LMFDB - Embedded newform 1680.2.f.j.881.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(881,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1680.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.f (of order $2$, degree $1$, 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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.2
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1680.881
Dual form			1680.2.f.j.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.618034 + 1.61803i) q^{3} +1.00000 q^{5} +(2.61803 - 0.381966i) q^{7} +(-2.23607 - 2.00000i) q^{9} +O(q^{10})$	\(q+(-0.618034 + 1.61803i) q^{3} +1.00000 q^{5} +(2.61803 - 0.381966i) q^{7} +(-2.23607 - 2.00000i) q^{9} +5.23607i q^{11} -3.23607i q^{13} +(-0.618034 + 1.61803i) q^{15} +4.47214 q^{17} -2.76393i q^{19} +(-1.00000 + 4.47214i) q^{21} +5.70820i q^{23} +1.00000 q^{25} +(4.61803 - 2.38197i) q^{27} +4.00000i q^{29} -1.23607i q^{31} +(-8.47214 - 3.23607i) q^{33} +(2.61803 - 0.381966i) q^{35} -4.47214 q^{37} +(5.23607 + 2.00000i) q^{39} +12.4721 q^{41} +2.76393 q^{43} +(-2.23607 - 2.00000i) q^{45} +1.23607 q^{47} +(6.70820 - 2.00000i) q^{49} +(-2.76393 + 7.23607i) q^{51} +4.76393i q^{53} +5.23607i q^{55} +(4.47214 + 1.70820i) q^{57} -8.94427 q^{59} +12.9443i q^{61} +(-6.61803 - 4.38197i) q^{63} -3.23607i q^{65} +3.70820 q^{67} +(-9.23607 - 3.52786i) q^{69} -10.1803i q^{71} -11.2361i q^{73} +(-0.618034 + 1.61803i) q^{75} +(2.00000 + 13.7082i) q^{77} -1.52786 q^{79} +(1.00000 + 8.94427i) q^{81} -2.76393 q^{83} +4.47214 q^{85} +(-6.47214 - 2.47214i) q^{87} +3.52786 q^{89} +(-1.23607 - 8.47214i) q^{91} +(2.00000 + 0.763932i) q^{93} -2.76393i q^{95} +8.18034i q^{97} +(10.4721 - 11.7082i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 4 q^{5} + 6 q^{7}+O(q^{10}) $ 4 * q + 2 * q^3 + 4 * q^5 + 6 * q^7	$ 4 q + 2 q^{3} + 4 q^{5} + 6 q^{7} + 2 q^{15} - 4 q^{21} + 4 q^{25} + 14 q^{27} - 16 q^{33} + 6 q^{35} + 12 q^{39} + 32 q^{41} + 20 q^{43} - 4 q^{47} - 20 q^{51} - 22 q^{63} - 12 q^{67} - 28 q^{69} + 2 q^{75} + 8 q^{77} - 24 q^{79} + 4 q^{81} - 20 q^{83} - 8 q^{87} + 32 q^{89} + 4 q^{91} + 8 q^{93} + 24 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 4 * q^5 + 6 * q^7 + 2 * q^15 - 4 * q^21 + 4 * q^25 + 14 * q^27 - 16 * q^33 + 6 * q^35 + 12 * q^39 + 32 * q^41 + 20 * q^43 - 4 * q^47 - 20 * q^51 - 22 * q^63 - 12 * q^67 - 28 * q^69 + 2 * q^75 + 8 * q^77 - 24 * q^79 + 4 * q^81 - 20 * q^83 - 8 * q^87 + 32 * q^89 + 4 * q^91 + 8 * q^93 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.618034	+	1.61803i	−0.356822	+	0.934172i
$3$	−0.618034	+	1.61803i	−0.356822	+	0.934172i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	2.61803	−	0.381966i	0.989524	−	0.144370i
$7$	2.61803	−	0.381966i	0.989524	−	0.144370i
$8$		0			0
$9$	−2.23607	−	2.00000i	−0.745356	−	0.666667i
$10$		0			0
$11$			5.23607i			1.57873i	0.613922	+	0.789367i	$0.289591\pi$
$11$			5.23607i			1.57873i	−0.613922	+	0.789367i	$0.710409\pi$
$12$		0			0
$13$		−	3.23607i		−	0.897524i	−0.893651	−	0.448762i	$-0.851865\pi$
$13$		−	3.23607i		−	0.897524i	0.893651	−	0.448762i	$-0.148135\pi$
$14$		0			0
$15$	−0.618034	+	1.61803i	−0.159576	+	0.417775i
$16$		0			0
$17$	4.47214			1.08465			0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214			1.08465			0.542326	+	0.840168i	$0.317544\pi$
$18$		0			0
$19$		−	2.76393i		−	0.634089i	−0.948411	−	0.317045i	$-0.897309\pi$
$19$		−	2.76393i		−	0.634089i	0.948411	−	0.317045i	$-0.102691\pi$
$20$		0			0
$21$	−1.00000	+	4.47214i	−0.218218	+	0.975900i
$22$		0			0
$23$			5.70820i			1.19024i	0.803636	+	0.595121i	$0.202896\pi$
$23$			5.70820i			1.19024i	−0.803636	+	0.595121i	$0.797104\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	4.61803	−	2.38197i	0.888741	−	0.458410i
$28$		0			0
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		−	1.23607i		−	0.222004i	−0.993820	−	0.111002i	$-0.964594\pi$
$31$		−	1.23607i		−	0.222004i	0.993820	−	0.111002i	$-0.0354061\pi$
$32$		0			0
$33$	−8.47214	−	3.23607i	−1.47481	−	0.563327i
$34$		0			0
$35$	2.61803	−	0.381966i	0.442529	−	0.0645640i
$36$		0			0
$37$	−4.47214			−0.735215			−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214			−0.735215			−0.367607	+	0.929981i	$0.619823\pi$
$38$		0			0
$39$	5.23607	+	2.00000i	0.838442	+	0.320256i
$40$		0			0
$41$	12.4721			1.94782			0.973910	−	0.226934i	$-0.0728701\pi$
$41$	12.4721			1.94782			0.973910	+	0.226934i	$0.0728701\pi$
$42$		0			0
$43$	2.76393			0.421496			0.210748	−	0.977540i	$-0.432410\pi$
$43$	2.76393			0.421496			0.210748	+	0.977540i	$0.432410\pi$
$44$		0			0
$45$	−2.23607	−	2.00000i	−0.333333	−	0.298142i
$46$		0			0
$47$	1.23607			0.180299			0.0901495	−	0.995928i	$-0.471266\pi$
$47$	1.23607			0.180299			0.0901495	+	0.995928i	$0.471266\pi$
$48$		0			0
$49$	6.70820	−	2.00000i	0.958315	−	0.285714i
$50$		0			0
$51$	−2.76393	+	7.23607i	−0.387028	+	1.01325i
$52$		0			0
$53$			4.76393i			0.654376i	0.944959	+	0.327188i	$0.106101\pi$
$53$			4.76393i			0.654376i	−0.944959	+	0.327188i	$0.893899\pi$
$54$		0			0
$55$			5.23607i			0.706031i
$56$		0			0
$57$	4.47214	+	1.70820i	0.592349	+	0.226257i
$58$		0			0
$59$	−8.94427			−1.16445			−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427			−1.16445			−0.582223	+	0.813029i	$0.697817\pi$
$60$		0			0
$61$			12.9443i			1.65734i	0.559734	+	0.828672i	$0.310903\pi$
$61$			12.9443i			1.65734i	−0.559734	+	0.828672i	$0.689097\pi$
$62$		0			0
$63$	−6.61803	−	4.38197i	−0.833794	−	0.552076i
$64$		0			0
$65$		−	3.23607i		−	0.401385i
$66$		0			0
$67$	3.70820			0.453029			0.226515	−	0.974008i	$-0.427267\pi$
$67$	3.70820			0.453029			0.226515	+	0.974008i	$0.427267\pi$
$68$		0			0
$69$	−9.23607	−	3.52786i	−1.11189	−	0.424705i
$70$		0			0
$71$		−	10.1803i		−	1.20818i	−0.796915	−	0.604092i	$-0.793536\pi$
$71$		−	10.1803i		−	1.20818i	0.796915	−	0.604092i	$-0.206464\pi$
$72$		0			0
$73$		−	11.2361i		−	1.31508i	−0.753419	−	0.657541i	$-0.771597\pi$
$73$		−	11.2361i		−	1.31508i	0.753419	−	0.657541i	$-0.228403\pi$
$74$		0			0
$75$	−0.618034	+	1.61803i	−0.0713644	+	0.186834i
$76$		0			0
$77$	2.00000	+	13.7082i	0.227921	+	1.56219i
$78$		0			0
$79$	−1.52786			−0.171898			−0.0859491	−	0.996300i	$-0.527392\pi$
$79$	−1.52786			−0.171898			−0.0859491	+	0.996300i	$0.527392\pi$
$80$		0			0
$81$	1.00000	+	8.94427i	0.111111	+	0.993808i
$82$		0			0
$83$	−2.76393			−0.303381			−0.151690	−	0.988428i	$-0.548472\pi$
$83$	−2.76393			−0.303381			−0.151690	+	0.988428i	$0.548472\pi$
$84$		0			0
$85$	4.47214			0.485071
$86$		0			0
$87$	−6.47214	−	2.47214i	−0.693886	−	0.265041i
$88$		0			0
$89$	3.52786			0.373953			0.186976	−	0.982364i	$-0.440131\pi$
$89$	3.52786			0.373953			0.186976	+	0.982364i	$0.440131\pi$
$90$		0			0
$91$	−1.23607	−	8.47214i	−0.129575	−	0.888121i
$92$		0			0
$93$	2.00000	+	0.763932i	0.207390	+	0.0792161i
$94$		0			0
$95$		−	2.76393i		−	0.283573i
$96$		0			0
$97$			8.18034i			0.830588i	0.909687	+	0.415294i	$0.136321\pi$
$97$			8.18034i			0.830588i	−0.909687	+	0.415294i	$0.863679\pi$
$98$		0			0
$99$	10.4721	−	11.7082i	1.05249	−	1.17672i
$100$		0			0
$101$	−10.9443			−1.08900			−0.544498	−	0.838762i	$-0.683280\pi$
$101$	−10.9443			−1.08900			−0.544498	+	0.838762i	$0.683280\pi$
$102$		0			0
$103$			15.2361i			1.50125i	0.660726	+	0.750627i	$0.270249\pi$
$103$			15.2361i			1.50125i	−0.660726	+	0.750627i	$0.729751\pi$
$104$		0			0
$105$	−1.00000	+	4.47214i	−0.0975900	+	0.436436i
$106$		0			0
$107$			13.7082i			1.32522i	0.748964	+	0.662611i	$0.230552\pi$
$107$			13.7082i			1.32522i	−0.748964	+	0.662611i	$0.769448\pi$
$108$		0			0
$109$	14.9443			1.43140			0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443			1.43140			0.715701	+	0.698407i	$0.246107\pi$
$110$		0			0
$111$	2.76393	−	7.23607i	0.262341	−	0.686817i
$112$		0			0
$113$			12.7639i			1.20073i	0.799726	+	0.600365i	$0.204978\pi$
$113$			12.7639i			1.20073i	−0.799726	+	0.600365i	$0.795022\pi$
$114$		0			0
$115$			5.70820i			0.532293i
$116$		0			0
$117$	−6.47214	+	7.23607i	−0.598349	+	0.668975i
$118$		0			0
$119$	11.7082	−	1.70820i	1.07329	−	0.156591i
$120$		0			0
$121$	−16.4164			−1.49240
$122$		0			0
$123$	−7.70820	+	20.1803i	−0.695025	+	1.81960i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−19.1246			−1.69703			−0.848517	−	0.529167i	$-0.822504\pi$
$127$	−19.1246			−1.69703			−0.848517	+	0.529167i	$0.822504\pi$
$128$		0			0
$129$	−1.70820	+	4.47214i	−0.150399	+	0.393750i
$130$		0			0
$131$	−11.4164			−0.997456			−0.498728	−	0.866758i	$-0.666199\pi$
$131$	−11.4164			−0.997456			−0.498728	+	0.866758i	$0.666199\pi$
$132$		0			0
$133$	−1.05573	−	7.23607i	−0.0915432	−	0.627447i
$134$		0			0
$135$	4.61803	−	2.38197i	0.397457	−	0.205007i
$136$		0			0
$137$			3.23607i			0.276476i	0.990399	+	0.138238i	$0.0441439\pi$
$137$			3.23607i			0.276476i	−0.990399	+	0.138238i	$0.955856\pi$
$138$		0			0
$139$		−	10.7639i		−	0.912985i	−0.889727	−	0.456492i	$-0.849106\pi$
$139$		−	10.7639i		−	0.912985i	0.889727	−	0.456492i	$-0.150894\pi$
$140$		0			0
$141$	−0.763932	+	2.00000i	−0.0643347	+	0.168430i
$142$		0			0
$143$	16.9443			1.41695
$144$		0			0
$145$			4.00000i			0.332182i
$146$		0			0
$147$	−0.909830	+	12.0902i	−0.0750415	+	0.997180i
$148$		0			0
$149$		−	11.4164i		−	0.935269i	−0.883922	−	0.467634i	$-0.845106\pi$
$149$		−	11.4164i		−	0.935269i	0.883922	−	0.467634i	$-0.154894\pi$
$150$		0			0
$151$	8.94427			0.727875			0.363937	−	0.931423i	$-0.381432\pi$
$151$	8.94427			0.727875			0.363937	+	0.931423i	$0.381432\pi$
$152$		0			0
$153$	−10.0000	−	8.94427i	−0.808452	−	0.723102i
$154$		0			0
$155$		−	1.23607i		−	0.0992834i
$156$		0			0
$157$			24.1803i			1.92980i	0.262615	+	0.964901i	$0.415415\pi$
$157$			24.1803i			1.92980i	−0.262615	+	0.964901i	$0.584585\pi$
$158$		0			0
$159$	−7.70820	−	2.94427i	−0.611300	−	0.233496i
$160$		0			0
$161$	2.18034	+	14.9443i	0.171835	+	1.17777i
$162$		0			0
$163$	12.6525			0.991018			0.495509	−	0.868603i	$-0.334981\pi$
$163$	12.6525			0.991018			0.495509	+	0.868603i	$0.334981\pi$
$164$		0			0
$165$	−8.47214	−	3.23607i	−0.659555	−	0.251928i
$166$		0			0
$167$	−13.2361			−1.02424			−0.512119	−	0.858915i	$-0.671139\pi$
$167$	−13.2361			−1.02424			−0.512119	+	0.858915i	$0.671139\pi$
$168$		0			0
$169$	2.52786			0.194451
$170$		0			0
$171$	−5.52786	+	6.18034i	−0.422726	+	0.472622i
$172$		0			0
$173$	−2.00000			−0.152057			−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000			−0.152057			−0.0760286	+	0.997106i	$0.524224\pi$
$174$		0			0
$175$	2.61803	−	0.381966i	0.197905	−	0.0288739i
$176$		0			0
$177$	5.52786	−	14.4721i	0.415500	−	1.08779i
$178$		0			0
$179$			17.2361i			1.28828i	0.764906	+	0.644142i	$0.222785\pi$
$179$			17.2361i			1.28828i	−0.764906	+	0.644142i	$0.777215\pi$
$180$		0			0
$181$		−	2.47214i		−	0.183752i	−0.995770	−	0.0918762i	$-0.970714\pi$
$181$		−	2.47214i		−	0.183752i	0.995770	−	0.0918762i	$-0.0292864\pi$
$182$		0			0
$183$	−20.9443	−	8.00000i	−1.54825	−	0.591377i
$184$		0			0
$185$	−4.47214			−0.328798
$186$		0			0
$187$			23.4164i			1.71238i
$188$		0			0
$189$	11.1803	−	8.00000i	0.813250	−	0.581914i
$190$		0			0
$191$			2.76393i			0.199991i	0.994988	+	0.0999956i	$0.0318829\pi$
$191$			2.76393i			0.199991i	−0.994988	+	0.0999956i	$0.968117\pi$
$192$		0			0
$193$	−6.00000			−0.431889			−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000			−0.431889			−0.215945	+	0.976406i	$0.569283\pi$
$194$		0			0
$195$	5.23607	+	2.00000i	0.374963	+	0.143223i
$196$		0			0
$197$		−	20.7639i		−	1.47937i	−0.672954	−	0.739684i	$-0.734975\pi$
$197$		−	20.7639i		−	1.47937i	0.672954	−	0.739684i	$-0.265025\pi$
$198$		0			0
$199$		−	13.2361i		−	0.938280i	−0.883124	−	0.469140i	$-0.844564\pi$
$199$		−	13.2361i		−	0.938280i	0.883124	−	0.469140i	$-0.155436\pi$
$200$		0			0
$201$	−2.29180	+	6.00000i	−0.161651	+	0.423207i
$202$		0			0
$203$	1.52786	+	10.4721i	0.107235	+	0.735000i
$204$		0			0
$205$	12.4721			0.871092
$206$		0			0
$207$	11.4164	−	12.7639i	0.793495	−	0.887155i
$208$		0			0
$209$	14.4721			1.00106
$210$		0			0
$211$	8.00000			0.550743			0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000			0.550743			0.275371	+	0.961338i	$0.411199\pi$
$212$		0			0
$213$	16.4721	+	6.29180i	1.12865	+	0.431107i
$214$		0			0
$215$	2.76393			0.188499
$216$		0			0
$217$	−0.472136	−	3.23607i	−0.0320507	−	0.219679i
$218$		0			0
$219$	18.1803	+	6.94427i	1.22851	+	0.469250i
$220$		0			0
$221$		−	14.4721i		−	0.973501i
$222$		0			0
$223$		−	1.70820i		−	0.114390i	−0.998363	−	0.0571949i	$-0.981784\pi$
$223$		−	1.70820i		−	0.114390i	0.998363	−	0.0571949i	$-0.0182156\pi$
$224$		0			0
$225$	−2.23607	−	2.00000i	−0.149071	−	0.133333i
$226$		0			0
$227$	18.7639			1.24541			0.622703	−	0.782458i	$-0.286035\pi$
$227$	18.7639			1.24541			0.622703	+	0.782458i	$0.286035\pi$
$228$		0			0
$229$		−	21.8885i		−	1.44644i	−0.690620	−	0.723218i	$-0.742662\pi$
$229$		−	21.8885i		−	1.44644i	0.690620	−	0.723218i	$-0.257338\pi$
$230$		0			0
$231$	−23.4164	−	5.23607i	−1.54069	−	0.344508i
$232$		0			0
$233$		−	24.1803i		−	1.58411i	−0.610452	−	0.792053i	$-0.709012\pi$
$233$		−	24.1803i		−	1.58411i	0.610452	−	0.792053i	$-0.290988\pi$
$234$		0			0
$235$	1.23607			0.0806322
$236$		0			0
$237$	0.944272	−	2.47214i	0.0613371	−	0.160582i
$238$		0			0
$239$		−	22.1803i		−	1.43473i	−0.696699	−	0.717363i	$-0.745349\pi$
$239$		−	22.1803i		−	1.43473i	0.696699	−	0.717363i	$-0.254651\pi$
$240$		0			0
$241$		−	28.3607i		−	1.82687i	−0.406982	−	0.913436i	$-0.633419\pi$
$241$		−	28.3607i		−	1.82687i	0.406982	−	0.913436i	$-0.366581\pi$
$242$		0			0
$243$	−15.0902	−	3.90983i	−0.968035	−	0.250816i
$244$		0			0
$245$	6.70820	−	2.00000i	0.428571	−	0.127775i
$246$		0			0
$247$	−8.94427			−0.569110
$248$		0			0
$249$	1.70820	−	4.47214i	0.108253	−	0.283410i
$250$		0			0
$251$	24.3607			1.53763			0.768816	−	0.639470i	$-0.220846\pi$
$251$	24.3607			1.53763			0.768816	+	0.639470i	$0.220846\pi$
$252$		0			0
$253$	−29.8885			−1.87908
$254$		0			0
$255$	−2.76393	+	7.23607i	−0.173084	+	0.453140i
$256$		0			0
$257$	7.52786			0.469575			0.234788	−	0.972047i	$-0.424561\pi$
$257$	7.52786			0.469575			0.234788	+	0.972047i	$0.424561\pi$
$258$		0			0
$259$	−11.7082	+	1.70820i	−0.727512	+	0.106143i
$260$		0			0
$261$	8.00000	−	8.94427i	0.495188	−	0.553637i
$262$		0			0
$263$		−	0.180340i		−	0.0111202i	−0.999985	−	0.00556012i	$-0.998230\pi$
$263$		−	0.180340i		−	0.0111202i	0.999985	−	0.00556012i	$-0.00176985\pi$
$264$		0			0
$265$			4.76393i			0.292646i
$266$		0			0
$267$	−2.18034	+	5.70820i	−0.133435	+	0.349336i
$268$		0			0
$269$	−28.4721			−1.73598			−0.867988	−	0.496584i	$-0.834587\pi$
$269$	−28.4721			−1.73598			−0.867988	+	0.496584i	$0.834587\pi$
$270$		0			0
$271$		−	19.1246i		−	1.16174i	−0.813997	−	0.580869i	$-0.802713\pi$
$271$		−	19.1246i		−	1.16174i	0.813997	−	0.580869i	$-0.197287\pi$
$272$		0			0
$273$	14.4721	+	3.23607i	0.875894	+	0.195856i
$274$		0			0
$275$			5.23607i			0.315747i
$276$		0			0
$277$	−25.4164			−1.52712			−0.763562	−	0.645735i	$-0.776551\pi$
$277$	−25.4164			−1.52712			−0.763562	+	0.645735i	$0.776551\pi$
$278$		0			0
$279$	−2.47214	+	2.76393i	−0.148003	+	0.165472i
$280$		0			0
$281$		−	15.4164i		−	0.919666i	−0.888005	−	0.459833i	$-0.847909\pi$
$281$		−	15.4164i		−	0.919666i	0.888005	−	0.459833i	$-0.152091\pi$
$282$		0			0
$283$		−	22.6525i		−	1.34655i	−0.739392	−	0.673275i	$-0.764887\pi$
$283$		−	22.6525i		−	1.34655i	0.739392	−	0.673275i	$-0.235113\pi$
$284$		0			0
$285$	4.47214	+	1.70820i	0.264906	+	0.101185i
$286$		0			0
$287$	32.6525	−	4.76393i	1.92741	−	0.281206i
$288$		0			0
$289$	3.00000			0.176471
$290$		0			0
$291$	−13.2361	−	5.05573i	−0.775912	−	0.296372i
$292$		0			0
$293$	14.0000			0.817889			0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000			0.817889			0.408944	+	0.912559i	$0.365897\pi$
$294$		0			0
$295$	−8.94427			−0.520756
$296$		0			0
$297$	12.4721	+	24.1803i	0.723707	+	1.40309i
$298$		0			0
$299$	18.4721			1.06827
$300$		0			0
$301$	7.23607	−	1.05573i	0.417080	−	0.0608512i
$302$		0			0
$303$	6.76393	−	17.7082i	0.388578	−	1.01731i
$304$		0			0
$305$			12.9443i			0.741187i
$306$		0			0
$307$			4.18034i			0.238585i	0.992859	+	0.119292i	$0.0380626\pi$
$307$			4.18034i			0.238585i	−0.992859	+	0.119292i	$0.961937\pi$
$308$		0			0
$309$	−24.6525	−	9.41641i	−1.40243	−	0.535681i
$310$		0			0
$311$	12.3607			0.700910			0.350455	−	0.936580i	$-0.386027\pi$
$311$	12.3607			0.700910			0.350455	+	0.936580i	$0.386027\pi$
$312$		0			0
$313$			17.7082i			1.00093i	0.865758	+	0.500463i	$0.166837\pi$
$313$			17.7082i			1.00093i	−0.865758	+	0.500463i	$0.833163\pi$
$314$		0			0
$315$	−6.61803	−	4.38197i	−0.372884	−	0.246896i
$316$		0			0
$317$			16.1803i			0.908778i	0.890803	+	0.454389i	$0.150142\pi$
$317$			16.1803i			0.908778i	−0.890803	+	0.454389i	$0.849858\pi$
$318$		0			0
$319$	−20.9443			−1.17265
$320$		0			0
$321$	−22.1803	−	8.47214i	−1.23799	−	0.472869i
$322$		0			0
$323$		−	12.3607i		−	0.687767i
$324$		0			0
$325$		−	3.23607i		−	0.179505i
$326$		0			0
$327$	−9.23607	+	24.1803i	−0.510756	+	1.33718i
$328$		0			0
$329$	3.23607	−	0.472136i	0.178410	−	0.0260297i
$330$		0			0
$331$	3.05573			0.167958			0.0839790	−	0.996468i	$-0.473237\pi$
$331$	3.05573			0.167958			0.0839790	+	0.996468i	$0.473237\pi$
$332$		0			0
$333$	10.0000	+	8.94427i	0.547997	+	0.490143i
$334$		0			0
$335$	3.70820			0.202601
$336$		0			0
$337$	2.00000			0.108947			0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000			0.108947			0.0544735	+	0.998515i	$0.482652\pi$
$338$		0			0
$339$	−20.6525	−	7.88854i	−1.12169	−	0.428447i
$340$		0			0
$341$	6.47214			0.350486
$342$		0			0
$343$	16.7984	−	7.79837i	0.907027	−	0.421073i
$344$		0			0
$345$	−9.23607	−	3.52786i	−0.497253	−	0.189934i
$346$		0			0
$347$		−	1.34752i		−	0.0723389i	−0.999346	−	0.0361694i	$-0.988484\pi$
$347$		−	1.34752i		−	0.0723389i	0.999346	−	0.0361694i	$-0.0115156\pi$
$348$		0			0
$349$		−	8.94427i		−	0.478776i	−0.970924	−	0.239388i	$-0.923053\pi$
$349$		−	8.94427i		−	0.478776i	0.970924	−	0.239388i	$-0.0769468\pi$
$350$		0			0
$351$	−7.70820	−	14.9443i	−0.411433	−	0.797666i
$352$		0			0
$353$	1.41641			0.0753878			0.0376939	−	0.999289i	$-0.487999\pi$
$353$	1.41641			0.0753878			0.0376939	+	0.999289i	$0.487999\pi$
$354$		0			0
$355$		−	10.1803i		−	0.540316i
$356$		0			0
$357$	−4.47214	+	20.0000i	−0.236691	+	1.05851i
$358$		0			0
$359$		−	24.0689i		−	1.27031i	−0.772386	−	0.635154i	$-0.780937\pi$
$359$		−	24.0689i		−	1.27031i	0.772386	−	0.635154i	$-0.219063\pi$
$360$		0			0
$361$	11.3607			0.597931
$362$		0			0
$363$	10.1459	−	26.5623i	0.532522	−	1.39416i
$364$		0			0
$365$		−	11.2361i		−	0.588123i
$366$		0			0
$367$		−	6.65248i		−	0.347256i	−0.984811	−	0.173628i	$-0.944451\pi$
$367$		−	6.65248i		−	0.347256i	0.984811	−	0.173628i	$-0.0555491\pi$
$368$		0			0
$369$	−27.8885	−	24.9443i	−1.45182	−	1.29855i
$370$		0			0
$371$	1.81966	+	12.4721i	0.0944720	+	0.647521i
$372$		0			0
$373$	16.4721			0.852895			0.426447	−	0.904512i	$-0.359765\pi$
$373$	16.4721			0.852895			0.426447	+	0.904512i	$0.359765\pi$
$374$		0			0
$375$	−0.618034	+	1.61803i	−0.0319151	+	0.0835549i
$376$		0			0
$377$	12.9443			0.666664
$378$		0			0
$379$	−10.4721			−0.537917			−0.268959	−	0.963152i	$-0.586680\pi$
$379$	−10.4721			−0.537917			−0.268959	+	0.963152i	$0.586680\pi$
$380$		0			0
$381$	11.8197	−	30.9443i	0.605540	−	1.58532i
$382$		0			0
$383$	−10.7639			−0.550011			−0.275006	−	0.961443i	$-0.588680\pi$
$383$	−10.7639			−0.550011			−0.275006	+	0.961443i	$0.588680\pi$
$384$		0			0
$385$	2.00000	+	13.7082i	0.101929	+	0.698635i
$386$		0			0
$387$	−6.18034	−	5.52786i	−0.314164	−	0.280997i
$388$		0			0
$389$			6.47214i			0.328150i	0.986448	+	0.164075i	$0.0524640\pi$
$389$			6.47214i			0.328150i	−0.986448	+	0.164075i	$0.947536\pi$
$390$		0			0
$391$			25.5279i			1.29100i
$392$		0			0
$393$	7.05573	−	18.4721i	0.355914	−	0.931796i
$394$		0			0
$395$	−1.52786			−0.0768752
$396$		0			0
$397$		−	9.70820i		−	0.487241i	−0.969871	−	0.243620i	$-0.921665\pi$
$397$		−	9.70820i		−	0.487241i	0.969871	−	0.243620i	$-0.0783351\pi$
$398$		0			0
$399$	12.3607	+	2.76393i	0.618808	+	0.138370i
$400$		0			0
$401$			11.0557i			0.552097i	0.961144	+	0.276048i	$0.0890250\pi$
$401$			11.0557i			0.552097i	−0.961144	+	0.276048i	$0.910975\pi$
$402$		0			0
$403$	−4.00000			−0.199254
$404$		0			0
$405$	1.00000	+	8.94427i	0.0496904	+	0.444444i
$406$		0			0
$407$		−	23.4164i		−	1.16071i
$408$		0			0
$409$			12.0000i			0.593362i	0.954977	+	0.296681i	$0.0958798\pi$
$409$			12.0000i			0.593362i	−0.954977	+	0.296681i	$0.904120\pi$
$410$		0			0
$411$	−5.23607	−	2.00000i	−0.258276	−	0.0986527i
$412$		0			0
$413$	−23.4164	+	3.41641i	−1.15225	+	0.168110i
$414$		0			0
$415$	−2.76393			−0.135676
$416$		0			0
$417$	17.4164	+	6.65248i	0.852885	+	0.325773i
$418$		0			0
$419$	−28.0000			−1.36789			−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000			−1.36789			−0.683945	+	0.729534i	$0.739737\pi$
$420$		0			0
$421$	23.5279			1.14668			0.573339	−	0.819318i	$-0.305648\pi$
$421$	23.5279			1.14668			0.573339	+	0.819318i	$0.305648\pi$
$422$		0			0
$423$	−2.76393	−	2.47214i	−0.134387	−	0.120199i
$424$		0			0
$425$	4.47214			0.216930
$426$		0			0
$427$	4.94427	+	33.8885i	0.239270	+	1.63998i
$428$		0			0
$429$	−10.4721	+	27.4164i	−0.505599	+	1.32368i
$430$		0			0
$431$		−	5.23607i		−	0.252213i	−0.992017	−	0.126106i	$-0.959752\pi$
$431$		−	5.23607i		−	0.252213i	0.992017	−	0.126106i	$-0.0402480\pi$
$432$		0			0
$433$		−	1.34752i		−	0.0647579i	−0.999476	−	0.0323789i	$-0.989692\pi$
$433$		−	1.34752i		−	0.0647579i	0.999476	−	0.0323789i	$-0.0103083\pi$
$434$		0			0
$435$	−6.47214	−	2.47214i	−0.310315	−	0.118530i
$436$		0			0
$437$	15.7771			0.754720
$438$		0			0
$439$		−	18.1803i		−	0.867700i	−0.900985	−	0.433850i	$-0.857155\pi$
$439$		−	18.1803i		−	0.867700i	0.900985	−	0.433850i	$-0.142845\pi$
$440$		0			0
$441$	−19.0000	−	8.94427i	−0.904762	−	0.425918i
$442$		0			0
$443$		−	34.0689i		−	1.61866i	−0.587353	−	0.809331i	$-0.699830\pi$
$443$		−	34.0689i		−	1.61866i	0.587353	−	0.809331i	$-0.300170\pi$
$444$		0			0
$445$	3.52786			0.167237
$446$		0			0
$447$	18.4721	+	7.05573i	0.873702	+	0.333724i
$448$		0			0
$449$		−	5.88854i		−	0.277898i	−0.990300	−	0.138949i	$-0.955628\pi$
$449$		−	5.88854i		−	0.277898i	0.990300	−	0.138949i	$-0.0443723\pi$
$450$		0			0
$451$			65.3050i			3.07509i
$452$		0			0
$453$	−5.52786	+	14.4721i	−0.259722	+	0.679960i
$454$		0			0
$455$	−1.23607	−	8.47214i	−0.0579478	−	0.397180i
$456$		0			0
$457$	−26.9443			−1.26040			−0.630200	−	0.776433i	$-0.717027\pi$
$457$	−26.9443			−1.26040			−0.630200	+	0.776433i	$0.717027\pi$
$458$		0			0
$459$	20.6525	−	10.6525i	0.963975	−	0.497215i
$460$		0			0
$461$	15.5279			0.723205			0.361602	−	0.932332i	$-0.382230\pi$
$461$	15.5279			0.723205			0.361602	+	0.932332i	$0.382230\pi$
$462$		0			0
$463$	25.2361			1.17282			0.586410	−	0.810015i	$-0.300541\pi$
$463$	25.2361			1.17282			0.586410	+	0.810015i	$0.300541\pi$
$464$		0			0
$465$	2.00000	+	0.763932i	0.0927478	+	0.0354265i
$466$		0			0
$467$	30.5410			1.41327			0.706635	−	0.707578i	$-0.250212\pi$
$467$	30.5410			1.41327			0.706635	+	0.707578i	$0.250212\pi$
$468$		0			0
$469$	9.70820	−	1.41641i	0.448283	−	0.0654036i
$470$		0			0
$471$	−39.1246	−	14.9443i	−1.80277	−	0.688596i
$472$		0			0
$473$			14.4721i			0.665430i
$474$		0			0
$475$		−	2.76393i		−	0.126818i
$476$		0			0
$477$	9.52786	−	10.6525i	0.436251	−	0.487743i
$478$		0			0
$479$	−4.94427			−0.225910			−0.112955	−	0.993600i	$-0.536032\pi$
$479$	−4.94427			−0.225910			−0.112955	+	0.993600i	$0.536032\pi$
$480$		0			0
$481$			14.4721i			0.659873i
$482$		0			0
$483$	−25.5279	−	5.70820i	−1.16156	−	0.259732i
$484$		0			0
$485$			8.18034i			0.371450i
$486$		0			0
$487$	−1.23607			−0.0560116			−0.0280058	−	0.999608i	$-0.508916\pi$
$487$	−1.23607			−0.0560116			−0.0280058	+	0.999608i	$0.508916\pi$
$488$		0			0
$489$	−7.81966	+	20.4721i	−0.353617	+	0.925782i
$490$		0			0
$491$		−	0.875388i		−	0.0395057i	−0.999805	−	0.0197529i	$-0.993712\pi$
$491$		−	0.875388i		−	0.0395057i	0.999805	−	0.0197529i	$-0.00628794\pi$
$492$		0			0
$493$			17.8885i			0.805659i
$494$		0			0
$495$	10.4721	−	11.7082i	0.470688	−	0.526245i
$496$		0			0
$497$	−3.88854	−	26.6525i	−0.174425	−	1.19553i
$498$		0			0
$499$	−28.3607			−1.26960			−0.634799	−	0.772677i	$-0.718917\pi$
$499$	−28.3607			−1.26960			−0.634799	+	0.772677i	$0.718917\pi$
$500$		0			0
$501$	8.18034	−	21.4164i	0.365471	−	0.956815i
$502$		0			0
$503$	−12.6525			−0.564146			−0.282073	−	0.959393i	$-0.591022\pi$
$503$	−12.6525			−0.564146			−0.282073	+	0.959393i	$0.591022\pi$
$504$		0			0
$505$	−10.9443			−0.487014
$506$		0			0
$507$	−1.56231	+	4.09017i	−0.0693844	+	0.181651i
$508$		0			0
$509$	5.05573			0.224091			0.112046	−	0.993703i	$-0.464260\pi$
$509$	5.05573			0.224091			0.112046	+	0.993703i	$0.464260\pi$
$510$		0			0
$511$	−4.29180	−	29.4164i	−0.189858	−	1.30131i
$512$		0			0
$513$	−6.58359	−	12.7639i	−0.290673	−	0.563541i
$514$		0			0
$515$			15.2361i			0.671381i
$516$		0			0
$517$			6.47214i			0.284644i
$518$		0			0
$519$	1.23607	−	3.23607i	0.0542574	−	0.142048i
$520$		0			0
$521$	−27.8885			−1.22182			−0.610910	−	0.791700i	$-0.709196\pi$
$521$	−27.8885			−1.22182			−0.610910	+	0.791700i	$0.709196\pi$
$522$		0			0
$523$		−	21.7082i		−	0.949233i	−0.880193	−	0.474617i	$-0.842587\pi$
$523$		−	21.7082i		−	0.949233i	0.880193	−	0.474617i	$-0.157413\pi$
$524$		0			0
$525$	−1.00000	+	4.47214i	−0.0436436	+	0.195180i
$526$		0			0
$527$		−	5.52786i		−	0.240798i
$528$		0			0
$529$	−9.58359			−0.416678
$530$		0			0
$531$	20.0000	+	17.8885i	0.867926	+	0.776297i
$532$		0			0
$533$		−	40.3607i		−	1.74822i
$534$		0			0
$535$			13.7082i			0.592657i
$536$		0			0
$537$	−27.8885	−	10.6525i	−1.20348	−	0.459688i
$538$		0			0
$539$	10.4721	+	35.1246i	0.451067	+	1.51292i
$540$		0			0
$541$	−8.11146			−0.348739			−0.174369	−	0.984680i	$-0.555789\pi$
$541$	−8.11146			−0.348739			−0.174369	+	0.984680i	$0.555789\pi$
$542$		0			0
$543$	4.00000	+	1.52786i	0.171656	+	0.0655669i
$544$		0			0
$545$	14.9443			0.640142
$546$		0			0
$547$	−4.29180			−0.183504			−0.0917520	−	0.995782i	$-0.529247\pi$
$547$	−4.29180			−0.183504			−0.0917520	+	0.995782i	$0.529247\pi$
$548$		0			0
$549$	25.8885	−	28.9443i	1.10490	−	1.23531i
$550$		0			0
$551$	11.0557			0.470990
$552$		0			0
$553$	−4.00000	+	0.583592i	−0.170097	+	0.0248169i
$554$		0			0
$555$	2.76393	−	7.23607i	0.117322	−	0.307154i
$556$		0			0
$557$			8.18034i			0.346612i	0.984868	+	0.173306i	$0.0554450\pi$
$557$			8.18034i			0.346612i	−0.984868	+	0.173306i	$0.944555\pi$
$558$		0			0
$559$		−	8.94427i		−	0.378302i
$560$		0			0
$561$	−37.8885	−	14.4721i	−1.59966	−	0.611014i
$562$		0			0
$563$	−40.0689			−1.68870			−0.844351	−	0.535790i	$-0.820014\pi$
$563$	−40.0689			−1.68870			−0.844351	+	0.535790i	$0.820014\pi$
$564$		0			0
$565$			12.7639i			0.536983i
$566$		0			0
$567$	6.03444	+	23.0344i	0.253423	+	0.967356i
$568$		0			0
$569$		−	13.8885i		−	0.582238i	−0.956687	−	0.291119i	$-0.905972\pi$
$569$		−	13.8885i		−	0.582238i	0.956687	−	0.291119i	$-0.0940276\pi$
$570$		0			0
$571$	46.8328			1.95989			0.979946	−	0.199262i	$-0.0638546\pi$
$571$	46.8328			1.95989			0.979946	+	0.199262i	$0.0638546\pi$
$572$		0			0
$573$	−4.47214	−	1.70820i	−0.186826	−	0.0713612i
$574$		0			0
$575$			5.70820i			0.238049i
$576$		0			0
$577$			26.0689i			1.08526i	0.839971	+	0.542631i	$0.182572\pi$
$577$			26.0689i			1.08526i	−0.839971	+	0.542631i	$0.817428\pi$
$578$		0			0
$579$	3.70820	−	9.70820i	0.154108	−	0.403459i
$580$		0			0
$581$	−7.23607	+	1.05573i	−0.300203	+	0.0437990i
$582$		0			0
$583$	−24.9443			−1.03309
$584$		0			0
$585$	−6.47214	+	7.23607i	−0.267590	+	0.299175i
$586$		0			0
$587$	−19.7082			−0.813445			−0.406722	−	0.913552i	$-0.633328\pi$
$587$	−19.7082			−0.813445			−0.406722	+	0.913552i	$0.633328\pi$
$588$		0			0
$589$	−3.41641			−0.140771
$590$		0			0
$591$	33.5967	+	12.8328i	1.38199	+	0.527872i
$592$		0			0
$593$	28.4721			1.16921			0.584605	−	0.811318i	$-0.301249\pi$
$593$	28.4721			1.16921			0.584605	+	0.811318i	$0.301249\pi$
$594$		0			0
$595$	11.7082	−	1.70820i	0.479990	−	0.0700295i
$596$		0			0
$597$	21.4164	+	8.18034i	0.876515	+	0.334799i
$598$		0			0
$599$			0.652476i			0.0266594i	0.999911	+	0.0133297i	$0.00424311\pi$
$599$			0.652476i			0.0266594i	−0.999911	+	0.0133297i	$0.995757\pi$
$600$		0			0
$601$		−	17.3050i		−	0.705884i	−0.935645	−	0.352942i	$-0.885181\pi$
$601$		−	17.3050i		−	0.705884i	0.935645	−	0.352942i	$-0.114819\pi$
$602$		0			0
$603$	−8.29180	−	7.41641i	−0.337668	−	0.302019i
$604$		0			0
$605$	−16.4164			−0.667422
$606$		0			0
$607$			18.0689i			0.733393i	0.930341	+	0.366697i	$0.119511\pi$
$607$			18.0689i			0.733393i	−0.930341	+	0.366697i	$0.880489\pi$
$608$		0			0
$609$	−17.8885	−	4.00000i	−0.724880	−	0.162088i
$610$		0			0
$611$		−	4.00000i		−	0.161823i
$612$		0			0
$613$	−22.3607			−0.903139			−0.451570	−	0.892236i	$-0.649136\pi$
$613$	−22.3607			−0.903139			−0.451570	+	0.892236i	$0.649136\pi$
$614$		0			0
$615$	−7.70820	+	20.1803i	−0.310825	+	0.813750i
$616$		0			0
$617$		−	40.5410i		−	1.63212i	−0.577967	−	0.816060i	$-0.696154\pi$
$617$		−	40.5410i		−	1.63212i	0.577967	−	0.816060i	$-0.303846\pi$
$618$		0			0
$619$		−	7.70820i		−	0.309819i	−0.987929	−	0.154909i	$-0.950491\pi$
$619$		−	7.70820i		−	0.309819i	0.987929	−	0.154909i	$-0.0495086\pi$
$620$		0			0
$621$	13.5967	+	26.3607i	0.545619	+	1.05782i
$622$		0			0
$623$	9.23607	−	1.34752i	0.370035	−	0.0539874i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−8.94427	+	23.4164i	−0.357200	+	0.935161i
$628$		0			0
$629$	−20.0000			−0.797452
$630$		0			0
$631$	8.94427			0.356066			0.178033	−	0.984025i	$-0.443027\pi$
$631$	8.94427			0.356066			0.178033	+	0.984025i	$0.443027\pi$
$632$		0			0
$633$	−4.94427	+	12.9443i	−0.196517	+	0.514489i
$634$		0			0
$635$	−19.1246			−0.758937
$636$		0			0
$637$	−6.47214	−	21.7082i	−0.256435	−	0.860110i
$638$		0			0
$639$	−20.3607	+	22.7639i	−0.805456	+	0.900527i
$640$		0			0
$641$			39.4164i			1.55685i	0.627735	+	0.778427i	$0.283982\pi$
$641$			39.4164i			1.55685i	−0.627735	+	0.778427i	$0.716018\pi$
$642$		0			0
$643$		−	37.7082i		−	1.48707i	−0.668699	−	0.743533i	$-0.733149\pi$
$643$		−	37.7082i		−	1.48707i	0.668699	−	0.743533i	$-0.266851\pi$
$644$		0			0
$645$	−1.70820	+	4.47214i	−0.0672605	+	0.176090i
$646$		0			0
$647$	−31.1246			−1.22363			−0.611817	−	0.790999i	$-0.709561\pi$
$647$	−31.1246			−1.22363			−0.611817	+	0.790999i	$0.709561\pi$
$648$		0			0
$649$		−	46.8328i		−	1.83835i
$650$		0			0
$651$	5.52786	+	1.23607i	0.216654	+	0.0484453i
$652$		0			0
$653$			27.5967i			1.07994i	0.841683	+	0.539972i	$0.181565\pi$
$653$			27.5967i			1.07994i	−0.841683	+	0.539972i	$0.818435\pi$
$654$		0			0
$655$	−11.4164			−0.446076
$656$		0			0
$657$	−22.4721	+	25.1246i	−0.876722	+	0.980204i
$658$		0			0
$659$			41.2361i			1.60633i	0.595757	+	0.803165i	$0.296852\pi$
$659$			41.2361i			1.60633i	−0.595757	+	0.803165i	$0.703148\pi$
$660$		0			0
$661$		−	30.8328i		−	1.19926i	−0.800278	−	0.599629i	$-0.795315\pi$
$661$		−	30.8328i		−	1.19926i	0.800278	−	0.599629i	$-0.204685\pi$
$662$		0			0
$663$	23.4164	+	8.94427i	0.909418	+	0.347367i
$664$		0			0
$665$	−1.05573	−	7.23607i	−0.0409394	−	0.280603i
$666$		0			0
$667$	−22.8328			−0.884090
$668$		0			0
$669$	2.76393	+	1.05573i	0.106860	+	0.0408168i
$670$		0			0
$671$	−67.7771			−2.61651
$672$		0			0
$673$	−2.94427			−0.113493			−0.0567467	−	0.998389i	$-0.518073\pi$
$673$	−2.94427			−0.113493			−0.0567467	+	0.998389i	$0.518073\pi$
$674$		0			0
$675$	4.61803	−	2.38197i	0.177748	−	0.0916819i
$676$		0			0
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$	3.12461	+	21.4164i	0.119912	+	0.821886i
$680$		0			0
$681$	−11.5967	+	30.3607i	−0.444388	+	1.16342i
$682$		0			0
$683$			33.7082i			1.28981i	0.764263	+	0.644904i	$0.223103\pi$
$683$			33.7082i			1.28981i	−0.764263	+	0.644904i	$0.776897\pi$
$684$		0			0
$685$			3.23607i			0.123644i
$686$		0			0
$687$	35.4164	+	13.5279i	1.35122	+	0.516120i
$688$		0			0
$689$	15.4164			0.587318
$690$		0			0
$691$			14.1803i			0.539446i	0.962938	+	0.269723i	$0.0869321\pi$
$691$			14.1803i			0.539446i	−0.962938	+	0.269723i	$0.913068\pi$
$692$		0			0
$693$	22.9443	−	34.6525i	0.871581	−	1.31634i
$694$		0			0
$695$		−	10.7639i		−	0.408299i
$696$		0			0
$697$	55.7771			2.11271
$698$		0			0
$699$	39.1246	+	14.9443i	1.47983	+	0.565244i
$700$		0			0
$701$		−	20.9443i		−	0.791054i	−0.918454	−	0.395527i	$-0.870562\pi$
$701$		−	20.9443i		−	0.791054i	0.918454	−	0.395527i	$-0.129438\pi$
$702$		0			0
$703$			12.3607i			0.466192i
$704$		0			0
$705$	−0.763932	+	2.00000i	−0.0287713	+	0.0753244i
$706$		0			0
$707$	−28.6525	+	4.18034i	−1.07759	+	0.157218i
$708$		0			0
$709$	−5.05573			−0.189872			−0.0949359	−	0.995483i	$-0.530265\pi$
$709$	−5.05573			−0.189872			−0.0949359	+	0.995483i	$0.530265\pi$
$710$		0			0
$711$	3.41641	+	3.05573i	0.128125	+	0.114599i
$712$		0			0
$713$	7.05573			0.264239
$714$		0			0
$715$	16.9443			0.633680
$716$		0			0
$717$	35.8885	+	13.7082i	1.34028	+	0.511942i
$718$		0			0
$719$	−41.8885			−1.56218			−0.781090	−	0.624419i	$-0.785336\pi$
$719$	−41.8885			−1.56218			−0.781090	+	0.624419i	$0.785336\pi$
$720$		0			0
$721$	5.81966	+	39.8885i	0.216735	+	1.48553i
$722$		0			0
$723$	45.8885	+	17.5279i	1.70661	+	0.651868i
$724$		0			0
$725$			4.00000i			0.148556i
$726$		0			0
$727$		−	9.70820i		−	0.360057i	−0.983661	−	0.180029i	$-0.942381\pi$
$727$		−	9.70820i		−	0.360057i	0.983661	−	0.180029i	$-0.0576191\pi$
$728$		0			0
$729$	15.6525	−	22.0000i	0.579721	−	0.814815i
$730$		0			0
$731$	12.3607			0.457176
$732$		0			0
$733$		−	17.3475i		−	0.640745i	−0.947292	−	0.320373i	$-0.896192\pi$
$733$		−	17.3475i		−	0.640745i	0.947292	−	0.320373i	$-0.103808\pi$
$734$		0			0
$735$	−0.909830	+	12.0902i	−0.0335596	+	0.445953i
$736$		0			0
$737$			19.4164i			0.715213i
$738$		0			0
$739$	4.36068			0.160410			0.0802051	−	0.996778i	$-0.474442\pi$
$739$	4.36068			0.160410			0.0802051	+	0.996778i	$0.474442\pi$
$740$		0			0
$741$	5.52786	−	14.4721i	0.203071	−	0.531647i
$742$		0			0
$743$			6.65248i			0.244056i	0.992527	+	0.122028i	$0.0389397\pi$
$743$			6.65248i			0.244056i	−0.992527	+	0.122028i	$0.961060\pi$
$744$		0			0
$745$		−	11.4164i		−	0.418265i
$746$		0			0
$747$	6.18034	+	5.52786i	0.226127	+	0.202254i
$748$		0			0
$749$	5.23607	+	35.8885i	0.191322	+	1.31134i
$750$		0			0
$751$	44.0000			1.60558			0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000			1.60558			0.802791	+	0.596260i	$0.203347\pi$
$752$		0			0
$753$	−15.0557	+	39.4164i	−0.548661	+	1.43641i
$754$		0			0
$755$	8.94427			0.325515
$756$		0			0
$757$	−27.3050			−0.992415			−0.496208	−	0.868204i	$-0.665275\pi$
$757$	−27.3050			−0.992415			−0.496208	+	0.868204i	$0.665275\pi$
$758$		0			0
$759$	18.4721	−	48.3607i	0.670496	−	1.75538i
$760$		0			0
$761$	31.3050			1.13480			0.567402	−	0.823441i	$-0.307949\pi$
$761$	31.3050			1.13480			0.567402	+	0.823441i	$0.307949\pi$
$762$		0			0
$763$	39.1246	−	5.70820i	1.41641	−	0.206651i
$764$		0			0
$765$	−10.0000	−	8.94427i	−0.361551	−	0.323381i
$766$		0			0
$767$			28.9443i			1.04512i
$768$		0			0
$769$		−	4.58359i		−	0.165289i	−0.996579	−	0.0826443i	$-0.973663\pi$
$769$		−	4.58359i		−	0.165289i	0.996579	−	0.0826443i	$-0.0263365\pi$
$770$		0			0
$771$	−4.65248	+	12.1803i	−0.167555	+	0.438664i
$772$		0			0
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$		−	1.23607i		−	0.0444009i
$776$		0			0
$777$	4.47214	−	20.0000i	0.160437	−	0.717496i
$778$		0			0
$779$		−	34.4721i		−	1.23509i
$780$		0			0
$781$	53.3050			1.90740
$782$		0			0
$783$	9.52786	+	18.4721i	0.340498	+	0.660140i
$784$		0			0
$785$			24.1803i			0.863033i
$786$		0			0
$787$		−	29.7082i		−	1.05898i	−0.848315	−	0.529492i	$-0.822383\pi$
$787$		−	29.7082i		−	1.05898i	0.848315	−	0.529492i	$-0.177617\pi$
$788$		0			0
$789$	0.291796	+	0.111456i	0.0103882	+	0.00396795i
$790$		0			0
$791$	4.87539	+	33.4164i	0.173349	+	1.18815i
$792$		0			0
$793$	41.8885			1.48751
$794$		0			0
$795$	−7.70820	−	2.94427i	−0.273382	−	0.104423i
$796$		0			0
$797$	−24.8328			−0.879623			−0.439812	−	0.898090i	$-0.644955\pi$
$797$	−24.8328			−0.879623			−0.439812	+	0.898090i	$0.644955\pi$
$798$		0			0
$799$	5.52786			0.195562
$800$		0			0
$801$	−7.88854	−	7.05573i	−0.278728	−	0.249302i
$802$		0			0
$803$	58.8328			2.07616
$804$		0			0
$805$	2.18034	+	14.9443i	0.0768469	+	0.526716i
$806$		0			0
$807$	17.5967	−	46.0689i	0.619435	−	1.62170i
$808$		0			0
$809$			32.3607i			1.13774i	0.822427	+	0.568870i	$0.192619\pi$
$809$			32.3607i			1.13774i	−0.822427	+	0.568870i	$0.807381\pi$
$810$		0			0
$811$		−	11.7082i		−	0.411131i	−0.978643	−	0.205565i	$-0.934097\pi$
$811$		−	11.7082i		−	0.411131i	0.978643	−	0.205565i	$-0.0659033\pi$
$812$		0			0
$813$	30.9443	+	11.8197i	1.08526	+	0.414534i
$814$		0			0
$815$	12.6525			0.443197
$816$		0			0
$817$		−	7.63932i		−	0.267266i
$818$		0			0
$819$	−14.1803	+	21.4164i	−0.495501	+	0.748350i
$820$		0			0
$821$			6.83282i			0.238467i	0.992866	+	0.119233i	$0.0380437\pi$
$821$			6.83282i			0.238467i	−0.992866	+	0.119233i	$0.961956\pi$
$822$		0			0
$823$	−37.5967			−1.31054			−0.655270	−	0.755395i	$-0.727445\pi$
$823$	−37.5967			−1.31054			−0.655270	+	0.755395i	$0.727445\pi$
$824$		0			0
$825$	−8.47214	−	3.23607i	−0.294962	−	0.112665i
$826$		0			0
$827$		−	26.2918i		−	0.914255i	−0.889401	−	0.457128i	$-0.848878\pi$
$827$		−	26.2918i		−	0.914255i	0.889401	−	0.457128i	$-0.151122\pi$
$828$		0			0
$829$			24.3607i			0.846081i	0.906111	+	0.423041i	$0.139037\pi$
$829$			24.3607i			0.846081i	−0.906111	+	0.423041i	$0.860963\pi$
$830$		0			0
$831$	15.7082	−	41.1246i	0.544912	−	1.42660i
$832$		0			0
$833$	30.0000	−	8.94427i	1.03944	−	0.309901i
$834$		0			0
$835$	−13.2361			−0.458053
$836$		0			0
$837$	−2.94427	−	5.70820i	−0.101769	−	0.197304i
$838$		0			0
$839$	4.94427			0.170695			0.0853476	−	0.996351i	$-0.472800\pi$
$839$	4.94427			0.170695			0.0853476	+	0.996351i	$0.472800\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$		0			0
$843$	24.9443	+	9.52786i	0.859126	+	0.328157i
$844$		0			0
$845$	2.52786			0.0869612
$846$		0			0
$847$	−42.9787	+	6.27051i	−1.47677	+	0.215457i
$848$		0			0
$849$	36.6525	+	14.0000i	1.25791	+	0.480479i
$850$		0			0
$851$		−	25.5279i		−	0.875084i
$852$		0			0
$853$		−	21.1246i		−	0.723293i	−0.932315	−	0.361646i	$-0.882215\pi$
$853$		−	21.1246i		−	0.723293i	0.932315	−	0.361646i	$-0.117785\pi$
$854$		0			0
$855$	−5.52786	+	6.18034i	−0.189049	+	0.211363i
$856$		0			0
$857$	−19.5279			−0.667059			−0.333530	−	0.942740i	$-0.608240\pi$
$857$	−19.5279			−0.667059			−0.333530	+	0.942740i	$0.608240\pi$
$858$		0			0
$859$		−	44.6525i		−	1.52352i	−0.647858	−	0.761761i	$-0.724335\pi$
$859$		−	44.6525i		−	1.52352i	0.647858	−	0.761761i	$-0.275665\pi$
$860$		0			0
$861$	−12.4721	+	55.7771i	−0.425049	+	1.90088i
$862$		0			0
$863$		−	21.3475i		−	0.726678i	−0.931657	−	0.363339i	$-0.881637\pi$
$863$		−	21.3475i		−	0.726678i	0.931657	−	0.363339i	$-0.118363\pi$
$864$		0			0
$865$	−2.00000			−0.0680020
$866$		0			0
$867$	−1.85410	+	4.85410i	−0.0629686	+	0.164854i
$868$		0			0
$869$		−	8.00000i		−	0.271381i
$870$		0			0
$871$		−	12.0000i		−	0.406604i
$872$		0			0
$873$	16.3607	−	18.2918i	0.553725	−	0.619083i
$874$		0			0
$875$	2.61803	−	0.381966i	0.0885057	−	0.0129128i
$876$		0			0
$877$	−23.5279			−0.794480			−0.397240	−	0.917715i	$-0.630032\pi$
$877$	−23.5279			−0.794480			−0.397240	+	0.917715i	$0.630032\pi$
$878$		0			0
$879$	−8.65248	+	22.6525i	−0.291841	+	0.764049i
$880$		0			0
$881$	−36.8328			−1.24093			−0.620465	−	0.784234i	$-0.713056\pi$
$881$	−36.8328			−1.24093			−0.620465	+	0.784234i	$0.713056\pi$
$882$		0			0
$883$	−13.5967			−0.457567			−0.228783	−	0.973477i	$-0.573475\pi$
$883$	−13.5967			−0.457567			−0.228783	+	0.973477i	$0.573475\pi$
$884$		0			0
$885$	5.52786	−	14.4721i	0.185817	−	0.486476i
$886$		0			0
$887$	6.18034			0.207516			0.103758	−	0.994603i	$-0.466913\pi$
$887$	6.18034			0.207516			0.103758	+	0.994603i	$0.466913\pi$
$888$		0			0
$889$	−50.0689	+	7.30495i	−1.67926	+	0.245000i
$890$		0			0
$891$	−46.8328	+	5.23607i	−1.56896	+	0.175415i
$892$		0			0
$893$		−	3.41641i		−	0.114326i
$894$		0			0
$895$			17.2361i			0.576138i
$896$		0			0
$897$	−11.4164	+	29.8885i	−0.381183	+	0.997949i
$898$		0			0
$899$	4.94427			0.164901
$900$		0			0
$901$			21.3050i			0.709771i
$902$		0			0
$903$	−2.76393	+	12.3607i	−0.0919779	+	0.411338i
$904$		0			0
$905$		−	2.47214i		−	0.0821766i
$906$		0			0
$907$	22.7639			0.755864			0.377932	−	0.925833i	$-0.376635\pi$
$907$	22.7639			0.755864			0.377932	+	0.925833i	$0.376635\pi$
$908$		0			0
$909$	24.4721	+	21.8885i	0.811690	+	0.725997i
$910$		0			0
$911$		−	12.0689i		−	0.399860i	−0.979810	−	0.199930i	$-0.935929\pi$
$911$		−	12.0689i		−	0.399860i	0.979810	−	0.199930i	$-0.0640715\pi$
$912$		0			0
$913$		−	14.4721i		−	0.478958i
$914$		0			0
$915$	−20.9443	−	8.00000i	−0.692396	−	0.264472i
$916$		0			0
$917$	−29.8885	+	4.36068i	−0.987007	+	0.144002i
$918$		0			0
$919$	25.5279			0.842087			0.421043	−	0.907041i	$-0.361664\pi$
$919$	25.5279			0.842087			0.421043	+	0.907041i	$0.361664\pi$
$920$		0			0
$921$	−6.76393	−	2.58359i	−0.222879	−	0.0851323i
$922$		0			0
$923$	−32.9443			−1.08437
$924$		0			0
$925$	−4.47214			−0.147043
$926$		0			0
$927$	30.4721	−	34.0689i	1.00084	−	1.11897i
$928$		0			0
$929$	−15.8885			−0.521286			−0.260643	−	0.965435i	$-0.583935\pi$
$929$	−15.8885			−0.521286			−0.260643	+	0.965435i	$0.583935\pi$
$930$		0			0
$931$	−5.52786	−	18.5410i	−0.181168	−	0.607657i
$932$		0			0
$933$	−7.63932	+	20.0000i	−0.250100	+	0.654771i
$934$		0			0
$935$			23.4164i			0.765798i
$936$		0			0
$937$			19.2361i			0.628415i	0.949354	+	0.314207i	$0.101739\pi$
$937$			19.2361i			0.628415i	−0.949354	+	0.314207i	$0.898261\pi$
$938$		0			0
$939$	−28.6525	−	10.9443i	−0.935038	−	0.357153i
$940$		0			0
$941$	−36.2492			−1.18169			−0.590845	−	0.806785i	$-0.701206\pi$
$941$	−36.2492			−1.18169			−0.590845	+	0.806785i	$0.701206\pi$
$942$		0			0
$943$			71.1935i			2.31838i
$944$		0			0
$945$	11.1803	−	8.00000i	0.363696	−	0.260240i
$946$		0			0
$947$			20.7639i			0.674737i	0.941373	+	0.337369i	$0.109537\pi$
$947$			20.7639i			0.674737i	−0.941373	+	0.337369i	$0.890463\pi$
$948$		0			0
$949$	−36.3607			−1.18032
$950$		0			0
$951$	−26.1803	−	10.0000i	−0.848956	−	0.324272i
$952$		0			0
$953$			9.70820i			0.314480i	0.987560	+	0.157240i	$0.0502596\pi$
$953$			9.70820i			0.314480i	−0.987560	+	0.157240i	$0.949740\pi$
$954$		0			0
$955$			2.76393i			0.0894387i
$956$		0			0
$957$	12.9443	−	33.8885i	0.418429	−	1.09546i
$958$		0			0
$959$	1.23607	+	8.47214i	0.0399147	+	0.273580i
$960$		0			0
$961$	29.4721			0.950714
$962$		0			0
$963$	27.4164	−	30.6525i	0.883481	−	0.987762i
$964$		0			0
$965$	−6.00000			−0.193147
$966$		0			0
$967$	−21.8197			−0.701673			−0.350836	−	0.936437i	$-0.614103\pi$
$967$	−21.8197			−0.701673			−0.350836	+	0.936437i	$0.614103\pi$
$968$		0			0
$969$	20.0000	+	7.63932i	0.642493	+	0.245410i
$970$		0			0
$971$	−27.4164			−0.879834			−0.439917	−	0.898038i	$-0.644992\pi$
$971$	−27.4164			−0.879834			−0.439917	+	0.898038i	$0.644992\pi$
$972$		0			0
$973$	−4.11146	−	28.1803i	−0.131807	−	0.903420i
$974$		0			0
$975$	5.23607	+	2.00000i	0.167688	+	0.0640513i
$976$		0			0
$977$			43.9574i			1.40632i	0.711030	+	0.703161i	$0.248229\pi$
$977$			43.9574i			1.40632i	−0.711030	+	0.703161i	$0.751771\pi$
$978$		0			0
$979$			18.4721i			0.590372i
$980$		0			0
$981$	−33.4164	−	29.8885i	−1.06690	−	0.954268i
$982$		0			0
$983$	26.5410			0.846527			0.423264	−	0.906007i	$-0.360884\pi$
$983$	26.5410			0.846527			0.423264	+	0.906007i	$0.360884\pi$
$984$		0			0
$985$		−	20.7639i		−	0.661594i
$986$		0			0
$987$	−1.23607	+	5.52786i	−0.0393445	+	0.175954i
$988$		0			0
$989$			15.7771i			0.501682i
$990$		0			0
$991$	31.7771			1.00943			0.504716	−	0.863285i	$-0.331597\pi$
$991$	31.7771			1.00943			0.504716	+	0.863285i	$0.331597\pi$
$992$		0			0
$993$	−1.88854	+	4.94427i	−0.0599311	+	0.156902i
$994$		0			0
$995$		−	13.2361i		−	0.419612i
$996$		0			0
$997$			42.0689i			1.33233i	0.745802	+	0.666167i	$0.232066\pi$
$997$			42.0689i			1.33233i	−0.745802	+	0.666167i	$0.767934\pi$
$998$		0			0
$999$	−20.6525	+	10.6525i	−0.653415	+	0.337029i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.f.j.881.2		4
3.2	odd	2		1680.2.f.f.881.4		4
4.3	odd	2		420.2.d.c.41.3	✓	4
7.6	odd	2		1680.2.f.f.881.3		4
12.11	even	2		420.2.d.d.41.1	yes	4
20.3	even	4		2100.2.f.g.1049.4		4
20.7	even	4		2100.2.f.a.1049.1		4
20.19	odd	2		2100.2.d.h.1301.2		4
21.20	even	2	inner	1680.2.f.j.881.1		4
28.27	even	2		420.2.d.d.41.2	yes	4
60.23	odd	4		2100.2.f.h.1049.3		4
60.47	odd	4		2100.2.f.b.1049.2		4
60.59	even	2		2100.2.d.g.1301.4		4
84.83	odd	2		420.2.d.c.41.4	yes	4
140.27	odd	4		2100.2.f.h.1049.4		4
140.83	odd	4		2100.2.f.b.1049.1		4
140.139	even	2		2100.2.d.g.1301.3		4
420.83	even	4		2100.2.f.a.1049.2		4
420.167	even	4		2100.2.f.g.1049.3		4
420.419	odd	2		2100.2.d.h.1301.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.d.c.41.3	✓	4	4.3	odd	2
420.2.d.c.41.4	yes	4	84.83	odd	2
420.2.d.d.41.1	yes	4	12.11	even	2
420.2.d.d.41.2	yes	4	28.27	even	2
1680.2.f.f.881.3		4	7.6	odd	2
1680.2.f.f.881.4		4	3.2	odd	2
1680.2.f.j.881.1		4	21.20	even	2	inner
1680.2.f.j.881.2		4	1.1	even	1	trivial
2100.2.d.g.1301.3		4	140.139	even	2
2100.2.d.g.1301.4		4	60.59	even	2
2100.2.d.h.1301.1		4	420.419	odd	2
2100.2.d.h.1301.2		4	20.19	odd	2
2100.2.f.a.1049.1		4	20.7	even	4
2100.2.f.a.1049.2		4	420.83	even	4
2100.2.f.b.1049.1		4	140.83	odd	4
2100.2.f.b.1049.2		4	60.47	odd	4
2100.2.f.g.1049.3		4	420.167	even	4
2100.2.f.g.1049.4		4	20.3	even	4
2100.2.f.h.1049.3		4	60.23	odd	4
2100.2.f.h.1049.4		4	140.27	odd	4

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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.618034 + 1.61803i) q^{3} +1.00000 q^{5} +(2.61803 - 0.381966i) q^{7} +(-2.23607 - 2.00000i) q^{9} +O(q^{10})\)	\(q+(-0.618034 + 1.61803i) q^{3} +1.00000 q^{5} +(2.61803 - 0.381966i) q^{7} +(-2.23607 - 2.00000i) q^{9} +5.23607i q^{11} -3.23607i q^{13} +(-0.618034 + 1.61803i) q^{15} +4.47214 q^{17} -2.76393i q^{19} +(-1.00000 + 4.47214i) q^{21} +5.70820i q^{23} +1.00000 q^{25} +(4.61803 - 2.38197i) q^{27} +4.00000i q^{29} -1.23607i q^{31} +(-8.47214 - 3.23607i) q^{33} +(2.61803 - 0.381966i) q^{35} -4.47214 q^{37} +(5.23607 + 2.00000i) q^{39} +12.4721 q^{41} +2.76393 q^{43} +(-2.23607 - 2.00000i) q^{45} +1.23607 q^{47} +(6.70820 - 2.00000i) q^{49} +(-2.76393 + 7.23607i) q^{51} +4.76393i q^{53} +5.23607i q^{55} +(4.47214 + 1.70820i) q^{57} -8.94427 q^{59} +12.9443i q^{61} +(-6.61803 - 4.38197i) q^{63} -3.23607i q^{65} +3.70820 q^{67} +(-9.23607 - 3.52786i) q^{69} -10.1803i q^{71} -11.2361i q^{73} +(-0.618034 + 1.61803i) q^{75} +(2.00000 + 13.7082i) q^{77} -1.52786 q^{79} +(1.00000 + 8.94427i) q^{81} -2.76393 q^{83} +4.47214 q^{85} +(-6.47214 - 2.47214i) q^{87} +3.52786 q^{89} +(-1.23607 - 8.47214i) q^{91} +(2.00000 + 0.763932i) q^{93} -2.76393i q^{95} +8.18034i q^{97} +(10.4721 - 11.7082i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 4 q^{5} + 6 q^{7}+O(q^{10}) \) 4 * q + 2 * q^3 + 4 * q^5 + 6 * q^7	\( 4 q + 2 q^{3} + 4 q^{5} + 6 q^{7} + 2 q^{15} - 4 q^{21} + 4 q^{25} + 14 q^{27} - 16 q^{33} + 6 q^{35} + 12 q^{39} + 32 q^{41} + 20 q^{43} - 4 q^{47} - 20 q^{51} - 22 q^{63} - 12 q^{67} - 28 q^{69} + 2 q^{75} + 8 q^{77} - 24 q^{79} + 4 q^{81} - 20 q^{83} - 8 q^{87} + 32 q^{89} + 4 q^{91} + 8 q^{93} + 24 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 4 * q^5 + 6 * q^7 + 2 * q^15 - 4 * q^21 + 4 * q^25 + 14 * q^27 - 16 * q^33 + 6 * q^35 + 12 * q^39 + 32 * q^41 + 20 * q^43 - 4 * q^47 - 20 * q^51 - 22 * q^63 - 12 * q^67 - 28 * q^69 + 2 * q^75 + 8 * q^77 - 24 * q^79 + 4 * q^81 - 20 * q^83 - 8 * q^87 + 32 * q^89 + 4 * q^91 + 8 * q^93 + 24 * q^99

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