LMFDB - Embedded newform 2940.2.k.f.589.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2940,2,Mod(589,2940)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2940.589"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2940.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-2,0,0,0,-8,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.4760181943$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.31678304256.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 8x^{5} + 32x^{4} - 20x^{3} + 8x^{2} + 8x + 4 $ x^8 - 2x^7 + 2x^6 + 8x^5 + 32x^4 - 20x^3 + 8x^2 + 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			589.3
Root			$-0.285451 - 0.285451i$ of defining polynomial
Character	$\chi$	$=$	2940.589
Dual form			2940.2.k.f.589.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(0.285451 - 2.21777i) q^{5} -1.00000 q^{9} +2.93232 q^{11} +5.76936i q^{13} +(-2.21777 - 0.285451i) q^{15} +0.237092i q^{17} -4.74032 q^{19} -6.76936i q^{23} +(-4.83704 - 1.26613i) q^{25} +1.00000i q^{27} -6.03549 q^{29} -10.5387 q^{31} -2.93232i q^{33} -3.07413i q^{37} +5.76936 q^{39} +1.06768 q^{41} -9.79697i q^{43} +(-0.285451 + 2.21777i) q^{45} +3.71271i q^{47} +0.237092 q^{51} -11.0064i q^{53} +(0.837035 - 6.50322i) q^{55} +4.74032i q^{57} +9.31265 q^{59} -8.13722 q^{61} +(12.7951 + 1.64687i) q^{65} -2.24354i q^{67} -6.76936 q^{69} -11.8421 q^{71} -3.79697i q^{73} +(-1.26613 + 4.83704i) q^{75} -0.0174780 q^{79} +1.00000 q^{81} -7.30162i q^{83} +(0.525816 + 0.0676782i) q^{85} +6.03549i q^{87} -17.9517 q^{89} +10.5387i q^{93} +(-1.35313 + 10.5129i) q^{95} +2.31122i q^{97} -2.93232 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} - 8 q^{9} + 8 q^{11} + 2 q^{15} - 8 q^{19} - 12 q^{25} + 12 q^{29} + 4 q^{39} + 24 q^{41} + 2 q^{45} - 4 q^{51} - 20 q^{55} + 28 q^{59} + 32 q^{61} + 26 q^{65} - 12 q^{69} - 28 q^{71} + 8 q^{75}+ \cdots - 8 q^{99}+O(q^{100}) $ 8 * q - 2 * q^5 - 8 * q^9 + 8 * q^11 + 2 * q^15 - 8 * q^19 - 12 * q^25 + 12 * q^29 + 4 * q^39 + 24 * q^41 + 2 * q^45 - 4 * q^51 - 20 * q^55 + 28 * q^59 + 32 * q^61 + 26 * q^65 - 12 * q^69 - 28 * q^71 + 8 * q^75 + 16 * q^79 + 8 * q^81 + 16 * q^85 + 16 * q^89 - 22 * q^95 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times$.

$n$	$1081$	$1177$	$1471$	$1961$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	0.285451	−	2.21777i	0.127658	−	0.991818i
$5$	0.285451	−	2.21777i	0.127658	−	0.991818i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	2.93232			0.884128			0.442064	−	0.896983i	$-0.354246\pi$
$11$	2.93232			0.884128			0.442064	+	0.896983i	$0.354246\pi$
$12$		0			0
$13$			5.76936i			1.60013i	0.599912	+	0.800066i	$0.295202\pi$
$13$			5.76936i			1.60013i	−0.599912	+	0.800066i	$0.704798\pi$
$14$		0			0
$15$	−2.21777	−	0.285451i	−0.572627	−	0.0737032i
$16$		0			0
$17$			0.237092i			0.0575032i	0.999587	+	0.0287516i	$0.00915319\pi$
$17$			0.237092i			0.0575032i	−0.999587	+	0.0287516i	$0.990847\pi$
$18$		0			0
$19$	−4.74032			−1.08750			−0.543752	−	0.839246i	$-0.682997\pi$
$19$	−4.74032			−1.08750			−0.543752	+	0.839246i	$0.682997\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.76936i		−	1.41151i	−0.708457	−	0.705754i	$-0.750608\pi$
$23$		−	6.76936i		−	1.41151i	0.708457	−	0.705754i	$-0.249392\pi$
$24$		0			0
$25$	−4.83704	−	1.26613i	−0.967407	−	0.253226i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−6.03549			−1.12076			−0.560381	−	0.828235i	$-0.689345\pi$
$29$	−6.03549			−1.12076			−0.560381	+	0.828235i	$0.689345\pi$
$30$		0			0
$31$	−10.5387			−1.89281			−0.946404	−	0.322984i	$-0.895314\pi$
$31$	−10.5387			−1.89281			−0.946404	+	0.322984i	$0.895314\pi$
$32$		0			0
$33$		−	2.93232i		−	0.510452i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	3.07413i		−	0.505383i	−0.967547	−	0.252692i	$-0.918684\pi$
$37$		−	3.07413i		−	0.505383i	0.967547	−	0.252692i	$-0.0813158\pi$
$38$		0			0
$39$	5.76936			0.923836
$40$		0			0
$41$	1.06768			0.166743			0.0833717	−	0.996519i	$-0.473431\pi$
$41$	1.06768			0.166743			0.0833717	+	0.996519i	$0.473431\pi$
$42$		0			0
$43$		−	9.79697i		−	1.49402i	−0.664811	−	0.747012i	$-0.731488\pi$
$43$		−	9.79697i		−	1.49402i	0.664811	−	0.747012i	$-0.268512\pi$
$44$		0			0
$45$	−0.285451	+	2.21777i	−0.0425526	+	0.330606i
$46$		0			0
$47$			3.71271i			0.541554i	0.962642	+	0.270777i	$0.0872806\pi$
$47$			3.71271i			0.541554i	−0.962642	+	0.270777i	$0.912719\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	0.237092			0.0331995
$52$		0			0
$53$		−	11.0064i		−	1.51185i	−0.654657	−	0.755926i	$-0.727187\pi$
$53$		−	11.0064i		−	1.51185i	0.654657	−	0.755926i	$-0.272813\pi$
$54$		0			0
$55$	0.837035	−	6.50322i	0.112866	−	0.876895i
$56$		0			0
$57$			4.74032i			0.627870i
$58$		0			0
$59$	9.31265			1.21240			0.606202	−	0.795311i	$-0.292692\pi$
$59$	9.31265			1.21240			0.606202	+	0.795311i	$0.292692\pi$
$60$		0			0
$61$	−8.13722			−1.04186			−0.520932	−	0.853598i	$-0.674416\pi$
$61$	−8.13722			−1.04186			−0.520932	+	0.853598i	$0.674416\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	12.7951	+	1.64687i	1.58704	+	0.204269i
$66$		0			0
$67$		−	2.24354i		−	0.274092i	−0.990565	−	0.137046i	$-0.956239\pi$
$67$		−	2.24354i		−	0.274092i	0.990565	−	0.137046i	$-0.0437608\pi$
$68$		0			0
$69$	−6.76936			−0.814935
$70$		0			0
$71$	−11.8421			−1.40539			−0.702696	−	0.711490i	$-0.748021\pi$
$71$	−11.8421			−1.40539			−0.702696	+	0.711490i	$0.748021\pi$
$72$		0			0
$73$		−	3.79697i		−	0.444401i	−0.975001	−	0.222201i	$-0.928676\pi$
$73$		−	3.79697i		−	0.444401i	0.975001	−	0.222201i	$-0.0713240\pi$
$74$		0			0
$75$	−1.26613	+	4.83704i	−0.146200	+	0.558533i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.0174780			−0.00196643			−0.000983215	−	1.00000i	$-0.500313\pi$
$79$	−0.0174780			−0.00196643			−0.000983215		1.00000i	$0.500313\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	7.30162i		−	0.801457i	−0.916197	−	0.400729i	$-0.868757\pi$
$83$		−	7.30162i		−	0.801457i	0.916197	−	0.400729i	$-0.131243\pi$
$84$		0			0
$85$	0.525816	+	0.0676782i	0.0570328	+	0.00734073i
$86$		0			0
$87$			6.03549i			0.647072i
$88$		0			0
$89$	−17.9517			−1.90287			−0.951437	−	0.307845i	$-0.900392\pi$
$89$	−17.9517			−1.90287			−0.951437	+	0.307845i	$0.900392\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			10.5387i			1.09281i
$94$		0			0
$95$	−1.35313	+	10.5129i	−0.138828	+	1.07861i
$96$		0			0
$97$			2.31122i			0.234669i	0.993092	+	0.117334i	$0.0374349\pi$
$97$			2.31122i			0.234669i	−0.993092	+	0.117334i	$0.962565\pi$
$98$		0			0
$99$	−2.93232			−0.294709
$100$		0			0
$101$	−7.84205			−0.780313			−0.390157	−	0.920748i	$-0.627579\pi$
$101$	−7.84205			−0.780313			−0.390157	+	0.920748i	$0.627579\pi$
$102$		0			0
$103$			4.08058i			0.402071i	0.979584	+	0.201036i	$0.0644307\pi$
$103$			4.08058i			0.402071i	−0.979584	+	0.201036i	$0.935569\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			18.3467i			1.77364i	0.462112	+	0.886822i	$0.347092\pi$
$107$			18.3467i			1.77364i	−0.462112	+	0.886822i	$0.652908\pi$
$108$		0			0
$109$	10.4144			0.997517			0.498759	−	0.866741i	$-0.333789\pi$
$109$	10.4144			0.997517			0.498759	+	0.866741i	$0.333789\pi$
$110$		0			0
$111$	−3.07413			−0.291783
$112$		0			0
$113$			2.14825i			0.202091i	0.994882	+	0.101045i	$0.0322187\pi$
$113$			2.14825i			0.202091i	−0.994882	+	0.101045i	$0.967781\pi$
$114$		0			0
$115$	−15.0129	−	1.93232i	−1.39996	−	0.180190i
$116$		0			0
$117$		−	5.76936i		−	0.533377i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.40149			−0.218317
$122$		0			0
$123$		−	1.06768i		−	0.0962693i
$124$		0			0
$125$	−4.18873	+	10.3660i	−0.374652	+	0.927166i
$126$		0			0
$127$		−	1.82457i		−	0.161905i	−0.996718	−	0.0809524i	$-0.974204\pi$
$127$		−	1.82457i		−	0.161905i	0.996718	−	0.0809524i	$-0.0257962\pi$
$128$		0			0
$129$	−9.79697			−0.862575
$130$		0			0
$131$	2.70626			0.236447			0.118223	−	0.992987i	$-0.462280\pi$
$131$	2.70626			0.236447			0.118223	+	0.992987i	$0.462280\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.21777	+	0.285451i	0.190876	+	0.0245677i
$136$		0			0
$137$			10.0050i			0.854786i	0.904066	+	0.427393i	$0.140568\pi$
$137$			10.0050i			0.854786i	−0.904066	+	0.427393i	$0.859432\pi$
$138$		0			0
$139$	2.07269			0.175804			0.0879018	−	0.996129i	$-0.471984\pi$
$139$	2.07269			0.175804			0.0879018	+	0.996129i	$0.471984\pi$
$140$		0			0
$141$	3.71271			0.312666
$142$		0			0
$143$			16.9176i			1.41472i
$144$		0			0
$145$	−1.72284	+	13.3853i	−0.143074	+	1.11159i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.10317			0.0903750			0.0451875	−	0.998979i	$-0.485611\pi$
$149$	1.10317			0.0903750			0.0451875	+	0.998979i	$0.485611\pi$
$150$		0			0
$151$	10.4632			0.851479			0.425740	−	0.904846i	$-0.360014\pi$
$151$	10.4632			0.851479			0.425740	+	0.904846i	$0.360014\pi$
$152$		0			0
$153$		−	0.237092i		−	0.0191677i
$154$		0			0
$155$	−3.00829	+	23.3725i	−0.241632	+	1.87732i
$156$		0			0
$157$		−	9.06167i		−	0.723200i	−0.932333	−	0.361600i	$-0.882231\pi$
$157$		−	9.06167i		−	0.723200i	0.932333	−	0.361600i	$-0.117769\pi$
$158$		0			0
$159$	−11.0064			−0.872868
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	8.33883i		−	0.653147i	−0.945172	−	0.326574i	$-0.894106\pi$
$163$		−	8.33883i		−	0.653147i	0.945172	−	0.326574i	$-0.105894\pi$
$164$		0			0
$165$	−6.50322	−	0.837035i	−0.506275	−	0.0651631i
$166$		0			0
$167$		−	10.2886i		−	0.796158i	−0.917351	−	0.398079i	$-0.869677\pi$
$167$		−	10.2886i		−	0.796158i	0.917351	−	0.398079i	$-0.130323\pi$
$168$		0			0
$169$	−20.2855			−1.56042
$170$		0			0
$171$	4.74032			0.362501
$172$		0			0
$173$		−	17.2937i		−	1.31482i	−0.753534	−	0.657409i	$-0.771652\pi$
$173$		−	17.2937i		−	1.31482i	0.753534	−	0.657409i	$-0.228348\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	9.31265i		−	0.699982i
$178$		0			0
$179$	11.0193			0.823625			0.411812	−	0.911269i	$-0.364896\pi$
$179$	11.0193			0.823625			0.411812	+	0.911269i	$0.364896\pi$
$180$		0			0
$181$	4.53685			0.337221			0.168611	−	0.985683i	$-0.446072\pi$
$181$	4.53685			0.337221			0.168611	+	0.985683i	$0.446072\pi$
$182$		0			0
$183$			8.13722i			0.601521i
$184$		0			0
$185$	−6.81772	−	0.877514i	−0.501248	−	0.0645161i
$186$		0			0
$187$			0.695230i			0.0508402i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.68052			0.628100			0.314050	−	0.949406i	$-0.398314\pi$
$191$	8.68052			0.628100			0.314050	+	0.949406i	$0.398314\pi$
$192$		0			0
$193$		−	7.06123i		−	0.508278i	−0.967168	−	0.254139i	$-0.918208\pi$
$193$		−	7.06123i		−	0.508278i	0.967168	−	0.254139i	$-0.0817921\pi$
$194$		0			0
$195$	1.64687	−	12.7951i	0.117935	−	0.916278i
$196$		0			0
$197$			3.58891i			0.255700i	0.991794	+	0.127850i	$0.0408075\pi$
$197$			3.58891i			0.255700i	−0.991794	+	0.127850i	$0.959192\pi$
$198$		0			0
$199$	2.98252			0.211425			0.105713	−	0.994397i	$-0.466288\pi$
$199$	2.98252			0.211425			0.105713	+	0.994397i	$0.466288\pi$
$200$		0			0
$201$	−2.24354			−0.158247
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.304770	−	2.36787i	0.0212861	−	0.165379i
$206$		0			0
$207$			6.76936i			0.470503i
$208$		0			0
$209$	−13.9001			−0.961492
$210$		0			0
$211$	−21.4273			−1.47512			−0.737558	−	0.675284i	$-0.764021\pi$
$211$	−21.4273			−1.47512			−0.737558	+	0.675284i	$0.764021\pi$
$212$		0			0
$213$			11.8421i			0.811404i
$214$		0			0
$215$	−21.7274	−	2.79656i	−1.48180	−	0.190724i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−3.79697			−0.256575
$220$		0			0
$221$	−1.36787			−0.0920128
$222$		0			0
$223$			16.3969i			1.09802i	0.835816	+	0.549009i	$0.184995\pi$
$223$			16.3969i			1.09802i	−0.835816	+	0.549009i	$0.815005\pi$
$224$		0			0
$225$	4.83704	+	1.26613i	0.322469	+	0.0844088i
$226$		0			0
$227$		−	9.53370i		−	0.632774i	−0.948630	−	0.316387i	$-0.897530\pi$
$227$		−	9.53370i		−	0.632774i	0.948630	−	0.316387i	$-0.102470\pi$
$228$		0			0
$229$	−18.1197			−1.19739			−0.598693	−	0.800978i	$-0.704313\pi$
$229$	−18.1197			−1.19739			−0.598693	+	0.800978i	$0.704313\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	12.0544i		−	0.789710i	−0.918743	−	0.394855i	$-0.870795\pi$
$233$		−	12.0544i		−	0.789710i	0.918743	−	0.394855i	$-0.129205\pi$
$234$		0			0
$235$	8.23394	+	1.05980i	0.537123	+	0.0691336i
$236$		0			0
$237$			0.0174780i			0.00113532i
$238$		0			0
$239$	21.3868			1.38340			0.691698	−	0.722187i	$-0.256863\pi$
$239$	21.3868			1.38340			0.691698	+	0.722187i	$0.256863\pi$
$240$		0			0
$241$	17.1889			1.10723			0.553616	−	0.832772i	$-0.313248\pi$
$241$	17.1889			1.10723			0.553616	+	0.832772i	$0.313248\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	27.3486i		−	1.74015i
$248$		0			0
$249$	−7.30162			−0.462721
$250$		0			0
$251$	23.1939			1.46398			0.731992	−	0.681313i	$-0.238591\pi$
$251$	23.1939			1.46398			0.731992	+	0.681313i	$0.238591\pi$
$252$		0			0
$253$		−	19.8499i		−	1.24795i
$254$		0			0
$255$	0.0676782	−	0.525816i	0.00423817	−	0.0329279i
$256$		0			0
$257$			26.1561i			1.63157i	0.578352	+	0.815787i	$0.303696\pi$
$257$			26.1561i			1.63157i	−0.578352	+	0.815787i	$0.696304\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	6.03549			0.373587
$262$		0			0
$263$		−	7.69981i		−	0.474791i	−0.971413	−	0.237395i	$-0.923706\pi$
$263$		−	7.69981i		−	0.474791i	0.971413	−	0.237395i	$-0.0762937\pi$
$264$		0			0
$265$	−24.4098	−	3.14181i	−1.49948	−	0.193000i
$266$		0			0
$267$			17.9517i			1.09862i
$268$		0			0
$269$	−25.0388			−1.52664			−0.763321	−	0.646019i	$-0.776432\pi$
$269$	−25.0388			−1.52664			−0.763321	+	0.646019i	$0.776432\pi$
$270$		0			0
$271$	19.0885			1.15954			0.579771	−	0.814780i	$-0.303142\pi$
$271$	19.0885			1.15954			0.579771	+	0.814780i	$0.303142\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−14.1837	−	3.71271i	−0.855312	−	0.223885i
$276$		0			0
$277$			12.1387i			0.729341i	0.931137	+	0.364671i	$0.118818\pi$
$277$			12.1387i			0.729341i	−0.931137	+	0.364671i	$0.881182\pi$
$278$		0			0
$279$	10.5387			0.630936
$280$		0			0
$281$	−6.34537			−0.378533			−0.189267	−	0.981926i	$-0.560611\pi$
$281$	−6.34537			−0.378533			−0.189267	+	0.981926i	$0.560611\pi$
$282$		0			0
$283$		−	13.6193i		−	0.809583i	−0.914409	−	0.404791i	$-0.867344\pi$
$283$		−	13.6193i		−	0.809583i	0.914409	−	0.404791i	$-0.132656\pi$
$284$		0			0
$285$	10.5129	+	1.35313i	0.622733	+	0.0801525i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.9438			0.996693
$290$		0			0
$291$	2.31122			0.135486
$292$		0			0
$293$			15.5221i			0.906813i	0.891304	+	0.453406i	$0.149791\pi$
$293$			15.5221i			0.906813i	−0.891304	+	0.453406i	$0.850209\pi$
$294$		0			0
$295$	2.65831	−	20.6533i	0.154773	−	1.20248i
$296$		0			0
$297$			2.93232i			0.170151i
$298$		0			0
$299$	39.0548			2.25860
$300$		0			0
$301$		0			0
$302$		0			0
$303$			7.84205i			0.450514i
$304$		0			0
$305$	−2.32278	+	18.0465i	−0.133002	+	1.03334i
$306$		0			0
$307$			13.4406i			0.767093i	0.923521	+	0.383547i	$0.125297\pi$
$307$			13.4406i			0.767093i	−0.923521	+	0.383547i	$0.874703\pi$
$308$		0			0
$309$	4.08058			0.232136
$310$		0			0
$311$	2.45159			0.139017			0.0695085	−	0.997581i	$-0.477857\pi$
$311$	2.45159			0.139017			0.0695085	+	0.997581i	$0.477857\pi$
$312$		0			0
$313$		−	9.91942i		−	0.560679i	−0.959901	−	0.280340i	$-0.909553\pi$
$313$		−	9.91942i		−	0.560679i	0.959901	−	0.280340i	$-0.0904470\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			25.9950i			1.46003i	0.683433	+	0.730013i	$0.260486\pi$
$317$			25.9950i			1.46003i	−0.683433	+	0.730013i	$0.739514\pi$
$318$		0			0
$319$	−17.6980			−0.990898
$320$		0			0
$321$	18.3467			1.02401
$322$		0			0
$323$		−	1.12389i		−	0.0625350i
$324$		0			0
$325$	7.30477	−	27.9066i	0.405196	−	1.54798i
$326$		0			0
$327$		−	10.4144i		−	0.575917i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	7.53226			0.414011			0.207005	−	0.978340i	$-0.433628\pi$
$331$	7.53226			0.414011			0.207005	+	0.978340i	$0.433628\pi$
$332$		0			0
$333$			3.07413i			0.168461i
$334$		0			0
$335$	−4.97566	−	0.640422i	−0.271850	−	0.0349900i
$336$		0			0
$337$		−	15.7050i		−	0.855505i	−0.903896	−	0.427752i	$-0.859306\pi$
$337$		−	15.7050i		−	0.855505i	0.903896	−	0.427752i	$-0.140694\pi$
$338$		0			0
$339$	2.14825			0.116677
$340$		0			0
$341$	−30.9029			−1.67349
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−1.93232	+	15.0129i	−0.104033	+	0.808267i
$346$		0			0
$347$		−	9.21278i		−	0.494568i	−0.968943	−	0.247284i	$-0.920462\pi$
$347$		−	9.21278i		−	0.494568i	0.968943	−	0.247284i	$-0.0795381\pi$
$348$		0			0
$349$	−21.1518			−1.13223			−0.566116	−	0.824326i	$-0.691555\pi$
$349$	−21.1518			−1.13223			−0.566116	+	0.824326i	$0.691555\pi$
$350$		0			0
$351$	−5.76936			−0.307945
$352$		0			0
$353$			14.9794i			0.797272i	0.917109	+	0.398636i	$0.130516\pi$
$353$			14.9794i			0.797272i	−0.917109	+	0.398636i	$0.869484\pi$
$354$		0			0
$355$	−3.38033	+	26.2630i	−0.179409	+	1.39389i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−20.9647			−1.10647			−0.553236	−	0.833024i	$-0.686607\pi$
$359$	−20.9647			−1.10647			−0.553236	+	0.833024i	$0.686607\pi$
$360$		0			0
$361$	3.47060			0.182663
$362$		0			0
$363$			2.40149i			0.126045i
$364$		0			0
$365$	−8.42081	−	1.08385i	−0.440765	−	0.0567312i
$366$		0			0
$367$		−	19.1146i		−	0.997776i	−0.866667	−	0.498888i	$-0.833742\pi$
$367$		−	19.1146i		−	0.997776i	0.866667	−	0.498888i	$-0.166258\pi$
$368$		0			0
$369$	−1.06768			−0.0555811
$370$		0			0
$371$		0			0
$372$		0			0
$373$			6.51425i			0.337295i	0.985676	+	0.168648i	$0.0539400\pi$
$373$			6.51425i			0.337295i	−0.985676	+	0.168648i	$0.946060\pi$
$374$		0			0
$375$	10.3660	+	4.18873i	0.535299	+	0.216305i
$376$		0			0
$377$		−	34.8209i		−	1.79337i
$378$		0			0
$379$	−14.4254			−0.740984			−0.370492	−	0.928836i	$-0.620811\pi$
$379$	−14.4254			−0.740984			−0.370492	+	0.928836i	$0.620811\pi$
$380$		0			0
$381$	−1.82457			−0.0934758
$382$		0			0
$383$		−	25.2901i		−	1.29226i	−0.763227	−	0.646131i	$-0.776386\pi$
$383$		−	25.2901i		−	1.29226i	0.763227	−	0.646131i	$-0.223614\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			9.79697i			0.498008i
$388$		0			0
$389$	−26.9968			−1.36879			−0.684395	−	0.729112i	$-0.739934\pi$
$389$	−26.9968			−1.36879			−0.684395	+	0.729112i	$0.739934\pi$
$390$		0			0
$391$	1.60496			0.0811663
$392$		0			0
$393$		−	2.70626i		−	0.136513i
$394$		0			0
$395$	−0.00498912	+	0.0387622i	−0.000251030	+	0.00195034i
$396$		0			0
$397$			31.6164i			1.58678i	0.608712	+	0.793391i	$0.291687\pi$
$397$			31.6164i			1.58678i	−0.608712	+	0.793391i	$0.708313\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.4899			−0.623718			−0.311859	−	0.950128i	$-0.600952\pi$
$401$	−12.4899			−0.623718			−0.311859	+	0.950128i	$0.600952\pi$
$402$		0			0
$403$		−	60.8016i		−	3.02874i
$404$		0			0
$405$	0.285451	−	2.21777i	0.0141842	−	0.110202i
$406$		0			0
$407$		−	9.01433i		−	0.446824i
$408$		0			0
$409$	−12.5608			−0.621090			−0.310545	−	0.950559i	$-0.600512\pi$
$409$	−12.5608			−0.621090			−0.310545	+	0.950559i	$0.600512\pi$
$410$		0			0
$411$	10.0050			0.493511
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−16.1933	−	2.08426i	−0.794900	−	0.102312i
$416$		0			0
$417$		−	2.07269i		−	0.101500i
$418$		0			0
$419$	17.6520			0.862357			0.431179	−	0.902267i	$-0.358098\pi$
$419$	17.6520			0.862357			0.431179	+	0.902267i	$0.358098\pi$
$420$		0			0
$421$	29.2634			1.42621			0.713106	−	0.701056i	$-0.247288\pi$
$421$	29.2634			1.42621			0.713106	+	0.701056i	$0.247288\pi$
$422$		0			0
$423$		−	3.71271i		−	0.180518i
$424$		0			0
$425$	0.300190	−	1.14682i	0.0145613	−	0.0556290i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	16.9176			0.816790
$430$		0			0
$431$	28.9226			1.39315			0.696577	−	0.717482i	$-0.254705\pi$
$431$	28.9226			1.39315			0.696577	+	0.717482i	$0.254705\pi$
$432$		0			0
$433$		−	10.0338i		−	0.482192i	−0.970501	−	0.241096i	$-0.922493\pi$
$433$		−	10.0338i		−	0.482192i	0.970501	−	0.241096i	$-0.0775069\pi$
$434$		0			0
$435$	13.3853	+	1.72284i	0.641778	+	0.0826038i
$436$		0			0
$437$			32.0889i			1.53502i
$438$		0			0
$439$	16.8149			0.802530			0.401265	−	0.915962i	$-0.368571\pi$
$439$	16.8149			0.802530			0.401265	+	0.915962i	$0.368571\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			10.1806i			0.483695i	0.970314	+	0.241847i	$0.0777533\pi$
$443$			10.1806i			0.483695i	−0.970314	+	0.241847i	$0.922247\pi$
$444$		0			0
$445$	−5.12433	+	39.8127i	−0.242916	+	1.88730i
$446$		0			0
$447$		−	1.10317i		−	0.0521780i
$448$		0			0
$449$	33.9937			1.60426			0.802131	−	0.597148i	$-0.203700\pi$
$449$	33.9937			1.60426			0.802131	+	0.597148i	$0.203700\pi$
$450$		0			0
$451$	3.13078			0.147422
$452$		0			0
$453$		−	10.4632i		−	0.491602i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	23.7823i		−	1.11249i	−0.831019	−	0.556243i	$-0.812242\pi$
$457$		−	23.7823i		−	1.11249i	0.831019	−	0.556243i	$-0.187758\pi$
$458$		0			0
$459$	−0.237092			−0.0110665
$460$		0			0
$461$	−35.2649			−1.64245			−0.821224	−	0.570606i	$-0.806708\pi$
$461$	−35.2649			−1.64245			−0.821224	+	0.570606i	$0.806708\pi$
$462$		0			0
$463$		−	38.6505i		−	1.79624i	−0.439750	−	0.898120i	$-0.644933\pi$
$463$		−	38.6505i		−	1.79624i	0.439750	−	0.898120i	$-0.355067\pi$
$464$		0			0
$465$	23.3725	+	3.00829i	1.08387	+	0.139506i
$466$		0			0
$467$			8.85819i			0.409908i	0.978772	+	0.204954i	$0.0657045\pi$
$467$			8.85819i			0.409908i	−0.978772	+	0.204954i	$0.934295\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−9.06167			−0.417540
$472$		0			0
$473$		−	28.7279i		−	1.32091i
$474$		0			0
$475$	22.9291	+	6.00187i	1.05206	+	0.275385i
$476$		0			0
$477$			11.0064i			0.503951i
$478$		0			0
$479$	8.53595			0.390017			0.195009	−	0.980802i	$-0.437526\pi$
$479$	8.53595			0.390017			0.195009	+	0.980802i	$0.437526\pi$
$480$		0			0
$481$	17.7357			0.808680
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.12576	+	0.659740i	0.232749	+	0.0299573i
$486$		0			0
$487$			1.96909i			0.0892280i	0.999004	+	0.0446140i	$0.0142058\pi$
$487$			1.96909i			0.0892280i	−0.999004	+	0.0446140i	$0.985794\pi$
$488$		0			0
$489$	−8.33883			−0.377095
$490$		0			0
$491$	21.4549			0.968246			0.484123	−	0.875000i	$-0.339139\pi$
$491$	21.4549			0.968246			0.484123	+	0.875000i	$0.339139\pi$
$492$		0			0
$493$		−	1.43097i		−	0.0644475i
$494$		0			0
$495$	−0.837035	+	6.50322i	−0.0376219	+	0.292298i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	14.9448			0.669020			0.334510	−	0.942392i	$-0.391429\pi$
$499$	14.9448			0.669020			0.334510	+	0.942392i	$0.391429\pi$
$500$		0			0
$501$	−10.2886			−0.459662
$502$		0			0
$503$			10.9715i			0.489195i	0.969625	+	0.244597i	$0.0786558\pi$
$503$			10.9715i			0.489195i	−0.969625	+	0.244597i	$0.921344\pi$
$504$		0			0
$505$	−2.23852	+	17.3919i	−0.0996130	+	0.773929i
$506$		0			0
$507$			20.2855i			0.900910i
$508$		0			0
$509$	37.2099			1.64930			0.824650	−	0.565643i	$-0.191372\pi$
$509$	37.2099			1.64930			0.824650	+	0.565643i	$0.191372\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	4.74032i		−	0.209290i
$514$		0			0
$515$	9.04979	+	1.16481i	0.398781	+	0.0513275i
$516$		0			0
$517$			10.8869i			0.478803i
$518$		0			0
$519$	−17.2937			−0.759111
$520$		0			0
$521$	16.5000			0.722877			0.361439	−	0.932396i	$-0.382286\pi$
$521$	16.5000			0.722877			0.361439	+	0.932396i	$0.382286\pi$
$522$		0			0
$523$		−	42.3495i		−	1.85181i	−0.377752	−	0.925907i	$-0.623303\pi$
$523$		−	42.3495i		−	1.85181i	0.377752	−	0.925907i	$-0.376697\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	2.49864i		−	0.108843i
$528$		0			0
$529$	−22.8242			−0.992356
$530$		0			0
$531$	−9.31265			−0.404135
$532$		0			0
$533$			6.15982i			0.266811i
$534$		0			0
$535$	40.6888	+	5.23709i	1.75913	+	0.226419i
$536$		0			0
$537$		−	11.0193i		−	0.475520i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−42.3159			−1.81930			−0.909650	−	0.415375i	$-0.863650\pi$
$541$	−42.3159			−1.81930			−0.909650	+	0.415375i	$0.863650\pi$
$542$		0			0
$543$		−	4.53685i		−	0.194695i
$544$		0			0
$545$	2.97280	−	23.0967i	0.127341	−	0.989356i
$546$		0			0
$547$		−	1.34528i		−	0.0575199i	−0.999586	−	0.0287599i	$-0.990844\pi$
$547$		−	1.34528i		−	0.0575199i	0.999586	−	0.0287599i	$-0.00915583\pi$
$548$		0			0
$549$	8.13722			0.347288
$550$		0			0
$551$	28.6101			1.21883
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−0.877514	+	6.81772i	−0.0372484	+	0.289396i
$556$		0			0
$557$			2.98987i			0.126685i	0.997992	+	0.0633424i	$0.0201760\pi$
$557$			2.98987i			0.126685i	−0.997992	+	0.0633424i	$0.979824\pi$
$558$		0			0
$559$	56.5222			2.39063
$560$		0			0
$561$	0.695230			0.0293526
$562$		0			0
$563$			18.2035i			0.767185i	0.923502	+	0.383592i	$0.125313\pi$
$563$			18.2035i			0.767185i	−0.923502	+	0.383592i	$0.874687\pi$
$564$		0			0
$565$	4.76434	+	0.613222i	0.200437	+	0.0257984i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	19.7740			0.828969			0.414484	−	0.910056i	$-0.363962\pi$
$569$	19.7740			0.828969			0.414484	+	0.910056i	$0.363962\pi$
$570$		0			0
$571$	−19.7496			−0.826496			−0.413248	−	0.910618i	$-0.635606\pi$
$571$	−19.7496			−0.826496			−0.413248	+	0.910618i	$0.635606\pi$
$572$		0			0
$573$		−	8.68052i		−	0.362634i
$574$		0			0
$575$	−8.57090	+	32.7436i	−0.357431	+	1.36550i
$576$		0			0
$577$		−	17.0723i		−	0.710730i	−0.934728	−	0.355365i	$-0.884357\pi$
$577$		−	17.0723i		−	0.710730i	0.934728	−	0.355365i	$-0.115643\pi$
$578$		0			0
$579$	−7.06123			−0.293455
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	32.2744i		−	1.33667i
$584$		0			0
$585$	−12.7951	−	1.64687i	−0.529013	−	0.0680897i
$586$		0			0
$587$			17.5308i			0.723575i	0.932261	+	0.361787i	$0.117833\pi$
$587$			17.5308i			0.723575i	−0.932261	+	0.361787i	$0.882167\pi$
$588$		0			0
$589$	49.9568			2.05844
$590$		0			0
$591$	3.58891			0.147628
$592$		0			0
$593$		−	26.1561i		−	1.07410i	−0.843549	−	0.537052i	$-0.819538\pi$
$593$		−	26.1561i		−	1.07410i	0.843549	−	0.537052i	$-0.180462\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	2.98252i		−	0.122066i
$598$		0			0
$599$	−29.9190			−1.22246			−0.611229	−	0.791453i	$-0.709325\pi$
$599$	−29.9190			−1.22246			−0.611229	+	0.791453i	$0.709325\pi$
$600$		0			0
$601$	15.6887			0.639955			0.319977	−	0.947425i	$-0.396325\pi$
$601$	15.6887			0.639955			0.319977	+	0.947425i	$0.396325\pi$
$602$		0			0
$603$			2.24354i			0.0913640i
$604$		0			0
$605$	−0.685508	+	5.32596i	−0.0278699	+	0.216531i
$606$		0			0
$607$		−	34.7464i		−	1.41031i	−0.709052	−	0.705156i	$-0.750877\pi$
$607$		−	34.7464i		−	1.41031i	0.709052	−	0.705156i	$-0.249123\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−21.4199			−0.866558
$612$		0			0
$613$			6.35907i			0.256841i	0.991720	+	0.128420i	$0.0409906\pi$
$613$			6.35907i			0.256841i	−0.991720	+	0.128420i	$0.959009\pi$
$614$		0			0
$615$	−2.36787	−	0.304770i	−0.0954817	−	0.0122895i
$616$		0			0
$617$			25.1778i			1.01362i	0.862057	+	0.506811i	$0.169176\pi$
$617$			25.1778i			1.01362i	−0.862057	+	0.506811i	$0.830824\pi$
$618$		0			0
$619$	−6.71925			−0.270070			−0.135035	−	0.990841i	$-0.543115\pi$
$619$	−6.71925			−0.270070			−0.135035	+	0.990841i	$0.543115\pi$
$620$		0			0
$621$	6.76936			0.271645
$622$		0			0
$623$		0			0
$624$		0			0
$625$	21.7938	+	12.2487i	0.871753	+	0.489946i
$626$		0			0
$627$			13.9001i			0.555118i
$628$		0			0
$629$	0.728851			0.0290612
$630$		0			0
$631$	0.439228			0.0174854			0.00874269	−	0.999962i	$-0.497217\pi$
$631$	0.439228			0.0174854			0.00874269	+	0.999962i	$0.497217\pi$
$632$		0			0
$633$			21.4273i			0.851658i
$634$		0			0
$635$	−4.04649	−	0.520827i	−0.160580	−	0.0206684i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	11.8421			0.468464
$640$		0			0
$641$	−11.0777			−0.437543			−0.218772	−	0.975776i	$-0.570205\pi$
$641$	−11.0777			−0.437543			−0.218772	+	0.975776i	$0.570205\pi$
$642$		0			0
$643$			16.2243i			0.639826i	0.947447	+	0.319913i	$0.103654\pi$
$643$			16.2243i			0.639826i	−0.947447	+	0.319913i	$0.896346\pi$
$644$		0			0
$645$	−2.79656	+	21.7274i	−0.110114	+	0.855517i
$646$		0			0
$647$		−	25.9627i		−	1.02070i	−0.859967	−	0.510349i	$-0.829516\pi$
$647$		−	25.9627i		−	1.02070i	0.859967	−	0.510349i	$-0.170484\pi$
$648$		0			0
$649$	27.3077			1.07192
$650$		0			0
$651$		0			0
$652$		0			0
$653$			7.32368i			0.286598i	0.989679	+	0.143299i	$0.0457710\pi$
$653$			7.32368i			0.286598i	−0.989679	+	0.143299i	$0.954229\pi$
$654$		0			0
$655$	0.772505	−	6.00187i	0.0301843	−	0.234512i
$656$		0			0
$657$			3.79697i			0.148134i
$658$		0			0
$659$	−2.45499			−0.0956328			−0.0478164	−	0.998856i	$-0.515226\pi$
$659$	−2.45499			−0.0956328			−0.0478164	+	0.998856i	$0.515226\pi$
$660$		0			0
$661$	−15.2883			−0.594648			−0.297324	−	0.954777i	$-0.596094\pi$
$661$	−15.2883			−0.594648			−0.297324	+	0.954777i	$0.596094\pi$
$662$		0			0
$663$			1.36787i			0.0531236i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			40.8564i			1.58197i
$668$		0			0
$669$	16.3969			0.633941
$670$		0			0
$671$	−23.8610			−0.921142
$672$		0			0
$673$		−	39.7823i		−	1.53349i	−0.641950	−	0.766747i	$-0.721874\pi$
$673$		−	39.7823i		−	1.53349i	0.641950	−	0.766747i	$-0.278126\pi$
$674$		0			0
$675$	1.26613	−	4.83704i	0.0487335	−	0.186178i
$676$		0			0
$677$		−	32.4664i		−	1.24778i	−0.781511	−	0.623892i	$-0.785551\pi$
$677$		−	32.4664i		−	1.24778i	0.781511	−	0.623892i	$-0.214449\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−9.53370			−0.365332
$682$		0			0
$683$			10.3725i			0.396893i	0.980112	+	0.198446i	$0.0635896\pi$
$683$			10.3725i			0.396893i	−0.980112	+	0.198446i	$0.936410\pi$
$684$		0			0
$685$	22.1889	+	2.85595i	0.847793	+	0.109120i
$686$		0			0
$687$			18.1197i			0.691311i
$688$		0			0
$689$	63.5001			2.41916
$690$		0			0
$691$	−5.47705			−0.208357			−0.104178	−	0.994559i	$-0.533221\pi$
$691$	−5.47705			−0.208357			−0.104178	+	0.994559i	$0.533221\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.591654	−	4.59677i	0.0224427	−	0.174365i
$696$		0			0
$697$			0.253138i			0.00958828i
$698$		0			0
$699$	−12.0544			−0.455939
$700$		0			0
$701$	29.8292			1.12663			0.563317	−	0.826241i	$-0.309525\pi$
$701$	29.8292			1.12663			0.563317	+	0.826241i	$0.309525\pi$
$702$		0			0
$703$			14.5723i			0.549606i
$704$		0			0
$705$	1.05980	−	8.23394i	0.0399143	−	0.310108i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−40.1049			−1.50617			−0.753086	−	0.657922i	$-0.771436\pi$
$709$	−40.1049			−1.50617			−0.753086	+	0.657922i	$0.771436\pi$
$710$		0			0
$711$	0.0174780			0.000655476
$712$		0			0
$713$			71.3403i			2.67172i
$714$		0			0
$715$	37.5194	+	4.82915i	1.40315	+	0.180600i
$716$		0			0
$717$		−	21.3868i		−	0.798704i
$718$		0			0
$719$	21.1776			0.789790			0.394895	−	0.918726i	$-0.370781\pi$
$719$	21.1776			0.789790			0.394895	+	0.918726i	$0.370781\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	17.1889i		−	0.639260i
$724$		0			0
$725$	29.1939	+	7.64173i	1.08423	+	0.283807i
$726$		0			0
$727$		−	12.3642i		−	0.458562i	−0.973360	−	0.229281i	$-0.926362\pi$
$727$		−	12.3642i		−	0.458562i	0.973360	−	0.229281i	$-0.0736375\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	2.32278			0.0859112
$732$		0			0
$733$			36.4536i			1.34644i	0.739440	+	0.673222i	$0.235090\pi$
$733$			36.4536i			1.34644i	−0.739440	+	0.673222i	$0.764910\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	6.57878i		−	0.242333i
$738$		0			0
$739$	38.4273			1.41357			0.706785	−	0.707428i	$-0.250145\pi$
$739$	38.4273			1.41357			0.706785	+	0.707428i	$0.250145\pi$
$740$		0			0
$741$	−27.3486			−1.00468
$742$		0			0
$743$		−	38.1561i		−	1.39981i	−0.714235	−	0.699906i	$-0.753225\pi$
$743$		−	38.1561i		−	1.39981i	0.714235	−	0.699906i	$-0.246775\pi$
$744$		0			0
$745$	0.314901	−	2.44658i	0.0115371	−	0.0896356i
$746$		0			0
$747$			7.30162i			0.267152i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−13.1612			−0.480257			−0.240129	−	0.970741i	$-0.577190\pi$
$751$	−13.1612			−0.480257			−0.240129	+	0.970741i	$0.577190\pi$
$752$		0			0
$753$		−	23.1939i		−	0.845232i
$754$		0			0
$755$	2.98672	−	23.2049i	0.108698	−	0.844513i
$756$		0			0
$757$			1.92272i			0.0698826i	0.999389	+	0.0349413i	$0.0111244\pi$
$757$			1.92272i			0.0698826i	−0.999389	+	0.0349413i	$0.988876\pi$
$758$		0			0
$759$	−19.8499			−0.720507
$760$		0			0
$761$	−18.2132			−0.660227			−0.330113	−	0.943941i	$-0.607087\pi$
$761$	−18.2132			−0.660227			−0.330113	+	0.943941i	$0.607087\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.525816	−	0.0676782i	−0.0190109	−	0.00244691i
$766$		0			0
$767$			53.7280i			1.94001i
$768$		0			0
$769$	−32.1465			−1.15923			−0.579617	−	0.814889i	$-0.696798\pi$
$769$	−32.1465			−1.15923			−0.579617	+	0.814889i	$0.696798\pi$
$770$		0			0
$771$	26.1561			0.941990
$772$		0			0
$773$		−	29.2372i		−	1.05159i	−0.850611	−	0.525795i	$-0.823768\pi$
$773$		−	29.2372i		−	1.05159i	0.850611	−	0.525795i	$-0.176232\pi$
$774$		0			0
$775$	50.9761	+	13.3434i	1.83112	+	0.479309i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.06113			−0.181334
$780$		0			0
$781$	−34.7247			−1.24255
$782$		0			0
$783$		−	6.03549i		−	0.215691i
$784$		0			0
$785$	−20.0967	−	2.58666i	−0.717283	−	0.0923220i
$786$		0			0
$787$			7.21278i			0.257108i	0.991703	+	0.128554i	$0.0410336\pi$
$787$			7.21278i			0.257108i	−0.991703	+	0.128554i	$0.958966\pi$
$788$		0			0
$789$	−7.69981			−0.274121
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	46.9466i		−	1.66712i
$794$		0			0
$795$	−3.14181	+	24.4098i	−0.111428	+	0.865727i
$796$		0			0
$797$		−	32.3827i		−	1.14706i	−0.819186	−	0.573528i	$-0.805575\pi$
$797$		−	32.3827i		−	1.14706i	0.819186	−	0.573528i	$-0.194425\pi$
$798$		0			0
$799$	−0.880253			−0.0311411
$800$		0			0
$801$	17.9517			0.634291
$802$		0			0
$803$		−	11.1339i		−	0.392908i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			25.0388i			0.881407i
$808$		0			0
$809$	−32.4264			−1.14005			−0.570026	−	0.821627i	$-0.693067\pi$
$809$	−32.4264			−1.14005			−0.570026	+	0.821627i	$0.693067\pi$
$810$		0			0
$811$	5.01118			0.175966			0.0879832	−	0.996122i	$-0.471958\pi$
$811$	5.01118			0.175966			0.0879832	+	0.996122i	$0.471958\pi$
$812$		0			0
$813$		−	19.0885i		−	0.669461i
$814$		0			0
$815$	−18.4936	−	2.38033i	−0.647804	−	0.0833793i
$816$		0			0
$817$			46.4407i			1.62475i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	51.0512			1.78170			0.890849	−	0.454300i	$-0.150111\pi$
$821$	51.0512			1.78170			0.890849	+	0.454300i	$0.150111\pi$
$822$		0			0
$823$		−	51.4207i		−	1.79241i	−0.443636	−	0.896207i	$-0.646312\pi$
$823$		−	51.4207i		−	1.79241i	0.443636	−	0.896207i	$-0.353688\pi$
$824$		0			0
$825$	−3.71271	+	14.1837i	−0.129260	+	0.493815i
$826$		0			0
$827$			11.7679i			0.409211i	0.978845	+	0.204605i	$0.0655911\pi$
$827$			11.7679i			0.409211i	−0.978845	+	0.204605i	$0.934409\pi$
$828$		0			0
$829$	−42.1989			−1.46563			−0.732814	−	0.680429i	$-0.761793\pi$
$829$	−42.1989			−1.46563			−0.732814	+	0.680429i	$0.761793\pi$
$830$		0			0
$831$	12.1387			0.421085
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−22.8178	−	2.93690i	−0.789644	−	0.101636i
$836$		0			0
$837$		−	10.5387i		−	0.364271i
$838$		0			0
$839$	34.6127			1.19496			0.597482	−	0.801882i	$-0.296168\pi$
$839$	34.6127			1.19496			0.597482	+	0.801882i	$0.296168\pi$
$840$		0			0
$841$	7.42713			0.256108
$842$		0			0
$843$			6.34537i			0.218546i
$844$		0			0
$845$	−5.79052	+	44.9886i	−0.199200	+	1.54765i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−13.6193			−0.467413
$850$		0			0
$851$	−20.8099			−0.713353
$852$		0			0
$853$			7.87710i			0.269707i	0.990866	+	0.134853i	$0.0430564\pi$
$853$			7.87710i			0.269707i	−0.990866	+	0.134853i	$0.956944\pi$
$854$		0			0
$855$	1.35313	−	10.5129i	0.0462761	−	0.359535i
$856$		0			0
$857$		−	4.48206i		−	0.153104i	−0.997066	−	0.0765522i	$-0.975609\pi$
$857$		−	4.48206i		−	0.153104i	0.997066	−	0.0765522i	$-0.0243912\pi$
$858$		0			0
$859$	−26.4429			−0.902220			−0.451110	−	0.892468i	$-0.648972\pi$
$859$	−26.4429			−0.902220			−0.451110	+	0.892468i	$0.648972\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			38.0001i			1.29354i	0.762686	+	0.646768i	$0.223880\pi$
$863$			38.0001i			1.29354i	−0.762686	+	0.646768i	$0.776120\pi$
$864$		0			0
$865$	−38.3536	−	4.93652i	−1.30406	−	0.167847i
$866$		0			0
$867$		−	16.9438i		−	0.575441i
$868$		0			0
$869$	−0.0512511			−0.00173858
$870$		0			0
$871$	12.9438			0.438584
$872$		0			0
$873$		−	2.31122i		−	0.0782229i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	12.8582i		−	0.434190i	−0.976150	−	0.217095i	$-0.930342\pi$
$877$		−	12.8582i		−	0.434190i	0.976150	−	0.217095i	$-0.0696582\pi$
$878$		0			0
$879$	15.5221			0.523549
$880$		0			0
$881$	32.9744			1.11094			0.555468	−	0.831538i	$-0.312539\pi$
$881$	32.9744			1.11094			0.555468	+	0.831538i	$0.312539\pi$
$882$		0			0
$883$		−	20.3321i		−	0.684229i	−0.939658	−	0.342115i	$-0.888857\pi$
$883$		−	20.3321i		−	0.684229i	0.939658	−	0.342115i	$-0.111143\pi$
$884$		0			0
$885$	−20.6533	−	2.65831i	−0.694255	−	0.0893581i
$886$		0			0
$887$			40.2574i			1.35171i	0.737034	+	0.675856i	$0.236226\pi$
$887$			40.2574i			1.35171i	−0.737034	+	0.675856i	$0.763774\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	2.93232			0.0982365
$892$		0			0
$893$		−	17.5994i		−	0.588942i
$894$		0			0
$895$	3.14549	−	24.4384i	0.105142	−	0.816886i
$896$		0			0
$897$		−	39.0548i		−	1.30400i
$898$		0			0
$899$	63.6063			2.12139
$900$		0			0
$901$	2.60954			0.0869364
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.29505	−	10.0617i	0.0430489	−	0.334462i
$906$		0			0
$907$			48.1516i			1.59885i	0.600768	+	0.799423i	$0.294861\pi$
$907$			48.1516i			1.59885i	−0.600768	+	0.799423i	$0.705139\pi$
$908$		0			0
$909$	7.84205			0.260104
$910$		0			0
$911$	−30.2224			−1.00131			−0.500656	−	0.865646i	$-0.666908\pi$
$911$	−30.2224			−1.00131			−0.500656	+	0.865646i	$0.666908\pi$
$912$		0			0
$913$		−	21.4107i		−	0.708591i
$914$		0			0
$915$	18.0465	+	2.32278i	0.596600	+	0.0767888i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−25.2773			−0.833822			−0.416911	−	0.908947i	$-0.636887\pi$
$919$	−25.2773			−0.833822			−0.416911	+	0.908947i	$0.636887\pi$
$920$		0			0
$921$	13.4406			0.442882
$922$		0			0
$923$		−	68.3210i		−	2.24881i
$924$		0			0
$925$	−3.89225	+	14.8697i	−0.127976	+	0.488911i
$926$		0			0
$927$		−	4.08058i		−	0.134024i
$928$		0			0
$929$	27.7003			0.908819			0.454409	−	0.890793i	$-0.349850\pi$
$929$	27.7003			0.908819			0.454409	+	0.890793i	$0.349850\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	2.45159i		−	0.0802615i
$934$		0			0
$935$	1.54186	+	0.198454i	0.0504243	+	0.00649015i
$936$		0			0
$937$		−	30.8163i		−	1.00673i	−0.864075	−	0.503363i	$-0.832096\pi$
$937$		−	30.8163i		−	1.00673i	0.864075	−	0.503363i	$-0.167904\pi$
$938$		0			0
$939$	−9.91942			−0.323708
$940$		0			0
$941$	−28.2361			−0.920470			−0.460235	−	0.887797i	$-0.652235\pi$
$941$	−28.2361			−0.920470			−0.460235	+	0.887797i	$0.652235\pi$
$942$		0			0
$943$		−	7.22749i		−	0.235360i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			2.99570i			0.0973473i	0.998815	+	0.0486736i	$0.0154994\pi$
$947$			2.99570i			0.0973473i	−0.998815	+	0.0486736i	$0.984501\pi$
$948$		0			0
$949$	21.9060			0.711100
$950$		0			0
$951$	25.9950			0.842947
$952$		0			0
$953$		−	10.0423i		−	0.325303i	−0.986684	−	0.162651i	$-0.947995\pi$
$953$		−	10.0423i		−	0.325303i	0.986684	−	0.162651i	$-0.0520046\pi$
$954$		0			0
$955$	2.47787	−	19.2514i	0.0801818	−	0.622961i
$956$		0			0
$957$			17.6980i			0.572095i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	80.0645			2.58273
$962$		0			0
$963$		−	18.3467i		−	0.591214i
$964$		0			0
$965$	−15.6602	−	2.01564i	−0.504120	−	0.0648857i
$966$		0			0
$967$		−	21.3514i		−	0.686616i	−0.939223	−	0.343308i	$-0.888453\pi$
$967$		−	21.3514i		−	0.686616i	0.939223	−	0.343308i	$-0.111547\pi$
$968$		0			0
$969$	−1.12389			−0.0361046
$970$		0			0
$971$	38.0001			1.21948			0.609740	−	0.792602i	$-0.291274\pi$
$971$	38.0001			1.21948			0.609740	+	0.792602i	$0.291274\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−27.9066	−	7.30477i	−0.893726	−	0.233940i
$976$		0			0
$977$		−	42.7702i		−	1.36834i	−0.729322	−	0.684170i	$-0.760164\pi$
$977$		−	42.7702i		−	1.36834i	0.729322	−	0.684170i	$-0.239836\pi$
$978$		0			0
$979$	−52.6401			−1.68238
$980$		0			0
$981$	−10.4144			−0.332506
$982$		0			0
$983$		−	13.4794i		−	0.429925i	−0.976622	−	0.214962i	$-0.931037\pi$
$983$		−	13.4794i		−	0.429925i	0.976622	−	0.214962i	$-0.0689629\pi$
$984$		0			0
$985$	7.95940	+	1.02446i	0.253607	+	0.0326420i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−66.3192			−2.10883
$990$		0			0
$991$	26.5987			0.844934			0.422467	−	0.906378i	$-0.361164\pi$
$991$	26.5987			0.844934			0.422467	+	0.906378i	$0.361164\pi$
$992$		0			0
$993$		−	7.53226i		−	0.239029i
$994$		0			0
$995$	0.851365	−	6.61456i	0.0269901	−	0.209696i
$996$		0			0
$997$		−	14.3873i		−	0.455651i	−0.973702	−	0.227825i	$-0.926838\pi$
$997$		−	14.3873i		−	0.455651i	0.973702	−	0.227825i	$-0.0731615\pi$
$998$		0			0
$999$	3.07413			0.0972611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2940.2.k.f.589.3		8
5.4	even	2	inner	2940.2.k.f.589.7		8
7.2	even	3		420.2.bb.a.109.5	yes	16
7.3	odd	6		2940.2.bb.i.1549.5		16
7.4	even	3		420.2.bb.a.289.4	yes	16
7.5	odd	6		2940.2.bb.i.949.4		16
7.6	odd	2		2940.2.k.g.589.6		8
21.2	odd	6		1260.2.bm.c.109.7		16
21.11	odd	6		1260.2.bm.c.289.2		16
28.11	odd	6		1680.2.di.e.289.8		16
28.23	odd	6		1680.2.di.e.529.1		16
35.2	odd	12		2100.2.q.m.1201.2		8
35.4	even	6		420.2.bb.a.289.5	yes	16
35.9	even	6		420.2.bb.a.109.4	✓	16
35.18	odd	12		2100.2.q.l.1801.3		8
35.19	odd	6		2940.2.bb.i.949.5		16
35.23	odd	12		2100.2.q.l.1201.3		8
35.24	odd	6		2940.2.bb.i.1549.4		16
35.32	odd	12		2100.2.q.m.1801.2		8
35.34	odd	2		2940.2.k.g.589.2		8
105.44	odd	6		1260.2.bm.c.109.2		16
105.74	odd	6		1260.2.bm.c.289.7		16
140.39	odd	6		1680.2.di.e.289.1		16
140.79	odd	6		1680.2.di.e.529.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bb.a.109.4	✓	16	35.9	even	6
420.2.bb.a.109.5	yes	16	7.2	even	3
420.2.bb.a.289.4	yes	16	7.4	even	3
420.2.bb.a.289.5	yes	16	35.4	even	6
1260.2.bm.c.109.2		16	105.44	odd	6
1260.2.bm.c.109.7		16	21.2	odd	6
1260.2.bm.c.289.2		16	21.11	odd	6
1260.2.bm.c.289.7		16	105.74	odd	6
1680.2.di.e.289.1		16	140.39	odd	6
1680.2.di.e.289.8		16	28.11	odd	6
1680.2.di.e.529.1		16	28.23	odd	6
1680.2.di.e.529.8		16	140.79	odd	6
2100.2.q.l.1201.3		8	35.23	odd	12
2100.2.q.l.1801.3		8	35.18	odd	12
2100.2.q.m.1201.2		8	35.2	odd	12
2100.2.q.m.1801.2		8	35.32	odd	12
2940.2.k.f.589.3		8	1.1	even	1	trivial
2940.2.k.f.589.7		8	5.4	even	2	inner
2940.2.k.g.589.2		8	35.34	odd	2
2940.2.k.g.589.6		8	7.6	odd	2
2940.2.bb.i.949.4		16	7.5	odd	6
2940.2.bb.i.949.5		16	35.19	odd	6
2940.2.bb.i.1549.4		16	35.24	odd	6
2940.2.bb.i.1549.5		16	7.3	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2940.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(0.285451 - 2.21777i) q^{5} -1.00000 q^{9} +2.93232 q^{11} +5.76936i q^{13} +(-2.21777 - 0.285451i) q^{15} +0.237092i q^{17} -4.74032 q^{19} -6.76936i q^{23} +(-4.83704 - 1.26613i) q^{25} +1.00000i q^{27} -6.03549 q^{29} -10.5387 q^{31} -2.93232i q^{33} -3.07413i q^{37} +5.76936 q^{39} +1.06768 q^{41} -9.79697i q^{43} +(-0.285451 + 2.21777i) q^{45} +3.71271i q^{47} +0.237092 q^{51} -11.0064i q^{53} +(0.837035 - 6.50322i) q^{55} +4.74032i q^{57} +9.31265 q^{59} -8.13722 q^{61} +(12.7951 + 1.64687i) q^{65} -2.24354i q^{67} -6.76936 q^{69} -11.8421 q^{71} -3.79697i q^{73} +(-1.26613 + 4.83704i) q^{75} -0.0174780 q^{79} +1.00000 q^{81} -7.30162i q^{83} +(0.525816 + 0.0676782i) q^{85} +6.03549i q^{87} -17.9517 q^{89} +10.5387i q^{93} +(-1.35313 + 10.5129i) q^{95} +2.31122i q^{97} -2.93232 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} - 8 q^{9} + 8 q^{11} + 2 q^{15} - 8 q^{19} - 12 q^{25} + 12 q^{29} + 4 q^{39} + 24 q^{41} + 2 q^{45} - 4 q^{51} - 20 q^{55} + 28 q^{59} + 32 q^{61} + 26 q^{65} - 12 q^{69} - 28 q^{71} + 8 q^{75}+ \cdots - 8 q^{99}+O(q^{100}) \) 8 * q - 2 * q^5 - 8 * q^9 + 8 * q^11 + 2 * q^15 - 8 * q^19 - 12 * q^25 + 12 * q^29 + 4 * q^39 + 24 * q^41 + 2 * q^45 - 4 * q^51 - 20 * q^55 + 28 * q^59 + 32 * q^61 + 26 * q^65 - 12 * q^69 - 28 * q^71 + 8 * q^75 + 16 * q^79 + 8 * q^81 + 16 * q^85 + 16 * q^89 - 22 * q^95 - 8 * q^99

Introduction

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