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 1.1
Root \(-2.08613\) 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)
 
\(f(q)\) \(=\) \(q-2.08613 q^{2} +2.35194 q^{4} -5.08613 q^{7} -0.734191 q^{8} +4.17226 q^{11} +1.08613 q^{13} +10.6103 q^{14} -3.17226 q^{16} +0.648061 q^{17} -2.70388 q^{19} -8.70388 q^{22} +7.52420 q^{23} -2.26581 q^{26} -11.9623 q^{28} -5.90645 q^{29} +1.00000 q^{31} +8.08613 q^{32} -1.35194 q^{34} -0.913870 q^{37} +5.64064 q^{38} -2.17226 q^{41} -8.17226 q^{43} +9.81290 q^{44} -15.6965 q^{46} -10.2281 q^{47} +18.8687 q^{49} +2.55451 q^{52} +5.52420 q^{53} +3.73419 q^{56} +12.3216 q^{58} -0.438069 q^{59} -2.00000 q^{61} -2.08613 q^{62} -10.5242 q^{64} -9.79001 q^{67} +1.52420 q^{68} -2.96969 q^{71} +1.25839 q^{73} +1.90645 q^{74} -6.35936 q^{76} -21.2207 q^{77} +10.8203 q^{79} +4.53162 q^{82} +14.2281 q^{83} +17.0484 q^{86} -3.06324 q^{88} -18.1903 q^{89} -5.52420 q^{91} +17.6965 q^{92} +21.3371 q^{94} +13.2207 q^{97} -39.3626 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} - 8 q^{7} + 3 q^{8} - 2 q^{11} - 4 q^{13} + 8 q^{14} + 5 q^{16} + 4 q^{17} - 4 q^{19} - 22 q^{22} + 6 q^{23} - 12 q^{26} - 10 q^{28} + 2 q^{29} + 3 q^{31} + 17 q^{32} - 2 q^{34} - 10 q^{37}+ \cdots - 57 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.08613 −1.47512 −0.737558 − 0.675283i \(-0.764021\pi\)
−0.737558 + 0.675283i \(0.764021\pi\)
\(3\) 0 0
\(4\) 2.35194 1.17597
\(5\) 0 0
\(6\) 0 0
\(7\) −5.08613 −1.92238 −0.961188 − 0.275893i \(-0.911026\pi\)
−0.961188 + 0.275893i \(0.911026\pi\)
\(8\) −0.734191 −0.259576
\(9\) 0 0
\(10\) 0 0
\(11\) 4.17226 1.25798 0.628992 − 0.777412i \(-0.283468\pi\)
0.628992 + 0.777412i \(0.283468\pi\)
\(12\) 0 0
\(13\) 1.08613 0.301238 0.150619 − 0.988592i \(-0.451873\pi\)
0.150619 + 0.988592i \(0.451873\pi\)
\(14\) 10.6103 2.83573
\(15\) 0 0
\(16\) −3.17226 −0.793065
\(17\) 0.648061 0.157178 0.0785889 − 0.996907i \(-0.474959\pi\)
0.0785889 + 0.996907i \(0.474959\pi\)
\(18\) 0 0
\(19\) −2.70388 −0.620312 −0.310156 − 0.950686i \(-0.600381\pi\)
−0.310156 + 0.950686i \(0.600381\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −8.70388 −1.85567
\(23\) 7.52420 1.56890 0.784452 − 0.620189i \(-0.212944\pi\)
0.784452 + 0.620189i \(0.212944\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.26581 −0.444362
\(27\) 0 0
\(28\) −11.9623 −2.26066
\(29\) −5.90645 −1.09680 −0.548400 − 0.836216i \(-0.684763\pi\)
−0.548400 + 0.836216i \(0.684763\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 8.08613 1.42944
\(33\) 0 0
\(34\) −1.35194 −0.231856
\(35\) 0 0
\(36\) 0 0
\(37\) −0.913870 −0.150239 −0.0751196 − 0.997175i \(-0.523934\pi\)
−0.0751196 + 0.997175i \(0.523934\pi\)
\(38\) 5.64064 0.915033
\(39\) 0 0
\(40\) 0 0
\(41\) −2.17226 −0.339250 −0.169625 − 0.985509i \(-0.554256\pi\)
−0.169625 + 0.985509i \(0.554256\pi\)
\(42\) 0 0
\(43\) −8.17226 −1.24626 −0.623129 − 0.782119i \(-0.714139\pi\)
−0.623129 + 0.782119i \(0.714139\pi\)
\(44\) 9.81290 1.47935
\(45\) 0 0
\(46\) −15.6965 −2.31432
\(47\) −10.2281 −1.49192 −0.745959 − 0.665992i \(-0.768009\pi\)
−0.745959 + 0.665992i \(0.768009\pi\)
\(48\) 0 0
\(49\) 18.8687 2.69553
\(50\) 0 0
\(51\) 0 0
\(52\) 2.55451 0.354247
\(53\) 5.52420 0.758807 0.379404 − 0.925231i \(-0.376129\pi\)
0.379404 + 0.925231i \(0.376129\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.73419 0.499002
\(57\) 0 0
\(58\) 12.3216 1.61791
\(59\) −0.438069 −0.0570318 −0.0285159 − 0.999593i \(-0.509078\pi\)
−0.0285159 + 0.999593i \(0.509078\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −2.08613 −0.264939
\(63\) 0 0
\(64\) −10.5242 −1.31552
\(65\) 0 0
\(66\) 0 0
\(67\) −9.79001 −1.19604 −0.598020 − 0.801481i \(-0.704046\pi\)
−0.598020 + 0.801481i \(0.704046\pi\)
\(68\) 1.52420 0.184836
\(69\) 0 0
\(70\) 0 0
\(71\) −2.96969 −0.352437 −0.176219 − 0.984351i \(-0.556387\pi\)
−0.176219 + 0.984351i \(0.556387\pi\)
\(72\) 0 0
\(73\) 1.25839 0.147283 0.0736417 − 0.997285i \(-0.476538\pi\)
0.0736417 + 0.997285i \(0.476538\pi\)
\(74\) 1.90645 0.221620
\(75\) 0 0
\(76\) −6.35936 −0.729468
\(77\) −21.2207 −2.41832
\(78\) 0 0
\(79\) 10.8203 1.21738 0.608691 − 0.793408i \(-0.291695\pi\)
0.608691 + 0.793408i \(0.291695\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.53162 0.500433
\(83\) 14.2281 1.56173 0.780867 − 0.624697i \(-0.214778\pi\)
0.780867 + 0.624697i \(0.214778\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 17.0484 1.83838
\(87\) 0 0
\(88\) −3.06324 −0.326542
\(89\) −18.1903 −1.92817 −0.964086 − 0.265589i \(-0.914434\pi\)
−0.964086 + 0.265589i \(0.914434\pi\)
\(90\) 0 0
\(91\) −5.52420 −0.579093
\(92\) 17.6965 1.84498
\(93\) 0 0
\(94\) 21.3371 2.20075
\(95\) 0 0
\(96\) 0 0
\(97\) 13.2207 1.34235 0.671177 − 0.741297i \(-0.265789\pi\)
0.671177 + 0.741297i \(0.265789\pi\)
\(98\) −39.3626 −3.97622
\(99\) 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.1.1 3
3.2 odd 2 2325.2.a.r.1.3 3
5.4 even 2 1395.2.a.j.1.3 3
15.2 even 4 2325.2.c.k.1024.5 6
15.8 even 4 2325.2.c.k.1024.2 6
15.14 odd 2 465.2.a.e.1.1 ✓ 3
60.59 even 2 7440.2.a.bs.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.e.1.1 ✓ 3 15.14 odd 2
1395.2.a.j.1.3 3 5.4 even 2
2325.2.a.r.1.3 3 3.2 odd 2
2325.2.c.k.1024.2 6 15.8 even 4
2325.2.c.k.1024.5 6 15.2 even 4
6975.2.a.bf.1.1 3 1.1 even 1 trivial
7440.2.a.bs.1.1 3 60.59 even 2