Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [605,2,Mod(112,605)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("605.112"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([5, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.m (of order \(20\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,2,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: 16.0.3429742096000000000000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{14} + 16x^{12} + 35x^{10} + 31x^{8} + 315x^{6} + 1296x^{4} + 3645x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{U}(1)[D_{20}]$

Embedding invariants

Embedding label 602.1
Root \(-0.0369185 - 1.73166i\) of defining polynomial
Character \(\chi\) \(=\) 605.602
Dual form 605.2.m.a.403.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.71963 - 1.38572i) q^{3} +(1.90211 + 0.618034i) q^{4} +(-2.22328 + 0.238794i) q^{5} +(3.71282 + 5.11026i) q^{9} +(-4.31662 - 4.31662i) q^{12} +(6.37741 + 2.43142i) q^{15} +(3.23607 + 2.35114i) q^{16} +(-4.37651 - 0.919850i) q^{20} +(2.84169 - 2.84169i) q^{23} +(4.88595 - 1.06181i) q^{25} +(-1.58365 - 9.99875i) q^{27} +(8.04962 - 5.84839i) q^{31} +(3.90389 + 12.0149i) q^{36} +(1.85853 - 0.946968i) q^{37} +(-9.47494 - 10.4749i) q^{45} +(5.98164 - 11.7396i) q^{47} +(-5.54289 - 10.8785i) q^{48} +(4.11450 - 5.66312i) q^{49} +(-0.803790 + 5.07493i) q^{53} +(-3.15430 - 1.02489i) q^{59} +(10.6279 + 8.56629i) q^{60} +(4.70228 + 6.47214i) q^{64} +(11.4749 + 11.4749i) q^{67} +(-11.6661 + 3.79056i) q^{69} +(2.42705 + 1.76336i) q^{71} +(-14.7594 - 3.88284i) q^{75} +(-7.75613 - 4.45449i) q^{80} +(-3.69271 + 11.3650i) q^{81} +9.00000i q^{89} +(7.16147 - 3.64895i) q^{92} +(-29.9962 + 4.75094i) q^{93} +(4.92382 + 0.779856i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{3} - 16 q^{12} + 8 q^{15} + 16 q^{16} - 12 q^{20} + 72 q^{23} - 2 q^{25} - 22 q^{27} - 24 q^{36} + 14 q^{37} - 72 q^{45} + 24 q^{47} - 8 q^{48} + 12 q^{53} + 28 q^{60} + 104 q^{67} + 12 q^{71}+ \cdots + 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{10}\right)\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 −0.987688 − 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(3\) −2.71963 − 1.38572i −1.57018 − 0.800047i −0.570408 − 0.821362i \(-0.693215\pi\)
−0.999773 + 0.0213149i \(0.993215\pi\)
\(4\) 1.90211 + 0.618034i 0.951057 + 0.309017i
\(5\) −2.22328 + 0.238794i −0.994281 + 0.106792i
\(6\) 0 0
\(7\) 0 0 0.891007 − 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(8\) 0 0
\(9\) 3.71282 + 5.11026i 1.23761 + 1.70342i
\(10\) 0 0
\(11\) 0 0
\(12\) −4.31662 − 4.31662i −1.24610 − 1.24610i
\(13\) 0 0 0.156434 − 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(14\) 0 0
\(15\) 6.37741 + 2.43142i 1.64664 + 0.627789i
\(16\) 3.23607 + 2.35114i 0.809017 + 0.587785i
\(17\) 0 0 −0.156434 − 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(18\) 0 0
\(19\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(20\) −4.37651 − 0.919850i −0.978618 − 0.205685i
\(21\) 0 0
\(22\) 0 0
\(23\) 2.84169 − 2.84169i 0.592533 − 0.592533i −0.345782 − 0.938315i \(-0.612386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) 0 0
\(25\) 4.88595 − 1.06181i 0.977191 − 0.212362i
\(26\) 0 0
\(27\) −1.58365 − 9.99875i −0.304773 − 1.92426i
\(28\) 0 0
\(29\) 0 0 −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(30\) 0 0
\(31\) 8.04962 − 5.84839i 1.44575 − 1.05040i 0.458954 − 0.888460i \(-0.348224\pi\)
0.986800 − 0.161942i \(-0.0517756\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 3.90389 + 12.0149i 0.650648 + 2.00249i
\(37\) 1.85853 − 0.946968i 0.305540 − 0.155681i −0.294497 − 0.955652i \(-0.595152\pi\)
0.600038 + 0.799972i \(0.295152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) −9.47494 − 10.4749i −1.41244 − 1.56151i
\(46\) 0 0
\(47\) 5.98164 − 11.7396i 0.872512 − 1.71240i 0.189653 − 0.981851i \(-0.439264\pi\)
0.682859 − 0.730550i \(-0.260736\pi\)
\(48\) −5.54289 − 10.8785i −0.800047 − 1.57018i
\(49\) 4.11450 − 5.66312i 0.587785 − 0.809017i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.803790 + 5.07493i −0.110409 + 0.697095i 0.868940 + 0.494918i \(0.164802\pi\)
−0.979349 + 0.202178i \(0.935198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.15430 − 1.02489i −0.410655 − 0.133430i 0.0964021 − 0.995342i \(-0.469267\pi\)
−0.507057 + 0.861913i \(0.669267\pi\)
\(60\) 10.6279 + 8.56629i 1.37205 + 1.10590i
\(61\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4.70228 + 6.47214i 0.587785 + 0.809017i
\(65\) 0 0
\(66\) 0 0
\(67\) 11.4749 + 11.4749i 1.40189 + 1.40189i 0.794101 + 0.607785i \(0.207942\pi\)
0.607785 + 0.794101i \(0.292058\pi\)
\(68\) 0 0
\(69\) −11.6661 + 3.79056i −1.40444 + 0.456329i
\(70\) 0 0
\(71\) 2.42705 + 1.76336i 0.288038 + 0.209272i 0.722416 − 0.691459i \(-0.243032\pi\)
−0.434378 + 0.900731i \(0.643032\pi\)
\(72\) 0 0
\(73\) 0 0 −0.453990 − 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(74\) 0 0
\(75\) −14.7594 − 3.88284i −1.70427 − 0.448351i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(80\) −7.75613 − 4.45449i −0.867161 − 0.498027i
\(81\) −3.69271 + 11.3650i −0.410302 + 1.26278i
\(82\) 0 0
\(83\) 0 0 0.987688 − 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000i 0.953998i 0.878904 + 0.476999i \(0.158275\pi\)
−0.878904 + 0.476999i \(0.841725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 7.16147 − 3.64895i 0.746635 − 0.380429i
\(93\) −29.9962 + 4.75094i −3.11047 + 0.492649i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.92382 + 0.779856i 0.499938 + 0.0791824i 0.401310 − 0.915942i \(-0.368555\pi\)
0.0986273 + 0.995124i \(0.468555\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 605.2.m.a.602.1 16
5.3 odd 4 inner 605.2.m.a.118.2 16
11.2 odd 10 55.2.e.b.32.1 ✓ 4
11.3 even 5 inner 605.2.m.a.457.2 16
11.4 even 5 inner 605.2.m.a.282.2 16
11.5 even 5 inner 605.2.m.a.112.2 16
11.6 odd 10 inner 605.2.m.a.112.2 16
11.7 odd 10 inner 605.2.m.a.282.2 16
11.8 odd 10 inner 605.2.m.a.457.2 16
11.9 even 5 55.2.e.b.32.1 ✓ 4
11.10 odd 2 CM 605.2.m.a.602.1 16
33.2 even 10 495.2.k.a.307.1 4
33.20 odd 10 495.2.k.a.307.1 4
44.31 odd 10 880.2.bd.d.417.2 4
44.35 even 10 880.2.bd.d.417.2 4
55.2 even 20 275.2.e.a.43.2 4
55.3 odd 20 inner 605.2.m.a.578.2 16
55.8 even 20 inner 605.2.m.a.578.2 16
55.9 even 10 275.2.e.a.32.2 4
55.13 even 20 55.2.e.b.43.1 yes 4
55.18 even 20 inner 605.2.m.a.403.1 16
55.24 odd 10 275.2.e.a.32.2 4
55.28 even 20 inner 605.2.m.a.233.2 16
55.38 odd 20 inner 605.2.m.a.233.2 16
55.42 odd 20 275.2.e.a.43.2 4
55.43 even 4 inner 605.2.m.a.118.2 16
55.48 odd 20 inner 605.2.m.a.403.1 16
55.53 odd 20 55.2.e.b.43.1 yes 4
165.53 even 20 495.2.k.a.208.1 4
165.68 odd 20 495.2.k.a.208.1 4
220.123 odd 20 880.2.bd.d.593.2 4
220.163 even 20 880.2.bd.d.593.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.e.b.32.1 ✓ 4 11.2 odd 10
55.2.e.b.32.1 ✓ 4 11.9 even 5
55.2.e.b.43.1 yes 4 55.13 even 20
55.2.e.b.43.1 yes 4 55.53 odd 20
275.2.e.a.32.2 4 55.9 even 10
275.2.e.a.32.2 4 55.24 odd 10
275.2.e.a.43.2 4 55.2 even 20
275.2.e.a.43.2 4 55.42 odd 20
495.2.k.a.208.1 4 165.53 even 20
495.2.k.a.208.1 4 165.68 odd 20
495.2.k.a.307.1 4 33.2 even 10
495.2.k.a.307.1 4 33.20 odd 10
605.2.m.a.112.2 16 11.5 even 5 inner
605.2.m.a.112.2 16 11.6 odd 10 inner
605.2.m.a.118.2 16 5.3 odd 4 inner
605.2.m.a.118.2 16 55.43 even 4 inner
605.2.m.a.233.2 16 55.28 even 20 inner
605.2.m.a.233.2 16 55.38 odd 20 inner
605.2.m.a.282.2 16 11.4 even 5 inner
605.2.m.a.282.2 16 11.7 odd 10 inner
605.2.m.a.403.1 16 55.18 even 20 inner
605.2.m.a.403.1 16 55.48 odd 20 inner
605.2.m.a.457.2 16 11.3 even 5 inner
605.2.m.a.457.2 16 11.8 odd 10 inner
605.2.m.a.578.2 16 55.3 odd 20 inner
605.2.m.a.578.2 16 55.8 even 20 inner
605.2.m.a.602.1 16 1.1 even 1 trivial
605.2.m.a.602.1 16 11.10 odd 2 CM
880.2.bd.d.417.2 4 44.31 odd 10
880.2.bd.d.417.2 4 44.35 even 10
880.2.bd.d.593.2 4 220.123 odd 20
880.2.bd.d.593.2 4 220.163 even 20