LMFDB - Embedded newform 2106.2.b.c.649.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2106,2,Mod(649,2106)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2106, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2106.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2106 = 2 \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2106.b (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:	$16.8164946657$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 34x^{12} + 435x^{10} + 2617x^{8} + 7651x^{6} + 10260x^{4} + 5589x^{2} + 729 $ x^14 + 34x^12 + 435x^10 + 2617x^8 + 7651x^6 + 10260x^4 + 5589x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.5
Root			$3.32820i$ of defining polynomial
Character	$\chi$	$=$	2106.649
Dual form			2106.2.b.c.649.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} +0.594758i q^{5} +1.67682i q^{7} +1.00000i q^{8} +0.594758 q^{10} +0.480744i q^{11} +(-1.28905 - 3.36725i) q^{13} +1.67682 q^{14} +1.00000 q^{16} -2.09349 q^{17} +0.480744i q^{19} -0.594758i q^{20} +0.480744 q^{22} +3.66679 q^{23} +4.64626 q^{25} +(-3.36725 + 1.28905i) q^{26} -1.67682i q^{28} -2.46678 q^{29} -1.14753i q^{31} -1.00000i q^{32} +2.09349i q^{34} -0.997301 q^{35} +3.65012i q^{37} +0.480744 q^{38} -0.594758 q^{40} +9.91005i q^{41} +6.91644 q^{43} -0.480744i q^{44} -3.66679i q^{46} +6.24102i q^{47} +4.18828 q^{49} -4.64626i q^{50} +(1.28905 + 3.36725i) q^{52} -5.08592 q^{53} -0.285927 q^{55} -1.67682 q^{56} +2.46678i q^{58} +9.38985i q^{59} +7.81270 q^{61} -1.14753 q^{62} -1.00000 q^{64} +(2.00270 - 0.766671i) q^{65} +14.3819i q^{67} +2.09349 q^{68} +0.997301i q^{70} -6.51028i q^{71} +5.91514i q^{73} +3.65012 q^{74} -0.480744i q^{76} -0.806121 q^{77} -2.05790 q^{79} +0.594758i q^{80} +9.91005 q^{82} +11.0601i q^{83} -1.24512i q^{85} -6.91644i q^{86} -0.480744 q^{88} +9.48720i q^{89} +(5.64626 - 2.16150i) q^{91} -3.66679 q^{92} +6.24102 q^{94} -0.285927 q^{95} -9.71535i q^{97} -4.18828i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{4} - 2 q^{13} - 8 q^{14} + 14 q^{16} + 8 q^{17} + 8 q^{23} - 14 q^{25} + 4 q^{26} + 16 q^{29} - 34 q^{35} + 4 q^{43} - 10 q^{49} + 2 q^{52} - 60 q^{53} + 8 q^{56} - 28 q^{61} + 34 q^{62} - 14 q^{64}+ \cdots - 8 q^{92}+O(q^{100}) $ 14 * q - 14 * q^4 - 2 * q^13 - 8 * q^14 + 14 * q^16 + 8 * q^17 + 8 * q^23 - 14 * q^25 + 4 * q^26 + 16 * q^29 - 34 * q^35 + 4 * q^43 - 10 * q^49 + 2 * q^52 - 60 * q^53 + 8 * q^56 - 28 * q^61 + 34 * q^62 - 14 * q^64 + 8 * q^65 - 8 * q^68 - 16 * q^74 + 24 * q^77 - 28 * q^79 - 24 * q^82 - 8 * q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2106\mathbb{Z}\right)^\times$.

$n$	$1379$	$1783$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			0.594758i			0.265984i	0.991117	+	0.132992i	$0.0424585\pi$
$5$			0.594758i			0.265984i	−0.991117	+	0.132992i	$0.957542\pi$
$6$		0			0
$7$			1.67682i			0.633778i	0.948463	+	0.316889i	$0.102638\pi$
$7$			1.67682i			0.633778i	−0.948463	+	0.316889i	$0.897362\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	0.594758			0.188079
$11$			0.480744i			0.144950i	0.997370	+	0.0724750i	$0.0230897\pi$
$11$			0.480744i			0.144950i	−0.997370	+	0.0724750i	$0.976910\pi$
$12$		0			0
$13$	−1.28905	−	3.36725i	−0.357517	−	0.933907i
$13$	−1.28905	−	3.36725i	−0.357517	−	0.933907i
$14$	1.67682			0.448149
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.09349			−0.507746			−0.253873	−	0.967238i	$-0.581705\pi$
$17$	−2.09349			−0.507746			−0.253873	+	0.967238i	$0.581705\pi$
$18$		0			0
$19$			0.480744i			0.110290i	0.998478	+	0.0551452i	$0.0175622\pi$
$19$			0.480744i			0.110290i	−0.998478	+	0.0551452i	$0.982438\pi$
$20$		−	0.594758i		−	0.132992i
$21$		0			0
$22$	0.480744			0.102495
$23$	3.66679			0.764578			0.382289	−	0.924043i	$-0.375136\pi$
$23$	3.66679			0.764578			0.382289	+	0.924043i	$0.375136\pi$
$24$		0			0
$25$	4.64626			0.929253
$26$	−3.36725	+	1.28905i	−0.660372	+	0.252803i
$27$		0			0
$28$		−	1.67682i		−	0.316889i
$29$	−2.46678			−0.458069			−0.229035	−	0.973418i	$-0.573557\pi$
$29$	−2.46678			−0.458069			−0.229035	+	0.973418i	$0.573557\pi$
$30$		0			0
$31$		−	1.14753i		−	0.206103i	−0.994676	−	0.103051i	$-0.967139\pi$
$31$		−	1.14753i		−	0.206103i	0.994676	−	0.103051i	$-0.0328606\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			2.09349i			0.359031i
$35$	−0.997301			−0.168575
$36$		0			0
$37$			3.65012i			0.600076i	0.953927	+	0.300038i	$0.0969994\pi$
$37$			3.65012i			0.600076i	−0.953927	+	0.300038i	$0.903001\pi$
$38$	0.480744			0.0779870
$39$		0			0
$40$	−0.594758			−0.0940395
$41$			9.91005i			1.54769i	0.633376	+	0.773845i	$0.281669\pi$
$41$			9.91005i			1.54769i	−0.633376	+	0.773845i	$0.718331\pi$
$42$		0			0
$43$	6.91644			1.05475			0.527374	−	0.849633i	$-0.323177\pi$
$43$	6.91644			1.05475			0.527374	+	0.849633i	$0.323177\pi$
$44$		−	0.480744i		−	0.0724750i
$45$		0			0
$46$		−	3.66679i		−	0.540638i
$47$			6.24102i			0.910346i	0.890403	+	0.455173i	$0.150423\pi$
$47$			6.24102i			0.910346i	−0.890403	+	0.455173i	$0.849577\pi$
$48$		0			0
$49$	4.18828			0.598326
$50$		−	4.64626i		−	0.657081i
$51$		0			0
$52$	1.28905	+	3.36725i	0.178759	+	0.466953i
$53$	−5.08592			−0.698605			−0.349302	−	0.937010i	$-0.613581\pi$
$53$	−5.08592			−0.698605			−0.349302	+	0.937010i	$0.613581\pi$
$54$		0			0
$55$	−0.285927			−0.0385544
$56$	−1.67682			−0.224074
$57$		0			0
$58$			2.46678i			0.323904i
$59$			9.38985i			1.22245i	0.791455	+	0.611227i	$0.209324\pi$
$59$			9.38985i			1.22245i	−0.791455	+	0.611227i	$0.790676\pi$
$60$		0			0
$61$	7.81270			1.00031			0.500157	−	0.865935i	$-0.333276\pi$
$61$	7.81270			1.00031			0.500157	+	0.865935i	$0.333276\pi$
$62$	−1.14753			−0.145737
$63$		0			0
$64$	−1.00000			−0.125000
$65$	2.00270	−	0.766671i	0.248404	−	0.0950939i
$66$		0			0
$67$			14.3819i			1.75703i	0.477714	+	0.878516i	$0.341466\pi$
$67$			14.3819i			1.75703i	−0.477714	+	0.878516i	$0.658534\pi$
$68$	2.09349			0.253873
$69$		0			0
$70$			0.997301i			0.119200i
$71$		−	6.51028i		−	0.772628i	−0.922367	−	0.386314i	$-0.873748\pi$
$71$		−	6.51028i		−	0.772628i	0.922367	−	0.386314i	$-0.126252\pi$
$72$		0			0
$73$			5.91514i			0.692315i	0.938176	+	0.346157i	$0.112514\pi$
$73$			5.91514i			0.692315i	−0.938176	+	0.346157i	$0.887486\pi$
$74$	3.65012			0.424318
$75$		0			0
$76$		−	0.480744i		−	0.0551452i
$77$	−0.806121			−0.0918660
$78$		0			0
$79$	−2.05790			−0.231532			−0.115766	−	0.993277i	$-0.536932\pi$
$79$	−2.05790			−0.231532			−0.115766	+	0.993277i	$0.536932\pi$
$80$			0.594758i			0.0664960i
$81$		0			0
$82$	9.91005			1.09438
$83$			11.0601i			1.21401i	0.794700	+	0.607003i	$0.207628\pi$
$83$			11.0601i			1.21401i	−0.794700	+	0.607003i	$0.792372\pi$
$84$		0			0
$85$		−	1.24512i		−	0.135052i
$86$		−	6.91644i		−	0.745819i
$87$		0			0
$88$	−0.480744			−0.0512475
$89$			9.48720i			1.00564i	0.864391	+	0.502821i	$0.167704\pi$
$89$			9.48720i			1.00564i	−0.864391	+	0.502821i	$0.832296\pi$
$90$		0			0
$91$	5.64626	−	2.16150i	0.591889	−	0.226586i
$92$	−3.66679			−0.382289
$93$		0			0
$94$	6.24102			0.643712
$95$	−0.285927			−0.0293355
$96$		0			0
$97$		−	9.71535i		−	0.986444i	−0.869903	−	0.493222i	$-0.835819\pi$
$97$		−	9.71535i		−	0.986444i	0.869903	−	0.493222i	$-0.164181\pi$
$98$		−	4.18828i		−	0.423080i
$99$		0			0
$100$	−4.64626			−0.464626
$101$	−5.79495			−0.576619			−0.288310	−	0.957537i	$-0.593093\pi$
$101$	−5.79495			−0.576619			−0.288310	+	0.957537i	$0.593093\pi$
$102$		0			0
$103$	1.71407			0.168893			0.0844463	−	0.996428i	$-0.473088\pi$
$103$	1.71407			0.168893			0.0844463	+	0.996428i	$0.473088\pi$
$104$	3.36725	−	1.28905i	0.330186	−	0.126401i
$105$		0			0
$106$			5.08592i			0.493988i
$107$	14.1021			1.36330			0.681651	−	0.731678i	$-0.261262\pi$
$107$	14.1021			1.36330			0.681651	+	0.731678i	$0.261262\pi$
$108$		0			0
$109$			18.3474i			1.75736i	0.477413	+	0.878679i	$0.341575\pi$
$109$			18.3474i			1.75736i	−0.477413	+	0.878679i	$0.658425\pi$
$110$			0.285927i			0.0272620i
$111$		0			0
$112$			1.67682i			0.158444i
$113$	−20.2779			−1.90759			−0.953793	−	0.300465i	$-0.902858\pi$
$113$	−20.2779			−1.90759			−0.953793	+	0.300465i	$0.902858\pi$
$114$		0			0
$115$			2.18085i			0.203365i
$116$	2.46678			0.229035
$117$		0			0
$118$	9.38985			0.864406
$119$		−	3.51040i		−	0.321798i
$120$		0			0
$121$	10.7689			0.978990
$122$		−	7.81270i		−	0.707328i
$123$		0			0
$124$			1.14753i			0.103051i
$125$			5.73719i			0.513150i
$126$		0			0
$127$	18.5473			1.64581			0.822904	−	0.568180i	$-0.192352\pi$
$127$	18.5473			1.64581			0.822904	+	0.568180i	$0.192352\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−0.766671	−	2.00270i	−0.0672415	−	0.175648i
$131$	−4.64513			−0.405847			−0.202923	−	0.979195i	$-0.565044\pi$
$131$	−4.64513			−0.405847			−0.202923	+	0.979195i	$0.565044\pi$
$132$		0			0
$133$	−0.806121			−0.0698996
$134$	14.3819			1.24241
$135$		0			0
$136$		−	2.09349i		−	0.179515i
$137$		−	13.0207i		−	1.11243i	−0.831038	−	0.556216i	$-0.812253\pi$
$137$		−	13.0207i		−	1.11243i	0.831038	−	0.556216i	$-0.187747\pi$
$138$		0			0
$139$	10.1805			0.863502			0.431751	−	0.901993i	$-0.357896\pi$
$139$	10.1805			0.863502			0.431751	+	0.901993i	$0.357896\pi$
$140$	0.997301			0.0842874
$141$		0			0
$142$	−6.51028			−0.546331
$143$	1.61879	−	0.619702i	0.135370	−	0.0518221i
$144$		0			0
$145$		−	1.46714i		−	0.121839i
$146$	5.91514			0.489541
$147$		0			0
$148$		−	3.65012i		−	0.300038i
$149$			5.25274i			0.430321i	0.976579	+	0.215161i	$0.0690275\pi$
$149$			5.25274i			0.430321i	−0.976579	+	0.215161i	$0.930973\pi$
$150$		0			0
$151$		−	11.9232i		−	0.970299i	−0.874431	−	0.485150i	$-0.838765\pi$
$151$		−	11.9232i		−	0.970299i	0.874431	−	0.485150i	$-0.161235\pi$
$152$	−0.480744			−0.0389935
$153$		0			0
$154$			0.806121i			0.0649591i
$155$	0.682503			0.0548200
$156$		0			0
$157$	12.3908			0.988893			0.494446	−	0.869208i	$-0.335371\pi$
$157$	12.3908			0.988893			0.494446	+	0.869208i	$0.335371\pi$
$158$			2.05790i			0.163718i
$159$		0			0
$160$	0.594758			0.0470198
$161$			6.14853i			0.484572i
$162$		0			0
$163$		−	21.3629i		−	1.67327i	−0.547760	−	0.836635i	$-0.684519\pi$
$163$		−	21.3629i		−	1.67327i	0.547760	−	0.836635i	$-0.315481\pi$
$164$		−	9.91005i		−	0.773845i
$165$		0			0
$166$	11.0601			0.858431
$167$		−	14.7356i		−	1.14027i	−0.821551	−	0.570136i	$-0.806891\pi$
$167$		−	14.7356i		−	1.14027i	0.821551	−	0.570136i	$-0.193109\pi$
$168$		0			0
$169$	−9.67672	+	8.68108i	−0.744363	+	0.667775i
$170$	−1.24512			−0.0954964
$171$		0			0
$172$	−6.91644			−0.527374
$173$	11.6487			0.885634			0.442817	−	0.896612i	$-0.353979\pi$
$173$	11.6487			0.885634			0.442817	+	0.896612i	$0.353979\pi$
$174$		0			0
$175$			7.79094i			0.588940i
$176$			0.480744i			0.0362375i
$177$		0			0
$178$	9.48720			0.711096
$179$	−0.896158			−0.0669820			−0.0334910	−	0.999439i	$-0.510663\pi$
$179$	−0.896158			−0.0669820			−0.0334910	+	0.999439i	$0.510663\pi$
$180$		0			0
$181$	1.42191			0.105689			0.0528447	−	0.998603i	$-0.483171\pi$
$181$	1.42191			0.105689			0.0528447	+	0.998603i	$0.483171\pi$
$182$	−2.16150	−	5.64626i	−0.160221	−	0.418529i
$183$		0			0
$184$			3.66679i			0.270319i
$185$	−2.17094			−0.159611
$186$		0			0
$187$		−	1.00643i		−	0.0735977i
$188$		−	6.24102i		−	0.455173i
$189$		0			0
$190$			0.285927i			0.0207433i
$191$	−14.8407			−1.07384			−0.536918	−	0.843634i	$-0.680412\pi$
$191$	−14.8407			−1.07384			−0.536918	+	0.843634i	$0.680412\pi$
$192$		0			0
$193$		−	14.1075i		−	1.01548i	−0.861510	−	0.507741i	$-0.830481\pi$
$193$		−	14.1075i		−	1.01548i	0.861510	−	0.507741i	$-0.169519\pi$
$194$	−9.71535			−0.697521
$195$		0			0
$196$	−4.18828			−0.299163
$197$		−	0.442517i		−	0.0315280i	−0.999876	−	0.0157640i	$-0.994982\pi$
$197$		−	0.442517i		−	0.0315280i	0.999876	−	0.0157640i	$-0.00501805\pi$
$198$		0			0
$199$	−13.2925			−0.942282			−0.471141	−	0.882058i	$-0.656158\pi$
$199$	−13.2925			−0.942282			−0.471141	+	0.882058i	$0.656158\pi$
$200$			4.64626i			0.328540i
$201$		0			0
$202$			5.79495i			0.407732i
$203$		−	4.13634i		−	0.290314i
$204$		0			0
$205$	−5.89408			−0.411661
$206$		−	1.71407i		−	0.119425i
$207$		0			0
$208$	−1.28905	−	3.36725i	−0.0893793	−	0.233477i
$209$	−0.231115			−0.0159866
$210$		0			0
$211$	−9.64064			−0.663689			−0.331845	−	0.943334i	$-0.607671\pi$
$211$	−9.64064			−0.663689			−0.331845	+	0.943334i	$0.607671\pi$
$212$	5.08592			0.349302
$213$		0			0
$214$		−	14.1021i		−	0.964000i
$215$			4.11361i			0.280546i
$216$		0			0
$217$	1.92420			0.130623
$218$	18.3474			1.24264
$219$		0			0
$220$	0.285927			0.0192772
$221$	2.69861	+	7.04930i	0.181528	+	0.474187i
$222$		0			0
$223$			23.1333i			1.54912i	0.632501	+	0.774559i	$0.282028\pi$
$223$			23.1333i			1.54912i	−0.632501	+	0.774559i	$0.717972\pi$
$224$	1.67682			0.112037
$225$		0			0
$226$			20.2779i			1.34887i
$227$			13.3066i			0.883191i	0.897214	+	0.441595i	$0.145587\pi$
$227$			13.3066i			0.883191i	−0.897214	+	0.441595i	$0.854413\pi$
$228$		0			0
$229$		−	0.809405i		−	0.0534870i	−0.999642	−	0.0267435i	$-0.991486\pi$
$229$		−	0.809405i		−	0.0534870i	0.999642	−	0.0267435i	$-0.00851373\pi$
$230$	2.18085			0.143801
$231$		0			0
$232$		−	2.46678i		−	0.161952i
$233$	−15.4712			−1.01355			−0.506776	−	0.862078i	$-0.669163\pi$
$233$	−15.4712			−1.01355			−0.506776	+	0.862078i	$0.669163\pi$
$234$		0			0
$235$	−3.71190			−0.242138
$236$		−	9.38985i		−	0.611227i
$237$		0			0
$238$	−3.51040			−0.227546
$239$		−	29.1575i		−	1.88604i	−0.332733	−	0.943021i	$-0.607971\pi$
$239$		−	29.1575i		−	1.88604i	0.332733	−	0.943021i	$-0.392029\pi$
$240$		0			0
$241$			15.5947i			1.00454i	0.864710	+	0.502271i	$0.167502\pi$
$241$			15.5947i			1.00454i	−0.864710	+	0.502271i	$0.832498\pi$
$242$		−	10.7689i		−	0.692250i
$243$		0			0
$244$	−7.81270			−0.500157
$245$			2.49101i			0.159145i
$246$		0			0
$247$	1.61879	−	0.619702i	0.103001	−	0.0394307i
$248$	1.14753			0.0728683
$249$		0			0
$250$	5.73719			0.362852
$251$	−2.79495			−0.176416			−0.0882079	−	0.996102i	$-0.528114\pi$
$251$	−2.79495			−0.176416			−0.0882079	+	0.996102i	$0.528114\pi$
$252$		0			0
$253$			1.76279i			0.110825i
$254$		−	18.5473i		−	1.16376i
$255$		0			0
$256$	1.00000			0.0625000
$257$	20.9596			1.30742			0.653712	−	0.756743i	$-0.273211\pi$
$257$	20.9596			1.30742			0.653712	+	0.756743i	$0.273211\pi$
$258$		0			0
$259$	−6.12059			−0.380315
$260$	−2.00270	+	0.766671i	−0.124202	+	0.0475469i
$261$		0			0
$262$			4.64513i			0.286977i
$263$	15.8484			0.977252			0.488626	−	0.872493i	$-0.337498\pi$
$263$	15.8484			0.977252			0.488626	+	0.872493i	$0.337498\pi$
$264$		0			0
$265$		−	3.02489i		−	0.185818i
$266$			0.806121i			0.0494265i
$267$		0			0
$268$		−	14.3819i		−	0.878516i
$269$	18.9028			1.15252			0.576261	−	0.817266i	$-0.304511\pi$
$269$	18.9028			1.15252			0.576261	+	0.817266i	$0.304511\pi$
$270$		0			0
$271$		−	9.63804i		−	0.585469i	−0.956194	−	0.292735i	$-0.905435\pi$
$271$		−	9.63804i		−	0.585469i	0.956194	−	0.292735i	$-0.0945653\pi$
$272$	−2.09349			−0.126936
$273$		0			0
$274$	−13.0207			−0.786608
$275$			2.23366i			0.134695i
$276$		0			0
$277$	2.70020			0.162239			0.0811197	−	0.996704i	$-0.474150\pi$
$277$	2.70020			0.162239			0.0811197	+	0.996704i	$0.474150\pi$
$278$		−	10.1805i		−	0.610588i
$279$		0			0
$280$		−	0.997301i		−	0.0596002i
$281$			10.7944i			0.643938i	0.946750	+	0.321969i	$0.104345\pi$
$281$			10.7944i			0.643938i	−0.946750	+	0.321969i	$0.895655\pi$
$282$		0			0
$283$	−29.2176			−1.73681			−0.868403	−	0.495859i	$-0.834853\pi$
$283$	−29.2176			−1.73681			−0.868403	+	0.495859i	$0.834853\pi$
$284$			6.51028i			0.386314i
$285$		0			0
$286$	−0.619702	−	1.61879i	−0.0366437	−	0.0957208i
$287$	−16.6173			−0.980891
$288$		0			0
$289$	−12.6173			−0.742194
$290$	−1.46714			−0.0861532
$291$		0			0
$292$		−	5.91514i		−	0.346157i
$293$			23.6944i			1.38424i	0.721783	+	0.692120i	$0.243323\pi$
$293$			23.6944i			1.38424i	−0.721783	+	0.692120i	$0.756677\pi$
$294$		0			0
$295$	−5.58469			−0.325153
$296$	−3.65012			−0.212159
$297$		0			0
$298$	5.25274			0.304283
$299$	−4.72666	−	12.3470i	−0.273350	−	0.714044i
$300$		0			0
$301$			11.5976i			0.668475i
$302$	−11.9232			−0.686105
$303$		0			0
$304$			0.480744i			0.0275726i
$305$			4.64667i			0.266067i
$306$		0			0
$307$		−	6.55237i		−	0.373963i	−0.982363	−	0.186982i	$-0.940129\pi$
$307$		−	6.55237i		−	0.373963i	0.982363	−	0.186982i	$-0.0598705\pi$
$308$	0.806121			0.0459330
$309$		0			0
$310$		−	0.682503i		−	0.0387636i
$311$	25.4230			1.44160			0.720802	−	0.693141i	$-0.243774\pi$
$311$	25.4230			1.44160			0.720802	+	0.693141i	$0.243774\pi$
$312$		0			0
$313$	1.18146			0.0667799			0.0333899	−	0.999442i	$-0.489370\pi$
$313$	1.18146			0.0667799			0.0333899	+	0.999442i	$0.489370\pi$
$314$		−	12.3908i		−	0.699253i
$315$		0			0
$316$	2.05790			0.115766
$317$			27.9448i			1.56954i	0.619789	+	0.784769i	$0.287218\pi$
$317$			27.9448i			1.56954i	−0.619789	+	0.784769i	$0.712782\pi$
$318$		0			0
$319$		−	1.18589i		−	0.0663971i
$320$		−	0.594758i		−	0.0332480i
$321$		0			0
$322$	6.14853			0.342644
$323$		−	1.00643i		−	0.0559995i
$324$		0			0
$325$	−5.98925	−	15.6451i	−0.332224	−	0.867835i
$326$	−21.3629			−1.18318
$327$		0			0
$328$	−9.91005			−0.547191
$329$	−10.4651			−0.576957
$330$		0			0
$331$		−	24.6531i		−	1.35506i	−0.735495	−	0.677530i	$-0.763051\pi$
$331$		−	24.6531i		−	1.35506i	0.735495	−	0.677530i	$-0.236949\pi$
$332$		−	11.0601i		−	0.607003i
$333$		0			0
$334$	−14.7356			−0.806294
$335$	−8.55376			−0.467342
$336$		0			0
$337$	−15.2788			−0.832290			−0.416145	−	0.909298i	$-0.636619\pi$
$337$	−15.2788			−0.832290			−0.416145	+	0.909298i	$0.636619\pi$
$338$	8.68108	+	9.67672i	0.472188	+	0.526344i
$339$		0			0
$340$			1.24512i			0.0675261i
$341$	0.551669			0.0298745
$342$		0			0
$343$			18.7607i			1.01298i
$344$			6.91644i			0.372909i
$345$		0			0
$346$		−	11.6487i		−	0.626238i
$347$	−1.21538			−0.0652448			−0.0326224	−	0.999468i	$-0.510386\pi$
$347$	−1.21538			−0.0652448			−0.0326224	+	0.999468i	$0.510386\pi$
$348$		0			0
$349$			3.81160i			0.204030i	0.994783	+	0.102015i	$0.0325290\pi$
$349$			3.81160i			0.204030i	−0.994783	+	0.102015i	$0.967471\pi$
$350$	7.79094			0.416443
$351$		0			0
$352$	0.480744			0.0256238
$353$		−	20.0171i		−	1.06540i	−0.846303	−	0.532702i	$-0.821177\pi$
$353$		−	20.0171i		−	1.06540i	0.846303	−	0.532702i	$-0.178823\pi$
$354$		0			0
$355$	3.87204			0.205507
$356$		−	9.48720i		−	0.502821i
$357$		0			0
$358$			0.896158i			0.0473634i
$359$		−	9.96717i		−	0.526047i	−0.964789	−	0.263024i	$-0.915280\pi$
$359$		−	9.96717i		−	0.526047i	0.964789	−	0.263024i	$-0.0847197\pi$
$360$		0			0
$361$	18.7689			0.987836
$362$		−	1.42191i		−	0.0747338i
$363$		0			0
$364$	−5.64626	+	2.16150i	−0.295945	+	0.113293i
$365$	−3.51808			−0.184145
$366$		0			0
$367$	14.3267			0.747846			0.373923	−	0.927460i	$-0.378012\pi$
$367$	14.3267			0.747846			0.373923	+	0.927460i	$0.378012\pi$
$368$	3.66679			0.191144
$369$		0			0
$370$			2.17094i			0.112862i
$371$		−	8.52816i		−	0.442760i
$372$		0			0
$373$	−27.9755			−1.44851			−0.724257	−	0.689530i	$-0.757817\pi$
$373$	−27.9755			−1.44851			−0.724257	+	0.689530i	$0.757817\pi$
$374$	−1.00643			−0.0520415
$375$		0			0
$376$	−6.24102			−0.321856
$377$	3.17979	+	8.30625i	0.163768	+	0.427794i
$378$		0			0
$379$		−	4.05068i		−	0.208069i	−0.994574	−	0.104035i	$-0.966825\pi$
$379$		−	4.05068i		−	0.208069i	0.994574	−	0.104035i	$-0.0331753\pi$
$380$	0.285927			0.0146677
$381$		0			0
$382$			14.8407i			0.759317i
$383$		−	31.2962i		−	1.59916i	−0.600558	−	0.799581i	$-0.705055\pi$
$383$		−	31.2962i		−	1.59916i	0.600558	−	0.799581i	$-0.294945\pi$
$384$		0			0
$385$		−	0.479447i		−	0.0244349i
$386$	−14.1075			−0.718054
$387$		0			0
$388$			9.71535i			0.493222i
$389$	−15.3393			−0.777733			−0.388866	−	0.921294i	$-0.627133\pi$
$389$	−15.3393			−0.777733			−0.388866	+	0.921294i	$0.627133\pi$
$390$		0			0
$391$	−7.67638			−0.388211
$392$			4.18828i			0.211540i
$393$		0			0
$394$	−0.442517			−0.0222937
$395$		−	1.22395i		−	0.0615837i
$396$		0			0
$397$		−	24.1832i		−	1.21372i	−0.794808	−	0.606860i	$-0.792429\pi$
$397$		−	24.1832i		−	1.21372i	0.794808	−	0.606860i	$-0.207571\pi$
$398$			13.2925i			0.666294i
$399$		0			0
$400$	4.64626			0.232313
$401$			28.7378i			1.43510i	0.696509	+	0.717548i	$0.254736\pi$
$401$			28.7378i			1.43510i	−0.696509	+	0.717548i	$0.745264\pi$
$402$		0			0
$403$	−3.86402	+	1.47922i	−0.192481	+	0.0736852i
$404$	5.79495			0.288310
$405$		0			0
$406$	−4.13634			−0.205283
$407$	−1.75478			−0.0869810
$408$		0			0
$409$		−	29.9183i		−	1.47936i	−0.672956	−	0.739682i	$-0.734976\pi$
$409$		−	29.9183i		−	1.47936i	0.672956	−	0.739682i	$-0.265024\pi$
$410$			5.89408i			0.291088i
$411$		0			0
$412$	−1.71407			−0.0844463
$413$	−15.7451			−0.774764
$414$		0			0
$415$	−6.57809			−0.322906
$416$	−3.36725	+	1.28905i	−0.165093	+	0.0632007i
$417$		0			0
$418$			0.231115i			0.0113042i
$419$	24.5716			1.20040			0.600200	−	0.799850i	$-0.295088\pi$
$419$	24.5716			1.20040			0.600200	+	0.799850i	$0.295088\pi$
$420$		0			0
$421$		−	28.4905i		−	1.38854i	−0.719714	−	0.694271i	$-0.755727\pi$
$421$		−	28.4905i		−	1.38854i	0.719714	−	0.694271i	$-0.244273\pi$
$422$			9.64064i			0.469299i
$423$		0			0
$424$		−	5.08592i		−	0.246994i
$425$	−9.72691			−0.471824
$426$		0			0
$427$			13.1005i			0.633976i
$428$	−14.1021			−0.681651
$429$		0			0
$430$	4.11361			0.198376
$431$			7.13767i			0.343809i	0.985114	+	0.171905i	$0.0549921\pi$
$431$			7.13767i			0.343809i	−0.985114	+	0.171905i	$0.945008\pi$
$432$		0			0
$433$	−7.85282			−0.377382			−0.188691	−	0.982036i	$-0.560424\pi$
$433$	−7.85282			−0.377382			−0.188691	+	0.982036i	$0.560424\pi$
$434$		−	1.92420i		−	0.0923646i
$435$		0			0
$436$		−	18.3474i		−	0.878679i
$437$			1.76279i			0.0843255i
$438$		0			0
$439$	−4.14853			−0.197999			−0.0989994	−	0.995087i	$-0.531564\pi$
$439$	−4.14853			−0.197999			−0.0989994	+	0.995087i	$0.531564\pi$
$440$		−	0.285927i		−	0.0136310i
$441$		0			0
$442$	7.04930	−	2.69861i	0.335301	−	0.128360i
$443$	17.8470			0.847936			0.423968	−	0.905677i	$-0.360637\pi$
$443$	17.8470			0.847936			0.423968	+	0.905677i	$0.360637\pi$
$444$		0			0
$445$	−5.64259			−0.267484
$446$	23.1333			1.09539
$447$		0			0
$448$		−	1.67682i		−	0.0792222i
$449$			19.7308i			0.931152i	0.885008	+	0.465576i	$0.154153\pi$
$449$			19.7308i			0.931152i	−0.885008	+	0.465576i	$0.845847\pi$
$450$		0			0
$451$	−4.76420			−0.224337
$452$	20.2779			0.953793
$453$		0			0
$454$	13.3066			0.624510
$455$	1.28557	+	3.35816i	0.0602684	+	0.157433i
$456$		0			0
$457$			36.3456i			1.70018i	0.526641	+	0.850088i	$0.323451\pi$
$457$			36.3456i			1.70018i	−0.526641	+	0.850088i	$0.676549\pi$
$458$	−0.809405			−0.0378210
$459$		0			0
$460$		−	2.18085i		−	0.101683i
$461$			22.9358i			1.06823i	0.845413	+	0.534114i	$0.179355\pi$
$461$			22.9358i			1.06823i	−0.845413	+	0.534114i	$0.820645\pi$
$462$		0			0
$463$		−	29.5769i		−	1.37455i	−0.726395	−	0.687277i	$-0.758806\pi$
$463$		−	29.5769i		−	1.37455i	0.726395	−	0.687277i	$-0.241194\pi$
$464$	−2.46678			−0.114517
$465$		0			0
$466$			15.4712i			0.716689i
$467$	−12.5970			−0.582920			−0.291460	−	0.956583i	$-0.594141\pi$
$467$	−12.5970			−0.582920			−0.291460	+	0.956583i	$0.594141\pi$
$468$		0			0
$469$	−24.1159			−1.11357
$470$			3.71190i			0.171217i
$471$		0			0
$472$	−9.38985			−0.432203
$473$			3.32504i			0.152886i
$474$		0			0
$475$			2.23366i			0.102488i
$476$			3.51040i			0.160899i
$477$		0			0
$478$	−29.1575			−1.33363
$479$		−	12.7483i		−	0.582486i	−0.956649	−	0.291243i	$-0.905931\pi$
$479$		−	12.7483i		−	0.582486i	0.956649	−	0.291243i	$-0.0940689\pi$
$480$		0			0
$481$	12.2909	−	4.70518i	0.560415	−	0.214538i
$482$	15.5947			0.710319
$483$		0			0
$484$	−10.7689			−0.489495
$485$	5.77828			0.262378
$486$		0			0
$487$			20.8054i			0.942782i	0.881924	+	0.471391i	$0.156248\pi$
$487$			20.8054i			0.942782i	−0.881924	+	0.471391i	$0.843752\pi$
$488$			7.81270i			0.353664i
$489$		0			0
$490$	2.49101			0.112533
$491$	−14.7472			−0.665534			−0.332767	−	0.943009i	$-0.607982\pi$
$491$	−14.7472			−0.665534			−0.332767	+	0.943009i	$0.607982\pi$
$492$		0			0
$493$	5.16418			0.232583
$494$	−0.619702	−	1.61879i	−0.0278817	−	0.0728326i
$495$		0			0
$496$		−	1.14753i		−	0.0515256i
$497$	10.9166			0.489675
$498$		0			0
$499$			29.7482i			1.33171i	0.746081	+	0.665856i	$0.231933\pi$
$499$			29.7482i			1.33171i	−0.746081	+	0.665856i	$0.768067\pi$
$500$		−	5.73719i		−	0.256575i
$501$		0			0
$502$			2.79495i			0.124745i
$503$	−26.1270			−1.16495			−0.582473	−	0.812850i	$-0.697915\pi$
$503$	−26.1270			−1.16495			−0.582473	+	0.812850i	$0.697915\pi$
$504$		0			0
$505$		−	3.44660i		−	0.153372i
$506$	1.76279			0.0783654
$507$		0			0
$508$	−18.5473			−0.822904
$509$			17.8759i			0.792335i	0.918178	+	0.396168i	$0.129660\pi$
$509$			17.8759i			0.792335i	−0.918178	+	0.396168i	$0.870340\pi$
$510$		0			0
$511$	−9.91862			−0.438774
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	20.9596i		−	0.924488i
$515$			1.01946i			0.0449227i
$516$		0			0
$517$	−3.00034			−0.131955
$518$			6.12059i			0.268923i
$519$		0			0
$520$	0.766671	+	2.00270i	0.0336208	+	0.0878241i
$521$	−13.7904			−0.604168			−0.302084	−	0.953281i	$-0.597682\pi$
$521$	−13.7904			−0.604168			−0.302084	+	0.953281i	$0.597682\pi$
$522$		0			0
$523$	−0.523640			−0.0228972			−0.0114486	−	0.999934i	$-0.503644\pi$
$523$	−0.523640			−0.0228972			−0.0114486	+	0.999934i	$0.503644\pi$
$524$	4.64513			0.202923
$525$		0			0
$526$		−	15.8484i		−	0.691022i
$527$			2.40234i			0.104648i
$528$		0			0
$529$	−9.55468			−0.415421
$530$	−3.02489			−0.131393
$531$		0			0
$532$	0.806121			0.0349498
$533$	33.3696	−	12.7745i	1.44540	−	0.553326i
$534$		0			0
$535$			8.38734i			0.362616i
$536$	−14.3819			−0.621204
$537$		0			0
$538$		−	18.9028i		−	0.814957i
$539$			2.01349i			0.0867273i
$540$		0			0
$541$		−	11.3636i		−	0.488558i	−0.969705	−	0.244279i	$-0.921449\pi$
$541$		−	11.3636i		−	0.488558i	0.969705	−	0.244279i	$-0.0785513\pi$
$542$	−9.63804			−0.413989
$543$		0			0
$544$			2.09349i			0.0897577i
$545$	−10.9122			−0.467429
$546$		0			0
$547$	−8.28479			−0.354232			−0.177116	−	0.984190i	$-0.556677\pi$
$547$	−8.28479			−0.354232			−0.177116	+	0.984190i	$0.556677\pi$
$548$			13.0207i			0.556216i
$549$		0			0
$550$	2.23366			0.0952438
$551$		−	1.18589i		−	0.0505206i
$552$		0			0
$553$		−	3.45072i		−	0.146740i
$554$		−	2.70020i		−	0.114721i
$555$		0			0
$556$	−10.1805			−0.431751
$557$		−	26.7334i		−	1.13273i	−0.824154	−	0.566365i	$-0.808349\pi$
$557$		−	26.7334i		−	1.13273i	0.824154	−	0.566365i	$-0.191651\pi$
$558$		0			0
$559$	−8.91562	−	23.2894i	−0.377090	−	0.985035i
$560$	−0.997301			−0.0421437
$561$		0			0
$562$	10.7944			0.455333
$563$	4.72143			0.198984			0.0994922	−	0.995038i	$-0.468278\pi$
$563$	4.72143			0.198984			0.0994922	+	0.995038i	$0.468278\pi$
$564$		0			0
$565$		−	12.0605i		−	0.507387i
$566$			29.2176i			1.22811i
$567$		0			0
$568$	6.51028			0.273165
$569$	−42.1091			−1.76531			−0.882653	−	0.470025i	$-0.844245\pi$
$569$	−42.1091			−1.76531			−0.882653	+	0.470025i	$0.844245\pi$
$570$		0			0
$571$	−11.7844			−0.493161			−0.246581	−	0.969122i	$-0.579307\pi$
$571$	−11.7844			−0.493161			−0.246581	+	0.969122i	$0.579307\pi$
$572$	−1.61879	+	0.619702i	−0.0676848	+	0.0259110i
$573$		0			0
$574$			16.6173i			0.693595i
$575$	17.0369			0.710486
$576$		0			0
$577$			21.5466i			0.896997i	0.893784	+	0.448498i	$0.148041\pi$
$577$			21.5466i			0.896997i	−0.893784	+	0.448498i	$0.851959\pi$
$578$			12.6173i			0.524810i
$579$		0			0
$580$			1.46714i			0.0609195i
$581$	−18.5458			−0.769409
$582$		0			0
$583$		−	2.44503i		−	0.101263i
$584$	−5.91514			−0.244770
$585$		0			0
$586$	23.6944			0.978805
$587$		−	8.80990i		−	0.363624i	−0.983333	−	0.181812i	$-0.941804\pi$
$587$		−	8.80990i		−	0.363624i	0.983333	−	0.181812i	$-0.0581962\pi$
$588$		0			0
$589$	0.551669			0.0227311
$590$			5.58469i			0.229918i
$591$		0			0
$592$			3.65012i			0.150019i
$593$			26.7483i			1.09842i	0.835685	+	0.549210i	$0.185071\pi$
$593$			26.7483i			1.09842i	−0.835685	+	0.549210i	$0.814929\pi$
$594$		0			0
$595$	2.08784			0.0855931
$596$		−	5.25274i		−	0.215161i
$597$		0			0
$598$	−12.3470	+	4.72666i	−0.504905	+	0.193287i
$599$	−47.8449			−1.95489			−0.977445	−	0.211192i	$-0.932266\pi$
$599$	−47.8449			−1.95489			−0.977445	+	0.211192i	$0.932266\pi$
$600$		0			0
$601$	−21.0386			−0.858181			−0.429091	−	0.903261i	$-0.641166\pi$
$601$	−21.0386			−0.858181			−0.429091	+	0.903261i	$0.641166\pi$
$602$	11.5976			0.472683
$603$		0			0
$604$			11.9232i			0.485150i
$605$			6.40488i			0.260396i
$606$		0			0
$607$	23.5367			0.955323			0.477662	−	0.878544i	$-0.341485\pi$
$607$	23.5367			0.955323			0.477662	+	0.878544i	$0.341485\pi$
$608$	0.480744			0.0194968
$609$		0			0
$610$	4.64667			0.188138
$611$	21.0151	−	8.04497i	0.850179	−	0.325465i
$612$		0			0
$613$			40.7009i			1.64390i	0.569563	+	0.821948i	$0.307112\pi$
$613$			40.7009i			1.64390i	−0.569563	+	0.821948i	$0.692888\pi$
$614$	−6.55237			−0.264432
$615$		0			0
$616$		−	0.806121i		−	0.0324795i
$617$		−	39.3665i		−	1.58483i	−0.609980	−	0.792417i	$-0.708822\pi$
$617$		−	39.3665i		−	1.58483i	0.609980	−	0.792417i	$-0.291178\pi$
$618$		0			0
$619$		−	24.4414i		−	0.982384i	−0.871051	−	0.491192i	$-0.836561\pi$
$619$		−	24.4414i		−	0.982384i	0.871051	−	0.491192i	$-0.163439\pi$
$620$	−0.682503			−0.0274100
$621$		0			0
$622$		−	25.4230i		−	1.01937i
$623$	−15.9083			−0.637353
$624$		0			0
$625$	19.8191			0.792763
$626$		−	1.18146i		−	0.0472205i
$627$		0			0
$628$	−12.3908			−0.494446
$629$		−	7.64149i		−	0.304686i
$630$		0			0
$631$		−	5.70930i		−	0.227284i	−0.993522	−	0.113642i	$-0.963748\pi$
$631$		−	5.70930i		−	0.227284i	0.993522	−	0.113642i	$-0.0362516\pi$
$632$		−	2.05790i		−	0.0818588i
$633$		0			0
$634$	27.9448			1.10983
$635$			11.0312i			0.437759i
$636$		0			0
$637$	−5.39889	−	14.1030i	−0.213912	−	0.558780i
$638$	−1.18589			−0.0469498
$639$		0			0
$640$	−0.594758			−0.0235099
$641$	39.4821			1.55945			0.779725	−	0.626122i	$-0.215359\pi$
$641$	39.4821			1.55945			0.779725	+	0.626122i	$0.215359\pi$
$642$		0			0
$643$		−	29.4988i		−	1.16332i	−0.813432	−	0.581660i	$-0.802403\pi$
$643$		−	29.4988i		−	1.16332i	0.813432	−	0.581660i	$-0.197597\pi$
$644$		−	6.14853i		−	0.242286i
$645$		0			0
$646$	−1.00643			−0.0395976
$647$	47.5940			1.87111			0.935556	−	0.353178i	$-0.114899\pi$
$647$	47.5940			1.87111			0.935556	+	0.353178i	$0.114899\pi$
$648$		0			0
$649$	−4.51412			−0.177195
$650$	−15.6451	+	5.98925i	−0.613652	+	0.234918i
$651$		0			0
$652$			21.3629i			0.836635i
$653$	12.4748			0.488175			0.244088	−	0.969753i	$-0.421512\pi$
$653$	12.4748			0.488175			0.244088	+	0.969753i	$0.421512\pi$
$654$		0			0
$655$		−	2.76273i		−	0.107949i
$656$			9.91005i			0.386922i
$657$		0			0
$658$			10.4651i			0.407970i
$659$	3.74611			0.145928			0.0729639	−	0.997335i	$-0.476754\pi$
$659$	3.74611			0.145928			0.0729639	+	0.997335i	$0.476754\pi$
$660$		0			0
$661$			33.0703i			1.28629i	0.765746	+	0.643143i	$0.222370\pi$
$661$			33.0703i			1.28629i	−0.765746	+	0.643143i	$0.777630\pi$
$662$	−24.6531			−0.958172
$663$		0			0
$664$	−11.0601			−0.429216
$665$		−	0.479447i		−	0.0185922i
$666$		0			0
$667$	−9.04515			−0.350230
$668$			14.7356i			0.570136i
$669$		0			0
$670$			8.55376i			0.330461i
$671$			3.75591i			0.144995i
$672$		0			0
$673$	−14.1436			−0.545194			−0.272597	−	0.962128i	$-0.587883\pi$
$673$	−14.1436			−0.545194			−0.272597	+	0.962128i	$0.587883\pi$
$674$			15.2788i			0.588518i
$675$		0			0
$676$	9.67672	−	8.68108i	0.372181	−	0.333888i
$677$	−34.2127			−1.31490			−0.657451	−	0.753498i	$-0.728365\pi$
$677$	−34.2127			−1.31490			−0.657451	+	0.753498i	$0.728365\pi$
$678$		0			0
$679$	16.2909			0.625186
$680$	1.24512			0.0477482
$681$		0			0
$682$		−	0.551669i		−	0.0211245i
$683$		−	16.3144i		−	0.624255i	−0.950040	−	0.312127i	$-0.898958\pi$
$683$		−	16.3144i		−	0.624255i	0.950040	−	0.312127i	$-0.101042\pi$
$684$		0			0
$685$	7.74416			0.295889
$686$	18.7607			0.716287
$687$		0			0
$688$	6.91644			0.263687
$689$	6.55599	+	17.1255i	0.249763	+	0.652431i
$690$		0			0
$691$			25.3475i			0.964265i	0.876098	+	0.482132i	$0.160138\pi$
$691$			25.3475i			0.964265i	−0.876098	+	0.482132i	$0.839862\pi$
$692$	−11.6487			−0.442817
$693$		0			0
$694$			1.21538i			0.0461350i
$695$			6.05496i			0.229678i
$696$		0			0
$697$		−	20.7466i		−	0.785833i
$698$	3.81160			0.144271
$699$		0			0
$700$		−	7.79094i		−	0.294470i
$701$	35.4651			1.33950			0.669750	−	0.742587i	$-0.266401\pi$
$701$	35.4651			1.33950			0.669750	+	0.742587i	$0.266401\pi$
$702$		0			0
$703$	−1.75478			−0.0661826
$704$		−	0.480744i		−	0.0181187i
$705$		0			0
$706$	−20.0171			−0.753354
$707$		−	9.71708i		−	0.365449i
$708$		0			0
$709$		−	21.6722i		−	0.813918i	−0.913446	−	0.406959i	$-0.866589\pi$
$709$		−	21.6722i		−	0.813918i	0.913446	−	0.406959i	$-0.133411\pi$
$710$		−	3.87204i		−	0.145315i
$711$		0			0
$712$	−9.48720			−0.355548
$713$		−	4.20775i		−	0.157581i
$714$		0			0
$715$	0.368573	+	0.962786i	0.0137838	+	0.0360062i
$716$	0.896158			0.0334910
$717$		0			0
$718$	−9.96717			−0.371972
$719$	15.4819			0.577378			0.288689	−	0.957423i	$-0.406781\pi$
$719$	15.4819			0.577378			0.288689	+	0.957423i	$0.406781\pi$
$720$		0			0
$721$			2.87419i			0.107040i
$722$		−	18.7689i		−	0.698506i
$723$		0			0
$724$	−1.42191			−0.0528447
$725$	−11.4613			−0.425662
$726$		0			0
$727$	−5.46560			−0.202708			−0.101354	−	0.994850i	$-0.532317\pi$
$727$	−5.46560			−0.202708			−0.101354	+	0.994850i	$0.532317\pi$
$728$	2.16150	+	5.64626i	0.0801104	+	0.209264i
$729$		0			0
$730$			3.51808i			0.130210i
$731$	−14.4795			−0.535544
$732$		0			0
$733$			19.1340i			0.706730i	0.935486	+	0.353365i	$0.114963\pi$
$733$			19.1340i			0.706730i	−0.935486	+	0.353365i	$0.885037\pi$
$734$		−	14.3267i		−	0.528807i
$735$		0			0
$736$		−	3.66679i		−	0.135160i
$737$	−6.91403			−0.254681
$738$		0			0
$739$			3.40317i			0.125188i	0.998039	+	0.0625938i	$0.0199373\pi$
$739$			3.40317i			0.125188i	−0.998039	+	0.0625938i	$0.980063\pi$
$740$	2.17094			0.0798053
$741$		0			0
$742$	−8.52816			−0.313079
$743$		−	22.9155i		−	0.840689i	−0.907365	−	0.420345i	$-0.861909\pi$
$743$		−	22.9155i		−	0.840689i	0.907365	−	0.420345i	$-0.138091\pi$
$744$		0			0
$745$	−3.12411			−0.114459
$746$			27.9755i			1.02425i
$747$		0			0
$748$			1.00643i			0.0367989i
$749$			23.6467i			0.864030i
$750$		0			0
$751$	−15.3739			−0.561001			−0.280500	−	0.959854i	$-0.590500\pi$
$751$	−15.3739			−0.561001			−0.280500	+	0.959854i	$0.590500\pi$
$752$			6.24102i			0.227587i
$753$		0			0
$754$	8.30625	−	3.17979i	0.302496	−	0.115801i
$755$	7.09144			0.258084
$756$		0			0
$757$	41.9292			1.52394			0.761971	−	0.647611i	$-0.224232\pi$
$757$	41.9292			1.52394			0.761971	+	0.647611i	$0.224232\pi$
$758$	−4.05068			−0.147127
$759$		0			0
$760$		−	0.285927i		−	0.0103717i
$761$		−	2.54040i		−	0.0920895i	−0.998939	−	0.0460447i	$-0.985338\pi$
$761$		−	2.54040i		−	0.0920895i	0.998939	−	0.0460447i	$-0.0146617\pi$
$762$		0			0
$763$	−30.7652			−1.11377
$764$	14.8407			0.536918
$765$		0			0
$766$	−31.2962			−1.13078
$767$	31.6180	−	12.1040i	1.14166	−	0.437049i
$768$		0			0
$769$			9.98966i			0.360236i	0.983645	+	0.180118i	$0.0576480\pi$
$769$			9.98966i			0.360236i	−0.983645	+	0.180118i	$0.942352\pi$
$770$	−0.479447			−0.0172781
$771$		0			0
$772$			14.1075i			0.507741i
$773$		−	5.76343i		−	0.207296i	−0.994614	−	0.103648i	$-0.966948\pi$
$773$		−	5.76343i		−	0.207296i	0.994614	−	0.103648i	$-0.0330516\pi$
$774$		0			0
$775$		−	5.33173i		−	0.191521i
$776$	9.71535			0.348761
$777$		0			0
$778$			15.3393i			0.549940i
$779$	−4.76420			−0.170695
$780$		0			0
$781$	3.12978			0.111992
$782$			7.67638i			0.274507i
$783$		0			0
$784$	4.18828			0.149581
$785$			7.36953i			0.263030i
$786$		0			0
$787$		−	9.47613i		−	0.337787i	−0.985634	−	0.168894i	$-0.945981\pi$
$787$		−	9.47613i		−	0.337787i	0.985634	−	0.168894i	$-0.0540195\pi$
$788$			0.442517i			0.0157640i
$789$		0			0
$790$	−1.22395			−0.0435463
$791$		−	34.0024i		−	1.20899i
$792$		0			0
$793$	−10.0709	−	26.3073i	−0.357629	−	0.934199i
$794$	−24.1832			−0.858230
$795$		0			0
$796$	13.2925			0.471141
$797$	−3.09742			−0.109716			−0.0548582	−	0.998494i	$-0.517471\pi$
$797$	−3.09742			−0.109716			−0.0548582	+	0.998494i	$0.517471\pi$
$798$		0			0
$799$		−	13.0655i		−	0.462225i
$800$		−	4.64626i		−	0.164270i
$801$		0			0
$802$	28.7378			1.01477
$803$	−2.84367			−0.100351
$804$		0			0
$805$	−3.65689			−0.128888
$806$	1.47922	+	3.86402i	0.0521033	+	0.136104i
$807$		0			0
$808$		−	5.79495i		−	0.203866i
$809$	18.7613			0.659612			0.329806	−	0.944049i	$-0.393017\pi$
$809$	18.7613			0.659612			0.329806	+	0.944049i	$0.393017\pi$
$810$		0			0
$811$			35.6282i			1.25108i	0.780193	+	0.625538i	$0.215121\pi$
$811$			35.6282i			1.25108i	−0.780193	+	0.625538i	$0.784879\pi$
$812$			4.13634i			0.145157i
$813$		0			0
$814$			1.75478i			0.0615049i
$815$	12.7058			0.445063
$816$		0			0
$817$			3.32504i			0.116328i
$818$	−29.9183			−1.04607
$819$		0			0
$820$	5.89408			0.205830
$821$			15.4701i			0.539911i	0.962873	+	0.269956i	$0.0870090\pi$
$821$			15.4701i			0.539911i	−0.962873	+	0.269956i	$0.912991\pi$
$822$		0			0
$823$	−8.08000			−0.281651			−0.140825	−	0.990034i	$-0.544976\pi$
$823$	−8.08000			−0.281651			−0.140825	+	0.990034i	$0.544976\pi$
$824$			1.71407i			0.0597126i
$825$		0			0
$826$			15.7451i			0.547841i
$827$		−	27.7387i		−	0.964569i	−0.876015	−	0.482284i	$-0.839807\pi$
$827$		−	27.7387i		−	0.964569i	0.876015	−	0.482284i	$-0.160193\pi$
$828$		0			0
$829$	−38.2112			−1.32713			−0.663566	−	0.748118i	$-0.730958\pi$
$829$	−38.2112			−1.32713			−0.663566	+	0.748118i	$0.730958\pi$
$830$			6.57809i			0.228329i
$831$		0			0
$832$	1.28905	+	3.36725i	0.0446897	+	0.116738i
$833$	−8.76812			−0.303798
$834$		0			0
$835$	8.76409			0.303294
$836$	0.231115			0.00799329
$837$		0			0
$838$		−	24.5716i		−	0.848811i
$839$		−	37.5319i		−	1.29574i	−0.761749	−	0.647872i	$-0.775659\pi$
$839$		−	37.5319i		−	1.29574i	0.761749	−	0.647872i	$-0.224341\pi$
$840$		0			0
$841$	−22.9150			−0.790173
$842$	−28.4905			−0.981847
$843$		0			0
$844$	9.64064			0.331845
$845$	−5.16314	−	5.75531i	−0.177618	−	0.197989i
$846$		0			0
$847$			18.0575i			0.620462i
$848$	−5.08592			−0.174651
$849$		0			0
$850$			9.72691i			0.333630i
$851$			13.3842i			0.458805i
$852$		0			0
$853$		−	49.9129i		−	1.70899i	−0.519463	−	0.854493i	$-0.673868\pi$
$853$		−	49.9129i		−	1.70899i	0.519463	−	0.854493i	$-0.326132\pi$
$854$	13.1005			0.448289
$855$		0			0
$856$			14.1021i			0.482000i
$857$	8.72992			0.298208			0.149104	−	0.988821i	$-0.452361\pi$
$857$	8.72992			0.298208			0.149104	+	0.988821i	$0.452361\pi$
$858$		0			0
$859$	31.5176			1.07537			0.537683	−	0.843147i	$-0.319300\pi$
$859$	31.5176			1.07537			0.537683	+	0.843147i	$0.319300\pi$
$860$		−	4.11361i		−	0.140273i
$861$		0			0
$862$	7.13767			0.243110
$863$		−	34.0790i		−	1.16006i	−0.814594	−	0.580031i	$-0.803040\pi$
$863$		−	34.0790i		−	1.16006i	0.814594	−	0.580031i	$-0.196960\pi$
$864$		0			0
$865$			6.92816i			0.235564i
$866$			7.85282i			0.266850i
$867$		0			0
$868$	−1.92420			−0.0653116
$869$		−	0.989324i		−	0.0335605i
$870$		0			0
$871$	48.4275	−	18.5390i	1.64090	−	0.628169i
$872$	−18.3474			−0.621320
$873$		0			0
$874$	1.76279			0.0596272
$875$	−9.62023			−0.325223
$876$		0			0
$877$			8.99660i			0.303794i	0.988396	+	0.151897i	$0.0485381\pi$
$877$			8.99660i			0.303794i	−0.988396	+	0.151897i	$0.951462\pi$
$878$			4.14853i			0.140006i
$879$		0			0
$880$	−0.285927			−0.00963859
$881$	−40.3722			−1.36017			−0.680086	−	0.733132i	$-0.738058\pi$
$881$	−40.3722			−1.36017			−0.680086	+	0.733132i	$0.738058\pi$
$882$		0			0
$883$	32.2736			1.08609			0.543046	−	0.839703i	$-0.317271\pi$
$883$	32.2736			1.08609			0.543046	+	0.839703i	$0.317271\pi$
$884$	−2.69861	−	7.04930i	−0.0907640	−	0.237094i
$885$		0			0
$886$		−	17.8470i		−	0.599581i
$887$	53.6494			1.80137			0.900685	−	0.434472i	$-0.143065\pi$
$887$	53.6494			1.80137			0.900685	+	0.434472i	$0.143065\pi$
$888$		0			0
$889$			31.1005i			1.04308i
$890$			5.64259i			0.189140i
$891$		0			0
$892$		−	23.1333i		−	0.774559i
$893$	−3.00034			−0.100402
$894$		0			0
$895$		−	0.532998i		−	0.0178161i
$896$	−1.67682			−0.0560186
$897$		0			0
$898$	19.7308			0.658424
$899$			2.83070i			0.0944092i
$900$		0			0
$901$	10.6473			0.354714
$902$			4.76420i			0.158630i
$903$		0			0
$904$		−	20.2779i		−	0.674433i
$905$			0.845691i			0.0281117i
$906$		0			0
$907$	44.2036			1.46776			0.733878	−	0.679281i	$-0.237708\pi$
$907$	44.2036			1.46776			0.733878	+	0.679281i	$0.237708\pi$
$908$		−	13.3066i		−	0.441595i
$909$		0			0
$910$	3.35816	−	1.28557i	0.111322	−	0.0426162i
$911$	54.9633			1.82102			0.910508	−	0.413492i	$-0.135691\pi$
$911$	54.9633			1.82102			0.910508	+	0.413492i	$0.135691\pi$
$912$		0			0
$913$	−5.31709			−0.175970
$914$	36.3456			1.20221
$915$		0			0
$916$			0.809405i			0.0267435i
$917$		−	7.78903i		−	0.257217i
$918$		0			0
$919$	−21.0595			−0.694690			−0.347345	−	0.937737i	$-0.612917\pi$
$919$	−21.0595			−0.694690			−0.347345	+	0.937737i	$0.612917\pi$
$920$	−2.18085			−0.0719005
$921$		0			0
$922$	22.9358			0.755351
$923$	−21.9217	+	8.39206i	−0.721563	+	0.276228i
$924$		0			0
$925$			16.9594i			0.557622i
$926$	−29.5769			−0.971957
$927$		0			0
$928$			2.46678i			0.0809760i
$929$		−	20.5891i		−	0.675506i	−0.941235	−	0.337753i	$-0.890333\pi$
$929$		−	20.5891i		−	0.675506i	0.941235	−	0.337753i	$-0.109667\pi$
$930$		0			0
$931$			2.01349i			0.0659896i
$932$	15.4712			0.506776
$933$		0			0
$934$			12.5970i			0.412187i
$935$	0.598585			0.0195758
$936$		0			0
$937$	−23.8463			−0.779023			−0.389512	−	0.921022i	$-0.627356\pi$
$937$	−23.8463			−0.779023			−0.389512	+	0.921022i	$0.627356\pi$
$938$			24.1159i			0.787411i
$939$		0			0
$940$	3.71190			0.121069
$941$		−	3.65686i		−	0.119210i	−0.998222	−	0.0596051i	$-0.981016\pi$
$941$		−	3.65686i		−	0.119210i	0.998222	−	0.0596051i	$-0.0189841\pi$
$942$		0			0
$943$			36.3380i			1.18333i
$944$			9.38985i			0.305614i
$945$		0			0
$946$	3.32504			0.108106
$947$		−	20.4278i		−	0.663813i	−0.943312	−	0.331907i	$-0.892308\pi$
$947$		−	20.4278i		−	0.663813i	0.943312	−	0.331907i	$-0.107692\pi$
$948$		0			0
$949$	19.9177	−	7.62489i	0.646557	−	0.247514i
$950$	2.23366			0.0724697
$951$		0			0
$952$	3.51040			0.113773
$953$	−40.9906			−1.32782			−0.663908	−	0.747814i	$-0.731104\pi$
$953$	−40.9906			−1.32782			−0.663908	+	0.747814i	$0.731104\pi$
$954$		0			0
$955$		−	8.82664i		−	0.285623i
$956$			29.1575i			0.943021i
$957$		0			0
$958$	−12.7483			−0.411880
$959$	21.8333			0.705035
$960$		0			0
$961$	29.6832			0.957522
$962$	−4.70518	−	12.2909i	−0.151701	−	0.396273i
$963$		0			0
$964$		−	15.5947i		−	0.502271i
$965$	8.39056			0.270102
$966$		0			0
$967$			3.74995i			0.120590i	0.998181	+	0.0602951i	$0.0192042\pi$
$967$			3.74995i			0.120590i	−0.998181	+	0.0602951i	$0.980796\pi$
$968$			10.7689i			0.346125i
$969$		0			0
$970$		−	5.77828i		−	0.185530i
$971$	−48.6853			−1.56239			−0.781193	−	0.624289i	$-0.785389\pi$
$971$	−48.6853			−1.56239			−0.781193	+	0.624289i	$0.785389\pi$
$972$		0			0
$973$			17.0709i			0.547269i
$974$	20.8054			0.666648
$975$		0			0
$976$	7.81270			0.250078
$977$		−	22.1641i		−	0.709091i	−0.935039	−	0.354546i	$-0.884636\pi$
$977$		−	22.1641i		−	0.709091i	0.935039	−	0.354546i	$-0.115364\pi$
$978$		0			0
$979$	−4.56092			−0.145768
$980$		−	2.49101i		−	0.0795725i
$981$		0			0
$982$			14.7472i			0.470604i
$983$		−	56.1331i		−	1.79037i	−0.445698	−	0.895183i	$-0.647044\pi$
$983$		−	56.1331i		−	1.79037i	0.445698	−	0.895183i	$-0.352956\pi$
$984$		0			0
$985$	0.263191			0.00838595
$986$		−	5.16418i		−	0.164461i
$987$		0			0
$988$	−1.61879	+	0.619702i	−0.0515004	+	0.0197153i
$989$	25.3611			0.806436
$990$		0			0
$991$	36.3755			1.15551			0.577753	−	0.816212i	$-0.303930\pi$
$991$	36.3755			1.15551			0.577753	+	0.816212i	$0.303930\pi$
$992$	−1.14753			−0.0364341
$993$		0			0
$994$		−	10.9166i		−	0.346252i
$995$		−	7.90584i		−	0.250632i
$996$		0			0
$997$	57.3475			1.81621			0.908107	−	0.418737i	$-0.137527\pi$
$997$	57.3475			1.81621			0.908107	+	0.418737i	$0.137527\pi$
$998$	29.7482			0.941662
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2106.2.b.c.649.5		14
3.2	odd	2		2106.2.b.d.649.10		14
9.2	odd	6		702.2.t.a.415.12		28
9.4	even	3		234.2.t.a.25.14	yes	28
9.5	odd	6		702.2.t.a.181.3		28
9.7	even	3		234.2.t.a.103.7	yes	28
13.12	even	2	inner	2106.2.b.c.649.10		14
39.38	odd	2		2106.2.b.d.649.5		14
117.25	even	6		234.2.t.a.103.14	yes	28
117.38	odd	6		702.2.t.a.415.3		28
117.77	odd	6		702.2.t.a.181.12		28
117.103	even	6		234.2.t.a.25.7	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.t.a.25.7	✓	28	117.103	even	6
234.2.t.a.25.14	yes	28	9.4	even	3
234.2.t.a.103.7	yes	28	9.7	even	3
234.2.t.a.103.14	yes	28	117.25	even	6
702.2.t.a.181.3		28	9.5	odd	6
702.2.t.a.181.12		28	117.77	odd	6
702.2.t.a.415.3		28	117.38	odd	6
702.2.t.a.415.12		28	9.2	odd	6
2106.2.b.c.649.5		14	1.1	even	1	trivial
2106.2.b.c.649.10		14	13.12	even	2	inner
2106.2.b.d.649.5		14	39.38	odd	2
2106.2.b.d.649.10		14	3.2	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:	\( N \)	\(=\)	\( 2106 = 2 \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2106.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +0.594758i q^{5} +1.67682i q^{7} +1.00000i q^{8} +0.594758 q^{10} +0.480744i q^{11} +(-1.28905 - 3.36725i) q^{13} +1.67682 q^{14} +1.00000 q^{16} -2.09349 q^{17} +0.480744i q^{19} -0.594758i q^{20} +0.480744 q^{22} +3.66679 q^{23} +4.64626 q^{25} +(-3.36725 + 1.28905i) q^{26} -1.67682i q^{28} -2.46678 q^{29} -1.14753i q^{31} -1.00000i q^{32} +2.09349i q^{34} -0.997301 q^{35} +3.65012i q^{37} +0.480744 q^{38} -0.594758 q^{40} +9.91005i q^{41} +6.91644 q^{43} -0.480744i q^{44} -3.66679i q^{46} +6.24102i q^{47} +4.18828 q^{49} -4.64626i q^{50} +(1.28905 + 3.36725i) q^{52} -5.08592 q^{53} -0.285927 q^{55} -1.67682 q^{56} +2.46678i q^{58} +9.38985i q^{59} +7.81270 q^{61} -1.14753 q^{62} -1.00000 q^{64} +(2.00270 - 0.766671i) q^{65} +14.3819i q^{67} +2.09349 q^{68} +0.997301i q^{70} -6.51028i q^{71} +5.91514i q^{73} +3.65012 q^{74} -0.480744i q^{76} -0.806121 q^{77} -2.05790 q^{79} +0.594758i q^{80} +9.91005 q^{82} +11.0601i q^{83} -1.24512i q^{85} -6.91644i q^{86} -0.480744 q^{88} +9.48720i q^{89} +(5.64626 - 2.16150i) q^{91} -3.66679 q^{92} +6.24102 q^{94} -0.285927 q^{95} -9.71535i q^{97} -4.18828i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{4} - 2 q^{13} - 8 q^{14} + 14 q^{16} + 8 q^{17} + 8 q^{23} - 14 q^{25} + 4 q^{26} + 16 q^{29} - 34 q^{35} + 4 q^{43} - 10 q^{49} + 2 q^{52} - 60 q^{53} + 8 q^{56} - 28 q^{61} + 34 q^{62} - 14 q^{64}+ \cdots - 8 q^{92}+O(q^{100}) \) 14 * q - 14 * q^4 - 2 * q^13 - 8 * q^14 + 14 * q^16 + 8 * q^17 + 8 * q^23 - 14 * q^25 + 4 * q^26 + 16 * q^29 - 34 * q^35 + 4 * q^43 - 10 * q^49 + 2 * q^52 - 60 * q^53 + 8 * q^56 - 28 * q^61 + 34 * q^62 - 14 * q^64 + 8 * q^65 - 8 * q^68 - 16 * q^74 + 24 * q^77 - 28 * q^79 - 24 * q^82 - 8 * q^92

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