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 1.2
Root \(0.311108\) 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)
 
\(f(q)\) \(=\) \(q+0.311108 q^{2} -1.90321 q^{4} -0.688892 q^{7} -1.21432 q^{8} -2.00000 q^{11} -3.73975 q^{13} -0.214320 q^{14} +3.42864 q^{16} +2.28100 q^{17} -0.622216 q^{22} +2.90321 q^{23} -1.16346 q^{26} +1.31111 q^{28} +4.02074 q^{29} +1.00000 q^{31} +3.49532 q^{32} +0.709636 q^{34} +8.79060 q^{37} -2.75557 q^{41} -1.05086 q^{43} +3.80642 q^{44} +0.903212 q^{46} +4.70964 q^{47} -6.52543 q^{49} +7.11753 q^{52} +2.14764 q^{53} +0.836535 q^{56} +1.25088 q^{58} +6.44938 q^{59} -5.05086 q^{61} +0.311108 q^{62} -5.76986 q^{64} +5.44446 q^{67} -4.34122 q^{68} -8.96989 q^{71} +8.92396 q^{73} +2.73483 q^{74} +1.37778 q^{77} -8.38271 q^{79} -0.857279 q^{82} -10.5161 q^{83} -0.326929 q^{86} +2.42864 q^{88} -13.5002 q^{89} +2.57628 q^{91} -5.52543 q^{92} +1.46520 q^{94} +12.0000 q^{97} -2.03011 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + q^{4} - 2 q^{7} + 3 q^{8} - 6 q^{11} + 2 q^{13} + 6 q^{14} - 3 q^{16} - 2 q^{22} + 2 q^{23} - 10 q^{26} + 4 q^{28} - 8 q^{29} + 3 q^{31} - 3 q^{32} - 18 q^{34} - 8 q^{41} + 10 q^{43} - 2 q^{44}+ \cdots - 13 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.311108 0.219986 0.109993 − 0.993932i \(-0.464917\pi\)
0.109993 + 0.993932i \(0.464917\pi\)
\(3\) 0 0
\(4\) −1.90321 −0.951606
\(5\) 0 0
\(6\) 0 0
\(7\) −0.688892 −0.260377 −0.130188 − 0.991489i \(-0.541558\pi\)
−0.130188 + 0.991489i \(0.541558\pi\)
\(8\) −1.21432 −0.429327
\(9\) 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
\(13\) −3.73975 −1.03722 −0.518610 − 0.855011i \(-0.673550\pi\)
−0.518610 + 0.855011i \(0.673550\pi\)
\(14\) −0.214320 −0.0572794
\(15\) 0 0
\(16\) 3.42864 0.857160
\(17\) 2.28100 0.553223 0.276611 − 0.960982i \(-0.410789\pi\)
0.276611 + 0.960982i \(0.410789\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.622216 −0.132657
\(23\) 2.90321 0.605362 0.302681 − 0.953092i \(-0.402118\pi\)
0.302681 + 0.953092i \(0.402118\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.16346 −0.228174
\(27\) 0 0
\(28\) 1.31111 0.247776
\(29\) 4.02074 0.746633 0.373317 − 0.927704i \(-0.378220\pi\)
0.373317 + 0.927704i \(0.378220\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 3.49532 0.617890
\(33\) 0 0
\(34\) 0.709636 0.121702
\(35\) 0 0
\(36\) 0 0
\(37\) 8.79060 1.44517 0.722583 − 0.691284i \(-0.242955\pi\)
0.722583 + 0.691284i \(0.242955\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.75557 −0.430348 −0.215174 − 0.976576i \(-0.569032\pi\)
−0.215174 + 0.976576i \(0.569032\pi\)
\(42\) 0 0
\(43\) −1.05086 −0.160254 −0.0801270 − 0.996785i \(-0.525533\pi\)
−0.0801270 + 0.996785i \(0.525533\pi\)
\(44\) 3.80642 0.573840
\(45\) 0 0
\(46\) 0.903212 0.133171
\(47\) 4.70964 0.686971 0.343485 − 0.939158i \(-0.388392\pi\)
0.343485 + 0.939158i \(0.388392\pi\)
\(48\) 0 0
\(49\) −6.52543 −0.932204
\(50\) 0 0
\(51\) 0 0
\(52\) 7.11753 0.987024
\(53\) 2.14764 0.295001 0.147501 − 0.989062i \(-0.452877\pi\)
0.147501 + 0.989062i \(0.452877\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.836535 0.111787
\(57\) 0 0
\(58\) 1.25088 0.164249
\(59\) 6.44938 0.839638 0.419819 − 0.907608i \(-0.362094\pi\)
0.419819 + 0.907608i \(0.362094\pi\)
\(60\) 0 0
\(61\) −5.05086 −0.646696 −0.323348 − 0.946280i \(-0.604808\pi\)
−0.323348 + 0.946280i \(0.604808\pi\)
\(62\) 0.311108 0.0395107
\(63\) 0 0
\(64\) −5.76986 −0.721232
\(65\) 0 0
\(66\) 0 0
\(67\) 5.44446 0.665147 0.332573 − 0.943077i \(-0.392083\pi\)
0.332573 + 0.943077i \(0.392083\pi\)
\(68\) −4.34122 −0.526450
\(69\) 0 0
\(70\) 0 0
\(71\) −8.96989 −1.06453 −0.532265 − 0.846578i \(-0.678659\pi\)
−0.532265 + 0.846578i \(0.678659\pi\)
\(72\) 0 0
\(73\) 8.92396 1.04447 0.522235 − 0.852802i \(-0.325098\pi\)
0.522235 + 0.852802i \(0.325098\pi\)
\(74\) 2.73483 0.317917
\(75\) 0 0
\(76\) 0 0
\(77\) 1.37778 0.157013
\(78\) 0 0
\(79\) −8.38271 −0.943128 −0.471564 − 0.881832i \(-0.656310\pi\)
−0.471564 + 0.881832i \(0.656310\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.857279 −0.0946706
\(83\) −10.5161 −1.15429 −0.577144 − 0.816643i \(-0.695833\pi\)
−0.577144 + 0.816643i \(0.695833\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.326929 −0.0352537
\(87\) 0 0
\(88\) 2.42864 0.258894
\(89\) −13.5002 −1.43102 −0.715511 − 0.698601i \(-0.753806\pi\)
−0.715511 + 0.698601i \(0.753806\pi\)
\(90\) 0 0
\(91\) 2.57628 0.270068
\(92\) −5.52543 −0.576066
\(93\) 0 0
\(94\) 1.46520 0.151124
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −2.03011 −0.205072
\(99\) 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.1.2 3
3.2 odd 2 2325.2.a.q.1.2 3
5.4 even 2 1395.2.a.i.1.2 3
15.2 even 4 2325.2.c.o.1024.3 6
15.8 even 4 2325.2.c.o.1024.4 6
15.14 odd 2 465.2.a.f.1.2 ✓ 3
60.59 even 2 7440.2.a.bp.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.f.1.2 ✓ 3 15.14 odd 2
1395.2.a.i.1.2 3 5.4 even 2
2325.2.a.q.1.2 3 3.2 odd 2
2325.2.c.o.1024.3 6 15.2 even 4
2325.2.c.o.1024.4 6 15.8 even 4
6975.2.a.be.1.2 3 1.1 even 1 trivial
7440.2.a.bp.1.2 3 60.59 even 2