LMFDB - Embedded newform 840.2.f.b.41.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [840,2,Mod(41,840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(840, 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("840.41");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	840.f (of order $2$, degree $1$, 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:	$6.70743376979$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + x^{14} - 4 x^{13} + 10 x^{12} - 32 x^{11} + 71 x^{10} - 70 x^{9} + 74 x^{8} + \cdots + 6561 $ x^16 - 2x^15 + x^14 - 4x^13 + 10x^12 - 32x^11 + 71x^10 - 70x^9 + 74x^8 - 210x^7 + 639x^6 - 864x^5 + 810x^4 - 972x^3 + 729x^2 - 4374x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.6
Root			$-1.49826 - 0.869033i$ of defining polynomial
Character	$\chi$	$=$	840.41
Dual form			840.2.f.b.41.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.869033 + 1.49826i) q^{3} +1.00000 q^{5} +(-0.807952 - 2.51937i) q^{7} +(-1.48956 - 2.60407i) q^{9} +O(q^{10})$	\(q+(-0.869033 + 1.49826i) q^{3} +1.00000 q^{5} +(-0.807952 - 2.51937i) q^{7} +(-1.48956 - 2.60407i) q^{9} +0.541366i q^{11} +5.85348i q^{13} +(-0.869033 + 1.49826i) q^{15} +6.22086 q^{17} +5.74951i q^{19} +(4.47680 + 0.978892i) q^{21} -6.55465i q^{23} +1.00000 q^{25} +(5.19606 + 0.0312759i) q^{27} +5.28395i q^{29} +8.99125i q^{31} +(-0.811106 - 0.470465i) q^{33} +(-0.807952 - 2.51937i) q^{35} +7.33408 q^{37} +(-8.77004 - 5.08687i) q^{39} +5.16593 q^{41} -2.48280 q^{43} +(-1.48956 - 2.60407i) q^{45} -2.09597 q^{47} +(-5.69443 + 4.07106i) q^{49} +(-5.40614 + 9.32047i) q^{51} +9.75661i q^{53} +0.541366i q^{55} +(-8.61426 - 4.99652i) q^{57} -9.76727 q^{59} -0.433188i q^{61} +(-5.35712 + 5.85672i) q^{63} +5.85348i q^{65} -8.26790 q^{67} +(9.82057 + 5.69621i) q^{69} +5.25045i q^{71} -9.42155i q^{73} +(-0.869033 + 1.49826i) q^{75} +(1.36390 - 0.437398i) q^{77} +17.3885 q^{79} +(-4.56241 + 7.75786i) q^{81} +14.8593 q^{83} +6.22086 q^{85} +(-7.91673 - 4.59193i) q^{87} +7.12418 q^{89} +(14.7471 - 4.72933i) q^{91} +(-13.4712 - 7.81370i) q^{93} +5.74951i q^{95} -3.24620i q^{97} +(1.40976 - 0.806398i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) $ 16 * q + 16 * q^5 + 2 * q^7 - 2 * q^9	$ 16 q + 16 q^{5} + 2 q^{7} - 2 q^{9} + 10 q^{21} + 16 q^{25} + 6 q^{27} + 6 q^{33} + 2 q^{35} + 12 q^{37} + 6 q^{39} + 32 q^{41} + 32 q^{43} - 2 q^{45} + 4 q^{47} - 4 q^{49} + 6 q^{51} - 24 q^{59} - 24 q^{63} + 8 q^{69} - 32 q^{77} - 4 q^{79} - 6 q^{81} + 20 q^{83} + 6 q^{87} - 24 q^{89} + 20 q^{91} - 32 q^{93} - 58 q^{99}+O(q^{100}) $ 16 * q + 16 * q^5 + 2 * q^7 - 2 * q^9 + 10 * q^21 + 16 * q^25 + 6 * q^27 + 6 * q^33 + 2 * q^35 + 12 * q^37 + 6 * q^39 + 32 * q^41 + 32 * q^43 - 2 * q^45 + 4 * q^47 - 4 * q^49 + 6 * q^51 - 24 * q^59 - 24 * q^63 + 8 * q^69 - 32 * q^77 - 4 * q^79 - 6 * q^81 + 20 * q^83 + 6 * q^87 - 24 * q^89 + 20 * q^91 - 32 * q^93 - 58 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$.

$n$	$241$	$281$	$337$	$421$	$631$
$\chi(n)$	$-1$	$-1$	$1$	$1$	$1$

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
$3$	−0.869033	+	1.49826i	−0.501737	+	0.865020i
$3$	−0.869033	+	1.49826i	−0.501737	+	0.865020i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−0.807952	−	2.51937i	−0.305377	−	0.952231i
$7$	−0.807952	−	2.51937i	−0.305377	−	0.952231i
$8$		0			0
$9$	−1.48956	−	2.60407i	−0.496521	−	0.868025i
$10$		0			0
$11$			0.541366i			0.163228i	0.996664	+	0.0816139i	$0.0260075\pi$
$11$			0.541366i			0.163228i	−0.996664	+	0.0816139i	$0.973993\pi$
$12$		0			0
$13$			5.85348i			1.62346i	0.584030	+	0.811732i	$0.301475\pi$
$13$			5.85348i			1.62346i	−0.584030	+	0.811732i	$0.698525\pi$
$14$		0			0
$15$	−0.869033	+	1.49826i	−0.224383	+	0.386849i
$16$		0			0
$17$	6.22086			1.50878			0.754390	−	0.656426i	$-0.227933\pi$
$17$	6.22086			1.50878			0.754390	+	0.656426i	$0.227933\pi$
$18$		0			0
$19$			5.74951i			1.31903i	0.751692	+	0.659515i	$0.229238\pi$
$19$			5.74951i			1.31903i	−0.751692	+	0.659515i	$0.770762\pi$
$20$		0			0
$21$	4.47680	+	0.978892i	0.976919	+	0.213612i
$22$		0			0
$23$		−	6.55465i		−	1.36674i	−0.730072	−	0.683370i	$-0.760514\pi$
$23$		−	6.55465i		−	1.36674i	0.730072	−	0.683370i	$-0.239486\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	5.19606	+	0.0312759i	0.999982	+	0.00601904i
$28$		0			0
$29$			5.28395i			0.981206i	0.871383	+	0.490603i	$0.163223\pi$
$29$			5.28395i			0.981206i	−0.871383	+	0.490603i	$0.836777\pi$
$30$		0			0
$31$			8.99125i			1.61488i	0.589952	+	0.807438i	$0.299147\pi$
$31$			8.99125i			1.61488i	−0.589952	+	0.807438i	$0.700853\pi$
$32$		0			0
$33$	−0.811106	−	0.470465i	−0.141195	−	0.0818974i
$34$		0			0
$35$	−0.807952	−	2.51937i	−0.136569	−	0.425851i
$36$		0			0
$37$	7.33408			1.20572			0.602858	−	0.797849i	$-0.294029\pi$
$37$	7.33408			1.20572			0.602858	+	0.797849i	$0.294029\pi$
$38$		0			0
$39$	−8.77004	−	5.08687i	−1.40433	−	0.814551i
$40$		0			0
$41$	5.16593			0.806783			0.403392	−	0.915027i	$-0.367831\pi$
$41$	5.16593			0.806783			0.403392	+	0.915027i	$0.367831\pi$
$42$		0			0
$43$	−2.48280			−0.378623			−0.189311	−	0.981917i	$-0.560626\pi$
$43$	−2.48280			−0.378623			−0.189311	+	0.981917i	$0.560626\pi$
$44$		0			0
$45$	−1.48956	−	2.60407i	−0.222051	−	0.388192i
$46$		0			0
$47$	−2.09597			−0.305728			−0.152864	−	0.988247i	$-0.548850\pi$
$47$	−2.09597			−0.305728			−0.152864	+	0.988247i	$0.548850\pi$
$48$		0			0
$49$	−5.69443	+	4.07106i	−0.813489	+	0.581580i
$50$		0			0
$51$	−5.40614	+	9.32047i	−0.757010	+	1.30513i
$52$		0			0
$53$			9.75661i			1.34017i	0.742283	+	0.670086i	$0.233743\pi$
$53$			9.75661i			1.34017i	−0.742283	+	0.670086i	$0.766257\pi$
$54$		0			0
$55$			0.541366i			0.0729977i
$56$		0			0
$57$	−8.61426	−	4.99652i	−1.14099	−	0.661805i
$58$		0			0
$59$	−9.76727			−1.27159			−0.635795	−	0.771858i	$-0.719328\pi$
$59$	−9.76727			−1.27159			−0.635795	+	0.771858i	$0.719328\pi$
$60$		0			0
$61$		−	0.433188i		−	0.0554640i	−0.999615	−	0.0277320i	$-0.991171\pi$
$61$		−	0.433188i		−	0.0554640i	0.999615	−	0.0277320i	$-0.00882850\pi$
$62$		0			0
$63$	−5.35712	+	5.85672i	−0.674934	+	0.737878i
$64$		0			0
$65$			5.85348i			0.726035i
$66$		0			0
$67$	−8.26790			−1.01008			−0.505042	−	0.863095i	$-0.668523\pi$
$67$	−8.26790			−1.01008			−0.505042	+	0.863095i	$0.668523\pi$
$68$		0			0
$69$	9.82057	+	5.69621i	1.18226	+	0.685743i
$70$		0			0
$71$			5.25045i			0.623114i	0.950228	+	0.311557i	$0.100850\pi$
$71$			5.25045i			0.623114i	−0.950228	+	0.311557i	$0.899150\pi$
$72$		0			0
$73$		−	9.42155i		−	1.10271i	−0.834271	−	0.551354i	$-0.814111\pi$
$73$		−	9.42155i		−	1.10271i	0.834271	−	0.551354i	$-0.185889\pi$
$74$		0			0
$75$	−0.869033	+	1.49826i	−0.100347	+	0.173004i
$76$		0			0
$77$	1.36390	−	0.437398i	0.155431	−	0.0498461i
$78$		0			0
$79$	17.3885			1.95635			0.978177	−	0.207772i	$-0.0666212\pi$
$79$	17.3885			1.95635			0.978177	+	0.207772i	$0.0666212\pi$
$80$		0			0
$81$	−4.56241	+	7.75786i	−0.506934	+	0.861985i
$82$		0			0
$83$	14.8593			1.63102			0.815510	−	0.578742i	$-0.196456\pi$
$83$	14.8593			1.63102			0.815510	+	0.578742i	$0.196456\pi$
$84$		0			0
$85$	6.22086			0.674747
$86$		0			0
$87$	−7.91673	−	4.59193i	−0.848763	−	0.492307i
$88$		0			0
$89$	7.12418			0.755162			0.377581	−	0.925977i	$-0.376756\pi$
$89$	7.12418			0.755162			0.377581	+	0.925977i	$0.376756\pi$
$90$		0			0
$91$	14.7471	−	4.72933i	1.54591	−	0.495769i
$92$		0			0
$93$	−13.4712	−	7.81370i	−1.39690	−	0.810243i
$94$		0			0
$95$			5.74951i			0.589888i
$96$		0			0
$97$		−	3.24620i		−	0.329602i	−0.986327	−	0.164801i	$-0.947302\pi$
$97$		−	3.24620i		−	0.329602i	0.986327	−	0.164801i	$-0.0526982\pi$
$98$		0			0
$99$	1.40976	−	0.806398i	0.141686	−	0.0810460i
$100$		0			0
$101$	−14.9842			−1.49098			−0.745492	−	0.666515i	$-0.767785\pi$
$101$	−14.9842			−1.49098			−0.745492	+	0.666515i	$0.767785\pi$
$102$		0			0
$103$			13.0115i			1.28206i	0.767515	+	0.641031i	$0.221493\pi$
$103$			13.0115i			1.28206i	−0.767515	+	0.641031i	$0.778507\pi$
$104$		0			0
$105$	4.47680	+	0.978892i	0.436891	+	0.0955301i
$106$		0			0
$107$		−	12.1046i		−	1.17019i	−0.810963	−	0.585097i	$-0.801057\pi$
$107$		−	12.1046i		−	1.17019i	0.810963	−	0.585097i	$-0.198943\pi$
$108$		0			0
$109$	−1.41160			−0.135207			−0.0676033	−	0.997712i	$-0.521535\pi$
$109$	−1.41160			−0.135207			−0.0676033	+	0.997712i	$0.521535\pi$
$110$		0			0
$111$	−6.37356	+	10.9884i	−0.604952	+	1.04297i
$112$		0			0
$113$		−	8.79738i		−	0.827588i	−0.910371	−	0.413794i	$-0.864203\pi$
$113$		−	8.79738i		−	0.827588i	0.910371	−	0.413794i	$-0.135797\pi$
$114$		0			0
$115$		−	6.55465i		−	0.611225i
$116$		0			0
$117$	15.2429	−	8.71913i	1.40921	−	0.806084i
$118$		0			0
$119$	−5.02616	−	15.6726i	−0.460747	−	1.43671i
$120$		0			0
$121$	10.7069			0.973357
$122$		0			0
$123$	−4.48937	+	7.73991i	−0.404793	+	0.697884i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	0.861929			0.0764839			0.0382419	−	0.999269i	$-0.487824\pi$
$127$	0.861929			0.0764839			0.0382419	+	0.999269i	$0.487824\pi$
$128$		0			0
$129$	2.15763	−	3.71987i	0.189969	−	0.327517i
$130$		0			0
$131$	−9.94338			−0.868757			−0.434378	−	0.900730i	$-0.643032\pi$
$131$	−9.94338			−0.868757			−0.434378	+	0.900730i	$0.643032\pi$
$132$		0			0
$133$	14.4851	−	4.64533i	1.25602	−	0.402802i
$134$		0			0
$135$	5.19606	+	0.0312759i	0.447205	+	0.00269180i
$136$		0			0
$137$		−	12.4338i		−	1.06229i	−0.847280	−	0.531147i	$-0.821761\pi$
$137$		−	12.4338i		−	1.06229i	0.847280	−	0.531147i	$-0.178239\pi$
$138$		0			0
$139$		−	5.11832i		−	0.434130i	−0.976157	−	0.217065i	$-0.930352\pi$
$139$		−	5.11832i		−	0.434130i	0.976157	−	0.217065i	$-0.0696484\pi$
$140$		0			0
$141$	1.82146	−	3.14030i	0.153395	−	0.264461i
$142$		0			0
$143$	−3.16887			−0.264995
$144$		0			0
$145$			5.28395i			0.438808i
$146$		0			0
$147$	−1.15086	−	12.0696i	−0.0949210	−	0.995485i
$148$		0			0
$149$			3.74937i			0.307161i	0.988136	+	0.153580i	$0.0490804\pi$
$149$			3.74937i			0.307161i	−0.988136	+	0.153580i	$0.950920\pi$
$150$		0			0
$151$	−3.33203			−0.271156			−0.135578	−	0.990767i	$-0.543289\pi$
$151$	−3.33203			−0.271156			−0.135578	+	0.990767i	$0.543289\pi$
$152$		0			0
$153$	−9.26636	−	16.1996i	−0.749141	−	1.30966i
$154$		0			0
$155$			8.99125i			0.722195i
$156$		0			0
$157$		−	2.75551i		−	0.219914i	−0.993936	−	0.109957i	$-0.964929\pi$
$157$		−	2.75551i		−	0.219914i	0.993936	−	0.109957i	$-0.0350713\pi$
$158$		0			0
$159$	−14.6179	−	8.47881i	−1.15928	−	0.672414i
$160$		0			0
$161$	−16.5136	+	5.29585i	−1.30145	+	0.417371i
$162$		0			0
$163$	11.6131			0.909611			0.454806	−	0.890591i	$-0.349709\pi$
$163$	11.6131			0.909611			0.454806	+	0.890591i	$0.349709\pi$
$164$		0			0
$165$	−0.811106	−	0.470465i	−0.0631445	−	0.0366256i
$166$		0			0
$167$	−7.85657			−0.607960			−0.303980	−	0.952678i	$-0.598316\pi$
$167$	−7.85657			−0.607960			−0.303980	+	0.952678i	$0.598316\pi$
$168$		0			0
$169$	−21.2633			−1.63564
$170$		0			0
$171$	14.9722	−	8.56426i	1.14495	−	0.654925i
$172$		0			0
$173$	5.82479			0.442850			0.221425	−	0.975177i	$-0.428929\pi$
$173$	5.82479			0.442850			0.221425	+	0.975177i	$0.428929\pi$
$174$		0			0
$175$	−0.807952	−	2.51937i	−0.0610755	−	0.190446i
$176$		0			0
$177$	8.48808	−	14.6339i	0.638003	−	1.09995i
$178$		0			0
$179$		−	10.5161i		−	0.786011i	−0.919536	−	0.393005i	$-0.871435\pi$
$179$		−	10.5161i		−	0.786011i	0.919536	−	0.393005i	$-0.128565\pi$
$180$		0			0
$181$		−	2.28411i		−	0.169777i	−0.996390	−	0.0848883i	$-0.972947\pi$
$181$		−	2.28411i		−	0.169777i	0.996390	−	0.0848883i	$-0.0270533\pi$
$182$		0			0
$183$	0.649028	+	0.376454i	0.0479775	+	0.0278283i
$184$		0			0
$185$	7.33408			0.539212
$186$		0			0
$187$			3.36776i			0.246275i
$188$		0			0
$189$	−4.11937	−	13.1161i	−0.299640	−	0.954052i
$190$		0			0
$191$			4.26399i			0.308531i	0.988029	+	0.154266i	$0.0493012\pi$
$191$			4.26399i			0.308531i	−0.988029	+	0.154266i	$0.950699\pi$
$192$		0			0
$193$	−8.02782			−0.577855			−0.288928	−	0.957351i	$-0.593299\pi$
$193$	−8.02782			−0.577855			−0.288928	+	0.957351i	$0.593299\pi$
$194$		0			0
$195$	−8.77004	−	5.08687i	−0.628035	−	0.364278i
$196$		0			0
$197$		−	11.3017i		−	0.805210i	−0.915374	−	0.402605i	$-0.868105\pi$
$197$		−	11.3017i		−	0.805210i	0.915374	−	0.402605i	$-0.131895\pi$
$198$		0			0
$199$			8.86775i			0.628618i	0.949321	+	0.314309i	$0.101773\pi$
$199$			8.86775i			0.628618i	−0.949321	+	0.314309i	$0.898227\pi$
$200$		0			0
$201$	7.18508	−	12.3875i	0.506796	−	0.873744i
$202$		0			0
$203$	13.3122	−	4.26918i	0.934335	−	0.299638i
$204$		0			0
$205$	5.16593			0.360804
$206$		0			0
$207$	−17.0688	+	9.76357i	−1.18636	+	0.678615i
$208$		0			0
$209$	−3.11259			−0.215302
$210$		0			0
$211$	21.5275			1.48201			0.741006	−	0.671499i	$-0.234349\pi$
$211$	21.5275			1.48201			0.741006	+	0.671499i	$0.234349\pi$
$212$		0			0
$213$	−7.86654	−	4.56282i	−0.539006	−	0.312639i
$214$		0			0
$215$	−2.48280			−0.169325
$216$		0			0
$217$	22.6523	−	7.26450i	1.53774	−	0.493147i
$218$		0			0
$219$	14.1159	+	8.18764i	0.953865	+	0.553269i
$220$		0			0
$221$			36.4137i			2.44945i
$222$		0			0
$223$		−	10.9108i		−	0.730640i	−0.930882	−	0.365320i	$-0.880960\pi$
$223$		−	10.9108i		−	0.730640i	0.930882	−	0.365320i	$-0.119040\pi$
$224$		0			0
$225$	−1.48956	−	2.60407i	−0.0993042	−	0.173605i
$226$		0			0
$227$	19.4610			1.29167			0.645835	−	0.763477i	$-0.276509\pi$
$227$	19.4610			1.29167			0.645835	+	0.763477i	$0.276509\pi$
$228$		0			0
$229$		−	18.8113i		−	1.24308i	−0.783381	−	0.621542i	$-0.786507\pi$
$229$		−	18.8113i		−	1.24308i	0.783381	−	0.621542i	$-0.213493\pi$
$230$		0			0
$231$	−0.529938	+	2.42359i	−0.0348674	+	0.159460i
$232$		0			0
$233$			7.01806i			0.459769i	0.973218	+	0.229884i	$0.0738348\pi$
$233$			7.01806i			0.459769i	−0.973218	+	0.229884i	$0.926165\pi$
$234$		0			0
$235$	−2.09597			−0.136726
$236$		0			0
$237$	−15.1111	+	26.0524i	−0.981575	+	1.69229i
$238$		0			0
$239$		−	8.60528i		−	0.556629i	−0.960490	−	0.278315i	$-0.910224\pi$
$239$		−	8.60528i		−	0.556629i	0.960490	−	0.278315i	$-0.0897758\pi$
$240$		0			0
$241$			12.4188i			0.799967i	0.916522	+	0.399984i	$0.130984\pi$
$241$			12.4188i			0.799967i	−0.916522	+	0.399984i	$0.869016\pi$
$242$		0			0
$243$	−7.65841	−	13.5775i	−0.491287	−	0.870998i
$244$		0			0
$245$	−5.69443	+	4.07106i	−0.363804	+	0.260090i
$246$		0			0
$247$	−33.6547			−2.14140
$248$		0			0
$249$	−12.9132	+	22.2631i	−0.818343	+	1.41087i
$250$		0			0
$251$	−7.77791			−0.490938			−0.245469	−	0.969405i	$-0.578942\pi$
$251$	−7.77791			−0.490938			−0.245469	+	0.969405i	$0.578942\pi$
$252$		0			0
$253$	3.54846			0.223090
$254$		0			0
$255$	−5.40614	+	9.32047i	−0.338545	+	0.583670i
$256$		0			0
$257$	−24.1332			−1.50539			−0.752694	−	0.658371i	$-0.771246\pi$
$257$	−24.1332			−1.50539			−0.752694	+	0.658371i	$0.771246\pi$
$258$		0			0
$259$	−5.92559	−	18.4772i	−0.368198	−	1.14812i
$260$		0			0
$261$	13.7598	−	7.87078i	0.851711	−	0.487189i
$262$		0			0
$263$		−	10.7824i		−	0.664874i	−0.943125	−	0.332437i	$-0.892129\pi$
$263$		−	10.7824i		−	0.664874i	0.943125	−	0.332437i	$-0.107871\pi$
$264$		0			0
$265$			9.75661i			0.599343i
$266$		0			0
$267$	−6.19115	+	10.6739i	−0.378892	+	0.653230i
$268$		0			0
$269$	−4.23229			−0.258047			−0.129024	−	0.991642i	$-0.541184\pi$
$269$	−4.23229			−0.258047			−0.129024	+	0.991642i	$0.541184\pi$
$270$		0			0
$271$		−	2.11109i		−	0.128240i	−0.997942	−	0.0641199i	$-0.979576\pi$
$271$		−	2.11109i		−	0.128240i	0.997942	−	0.0641199i	$-0.0204240\pi$
$272$		0			0
$273$	−5.72993	+	26.2049i	−0.346791	+	1.58599i
$274$		0			0
$275$			0.541366i			0.0326456i
$276$		0			0
$277$	−10.9024			−0.655062			−0.327531	−	0.944840i	$-0.606217\pi$
$277$	−10.9024			−0.655062			−0.327531	+	0.944840i	$0.606217\pi$
$278$		0			0
$279$	23.4139	−	13.3930i	1.40175	−	0.801820i
$280$		0			0
$281$			21.6151i			1.28945i	0.764415	+	0.644724i	$0.223028\pi$
$281$			21.6151i			1.28945i	−0.764415	+	0.644724i	$0.776972\pi$
$282$		0			0
$283$		−	13.9033i		−	0.826463i	−0.910626	−	0.413231i	$-0.864400\pi$
$283$		−	13.9033i		−	0.826463i	0.910626	−	0.413231i	$-0.135600\pi$
$284$		0			0
$285$	−8.61426	−	4.99652i	−0.510265	−	0.295968i
$286$		0			0
$287$	−4.17383	−	13.0149i	−0.246373	−	0.768244i
$288$		0			0
$289$	21.6991			1.27642
$290$		0			0
$291$	4.86366	+	2.82106i	0.285113	+	0.165373i
$292$		0			0
$293$	3.86654			0.225885			0.112943	−	0.993602i	$-0.463972\pi$
$293$	3.86654			0.225885			0.112943	+	0.993602i	$0.463972\pi$
$294$		0			0
$295$	−9.76727			−0.568672
$296$		0			0
$297$	−0.0169317	+	2.81297i	−0.000982476	+	0.163225i
$298$		0			0
$299$	38.3676			2.21885
$300$		0			0
$301$	2.00598	+	6.25508i	0.115623	+	0.360537i
$302$		0			0
$303$	13.0218	−	22.4502i	0.748081	−	1.28973i
$304$		0			0
$305$		−	0.433188i		−	0.0248043i
$306$		0			0
$307$			5.12531i			0.292517i	0.989246	+	0.146258i	$0.0467231\pi$
$307$			5.12531i			0.292517i	−0.989246	+	0.146258i	$0.953277\pi$
$308$		0			0
$309$	−19.4946	−	11.3074i	−1.10901	−	0.643257i
$310$		0			0
$311$	−18.8443			−1.06856			−0.534280	−	0.845307i	$-0.679417\pi$
$311$	−18.8443			−1.06856			−0.534280	+	0.845307i	$0.679417\pi$
$312$		0			0
$313$			9.36435i			0.529304i	0.964344	+	0.264652i	$0.0852571\pi$
$313$			9.36435i			0.529304i	−0.964344	+	0.264652i	$0.914743\pi$
$314$		0			0
$315$	−5.35712	+	5.85672i	−0.301840	+	0.329989i
$316$		0			0
$317$		−	19.1203i		−	1.07390i	−0.843613	−	0.536952i	$-0.819576\pi$
$317$		−	19.1203i		−	1.07390i	0.843613	−	0.536952i	$-0.180424\pi$
$318$		0			0
$319$	−2.86055			−0.160160
$320$		0			0
$321$	18.1358	+	10.5193i	1.01224	+	0.587129i
$322$		0			0
$323$			35.7669i			1.99013i
$324$		0			0
$325$			5.85348i			0.324693i
$326$		0			0
$327$	1.22673	−	2.11494i	0.0678381	−	0.116956i
$328$		0			0
$329$	1.69344	+	5.28051i	0.0933624	+	0.291124i
$330$		0			0
$331$	7.94896			0.436915			0.218457	−	0.975847i	$-0.429898\pi$
$331$	7.94896			0.436915			0.218457	+	0.975847i	$0.429898\pi$
$332$		0			0
$333$	−10.9246	−	19.0985i	−0.598663	−	1.04659i
$334$		0			0
$335$	−8.26790			−0.451724
$336$		0			0
$337$	−5.34820			−0.291335			−0.145667	−	0.989334i	$-0.546533\pi$
$337$	−5.34820			−0.291335			−0.145667	+	0.989334i	$0.546533\pi$
$338$		0			0
$339$	13.1808	+	7.64521i	0.715880	+	0.415231i
$340$		0			0
$341$	−4.86755			−0.263593
$342$		0			0
$343$	14.8573	+	11.0571i	0.802220	+	0.597029i
$344$		0			0
$345$	9.82057	+	5.69621i	0.528722	+	0.306674i
$346$		0			0
$347$		−	7.49471i		−	0.402337i	−0.979557	−	0.201169i	$-0.935526\pi$
$347$		−	7.49471i		−	0.402337i	0.979557	−	0.201169i	$-0.0644739\pi$
$348$		0			0
$349$		−	25.1595i		−	1.34676i	−0.739297	−	0.673379i	$-0.764842\pi$
$349$		−	25.1595i		−	1.34676i	0.739297	−	0.673379i	$-0.235158\pi$
$350$		0			0
$351$	−0.183073	+	30.4150i	−0.00977170	+	1.62343i
$352$		0			0
$353$	7.26591			0.386726			0.193363	−	0.981127i	$-0.438061\pi$
$353$	7.26591			0.386726			0.193363	+	0.981127i	$0.438061\pi$
$354$		0			0
$355$			5.25045i			0.278665i
$356$		0			0
$357$	27.8496	+	6.08955i	1.47396	+	0.322293i
$358$		0			0
$359$		−	5.72746i		−	0.302284i	−0.988512	−	0.151142i	$-0.951705\pi$
$359$		−	5.72746i		−	0.302284i	0.988512	−	0.151142i	$-0.0482950\pi$
$360$		0			0
$361$	−14.0569			−0.739838
$362$		0			0
$363$	−9.30467	+	16.0417i	−0.488369	+	0.841973i
$364$		0			0
$365$		−	9.42155i		−	0.493146i
$366$		0			0
$367$			13.2019i			0.689132i	0.938762	+	0.344566i	$0.111974\pi$
$367$			13.2019i			0.689132i	−0.938762	+	0.344566i	$0.888026\pi$
$368$		0			0
$369$	−7.69498	−	13.4525i	−0.400585	−	0.700308i
$370$		0			0
$371$	24.5805	−	7.88287i	1.27615	−	0.409258i
$372$		0			0
$373$	−37.5328			−1.94338			−0.971688	−	0.236266i	$-0.924076\pi$
$373$	−37.5328			−1.94338			−0.971688	+	0.236266i	$0.924076\pi$
$374$		0			0
$375$	−0.869033	+	1.49826i	−0.0448767	+	0.0773698i
$376$		0			0
$377$	−30.9295			−1.59295
$378$		0			0
$379$	6.85408			0.352070			0.176035	−	0.984384i	$-0.443673\pi$
$379$	6.85408			0.352070			0.176035	+	0.984384i	$0.443673\pi$
$380$		0			0
$381$	−0.749045	+	1.29139i	−0.0383748	+	0.0661601i
$382$		0			0
$383$	19.8927			1.01647			0.508234	−	0.861219i	$-0.330298\pi$
$383$	19.8927			1.01647			0.508234	+	0.861219i	$0.330298\pi$
$384$		0			0
$385$	1.36390	−	0.437398i	0.0695107	−	0.0222918i
$386$		0			0
$387$	3.69828	+	6.46539i	0.187994	+	0.328654i
$388$		0			0
$389$			25.4577i			1.29076i	0.763864	+	0.645378i	$0.223300\pi$
$389$			25.4577i			1.29076i	−0.763864	+	0.645378i	$0.776700\pi$
$390$		0			0
$391$		−	40.7756i		−	2.06211i
$392$		0			0
$393$	8.64112	−	14.8978i	0.435887	−	0.751492i
$394$		0			0
$395$	17.3885			0.874908
$396$		0			0
$397$			28.3280i			1.42174i	0.703322	+	0.710871i	$0.251699\pi$
$397$			28.3280i			1.42174i	−0.703322	+	0.710871i	$0.748301\pi$
$398$		0			0
$399$	−5.62815	+	25.7394i	−0.281760	+	1.28858i
$400$		0			0
$401$		−	37.1327i		−	1.85432i	−0.374666	−	0.927160i	$-0.622243\pi$
$401$		−	37.1327i		−	1.85432i	0.374666	−	0.927160i	$-0.377757\pi$
$402$		0			0
$403$	−52.6301			−2.62169
$404$		0			0
$405$	−4.56241	+	7.75786i	−0.226708	+	0.385491i
$406$		0			0
$407$			3.97042i			0.196806i
$408$		0			0
$409$			9.94224i			0.491612i	0.969319	+	0.245806i	$0.0790527\pi$
$409$			9.94224i			0.491612i	−0.969319	+	0.245806i	$0.920947\pi$
$410$		0			0
$411$	18.6291	+	10.8054i	0.918906	+	0.532992i
$412$		0			0
$413$	7.89149	+	24.6073i	0.388315	+	1.21085i
$414$		0			0
$415$	14.8593			0.729415
$416$		0			0
$417$	7.66857	+	4.44799i	0.375531	+	0.217819i
$418$		0			0
$419$	13.7361			0.671052			0.335526	−	0.942031i	$-0.391086\pi$
$419$	13.7361			0.671052			0.335526	+	0.942031i	$0.391086\pi$
$420$		0			0
$421$	−23.8838			−1.16402			−0.582012	−	0.813181i	$-0.697734\pi$
$421$	−23.8838			−1.16402			−0.582012	+	0.813181i	$0.697734\pi$
$422$		0			0
$423$	3.12207	+	5.45805i	0.151800	+	0.265379i
$424$		0			0
$425$	6.22086			0.301756
$426$		0			0
$427$	−1.09136	+	0.349995i	−0.0528146	+	0.0169374i
$428$		0			0
$429$	2.75386	−	4.74780i	0.132957	−	0.229226i
$430$		0			0
$431$		−	11.2319i		−	0.541022i	−0.962717	−	0.270511i	$-0.912807\pi$
$431$		−	11.2319i		−	0.541022i	0.962717	−	0.270511i	$-0.0871927\pi$
$432$		0			0
$433$		−	5.14457i		−	0.247232i	−0.992330	−	0.123616i	$-0.960551\pi$
$433$		−	5.14457i		−	0.247232i	0.992330	−	0.123616i	$-0.0394492\pi$
$434$		0			0
$435$	−7.91673	−	4.59193i	−0.379578	−	0.220166i
$436$		0			0
$437$	37.6861			1.80277
$438$		0			0
$439$			39.7058i			1.89506i	0.319672	+	0.947528i	$0.396427\pi$
$439$			39.7058i			1.89506i	−0.319672	+	0.947528i	$0.603573\pi$
$440$		0			0
$441$	19.0835	+	8.76461i	0.908740	+	0.417363i
$442$		0			0
$443$		−	13.7817i		−	0.654789i	−0.944888	−	0.327395i	$-0.893829\pi$
$443$		−	13.7817i		−	0.654789i	0.944888	−	0.327395i	$-0.106171\pi$
$444$		0			0
$445$	7.12418			0.337719
$446$		0			0
$447$	−5.61753	−	3.25833i	−0.265700	−	0.154114i
$448$		0			0
$449$		−	40.7969i		−	1.92532i	−0.270707	−	0.962662i	$-0.587257\pi$
$449$		−	40.7969i		−	1.92532i	0.270707	−	0.962662i	$-0.412743\pi$
$450$		0			0
$451$			2.79666i			0.131689i
$452$		0			0
$453$	2.89564	−	4.99224i	0.136049	−	0.234556i
$454$		0			0
$455$	14.7471	−	4.72933i	0.691353	−	0.221715i
$456$		0			0
$457$	10.4973			0.491044			0.245522	−	0.969391i	$-0.421041\pi$
$457$	10.4973			0.491044			0.245522	+	0.969391i	$0.421041\pi$
$458$		0			0
$459$	32.3240	+	0.194563i	1.50875	+	0.00908142i
$460$		0			0
$461$	−39.7172			−1.84982			−0.924908	−	0.380192i	$-0.875858\pi$
$461$	−39.7172			−1.84982			−0.924908	+	0.380192i	$0.875858\pi$
$462$		0			0
$463$	35.3647			1.64353			0.821767	−	0.569823i	$-0.192988\pi$
$463$	35.3647			1.64353			0.821767	+	0.569823i	$0.192988\pi$
$464$		0			0
$465$	−13.4712	−	7.81370i	−0.624713	−	0.362351i
$466$		0			0
$467$	−9.44968			−0.437279			−0.218640	−	0.975806i	$-0.570162\pi$
$467$	−9.44968			−0.437279			−0.218640	+	0.975806i	$0.570162\pi$
$468$		0			0
$469$	6.68007	+	20.8299i	0.308457	+	0.961834i
$470$		0			0
$471$	4.12847	+	2.39463i	0.190230	+	0.110339i
$472$		0			0
$473$		−	1.34410i		−	0.0618018i
$474$		0			0
$475$			5.74951i			0.263806i
$476$		0			0
$477$	25.4069	−	14.5331i	1.16330	−	0.665424i
$478$		0			0
$479$	−15.2364			−0.696168			−0.348084	−	0.937463i	$-0.613168\pi$
$479$	−15.2364			−0.696168			−0.348084	+	0.937463i	$0.613168\pi$
$480$		0			0
$481$			42.9299i			1.95744i
$482$		0			0
$483$	6.41630	−	29.3439i	0.291952	−	1.33519i
$484$		0			0
$485$		−	3.24620i		−	0.147403i
$486$		0			0
$487$	−24.3328			−1.10263			−0.551313	−	0.834299i	$-0.685873\pi$
$487$	−24.3328			−1.10263			−0.551313	+	0.834299i	$0.685873\pi$
$488$		0			0
$489$	−10.0922	+	17.3995i	−0.456385	+	0.786832i
$490$		0			0
$491$			2.14825i			0.0969494i	0.998824	+	0.0484747i	$0.0154360\pi$
$491$			2.14825i			0.0969494i	−0.998824	+	0.0484747i	$0.984564\pi$
$492$		0			0
$493$			32.8707i			1.48042i
$494$		0			0
$495$	1.40976	−	0.806398i	0.0633638	−	0.0362449i
$496$		0			0
$497$	13.2278	−	4.24211i	0.593348	−	0.190285i
$498$		0			0
$499$	6.22753			0.278783			0.139391	−	0.990237i	$-0.455485\pi$
$499$	6.22753			0.278783			0.139391	+	0.990237i	$0.455485\pi$
$500$		0			0
$501$	6.82762	−	11.7712i	0.305036	−	0.525898i
$502$		0			0
$503$	19.1213			0.852575			0.426288	−	0.904588i	$-0.359821\pi$
$503$	19.1213			0.852575			0.426288	+	0.904588i	$0.359821\pi$
$504$		0			0
$505$	−14.9842			−0.666788
$506$		0			0
$507$	18.4785	−	31.8579i	0.820658	−	1.41486i
$508$		0			0
$509$	−32.3463			−1.43372			−0.716861	−	0.697216i	$-0.754422\pi$
$509$	−32.3463			−1.43372			−0.716861	+	0.697216i	$0.754422\pi$
$510$		0			0
$511$	−23.7363	+	7.61216i	−1.05003	+	0.336742i
$512$		0			0
$513$	−0.179821	+	29.8748i	−0.00793929	+	1.31901i
$514$		0			0
$515$			13.0115i			0.573355i
$516$		0			0
$517$		−	1.13468i		−	0.0499033i
$518$		0			0
$519$	−5.06193	+	8.72704i	−0.222194	+	0.383075i
$520$		0			0
$521$	23.3045			1.02099			0.510494	−	0.859881i	$-0.329463\pi$
$521$	23.3045			1.02099			0.510494	+	0.859881i	$0.329463\pi$
$522$		0			0
$523$		−	8.56564i		−	0.374549i	−0.982308	−	0.187275i	$-0.940035\pi$
$523$		−	8.56564i		−	0.374549i	0.982308	−	0.187275i	$-0.0599654\pi$
$524$		0			0
$525$	4.47680	+	0.978892i	0.195384	+	0.0427223i
$526$		0			0
$527$			55.9333i			2.43649i
$528$		0			0
$529$	−19.9635			−0.867978
$530$		0			0
$531$	14.5490	+	25.4347i	0.631371	+	1.10377i
$532$		0			0
$533$			30.2387i			1.30978i
$534$		0			0
$535$		−	12.1046i		−	0.523326i
$536$		0			0
$537$	15.7559	+	9.13884i	0.679915	+	0.394370i
$538$		0			0
$539$	−2.20393	−	3.08277i	−0.0949300	−	0.132784i
$540$		0			0
$541$	−12.3978			−0.533021			−0.266511	−	0.963832i	$-0.585871\pi$
$541$	−12.3978			−0.533021			−0.266511	+	0.963832i	$0.585871\pi$
$542$		0			0
$543$	3.42219	+	1.98497i	0.146860	+	0.0851831i
$544$		0			0
$545$	−1.41160			−0.0604662
$546$		0			0
$547$	−16.9082			−0.722944			−0.361472	−	0.932383i	$-0.617726\pi$
$547$	−16.9082			−0.722944			−0.361472	+	0.932383i	$0.617726\pi$
$548$		0			0
$549$	−1.12805	+	0.645260i	−0.0481441	+	0.0275390i
$550$		0			0
$551$	−30.3802			−1.29424
$552$		0			0
$553$	−14.0490	−	43.8079i	−0.597426	−	1.86290i
$554$		0			0
$555$	−6.37356	+	10.9884i	−0.270543	+	0.466430i
$556$		0			0
$557$			45.7000i			1.93637i	0.250235	+	0.968185i	$0.419492\pi$
$557$			45.7000i			1.93637i	−0.250235	+	0.968185i	$0.580508\pi$
$558$		0			0
$559$		−	14.5330i		−	0.614681i
$560$		0			0
$561$	−5.04578	−	2.92670i	−0.213033	−	0.123565i
$562$		0			0
$563$	2.39076			0.100759			0.0503793	−	0.998730i	$-0.483957\pi$
$563$	2.39076			0.100759			0.0503793	+	0.998730i	$0.483957\pi$
$564$		0			0
$565$		−	8.79738i		−	0.370108i
$566$		0			0
$567$	23.2311	+	5.22640i	0.975615	+	0.219488i
$568$		0			0
$569$			21.3765i			0.896151i	0.893996	+	0.448075i	$0.147890\pi$
$569$			21.3765i			0.896151i	−0.893996	+	0.448075i	$0.852110\pi$
$570$		0			0
$571$	−10.2563			−0.429214			−0.214607	−	0.976701i	$-0.568847\pi$
$571$	−10.2563			−0.429214			−0.214607	+	0.976701i	$0.568847\pi$
$572$		0			0
$573$	−6.38856	−	3.70555i	−0.266886	−	0.154801i
$574$		0			0
$575$		−	6.55465i		−	0.273348i
$576$		0			0
$577$		−	15.6181i		−	0.650189i	−0.945682	−	0.325094i	$-0.894604\pi$
$577$		−	15.6181i		−	0.650189i	0.945682	−	0.325094i	$-0.105396\pi$
$578$		0			0
$579$	6.97645	−	12.0278i	0.289931	−	0.499857i
$580$		0			0
$581$	−12.0056	−	37.4361i	−0.498077	−	1.55311i
$582$		0			0
$583$	−5.28189			−0.218754
$584$		0			0
$585$	15.2429	−	8.71913i	0.630216	−	0.360492i
$586$		0			0
$587$	−34.1826			−1.41087			−0.705433	−	0.708777i	$-0.749247\pi$
$587$	−34.1826			−1.41087			−0.705433	+	0.708777i	$0.749247\pi$
$588$		0			0
$589$	−51.6953			−2.13007
$590$		0			0
$591$	16.9328	+	9.82152i	0.696524	+	0.404004i
$592$		0			0
$593$	13.3917			0.549932			0.274966	−	0.961454i	$-0.411333\pi$
$593$	13.3917			0.549932			0.274966	+	0.961454i	$0.411333\pi$
$594$		0			0
$595$	−5.02616	−	15.6726i	−0.206052	−	0.642516i
$596$		0			0
$597$	−13.2862	−	7.70637i	−0.543767	−	0.315401i
$598$		0			0
$599$			13.1768i			0.538388i	0.963086	+	0.269194i	$0.0867572\pi$
$599$			13.1768i			0.538388i	−0.963086	+	0.269194i	$0.913243\pi$
$600$		0			0
$601$		−	3.11187i		−	0.126936i	−0.997984	−	0.0634678i	$-0.979784\pi$
$601$		−	3.11187i		−	0.126936i	0.997984	−	0.0634678i	$-0.0202160\pi$
$602$		0			0
$603$	12.3156	+	21.5302i	0.501528	+	0.876778i
$604$		0			0
$605$	10.7069			0.435298
$606$		0			0
$607$		−	6.97466i		−	0.283093i	−0.989932	−	0.141546i	$-0.954793\pi$
$607$		−	6.97466i		−	0.283093i	0.989932	−	0.141546i	$-0.0452075\pi$
$608$		0			0
$609$	−5.17242	+	23.6552i	−0.209597	+	0.958558i
$610$		0			0
$611$		−	12.2687i		−	0.496338i
$612$		0			0
$613$	26.1829			1.05752			0.528759	−	0.848772i	$-0.322658\pi$
$613$	26.1829			1.05752			0.528759	+	0.848772i	$0.322658\pi$
$614$		0			0
$615$	−4.48937	+	7.73991i	−0.181029	+	0.312103i
$616$		0			0
$617$		−	15.1290i		−	0.609070i	−0.952501	−	0.304535i	$-0.901499\pi$
$617$		−	15.1290i		−	0.609070i	0.952501	−	0.304535i	$-0.0985010\pi$
$618$		0			0
$619$			8.76028i			0.352105i	0.984381	+	0.176053i	$0.0563329\pi$
$619$			8.76028i			0.352105i	−0.984381	+	0.176053i	$0.943667\pi$
$620$		0			0
$621$	0.205002	−	34.0584i	0.00822647	−	1.36672i
$622$		0			0
$623$	−5.75600	−	17.9484i	−0.230609	−	0.719089i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	2.70494	−	4.66347i	0.108025	−	0.186241i
$628$		0			0
$629$	45.6243			1.81916
$630$		0			0
$631$	4.79895			0.191043			0.0955215	−	0.995427i	$-0.469548\pi$
$631$	4.79895			0.191043			0.0955215	+	0.995427i	$0.469548\pi$
$632$		0			0
$633$	−18.7081	+	32.2537i	−0.743579	+	1.28197i
$634$		0			0
$635$	0.861929			0.0342046
$636$		0			0
$637$	−23.8299	−	33.3322i	−0.944174	−	1.32067i
$638$		0			0
$639$	13.6726	−	7.82087i	0.540878	−	0.309389i
$640$		0			0
$641$		−	0.836584i		−	0.0330431i	−0.999864	−	0.0165215i	$-0.994741\pi$
$641$		−	0.836584i		−	0.0330431i	0.999864	−	0.0165215i	$-0.00525921\pi$
$642$		0			0
$643$		−	8.91098i		−	0.351415i	−0.984442	−	0.175707i	$-0.943779\pi$
$643$		−	8.91098i		−	0.351415i	0.984442	−	0.175707i	$-0.0562212\pi$
$644$		0			0
$645$	2.15763	−	3.71987i	0.0849567	−	0.146470i
$646$		0			0
$647$	−25.8709			−1.01709			−0.508544	−	0.861036i	$-0.669816\pi$
$647$	−25.8709			−1.01709			−0.508544	+	0.861036i	$0.669816\pi$
$648$		0			0
$649$		−	5.28766i		−	0.207559i
$650$		0			0
$651$	−8.80146	+	40.2521i	−0.344957	+	1.57760i
$652$		0			0
$653$		−	18.2837i		−	0.715495i	−0.933818	−	0.357747i	$-0.883545\pi$
$653$		−	18.2837i		−	0.715495i	0.933818	−	0.357747i	$-0.116455\pi$
$654$		0			0
$655$	−9.94338			−0.388520
$656$		0			0
$657$	−24.5344	+	14.0340i	−0.957178	+	0.547518i
$658$		0			0
$659$		−	9.66460i		−	0.376479i	−0.982123	−	0.188240i	$-0.939722\pi$
$659$		−	9.66460i		−	0.376479i	0.982123	−	0.188240i	$-0.0602782\pi$
$660$		0			0
$661$		−	19.6253i		−	0.763336i	−0.924299	−	0.381668i	$-0.875350\pi$
$661$		−	19.6253i		−	0.763336i	0.924299	−	0.381668i	$-0.124650\pi$
$662$		0			0
$663$	−54.5572	−	31.6447i	−2.11883	−	1.22898i
$664$		0			0
$665$	14.4851	−	4.64533i	0.561710	−	0.180138i
$666$		0			0
$667$	34.6345			1.34105
$668$		0			0
$669$	16.3472	+	9.48184i	0.632019	+	0.366589i
$670$		0			0
$671$	0.234513			0.00905327
$672$		0			0
$673$	−11.8787			−0.457889			−0.228945	−	0.973439i	$-0.573527\pi$
$673$	−11.8787			−0.457889			−0.228945	+	0.973439i	$0.573527\pi$
$674$		0			0
$675$	5.19606	+	0.0312759i	0.199996	+	0.00120381i
$676$		0			0
$677$	−32.7301			−1.25792			−0.628961	−	0.777437i	$-0.716519\pi$
$677$	−32.7301			−1.25792			−0.628961	+	0.777437i	$0.716519\pi$
$678$		0			0
$679$	−8.17838	+	2.62278i	−0.313858	+	0.100653i
$680$		0			0
$681$	−16.9122	+	29.1576i	−0.648078	+	1.11732i
$682$		0			0
$683$			18.9837i			0.726391i	0.931713	+	0.363196i	$0.118314\pi$
$683$			18.9837i			0.726391i	−0.931713	+	0.363196i	$0.881686\pi$
$684$		0			0
$685$		−	12.4338i		−	0.475072i
$686$		0			0
$687$	28.1842	+	16.3476i	1.07529	+	0.623701i
$688$		0			0
$689$	−57.1101			−2.17572
$690$		0			0
$691$		−	25.8425i		−	0.983095i	−0.870851	−	0.491547i	$-0.836431\pi$
$691$		−	25.8425i		−	0.983095i	0.870851	−	0.491547i	$-0.163569\pi$
$692$		0			0
$693$	−3.17063	−	2.90016i	−0.120442	−	0.110168i
$694$		0			0
$695$		−	5.11832i		−	0.194149i
$696$		0			0
$697$	32.1366			1.21726
$698$		0			0
$699$	−10.5149	−	6.09893i	−0.397709	−	0.230683i
$700$		0			0
$701$		−	34.9210i		−	1.31895i	−0.751727	−	0.659475i	$-0.770779\pi$
$701$		−	34.9210i		−	1.31895i	0.751727	−	0.659475i	$-0.229221\pi$
$702$		0			0
$703$			42.1674i			1.59037i
$704$		0			0
$705$	1.82146	−	3.14030i	0.0686003	−	0.118271i
$706$		0			0
$707$	12.1065	+	37.7507i	0.455313	+	1.41976i
$708$		0			0
$709$	−2.52065			−0.0946648			−0.0473324	−	0.998879i	$-0.515072\pi$
$709$	−2.52065			−0.0946648			−0.0473324	+	0.998879i	$0.515072\pi$
$710$		0			0
$711$	−25.9012	−	45.2808i	−0.971371	−	1.69816i
$712$		0			0
$713$	58.9345			2.20712
$714$		0			0
$715$	−3.16887			−0.118509
$716$		0			0
$717$	12.8929	+	7.47827i	0.481496	+	0.279281i
$718$		0			0
$719$	22.1230			0.825049			0.412524	−	0.910947i	$-0.364647\pi$
$719$	22.1230			0.825049			0.412524	+	0.910947i	$0.364647\pi$
$720$		0			0
$721$	32.7808	−	10.5127i	1.22082	−	0.391512i
$722$		0			0
$723$	−18.6066	−	10.7924i	−0.691988	−	0.401373i
$724$		0			0
$725$			5.28395i			0.196241i
$726$		0			0
$727$			15.2639i			0.566107i	0.959104	+	0.283054i	$0.0913474\pi$
$727$			15.2639i			0.566107i	−0.959104	+	0.283054i	$0.908653\pi$
$728$		0			0
$729$	26.9980	+	0.325022i	0.999928	+	0.0120379i
$730$		0			0
$731$	−15.4451			−0.571259
$732$		0			0
$733$			18.2260i			0.673191i	0.941649	+	0.336596i	$0.109276\pi$
$733$			18.2260i			0.673191i	−0.941649	+	0.336596i	$0.890724\pi$
$734$		0			0
$735$	−1.15086	−	12.0696i	−0.0424499	−	0.445194i
$736$		0			0
$737$		−	4.47596i		−	0.164874i
$738$		0			0
$739$	3.67453			0.135170			0.0675849	−	0.997714i	$-0.478471\pi$
$739$	3.67453			0.135170			0.0675849	+	0.997714i	$0.478471\pi$
$740$		0			0
$741$	29.2470	−	50.4234i	1.07442	−	1.85235i
$742$		0			0
$743$		−	1.79110i		−	0.0657090i	−0.999460	−	0.0328545i	$-0.989540\pi$
$743$		−	1.79110i		−	0.0657090i	0.999460	−	0.0328545i	$-0.0104598\pi$
$744$		0			0
$745$			3.74937i			0.137366i
$746$		0			0
$747$	−22.1339	−	38.6947i	−0.809836	−	1.41577i
$748$		0			0
$749$	−30.4959	+	9.77992i	−1.11430	+	0.357351i
$750$		0			0
$751$	−39.0699			−1.42568			−0.712840	−	0.701327i	$-0.752591\pi$
$751$	−39.0699			−1.42568			−0.712840	+	0.701327i	$0.752591\pi$
$752$		0			0
$753$	6.75926	−	11.6533i	0.246321	−	0.424671i
$754$		0			0
$755$	−3.33203			−0.121265
$756$		0			0
$757$	42.9534			1.56117			0.780583	−	0.625052i	$-0.214922\pi$
$757$	42.9534			1.56117			0.780583	+	0.625052i	$0.214922\pi$
$758$		0			0
$759$	−3.08373	+	5.31652i	−0.111932	+	0.192977i
$760$		0			0
$761$	19.0235			0.689603			0.344801	−	0.938676i	$-0.387946\pi$
$761$	19.0235			0.689603			0.344801	+	0.938676i	$0.387946\pi$
$762$		0			0
$763$	1.14050	+	3.55633i	0.0412890	+	0.128748i
$764$		0			0
$765$	−9.26636	−	16.1996i	−0.335026	−	0.585697i
$766$		0			0
$767$		−	57.1725i		−	2.06438i
$768$		0			0
$769$		−	48.9415i		−	1.76488i	−0.470429	−	0.882438i	$-0.655901\pi$
$769$		−	48.9415i		−	1.76488i	0.470429	−	0.882438i	$-0.344099\pi$
$770$		0			0
$771$	20.9725	−	36.1578i	0.755308	−	1.30219i
$772$		0			0
$773$	14.2306			0.511840			0.255920	−	0.966698i	$-0.417622\pi$
$773$	14.2306			0.511840			0.255920	+	0.966698i	$0.417622\pi$
$774$		0			0
$775$			8.99125i			0.322975i
$776$		0			0
$777$	32.8332	+	7.17927i	1.17789	+	0.257555i
$778$		0			0
$779$			29.7016i			1.06417i
$780$		0			0
$781$	−2.84241			−0.101710
$782$		0			0
$783$	−0.165260	+	27.4557i	−0.00590592	+	0.981188i
$784$		0			0
$785$		−	2.75551i		−	0.0983484i
$786$		0			0
$787$		−	39.6962i		−	1.41502i	−0.706705	−	0.707508i	$-0.749819\pi$
$787$		−	39.6962i		−	1.41502i	0.706705	−	0.707508i	$-0.250181\pi$
$788$		0			0
$789$	16.1549	+	9.37030i	0.575130	+	0.333592i
$790$		0			0
$791$	−22.1638	+	7.10786i	−0.788055	+	0.252726i
$792$		0			0
$793$	2.53566			0.0900438
$794$		0			0
$795$	−14.6179	−	8.47881i	−0.518444	−	0.300713i
$796$		0			0
$797$	4.31625			0.152890			0.0764448	−	0.997074i	$-0.475643\pi$
$797$	4.31625			0.152890			0.0764448	+	0.997074i	$0.475643\pi$
$798$		0			0
$799$	−13.0387			−0.461276
$800$		0			0
$801$	−10.6119	−	18.5519i	−0.374954	−	0.655499i
$802$		0			0
$803$	5.10050			0.179993
$804$		0			0
$805$	−16.5136	+	5.29585i	−0.582027	+	0.186654i
$806$		0			0
$807$	3.67800	−	6.34107i	0.129472	−	0.223216i
$808$		0			0
$809$		−	24.0182i		−	0.844434i	−0.906495	−	0.422217i	$-0.861252\pi$
$809$		−	24.0182i		−	0.844434i	0.906495	−	0.422217i	$-0.138748\pi$
$810$		0			0
$811$			12.3193i			0.432589i	0.976328	+	0.216294i	$0.0693971\pi$
$811$			12.3193i			0.432589i	−0.976328	+	0.216294i	$0.930603\pi$
$812$		0			0
$813$	3.16297	+	1.83461i	0.110930	+	0.0643426i
$814$		0			0
$815$	11.6131			0.406791
$816$		0			0
$817$		−	14.2749i		−	0.499415i
$818$		0			0
$819$	−34.2822	−	31.3578i	−1.19792	−	1.09573i
$820$		0			0
$821$		−	7.25501i		−	0.253202i	−0.991954	−	0.126601i	$-0.959593\pi$
$821$		−	7.25501i		−	0.253202i	0.991954	−	0.126601i	$-0.0404067\pi$
$822$		0			0
$823$	53.1582			1.85298			0.926489	−	0.376321i	$-0.122811\pi$
$823$	53.1582			1.85298			0.926489	+	0.376321i	$0.122811\pi$
$824$		0			0
$825$	−0.811106	−	0.470465i	−0.0282391	−	0.0163795i
$826$		0			0
$827$			41.3798i			1.43892i	0.694536	+	0.719458i	$0.255610\pi$
$827$			41.3798i			1.43892i	−0.694536	+	0.719458i	$0.744390\pi$
$828$		0			0
$829$		−	1.37838i		−	0.0478730i	−0.999713	−	0.0239365i	$-0.992380\pi$
$829$		−	1.37838i		−	0.0478730i	0.999713	−	0.0239365i	$-0.00761996\pi$
$830$		0			0
$831$	9.47455	−	16.3346i	0.328668	−	0.566642i
$832$		0			0
$833$	−35.4242	+	25.3255i	−1.22738	+	0.877476i
$834$		0			0
$835$	−7.85657			−0.271888
$836$		0			0
$837$	−0.281209	+	46.7191i	−0.00972001	+	1.61485i
$838$		0			0
$839$	−0.647037			−0.0223382			−0.0111691	−	0.999938i	$-0.503555\pi$
$839$	−0.647037			−0.0223382			−0.0111691	+	0.999938i	$0.503555\pi$
$840$		0			0
$841$	1.07983			0.0372357
$842$		0			0
$843$	−32.3850	−	18.7842i	−1.11540	−	0.646963i
$844$		0			0
$845$	−21.2633			−0.731478
$846$		0			0
$847$	−8.65068	−	26.9747i	−0.297241	−	0.926861i
$848$		0			0
$849$	20.8307	+	12.0824i	0.714907	+	0.414667i
$850$		0			0
$851$		−	48.0724i		−	1.64790i
$852$		0			0
$853$		−	43.9133i		−	1.50356i	−0.659412	−	0.751782i	$-0.729195\pi$
$853$		−	43.9133i		−	1.50356i	0.659412	−	0.751782i	$-0.270805\pi$
$854$		0			0
$855$	14.9722	−	8.56426i	0.512037	−	0.292892i
$856$		0			0
$857$	27.6363			0.944039			0.472019	−	0.881588i	$-0.343525\pi$
$857$	27.6363			0.944039			0.472019	+	0.881588i	$0.343525\pi$
$858$		0			0
$859$			43.0054i			1.46733i	0.679513	+	0.733663i	$0.262191\pi$
$859$			43.0054i			1.46733i	−0.679513	+	0.733663i	$0.737809\pi$
$860$		0			0
$861$	23.1269	+	5.05689i	0.788161	+	0.172338i
$862$		0			0
$863$		−	5.18506i		−	0.176501i	−0.996098	−	0.0882507i	$-0.971872\pi$
$863$		−	5.18506i		−	0.176501i	0.996098	−	0.0882507i	$-0.0281277\pi$
$864$		0			0
$865$	5.82479			0.198049
$866$		0			0
$867$	−18.8573	+	32.5109i	−0.640426	+	1.10413i
$868$		0			0
$869$			9.41351i			0.319332i
$870$		0			0
$871$		−	48.3960i		−	1.63984i
$872$		0			0
$873$	−8.45336	+	4.83542i	−0.286103	+	0.163654i
$874$		0			0
$875$	−0.807952	−	2.51937i	−0.0273138	−	0.0851702i
$876$		0			0
$877$	31.8853			1.07669			0.538346	−	0.842724i	$-0.319049\pi$
$877$	31.8853			1.07669			0.538346	+	0.842724i	$0.319049\pi$
$878$		0			0
$879$	−3.36015	+	5.79307i	−0.113335	+	0.195396i
$880$		0			0
$881$	22.8971			0.771422			0.385711	−	0.922620i	$-0.373956\pi$
$881$	22.8971			0.771422			0.385711	+	0.922620i	$0.373956\pi$
$882$		0			0
$883$	0.815622			0.0274479			0.0137239	−	0.999906i	$-0.495631\pi$
$883$	0.815622			0.0274479			0.0137239	+	0.999906i	$0.495631\pi$
$884$		0			0
$885$	8.48808	−	14.6339i	0.285324	−	0.491913i
$886$		0			0
$887$	29.9931			1.00707			0.503535	−	0.863975i	$-0.332033\pi$
$887$	29.9931			1.00707			0.503535	+	0.863975i	$0.332033\pi$
$888$		0			0
$889$	−0.696398	−	2.17152i	−0.0233564	−	0.0728303i
$890$		0			0
$891$	−4.19984	−	2.46993i	−0.140700	−	0.0827458i
$892$		0			0
$893$		−	12.0508i		−	0.403264i
$894$		0			0
$895$		−	10.5161i		−	0.351515i
$896$		0			0
$897$	−33.3427	+	57.4845i	−1.11328	+	1.91935i
$898$		0			0
$899$	−47.5094			−1.58453
$900$		0			0
$901$			60.6945i			2.02203i
$902$		0			0
$903$	−11.1150	−	2.43039i	−0.369884	−	0.0808783i
$904$		0			0
$905$		−	2.28411i		−	0.0759264i
$906$		0			0
$907$	34.0918			1.13200			0.566000	−	0.824405i	$-0.308490\pi$
$907$	34.0918			1.13200			0.566000	+	0.824405i	$0.308490\pi$
$908$		0			0
$909$	22.3199	+	39.0200i	0.740305	+	1.29421i
$910$		0			0
$911$		−	14.3838i		−	0.476558i	−0.971197	−	0.238279i	$-0.923417\pi$
$911$		−	14.3838i		−	0.476558i	0.971197	−	0.238279i	$-0.0765832\pi$
$912$		0			0
$913$			8.04432i			0.266228i
$914$		0			0
$915$	0.649028	+	0.376454i	0.0214562	+	0.0124452i
$916$		0			0
$917$	8.03377	+	25.0510i	0.265299	+	0.827258i
$918$		0			0
$919$	32.0149			1.05608			0.528038	−	0.849221i	$-0.322928\pi$
$919$	32.0149			1.05608			0.528038	+	0.849221i	$0.322928\pi$
$920$		0			0
$921$	−7.67904	−	4.45406i	−0.253033	−	0.146766i
$922$		0			0
$923$	−30.7334			−1.01160
$924$		0			0
$925$	7.33408			0.241143
$926$		0			0
$927$	33.8829	−	19.3814i	1.11286	−	0.636570i
$928$		0			0
$929$	−19.3260			−0.634064			−0.317032	−	0.948415i	$-0.602686\pi$
$929$	−19.3260			−0.634064			−0.317032	+	0.948415i	$0.602686\pi$
$930$		0			0
$931$	−23.4066	−	32.7402i	−0.767121	−	1.07302i
$932$		0			0
$933$	16.3763	−	28.2336i	0.536136	−	0.924327i
$934$		0			0
$935$			3.36776i			0.110138i
$936$		0			0
$937$		−	7.81494i		−	0.255303i	−0.991819	−	0.127651i	$-0.959256\pi$
$937$		−	7.81494i		−	0.255303i	0.991819	−	0.127651i	$-0.0407439\pi$
$938$		0			0
$939$	−14.0302	−	8.13793i	−0.457859	−	0.265571i
$940$		0			0
$941$	−5.16553			−0.168391			−0.0841957	−	0.996449i	$-0.526832\pi$
$941$	−5.16553			−0.168391			−0.0841957	+	0.996449i	$0.526832\pi$
$942$		0			0
$943$		−	33.8609i		−	1.10266i
$944$		0			0
$945$	−4.11937	−	13.1161i	−0.134003	−	0.426665i
$946$		0			0
$947$		−	8.58998i		−	0.279137i	−0.990212	−	0.139568i	$-0.955428\pi$
$947$		−	8.58998i		−	0.279137i	0.990212	−	0.139568i	$-0.0445715\pi$
$948$		0			0
$949$	55.1489			1.79021
$950$		0			0
$951$	28.6472	+	16.6162i	0.928949	+	0.538817i
$952$		0			0
$953$		−	55.0367i		−	1.78281i	−0.453205	−	0.891406i	$-0.649719\pi$
$953$		−	55.0367i		−	1.78281i	0.453205	−	0.891406i	$-0.350281\pi$
$954$		0			0
$955$			4.26399i			0.137979i
$956$		0			0
$957$	2.48591	−	4.28585i	0.0803582	−	0.138542i
$958$		0			0
$959$	−31.3254	+	10.0459i	−1.01155	+	0.324400i
$960$		0			0
$961$	−49.8426			−1.60783
$962$		0			0
$963$	−31.5212	+	18.0305i	−1.01576	+	0.581026i
$964$		0			0
$965$	−8.02782			−0.258425
$966$		0			0
$967$	−44.3877			−1.42741			−0.713707	−	0.700445i	$-0.752985\pi$
$967$	−44.3877			−1.42741			−0.713707	+	0.700445i	$0.752985\pi$
$968$		0			0
$969$	−53.5882	−	31.0827i	−1.72150	−	0.998519i
$970$		0			0
$971$	12.7190			0.408172			0.204086	−	0.978953i	$-0.434578\pi$
$971$	12.7190			0.408172			0.204086	+	0.978953i	$0.434578\pi$
$972$		0			0
$973$	−12.8949	+	4.13536i	−0.413392	+	0.132573i
$974$		0			0
$975$	−8.77004	−	5.08687i	−0.280866	−	0.162910i
$976$		0			0
$977$		−	33.1021i		−	1.05903i	−0.848300	−	0.529516i	$-0.822374\pi$
$977$		−	33.1021i		−	1.05903i	0.848300	−	0.529516i	$-0.177626\pi$
$978$		0			0
$979$			3.85679i			0.123263i
$980$		0			0
$981$	2.10266	+	3.67590i	0.0671329	+	0.117363i
$982$		0			0
$983$	−33.7287			−1.07578			−0.537889	−	0.843016i	$-0.680778\pi$
$983$	−33.7287			−1.07578			−0.537889	+	0.843016i	$0.680778\pi$
$984$		0			0
$985$		−	11.3017i		−	0.360101i
$986$		0			0
$987$	−9.38322	−	2.05172i	−0.298671	−	0.0653071i
$988$		0			0
$989$			16.2739i			0.517479i
$990$		0			0
$991$	24.4912			0.777990			0.388995	−	0.921240i	$-0.372822\pi$
$991$	24.4912			0.777990			0.388995	+	0.921240i	$0.372822\pi$
$992$		0			0
$993$	−6.90791	+	11.9096i	−0.219216	+	0.377940i
$994$		0			0
$995$			8.86775i			0.281127i
$996$		0			0
$997$		−	43.8446i		−	1.38857i	−0.719700	−	0.694286i	$-0.755720\pi$
$997$		−	43.8446i		−	1.38857i	0.719700	−	0.694286i	$-0.244280\pi$
$998$		0			0
$999$	38.1083	+	0.229380i	1.20569	+	0.00725726i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.f.b.41.6	yes	16
3.2	odd	2		840.2.f.a.41.12	yes	16
4.3	odd	2		1680.2.f.l.881.11		16
7.6	odd	2		840.2.f.a.41.11	✓	16
12.11	even	2		1680.2.f.k.881.5		16
21.20	even	2	inner	840.2.f.b.41.5	yes	16
28.27	even	2		1680.2.f.k.881.6		16
84.83	odd	2		1680.2.f.l.881.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.f.a.41.11	✓	16	7.6	odd	2
840.2.f.a.41.12	yes	16	3.2	odd	2
840.2.f.b.41.5	yes	16	21.20	even	2	inner
840.2.f.b.41.6	yes	16	1.1	even	1	trivial
1680.2.f.k.881.5		16	12.11	even	2
1680.2.f.k.881.6		16	28.27	even	2
1680.2.f.l.881.11		16	4.3	odd	2
1680.2.f.l.881.12		16	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.869033 + 1.49826i) q^{3} +1.00000 q^{5} +(-0.807952 - 2.51937i) q^{7} +(-1.48956 - 2.60407i) q^{9} +O(q^{10})\)	\(q+(-0.869033 + 1.49826i) q^{3} +1.00000 q^{5} +(-0.807952 - 2.51937i) q^{7} +(-1.48956 - 2.60407i) q^{9} +0.541366i q^{11} +5.85348i q^{13} +(-0.869033 + 1.49826i) q^{15} +6.22086 q^{17} +5.74951i q^{19} +(4.47680 + 0.978892i) q^{21} -6.55465i q^{23} +1.00000 q^{25} +(5.19606 + 0.0312759i) q^{27} +5.28395i q^{29} +8.99125i q^{31} +(-0.811106 - 0.470465i) q^{33} +(-0.807952 - 2.51937i) q^{35} +7.33408 q^{37} +(-8.77004 - 5.08687i) q^{39} +5.16593 q^{41} -2.48280 q^{43} +(-1.48956 - 2.60407i) q^{45} -2.09597 q^{47} +(-5.69443 + 4.07106i) q^{49} +(-5.40614 + 9.32047i) q^{51} +9.75661i q^{53} +0.541366i q^{55} +(-8.61426 - 4.99652i) q^{57} -9.76727 q^{59} -0.433188i q^{61} +(-5.35712 + 5.85672i) q^{63} +5.85348i q^{65} -8.26790 q^{67} +(9.82057 + 5.69621i) q^{69} +5.25045i q^{71} -9.42155i q^{73} +(-0.869033 + 1.49826i) q^{75} +(1.36390 - 0.437398i) q^{77} +17.3885 q^{79} +(-4.56241 + 7.75786i) q^{81} +14.8593 q^{83} +6.22086 q^{85} +(-7.91673 - 4.59193i) q^{87} +7.12418 q^{89} +(14.7471 - 4.72933i) q^{91} +(-13.4712 - 7.81370i) q^{93} +5.74951i q^{95} -3.24620i q^{97} +(1.40976 - 0.806398i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 16 * q + 16 * q^5 + 2 * q^7 - 2 * q^9	\( 16 q + 16 q^{5} + 2 q^{7} - 2 q^{9} + 10 q^{21} + 16 q^{25} + 6 q^{27} + 6 q^{33} + 2 q^{35} + 12 q^{37} + 6 q^{39} + 32 q^{41} + 32 q^{43} - 2 q^{45} + 4 q^{47} - 4 q^{49} + 6 q^{51} - 24 q^{59} - 24 q^{63} + 8 q^{69} - 32 q^{77} - 4 q^{79} - 6 q^{81} + 20 q^{83} + 6 q^{87} - 24 q^{89} + 20 q^{91} - 32 q^{93} - 58 q^{99}+O(q^{100}) \) 16 * q + 16 * q^5 + 2 * q^7 - 2 * q^9 + 10 * q^21 + 16 * q^25 + 6 * q^27 + 6 * q^33 + 2 * q^35 + 12 * q^37 + 6 * q^39 + 32 * q^41 + 32 * q^43 - 2 * q^45 + 4 * q^47 - 4 * q^49 + 6 * q^51 - 24 * q^59 - 24 * q^63 + 8 * q^69 - 32 * q^77 - 4 * q^79 - 6 * q^81 + 20 * q^83 + 6 * q^87 - 24 * q^89 + 20 * q^91 - 32 * q^93 - 58 * q^99

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