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 1.3
Root \(2.17009\) 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)
 
\(f(q)\) \(=\) \(q+2.17009 q^{2} +2.70928 q^{4} +1.17009 q^{7} +1.53919 q^{8} -2.00000 q^{11} -0.0917087 q^{13} +2.53919 q^{14} -2.07838 q^{16} -6.04945 q^{17} -4.34017 q^{22} -1.70928 q^{23} -0.199016 q^{26} +3.17009 q^{28} -7.95774 q^{29} +1.00000 q^{31} -7.58864 q^{32} -13.1278 q^{34} +3.35350 q^{37} +4.68035 q^{41} +0.738205 q^{43} -5.41855 q^{44} -3.70928 q^{46} -9.12783 q^{47} -5.63090 q^{49} -0.248464 q^{52} +4.97107 q^{53} +1.80098 q^{56} -17.2690 q^{58} -11.0361 q^{59} -3.26180 q^{61} +2.17009 q^{62} -12.3112 q^{64} -3.85043 q^{67} -16.3896 q^{68} +1.21953 q^{71} -7.66701 q^{73} +7.27739 q^{74} -2.34017 q^{77} +3.52586 q^{79} +10.1568 q^{82} +12.5464 q^{83} +1.60197 q^{86} -3.07838 q^{88} +5.77432 q^{89} -0.107307 q^{91} -4.63090 q^{92} -19.8082 q^{94} +12.0000 q^{97} -12.2195 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + q^{4} - 2 q^{7} + 3 q^{8} - 6 q^{11} + 2 q^{13} + 6 q^{14} - 3 q^{16} - 2 q^{22} + 2 q^{23} - 10 q^{26} + 4 q^{28} - 8 q^{29} + 3 q^{31} - 3 q^{32} - 18 q^{34} - 8 q^{41} + 10 q^{43} - 2 q^{44}+ \cdots - 13 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.17009 1.53448 0.767241 − 0.641358i \(-0.221629\pi\)
0.767241 + 0.641358i \(0.221629\pi\)
\(3\) 0 0
\(4\) 2.70928 1.35464
\(5\) 0 0
\(6\) 0 0
\(7\) 1.17009 0.442251 0.221126 − 0.975245i \(-0.429027\pi\)
0.221126 + 0.975245i \(0.429027\pi\)
\(8\) 1.53919 0.544185
\(9\) 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
\(13\) −0.0917087 −0.0254354 −0.0127177 − 0.999919i \(-0.504048\pi\)
−0.0127177 + 0.999919i \(0.504048\pi\)
\(14\) 2.53919 0.678627
\(15\) 0 0
\(16\) −2.07838 −0.519594
\(17\) −6.04945 −1.46721 −0.733603 − 0.679578i \(-0.762163\pi\)
−0.733603 + 0.679578i \(0.762163\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.34017 −0.925328
\(23\) −1.70928 −0.356409 −0.178204 − 0.983994i \(-0.557029\pi\)
−0.178204 + 0.983994i \(0.557029\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.199016 −0.0390302
\(27\) 0 0
\(28\) 3.17009 0.599090
\(29\) −7.95774 −1.47772 −0.738858 − 0.673862i \(-0.764634\pi\)
−0.738858 + 0.673862i \(0.764634\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −7.58864 −1.34149
\(33\) 0 0
\(34\) −13.1278 −2.25140
\(35\) 0 0
\(36\) 0 0
\(37\) 3.35350 0.551313 0.275656 − 0.961256i \(-0.411105\pi\)
0.275656 + 0.961256i \(0.411105\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.68035 0.730947 0.365474 − 0.930822i \(-0.380907\pi\)
0.365474 + 0.930822i \(0.380907\pi\)
\(42\) 0 0
\(43\) 0.738205 0.112575 0.0562876 − 0.998415i \(-0.482074\pi\)
0.0562876 + 0.998415i \(0.482074\pi\)
\(44\) −5.41855 −0.816877
\(45\) 0 0
\(46\) −3.70928 −0.546903
\(47\) −9.12783 −1.33143 −0.665715 − 0.746206i \(-0.731873\pi\)
−0.665715 + 0.746206i \(0.731873\pi\)
\(48\) 0 0
\(49\) −5.63090 −0.804414
\(50\) 0 0
\(51\) 0 0
\(52\) −0.248464 −0.0344558
\(53\) 4.97107 0.682829 0.341415 − 0.939913i \(-0.389094\pi\)
0.341415 + 0.939913i \(0.389094\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.80098 0.240667
\(57\) 0 0
\(58\) −17.2690 −2.26753
\(59\) −11.0361 −1.43678 −0.718390 − 0.695641i \(-0.755121\pi\)
−0.718390 + 0.695641i \(0.755121\pi\)
\(60\) 0 0
\(61\) −3.26180 −0.417630 −0.208815 − 0.977955i \(-0.566961\pi\)
−0.208815 + 0.977955i \(0.566961\pi\)
\(62\) 2.17009 0.275601
\(63\) 0 0
\(64\) −12.3112 −1.53891
\(65\) 0 0
\(66\) 0 0
\(67\) −3.85043 −0.470405 −0.235203 − 0.971946i \(-0.575575\pi\)
−0.235203 + 0.971946i \(0.575575\pi\)
\(68\) −16.3896 −1.98753
\(69\) 0 0
\(70\) 0 0
\(71\) 1.21953 0.144732 0.0723661 − 0.997378i \(-0.476945\pi\)
0.0723661 + 0.997378i \(0.476945\pi\)
\(72\) 0 0
\(73\) −7.66701 −0.897356 −0.448678 − 0.893693i \(-0.648105\pi\)
−0.448678 + 0.893693i \(0.648105\pi\)
\(74\) 7.27739 0.845980
\(75\) 0 0
\(76\) 0 0
\(77\) −2.34017 −0.266687
\(78\) 0 0
\(79\) 3.52586 0.396690 0.198345 − 0.980132i \(-0.436443\pi\)
0.198345 + 0.980132i \(0.436443\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 10.1568 1.12163
\(83\) 12.5464 1.37714 0.688572 − 0.725168i \(-0.258238\pi\)
0.688572 + 0.725168i \(0.258238\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.60197 0.172745
\(87\) 0 0
\(88\) −3.07838 −0.328156
\(89\) 5.77432 0.612077 0.306038 − 0.952019i \(-0.400996\pi\)
0.306038 + 0.952019i \(0.400996\pi\)
\(90\) 0 0
\(91\) −0.107307 −0.0112488
\(92\) −4.63090 −0.482804
\(93\) 0 0
\(94\) −19.8082 −2.04306
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) −12.2195 −1.23436
\(99\) 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.1.3 3
3.2 odd 2 2325.2.a.q.1.1 3
5.4 even 2 1395.2.a.i.1.1 3
15.2 even 4 2325.2.c.o.1024.1 6
15.8 even 4 2325.2.c.o.1024.6 6
15.14 odd 2 465.2.a.f.1.3 ✓ 3
60.59 even 2 7440.2.a.bp.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.f.1.3 ✓ 3 15.14 odd 2
1395.2.a.i.1.1 3 5.4 even 2
2325.2.a.q.1.1 3 3.2 odd 2
2325.2.c.o.1024.1 6 15.2 even 4
2325.2.c.o.1024.6 6 15.8 even 4
6975.2.a.be.1.3 3 1.1 even 1 trivial
7440.2.a.bp.1.3 3 60.59 even 2