LMFDB - Embedded newform 896.3.k.b.799.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,3,Mod(351,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("896.351");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	896.k (of order $4$, 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:	$24.4142316867$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			799.5
Character	$\chi$	$=$	896.799
Dual form			896.3.k.b.351.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.60485 - 2.60485i) q^{3} +(5.26925 + 5.26925i) q^{5} +2.64575 q^{7} +4.57046i q^{9} +O(q^{10})$	\(q+(-2.60485 - 2.60485i) q^{3} +(5.26925 + 5.26925i) q^{5} +2.64575 q^{7} +4.57046i q^{9} +(-0.0342750 + 0.0342750i) q^{11} +(-5.31088 + 5.31088i) q^{13} -27.4512i q^{15} +27.5699 q^{17} +(-1.06310 - 1.06310i) q^{19} +(-6.89178 - 6.89178i) q^{21} +8.59472 q^{23} +30.5299i q^{25} +(-11.5383 + 11.5383i) q^{27} +(-35.6808 + 35.6808i) q^{29} -49.7626i q^{31} +0.178562 q^{33} +(13.9411 + 13.9411i) q^{35} +(28.9728 + 28.9728i) q^{37} +27.6681 q^{39} +47.8460i q^{41} +(-8.62924 + 8.62924i) q^{43} +(-24.0829 + 24.0829i) q^{45} +79.5643i q^{47} +7.00000 q^{49} +(-71.8154 - 71.8154i) q^{51} +(-26.1653 - 26.1653i) q^{53} -0.361207 q^{55} +5.53844i q^{57} +(28.4333 - 28.4333i) q^{59} +(39.9332 - 39.9332i) q^{61} +12.0923i q^{63} -55.9687 q^{65} +(59.3574 + 59.3574i) q^{67} +(-22.3879 - 22.3879i) q^{69} +76.2561 q^{71} -103.027i q^{73} +(79.5258 - 79.5258i) q^{75} +(-0.0906831 + 0.0906831i) q^{77} +45.1896i q^{79} +101.245 q^{81} +(55.5063 + 55.5063i) q^{83} +(145.273 + 145.273i) q^{85} +185.886 q^{87} +51.0768i q^{89} +(-14.0513 + 14.0513i) q^{91} +(-129.624 + 129.624i) q^{93} -11.2035i q^{95} -27.5785 q^{97} +(-0.156653 - 0.156653i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 16 q^{11} - 64 q^{19} + 64 q^{23} - 96 q^{27} - 16 q^{29} + 48 q^{37} - 384 q^{39} + 176 q^{43} + 336 q^{49} - 192 q^{51} + 80 q^{53} + 512 q^{55} - 288 q^{59} + 64 q^{61} - 32 q^{65} + 80 q^{67} - 192 q^{69} + 608 q^{75} - 112 q^{77} - 432 q^{81} - 160 q^{83} - 320 q^{85} + 896 q^{87} - 96 q^{93} + 496 q^{99}+O(q^{100}) $ 48 * q + 16 * q^11 - 64 * q^19 + 64 * q^23 - 96 * q^27 - 16 * q^29 + 48 * q^37 - 384 * q^39 + 176 * q^43 + 336 * q^49 - 192 * q^51 + 80 * q^53 + 512 * q^55 - 288 * q^59 + 64 * q^61 - 32 * q^65 + 80 * q^67 - 192 * q^69 + 608 * q^75 - 112 * q^77 - 432 * q^81 - 160 * q^83 - 320 * q^85 + 896 * q^87 - 96 * q^93 + 496 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$645$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	−2.60485	−	2.60485i	−0.868283	−	0.868283i	0.124000	−	0.992282i	$-0.460428\pi$
$3$	−2.60485	−	2.60485i	−0.868283	−	0.868283i	−0.992282	+	0.124000i	$0.960428\pi$
$4$		0			0
$5$	5.26925	+	5.26925i	1.05385	+	1.05385i	0.998465	+	0.0553843i	$0.0176384\pi$
$5$	5.26925	+	5.26925i	1.05385	+	1.05385i	0.0553843	+	0.998465i	$0.482362\pi$
$6$		0			0
$7$	2.64575			0.377964
$7$	2.64575			0.377964
$8$		0			0
$9$			4.57046i			0.507829i
$10$		0			0
$11$	−0.0342750	+	0.0342750i	−0.00311591	+	0.00311591i	−0.708663	−	0.705547i	$-0.750701\pi$
$11$	−0.0342750	+	0.0342750i	−0.00311591	+	0.00311591i	0.705547	+	0.708663i	$0.250701\pi$
$12$		0			0
$13$	−5.31088	+	5.31088i	−0.408529	+	0.408529i	−0.881225	−	0.472696i	$-0.843281\pi$
$13$	−5.31088	+	5.31088i	−0.408529	+	0.408529i	0.472696	+	0.881225i	$0.343281\pi$
$14$		0			0
$15$		−	27.4512i		−	1.83008i
$16$		0			0
$17$	27.5699			1.62176			0.810880	−	0.585213i	$-0.198989\pi$
$17$	27.5699			1.62176			0.810880	+	0.585213i	$0.198989\pi$
$18$		0			0
$19$	−1.06310	−	1.06310i	−0.0559528	−	0.0559528i	0.678577	−	0.734530i	$-0.262597\pi$
$19$	−1.06310	−	1.06310i	−0.0559528	−	0.0559528i	−0.734530	+	0.678577i	$0.762597\pi$
$20$		0			0
$21$	−6.89178	−	6.89178i	−0.328180	−	0.328180i
$22$		0			0
$23$	8.59472			0.373684			0.186842	−	0.982390i	$-0.440175\pi$
$23$	8.59472			0.373684			0.186842	+	0.982390i	$0.440175\pi$
$24$		0			0
$25$			30.5299i			1.22120i
$26$		0			0
$27$	−11.5383	+	11.5383i	−0.427343	+	0.427343i
$28$		0			0
$29$	−35.6808	+	35.6808i	−1.23037	+	1.23037i	−0.266553	+	0.963820i	$0.585885\pi$
$29$	−35.6808	+	35.6808i	−1.23037	+	1.23037i	−0.963820	+	0.266553i	$0.914115\pi$
$30$		0			0
$31$		−	49.7626i		−	1.60525i	−0.596487	−	0.802623i	$-0.703437\pi$
$31$		−	49.7626i		−	1.60525i	0.596487	−	0.802623i	$-0.296563\pi$
$32$		0			0
$33$	0.178562			0.00541098
$34$		0			0
$35$	13.9411	+	13.9411i	0.398318	+	0.398318i
$36$		0			0
$37$	28.9728	+	28.9728i	0.783047	+	0.783047i	0.980344	−	0.197296i	$-0.0632162\pi$
$37$	28.9728	+	28.9728i	0.783047	+	0.783047i	−0.197296	+	0.980344i	$0.563216\pi$
$38$		0			0
$39$	27.6681			0.709437
$40$		0			0
$41$			47.8460i			1.16698i	0.812122	+	0.583488i	$0.198312\pi$
$41$			47.8460i			1.16698i	−0.812122	+	0.583488i	$0.801688\pi$
$42$		0			0
$43$	−8.62924	+	8.62924i	−0.200680	+	0.200680i	−0.800291	−	0.599611i	$-0.795322\pi$
$43$	−8.62924	+	8.62924i	−0.200680	+	0.200680i	0.599611	+	0.800291i	$0.295322\pi$
$44$		0			0
$45$	−24.0829	+	24.0829i	−0.535176	+	0.535176i
$46$		0			0
$47$			79.5643i			1.69286i	0.532503	+	0.846428i	$0.321252\pi$
$47$			79.5643i			1.69286i	−0.532503	+	0.846428i	$0.678748\pi$
$48$		0			0
$49$	7.00000			0.142857
$50$		0			0
$51$	−71.8154	−	71.8154i	−1.40815	−	1.40815i
$52$		0			0
$53$	−26.1653	−	26.1653i	−0.493686	−	0.493686i	0.415780	−	0.909465i	$-0.363509\pi$
$53$	−26.1653	−	26.1653i	−0.493686	−	0.493686i	−0.909465	+	0.415780i	$0.863509\pi$
$54$		0			0
$55$	−0.361207			−0.00656740
$56$		0			0
$57$			5.53844i			0.0971656i
$58$		0			0
$59$	28.4333	−	28.4333i	0.481921	−	0.481921i	−0.423824	−	0.905745i	$-0.639312\pi$
$59$	28.4333	−	28.4333i	0.481921	−	0.481921i	0.905745	+	0.423824i	$0.139312\pi$
$60$		0			0
$61$	39.9332	−	39.9332i	0.654642	−	0.654642i	−0.299465	−	0.954107i	$-0.596808\pi$
$61$	39.9332	−	39.9332i	0.654642	−	0.654642i	0.954107	+	0.299465i	$0.0968083\pi$
$62$		0			0
$63$			12.0923i			0.191941i
$64$		0			0
$65$	−55.9687			−0.861056
$66$		0			0
$67$	59.3574	+	59.3574i	0.885932	+	0.885932i	0.994129	−	0.108197i	$-0.0345079\pi$
$67$	59.3574	+	59.3574i	0.885932	+	0.885932i	−0.108197	+	0.994129i	$0.534508\pi$
$68$		0			0
$69$	−22.3879	−	22.3879i	−0.324463	−	0.324463i
$70$		0			0
$71$	76.2561			1.07403			0.537015	−	0.843573i	$-0.319552\pi$
$71$	76.2561			1.07403			0.537015	+	0.843573i	$0.319552\pi$
$72$		0			0
$73$		−	103.027i		−	1.41133i	−0.708546	−	0.705664i	$-0.750649\pi$
$73$		−	103.027i		−	1.41133i	0.708546	−	0.705664i	$-0.249351\pi$
$74$		0			0
$75$	79.5258	−	79.5258i	1.06034	−	1.06034i
$76$		0			0
$77$	−0.0906831	+	0.0906831i	−0.00117770	+	0.00117770i
$78$		0			0
$79$			45.1896i			0.572020i	0.958227	+	0.286010i	$0.0923291\pi$
$79$			45.1896i			0.572020i	−0.958227	+	0.286010i	$0.907671\pi$
$80$		0			0
$81$	101.245			1.24994
$82$		0			0
$83$	55.5063	+	55.5063i	0.668750	+	0.668750i	0.957427	−	0.288676i	$-0.0932152\pi$
$83$	55.5063	+	55.5063i	0.668750	+	0.668750i	−0.288676	+	0.957427i	$0.593215\pi$
$84$		0			0
$85$	145.273	+	145.273i	1.70909	+	1.70909i
$86$		0			0
$87$	185.886			2.13662
$88$		0			0
$89$			51.0768i			0.573896i	0.957946	+	0.286948i	$0.0926408\pi$
$89$			51.0768i			0.573896i	−0.957946	+	0.286948i	$0.907359\pi$
$90$		0			0
$91$	−14.0513	+	14.0513i	−0.154409	+	0.154409i
$92$		0			0
$93$	−129.624	+	129.624i	−1.39381	+	1.39381i
$94$		0			0
$95$		−	11.2035i		−	0.117932i
$96$		0			0
$97$	−27.5785			−0.284315			−0.142157	−	0.989844i	$-0.545404\pi$
$97$	−27.5785			−0.284315			−0.142157	+	0.989844i	$0.545404\pi$
$98$		0			0
$99$	−0.156653	−	0.156653i	−0.00158235	−	0.00158235i
$100$		0			0
$101$	63.1521	+	63.1521i	0.625269	+	0.625269i	0.946874	−	0.321605i	$-0.104222\pi$
$101$	63.1521	+	63.1521i	0.625269	+	0.625269i	−0.321605	+	0.946874i	$0.604222\pi$
$102$		0			0
$103$	−57.5674			−0.558907			−0.279454	−	0.960159i	$-0.590153\pi$
$103$	−57.5674			−0.558907			−0.279454	+	0.960159i	$0.590153\pi$
$104$		0			0
$105$		−	72.6290i		−	0.691705i
$106$		0			0
$107$	17.4928	−	17.4928i	0.163485	−	0.163485i	−0.620624	−	0.784108i	$-0.713121\pi$
$107$	17.4928	−	17.4928i	0.163485	−	0.163485i	0.784108	+	0.620624i	$0.213121\pi$
$108$		0			0
$109$	136.211	−	136.211i	1.24964	−	1.24964i	0.293767	−	0.955877i	$-0.405091\pi$
$109$	136.211	−	136.211i	1.24964	−	1.24964i	0.955877	−	0.293767i	$-0.0949089\pi$
$110$		0			0
$111$		−	150.939i		−	1.35981i
$112$		0			0
$113$	198.803			1.75932			0.879660	−	0.475603i	$-0.157770\pi$
$113$	198.803			1.75932			0.879660	+	0.475603i	$0.157770\pi$
$114$		0			0
$115$	45.2877	+	45.2877i	0.393806	+	0.393806i
$116$		0			0
$117$	−24.2732	−	24.2732i	−0.207463	−	0.207463i
$118$		0			0
$119$	72.9431			0.612968
$120$		0			0
$121$			120.998i			0.999981i
$122$		0			0
$123$	124.631	−	124.631i	1.01326	−	1.01326i
$124$		0			0
$125$	−29.1386	+	29.1386i	−0.233108	+	0.233108i
$126$		0			0
$127$		−	115.874i		−	0.912397i	−0.889878	−	0.456198i	$-0.849211\pi$
$127$		−	115.874i		−	0.912397i	0.889878	−	0.456198i	$-0.150789\pi$
$128$		0			0
$129$	44.9557			0.348494
$130$		0			0
$131$	116.621	+	116.621i	0.890234	+	0.890234i	0.994545	−	0.104311i	$-0.0332637\pi$
$131$	116.621	+	116.621i	0.890234	+	0.890234i	−0.104311	+	0.994545i	$0.533264\pi$
$132$		0			0
$133$	−2.81271	−	2.81271i	−0.0211482	−	0.0211482i
$134$		0			0
$135$	−121.596			−0.900711
$136$		0			0
$137$			79.6980i			0.581737i	0.956763	+	0.290869i	$0.0939443\pi$
$137$			79.6980i			0.581737i	−0.956763	+	0.290869i	$0.906056\pi$
$138$		0			0
$139$	−122.108	+	122.108i	−0.878472	+	0.878472i	−0.993377	−	0.114905i	$-0.963344\pi$
$139$	−122.108	+	122.108i	−0.878472	+	0.878472i	0.114905	+	0.993377i	$0.463344\pi$
$140$		0			0
$141$	207.253	−	207.253i	1.46988	−	1.46988i
$142$		0			0
$143$		−	0.364061i		−	0.00254588i
$144$		0			0
$145$	−376.022			−2.59326
$146$		0			0
$147$	−18.2339	−	18.2339i	−0.124040	−	0.124040i
$148$		0			0
$149$	−32.1547	−	32.1547i	−0.215803	−	0.215803i	0.590924	−	0.806727i	$-0.298763\pi$
$149$	−32.1547	−	32.1547i	−0.215803	−	0.215803i	−0.806727	+	0.590924i	$0.798763\pi$
$150$		0			0
$151$	115.195			0.762878			0.381439	−	0.924394i	$-0.375429\pi$
$151$	115.195			0.762878			0.381439	+	0.924394i	$0.375429\pi$
$152$		0			0
$153$			126.007i			0.823577i
$154$		0			0
$155$	262.211	−	262.211i	1.69169	−	1.69169i
$156$		0			0
$157$	−16.5196	+	16.5196i	−0.105220	+	0.105220i	−0.757757	−	0.652537i	$-0.773705\pi$
$157$	−16.5196	+	16.5196i	−0.105220	+	0.105220i	0.652537	+	0.757757i	$0.273705\pi$
$158$		0			0
$159$			136.313i			0.857317i
$160$		0			0
$161$	22.7395			0.141239
$162$		0			0
$163$	−95.7869	−	95.7869i	−0.587650	−	0.587650i	0.349345	−	0.936994i	$-0.386404\pi$
$163$	−95.7869	−	95.7869i	−0.587650	−	0.587650i	−0.936994	+	0.349345i	$0.886404\pi$
$164$		0			0
$165$	0.940889	+	0.940889i	0.00570236	+	0.00570236i
$166$		0			0
$167$	−290.923			−1.74205			−0.871027	−	0.491235i	$-0.836545\pi$
$167$	−290.923			−1.74205			−0.871027	+	0.491235i	$0.836545\pi$
$168$		0			0
$169$			112.589i			0.666208i
$170$		0			0
$171$	4.85887	−	4.85887i	0.0284145	−	0.0284145i
$172$		0			0
$173$	84.4773	−	84.4773i	0.488308	−	0.488308i	−0.419464	−	0.907772i	$-0.637782\pi$
$173$	84.4773	−	84.4773i	0.488308	−	0.488308i	0.907772	+	0.419464i	$0.137782\pi$
$174$		0			0
$175$			80.7746i			0.461569i
$176$		0			0
$177$	−148.129			−0.836887
$178$		0			0
$179$	−83.2957	−	83.2957i	−0.465339	−	0.465339i	0.435062	−	0.900401i	$-0.356727\pi$
$179$	−83.2957	−	83.2957i	−0.465339	−	0.465339i	−0.900401	+	0.435062i	$0.856727\pi$
$180$		0			0
$181$	−66.8607	−	66.8607i	−0.369396	−	0.369396i	0.497861	−	0.867257i	$-0.334119\pi$
$181$	−66.8607	−	66.8607i	−0.369396	−	0.369396i	−0.867257	+	0.497861i	$0.834119\pi$
$182$		0			0
$183$	−208.040			−1.13683
$184$		0			0
$185$			305.329i			1.65043i
$186$		0			0
$187$	−0.944959	+	0.944959i	−0.00505325	+	0.00505325i
$188$		0			0
$189$	−30.5274	+	30.5274i	−0.161521	+	0.161521i
$190$		0			0
$191$		−	78.8368i		−	0.412758i	−0.978472	−	0.206379i	$-0.933832\pi$
$191$		−	78.8368i		−	0.412758i	0.978472	−	0.206379i	$-0.0661680\pi$
$192$		0			0
$193$	99.3044			0.514531			0.257265	−	0.966341i	$-0.417179\pi$
$193$	99.3044			0.514531			0.257265	+	0.966341i	$0.417179\pi$
$194$		0			0
$195$	145.790	+	145.790i	0.747640	+	0.747640i
$196$		0			0
$197$	108.838	+	108.838i	0.552478	+	0.552478i	0.927155	−	0.374678i	$-0.122247\pi$
$197$	108.838	+	108.838i	0.552478	+	0.552478i	−0.374678	+	0.927155i	$0.622247\pi$
$198$		0			0
$199$	−183.004			−0.919618			−0.459809	−	0.888018i	$-0.652082\pi$
$199$	−183.004			−0.919618			−0.459809	+	0.888018i	$0.652082\pi$
$200$		0			0
$201$		−	309.234i		−	1.53848i
$202$		0			0
$203$	−94.4026	+	94.4026i	−0.465037	+	0.465037i
$204$		0			0
$205$	−252.112	+	252.112i	−1.22982	+	1.22982i
$206$		0			0
$207$			39.2819i			0.189768i
$208$		0			0
$209$	0.0728757			0.000348687
$210$		0			0
$211$	−67.3257	−	67.3257i	−0.319079	−	0.319079i	0.529334	−	0.848413i	$-0.322442\pi$
$211$	−67.3257	−	67.3257i	−0.319079	−	0.319079i	−0.848413	+	0.529334i	$0.822442\pi$
$212$		0			0
$213$	−198.636	−	198.636i	−0.932561	−	0.932561i
$214$		0			0
$215$	−90.9391			−0.422973
$216$		0			0
$217$		−	131.659i		−	0.606726i
$218$		0			0
$219$	−268.370	+	268.370i	−1.22543	+	1.22543i
$220$		0			0
$221$	−146.420	+	146.420i	−0.662536	+	0.662536i
$222$		0			0
$223$			261.480i			1.17256i	0.810110	+	0.586278i	$0.199407\pi$
$223$			261.480i			1.17256i	−0.810110	+	0.586278i	$0.800593\pi$
$224$		0			0
$225$	−139.536			−0.620160
$226$		0			0
$227$	−232.278	−	232.278i	−1.02325	−	1.02325i	−0.999723	−	0.0235289i	$-0.992510\pi$
$227$	−232.278	−	232.278i	−1.02325	−	1.02325i	−0.0235289	−	0.999723i	$-0.507490\pi$
$228$		0			0
$229$	−4.69312	−	4.69312i	−0.0204940	−	0.0204940i	0.696786	−	0.717280i	$-0.254613\pi$
$229$	−4.69312	−	4.69312i	−0.0204940	−	0.0204940i	−0.717280	+	0.696786i	$0.754613\pi$
$230$		0			0
$231$	0.472431			0.00204516
$232$		0			0
$233$		−	262.715i		−	1.12753i	−0.825934	−	0.563766i	$-0.809352\pi$
$233$		−	262.715i		−	1.12753i	0.825934	−	0.563766i	$-0.190648\pi$
$234$		0			0
$235$	−419.244	+	419.244i	−1.78402	+	1.78402i
$236$		0			0
$237$	117.712	−	117.712i	0.496675	−	0.496675i
$238$		0			0
$239$			451.256i			1.88810i	0.329801	+	0.944051i	$0.393018\pi$
$239$			451.256i			1.88810i	−0.329801	+	0.944051i	$0.606982\pi$
$240$		0			0
$241$	−186.818			−0.775180			−0.387590	−	0.921832i	$-0.626692\pi$
$241$	−186.818			−0.775180			−0.387590	+	0.921832i	$0.626692\pi$
$242$		0			0
$243$	−159.883	−	159.883i	−0.657957	−	0.657957i
$244$		0			0
$245$	36.8847	+	36.8847i	0.150550	+	0.150550i
$246$		0			0
$247$	11.2920			0.0457167
$248$		0			0
$249$		−	289.171i		−	1.16133i
$250$		0			0
$251$	−91.8783	+	91.8783i	−0.366049	+	0.366049i	−0.866034	−	0.499985i	$-0.833339\pi$
$251$	−91.8783	+	91.8783i	−0.366049	+	0.366049i	0.499985	+	0.866034i	$0.333339\pi$
$252$		0			0
$253$	−0.294584	+	0.294584i	−0.00116436	+	0.00116436i
$254$		0			0
$255$		−	756.826i		−	2.96795i
$256$		0			0
$257$	−151.189			−0.588284			−0.294142	−	0.955762i	$-0.595034\pi$
$257$	−151.189			−0.588284			−0.294142	+	0.955762i	$0.595034\pi$
$258$		0			0
$259$	76.6547	+	76.6547i	0.295964	+	0.295964i
$260$		0			0
$261$	−163.078	−	163.078i	−0.624820	−	0.624820i
$262$		0			0
$263$	144.762			0.550427			0.275214	−	0.961383i	$-0.411251\pi$
$263$	144.762			0.550427			0.275214	+	0.961383i	$0.411251\pi$
$264$		0			0
$265$		−	275.743i		−	1.04054i
$266$		0			0
$267$	133.047	−	133.047i	0.498304	−	0.498304i
$268$		0			0
$269$	−107.045	+	107.045i	−0.397936	+	0.397936i	−0.877504	−	0.479569i	$-0.840793\pi$
$269$	−107.045	+	107.045i	−0.397936	+	0.397936i	0.479569	+	0.877504i	$0.340793\pi$
$270$		0			0
$271$			85.1410i			0.314173i	0.987585	+	0.157087i	$0.0502102\pi$
$271$			85.1410i			0.314173i	−0.987585	+	0.157087i	$0.949790\pi$
$272$		0			0
$273$	73.2028			0.268142
$274$		0			0
$275$	−1.04641	−	1.04641i	−0.00380514	−	0.00380514i
$276$		0			0
$277$	116.995	+	116.995i	0.422365	+	0.422365i	0.886017	−	0.463652i	$-0.153461\pi$
$277$	116.995	+	116.995i	0.422365	+	0.422365i	−0.463652	+	0.886017i	$0.653461\pi$
$278$		0			0
$279$	227.438			0.815191
$280$		0			0
$281$		−	40.2996i		−	0.143415i	−0.997426	−	0.0717075i	$-0.977155\pi$
$281$		−	40.2996i		−	0.143415i	0.997426	−	0.0717075i	$-0.0228448\pi$
$282$		0			0
$283$	150.943	−	150.943i	0.533366	−	0.533366i	−0.388207	−	0.921572i	$-0.626905\pi$
$283$	150.943	−	150.943i	0.533366	−	0.533366i	0.921572	+	0.388207i	$0.126905\pi$
$284$		0			0
$285$	−29.1834	+	29.1834i	−0.102398	+	0.102398i
$286$		0			0
$287$			126.589i			0.441075i
$288$		0			0
$289$	471.100			1.63010
$290$		0			0
$291$	71.8379	+	71.8379i	0.246866	+	0.246866i
$292$		0			0
$293$	−237.816	−	237.816i	−0.811659	−	0.811659i	0.173224	−	0.984883i	$-0.444582\pi$
$293$	−237.816	−	237.816i	−0.811659	−	0.811659i	−0.984883	+	0.173224i	$0.944582\pi$
$294$		0			0
$295$	299.644			1.01574
$296$		0			0
$297$		−	0.790948i		−	0.00266312i
$298$		0			0
$299$	−45.6455	+	45.6455i	−0.152661	+	0.152661i
$300$		0			0
$301$	−22.8308	+	22.8308i	−0.0758499	+	0.0758499i
$302$		0			0
$303$		−	329.003i		−	1.08582i
$304$		0			0
$305$	420.835			1.37979
$306$		0			0
$307$	20.6127	+	20.6127i	0.0671424	+	0.0671424i	0.739881	−	0.672738i	$-0.234882\pi$
$307$	20.6127	+	20.6127i	0.0671424	+	0.0671424i	−0.672738	+	0.739881i	$0.734882\pi$
$308$		0			0
$309$	149.954	+	149.954i	0.485289	+	0.485289i
$310$		0			0
$311$	−153.723			−0.494286			−0.247143	−	0.968979i	$-0.579492\pi$
$311$	−153.723			−0.494286			−0.247143	+	0.968979i	$0.579492\pi$
$312$		0			0
$313$			119.591i			0.382081i	0.981582	+	0.191040i	$0.0611862\pi$
$313$			119.591i			0.382081i	−0.981582	+	0.191040i	$0.938814\pi$
$314$		0			0
$315$	−63.7174	+	63.7174i	−0.202277	+	0.202277i
$316$		0			0
$317$	250.641	−	250.641i	0.790665	−	0.790665i	−0.190937	−	0.981602i	$-0.561153\pi$
$317$	250.641	−	250.641i	0.790665	−	0.790665i	0.981602	+	0.190937i	$0.0611528\pi$
$318$		0			0
$319$		−	2.44592i		−	0.00766746i
$320$		0			0
$321$	−91.1324			−0.283902
$322$		0			0
$323$	−29.3097	−	29.3097i	−0.0907420	−	0.0907420i
$324$		0			0
$325$	−162.141	−	162.141i	−0.498895	−	0.498895i
$326$		0			0
$327$	−709.619			−2.17009
$328$		0			0
$329$			210.507i			0.639840i
$330$		0			0
$331$	−67.5747	+	67.5747i	−0.204153	+	0.204153i	−0.801777	−	0.597624i	$-0.796112\pi$
$331$	−67.5747	+	67.5747i	−0.204153	+	0.204153i	0.597624	+	0.801777i	$0.296112\pi$
$332$		0			0
$333$	−132.419	+	132.419i	−0.397654	+	0.397654i
$334$		0			0
$335$			625.538i			1.86728i
$336$		0			0
$337$	149.179			0.442668			0.221334	−	0.975198i	$-0.428959\pi$
$337$	149.179			0.442668			0.221334	+	0.975198i	$0.428959\pi$
$338$		0			0
$339$	−517.852	−	517.852i	−1.52759	−	1.52759i
$340$		0			0
$341$	1.70561	+	1.70561i	0.00500180	+	0.00500180i
$342$		0			0
$343$	18.5203			0.0539949
$344$		0			0
$345$		−	235.935i		−	0.683870i
$346$		0			0
$347$	−151.702	+	151.702i	−0.437182	+	0.437182i	−0.891062	−	0.453881i	$-0.850039\pi$
$347$	−151.702	+	151.702i	−0.437182	+	0.437182i	0.453881	+	0.891062i	$0.350039\pi$
$348$		0			0
$349$	232.328	−	232.328i	0.665698	−	0.665698i	−0.291020	−	0.956717i	$-0.593994\pi$
$349$	232.328	−	232.328i	0.665698	−	0.665698i	0.956717	+	0.291020i	$0.0939945\pi$
$350$		0			0
$351$		−	122.557i		−	0.349164i
$352$		0			0
$353$	205.431			0.581956			0.290978	−	0.956730i	$-0.406019\pi$
$353$	205.431			0.581956			0.290978	+	0.956730i	$0.406019\pi$
$354$		0			0
$355$	401.812	+	401.812i	1.13187	+	1.13187i
$356$		0			0
$357$	−190.006	−	190.006i	−0.532229	−	0.532229i
$358$		0			0
$359$	607.058			1.69097			0.845485	−	0.534000i	$-0.179312\pi$
$359$	607.058			1.69097			0.845485	+	0.534000i	$0.179312\pi$
$360$		0			0
$361$		−	358.740i		−	0.993739i
$362$		0			0
$363$	315.180	−	315.180i	0.868266	−	0.868266i
$364$		0			0
$365$	542.875	−	542.875i	1.48733	−	1.48733i
$366$		0			0
$367$		−	103.467i		−	0.281927i	−0.990015	−	0.140964i	$-0.954980\pi$
$367$		−	103.467i		−	0.281927i	0.990015	−	0.140964i	$-0.0450201\pi$
$368$		0			0
$369$	−218.678			−0.592624
$370$		0			0
$371$	−69.2270	−	69.2270i	−0.186596	−	0.186596i
$372$		0			0
$373$	−247.603	−	247.603i	−0.663816	−	0.663816i	0.292461	−	0.956277i	$-0.405526\pi$
$373$	−247.603	−	247.603i	−0.663816	−	0.663816i	−0.956277	+	0.292461i	$0.905526\pi$
$374$		0			0
$375$	151.803			0.404808
$376$		0			0
$377$		−	378.993i		−	1.00529i
$378$		0			0
$379$	−262.881	+	262.881i	−0.693618	+	0.693618i	−0.963026	−	0.269408i	$-0.913172\pi$
$379$	−262.881	+	262.881i	−0.693618	+	0.693618i	0.269408	+	0.963026i	$0.413172\pi$
$380$		0			0
$381$	−301.835	+	301.835i	−0.792218	+	0.792218i
$382$		0			0
$383$		−	468.920i		−	1.22434i	−0.790728	−	0.612168i	$-0.790298\pi$
$383$		−	468.920i		−	1.22434i	0.790728	−	0.612168i	$-0.209702\pi$
$384$		0			0
$385$	−0.955663			−0.00248224
$386$		0			0
$387$	−39.4396	−	39.4396i	−0.101911	−	0.101911i
$388$		0			0
$389$	−278.401	−	278.401i	−0.715685	−	0.715685i	0.252034	−	0.967718i	$-0.418901\pi$
$389$	−278.401	−	278.401i	−0.715685	−	0.715685i	−0.967718	+	0.252034i	$0.918901\pi$
$390$		0			0
$391$	236.956			0.606025
$392$		0			0
$393$		−	607.558i		−	1.54595i
$394$		0			0
$395$	−238.115	+	238.115i	−0.602823	+	0.602823i
$396$		0			0
$397$	−29.5239	+	29.5239i	−0.0743676	+	0.0743676i	−0.743312	−	0.668945i	$-0.766746\pi$
$397$	−29.5239	+	29.5239i	−0.0743676	+	0.0743676i	0.668945	+	0.743312i	$0.266746\pi$
$398$		0			0
$399$			14.6533i			0.0367252i
$400$		0			0
$401$	−569.183			−1.41941			−0.709705	−	0.704499i	$-0.751172\pi$
$401$	−569.183			−1.41941			−0.709705	+	0.704499i	$0.751172\pi$
$402$		0			0
$403$	264.283	+	264.283i	0.655789	+	0.655789i
$404$		0			0
$405$	533.485	+	533.485i	1.31725	+	1.31725i
$406$		0			0
$407$	−1.98608			−0.00487981
$408$		0			0
$409$			442.593i			1.08214i	0.840979	+	0.541068i	$0.181980\pi$
$409$			442.593i			1.08214i	−0.840979	+	0.541068i	$0.818020\pi$
$410$		0			0
$411$	207.601	−	207.601i	0.505112	−	0.505112i
$412$		0			0
$413$	75.2275	−	75.2275i	0.182149	−	0.182149i
$414$		0			0
$415$			584.953i			1.40952i
$416$		0			0
$417$	636.143			1.52552
$418$		0			0
$419$	−154.181	−	154.181i	−0.367974	−	0.367974i	0.498764	−	0.866738i	$-0.333787\pi$
$419$	−154.181	−	154.181i	−0.367974	−	0.367974i	−0.866738	+	0.498764i	$0.833787\pi$
$420$		0			0
$421$	−174.166	−	174.166i	−0.413696	−	0.413696i	0.469328	−	0.883024i	$-0.344496\pi$
$421$	−174.166	−	174.166i	−0.413696	−	0.413696i	−0.883024	+	0.469328i	$0.844496\pi$
$422$		0			0
$423$	−363.646			−0.859682
$424$		0			0
$425$			841.708i			1.98049i
$426$		0			0
$427$	105.653	−	105.653i	0.247431	−	0.247431i
$428$		0			0
$429$	−0.948323	+	0.948323i	−0.00221054	+	0.00221054i
$430$		0			0
$431$		−	108.471i		−	0.251673i	−0.992051	−	0.125837i	$-0.959838\pi$
$431$		−	108.471i		−	0.251673i	0.992051	−	0.125837i	$-0.0401615\pi$
$432$		0			0
$433$	−457.359			−1.05626			−0.528128	−	0.849165i	$-0.677106\pi$
$433$	−457.359			−1.05626			−0.528128	+	0.849165i	$0.677106\pi$
$434$		0			0
$435$	979.481	+	979.481i	2.25168	+	2.25168i
$436$		0			0
$437$	−9.13708	−	9.13708i	−0.0209086	−	0.0209086i
$438$		0			0
$439$	−287.953			−0.655930			−0.327965	−	0.944690i	$-0.606363\pi$
$439$	−287.953			−0.655930			−0.327965	+	0.944690i	$0.606363\pi$
$440$		0			0
$441$			31.9932i			0.0725470i
$442$		0			0
$443$	175.112	−	175.112i	0.395286	−	0.395286i	−0.481281	−	0.876567i	$-0.659828\pi$
$443$	175.112	−	175.112i	0.395286	−	0.395286i	0.876567	+	0.481281i	$0.159828\pi$
$444$		0			0
$445$	−269.136	+	269.136i	−0.604800	+	0.604800i
$446$		0			0
$447$			167.516i			0.374756i
$448$		0			0
$449$	628.471			1.39971			0.699856	−	0.714284i	$-0.253247\pi$
$449$	628.471			1.39971			0.699856	+	0.714284i	$0.253247\pi$
$450$		0			0
$451$	−1.63992	−	1.63992i	−0.00363619	−	0.00363619i
$452$		0			0
$453$	−300.064	−	300.064i	−0.662394	−	0.662394i
$454$		0			0
$455$	−148.079			−0.325449
$456$		0			0
$457$		−	89.4733i		−	0.195784i	−0.995197	−	0.0978920i	$-0.968790\pi$
$457$		−	89.4733i		−	0.195784i	0.995197	−	0.0978920i	$-0.0312100\pi$
$458$		0			0
$459$	−318.109	+	318.109i	−0.693048	+	0.693048i
$460$		0			0
$461$	452.303	−	452.303i	0.981134	−	0.981134i	−0.0186911	−	0.999825i	$-0.505950\pi$
$461$	452.303	−	452.303i	0.981134	−	0.981134i	0.999825	+	0.0186911i	$0.00594990\pi$
$462$		0			0
$463$			306.653i			0.662317i	0.943575	+	0.331159i	$0.107440\pi$
$463$			306.653i			0.662317i	−0.943575	+	0.331159i	$0.892560\pi$
$464$		0			0
$465$	−1366.04			−2.93772
$466$		0			0
$467$	−630.902	−	630.902i	−1.35097	−	1.35097i	−0.884580	−	0.466389i	$-0.845555\pi$
$467$	−630.902	−	630.902i	−1.35097	−	1.35097i	−0.466389	−	0.884580i	$-0.654445\pi$
$468$		0			0
$469$	157.045	+	157.045i	0.334851	+	0.334851i
$470$		0			0
$471$	86.0618			0.182722
$472$		0			0
$473$		−	0.591534i		−	0.00125060i
$474$		0			0
$475$	32.4565	−	32.4565i	0.0683294	−	0.0683294i
$476$		0			0
$477$	119.588	−	119.588i	0.250708	−	0.250708i
$478$		0			0
$479$			235.666i			0.491996i	0.969270	+	0.245998i	$0.0791157\pi$
$479$			235.666i			0.491996i	−0.969270	+	0.245998i	$0.920884\pi$
$480$		0			0
$481$	−307.742			−0.639795
$482$		0			0
$483$	−59.2329	−	59.2329i	−0.122636	−	0.122636i
$484$		0			0
$485$	−145.318	−	145.318i	−0.299625	−	0.299625i
$486$		0			0
$487$	−20.2420			−0.0415646			−0.0207823	−	0.999784i	$-0.506616\pi$
$487$	−20.2420			−0.0415646			−0.0207823	+	0.999784i	$0.506616\pi$
$488$		0			0
$489$			499.021i			1.02049i
$490$		0			0
$491$	−375.295	+	375.295i	−0.764349	+	0.764349i	−0.977105	−	0.212756i	$-0.931756\pi$
$491$	−375.295	+	375.295i	−0.764349	+	0.764349i	0.212756	+	0.977105i	$0.431756\pi$
$492$		0			0
$493$	−983.717	+	983.717i	−1.99537	+	1.99537i
$494$		0			0
$495$		−	1.65088i		−	0.00333512i
$496$		0			0
$497$	201.755			0.405945
$498$		0			0
$499$	318.122	+	318.122i	0.637518	+	0.637518i	0.949943	−	0.312424i	$-0.101141\pi$
$499$	318.122	+	318.122i	0.637518	+	0.637518i	−0.312424	+	0.949943i	$0.601141\pi$
$500$		0			0
$501$	757.810	+	757.810i	1.51259	+	1.51259i
$502$		0			0
$503$	−645.420			−1.28314			−0.641571	−	0.767064i	$-0.721717\pi$
$503$	−645.420			−1.28314			−0.641571	+	0.767064i	$0.721717\pi$
$504$		0			0
$505$			665.529i			1.31788i
$506$		0			0
$507$	293.278	−	293.278i	0.578457	−	0.578457i
$508$		0			0
$509$	27.0881	−	27.0881i	0.0532183	−	0.0532183i	−0.679997	−	0.733215i	$-0.738019\pi$
$509$	27.0881	−	27.0881i	0.0532183	−	0.0532183i	0.733215	+	0.679997i	$0.238019\pi$
$510$		0			0
$511$		−	272.584i		−	0.533432i
$512$		0			0
$513$	24.5327			0.0478221
$514$		0			0
$515$	−303.337	−	303.337i	−0.589004	−	0.589004i
$516$		0			0
$517$	−2.72706	−	2.72706i	−0.00527479	−	0.00527479i
$518$		0			0
$519$	−440.101			−0.847979
$520$		0			0
$521$		−	414.463i		−	0.795513i	−0.917491	−	0.397757i	$-0.869789\pi$
$521$		−	414.463i		−	0.795513i	0.917491	−	0.397757i	$-0.130211\pi$
$522$		0			0
$523$	121.736	−	121.736i	0.232764	−	0.232764i	−0.581082	−	0.813845i	$-0.697370\pi$
$523$	121.736	−	121.736i	0.232764	−	0.232764i	0.813845	+	0.581082i	$0.197370\pi$
$524$		0			0
$525$	210.406	−	210.406i	0.400772	−	0.400772i
$526$		0			0
$527$		−	1371.95i		−	2.60332i
$528$		0			0
$529$	−455.131			−0.860361
$530$		0			0
$531$	129.953	+	129.953i	0.244733	+	0.244733i
$532$		0			0
$533$	−254.104	−	254.104i	−0.476743	−	0.476743i
$534$		0			0
$535$	184.348			0.344576
$536$		0			0
$537$			433.945i			0.808091i
$538$		0			0
$539$	−0.239925	+	0.239925i	−0.000445130	+	0.000445130i
$540$		0			0
$541$	−315.972	+	315.972i	−0.584052	+	0.584052i	−0.936014	−	0.351962i	$-0.885515\pi$
$541$	−315.972	+	315.972i	−0.584052	+	0.584052i	0.351962	+	0.936014i	$0.385515\pi$
$542$		0			0
$543$			348.324i			0.641480i
$544$		0			0
$545$	1435.46			2.63387
$546$		0			0
$547$	194.242	+	194.242i	0.355104	+	0.355104i	0.862005	−	0.506900i	$-0.169209\pi$
$547$	194.242	+	194.242i	0.355104	+	0.355104i	−0.506900	+	0.862005i	$0.669209\pi$
$548$		0			0
$549$	182.513	+	182.513i	0.332446	+	0.332446i
$550$		0			0
$551$	75.8648			0.137686
$552$		0			0
$553$			119.560i			0.216203i
$554$		0			0
$555$	795.336	−	795.336i	1.43304	−	1.43304i
$556$		0			0
$557$	611.421	−	611.421i	1.09770	−	1.09770i	0.103025	−	0.994679i	$-0.467148\pi$
$557$	611.421	−	611.421i	1.09770	−	1.09770i	0.994679	−	0.103025i	$-0.0328523\pi$
$558$		0			0
$559$		−	91.6576i		−	0.163967i
$560$		0			0
$561$	4.92295			0.00877531
$562$		0			0
$563$	−587.190	−	587.190i	−1.04297	−	1.04297i	−0.999035	−	0.0439324i	$-0.986011\pi$
$563$	−587.190	−	587.190i	−1.04297	−	1.04297i	−0.0439324	−	0.999035i	$-0.513989\pi$
$564$		0			0
$565$	1047.54	+	1047.54i	1.85406	+	1.85406i
$566$		0			0
$567$	267.869			0.472432
$568$		0			0
$569$			462.064i			0.812063i	0.913859	+	0.406031i	$0.133088\pi$
$569$			462.064i			0.812063i	−0.913859	+	0.406031i	$0.866912\pi$
$570$		0			0
$571$	397.152	−	397.152i	0.695538	−	0.695538i	−0.267907	−	0.963445i	$-0.586332\pi$
$571$	397.152	−	397.152i	0.695538	−	0.695538i	0.963445	+	0.267907i	$0.0863319\pi$
$572$		0			0
$573$	−205.358	+	205.358i	−0.358391	+	0.358391i
$574$		0			0
$575$			262.396i			0.456341i
$576$		0			0
$577$	−871.434			−1.51028			−0.755142	−	0.655561i	$-0.772432\pi$
$577$	−871.434			−1.51028			−0.755142	+	0.655561i	$0.772432\pi$
$578$		0			0
$579$	−258.673	−	258.673i	−0.446758	−	0.446758i
$580$		0			0
$581$	146.856	+	146.856i	0.252764	+	0.252764i
$582$		0			0
$583$	1.79363			0.00307656
$584$		0			0
$585$		−	255.803i		−	0.437270i
$586$		0			0
$587$	628.244	−	628.244i	1.07026	−	1.07026i	0.0729252	−	0.997337i	$-0.476767\pi$
$587$	628.244	−	628.244i	1.07026	−	1.07026i	0.997337	−	0.0729252i	$-0.0232334\pi$
$588$		0			0
$589$	−52.9028	+	52.9028i	−0.0898179	+	0.0898179i
$590$		0			0
$591$		−	567.013i		−	0.959414i
$592$		0			0
$593$	−400.187			−0.674852			−0.337426	−	0.941352i	$-0.609556\pi$
$593$	−400.187			−0.674852			−0.337426	+	0.941352i	$0.609556\pi$
$594$		0			0
$595$	384.355	+	384.355i	0.645975	+	0.645975i
$596$		0			0
$597$	476.698	+	476.698i	0.798488	+	0.798488i
$598$		0			0
$599$	437.482			0.730355			0.365177	−	0.930938i	$-0.381008\pi$
$599$	437.482			0.730355			0.365177	+	0.930938i	$0.381008\pi$
$600$		0			0
$601$		−	374.850i		−	0.623710i	−0.950130	−	0.311855i	$-0.899050\pi$
$601$		−	374.850i		−	0.623710i	0.950130	−	0.311855i	$-0.100950\pi$
$602$		0			0
$603$	−271.291	+	271.291i	−0.449902	+	0.449902i
$604$		0			0
$605$	−637.567	+	637.567i	−1.05383	+	1.05383i
$606$		0			0
$607$		−	326.106i		−	0.537242i	−0.963246	−	0.268621i	$-0.913432\pi$
$607$		−	326.106i		−	0.537242i	0.963246	−	0.268621i	$-0.0865679\pi$
$608$		0			0
$609$	491.809			0.807568
$610$		0			0
$611$	−422.556	−	422.556i	−0.691581	−	0.691581i
$612$		0			0
$613$	17.5228	+	17.5228i	0.0285853	+	0.0285853i	0.721255	−	0.692670i	$-0.243566\pi$
$613$	17.5228	+	17.5228i	0.0285853	+	0.0285853i	−0.692670	+	0.721255i	$0.743566\pi$
$614$		0			0
$615$	1313.43			2.13566
$616$		0			0
$617$			51.8009i			0.0839562i	0.999119	+	0.0419781i	$0.0133660\pi$
$617$			51.8009i			0.0839562i	−0.999119	+	0.0419781i	$0.986634\pi$
$618$		0			0
$619$	360.793	−	360.793i	0.582864	−	0.582864i	−0.352825	−	0.935689i	$-0.614779\pi$
$619$	360.793	−	360.793i	0.582864	−	0.582864i	0.935689	+	0.352825i	$0.114779\pi$
$620$		0			0
$621$	−99.1682	+	99.1682i	−0.159691	+	0.159691i
$622$		0			0
$623$			135.136i			0.216912i
$624$		0			0
$625$	456.172			0.729875
$626$		0			0
$627$	−0.189830	−	0.189830i	−0.000302759	−	0.000302759i
$628$		0			0
$629$	798.776	+	798.776i	1.26991	+	1.26991i
$630$		0			0
$631$	7.85965			0.0124559			0.00622793	−	0.999981i	$-0.498018\pi$
$631$	7.85965			0.0124559			0.00622793	+	0.999981i	$0.498018\pi$
$632$		0			0
$633$			350.747i			0.554102i
$634$		0			0
$635$	610.571	−	610.571i	0.961529	−	0.961529i
$636$		0			0
$637$	−37.1761	+	37.1761i	−0.0583613	+	0.0583613i
$638$		0			0
$639$			348.526i			0.545424i
$640$		0			0
$641$	−655.356			−1.02240			−0.511198	−	0.859463i	$-0.670798\pi$
$641$	−655.356			−1.02240			−0.511198	+	0.859463i	$0.670798\pi$
$642$		0			0
$643$	115.057	+	115.057i	0.178938	+	0.178938i	0.790893	−	0.611955i	$-0.209617\pi$
$643$	115.057	+	115.057i	0.178938	+	0.178938i	−0.611955	+	0.790893i	$0.709617\pi$
$644$		0			0
$645$	236.883	+	236.883i	0.367260	+	0.367260i
$646$		0			0
$647$	−51.9916			−0.0803580			−0.0401790	−	0.999192i	$-0.512793\pi$
$647$	−51.9916			−0.0803580			−0.0401790	+	0.999192i	$0.512793\pi$
$648$		0			0
$649$			1.94910i			0.00300324i
$650$		0			0
$651$	−342.953	+	342.953i	−0.526809	+	0.526809i
$652$		0			0
$653$	286.170	−	286.170i	0.438238	−	0.438238i	−0.453180	−	0.891419i	$-0.649711\pi$
$653$	286.170	−	286.170i	0.438238	−	0.438238i	0.891419	+	0.453180i	$0.149711\pi$
$654$		0			0
$655$			1229.01i			1.87635i
$656$		0			0
$657$	470.881			0.716714
$658$		0			0
$659$	−64.6005	−	64.6005i	−0.0980281	−	0.0980281i	0.656392	−	0.754420i	$-0.272082\pi$
$659$	−64.6005	−	64.6005i	−0.0980281	−	0.0980281i	−0.754420	+	0.656392i	$0.772082\pi$
$660$		0			0
$661$	−120.408	−	120.408i	−0.182160	−	0.182160i	0.610136	−	0.792297i	$-0.291115\pi$
$661$	−120.408	−	120.408i	−0.182160	−	0.182160i	−0.792297	+	0.610136i	$0.791115\pi$
$662$		0			0
$663$	762.806			1.15054
$664$		0			0
$665$		−	29.6417i		−	0.0445740i
$666$		0			0
$667$	−306.667	+	306.667i	−0.459770	+	0.459770i
$668$		0			0
$669$	681.116	−	681.116i	1.01811	−	1.01811i
$670$		0			0
$671$			2.73742i			0.00407961i
$672$		0			0
$673$	−772.248			−1.14747			−0.573736	−	0.819041i	$-0.694506\pi$
$673$	−772.248			−1.14747			−0.573736	+	0.819041i	$0.694506\pi$
$674$		0			0
$675$	−352.262	−	352.262i	−0.521870	−	0.521870i
$676$		0			0
$677$	−143.226	−	143.226i	−0.211560	−	0.211560i	0.593370	−	0.804930i	$-0.297797\pi$
$677$	−143.226	−	143.226i	−0.211560	−	0.211560i	−0.804930	+	0.593370i	$0.797797\pi$
$678$		0			0
$679$	−72.9660			−0.107461
$680$		0			0
$681$			1210.10i			1.77694i
$682$		0			0
$683$	855.173	−	855.173i	1.25208	−	1.25208i	0.297299	−	0.954784i	$-0.403914\pi$
$683$	855.173	−	855.173i	1.25208	−	1.25208i	0.954784	−	0.297299i	$-0.0960859\pi$
$684$		0			0
$685$	−419.948	+	419.948i	−0.613063	+	0.613063i
$686$		0			0
$687$			24.4497i			0.0355891i
$688$		0			0
$689$	277.922			0.403370
$690$		0			0
$691$	579.764	+	579.764i	0.839022	+	0.839022i	0.988730	−	0.149709i	$-0.0478335\pi$
$691$	579.764	+	579.764i	0.839022	+	0.839022i	−0.149709	+	0.988730i	$0.547834\pi$
$692$		0			0
$693$	−0.414464	−	0.414464i	−0.000598072	−	0.000598072i
$694$		0			0
$695$	−1286.83			−1.85155
$696$		0			0
$697$			1319.11i			1.89255i
$698$		0			0
$699$	−684.333	+	684.333i	−0.979017	+	0.979017i
$700$		0			0
$701$	−339.029	+	339.029i	−0.483637	+	0.483637i	−0.906291	−	0.422654i	$-0.861098\pi$
$701$	−339.029	+	339.029i	−0.483637	+	0.483637i	0.422654	+	0.906291i	$0.361098\pi$
$702$		0			0
$703$		−	61.6020i		−	0.0876274i
$704$		0			0
$705$	2184.13			3.09806
$706$		0			0
$707$	167.085	+	167.085i	0.236329	+	0.236329i
$708$		0			0
$709$	−650.641	−	650.641i	−0.917688	−	0.917688i	0.0791726	−	0.996861i	$-0.474772\pi$
$709$	−650.641	−	650.641i	−0.917688	−	0.917688i	−0.996861	+	0.0791726i	$0.974772\pi$
$710$		0			0
$711$	−206.537			−0.290489
$712$		0			0
$713$		−	427.696i		−	0.599854i
$714$		0			0
$715$	1.91833	−	1.91833i	0.00268297	−	0.00268297i
$716$		0			0
$717$	1175.45	−	1175.45i	1.63941	−	1.63941i
$718$		0			0
$719$		−	498.150i		−	0.692838i	−0.938080	−	0.346419i	$-0.887398\pi$
$719$		−	498.150i		−	0.692838i	0.938080	−	0.346419i	$-0.112602\pi$
$720$		0			0
$721$	−152.309			−0.211247
$722$		0			0
$723$	486.634	+	486.634i	0.673076	+	0.673076i
$724$		0			0
$725$	−1089.33	−	1089.33i	−1.50253	−	1.50253i
$726$		0			0
$727$	481.762			0.662671			0.331335	−	0.943513i	$-0.392501\pi$
$727$	481.762			0.662671			0.331335	+	0.943513i	$0.392501\pi$
$728$		0			0
$729$		−	78.2610i		−	0.107354i
$730$		0			0
$731$	−237.907	+	237.907i	−0.325455	+	0.325455i
$732$		0			0
$733$	152.972	−	152.972i	0.208693	−	0.208693i	−0.595019	−	0.803712i	$-0.702855\pi$
$733$	152.972	−	152.972i	0.208693	−	0.208693i	0.803712	+	0.595019i	$0.202855\pi$
$734$		0			0
$735$		−	192.158i		−	0.261440i
$736$		0			0
$737$	−4.06895			−0.00552097
$738$		0			0
$739$	1028.07	+	1028.07i	1.39117	+	1.39117i	0.822716	+	0.568452i	$0.192458\pi$
$739$	1028.07	+	1028.07i	1.39117	+	1.39117i	0.568452	+	0.822716i	$0.307542\pi$
$740$		0			0
$741$	−29.4140	−	29.4140i	−0.0396950	−	0.0396950i
$742$		0			0
$743$	701.518			0.944169			0.472085	−	0.881553i	$-0.343502\pi$
$743$	701.518			0.944169			0.472085	+	0.881553i	$0.343502\pi$
$744$		0			0
$745$		−	338.862i		−	0.454848i
$746$		0			0
$747$	−253.689	+	253.689i	−0.339611	+	0.339611i
$748$		0			0
$749$	46.2817	−	46.2817i	0.0617914	−	0.0617914i
$750$		0			0
$751$		−	1256.03i		−	1.67248i	−0.548367	−	0.836238i	$-0.684750\pi$
$751$		−	1256.03i		−	1.67248i	0.548367	−	0.836238i	$-0.315250\pi$
$752$		0			0
$753$	478.658			0.635668
$754$		0			0
$755$	606.989	+	606.989i	0.803958	+	0.803958i
$756$		0			0
$757$	−1001.27	−	1001.27i	−1.32268	−	1.32268i	−0.911600	−	0.411078i	$-0.865152\pi$
$757$	−1001.27	−	1001.27i	−1.32268	−	1.32268i	−0.411078	−	0.911600i	$-0.634848\pi$
$758$		0			0
$759$	1.53469			0.00202199
$760$		0			0
$761$		−	1399.88i		−	1.83952i	−0.392481	−	0.919760i	$-0.628383\pi$
$761$		−	1399.88i		−	1.83952i	0.392481	−	0.919760i	$-0.371617\pi$
$762$		0			0
$763$	360.381	−	360.381i	0.472321	−	0.472321i
$764$		0			0
$765$	−663.964	+	663.964i	−0.867926	+	0.867926i
$766$		0			0
$767$			302.012i			0.393757i
$768$		0			0
$769$	−160.348			−0.208515			−0.104257	−	0.994550i	$-0.533247\pi$
$769$	−160.348			−0.208515			−0.104257	+	0.994550i	$0.533247\pi$
$770$		0			0
$771$	393.825	+	393.825i	0.510797	+	0.510797i
$772$		0			0
$773$	−228.104	−	228.104i	−0.295089	−	0.295089i	0.543998	−	0.839087i	$-0.316910\pi$
$773$	−228.104	−	228.104i	−0.295089	−	0.295089i	−0.839087	+	0.543998i	$0.816910\pi$
$774$		0			0
$775$	1519.25			1.96032
$776$		0			0
$777$		−	399.348i		−	0.513961i
$778$		0			0
$779$	50.8652	−	50.8652i	0.0652955	−	0.0652955i
$780$		0			0
$781$	−2.61368	+	2.61368i	−0.00334658	+	0.00334658i
$782$		0			0
$783$		−	823.390i		−	1.05158i
$784$		0			0
$785$	−174.091			−0.221772
$786$		0			0
$787$	−249.697	−	249.697i	−0.317278	−	0.317278i	0.530443	−	0.847721i	$-0.322026\pi$
$787$	−249.697	−	249.697i	−0.317278	−	0.317278i	−0.847721	+	0.530443i	$0.822026\pi$
$788$		0			0
$789$	−377.084	−	377.084i	−0.477926	−	0.477926i
$790$		0			0
$791$	525.984			0.664961
$792$		0			0
$793$			424.160i			0.534881i
$794$		0			0
$795$	−718.269	+	718.269i	−0.903483	+	0.903483i
$796$		0			0
$797$	−252.225	+	252.225i	−0.316468	+	0.316468i	−0.847409	−	0.530941i	$-0.821839\pi$
$797$	−252.225	+	252.225i	−0.316468	+	0.316468i	0.530941	+	0.847409i	$0.321839\pi$
$798$		0			0
$799$			2193.58i			2.74541i
$800$		0			0
$801$	−233.445			−0.291441
$802$		0			0
$803$	3.53125	+	3.53125i	0.00439757	+	0.00439757i
$804$		0			0
$805$	119.820	+	119.820i	0.148845	+	0.148845i
$806$		0			0
$807$	557.670			0.691041
$808$		0			0
$809$		−	264.570i		−	0.327033i	−0.986541	−	0.163517i	$-0.947716\pi$
$809$		−	264.570i		−	0.327033i	0.986541	−	0.163517i	$-0.0522837\pi$
$810$		0			0
$811$	−342.030	+	342.030i	−0.421739	+	0.421739i	−0.885802	−	0.464063i	$-0.846391\pi$
$811$	−342.030	+	342.030i	−0.421739	+	0.421739i	0.464063	+	0.885802i	$0.346391\pi$
$812$		0			0
$813$	221.779	−	221.779i	0.272791	−	0.272791i
$814$		0			0
$815$		−	1009.45i		−	1.23859i
$816$		0			0
$817$	18.3475			0.0224572
$818$		0			0
$819$	−64.2208	−	64.2208i	−0.0784137	−	0.0784137i
$820$		0			0
$821$	1063.31	+	1063.31i	1.29514	+	1.29514i	0.931563	+	0.363580i	$0.118446\pi$
$821$	1063.31	+	1063.31i	1.29514	+	1.29514i	0.363580	+	0.931563i	$0.381554\pi$
$822$		0			0
$823$	−604.202			−0.734145			−0.367073	−	0.930192i	$-0.619640\pi$
$823$	−604.202			−0.734145			−0.367073	+	0.930192i	$0.619640\pi$
$824$		0			0
$825$			5.45149i			0.00660787i
$826$		0			0
$827$	−908.534	+	908.534i	−1.09859	+	1.09859i	−0.104014	+	0.994576i	$0.533169\pi$
$827$	−908.534	+	908.534i	−1.09859	+	1.09859i	−0.994576	+	0.104014i	$0.966831\pi$
$828$		0			0
$829$	−608.698	+	608.698i	−0.734255	+	0.734255i	−0.971460	−	0.237205i	$-0.923769\pi$
$829$	−608.698	+	608.698i	−0.734255	+	0.734255i	0.237205	+	0.971460i	$0.423769\pi$
$830$		0			0
$831$		−	609.509i		−	0.733464i
$832$		0			0
$833$	192.989			0.231680
$834$		0			0
$835$	−1532.94	−	1532.94i	−1.83586	−	1.83586i
$836$		0			0
$837$	574.174	+	574.174i	0.685991	+	0.685991i
$838$		0			0
$839$	−70.3128			−0.0838055			−0.0419027	−	0.999122i	$-0.513342\pi$
$839$	−70.3128			−0.0838055			−0.0419027	+	0.999122i	$0.513342\pi$
$840$		0			0
$841$		−	1705.24i		−	2.02764i
$842$		0			0
$843$	−104.974	+	104.974i	−0.124525	+	0.124525i
$844$		0			0
$845$	−593.260	+	593.260i	−0.702083	+	0.702083i
$846$		0			0
$847$			320.130i			0.377957i
$848$		0			0
$849$	−786.365			−0.926225
$850$		0			0
$851$	249.013	+	249.013i	0.292612	+	0.292612i
$852$		0			0
$853$	−105.286	−	105.286i	−0.123430	−	0.123430i	0.642693	−	0.766124i	$-0.277817\pi$
$853$	−105.286	−	105.286i	−0.123430	−	0.123430i	−0.766124	+	0.642693i	$0.777817\pi$
$854$		0			0
$855$	51.2052			0.0598891
$856$		0			0
$857$		−	42.0246i		−	0.0490369i	−0.999699	−	0.0245184i	$-0.992195\pi$
$857$		−	42.0246i		−	0.0490369i	0.999699	−	0.0245184i	$-0.00780524\pi$
$858$		0			0
$859$	−551.162	+	551.162i	−0.641632	+	0.641632i	−0.950956	−	0.309325i	$-0.899897\pi$
$859$	−551.162	+	551.162i	−0.641632	+	0.641632i	0.309325	+	0.950956i	$0.399897\pi$
$860$		0			0
$861$	329.744	−	329.744i	0.382978	−	0.382978i
$862$		0			0
$863$		−	332.787i		−	0.385617i	−0.981236	−	0.192809i	$-0.938240\pi$
$863$		−	332.787i		−	0.385617i	0.981236	−	0.192809i	$-0.0617596\pi$
$864$		0			0
$865$	890.264			1.02921
$866$		0			0
$867$	−1227.14	−	1227.14i	−1.41539	−	1.41539i
$868$		0			0
$869$	−1.54887	−	1.54887i	−0.00178236	−	0.00178236i
$870$		0			0
$871$	−630.480			−0.723858
$872$		0			0
$873$		−	126.047i		−	0.144383i
$874$		0			0
$875$	−77.0934	+	77.0934i	−0.0881067	+	0.0881067i
$876$		0			0
$877$	−112.953	+	112.953i	−0.128795	+	0.128795i	−0.768566	−	0.639771i	$-0.779029\pi$
$877$	−112.953	+	112.953i	−0.128795	+	0.128795i	0.639771	+	0.768566i	$0.279029\pi$
$878$		0			0
$879$			1238.95i			1.40950i
$880$		0			0
$881$	232.822			0.264270			0.132135	−	0.991232i	$-0.457817\pi$
$881$	232.822			0.264270			0.132135	+	0.991232i	$0.457817\pi$
$882$		0			0
$883$	−12.0542	−	12.0542i	−0.0136514	−	0.0136514i	0.700248	−	0.713900i	$-0.253073\pi$
$883$	−12.0542	−	12.0542i	−0.0136514	−	0.0136514i	−0.713900	+	0.700248i	$0.753073\pi$
$884$		0			0
$885$	−780.528	−	780.528i	−0.881953	−	0.881953i
$886$		0			0
$887$	405.899			0.457608			0.228804	−	0.973472i	$-0.426518\pi$
$887$	405.899			0.457608			0.228804	+	0.973472i	$0.426518\pi$
$888$		0			0
$889$		−	306.575i		−	0.344854i
$890$		0			0
$891$	−3.47017	+	3.47017i	−0.00389469	+	0.00389469i
$892$		0			0
$893$	84.5850	−	84.5850i	0.0947200	−	0.0947200i
$894$		0			0
$895$		−	877.811i		−	0.980794i
$896$		0			0
$897$	237.799			0.265105
$898$		0			0
$899$	1775.57	+	1775.57i	1.97505	+	1.97505i
$900$		0			0
$901$	−721.376	−	721.376i	−0.800639	−	0.800639i
$902$		0			0
$903$	118.942			0.131718
$904$		0			0
$905$		−	704.611i		−	0.778576i
$906$		0			0
$907$	−269.543	+	269.543i	−0.297181	+	0.297181i	−0.839909	−	0.542728i	$-0.817391\pi$
$907$	−269.543	+	269.543i	−0.297181	+	0.297181i	0.542728	+	0.839909i	$0.317391\pi$
$908$		0			0
$909$	−288.635	+	288.635i	−0.317530	+	0.317530i
$910$		0			0
$911$			107.577i			0.118087i	0.998255	+	0.0590434i	$0.0188050\pi$
$911$			107.577i			0.118087i	−0.998255	+	0.0590434i	$0.981195\pi$
$912$		0			0
$913$	−3.80495			−0.00416753
$914$		0			0
$915$	−1096.21	−	1096.21i	−1.19805	−	1.19805i
$916$		0			0
$917$	308.549	+	308.549i	0.336477	+	0.336477i
$918$		0			0
$919$	−1479.46			−1.60986			−0.804930	−	0.593370i	$-0.797797\pi$
$919$	−1479.46			−1.60986			−0.804930	+	0.593370i	$0.797797\pi$
$920$		0			0
$921$		−	107.386i		−	0.116597i
$922$		0			0
$923$	−404.987	+	404.987i	−0.438772	+	0.438772i
$924$		0			0
$925$	−884.536	+	884.536i	−0.956255	+	0.956255i
$926$		0			0
$927$		−	263.110i		−	0.283829i
$928$		0			0
$929$	256.867			0.276498			0.138249	−	0.990397i	$-0.455853\pi$
$929$	256.867			0.276498			0.138249	+	0.990397i	$0.455853\pi$
$930$		0			0
$931$	−7.44172	−	7.44172i	−0.00799325	−	0.00799325i
$932$		0			0
$933$	400.425	+	400.425i	0.429180	+	0.429180i
$934$		0			0
$935$	−9.95844			−0.0106507
$936$		0			0
$937$			183.878i			0.196241i	0.995175	+	0.0981203i	$0.0312830\pi$
$937$			183.878i			0.196241i	−0.995175	+	0.0981203i	$0.968717\pi$
$938$		0			0
$939$	311.517	−	311.517i	0.331754	−	0.331754i
$940$		0			0
$941$	359.757	−	359.757i	0.382313	−	0.382313i	−0.489622	−	0.871935i	$-0.662865\pi$
$941$	359.757	−	359.757i	0.382313	−	0.382313i	0.871935	+	0.489622i	$0.162865\pi$
$942$		0			0
$943$			411.223i			0.436080i
$944$		0			0
$945$	−321.713			−0.340437
$946$		0			0
$947$	−636.682	−	636.682i	−0.672315	−	0.672315i	0.285934	−	0.958249i	$-0.407696\pi$
$947$	−636.682	−	636.682i	−0.672315	−	0.672315i	−0.958249	+	0.285934i	$0.907696\pi$
$948$		0			0
$949$	547.164	+	547.164i	0.576569	+	0.576569i
$950$		0			0
$951$	−1305.76			−1.37304
$952$		0			0
$953$			1086.51i			1.14010i	0.821612	+	0.570048i	$0.193075\pi$
$953$			1086.51i			1.14010i	−0.821612	+	0.570048i	$0.806925\pi$
$954$		0			0
$955$	415.411	−	415.411i	0.434985	−	0.434985i
$956$		0			0
$957$	−6.37125	+	6.37125i	−0.00665752	+	0.00665752i
$958$		0			0
$959$			210.861i			0.219876i
$960$		0			0
$961$	−1515.32			−1.57681
$962$		0			0
$963$	79.9504	+	79.9504i	0.0830223	+	0.0830223i
$964$		0			0
$965$	523.260	+	523.260i	0.542238	+	0.542238i
$966$		0			0
$967$	341.586			0.353243			0.176622	−	0.984279i	$-0.443483\pi$
$967$	341.586			0.353243			0.176622	+	0.984279i	$0.443483\pi$
$968$		0			0
$969$			152.694i			0.157579i
$970$		0			0
$971$	622.601	−	622.601i	0.641196	−	0.641196i	−0.309653	−	0.950849i	$-0.600213\pi$
$971$	622.601	−	622.601i	0.641196	−	0.641196i	0.950849	+	0.309653i	$0.100213\pi$
$972$		0			0
$973$	−323.066	+	323.066i	−0.332031	+	0.332031i
$974$		0			0
$975$			844.704i			0.866363i
$976$		0			0
$977$	−1065.93			−1.09103			−0.545514	−	0.838102i	$-0.683666\pi$
$977$	−1065.93			−1.09103			−0.545514	+	0.838102i	$0.683666\pi$
$978$		0			0
$979$	−1.75066	−	1.75066i	−0.00178821	−	0.00178821i
$980$		0			0
$981$	622.548	+	622.548i	0.634606	+	0.634606i
$982$		0			0
$983$	612.836			0.623434			0.311717	−	0.950175i	$-0.399096\pi$
$983$	612.836			0.623434			0.311717	+	0.950175i	$0.399096\pi$
$984$		0			0
$985$			1146.99i			1.16446i
$986$		0			0
$987$	548.339	−	548.339i	0.555562	−	0.555562i
$988$		0			0
$989$	−74.1659	+	74.1659i	−0.0749908	+	0.0749908i
$990$		0			0
$991$		−	49.8899i		−	0.0503430i	−0.999683	−	0.0251715i	$-0.991987\pi$
$991$		−	49.8899i		−	0.0503430i	0.999683	−	0.0251715i	$-0.00801319\pi$
$992$		0			0
$993$	352.044			0.354525
$994$		0			0
$995$	−964.293	−	964.293i	−0.969139	−	0.969139i
$996$		0			0
$997$	−1288.31	−	1288.31i	−1.29218	−	1.29218i	−0.933435	−	0.358747i	$-0.883204\pi$
$997$	−1288.31	−	1288.31i	−1.29218	−	1.29218i	−0.358747	−	0.933435i	$-0.616796\pi$
$998$		0			0
$999$	−668.591			−0.669260

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.3.k.b.799.5		48
4.3	odd	2		896.3.k.a.799.20		48
8.3	odd	2		112.3.k.a.43.24	✓	48
8.5	even	2		448.3.k.a.15.20		48
16.3	odd	4	inner	896.3.k.b.351.5		48
16.5	even	4		112.3.k.a.99.24	yes	48
16.11	odd	4		448.3.k.a.239.20		48
16.13	even	4		896.3.k.a.351.20		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.k.a.43.24	✓	48	8.3	odd	2
112.3.k.a.99.24	yes	48	16.5	even	4
448.3.k.a.15.20		48	8.5	even	2
448.3.k.a.239.20		48	16.11	odd	4
896.3.k.a.351.20		48	16.13	even	4
896.3.k.a.799.20		48	4.3	odd	2
896.3.k.b.351.5		48	16.3	odd	4	inner
896.3.k.b.799.5		48	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	896.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.60485 - 2.60485i) q^{3} +(5.26925 + 5.26925i) q^{5} +2.64575 q^{7} +4.57046i q^{9} +O(q^{10})\)	\(q+(-2.60485 - 2.60485i) q^{3} +(5.26925 + 5.26925i) q^{5} +2.64575 q^{7} +4.57046i q^{9} +(-0.0342750 + 0.0342750i) q^{11} +(-5.31088 + 5.31088i) q^{13} -27.4512i q^{15} +27.5699 q^{17} +(-1.06310 - 1.06310i) q^{19} +(-6.89178 - 6.89178i) q^{21} +8.59472 q^{23} +30.5299i q^{25} +(-11.5383 + 11.5383i) q^{27} +(-35.6808 + 35.6808i) q^{29} -49.7626i q^{31} +0.178562 q^{33} +(13.9411 + 13.9411i) q^{35} +(28.9728 + 28.9728i) q^{37} +27.6681 q^{39} +47.8460i q^{41} +(-8.62924 + 8.62924i) q^{43} +(-24.0829 + 24.0829i) q^{45} +79.5643i q^{47} +7.00000 q^{49} +(-71.8154 - 71.8154i) q^{51} +(-26.1653 - 26.1653i) q^{53} -0.361207 q^{55} +5.53844i q^{57} +(28.4333 - 28.4333i) q^{59} +(39.9332 - 39.9332i) q^{61} +12.0923i q^{63} -55.9687 q^{65} +(59.3574 + 59.3574i) q^{67} +(-22.3879 - 22.3879i) q^{69} +76.2561 q^{71} -103.027i q^{73} +(79.5258 - 79.5258i) q^{75} +(-0.0906831 + 0.0906831i) q^{77} +45.1896i q^{79} +101.245 q^{81} +(55.5063 + 55.5063i) q^{83} +(145.273 + 145.273i) q^{85} +185.886 q^{87} +51.0768i q^{89} +(-14.0513 + 14.0513i) q^{91} +(-129.624 + 129.624i) q^{93} -11.2035i q^{95} -27.5785 q^{97} +(-0.156653 - 0.156653i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 16 q^{11} - 64 q^{19} + 64 q^{23} - 96 q^{27} - 16 q^{29} + 48 q^{37} - 384 q^{39} + 176 q^{43} + 336 q^{49} - 192 q^{51} + 80 q^{53} + 512 q^{55} - 288 q^{59} + 64 q^{61} - 32 q^{65} + 80 q^{67} - 192 q^{69} + 608 q^{75} - 112 q^{77} - 432 q^{81} - 160 q^{83} - 320 q^{85} + 896 q^{87} - 96 q^{93} + 496 q^{99}+O(q^{100}) \) 48 * q + 16 * q^11 - 64 * q^19 + 64 * q^23 - 96 * q^27 - 16 * q^29 + 48 * q^37 - 384 * q^39 + 176 * q^43 + 336 * q^49 - 192 * q^51 + 80 * q^53 + 512 * q^55 - 288 * q^59 + 64 * q^61 - 32 * q^65 + 80 * q^67 - 192 * q^69 + 608 * q^75 - 112 * q^77 - 432 * q^81 - 160 * q^83 - 320 * q^85 + 896 * q^87 - 96 * q^93 + 496 * q^99

Introduction

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