LMFDB - Embedded newform 108.6.e.a.73.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,6,Mod(37,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("108.37");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	108.e (of order $3$, degree $2$, not 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:	$17.3214525398$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 175x^{8} + 8800x^{6} + 124623x^{4} + 498609x^{2} + 442368 $ x^10 + 175x^8 + 8800x^6 + 124623x^4 + 498609x^2 + 442368
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{16} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			73.5
Root			$1.11227i$ of defining polynomial
Character	$\chi$	$=$	108.73
Dual form			108.6.e.a.37.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(55.1996 + 95.6086i) q^{5} +(-50.8724 + 88.1135i) q^{7} +O(q^{10})$	$q+(55.1996 + 95.6086i) q^{5} +(-50.8724 + 88.1135i) q^{7} +(-75.1560 + 130.174i) q^{11} +(-317.712 - 550.293i) q^{13} -1498.54 q^{17} +1437.69 q^{19} +(-632.053 - 1094.75i) q^{23} +(-4531.50 + 7848.79i) q^{25} +(-1388.75 + 2405.38i) q^{29} +(3484.34 + 6035.05i) q^{31} -11232.5 q^{35} -7950.71 q^{37} +(-1013.77 - 1755.90i) q^{41} +(6261.65 - 10845.5i) q^{43} +(-3241.17 + 5613.87i) q^{47} +(3227.51 + 5590.21i) q^{49} -9827.54 q^{53} -16594.3 q^{55} +(23544.0 + 40779.3i) q^{59} +(-4168.92 + 7220.78i) q^{61} +(35075.2 - 60752.0i) q^{65} +(3630.45 + 6288.12i) q^{67} -3582.33 q^{71} +58077.5 q^{73} +(-7646.73 - 13244.5i) q^{77} +(31871.4 - 55202.9i) q^{79} +(41423.3 - 71747.2i) q^{83} +(-82718.9 - 143273. i) q^{85} +3861.51 q^{89} +64651.0 q^{91} +(79359.8 + 137455. i) q^{95} +(-34638.6 + 59995.8i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 21 q^{5} + 29 q^{7}+O(q^{10}) $ 10 * q + 21 * q^5 + 29 * q^7	$ 10 q + 21 q^{5} + 29 q^{7} - 177 q^{11} - 181 q^{13} - 2280 q^{17} - 832 q^{19} - 399 q^{23} - 4778 q^{25} + 6033 q^{29} + 2759 q^{31} - 37146 q^{35} - 15172 q^{37} + 18435 q^{41} + 1469 q^{43} + 25155 q^{47} - 4056 q^{49} - 116844 q^{53} + 14778 q^{55} + 90537 q^{59} + 1403 q^{61} + 148407 q^{65} + 13907 q^{67} - 229368 q^{71} + 15200 q^{73} + 211983 q^{77} + 29993 q^{79} + 228951 q^{83} - 49662 q^{85} - 598332 q^{89} + 124930 q^{91} + 394764 q^{95} + 40541 q^{97}+O(q^{100}) $ 10 * q + 21 * q^5 + 29 * q^7 - 177 * q^11 - 181 * q^13 - 2280 * q^17 - 832 * q^19 - 399 * q^23 - 4778 * q^25 + 6033 * q^29 + 2759 * q^31 - 37146 * q^35 - 15172 * q^37 + 18435 * q^41 + 1469 * q^43 + 25155 * q^47 - 4056 * q^49 - 116844 * q^53 + 14778 * q^55 + 90537 * q^59 + 1403 * q^61 + 148407 * q^65 + 13907 * q^67 - 229368 * q^71 + 15200 * q^73 + 211983 * q^77 + 29993 * q^79 + 228951 * q^83 - 49662 * q^85 - 598332 * q^89 + 124930 * q^91 + 394764 * q^95 + 40541 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$	55.1996	+	95.6086i	0.987441	+	1.71030i	0.630542	+	0.776155i	$0.282833\pi$
$5$	55.1996	+	95.6086i	0.987441	+	1.71030i	0.356899	+	0.934143i	$0.383834\pi$
$6$		0			0
$7$	−50.8724	+	88.1135i	−0.392407	+	0.679669i	−0.992766	−	0.120061i	$-0.961691\pi$
$7$	−50.8724	+	88.1135i	−0.392407	+	0.679669i	0.600359	+	0.799730i	$0.295024\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−75.1560	+	130.174i	−0.187276	+	0.324371i	−0.944341	−	0.328968i	$-0.893299\pi$
$11$	−75.1560	+	130.174i	−0.187276	+	0.324371i	0.757065	+	0.653339i	$0.226633\pi$
$12$		0			0
$13$	−317.712	−	550.293i	−0.521405	−	0.903100i	−0.999690	−	0.0248953i	$-0.992075\pi$
$13$	−317.712	−	550.293i	−0.521405	−	0.903100i	0.478285	−	0.878205i	$-0.341259\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1498.54			−1.25761			−0.628806	−	0.777562i	$-0.716456\pi$
$17$	−1498.54			−1.25761			−0.628806	+	0.777562i	$0.716456\pi$
$18$		0			0
$19$	1437.69			0.913651			0.456825	−	0.889556i	$-0.348986\pi$
$19$	1437.69			0.913651			0.456825	+	0.889556i	$0.348986\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−632.053	−	1094.75i	−0.249134	−	0.431513i	0.714151	−	0.699991i	$-0.246813\pi$
$23$	−632.053	−	1094.75i	−0.249134	−	0.431513i	−0.963286	+	0.268478i	$0.913479\pi$
$24$		0			0
$25$	−4531.50	+	7848.79i	−1.45008	+	2.51161i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1388.75	+	2405.38i	−0.306640	+	0.531116i	−0.977625	−	0.210355i	$-0.932538\pi$
$29$	−1388.75	+	2405.38i	−0.306640	+	0.531116i	0.670985	+	0.741471i	$0.265871\pi$
$30$		0			0
$31$	3484.34	+	6035.05i	0.651203	+	1.12792i	0.982831	+	0.184506i	$0.0590686\pi$
$31$	3484.34	+	6035.05i	0.651203	+	1.12792i	−0.331628	+	0.943410i	$0.607598\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−11232.5			−1.54992
$36$		0			0
$37$	−7950.71			−0.954776			−0.477388	−	0.878693i	$-0.658416\pi$
$37$	−7950.71			−0.954776			−0.477388	+	0.878693i	$0.658416\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1013.77	−	1755.90i	−0.0941843	−	0.163132i	0.815084	−	0.579343i	$-0.196691\pi$
$41$	−1013.77	−	1755.90i	−0.0941843	−	0.163132i	−0.909268	+	0.416211i	$0.863358\pi$
$42$		0			0
$43$	6261.65	−	10845.5i	0.516438	−	0.894496i	−0.483380	−	0.875410i	$-0.660591\pi$
$43$	6261.65	−	10845.5i	0.516438	−	0.894496i	0.999818	−	0.0190856i	$-0.00607551\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3241.17	+	5613.87i	−0.214021	+	0.370696i	−0.952969	−	0.303066i	$-0.901990\pi$
$47$	−3241.17	+	5613.87i	−0.214021	+	0.370696i	0.738948	+	0.673763i	$0.235323\pi$
$48$		0			0
$49$	3227.51	+	5590.21i	0.192034	+	0.332612i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9827.54			−0.480568			−0.240284	−	0.970703i	$-0.577241\pi$
$53$	−9827.54			−0.480568			−0.240284	+	0.970703i	$0.577241\pi$
$54$		0			0
$55$	−16594.3			−0.739696
$56$		0			0
$57$		0			0
$58$		0			0
$59$	23544.0	+	40779.3i	0.880541	+	1.52514i	0.850741	+	0.525586i	$0.176154\pi$
$59$	23544.0	+	40779.3i	0.880541	+	1.52514i	0.0298005	+	0.999556i	$0.490513\pi$
$60$		0			0
$61$	−4168.92	+	7220.78i	−0.143450	+	0.248462i	−0.928793	−	0.370598i	$-0.879153\pi$
$61$	−4168.92	+	7220.78i	−0.143450	+	0.248462i	0.785344	+	0.619060i	$0.212486\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	35075.2	−	60752.0i	1.02971	−	1.78352i
$66$		0			0
$67$	3630.45	+	6288.12i	0.0988036	+	0.171133i	0.911190	−	0.411987i	$-0.135165\pi$
$67$	3630.45	+	6288.12i	0.0988036	+	0.171133i	−0.812386	+	0.583120i	$0.801832\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3582.33			−0.0843372			−0.0421686	−	0.999111i	$-0.513427\pi$
$71$	−3582.33			−0.0843372			−0.0421686	+	0.999111i	$0.513427\pi$
$72$		0			0
$73$	58077.5			1.27556			0.637780	−	0.770218i	$-0.279853\pi$
$73$	58077.5			1.27556			0.637780	+	0.770218i	$0.279853\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7646.73	−	13244.5i	−0.146977	−	0.254571i
$78$		0			0
$79$	31871.4	−	55202.9i	0.574558	−	0.995163i	−0.421532	−	0.906814i	$-0.638507\pi$
$79$	31871.4	−	55202.9i	0.574558	−	0.995163i	0.996090	−	0.0883495i	$-0.0281592\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	41423.3	−	71747.2i	0.660008	−	1.14317i	−0.320605	−	0.947213i	$-0.603886\pi$
$83$	41423.3	−	71747.2i	0.660008	−	1.14317i	0.980613	−	0.195954i	$-0.0627804\pi$
$84$		0			0
$85$	−82718.9	−	143273.i	−1.24182	−	2.15089i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3861.51			0.0516752			0.0258376	−	0.999666i	$-0.491775\pi$
$89$	3861.51			0.0516752			0.0258376	+	0.999666i	$0.491775\pi$
$90$		0			0
$91$	64651.0			0.818412
$92$		0			0
$93$		0			0
$94$		0			0
$95$	79359.8	+	137455.i	0.902176	+	1.56262i
$96$		0			0
$97$	−34638.6	+	59995.8i	−0.373793	+	0.647428i	−0.990146	−	0.140042i	$-0.955276\pi$
$97$	−34638.6	+	59995.8i	−0.373793	+	0.647428i	0.616353	+	0.787470i	$0.288610\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12722.8	+	22036.6i	−0.124103	+	0.214952i	−0.921382	−	0.388659i	$-0.872939\pi$
$101$	−12722.8	+	22036.6i	−0.124103	+	0.214952i	0.797279	+	0.603611i	$0.206272\pi$
$102$		0			0
$103$	31693.9	+	54895.4i	0.294363	+	0.509851i	0.974836	−	0.222922i	$-0.0715594\pi$
$103$	31693.9	+	54895.4i	0.294363	+	0.509851i	−0.680474	+	0.732772i	$0.738226\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	61158.9			0.516417			0.258208	−	0.966089i	$-0.416868\pi$
$107$	61158.9			0.516417			0.258208	+	0.966089i	$0.416868\pi$
$108$		0			0
$109$	−124036.			−0.999957			−0.499979	−	0.866038i	$-0.666659\pi$
$109$	−124036.			−0.999957			−0.499979	+	0.866038i	$0.666659\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	58718.5	+	101703.i	0.432592	+	0.749272i	0.997096	−	0.0761589i	$-0.0242656\pi$
$113$	58718.5	+	101703.i	0.432592	+	0.749272i	−0.564503	+	0.825431i	$0.690932\pi$
$114$		0			0
$115$	69778.2	−	120859.i	0.492011	−	0.852188i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	76234.3	−	132042.i	0.493495	−	0.854759i
$120$		0			0
$121$	69228.6	+	119908.i	0.429855	+	0.744531i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−655551.			−3.75259
$126$		0			0
$127$	−29838.6			−0.164161			−0.0820803	−	0.996626i	$-0.526156\pi$
$127$	−29838.6			−0.164161			−0.0820803	+	0.996626i	$0.526156\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−86028.8	−	149006.i	−0.437992	−	0.758624i	0.559543	−	0.828801i	$-0.310977\pi$
$131$	−86028.8	−	149006.i	−0.437992	−	0.758624i	−0.997535	+	0.0701777i	$0.977643\pi$
$132$		0			0
$133$	−73138.5	+	126680.i	−0.358523	+	0.620980i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	95180.7	−	164858.i	0.433259	−	0.750426i	−0.563893	−	0.825848i	$-0.690697\pi$
$137$	95180.7	−	164858.i	0.433259	−	0.750426i	0.997152	+	0.0754216i	$0.0240303\pi$
$138$		0			0
$139$	168532.	+	291905.i	0.739852	+	1.28146i	0.952562	+	0.304345i	$0.0984376\pi$
$139$	168532.	+	291905.i	0.739852	+	1.28146i	−0.212710	+	0.977115i	$0.568229\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	95511.9			0.390586
$144$		0			0
$145$	−306634.			−1.21115
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−27549.4	−	47716.9i	−0.101659	−	0.176079i	0.810709	−	0.585449i	$-0.199082\pi$
$149$	−27549.4	−	47716.9i	−0.101659	−	0.176079i	−0.912368	+	0.409370i	$0.865748\pi$
$150$		0			0
$151$	−167032.	+	289308.i	−0.596152	+	1.03257i	0.397231	+	0.917719i	$0.369971\pi$
$151$	−167032.	+	289308.i	−0.596152	+	1.03257i	−0.993383	+	0.114847i	$0.963362\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−384669.	+	666266.i	−1.28605	+	2.22750i
$156$		0			0
$157$	60053.4	+	104015.i	0.194441	+	0.336782i	0.946717	−	0.322066i	$-0.104377\pi$
$157$	60053.4	+	104015.i	0.194441	+	0.336782i	−0.752276	+	0.658848i	$0.771044\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	128616.			0.391048
$162$		0			0
$163$	367083.			1.08217			0.541085	−	0.840968i	$-0.318014\pi$
$163$	367083.			1.08217			0.541085	+	0.840968i	$0.318014\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	294724.	+	510477.i	0.817758	+	1.41640i	0.907331	+	0.420418i	$0.138116\pi$
$167$	294724.	+	510477.i	0.817758	+	1.41640i	−0.0895730	+	0.995980i	$0.528550\pi$
$168$		0			0
$169$	−16235.3	+	28120.4i	−0.0437264	+	0.0757363i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	218229.	−	377983.i	0.554366	−	0.960190i	−0.443587	−	0.896231i	$-0.646294\pi$
$173$	218229.	−	377983.i	0.554366	−	0.960190i	0.997953	−	0.0639583i	$-0.0203725\pi$
$174$		0			0
$175$	−461056.	−	798572.i	−1.13804	−	1.97115i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−241064.			−0.562341			−0.281171	−	0.959658i	$-0.590723\pi$
$179$	−241064.			−0.562341			−0.281171	+	0.959658i	$0.590723\pi$
$180$		0			0
$181$	27128.8			0.0615510			0.0307755	−	0.999526i	$-0.490202\pi$
$181$	27128.8			0.0615510			0.0307755	+	0.999526i	$0.490202\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−438876.	−	760156.i	−0.942785	−	1.63295i
$186$		0			0
$187$	112624.	−	195071.i	0.235520	−	0.407933i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−28036.3	+	48560.2i	−0.0556079	+	0.0963157i	−0.892489	−	0.451068i	$-0.851043\pi$
$191$	−28036.3	+	48560.2i	−0.0556079	+	0.0963157i	0.836881	+	0.547384i	$0.184376\pi$
$192$		0			0
$193$	−177242.	−	306992.i	−0.342510	−	0.593244i	0.642388	−	0.766379i	$-0.277944\pi$
$193$	−177242.	−	306992.i	−0.342510	−	0.593244i	−0.984898	+	0.173135i	$0.944610\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	816895.			1.49969			0.749844	−	0.661615i	$-0.230129\pi$
$197$	816895.			1.49969			0.749844	+	0.661615i	$0.230129\pi$
$198$		0			0
$199$	860396.			1.54016			0.770079	−	0.637948i	$-0.220217\pi$
$199$	860396.			1.54016			0.770079	+	0.637948i	$0.220217\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−141298.	−	244735.i	−0.240655	−	0.416827i
$204$		0			0
$205$	111919.	−	193850.i	0.186003	−	0.322166i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−108051.	+	187149.i	−0.171105	+	0.296362i
$210$		0			0
$211$	193586.	+	335300.i	0.299341	+	0.518475i	0.975985	−	0.217836i	$-0.0698997\pi$
$211$	193586.	+	335300.i	0.299341	+	0.518475i	−0.676644	+	0.736310i	$0.736566\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.38256e6			2.03981
$216$		0			0
$217$	−709026.			−1.02215
$218$		0			0
$219$		0			0
$220$		0			0
$221$	476105.	+	824637.i	0.655725	+	1.13575i
$222$		0			0
$223$	147497.	−	255472.i	0.198619	−	0.344018i	−0.749462	−	0.662047i	$-0.769688\pi$
$223$	147497.	−	255472.i	0.198619	−	0.344018i	0.948081	+	0.318029i	$0.103021\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	343990.	−	595808.i	0.443079	−	0.767435i	−0.554837	−	0.831959i	$-0.687220\pi$
$227$	343990.	−	595808.i	0.443079	−	0.767435i	0.997916	+	0.0645238i	$0.0205529\pi$
$228$		0			0
$229$	543233.	+	940908.i	0.684538	+	1.18565i	0.973582	+	0.228339i	$0.0733294\pi$
$229$	543233.	+	940908.i	0.684538	+	1.18565i	−0.289044	+	0.957316i	$0.593337\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−168058.			−0.202801			−0.101400	−	0.994846i	$-0.532332\pi$
$233$	−168058.			−0.202801			−0.101400	+	0.994846i	$0.532332\pi$
$234$		0			0
$235$	−715646.			−0.845334
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−773138.	−	1.33911e6i	−0.875512	−	1.51643i	−0.856217	−	0.516617i	$-0.827191\pi$
$239$	−773138.	−	1.33911e6i	−0.875512	−	1.51643i	−0.0192952	−	0.999814i	$-0.506142\pi$
$240$		0			0
$241$	576865.	−	999160.i	0.639782	−	1.10813i	−0.345699	−	0.938346i	$-0.612358\pi$
$241$	576865.	−	999160.i	0.639782	−	1.10813i	0.985480	−	0.169789i	$-0.0543086\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−356314.	+	617155.i	−0.379244	+	0.656869i
$246$		0			0
$247$	−456770.	−	791149.i	−0.476382	−	0.825118i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	586711.			0.587814			0.293907	−	0.955834i	$-0.405044\pi$
$251$	586711.			0.587814			0.293907	+	0.955834i	$0.405044\pi$
$252$		0			0
$253$	190010.			0.186628
$254$		0			0
$255$		0			0
$256$		0			0
$257$	54171.0	+	93826.9i	0.0511604	+	0.0886124i	0.890471	−	0.455039i	$-0.150375\pi$
$257$	54171.0	+	93826.9i	0.0511604	+	0.0886124i	−0.839311	+	0.543651i	$0.817041\pi$
$258$		0			0
$259$	404471.	−	700565.i	0.374661	−	0.648931i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−868353.	+	1.50403e6i	−0.774117	+	1.34081i	0.161172	+	0.986926i	$0.448473\pi$
$263$	−868353.	+	1.50403e6i	−0.774117	+	1.34081i	−0.935289	+	0.353884i	$0.884861\pi$
$264$		0			0
$265$	−542476.	−	939597.i	−0.474533	−	0.821915i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−18297.3			−0.0154172			−0.00770860	−	0.999970i	$-0.502454\pi$
$269$	−18297.3			−0.0154172			−0.00770860	+	0.999970i	$0.502454\pi$
$270$		0			0
$271$	7105.09			0.00587688			0.00293844	−	0.999996i	$-0.499065\pi$
$271$	7105.09			0.00587688			0.00293844	+	0.999996i	$0.499065\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−681139.	−	1.17977e6i	−0.543130	−	0.940729i
$276$		0			0
$277$	−355202.	+	615227.i	−0.278148	+	0.481766i	−0.970924	−	0.239386i	$-0.923054\pi$
$277$	−355202.	+	615227.i	−0.278148	+	0.481766i	0.692777	+	0.721152i	$0.256387\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−514224.	+	890661.i	−0.388496	+	0.672894i	−0.992247	−	0.124278i	$-0.960338\pi$
$281$	−514224.	+	890661.i	−0.388496	+	0.672894i	0.603752	+	0.797172i	$0.293672\pi$
$282$		0			0
$283$	922317.	+	1.59750e6i	0.684564	+	1.18570i	0.973574	+	0.228374i	$0.0733407\pi$
$283$	922317.	+	1.59750e6i	0.684564	+	1.18570i	−0.289010	+	0.957326i	$0.593326\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	206291.			0.147834
$288$		0			0
$289$	825769.			0.581586
$290$		0			0
$291$		0			0
$292$		0			0
$293$	470486.	+	814906.i	0.320168	+	0.554547i	0.980522	−	0.196407i	$-0.0629274\pi$
$293$	470486.	+	814906.i	0.320168	+	0.554547i	−0.660355	+	0.750954i	$0.729594\pi$
$294$		0			0
$295$	−2.59924e6	+	4.50201e6i	−1.73896	+	3.01198i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−401621.	+	695629.i	−0.259800	+	0.449987i
$300$		0			0
$301$	637090.	+	1.10347e6i	0.405307	+	0.702013i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−920492.			−0.566592
$306$		0			0
$307$	−2.93094e6			−1.77485			−0.887425	−	0.460952i	$-0.847508\pi$
$307$	−2.93094e6			−1.77485			−0.887425	+	0.460952i	$0.847508\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.14126e6	−	1.97671e6i	−0.669086	−	1.15889i	−0.978160	−	0.207853i	$-0.933352\pi$
$311$	−1.14126e6	−	1.97671e6i	−0.669086	−	1.15889i	0.309074	−	0.951038i	$-0.399981\pi$
$312$		0			0
$313$	−401324.	+	695114.i	−0.231544	+	0.401047i	−0.958263	−	0.285889i	$-0.907711\pi$
$313$	−401324.	+	695114.i	−0.231544	+	0.401047i	0.726718	+	0.686936i	$0.241045\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	555982.	−	962990.i	0.310751	−	0.538237i	−0.667774	−	0.744364i	$-0.732753\pi$
$317$	555982.	−	962990.i	0.310751	−	0.538237i	0.978525	+	0.206127i	$0.0660861\pi$
$318$		0			0
$319$	−208746.	−	361558.i	−0.114853	−	0.198930i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.15443e6			−1.14902
$324$		0			0
$325$	5.75885e6			3.02432
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−329772.	−	571182.i	−0.167967	−	0.290927i
$330$		0			0
$331$	613290.	−	1.06225e6i	0.307678	−	0.532913i	−0.670176	−	0.742202i	$-0.733782\pi$
$331$	613290.	−	1.06225e6i	0.307678	−	0.532913i	0.977854	+	0.209289i	$0.0671149\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−400799.	+	694203.i	−0.195126	+	0.337967i
$336$		0			0
$337$	−1.15064e6	−	1.99297e6i	−0.551907	−	0.955930i	−0.998137	−	0.0610122i	$-0.980567\pi$
$337$	−1.15064e6	−	1.99297e6i	−0.551907	−	0.955930i	0.446230	−	0.894918i	$-0.352766\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.04748e6			−0.487818
$342$		0			0
$343$	−2.36679e6			−1.08624
$344$		0			0
$345$		0			0
$346$		0			0
$347$	809651.	+	1.40236e6i	0.360973	+	0.625223i	0.988121	−	0.153677i	$-0.0491115\pi$
$347$	809651.	+	1.40236e6i	0.360973	+	0.625223i	−0.627149	+	0.778900i	$0.715778\pi$
$348$		0			0
$349$	139307.	−	241287.i	0.0612223	−	0.106040i	−0.833790	−	0.552082i	$-0.813833\pi$
$349$	139307.	−	241287.i	0.0612223	−	0.106040i	0.895012	+	0.446042i	$0.147167\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.68496e6	−	2.91844e6i	0.719704	−	1.24656i	−0.241413	−	0.970422i	$-0.577611\pi$
$353$	1.68496e6	−	2.91844e6i	0.719704	−	1.24656i	0.961117	−	0.276142i	$-0.0890559\pi$
$354$		0			0
$355$	−197743.	−	342501.i	−0.0832781	−	0.144242i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−272571.			−0.111621			−0.0558103	−	0.998441i	$-0.517774\pi$
$359$	−272571.			−0.111621			−0.0558103	+	0.998441i	$0.517774\pi$
$360$		0			0
$361$	−409156.			−0.165242
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.20586e6	+	5.55271e6i	1.25954	+	2.18159i
$366$		0			0
$367$	1.97882e6	−	3.42743e6i	0.766906	−	1.32832i	−0.172327	−	0.985040i	$-0.555129\pi$
$367$	1.97882e6	−	3.42743e6i	0.766906	−	1.32832i	0.939233	−	0.343281i	$-0.111538\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	499950.	−	865939.i	0.188578	−	0.326627i
$372$		0			0
$373$	961939.	+	1.66613e6i	0.357994	+	0.620063i	0.987626	−	0.156830i	$-0.0501276\pi$
$373$	961939.	+	1.66613e6i	0.357994	+	0.620063i	−0.629632	+	0.776894i	$0.716794\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.76489e6			0.639534
$378$		0			0
$379$	−3.11015e6			−1.11220			−0.556101	−	0.831115i	$-0.687703\pi$
$379$	−3.11015e6			−1.11220			−0.556101	+	0.831115i	$0.687703\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.41711e6	−	2.45450e6i	−0.493634	−	0.855000i	0.506339	−	0.862335i	$-0.330999\pi$
$383$	−1.41711e6	−	2.45450e6i	−0.493634	−	0.855000i	−0.999973	+	0.00733479i	$0.997665\pi$
$384$		0			0
$385$	844193.	−	1.46218e6i	0.290262	−	0.502748i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.02692e6	+	1.77867e6i	−0.344081	+	0.595966i	−0.985186	−	0.171487i	$-0.945143\pi$
$389$	−1.02692e6	+	1.77867e6i	−0.344081	+	0.595966i	0.641105	+	0.767453i	$0.278476\pi$
$390$		0			0
$391$	947157.	+	1.64052e6i	0.313314	+	0.542676i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.03716e6			2.26937
$396$		0			0
$397$	−3.82167e6			−1.21696			−0.608480	−	0.793569i	$-0.708220\pi$
$397$	−3.82167e6			−1.21696			−0.608480	+	0.793569i	$0.708220\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.65864e6	+	4.60491e6i	0.825656	+	1.43008i	0.901417	+	0.432953i	$0.142528\pi$
$401$	2.65864e6	+	4.60491e6i	0.825656	+	1.43008i	−0.0757604	+	0.997126i	$0.524138\pi$
$402$		0			0
$403$	2.21403e6	−	3.83482e6i	0.679081	−	1.17620i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	597543.	−	1.03498e6i	0.178807	−	0.309702i
$408$		0			0
$409$	−1.57922e6	−	2.73528e6i	−0.466803	−	0.808526i	0.532478	−	0.846444i	$-0.321261\pi$
$409$	−1.57922e6	−	2.73528e6i	−0.466803	−	0.808526i	−0.999281	+	0.0379179i	$0.987927\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.79095e6			−1.38212
$414$		0			0
$415$	9.14620e6			2.60688
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.80782e6	−	3.13124e6i	−0.503060	−	0.871326i	−0.999994	−	0.00353739i	$-0.998874\pi$
$419$	−1.80782e6	−	3.13124e6i	−0.503060	−	0.871326i	0.496933	−	0.867789i	$-0.334459\pi$
$420$		0			0
$421$	−514445.	+	891044.i	−0.141460	+	0.245016i	−0.928047	−	0.372464i	$-0.878513\pi$
$421$	−514445.	+	891044.i	−0.141460	+	0.245016i	0.786587	+	0.617480i	$0.211846\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.79064e6	−	1.17617e7i	1.82364	−	3.15863i
$426$		0			0
$427$	−424166.	−	734677.i	−0.112581	−	0.194996i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.21668e6			−1.61200			−0.806001	−	0.591914i	$-0.798372\pi$
$431$	−6.21668e6			−1.61200			−0.806001	+	0.591914i	$0.798372\pi$
$432$		0			0
$433$	−598070.			−0.153297			−0.0766483	−	0.997058i	$-0.524422\pi$
$433$	−598070.			−0.153297			−0.0766483	+	0.997058i	$0.524422\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−908694.	−	1.57390e6i	−0.227622	−	0.394253i
$438$		0			0
$439$	246548.	−	427034.i	0.0610577	−	0.105755i	−0.833881	−	0.551945i	$-0.813886\pi$
$439$	246548.	−	427034.i	0.0610577	−	0.105755i	0.894938	+	0.446190i	$0.147219\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−97755.8	+	169318.i	−0.0236665	+	0.0409915i	−0.877616	−	0.479364i	$-0.840867\pi$
$443$	−97755.8	+	169318.i	−0.0236665	+	0.0409915i	0.853950	+	0.520356i	$0.174201\pi$
$444$		0			0
$445$	213154.	+	369194.i	0.0510263	+	0.0883801i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.51225e6			0.354004			0.177002	−	0.984210i	$-0.443360\pi$
$449$	1.51225e6			0.354004			0.177002	+	0.984210i	$0.443360\pi$
$450$		0			0
$451$	304763.			0.0705538
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.56871e6	+	6.18119e6i	0.808133	+	1.39973i
$456$		0			0
$457$	−1.28236e6	+	2.22111e6i	−0.287223	+	0.497484i	−0.973146	−	0.230190i	$-0.926065\pi$
$457$	−1.28236e6	+	2.22111e6i	−0.287223	+	0.497484i	0.685923	+	0.727674i	$0.259399\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.68336e6	+	4.64772e6i	−0.588067	+	1.01856i	0.406418	+	0.913687i	$0.366778\pi$
$461$	−2.68336e6	+	4.64772e6i	−0.588067	+	1.01856i	−0.994485	+	0.104875i	$0.966556\pi$
$462$		0			0
$463$	−2.75107e6	−	4.76500e6i	−0.596417	−	1.03302i	−0.993345	−	0.115175i	$-0.963257\pi$
$463$	−2.75107e6	−	4.76500e6i	−0.596417	−	1.03302i	0.396928	−	0.917850i	$-0.370076\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−964114.			−0.204567			−0.102284	−	0.994755i	$-0.532615\pi$
$467$	−964114.			−0.204567			−0.102284	+	0.994755i	$0.532615\pi$
$468$		0			0
$469$	−738757.			−0.155085
$470$		0			0
$471$		0			0
$472$		0			0
$473$	941202.	+	1.63021e6i	0.193433	+	0.335035i
$474$		0			0
$475$	−6.51488e6	+	1.12841e7i	−1.32487	+	2.29474i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−281267.	+	487168.i	−0.0560118	+	0.0970153i	−0.892672	−	0.450707i	$-0.851172\pi$
$479$	−281267.	+	487168.i	−0.0560118	+	0.0970153i	0.836660	+	0.547723i	$0.184505\pi$
$480$		0			0
$481$	2.52603e6	+	4.37522e6i	0.497825	+	0.862258i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.64815e6			−1.47639
$486$		0			0
$487$	3.14185e6			0.600292			0.300146	−	0.953893i	$-0.402965\pi$
$487$	3.14185e6			0.600292			0.300146	+	0.953893i	$0.402965\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.86681e6	+	4.96545e6i	0.536654	+	0.929512i	0.999081	+	0.0428549i	$0.0136453\pi$
$491$	2.86681e6	+	4.96545e6i	0.536654	+	0.929512i	−0.462427	+	0.886657i	$0.653021\pi$
$492$		0			0
$493$	2.08110e6	−	3.60457e6i	0.385634	−	0.667937i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	182241.	−	315651.i	0.0330945	−	0.0573214i
$498$		0			0
$499$	−857810.	−	1.48577e6i	−0.154220	−	0.267116i	0.778555	−	0.627576i	$-0.215953\pi$
$499$	−857810.	−	1.48577e6i	−0.154220	−	0.267116i	−0.932775	+	0.360460i	$0.882620\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5.15229e6			0.907989			0.453995	−	0.891004i	$-0.349999\pi$
$503$	5.15229e6			0.907989			0.453995	+	0.891004i	$0.349999\pi$
$504$		0			0
$505$	−2.80919e6			−0.490176
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−521280.	−	902884.i	−0.0891819	−	0.154468i	0.817984	−	0.575242i	$-0.195092\pi$
$509$	−521280.	−	902884.i	−0.0891819	−	0.154468i	−0.907166	+	0.420774i	$0.861759\pi$
$510$		0			0
$511$	−2.95454e6	+	5.11742e6i	−0.500539	+	0.866959i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.49898e6	+	6.06041e6i	−0.581331	+	1.00690i
$516$		0			0
$517$	−487187.	−	843833.i	−0.0801621	−	0.138845i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.52239e6			−1.53692			−0.768461	−	0.639897i	$-0.778977\pi$
$521$	−9.52239e6			−1.53692			−0.768461	+	0.639897i	$0.778977\pi$
$522$		0			0
$523$	−2.97573e6			−0.475706			−0.237853	−	0.971301i	$-0.576444\pi$
$523$	−2.97573e6			−0.475706			−0.237853	+	0.971301i	$0.576444\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.22143e6	−	9.04378e6i	−0.818960	−	1.41848i
$528$		0			0
$529$	2.41919e6	−	4.19016e6i	0.375864	−	0.651016i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−644172.	+	1.11574e6i	−0.0982163	+	0.170116i
$534$		0			0
$535$	3.37595e6	+	5.84731e6i	0.509931	+	0.883226i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−970266.			−0.143853
$540$		0			0
$541$	1.80579e6			0.265261			0.132631	−	0.991166i	$-0.457658\pi$
$541$	1.80579e6			0.265261			0.132631	+	0.991166i	$0.457658\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.84674e6	−	1.18589e7i	−0.987399	−	1.71023i
$546$		0			0
$547$	4.68486e6	−	8.11442e6i	0.669466	−	1.15955i	−0.308587	−	0.951196i	$-0.599856\pi$
$547$	4.68486e6	−	8.11442e6i	0.669466	−	1.15955i	0.978054	−	0.208353i	$-0.0668104\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.99658e6	+	3.45819e6i	−0.280162	+	0.485254i
$552$		0			0
$553$	3.24275e6	+	5.61660e6i	0.450921	+	0.781018i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.53733e6			−0.483101			−0.241550	−	0.970388i	$-0.577656\pi$
$557$	−3.53733e6			−0.483101			−0.241550	+	0.970388i	$0.577656\pi$
$558$		0			0
$559$	−7.95761e6			−1.07709
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.39030e6	+	2.40807e6i	0.184858	+	0.320183i	0.943529	−	0.331291i	$-0.107484\pi$
$563$	1.39030e6	+	2.40807e6i	0.184858	+	0.320183i	−0.758671	+	0.651474i	$0.774151\pi$
$564$		0			0
$565$	−6.48248e6	+	1.12280e7i	−0.854319	+	1.47972i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.67647e6	−	6.36782e6i	0.476047	−	0.824537i	−0.523576	−	0.851979i	$-0.675403\pi$
$569$	3.67647e6	−	6.36782e6i	0.476047	−	0.824537i	0.999623	+	0.0274412i	$0.00873590\pi$
$570$		0			0
$571$	2.87585e6	+	4.98112e6i	0.369128	+	0.639348i	0.989429	−	0.145016i	$-0.0463232\pi$
$571$	2.87585e6	+	4.98112e6i	0.369128	+	0.639348i	−0.620302	+	0.784363i	$0.712990\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.14566e7			1.44506
$576$		0			0
$577$	−1.30488e7			−1.63167			−0.815835	−	0.578285i	$-0.803722\pi$
$577$	−1.30488e7			−1.63167			−0.815835	+	0.578285i	$0.803722\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.21460e6	+	7.29990e6i	0.517983	+	0.897173i
$582$		0			0
$583$	738598.	−	1.27929e6i	0.0899989	−	0.155883i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	623618.	−	1.08014e6i	0.0747005	−	0.129385i	−0.826255	−	0.563296i	$-0.809533\pi$
$587$	623618.	−	1.08014e6i	0.0747005	−	0.129385i	0.900956	+	0.433910i	$0.142867\pi$
$588$		0			0
$589$	5.00939e6	+	8.67652e6i	0.594972	+	1.03052i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.32769e7			1.55045			0.775226	−	0.631684i	$-0.217636\pi$
$593$	1.32769e7			1.55045			0.775226	+	0.631684i	$0.217636\pi$
$594$		0			0
$595$	1.68324e7			1.94919
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.02457e6	+	3.50666e6i	0.230551	+	0.399325i	0.957970	−	0.286868i	$-0.0926140\pi$
$599$	2.02457e6	+	3.50666e6i	0.230551	+	0.399325i	−0.727420	+	0.686193i	$0.759281\pi$
$600$		0			0
$601$	3.41857e6	−	5.92113e6i	0.386063	−	0.668681i	−0.605853	−	0.795577i	$-0.707168\pi$
$601$	3.41857e6	−	5.92113e6i	0.386063	−	0.668681i	0.991916	+	0.126896i	$0.0405014\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.64279e6	+	1.32377e7i	−0.848914	+	1.47036i
$606$		0			0
$607$	−348100.	−	602927.i	−0.0383471	−	0.0664191i	0.846215	−	0.532842i	$-0.178876\pi$
$607$	−348100.	−	602927.i	−0.0383471	−	0.0664191i	−0.884562	+	0.466423i	$0.845543\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.11904e6			0.446367
$612$		0			0
$613$	−1.42785e6			−0.153473			−0.0767363	−	0.997051i	$-0.524450\pi$
$613$	−1.42785e6			−0.153473			−0.0767363	+	0.997051i	$0.524450\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.51675e6	−	1.30194e7i	−0.794908	−	1.37682i	−0.922897	−	0.385046i	$-0.874186\pi$
$617$	−7.51675e6	−	1.30194e7i	−0.794908	−	1.37682i	0.127989	−	0.991776i	$-0.459148\pi$
$618$		0			0
$619$	3.58151e6	−	6.20336e6i	0.375699	−	0.650730i	−0.614732	−	0.788736i	$-0.710736\pi$
$619$	3.58151e6	−	6.20336e6i	0.375699	−	0.650730i	0.990431	+	0.138006i	$0.0440693\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−196444.	+	340251.i	−0.0202777	+	0.0351221i
$624$		0			0
$625$	−2.20252e7	−	3.81488e7i	−2.25538	−	3.90644i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.19145e7			1.20074
$630$		0			0
$631$	1.38021e7			1.37998			0.689989	−	0.723820i	$-0.257615\pi$
$631$	1.38021e7			1.37998			0.689989	+	0.723820i	$0.257615\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.64708e6	−	2.85283e6i	−0.162099	−	0.280764i
$636$		0			0
$637$	2.05084e6	−	3.55215e6i	0.200254	−	0.346851i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.02329e6	+	3.50445e6i	−0.194497	+	0.336879i	−0.946736	−	0.322012i	$-0.895641\pi$
$641$	−2.02329e6	+	3.50445e6i	−0.194497	+	0.336879i	0.752238	+	0.658891i	$0.228974\pi$
$642$		0			0
$643$	−5.44381e6	−	9.42896e6i	−0.519249	−	0.899366i	−0.999750	−	0.0223713i	$-0.992878\pi$
$643$	−5.44381e6	−	9.42896e6i	−0.519249	−	0.899366i	0.480501	−	0.876994i	$-0.340455\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.45797e6			0.794339			0.397169	−	0.917745i	$-0.369993\pi$
$647$	8.45797e6			0.794339			0.397169	+	0.917745i	$0.369993\pi$
$648$		0			0
$649$	−7.07788e6			−0.659617
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.47774e6	+	4.29157e6i	0.227391	+	0.393853i	0.957034	−	0.289975i	$-0.0936472\pi$
$653$	2.47774e6	+	4.29157e6i	0.227391	+	0.393853i	−0.729643	+	0.683828i	$0.760314\pi$
$654$		0			0
$655$	9.49752e6	−	1.64502e7i	0.864982	−	1.49819i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	8.37152e6	−	1.44999e7i	0.750915	−	1.30062i	−0.196464	−	0.980511i	$-0.562946\pi$
$659$	8.37152e6	−	1.44999e7i	0.750915	−	1.30062i	0.947380	−	0.320113i	$-0.103721\pi$
$660$		0			0
$661$	7.53992e6	+	1.30595e7i	0.671217	+	1.16258i	0.977559	+	0.210661i	$0.0675617\pi$
$661$	7.53992e6	+	1.30595e7i	0.671217	+	1.16258i	−0.306342	+	0.951922i	$0.599105\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.61489e7			−1.41608
$666$		0			0
$667$	3.51105e6			0.305578
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−626639.	−	1.08537e6i	−0.0537293	−	0.0930619i
$672$		0			0
$673$	−7.07096e6	+	1.22473e7i	−0.601784	+	1.04232i	0.390767	+	0.920490i	$0.372210\pi$
$673$	−7.07096e6	+	1.22473e7i	−0.601784	+	1.04232i	−0.992551	+	0.121831i	$0.961123\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.29397e6	−	1.26335e7i	0.611635	−	1.05938i	−0.379330	−	0.925262i	$-0.623845\pi$
$677$	7.29397e6	−	1.26335e7i	0.611635	−	1.05938i	0.990965	−	0.134122i	$-0.0428213\pi$
$678$		0			0
$679$	−3.52429e6	−	6.10425e6i	−0.293358	−	0.508110i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.10096e7			0.903065			0.451532	−	0.892255i	$-0.350878\pi$
$683$	1.10096e7			0.903065			0.451532	+	0.892255i	$0.350878\pi$
$684$		0			0
$685$	2.10158e7			1.71127
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.12233e6	+	5.40803e6i	0.250571	+	0.434001i
$690$		0			0
$691$	−5.51793e6	+	9.55733e6i	−0.439623	+	0.761450i	−0.997660	−	0.0683661i	$-0.978221\pi$
$691$	−5.51793e6	+	9.55733e6i	−0.439623	+	0.761450i	0.558037	+	0.829816i	$0.311555\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.86058e7	+	3.22261e7i	−1.46112	+	2.53073i
$696$		0			0
$697$	1.51917e6	+	2.63128e6i	0.118447	+	0.205157i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	2.51294e7			1.93146			0.965732	−	0.259542i	$-0.0835717\pi$
$701$	2.51294e7			1.93146			0.965732	+	0.259542i	$0.0835717\pi$
$702$		0			0
$703$	−1.14306e7			−0.872332
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.29448e6	−	2.24211e6i	−0.0973974	−	0.168697i
$708$		0			0
$709$	−6.26506e6	+	1.08514e7i	−0.468068	+	0.810718i	−0.999334	−	0.0364869i	$-0.988383\pi$
$709$	−6.26506e6	+	1.08514e7i	−0.468068	+	0.810718i	0.531266	+	0.847205i	$0.321717\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.40457e6	−	7.62894e6i	0.324474	−	0.562006i
$714$		0			0
$715$	5.27222e6	+	9.13175e6i	0.385681	+	0.668019i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.72246e6			0.124258			0.0621292	−	0.998068i	$-0.480211\pi$
$719$	1.72246e6			0.124258			0.0621292	+	0.998068i	$0.480211\pi$
$720$		0			0
$721$	−6.44937e6			−0.462040
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.25862e7	−	2.18000e7i	−0.889304	−	1.54032i
$726$		0			0
$727$	−3.49425e6	+	6.05221e6i	−0.245198	+	0.424696i	−0.962187	−	0.272389i	$-0.912186\pi$
$727$	−3.49425e6	+	6.05221e6i	−0.245198	+	0.424696i	0.716989	+	0.697084i	$0.245520\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.38335e6	+	1.62524e7i	−0.649478	+	1.12493i
$732$		0			0
$733$	2.96305e6	+	5.13214e6i	0.203694	+	0.352808i	0.949716	−	0.313113i	$-0.101372\pi$
$733$	2.96305e6	+	5.13214e6i	0.203694	+	0.352808i	−0.746022	+	0.665922i	$0.768039\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.09140e6			−0.0740142
$738$		0			0
$739$	−1.06632e7			−0.718254			−0.359127	−	0.933289i	$-0.616925\pi$
$739$	−1.06632e7			−0.718254			−0.359127	+	0.933289i	$0.616925\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9.24932e6	−	1.60203e7i	−0.614664	−	1.06463i	−0.990443	−	0.137920i	$-0.955958\pi$
$743$	−9.24932e6	−	1.60203e7i	−0.614664	−	1.06463i	0.375779	−	0.926709i	$-0.377375\pi$
$744$		0			0
$745$	3.04143e6	−	5.26791e6i	0.200765	−	0.347734i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.11130e6	+	5.38892e6i	−0.202645	+	0.350992i
$750$		0			0
$751$	1.30624e7	+	2.26247e7i	0.845127	+	1.46380i	0.885511	+	0.464618i	$0.153808\pi$
$751$	1.30624e7	+	2.26247e7i	0.845127	+	1.46380i	−0.0403845	+	0.999184i	$0.512858\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.68804e7			−2.35466
$756$		0			0
$757$	2.49174e7			1.58039			0.790193	−	0.612858i	$-0.209980\pi$
$757$	2.49174e7			1.58039			0.790193	+	0.612858i	$0.209980\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.62194e6	+	9.73748e6i	0.351904	+	0.609516i	0.986583	−	0.163260i	$-0.0522010\pi$
$761$	5.62194e6	+	9.73748e6i	0.351904	+	0.609516i	−0.634679	+	0.772776i	$0.718868\pi$
$762$		0			0
$763$	6.31000e6	−	1.09292e7i	0.392390	−	0.679640i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.49604e7	−	2.59122e7i	0.918237	−	1.59043i
$768$		0			0
$769$	−4.33731e6	−	7.51244e6i	−0.264487	−	0.458105i	0.702942	−	0.711247i	$-0.251869\pi$
$769$	−4.33731e6	−	7.51244e6i	−0.264487	−	0.458105i	−0.967429	+	0.253142i	$0.918536\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2.17066e7			−1.30660			−0.653300	−	0.757099i	$-0.726616\pi$
$773$	−2.17066e7			−1.30660			−0.653300	+	0.757099i	$0.726616\pi$
$774$		0			0
$775$	−6.31571e7			−3.77718
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.45748e6	−	2.52443e6i	−0.0860516	−	0.149046i
$780$		0			0
$781$	269233.	−	466326.i	0.0157943	−	0.0273566i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.62985e6	+	1.14832e7i	−0.383998	+	0.665105i
$786$		0			0
$787$	−3.93215e6	−	6.81068e6i	−0.226304	−	0.391971i	0.730406	−	0.683014i	$-0.239331\pi$
$787$	−3.93215e6	−	6.81068e6i	−0.226304	−	0.391971i	−0.956710	+	0.291043i	$0.905998\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.19486e7			−0.679009
$792$		0			0
$793$	5.29807e6			0.299181
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9.48089e6	+	1.64214e7i	0.528693	+	0.915723i	0.999440	+	0.0334547i	$0.0106509\pi$
$797$	9.48089e6	+	1.64214e7i	0.528693	+	0.915723i	−0.470747	+	0.882268i	$0.656016\pi$
$798$		0			0
$799$	4.85703e6	−	8.41262e6i	0.269156	−	0.466192i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−4.36488e6	+	7.56019e6i	−0.238882	+	0.413755i
$804$		0			0
$805$	7.09956e6	+	1.22968e7i	0.386137	+	0.668809i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.09408e7			1.12492			0.562459	−	0.826825i	$-0.309855\pi$
$809$	2.09408e7			1.12492			0.562459	+	0.826825i	$0.309855\pi$
$810$		0			0
$811$	3.21466e7			1.71626			0.858130	−	0.513432i	$-0.171626\pi$
$811$	3.21466e7			1.71626			0.858130	+	0.513432i	$0.171626\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.02629e7	+	3.50963e7i	1.06858	+	1.85083i
$816$		0			0
$817$	9.00229e6	−	1.55924e7i	0.471844	−	0.817257i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.48802e7	+	2.57733e7i	−0.770463	+	1.33448i	0.166846	+	0.985983i	$0.446642\pi$
$821$	−1.48802e7	+	2.57733e7i	−0.770463	+	1.33448i	−0.937309	+	0.348498i	$0.886692\pi$
$822$		0			0
$823$	8.26102e6	+	1.43085e7i	0.425142	+	0.736368i	0.996434	−	0.0843793i	$-0.0268907\pi$
$823$	8.26102e6	+	1.43085e7i	0.425142	+	0.736368i	−0.571291	+	0.820747i	$0.693557\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.49588e7			−1.26900			−0.634498	−	0.772925i	$-0.718793\pi$
$827$	−2.49588e7			−1.26900			−0.634498	+	0.772925i	$0.718793\pi$
$828$		0			0
$829$	−2.92909e7			−1.48029			−0.740145	−	0.672447i	$-0.765243\pi$
$829$	−2.92909e7			−1.48029			−0.740145	+	0.672447i	$0.765243\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.83655e6	−	8.37716e6i	−0.241504	−	0.418296i
$834$		0			0
$835$	−3.25373e7	+	5.63563e7i	−1.61498	+	2.79722i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.87417e7	−	3.24616e7i	0.919189	−	1.59208i	0.118539	−	0.992949i	$-0.462179\pi$
$839$	1.87417e7	−	3.24616e7i	0.919189	−	1.59208i	0.800650	−	0.599133i	$-0.204488\pi$
$840$		0			0
$841$	6.39833e6	+	1.10822e7i	0.311944	+	0.540303i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.58473e6			−0.172709
$846$		0			0
$847$	−1.40873e7			−0.674713
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.02527e6	+	8.70402e6i	0.237868	+	0.411999i
$852$		0			0
$853$	−7.41357e6	+	1.28407e7i	−0.348863	+	0.604248i	−0.986048	−	0.166463i	$-0.946765\pi$
$853$	−7.41357e6	+	1.28407e7i	−0.348863	+	0.604248i	0.637185	+	0.770711i	$0.280099\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.66185e6	−	4.61046e6i	0.123803	−	0.214433i	−0.797461	−	0.603370i	$-0.793824\pi$
$857$	2.66185e6	−	4.61046e6i	0.123803	−	0.214433i	0.921264	+	0.388937i	$0.127158\pi$
$858$		0			0
$859$	−1.34434e7	−	2.32846e7i	−0.621621	−	1.07668i	−0.989184	−	0.146680i	$-0.953141\pi$
$859$	−1.34434e7	−	2.32846e7i	−0.621621	−	1.07668i	0.367563	−	0.929999i	$-0.380192\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.21441e6			0.101212			0.0506060	−	0.998719i	$-0.483885\pi$
$863$	2.21441e6			0.101212			0.0506060	+	0.998719i	$0.483885\pi$
$864$		0			0
$865$	4.81846e7			2.18961
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.79066e6	+	8.29766e6i	0.215202	+	0.372740i
$870$		0			0
$871$	2.30687e6	−	3.99562e6i	0.103033	−	0.178459i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.33494e7	−	5.77629e7i	1.47254	−	2.55052i
$876$		0			0
$877$	−1.62323e7	−	2.81151e7i	−0.712657	−	1.23436i	−0.963856	−	0.266423i	$-0.914158\pi$
$877$	−1.62323e7	−	2.81151e7i	−0.712657	−	1.23436i	0.251199	−	0.967935i	$-0.419175\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−7.78418e6			−0.337888			−0.168944	−	0.985626i	$-0.554036\pi$
$881$	−7.78418e6			−0.337888			−0.168944	+	0.985626i	$0.554036\pi$
$882$		0			0
$883$	1.71890e7			0.741906			0.370953	−	0.928652i	$-0.379031\pi$
$883$	1.71890e7			0.741906			0.370953	+	0.928652i	$0.379031\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.81716e7	−	3.14742e7i	−0.775506	−	1.34322i	−0.934510	−	0.355938i	$-0.884162\pi$
$887$	−1.81716e7	−	3.14742e7i	−0.775506	−	1.34322i	0.159004	−	0.987278i	$-0.449172\pi$
$888$		0			0
$889$	1.51796e6	−	2.62918e6i	0.0644178	−	0.111575i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.65979e6	+	8.07099e6i	−0.195541	+	0.338687i
$894$		0			0
$895$	−1.33067e7	−	2.30478e7i	−0.555279	−	0.961771i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.93555e7			−0.798739
$900$		0			0
$901$	1.47270e7			0.604368
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.49750e6	+	2.59375e6i	0.0607779	+	0.105270i
$906$		0			0
$907$	4.39209e6	−	7.60732e6i	0.177277	−	0.307053i	−0.763670	−	0.645607i	$-0.776604\pi$
$907$	4.39209e6	−	7.60732e6i	0.177277	−	0.307053i	0.940947	+	0.338554i	$0.109938\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.83503e6	+	3.17836e6i	−0.0732566	+	0.126884i	−0.900327	−	0.435214i	$-0.856673\pi$
$911$	−1.83503e6	+	3.17836e6i	−0.0732566	+	0.126884i	0.827070	+	0.562099i	$0.190006\pi$
$912$		0			0
$913$	6.22641e6	+	1.07845e7i	0.247207	+	0.428175i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.75060e7			0.687484
$918$		0			0
$919$	3.38118e7			1.32063			0.660313	−	0.750990i	$-0.270424\pi$
$919$	3.38118e7			1.32063			0.660313	+	0.750990i	$0.270424\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.13815e6	+	1.97133e6i	0.0439739	+	0.0761650i
$924$		0			0
$925$	3.60286e7	−	6.24034e7i	1.38450	−	2.39803i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5.89173e6	+	1.02048e7i	−0.223977	+	0.387939i	−0.956012	−	0.293327i	$-0.905237\pi$
$929$	−5.89173e6	+	1.02048e7i	−0.223977	+	0.387939i	0.732035	+	0.681267i	$0.238571\pi$
$930$		0			0
$931$	4.64014e6	+	8.03697e6i	0.175452	+	0.303891i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.48673e7			0.930250
$936$		0			0
$937$	−3.78550e7			−1.40856			−0.704278	−	0.709924i	$-0.748729\pi$
$937$	−3.78550e7			−1.40856			−0.704278	+	0.709924i	$0.748729\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	2.28136e7	+	3.95143e7i	0.839884	+	1.45472i	0.889991	+	0.455978i	$0.150710\pi$
$941$	2.28136e7	+	3.95143e7i	0.839884	+	1.45472i	−0.0501075	+	0.998744i	$0.515956\pi$
$942$		0			0
$943$	−1.28151e6	+	2.21964e6i	−0.0469291	+	0.0812836i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.07942e7	−	3.60166e7i	0.753472	−	1.30505i	−0.192659	−	0.981266i	$-0.561711\pi$
$947$	2.07942e7	−	3.60166e7i	0.753472	−	1.30505i	0.946130	−	0.323786i	$-0.104956\pi$
$948$		0			0
$949$	−1.84519e7	−	3.19597e7i	−0.665084	−	1.15196i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	5.40042e7			1.92617			0.963087	−	0.269190i	$-0.0867560\pi$
$953$	5.40042e7			1.92617			0.963087	+	0.269190i	$0.0867560\pi$
$954$		0			0
$955$	−6.19036e6			−0.219638
$956$		0			0
$957$		0			0
$958$		0			0
$959$	9.68413e6	+	1.67734e7i	0.340028	+	0.588945i
$960$		0			0
$961$	−9.96668e6	+	1.72628e7i	−0.348130	+	0.602979i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.95674e7	−	3.38917e7i	0.676416	−	1.17159i
$966$		0			0
$967$	−8.64673e6	−	1.49766e7i	−0.297362	−	0.515046i	0.678169	−	0.734906i	$-0.262774\pi$
$967$	−8.64673e6	−	1.49766e7i	−0.297362	−	0.515046i	−0.975532	+	0.219859i	$0.929440\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.55169e7			−1.54926			−0.774630	−	0.632415i	$-0.782064\pi$
$971$	−4.55169e7			−1.54926			−0.774630	+	0.632415i	$0.782064\pi$
$972$		0			0
$973$	−3.42944e7			−1.16129
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.29826e6	+	5.71275e6i	0.110547	+	0.191474i	0.915991	−	0.401199i	$-0.131406\pi$
$977$	3.29826e6	+	5.71275e6i	0.110547	+	0.191474i	−0.805444	+	0.592672i	$0.798073\pi$
$978$		0			0
$979$	−290216.	+	502669.i	−0.00967753	+	0.0167620i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.84729e7	+	3.19959e7i	−0.609748	+	1.05611i	0.381534	+	0.924355i	$0.375396\pi$
$983$	−1.84729e7	+	3.19959e7i	−0.609748	+	1.05611i	−0.991282	+	0.131760i	$0.957937\pi$
$984$		0			0
$985$	4.50923e7	+	7.81022e7i	1.48085	+	2.56491i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.58308e7			−0.514649
$990$		0			0
$991$	−7.56015e6			−0.244538			−0.122269	−	0.992497i	$-0.539017\pi$
$991$	−7.56015e6			−0.244538			−0.122269	+	0.992497i	$0.539017\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.74935e7	+	8.22612e7i	1.52082	+	2.63413i
$996$		0			0
$997$	7.80763e6	−	1.35232e7i	0.248761	−	0.430866i	−0.714422	−	0.699715i	$-0.753310\pi$
$997$	7.80763e6	−	1.35232e7i	0.248761	−	0.430866i	0.963182	+	0.268850i	$0.0866434\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.e.a.73.5		10
3.2	odd	2		36.6.e.a.25.5	yes	10
4.3	odd	2		432.6.i.d.289.5		10
9.2	odd	6		324.6.a.e.1.5		5
9.4	even	3	inner	108.6.e.a.37.5		10
9.5	odd	6		36.6.e.a.13.5	✓	10
9.7	even	3		324.6.a.d.1.1		5
12.11	even	2		144.6.i.d.97.1		10
36.23	even	6		144.6.i.d.49.1		10
36.31	odd	6		432.6.i.d.145.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.e.a.13.5	✓	10	9.5	odd	6
36.6.e.a.25.5	yes	10	3.2	odd	2
108.6.e.a.37.5		10	9.4	even	3	inner
108.6.e.a.73.5		10	1.1	even	1	trivial
144.6.i.d.49.1		10	36.23	even	6
144.6.i.d.97.1		10	12.11	even	2
324.6.a.d.1.1		5	9.7	even	3
324.6.a.e.1.5		5	9.2	odd	6
432.6.i.d.145.5		10	36.31	odd	6
432.6.i.d.289.5		10	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	108.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(55.1996 + 95.6086i) q^{5} +(-50.8724 + 88.1135i) q^{7} +O(q^{10})\)	\(q+(55.1996 + 95.6086i) q^{5} +(-50.8724 + 88.1135i) q^{7} +(-75.1560 + 130.174i) q^{11} +(-317.712 - 550.293i) q^{13} -1498.54 q^{17} +1437.69 q^{19} +(-632.053 - 1094.75i) q^{23} +(-4531.50 + 7848.79i) q^{25} +(-1388.75 + 2405.38i) q^{29} +(3484.34 + 6035.05i) q^{31} -11232.5 q^{35} -7950.71 q^{37} +(-1013.77 - 1755.90i) q^{41} +(6261.65 - 10845.5i) q^{43} +(-3241.17 + 5613.87i) q^{47} +(3227.51 + 5590.21i) q^{49} -9827.54 q^{53} -16594.3 q^{55} +(23544.0 + 40779.3i) q^{59} +(-4168.92 + 7220.78i) q^{61} +(35075.2 - 60752.0i) q^{65} +(3630.45 + 6288.12i) q^{67} -3582.33 q^{71} +58077.5 q^{73} +(-7646.73 - 13244.5i) q^{77} +(31871.4 - 55202.9i) q^{79} +(41423.3 - 71747.2i) q^{83} +(-82718.9 - 143273. i) q^{85} +3861.51 q^{89} +64651.0 q^{91} +(79359.8 + 137455. i) q^{95} +(-34638.6 + 59995.8i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 21 q^{5} + 29 q^{7}+O(q^{10}) \) 10 * q + 21 * q^5 + 29 * q^7	\( 10 q + 21 q^{5} + 29 q^{7} - 177 q^{11} - 181 q^{13} - 2280 q^{17} - 832 q^{19} - 399 q^{23} - 4778 q^{25} + 6033 q^{29} + 2759 q^{31} - 37146 q^{35} - 15172 q^{37} + 18435 q^{41} + 1469 q^{43} + 25155 q^{47} - 4056 q^{49} - 116844 q^{53} + 14778 q^{55} + 90537 q^{59} + 1403 q^{61} + 148407 q^{65} + 13907 q^{67} - 229368 q^{71} + 15200 q^{73} + 211983 q^{77} + 29993 q^{79} + 228951 q^{83} - 49662 q^{85} - 598332 q^{89} + 124930 q^{91} + 394764 q^{95} + 40541 q^{97}+O(q^{100}) \) 10 * q + 21 * q^5 + 29 * q^7 - 177 * q^11 - 181 * q^13 - 2280 * q^17 - 832 * q^19 - 399 * q^23 - 4778 * q^25 + 6033 * q^29 + 2759 * q^31 - 37146 * q^35 - 15172 * q^37 + 18435 * q^41 + 1469 * q^43 + 25155 * q^47 - 4056 * q^49 - 116844 * q^53 + 14778 * q^55 + 90537 * q^59 + 1403 * q^61 + 148407 * q^65 + 13907 * q^67 - 229368 * q^71 + 15200 * q^73 + 211983 * q^77 + 29993 * q^79 + 228951 * q^83 - 49662 * q^85 - 598332 * q^89 + 124930 * q^91 + 394764 * q^95 + 40541 * q^97

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