LMFDB - Embedded newform 4140.2.s.a.737.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(737,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))

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

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

chi := DirichletCharacter("4140.737");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.s (of order $4$, degree $2$, 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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$22$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			737.11
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.a.2393.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.00942680 + 2.23605i) q^{5} +(-1.55392 - 1.55392i) q^{7} +O(q^{10})$	$q+(-0.00942680 + 2.23605i) q^{5} +(-1.55392 - 1.55392i) q^{7} +6.24582i q^{11} +(-4.66033 + 4.66033i) q^{13} +(-2.07495 + 2.07495i) q^{17} +3.70071i q^{19} +(-0.707107 - 0.707107i) q^{23} +(-4.99982 - 0.0421576i) q^{25} +7.12296 q^{29} -2.07962 q^{31} +(3.48928 - 3.45999i) q^{35} +(-1.56384 - 1.56384i) q^{37} -0.905077i q^{41} +(5.14434 - 5.14434i) q^{43} +(1.35961 - 1.35961i) q^{47} -2.17068i q^{49} +(-7.81185 - 7.81185i) q^{53} +(-13.9660 - 0.0588782i) q^{55} +9.74737 q^{59} -1.55919 q^{61} +(-10.3768 - 10.4647i) q^{65} +(-3.97119 - 3.97119i) q^{67} -2.59463i q^{71} +(5.20253 - 5.20253i) q^{73} +(9.70549 - 9.70549i) q^{77} +1.96587i q^{79} +(-8.81051 - 8.81051i) q^{83} +(-4.62013 - 4.65925i) q^{85} -1.58239 q^{89} +14.4835 q^{91} +(-8.27496 - 0.0348858i) q^{95} +(5.50862 + 5.50862i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 12 q^{7}+O(q^{10}) $ 44 * q - 12 * q^7	$ 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) $ 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.00942680	+	2.23605i	−0.00421579	+	0.999991i
$5$	−0.00942680	+	2.23605i	−0.00421579	+	0.999991i
$6$		0			0
$7$	−1.55392	−	1.55392i	−0.587326	−	0.587326i	0.349581	−	0.936906i	$-0.386324\pi$
$7$	−1.55392	−	1.55392i	−0.587326	−	0.587326i	−0.936906	+	0.349581i	$0.886324\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			6.24582i			1.88319i	0.336752	+	0.941593i	$0.390672\pi$
$11$			6.24582i			1.88319i	−0.336752	+	0.941593i	$0.609328\pi$
$12$		0			0
$13$	−4.66033	+	4.66033i	−1.29254	+	1.29254i	−0.359333	+	0.933209i	$0.616996\pi$
$13$	−4.66033	+	4.66033i	−1.29254	+	1.29254i	−0.933209	+	0.359333i	$0.883004\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.07495	+	2.07495i	−0.503250	+	0.503250i	−0.912446	−	0.409196i	$-0.865809\pi$
$17$	−2.07495	+	2.07495i	−0.503250	+	0.503250i	0.409196	+	0.912446i	$0.365809\pi$
$18$		0			0
$19$			3.70071i			0.849000i	0.905428	+	0.424500i	$0.139550\pi$
$19$			3.70071i			0.849000i	−0.905428	+	0.424500i	$0.860450\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.707107	−	0.707107i	−0.147442	−	0.147442i
$23$	−0.707107	−	0.707107i	−0.147442	−	0.147442i
$24$		0			0
$25$	−4.99982	−	0.0421576i	−0.999964	−	0.00843151i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.12296			1.32270			0.661350	−	0.750077i	$-0.269984\pi$
$29$	7.12296			1.32270			0.661350	+	0.750077i	$0.269984\pi$
$30$		0			0
$31$	−2.07962			−0.373512			−0.186756	−	0.982406i	$-0.559797\pi$
$31$	−2.07962			−0.373512			−0.186756	+	0.982406i	$0.559797\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.48928	−	3.45999i	0.589796	−	0.584844i
$36$		0			0
$37$	−1.56384	−	1.56384i	−0.257094	−	0.257094i	0.566777	−	0.823871i	$-0.308190\pi$
$37$	−1.56384	−	1.56384i	−0.257094	−	0.257094i	−0.823871	+	0.566777i	$0.808190\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	0.905077i		−	0.141349i	−0.997499	−	0.0706747i	$-0.977485\pi$
$41$		−	0.905077i		−	0.141349i	0.997499	−	0.0706747i	$-0.0225152\pi$
$42$		0			0
$43$	5.14434	−	5.14434i	0.784504	−	0.784504i	−0.196083	−	0.980587i	$-0.562822\pi$
$43$	5.14434	−	5.14434i	0.784504	−	0.784504i	0.980587	+	0.196083i	$0.0628222\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.35961	−	1.35961i	0.198320	−	0.198320i	−0.600959	−	0.799280i	$-0.705215\pi$
$47$	1.35961	−	1.35961i	0.198320	−	0.198320i	0.799280	+	0.600959i	$0.205215\pi$
$48$		0			0
$49$		−	2.17068i		−	0.310097i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.81185	−	7.81185i	−1.07304	−	1.07304i	−0.997113	−	0.0759272i	$-0.975808\pi$
$53$	−7.81185	−	7.81185i	−1.07304	−	1.07304i	−0.0759272	−	0.997113i	$-0.524192\pi$
$54$		0			0
$55$	−13.9660	−	0.0588782i	−1.88317	−	0.00793913i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.74737			1.26900			0.634500	−	0.772923i	$-0.281206\pi$
$59$	9.74737			1.26900			0.634500	+	0.772923i	$0.281206\pi$
$60$		0			0
$61$	−1.55919			−0.199634			−0.0998169	−	0.995006i	$-0.531826\pi$
$61$	−1.55919			−0.199634			−0.0998169	+	0.995006i	$0.531826\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−10.3768	−	10.4647i	−1.28708	−	1.29798i
$66$		0			0
$67$	−3.97119	−	3.97119i	−0.485158	−	0.485158i	0.421616	−	0.906774i	$-0.361463\pi$
$67$	−3.97119	−	3.97119i	−0.485158	−	0.485158i	−0.906774	+	0.421616i	$0.861463\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2.59463i		−	0.307926i	−0.988077	−	0.153963i	$-0.950796\pi$
$71$		−	2.59463i		−	0.307926i	0.988077	−	0.153963i	$-0.0492037\pi$
$72$		0			0
$73$	5.20253	−	5.20253i	0.608910	−	0.608910i	−0.333751	−	0.942661i	$-0.608314\pi$
$73$	5.20253	−	5.20253i	0.608910	−	0.608910i	0.942661	+	0.333751i	$0.108314\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.70549	−	9.70549i	1.10604	−	1.10604i
$78$		0			0
$79$			1.96587i			0.221177i	0.993866	+	0.110589i	$0.0352736\pi$
$79$			1.96587i			0.221177i	−0.993866	+	0.110589i	$0.964726\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.81051	−	8.81051i	−0.967079	−	0.967079i	0.0323962	−	0.999475i	$-0.489686\pi$
$83$	−8.81051	−	8.81051i	−0.967079	−	0.967079i	−0.999475	+	0.0323962i	$0.989686\pi$
$84$		0			0
$85$	−4.62013	−	4.65925i	−0.501124	−	0.505367i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.58239			−0.167733			−0.0838667	−	0.996477i	$-0.526727\pi$
$89$	−1.58239			−0.167733			−0.0838667	+	0.996477i	$0.526727\pi$
$90$		0			0
$91$	14.4835			1.51829
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.27496	−	0.0348858i	−0.848993	−	0.00357921i
$96$		0			0
$97$	5.50862	+	5.50862i	0.559315	+	0.559315i	0.929112	−	0.369797i	$-0.120573\pi$
$97$	5.50862	+	5.50862i	0.559315	+	0.559315i	−0.369797	+	0.929112i	$0.620573\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	12.9408i		−	1.28766i	−0.765170	−	0.643829i	$-0.777345\pi$
$101$		−	12.9408i		−	1.28766i	0.765170	−	0.643829i	$-0.222655\pi$
$102$		0			0
$103$	−2.83379	+	2.83379i	−0.279222	+	0.279222i	−0.832798	−	0.553577i	$-0.813263\pi$
$103$	−2.83379	+	2.83379i	−0.279222	+	0.279222i	0.553577	+	0.832798i	$0.313263\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.9019	+	11.9019i	−1.15060	+	1.15060i	−0.164168	+	0.986432i	$0.552494\pi$
$107$	−11.9019	+	11.9019i	−1.15060	+	1.15060i	−0.986432	+	0.164168i	$0.947506\pi$
$108$		0			0
$109$			17.0357i			1.63173i	0.578246	+	0.815863i	$0.303738\pi$
$109$			17.0357i			1.63173i	−0.578246	+	0.815863i	$0.696262\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.26722	+	6.26722i	0.589571	+	0.589571i	0.937515	−	0.347944i	$-0.113120\pi$
$113$	6.26722	+	6.26722i	0.589571	+	0.589571i	−0.347944	+	0.937515i	$0.613120\pi$
$114$		0			0
$115$	1.58779	−	1.57446i	0.148062	−	0.146819i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.44861			0.591143
$120$		0			0
$121$	−28.0103			−2.54639
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.141399	−	11.1794i	0.0126471	−	0.999920i
$126$		0			0
$127$	7.29049	+	7.29049i	0.646926	+	0.646926i	0.952249	−	0.305323i	$-0.0987644\pi$
$127$	7.29049	+	7.29049i	0.646926	+	0.646926i	−0.305323	+	0.952249i	$0.598764\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	10.5662i		−	0.923176i	−0.887094	−	0.461588i	$-0.847280\pi$
$131$		−	10.5662i		−	0.923176i	0.887094	−	0.461588i	$-0.152720\pi$
$132$		0			0
$133$	5.75059	−	5.75059i	0.498640	−	0.498640i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.12530	−	1.12530i	0.0961407	−	0.0961407i	−0.657401	−	0.753541i	$-0.728344\pi$
$137$	1.12530	−	1.12530i	0.0961407	−	0.0961407i	0.753541	+	0.657401i	$0.228344\pi$
$138$		0			0
$139$			20.6527i			1.75174i	0.482545	+	0.875871i	$0.339712\pi$
$139$			20.6527i			1.75174i	−0.482545	+	0.875871i	$0.660288\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−29.1076	−	29.1076i	−2.43410	−	2.43410i
$144$		0			0
$145$	−0.0671467	+	15.9273i	−0.00557623	+	1.32269i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.60074			0.786523			0.393262	−	0.919427i	$-0.371347\pi$
$149$	9.60074			0.786523			0.393262	+	0.919427i	$0.371347\pi$
$150$		0			0
$151$	−3.93774			−0.320449			−0.160224	−	0.987081i	$-0.551222\pi$
$151$	−3.93774			−0.320449			−0.160224	+	0.987081i	$0.551222\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.0196042	−	4.65014i	0.00157465	−	0.373508i
$156$		0			0
$157$	−10.4062	−	10.4062i	−0.830507	−	0.830507i	0.157079	−	0.987586i	$-0.449792\pi$
$157$	−10.4062	−	10.4062i	−0.830507	−	0.830507i	−0.987586	+	0.157079i	$0.949792\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			2.19757i			0.173193i
$162$		0			0
$163$	−2.18375	+	2.18375i	−0.171044	+	0.171044i	−0.787438	−	0.616394i	$-0.788593\pi$
$163$	−2.18375	+	2.18375i	−0.171044	+	0.171044i	0.616394	+	0.787438i	$0.288593\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−14.9541	+	14.9541i	−1.15718	+	1.15718i	−0.172105	+	0.985079i	$0.555057\pi$
$167$	−14.9541	+	14.9541i	−1.15718	+	1.15718i	−0.985079	+	0.172105i	$0.944943\pi$
$168$		0			0
$169$		−	30.4373i		−	2.34133i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.40271	−	2.40271i	−0.182675	−	0.182675i	0.609846	−	0.792520i	$-0.291231\pi$
$173$	−2.40271	−	2.40271i	−0.182675	−	0.182675i	−0.792520	+	0.609846i	$0.791231\pi$
$174$		0			0
$175$	7.70380	+	7.83482i	0.582353	+	0.592257i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.2490			0.840791			0.420396	−	0.907341i	$-0.361891\pi$
$179$	11.2490			0.840791			0.420396	+	0.907341i	$0.361891\pi$
$180$		0			0
$181$	−10.5481			−0.784037			−0.392019	−	0.919957i	$-0.628223\pi$
$181$	−10.5481			−0.784037			−0.392019	+	0.919957i	$0.628223\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.51157	−	3.48208i	0.258176	−	0.256008i
$186$		0			0
$187$	−12.9598	−	12.9598i	−0.947714	−	0.947714i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			15.3707i			1.11219i	0.831119	+	0.556094i	$0.187701\pi$
$191$			15.3707i			1.11219i	−0.831119	+	0.556094i	$0.812299\pi$
$192$		0			0
$193$	−5.83680	+	5.83680i	−0.420142	+	0.420142i	−0.885253	−	0.465111i	$-0.846015\pi$
$193$	−5.83680	+	5.83680i	−0.420142	+	0.420142i	0.465111	+	0.885253i	$0.346015\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.92449	+	1.92449i	−0.137114	+	0.137114i	−0.772333	−	0.635218i	$-0.780910\pi$
$197$	−1.92449	+	1.92449i	−0.137114	+	0.137114i	0.635218	+	0.772333i	$0.280910\pi$
$198$		0			0
$199$			17.8337i			1.26420i	0.774887	+	0.632099i	$0.217807\pi$
$199$			17.8337i			1.26420i	−0.774887	+	0.632099i	$0.782193\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−11.0685	−	11.0685i	−0.776856	−	0.776856i
$204$		0			0
$205$	2.02380	+	0.00853199i	0.141348	+	0.000595900i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−23.1140			−1.59883
$210$		0			0
$211$	22.5611			1.55317			0.776584	−	0.630013i	$-0.216951\pi$
$211$	22.5611			1.55317			0.776584	+	0.630013i	$0.216951\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	11.4545	+	11.5515i	0.781190	+	0.787805i
$216$		0			0
$217$	3.23156	+	3.23156i	0.219373	+	0.219373i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	19.3399i		−	1.30094i
$222$		0			0
$223$	−6.73326	+	6.73326i	−0.450892	+	0.450892i	−0.895651	−	0.444758i	$-0.853290\pi$
$223$	−6.73326	+	6.73326i	−0.450892	+	0.450892i	0.444758	+	0.895651i	$0.353290\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.99799	−	9.99799i	0.663590	−	0.663590i	−0.292634	−	0.956224i	$-0.594532\pi$
$227$	9.99799	−	9.99799i	0.663590	−	0.663590i	0.956224	+	0.292634i	$0.0945319\pi$
$228$		0			0
$229$		−	25.6826i		−	1.69715i	−0.529072	−	0.848577i	$-0.677460\pi$
$229$		−	25.6826i		−	1.69715i	0.529072	−	0.848577i	$-0.322540\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.9601	+	10.9601i	0.718017	+	0.718017i	0.968199	−	0.250182i	$-0.0804903\pi$
$233$	10.9601	+	10.9601i	0.718017	+	0.718017i	−0.250182	+	0.968199i	$0.580490\pi$
$234$		0			0
$235$	3.02735	+	3.05298i	0.197482	+	0.199154i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.95934			−0.644216			−0.322108	−	0.946703i	$-0.604391\pi$
$239$	−9.95934			−0.644216			−0.322108	+	0.946703i	$0.604391\pi$
$240$		0			0
$241$	−8.83203			−0.568921			−0.284461	−	0.958688i	$-0.591814\pi$
$241$	−8.83203			−0.568921			−0.284461	+	0.958688i	$0.591814\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.85375	+	0.0204626i	0.310095	+	0.00130731i
$246$		0			0
$247$	−17.2465	−	17.2465i	−1.09737	−	1.09737i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	7.77255i		−	0.490599i	−0.969447	−	0.245299i	$-0.921114\pi$
$251$		−	7.77255i		−	0.490599i	0.969447	−	0.245299i	$-0.0788863\pi$
$252$		0			0
$253$	4.41646	−	4.41646i	0.277661	−	0.277661i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.72628	−	6.72628i	0.419574	−	0.419574i	−0.465483	−	0.885057i	$-0.654119\pi$
$257$	6.72628	−	6.72628i	0.419574	−	0.419574i	0.885057	+	0.465483i	$0.154119\pi$
$258$		0			0
$259$			4.86016i			0.301996i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	16.0838	+	16.0838i	0.991770	+	0.991770i	0.999966	−	0.00819665i	$-0.00260910\pi$
$263$	16.0838	+	16.0838i	0.991770	+	0.991770i	−0.00819665	+	0.999966i	$0.502609\pi$
$264$		0			0
$265$	17.5413	−	17.3940i	1.07755	−	1.06851i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.54327			0.216037			0.108019	−	0.994149i	$-0.465549\pi$
$269$	3.54327			0.216037			0.108019	+	0.994149i	$0.465549\pi$
$270$		0			0
$271$	−25.8290			−1.56900			−0.784499	−	0.620130i	$-0.787080\pi$
$271$	−25.8290			−1.56900			−0.784499	+	0.620130i	$0.787080\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.263309	−	31.2280i	0.0158781	−	1.88312i
$276$		0			0
$277$	3.35385	+	3.35385i	0.201513	+	0.201513i	0.800648	−	0.599135i	$-0.204489\pi$
$277$	3.35385	+	3.35385i	0.201513	+	0.201513i	−0.599135	+	0.800648i	$0.704489\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	21.2289i		−	1.26641i	−0.773983	−	0.633206i	$-0.781738\pi$
$281$		−	21.2289i		−	1.26641i	0.773983	−	0.633206i	$-0.218262\pi$
$282$		0			0
$283$	−8.44359	+	8.44359i	−0.501920	+	0.501920i	−0.912034	−	0.410114i	$-0.865489\pi$
$283$	−8.44359	+	8.44359i	−0.501920	+	0.501920i	0.410114	+	0.912034i	$0.365489\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.40642	+	1.40642i	−0.0830181	+	0.0830181i
$288$		0			0
$289$			8.38914i			0.493479i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.06565	−	5.06565i	−0.295938	−	0.295938i	0.543482	−	0.839421i	$-0.317105\pi$
$293$	−5.06565	−	5.06565i	−0.295938	−	0.295938i	−0.839421	+	0.543482i	$0.817105\pi$
$294$		0			0
$295$	−0.0918865	+	21.7956i	−0.00534984	+	1.26899i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.59070			0.381150
$300$		0			0
$301$	−15.9878			−0.921519
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.0146982	−	3.48642i	0.000841615	−	0.199632i
$306$		0			0
$307$	−7.08165	−	7.08165i	−0.404171	−	0.404171i	0.475529	−	0.879700i	$-0.342257\pi$
$307$	−7.08165	−	7.08165i	−0.404171	−	0.404171i	−0.879700	+	0.475529i	$0.842257\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	12.0670i		−	0.684256i	−0.939653	−	0.342128i	$-0.888852\pi$
$311$		−	12.0670i		−	0.684256i	0.939653	−	0.342128i	$-0.111148\pi$
$312$		0			0
$313$	19.1588	−	19.1588i	1.08292	−	1.08292i	0.0866848	−	0.996236i	$-0.472373\pi$
$313$	19.1588	−	19.1588i	1.08292	−	1.08292i	0.996236	−	0.0866848i	$-0.0276273\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.2910	−	10.2910i	0.578001	−	0.578001i	−0.356351	−	0.934352i	$-0.615979\pi$
$317$	10.2910	−	10.2910i	0.578001	−	0.578001i	0.934352	+	0.356351i	$0.115979\pi$
$318$		0			0
$319$			44.4888i			2.49089i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.67879	−	7.67879i	−0.427259	−	0.427259i
$324$		0			0
$325$	23.4973	−	23.1043i	1.30339	−	1.28160i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.22546			−0.232957
$330$		0			0
$331$	2.44901			0.134610			0.0673050	−	0.997732i	$-0.478560\pi$
$331$	2.44901			0.134610			0.0673050	+	0.997732i	$0.478560\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.91721	−	8.84234i	0.487199	−	0.483109i
$336$		0			0
$337$	−22.3445	−	22.3445i	−1.21718	−	1.21718i	−0.968614	−	0.248569i	$-0.920040\pi$
$337$	−22.3445	−	22.3445i	−1.21718	−	1.21718i	−0.248569	−	0.968614i	$-0.579960\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	12.9890i		−	0.703392i
$342$		0			0
$343$	−14.2505	+	14.2505i	−0.769454	+	0.769454i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.6959	−	10.6959i	0.574186	−	0.574186i	−0.359110	−	0.933295i	$-0.616920\pi$
$347$	10.6959	−	10.6959i	0.574186	−	0.574186i	0.933295	+	0.359110i	$0.116920\pi$
$348$		0			0
$349$		−	10.7099i		−	0.573286i	−0.958037	−	0.286643i	$-0.907461\pi$
$349$		−	10.7099i		−	0.573286i	0.958037	−	0.286643i	$-0.0925394\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	19.6805	+	19.6805i	1.04749	+	1.04749i	0.998815	+	0.0486718i	$0.0154988\pi$
$353$	19.6805	+	19.6805i	1.04749	+	1.04749i	0.0486718	+	0.998815i	$0.484501\pi$
$354$		0			0
$355$	5.80172	+	0.0244591i	0.307924	+	0.00129815i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−19.6169			−1.03534			−0.517670	−	0.855580i	$-0.673201\pi$
$359$	−19.6169			−1.03534			−0.517670	+	0.855580i	$0.673201\pi$
$360$		0			0
$361$	5.30477			0.279198
$362$		0			0
$363$		0			0
$364$		0			0
$365$	11.5841	+	11.6822i	0.606338	+	0.611472i
$366$		0			0
$367$	1.62380	+	1.62380i	0.0847616	+	0.0847616i	0.748216	−	0.663455i	$-0.230911\pi$
$367$	1.62380	+	1.62380i	0.0847616	+	0.0847616i	−0.663455	+	0.748216i	$0.730911\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			24.2779i			1.26045i
$372$		0			0
$373$	0.529075	−	0.529075i	0.0273945	−	0.0273945i	−0.693277	−	0.720671i	$-0.743834\pi$
$373$	0.529075	−	0.529075i	0.0273945	−	0.0273945i	0.720671	+	0.693277i	$0.243834\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−33.1953	+	33.1953i	−1.70965	+	1.70965i
$378$		0			0
$379$		−	10.3217i		−	0.530191i	−0.964222	−	0.265095i	$-0.914597\pi$
$379$		−	10.3217i		−	0.530191i	0.964222	−	0.265095i	$-0.0854035\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.0688	−	13.0688i	−0.667783	−	0.667783i	0.289419	−	0.957202i	$-0.406538\pi$
$383$	−13.0688	−	13.0688i	−0.667783	−	0.667783i	−0.957202	+	0.289419i	$0.906538\pi$
$384$		0			0
$385$	21.6105	+	21.7934i	1.10137	+	1.11070i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.4773			0.835433			0.417716	−	0.908577i	$-0.362831\pi$
$389$	16.4773			0.835433			0.417716	+	0.908577i	$0.362831\pi$
$390$		0			0
$391$	2.93443			0.148400
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.39577	−	0.0185318i	−0.221175	−	0.000932438i
$396$		0			0
$397$	−3.87951	−	3.87951i	−0.194707	−	0.194707i	0.603019	−	0.797727i	$-0.293964\pi$
$397$	−3.87951	−	3.87951i	−0.194707	−	0.194707i	−0.797727	+	0.603019i	$0.793964\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			30.3449i			1.51535i	0.652632	+	0.757675i	$0.273665\pi$
$401$			30.3449i			1.51535i	−0.652632	+	0.757675i	$0.726335\pi$
$402$		0			0
$403$	9.69173	−	9.69173i	0.482780	−	0.482780i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.76748	−	9.76748i	0.484156	−	0.484156i
$408$		0			0
$409$			7.61720i			0.376647i	0.982107	+	0.188323i	$0.0603053\pi$
$409$			7.61720i			0.376647i	−0.982107	+	0.188323i	$0.939695\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−15.1466	−	15.1466i	−0.745316	−	0.745316i
$414$		0			0
$415$	19.7838	−	19.6177i	0.971147	−	0.962993i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−35.2723			−1.72316			−0.861582	−	0.507619i	$-0.830526\pi$
$419$	−35.2723			−1.72316			−0.861582	+	0.507619i	$0.830526\pi$
$420$		0			0
$421$	−38.5456			−1.87860			−0.939298	−	0.343102i	$-0.888522\pi$
$421$	−38.5456			−1.87860			−0.939298	+	0.343102i	$0.888522\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	10.4619	−	10.2869i	0.507475	−	0.498989i
$426$		0			0
$427$	2.42285	+	2.42285i	0.117250	+	0.117250i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	20.4909i		−	0.987013i	−0.869742	−	0.493507i	$-0.835715\pi$
$431$		−	20.4909i		−	0.987013i	0.869742	−	0.493507i	$-0.164285\pi$
$432$		0			0
$433$	−25.3083	+	25.3083i	−1.21624	+	1.21624i	−0.247302	+	0.968938i	$0.579544\pi$
$433$	−25.3083	+	25.3083i	−1.21624	+	1.21624i	−0.968938	+	0.247302i	$0.920456\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.61679	−	2.61679i	0.125178	−	0.125178i
$438$		0			0
$439$			26.1881i			1.24989i	0.780668	+	0.624946i	$0.214879\pi$
$439$			26.1881i			1.24989i	−0.780668	+	0.624946i	$0.785121\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.54960	+	6.54960i	0.311181	+	0.311181i	0.845367	−	0.534186i	$-0.179382\pi$
$443$	6.54960	+	6.54960i	0.311181	+	0.311181i	−0.534186	+	0.845367i	$0.679382\pi$
$444$		0			0
$445$	0.0149169	−	3.53831i	0.000707130	−	0.167732i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	36.8118			1.73725			0.868627	−	0.495466i	$-0.165003\pi$
$449$	36.8118			1.73725			0.868627	+	0.495466i	$0.165003\pi$
$450$		0			0
$451$	5.65295			0.266187
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.136533	+	32.3859i	−0.00640078	+	1.51827i
$456$		0			0
$457$	−24.4785	−	24.4785i	−1.14506	−	1.14506i	−0.987512	−	0.157546i	$-0.949642\pi$
$457$	−24.4785	−	24.4785i	−1.14506	−	1.14506i	−0.157546	−	0.987512i	$-0.550358\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	16.3941i		−	0.763548i	−0.924256	−	0.381774i	$-0.875313\pi$
$461$		−	16.3941i		−	0.763548i	0.924256	−	0.381774i	$-0.124687\pi$
$462$		0			0
$463$	4.23083	−	4.23083i	0.196623	−	0.196623i	−0.601927	−	0.798551i	$-0.705600\pi$
$463$	4.23083	−	4.23083i	0.196623	−	0.196623i	0.798551	+	0.601927i	$0.205600\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.0495	+	16.0495i	−0.742684	+	0.742684i	−0.973094	−	0.230410i	$-0.925993\pi$
$467$	−16.0495	+	16.0495i	−0.742684	+	0.742684i	0.230410	+	0.973094i	$0.425993\pi$
$468$		0			0
$469$			12.3418i			0.569892i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	32.1306	+	32.1306i	1.47737	+	1.47737i
$474$		0			0
$475$	0.156013	−	18.5029i	0.00715836	−	0.848970i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.7518			−0.902484			−0.451242	−	0.892402i	$-0.649019\pi$
$479$	−19.7518			−0.902484			−0.451242	+	0.892402i	$0.649019\pi$
$480$		0			0
$481$	14.5760			0.664610
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.3695	+	12.2656i	−0.561668	+	0.556952i
$486$		0			0
$487$	−6.32522	−	6.32522i	−0.286623	−	0.286623i	0.549120	−	0.835743i	$-0.314963\pi$
$487$	−6.32522	−	6.32522i	−0.286623	−	0.286623i	−0.835743	+	0.549120i	$0.814963\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			19.6085i			0.884919i	0.896788	+	0.442460i	$0.145894\pi$
$491$			19.6085i			0.884919i	−0.896788	+	0.442460i	$0.854106\pi$
$492$		0			0
$493$	−14.7798	+	14.7798i	−0.665649	+	0.665649i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.03184	+	4.03184i	−0.180853	+	0.180853i
$498$		0			0
$499$			6.62210i			0.296446i	0.988954	+	0.148223i	$0.0473553\pi$
$499$			6.62210i			0.296446i	−0.988954	+	0.148223i	$0.952645\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5.61934	+	5.61934i	0.250554	+	0.250554i	0.821198	−	0.570644i	$-0.193306\pi$
$503$	5.61934	+	5.61934i	0.250554	+	0.250554i	−0.570644	+	0.821198i	$0.693306\pi$
$504$		0			0
$505$	28.9362	+	0.121990i	1.28765	+	0.00542850i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−15.3535			−0.680534			−0.340267	−	0.940329i	$-0.610517\pi$
$509$	−15.3535			−0.680534			−0.340267	+	0.940329i	$0.610517\pi$
$510$		0			0
$511$	−16.1686			−0.715257
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.30978	−	6.36320i	−0.278042	−	0.280396i
$516$		0			0
$517$	8.49191	+	8.49191i	0.373474	+	0.373474i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			35.9502i			1.57501i	0.616310	+	0.787503i	$0.288627\pi$
$521$			35.9502i			1.57501i	−0.616310	+	0.787503i	$0.711373\pi$
$522$		0			0
$523$	−18.1551	+	18.1551i	−0.793868	+	0.793868i	−0.982121	−	0.188252i	$-0.939718\pi$
$523$	−18.1551	+	18.1551i	−0.793868	+	0.793868i	0.188252	+	0.982121i	$0.439718\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.31512	−	4.31512i	0.187970	−	0.187970i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.21796	+	4.21796i	0.182700	+	0.182700i
$534$		0			0
$535$	−26.5010	−	26.7254i	−1.14574	−	1.15544i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	13.5577			0.583972
$540$		0			0
$541$	17.7818			0.764501			0.382250	−	0.924059i	$-0.375149\pi$
$541$	17.7818			0.764501			0.382250	+	0.924059i	$0.375149\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−38.0927	−	0.160592i	−1.63171	−	0.00687902i
$546$		0			0
$547$	20.6855	+	20.6855i	0.884448	+	0.884448i	0.993983	−	0.109535i	$-0.0349360\pi$
$547$	20.6855	+	20.6855i	0.884448	+	0.884448i	−0.109535	+	0.993983i	$0.534936\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			26.3600i			1.12297i
$552$		0			0
$553$	3.05479	−	3.05479i	0.129903	−	0.129903i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.0909	−	18.0909i	0.766537	−	0.766537i	−0.210958	−	0.977495i	$-0.567658\pi$
$557$	18.0909	−	18.0909i	0.766537	−	0.766537i	0.977495	+	0.210958i	$0.0676583\pi$
$558$		0			0
$559$			47.9486i			2.02801i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.53288	−	4.53288i	−0.191038	−	0.191038i	0.605106	−	0.796145i	$-0.293131\pi$
$563$	−4.53288	−	4.53288i	−0.191038	−	0.191038i	−0.796145	+	0.605106i	$0.793131\pi$
$564$		0			0
$565$	−14.0729	+	13.9547i	−0.592051	+	0.587080i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.7779			−0.493756			−0.246878	−	0.969047i	$-0.579405\pi$
$569$	−11.7779			−0.493756			−0.246878	+	0.969047i	$0.579405\pi$
$570$		0			0
$571$	7.88190			0.329847			0.164924	−	0.986306i	$-0.447262\pi$
$571$	7.88190			0.329847			0.164924	+	0.986306i	$0.447262\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.50560	+	3.56522i	0.146194	+	0.148680i
$576$		0			0
$577$	−5.52889	−	5.52889i	−0.230171	−	0.230171i	0.582593	−	0.812764i	$-0.302038\pi$
$577$	−5.52889	−	5.52889i	−0.230171	−	0.230171i	−0.812764	+	0.582593i	$0.802038\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			27.3816i			1.13598i
$582$		0			0
$583$	48.7915	−	48.7915i	2.02074	−	2.02074i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.39162	−	6.39162i	0.263810	−	0.263810i	−0.562790	−	0.826600i	$-0.690272\pi$
$587$	6.39162	−	6.39162i	0.263810	−	0.263810i	0.826600	+	0.562790i	$0.190272\pi$
$588$		0			0
$589$		−	7.69608i		−	0.317112i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	27.2569	+	27.2569i	1.11931	+	1.11931i	0.991843	+	0.127465i	$0.0406840\pi$
$593$	27.2569	+	27.2569i	1.11931	+	1.11931i	0.127465	+	0.991843i	$0.459316\pi$
$594$		0			0
$595$	−0.0607898	+	14.4194i	−0.00249214	+	0.591138i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.3475			−0.504503			−0.252252	−	0.967662i	$-0.581171\pi$
$599$	−12.3475			−0.504503			−0.252252	+	0.967662i	$0.581171\pi$
$600$		0			0
$601$	−42.8282			−1.74700			−0.873500	−	0.486824i	$-0.838155\pi$
$601$	−42.8282			−1.74700			−0.873500	+	0.486824i	$0.838155\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.264048	−	62.6324i	0.0107351	−	2.54637i
$606$		0			0
$607$	−26.3194	−	26.3194i	−1.06827	−	1.06827i	−0.997492	−	0.0707784i	$-0.977452\pi$
$607$	−26.3194	−	26.3194i	−1.06827	−	1.06827i	−0.0707784	−	0.997492i	$-0.522548\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			12.6725i			0.512674i
$612$		0			0
$613$	23.8991	−	23.8991i	0.965276	−	0.965276i	−0.0341413	−	0.999417i	$-0.510870\pi$
$613$	23.8991	−	23.8991i	0.965276	−	0.965276i	0.999417	+	0.0341413i	$0.0108696\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−20.0064	+	20.0064i	−0.805428	+	0.805428i	−0.983938	−	0.178510i	$-0.942872\pi$
$617$	−20.0064	+	20.0064i	−0.805428	+	0.805428i	0.178510	+	0.983938i	$0.442872\pi$
$618$		0			0
$619$			13.2217i			0.531426i	0.964052	+	0.265713i	$0.0856074\pi$
$619$			13.2217i			0.531426i	−0.964052	+	0.265713i	$0.914393\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.45891	+	2.45891i	0.0985141	+	0.0985141i
$624$		0			0
$625$	24.9964	+	0.421561i	0.999858	+	0.0168624i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.48980			0.258765
$630$		0			0
$631$	−28.3707			−1.12942			−0.564710	−	0.825290i	$-0.691012\pi$
$631$	−28.3707			−1.12942			−0.564710	+	0.825290i	$0.691012\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−16.3706	+	16.2332i	−0.649648	+	0.644193i
$636$		0			0
$637$	10.1161	+	10.1161i	0.400814	+	0.400814i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	23.8713i		−	0.942861i	−0.881903	−	0.471431i	$-0.843738\pi$
$641$		−	23.8713i		−	0.942861i	0.881903	−	0.471431i	$-0.156262\pi$
$642$		0			0
$643$	−6.51492	+	6.51492i	−0.256923	+	0.256923i	−0.823802	−	0.566878i	$-0.808151\pi$
$643$	−6.51492	+	6.51492i	−0.256923	+	0.256923i	0.566878	+	0.823802i	$0.308151\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11.3703	−	11.3703i	0.447013	−	0.447013i	−0.447347	−	0.894360i	$-0.647631\pi$
$647$	11.3703	−	11.3703i	0.447013	−	0.447013i	0.894360	+	0.447347i	$0.147631\pi$
$648$		0			0
$649$			60.8803i			2.38976i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−28.7343	−	28.7343i	−1.12446	−	1.12446i	−0.991063	−	0.133398i	$-0.957411\pi$
$653$	−28.7343	−	28.7343i	−1.12446	−	1.12446i	−0.133398	−	0.991063i	$-0.542589\pi$
$654$		0			0
$655$	23.6266	+	0.0996058i	0.923168	+	0.00389192i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	48.8690			1.90366			0.951832	−	0.306619i	$-0.0991979\pi$
$659$	48.8690			1.90366			0.951832	+	0.306619i	$0.0991979\pi$
$660$		0			0
$661$	−12.7332			−0.495262			−0.247631	−	0.968854i	$-0.579652\pi$
$661$	−12.7332			−0.495262			−0.247631	+	0.968854i	$0.579652\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.8044	+	12.9128i	0.496533	+	0.500737i
$666$		0			0
$667$	−5.03669	−	5.03669i	−0.195022	−	0.195022i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	9.73842i		−	0.375948i
$672$		0			0
$673$	−20.3216	+	20.3216i	−0.783342	+	0.783342i	−0.980393	−	0.197051i	$-0.936863\pi$
$673$	−20.3216	+	20.3216i	−0.783342	+	0.783342i	0.197051	+	0.980393i	$0.436863\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.66679	−	4.66679i	0.179359	−	0.179359i	−0.611717	−	0.791077i	$-0.709521\pi$
$677$	4.66679	−	4.66679i	0.179359	−	0.179359i	0.791077	+	0.611717i	$0.209521\pi$
$678$		0			0
$679$		−	17.1199i		−	0.657000i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−26.1459	−	26.1459i	−1.00045	−	1.00045i	−1.00000		0.000447001i	$-0.999858\pi$
$683$	−26.1459	−	26.1459i	−1.00045	−	1.00045i	−0.000447001	−	1.00000i	$-0.500142\pi$
$684$		0			0
$685$	2.50561	+	2.52683i	0.0957345	+	0.0965451i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	72.8116			2.77390
$690$		0			0
$691$	−30.7516			−1.16985			−0.584923	−	0.811089i	$-0.698875\pi$
$691$	−30.7516			−1.16985			−0.584923	+	0.811089i	$0.698875\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−46.1805	−	0.194689i	−1.75173	−	0.00738499i
$696$		0			0
$697$	1.87799	+	1.87799i	0.0711341	+	0.0711341i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			23.1492i			0.874333i	0.899381	+	0.437167i	$0.144018\pi$
$701$			23.1492i			0.874333i	−0.899381	+	0.437167i	$0.855982\pi$
$702$		0			0
$703$	5.78732	−	5.78732i	0.218273	−	0.218273i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−20.1089	+	20.1089i	−0.756274	+	0.756274i
$708$		0			0
$709$		−	39.5299i		−	1.48458i	−0.670081	−	0.742288i	$-0.733741\pi$
$709$		−	39.5299i		−	1.48458i	0.670081	−	0.742288i	$-0.266259\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.47052	+	1.47052i	0.0550713	+	0.0550713i
$714$		0			0
$715$	65.3604	−	64.8116i	2.44434	−	2.42382i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	0.0442843			0.00165153			0.000825763	−	1.00000i	$-0.499737\pi$
$719$	0.0442843			0.00165153			0.000825763		1.00000i	$0.499737\pi$
$720$		0			0
$721$	8.80695			0.327988
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−35.6135	−	0.300287i	−1.32265	−	0.0111524i
$726$		0			0
$727$	−20.1613	−	20.1613i	−0.747741	−	0.747741i	0.226313	−	0.974055i	$-0.427333\pi$
$727$	−20.1613	−	20.1613i	−0.747741	−	0.747741i	−0.974055	+	0.226313i	$0.927333\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			21.3485i			0.789603i
$732$		0			0
$733$	19.0075	−	19.0075i	0.702057	−	0.702057i	−0.262795	−	0.964852i	$-0.584644\pi$
$733$	19.0075	−	19.0075i	0.702057	−	0.702057i	0.964852	+	0.262795i	$0.0846442\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.8034	−	24.8034i	0.913643	−	0.913643i
$738$		0			0
$739$		−	3.47447i		−	0.127811i	−0.997956	−	0.0639053i	$-0.979644\pi$
$739$		−	3.47447i		−	0.127811i	0.997956	−	0.0639053i	$-0.0203555\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	9.90194	+	9.90194i	0.363267	+	0.363267i	0.865014	−	0.501747i	$-0.167309\pi$
$743$	9.90194	+	9.90194i	0.363267	+	0.363267i	−0.501747	+	0.865014i	$0.667309\pi$
$744$		0			0
$745$	−0.0905043	+	21.4677i	−0.00331582	+	0.786516i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	36.9891			1.35155
$750$		0			0
$751$	25.5210			0.931276			0.465638	−	0.884975i	$-0.345825\pi$
$751$	25.5210			0.931276			0.465638	+	0.884975i	$0.345825\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.0371203	−	8.80498i	0.00135095	−	0.320446i
$756$		0			0
$757$	28.6371	+	28.6371i	1.04083	+	1.04083i	0.999130	+	0.0417035i	$0.0132785\pi$
$757$	28.6371	+	28.6371i	1.04083	+	1.04083i	0.0417035	+	0.999130i	$0.486721\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	13.3747i		−	0.484834i	−0.970172	−	0.242417i	$-0.922060\pi$
$761$		−	13.3747i		−	0.484834i	0.970172	−	0.242417i	$-0.0779402\pi$
$762$		0			0
$763$	26.4721	−	26.4721i	0.958354	−	0.958354i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−45.4259	+	45.4259i	−1.64024	+	1.64024i
$768$		0			0
$769$		−	33.3559i		−	1.20284i	−0.798931	−	0.601422i	$-0.794601\pi$
$769$		−	33.3559i		−	1.20284i	0.798931	−	0.601422i	$-0.205399\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.2351	−	12.2351i	−0.440066	−	0.440066i	0.451968	−	0.892034i	$-0.350722\pi$
$773$	−12.2351	−	12.2351i	−0.440066	−	0.440066i	−0.892034	+	0.451968i	$0.850722\pi$
$774$		0			0
$775$	10.3978	+	0.0876719i	0.373498	+	0.00314927i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.34943			0.120006
$780$		0			0
$781$	16.2056			0.579883
$782$		0			0
$783$		0			0
$784$		0			0
$785$	23.3669	−	23.1707i	0.834001	−	0.826998i
$786$		0			0
$787$	15.2931	+	15.2931i	0.545141	+	0.545141i	0.925032	−	0.379890i	$-0.124038\pi$
$787$	15.2931	+	15.2931i	0.545141	+	0.545141i	−0.379890	+	0.925032i	$0.624038\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	19.4775i		−	0.692540i
$792$		0			0
$793$	7.26633	−	7.26633i	0.258035	−	0.258035i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.69592	−	6.69592i	0.237182	−	0.237182i	−0.578501	−	0.815682i	$-0.696362\pi$
$797$	6.69592	−	6.69592i	0.237182	−	0.237182i	0.815682	+	0.578501i	$0.196362\pi$
$798$		0			0
$799$			5.64227i			0.199609i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	32.4941	+	32.4941i	1.14669	+	1.14669i
$804$		0			0
$805$	−4.91387	−	0.0207161i	−0.173191	−	0.000730145i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	40.4880			1.42348			0.711741	−	0.702442i	$-0.247907\pi$
$809$	40.4880			1.42348			0.711741	+	0.702442i	$0.247907\pi$
$810$		0			0
$811$	31.3363			1.10037			0.550184	−	0.835044i	$-0.314558\pi$
$811$	31.3363			1.10037			0.550184	+	0.835044i	$0.314558\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.86238	−	4.90355i	−0.170322	−	0.171764i
$816$		0			0
$817$	19.0377	+	19.0377i	0.666044	+	0.666044i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			34.1074i			1.19036i	0.803594	+	0.595178i	$0.202918\pi$
$821$			34.1074i			1.19036i	−0.803594	+	0.595178i	$0.797082\pi$
$822$		0			0
$823$	21.6885	−	21.6885i	0.756014	−	0.756014i	−0.219580	−	0.975594i	$-0.570469\pi$
$823$	21.6885	−	21.6885i	0.756014	−	0.756014i	0.975594	+	0.219580i	$0.0704688\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.4940	+	35.4940i	−1.23425	+	1.23425i	−0.271930	+	0.962317i	$0.587662\pi$
$827$	−35.4940	+	35.4940i	−1.23425	+	1.23425i	−0.962317	+	0.271930i	$0.912338\pi$
$828$		0			0
$829$			32.9345i			1.14386i	0.820302	+	0.571931i	$0.193805\pi$
$829$			32.9345i			1.14386i	−0.820302	+	0.571931i	$0.806195\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.50406	+	4.50406i	0.156057	+	0.156057i
$834$		0			0
$835$	−33.2971	−	33.5791i	−1.15230	−	1.16205i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.1787			0.351407			0.175703	−	0.984443i	$-0.443780\pi$
$839$	10.1787			0.351407			0.175703	+	0.984443i	$0.443780\pi$
$840$		0			0
$841$	21.7366			0.749536
$842$		0			0
$843$		0			0
$844$		0			0
$845$	68.0593	+	0.286927i	2.34131	+	0.00987058i
$846$		0			0
$847$	43.5257	+	43.5257i	1.49556	+	1.49556i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			2.21161i			0.0758129i
$852$		0			0
$853$	27.1540	−	27.1540i	0.929736	−	0.929736i	−0.0679529	−	0.997689i	$-0.521647\pi$
$853$	27.1540	−	27.1540i	0.929736	−	0.929736i	0.997689	+	0.0679529i	$0.0216468\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	26.7404	−	26.7404i	0.913436	−	0.913436i	−0.0831052	−	0.996541i	$-0.526484\pi$
$857$	26.7404	−	26.7404i	0.913436	−	0.913436i	0.996541	+	0.0831052i	$0.0264838\pi$
$858$		0			0
$859$			8.58561i			0.292937i	0.989215	+	0.146469i	$0.0467907\pi$
$859$			8.58561i			0.292937i	−0.989215	+	0.146469i	$0.953209\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.85618	−	1.85618i	−0.0631851	−	0.0631851i	0.674808	−	0.737993i	$-0.264226\pi$
$863$	−1.85618	−	1.85618i	−0.0631851	−	0.0631851i	−0.737993	+	0.674808i	$0.764226\pi$
$864$		0			0
$865$	5.39523	−	5.34993i	0.183443	−	0.181903i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.2785			−0.416518
$870$		0			0
$871$	37.0141			1.25418
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−17.5917	+	17.1522i	−0.594706	+	0.579851i
$876$		0			0
$877$	4.18223	+	4.18223i	0.141224	+	0.141224i	0.774184	−	0.632960i	$-0.218160\pi$
$877$	4.18223	+	4.18223i	0.141224	+	0.141224i	−0.632960	+	0.774184i	$0.718160\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			16.2981i			0.549098i	0.961573	+	0.274549i	$0.0885285\pi$
$881$			16.2981i			0.549098i	−0.961573	+	0.274549i	$0.911471\pi$
$882$		0			0
$883$	19.5723	−	19.5723i	0.658660	−	0.658660i	−0.296403	−	0.955063i	$-0.595787\pi$
$883$	19.5723	−	19.5723i	0.658660	−	0.658660i	0.955063	+	0.296403i	$0.0957872\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.69446	+	8.69446i	−0.291931	+	0.291931i	−0.837843	−	0.545912i	$-0.816183\pi$
$887$	−8.69446	+	8.69446i	−0.291931	+	0.291931i	0.545912	+	0.837843i	$0.316183\pi$
$888$		0			0
$889$		−	22.6576i		−	0.759912i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.03153	+	5.03153i	0.168374	+	0.168374i
$894$		0			0
$895$	−0.106042	+	25.1533i	−0.00354460	+	0.840784i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−14.8131			−0.494044
$900$		0			0
$901$	32.4185			1.08002
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.0994352	−	23.5861i	0.00330534	−	0.784030i
$906$		0			0
$907$	35.7535	+	35.7535i	1.18717	+	1.18717i	0.977845	+	0.209329i	$0.0671280\pi$
$907$	35.7535	+	35.7535i	1.18717	+	1.18717i	0.209329	+	0.977845i	$0.432872\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			14.1361i			0.468349i	0.972195	+	0.234174i	$0.0752387\pi$
$911$			14.1361i			0.468349i	−0.972195	+	0.234174i	$0.924761\pi$
$912$		0			0
$913$	55.0289	−	55.0289i	1.82119	−	1.82119i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−16.4190	+	16.4190i	−0.542205	+	0.542205i
$918$		0			0
$919$		−	33.6518i		−	1.11007i	−0.831827	−	0.555035i	$-0.812705\pi$
$919$		−	33.6518i		−	1.11007i	0.831827	−	0.555035i	$-0.187295\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.0918	+	12.0918i	0.398008	+	0.398008i
$924$		0			0
$925$	7.75300	+	7.88486i	0.254917	+	0.259253i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.08154			−0.0354841			−0.0177420	−	0.999843i	$-0.505648\pi$
$929$	−1.08154			−0.0354841			−0.0177420	+	0.999843i	$0.505648\pi$
$930$		0			0
$931$	8.03306			0.263273
$932$		0			0
$933$		0			0
$934$		0			0
$935$	29.1009	−	28.8565i	0.951701	−	0.943710i
$936$		0			0
$937$	3.45593	+	3.45593i	0.112900	+	0.112900i	0.761300	−	0.648400i	$-0.224561\pi$
$937$	3.45593	+	3.45593i	0.112900	+	0.112900i	−0.648400	+	0.761300i	$0.724561\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			40.5009i			1.32029i	0.751137	+	0.660146i	$0.229506\pi$
$941$			40.5009i			1.32029i	−0.751137	+	0.660146i	$0.770494\pi$
$942$		0			0
$943$	−0.639986	+	0.639986i	−0.0208408	+	0.0208408i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−35.6114	+	35.6114i	−1.15721	+	1.15721i	−0.172142	+	0.985072i	$0.555069\pi$
$947$	−35.6114	+	35.6114i	−1.15721	+	1.15721i	−0.985072	+	0.172142i	$0.944931\pi$
$948$		0			0
$949$			48.4910i			1.57409i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.46410	−	9.46410i	−0.306572	−	0.306572i	0.537006	−	0.843578i	$-0.319555\pi$
$953$	−9.46410	−	9.46410i	−0.306572	−	0.306572i	−0.843578	+	0.537006i	$0.819555\pi$
$954$		0			0
$955$	−34.3697	−	0.144897i	−1.11218	−	0.00468876i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.49724			−0.112932
$960$		0			0
$961$	−26.6752			−0.860489
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12.9963	−	13.1064i	−0.418367	−	0.421909i
$966$		0			0
$967$	−38.4628	−	38.4628i	−1.23688	−	1.23688i	−0.961270	−	0.275609i	$-0.911120\pi$
$967$	−38.4628	−	38.4628i	−1.23688	−	1.23688i	−0.275609	−	0.961270i	$-0.588880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			55.3325i			1.77571i	0.460128	+	0.887853i	$0.347804\pi$
$971$			55.3325i			1.77571i	−0.460128	+	0.887853i	$0.652196\pi$
$972$		0			0
$973$	32.0926	−	32.0926i	1.02884	−	1.02884i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.63009	+	5.63009i	−0.180122	+	0.180122i	−0.791409	−	0.611287i	$-0.790652\pi$
$977$	−5.63009	+	5.63009i	−0.180122	+	0.180122i	0.611287	+	0.791409i	$0.290652\pi$
$978$		0			0
$979$		−	9.88336i		−	0.315873i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−31.5689	−	31.5689i	−1.00689	−	1.00689i	−0.999976	−	0.00691570i	$-0.997799\pi$
$983$	−31.5689	−	31.5689i	−1.00689	−	1.00689i	−0.00691570	−	0.999976i	$-0.502201\pi$
$984$		0			0
$985$	−4.28511	−	4.32139i	−0.136535	−	0.137691i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7.27519			−0.231338
$990$		0			0
$991$	−27.0615			−0.859638			−0.429819	−	0.902915i	$-0.641423\pi$
$991$	−27.0615			−0.859638			−0.429819	+	0.902915i	$0.641423\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−39.8771	−	0.168115i	−1.26419	−	0.00532960i
$996$		0			0
$997$	−31.7994	−	31.7994i	−1.00710	−	1.00710i	−0.999975	−	0.00712195i	$-0.997733\pi$
$997$	−31.7994	−	31.7994i	−1.00710	−	1.00710i	−0.00712195	−	0.999975i	$-0.502267\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.s.a.737.11	✓	44
3.2	odd	2	inner	4140.2.s.a.737.12	yes	44
5.3	odd	4	inner	4140.2.s.a.2393.12	yes	44
15.8	even	4	inner	4140.2.s.a.2393.11	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.a.737.11	✓	44	1.1	even	1	trivial
4140.2.s.a.737.12	yes	44	3.2	odd	2	inner
4140.2.s.a.2393.11	yes	44	15.8	even	4	inner
4140.2.s.a.2393.12	yes	44	5.3	odd	4	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.00942680 + 2.23605i) q^{5} +(-1.55392 - 1.55392i) q^{7} +O(q^{10})\)	\(q+(-0.00942680 + 2.23605i) q^{5} +(-1.55392 - 1.55392i) q^{7} +6.24582i q^{11} +(-4.66033 + 4.66033i) q^{13} +(-2.07495 + 2.07495i) q^{17} +3.70071i q^{19} +(-0.707107 - 0.707107i) q^{23} +(-4.99982 - 0.0421576i) q^{25} +7.12296 q^{29} -2.07962 q^{31} +(3.48928 - 3.45999i) q^{35} +(-1.56384 - 1.56384i) q^{37} -0.905077i q^{41} +(5.14434 - 5.14434i) q^{43} +(1.35961 - 1.35961i) q^{47} -2.17068i q^{49} +(-7.81185 - 7.81185i) q^{53} +(-13.9660 - 0.0588782i) q^{55} +9.74737 q^{59} -1.55919 q^{61} +(-10.3768 - 10.4647i) q^{65} +(-3.97119 - 3.97119i) q^{67} -2.59463i q^{71} +(5.20253 - 5.20253i) q^{73} +(9.70549 - 9.70549i) q^{77} +1.96587i q^{79} +(-8.81051 - 8.81051i) q^{83} +(-4.62013 - 4.65925i) q^{85} -1.58239 q^{89} +14.4835 q^{91} +(-8.27496 - 0.0348858i) q^{95} +(5.50862 + 5.50862i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 12 q^{7}+O(q^{10}) \) 44 * q - 12 * q^7	\( 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) \) 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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