Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,2,Mod(32,275)] 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("275.32"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 32.1
Root \(-1.58114 + 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 275.32
Dual form 275.2.e.b.43.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.58114 + 1.58114i) q^{2} +(1.00000 - 1.00000i) q^{3} -3.00000i q^{4} +3.16228i q^{6} +(1.58114 + 1.58114i) q^{8} +1.00000i q^{9} +(1.00000 - 3.16228i) q^{11} +(-3.00000 - 3.00000i) q^{12} +(3.16228 + 3.16228i) q^{13} +1.00000 q^{16} +(3.16228 - 3.16228i) q^{17} +(-1.58114 - 1.58114i) q^{18} +6.32456 q^{19} +(3.41886 + 6.58114i) q^{22} +(1.00000 - 1.00000i) q^{23} +3.16228 q^{24} -10.0000 q^{26} +(4.00000 + 4.00000i) q^{27} -6.32456 q^{29} +2.00000 q^{31} +(-4.74342 + 4.74342i) q^{32} +(-2.16228 - 4.16228i) q^{33} +10.0000i q^{34} +3.00000 q^{36} +(-3.00000 - 3.00000i) q^{37} +(-10.0000 + 10.0000i) q^{38} +6.32456 q^{39} +6.32456i q^{41} +(-9.48683 - 3.00000i) q^{44} +3.16228i q^{46} +(-3.00000 - 3.00000i) q^{47} +(1.00000 - 1.00000i) q^{48} +7.00000i q^{49} -6.32456i q^{51} +(9.48683 - 9.48683i) q^{52} +(1.00000 - 1.00000i) q^{53} -12.6491 q^{54} +(6.32456 - 6.32456i) q^{57} +(10.0000 - 10.0000i) q^{58} -6.00000i q^{59} -6.32456i q^{61} +(-3.16228 + 3.16228i) q^{62} -13.0000i q^{64} +(10.0000 + 3.16228i) q^{66} +(-3.00000 - 3.00000i) q^{67} +(-9.48683 - 9.48683i) q^{68} -2.00000i q^{69} -8.00000 q^{71} +(-1.58114 + 1.58114i) q^{72} +(-3.16228 - 3.16228i) q^{73} +9.48683 q^{74} -18.9737i q^{76} +(-10.0000 + 10.0000i) q^{78} -6.32456 q^{79} +5.00000 q^{81} +(-10.0000 - 10.0000i) q^{82} +(6.32456 + 6.32456i) q^{83} +(-6.32456 + 6.32456i) q^{87} +(6.58114 - 3.41886i) q^{88} -6.00000i q^{89} +(-3.00000 - 3.00000i) q^{92} +(2.00000 - 2.00000i) q^{93} +9.48683 q^{94} +9.48683i q^{96} +(7.00000 + 7.00000i) q^{97} +(-11.0680 - 11.0680i) q^{98} +(3.16228 + 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{11} - 12 q^{12} + 4 q^{16} + 20 q^{22} + 4 q^{23} - 40 q^{26} + 16 q^{27} + 8 q^{31} + 4 q^{33} + 12 q^{36} - 12 q^{37} - 40 q^{38} - 12 q^{47} + 4 q^{48} + 4 q^{53} + 40 q^{58} + 40 q^{66}+ \cdots + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.58114 + 1.58114i −1.11803 + 1.11803i −0.126004 + 0.992030i \(0.540215\pi\)
−0.992030 + 0.126004i \(0.959785\pi\)
\(3\) 1.00000 − 1.00000i 0.577350 − 0.577350i −0.356822 − 0.934172i \(-0.616140\pi\)
0.934172 + 0.356822i \(0.116140\pi\)
\(4\) − 3.00000i − 1.50000i
\(5\) 0 0
\(6\) 3.16228i 1.29099i
\(7\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 1.58114 + 1.58114i 0.559017 + 0.559017i
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 1.00000 − 3.16228i 0.301511 − 0.953463i
\(12\) −3.00000 − 3.00000i −0.866025 − 0.866025i
\(13\) 3.16228 + 3.16228i 0.877058 + 0.877058i 0.993229 − 0.116171i \(-0.0370621\pi\)
−0.116171 + 0.993229i \(0.537062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.16228 − 3.16228i 0.766965 − 0.766965i −0.210606 − 0.977571i \(-0.567544\pi\)
0.977571 + 0.210606i \(0.0675437\pi\)
\(18\) −1.58114 − 1.58114i −0.372678 − 0.372678i
\(19\) 6.32456 1.45095 0.725476 − 0.688247i \(-0.241620\pi\)
0.725476 + 0.688247i \(0.241620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.41886 + 6.58114i 0.728904 + 1.40310i
\(23\) 1.00000 − 1.00000i 0.208514 − 0.208514i −0.595121 − 0.803636i \(-0.702896\pi\)
0.803636 + 0.595121i \(0.202896\pi\)
\(24\) 3.16228 0.645497
\(25\) 0 0
\(26\) −10.0000 −1.96116
\(27\) 4.00000 + 4.00000i 0.769800 + 0.769800i
\(28\) 0 0
\(29\) −6.32456 −1.17444 −0.587220 − 0.809427i \(-0.699778\pi\)
−0.587220 + 0.809427i \(0.699778\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −4.74342 + 4.74342i −0.838525 + 0.838525i
\(33\) −2.16228 − 4.16228i −0.376404 − 0.724560i
\(34\) 10.0000i 1.71499i
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) −3.00000 − 3.00000i −0.493197 − 0.493197i 0.416115 − 0.909312i \(-0.363391\pi\)
−0.909312 + 0.416115i \(0.863391\pi\)
\(38\) −10.0000 + 10.0000i −1.62221 + 1.62221i
\(39\) 6.32456 1.01274
\(40\) 0 0
\(41\) 6.32456i 0.987730i 0.869539 + 0.493865i \(0.164416\pi\)
−0.869539 + 0.493865i \(0.835584\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) −9.48683 − 3.00000i −1.43019 − 0.452267i
\(45\) 0 0
\(46\) 3.16228i 0.466252i
\(47\) −3.00000 − 3.00000i −0.437595 − 0.437595i 0.453607 − 0.891202i \(-0.350137\pi\)
−0.891202 + 0.453607i \(0.850137\pi\)
\(48\) 1.00000 − 1.00000i 0.144338 − 0.144338i
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) − 6.32456i − 0.885615i
\(52\) 9.48683 − 9.48683i 1.31559 − 1.31559i
\(53\) 1.00000 − 1.00000i 0.137361 − 0.137361i −0.635083 − 0.772444i \(-0.719034\pi\)
0.772444 + 0.635083i \(0.219034\pi\)
\(54\) −12.6491 −1.72133
\(55\) 0 0
\(56\) 0 0
\(57\) 6.32456 − 6.32456i 0.837708 − 0.837708i
\(58\) 10.0000 − 10.0000i 1.31306 − 1.31306i
\(59\) − 6.00000i − 0.781133i −0.920575 − 0.390567i \(-0.872279\pi\)
0.920575 − 0.390567i \(-0.127721\pi\)
\(60\) 0 0
\(61\) − 6.32456i − 0.809776i −0.914366 − 0.404888i \(-0.867310\pi\)
0.914366 − 0.404888i \(-0.132690\pi\)
\(62\) −3.16228 + 3.16228i −0.401610 + 0.401610i
\(63\) 0 0
\(64\) − 13.0000i − 1.62500i
\(65\) 0 0
\(66\) 10.0000 + 3.16228i 1.23091 + 0.389249i
\(67\) −3.00000 − 3.00000i −0.366508 − 0.366508i 0.499694 − 0.866202i \(-0.333446\pi\)
−0.866202 + 0.499694i \(0.833446\pi\)
\(68\) −9.48683 − 9.48683i −1.15045 − 1.15045i
\(69\) − 2.00000i − 0.240772i
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.58114 + 1.58114i −0.186339 + 0.186339i
\(73\) −3.16228 − 3.16228i −0.370117 − 0.370117i 0.497403 − 0.867520i \(-0.334287\pi\)
−0.867520 + 0.497403i \(0.834287\pi\)
\(74\) 9.48683 1.10282
\(75\) 0 0
\(76\) − 18.9737i − 2.17643i
\(77\) 0 0
\(78\) −10.0000 + 10.0000i −1.13228 + 1.13228i
\(79\) −6.32456 −0.711568 −0.355784 − 0.934568i \(-0.615786\pi\)
−0.355784 + 0.934568i \(0.615786\pi\)
\(80\) 0 0
\(81\) 5.00000 0.555556
\(82\) −10.0000 − 10.0000i −1.10432 − 1.10432i
\(83\) 6.32456 + 6.32456i 0.694210 + 0.694210i 0.963155 − 0.268945i \(-0.0866751\pi\)
−0.268945 + 0.963155i \(0.586675\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.32456 + 6.32456i −0.678064 + 0.678064i
\(88\) 6.58114 − 3.41886i 0.701552 − 0.364452i
\(89\) − 6.00000i − 0.635999i −0.948091 − 0.317999i \(-0.896989\pi\)
0.948091 − 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.00000 − 3.00000i −0.312772 − 0.312772i
\(93\) 2.00000 − 2.00000i 0.207390 − 0.207390i
\(94\) 9.48683 0.978492
\(95\) 0 0
\(96\) 9.48683i 0.968246i
\(97\) 7.00000 + 7.00000i 0.710742 + 0.710742i 0.966691 − 0.255948i \(-0.0823876\pi\)
−0.255948 + 0.966691i \(0.582388\pi\)
\(98\) −11.0680 − 11.0680i −1.11803 − 1.11803i
\(99\) 3.16228 + 1.00000i 0.317821 + 0.100504i
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 275.2.e.b.32.1 4
5.2 odd 4 55.2.e.a.43.1 yes 4
5.3 odd 4 inner 275.2.e.b.43.2 4
5.4 even 2 55.2.e.a.32.2 yes 4
11.10 odd 2 inner 275.2.e.b.32.2 4
15.2 even 4 495.2.k.b.208.2 4
15.14 odd 2 495.2.k.b.307.1 4
20.7 even 4 880.2.bd.e.593.2 4
20.19 odd 2 880.2.bd.e.417.2 4
55.2 even 20 605.2.m.b.403.1 16
55.4 even 10 605.2.m.b.457.2 16
55.7 even 20 605.2.m.b.578.1 16
55.9 even 10 605.2.m.b.282.1 16
55.14 even 10 605.2.m.b.112.1 16
55.17 even 20 605.2.m.b.118.1 16
55.19 odd 10 605.2.m.b.112.2 16
55.24 odd 10 605.2.m.b.282.2 16
55.27 odd 20 605.2.m.b.118.2 16
55.29 odd 10 605.2.m.b.457.1 16
55.32 even 4 55.2.e.a.43.2 yes 4
55.37 odd 20 605.2.m.b.578.2 16
55.39 odd 10 605.2.m.b.602.2 16
55.42 odd 20 605.2.m.b.403.2 16
55.43 even 4 inner 275.2.e.b.43.1 4
55.47 odd 20 605.2.m.b.233.1 16
55.49 even 10 605.2.m.b.602.1 16
55.52 even 20 605.2.m.b.233.2 16
55.54 odd 2 55.2.e.a.32.1 ✓ 4
165.32 odd 4 495.2.k.b.208.1 4
165.164 even 2 495.2.k.b.307.2 4
220.87 odd 4 880.2.bd.e.593.1 4
220.219 even 2 880.2.bd.e.417.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.e.a.32.1 ✓ 4 55.54 odd 2
55.2.e.a.32.2 yes 4 5.4 even 2
55.2.e.a.43.1 yes 4 5.2 odd 4
55.2.e.a.43.2 yes 4 55.32 even 4
275.2.e.b.32.1 4 1.1 even 1 trivial
275.2.e.b.32.2 4 11.10 odd 2 inner
275.2.e.b.43.1 4 55.43 even 4 inner
275.2.e.b.43.2 4 5.3 odd 4 inner
495.2.k.b.208.1 4 165.32 odd 4
495.2.k.b.208.2 4 15.2 even 4
495.2.k.b.307.1 4 15.14 odd 2
495.2.k.b.307.2 4 165.164 even 2
605.2.m.b.112.1 16 55.14 even 10
605.2.m.b.112.2 16 55.19 odd 10
605.2.m.b.118.1 16 55.17 even 20
605.2.m.b.118.2 16 55.27 odd 20
605.2.m.b.233.1 16 55.47 odd 20
605.2.m.b.233.2 16 55.52 even 20
605.2.m.b.282.1 16 55.9 even 10
605.2.m.b.282.2 16 55.24 odd 10
605.2.m.b.403.1 16 55.2 even 20
605.2.m.b.403.2 16 55.42 odd 20
605.2.m.b.457.1 16 55.29 odd 10
605.2.m.b.457.2 16 55.4 even 10
605.2.m.b.578.1 16 55.7 even 20
605.2.m.b.578.2 16 55.37 odd 20
605.2.m.b.602.1 16 55.49 even 10
605.2.m.b.602.2 16 55.39 odd 10
880.2.bd.e.417.1 4 220.219 even 2
880.2.bd.e.417.2 4 20.19 odd 2
880.2.bd.e.593.1 4 220.87 odd 4
880.2.bd.e.593.2 4 20.7 even 4