LMFDB - Embedded newform 840.2.t.d.169.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [840,2,Mod(169,840)]

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, 0, 1, 0]))

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

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

chi := DirichletCharacter("840.169");

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.t (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:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.1
Root			$-0.854638 + 0.854638i$ of defining polynomial
Character	$\chi$	$=$	840.169
Dual form			840.2.t.d.169.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -3.26180 q^{11} -0.340173i q^{13} +(-0.539189 + 2.17009i) q^{15} +5.75872i q^{17} +6.49693 q^{19} +1.00000 q^{21} +8.49693i q^{23} +(4.41855 + 2.34017i) q^{25} +1.00000i q^{27} +2.00000 q^{29} -8.34017 q^{31} +3.26180i q^{33} +(0.539189 - 2.17009i) q^{35} -6.15676i q^{37} -0.340173 q^{39} +0.340173 q^{41} +8.68035i q^{43} +(2.17009 + 0.539189i) q^{45} -1.00000 q^{49} +5.75872 q^{51} +8.34017i q^{53} +(7.07838 + 1.75872i) q^{55} -6.49693i q^{57} -6.83710 q^{59} +15.3607 q^{61} -1.00000i q^{63} +(-0.183417 + 0.738205i) q^{65} +14.8371i q^{67} +8.49693 q^{69} -15.9421 q^{71} +1.50307i q^{73} +(2.34017 - 4.41855i) q^{75} -3.26180i q^{77} -8.68035 q^{79} +1.00000 q^{81} -6.83710i q^{83} +(3.10504 - 12.4969i) q^{85} -2.00000i q^{87} +15.1773 q^{89} +0.340173 q^{91} +8.34017i q^{93} +(-14.0989 - 3.50307i) q^{95} -6.49693i q^{97} +3.26180 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^5 - 6 * q^9	$ 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{21} - 2 q^{25} + 12 q^{29} - 28 q^{31} + 20 q^{39} - 20 q^{41} + 2 q^{45} - 6 q^{49} - 16 q^{51} + 36 q^{55} + 16 q^{59} + 4 q^{61} + 8 q^{65} + 16 q^{69} - 36 q^{71} - 8 q^{75} - 8 q^{79} + 6 q^{81} + 16 q^{85} + 12 q^{89} - 20 q^{91} - 12 q^{95} + 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^5 - 6 * q^9 - 4 * q^11 + 4 * q^19 + 6 * q^21 - 2 * q^25 + 12 * q^29 - 28 * q^31 + 20 * q^39 - 20 * q^41 + 2 * q^45 - 6 * q^49 - 16 * q^51 + 36 * q^55 + 16 * q^59 + 4 * q^61 + 8 * q^65 + 16 * q^69 - 36 * q^71 - 8 * q^75 - 8 * q^79 + 6 * q^81 + 16 * q^85 + 12 * q^89 - 20 * q^91 - 12 * q^95 + 4 * 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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−2.17009	−	0.539189i	−0.970492	−	0.241133i
$5$	−2.17009	−	0.539189i	−0.970492	−	0.241133i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−3.26180			−0.983468			−0.491734	−	0.870745i	$-0.663637\pi$
$11$	−3.26180			−0.983468			−0.491734	+	0.870745i	$0.663637\pi$
$12$		0			0
$13$		−	0.340173i		−	0.0943470i	−0.998887	−	0.0471735i	$-0.984979\pi$
$13$		−	0.340173i		−	0.0943470i	0.998887	−	0.0471735i	$-0.0150214\pi$
$14$		0			0
$15$	−0.539189	+	2.17009i	−0.139218	+	0.560314i
$16$		0			0
$17$			5.75872i			1.39670i	0.715759	+	0.698348i	$0.246081\pi$
$17$			5.75872i			1.39670i	−0.715759	+	0.698348i	$0.753919\pi$
$18$		0			0
$19$	6.49693			1.49050			0.745249	−	0.666786i	$-0.232331\pi$
$19$	6.49693			1.49050			0.745249	+	0.666786i	$0.232331\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$			8.49693i			1.77173i	0.463941	+	0.885866i	$0.346435\pi$
$23$			8.49693i			1.77173i	−0.463941	+	0.885866i	$0.653565\pi$
$24$		0			0
$25$	4.41855	+	2.34017i	0.883710	+	0.468035i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	−8.34017			−1.49794			−0.748970	−	0.662604i	$-0.769451\pi$
$31$	−8.34017			−1.49794			−0.748970	+	0.662604i	$0.769451\pi$
$32$		0			0
$33$			3.26180i			0.567806i
$34$		0			0
$35$	0.539189	−	2.17009i	0.0911396	−	0.366812i
$36$		0			0
$37$		−	6.15676i		−	1.01216i	−0.862485	−	0.506082i	$-0.831093\pi$
$37$		−	6.15676i		−	1.01216i	0.862485	−	0.506082i	$-0.168907\pi$
$38$		0			0
$39$	−0.340173			−0.0544713
$40$		0			0
$41$	0.340173			0.0531261			0.0265630	−	0.999647i	$-0.491544\pi$
$41$	0.340173			0.0531261			0.0265630	+	0.999647i	$0.491544\pi$
$42$		0			0
$43$			8.68035i			1.32374i	0.749618	+	0.661870i	$0.230237\pi$
$43$			8.68035i			1.32374i	−0.749618	+	0.661870i	$0.769763\pi$
$44$		0			0
$45$	2.17009	+	0.539189i	0.323497	+	0.0803775i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	5.75872			0.806383
$52$		0			0
$53$			8.34017i			1.14561i	0.819691	+	0.572805i	$0.194145\pi$
$53$			8.34017i			1.14561i	−0.819691	+	0.572805i	$0.805855\pi$
$54$		0			0
$55$	7.07838	+	1.75872i	0.954448	+	0.237146i
$56$		0			0
$57$		−	6.49693i		−	0.860539i
$58$		0			0
$59$	−6.83710			−0.890115			−0.445057	−	0.895502i	$-0.646817\pi$
$59$	−6.83710			−0.890115			−0.445057	+	0.895502i	$0.646817\pi$
$60$		0			0
$61$	15.3607			1.96674			0.983368	−	0.181627i	$-0.0581363\pi$
$61$	15.3607			1.96674			0.983368	+	0.181627i	$0.0581363\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	−0.183417	+	0.738205i	−0.0227501	+	0.0915630i
$66$		0			0
$67$			14.8371i			1.81264i	0.422592	+	0.906320i	$0.361120\pi$
$67$			14.8371i			1.81264i	−0.422592	+	0.906320i	$0.638880\pi$
$68$		0			0
$69$	8.49693			1.02291
$70$		0			0
$71$	−15.9421			−1.89198			−0.945992	−	0.324190i	$-0.894908\pi$
$71$	−15.9421			−1.89198			−0.945992	+	0.324190i	$0.894908\pi$
$72$		0			0
$73$			1.50307i			0.175921i	0.996124	+	0.0879606i	$0.0280350\pi$
$73$			1.50307i			0.175921i	−0.996124	+	0.0879606i	$0.971965\pi$
$74$		0			0
$75$	2.34017	−	4.41855i	0.270220	−	0.510210i
$76$		0			0
$77$		−	3.26180i		−	0.371716i
$78$		0			0
$79$	−8.68035			−0.976615			−0.488308	−	0.872672i	$-0.662386\pi$
$79$	−8.68035			−0.976615			−0.488308	+	0.872672i	$0.662386\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	6.83710i		−	0.750469i	−0.926930	−	0.375235i	$-0.877562\pi$
$83$		−	6.83710i		−	0.750469i	0.926930	−	0.375235i	$-0.122438\pi$
$84$		0			0
$85$	3.10504	−	12.4969i	0.336789	−	1.35548i
$86$		0			0
$87$		−	2.00000i		−	0.214423i
$88$		0			0
$89$	15.1773			1.60879			0.804394	−	0.594096i	$-0.202490\pi$
$89$	15.1773			1.60879			0.804394	+	0.594096i	$0.202490\pi$
$90$		0			0
$91$	0.340173			0.0356598
$92$		0			0
$93$			8.34017i			0.864836i
$94$		0			0
$95$	−14.0989	−	3.50307i	−1.44652	−	0.359408i
$96$		0			0
$97$		−	6.49693i		−	0.659663i	−0.944040	−	0.329832i	$-0.893008\pi$
$97$		−	6.49693i		−	0.659663i	0.944040	−	0.329832i	$-0.106992\pi$
$98$		0			0
$99$	3.26180			0.327823
$100$		0			0
$101$	−2.18342			−0.217258			−0.108629	−	0.994082i	$-0.534646\pi$
$101$	−2.18342			−0.217258			−0.108629	+	0.994082i	$0.534646\pi$
$102$		0			0
$103$			5.84324i			0.575752i	0.957668	+	0.287876i	$0.0929491\pi$
$103$			5.84324i			0.575752i	−0.957668	+	0.287876i	$0.907051\pi$
$104$		0			0
$105$	−2.17009	−	0.539189i	−0.211779	−	0.0526194i
$106$		0			0
$107$		−	8.18342i		−	0.791121i	−0.918440	−	0.395560i	$-0.870550\pi$
$107$		−	8.18342i		−	0.791121i	0.918440	−	0.395560i	$-0.129450\pi$
$108$		0			0
$109$	−16.8371			−1.61270			−0.806351	−	0.591437i	$-0.798561\pi$
$109$	−16.8371			−1.61270			−0.806351	+	0.591437i	$0.798561\pi$
$110$		0			0
$111$	−6.15676			−0.584373
$112$		0			0
$113$			13.0205i			1.22487i	0.790522	+	0.612434i	$0.209809\pi$
$113$			13.0205i			1.22487i	−0.790522	+	0.612434i	$0.790191\pi$
$114$		0			0
$115$	4.58145	−	18.4391i	0.427222	−	1.71945i
$116$		0			0
$117$			0.340173i			0.0314490i
$118$		0			0
$119$	−5.75872			−0.527901
$120$		0			0
$121$	−0.360692			−0.0327902
$122$		0			0
$123$		−	0.340173i		−	0.0306724i
$124$		0			0
$125$	−8.32684	−	7.46081i	−0.744775	−	0.667315i
$126$		0			0
$127$		−	1.84324i		−	0.163562i	−0.996650	−	0.0817808i	$-0.973939\pi$
$127$		−	1.84324i		−	0.163562i	0.996650	−	0.0817808i	$-0.0260607\pi$
$128$		0			0
$129$	8.68035			0.764262
$130$		0			0
$131$	−0.313511			−0.0273916			−0.0136958	−	0.999906i	$-0.504360\pi$
$131$	−0.313511			−0.0273916			−0.0136958	+	0.999906i	$0.504360\pi$
$132$		0			0
$133$			6.49693i			0.563355i
$134$		0			0
$135$	0.539189	−	2.17009i	0.0464060	−	0.186771i
$136$		0			0
$137$		−	2.18342i		−	0.186542i	−0.995641	−	0.0932710i	$-0.970268\pi$
$137$		−	2.18342i		−	0.186542i	0.995641	−	0.0932710i	$-0.0297323\pi$
$138$		0			0
$139$	1.02052			0.0865593			0.0432796	−	0.999063i	$-0.486219\pi$
$139$	1.02052			0.0865593			0.0432796	+	0.999063i	$0.486219\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1.10957i			0.0927873i
$144$		0			0
$145$	−4.34017	−	1.07838i	−0.360432	−	0.0895544i
$146$		0			0
$147$			1.00000i			0.0824786i
$148$		0			0
$149$	−14.6803			−1.20266			−0.601330	−	0.799000i	$-0.705362\pi$
$149$	−14.6803			−1.20266			−0.601330	+	0.799000i	$0.705362\pi$
$150$		0			0
$151$	2.15676			0.175514			0.0877571	−	0.996142i	$-0.472030\pi$
$151$	2.15676			0.175514			0.0877571	+	0.996142i	$0.472030\pi$
$152$		0			0
$153$		−	5.75872i		−	0.465565i
$154$		0			0
$155$	18.0989	+	4.49693i	1.45374	+	0.361202i
$156$		0			0
$157$		−	6.18342i		−	0.493490i	−0.969080	−	0.246745i	$-0.920639\pi$
$157$		−	6.18342i		−	0.493490i	0.969080	−	0.246745i	$-0.0793611\pi$
$158$		0			0
$159$	8.34017			0.661419
$160$		0			0
$161$	−8.49693			−0.669652
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$	1.75872	−	7.07838i	0.136916	−	0.551051i
$166$		0			0
$167$			1.47641i			0.114248i	0.998367	+	0.0571240i	$0.0181930\pi$
$167$			1.47641i			0.114248i	−0.998367	+	0.0571240i	$0.981807\pi$
$168$		0			0
$169$	12.8843			0.991099
$170$		0			0
$171$	−6.49693			−0.496833
$172$		0			0
$173$		−	1.75872i		−	0.133713i	−0.997763	−	0.0668566i	$-0.978703\pi$
$173$		−	1.75872i		−	0.133713i	0.997763	−	0.0668566i	$-0.0212970\pi$
$174$		0			0
$175$	−2.34017	+	4.41855i	−0.176900	+	0.334011i
$176$		0			0
$177$			6.83710i			0.513908i
$178$		0			0
$179$	−0.424694			−0.0317431			−0.0158716	−	0.999874i	$-0.505052\pi$
$179$	−0.424694			−0.0317431			−0.0158716	+	0.999874i	$0.505052\pi$
$180$		0			0
$181$	10.3668			0.770561			0.385280	−	0.922800i	$-0.374105\pi$
$181$	10.3668			0.770561			0.385280	+	0.922800i	$0.374105\pi$
$182$		0			0
$183$		−	15.3607i		−	1.13550i
$184$		0			0
$185$	−3.31965	+	13.3607i	−0.244066	+	0.982298i
$186$		0			0
$187$		−	18.7838i		−	1.37361i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	−21.7321			−1.57248			−0.786238	−	0.617923i	$-0.787974\pi$
$191$	−21.7321			−1.57248			−0.786238	+	0.617923i	$0.787974\pi$
$192$		0			0
$193$		−	8.36683i		−	0.602258i	−0.953583	−	0.301129i	$-0.902637\pi$
$193$		−	8.36683i		−	0.602258i	0.953583	−	0.301129i	$-0.0973635\pi$
$194$		0			0
$195$	0.738205	+	0.183417i	0.0528639	+	0.0131348i
$196$		0			0
$197$		−	13.3340i		−	0.950010i	−0.879983	−	0.475005i	$-0.842446\pi$
$197$		−	13.3340i		−	0.950010i	0.879983	−	0.475005i	$-0.157554\pi$
$198$		0			0
$199$	−6.49693			−0.460555			−0.230278	−	0.973125i	$-0.573963\pi$
$199$	−6.49693			−0.460555			−0.230278	+	0.973125i	$0.573963\pi$
$200$		0			0
$201$	14.8371			1.04653
$202$		0			0
$203$			2.00000i			0.140372i
$204$		0			0
$205$	−0.738205	−	0.183417i	−0.0515585	−	0.0128104i
$206$		0			0
$207$		−	8.49693i		−	0.590577i
$208$		0			0
$209$	−21.1917			−1.46586
$210$		0			0
$211$	6.83710			0.470685			0.235343	−	0.971912i	$-0.424379\pi$
$211$	6.83710			0.470685			0.235343	+	0.971912i	$0.424379\pi$
$212$		0			0
$213$			15.9421i			1.09234i
$214$		0			0
$215$	4.68035	−	18.8371i	0.319197	−	1.28468i
$216$		0			0
$217$		−	8.34017i		−	0.566168i
$218$		0			0
$219$	1.50307			0.101568
$220$		0			0
$221$	1.95896			0.131774
$222$		0			0
$223$			17.3607i			1.16256i	0.813704	+	0.581279i	$0.197447\pi$
$223$			17.3607i			1.16256i	−0.813704	+	0.581279i	$0.802553\pi$
$224$		0			0
$225$	−4.41855	−	2.34017i	−0.294570	−	0.156012i
$226$		0			0
$227$		−	10.1568i		−	0.674128i	−0.941482	−	0.337064i	$-0.890566\pi$
$227$		−	10.1568i		−	0.674128i	0.941482	−	0.337064i	$-0.109434\pi$
$228$		0			0
$229$	−14.9939			−0.990822			−0.495411	−	0.868659i	$-0.664983\pi$
$229$	−14.9939			−0.990822			−0.495411	+	0.868659i	$0.664983\pi$
$230$		0			0
$231$	−3.26180			−0.214610
$232$		0			0
$233$		−	11.5441i		−	0.756280i	−0.925748	−	0.378140i	$-0.876564\pi$
$233$		−	11.5441i		−	0.756280i	0.925748	−	0.378140i	$-0.123436\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			8.68035i			0.563849i
$238$		0			0
$239$	10.8950			0.704736			0.352368	−	0.935861i	$-0.385376\pi$
$239$	10.8950			0.704736			0.352368	+	0.935861i	$0.385376\pi$
$240$		0			0
$241$	−6.68035			−0.430319			−0.215159	−	0.976579i	$-0.569027\pi$
$241$	−6.68035			−0.430319			−0.215159	+	0.976579i	$0.569027\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	2.17009	+	0.539189i	0.138642	+	0.0344475i
$246$		0			0
$247$		−	2.21008i		−	0.140624i
$248$		0			0
$249$	−6.83710			−0.433284
$250$		0			0
$251$	24.1978			1.52735			0.763676	−	0.645600i	$-0.223393\pi$
$251$	24.1978			1.52735			0.763676	+	0.645600i	$0.223393\pi$
$252$		0			0
$253$		−	27.7152i		−	1.74244i
$254$		0			0
$255$	−12.4969	−	3.10504i	−0.782588	−	0.194445i
$256$		0			0
$257$		−	15.2351i		−	0.950342i	−0.879894	−	0.475171i	$-0.842386\pi$
$257$		−	15.2351i		−	0.950342i	0.879894	−	0.475171i	$-0.157614\pi$
$258$		0			0
$259$	6.15676			0.382562
$260$		0			0
$261$	−2.00000			−0.123797
$262$		0			0
$263$			29.1773i			1.79915i	0.436769	+	0.899574i	$0.356123\pi$
$263$			29.1773i			1.79915i	−0.436769	+	0.899574i	$0.643877\pi$
$264$		0			0
$265$	4.49693	−	18.0989i	0.276244	−	1.11181i
$266$		0			0
$267$		−	15.1773i		−	0.928834i
$268$		0			0
$269$	6.13009			0.373758			0.186879	−	0.982383i	$-0.440163\pi$
$269$	6.13009			0.373758			0.186879	+	0.982383i	$0.440163\pi$
$270$		0			0
$271$	25.7009			1.56122			0.780608	−	0.625021i	$-0.214909\pi$
$271$	25.7009			1.56122			0.780608	+	0.625021i	$0.214909\pi$
$272$		0			0
$273$		−	0.340173i		−	0.0205882i
$274$		0			0
$275$	−14.4124	−	7.63317i	−0.869101	−	0.460297i
$276$		0			0
$277$		−	11.2039i		−	0.673179i	−0.941651	−	0.336590i	$-0.890726\pi$
$277$		−	11.2039i		−	0.673179i	0.941651	−	0.336590i	$-0.109274\pi$
$278$		0			0
$279$	8.34017			0.499313
$280$		0			0
$281$	24.3545			1.45287			0.726435	−	0.687235i	$-0.241176\pi$
$281$	24.3545			1.45287			0.726435	+	0.687235i	$0.241176\pi$
$282$		0			0
$283$		−	6.15676i		−	0.365981i	−0.983115	−	0.182991i	$-0.941422\pi$
$283$		−	6.15676i		−	0.365981i	0.983115	−	0.182991i	$-0.0585778\pi$
$284$		0			0
$285$	−3.50307	+	14.0989i	−0.207504	+	0.835147i
$286$		0			0
$287$			0.340173i			0.0200798i
$288$		0			0
$289$	−16.1629			−0.950759
$290$		0			0
$291$	−6.49693			−0.380857
$292$		0			0
$293$			1.75872i			0.102746i	0.998680	+	0.0513729i	$0.0163597\pi$
$293$			1.75872i			0.102746i	−0.998680	+	0.0513729i	$0.983640\pi$
$294$		0			0
$295$	14.8371	+	3.68649i	0.863849	+	0.214636i
$296$		0			0
$297$		−	3.26180i		−	0.189269i
$298$		0			0
$299$	2.89043			0.167158
$300$		0			0
$301$	−8.68035			−0.500327
$302$		0			0
$303$			2.18342i			0.125434i
$304$		0			0
$305$	−33.3340	−	8.28231i	−1.90870	−	0.474244i
$306$		0			0
$307$			27.2039i			1.55261i	0.630357	+	0.776305i	$0.282908\pi$
$307$			27.2039i			1.55261i	−0.630357	+	0.776305i	$0.717092\pi$
$308$		0			0
$309$	5.84324			0.332411
$310$		0			0
$311$	−21.8432			−1.23862			−0.619308	−	0.785148i	$-0.712587\pi$
$311$	−21.8432			−1.23862			−0.619308	+	0.785148i	$0.712587\pi$
$312$		0			0
$313$			25.3874i			1.43498i	0.696570	+	0.717489i	$0.254709\pi$
$313$			25.3874i			1.43498i	−0.696570	+	0.717489i	$0.745291\pi$
$314$		0			0
$315$	−0.539189	+	2.17009i	−0.0303799	+	0.122271i
$316$		0			0
$317$			22.6947i			1.27466i	0.770590	+	0.637331i	$0.219962\pi$
$317$			22.6947i			1.27466i	−0.770590	+	0.637331i	$0.780038\pi$
$318$		0			0
$319$	−6.52359			−0.365251
$320$		0			0
$321$	−8.18342			−0.456754
$322$		0			0
$323$			37.4140i			2.08177i
$324$		0			0
$325$	0.796064	−	1.50307i	0.0441577	−	0.0833754i
$326$		0			0
$327$			16.8371i			0.931094i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	5.47641			0.301011			0.150505	−	0.988609i	$-0.451910\pi$
$331$	5.47641			0.301011			0.150505	+	0.988609i	$0.451910\pi$
$332$		0			0
$333$			6.15676i			0.337388i
$334$		0			0
$335$	8.00000	−	32.1978i	0.437087	−	1.75915i
$336$		0			0
$337$			13.6742i			0.744881i	0.928056	+	0.372441i	$0.121479\pi$
$337$			13.6742i			0.744881i	−0.928056	+	0.372441i	$0.878521\pi$
$338$		0			0
$339$	13.0205			0.707178
$340$		0			0
$341$	27.2039			1.47318
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−18.4391	−	4.58145i	−0.992726	−	0.246657i
$346$		0			0
$347$		−	0.183417i		−	0.00984637i	−0.999988	−	0.00492318i	$-0.998433\pi$
$347$		−	0.183417i		−	0.00984637i	0.999988	−	0.00492318i	$-0.00156710\pi$
$348$		0			0
$349$	7.67420			0.410791			0.205395	−	0.978679i	$-0.434152\pi$
$349$	7.67420			0.410791			0.205395	+	0.978679i	$0.434152\pi$
$350$		0			0
$351$	0.340173			0.0181571
$352$		0			0
$353$			19.4329i			1.03431i	0.855892	+	0.517155i	$0.173009\pi$
$353$			19.4329i			1.03431i	−0.855892	+	0.517155i	$0.826991\pi$
$354$		0			0
$355$	34.5958	+	8.59583i	1.83616	+	0.456219i
$356$		0			0
$357$			5.75872i			0.304784i
$358$		0			0
$359$	25.4186			1.34154			0.670770	−	0.741666i	$-0.265964\pi$
$359$	25.4186			1.34154			0.670770	+	0.741666i	$0.265964\pi$
$360$		0			0
$361$	23.2101			1.22158
$362$		0			0
$363$			0.360692i			0.0189314i
$364$		0			0
$365$	0.810439	−	3.26180i	0.0424203	−	0.170730i
$366$		0			0
$367$		−	9.36069i		−	0.488624i	−0.969697	−	0.244312i	$-0.921438\pi$
$367$		−	9.36069i		−	0.488624i	0.969697	−	0.244312i	$-0.0785621\pi$
$368$		0			0
$369$	−0.340173			−0.0177087
$370$		0			0
$371$	−8.34017			−0.433000
$372$		0			0
$373$			27.5174i			1.42480i	0.701774	+	0.712400i	$0.252392\pi$
$373$			27.5174i			1.42480i	−0.701774	+	0.712400i	$0.747608\pi$
$374$		0			0
$375$	−7.46081	+	8.32684i	−0.385275	+	0.429996i
$376$		0			0
$377$		−	0.680346i		−	0.0350396i
$378$		0			0
$379$	−9.84324			−0.505614			−0.252807	−	0.967517i	$-0.581354\pi$
$379$	−9.84324			−0.505614			−0.252807	+	0.967517i	$0.581354\pi$
$380$		0			0
$381$	−1.84324			−0.0944323
$382$		0			0
$383$		−	27.8310i		−	1.42210i	−0.703144	−	0.711048i	$-0.748221\pi$
$383$		−	27.8310i		−	1.42210i	0.703144	−	0.711048i	$-0.251779\pi$
$384$		0			0
$385$	−1.75872	+	7.07838i	−0.0896329	+	0.360748i
$386$		0			0
$387$		−	8.68035i		−	0.441247i
$388$		0			0
$389$	3.67420			0.186289			0.0931447	−	0.995653i	$-0.470308\pi$
$389$	3.67420			0.186289			0.0931447	+	0.995653i	$0.470308\pi$
$390$		0			0
$391$	−48.9315			−2.47457
$392$		0			0
$393$			0.313511i			0.0158145i
$394$		0			0
$395$	18.8371	+	4.68035i	0.947797	+	0.235494i
$396$		0			0
$397$		−	22.8638i		−	1.14750i	−0.819031	−	0.573750i	$-0.805488\pi$
$397$		−	22.8638i		−	1.14750i	0.819031	−	0.573750i	$-0.194512\pi$
$398$		0			0
$399$	6.49693			0.325253
$400$		0			0
$401$	5.31965			0.265651			0.132825	−	0.991139i	$-0.457595\pi$
$401$	5.31965			0.265651			0.132825	+	0.991139i	$0.457595\pi$
$402$		0			0
$403$			2.83710i			0.141326i
$404$		0			0
$405$	−2.17009	−	0.539189i	−0.107832	−	0.0267925i
$406$		0			0
$407$			20.0821i			0.995432i
$408$		0			0
$409$	−27.7275			−1.37104			−0.685519	−	0.728055i	$-0.740425\pi$
$409$	−27.7275			−1.37104			−0.685519	+	0.728055i	$0.740425\pi$
$410$		0			0
$411$	−2.18342			−0.107700
$412$		0			0
$413$		−	6.83710i		−	0.336432i
$414$		0			0
$415$	−3.68649	+	14.8371i	−0.180963	+	0.728325i
$416$		0			0
$417$		−	1.02052i		−	0.0499750i
$418$		0			0
$419$	23.5174			1.14890			0.574451	−	0.818539i	$-0.305215\pi$
$419$	23.5174			1.14890			0.574451	+	0.818539i	$0.305215\pi$
$420$		0			0
$421$	−1.15061			−0.0560774			−0.0280387	−	0.999607i	$-0.508926\pi$
$421$	−1.15061			−0.0560774			−0.0280387	+	0.999607i	$0.508926\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−13.4764	+	25.4452i	−0.653702	+	1.23427i
$426$		0			0
$427$			15.3607i			0.743356i
$428$		0			0
$429$	1.10957			0.0535708
$430$		0			0
$431$	17.4186			0.839022			0.419511	−	0.907750i	$-0.362202\pi$
$431$	17.4186			0.839022			0.419511	+	0.907750i	$0.362202\pi$
$432$		0			0
$433$			26.0144i			1.25017i	0.780556	+	0.625086i	$0.214936\pi$
$433$			26.0144i			1.25017i	−0.780556	+	0.625086i	$0.785064\pi$
$434$		0			0
$435$	−1.07838	+	4.34017i	−0.0517043	+	0.208095i
$436$		0			0
$437$			55.2039i			2.64076i
$438$		0			0
$439$	−5.13624			−0.245139			−0.122570	−	0.992460i	$-0.539113\pi$
$439$	−5.13624			−0.245139			−0.122570	+	0.992460i	$0.539113\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	18.5380i		−	0.880765i	−0.897810	−	0.440383i	$-0.854843\pi$
$443$		−	18.5380i		−	0.880765i	0.897810	−	0.440383i	$-0.145157\pi$
$444$		0			0
$445$	−32.9360	−	8.18342i	−1.56132	−	0.387931i
$446$		0			0
$447$			14.6803i			0.694357i
$448$		0			0
$449$	10.3135			0.486725			0.243362	−	0.969935i	$-0.421750\pi$
$449$	10.3135			0.486725			0.243362	+	0.969935i	$0.421750\pi$
$450$		0			0
$451$	−1.10957			−0.0522478
$452$		0			0
$453$		−	2.15676i		−	0.101333i
$454$		0			0
$455$	−0.738205	−	0.183417i	−0.0346076	−	0.00859874i
$456$		0			0
$457$			4.36683i			0.204272i	0.994770	+	0.102136i	$0.0325677\pi$
$457$			4.36683i			0.204272i	−0.994770	+	0.102136i	$0.967432\pi$
$458$		0			0
$459$	−5.75872			−0.268794
$460$		0			0
$461$	−33.7009			−1.56961			−0.784803	−	0.619745i	$-0.787236\pi$
$461$	−33.7009			−1.56961			−0.784803	+	0.619745i	$0.787236\pi$
$462$		0			0
$463$			10.4703i			0.486595i	0.969952	+	0.243297i	$0.0782291\pi$
$463$			10.4703i			0.486595i	−0.969952	+	0.243297i	$0.921771\pi$
$464$		0			0
$465$	4.49693	−	18.0989i	0.208540	−	0.839316i
$466$		0			0
$467$		−	3.51745i		−	0.162768i	−0.996683	−	0.0813840i	$-0.974066\pi$
$467$		−	3.51745i		−	0.162768i	0.996683	−	0.0813840i	$-0.0259340\pi$
$468$		0			0
$469$	−14.8371			−0.685114
$470$		0			0
$471$	−6.18342			−0.284917
$472$		0			0
$473$		−	28.3135i		−	1.30186i
$474$		0			0
$475$	28.7070	+	15.2039i	1.31717	+	0.697604i
$476$		0			0
$477$		−	8.34017i		−	0.381870i
$478$		0			0
$479$	21.8432			0.998043			0.499022	−	0.866590i	$-0.333693\pi$
$479$	21.8432			0.998043			0.499022	+	0.866590i	$0.333693\pi$
$480$		0			0
$481$	−2.09436			−0.0954947
$482$		0			0
$483$			8.49693i			0.386624i
$484$		0			0
$485$	−3.50307	+	14.0989i	−0.159066	+	0.640198i
$486$		0			0
$487$			7.88428i			0.357271i	0.983915	+	0.178635i	$0.0571683\pi$
$487$			7.88428i			0.357271i	−0.983915	+	0.178635i	$0.942832\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$	19.7731			0.892347			0.446174	−	0.894946i	$-0.352786\pi$
$491$	19.7731			0.892347			0.446174	+	0.894946i	$0.352786\pi$
$492$		0			0
$493$			11.5174i			0.518720i
$494$		0			0
$495$	−7.07838	−	1.75872i	−0.318149	−	0.0790488i
$496$		0			0
$497$		−	15.9421i		−	0.715103i
$498$		0			0
$499$	−39.5174			−1.76904			−0.884522	−	0.466499i	$-0.845515\pi$
$499$	−39.5174			−1.76904			−0.884522	+	0.466499i	$0.845515\pi$
$500$		0			0
$501$	1.47641			0.0659611
$502$		0			0
$503$			11.2039i			0.499559i	0.968303	+	0.249779i	$0.0803581\pi$
$503$			11.2039i			0.499559i	−0.968303	+	0.249779i	$0.919642\pi$
$504$		0			0
$505$	4.73820	+	1.17727i	0.210847	+	0.0523880i
$506$		0			0
$507$		−	12.8843i		−	0.572211i
$508$		0			0
$509$	23.5441			1.04357			0.521787	−	0.853076i	$-0.325266\pi$
$509$	23.5441			1.04357			0.521787	+	0.853076i	$0.325266\pi$
$510$		0			0
$511$	−1.50307			−0.0664920
$512$		0			0
$513$			6.49693i			0.286846i
$514$		0			0
$515$	3.15061	−	12.6803i	0.138833	−	0.558763i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−1.75872			−0.0771994
$520$		0			0
$521$	24.6537			1.08010			0.540049	−	0.841634i	$-0.318406\pi$
$521$	24.6537			1.08010			0.540049	+	0.841634i	$0.318406\pi$
$522$		0			0
$523$			2.63931i			0.115409i	0.998334	+	0.0577044i	$0.0183781\pi$
$523$			2.63931i			0.115409i	−0.998334	+	0.0577044i	$0.981622\pi$
$524$		0			0
$525$	4.41855	+	2.34017i	0.192841	+	0.102134i
$526$		0			0
$527$		−	48.0288i		−	2.09217i
$528$		0			0
$529$	−49.1978			−2.13903
$530$		0			0
$531$	6.83710			0.296705
$532$		0			0
$533$		−	0.115718i		−	0.00501229i
$534$		0			0
$535$	−4.41241	+	17.7587i	−0.190765	+	0.767777i
$536$		0			0
$537$			0.424694i			0.0183269i
$538$		0			0
$539$	3.26180			0.140495
$540$		0			0
$541$	25.1506			1.08131			0.540655	−	0.841245i	$-0.318177\pi$
$541$	25.1506			1.08131			0.540655	+	0.841245i	$0.318177\pi$
$542$		0			0
$543$		−	10.3668i		−	0.444883i
$544$		0			0
$545$	36.5380	+	9.07838i	1.56511	+	0.388875i
$546$		0			0
$547$		−	35.1506i		−	1.50293i	−0.659772	−	0.751466i	$-0.729347\pi$
$547$		−	35.1506i		−	1.50293i	0.659772	−	0.751466i	$-0.270653\pi$
$548$		0			0
$549$	−15.3607			−0.655578
$550$		0			0
$551$	12.9939			0.553557
$552$		0			0
$553$		−	8.68035i		−	0.369126i
$554$		0			0
$555$	13.3607	+	3.31965i	0.567130	+	0.140911i
$556$		0			0
$557$			26.3812i			1.11781i	0.829232	+	0.558904i	$0.188778\pi$
$557$			26.3812i			1.11781i	−0.829232	+	0.558904i	$0.811222\pi$
$558$		0			0
$559$	2.95282			0.124891
$560$		0			0
$561$	−18.7838			−0.793052
$562$		0			0
$563$		−	12.9939i		−	0.547626i	−0.961783	−	0.273813i	$-0.911715\pi$
$563$		−	12.9939i		−	0.547626i	0.961783	−	0.273813i	$-0.0882849\pi$
$564$		0			0
$565$	7.02052	−	28.2557i	0.295355	−	1.18872i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	6.36683			0.266912			0.133456	−	0.991055i	$-0.457393\pi$
$569$	6.36683			0.266912			0.133456	+	0.991055i	$0.457393\pi$
$570$		0			0
$571$	2.63931			0.110452			0.0552258	−	0.998474i	$-0.482412\pi$
$571$	2.63931			0.110452			0.0552258	+	0.998474i	$0.482412\pi$
$572$		0			0
$573$			21.7321i			0.907870i
$574$		0			0
$575$	−19.8843	+	37.5441i	−0.829232	+	1.56570i
$576$		0			0
$577$			33.3340i			1.38771i	0.720113	+	0.693857i	$0.244090\pi$
$577$			33.3340i			1.38771i	−0.720113	+	0.693857i	$0.755910\pi$
$578$		0			0
$579$	−8.36683			−0.347714
$580$		0			0
$581$	6.83710			0.283651
$582$		0			0
$583$		−	27.2039i		−	1.12667i
$584$		0			0
$585$	0.183417	−	0.738205i	0.00758338	−	0.0305210i
$586$		0			0
$587$		−	17.3074i		−	0.714352i	−0.934037	−	0.357176i	$-0.883740\pi$
$587$		−	17.3074i		−	0.714352i	0.934037	−	0.357176i	$-0.116260\pi$
$588$		0			0
$589$	−54.1855			−2.23267
$590$		0			0
$591$	−13.3340			−0.548489
$592$		0			0
$593$		−	47.1194i		−	1.93496i	−0.252943	−	0.967481i	$-0.581398\pi$
$593$		−	47.1194i		−	1.93496i	0.252943	−	0.967481i	$-0.418602\pi$
$594$		0			0
$595$	12.4969	+	3.10504i	0.512324	+	0.127294i
$596$		0			0
$597$			6.49693i			0.265902i
$598$		0			0
$599$	13.1050			0.535457			0.267729	−	0.963494i	$-0.413727\pi$
$599$	13.1050			0.535457			0.267729	+	0.963494i	$0.413727\pi$
$600$		0			0
$601$	29.0349			1.18436			0.592179	−	0.805806i	$-0.298268\pi$
$601$	29.0349			1.18436			0.592179	+	0.805806i	$0.298268\pi$
$602$		0			0
$603$		−	14.8371i		−	0.604213i
$604$		0			0
$605$	0.782733	+	0.194481i	0.0318226	+	0.00790678i
$606$		0			0
$607$			25.9877i			1.05481i	0.849614	+	0.527404i	$0.176835\pi$
$607$			25.9877i			1.05481i	−0.849614	+	0.527404i	$0.823165\pi$
$608$		0			0
$609$	2.00000			0.0810441
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	20.1978i		−	0.815781i	−0.913031	−	0.407891i	$-0.866264\pi$
$613$		−	20.1978i		−	0.815781i	0.913031	−	0.407891i	$-0.133736\pi$
$614$		0			0
$615$	−0.183417	+	0.738205i	−0.00739611	+	0.0297673i
$616$		0			0
$617$			47.8576i			1.92668i	0.268292	+	0.963338i	$0.413541\pi$
$617$			47.8576i			1.92668i	−0.268292	+	0.963338i	$0.586459\pi$
$618$		0			0
$619$	18.3812			0.738803			0.369402	−	0.929270i	$-0.379563\pi$
$619$	18.3812			0.738803			0.369402	+	0.929270i	$0.379563\pi$
$620$		0			0
$621$	−8.49693			−0.340970
$622$		0			0
$623$			15.1773i			0.608065i
$624$		0			0
$625$	14.0472	+	20.6803i	0.561887	+	0.827214i
$626$		0			0
$627$			21.1917i			0.846313i
$628$		0			0
$629$	35.4551			1.41369
$630$		0			0
$631$	−7.94668			−0.316352			−0.158176	−	0.987411i	$-0.550561\pi$
$631$	−7.94668			−0.316352			−0.158176	+	0.987411i	$0.550561\pi$
$632$		0			0
$633$		−	6.83710i		−	0.271750i
$634$		0			0
$635$	−0.993857	+	4.00000i	−0.0394400	+	0.158735i
$636$		0			0
$637$			0.340173i			0.0134781i
$638$		0			0
$639$	15.9421			0.630661
$640$		0			0
$641$	15.7275			0.621200			0.310600	−	0.950541i	$-0.399470\pi$
$641$	15.7275			0.621200			0.310600	+	0.950541i	$0.399470\pi$
$642$		0			0
$643$		−	44.5646i		−	1.75746i	−0.477322	−	0.878729i	$-0.658392\pi$
$643$		−	44.5646i		−	1.75746i	0.477322	−	0.878729i	$-0.341608\pi$
$644$		0			0
$645$	−18.8371	−	4.68035i	−0.741710	−	0.184288i
$646$		0			0
$647$		−	5.16290i		−	0.202974i	−0.994837	−	0.101487i	$-0.967640\pi$
$647$		−	5.16290i		−	0.202974i	0.994837	−	0.101487i	$-0.0323601\pi$
$648$		0			0
$649$	22.3012			0.875400
$650$		0			0
$651$	−8.34017			−0.326877
$652$		0			0
$653$			7.49079i			0.293137i	0.989201	+	0.146569i	$0.0468229\pi$
$653$			7.49079i			0.293137i	−0.989201	+	0.146569i	$0.953177\pi$
$654$		0			0
$655$	0.680346	+	0.169042i	0.0265833	+	0.00660500i
$656$		0			0
$657$		−	1.50307i		−	0.0586404i
$658$		0			0
$659$	22.7259			0.885276			0.442638	−	0.896700i	$-0.354043\pi$
$659$	22.7259			0.885276			0.442638	+	0.896700i	$0.354043\pi$
$660$		0			0
$661$	−12.3258			−0.479418			−0.239709	−	0.970845i	$-0.577052\pi$
$661$	−12.3258			−0.479418			−0.239709	+	0.970845i	$0.577052\pi$
$662$		0			0
$663$		−	1.95896i		−	0.0760798i
$664$		0			0
$665$	3.50307	−	14.0989i	0.135843	−	0.546732i
$666$		0			0
$667$			16.9939i			0.658005i
$668$		0			0
$669$	17.3607			0.671203
$670$		0			0
$671$	−50.1034			−1.93422
$672$		0			0
$673$		−	25.6742i		−	0.989668i	−0.868988	−	0.494834i	$-0.835229\pi$
$673$		−	25.6742i		−	0.989668i	0.868988	−	0.494834i	$-0.164771\pi$
$674$		0			0
$675$	−2.34017	+	4.41855i	−0.0900733	+	0.170070i
$676$		0			0
$677$		−	29.2762i		−	1.12517i	−0.826738	−	0.562587i	$-0.809806\pi$
$677$		−	29.2762i		−	1.12517i	0.826738	−	0.562587i	$-0.190194\pi$
$678$		0			0
$679$	6.49693			0.249329
$680$		0			0
$681$	−10.1568			−0.389208
$682$		0			0
$683$			25.5441i			0.977418i	0.872447	+	0.488709i	$0.162532\pi$
$683$			25.5441i			0.977418i	−0.872447	+	0.488709i	$0.837468\pi$
$684$		0			0
$685$	−1.17727	+	4.73820i	−0.0449813	+	0.181037i
$686$		0			0
$687$			14.9939i			0.572051i
$688$		0			0
$689$	2.83710			0.108085
$690$		0			0
$691$	26.3812			1.00359			0.501794	−	0.864987i	$-0.332673\pi$
$691$	26.3812			1.00359			0.501794	+	0.864987i	$0.332673\pi$
$692$		0			0
$693$			3.26180i			0.123905i
$694$		0			0
$695$	−2.21461	−	0.550252i	−0.0840051	−	0.0208723i
$696$		0			0
$697$			1.95896i			0.0742010i
$698$		0			0
$699$	−11.5441			−0.436638
$700$		0			0
$701$	−23.6742			−0.894162			−0.447081	−	0.894493i	$-0.647536\pi$
$701$	−23.6742			−0.894162			−0.447081	+	0.894493i	$0.647536\pi$
$702$		0			0
$703$		−	40.0000i		−	1.50863i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	2.18342i		−	0.0821159i
$708$		0			0
$709$	40.7214			1.52932			0.764662	−	0.644432i	$-0.222906\pi$
$709$	40.7214			1.52932			0.764662	+	0.644432i	$0.222906\pi$
$710$		0			0
$711$	8.68035			0.325538
$712$		0			0
$713$		−	70.8659i		−	2.65395i
$714$		0			0
$715$	0.598270	−	2.40787i	0.0223740	−	0.0900493i
$716$		0			0
$717$		−	10.8950i		−	0.406880i
$718$		0			0
$719$	−22.3545			−0.833684			−0.416842	−	0.908979i	$-0.636863\pi$
$719$	−22.3545			−0.833684			−0.416842	+	0.908979i	$0.636863\pi$
$720$		0			0
$721$	−5.84324			−0.217614
$722$		0			0
$723$			6.68035i			0.248445i
$724$		0			0
$725$	8.83710	+	4.68035i	0.328202	+	0.173824i
$726$		0			0
$727$			5.10957i			0.189504i	0.995501	+	0.0947518i	$0.0302058\pi$
$727$			5.10957i			0.189504i	−0.995501	+	0.0947518i	$0.969794\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−49.9877			−1.84886
$732$		0			0
$733$			41.6475i			1.53829i	0.639076	+	0.769144i	$0.279317\pi$
$733$			41.6475i			1.53829i	−0.639076	+	0.769144i	$0.720683\pi$
$734$		0			0
$735$	0.539189	−	2.17009i	0.0198883	−	0.0800448i
$736$		0			0
$737$		−	48.3956i		−	1.78267i
$738$		0			0
$739$	−32.3135			−1.18867			−0.594336	−	0.804217i	$-0.702585\pi$
$739$	−32.3135			−1.18867			−0.594336	+	0.804217i	$0.702585\pi$
$740$		0			0
$741$	−2.21008			−0.0811893
$742$		0			0
$743$			30.1711i			1.10687i	0.832892	+	0.553436i	$0.186684\pi$
$743$			30.1711i			1.10687i	−0.832892	+	0.553436i	$0.813316\pi$
$744$		0			0
$745$	31.8576	+	7.91548i	1.16717	+	0.290001i
$746$		0			0
$747$			6.83710i			0.250156i
$748$		0			0
$749$	8.18342			0.299016
$750$		0			0
$751$	−46.1855			−1.68533			−0.842667	−	0.538436i	$-0.819015\pi$
$751$	−46.1855			−1.68533			−0.842667	+	0.538436i	$0.819015\pi$
$752$		0			0
$753$		−	24.1978i		−	0.881817i
$754$		0			0
$755$	−4.68035	−	1.16290i	−0.170335	−	0.0423222i
$756$		0			0
$757$		−	50.5523i		−	1.83736i	−0.395007	−	0.918678i	$-0.629258\pi$
$757$		−	50.5523i		−	1.83736i	0.395007	−	0.918678i	$-0.370742\pi$
$758$		0			0
$759$	−27.7152			−1.00600
$760$		0			0
$761$	−12.1711			−0.441203			−0.220602	−	0.975364i	$-0.570802\pi$
$761$	−12.1711			−0.441203			−0.220602	+	0.975364i	$0.570802\pi$
$762$		0			0
$763$		−	16.8371i		−	0.609544i
$764$		0			0
$765$	−3.10504	+	12.4969i	−0.112263	+	0.451827i
$766$		0			0
$767$			2.32580i			0.0839797i
$768$		0			0
$769$	−7.36069			−0.265433			−0.132717	−	0.991154i	$-0.542370\pi$
$769$	−7.36069			−0.265433			−0.132717	+	0.991154i	$0.542370\pi$
$770$		0			0
$771$	−15.2351			−0.548680
$772$		0			0
$773$		−	10.5548i		−	0.379629i	−0.981820	−	0.189815i	$-0.939211\pi$
$773$		−	10.5548i		−	0.379629i	0.981820	−	0.189815i	$-0.0607887\pi$
$774$		0			0
$775$	−36.8515	−	19.5174i	−1.32374	−	0.701087i
$776$		0			0
$777$		−	6.15676i		−	0.220872i
$778$		0			0
$779$	2.21008			0.0791843
$780$		0			0
$781$	52.0000			1.86071
$782$		0			0
$783$			2.00000i			0.0714742i
$784$		0			0
$785$	−3.33403	+	13.4186i	−0.118997	+	0.478929i
$786$		0			0
$787$		−	9.04718i		−	0.322497i	−0.986914	−	0.161249i	$-0.948448\pi$
$787$		−	9.04718i		−	0.322497i	0.986914	−	0.161249i	$-0.0515521\pi$
$788$		0			0
$789$	29.1773			1.03874
$790$		0			0
$791$	−13.0205			−0.462956
$792$		0			0
$793$		−	5.22529i		−	0.185556i
$794$		0			0
$795$	−18.0989	−	4.49693i	−0.641902	−	0.159490i
$796$		0			0
$797$		−	16.9627i		−	0.600848i	−0.953806	−	0.300424i	$-0.902872\pi$
$797$		−	16.9627i		−	0.600848i	0.953806	−	0.300424i	$-0.0971282\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−15.1773			−0.536263
$802$		0			0
$803$		−	4.90271i		−	0.173013i
$804$		0			0
$805$	18.4391	+	4.58145i	0.649892	+	0.161475i
$806$		0			0
$807$		−	6.13009i		−	0.215790i
$808$		0			0
$809$	−15.9877			−0.562098			−0.281049	−	0.959693i	$-0.590682\pi$
$809$	−15.9877			−0.562098			−0.281049	+	0.959693i	$0.590682\pi$
$810$		0			0
$811$	−8.39350			−0.294736			−0.147368	−	0.989082i	$-0.547080\pi$
$811$	−8.39350			−0.294736			−0.147368	+	0.989082i	$0.547080\pi$
$812$		0			0
$813$		−	25.7009i		−	0.901369i
$814$		0			0
$815$	2.15676	−	8.68035i	0.0755478	−	0.304059i
$816$		0			0
$817$			56.3956i			1.97303i
$818$		0			0
$819$	−0.340173			−0.0118866
$820$		0			0
$821$	38.0288			1.32721			0.663606	−	0.748082i	$-0.269025\pi$
$821$	38.0288			1.32721			0.663606	+	0.748082i	$0.269025\pi$
$822$		0			0
$823$		−	22.1568i		−	0.772336i	−0.922428	−	0.386168i	$-0.873799\pi$
$823$		−	22.1568i		−	0.772336i	0.922428	−	0.386168i	$-0.126201\pi$
$824$		0			0
$825$	−7.63317	+	14.4124i	−0.265753	+	0.501776i
$826$		0			0
$827$		−	14.8227i		−	0.515437i	−0.966220	−	0.257718i	$-0.917029\pi$
$827$		−	14.8227i		−	0.515437i	0.966220	−	0.257718i	$-0.0829707\pi$
$828$		0			0
$829$	−13.9467			−0.484388			−0.242194	−	0.970228i	$-0.577867\pi$
$829$	−13.9467			−0.484388			−0.242194	+	0.970228i	$0.577867\pi$
$830$		0			0
$831$	−11.2039			−0.388660
$832$		0			0
$833$		−	5.75872i		−	0.199528i
$834$		0			0
$835$	0.796064	−	3.20394i	0.0275489	−	0.110877i
$836$		0			0
$837$		−	8.34017i		−	0.288279i
$838$		0			0
$839$	25.4764			0.879543			0.439772	−	0.898110i	$-0.355059\pi$
$839$	25.4764			0.879543			0.439772	+	0.898110i	$0.355059\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		−	24.3545i		−	0.838815i
$844$		0			0
$845$	−27.9600	−	6.94706i	−0.961853	−	0.238986i
$846$		0			0
$847$		−	0.360692i		−	0.0123935i
$848$		0			0
$849$	−6.15676			−0.211299
$850$		0			0
$851$	52.3135			1.79328
$852$		0			0
$853$		−	58.1588i		−	1.99132i	−0.0930604	−	0.995660i	$-0.529665\pi$
$853$		−	58.1588i		−	1.99132i	0.0930604	−	0.995660i	$-0.470335\pi$
$854$		0			0
$855$	14.0989	+	3.50307i	0.482172	+	0.119803i
$856$		0			0
$857$		−	28.9093i		−	0.987524i	−0.869597	−	0.493762i	$-0.835621\pi$
$857$		−	28.9093i		−	0.987524i	0.869597	−	0.493762i	$-0.164379\pi$
$858$		0			0
$859$	−50.1834			−1.71224			−0.856118	−	0.516780i	$-0.827130\pi$
$859$	−50.1834			−1.71224			−0.856118	+	0.516780i	$0.827130\pi$
$860$		0			0
$861$	0.340173			0.0115931
$862$		0			0
$863$		−	8.13009i		−	0.276752i	−0.990380	−	0.138376i	$-0.955812\pi$
$863$		−	8.13009i		−	0.276752i	0.990380	−	0.138376i	$-0.0441882\pi$
$864$		0			0
$865$	−0.948284	+	3.81658i	−0.0322426	+	0.129768i
$866$		0			0
$867$			16.1629i			0.548921i
$868$		0			0
$869$	28.3135			0.960470
$870$		0			0
$871$	5.04718			0.171017
$872$		0			0
$873$			6.49693i			0.219888i
$874$		0			0
$875$	7.46081	−	8.32684i	0.252221	−	0.281499i
$876$		0			0
$877$		−	26.8371i		−	0.906225i	−0.891453	−	0.453112i	$-0.850314\pi$
$877$		−	26.8371i		−	0.906225i	0.891453	−	0.453112i	$-0.149686\pi$
$878$		0			0
$879$	1.75872			0.0593203
$880$		0			0
$881$	−44.4846			−1.49873			−0.749363	−	0.662160i	$-0.769640\pi$
$881$	−44.4846			−1.49873			−0.749363	+	0.662160i	$0.769640\pi$
$882$		0			0
$883$			20.8781i			0.702605i	0.936262	+	0.351303i	$0.114261\pi$
$883$			20.8781i			0.702605i	−0.936262	+	0.351303i	$0.885739\pi$
$884$		0			0
$885$	3.68649	−	14.8371i	0.123920	−	0.498744i
$886$		0			0
$887$		−	22.4079i		−	0.752383i	−0.926542	−	0.376191i	$-0.877234\pi$
$887$		−	22.4079i		−	0.752383i	0.926542	−	0.376191i	$-0.122766\pi$
$888$		0			0
$889$	1.84324			0.0618204
$890$		0			0
$891$	−3.26180			−0.109274
$892$		0			0
$893$		0			0
$894$		0			0
$895$	0.921622	+	0.228990i	0.0308064	+	0.00765430i
$896$		0			0
$897$		−	2.89043i		−	0.0965085i
$898$		0			0
$899$	−16.6803			−0.556321
$900$		0			0
$901$	−48.0288			−1.60007
$902$		0			0
$903$			8.68035i			0.288864i
$904$		0			0
$905$	−22.4969	−	5.58968i	−0.747823	−	0.185807i
$906$		0			0
$907$		−	20.8781i		−	0.693247i	−0.938004	−	0.346624i	$-0.887328\pi$
$907$		−	20.8781i		−	0.693247i	0.938004	−	0.346624i	$-0.112672\pi$
$908$		0			0
$909$	2.18342			0.0724194
$910$		0			0
$911$	3.52198			0.116688			0.0583442	−	0.998297i	$-0.481418\pi$
$911$	3.52198			0.116688			0.0583442	+	0.998297i	$0.481418\pi$
$912$		0			0
$913$			22.3012i			0.738063i
$914$		0			0
$915$	−8.28231	+	33.3340i	−0.273805	+	1.10199i
$916$		0			0
$917$		−	0.313511i		−	0.0103530i
$918$		0			0
$919$	−12.1978			−0.402368			−0.201184	−	0.979553i	$-0.564479\pi$
$919$	−12.1978			−0.402368			−0.201184	+	0.979553i	$0.564479\pi$
$920$		0			0
$921$	27.2039			0.896400
$922$		0			0
$923$			5.42309i			0.178503i
$924$		0			0
$925$	14.4079	−	27.2039i	0.473728	−	0.894460i
$926$		0			0
$927$		−	5.84324i		−	0.191917i
$928$		0			0
$929$	−2.86376			−0.0939570			−0.0469785	−	0.998896i	$-0.514959\pi$
$929$	−2.86376			−0.0939570			−0.0469785	+	0.998896i	$0.514959\pi$
$930$		0			0
$931$	−6.49693			−0.212928
$932$		0			0
$933$			21.8432i			0.715116i
$934$		0			0
$935$	−10.1280	+	40.7624i	−0.331221	+	1.33307i
$936$		0			0
$937$			26.7480i			0.873821i	0.899505	+	0.436910i	$0.143927\pi$
$937$			26.7480i			0.873821i	−0.899505	+	0.436910i	$0.856073\pi$
$938$		0			0
$939$	25.3874			0.828485
$940$		0			0
$941$	10.0144			0.326459			0.163230	−	0.986588i	$-0.447809\pi$
$941$	10.0144			0.326459			0.163230	+	0.986588i	$0.447809\pi$
$942$		0			0
$943$			2.89043i			0.0941252i
$944$		0			0
$945$	2.17009	+	0.539189i	0.0705929	+	0.0175398i
$946$		0			0
$947$			47.2183i			1.53439i	0.641415	+	0.767194i	$0.278348\pi$
$947$			47.2183i			1.53439i	−0.641415	+	0.767194i	$0.721652\pi$
$948$		0			0
$949$	0.511304			0.0165976
$950$		0			0
$951$	22.6947			0.735927
$952$		0			0
$953$		−	14.6660i		−	0.475077i	−0.971378	−	0.237539i	$-0.923659\pi$
$953$		−	14.6660i		−	0.475077i	0.971378	−	0.237539i	$-0.0763407\pi$
$954$		0			0
$955$	47.1605	+	11.7177i	1.52608	+	0.379175i
$956$		0			0
$957$			6.52359i			0.210878i
$958$		0			0
$959$	2.18342			0.0705062
$960$		0			0
$961$	38.5585			1.24382
$962$		0			0
$963$			8.18342i			0.263707i
$964$		0			0
$965$	−4.51130	+	18.1568i	−0.145224	+	0.584487i
$966$		0			0
$967$		−	57.3894i		−	1.84552i	−0.385375	−	0.922760i	$-0.625928\pi$
$967$		−	57.3894i		−	1.84552i	0.385375	−	0.922760i	$-0.374072\pi$
$968$		0			0
$969$	37.4140			1.20191
$970$		0			0
$971$	−23.4017			−0.750997			−0.375499	−	0.926823i	$-0.622529\pi$
$971$	−23.4017			−0.750997			−0.375499	+	0.926823i	$0.622529\pi$
$972$		0			0
$973$			1.02052i			0.0327163i
$974$		0			0
$975$	−1.50307	−	0.796064i	−0.0481368	−	0.0254944i
$976$		0			0
$977$			3.54411i			0.113386i	0.998392	+	0.0566931i	$0.0180556\pi$
$977$			3.54411i			0.113386i	−0.998392	+	0.0566931i	$0.981944\pi$
$978$		0			0
$979$	−49.5052			−1.58219
$980$		0			0
$981$	16.8371			0.537567
$982$		0			0
$983$		−	11.7152i		−	0.373658i	−0.982392	−	0.186829i	$-0.940179\pi$
$983$		−	11.7152i		−	0.373658i	0.982392	−	0.186829i	$-0.0598211\pi$
$984$		0			0
$985$	−7.18956	+	28.9360i	−0.229078	+	0.921978i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−73.7563			−2.34531
$990$		0			0
$991$	56.5113			1.79514			0.897570	−	0.440871i	$-0.145330\pi$
$991$	56.5113			1.79514			0.897570	+	0.440871i	$0.145330\pi$
$992$		0			0
$993$		−	5.47641i		−	0.173789i
$994$		0			0
$995$	14.0989	+	3.50307i	0.446965	+	0.111055i
$996$		0			0
$997$			40.9048i			1.29547i	0.761867	+	0.647734i	$0.224283\pi$
$997$			40.9048i			1.29547i	−0.761867	+	0.647734i	$0.775717\pi$
$998$		0			0
$999$	6.15676			0.194791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.t.d.169.1	✓	6
3.2	odd	2		2520.2.t.k.1009.6		6
4.3	odd	2		1680.2.t.j.1009.4		6
5.2	odd	4		4200.2.a.bn.1.2		3
5.3	odd	4		4200.2.a.bp.1.2		3
5.4	even	2	inner	840.2.t.d.169.4	yes	6
12.11	even	2		5040.2.t.z.1009.6		6
15.14	odd	2		2520.2.t.k.1009.5		6
20.3	even	4		8400.2.a.di.1.2		3
20.7	even	4		8400.2.a.dl.1.2		3
20.19	odd	2		1680.2.t.j.1009.1		6
60.59	even	2		5040.2.t.z.1009.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.d.169.1	✓	6	1.1	even	1	trivial
840.2.t.d.169.4	yes	6	5.4	even	2	inner
1680.2.t.j.1009.1		6	20.19	odd	2
1680.2.t.j.1009.4		6	4.3	odd	2
2520.2.t.k.1009.5		6	15.14	odd	2
2520.2.t.k.1009.6		6	3.2	odd	2
4200.2.a.bn.1.2		3	5.2	odd	4
4200.2.a.bp.1.2		3	5.3	odd	4
5040.2.t.z.1009.5		6	60.59	even	2
5040.2.t.z.1009.6		6	12.11	even	2
8400.2.a.di.1.2		3	20.3	even	4
8400.2.a.dl.1.2		3	20.7	even	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:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.t (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -3.26180 q^{11} -0.340173i q^{13} +(-0.539189 + 2.17009i) q^{15} +5.75872i q^{17} +6.49693 q^{19} +1.00000 q^{21} +8.49693i q^{23} +(4.41855 + 2.34017i) q^{25} +1.00000i q^{27} +2.00000 q^{29} -8.34017 q^{31} +3.26180i q^{33} +(0.539189 - 2.17009i) q^{35} -6.15676i q^{37} -0.340173 q^{39} +0.340173 q^{41} +8.68035i q^{43} +(2.17009 + 0.539189i) q^{45} -1.00000 q^{49} +5.75872 q^{51} +8.34017i q^{53} +(7.07838 + 1.75872i) q^{55} -6.49693i q^{57} -6.83710 q^{59} +15.3607 q^{61} -1.00000i q^{63} +(-0.183417 + 0.738205i) q^{65} +14.8371i q^{67} +8.49693 q^{69} -15.9421 q^{71} +1.50307i q^{73} +(2.34017 - 4.41855i) q^{75} -3.26180i q^{77} -8.68035 q^{79} +1.00000 q^{81} -6.83710i q^{83} +(3.10504 - 12.4969i) q^{85} -2.00000i q^{87} +15.1773 q^{89} +0.340173 q^{91} +8.34017i q^{93} +(-14.0989 - 3.50307i) q^{95} -6.49693i q^{97} +3.26180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^5 - 6 * q^9	\( 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{21} - 2 q^{25} + 12 q^{29} - 28 q^{31} + 20 q^{39} - 20 q^{41} + 2 q^{45} - 6 q^{49} - 16 q^{51} + 36 q^{55} + 16 q^{59} + 4 q^{61} + 8 q^{65} + 16 q^{69} - 36 q^{71} - 8 q^{75} - 8 q^{79} + 6 q^{81} + 16 q^{85} + 12 q^{89} - 20 q^{91} - 12 q^{95} + 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^5 - 6 * q^9 - 4 * q^11 + 4 * q^19 + 6 * q^21 - 2 * q^25 + 12 * q^29 - 28 * q^31 + 20 * q^39 - 20 * q^41 + 2 * q^45 - 6 * q^49 - 16 * q^51 + 36 * q^55 + 16 * q^59 + 4 * q^61 + 8 * q^65 + 16 * q^69 - 36 * q^71 - 8 * q^75 - 8 * q^79 + 6 * q^81 + 16 * q^85 + 12 * q^89 - 20 * q^91 - 12 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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