Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6975.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,1,0,1,0,0,-2,3,0,0,-6,0,2,6,0,-3,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(55.6956554098\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 465)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 6975.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.48119 q^{2} +0.193937 q^{4} -2.48119 q^{7} +2.67513 q^{8} -2.00000 q^{11} +5.83146 q^{13} +3.67513 q^{14} -4.35026 q^{16} +3.76845 q^{17} +2.96239 q^{22} +0.806063 q^{23} -8.63752 q^{26} -0.481194 q^{28} -4.06300 q^{29} +1.00000 q^{31} +1.09332 q^{32} -5.58181 q^{34} -12.1441 q^{37} -9.92478 q^{41} +10.3127 q^{43} -0.387873 q^{44} -1.19394 q^{46} -1.58181 q^{47} -0.843675 q^{49} +1.13093 q^{52} -7.11871 q^{53} -6.63752 q^{56} +6.01810 q^{58} -9.41327 q^{59} +6.31265 q^{61} -1.48119 q^{62} +7.08110 q^{64} +14.4060 q^{67} +0.730841 q^{68} -12.2496 q^{71} -1.25694 q^{73} +17.9878 q^{74} +4.96239 q^{77} +12.8568 q^{79} +14.7005 q^{82} -0.0303172 q^{83} -15.2750 q^{86} -5.35026 q^{88} +13.7259 q^{89} -14.4690 q^{91} +0.156325 q^{92} +2.34297 q^{94} +12.0000 q^{97} +1.24965 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + q^{4} - 2 q^{7} + 3 q^{8} - 6 q^{11} + 2 q^{13} + 6 q^{14} - 3 q^{16} - 2 q^{22} + 2 q^{23} - 10 q^{26} + 4 q^{28} - 8 q^{29} + 3 q^{31} - 3 q^{32} - 18 q^{34} - 8 q^{41} + 10 q^{43} - 2 q^{44}+ \cdots - 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.48119 −1.04736 −0.523681 − 0.851914i \(-0.675442\pi\)
−0.523681 + 0.851914i \(0.675442\pi\)
\(3\) 0 0
\(4\) 0.193937 0.0969683
\(5\) 0 0
\(6\) 0 0
\(7\) −2.48119 −0.937803 −0.468902 − 0.883250i \(-0.655350\pi\)
−0.468902 + 0.883250i \(0.655350\pi\)
\(8\) 2.67513 0.945802
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 5.83146 1.61735 0.808677 − 0.588252i \(-0.200184\pi\)
0.808677 + 0.588252i \(0.200184\pi\)
\(14\) 3.67513 0.982220
\(15\) 0 0
\(16\) −4.35026 −1.08757
\(17\) 3.76845 0.913984 0.456992 − 0.889471i \(-0.348927\pi\)
0.456992 + 0.889471i \(0.348927\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.96239 0.631583
\(23\) 0.806063 0.168076 0.0840379 − 0.996463i \(-0.473218\pi\)
0.0840379 + 0.996463i \(0.473218\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −8.63752 −1.69396
\(27\) 0 0
\(28\) −0.481194 −0.0909372
\(29\) −4.06300 −0.754481 −0.377240 − 0.926115i \(-0.623127\pi\)
−0.377240 + 0.926115i \(0.623127\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.09332 0.193274
\(33\) 0 0
\(34\) −5.58181 −0.957272
\(35\) 0 0
\(36\) 0 0
\(37\) −12.1441 −1.99648 −0.998239 − 0.0593136i \(-0.981109\pi\)
−0.998239 + 0.0593136i \(0.981109\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.92478 −1.54999 −0.774995 − 0.631967i \(-0.782248\pi\)
−0.774995 + 0.631967i \(0.782248\pi\)
\(42\) 0 0
\(43\) 10.3127 1.57266 0.786332 − 0.617804i \(-0.211977\pi\)
0.786332 + 0.617804i \(0.211977\pi\)
\(44\) −0.387873 −0.0584741
\(45\) 0 0
\(46\) −1.19394 −0.176036
\(47\) −1.58181 −0.230731 −0.115365 − 0.993323i \(-0.536804\pi\)
−0.115365 + 0.993323i \(0.536804\pi\)
\(48\) 0 0
\(49\) −0.843675 −0.120525
\(50\) 0 0
\(51\) 0 0
\(52\) 1.13093 0.156832
\(53\) −7.11871 −0.977831 −0.488915 − 0.872331i \(-0.662607\pi\)
−0.488915 + 0.872331i \(0.662607\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −6.63752 −0.886976
\(57\) 0 0
\(58\) 6.01810 0.790215
\(59\) −9.41327 −1.22550 −0.612751 − 0.790276i \(-0.709937\pi\)
−0.612751 + 0.790276i \(0.709937\pi\)
\(60\) 0 0
\(61\) 6.31265 0.808252 0.404126 − 0.914703i \(-0.367576\pi\)
0.404126 + 0.914703i \(0.367576\pi\)
\(62\) −1.48119 −0.188112
\(63\) 0 0
\(64\) 7.08110 0.885138
\(65\) 0 0
\(66\) 0 0
\(67\) 14.4060 1.75997 0.879985 − 0.475002i \(-0.157553\pi\)
0.879985 + 0.475002i \(0.157553\pi\)
\(68\) 0.730841 0.0886274
\(69\) 0 0
\(70\) 0 0
\(71\) −12.2496 −1.45377 −0.726883 − 0.686762i \(-0.759032\pi\)
−0.726883 + 0.686762i \(0.759032\pi\)
\(72\) 0 0
\(73\) −1.25694 −0.147114 −0.0735569 − 0.997291i \(-0.523435\pi\)
−0.0735569 + 0.997291i \(0.523435\pi\)
\(74\) 17.9878 2.09104
\(75\) 0 0
\(76\) 0 0
\(77\) 4.96239 0.565517
\(78\) 0 0
\(79\) 12.8568 1.44651 0.723254 − 0.690582i \(-0.242645\pi\)
0.723254 + 0.690582i \(0.242645\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 14.7005 1.62340
\(83\) −0.0303172 −0.00332774 −0.00166387 − 0.999999i \(-0.500530\pi\)
−0.00166387 + 0.999999i \(0.500530\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −15.2750 −1.64715
\(87\) 0 0
\(88\) −5.35026 −0.570340
\(89\) 13.7259 1.45494 0.727472 − 0.686137i \(-0.240695\pi\)
0.727472 + 0.686137i \(0.240695\pi\)
\(90\) 0 0
\(91\) −14.4690 −1.51676
\(92\) 0.156325 0.0162980
\(93\) 0 0
\(94\) 2.34297 0.241659
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 1.24965 0.126233
\(99\) 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 6975.2.a.be.1.1 3
3.2 odd 2 2325.2.a.q.1.3 3
5.4 even 2 1395.2.a.i.1.3 3
15.2 even 4 2325.2.c.o.1024.5 6
15.8 even 4 2325.2.c.o.1024.2 6
15.14 odd 2 465.2.a.f.1.1 ✓ 3
60.59 even 2 7440.2.a.bp.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.f.1.1 ✓ 3 15.14 odd 2
1395.2.a.i.1.3 3 5.4 even 2
2325.2.a.q.1.3 3 3.2 odd 2
2325.2.c.o.1024.2 6 15.8 even 4
2325.2.c.o.1024.5 6 15.2 even 4
6975.2.a.be.1.1 3 1.1 even 1 trivial
7440.2.a.bp.1.1 3 60.59 even 2