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 = [3,1,0,5,0,0,-8,3,0,0,-2,0,-4,8,0,5,4] 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: 3.3.564.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) 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 \(2.51414\) 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)
 
\(f(q)\) \(=\) \(q+2.51414 q^{2} +4.32088 q^{4} -0.485863 q^{7} +5.83502 q^{8} -5.02827 q^{11} -3.51414 q^{13} -1.22153 q^{14} +6.02827 q^{16} -1.32088 q^{17} -6.64177 q^{19} -12.6418 q^{22} +0.292611 q^{23} -8.83502 q^{26} -2.09936 q^{28} +9.86330 q^{29} +1.00000 q^{31} +3.48586 q^{32} -3.32088 q^{34} -5.51414 q^{37} -16.6983 q^{38} +7.02827 q^{41} +1.02827 q^{43} -21.7266 q^{44} +0.735663 q^{46} -6.93438 q^{47} -6.76394 q^{49} -15.1842 q^{52} -1.70739 q^{53} -2.83502 q^{56} +24.7977 q^{58} +2.19325 q^{59} -2.00000 q^{61} +2.51414 q^{62} -3.29261 q^{64} -9.12763 q^{67} -5.70739 q^{68} -13.4768 q^{71} -12.5424 q^{73} -13.8633 q^{74} -28.6983 q^{76} +2.44305 q^{77} -0.349158 q^{79} +17.6700 q^{82} +10.9344 q^{83} +2.58522 q^{86} -29.3401 q^{88} -5.03374 q^{89} +1.70739 q^{91} +1.26434 q^{92} -17.4340 q^{94} -10.4431 q^{97} -17.0055 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} - 8 q^{7} + 3 q^{8} - 2 q^{11} - 4 q^{13} + 8 q^{14} + 5 q^{16} + 4 q^{17} - 4 q^{19} - 22 q^{22} + 6 q^{23} - 12 q^{26} - 10 q^{28} + 2 q^{29} + 3 q^{31} + 17 q^{32} - 2 q^{34} - 10 q^{37}+ \cdots - 57 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.51414 1.77776 0.888882 − 0.458137i \(-0.151483\pi\)
0.888882 + 0.458137i \(0.151483\pi\)
\(3\) 0 0
\(4\) 4.32088 2.16044
\(5\) 0 0
\(6\) 0 0
\(7\) −0.485863 −0.183639 −0.0918195 − 0.995776i \(-0.529268\pi\)
−0.0918195 + 0.995776i \(0.529268\pi\)
\(8\) 5.83502 2.06299
\(9\) 0 0
\(10\) 0 0
\(11\) −5.02827 −1.51608 −0.758041 − 0.652207i \(-0.773843\pi\)
−0.758041 + 0.652207i \(0.773843\pi\)
\(12\) 0 0
\(13\) −3.51414 −0.974646 −0.487323 − 0.873222i \(-0.662027\pi\)
−0.487323 + 0.873222i \(0.662027\pi\)
\(14\) −1.22153 −0.326467
\(15\) 0 0
\(16\) 6.02827 1.50707
\(17\) −1.32088 −0.320362 −0.160181 − 0.987088i \(-0.551208\pi\)
−0.160181 + 0.987088i \(0.551208\pi\)
\(18\) 0 0
\(19\) −6.64177 −1.52373 −0.761863 − 0.647738i \(-0.775715\pi\)
−0.761863 + 0.647738i \(0.775715\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −12.6418 −2.69523
\(23\) 0.292611 0.0610135 0.0305068 − 0.999535i \(-0.490288\pi\)
0.0305068 + 0.999535i \(0.490288\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −8.83502 −1.73269
\(27\) 0 0
\(28\) −2.09936 −0.396741
\(29\) 9.86330 1.83157 0.915784 − 0.401671i \(-0.131571\pi\)
0.915784 + 0.401671i \(0.131571\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 3.48586 0.616219
\(33\) 0 0
\(34\) −3.32088 −0.569527
\(35\) 0 0
\(36\) 0 0
\(37\) −5.51414 −0.906519 −0.453259 − 0.891379i \(-0.649739\pi\)
−0.453259 + 0.891379i \(0.649739\pi\)
\(38\) −16.6983 −2.70882
\(39\) 0 0
\(40\) 0 0
\(41\) 7.02827 1.09763 0.548816 − 0.835943i \(-0.315079\pi\)
0.548816 + 0.835943i \(0.315079\pi\)
\(42\) 0 0
\(43\) 1.02827 0.156810 0.0784051 − 0.996922i \(-0.475017\pi\)
0.0784051 + 0.996922i \(0.475017\pi\)
\(44\) −21.7266 −3.27541
\(45\) 0 0
\(46\) 0.735663 0.108468
\(47\) −6.93438 −1.01148 −0.505742 − 0.862685i \(-0.668781\pi\)
−0.505742 + 0.862685i \(0.668781\pi\)
\(48\) 0 0
\(49\) −6.76394 −0.966277
\(50\) 0 0
\(51\) 0 0
\(52\) −15.1842 −2.10567
\(53\) −1.70739 −0.234528 −0.117264 − 0.993101i \(-0.537412\pi\)
−0.117264 + 0.993101i \(0.537412\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.83502 −0.378846
\(57\) 0 0
\(58\) 24.7977 3.25609
\(59\) 2.19325 0.285537 0.142769 − 0.989756i \(-0.454400\pi\)
0.142769 + 0.989756i \(0.454400\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 2.51414 0.319296
\(63\) 0 0
\(64\) −3.29261 −0.411576
\(65\) 0 0
\(66\) 0 0
\(67\) −9.12763 −1.11512 −0.557559 − 0.830137i \(-0.688262\pi\)
−0.557559 + 0.830137i \(0.688262\pi\)
\(68\) −5.70739 −0.692123
\(69\) 0 0
\(70\) 0 0
\(71\) −13.4768 −1.59940 −0.799700 − 0.600399i \(-0.795008\pi\)
−0.799700 + 0.600399i \(0.795008\pi\)
\(72\) 0 0
\(73\) −12.5424 −1.46798 −0.733989 − 0.679161i \(-0.762344\pi\)
−0.733989 + 0.679161i \(0.762344\pi\)
\(74\) −13.8633 −1.61158
\(75\) 0 0
\(76\) −28.6983 −3.29192
\(77\) 2.44305 0.278412
\(78\) 0 0
\(79\) −0.349158 −0.0392834 −0.0196417 − 0.999807i \(-0.506253\pi\)
−0.0196417 + 0.999807i \(0.506253\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 17.6700 1.95133
\(83\) 10.9344 1.20020 0.600102 − 0.799923i \(-0.295127\pi\)
0.600102 + 0.799923i \(0.295127\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.58522 0.278772
\(87\) 0 0
\(88\) −29.3401 −3.12766
\(89\) −5.03374 −0.533575 −0.266788 − 0.963755i \(-0.585962\pi\)
−0.266788 + 0.963755i \(0.585962\pi\)
\(90\) 0 0
\(91\) 1.70739 0.178983
\(92\) 1.26434 0.131816
\(93\) 0 0
\(94\) −17.4340 −1.79818
\(95\) 0 0
\(96\) 0 0
\(97\) −10.4431 −1.06033 −0.530166 − 0.847894i \(-0.677870\pi\)
−0.530166 + 0.847894i \(0.677870\pi\)
\(98\) −17.0055 −1.71781
\(99\) 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.bf.1.3 3
3.2 odd 2 2325.2.a.r.1.1 3
5.4 even 2 1395.2.a.j.1.1 3
15.2 even 4 2325.2.c.k.1024.1 6
15.8 even 4 2325.2.c.k.1024.6 6
15.14 odd 2 465.2.a.e.1.3 ✓ 3
60.59 even 2 7440.2.a.bs.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.e.1.3 ✓ 3 15.14 odd 2
1395.2.a.j.1.1 3 5.4 even 2
2325.2.a.r.1.1 3 3.2 odd 2
2325.2.c.k.1024.1 6 15.2 even 4
2325.2.c.k.1024.6 6 15.8 even 4
6975.2.a.bf.1.3 3 1.1 even 1 trivial
7440.2.a.bs.1.3 3 60.59 even 2