - 23719.2i) q^{67} +(16750.0 - 29011.9i) q^{71} +(-8986.08 + 15564.3i) q^{73} -22211.9 q^{77} +(41242.9 - 71434.8i) q^{79} -40274.4 q^{83} +(-20353.5 - 35253.4i) q^{85} +(27641.5 + 47876.5i) q^{89} +(77678.2 + 134543. i) q^{91} +(-107423. - 32264.7i) q^{95} +(13348.8 - 23120.7i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 11 q^{5} + 336 q^{7}+O(q^{10}) \) 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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