LMFDB - Embedded newform 4140.2.s.b.737.9

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			737.9
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.b.2393.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.994073 + 2.00295i) q^{5} +(-2.91813 - 2.91813i) q^{7} +O(q^{10})$	$q+(-0.994073 + 2.00295i) q^{5} +(-2.91813 - 2.91813i) q^{7} -4.72853i q^{11} +(0.839370 - 0.839370i) q^{13} +(-5.22154 + 5.22154i) q^{17} -3.12186i q^{19} +(-0.707107 - 0.707107i) q^{23} +(-3.02364 - 3.98216i) q^{25} +4.03151 q^{29} -6.68988 q^{31} +(8.74572 - 2.94404i) q^{35} +(2.81976 + 2.81976i) q^{37} +9.02254i q^{41} +(1.64743 - 1.64743i) q^{43} +(-1.14788 + 1.14788i) q^{47} +10.0310i q^{49} +(-0.243668 - 0.243668i) q^{53} +(9.47101 + 4.70050i) q^{55} +3.07448 q^{59} +8.53710 q^{61} +(0.846823 + 2.51561i) q^{65} +(-4.44459 - 4.44459i) q^{67} -5.08047i q^{71} +(-1.30895 + 1.30895i) q^{73} +(-13.7985 + 13.7985i) q^{77} -0.837744i q^{79} +(1.87295 + 1.87295i) q^{83} +(-5.26791 - 15.6491i) q^{85} +13.6034 q^{89} -4.89878 q^{91} +(6.25293 + 3.10335i) q^{95} +(5.02715 + 5.02715i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{7}+O(q^{10}) $ 44 * q - 4 * q^7	$ 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) $ 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * 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
$5$	−0.994073	+	2.00295i	−0.444563	+	0.895748i
$5$	−0.994073	+	2.00295i	−0.444563	+	0.895748i
$6$		0			0
$7$	−2.91813	−	2.91813i	−1.10295	−	1.10295i	−0.994053	−	0.108897i	$-0.965268\pi$
$7$	−2.91813	−	2.91813i	−1.10295	−	1.10295i	−0.108897	−	0.994053i	$-0.534732\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.72853i		−	1.42570i	−0.701315	−	0.712852i	$-0.747403\pi$
$11$		−	4.72853i		−	1.42570i	0.701315	−	0.712852i	$-0.252597\pi$
$12$		0			0
$13$	0.839370	−	0.839370i	0.232799	−	0.232799i	−0.581061	−	0.813860i	$-0.697362\pi$
$13$	0.839370	−	0.839370i	0.232799	−	0.232799i	0.813860	+	0.581061i	$0.197362\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.22154	+	5.22154i	−1.26641	+	1.26641i	−0.318481	+	0.947929i	$0.603173\pi$
$17$	−5.22154	+	5.22154i	−1.26641	+	1.26641i	−0.947929	+	0.318481i	$0.896827\pi$
$18$		0			0
$19$		−	3.12186i		−	0.716203i	−0.933683	−	0.358102i	$-0.883424\pi$
$19$		−	3.12186i		−	0.716203i	0.933683	−	0.358102i	$-0.116576\pi$
$20$		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
$25$	−3.02364	−	3.98216i	−0.604728	−	0.796432i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.03151			0.748632			0.374316	−	0.927301i	$-0.377878\pi$
$29$	4.03151			0.748632			0.374316	+	0.927301i	$0.377878\pi$
$30$		0			0
$31$	−6.68988			−1.20154			−0.600769	−	0.799423i	$-0.705139\pi$
$31$	−6.68988			−1.20154			−0.600769	+	0.799423i	$0.705139\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.74572	−	2.94404i	1.47830	−	0.497634i
$36$		0			0
$37$	2.81976	+	2.81976i	0.463566	+	0.463566i	0.899822	−	0.436257i	$-0.143696\pi$
$37$	2.81976	+	2.81976i	0.463566	+	0.463566i	−0.436257	+	0.899822i	$0.643696\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			9.02254i			1.40908i	0.709662	+	0.704542i	$0.248848\pi$
$41$			9.02254i			1.40908i	−0.709662	+	0.704542i	$0.751152\pi$
$42$		0			0
$43$	1.64743	−	1.64743i	0.251231	−	0.251231i	−0.570244	−	0.821475i	$-0.693151\pi$
$43$	1.64743	−	1.64743i	0.251231	−	0.251231i	0.821475	+	0.570244i	$0.193151\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.14788	+	1.14788i	−0.167436	+	0.167436i	−0.785851	−	0.618416i	$-0.787775\pi$
$47$	−1.14788	+	1.14788i	−0.167436	+	0.167436i	0.618416	+	0.785851i	$0.287775\pi$
$48$		0			0
$49$			10.0310i			1.43300i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.243668	−	0.243668i	−0.0334704	−	0.0334704i	0.690174	−	0.723644i	$-0.257534\pi$
$53$	−0.243668	−	0.243668i	−0.0334704	−	0.0334704i	−0.723644	+	0.690174i	$0.757534\pi$
$54$		0			0
$55$	9.47101	+	4.70050i	1.27707	+	0.633815i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.07448			0.400263			0.200131	−	0.979769i	$-0.435863\pi$
$59$	3.07448			0.400263			0.200131	+	0.979769i	$0.435863\pi$
$60$		0			0
$61$	8.53710			1.09306			0.546532	−	0.837438i	$-0.315948\pi$
$61$	8.53710			1.09306			0.546532	+	0.837438i	$0.315948\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.846823	+	2.51561i	0.105035	+	0.312023i
$66$		0			0
$67$	−4.44459	−	4.44459i	−0.542994	−	0.542994i	0.381412	−	0.924405i	$-0.375438\pi$
$67$	−4.44459	−	4.44459i	−0.542994	−	0.542994i	−0.924405	+	0.381412i	$0.875438\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.08047i		−	0.602941i	−0.953476	−	0.301470i	$-0.902523\pi$
$71$		−	5.08047i		−	0.602941i	0.953476	−	0.301470i	$-0.0974775\pi$
$72$		0			0
$73$	−1.30895	+	1.30895i	−0.153201	+	0.153201i	−0.779546	−	0.626345i	$-0.784550\pi$
$73$	−1.30895	+	1.30895i	−0.153201	+	0.153201i	0.626345	+	0.779546i	$0.284550\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−13.7985	+	13.7985i	−1.57248	+	1.57248i
$78$		0			0
$79$		−	0.837744i		−	0.0942535i	−0.998889	−	0.0471268i	$-0.984994\pi$
$79$		−	0.837744i		−	0.0942535i	0.998889	−	0.0471268i	$-0.0150065\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.87295	+	1.87295i	0.205583	+	0.205583i	0.802387	−	0.596804i	$-0.203563\pi$
$83$	1.87295	+	1.87295i	0.205583	+	0.205583i	−0.596804	+	0.802387i	$0.703563\pi$
$84$		0			0
$85$	−5.26791	−	15.6491i	−0.571385	−	1.69738i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.6034			1.44196			0.720978	−	0.692957i	$-0.243693\pi$
$89$	13.6034			1.44196			0.720978	+	0.692957i	$0.243693\pi$
$90$		0			0
$91$	−4.89878			−0.513532
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.25293	+	3.10335i	0.641537	+	0.318397i
$96$		0			0
$97$	5.02715	+	5.02715i	0.510429	+	0.510429i	0.914658	−	0.404229i	$-0.132460\pi$
$97$	5.02715	+	5.02715i	0.510429	+	0.510429i	−0.404229	+	0.914658i	$0.632460\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			12.3541i			1.22928i	0.788810	+	0.614638i	$0.210698\pi$
$101$			12.3541i			1.22928i	−0.788810	+	0.614638i	$0.789302\pi$
$102$		0			0
$103$	−13.2482	+	13.2482i	−1.30538	+	1.30538i	−0.380676	+	0.924708i	$0.624309\pi$
$103$	−13.2482	+	13.2482i	−1.30538	+	1.30538i	−0.924708	+	0.380676i	$0.875691\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.81864	−	8.81864i	0.852530	−	0.852530i	−0.137914	−	0.990444i	$-0.544040\pi$
$107$	8.81864	−	8.81864i	0.852530	−	0.852530i	0.990444	+	0.137914i	$0.0440397\pi$
$108$		0			0
$109$			5.58780i			0.535214i	0.963528	+	0.267607i	$0.0862328\pi$
$109$			5.58780i			0.535214i	−0.963528	+	0.267607i	$0.913767\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.20634	+	3.20634i	0.301627	+	0.301627i	0.841650	−	0.540023i	$-0.181585\pi$
$113$	3.20634	+	3.20634i	0.301627	+	0.301627i	−0.540023	+	0.841650i	$0.681585\pi$
$114$		0			0
$115$	2.11922	−	0.713386i	0.197618	−	0.0665236i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	30.4743			2.79358
$120$		0			0
$121$	−11.3590			−1.03263
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.9818	−	2.09765i	0.982242	−	0.187619i
$126$		0			0
$127$	7.49795	+	7.49795i	0.665335	+	0.665335i	0.956633	−	0.291297i	$-0.0940869\pi$
$127$	7.49795	+	7.49795i	0.665335	+	0.665335i	−0.291297	+	0.956633i	$0.594087\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			17.2677i			1.50869i	0.656479	+	0.754344i	$0.272045\pi$
$131$			17.2677i			1.50869i	−0.656479	+	0.754344i	$0.727955\pi$
$132$		0			0
$133$	−9.10999	+	9.10999i	−0.789936	+	0.789936i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.3389	+	12.3389i	−1.05419	+	1.05419i	−0.0557408	+	0.998445i	$0.517752\pi$
$137$	−12.3389	+	12.3389i	−1.05419	+	1.05419i	−0.998445	+	0.0557408i	$0.982248\pi$
$138$		0			0
$139$		−	16.2221i		−	1.37594i	−0.725738	−	0.687971i	$-0.758502\pi$
$139$		−	16.2221i		−	1.37594i	0.725738	−	0.687971i	$-0.241498\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.96898	−	3.96898i	−0.331903	−	0.331903i
$144$		0			0
$145$	−4.00761	+	8.07492i	−0.332814	+	0.670585i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	22.3864			1.83396			0.916981	−	0.398930i	$-0.130619\pi$
$149$	22.3864			1.83396			0.916981	+	0.398930i	$0.130619\pi$
$150$		0			0
$151$	−13.4827			−1.09721			−0.548604	−	0.836082i	$-0.684841\pi$
$151$	−13.4827			−1.09721			−0.548604	+	0.836082i	$0.684841\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.65022	−	13.3995i	0.534159	−	1.07627i
$156$		0			0
$157$	1.19044	+	1.19044i	0.0950074	+	0.0950074i	0.753013	−	0.658006i	$-0.228600\pi$
$157$	1.19044	+	1.19044i	0.0950074	+	0.0950074i	−0.658006	+	0.753013i	$0.728600\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			4.12686i			0.325242i
$162$		0			0
$163$	0.374027	−	0.374027i	0.0292960	−	0.0292960i	−0.692307	−	0.721603i	$-0.743406\pi$
$163$	0.374027	−	0.374027i	0.0292960	−	0.0292960i	0.721603	+	0.692307i	$0.243406\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.0979	+	11.0979i	−0.858782	+	0.858782i	−0.991195	−	0.132413i	$-0.957728\pi$
$167$	−11.0979	+	11.0979i	−0.858782	+	0.858782i	0.132413	+	0.991195i	$0.457728\pi$
$168$		0			0
$169$			11.5909i			0.891609i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.02031	+	6.02031i	0.457716	+	0.457716i	0.897905	−	0.440189i	$-0.145089\pi$
$173$	6.02031	+	6.02031i	0.457716	+	0.457716i	−0.440189	+	0.897905i	$0.645089\pi$
$174$		0			0
$175$	−2.79710	+	20.4438i	−0.211441	+	1.54541i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.60781			−0.419148			−0.209574	−	0.977793i	$-0.567208\pi$
$179$	−5.60781			−0.419148			−0.209574	+	0.977793i	$0.567208\pi$
$180$		0			0
$181$	−18.0468			−1.34141			−0.670703	−	0.741726i	$-0.734008\pi$
$181$	−18.0468			−1.34141			−0.670703	+	0.741726i	$0.734008\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.45089	+	2.84480i	−0.621322	+	0.209154i
$186$		0			0
$187$	24.6902	+	24.6902i	1.80553	+	1.80553i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	14.9521i		−	1.08190i	−0.841056	−	0.540948i	$-0.818065\pi$
$191$		−	14.9521i		−	1.08190i	0.841056	−	0.540948i	$-0.181935\pi$
$192$		0			0
$193$	−16.0627	+	16.0627i	−1.15622	+	1.15622i	−0.170940	+	0.985281i	$0.554681\pi$
$193$	−16.0627	+	16.0627i	−1.15622	+	1.15622i	−0.985281	+	0.170940i	$0.945319\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.58286	+	2.58286i	−0.184021	+	0.184021i	−0.793105	−	0.609084i	$-0.791537\pi$
$197$	−2.58286	+	2.58286i	−0.184021	+	0.184021i	0.609084	+	0.793105i	$0.291537\pi$
$198$		0			0
$199$			13.4614i			0.954253i	0.878835	+	0.477127i	$0.158322\pi$
$199$			13.4614i			0.954253i	−0.878835	+	0.477127i	$0.841678\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−11.7645	−	11.7645i	−0.825704	−	0.825704i
$204$		0			0
$205$	−18.0717	−	8.96906i	−1.26218	−	0.626427i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−14.7618			−1.02109
$210$		0			0
$211$	−7.31897			−0.503859			−0.251929	−	0.967746i	$-0.581065\pi$
$211$	−7.31897			−0.503859			−0.251929	+	0.967746i	$0.581065\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.66206	+	4.93739i	0.113352	+	0.336727i
$216$		0			0
$217$	19.5219	+	19.5219i	1.32524	+	1.32524i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			8.76561i			0.589639i
$222$		0			0
$223$	−18.7435	+	18.7435i	−1.25516	+	1.25516i	−0.301778	+	0.953378i	$0.597580\pi$
$223$	−18.7435	+	18.7435i	−1.25516	+	1.25516i	−0.953378	+	0.301778i	$0.902420\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.80427	−	3.80427i	0.252498	−	0.252498i	−0.569496	−	0.821994i	$-0.692862\pi$
$227$	3.80427	−	3.80427i	0.252498	−	0.252498i	0.821994	+	0.569496i	$0.192862\pi$
$228$		0			0
$229$		−	15.5644i		−	1.02852i	−0.857633	−	0.514262i	$-0.828066\pi$
$229$		−	15.5644i		−	1.02852i	0.857633	−	0.514262i	$-0.171934\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.33052	+	3.33052i	0.218189	+	0.218189i	0.807735	−	0.589546i	$-0.200693\pi$
$233$	3.33052	+	3.33052i	0.218189	+	0.218189i	−0.589546	+	0.807735i	$0.700693\pi$
$234$		0			0
$235$	−1.15807	−	3.44023i	−0.0755444	−	0.224416i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.79460			−0.245453			−0.122726	−	0.992441i	$-0.539164\pi$
$239$	−3.79460			−0.245453			−0.122726	+	0.992441i	$0.539164\pi$
$240$		0			0
$241$	−0.0211915			−0.00136506			−0.000682531	−	1.00000i	$-0.500217\pi$
$241$	−0.0211915			−0.00136506			−0.000682531		1.00000i	$0.500217\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−20.0916	−	9.97153i	−1.28360	−	0.637058i
$246$		0			0
$247$	−2.62039	−	2.62039i	−0.166732	−	0.166732i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			4.23317i			0.267195i	0.991036	+	0.133598i	$0.0426530\pi$
$251$			4.23317i			0.267195i	−0.991036	+	0.133598i	$0.957347\pi$
$252$		0			0
$253$	−3.34357	+	3.34357i	−0.210209	+	0.210209i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.57287	−	6.57287i	0.410005	−	0.410005i	−0.471736	−	0.881740i	$-0.656372\pi$
$257$	6.57287	−	6.57287i	0.410005	−	0.410005i	0.881740	+	0.471736i	$0.156372\pi$
$258$		0			0
$259$		−	16.4569i		−	1.02258i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	20.9642	+	20.9642i	1.29271	+	1.29271i	0.933106	+	0.359601i	$0.117087\pi$
$263$	20.9642	+	20.9642i	1.29271	+	1.29271i	0.359601	+	0.933106i	$0.382913\pi$
$264$		0			0
$265$	0.730279	−	0.245832i	0.0448607	−	0.0151013i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−23.9852			−1.46241			−0.731203	−	0.682160i	$-0.761041\pi$
$269$	−23.9852			−1.46241			−0.731203	+	0.682160i	$0.761041\pi$
$270$		0			0
$271$	7.03231			0.427183			0.213591	−	0.976923i	$-0.431484\pi$
$271$	7.03231			0.427183			0.213591	+	0.976923i	$0.431484\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−18.8298	+	14.2974i	−1.13548	+	0.862163i
$276$		0			0
$277$	−10.4202	−	10.4202i	−0.626086	−	0.626086i	0.320995	−	0.947081i	$-0.395983\pi$
$277$	−10.4202	−	10.4202i	−0.626086	−	0.626086i	−0.947081	+	0.320995i	$0.895983\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	10.4915i		−	0.625867i	−0.949775	−	0.312934i	$-0.898688\pi$
$281$		−	10.4915i		−	0.625867i	0.949775	−	0.312934i	$-0.101312\pi$
$282$		0			0
$283$	−0.702881	+	0.702881i	−0.0417819	+	0.0417819i	−0.727689	−	0.685907i	$-0.759406\pi$
$283$	−0.702881	+	0.702881i	−0.0417819	+	0.0417819i	0.685907	+	0.727689i	$0.259406\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	26.3290	−	26.3290i	1.55415	−	1.55415i
$288$		0			0
$289$		−	37.5290i		−	2.20759i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.94197	+	3.94197i	0.230292	+	0.230292i	0.812815	−	0.582522i	$-0.197934\pi$
$293$	3.94197	+	3.94197i	0.230292	+	0.230292i	−0.582522	+	0.812815i	$0.697934\pi$
$294$		0			0
$295$	−3.05625	+	6.15803i	−0.177942	+	0.358534i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.18705			−0.0686488
$300$		0			0
$301$	−9.61484			−0.554190
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.48650	+	17.0994i	−0.485936	+	0.979110i
$306$		0			0
$307$	4.68784	+	4.68784i	0.267549	+	0.267549i	0.828112	−	0.560563i	$-0.189415\pi$
$307$	4.68784	+	4.68784i	0.267549	+	0.267549i	−0.560563	+	0.828112i	$0.689415\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			15.1260i			0.857720i	0.903371	+	0.428860i	$0.141085\pi$
$311$			15.1260i			0.857720i	−0.903371	+	0.428860i	$0.858915\pi$
$312$		0			0
$313$	18.0514	−	18.0514i	1.02033	−	1.02033i	0.0205376	−	0.999789i	$-0.493462\pi$
$313$	18.0514	−	18.0514i	1.02033	−	1.02033i	0.999789	−	0.0205376i	$-0.00653778\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.34109	−	5.34109i	0.299985	−	0.299985i	−0.541023	−	0.841008i	$-0.681963\pi$
$317$	5.34109	−	5.34109i	0.299985	−	0.299985i	0.841008	+	0.541023i	$0.181963\pi$
$318$		0			0
$319$		−	19.0631i		−	1.06733i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.3009	+	16.3009i	0.907007	+	0.907007i
$324$		0			0
$325$	−5.88046	−	0.804555i	−0.326189	−	0.0446287i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.69933			0.369346
$330$		0			0
$331$	26.4675			1.45478			0.727391	−	0.686223i	$-0.240733\pi$
$331$	26.4675			1.45478			0.727391	+	0.686223i	$0.240733\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	13.3206	−	4.48406i	0.727780	−	0.244990i
$336$		0			0
$337$	−8.69148	−	8.69148i	−0.473455	−	0.473455i	0.429576	−	0.903031i	$-0.358663\pi$
$337$	−8.69148	−	8.69148i	−0.473455	−	0.473455i	−0.903031	+	0.429576i	$0.858663\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			31.6332i			1.71304i
$342$		0			0
$343$	8.84482	−	8.84482i	0.477575	−	0.477575i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−20.0696	+	20.0696i	−1.07739	+	1.07739i	−0.0806486	+	0.996743i	$0.525699\pi$
$347$	−20.0696	+	20.0696i	−1.07739	+	1.07739i	−0.996743	+	0.0806486i	$0.974301\pi$
$348$		0			0
$349$			13.4197i			0.718341i	0.933272	+	0.359170i	$0.116940\pi$
$349$			13.4197i			0.718341i	−0.933272	+	0.359170i	$0.883060\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.40408	+	3.40408i	0.181181	+	0.181181i	0.791870	−	0.610689i	$-0.209108\pi$
$353$	3.40408	+	3.40408i	0.181181	+	0.181181i	−0.610689	+	0.791870i	$0.709108\pi$
$354$		0			0
$355$	10.1759	+	5.05036i	0.540083	+	0.268045i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.7347			−0.672109			−0.336055	−	0.941842i	$-0.609093\pi$
$359$	−12.7347			−0.672109			−0.336055	+	0.941842i	$0.609093\pi$
$360$		0			0
$361$	9.25401			0.487053
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.32057	−	3.92296i	−0.0691220	−	0.205337i
$366$		0			0
$367$	−7.91038	−	7.91038i	−0.412918	−	0.412918i	0.469836	−	0.882754i	$-0.344313\pi$
$367$	−7.91038	−	7.91038i	−0.412918	−	0.412918i	−0.882754	+	0.469836i	$0.844313\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			1.42211i			0.0738323i
$372$		0			0
$373$	−3.04795	+	3.04795i	−0.157817	+	0.157817i	−0.781599	−	0.623782i	$-0.785595\pi$
$373$	−3.04795	+	3.04795i	−0.157817	+	0.157817i	0.623782	+	0.781599i	$0.285595\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.38393	−	3.38393i	0.174281	−	0.174281i
$378$		0			0
$379$			11.1528i			0.572879i	0.958098	+	0.286439i	$0.0924717\pi$
$379$			11.1528i			0.572879i	−0.958098	+	0.286439i	$0.907528\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.69911	−	2.69911i	−0.137918	−	0.137918i	0.634777	−	0.772695i	$-0.281092\pi$
$383$	−2.69911	−	2.69911i	−0.137918	−	0.137918i	−0.772695	+	0.634777i	$0.781092\pi$
$384$		0			0
$385$	−13.9210	−	41.3543i	−0.709479	−	2.10761i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	20.8945			1.05940			0.529698	−	0.848187i	$-0.322305\pi$
$389$	20.8945			1.05940			0.529698	+	0.848187i	$0.322305\pi$
$390$		0			0
$391$	7.38438			0.373444
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.67796	+	0.832778i	0.0844274	+	0.0419016i
$396$		0			0
$397$	21.7698	+	21.7698i	1.09260	+	1.09260i	0.995251	+	0.0973450i	$0.0310350\pi$
$397$	21.7698	+	21.7698i	1.09260	+	1.09260i	0.0973450	+	0.995251i	$0.468965\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	31.1024i		−	1.55318i	−0.630008	−	0.776589i	$-0.716948\pi$
$401$		−	31.1024i		−	1.55318i	0.630008	−	0.776589i	$-0.283052\pi$
$402$		0			0
$403$	−5.61528	+	5.61528i	−0.279717	+	0.279717i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	13.3333	−	13.3333i	0.660908	−	0.660908i
$408$		0			0
$409$		−	34.6501i		−	1.71334i	−0.515868	−	0.856668i	$-0.672530\pi$
$409$		−	34.6501i		−	1.71334i	0.515868	−	0.856668i	$-0.327470\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.97172	−	8.97172i	−0.441470	−	0.441470i
$414$		0			0
$415$	−5.61329	+	1.88958i	−0.275545	+	0.0927561i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	32.3996			1.58283			0.791413	−	0.611282i	$-0.209346\pi$
$419$	32.3996			1.58283			0.791413	+	0.611282i	$0.209346\pi$
$420$		0			0
$421$	34.2519			1.66934			0.834668	−	0.550753i	$-0.185660\pi$
$421$	34.2519			1.66934			0.834668	+	0.550753i	$0.185660\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	36.5811	+	5.00497i	1.77444	+	0.242777i
$426$		0			0
$427$	−24.9124	−	24.9124i	−1.20560	−	1.20560i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			16.2052i			0.780578i	0.920692	+	0.390289i	$0.127625\pi$
$431$			16.2052i			0.780578i	−0.920692	+	0.390289i	$0.872375\pi$
$432$		0			0
$433$	29.3648	−	29.3648i	1.41118	−	1.41118i	0.659312	−	0.751870i	$-0.270848\pi$
$433$	29.3648	−	29.3648i	1.41118	−	1.41118i	0.751870	−	0.659312i	$-0.229152\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.20749	+	2.20749i	−0.105598	+	0.105598i
$438$		0			0
$439$		−	20.8799i		−	0.996542i	−0.867021	−	0.498271i	$-0.833969\pi$
$439$		−	20.8799i		−	0.996542i	0.867021	−	0.498271i	$-0.166031\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8.56069	−	8.56069i	−0.406731	−	0.406731i	0.473866	−	0.880597i	$-0.342858\pi$
$443$	−8.56069	−	8.56069i	−0.406731	−	0.406731i	−0.880597	+	0.473866i	$0.842858\pi$
$444$		0			0
$445$	−13.5228	+	27.2470i	−0.641041	+	1.29163i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.11306			0.0997217			0.0498608	−	0.998756i	$-0.484122\pi$
$449$	2.11306			0.0997217			0.0498608	+	0.998756i	$0.484122\pi$
$450$		0			0
$451$	42.6633			2.00894
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.86975	−	9.81203i	0.228297	−	0.459995i
$456$		0			0
$457$	−17.1766	−	17.1766i	−0.803486	−	0.803486i	0.180153	−	0.983639i	$-0.442341\pi$
$457$	−17.1766	−	17.1766i	−0.803486	−	0.803486i	−0.983639	+	0.180153i	$0.942341\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	25.3697i		−	1.18158i	−0.806824	−	0.590792i	$-0.798816\pi$
$461$		−	25.3697i		−	1.18158i	0.806824	−	0.590792i	$-0.201184\pi$
$462$		0			0
$463$	−26.7930	+	26.7930i	−1.24517	+	1.24517i	−0.287349	+	0.957826i	$0.592774\pi$
$463$	−26.7930	+	26.7930i	−1.24517	+	1.24517i	−0.957826	+	0.287349i	$0.907226\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−29.5236	+	29.5236i	−1.36619	+	1.36619i	−0.500393	+	0.865799i	$0.666811\pi$
$467$	−29.5236	+	29.5236i	−1.36619	+	1.36619i	−0.865799	+	0.500393i	$0.833189\pi$
$468$		0			0
$469$			25.9398i			1.19779i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.78992	−	7.78992i	−0.358181	−	0.358181i
$474$		0			0
$475$	−12.4317	+	9.43937i	−0.570407	+	0.433108i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−22.7444			−1.03922			−0.519608	−	0.854405i	$-0.673922\pi$
$479$	−22.7444			−1.03922			−0.519608	+	0.854405i	$0.673922\pi$
$480$		0			0
$481$	4.73364			0.215836
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−15.0665	+	5.07179i	−0.684134	+	0.230298i
$486$		0			0
$487$	27.8042	+	27.8042i	1.25993	+	1.25993i	0.951126	+	0.308803i	$0.0999283\pi$
$487$	27.8042	+	27.8042i	1.25993	+	1.25993i	0.308803	+	0.951126i	$0.400072\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			30.7081i			1.38584i	0.721015	+	0.692919i	$0.243676\pi$
$491$			30.7081i			1.38584i	−0.721015	+	0.692919i	$0.756324\pi$
$492$		0			0
$493$	−21.0507	+	21.0507i	−0.948076	+	0.948076i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−14.8255	+	14.8255i	−0.665014	+	0.665014i
$498$		0			0
$499$		−	15.8842i		−	0.711076i	−0.934662	−	0.355538i	$-0.884298\pi$
$499$		−	15.8842i		−	0.711076i	0.934662	−	0.355538i	$-0.115702\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	17.7959	+	17.7959i	0.793478	+	0.793478i	0.982058	−	0.188580i	$-0.0603885\pi$
$503$	17.7959	+	17.7959i	0.793478	+	0.793478i	−0.188580	+	0.982058i	$0.560388\pi$
$504$		0			0
$505$	−24.7446	−	12.2808i	−1.10112	−	0.546490i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.19383			0.0529156			0.0264578	−	0.999650i	$-0.491577\pi$
$509$	1.19383			0.0529156			0.0264578	+	0.999650i	$0.491577\pi$
$510$		0			0
$511$	7.63938			0.337946
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−13.3658	−	39.7052i	−0.588970	−	1.74962i
$516$		0			0
$517$	5.42778	+	5.42778i	0.238714	+	0.238714i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			34.9789i			1.53245i	0.642571	+	0.766227i	$0.277868\pi$
$521$			34.9789i			1.53245i	−0.642571	+	0.766227i	$0.722132\pi$
$522$		0			0
$523$	−5.39807	+	5.39807i	−0.236041	+	0.236041i	−0.815209	−	0.579167i	$-0.803378\pi$
$523$	−5.39807	+	5.39807i	−0.236041	+	0.236041i	0.579167	+	0.815209i	$0.303378\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	34.9315	−	34.9315i	1.52164	−	1.52164i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	7.57325	+	7.57325i	0.328034	+	0.328034i
$534$		0			0
$535$	8.89695	+	26.4297i	0.384649	+	1.14266i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	47.4318			2.04303
$540$		0			0
$541$	−19.8971			−0.855444			−0.427722	−	0.903910i	$-0.640684\pi$
$541$	−19.8971			−0.855444			−0.427722	+	0.903910i	$0.640684\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−11.1921	−	5.55468i	−0.479416	−	0.237936i
$546$		0			0
$547$	−26.2679	−	26.2679i	−1.12313	−	1.12313i	−0.991268	−	0.131866i	$-0.957903\pi$
$547$	−26.2679	−	26.2679i	−1.12313	−	1.12313i	−0.131866	−	0.991268i	$-0.542097\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	12.5858i		−	0.536173i
$552$		0			0
$553$	−2.44465	+	2.44465i	−0.103957	+	0.103957i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.72293	−	9.72293i	0.411974	−	0.411974i	−0.470452	−	0.882426i	$-0.655909\pi$
$557$	9.72293	−	9.72293i	0.411974	−	0.411974i	0.882426	+	0.470452i	$0.155909\pi$
$558$		0			0
$559$		−	2.76561i		−	0.116973i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.0543	−	12.0543i	−0.508029	−	0.508029i	0.405892	−	0.913921i	$-0.366961\pi$
$563$	−12.0543	−	12.0543i	−0.508029	−	0.508029i	−0.913921	+	0.405892i	$0.866961\pi$
$564$		0			0
$565$	−9.60948	+	3.23481i	−0.404274	+	0.136090i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.29918			−0.0963865			−0.0481932	−	0.998838i	$-0.515346\pi$
$569$	−2.29918			−0.0963865			−0.0481932	+	0.998838i	$0.515346\pi$
$570$		0			0
$571$	−4.29641			−0.179799			−0.0898996	−	0.995951i	$-0.528655\pi$
$571$	−4.29641			−0.179799			−0.0898996	+	0.995951i	$0.528655\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.677778	+	4.95385i	−0.0282653	+	0.206590i
$576$		0			0
$577$	23.5982	+	23.5982i	0.982407	+	0.982407i	0.999848	−	0.0174410i	$-0.00555191\pi$
$577$	23.5982	+	23.5982i	0.982407	+	0.982407i	−0.0174410	+	0.999848i	$0.505552\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	10.9310i		−	0.453496i
$582$		0			0
$583$	−1.15219	+	1.15219i	−0.0477189	+	0.0477189i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.9756	−	10.9756i	0.453010	−	0.453010i	−0.443342	−	0.896352i	$-0.646207\pi$
$587$	10.9756	−	10.9756i	0.453010	−	0.453010i	0.896352	+	0.443342i	$0.146207\pi$
$588$		0			0
$589$			20.8848i			0.860545i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16.5040	−	16.5040i	−0.677738	−	0.677738i	0.281750	−	0.959488i	$-0.409085\pi$
$593$	−16.5040	−	16.5040i	−0.677738	−	0.677738i	−0.959488	+	0.281750i	$0.909085\pi$
$594$		0			0
$595$	−30.2937	+	61.0386i	−1.24192	+	2.50234i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.9599			0.652102			0.326051	−	0.945352i	$-0.394282\pi$
$599$	15.9599			0.652102			0.326051	+	0.945352i	$0.394282\pi$
$600$		0			0
$601$	26.7609			1.09160			0.545800	−	0.837916i	$-0.316226\pi$
$601$	26.7609			1.09160			0.545800	+	0.837916i	$0.316226\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.2916	−	22.7514i	0.459070	−	0.924978i
$606$		0			0
$607$	31.2977	+	31.2977i	1.27033	+	1.27033i	0.945912	+	0.324422i	$0.105170\pi$
$607$	31.2977	+	31.2977i	1.27033	+	1.27033i	0.324422	+	0.945912i	$0.394830\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1.92699i			0.0779578i
$612$		0			0
$613$	5.57916	−	5.57916i	0.225340	−	0.225340i	−0.585403	−	0.810743i	$-0.699064\pi$
$613$	5.57916	−	5.57916i	0.225340	−	0.225340i	0.810743	+	0.585403i	$0.199064\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−9.22980	+	9.22980i	−0.371578	+	0.371578i	−0.868052	−	0.496474i	$-0.834628\pi$
$617$	−9.22980	+	9.22980i	−0.371578	+	0.371578i	0.496474	+	0.868052i	$0.334628\pi$
$618$		0			0
$619$		−	20.4854i		−	0.823376i	−0.911325	−	0.411688i	$-0.864939\pi$
$619$		−	20.4854i		−	0.823376i	0.911325	−	0.411688i	$-0.135061\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−39.6965	−	39.6965i	−1.59041	−	1.59041i
$624$		0			0
$625$	−6.71522	+	24.0812i	−0.268609	+	0.963249i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−29.4470			−1.17413
$630$		0			0
$631$	−4.02868			−0.160379			−0.0801895	−	0.996780i	$-0.525553\pi$
$631$	−4.02868			−0.160379			−0.0801895	+	0.996780i	$0.525553\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−22.4715	+	7.56453i	−0.891756	+	0.300189i
$636$		0			0
$637$	8.41971	+	8.41971i	0.333601	+	0.333601i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	30.1882i		−	1.19236i	−0.802851	−	0.596180i	$-0.796684\pi$
$641$		−	30.1882i		−	1.19236i	0.802851	−	0.596180i	$-0.203316\pi$
$642$		0			0
$643$	8.01355	−	8.01355i	0.316024	−	0.316024i	−0.531214	−	0.847238i	$-0.678264\pi$
$643$	8.01355	−	8.01355i	0.316024	−	0.316024i	0.847238	+	0.531214i	$0.178264\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−25.5778	+	25.5778i	−1.00557	+	1.00557i	−0.00558384	+	0.999984i	$0.501777\pi$
$647$	−25.5778	+	25.5778i	−1.00557	+	1.00557i	−0.999984	+	0.00558384i	$0.998223\pi$
$648$		0			0
$649$		−	14.5377i		−	0.570656i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.8797	+	25.8797i	1.01275	+	1.01275i	0.999918	+	0.0128336i	$0.00408517\pi$
$653$	25.8797	+	25.8797i	1.01275	+	1.01275i	0.0128336	+	0.999918i	$0.495915\pi$
$654$		0			0
$655$	−34.5865	−	17.1654i	−1.35140	−	0.670707i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−25.8314			−1.00625			−0.503123	−	0.864215i	$-0.667816\pi$
$659$	−25.8314			−1.00625			−0.503123	+	0.864215i	$0.667816\pi$
$660$		0			0
$661$	31.7297			1.23414			0.617071	−	0.786907i	$-0.288319\pi$
$661$	31.7297			1.23414			0.617071	+	0.786907i	$0.288319\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−9.19089	−	27.3029i	−0.356407	−	1.05876i
$666$		0			0
$667$	−2.85071	−	2.85071i	−0.110380	−	0.110380i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	40.3679i		−	1.55839i
$672$		0			0
$673$	6.75705	−	6.75705i	0.260465	−	0.260465i	−0.564778	−	0.825243i	$-0.691038\pi$
$673$	6.75705	−	6.75705i	0.260465	−	0.260465i	0.825243	+	0.564778i	$0.191038\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9.17720	+	9.17720i	−0.352709	+	0.352709i	−0.861116	−	0.508408i	$-0.830234\pi$
$677$	−9.17720	+	9.17720i	−0.352709	+	0.352709i	0.508408	+	0.861116i	$0.330234\pi$
$678$		0			0
$679$		−	29.3398i		−	1.12596i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18.3457	−	18.3457i	−0.701978	−	0.701978i	0.262857	−	0.964835i	$-0.415335\pi$
$683$	−18.3457	−	18.3457i	−0.701978	−	0.701978i	−0.964835	+	0.262857i	$0.915335\pi$
$684$		0			0
$685$	−12.4485	−	36.9801i	−0.475633	−	1.41294i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.409055			−0.0155838
$690$		0			0
$691$	17.9714			0.683665			0.341832	−	0.939761i	$-0.388952\pi$
$691$	17.9714			0.683665			0.341832	+	0.939761i	$0.388952\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	32.4921	+	16.1260i	1.23250	+	0.611692i
$696$		0			0
$697$	−47.1116	−	47.1116i	−1.78448	−	1.78448i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			40.7462i			1.53896i	0.638670	+	0.769481i	$0.279485\pi$
$701$			40.7462i			1.53896i	−0.638670	+	0.769481i	$0.720515\pi$
$702$		0			0
$703$	8.80289	−	8.80289i	0.332007	−	0.332007i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	36.0508	−	36.0508i	1.35583	−	1.35583i
$708$		0			0
$709$		−	1.89591i		−	0.0712023i	−0.999366	−	0.0356011i	$-0.988665\pi$
$709$		−	1.89591i		−	0.0712023i	0.999366	−	0.0356011i	$-0.0113346\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.73046	+	4.73046i	0.177157	+	0.177157i
$714$		0			0
$715$	11.8951	−	4.00422i	0.444853	−	0.149750i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	34.4644			1.28530			0.642652	−	0.766158i	$-0.277834\pi$
$719$	34.4644			1.28530			0.642652	+	0.766158i	$0.277834\pi$
$720$		0			0
$721$	77.3200			2.87955
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12.1898	−	16.0541i	−0.452719	−	0.596235i
$726$		0			0
$727$	3.90227	+	3.90227i	0.144727	+	0.144727i	0.775758	−	0.631031i	$-0.217368\pi$
$727$	3.90227	+	3.90227i	0.144727	+	0.144727i	−0.631031	+	0.775758i	$0.717368\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			17.2043i			0.636323i
$732$		0			0
$733$	37.0869	−	37.0869i	1.36984	−	1.36984i	0.509168	−	0.860667i	$-0.329953\pi$
$733$	37.0869	−	37.0869i	1.36984	−	1.36984i	0.860667	−	0.509168i	$-0.170047\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−21.0164	+	21.0164i	−0.774148	+	0.774148i
$738$		0			0
$739$			27.9216i			1.02711i	0.858056	+	0.513556i	$0.171672\pi$
$739$			27.9216i			1.02711i	−0.858056	+	0.513556i	$0.828328\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.18841	+	3.18841i	0.116972	+	0.116972i	0.763170	−	0.646198i	$-0.223642\pi$
$743$	3.18841	+	3.18841i	0.116972	+	0.116972i	−0.646198	+	0.763170i	$0.723642\pi$
$744$		0			0
$745$	−22.2537	+	44.8388i	−0.815312	+	1.64277i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−51.4679			−1.88060
$750$		0			0
$751$	8.73724			0.318827			0.159413	−	0.987212i	$-0.449040\pi$
$751$	8.73724			0.318827			0.159413	+	0.987212i	$0.449040\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	13.4028	−	27.0052i	0.487778	−	0.982821i
$756$		0			0
$757$	−35.4107	−	35.4107i	−1.28703	−	1.28703i	−0.936585	−	0.350440i	$-0.886032\pi$
$757$	−35.4107	−	35.4107i	−1.28703	−	1.28703i	−0.350440	−	0.936585i	$-0.613968\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			6.99316i			0.253502i	0.991935	+	0.126751i	$0.0404549\pi$
$761$			6.99316i			0.253502i	−0.991935	+	0.126751i	$0.959545\pi$
$762$		0			0
$763$	16.3059	−	16.3059i	0.590314	−	0.590314i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.58062	−	2.58062i	0.0931808	−	0.0931808i
$768$		0			0
$769$		−	49.3679i		−	1.78025i	−0.455716	−	0.890125i	$-0.650617\pi$
$769$		−	49.3679i		−	1.78025i	0.455716	−	0.890125i	$-0.349383\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−26.8485	−	26.8485i	−0.965672	−	0.965672i	0.0337576	−	0.999430i	$-0.489253\pi$
$773$	−26.8485	−	26.8485i	−0.965672	−	0.965672i	−0.999430	+	0.0337576i	$0.989253\pi$
$774$		0			0
$775$	20.2278	+	26.6402i	0.726603	+	0.956943i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28.1671			1.00919
$780$		0			0
$781$	−24.0231			−0.859615
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3.56778	+	1.20101i	−0.127339	+	0.0428659i
$786$		0			0
$787$	2.07999	+	2.07999i	0.0741438	+	0.0741438i	0.743206	−	0.669062i	$-0.233304\pi$
$787$	2.07999	+	2.07999i	0.0741438	+	0.0741438i	−0.669062	+	0.743206i	$0.733304\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	18.7130i		−	0.665359i
$792$		0			0
$793$	7.16579	−	7.16579i	0.254465	−	0.254465i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−37.5939	+	37.5939i	−1.33164	+	1.33164i	−0.427741	+	0.903901i	$0.640691\pi$
$797$	−37.5939	+	37.5939i	−1.33164	+	1.33164i	−0.903901	+	0.427741i	$0.859309\pi$
$798$		0			0
$799$		−	11.9874i		−	0.424084i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6.18941	+	6.18941i	0.218419	+	0.218419i
$804$		0			0
$805$	−8.26591	−	4.10240i	−0.291335	−	0.144591i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	4.18246			0.147047			0.0735237	−	0.997293i	$-0.476576\pi$
$809$	4.18246			0.147047			0.0735237	+	0.997293i	$0.476576\pi$
$810$		0			0
$811$	34.3768			1.20713			0.603567	−	0.797312i	$-0.293746\pi$
$811$	34.3768			1.20713			0.603567	+	0.797312i	$0.293746\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.377348	+	1.12097i	0.0132179	+	0.0392658i
$816$		0			0
$817$	−5.14304	−	5.14304i	−0.179932	−	0.179932i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			3.79087i			0.132302i	0.997810	+	0.0661512i	$0.0210720\pi$
$821$			3.79087i			0.132302i	−0.997810	+	0.0661512i	$0.978928\pi$
$822$		0			0
$823$	28.2508	−	28.2508i	0.984759	−	0.984759i	−0.0151263	−	0.999886i	$-0.504815\pi$
$823$	28.2508	−	28.2508i	0.984759	−	0.984759i	0.999886	+	0.0151263i	$0.00481504\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.40446	+	4.40446i	−0.153158	+	0.153158i	−0.779527	−	0.626369i	$-0.784540\pi$
$827$	−4.40446	+	4.40446i	−0.153158	+	0.153158i	0.626369	+	0.779527i	$0.284540\pi$
$828$		0			0
$829$			17.9363i			0.622952i	0.950254	+	0.311476i	$0.100823\pi$
$829$			17.9363i			0.622952i	−0.950254	+	0.311476i	$0.899177\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−52.3772	−	52.3772i	−1.81476	−	1.81476i
$834$		0			0
$835$	−11.1965	−	33.2607i	−0.387469	−	1.15103i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3.15968			0.109084			0.0545421	−	0.998511i	$-0.482630\pi$
$839$	3.15968			0.109084			0.0545421	+	0.998511i	$0.482630\pi$
$840$		0			0
$841$	−12.7469			−0.439550
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−23.2161	−	11.5222i	−0.798657	−	0.396376i
$846$		0			0
$847$	33.1469	+	33.1469i	1.13894	+	1.13894i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	3.98774i		−	0.136698i
$852$		0			0
$853$	−28.7919	+	28.7919i	−0.985817	+	0.985817i	−0.999901	−	0.0140843i	$-0.995517\pi$
$853$	−28.7919	+	28.7919i	−0.985817	+	0.985817i	0.0140843	+	0.999901i	$0.495517\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.1550	+	19.1550i	−0.654321	+	0.654321i	−0.954031	−	0.299709i	$-0.903110\pi$
$857$	−19.1550	+	19.1550i	−0.654321	+	0.654321i	0.299709	+	0.954031i	$0.403110\pi$
$858$		0			0
$859$			42.6427i			1.45495i	0.686134	+	0.727475i	$0.259306\pi$
$859$			42.6427i			1.45495i	−0.686134	+	0.727475i	$0.740694\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−28.5618	−	28.5618i	−0.972255	−	0.972255i	0.0273708	−	0.999625i	$-0.491287\pi$
$863$	−28.5618	−	28.5618i	−0.972255	−	0.972255i	−0.999625	+	0.0273708i	$0.991287\pi$
$864$		0			0
$865$	−18.0430	+	6.07377i	−0.613481	+	0.206514i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.96129			−0.134378
$870$		0			0
$871$	−7.46132			−0.252817
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−38.1675	−	25.9251i	−1.29030	−	0.876429i
$876$		0			0
$877$	18.8140	+	18.8140i	0.635303	+	0.635303i	0.949393	−	0.314090i	$-0.101699\pi$
$877$	18.8140	+	18.8140i	0.635303	+	0.635303i	−0.314090	+	0.949393i	$0.601699\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	31.8268i		−	1.07227i	−0.844131	−	0.536137i	$-0.819883\pi$
$881$		−	31.8268i		−	1.07227i	0.844131	−	0.536137i	$-0.180117\pi$
$882$		0			0
$883$	21.9780	−	21.9780i	0.739619	−	0.739619i	−0.232885	−	0.972504i	$-0.574817\pi$
$883$	21.9780	−	21.9780i	0.739619	−	0.739619i	0.972504	+	0.232885i	$0.0748166\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.4650	+	12.4650i	−0.418535	+	0.418535i	−0.884699	−	0.466163i	$-0.845636\pi$
$887$	−12.4650	+	12.4650i	−0.418535	+	0.418535i	0.466163	+	0.884699i	$0.345636\pi$
$888$		0			0
$889$		−	43.7600i		−	1.46766i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.58352	+	3.58352i	0.119918	+	0.119918i
$894$		0			0
$895$	5.57457	−	11.2322i	0.186337	−	0.375450i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−26.9703			−0.899509
$900$		0			0
$901$	2.54465			0.0847745
$902$		0			0
$903$		0			0
$904$		0			0
$905$	17.9398	−	36.1468i	0.596340	−	1.20156i
$906$		0			0
$907$	−3.14029	−	3.14029i	−0.104272	−	0.104272i	0.653046	−	0.757318i	$-0.273491\pi$
$907$	−3.14029	−	3.14029i	−0.104272	−	0.104272i	−0.757318	+	0.653046i	$0.773491\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			26.2589i			0.869997i	0.900431	+	0.434998i	$0.143251\pi$
$911$			26.2589i			0.869997i	−0.900431	+	0.434998i	$0.856749\pi$
$912$		0			0
$913$	8.85631	−	8.85631i	0.293101	−	0.293101i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	50.3895	−	50.3895i	1.66401	−	1.66401i
$918$		0			0
$919$			30.1612i			0.994927i	0.867485	+	0.497464i	$0.165735\pi$
$919$			30.1612i			0.994927i	−0.867485	+	0.497464i	$0.834265\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.26439	−	4.26439i	−0.140364	−	0.140364i
$924$		0			0
$925$	2.70280	−	19.7547i	0.0888677	−	0.649530i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−59.8424			−1.96337			−0.981683	−	0.190521i	$-0.938982\pi$
$929$	−59.8424			−1.96337			−0.981683	+	0.190521i	$0.938982\pi$
$930$		0			0
$931$	31.3153			1.02632
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−73.9972	+	24.9094i	−2.41997	+	0.814626i
$936$		0			0
$937$	−13.5641	−	13.5641i	−0.443121	−	0.443121i	0.449939	−	0.893059i	$-0.351446\pi$
$937$	−13.5641	−	13.5641i	−0.443121	−	0.443121i	−0.893059	+	0.449939i	$0.851446\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			13.3231i			0.434322i	0.976136	+	0.217161i	$0.0696796\pi$
$941$			13.3231i			0.434322i	−0.976136	+	0.217161i	$0.930320\pi$
$942$		0			0
$943$	6.37990	−	6.37990i	0.207758	−	0.207758i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−34.0173	+	34.0173i	−1.10541	+	1.10541i	−0.111669	+	0.993745i	$0.535620\pi$
$947$	−34.0173	+	34.0173i	−1.10541	+	1.10541i	−0.993745	+	0.111669i	$0.964380\pi$
$948$		0			0
$949$			2.19739i			0.0713302i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−22.4578	−	22.4578i	−0.727480	−	0.727480i	0.242637	−	0.970117i	$-0.421988\pi$
$953$	−22.4578	−	22.4578i	−0.727480	−	0.727480i	−0.970117	+	0.242637i	$0.921988\pi$
$954$		0			0
$955$	29.9484	+	14.8635i	0.969106	+	0.480971i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	72.0133			2.32543
$960$		0			0
$961$	13.7544			0.443692
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16.2054	−	48.1405i	−0.521670	−	1.54970i
$966$		0			0
$967$	22.8718	+	22.8718i	0.735507	+	0.735507i	0.971705	−	0.236198i	$-0.0759013\pi$
$967$	22.8718	+	22.8718i	0.735507	+	0.735507i	−0.236198	+	0.971705i	$0.575901\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			15.0815i			0.483989i	0.970278	+	0.241994i	$0.0778016\pi$
$971$			15.0815i			0.483989i	−0.970278	+	0.241994i	$0.922198\pi$
$972$		0			0
$973$	−47.3383	+	47.3383i	−1.51759	+	1.51759i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.5729	−	14.5729i	0.466229	−	0.466229i	−0.434462	−	0.900690i	$-0.643061\pi$
$977$	14.5729	−	14.5729i	0.466229	−	0.466229i	0.900690	+	0.434462i	$0.143061\pi$
$978$		0			0
$979$		−	64.3240i		−	2.05580i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	28.1838	+	28.1838i	0.898924	+	0.898924i	0.995341	−	0.0964169i	$-0.0307382\pi$
$983$	28.1838	+	28.1838i	0.898924	+	0.898924i	−0.0964169	+	0.995341i	$0.530738\pi$
$984$		0			0
$985$	−2.60579	−	7.74089i	−0.0830275	−	0.246645i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.32982			−0.0740839
$990$		0			0
$991$	−55.4878			−1.76263			−0.881314	−	0.472531i	$-0.843340\pi$
$991$	−55.4878			−1.76263			−0.881314	+	0.472531i	$0.843340\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−26.9625	−	13.3816i	−0.854770	−	0.424226i
$996$		0			0
$997$	−2.84348	−	2.84348i	−0.0900539	−	0.0900539i	0.660645	−	0.750699i	$-0.270283\pi$
$997$	−2.84348	−	2.84348i	−0.0900539	−	0.0900539i	−0.750699	+	0.660645i	$0.770283\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.s.b.737.9	✓	44
3.2	odd	2	inner	4140.2.s.b.737.14	yes	44
5.3	odd	4	inner	4140.2.s.b.2393.14	yes	44
15.8	even	4	inner	4140.2.s.b.2393.9	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.b.737.9	✓	44	1.1	even	1	trivial
4140.2.s.b.737.14	yes	44	3.2	odd	2	inner
4140.2.s.b.2393.9	yes	44	15.8	even	4	inner
4140.2.s.b.2393.14	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)

\(f(q)\)	\(=\)	\(q+(-0.994073 + 2.00295i) q^{5} +(-2.91813 - 2.91813i) q^{7} +O(q^{10})\)	\(q+(-0.994073 + 2.00295i) q^{5} +(-2.91813 - 2.91813i) q^{7} -4.72853i q^{11} +(0.839370 - 0.839370i) q^{13} +(-5.22154 + 5.22154i) q^{17} -3.12186i q^{19} +(-0.707107 - 0.707107i) q^{23} +(-3.02364 - 3.98216i) q^{25} +4.03151 q^{29} -6.68988 q^{31} +(8.74572 - 2.94404i) q^{35} +(2.81976 + 2.81976i) q^{37} +9.02254i q^{41} +(1.64743 - 1.64743i) q^{43} +(-1.14788 + 1.14788i) q^{47} +10.0310i q^{49} +(-0.243668 - 0.243668i) q^{53} +(9.47101 + 4.70050i) q^{55} +3.07448 q^{59} +8.53710 q^{61} +(0.846823 + 2.51561i) q^{65} +(-4.44459 - 4.44459i) q^{67} -5.08047i q^{71} +(-1.30895 + 1.30895i) q^{73} +(-13.7985 + 13.7985i) q^{77} -0.837744i q^{79} +(1.87295 + 1.87295i) q^{83} +(-5.26791 - 15.6491i) q^{85} +13.6034 q^{89} -4.89878 q^{91} +(6.25293 + 3.10335i) q^{95} +(5.02715 + 5.02715i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{7}+O(q^{10}) \) 44 * q - 4 * q^7	\( 44 q - 4 q^{7} - 4 q^{13} - 24 q^{25} + 32 q^{31} + 40 q^{37} - 8 q^{43} - 24 q^{55} + 64 q^{61} + 12 q^{67} - 84 q^{73} - 104 q^{85} - 48 q^{91} + 44 q^{97}+O(q^{100}) \) 44 * q - 4 * q^7 - 4 * q^13 - 24 * q^25 + 32 * q^31 + 40 * q^37 - 8 * q^43 - 24 * q^55 + 64 * q^61 + 12 * q^67 - 84 * q^73 - 104 * q^85 - 48 * q^91 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more