LMFDB - Embedded newform 1300.2.m.b.593.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1300,2,Mod(57,1300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1300, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1300.57"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.m (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-2,0,0,0,0,0,0,0,-6] 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:	$10.3805522628$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.2
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1300.593
Dual form			1300.2.m.b.57.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.618034 + 0.618034i) q^{3} -2.23607i q^{9} +(-2.61803 + 2.61803i) q^{11} +(2.00000 + 3.00000i) q^{13} +(2.23607 + 2.23607i) q^{17} +(5.85410 - 5.85410i) q^{19} +(-3.38197 + 3.38197i) q^{23} +(3.23607 - 3.23607i) q^{27} +5.23607i q^{29} +(5.85410 + 5.85410i) q^{31} -3.23607 q^{33} +9.70820i q^{37} +(-0.618034 + 3.09017i) q^{39} +(3.76393 + 3.76393i) q^{41} +(3.85410 - 3.85410i) q^{43} -8.94427i q^{47} +7.00000 q^{49} +2.76393i q^{51} +(-1.47214 - 1.47214i) q^{53} +7.23607 q^{57} +(-3.38197 - 3.38197i) q^{59} +5.70820 q^{61} +0.291796 q^{67} -4.18034 q^{69} +(2.61803 + 2.61803i) q^{71} -11.7082 q^{73} +3.70820i q^{79} -2.70820 q^{81} +10.4721i q^{83} +(-3.23607 + 3.23607i) q^{87} +(-2.23607 - 2.23607i) q^{89} +7.23607i q^{93} -15.4164 q^{97} +(5.85410 + 5.85410i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 6 q^{11} + 8 q^{13} + 10 q^{19} - 18 q^{23} + 4 q^{27} + 10 q^{31} - 4 q^{33} + 2 q^{39} + 24 q^{41} + 2 q^{43} + 28 q^{49} + 12 q^{53} + 20 q^{57} - 18 q^{59} - 4 q^{61} + 28 q^{67} + 28 q^{69}+ \cdots + 10 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 6 * q^11 + 8 * q^13 + 10 * q^19 - 18 * q^23 + 4 * q^27 + 10 * q^31 - 4 * q^33 + 2 * q^39 + 24 * q^41 + 2 * q^43 + 28 * q^49 + 12 * q^53 + 20 * q^57 - 18 * q^59 - 4 * q^61 + 28 * q^67 + 28 * q^69 + 6 * q^71 - 20 * q^73 + 16 * q^81 - 4 * q^87 - 8 * q^97 + 10 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$.

$n$	$301$	$651$	$677$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$	0.618034	+	0.618034i	0.356822	+	0.356822i	0.862640	−	0.505818i	$-0.168809\pi$
$3$	0.618034	+	0.618034i	0.356822	+	0.356822i	−0.505818	+	0.862640i	$0.668809\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$		−	2.23607i		−	0.745356i
$10$		0			0
$11$	−2.61803	+	2.61803i	−0.789367	+	0.789367i	−0.981390	−	0.192023i	$-0.938495\pi$
$11$	−2.61803	+	2.61803i	−0.789367	+	0.789367i	0.192023	+	0.981390i	$0.438495\pi$
$12$		0			0
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.23607	+	2.23607i	0.542326	+	0.542326i	0.924210	−	0.381884i	$-0.124725\pi$
$17$	2.23607	+	2.23607i	0.542326	+	0.542326i	−0.381884	+	0.924210i	$0.624725\pi$
$18$		0			0
$19$	5.85410	−	5.85410i	1.34302	−	1.34302i	0.449989	−	0.893034i	$-0.351428\pi$
$19$	5.85410	−	5.85410i	1.34302	−	1.34302i	0.893034	−	0.449989i	$-0.148572\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.38197	+	3.38197i	−0.705189	+	0.705189i	−0.965519	−	0.260331i	$-0.916168\pi$
$23$	−3.38197	+	3.38197i	−0.705189	+	0.705189i	0.260331	+	0.965519i	$0.416168\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.23607	−	3.23607i	0.622782	−	0.622782i
$28$		0			0
$29$			5.23607i			0.972313i	0.873872	+	0.486157i	$0.161602\pi$
$29$			5.23607i			0.972313i	−0.873872	+	0.486157i	$0.838398\pi$
$30$		0			0
$31$	5.85410	+	5.85410i	1.05143	+	1.05143i	0.998604	+	0.0528239i	$0.0168222\pi$
$31$	5.85410	+	5.85410i	1.05143	+	1.05143i	0.0528239	+	0.998604i	$0.483178\pi$
$32$		0			0
$33$	−3.23607			−0.563327
$34$		0			0
$35$		0			0
$36$		0			0
$37$			9.70820i			1.59602i	0.602645	+	0.798009i	$0.294114\pi$
$37$			9.70820i			1.59602i	−0.602645	+	0.798009i	$0.705886\pi$
$38$		0			0
$39$	−0.618034	+	3.09017i	−0.0989646	+	0.494823i
$40$		0			0
$41$	3.76393	+	3.76393i	0.587827	+	0.587827i	0.937043	−	0.349215i	$-0.113552\pi$
$41$	3.76393	+	3.76393i	0.587827	+	0.587827i	−0.349215	+	0.937043i	$0.613552\pi$
$42$		0			0
$43$	3.85410	−	3.85410i	0.587745	−	0.587745i	−0.349275	−	0.937020i	$-0.613572\pi$
$43$	3.85410	−	3.85410i	0.587745	−	0.587745i	0.937020	+	0.349275i	$0.113572\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	8.94427i		−	1.30466i	−0.757937	−	0.652328i	$-0.773792\pi$
$47$		−	8.94427i		−	1.30466i	0.757937	−	0.652328i	$-0.226208\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$			2.76393i			0.387028i
$52$		0			0
$53$	−1.47214	−	1.47214i	−0.202213	−	0.202213i	0.598734	−	0.800948i	$-0.295671\pi$
$53$	−1.47214	−	1.47214i	−0.202213	−	0.202213i	−0.800948	+	0.598734i	$0.795671\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	7.23607			0.958441
$58$		0			0
$59$	−3.38197	−	3.38197i	−0.440294	−	0.440294i	0.451816	−	0.892111i	$-0.350776\pi$
$59$	−3.38197	−	3.38197i	−0.440294	−	0.440294i	−0.892111	+	0.451816i	$0.850776\pi$
$60$		0			0
$61$	5.70820			0.730861			0.365430	−	0.930839i	$-0.380922\pi$
$61$	5.70820			0.730861			0.365430	+	0.930839i	$0.380922\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.291796			0.0356486			0.0178243	−	0.999841i	$-0.494326\pi$
$67$	0.291796			0.0356486			0.0178243	+	0.999841i	$0.494326\pi$
$68$		0			0
$69$	−4.18034			−0.503254
$70$		0			0
$71$	2.61803	+	2.61803i	0.310703	+	0.310703i	0.845182	−	0.534479i	$-0.179492\pi$
$71$	2.61803	+	2.61803i	0.310703	+	0.310703i	−0.534479	+	0.845182i	$0.679492\pi$
$72$		0			0
$73$	−11.7082			−1.37034			−0.685171	−	0.728382i	$-0.740272\pi$
$73$	−11.7082			−1.37034			−0.685171	+	0.728382i	$0.740272\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$			3.70820i			0.417206i	0.978000	+	0.208603i	$0.0668916\pi$
$79$			3.70820i			0.417206i	−0.978000	+	0.208603i	$0.933108\pi$
$80$		0			0
$81$	−2.70820			−0.300912
$82$		0			0
$83$			10.4721i			1.14947i	0.818341	+	0.574733i	$0.194894\pi$
$83$			10.4721i			1.14947i	−0.818341	+	0.574733i	$0.805106\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−3.23607	+	3.23607i	−0.346943	+	0.346943i
$88$		0			0
$89$	−2.23607	−	2.23607i	−0.237023	−	0.237023i	0.578593	−	0.815616i	$-0.303602\pi$
$89$	−2.23607	−	2.23607i	−0.237023	−	0.237023i	−0.815616	+	0.578593i	$0.803602\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			7.23607i			0.750345i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−15.4164			−1.56530			−0.782650	−	0.622463i	$-0.786132\pi$
$97$	−15.4164			−1.56530			−0.782650	+	0.622463i	$0.786132\pi$
$98$		0			0
$99$	5.85410	+	5.85410i	0.588359	+	0.588359i
$100$		0			0
$101$			1.52786i			0.152028i	0.997107	+	0.0760141i	$0.0242194\pi$
$101$			1.52786i			0.152028i	−0.997107	+	0.0760141i	$0.975781\pi$
$102$		0			0
$103$	3.85410	−	3.85410i	0.379756	−	0.379756i	−0.491258	−	0.871014i	$-0.663463\pi$
$103$	3.85410	−	3.85410i	0.379756	−	0.379756i	0.871014	+	0.491258i	$0.163463\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.61803	−	2.61803i	0.253095	−	0.253095i	−0.569143	−	0.822238i	$-0.692725\pi$
$107$	2.61803	−	2.61803i	0.253095	−	0.253095i	0.822238	+	0.569143i	$0.192725\pi$
$108$		0			0
$109$	4.70820	−	4.70820i	0.450964	−	0.450964i	−0.444710	−	0.895674i	$-0.646693\pi$
$109$	4.70820	−	4.70820i	0.450964	−	0.450964i	0.895674	+	0.444710i	$0.146693\pi$
$110$		0			0
$111$	−6.00000	+	6.00000i	−0.569495	+	0.569495i
$112$		0			0
$113$	11.9443	+	11.9443i	1.12362	+	1.12362i	0.991192	+	0.132430i	$0.0422781\pi$
$113$	11.9443	+	11.9443i	1.12362	+	1.12362i	0.132430	+	0.991192i	$0.457722\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	6.70820	−	4.47214i	0.620174	−	0.413449i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	2.70820i		−	0.246200i
$122$		0			0
$123$			4.65248i			0.419500i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	0.145898	+	0.145898i	0.0129464	+	0.0129464i	0.713550	−	0.700604i	$-0.247086\pi$
$127$	0.145898	+	0.145898i	0.0129464	+	0.0129464i	−0.700604	+	0.713550i	$0.747086\pi$
$128$		0			0
$129$	4.76393			0.419441
$130$		0			0
$131$	−7.41641			−0.647975			−0.323987	−	0.946061i	$-0.605024\pi$
$131$	−7.41641			−0.647975			−0.323987	+	0.946061i	$0.605024\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			16.4721i			1.40731i	0.710542	+	0.703655i	$0.248450\pi$
$137$			16.4721i			1.40731i	−0.710542	+	0.703655i	$0.751550\pi$
$138$		0			0
$139$		−	23.1246i		−	1.96140i	−0.195509	−	0.980702i	$-0.562636\pi$
$139$		−	23.1246i		−	1.96140i	0.195509	−	0.980702i	$-0.437364\pi$
$140$		0			0
$141$	5.52786	−	5.52786i	0.465530	−	0.465530i
$142$		0			0
$143$	−13.0902	−	2.61803i	−1.09466	−	0.218931i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.32624	+	4.32624i	0.356822	+	0.356822i
$148$		0			0
$149$	5.94427	−	5.94427i	0.486974	−	0.486974i	−0.420376	−	0.907350i	$-0.638102\pi$
$149$	5.94427	−	5.94427i	0.486974	−	0.486974i	0.907350	+	0.420376i	$0.138102\pi$
$150$		0			0
$151$	−0.145898	+	0.145898i	−0.0118730	+	0.0118730i	−0.713018	−	0.701145i	$-0.752672\pi$
$151$	−0.145898	+	0.145898i	−0.0118730	+	0.0118730i	0.701145	+	0.713018i	$0.252672\pi$
$152$		0			0
$153$	5.00000	−	5.00000i	0.404226	−	0.404226i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.00000	−	5.00000i	0.399043	−	0.399043i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	5.00000	−	5.00000i	0.399043	−	0.399043i	0.877896	+	0.478852i	$0.158947\pi$
$158$		0			0
$159$		−	1.81966i		−	0.144308i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.29180			−0.336159			−0.168080	−	0.985773i	$-0.553757\pi$
$163$	−4.29180			−0.336159			−0.168080	+	0.985773i	$0.553757\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	13.5279i		−	1.04682i	−0.852082	−	0.523409i	$-0.824660\pi$
$167$		−	13.5279i		−	1.04682i	0.852082	−	0.523409i	$-0.175340\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$	−13.0902	−	13.0902i	−1.00103	−	1.00103i
$172$		0			0
$173$	17.1803	−	17.1803i	1.30620	−	1.30620i	0.382059	−	0.924138i	$-0.375215\pi$
$173$	17.1803	−	17.1803i	1.30620	−	1.30620i	0.924138	−	0.382059i	$-0.124785\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	4.18034i		−	0.314214i
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$		−	8.29180i		−	0.616324i	−0.951334	−	0.308162i	$-0.900286\pi$
$181$		−	8.29180i		−	0.616324i	0.951334	−	0.308162i	$-0.0997139\pi$
$182$		0			0
$183$	3.52786	+	3.52786i	0.260787	+	0.260787i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−11.7082			−0.856189
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.41641			0.536632			0.268316	−	0.963331i	$-0.413533\pi$
$191$	7.41641			0.536632			0.268316	+	0.963331i	$0.413533\pi$
$192$		0			0
$193$	−3.41641			−0.245918			−0.122959	−	0.992412i	$-0.539238\pi$
$193$	−3.41641			−0.245918			−0.122959	+	0.992412i	$0.539238\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	−11.4164			−0.809288			−0.404644	−	0.914474i	$-0.632605\pi$
$199$	−11.4164			−0.809288			−0.404644	+	0.914474i	$0.632605\pi$
$200$		0			0
$201$	0.180340	+	0.180340i	0.0127202	+	0.0127202i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	7.56231	+	7.56231i	0.525617	+	0.525617i
$208$		0			0
$209$			30.6525i			2.12028i
$210$		0			0
$211$	3.41641			0.235195			0.117598	−	0.993061i	$-0.462481\pi$
$211$	3.41641			0.235195			0.117598	+	0.993061i	$0.462481\pi$
$212$		0			0
$213$			3.23607i			0.221732i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−7.23607	−	7.23607i	−0.488968	−	0.488968i
$220$		0			0
$221$	−2.23607	+	11.1803i	−0.150414	+	0.752071i
$222$		0			0
$223$		−	7.41641i		−	0.496639i	−0.968678	−	0.248320i	$-0.920122\pi$
$223$		−	7.41641i		−	0.496639i	0.968678	−	0.248320i	$-0.0798784\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.29180			−0.550346			−0.275173	−	0.961395i	$-0.588735\pi$
$227$	−8.29180			−0.550346			−0.275173	+	0.961395i	$0.588735\pi$
$228$		0			0
$229$	8.41641	+	8.41641i	0.556172	+	0.556172i	0.928215	−	0.372043i	$-0.121343\pi$
$229$	8.41641	+	8.41641i	0.556172	+	0.556172i	−0.372043	+	0.928215i	$0.621343\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.47214	−	7.47214i	0.489516	−	0.489516i	−0.418638	−	0.908153i	$-0.637492\pi$
$233$	7.47214	−	7.47214i	0.489516	−	0.489516i	0.908153	+	0.418638i	$0.137492\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2.29180	+	2.29180i	−0.148868	+	0.148868i
$238$		0			0
$239$	−8.61803	+	8.61803i	−0.557454	+	0.557454i	−0.928582	−	0.371128i	$-0.878971\pi$
$239$	−8.61803	+	8.61803i	−0.557454	+	0.557454i	0.371128	+	0.928582i	$0.378971\pi$
$240$		0			0
$241$	−2.70820	+	2.70820i	−0.174451	+	0.174451i	−0.788932	−	0.614481i	$-0.789365\pi$
$241$	−2.70820	+	2.70820i	−0.174451	+	0.174451i	0.614481	+	0.788932i	$0.289365\pi$
$242$		0			0
$243$	−11.3820	−	11.3820i	−0.730153	−	0.730153i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	29.2705	+	5.85410i	1.86244	+	0.372488i
$248$		0			0
$249$	−6.47214	+	6.47214i	−0.410155	+	0.410155i
$250$		0			0
$251$		−	26.1803i		−	1.65249i	−0.563312	−	0.826244i	$-0.690473\pi$
$251$		−	26.1803i		−	1.65249i	0.563312	−	0.826244i	$-0.309527\pi$
$252$		0			0
$253$		−	17.7082i		−	1.11331i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−17.1803	−	17.1803i	−1.07168	−	1.07168i	−0.997224	−	0.0744558i	$-0.976278\pi$
$257$	−17.1803	−	17.1803i	−1.07168	−	1.07168i	−0.0744558	−	0.997224i	$-0.523722\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	11.7082			0.724720
$262$		0			0
$263$	−22.0344	−	22.0344i	−1.35870	−	1.35870i	−0.875520	−	0.483182i	$-0.839481\pi$
$263$	−22.0344	−	22.0344i	−1.35870	−	1.35870i	−0.483182	−	0.875520i	$-0.660519\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	2.76393i		−	0.169150i
$268$		0			0
$269$			13.5279i			0.824808i	0.911001	+	0.412404i	$0.135311\pi$
$269$			13.5279i			0.824808i	−0.911001	+	0.412404i	$0.864689\pi$
$270$		0			0
$271$	−7.56231	+	7.56231i	−0.459377	+	0.459377i	−0.898451	−	0.439074i	$-0.855307\pi$
$271$	−7.56231	+	7.56231i	−0.459377	+	0.459377i	0.439074	+	0.898451i	$0.355307\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.70820	+	2.70820i	0.162720	+	0.162720i	0.783771	−	0.621050i	$-0.213294\pi$
$277$	2.70820	+	2.70820i	0.162720	+	0.162720i	−0.621050	+	0.783771i	$0.713294\pi$
$278$		0			0
$279$	13.0902	−	13.0902i	0.783688	−	0.783688i
$280$		0			0
$281$	2.23607	−	2.23607i	0.133393	−	0.133393i	−0.637258	−	0.770651i	$-0.719931\pi$
$281$	2.23607	−	2.23607i	0.133393	−	0.133393i	0.770651	+	0.637258i	$0.219931\pi$
$282$		0			0
$283$	15.8541	−	15.8541i	0.942429	−	0.942429i	−0.0560021	−	0.998431i	$-0.517835\pi$
$283$	15.8541	−	15.8541i	0.942429	−	0.942429i	0.998431	+	0.0560021i	$0.0178354\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	7.00000i		−	0.411765i
$290$		0			0
$291$	−9.52786	−	9.52786i	−0.558533	−	0.558533i
$292$		0			0
$293$	−27.7082			−1.61873			−0.809365	−	0.587306i	$-0.800189\pi$
$293$	−27.7082			−1.61873			−0.809365	+	0.587306i	$0.800189\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			16.9443i			0.983206i
$298$		0			0
$299$	−16.9098	−	3.38197i	−0.977921	−	0.195584i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−0.944272	+	0.944272i	−0.0542470	+	0.0542470i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	12.0000i		−	0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$		−	12.0000i		−	0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$		0			0
$309$	4.76393			0.271011
$310$		0			0
$311$		−	2.18034i		−	0.123636i	−0.998087	−	0.0618179i	$-0.980310\pi$
$311$		−	2.18034i		−	0.123636i	0.998087	−	0.0618179i	$-0.0196898\pi$
$312$		0			0
$313$	14.7082	+	14.7082i	0.831357	+	0.831357i	0.987702	−	0.156346i	$-0.0499714\pi$
$313$	14.7082	+	14.7082i	0.831357	+	0.831357i	−0.156346	+	0.987702i	$0.549971\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.70820			0.208273			0.104137	−	0.994563i	$-0.466792\pi$
$317$	3.70820			0.208273			0.104137	+	0.994563i	$0.466792\pi$
$318$		0			0
$319$	−13.7082	−	13.7082i	−0.767512	−	0.767512i
$320$		0			0
$321$	3.23607			0.180620
$322$		0			0
$323$	26.1803			1.45671
$324$		0			0
$325$		0			0
$326$		0			0
$327$	5.81966			0.321828
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−15.8541	−	15.8541i	−0.871420	−	0.871420i	0.121207	−	0.992627i	$-0.461324\pi$
$331$	−15.8541	−	15.8541i	−0.871420	−	0.871420i	−0.992627	+	0.121207i	$0.961324\pi$
$332$		0			0
$333$	21.7082			1.18960
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.4164	−	14.4164i	−0.785312	−	0.785312i	0.195410	−	0.980722i	$-0.437396\pi$
$337$	−14.4164	−	14.4164i	−0.785312	−	0.785312i	−0.980722	+	0.195410i	$0.937396\pi$
$338$		0			0
$339$			14.7639i			0.801867i
$340$		0			0
$341$	−30.6525			−1.65992
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.09017	+	7.09017i	−0.380620	+	0.380620i	−0.871326	−	0.490705i	$-0.836739\pi$
$347$	−7.09017	+	7.09017i	−0.380620	+	0.380620i	0.490705	+	0.871326i	$0.336739\pi$
$348$		0			0
$349$	16.7082	+	16.7082i	0.894370	+	0.894370i	0.994931	−	0.100561i	$-0.0320638\pi$
$349$	16.7082	+	16.7082i	0.894370	+	0.894370i	−0.100561	+	0.994931i	$0.532064\pi$
$350$		0			0
$351$	16.1803	+	3.23607i	0.863643	+	0.172729i
$352$		0			0
$353$			12.7639i			0.679356i	0.940542	+	0.339678i	$0.110318\pi$
$353$			12.7639i			0.679356i	−0.940542	+	0.339678i	$0.889682\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−13.0902	−	13.0902i	−0.690873	−	0.690873i	0.271551	−	0.962424i	$-0.412463\pi$
$359$	−13.0902	−	13.0902i	−0.690873	−	0.690873i	−0.962424	+	0.271551i	$0.912463\pi$
$360$		0			0
$361$		−	49.5410i		−	2.60742i
$362$		0			0
$363$	1.67376	−	1.67376i	0.0878497	−	0.0878497i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−11.8541	+	11.8541i	−0.618779	+	0.618779i	−0.945218	−	0.326439i	$-0.894151\pi$
$367$	−11.8541	+	11.8541i	−0.618779	+	0.618779i	0.326439	+	0.945218i	$0.394151\pi$
$368$		0			0
$369$	8.41641	−	8.41641i	0.438141	−	0.438141i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.416408	+	0.416408i	0.0215608	+	0.0215608i	0.717805	−	0.696244i	$-0.245147\pi$
$373$	0.416408	+	0.416408i	0.0215608	+	0.0215608i	−0.696244	+	0.717805i	$0.745147\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−15.7082	+	10.4721i	−0.809014	+	0.539342i
$378$		0			0
$379$	−3.85410	+	3.85410i	−0.197972	+	0.197972i	−0.799130	−	0.601158i	$-0.794706\pi$
$379$	−3.85410	+	3.85410i	−0.197972	+	0.197972i	0.601158	+	0.799130i	$0.294706\pi$
$380$		0			0
$381$			0.180340i			0.00923909i
$382$		0			0
$383$			17.8885i			0.914062i	0.889451	+	0.457031i	$0.151087\pi$
$383$			17.8885i			0.914062i	−0.889451	+	0.457031i	$0.848913\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−8.61803	−	8.61803i	−0.438079	−	0.438079i
$388$		0			0
$389$	−25.4164			−1.28866			−0.644332	−	0.764746i	$-0.722864\pi$
$389$	−25.4164			−1.28866			−0.644332	+	0.764746i	$0.722864\pi$
$390$		0			0
$391$	−15.1246			−0.764884
$392$		0			0
$393$	−4.58359	−	4.58359i	−0.231212	−	0.231212i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	5.12461i		−	0.257197i	−0.991697	−	0.128598i	$-0.958952\pi$
$397$		−	5.12461i		−	0.257197i	0.991697	−	0.128598i	$-0.0410478\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.23607	−	2.23607i	0.111664	−	0.111664i	−0.649067	−	0.760731i	$-0.724841\pi$
$401$	2.23607	−	2.23607i	0.111664	−	0.111664i	0.760731	+	0.649067i	$0.224841\pi$
$402$		0			0
$403$	−5.85410	+	29.2705i	−0.291614	+	1.45807i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−25.4164	−	25.4164i	−1.25984	−	1.25984i
$408$		0			0
$409$	3.29180	−	3.29180i	0.162769	−	0.162769i	−0.621023	−	0.783792i	$-0.713283\pi$
$409$	3.29180	−	3.29180i	0.162769	−	0.162769i	0.783792	+	0.621023i	$0.213283\pi$
$410$		0			0
$411$	−10.1803	+	10.1803i	−0.502159	+	0.502159i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	14.2918	−	14.2918i	0.699872	−	0.699872i
$418$		0			0
$419$		−	21.5967i		−	1.05507i	−0.849533	−	0.527535i	$-0.823116\pi$
$419$		−	21.5967i		−	1.05507i	0.849533	−	0.527535i	$-0.176884\pi$
$420$		0			0
$421$	3.29180	+	3.29180i	0.160432	+	0.160432i	0.782758	−	0.622326i	$-0.213812\pi$
$421$	3.29180	+	3.29180i	0.160432	+	0.160432i	−0.622326	+	0.782758i	$0.713812\pi$
$422$		0			0
$423$	−20.0000			−0.972433
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−6.47214	−	9.70820i	−0.312478	−	0.468717i
$430$		0			0
$431$	16.9098	+	16.9098i	0.814518	+	0.814518i	0.985308	−	0.170790i	$-0.0546319\pi$
$431$	16.9098	+	16.9098i	0.814518	+	0.814518i	−0.170790	+	0.985308i	$0.554632\pi$
$432$		0			0
$433$	17.0000	−	17.0000i	0.816968	−	0.816968i	−0.168700	−	0.985668i	$-0.553957\pi$
$433$	17.0000	−	17.0000i	0.816968	−	0.816968i	0.985668	+	0.168700i	$0.0539568\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			39.5967i			1.89417i
$438$		0			0
$439$	20.0000			0.954548			0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000			0.954548			0.477274	+	0.878755i	$0.341625\pi$
$440$		0			0
$441$		−	15.6525i		−	0.745356i
$442$		0			0
$443$	−0.326238	−	0.326238i	−0.0155000	−	0.0155000i	0.699314	−	0.714814i	$-0.253489\pi$
$443$	−0.326238	−	0.326238i	−0.0155000	−	0.0155000i	−0.714814	+	0.699314i	$0.753489\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	7.34752			0.347526
$448$		0			0
$449$	−10.5279	−	10.5279i	−0.496841	−	0.496841i	0.413612	−	0.910453i	$-0.364267\pi$
$449$	−10.5279	−	10.5279i	−0.496841	−	0.496841i	−0.910453	+	0.413612i	$0.864267\pi$
$450$		0			0
$451$	−19.7082			−0.928023
$452$		0			0
$453$	−0.180340			−0.00847311
$454$		0			0
$455$		0			0
$456$		0			0
$457$	23.4164			1.09537			0.547687	−	0.836684i	$-0.315509\pi$
$457$	23.4164			1.09537			0.547687	+	0.836684i	$0.315509\pi$
$458$		0			0
$459$	14.4721			0.675501
$460$		0			0
$461$	17.1803	+	17.1803i	0.800168	+	0.800168i	0.983122	−	0.182953i	$-0.0585657\pi$
$461$	17.1803	+	17.1803i	0.800168	+	0.800168i	−0.182953	+	0.983122i	$0.558566\pi$
$462$		0			0
$463$	−4.29180			−0.199457			−0.0997283	−	0.995015i	$-0.531797\pi$
$463$	−4.29180			−0.199457			−0.0997283	+	0.995015i	$0.531797\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.0344	−	16.0344i	−0.741985	−	0.741985i	0.230974	−	0.972960i	$-0.425809\pi$
$467$	−16.0344	−	16.0344i	−0.741985	−	0.741985i	−0.972960	+	0.230974i	$0.925809\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	6.18034			0.284775
$472$		0			0
$473$			20.1803i			0.927893i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−3.29180	+	3.29180i	−0.150721	+	0.150721i
$478$		0			0
$479$	25.7426	+	25.7426i	1.17621	+	1.17621i	0.980702	+	0.195510i	$0.0626361\pi$
$479$	25.7426	+	25.7426i	1.17621	+	1.17621i	0.195510	+	0.980702i	$0.437364\pi$
$480$		0			0
$481$	−29.1246	+	19.4164i	−1.32797	+	0.885312i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	15.1246			0.685362			0.342681	−	0.939452i	$-0.388665\pi$
$487$	15.1246			0.685362			0.342681	+	0.939452i	$0.388665\pi$
$488$		0			0
$489$	−2.65248	−	2.65248i	−0.119949	−	0.119949i
$490$		0			0
$491$			24.6525i			1.11255i	0.830998	+	0.556275i	$0.187770\pi$
$491$			24.6525i			1.11255i	−0.830998	+	0.556275i	$0.812230\pi$
$492$		0			0
$493$	−11.7082	+	11.7082i	−0.527311	+	0.527311i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−20.9787	+	20.9787i	−0.939136	+	0.939136i	−0.998251	−	0.0591150i	$-0.981172\pi$
$499$	−20.9787	+	20.9787i	−0.939136	+	0.939136i	0.0591150	+	0.998251i	$0.481172\pi$
$500$		0			0
$501$	8.36068	−	8.36068i	0.373528	−	0.373528i
$502$		0			0
$503$	11.6738	+	11.6738i	0.520507	+	0.520507i	0.917725	−	0.397217i	$-0.130024\pi$
$503$	11.6738	+	11.6738i	0.520507	+	0.520507i	−0.397217	+	0.917725i	$0.630024\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−10.5066	+	4.32624i	−0.466614	+	0.192135i
$508$		0			0
$509$	−9.76393	+	9.76393i	−0.432779	+	0.432779i	−0.889573	−	0.456794i	$-0.848998\pi$
$509$	−9.76393	+	9.76393i	−0.432779	+	0.432779i	0.456794	+	0.889573i	$0.348998\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	37.8885i		−	1.67282i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	23.4164	+	23.4164i	1.02985	+	1.02985i
$518$		0			0
$519$	21.2361			0.932160
$520$		0			0
$521$	41.1246			1.80170			0.900851	−	0.434128i	$-0.142944\pi$
$521$	41.1246			1.80170			0.900851	+	0.434128i	$0.142944\pi$
$522$		0			0
$523$	−25.2705	−	25.2705i	−1.10500	−	1.10500i	−0.993798	−	0.111205i	$-0.964529\pi$
$523$	−25.2705	−	25.2705i	−1.10500	−	1.10500i	−0.111205	−	0.993798i	$-0.535471\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			26.1803i			1.14043i
$528$		0			0
$529$			0.124612i			0.00541790i
$530$		0			0
$531$	−7.56231	+	7.56231i	−0.328176	+	0.328176i
$532$		0			0
$533$	−3.76393	+	18.8197i	−0.163034	+	0.815170i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−7.41641	−	7.41641i	−0.320042	−	0.320042i
$538$		0			0
$539$	−18.3262	+	18.3262i	−0.789367	+	0.789367i
$540$		0			0
$541$	14.4164	−	14.4164i	0.619810	−	0.619810i	−0.325673	−	0.945483i	$-0.605591\pi$
$541$	14.4164	−	14.4164i	0.619810	−	0.619810i	0.945483	+	0.325673i	$0.105591\pi$
$542$		0			0
$543$	5.12461	−	5.12461i	0.219918	−	0.219918i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	7.56231	−	7.56231i	0.323341	−	0.323341i	−0.526706	−	0.850047i	$-0.676573\pi$
$547$	7.56231	−	7.56231i	0.323341	−	0.323341i	0.850047	+	0.526706i	$0.176573\pi$
$548$		0			0
$549$		−	12.7639i		−	0.544751i
$550$		0			0
$551$	30.6525	+	30.6525i	1.30584	+	1.30584i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	35.2361i		−	1.49300i	−0.665385	−	0.746500i	$-0.731733\pi$
$557$		−	35.2361i		−	1.49300i	0.665385	−	0.746500i	$-0.268267\pi$
$558$		0			0
$559$	19.2705	+	3.85410i	0.815056	+	0.163011i
$560$		0			0
$561$	−7.23607	−	7.23607i	−0.305507	−	0.305507i
$562$		0			0
$563$	−1.09017	+	1.09017i	−0.0459452	+	0.0459452i	−0.729706	−	0.683761i	$-0.760343\pi$
$563$	−1.09017	+	1.09017i	−0.0459452	+	0.0459452i	0.683761	+	0.729706i	$0.260343\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−17.1246			−0.717901			−0.358951	−	0.933357i	$-0.616865\pi$
$569$	−17.1246			−0.717901			−0.358951	+	0.933357i	$0.616865\pi$
$570$		0			0
$571$		−	8.29180i		−	0.347001i	−0.984834	−	0.173500i	$-0.944492\pi$
$571$		−	8.29180i		−	0.347001i	0.984834	−	0.173500i	$-0.0555078\pi$
$572$		0			0
$573$	4.58359	+	4.58359i	0.191482	+	0.191482i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	27.1246			1.12921			0.564606	−	0.825360i	$-0.309028\pi$
$577$	27.1246			1.12921			0.564606	+	0.825360i	$0.309028\pi$
$578$		0			0
$579$	−2.11146	−	2.11146i	−0.0877491	−	0.0877491i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	7.70820			0.319241
$584$		0			0
$585$		0			0
$586$		0			0
$587$	23.1246			0.954455			0.477227	−	0.878780i	$-0.341642\pi$
$587$	23.1246			0.954455			0.477227	+	0.878780i	$0.341642\pi$
$588$		0			0
$589$	68.5410			2.82418
$590$		0			0
$591$	−7.41641	−	7.41641i	−0.305070	−	0.305070i
$592$		0			0
$593$	−16.5836			−0.681007			−0.340503	−	0.940243i	$-0.610597\pi$
$593$	−16.5836			−0.681007			−0.340503	+	0.940243i	$0.610597\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.05573	−	7.05573i	−0.288772	−	0.288772i
$598$		0			0
$599$		−	41.0132i		−	1.67575i	−0.545861	−	0.837876i	$-0.683797\pi$
$599$		−	41.0132i		−	1.67575i	0.545861	−	0.837876i	$-0.316203\pi$
$600$		0			0
$601$	14.0000			0.571072			0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000			0.571072			0.285536	+	0.958368i	$0.407828\pi$
$602$		0			0
$603$		−	0.652476i		−	0.0265709i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	29.2705	−	29.2705i	1.18805	−	1.18805i	0.210448	−	0.977605i	$-0.432508\pi$
$607$	29.2705	−	29.2705i	1.18805	−	1.18805i	0.977605	−	0.210448i	$-0.0674924\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	26.8328	−	17.8885i	1.08554	−	0.723693i
$612$		0			0
$613$		−	13.4164i		−	0.541884i	−0.962596	−	0.270942i	$-0.912665\pi$
$613$		−	13.4164i		−	0.541884i	0.962596	−	0.270942i	$-0.0873351\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.58359			−0.184528			−0.0922642	−	0.995735i	$-0.529410\pi$
$617$	−4.58359			−0.184528			−0.0922642	+	0.995735i	$0.529410\pi$
$618$		0			0
$619$	−17.2705	−	17.2705i	−0.694160	−	0.694160i	0.268984	−	0.963145i	$-0.413312\pi$
$619$	−17.2705	−	17.2705i	−0.694160	−	0.694160i	−0.963145	+	0.268984i	$0.913312\pi$
$620$		0			0
$621$			21.8885i			0.878357i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−18.9443	+	18.9443i	−0.756561	+	0.756561i
$628$		0			0
$629$	−21.7082	+	21.7082i	−0.865563	+	0.865563i
$630$		0			0
$631$	−29.2705	+	29.2705i	−1.16524	+	1.16524i	−0.181929	+	0.983312i	$0.558234\pi$
$631$	−29.2705	+	29.2705i	−1.16524	+	1.16524i	−0.983312	+	0.181929i	$0.941766\pi$
$632$		0			0
$633$	2.11146	+	2.11146i	0.0839228	+	0.0839228i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	14.0000	+	21.0000i	0.554700	+	0.832050i
$638$		0			0
$639$	5.85410	−	5.85410i	0.231585	−	0.231585i
$640$		0			0
$641$			32.9443i			1.30122i	0.759412	+	0.650610i	$0.225487\pi$
$641$			32.9443i			1.30122i	−0.759412	+	0.650610i	$0.774513\pi$
$642$		0			0
$643$		−	43.4164i		−	1.71218i	−0.516830	−	0.856088i	$-0.672888\pi$
$643$		−	43.4164i		−	1.71218i	0.516830	−	0.856088i	$-0.327112\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.09017	+	1.09017i	0.0428590	+	0.0428590i	0.728211	−	0.685352i	$-0.240352\pi$
$647$	1.09017	+	1.09017i	0.0428590	+	0.0428590i	−0.685352	+	0.728211i	$0.740352\pi$
$648$		0			0
$649$	17.7082			0.695108
$650$		0			0
$651$		0			0
$652$		0			0
$653$	29.0689	+	29.0689i	1.13755	+	1.13755i	0.988887	+	0.148666i	$0.0474979\pi$
$653$	29.0689	+	29.0689i	1.13755	+	1.13755i	0.148666	+	0.988887i	$0.452502\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			26.1803i			1.02139i
$658$		0			0
$659$			32.0689i			1.24923i	0.780934	+	0.624613i	$0.214743\pi$
$659$			32.0689i			1.24923i	−0.780934	+	0.624613i	$0.785257\pi$
$660$		0			0
$661$	10.7082	−	10.7082i	0.416501	−	0.416501i	−0.467495	−	0.883996i	$-0.654843\pi$
$661$	10.7082	−	10.7082i	0.416501	−	0.416501i	0.883996	+	0.467495i	$0.154843\pi$
$662$		0			0
$663$	−8.29180	+	5.52786i	−0.322027	+	0.214684i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−17.7082	−	17.7082i	−0.685664	−	0.685664i
$668$		0			0
$669$	4.58359	−	4.58359i	0.177212	−	0.177212i
$670$		0			0
$671$	−14.9443	+	14.9443i	−0.576917	+	0.576917i
$672$		0			0
$673$	−30.1246	+	30.1246i	−1.16122	+	1.16122i	−0.177009	+	0.984209i	$0.556642\pi$
$673$	−30.1246	+	30.1246i	−1.16122	+	1.16122i	−0.984209	+	0.177009i	$0.943358\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	0.0557281	−	0.0557281i	0.00214180	−	0.00214180i	−0.706035	−	0.708177i	$-0.749518\pi$
$677$	0.0557281	−	0.0557281i	0.00214180	−	0.00214180i	0.708177	+	0.706035i	$0.249518\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−5.12461	−	5.12461i	−0.196376	−	0.196376i
$682$		0			0
$683$	−35.1246			−1.34401			−0.672003	−	0.740548i	$-0.734566\pi$
$683$	−35.1246			−1.34401			−0.672003	+	0.740548i	$0.734566\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			10.4033i			0.396909i
$688$		0			0
$689$	1.47214	−	7.36068i	0.0560839	−	0.280420i
$690$		0			0
$691$	34.9787	+	34.9787i	1.33065	+	1.33065i	0.904786	+	0.425867i	$0.140031\pi$
$691$	34.9787	+	34.9787i	1.33065	+	1.33065i	0.425867	+	0.904786i	$0.359969\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$			16.8328i			0.637588i
$698$		0			0
$699$	9.23607			0.349340
$700$		0			0
$701$			28.3607i			1.07117i	0.844482	+	0.535584i	$0.179908\pi$
$701$			28.3607i			1.07117i	−0.844482	+	0.535584i	$0.820092\pi$
$702$		0			0
$703$	56.8328	+	56.8328i	2.14349	+	2.14349i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−18.4164	−	18.4164i	−0.691643	−	0.691643i	0.270951	−	0.962593i	$-0.412662\pi$
$709$	−18.4164	−	18.4164i	−0.691643	−	0.691643i	−0.962593	+	0.270951i	$0.912662\pi$
$710$		0			0
$711$	8.29180			0.310967
$712$		0			0
$713$	−39.5967			−1.48291
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−10.6525			−0.397824
$718$		0			0
$719$	31.4164			1.17163			0.585817	−	0.810443i	$-0.300774\pi$
$719$	31.4164			1.17163			0.585817	+	0.810443i	$0.300774\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−3.34752			−0.124496
$724$		0			0
$725$		0			0
$726$		0			0
$727$	21.8541	+	21.8541i	0.810524	+	0.810524i	0.984712	−	0.174189i	$-0.0557302\pi$
$727$	21.8541	+	21.8541i	0.810524	+	0.810524i	−0.174189	+	0.984712i	$0.555730\pi$
$728$		0			0
$729$		−	5.94427i		−	0.220158i
$730$		0			0
$731$	17.2361			0.637499
$732$		0			0
$733$			30.0000i			1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$			30.0000i			1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.763932	+	0.763932i	−0.0281398	+	0.0281398i
$738$		0			0
$739$	−7.56231	−	7.56231i	−0.278184	−	0.278184i	0.554200	−	0.832384i	$-0.313024\pi$
$739$	−7.56231	−	7.56231i	−0.278184	−	0.278184i	−0.832384	+	0.554200i	$0.813024\pi$
$740$		0			0
$741$	14.4721	+	21.7082i	0.531647	+	0.797471i
$742$		0			0
$743$			37.3050i			1.36859i	0.729207	+	0.684293i	$0.239889\pi$
$743$			37.3050i			1.36859i	−0.729207	+	0.684293i	$0.760111\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	23.4164			0.856762
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	11.1246i		−	0.405943i	−0.979185	−	0.202971i	$-0.934940\pi$
$751$		−	11.1246i		−	0.405943i	0.979185	−	0.202971i	$-0.0650599\pi$
$752$		0			0
$753$	16.1803	−	16.1803i	0.589644	−	0.589644i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	13.2918	−	13.2918i	0.483099	−	0.483099i	−0.423021	−	0.906120i	$-0.639030\pi$
$757$	13.2918	−	13.2918i	0.483099	−	0.483099i	0.906120	+	0.423021i	$0.139030\pi$
$758$		0			0
$759$	10.9443	−	10.9443i	0.397252	−	0.397252i
$760$		0			0
$761$	−25.4721	+	25.4721i	−0.923364	+	0.923364i	−0.997266	−	0.0739013i	$-0.976455\pi$
$761$	−25.4721	+	25.4721i	−0.923364	+	0.923364i	0.0739013	+	0.997266i	$0.476455\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.38197	−	16.9098i	0.122116	−	0.610579i
$768$		0			0
$769$	−5.00000	+	5.00000i	−0.180305	+	0.180305i	−0.791489	−	0.611184i	$-0.790694\pi$
$769$	−5.00000	+	5.00000i	−0.180305	+	0.180305i	0.611184	+	0.791489i	$0.290694\pi$
$770$		0			0
$771$		−	21.2361i		−	0.764798i
$772$		0			0
$773$		−	30.6525i		−	1.10249i	−0.834342	−	0.551246i	$-0.814152\pi$
$773$		−	30.6525i		−	1.10249i	0.834342	−	0.551246i	$-0.185848\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	44.0689			1.57893
$780$		0			0
$781$	−13.7082			−0.490518
$782$		0			0
$783$	16.9443	+	16.9443i	0.605539	+	0.605539i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			16.5836i			0.591141i	0.955321	+	0.295571i	$0.0955098\pi$
$787$			16.5836i			0.591141i	−0.955321	+	0.295571i	$0.904490\pi$
$788$		0			0
$789$		−	27.2361i		−	0.969630i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	11.4164	+	17.1246i	0.405409	+	0.608113i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−26.8885	−	26.8885i	−0.952441	−	0.952441i	0.0464782	−	0.998919i	$-0.485200\pi$
$797$	−26.8885	−	26.8885i	−0.952441	−	0.952441i	−0.998919	+	0.0464782i	$0.985200\pi$
$798$		0			0
$799$	20.0000	−	20.0000i	0.707549	−	0.707549i
$800$		0			0
$801$	−5.00000	+	5.00000i	−0.176666	+	0.176666i
$802$		0			0
$803$	30.6525	−	30.6525i	1.08170	−	1.08170i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−8.36068	+	8.36068i	−0.294310	+	0.294310i
$808$		0			0
$809$		−	26.1803i		−	0.920452i	−0.887802	−	0.460226i	$-0.847768\pi$
$809$		−	26.1803i		−	0.920452i	0.887802	−	0.460226i	$-0.152232\pi$
$810$		0			0
$811$	3.56231	+	3.56231i	0.125089	+	0.125089i	0.766880	−	0.641791i	$-0.221808\pi$
$811$	3.56231	+	3.56231i	0.125089	+	0.125089i	−0.641791	+	0.766880i	$0.721808\pi$
$812$		0			0
$813$	−9.34752			−0.327832
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	45.1246i		−	1.57871i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−37.3607	−	37.3607i	−1.30390	−	1.30390i	−0.925743	−	0.378154i	$-0.876559\pi$
$821$	−37.3607	−	37.3607i	−1.30390	−	1.30390i	−0.378154	−	0.925743i	$-0.623441\pi$
$822$		0			0
$823$	−22.9787	+	22.9787i	−0.800988	+	0.800988i	−0.983250	−	0.182262i	$-0.941658\pi$
$823$	−22.9787	+	22.9787i	−0.800988	+	0.800988i	0.182262	+	0.983250i	$0.441658\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			22.4721i			0.781433i	0.920511	+	0.390716i	$0.127773\pi$
$827$			22.4721i			0.781433i	−0.920511	+	0.390716i	$0.872227\pi$
$828$		0			0
$829$	−13.7082			−0.476106			−0.238053	−	0.971252i	$-0.576509\pi$
$829$	−13.7082			−0.476106			−0.238053	+	0.971252i	$0.576509\pi$
$830$		0			0
$831$			3.34752i			0.116124i
$832$		0			0
$833$	15.6525	+	15.6525i	0.542326	+	0.542326i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	37.8885			1.30962
$838$		0			0
$839$	−22.7984	−	22.7984i	−0.787087	−	0.787087i	0.193928	−	0.981016i	$-0.437877\pi$
$839$	−22.7984	−	22.7984i	−0.787087	−	0.787087i	−0.981016	+	0.193928i	$0.937877\pi$
$840$		0			0
$841$	1.58359			0.0546066
$842$		0			0
$843$	2.76393			0.0951949
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	19.5967			0.672559
$850$		0			0
$851$	−32.8328	−	32.8328i	−1.12549	−	1.12549i
$852$		0			0
$853$	4.87539			0.166930			0.0834651	−	0.996511i	$-0.473401\pi$
$853$	4.87539			0.166930			0.0834651	+	0.996511i	$0.473401\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.65248	+	9.65248i	0.329722	+	0.329722i	0.852481	−	0.522759i	$-0.175097\pi$
$857$	9.65248	+	9.65248i	0.329722	+	0.329722i	−0.522759	+	0.852481i	$0.675097\pi$
$858$		0			0
$859$			51.7082i			1.76426i	0.471005	+	0.882131i	$0.343891\pi$
$859$			51.7082i			1.76426i	−0.471005	+	0.882131i	$0.656109\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			34.4721i			1.17345i	0.809788	+	0.586723i	$0.199582\pi$
$863$			34.4721i			1.17345i	−0.809788	+	0.586723i	$0.800418\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	4.32624	−	4.32624i	0.146927	−	0.146927i
$868$		0			0
$869$	−9.70820	−	9.70820i	−0.329328	−	0.329328i
$870$		0			0
$871$	0.583592	+	0.875388i	0.0197743	+	0.0296614i
$872$		0			0
$873$			34.4721i			1.16671i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−46.8328			−1.58143			−0.790716	−	0.612183i	$-0.790291\pi$
$877$	−46.8328			−1.58143			−0.790716	+	0.612183i	$0.790291\pi$
$878$		0			0
$879$	−17.1246	−	17.1246i	−0.577599	−	0.577599i
$880$		0			0
$881$			20.0689i			0.676138i	0.941121	+	0.338069i	$0.109774\pi$
$881$			20.0689i			0.676138i	−0.941121	+	0.338069i	$0.890226\pi$
$882$		0			0
$883$	27.8541	−	27.8541i	0.937365	−	0.937365i	−0.0607857	−	0.998151i	$-0.519361\pi$
$883$	27.8541	−	27.8541i	0.937365	−	0.937365i	0.998151	+	0.0607857i	$0.0193606\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21.3820	+	21.3820i	−0.717936	+	0.717936i	−0.968182	−	0.250246i	$-0.919488\pi$
$887$	−21.3820	+	21.3820i	−0.717936	+	0.717936i	0.250246	+	0.968182i	$0.419488\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	7.09017	−	7.09017i	0.237530	−	0.237530i
$892$		0			0
$893$	−52.3607	−	52.3607i	−1.75218	−	1.75218i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−8.36068	−	12.5410i	−0.279155	−	0.418732i
$898$		0			0
$899$	−30.6525	+	30.6525i	−1.02232	+	1.02232i
$900$		0			0
$901$		−	6.58359i		−	0.219331i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	5.27051	+	5.27051i	0.175004	+	0.175004i	0.789174	−	0.614170i	$-0.210509\pi$
$907$	5.27051	+	5.27051i	0.175004	+	0.175004i	−0.614170	+	0.789174i	$0.710509\pi$
$908$		0			0
$909$	3.41641			0.113315
$910$		0			0
$911$	50.8328			1.68417			0.842083	−	0.539348i	$-0.181329\pi$
$911$	50.8328			1.68417			0.842083	+	0.539348i	$0.181329\pi$
$912$		0			0
$913$	−27.4164	−	27.4164i	−0.907351	−	0.907351i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$			35.1246i			1.15865i	0.815095	+	0.579327i	$0.196685\pi$
$919$			35.1246i			1.15865i	−0.815095	+	0.579327i	$0.803315\pi$
$920$		0			0
$921$	7.41641	−	7.41641i	0.244379	−	0.244379i
$922$		0			0
$923$	−2.61803	+	13.0902i	−0.0861736	+	0.430868i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−8.61803	−	8.61803i	−0.283053	−	0.283053i
$928$		0			0
$929$	−2.88854	+	2.88854i	−0.0947700	+	0.0947700i	−0.752902	−	0.658132i	$-0.771347\pi$
$929$	−2.88854	+	2.88854i	−0.0947700	+	0.0947700i	0.658132	+	0.752902i	$0.271347\pi$
$930$		0			0
$931$	40.9787	−	40.9787i	1.34302	−	1.34302i
$932$		0			0
$933$	1.34752	−	1.34752i	0.0441160	−	0.0441160i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−32.4164	+	32.4164i	−1.05900	+	1.05900i	−0.0608510	+	0.998147i	$0.519381\pi$
$937$	−32.4164	+	32.4164i	−1.05900	+	1.05900i	−0.998147	+	0.0608510i	$0.980619\pi$
$938$		0			0
$939$			18.1803i			0.593293i
$940$		0			0
$941$	−13.3607	−	13.3607i	−0.435546	−	0.435546i	0.454964	−	0.890510i	$-0.349652\pi$
$941$	−13.3607	−	13.3607i	−0.435546	−	0.435546i	−0.890510	+	0.454964i	$0.849652\pi$
$942$		0			0
$943$	−25.4590			−0.829058
$944$		0			0
$945$		0			0
$946$		0			0
$947$			3.05573i			0.0992978i	0.998767	+	0.0496489i	$0.0158102\pi$
$947$			3.05573i			0.0992978i	−0.998767	+	0.0496489i	$0.984190\pi$
$948$		0			0
$949$	−23.4164	−	35.1246i	−0.760129	−	1.14019i
$950$		0			0
$951$	2.29180	+	2.29180i	0.0743166	+	0.0743166i
$952$		0			0
$953$	27.7639	−	27.7639i	0.899362	−	0.899362i	−0.0960177	−	0.995380i	$-0.530611\pi$
$953$	27.7639	−	27.7639i	0.899362	−	0.899362i	0.995380	+	0.0960177i	$0.0306105\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	16.9443i		−	0.547731i
$958$		0			0
$959$		0			0
$960$		0			0
$961$			37.5410i			1.21100i
$962$		0			0
$963$	−5.85410	−	5.85410i	−0.188646	−	0.188646i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−7.12461			−0.229112			−0.114556	−	0.993417i	$-0.536545\pi$
$967$	−7.12461			−0.229112			−0.114556	+	0.993417i	$0.536545\pi$
$968$		0			0
$969$	16.1803	+	16.1803i	0.519787	+	0.519787i
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.70820			−0.118636			−0.0593180	−	0.998239i	$-0.518893\pi$
$977$	−3.70820			−0.118636			−0.0593180	+	0.998239i	$0.518893\pi$
$978$		0			0
$979$	11.7082			0.374196
$980$		0			0
$981$	−10.5279	−	10.5279i	−0.336129	−	0.336129i
$982$		0			0
$983$	32.2918			1.02995			0.514974	−	0.857206i	$-0.327802\pi$
$983$	32.2918			1.02995			0.514974	+	0.857206i	$0.327802\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			26.0689i			0.828942i
$990$		0			0
$991$	−20.5836			−0.653859			−0.326930	−	0.945049i	$-0.606014\pi$
$991$	−20.5836			−0.653859			−0.326930	+	0.945049i	$0.606014\pi$
$992$		0			0
$993$		−	19.5967i		−	0.621884i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	29.0000	−	29.0000i	0.918439	−	0.918439i	−0.0784767	−	0.996916i	$-0.525006\pi$
$997$	29.0000	−	29.0000i	0.918439	−	0.918439i	0.996916	+	0.0784767i	$0.0250056\pi$
$998$		0			0
$999$	31.4164	+	31.4164i	0.993971	+	0.993971i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.m.b.593.2		4
5.2	odd	4		1300.2.r.b.957.2		4
5.3	odd	4		260.2.r.b.177.1	yes	4
5.4	even	2		260.2.m.b.73.1	yes	4
13.5	odd	4		1300.2.r.b.993.2		4
15.8	even	4		2340.2.bp.e.1477.2		4
15.14	odd	2		2340.2.u.f.73.2		4
20.3	even	4		1040.2.cd.j.177.2		4
20.19	odd	2		1040.2.bg.j.593.2		4
65.18	even	4		260.2.m.b.57.1	✓	4
65.44	odd	4		260.2.r.b.213.1	yes	4
65.57	even	4	inner	1300.2.m.b.57.2		4
195.44	even	4		2340.2.bp.e.1513.2		4
195.83	odd	4		2340.2.u.f.577.1		4
260.83	odd	4		1040.2.bg.j.577.2		4
260.239	even	4		1040.2.cd.j.993.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.m.b.57.1	✓	4	65.18	even	4
260.2.m.b.73.1	yes	4	5.4	even	2
260.2.r.b.177.1	yes	4	5.3	odd	4
260.2.r.b.213.1	yes	4	65.44	odd	4
1040.2.bg.j.577.2		4	260.83	odd	4
1040.2.bg.j.593.2		4	20.19	odd	2
1040.2.cd.j.177.2		4	20.3	even	4
1040.2.cd.j.993.2		4	260.239	even	4
1300.2.m.b.57.2		4	65.57	even	4	inner
1300.2.m.b.593.2		4	1.1	even	1	trivial
1300.2.r.b.957.2		4	5.2	odd	4
1300.2.r.b.993.2		4	13.5	odd	4
2340.2.u.f.73.2		4	15.14	odd	2
2340.2.u.f.577.1		4	195.83	odd	4
2340.2.bp.e.1477.2		4	15.8	even	4
2340.2.bp.e.1513.2		4	195.44	even	4

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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.m (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.618034 + 0.618034i) q^{3} -2.23607i q^{9} +(-2.61803 + 2.61803i) q^{11} +(2.00000 + 3.00000i) q^{13} +(2.23607 + 2.23607i) q^{17} +(5.85410 - 5.85410i) q^{19} +(-3.38197 + 3.38197i) q^{23} +(3.23607 - 3.23607i) q^{27} +5.23607i q^{29} +(5.85410 + 5.85410i) q^{31} -3.23607 q^{33} +9.70820i q^{37} +(-0.618034 + 3.09017i) q^{39} +(3.76393 + 3.76393i) q^{41} +(3.85410 - 3.85410i) q^{43} -8.94427i q^{47} +7.00000 q^{49} +2.76393i q^{51} +(-1.47214 - 1.47214i) q^{53} +7.23607 q^{57} +(-3.38197 - 3.38197i) q^{59} +5.70820 q^{61} +0.291796 q^{67} -4.18034 q^{69} +(2.61803 + 2.61803i) q^{71} -11.7082 q^{73} +3.70820i q^{79} -2.70820 q^{81} +10.4721i q^{83} +(-3.23607 + 3.23607i) q^{87} +(-2.23607 - 2.23607i) q^{89} +7.23607i q^{93} -15.4164 q^{97} +(5.85410 + 5.85410i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 6 q^{11} + 8 q^{13} + 10 q^{19} - 18 q^{23} + 4 q^{27} + 10 q^{31} - 4 q^{33} + 2 q^{39} + 24 q^{41} + 2 q^{43} + 28 q^{49} + 12 q^{53} + 20 q^{57} - 18 q^{59} - 4 q^{61} + 28 q^{67} + 28 q^{69}+ \cdots + 10 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 6 * q^11 + 8 * q^13 + 10 * q^19 - 18 * q^23 + 4 * q^27 + 10 * q^31 - 4 * q^33 + 2 * q^39 + 24 * q^41 + 2 * q^43 + 28 * q^49 + 12 * q^53 + 20 * q^57 - 18 * q^59 - 4 * q^61 + 28 * q^67 + 28 * q^69 + 6 * q^71 - 20 * q^73 + 16 * q^81 - 4 * q^87 - 8 * q^97 + 10 * q^99

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