LMFDB - Embedded newform 4140.2.s.a.737.20

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			737.20
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.a.2393.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.20494 - 0.371802i) q^{5} +(-3.41792 - 3.41792i) q^{7} +O(q^{10})$	$q+(2.20494 - 0.371802i) q^{5} +(-3.41792 - 3.41792i) q^{7} +4.26967i q^{11} +(-1.36964 + 1.36964i) q^{13} +(-3.76377 + 3.76377i) q^{17} -3.34766i q^{19} +(0.707107 + 0.707107i) q^{23} +(4.72353 - 1.63960i) q^{25} +4.28737 q^{29} +1.17527 q^{31} +(-8.80711 - 6.26553i) q^{35} +(-1.12240 - 1.12240i) q^{37} +10.1412i q^{41} +(8.84843 - 8.84843i) q^{43} +(4.85281 - 4.85281i) q^{47} +16.3644i q^{49} +(7.95219 + 7.95219i) q^{53} +(1.58747 + 9.41436i) q^{55} -4.65152 q^{59} -12.2709 q^{61} +(-2.51074 + 3.52920i) q^{65} +(9.27273 + 9.27273i) q^{67} +7.18378i q^{71} +(-0.225292 + 0.225292i) q^{73} +(14.5934 - 14.5934i) q^{77} +10.6943i q^{79} +(7.70519 + 7.70519i) q^{83} +(-6.89951 + 9.69826i) q^{85} -3.97729 q^{89} +9.36263 q^{91} +(-1.24467 - 7.38139i) q^{95} +(-0.874011 - 0.874011i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.20494	−	0.371802i	0.986079	−	0.166275i
$5$	2.20494	−	0.371802i	0.986079	−	0.166275i
$6$		0			0
$7$	−3.41792	−	3.41792i	−1.29185	−	1.29185i	−0.933639	−	0.358214i	$-0.883386\pi$
$7$	−3.41792	−	3.41792i	−1.29185	−	1.29185i	−0.358214	−	0.933639i	$-0.616614\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.26967i			1.28735i	0.765298	+	0.643677i	$0.222592\pi$
$11$			4.26967i			1.28735i	−0.765298	+	0.643677i	$0.777408\pi$
$12$		0			0
$13$	−1.36964	+	1.36964i	−0.379869	+	0.379869i	−0.871055	−	0.491186i	$-0.836564\pi$
$13$	−1.36964	+	1.36964i	−0.379869	+	0.379869i	0.491186	+	0.871055i	$0.336564\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.76377	+	3.76377i	−0.912848	+	0.912848i	−0.996495	−	0.0836476i	$-0.973343\pi$
$17$	−3.76377	+	3.76377i	−0.912848	+	0.912848i	0.0836476	+	0.996495i	$0.473343\pi$
$18$		0			0
$19$		−	3.34766i		−	0.768006i	−0.923332	−	0.384003i	$-0.874545\pi$
$19$		−	3.34766i		−	0.768006i	0.923332	−	0.384003i	$-0.125455\pi$
$20$		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
$25$	4.72353	−	1.63960i	0.944705	−	0.327920i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.28737			0.796145			0.398072	−	0.917354i	$-0.369679\pi$
$29$	4.28737			0.796145			0.398072	+	0.917354i	$0.369679\pi$
$30$		0			0
$31$	1.17527			0.211085			0.105542	−	0.994415i	$-0.466342\pi$
$31$	1.17527			0.211085			0.105542	+	0.994415i	$0.466342\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.80711	−	6.26553i	−1.48867	−	1.05907i
$36$		0			0
$37$	−1.12240	−	1.12240i	−0.184521	−	0.184521i	0.608802	−	0.793322i	$-0.291650\pi$
$37$	−1.12240	−	1.12240i	−0.184521	−	0.184521i	−0.793322	+	0.608802i	$0.791650\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.1412i			1.58378i	0.610661	+	0.791892i	$0.290904\pi$
$41$			10.1412i			1.58378i	−0.610661	+	0.791892i	$0.709096\pi$
$42$		0			0
$43$	8.84843	−	8.84843i	1.34937	−	1.34937i	0.463029	−	0.886343i	$-0.346763\pi$
$43$	8.84843	−	8.84843i	1.34937	−	1.34937i	0.886343	−	0.463029i	$-0.153237\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.85281	−	4.85281i	0.707855	−	0.707855i	−0.258229	−	0.966084i	$-0.583139\pi$
$47$	4.85281	−	4.85281i	0.707855	−	0.707855i	0.966084	+	0.258229i	$0.0831390\pi$
$48$		0			0
$49$			16.3644i			2.33777i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.95219	+	7.95219i	1.09232	+	1.09232i	0.995281	+	0.0970365i	$0.0309363\pi$
$53$	7.95219	+	7.95219i	1.09232	+	1.09232i	0.0970365	+	0.995281i	$0.469064\pi$
$54$		0			0
$55$	1.58747	+	9.41436i	0.214054	+	1.26943i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.65152			−0.605577			−0.302788	−	0.953058i	$-0.597918\pi$
$59$	−4.65152			−0.605577			−0.302788	+	0.953058i	$0.597918\pi$
$60$		0			0
$61$	−12.2709			−1.57113			−0.785566	−	0.618779i	$-0.787628\pi$
$61$	−12.2709			−1.57113			−0.785566	+	0.618779i	$0.787628\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.51074	+	3.52920i	−0.311419	+	0.437744i
$66$		0			0
$67$	9.27273	+	9.27273i	1.13284	+	1.13284i	0.989702	+	0.143142i	$0.0457207\pi$
$67$	9.27273	+	9.27273i	1.13284	+	1.13284i	0.143142	+	0.989702i	$0.454279\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.18378i			0.852558i	0.904592	+	0.426279i	$0.140176\pi$
$71$			7.18378i			0.852558i	−0.904592	+	0.426279i	$0.859824\pi$
$72$		0			0
$73$	−0.225292	+	0.225292i	−0.0263685	+	0.0263685i	−0.720168	−	0.693800i	$-0.755935\pi$
$73$	−0.225292	+	0.225292i	−0.0263685	+	0.0263685i	0.693800	+	0.720168i	$0.255935\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	14.5934	−	14.5934i	1.66307	−	1.66307i
$78$		0			0
$79$			10.6943i			1.20321i	0.798795	+	0.601603i	$0.205471\pi$
$79$			10.6943i			1.20321i	−0.798795	+	0.601603i	$0.794529\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.70519	+	7.70519i	0.845755	+	0.845755i	0.989600	−	0.143845i	$-0.0459468\pi$
$83$	7.70519	+	7.70519i	0.845755	+	0.845755i	−0.143845	+	0.989600i	$0.545947\pi$
$84$		0			0
$85$	−6.89951	+	9.69826i	−0.748357	+	1.05192i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.97729			−0.421592			−0.210796	−	0.977530i	$-0.567606\pi$
$89$	−3.97729			−0.421592			−0.210796	+	0.977530i	$0.567606\pi$
$90$		0			0
$91$	9.36263			0.981471
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.24467	−	7.38139i	−0.127700	−	0.757315i
$96$		0			0
$97$	−0.874011	−	0.874011i	−0.0887423	−	0.0887423i	0.661342	−	0.750084i	$-0.269987\pi$
$97$	−0.874011	−	0.874011i	−0.0887423	−	0.0887423i	−0.750084	+	0.661342i	$0.769987\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	13.0972i		−	1.30322i	−0.758556	−	0.651608i	$-0.774095\pi$
$101$		−	13.0972i		−	1.30322i	0.758556	−	0.651608i	$-0.225905\pi$
$102$		0			0
$103$	10.7391	−	10.7391i	1.05816	−	1.05816i	0.0599585	−	0.998201i	$-0.480903\pi$
$103$	10.7391	−	10.7391i	1.05816	−	1.05816i	0.998201	−	0.0599585i	$-0.0190968\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.92524	−	2.92524i	0.282794	−	0.282794i	−0.551428	−	0.834222i	$-0.685917\pi$
$107$	2.92524	−	2.92524i	0.282794	−	0.282794i	0.834222	+	0.551428i	$0.185917\pi$
$108$		0			0
$109$			9.50341i			0.910262i	0.890425	+	0.455131i	$0.150408\pi$
$109$			9.50341i			0.910262i	−0.890425	+	0.455131i	$0.849592\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.58874	+	8.58874i	0.807960	+	0.807960i	0.984325	−	0.176365i	$-0.0564338\pi$
$113$	8.58874	+	8.58874i	0.807960	+	0.807960i	−0.176365	+	0.984325i	$0.556434\pi$
$114$		0			0
$115$	1.82203	+	1.29623i	0.169905	+	0.120874i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	25.7285			2.35853
$120$		0			0
$121$	−7.23006			−0.657278
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.80549	−	5.37144i	0.877030	−	0.480436i
$126$		0			0
$127$	−4.92710	−	4.92710i	−0.437210	−	0.437210i	0.453862	−	0.891072i	$-0.350046\pi$
$127$	−4.92710	−	4.92710i	−0.437210	−	0.437210i	−0.891072	+	0.453862i	$0.850046\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	15.1467i		−	1.32337i	−0.749782	−	0.661685i	$-0.769842\pi$
$131$		−	15.1467i		−	1.32337i	0.749782	−	0.661685i	$-0.230158\pi$
$132$		0			0
$133$	−11.4420	+	11.4420i	−0.992151	+	0.992151i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.98387	−	7.98387i	0.682108	−	0.682108i	−0.278366	−	0.960475i	$-0.589793\pi$
$137$	7.98387	−	7.98387i	0.682108	−	0.682108i	0.960475	+	0.278366i	$0.0897930\pi$
$138$		0			0
$139$			11.1601i			0.946590i	0.880904	+	0.473295i	$0.156936\pi$
$139$			11.1601i			0.946590i	−0.880904	+	0.473295i	$0.843064\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.84790	−	5.84790i	−0.489026	−	0.489026i
$144$		0			0
$145$	9.45340	−	1.59405i	0.785062	−	0.132379i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.250755			−0.0205426			−0.0102713	−	0.999947i	$-0.503270\pi$
$149$	−0.250755			−0.0205426			−0.0102713	+	0.999947i	$0.503270\pi$
$150$		0			0
$151$	4.15636			0.338240			0.169120	−	0.985596i	$-0.445908\pi$
$151$	4.15636			0.338240			0.169120	+	0.985596i	$0.445908\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.59140	−	0.436967i	0.208146	−	0.0350980i
$156$		0			0
$157$	−8.45615	−	8.45615i	−0.674874	−	0.674874i	0.283961	−	0.958836i	$-0.408351\pi$
$157$	−8.45615	−	8.45615i	−0.674874	−	0.674874i	−0.958836	+	0.283961i	$0.908351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	4.83367i		−	0.380947i
$162$		0			0
$163$	2.29491	−	2.29491i	0.179751	−	0.179751i	−0.611496	−	0.791247i	$-0.709432\pi$
$163$	2.29491	−	2.29491i	0.179751	−	0.179751i	0.791247	+	0.611496i	$0.209432\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13.9321	−	13.9321i	1.07810	−	1.07810i	0.0814162	−	0.996680i	$-0.474056\pi$
$167$	13.9321	−	13.9321i	1.07810	−	1.07810i	0.996680	−	0.0814162i	$-0.0259443\pi$
$168$		0			0
$169$			9.24819i			0.711399i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.28865	+	9.28865i	0.706203	+	0.706203i	0.965735	−	0.259532i	$-0.0835682\pi$
$173$	9.28865	+	9.28865i	0.706203	+	0.706203i	−0.259532	+	0.965735i	$0.583568\pi$
$174$		0			0
$175$	−21.7487	−	10.5406i	−1.64405	−	0.796796i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.4124			0.927749			0.463874	−	0.885901i	$-0.346459\pi$
$179$	12.4124			0.927749			0.463874	+	0.885901i	$0.346459\pi$
$180$		0			0
$181$	22.1507			1.64645			0.823225	−	0.567715i	$-0.192173\pi$
$181$	22.1507			1.64645			0.823225	+	0.567715i	$0.192173\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.89212	−	2.05751i	−0.212633	−	0.151271i
$186$		0			0
$187$	−16.0700	−	16.0700i	−1.17516	−	1.17516i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			2.43431i			0.176141i	0.996114	+	0.0880703i	$0.0280700\pi$
$191$			2.43431i			0.176141i	−0.996114	+	0.0880703i	$0.971930\pi$
$192$		0			0
$193$	−12.2690	+	12.2690i	−0.883144	+	0.883144i	−0.993853	−	0.110709i	$-0.964688\pi$
$193$	−12.2690	+	12.2690i	−0.883144	+	0.883144i	0.110709	+	0.993853i	$0.464688\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.9955	+	10.9955i	−0.783395	+	0.783395i	−0.980402	−	0.197007i	$-0.936878\pi$
$197$	−10.9955	+	10.9955i	−0.783395	+	0.783395i	0.197007	+	0.980402i	$0.436878\pi$
$198$		0			0
$199$			14.8224i			1.05073i	0.850877	+	0.525365i	$0.176071\pi$
$199$			14.8224i			1.05073i	−0.850877	+	0.525365i	$0.823929\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−14.6539	−	14.6539i	−1.02850	−	1.02850i
$204$		0			0
$205$	3.77050	+	22.3607i	0.263343	+	1.56174i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	14.2934			0.988695
$210$		0			0
$211$	−10.0744			−0.693549			−0.346774	−	0.937949i	$-0.612723\pi$
$211$	−10.0744			−0.693549			−0.346774	+	0.937949i	$0.612723\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	16.2204	−	22.8001i	1.10622	−	1.55495i
$216$		0			0
$217$	−4.01698	−	4.01698i	−0.272691	−	0.272691i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	10.3100i		−	0.693525i
$222$		0			0
$223$	17.5428	−	17.5428i	1.17475	−	1.17475i	0.193687	−	0.981063i	$-0.437955\pi$
$223$	17.5428	−	17.5428i	1.17475	−	1.17475i	0.981063	−	0.193687i	$-0.0620447\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.09202	−	3.09202i	0.205225	−	0.205225i	−0.597009	−	0.802234i	$-0.703645\pi$
$227$	3.09202	−	3.09202i	0.205225	−	0.205225i	0.802234	+	0.597009i	$0.203645\pi$
$228$		0			0
$229$			4.02242i			0.265809i	0.991129	+	0.132905i	$0.0424304\pi$
$229$			4.02242i			0.265809i	−0.991129	+	0.132905i	$0.957570\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.85064	+	4.85064i	0.317776	+	0.317776i	0.847912	−	0.530136i	$-0.177859\pi$
$233$	4.85064	+	4.85064i	0.317776	+	0.317776i	−0.530136	+	0.847912i	$0.677859\pi$
$234$		0			0
$235$	8.89587	−	12.5044i	0.580303	−	0.815699i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−23.2005			−1.50072			−0.750359	−	0.661031i	$-0.770119\pi$
$239$	−23.2005			−1.50072			−0.750359	+	0.661031i	$0.770119\pi$
$240$		0			0
$241$	21.7481			1.40092			0.700458	−	0.713694i	$-0.252979\pi$
$241$	21.7481			1.40092			0.700458	+	0.713694i	$0.252979\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.08431	+	36.0825i	0.388712	+	2.30523i
$246$		0			0
$247$	4.58508	+	4.58508i	0.291742	+	0.291742i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			23.3240i			1.47220i	0.676875	+	0.736098i	$0.263334\pi$
$251$			23.3240i			1.47220i	−0.676875	+	0.736098i	$0.736666\pi$
$252$		0			0
$253$	−3.01911	+	3.01911i	−0.189810	+	0.189810i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.84479	−	3.84479i	0.239831	−	0.239831i	−0.576949	−	0.816780i	$-0.695757\pi$
$257$	3.84479	−	3.84479i	0.239831	−	0.239831i	0.816780	+	0.576949i	$0.195757\pi$
$258$		0			0
$259$			7.67252i			0.476747i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.16327	+	5.16327i	0.318381	+	0.318381i	0.848145	−	0.529764i	$-0.177720\pi$
$263$	5.16327	+	5.16327i	0.318381	+	0.318381i	−0.529764	+	0.848145i	$0.677720\pi$
$264$		0			0
$265$	20.4907	+	14.5775i	1.25874	+	0.895487i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.73810			0.410829			0.205415	−	0.978675i	$-0.434146\pi$
$269$	6.73810			0.410829			0.205415	+	0.978675i	$0.434146\pi$
$270$		0			0
$271$	28.0070			1.70131			0.850653	−	0.525727i	$-0.176207\pi$
$271$	28.0070			1.70131			0.850653	+	0.525727i	$0.176207\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.00055	+	20.1679i	0.422149	+	1.21617i
$276$		0			0
$277$	−18.9659	−	18.9659i	−1.13955	−	1.13955i	−0.988531	−	0.151020i	$-0.951744\pi$
$277$	−18.9659	−	18.9659i	−1.13955	−	1.13955i	−0.151020	−	0.988531i	$-0.548256\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			28.7383i			1.71438i	0.514999	+	0.857191i	$0.327792\pi$
$281$			28.7383i			1.71438i	−0.514999	+	0.857191i	$0.672208\pi$
$282$		0			0
$283$	−20.9490	+	20.9490i	−1.24529	+	1.24529i	−0.287515	+	0.957776i	$0.592829\pi$
$283$	−20.9490	+	20.9490i	−1.24529	+	1.24529i	−0.957776	+	0.287515i	$0.907171\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	34.6617	−	34.6617i	2.04602	−	2.04602i
$288$		0			0
$289$		−	11.3319i		−	0.666582i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.15026	+	3.15026i	0.184040	+	0.184040i	0.793114	−	0.609074i	$-0.208459\pi$
$293$	3.15026	+	3.15026i	0.184040	+	0.184040i	−0.609074	+	0.793114i	$0.708459\pi$
$294$		0			0
$295$	−10.2563	+	1.72944i	−0.597147	+	0.100692i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.93696			−0.112017
$300$		0			0
$301$	−60.4865			−3.48638
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−27.0567	+	4.56235i	−1.54926	+	0.261239i
$306$		0			0
$307$	−5.73372	−	5.73372i	−0.327241	−	0.327241i	0.524296	−	0.851536i	$-0.324329\pi$
$307$	−5.73372	−	5.73372i	−0.327241	−	0.327241i	−0.851536	+	0.524296i	$0.824329\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	4.19062i		−	0.237628i	−0.992916	−	0.118814i	$-0.962091\pi$
$311$		−	4.19062i		−	0.237628i	0.992916	−	0.118814i	$-0.0379093\pi$
$312$		0			0
$313$	−18.2983	+	18.2983i	−1.03428	+	1.03428i	−0.0348889	+	0.999391i	$0.511108\pi$
$313$	−18.2983	+	18.2983i	−1.03428	+	1.03428i	−0.999391	+	0.0348889i	$0.988892\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.3760	+	14.3760i	−0.807436	+	0.807436i	−0.984245	−	0.176809i	$-0.943422\pi$
$317$	−14.3760	+	14.3760i	−0.807436	+	0.807436i	0.176809	+	0.984245i	$0.443422\pi$
$318$		0			0
$319$			18.3056i			1.02492i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	12.5998	+	12.5998i	0.701073	+	0.701073i
$324$		0			0
$325$	−4.22386	+	8.71518i	−0.234298	+	0.483431i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−33.1730			−1.82889
$330$		0			0
$331$	−20.6622			−1.13570			−0.567849	−	0.823133i	$-0.692224\pi$
$331$	−20.6622			−1.13570			−0.567849	+	0.823133i	$0.692224\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	23.8934	+	16.9982i	1.30544	+	0.928711i
$336$		0			0
$337$	−17.7908	−	17.7908i	−0.969128	−	0.969128i	0.0304099	−	0.999538i	$-0.490319\pi$
$337$	−17.7908	−	17.7908i	−0.969128	−	0.969128i	−0.999538	+	0.0304099i	$0.990319\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			5.01801i			0.271741i
$342$		0			0
$343$	32.0068	−	32.0068i	1.72821	−	1.72821i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.0764	+	17.0764i	−0.916707	+	0.916707i	−0.996788	−	0.0800810i	$-0.974482\pi$
$347$	−17.0764	+	17.0764i	−0.916707	+	0.916707i	0.0800810	+	0.996788i	$0.474482\pi$
$348$		0			0
$349$		−	3.50127i		−	0.187419i	−0.995600	−	0.0937094i	$-0.970128\pi$
$349$		−	3.50127i		−	0.187419i	0.995600	−	0.0937094i	$-0.0298725\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−25.6060	−	25.6060i	−1.36287	−	1.36287i	−0.870240	−	0.492629i	$-0.836036\pi$
$353$	−25.6060	−	25.6060i	−1.36287	−	1.36287i	−0.492629	−	0.870240i	$-0.663964\pi$
$354$		0			0
$355$	2.67094	+	15.8398i	0.141759	+	0.840690i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−22.9666			−1.21213			−0.606066	−	0.795414i	$-0.707253\pi$
$359$	−22.9666			−1.21213			−0.606066	+	0.795414i	$0.707253\pi$
$360$		0			0
$361$	7.79317			0.410167
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.412992	+	0.580521i	−0.0216170	+	0.0303858i
$366$		0			0
$367$	1.62630	+	1.62630i	0.0848920	+	0.0848920i	0.748278	−	0.663386i	$-0.230881\pi$
$367$	1.62630	+	1.62630i	0.0848920	+	0.0848920i	−0.663386	+	0.748278i	$0.730881\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	54.3600i		−	2.82223i
$372$		0			0
$373$	8.89347	−	8.89347i	0.460487	−	0.460487i	−0.438328	−	0.898815i	$-0.644429\pi$
$373$	8.89347	−	8.89347i	0.460487	−	0.460487i	0.898815	+	0.438328i	$0.144429\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.87214	+	5.87214i	−0.302431	+	0.302431i
$378$		0			0
$379$			14.5853i			0.749196i	0.927187	+	0.374598i	$0.122219\pi$
$379$			14.5853i			0.749196i	−0.927187	+	0.374598i	$0.877781\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.06119	+	9.06119i	0.463005	+	0.463005i	0.899639	−	0.436634i	$-0.143830\pi$
$383$	9.06119	+	9.06119i	0.463005	+	0.463005i	−0.436634	+	0.899639i	$0.643830\pi$
$384$		0			0
$385$	26.7517	−	37.6034i	1.36339	−	1.91645i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.01516			0.304981			0.152490	−	0.988305i	$-0.451271\pi$
$389$	6.01516			0.304981			0.152490	+	0.988305i	$0.451271\pi$
$390$		0			0
$391$	−5.32277			−0.269184
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.97617	+	23.5804i	0.200063	+	1.18646i
$396$		0			0
$397$	8.87141	+	8.87141i	0.445243	+	0.445243i	0.893770	−	0.448526i	$-0.148051\pi$
$397$	8.87141	+	8.87141i	0.445243	+	0.445243i	−0.448526	+	0.893770i	$0.648051\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	4.87130i		−	0.243261i	−0.992575	−	0.121631i	$-0.961188\pi$
$401$		−	4.87130i		−	0.243261i	0.992575	−	0.121631i	$-0.0388123\pi$
$402$		0			0
$403$	−1.60969	+	1.60969i	−0.0801846	+	0.0801846i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.79225	−	4.79225i	0.237543	−	0.237543i
$408$		0			0
$409$			26.9407i			1.33213i	0.745893	+	0.666065i	$0.232023\pi$
$409$			26.9407i			1.33213i	−0.745893	+	0.666065i	$0.767977\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	15.8985	+	15.8985i	0.782316	+	0.782316i
$414$		0			0
$415$	19.8543	+	14.1247i	0.974609	+	0.693354i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−21.6331			−1.05685			−0.528423	−	0.848981i	$-0.677217\pi$
$419$	−21.6331			−1.05685			−0.528423	+	0.848981i	$0.677217\pi$
$420$		0			0
$421$	26.8620			1.30917			0.654587	−	0.755987i	$-0.272842\pi$
$421$	26.8620			1.30917			0.654587	+	0.755987i	$0.272842\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−11.6072	+	23.9493i	−0.563031	+	1.16171i
$426$		0			0
$427$	41.9411	+	41.9411i	2.02967	+	2.02967i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	22.3927i		−	1.07862i	−0.842107	−	0.539310i	$-0.818685\pi$
$431$		−	22.3927i		−	1.07862i	0.842107	−	0.539310i	$-0.181315\pi$
$432$		0			0
$433$	10.7684	−	10.7684i	0.517497	−	0.517497i	−0.399316	−	0.916813i	$-0.630752\pi$
$433$	10.7684	−	10.7684i	0.517497	−	0.517497i	0.916813	+	0.399316i	$0.130752\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.36715	−	2.36715i	0.113236	−	0.113236i
$438$		0			0
$439$			2.34035i			0.111699i	0.998439	+	0.0558495i	$0.0177867\pi$
$439$			2.34035i			0.111699i	−0.998439	+	0.0558495i	$0.982213\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−20.7252	−	20.7252i	−0.984682	−	0.984682i	0.0152022	−	0.999884i	$-0.495161\pi$
$443$	−20.7252	−	20.7252i	−0.984682	−	0.984682i	−0.999884	+	0.0152022i	$0.995161\pi$
$444$		0			0
$445$	−8.76970	+	1.47876i	−0.415724	+	0.0701001i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	26.9780			1.27317			0.636584	−	0.771207i	$-0.280347\pi$
$449$	26.9780			1.27317			0.636584	+	0.771207i	$0.280347\pi$
$450$		0			0
$451$	−43.2994			−2.03889
$452$		0			0
$453$		0			0
$454$		0			0
$455$	20.6441	−	3.48104i	0.967808	−	0.163194i
$456$		0			0
$457$	21.8386	+	21.8386i	1.02157	+	1.02157i	0.999762	+	0.0218045i	$0.00694112\pi$
$457$	21.8386	+	21.8386i	1.02157	+	1.02157i	0.0218045	+	0.999762i	$0.493059\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	8.84937i		−	0.412156i	−0.978536	−	0.206078i	$-0.933930\pi$
$461$		−	8.84937i		−	0.412156i	0.978536	−	0.206078i	$-0.0660701\pi$
$462$		0			0
$463$	6.78554	−	6.78554i	0.315351	−	0.315351i	−0.531628	−	0.846978i	$-0.678419\pi$
$463$	6.78554	−	6.78554i	0.315351	−	0.315351i	0.846978	+	0.531628i	$0.178419\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.50861	+	8.50861i	−0.393732	+	0.393732i	−0.876015	−	0.482284i	$-0.839807\pi$
$467$	−8.50861	+	8.50861i	−0.393732	+	0.393732i	0.482284	+	0.876015i	$0.339807\pi$
$468$		0			0
$469$		−	63.3870i		−	2.92694i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	37.7798	+	37.7798i	1.73712	+	1.73712i
$474$		0			0
$475$	−5.48883	−	15.8128i	−0.251845	−	0.725539i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.97105			−0.272824			−0.136412	−	0.990652i	$-0.543557\pi$
$479$	−5.97105			−0.272824			−0.136412	+	0.990652i	$0.543557\pi$
$480$		0			0
$481$	3.07455			0.140187
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.25210	−	1.60218i	−0.102263	−	0.0727514i
$486$		0			0
$487$	20.2838	+	20.2838i	0.919148	+	0.919148i	0.996967	−	0.0778196i	$-0.0247958\pi$
$487$	20.2838	+	20.2838i	0.919148	+	0.919148i	−0.0778196	+	0.996967i	$0.524796\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			25.3494i			1.14400i	0.820253	+	0.572001i	$0.193833\pi$
$491$			25.3494i			1.14400i	−0.820253	+	0.572001i	$0.806167\pi$
$492$		0			0
$493$	−16.1367	+	16.1367i	−0.726759	+	0.726759i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	24.5536	−	24.5536i	1.10138	−	1.10138i
$498$		0			0
$499$		−	22.8057i		−	1.02092i	−0.859901	−	0.510461i	$-0.829475\pi$
$499$		−	22.8057i		−	1.02092i	0.859901	−	0.510461i	$-0.170525\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.65002	+	3.65002i	0.162746	+	0.162746i	0.783782	−	0.621036i	$-0.213288\pi$
$503$	3.65002	+	3.65002i	0.162746	+	0.162746i	−0.621036	+	0.783782i	$0.713288\pi$
$504$		0			0
$505$	−4.86954	−	28.8784i	−0.216692	−	1.28507i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	18.6901			0.828425			0.414212	−	0.910180i	$-0.364057\pi$
$509$	18.6901			0.828425			0.414212	+	0.910180i	$0.364057\pi$
$510$		0			0
$511$	1.54006			0.0681285
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19.6863	−	27.6720i	0.867484	−	1.21937i
$516$		0			0
$517$	20.7199	+	20.7199i	0.911259	+	0.911259i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	10.1355i		−	0.444046i	−0.975041	−	0.222023i	$-0.928734\pi$
$521$		−	10.1355i		−	0.444046i	0.975041	−	0.222023i	$-0.0712659\pi$
$522$		0			0
$523$	0.892711	−	0.892711i	0.0390355	−	0.0390355i	−0.687320	−	0.726355i	$-0.741213\pi$
$523$	0.892711	−	0.892711i	0.0390355	−	0.0390355i	0.726355	+	0.687320i	$0.241213\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.42344	+	4.42344i	−0.192688	+	0.192688i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−13.8897	−	13.8897i	−0.601631	−	0.601631i
$534$		0			0
$535$	5.36238	−	7.53759i	0.231836	−	0.325879i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−69.8705			−3.00954
$540$		0			0
$541$	10.3981			0.447048			0.223524	−	0.974698i	$-0.428244\pi$
$541$	10.3981			0.447048			0.223524	+	0.974698i	$0.428244\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.53338	+	20.9545i	0.151354	+	0.897590i
$546$		0			0
$547$	−5.93355	−	5.93355i	−0.253700	−	0.253700i	0.568786	−	0.822486i	$-0.307413\pi$
$547$	−5.93355	−	5.93355i	−0.253700	−	0.253700i	−0.822486	+	0.568786i	$0.807413\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	14.3527i		−	0.611444i
$552$		0			0
$553$	36.5524	−	36.5524i	1.55437	−	1.55437i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.48712	−	1.48712i	0.0630112	−	0.0630112i	−0.674899	−	0.737910i	$-0.735813\pi$
$557$	1.48712	−	1.48712i	0.0630112	−	0.0630112i	0.737910	+	0.674899i	$0.235813\pi$
$558$		0			0
$559$			24.2383i			1.02517i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.34390	−	1.34390i	−0.0566386	−	0.0566386i	0.678220	−	0.734859i	$-0.262752\pi$
$563$	−1.34390	−	1.34390i	−0.0566386	−	0.0566386i	−0.734859	+	0.678220i	$0.762752\pi$
$564$		0			0
$565$	22.1310	+	15.7443i	0.931056	+	0.662370i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.59881			−0.150870			−0.0754351	−	0.997151i	$-0.524035\pi$
$569$	−3.59881			−0.150870			−0.0754351	+	0.997151i	$0.524035\pi$
$570$		0			0
$571$	−36.1665			−1.51352			−0.756760	−	0.653693i	$-0.773219\pi$
$571$	−36.1665			−1.51352			−0.756760	+	0.653693i	$0.773219\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.49941	+	2.18067i	0.187638	+	0.0909400i
$576$		0			0
$577$	−6.91060	−	6.91060i	−0.287692	−	0.287692i	0.548475	−	0.836167i	$-0.315209\pi$
$577$	−6.91060	−	6.91060i	−0.287692	−	0.287692i	−0.836167	+	0.548475i	$0.815209\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	52.6715i		−	2.18518i
$582$		0			0
$583$	−33.9532	+	33.9532i	−1.40620	+	1.40620i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.0170	−	16.0170i	0.661094	−	0.661094i	−0.294544	−	0.955638i	$-0.595168\pi$
$587$	16.0170	−	16.0170i	0.661094	−	0.661094i	0.955638	+	0.294544i	$0.0951678\pi$
$588$		0			0
$589$		−	3.93440i		−	0.162114i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.91800	+	5.91800i	0.243023	+	0.243023i	0.818100	−	0.575076i	$-0.195028\pi$
$593$	5.91800	+	5.91800i	0.243023	+	0.243023i	−0.575076	+	0.818100i	$0.695028\pi$
$594$		0			0
$595$	56.7299	−	9.56591i	2.32570	−	0.392164i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−20.5141			−0.838184			−0.419092	−	0.907944i	$-0.637652\pi$
$599$	−20.5141			−0.838184			−0.419092	+	0.907944i	$0.637652\pi$
$600$		0			0
$601$	−22.4713			−0.916625			−0.458312	−	0.888791i	$-0.651546\pi$
$601$	−22.4713			−0.916625			−0.458312	+	0.888791i	$0.651546\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−15.9418	+	2.68815i	−0.648128	+	0.109289i
$606$		0			0
$607$	−9.64464	−	9.64464i	−0.391464	−	0.391464i	0.483745	−	0.875209i	$-0.339276\pi$
$607$	−9.64464	−	9.64464i	−0.391464	−	0.391464i	−0.875209	+	0.483745i	$0.839276\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			13.2932i			0.537784i
$612$		0			0
$613$	−26.7423	+	26.7423i	−1.08011	+	1.08011i	−0.0836136	+	0.996498i	$0.526646\pi$
$613$	−26.7423	+	26.7423i	−1.08011	+	1.08011i	−0.996498	+	0.0836136i	$0.973354\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.28593	+	4.28593i	−0.172545	+	0.172545i	−0.788097	−	0.615551i	$-0.788933\pi$
$617$	−4.28593	+	4.28593i	−0.172545	+	0.172545i	0.615551	+	0.788097i	$0.288933\pi$
$618$		0			0
$619$			3.08175i			0.123866i	0.998080	+	0.0619330i	$0.0197265\pi$
$619$			3.08175i			0.123866i	−0.998080	+	0.0619330i	$0.980274\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	13.5941	+	13.5941i	0.544636	+	0.544636i
$624$		0			0
$625$	19.6234	−	15.4894i	0.784937	−	0.619576i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.44887			0.336879
$630$		0			0
$631$	33.7808			1.34479			0.672396	−	0.740192i	$-0.265265\pi$
$631$	33.7808			1.34479			0.672396	+	0.740192i	$0.265265\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.6959	−	9.03207i	−0.503820	−	0.358427i
$636$		0			0
$637$	−22.4133	−	22.4133i	−0.888047	−	0.888047i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	6.27083i		−	0.247683i	−0.992302	−	0.123841i	$-0.960479\pi$
$641$		−	6.27083i		−	0.247683i	0.992302	−	0.123841i	$-0.0395214\pi$
$642$		0			0
$643$	0.401964	−	0.401964i	0.0158519	−	0.0158519i	−0.699136	−	0.714988i	$-0.746432\pi$
$643$	0.401964	−	0.401964i	0.0158519	−	0.0158519i	0.714988	+	0.699136i	$0.246432\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	18.0254	−	18.0254i	0.708650	−	0.708650i	−0.257602	−	0.966251i	$-0.582932\pi$
$647$	18.0254	−	18.0254i	0.708650	−	0.708650i	0.966251	+	0.257602i	$0.0829322\pi$
$648$		0			0
$649$		−	19.8604i		−	0.779591i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10.4282	−	10.4282i	−0.408087	−	0.408087i	0.472984	−	0.881071i	$-0.343177\pi$
$653$	−10.4282	−	10.4282i	−0.408087	−	0.408087i	−0.881071	+	0.472984i	$0.843177\pi$
$654$		0			0
$655$	−5.63155	−	33.3975i	−0.220043	−	1.30495i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	34.7163			1.35235			0.676177	−	0.736740i	$-0.263636\pi$
$659$	34.7163			1.35235			0.676177	+	0.736740i	$0.263636\pi$
$660$		0			0
$661$	2.16381			0.0841626			0.0420813	−	0.999114i	$-0.486601\pi$
$661$	2.16381			0.0841626			0.0420813	+	0.999114i	$0.486601\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−20.9749	+	29.4832i	−0.813370	+	1.14331i
$666$		0			0
$667$	3.03163	+	3.03163i	0.117385	+	0.117385i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	52.3928i		−	2.02260i
$672$		0			0
$673$	−24.4973	+	24.4973i	−0.944301	+	0.944301i	−0.998529	−	0.0542280i	$-0.982730\pi$
$673$	−24.4973	+	24.4973i	−0.944301	+	0.944301i	0.0542280	+	0.998529i	$0.482730\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17.3472	−	17.3472i	0.666707	−	0.666707i	−0.290245	−	0.956952i	$-0.593737\pi$
$677$	17.3472	−	17.3472i	0.666707	−	0.666707i	0.956952	+	0.290245i	$0.0937369\pi$
$678$		0			0
$679$			5.97460i			0.229284i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−28.2852	−	28.2852i	−1.08230	−	1.08230i	−0.996294	−	0.0860078i	$-0.972589\pi$
$683$	−28.2852	−	28.2852i	−1.08230	−	1.08230i	−0.0860078	−	0.996294i	$-0.527411\pi$
$684$		0			0
$685$	14.6356	−	20.5724i	0.559196	−	0.786030i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−21.7832			−0.829875
$690$		0			0
$691$	4.15648			0.158120			0.0790600	−	0.996870i	$-0.474808\pi$
$691$	4.15648			0.158120			0.0790600	+	0.996870i	$0.474808\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4.14935	+	24.6074i	0.157394	+	0.933413i
$696$		0			0
$697$	−38.1690	−	38.1690i	−1.44575	−	1.44575i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	2.46884i		−	0.0932469i	−0.998913	−	0.0466235i	$-0.985154\pi$
$701$		−	2.46884i		−	0.0932469i	0.998913	−	0.0466235i	$-0.0148461\pi$
$702$		0			0
$703$	−3.75740	+	3.75740i	−0.141713	+	0.141713i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−44.7651	+	44.7651i	−1.68356	+	1.68356i
$708$		0			0
$709$			30.8058i			1.15694i	0.815705	+	0.578468i	$0.196349\pi$
$709$			30.8058i			1.15694i	−0.815705	+	0.578468i	$0.803651\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.831041	+	0.831041i	0.0311227	+	0.0311227i
$714$		0			0
$715$	−15.0685	−	10.7200i	−0.563531	−	0.400906i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.6337			−0.806803			−0.403401	−	0.915023i	$-0.632172\pi$
$719$	−21.6337			−0.806803			−0.403401	+	0.915023i	$0.632172\pi$
$720$		0			0
$721$	−73.4111			−2.73397
$722$		0			0
$723$		0			0
$724$		0			0
$725$	20.2515	−	7.02958i	0.752122	−	0.261072i
$726$		0			0
$727$	−17.2014	−	17.2014i	−0.637964	−	0.637964i	0.312089	−	0.950053i	$-0.398971\pi$
$727$	−17.2014	−	17.2014i	−0.637964	−	0.637964i	−0.950053	+	0.312089i	$0.898971\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			66.6068i			2.46354i
$732$		0			0
$733$	24.5444	−	24.5444i	0.906568	−	0.906568i	−0.0894254	−	0.995994i	$-0.528503\pi$
$733$	24.5444	−	24.5444i	0.906568	−	0.906568i	0.995994	+	0.0894254i	$0.0285031\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−39.5915	+	39.5915i	−1.45837	+	1.45837i
$738$		0			0
$739$			35.7299i			1.31435i	0.753740	+	0.657173i	$0.228248\pi$
$739$			35.7299i			1.31435i	−0.753740	+	0.657173i	$0.771752\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	19.2253	+	19.2253i	0.705308	+	0.705308i	0.965545	−	0.260237i	$-0.0838007\pi$
$743$	19.2253	+	19.2253i	0.705308	+	0.705308i	−0.260237	+	0.965545i	$0.583801\pi$
$744$		0			0
$745$	−0.552899	+	0.0932310i	−0.0202567	+	0.00341572i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−19.9965			−0.730656
$750$		0			0
$751$	−25.8862			−0.944601			−0.472301	−	0.881438i	$-0.656576\pi$
$751$	−25.8862			−0.944601			−0.472301	+	0.881438i	$0.656576\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9.16452	−	1.54534i	0.333531	−	0.0562407i
$756$		0			0
$757$	12.2623	+	12.2623i	0.445683	+	0.445683i	0.893916	−	0.448234i	$-0.147947\pi$
$757$	12.2623	+	12.2623i	0.445683	+	0.445683i	−0.448234	+	0.893916i	$0.647947\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			32.6802i			1.18466i	0.805697	+	0.592328i	$0.201791\pi$
$761$			32.6802i			1.18466i	−0.805697	+	0.592328i	$0.798209\pi$
$762$		0			0
$763$	32.4819	−	32.4819i	1.17592	−	1.17592i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.37090	−	6.37090i	0.230040	−	0.230040i
$768$		0			0
$769$		−	28.8429i		−	1.04010i	−0.854135	−	0.520051i	$-0.825913\pi$
$769$		−	28.8429i		−	1.04010i	0.854135	−	0.520051i	$-0.174087\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−18.7783	−	18.7783i	−0.675410	−	0.675410i	0.283548	−	0.958958i	$-0.408489\pi$
$773$	−18.7783	−	18.7783i	−0.675410	−	0.675410i	−0.958958	+	0.283548i	$0.908489\pi$
$774$		0			0
$775$	5.55142	−	1.92697i	0.199413	−	0.0692189i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	33.9492			1.21636
$780$		0			0
$781$	−30.6723			−1.09754
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−21.7893	−	15.5013i	−0.777694	−	0.553265i
$786$		0			0
$787$	−18.1501	−	18.1501i	−0.646980	−	0.646980i	0.305282	−	0.952262i	$-0.401249\pi$
$787$	−18.1501	−	18.1501i	−0.646980	−	0.646980i	−0.952262	+	0.305282i	$0.901249\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	58.7113i		−	2.08753i
$792$		0			0
$793$	16.8067	−	16.8067i	0.596824	−	0.596824i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−3.17940	+	3.17940i	−0.112620	+	0.112620i	−0.761171	−	0.648551i	$-0.775375\pi$
$797$	−3.17940	+	3.17940i	−0.112620	+	0.112620i	0.648551	+	0.761171i	$0.275375\pi$
$798$		0			0
$799$			36.5297i			1.29233i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.961924	−	0.961924i	−0.0339456	−	0.0339456i
$804$		0			0
$805$	−1.79717	−	10.6580i	−0.0633418	−	0.375644i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.71067			−0.0601440			−0.0300720	−	0.999548i	$-0.509574\pi$
$809$	−1.71067			−0.0601440			−0.0300720	+	0.999548i	$0.509574\pi$
$810$		0			0
$811$	−37.8731			−1.32991			−0.664953	−	0.746885i	$-0.731548\pi$
$811$	−37.8731			−1.32991			−0.664953	+	0.746885i	$0.731548\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.20689	−	5.91340i	0.147361	−	0.207137i
$816$		0			0
$817$	−29.6215	−	29.6215i	−1.03633	−	1.03633i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			22.4969i			0.785148i	0.919720	+	0.392574i	$0.128415\pi$
$821$			22.4969i			0.785148i	−0.919720	+	0.392574i	$0.871585\pi$
$822$		0			0
$823$	4.38947	−	4.38947i	0.153007	−	0.153007i	−0.626452	−	0.779460i	$-0.715494\pi$
$823$	4.38947	−	4.38947i	0.153007	−	0.153007i	0.779460	+	0.626452i	$0.215494\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	21.4863	−	21.4863i	0.747152	−	0.747152i	−0.226792	−	0.973943i	$-0.572824\pi$
$827$	21.4863	−	21.4863i	0.747152	−	0.747152i	0.973943	+	0.226792i	$0.0728236\pi$
$828$		0			0
$829$			32.9243i			1.14351i	0.820425	+	0.571755i	$0.193737\pi$
$829$			32.9243i			1.14351i	−0.820425	+	0.571755i	$0.806263\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−61.5918	−	61.5918i	−2.13403	−	2.13403i
$834$		0			0
$835$	25.5394	−	35.8994i	0.883829	−	1.24235i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−42.9008			−1.48110			−0.740550	−	0.672002i	$-0.765435\pi$
$839$	−42.9008			−1.48110			−0.740550	+	0.672002i	$0.765435\pi$
$840$		0			0
$841$	−10.6185			−0.366153
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.43849	+	20.3917i	0.118288	+	0.701496i
$846$		0			0
$847$	24.7118	+	24.7118i	0.849107	+	0.849107i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1.58731i		−	0.0544122i
$852$		0			0
$853$	1.30459	−	1.30459i	0.0446684	−	0.0446684i	−0.684420	−	0.729088i	$-0.739944\pi$
$853$	1.30459	−	1.30459i	0.0446684	−	0.0446684i	0.729088	+	0.684420i	$0.239944\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.0277	+	20.0277i	−0.684132	+	0.684132i	−0.960929	−	0.276797i	$-0.910727\pi$
$857$	−20.0277	+	20.0277i	−0.684132	+	0.684132i	0.276797	+	0.960929i	$0.410727\pi$
$858$		0			0
$859$		−	25.4308i		−	0.867689i	−0.900988	−	0.433844i	$-0.857157\pi$
$859$		−	25.4308i		−	0.867689i	0.900988	−	0.433844i	$-0.142843\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.93715	−	7.93715i	−0.270184	−	0.270184i	0.558990	−	0.829174i	$-0.311189\pi$
$863$	−7.93715	−	7.93715i	−0.270184	−	0.270184i	−0.829174	+	0.558990i	$0.811189\pi$
$864$		0			0
$865$	23.9345	+	17.0274i	0.813796	+	0.578949i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−45.6612			−1.54895
$870$		0			0
$871$	−25.4006			−0.860665
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−51.8736	−	15.1553i	−1.75365	−	0.512341i
$876$		0			0
$877$	−38.3803	−	38.3803i	−1.29601	−	1.29601i	−0.931005	−	0.365006i	$-0.881067\pi$
$877$	−38.3803	−	38.3803i	−1.29601	−	1.29601i	−0.365006	−	0.931005i	$-0.618933\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	39.3146i		−	1.32454i	−0.749264	−	0.662272i	$-0.769593\pi$
$881$		−	39.3146i		−	1.32454i	0.749264	−	0.662272i	$-0.230407\pi$
$882$		0			0
$883$	12.4985	−	12.4985i	0.420608	−	0.420608i	−0.464805	−	0.885413i	$-0.653876\pi$
$883$	12.4985	−	12.4985i	0.420608	−	0.420608i	0.885413	+	0.464805i	$0.153876\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−30.6349	+	30.6349i	−1.02862	+	1.02862i	−0.0290398	+	0.999578i	$0.509245\pi$
$887$	−30.6349	+	30.6349i	−1.02862	+	1.02862i	−0.999578	+	0.0290398i	$0.990755\pi$
$888$		0			0
$889$			33.6809i			1.12962i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−16.2455	−	16.2455i	−0.543636	−	0.543636i
$894$		0			0
$895$	27.3687	−	4.61496i	0.914834	−	0.154261i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.03882			0.168054
$900$		0			0
$901$	−59.8604			−1.99424
$902$		0			0
$903$		0			0
$904$		0			0
$905$	48.8410	−	8.23567i	1.62353	−	0.273763i
$906$		0			0
$907$	15.5458	+	15.5458i	0.516190	+	0.516190i	0.916416	−	0.400226i	$-0.131068\pi$
$907$	15.5458	+	15.5458i	0.516190	+	0.516190i	−0.400226	+	0.916416i	$0.631068\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			16.9576i			0.561830i	0.959733	+	0.280915i	$0.0906379\pi$
$911$			16.9576i			0.561830i	−0.959733	+	0.280915i	$0.909362\pi$
$912$		0			0
$913$	−32.8986	+	32.8986i	−1.08878	+	1.08878i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−51.7701	+	51.7701i	−1.70960	+	1.70960i
$918$		0			0
$919$		−	23.4546i		−	0.773695i	−0.922144	−	0.386847i	$-0.873564\pi$
$919$		−	23.4546i		−	0.773695i	0.922144	−	0.386847i	$-0.126436\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−9.83917	−	9.83917i	−0.323860	−	0.323860i
$924$		0			0
$925$	−7.14195	−	3.46138i	−0.234826	−	0.113810i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−20.7528			−0.680878			−0.340439	−	0.940267i	$-0.610576\pi$
$929$	−20.7528			−0.680878			−0.340439	+	0.940267i	$0.610576\pi$
$930$		0			0
$931$	54.7825			1.79542
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−41.4083	−	29.4586i	−1.35420	−	0.963400i
$936$		0			0
$937$	16.0717	+	16.0717i	0.525039	+	0.525039i	0.919089	−	0.394050i	$-0.128926\pi$
$937$	16.0717	+	16.0717i	0.525039	+	0.525039i	−0.394050	+	0.919089i	$0.628926\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			41.8188i			1.36325i	0.731700	+	0.681627i	$0.238727\pi$
$941$			41.8188i			1.36325i	−0.731700	+	0.681627i	$0.761273\pi$
$942$		0			0
$943$	−7.17089	+	7.17089i	−0.233516	+	0.233516i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−31.8172	+	31.8172i	−1.03392	+	1.03392i	−0.0345160	+	0.999404i	$0.510989\pi$
$947$	−31.8172	+	31.8172i	−1.03392	+	1.03392i	−0.999404	+	0.0345160i	$0.989011\pi$
$948$		0			0
$949$		−	0.617138i		−	0.0200332i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	23.7060	+	23.7060i	0.767914	+	0.767914i	0.977739	−	0.209825i	$-0.0672895\pi$
$953$	23.7060	+	23.7060i	0.767914	+	0.767914i	−0.209825	+	0.977739i	$0.567290\pi$
$954$		0			0
$955$	0.905081	+	5.36751i	0.0292877	+	0.173689i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−54.5765			−1.76237
$960$		0			0
$961$	−29.6187			−0.955443
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−22.4908	+	31.6141i	−0.724005	+	1.01769i
$966$		0			0
$967$	2.79750	+	2.79750i	0.0899614	+	0.0899614i	0.750655	−	0.660694i	$-0.229738\pi$
$967$	2.79750	+	2.79750i	0.0899614	+	0.0899614i	−0.660694	+	0.750655i	$0.729738\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	42.0193i		−	1.34846i	−0.738520	−	0.674231i	$-0.764475\pi$
$971$		−	42.0193i		−	1.34846i	0.738520	−	0.674231i	$-0.235525\pi$
$972$		0			0
$973$	38.1445	−	38.1445i	1.22286	−	1.22286i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.9684	+	18.9684i	−0.606853	+	0.606853i	−0.942122	−	0.335270i	$-0.891173\pi$
$977$	−18.9684	+	18.9684i	−0.606853	+	0.606853i	0.335270	+	0.942122i	$0.391173\pi$
$978$		0			0
$979$		−	16.9817i		−	0.542738i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−33.3002	−	33.3002i	−1.06211	−	1.06211i	−0.997939	−	0.0641732i	$-0.979559\pi$
$983$	−33.3002	−	33.3002i	−1.06211	−	1.06211i	−0.0641732	−	0.997939i	$-0.520441\pi$
$984$		0			0
$985$	−20.1562	+	28.3325i	−0.642231	+	0.902749i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.5136			0.397908
$990$		0			0
$991$	39.9998			1.27064			0.635318	−	0.772250i	$-0.280869\pi$
$991$	39.9998			1.27064			0.635318	+	0.772250i	$0.280869\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.51098	+	32.6824i	0.174710	+	1.03610i
$996$		0			0
$997$	−2.72877	−	2.72877i	−0.0864211	−	0.0864211i	0.662575	−	0.748996i	$-0.269464\pi$
$997$	−2.72877	−	2.72877i	−0.0864211	−	0.0864211i	−0.748996	+	0.662575i	$0.769464\pi$
$998$		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.737.20	yes	44
3.2	odd	2	inner	4140.2.s.a.737.3	✓	44
5.3	odd	4	inner	4140.2.s.a.2393.3	yes	44
15.8	even	4	inner	4140.2.s.a.2393.20	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.a.737.3	✓	44	3.2	odd	2	inner
4140.2.s.a.737.20	yes	44	1.1	even	1	trivial
4140.2.s.a.2393.3	yes	44	5.3	odd	4	inner
4140.2.s.a.2393.20	yes	44	15.8	even	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)

\(f(q)\)	\(=\)	\(q+(2.20494 - 0.371802i) q^{5} +(-3.41792 - 3.41792i) q^{7} +O(q^{10})\)	\(q+(2.20494 - 0.371802i) q^{5} +(-3.41792 - 3.41792i) q^{7} +4.26967i q^{11} +(-1.36964 + 1.36964i) q^{13} +(-3.76377 + 3.76377i) q^{17} -3.34766i q^{19} +(0.707107 + 0.707107i) q^{23} +(4.72353 - 1.63960i) q^{25} +4.28737 q^{29} +1.17527 q^{31} +(-8.80711 - 6.26553i) q^{35} +(-1.12240 - 1.12240i) q^{37} +10.1412i q^{41} +(8.84843 - 8.84843i) q^{43} +(4.85281 - 4.85281i) q^{47} +16.3644i q^{49} +(7.95219 + 7.95219i) q^{53} +(1.58747 + 9.41436i) q^{55} -4.65152 q^{59} -12.2709 q^{61} +(-2.51074 + 3.52920i) q^{65} +(9.27273 + 9.27273i) q^{67} +7.18378i q^{71} +(-0.225292 + 0.225292i) q^{73} +(14.5934 - 14.5934i) q^{77} +10.6943i q^{79} +(7.70519 + 7.70519i) q^{83} +(-6.89951 + 9.69826i) q^{85} -3.97729 q^{89} +9.36263 q^{91} +(-1.24467 - 7.38139i) q^{95} +(-0.874011 - 0.874011i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 12 q^{7}+O(q^{10}) \) 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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