LMFDB - Embedded newform 4014.2.d.a.4013.37

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(4013,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4014.4013");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.d (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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4013.37
Character	$\chi$	$=$	4014.4013
Dual form			4014.2.d.a.4013.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -3.59603 q^{5} +4.66381 q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -3.59603 q^{5} +4.66381 q^{7} +1.00000i q^{8} +3.59603i q^{10} +0.902432 q^{11} +3.90912i q^{13} -4.66381i q^{14} +1.00000 q^{16} +3.88874i q^{17} +2.00606 q^{19} +3.59603 q^{20} -0.902432i q^{22} -9.11811 q^{23} +7.93141 q^{25} +3.90912 q^{26} -4.66381 q^{28} +5.06669i q^{29} -7.64590 q^{31} -1.00000i q^{32} +3.88874 q^{34} -16.7712 q^{35} +2.27714 q^{37} -2.00606i q^{38} -3.59603i q^{40} -1.40849i q^{41} -5.26130 q^{43} -0.902432 q^{44} +9.11811i q^{46} -5.68266i q^{47} +14.7511 q^{49} -7.93141i q^{50} -3.90912i q^{52} -12.3390i q^{53} -3.24517 q^{55} +4.66381i q^{56} +5.06669 q^{58} +1.09050 q^{59} +1.98011i q^{61} +7.64590i q^{62} -1.00000 q^{64} -14.0573i q^{65} -13.9899i q^{67} -3.88874i q^{68} +16.7712i q^{70} +6.23553 q^{71} -15.3293 q^{73} -2.27714i q^{74} -2.00606 q^{76} +4.20877 q^{77} -5.68601i q^{79} -3.59603 q^{80} -1.40849 q^{82} +2.04815i q^{83} -13.9840i q^{85} +5.26130i q^{86} +0.902432i q^{88} +1.95797i q^{89} +18.2314i q^{91} +9.11811 q^{92} -5.68266 q^{94} -7.21384 q^{95} -10.0724i q^{97} -14.7511i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) $ 72 * q - 72 * q^4 + 16 * q^7	$ 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) $ 72 * q - 72 * q^4 + 16 * q^7 + 72 * q^16 - 40 * q^19 + 96 * q^25 - 16 * q^28 - 24 * q^37 - 8 * q^43 + 56 * q^49 + 40 * q^58 - 72 * q^64 - 32 * q^73 + 40 * q^76 + 16 * q^82

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4014\mathbb{Z}\right)^\times$.

$n$	$893$	$2233$
$\chi(n)$	$-1$	$-1$

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
$5$	−3.59603			−1.60819			−0.804096	−	0.594499i	$-0.797350\pi$
$5$	−3.59603			−1.60819			−0.804096	+	0.594499i	$0.797350\pi$
$6$		0			0
$7$	4.66381			1.76276			0.881378	−	0.472412i	$-0.156617\pi$
$7$	4.66381			1.76276			0.881378	+	0.472412i	$0.156617\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$			3.59603i			1.13716i
$11$	0.902432			0.272093			0.136047	−	0.990702i	$-0.456560\pi$
$11$	0.902432			0.272093			0.136047	+	0.990702i	$0.456560\pi$
$12$		0			0
$13$			3.90912i			1.08419i	0.840316	+	0.542097i	$0.182369\pi$
$13$			3.90912i			1.08419i	−0.840316	+	0.542097i	$0.817631\pi$
$14$		−	4.66381i		−	1.24646i
$15$		0			0
$16$	1.00000			0.250000
$17$			3.88874i			0.943158i	0.881824	+	0.471579i	$0.156316\pi$
$17$			3.88874i			0.943158i	−0.881824	+	0.471579i	$0.843684\pi$
$18$		0			0
$19$	2.00606			0.460222			0.230111	−	0.973164i	$-0.426091\pi$
$19$	2.00606			0.460222			0.230111	+	0.973164i	$0.426091\pi$
$20$	3.59603			0.804096
$21$		0			0
$22$		−	0.902432i		−	0.192399i
$23$	−9.11811			−1.90126			−0.950629	−	0.310331i	$-0.899560\pi$
$23$	−9.11811			−1.90126			−0.950629	+	0.310331i	$0.899560\pi$
$24$		0			0
$25$	7.93141			1.58628
$26$	3.90912			0.766641
$27$		0			0
$28$	−4.66381			−0.881378
$29$			5.06669i			0.940861i	0.882437	+	0.470431i	$0.155901\pi$
$29$			5.06669i			0.940861i	−0.882437	+	0.470431i	$0.844099\pi$
$30$		0			0
$31$	−7.64590			−1.37324			−0.686622	−	0.727014i	$-0.740907\pi$
$31$	−7.64590			−1.37324			−0.686622	+	0.727014i	$0.740907\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	3.88874			0.666914
$35$	−16.7712			−2.83485
$36$		0			0
$37$	2.27714			0.374359			0.187180	−	0.982326i	$-0.440065\pi$
$37$	2.27714			0.374359			0.187180	+	0.982326i	$0.440065\pi$
$38$		−	2.00606i		−	0.325426i
$39$		0			0
$40$		−	3.59603i		−	0.568582i
$41$		−	1.40849i		−	0.219969i	−0.993933	−	0.109984i	$-0.964920\pi$
$41$		−	1.40849i		−	0.219969i	0.993933	−	0.109984i	$-0.0350801\pi$
$42$		0			0
$43$	−5.26130			−0.802341			−0.401170	−	0.916003i	$-0.631396\pi$
$43$	−5.26130			−0.802341			−0.401170	+	0.916003i	$0.631396\pi$
$44$	−0.902432			−0.136047
$45$		0			0
$46$			9.11811i			1.34439i
$47$		−	5.68266i		−	0.828901i	−0.910072	−	0.414451i	$-0.863974\pi$
$47$		−	5.68266i		−	0.828901i	0.910072	−	0.414451i	$-0.136026\pi$
$48$		0			0
$49$	14.7511			2.10731
$50$		−	7.93141i		−	1.12167i
$51$		0			0
$52$		−	3.90912i		−	0.542097i
$53$		−	12.3390i		−	1.69490i	−0.530878	−	0.847448i	$-0.678138\pi$
$53$		−	12.3390i		−	1.69490i	0.530878	−	0.847448i	$-0.321862\pi$
$54$		0			0
$55$	−3.24517			−0.437579
$56$			4.66381i			0.623228i
$57$		0			0
$58$	5.06669			0.665289
$59$	1.09050			0.141972			0.0709858	−	0.997477i	$-0.477386\pi$
$59$	1.09050			0.141972			0.0709858	+	0.997477i	$0.477386\pi$
$60$		0			0
$61$			1.98011i			0.253527i	0.991933	+	0.126763i	$0.0404589\pi$
$61$			1.98011i			0.253527i	−0.991933	+	0.126763i	$0.959541\pi$
$62$			7.64590i			0.971030i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	14.0573i		−	1.74359i
$66$		0			0
$67$		−	13.9899i		−	1.70914i	−0.519340	−	0.854568i	$-0.673822\pi$
$67$		−	13.9899i		−	1.70914i	0.519340	−	0.854568i	$-0.326178\pi$
$68$		−	3.88874i		−	0.471579i
$69$		0			0
$70$			16.7712i			2.00454i
$71$	6.23553			0.740021			0.370010	−	0.929028i	$-0.379354\pi$
$71$	6.23553			0.740021			0.370010	+	0.929028i	$0.379354\pi$
$72$		0			0
$73$	−15.3293			−1.79415			−0.897077	−	0.441875i	$-0.854313\pi$
$73$	−15.3293			−1.79415			−0.897077	+	0.441875i	$0.854313\pi$
$74$		−	2.27714i		−	0.264712i
$75$		0			0
$76$	−2.00606			−0.230111
$77$	4.20877			0.479634
$78$		0			0
$79$		−	5.68601i		−	0.639726i	−0.947464	−	0.319863i	$-0.896363\pi$
$79$		−	5.68601i		−	0.639726i	0.947464	−	0.319863i	$-0.103637\pi$
$80$	−3.59603			−0.402048
$81$		0			0
$82$	−1.40849			−0.155541
$83$			2.04815i			0.224813i	0.993662	+	0.112407i	$0.0358559\pi$
$83$			2.04815i			0.224813i	−0.993662	+	0.112407i	$0.964144\pi$
$84$		0			0
$85$		−	13.9840i		−	1.51678i
$86$			5.26130i			0.567341i
$87$		0			0
$88$			0.902432i			0.0961996i
$89$			1.95797i			0.207545i	0.994601	+	0.103772i	$0.0330914\pi$
$89$			1.95797i			0.207545i	−0.994601	+	0.103772i	$0.966909\pi$
$90$		0			0
$91$			18.2314i			1.91117i
$92$	9.11811			0.950629
$93$		0			0
$94$	−5.68266			−0.586122
$95$	−7.21384			−0.740125
$96$		0			0
$97$		−	10.0724i		−	1.02270i	−0.859374	−	0.511348i	$-0.829146\pi$
$97$		−	10.0724i		−	1.02270i	0.859374	−	0.511348i	$-0.170854\pi$
$98$		−	14.7511i		−	1.49009i
$99$		0			0
$100$	−7.93141			−0.793141
$101$			11.9250i			1.18658i	0.804988	+	0.593291i	$0.202172\pi$
$101$			11.9250i			1.18658i	−0.804988	+	0.593291i	$0.797828\pi$
$102$		0			0
$103$			15.4768i			1.52498i	0.647003	+	0.762488i	$0.276022\pi$
$103$			15.4768i			1.52498i	−0.647003	+	0.762488i	$0.723978\pi$
$104$	−3.90912			−0.383320
$105$		0			0
$106$	−12.3390			−1.19847
$107$	−18.9884			−1.83568			−0.917841	−	0.396948i	$-0.870069\pi$
$107$	−18.9884			−1.83568			−0.917841	+	0.396948i	$0.870069\pi$
$108$		0			0
$109$	−4.80760			−0.460484			−0.230242	−	0.973133i	$-0.573952\pi$
$109$	−4.80760			−0.460484			−0.230242	+	0.973133i	$0.573952\pi$
$110$			3.24517i			0.309415i
$111$		0			0
$112$	4.66381			0.440689
$113$	−18.1589			−1.70825			−0.854125	−	0.520068i	$-0.825907\pi$
$113$	−18.1589			−1.70825			−0.854125	+	0.520068i	$0.825907\pi$
$114$		0			0
$115$	32.7890			3.05759
$116$		−	5.06669i		−	0.470431i
$117$		0			0
$118$		−	1.09050i		−	0.100389i
$119$			18.1364i			1.66256i
$120$		0			0
$121$	−10.1856			−0.925965
$122$	1.98011			0.179270
$123$		0			0
$124$	7.64590			0.686622
$125$	−10.5414			−0.942853
$126$		0			0
$127$	−3.78190			−0.335590			−0.167795	−	0.985822i	$-0.553665\pi$
$127$	−3.78190			−0.335590			−0.167795	+	0.985822i	$0.553665\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−14.0573			−1.23291
$131$			9.66672i			0.844585i	0.906460	+	0.422293i	$0.138775\pi$
$131$			9.66672i			0.844585i	−0.906460	+	0.422293i	$0.861225\pi$
$132$		0			0
$133$	9.35589			0.811258
$134$	−13.9899			−1.20854
$135$		0			0
$136$	−3.88874			−0.333457
$137$	0.470346			0.0401844			0.0200922	−	0.999798i	$-0.493604\pi$
$137$	0.470346			0.0401844			0.0200922	+	0.999798i	$0.493604\pi$
$138$		0			0
$139$	−14.4688			−1.22722			−0.613612	−	0.789607i	$-0.710284\pi$
$139$	−14.4688			−1.22722			−0.613612	+	0.789607i	$0.710284\pi$
$140$	16.7712			1.41742
$141$		0			0
$142$		−	6.23553i		−	0.523274i
$143$			3.52771i			0.295002i
$144$		0			0
$145$		−	18.2200i		−	1.51309i
$146$			15.3293i			1.26866i
$147$		0			0
$148$	−2.27714			−0.187180
$149$	−0.978503			−0.0801621			−0.0400810	−	0.999196i	$-0.512762\pi$
$149$	−0.978503			−0.0801621			−0.0400810	+	0.999196i	$0.512762\pi$
$150$		0			0
$151$		−	3.93932i		−	0.320577i	−0.987070	−	0.160289i	$-0.948757\pi$
$151$		−	3.93932i		−	0.320577i	0.987070	−	0.160289i	$-0.0512425\pi$
$152$			2.00606i			0.162713i
$153$		0			0
$154$		−	4.20877i		−	0.339153i
$155$	27.4949			2.20844
$156$		0			0
$157$			11.1663i			0.891167i	0.895240	+	0.445583i	$0.147004\pi$
$157$			11.1663i			0.891167i	−0.895240	+	0.445583i	$0.852996\pi$
$158$	−5.68601			−0.452355
$159$		0			0
$160$			3.59603i			0.284291i
$161$	−42.5251			−3.35145
$162$		0			0
$163$			15.5956i			1.22154i	0.791807	+	0.610771i	$0.209141\pi$
$163$			15.5956i			1.22154i	−0.791807	+	0.610771i	$0.790859\pi$
$164$			1.40849i			0.109984i
$165$		0			0
$166$	2.04815			0.158967
$167$	7.90448			0.611667			0.305833	−	0.952085i	$-0.401065\pi$
$167$	7.90448			0.611667			0.305833	+	0.952085i	$0.401065\pi$
$168$		0			0
$169$	−2.28120			−0.175477
$170$	−13.9840			−1.07253
$171$		0			0
$172$	5.26130			0.401170
$173$	15.7797			1.19971			0.599856	−	0.800108i	$-0.295225\pi$
$173$	15.7797			1.19971			0.599856	+	0.800108i	$0.295225\pi$
$174$		0			0
$175$	36.9906			2.79623
$176$	0.902432			0.0680234
$177$		0			0
$178$	1.95797			0.146756
$179$		−	4.39415i		−	0.328434i	−0.986424	−	0.164217i	$-0.947490\pi$
$179$		−	4.39415i		−	0.328434i	0.986424	−	0.164217i	$-0.0525098\pi$
$180$		0			0
$181$	−5.91690			−0.439800			−0.219900	−	0.975522i	$-0.570573\pi$
$181$	−5.91690			−0.439800			−0.219900	+	0.975522i	$0.570573\pi$
$182$	18.2314			1.35140
$183$		0			0
$184$		−	9.11811i		−	0.672196i
$185$	−8.18865			−0.602042
$186$		0			0
$187$			3.50932i			0.256627i
$188$			5.68266i			0.414451i
$189$		0			0
$190$			7.21384i			0.523347i
$191$	−18.1304			−1.31187			−0.655936	−	0.754816i	$-0.727726\pi$
$191$	−18.1304			−1.31187			−0.655936	+	0.754816i	$0.727726\pi$
$192$		0			0
$193$		−	12.2241i		−	0.879910i	−0.898020	−	0.439955i	$-0.854994\pi$
$193$		−	12.2241i		−	0.879910i	0.898020	−	0.439955i	$-0.145006\pi$
$194$	−10.0724			−0.723155
$195$		0			0
$196$	−14.7511			−1.05365
$197$			2.60238i			0.185412i	0.995694	+	0.0927061i	$0.0295517\pi$
$197$			2.60238i			0.185412i	−0.995694	+	0.0927061i	$0.970448\pi$
$198$		0			0
$199$	−14.9269			−1.05814			−0.529071	−	0.848578i	$-0.677459\pi$
$199$	−14.9269			−1.05814			−0.529071	+	0.848578i	$0.677459\pi$
$200$			7.93141i			0.560835i
$201$		0			0
$202$	11.9250			0.839040
$203$			23.6301i			1.65851i
$204$		0			0
$205$			5.06496i			0.353752i
$206$	15.4768			1.07832
$207$		0			0
$208$			3.90912i			0.271049i
$209$	1.81033			0.125223
$210$		0			0
$211$	16.4855			1.13491			0.567455	−	0.823405i	$-0.307928\pi$
$211$	16.4855			1.13491			0.567455	+	0.823405i	$0.307928\pi$
$212$			12.3390i			0.847448i
$213$		0			0
$214$			18.9884i			1.29802i
$215$	18.9198			1.29032
$216$		0			0
$217$	−35.6591			−2.42069
$218$			4.80760i			0.325612i
$219$		0			0
$220$	3.24517			0.218789
$221$	−15.2015			−1.02257
$222$		0			0
$223$	0.326717	+	14.9296i	0.0218786	+	0.999761i
$223$	0.326717	+	14.9296i	0.0218786	+	0.999761i
$224$		−	4.66381i		−	0.311614i
$225$		0			0
$226$			18.1589i			1.20792i
$227$		−	3.91525i		−	0.259865i	−0.991523	−	0.129932i	$-0.958524\pi$
$227$		−	3.91525i		−	0.259865i	0.991523	−	0.129932i	$-0.0414760\pi$
$228$		0			0
$229$		−	5.35015i		−	0.353548i	−0.984251	−	0.176774i	$-0.943434\pi$
$229$		−	5.35015i		−	0.353548i	0.984251	−	0.176774i	$-0.0565662\pi$
$230$		−	32.7890i		−	2.16204i
$231$		0			0
$232$	−5.06669			−0.332645
$233$	−15.0088			−0.983258			−0.491629	−	0.870805i	$-0.663598\pi$
$233$	−15.0088			−0.983258			−0.491629	+	0.870805i	$0.663598\pi$
$234$		0			0
$235$			20.4350i			1.33303i
$236$	−1.09050			−0.0709858
$237$		0			0
$238$	18.1364			1.17561
$239$			22.7374i			1.47076i	0.677656	+	0.735379i	$0.262996\pi$
$239$			22.7374i			1.47076i	−0.677656	+	0.735379i	$0.737004\pi$
$240$		0			0
$241$	−15.5649			−1.00263			−0.501313	−	0.865266i	$-0.667150\pi$
$241$	−15.5649			−1.00263			−0.501313	+	0.865266i	$0.667150\pi$
$242$			10.1856i			0.654756i
$243$		0			0
$244$		−	1.98011i		−	0.126763i
$245$	−53.0455			−3.38895
$246$		0			0
$247$			7.84192i			0.498970i
$248$		−	7.64590i		−	0.485515i
$249$		0			0
$250$			10.5414i			0.666698i
$251$			25.2944i			1.59657i	0.602280	+	0.798285i	$0.294259\pi$
$251$			25.2944i			1.59657i	−0.602280	+	0.798285i	$0.705741\pi$
$252$		0			0
$253$	−8.22847			−0.517320
$254$			3.78190i			0.237298i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	27.5846i		−	1.72068i	−0.509720	−	0.860340i	$-0.670251\pi$
$257$		−	27.5846i		−	1.72068i	0.509720	−	0.860340i	$-0.329749\pi$
$258$		0			0
$259$	10.6201			0.659904
$260$			14.0573i			0.871796i
$261$		0			0
$262$	9.66672			0.597212
$263$	−7.69202			−0.474310			−0.237155	−	0.971472i	$-0.576215\pi$
$263$	−7.69202			−0.474310			−0.237155	+	0.971472i	$0.576215\pi$
$264$		0			0
$265$			44.3715i			2.72572i
$266$		−	9.35589i		−	0.573646i
$267$		0			0
$268$			13.9899i			0.854568i
$269$	31.7587			1.93636			0.968181	−	0.250251i	$-0.0805130\pi$
$269$	31.7587			1.93636			0.968181	+	0.250251i	$0.0805130\pi$
$270$		0			0
$271$		−	8.67132i		−	0.526745i	−0.964694	−	0.263373i	$-0.915165\pi$
$271$		−	8.67132i		−	0.526745i	0.964694	−	0.263373i	$-0.0848348\pi$
$272$			3.88874i			0.235790i
$273$		0			0
$274$		−	0.470346i		−	0.0284147i
$275$	7.15756			0.431617
$276$		0			0
$277$			28.7339i			1.72645i	0.504815	+	0.863227i	$0.331561\pi$
$277$			28.7339i			1.72645i	−0.504815	+	0.863227i	$0.668439\pi$
$278$			14.4688i			0.867779i
$279$		0			0
$280$		−	16.7712i		−	1.00227i
$281$		−	23.8796i		−	1.42454i	−0.701906	−	0.712270i	$-0.747667\pi$
$281$		−	23.8796i		−	1.42454i	0.701906	−	0.712270i	$-0.252333\pi$
$282$		0			0
$283$	−18.3009			−1.08787			−0.543937	−	0.839126i	$-0.683067\pi$
$283$	−18.3009			−1.08787			−0.543937	+	0.839126i	$0.683067\pi$
$284$	−6.23553			−0.370010
$285$		0			0
$286$	3.52771			0.208598
$287$		−	6.56892i		−	0.387751i
$288$		0			0
$289$	1.87769			0.110452
$290$	−18.2200			−1.06991
$291$		0			0
$292$	15.3293			0.897077
$293$	18.0681			1.05555			0.527776	−	0.849384i	$-0.323026\pi$
$293$	18.0681			1.05555			0.527776	+	0.849384i	$0.323026\pi$
$294$		0			0
$295$	−3.92148			−0.228317
$296$			2.27714i			0.132356i
$297$		0			0
$298$			0.978503i			0.0566832i
$299$		−	35.6438i		−	2.06133i
$300$		0			0
$301$	−24.5377			−1.41433
$302$	−3.93932			−0.226682
$303$		0			0
$304$	2.00606			0.115055
$305$		−	7.12051i		−	0.407719i
$306$		0			0
$307$		−	4.70645i		−	0.268611i	−0.990940	−	0.134306i	$-0.957120\pi$
$307$		−	4.70645i		−	0.268611i	0.990940	−	0.134306i	$-0.0428804\pi$
$308$	−4.20877			−0.239817
$309$		0			0
$310$		−	27.4949i		−	1.56160i
$311$	−15.0235			−0.851904			−0.425952	−	0.904746i	$-0.640061\pi$
$311$	−15.0235			−0.851904			−0.425952	+	0.904746i	$0.640061\pi$
$312$		0			0
$313$		−	4.54957i		−	0.257157i	−0.991699	−	0.128578i	$-0.958959\pi$
$313$		−	4.54957i		−	0.257157i	0.991699	−	0.128578i	$-0.0410414\pi$
$314$	11.1663			0.630150
$315$		0			0
$316$			5.68601i			0.319863i
$317$		−	18.1436i		−	1.01905i	−0.860457	−	0.509524i	$-0.829822\pi$
$317$		−	18.1436i		−	1.01905i	0.860457	−	0.509524i	$-0.170178\pi$
$318$		0			0
$319$			4.57235i			0.256002i
$320$	3.59603			0.201024
$321$		0			0
$322$			42.5251i			2.36983i
$323$			7.80105i			0.434062i
$324$		0			0
$325$			31.0048i			1.71984i
$326$	15.5956			0.863761
$327$		0			0
$328$	1.40849			0.0777707
$329$		−	26.5029i		−	1.46115i
$330$		0			0
$331$			14.3644i			0.789537i	0.918781	+	0.394769i	$0.129175\pi$
$331$			14.3644i			0.789537i	−0.918781	+	0.394769i	$0.870825\pi$
$332$		−	2.04815i		−	0.112407i
$333$		0			0
$334$		−	7.90448i		−	0.432514i
$335$			50.3080i			2.74862i
$336$		0			0
$337$			28.9673i			1.57795i	0.614427	+	0.788974i	$0.289387\pi$
$337$			28.9673i			1.57795i	−0.614427	+	0.788974i	$0.710613\pi$
$338$			2.28120i			0.124081i
$339$		0			0
$340$			13.9840i			0.758390i
$341$	−6.89991			−0.373651
$342$		0			0
$343$	36.1499			1.95191
$344$		−	5.26130i		−	0.283670i
$345$		0			0
$346$		−	15.7797i		−	0.848324i
$347$			11.7543i			0.631003i	0.948925	+	0.315501i	$0.102173\pi$
$347$			11.7543i			0.631003i	−0.948925	+	0.315501i	$0.897827\pi$
$348$		0			0
$349$	−1.58365			−0.0847711			−0.0423855	−	0.999101i	$-0.513496\pi$
$349$	−1.58365			−0.0847711			−0.0423855	+	0.999101i	$0.513496\pi$
$350$		−	36.9906i		−	1.97723i
$351$		0			0
$352$		−	0.902432i		−	0.0480998i
$353$		−	25.8563i		−	1.37619i	−0.725619	−	0.688096i	$-0.758447\pi$
$353$		−	25.8563i		−	1.37619i	0.725619	−	0.688096i	$-0.241553\pi$
$354$		0			0
$355$	−22.4231			−1.19010
$356$		−	1.95797i		−	0.103772i
$357$		0			0
$358$	−4.39415			−0.232238
$359$		−	28.3204i		−	1.49469i	−0.664434	−	0.747347i	$-0.731327\pi$
$359$		−	28.3204i		−	1.49469i	0.664434	−	0.747347i	$-0.268673\pi$
$360$		0			0
$361$	−14.9757			−0.788196
$362$			5.91690i			0.310986i
$363$		0			0
$364$		−	18.2314i		−	0.955585i
$365$	55.1244			2.88534
$366$		0			0
$367$	11.9073			0.621557			0.310778	−	0.950482i	$-0.399410\pi$
$367$	11.9073			0.621557			0.310778	+	0.950482i	$0.399410\pi$
$368$	−9.11811			−0.475314
$369$		0			0
$370$			8.18865i			0.425708i
$371$		−	57.5469i		−	2.98769i
$372$		0			0
$373$			18.4501i			0.955309i	0.878548	+	0.477655i	$0.158513\pi$
$373$			18.4501i			0.955309i	−0.878548	+	0.477655i	$0.841487\pi$
$374$	3.50932			0.181463
$375$		0			0
$376$	5.68266			0.293061
$377$	−19.8063			−1.02008
$378$		0			0
$379$	−3.95014			−0.202905			−0.101453	−	0.994840i	$-0.532349\pi$
$379$	−3.95014			−0.202905			−0.101453	+	0.994840i	$0.532349\pi$
$380$	7.21384			0.370062
$381$		0			0
$382$			18.1304i			0.927634i
$383$	−2.90672			−0.148526			−0.0742632	−	0.997239i	$-0.523660\pi$
$383$	−2.90672			−0.148526			−0.0742632	+	0.997239i	$0.523660\pi$
$384$		0			0
$385$	−15.1349			−0.771344
$386$	−12.2241			−0.622191
$387$		0			0
$388$			10.0724i			0.511348i
$389$			13.6810i			0.693655i	0.937929	+	0.346827i	$0.112741\pi$
$389$			13.6810i			0.693655i	−0.937929	+	0.346827i	$0.887259\pi$
$390$		0			0
$391$		−	35.4580i		−	1.79319i
$392$			14.7511i			0.745045i
$393$		0			0
$394$	2.60238			0.131106
$395$			20.4471i			1.02880i
$396$		0			0
$397$		−	26.2699i		−	1.31845i	−0.751946	−	0.659225i	$-0.770885\pi$
$397$		−	26.2699i		−	1.31845i	0.751946	−	0.659225i	$-0.229115\pi$
$398$			14.9269i			0.748219i
$399$		0			0
$400$	7.93141			0.396570
$401$			1.64967i			0.0823808i	0.999151	+	0.0411904i	$0.0131150\pi$
$401$			1.64967i			0.0823808i	−0.999151	+	0.0411904i	$0.986885\pi$
$402$		0			0
$403$		−	29.8887i		−	1.48886i
$404$		−	11.9250i		−	0.593291i
$405$		0			0
$406$	23.6301			1.17274
$407$	2.05496			0.101861
$408$		0			0
$409$		−	15.3942i		−	0.761196i	−0.924741	−	0.380598i	$-0.875718\pi$
$409$		−	15.3942i		−	0.761196i	0.924741	−	0.380598i	$-0.124282\pi$
$410$	5.06496			0.250141
$411$		0			0
$412$		−	15.4768i		−	0.762488i
$413$	5.08591			0.250261
$414$		0			0
$415$		−	7.36519i		−	0.361543i
$416$	3.90912			0.191660
$417$		0			0
$418$		−	1.81033i		−	0.0885463i
$419$			24.8933i			1.21612i	0.793891	+	0.608060i	$0.208052\pi$
$419$			24.8933i			1.21612i	−0.793891	+	0.608060i	$0.791948\pi$
$420$		0			0
$421$			1.89567i			0.0923894i	0.998932	+	0.0461947i	$0.0147095\pi$
$421$			1.89567i			0.0923894i	−0.998932	+	0.0461947i	$0.985291\pi$
$422$		−	16.4855i		−	0.802502i
$423$		0			0
$424$	12.3390			0.599236
$425$			30.8432i			1.49611i
$426$		0			0
$427$			9.23484i			0.446905i
$428$	18.9884			0.917841
$429$		0			0
$430$		−	18.9198i		−	0.912393i
$431$	−20.7721			−1.00056			−0.500279	−	0.865865i	$-0.666769\pi$
$431$	−20.7721			−1.00056			−0.500279	+	0.865865i	$0.666769\pi$
$432$		0			0
$433$	23.5538			1.13192			0.565961	−	0.824432i	$-0.308505\pi$
$433$	23.5538			1.13192			0.565961	+	0.824432i	$0.308505\pi$
$434$			35.6591i			1.71169i
$435$		0			0
$436$	4.80760			0.230242
$437$	−18.2915			−0.875000
$438$		0			0
$439$		−	27.5002i		−	1.31251i	−0.754538	−	0.656256i	$-0.772139\pi$
$439$		−	27.5002i		−	1.31251i	0.754538	−	0.656256i	$-0.227861\pi$
$440$		−	3.24517i		−	0.154707i
$441$		0			0
$442$			15.2015i			0.723064i
$443$		−	24.3687i		−	1.15779i	−0.815401	−	0.578896i	$-0.803484\pi$
$443$		−	24.3687i		−	1.15779i	0.815401	−	0.578896i	$-0.196516\pi$
$444$		0			0
$445$		−	7.04093i		−	0.333772i
$446$	14.9296	−	0.326717i	0.706938	−	0.0154705i
$447$		0			0
$448$	−4.66381			−0.220344
$449$	22.6953			1.07106			0.535529	−	0.844517i	$-0.320112\pi$
$449$	22.6953			1.07106			0.535529	+	0.844517i	$0.320112\pi$
$450$		0			0
$451$		−	1.27106i		−	0.0598521i
$452$	18.1589			0.854125
$453$		0			0
$454$	−3.91525			−0.183752
$455$		−	65.5606i		−	3.07353i
$456$		0			0
$457$			20.6720i			0.966996i	0.875345	+	0.483498i	$0.160634\pi$
$457$			20.6720i			0.966996i	−0.875345	+	0.483498i	$0.839366\pi$
$458$	−5.35015			−0.249996
$459$		0			0
$460$	−32.7890			−1.52879
$461$			25.4522i			1.18543i	0.805413	+	0.592714i	$0.201943\pi$
$461$			25.4522i			1.18543i	−0.805413	+	0.592714i	$0.798057\pi$
$462$		0			0
$463$	25.3610			1.17863			0.589313	−	0.807905i	$-0.299399\pi$
$463$	25.3610			1.17863			0.589313	+	0.807905i	$0.299399\pi$
$464$			5.06669i			0.235215i
$465$		0			0
$466$			15.0088i			0.695269i
$467$	−5.67695			−0.262698			−0.131349	−	0.991336i	$-0.541931\pi$
$467$	−5.67695			−0.262698			−0.131349	+	0.991336i	$0.541931\pi$
$468$		0			0
$469$		−	65.2462i		−	3.01279i
$470$	20.4350			0.942596
$471$		0			0
$472$			1.09050i			0.0501945i
$473$	−4.74797			−0.218312
$474$		0			0
$475$	15.9109			0.730041
$476$		−	18.1364i		−	0.831279i
$477$		0			0
$478$	22.7374			1.03998
$479$		−	7.99218i		−	0.365172i	−0.983190	−	0.182586i	$-0.941553\pi$
$479$		−	7.99218i		−	0.365172i	0.983190	−	0.182586i	$-0.0584468\pi$
$480$		0			0
$481$			8.90160i			0.405878i
$482$			15.5649i			0.708964i
$483$		0			0
$484$	10.1856			0.462983
$485$			36.2206i			1.64469i
$486$		0			0
$487$	17.2751			0.782810			0.391405	−	0.920219i	$-0.371989\pi$
$487$	17.2751			0.782810			0.391405	+	0.920219i	$0.371989\pi$
$488$	−1.98011			−0.0896352
$489$		0			0
$490$			53.0455i			2.39635i
$491$	−6.80166			−0.306955			−0.153477	−	0.988152i	$-0.549047\pi$
$491$	−6.80166			−0.306955			−0.153477	+	0.988152i	$0.549047\pi$
$492$		0			0
$493$	−19.7031			−0.887381
$494$	7.84192			0.352825
$495$		0			0
$496$	−7.64590			−0.343311
$497$	29.0813			1.30448
$498$		0			0
$499$	−34.3509			−1.53776			−0.768879	−	0.639395i	$-0.779185\pi$
$499$	−34.3509			−1.53776			−0.768879	+	0.639395i	$0.779185\pi$
$500$	10.5414			0.471427
$501$		0			0
$502$	25.2944			1.12895
$503$	−38.9506			−1.73672			−0.868360	−	0.495935i	$-0.834826\pi$
$503$	−38.9506			−1.73672			−0.868360	+	0.495935i	$0.834826\pi$
$504$		0			0
$505$		−	42.8826i		−	1.90825i
$506$			8.22847i			0.365800i
$507$		0			0
$508$	3.78190			0.167795
$509$		−	7.69102i		−	0.340898i	−0.985366	−	0.170449i	$-0.945478\pi$
$509$		−	7.69102i		−	0.340898i	0.985366	−	0.170449i	$-0.0545219\pi$
$510$		0			0
$511$	−71.4928			−3.16265
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−27.5846			−1.21671
$515$		−	55.6550i		−	2.45245i
$516$		0			0
$517$		−	5.12821i		−	0.225539i
$518$		−	10.6201i		−	0.466622i
$519$		0			0
$520$	14.0573			0.616453
$521$	−6.39420			−0.280135			−0.140067	−	0.990142i	$-0.544732\pi$
$521$	−6.39420			−0.280135			−0.140067	+	0.990142i	$0.544732\pi$
$522$		0			0
$523$		−	4.03803i		−	0.176571i	−0.996095	−	0.0882854i	$-0.971861\pi$
$523$		−	4.03803i		−	0.176571i	0.996095	−	0.0882854i	$-0.0281388\pi$
$524$		−	9.66672i		−	0.422293i
$525$		0			0
$526$			7.69202i			0.335388i
$527$		−	29.7329i		−	1.29519i
$528$		0			0
$529$	60.1399			2.61478
$530$	44.3715			1.92737
$531$		0			0
$532$	−9.35589			−0.405629
$533$	5.50595			0.238489
$534$		0			0
$535$	68.2830			2.95213
$536$	13.9899			0.604271
$537$		0			0
$538$		−	31.7587i		−	1.36921i
$539$	13.3119			0.573384
$540$		0			0
$541$			7.94748i			0.341689i	0.985298	+	0.170844i	$0.0546495\pi$
$541$			7.94748i			0.341689i	−0.985298	+	0.170844i	$0.945350\pi$
$542$	−8.67132			−0.372465
$543$		0			0
$544$	3.88874			0.166728
$545$	17.2883			0.740547
$546$		0			0
$547$	7.34285			0.313958			0.156979	−	0.987602i	$-0.449825\pi$
$547$	7.34285			0.313958			0.156979	+	0.987602i	$0.449825\pi$
$548$	−0.470346			−0.0200922
$549$		0			0
$550$		−	7.15756i		−	0.305199i
$551$			10.1641i			0.433005i
$552$		0			0
$553$		−	26.5185i		−	1.12768i
$554$	28.7339			1.22079
$555$		0			0
$556$	14.4688			0.613612
$557$	23.3969			0.991360			0.495680	−	0.868505i	$-0.334919\pi$
$557$	23.3969			0.991360			0.495680	+	0.868505i	$0.334919\pi$
$558$		0			0
$559$		−	20.5670i		−	0.869893i
$560$	−16.7712			−0.708712
$561$		0			0
$562$	−23.8796			−1.00730
$563$	9.73178			0.410146			0.205073	−	0.978747i	$-0.434257\pi$
$563$	9.73178			0.410146			0.205073	+	0.978747i	$0.434257\pi$
$564$		0			0
$565$	65.3001			2.74719
$566$			18.3009i			0.769243i
$567$		0			0
$568$			6.23553i			0.261637i
$569$	−4.19768			−0.175976			−0.0879879	−	0.996122i	$-0.528044\pi$
$569$	−4.19768			−0.175976			−0.0879879	+	0.996122i	$0.528044\pi$
$570$		0			0
$571$		−	35.4089i		−	1.48182i	−0.671606	−	0.740908i	$-0.734395\pi$
$571$		−	35.4089i		−	1.48182i	0.671606	−	0.740908i	$-0.265605\pi$
$572$		−	3.52771i		−	0.147501i
$573$		0			0
$574$	−6.56892			−0.274182
$575$	−72.3194			−3.01593
$576$		0			0
$577$	−24.2588			−1.00991			−0.504953	−	0.863147i	$-0.668490\pi$
$577$	−24.2588			−1.00991			−0.504953	+	0.863147i	$0.668490\pi$
$578$		−	1.87769i		−	0.0781017i
$579$		0			0
$580$			18.2200i			0.756543i
$581$			9.55217i			0.396291i
$582$		0			0
$583$		−	11.1351i		−	0.461170i
$584$		−	15.3293i		−	0.634329i
$585$		0			0
$586$		−	18.0681i		−	0.746388i
$587$	−14.2496			−0.588146			−0.294073	−	0.955783i	$-0.595011\pi$
$587$	−14.2496			−0.588146			−0.294073	+	0.955783i	$0.595011\pi$
$588$		0			0
$589$	−15.3381			−0.631997
$590$			3.92148i			0.161445i
$591$		0			0
$592$	2.27714			0.0935898
$593$	18.9380			0.777690			0.388845	−	0.921303i	$-0.372874\pi$
$593$	18.9380			0.777690			0.388845	+	0.921303i	$0.372874\pi$
$594$		0			0
$595$		−	65.2188i		−	2.67371i
$596$	0.978503			0.0400810
$597$		0			0
$598$	−35.6438			−1.45758
$599$			45.9774i			1.87858i	0.343118	+	0.939292i	$0.388517\pi$
$599$			45.9774i			1.87858i	−0.343118	+	0.939292i	$0.611483\pi$
$600$		0			0
$601$		−	19.7024i		−	0.803677i	−0.915711	−	0.401839i	$-0.868371\pi$
$601$		−	19.7024i		−	0.803677i	0.915711	−	0.401839i	$-0.131629\pi$
$602$			24.5377i			1.00008i
$603$		0			0
$604$			3.93932i			0.160289i
$605$	36.6277			1.48913
$606$		0			0
$607$			17.8892i			0.726099i	0.931770	+	0.363049i	$0.118264\pi$
$607$			17.8892i			0.726099i	−0.931770	+	0.363049i	$0.881736\pi$
$608$		−	2.00606i		−	0.0813565i
$609$		0			0
$610$	−7.12051			−0.288301
$611$	22.2142			0.898690
$612$		0			0
$613$		−	1.04687i		−	0.0422828i	−0.999776	−	0.0211414i	$-0.993270\pi$
$613$		−	1.04687i		−	0.0422828i	0.999776	−	0.0211414i	$-0.00673001\pi$
$614$	−4.70645			−0.189937
$615$		0			0
$616$			4.20877i			0.169576i
$617$			2.67707i			0.107775i	0.998547	+	0.0538873i	$0.0171612\pi$
$617$			2.67707i			0.107775i	−0.998547	+	0.0538873i	$0.982839\pi$
$618$		0			0
$619$			31.8865i			1.28163i	0.767696	+	0.640814i	$0.221403\pi$
$619$			31.8865i			1.28163i	−0.767696	+	0.640814i	$0.778597\pi$
$620$	−27.4949			−1.10422
$621$		0			0
$622$			15.0235i			0.602387i
$623$			9.13162i			0.365851i
$624$		0			0
$625$	−1.74981			−0.0699925
$626$	−4.54957			−0.181837
$627$		0			0
$628$		−	11.1663i		−	0.445583i
$629$			8.85520i			0.353080i
$630$		0			0
$631$			46.9308i			1.86828i	0.356901	+	0.934142i	$0.383833\pi$
$631$			46.9308i			1.86828i	−0.356901	+	0.934142i	$0.616167\pi$
$632$	5.68601			0.226177
$633$		0			0
$634$	−18.1436			−0.720576
$635$	13.5998			0.539693
$636$		0			0
$637$			57.6640i			2.28473i
$638$	4.57235			0.181021
$639$		0			0
$640$		−	3.59603i		−	0.142145i
$641$	10.6710			0.421480			0.210740	−	0.977542i	$-0.432413\pi$
$641$	10.6710			0.421480			0.210740	+	0.977542i	$0.432413\pi$
$642$		0			0
$643$	−2.07541			−0.0818463			−0.0409231	−	0.999162i	$-0.513030\pi$
$643$	−2.07541			−0.0818463			−0.0409231	+	0.999162i	$0.513030\pi$
$644$	42.5251			1.67573
$645$		0			0
$646$	7.80105			0.306928
$647$			25.0950i			0.986587i	0.869863	+	0.493294i	$0.164207\pi$
$647$			25.0950i			0.986587i	−0.869863	+	0.493294i	$0.835793\pi$
$648$		0			0
$649$	0.984106			0.0386295
$650$	31.0048			1.21611
$651$		0			0
$652$		−	15.5956i		−	0.610771i
$653$	13.3767			0.523470			0.261735	−	0.965140i	$-0.415705\pi$
$653$	13.3767			0.523470			0.261735	+	0.965140i	$0.415705\pi$
$654$		0			0
$655$		−	34.7618i		−	1.35826i
$656$		−	1.40849i		−	0.0549922i
$657$		0			0
$658$	−26.5029			−1.03319
$659$			44.7250i			1.74224i	0.491072	+	0.871119i	$0.336605\pi$
$659$			44.7250i			1.74224i	−0.491072	+	0.871119i	$0.663395\pi$
$660$		0			0
$661$		−	32.3751i		−	1.25924i	−0.776901	−	0.629622i	$-0.783210\pi$
$661$		−	32.3751i		−	1.25924i	0.776901	−	0.629622i	$-0.216790\pi$
$662$	14.3644			0.558287
$663$		0			0
$664$	−2.04815			−0.0794835
$665$	−33.6440			−1.30466
$666$		0			0
$667$		−	46.1987i		−	1.78882i
$668$	−7.90448			−0.305833
$669$		0			0
$670$	50.3080			1.94357
$671$			1.78691i			0.0689829i
$672$		0			0
$673$	−44.4949			−1.71515			−0.857575	−	0.514358i	$-0.828030\pi$
$673$	−44.4949			−1.71515			−0.857575	+	0.514358i	$0.828030\pi$
$674$	28.9673			1.11578
$675$		0			0
$676$	2.28120			0.0877384
$677$			5.75638i			0.221236i	0.993863	+	0.110618i	$0.0352830\pi$
$677$			5.75638i			0.221236i	−0.993863	+	0.110618i	$0.964717\pi$
$678$		0			0
$679$		−	46.9757i		−	1.80276i
$680$	13.9840			0.536263
$681$		0			0
$682$			6.89991i			0.264211i
$683$		−	36.0279i		−	1.37857i	−0.724490	−	0.689285i	$-0.757925\pi$
$683$		−	36.0279i		−	1.37857i	0.724490	−	0.689285i	$-0.242075\pi$
$684$		0			0
$685$	−1.69138			−0.0646242
$686$		−	36.1499i		−	1.38021i
$687$		0			0
$688$	−5.26130			−0.200585
$689$	48.2347			1.83760
$690$		0			0
$691$			31.5987i			1.20207i	0.799222	+	0.601036i	$0.205245\pi$
$691$			31.5987i			1.20207i	−0.799222	+	0.601036i	$0.794755\pi$
$692$	−15.7797			−0.599856
$693$		0			0
$694$	11.7543			0.446186
$695$	52.0301			1.97361
$696$		0			0
$697$	5.47725			0.207465
$698$			1.58365i			0.0599422i
$699$		0			0
$700$	−36.9906			−1.39811
$701$			24.6638i			0.931538i	0.884906	+	0.465769i	$0.154222\pi$
$701$			24.6638i			0.931538i	−0.884906	+	0.465769i	$0.845778\pi$
$702$		0			0
$703$	4.56808			0.172288
$704$	−0.902432			−0.0340117
$705$		0			0
$706$	−25.8563			−0.973115
$707$			55.6160i			2.09165i
$708$		0			0
$709$		−	14.3003i		−	0.537060i	−0.963271	−	0.268530i	$-0.913462\pi$
$709$		−	14.3003i		−	0.537060i	0.963271	−	0.268530i	$-0.0865378\pi$
$710$			22.4231i			0.841525i
$711$		0			0
$712$	−1.95797			−0.0733782
$713$	69.7162			2.61089
$714$		0			0
$715$		−	12.6857i		−	0.474420i
$716$			4.39415i			0.164217i
$717$		0			0
$718$	−28.3204			−1.05691
$719$			3.36040i			0.125322i	0.998035	+	0.0626609i	$0.0199586\pi$
$719$			3.36040i			0.125322i	−0.998035	+	0.0626609i	$0.980041\pi$
$720$		0			0
$721$			72.1809i			2.68816i
$722$			14.9757i			0.557339i
$723$		0			0
$724$	5.91690			0.219900
$725$			40.1860i			1.49247i
$726$		0			0
$727$	15.6392			0.580025			0.290013	−	0.957023i	$-0.406340\pi$
$727$	15.6392			0.580025			0.290013	+	0.957023i	$0.406340\pi$
$728$	−18.2314			−0.675700
$729$		0			0
$730$		−	55.1244i		−	2.04025i
$731$		−	20.4598i		−	0.756734i
$732$		0			0
$733$	−13.7550			−0.508051			−0.254025	−	0.967198i	$-0.581755\pi$
$733$	−13.7550			−0.508051			−0.254025	+	0.967198i	$0.581755\pi$
$734$		−	11.9073i		−	0.439507i
$735$		0			0
$736$			9.11811i			0.336098i
$737$		−	12.6249i		−	0.465045i
$738$		0			0
$739$		−	22.3492i		−	0.822127i	−0.911607	−	0.411064i	$-0.865157\pi$
$739$		−	22.3492i		−	0.822127i	0.911607	−	0.411064i	$-0.134843\pi$
$740$	8.18865			0.301021
$741$		0			0
$742$	−57.5469			−2.11261
$743$			43.1880i			1.58441i	0.610252	+	0.792207i	$0.291068\pi$
$743$			43.1880i			1.58441i	−0.610252	+	0.792207i	$0.708932\pi$
$744$		0			0
$745$	3.51872			0.128916
$746$	18.4501			0.675506
$747$		0			0
$748$		−	3.50932i		−	0.128314i
$749$	−88.5586			−3.23586
$750$		0			0
$751$	−27.0187			−0.985925			−0.492962	−	0.870051i	$-0.664086\pi$
$751$	−27.0187			−0.985925			−0.492962	+	0.870051i	$0.664086\pi$
$752$		−	5.68266i		−	0.207225i
$753$		0			0
$754$			19.8063i			0.721303i
$755$			14.1659i			0.515550i
$756$		0			0
$757$			32.1030i			1.16680i	0.812184	+	0.583402i	$0.198279\pi$
$757$			32.1030i			1.16680i	−0.812184	+	0.583402i	$0.801721\pi$
$758$			3.95014i			0.143476i
$759$		0			0
$760$		−	7.21384i		−	0.261674i
$761$	−22.6216			−0.820031			−0.410016	−	0.912078i	$-0.634477\pi$
$761$	−22.6216			−0.820031			−0.410016	+	0.912078i	$0.634477\pi$
$762$		0			0
$763$	−22.4217			−0.811721
$764$	18.1304			0.655936
$765$		0			0
$766$			2.90672i			0.105024i
$767$			4.26291i			0.153925i
$768$		0			0
$769$	−38.3830			−1.38413			−0.692063	−	0.721837i	$-0.743298\pi$
$769$	−38.3830			−1.38413			−0.692063	+	0.721837i	$0.743298\pi$
$770$			15.1349i			0.545423i
$771$		0			0
$772$			12.2241i			0.439955i
$773$	41.3477			1.48717			0.743586	−	0.668640i	$-0.233123\pi$
$773$	41.3477			1.48717			0.743586	+	0.668640i	$0.233123\pi$
$774$		0			0
$775$	−60.6428			−2.17835
$776$	10.0724			0.361577
$777$		0			0
$778$	13.6810			0.490488
$779$		−	2.82551i		−	0.101234i
$780$		0			0
$781$	5.62714			0.201355
$782$	−35.4580			−1.26797
$783$		0			0
$784$	14.7511			0.526827
$785$		−	40.1543i		−	1.43317i
$786$		0			0
$787$			30.4917i			1.08691i	0.839437	+	0.543456i	$0.182885\pi$
$787$			30.4917i			1.08691i	−0.839437	+	0.543456i	$0.817115\pi$
$788$		−	2.60238i		−	0.0927061i
$789$		0			0
$790$	20.4471			0.727473
$791$	−84.6899			−3.01123
$792$		0			0
$793$	−7.74047			−0.274872
$794$	−26.2699			−0.932284
$795$		0			0
$796$	14.9269			0.529071
$797$			1.28996i			0.0456928i	0.999739	+	0.0228464i	$0.00727286\pi$
$797$			1.28996i			0.0456928i	−0.999739	+	0.0228464i	$0.992727\pi$
$798$		0			0
$799$	22.0984			0.781785
$800$		−	7.93141i		−	0.280418i
$801$		0			0
$802$	1.64967			0.0582520
$803$	−13.8336			−0.488177
$804$		0			0
$805$	152.922			5.38978
$806$	−29.8887			−1.05279
$807$		0			0
$808$	−11.9250			−0.419520
$809$	−41.9767			−1.47582			−0.737911	−	0.674898i	$-0.764187\pi$
$809$	−41.9767			−1.47582			−0.737911	+	0.674898i	$0.764187\pi$
$810$		0			0
$811$			2.73208i			0.0959365i	0.998849	+	0.0479682i	$0.0152746\pi$
$811$			2.73208i			0.0959365i	−0.998849	+	0.0479682i	$0.984725\pi$
$812$		−	23.6301i		−	0.829254i
$813$		0			0
$814$		−	2.05496i		−	0.0720264i
$815$		−	56.0822i		−	1.96448i
$816$		0			0
$817$	−10.5545			−0.369255
$818$	−15.3942			−0.538247
$819$		0			0
$820$		−	5.06496i		−	0.176876i
$821$		−	52.4879i		−	1.83184i	−0.401359	−	0.915921i	$-0.631462\pi$
$821$		−	52.4879i		−	1.83184i	0.401359	−	0.915921i	$-0.368538\pi$
$822$		0			0
$823$		−	48.4730i		−	1.68966i	−0.535032	−	0.844832i	$-0.679701\pi$
$823$		−	48.4730i		−	1.68966i	0.535032	−	0.844832i	$-0.320299\pi$
$824$	−15.4768			−0.539160
$825$		0			0
$826$		−	5.08591i		−	0.176961i
$827$	−19.6698			−0.683987			−0.341994	−	0.939702i	$-0.611102\pi$
$827$	−19.6698			−0.683987			−0.341994	+	0.939702i	$0.611102\pi$
$828$		0			0
$829$		−	0.791876i		−	0.0275030i	−0.999905	−	0.0137515i	$-0.995623\pi$
$829$		−	0.791876i		−	0.0275030i	0.999905	−	0.0137515i	$-0.00437737\pi$
$830$	−7.36519			−0.255649
$831$		0			0
$832$		−	3.90912i		−	0.135524i
$833$			57.3634i			1.98752i
$834$		0			0
$835$	−28.4247			−0.983678
$836$	−1.81033			−0.0626117
$837$		0			0
$838$	24.8933			0.859926
$839$	21.2090			0.732215			0.366108	−	0.930573i	$-0.380690\pi$
$839$	21.2090			0.732215			0.366108	+	0.930573i	$0.380690\pi$
$840$		0			0
$841$	3.32862			0.114780
$842$	1.89567			0.0653292
$843$		0			0
$844$	−16.4855			−0.567455
$845$	8.20325			0.282200
$846$		0			0
$847$	−47.5038			−1.63225
$848$		−	12.3390i		−	0.423724i
$849$		0			0
$850$	30.8432			1.05791
$851$	−20.7632			−0.711753
$852$		0			0
$853$		−	12.6616i		−	0.433525i	−0.976224	−	0.216762i	$-0.930450\pi$
$853$		−	12.6616i		−	0.433525i	0.976224	−	0.216762i	$-0.0695497\pi$
$854$	9.23484			0.316010
$855$		0			0
$856$		−	18.9884i		−	0.649012i
$857$		−	27.3655i		−	0.934786i	−0.884050	−	0.467393i	$-0.845193\pi$
$857$		−	27.3655i		−	0.934786i	0.884050	−	0.467393i	$-0.154807\pi$
$858$		0			0
$859$			29.0721i			0.991927i	0.868343	+	0.495963i	$0.165185\pi$
$859$			29.0721i			0.991927i	−0.868343	+	0.495963i	$0.834815\pi$
$860$	−18.9198			−0.645159
$861$		0			0
$862$			20.7721i			0.707501i
$863$	−26.9203			−0.916379			−0.458190	−	0.888854i	$-0.651502\pi$
$863$	−26.9203			−0.916379			−0.458190	+	0.888854i	$0.651502\pi$
$864$		0			0
$865$	−56.7444			−1.92937
$866$		−	23.5538i		−	0.800389i
$867$		0			0
$868$	35.6591			1.21035
$869$		−	5.13124i		−	0.174065i
$870$		0			0
$871$	54.6881			1.85303
$872$		−	4.80760i		−	0.162806i
$873$		0			0
$874$			18.2915i			0.618718i
$875$	−49.1632			−1.66202
$876$		0			0
$877$			51.7787i			1.74844i	0.485529	+	0.874220i	$0.338627\pi$
$877$			51.7787i			1.74844i	−0.485529	+	0.874220i	$0.661373\pi$
$878$	−27.5002			−0.928086
$879$		0			0
$880$	−3.24517			−0.109395
$881$		−	4.09113i		−	0.137834i	−0.997622	−	0.0689168i	$-0.978046\pi$
$881$		−	4.09113i		−	0.137834i	0.997622	−	0.0689168i	$-0.0219543\pi$
$882$		0			0
$883$			3.63618i			0.122367i	0.998127	+	0.0611837i	$0.0194875\pi$
$883$			3.63618i			0.122367i	−0.998127	+	0.0611837i	$0.980512\pi$
$884$	15.2015			0.511283
$885$		0			0
$886$	−24.3687			−0.818683
$887$		−	7.58846i		−	0.254796i	−0.991852	−	0.127398i	$-0.959338\pi$
$887$		−	7.58846i		−	0.254796i	0.991852	−	0.127398i	$-0.0406625\pi$
$888$		0			0
$889$	−17.6381			−0.591563
$890$	−7.04093			−0.236012
$891$		0			0
$892$	−0.326717	−	14.9296i	−0.0109393	−	0.499880i
$893$		−	11.3998i		−	0.381478i
$894$		0			0
$895$			15.8015i			0.528186i
$896$			4.66381i			0.155807i
$897$		0			0
$898$		−	22.6953i		−	0.757352i
$899$		−	38.7394i		−	1.29203i
$900$		0			0
$901$	47.9833			1.59856
$902$	−1.27106			−0.0423218
$903$		0			0
$904$		−	18.1589i		−	0.603958i
$905$	21.2773			0.707283
$906$		0			0
$907$	−32.2633			−1.07129			−0.535643	−	0.844444i	$-0.679931\pi$
$907$	−32.2633			−1.07129			−0.535643	+	0.844444i	$0.679931\pi$
$908$			3.91525i			0.129932i
$909$		0			0
$910$	−65.5606			−2.17331
$911$			18.1569i			0.601564i	0.953693	+	0.300782i	$0.0972477\pi$
$911$			18.1569i			0.601564i	−0.953693	+	0.300782i	$0.902752\pi$
$912$		0			0
$913$			1.84831i			0.0611702i
$914$	20.6720			0.683769
$915$		0			0
$916$			5.35015i			0.176774i
$917$			45.0838i			1.48880i
$918$		0			0
$919$		−	49.5535i		−	1.63462i	−0.576200	−	0.817309i	$-0.695465\pi$
$919$		−	49.5535i		−	1.63462i	0.576200	−	0.817309i	$-0.304535\pi$
$920$			32.7890i			1.08102i
$921$		0			0
$922$	25.4522			0.838224
$923$			24.3754i			0.802326i
$924$		0			0
$925$	18.0609			0.593839
$926$		−	25.3610i		−	0.833414i
$927$		0			0
$928$	5.06669			0.166322
$929$			24.2071i			0.794208i	0.917774	+	0.397104i	$0.129985\pi$
$929$			24.2071i			0.794208i	−0.917774	+	0.397104i	$0.870015\pi$
$930$		0			0
$931$	29.5917			0.969828
$932$	15.0088			0.491629
$933$		0			0
$934$			5.67695i			0.185756i
$935$		−	12.6196i		−	0.412706i
$936$		0			0
$937$		−	22.3246i		−	0.729314i	−0.931142	−	0.364657i	$-0.881186\pi$
$937$		−	22.3246i		−	0.729314i	0.931142	−	0.364657i	$-0.118814\pi$
$938$	−65.2462			−2.13036
$939$		0			0
$940$		−	20.4350i		−	0.666516i
$941$		−	17.3826i		−	0.566657i	−0.959023	−	0.283328i	$-0.908561\pi$
$941$		−	17.3826i		−	0.566657i	0.959023	−	0.283328i	$-0.0914386\pi$
$942$		0			0
$943$			12.8427i			0.418217i
$944$	1.09050			0.0354929
$945$		0			0
$946$			4.74797i			0.154370i
$947$			13.8285i			0.449365i	0.974432	+	0.224682i	$0.0721344\pi$
$947$			13.8285i			0.449365i	−0.974432	+	0.224682i	$0.927866\pi$
$948$		0			0
$949$		−	59.9238i		−	1.94521i
$950$		−	15.9109i		−	0.516217i
$951$		0			0
$952$	−18.1364			−0.587803
$953$	23.3803			0.757363			0.378682	−	0.925527i	$-0.376378\pi$
$953$	23.3803			0.757363			0.378682	+	0.925527i	$0.376378\pi$
$954$		0			0
$955$	65.1975			2.10974
$956$		−	22.7374i		−	0.735379i
$957$		0			0
$958$	−7.99218			−0.258216
$959$	2.19361			0.0708353
$960$		0			0
$961$	27.4598			0.885800
$962$	8.90160			0.286999
$963$		0			0
$964$	15.5649			0.501313
$965$			43.9582i			1.41506i
$966$		0			0
$967$		−	31.7180i		−	1.01998i	−0.860179	−	0.509992i	$-0.829649\pi$
$967$		−	31.7180i		−	1.01998i	0.860179	−	0.509992i	$-0.170351\pi$
$968$		−	10.1856i		−	0.327378i
$969$		0			0
$970$	36.2206			1.16297
$971$	43.6042			1.39933			0.699663	−	0.714473i	$-0.253333\pi$
$971$	43.6042			1.39933			0.699663	+	0.714473i	$0.253333\pi$
$972$		0			0
$973$	−67.4796			−2.16330
$974$		−	17.2751i		−	0.553530i
$975$		0			0
$976$			1.98011i			0.0633816i
$977$	35.3637			1.13139			0.565693	−	0.824616i	$-0.308609\pi$
$977$	35.3637			1.13139			0.565693	+	0.824616i	$0.308609\pi$
$978$		0			0
$979$			1.76694i			0.0564716i
$980$	53.0455			1.69448
$981$		0			0
$982$			6.80166i			0.217050i
$983$	−42.9170			−1.36884			−0.684420	−	0.729088i	$-0.739944\pi$
$983$	−42.9170			−1.36884			−0.684420	+	0.729088i	$0.739944\pi$
$984$		0			0
$985$		−	9.35825i		−	0.298178i
$986$			19.7031i			0.627473i
$987$		0			0
$988$		−	7.84192i		−	0.249485i
$989$	47.9731			1.52546
$990$		0			0
$991$		−	31.2085i		−	0.991371i	−0.868502	−	0.495685i	$-0.834917\pi$
$991$		−	31.2085i		−	0.991371i	0.868502	−	0.495685i	$-0.165083\pi$
$992$			7.64590i			0.242758i
$993$		0			0
$994$		−	29.0813i		−	0.922404i
$995$	53.6776			1.70170
$996$		0			0
$997$	−17.6931			−0.560347			−0.280174	−	0.959949i	$-0.590392\pi$
$997$	−17.6931			−0.560347			−0.280174	+	0.959949i	$0.590392\pi$
$998$			34.3509i			1.08736i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.d.a.4013.37	yes	72
3.2	odd	2	inner	4014.2.d.a.4013.10	yes	72
223.222	odd	2	inner	4014.2.d.a.4013.9	✓	72
669.668	even	2	inner	4014.2.d.a.4013.38	yes	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.d.a.4013.9	✓	72	223.222	odd	2	inner
4014.2.d.a.4013.10	yes	72	3.2	odd	2	inner
4014.2.d.a.4013.37	yes	72	1.1	even	1	trivial
4014.2.d.a.4013.38	yes	72	669.668	even	2	inner

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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.59603 q^{5} +4.66381 q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.59603 q^{5} +4.66381 q^{7} +1.00000i q^{8} +3.59603i q^{10} +0.902432 q^{11} +3.90912i q^{13} -4.66381i q^{14} +1.00000 q^{16} +3.88874i q^{17} +2.00606 q^{19} +3.59603 q^{20} -0.902432i q^{22} -9.11811 q^{23} +7.93141 q^{25} +3.90912 q^{26} -4.66381 q^{28} +5.06669i q^{29} -7.64590 q^{31} -1.00000i q^{32} +3.88874 q^{34} -16.7712 q^{35} +2.27714 q^{37} -2.00606i q^{38} -3.59603i q^{40} -1.40849i q^{41} -5.26130 q^{43} -0.902432 q^{44} +9.11811i q^{46} -5.68266i q^{47} +14.7511 q^{49} -7.93141i q^{50} -3.90912i q^{52} -12.3390i q^{53} -3.24517 q^{55} +4.66381i q^{56} +5.06669 q^{58} +1.09050 q^{59} +1.98011i q^{61} +7.64590i q^{62} -1.00000 q^{64} -14.0573i q^{65} -13.9899i q^{67} -3.88874i q^{68} +16.7712i q^{70} +6.23553 q^{71} -15.3293 q^{73} -2.27714i q^{74} -2.00606 q^{76} +4.20877 q^{77} -5.68601i q^{79} -3.59603 q^{80} -1.40849 q^{82} +2.04815i q^{83} -13.9840i q^{85} +5.26130i q^{86} +0.902432i q^{88} +1.95797i q^{89} +18.2314i q^{91} +9.11811 q^{92} -5.68266 q^{94} -7.21384 q^{95} -10.0724i q^{97} -14.7511i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) \) 72 * q - 72 * q^4 + 16 * q^7	\( 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) \) 72 * q - 72 * q^4 + 16 * q^7 + 72 * q^16 - 40 * q^19 + 96 * q^25 - 16 * q^28 - 24 * q^37 - 8 * q^43 + 56 * q^49 + 40 * q^58 - 72 * q^64 - 32 * q^73 + 40 * q^76 + 16 * q^82

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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