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 1.2
Root \(0.571993\) 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)
 
\(f(q)\) \(=\) \(q+0.571993 q^{2} -1.67282 q^{4} -2.42801 q^{7} -2.10083 q^{8} -1.14399 q^{11} -1.57199 q^{13} -1.38880 q^{14} +2.14399 q^{16} +4.67282 q^{17} +5.34565 q^{19} -0.654353 q^{22} -1.81681 q^{23} -0.899170 q^{26} +4.06163 q^{28} -1.95684 q^{29} +1.00000 q^{31} +5.42801 q^{32} +2.67282 q^{34} -3.57199 q^{37} +3.05767 q^{38} +3.14399 q^{41} -2.85601 q^{43} +1.91369 q^{44} -1.03920 q^{46} +7.16246 q^{47} -1.10478 q^{49} +2.62967 q^{52} -3.81681 q^{53} +5.10083 q^{56} -1.11930 q^{58} +6.24482 q^{59} -2.00000 q^{61} +0.571993 q^{62} -1.18319 q^{64} +0.917641 q^{67} -7.81681 q^{68} +6.44648 q^{71} -6.71598 q^{73} -2.04316 q^{74} -8.94233 q^{76} +2.77761 q^{77} +9.52884 q^{79} +1.79834 q^{82} -3.16246 q^{83} -1.63362 q^{86} +2.40332 q^{88} +15.2241 q^{89} +3.81681 q^{91} +3.03920 q^{92} +4.09688 q^{94} -10.7776 q^{97} -0.631929 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} - 8 q^{7} + 3 q^{8} - 2 q^{11} - 4 q^{13} + 8 q^{14} + 5 q^{16} + 4 q^{17} - 4 q^{19} - 22 q^{22} + 6 q^{23} - 12 q^{26} - 10 q^{28} + 2 q^{29} + 3 q^{31} + 17 q^{32} - 2 q^{34} - 10 q^{37}+ \cdots - 57 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.571993 0.404460 0.202230 − 0.979338i \(-0.435181\pi\)
0.202230 + 0.979338i \(0.435181\pi\)
\(3\) 0 0
\(4\) −1.67282 −0.836412
\(5\) 0 0
\(6\) 0 0
\(7\) −2.42801 −0.917700 −0.458850 − 0.888514i \(-0.651739\pi\)
−0.458850 + 0.888514i \(0.651739\pi\)
\(8\) −2.10083 −0.742756
\(9\) 0 0
\(10\) 0 0
\(11\) −1.14399 −0.344925 −0.172462 − 0.985016i \(-0.555172\pi\)
−0.172462 + 0.985016i \(0.555172\pi\)
\(12\) 0 0
\(13\) −1.57199 −0.435992 −0.217996 − 0.975950i \(-0.569952\pi\)
−0.217996 + 0.975950i \(0.569952\pi\)
\(14\) −1.38880 −0.371173
\(15\) 0 0
\(16\) 2.14399 0.535997
\(17\) 4.67282 1.13333 0.566663 − 0.823950i \(-0.308234\pi\)
0.566663 + 0.823950i \(0.308234\pi\)
\(18\) 0 0
\(19\) 5.34565 1.22638 0.613188 − 0.789937i \(-0.289887\pi\)
0.613188 + 0.789937i \(0.289887\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.654353 −0.139508
\(23\) −1.81681 −0.378831 −0.189416 − 0.981897i \(-0.560659\pi\)
−0.189416 + 0.981897i \(0.560659\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.899170 −0.176342
\(27\) 0 0
\(28\) 4.06163 0.767575
\(29\) −1.95684 −0.363377 −0.181688 − 0.983356i \(-0.558156\pi\)
−0.181688 + 0.983356i \(0.558156\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 5.42801 0.959545
\(33\) 0 0
\(34\) 2.67282 0.458385
\(35\) 0 0
\(36\) 0 0
\(37\) −3.57199 −0.587232 −0.293616 − 0.955923i \(-0.594859\pi\)
−0.293616 + 0.955923i \(0.594859\pi\)
\(38\) 3.05767 0.496020
\(39\) 0 0
\(40\) 0 0
\(41\) 3.14399 0.491008 0.245504 − 0.969396i \(-0.421047\pi\)
0.245504 + 0.969396i \(0.421047\pi\)
\(42\) 0 0
\(43\) −2.85601 −0.435538 −0.217769 − 0.976000i \(-0.569878\pi\)
−0.217769 + 0.976000i \(0.569878\pi\)
\(44\) 1.91369 0.288499
\(45\) 0 0
\(46\) −1.03920 −0.153222
\(47\) 7.16246 1.04475 0.522376 − 0.852715i \(-0.325046\pi\)
0.522376 + 0.852715i \(0.325046\pi\)
\(48\) 0 0
\(49\) −1.10478 −0.157826
\(50\) 0 0
\(51\) 0 0
\(52\) 2.62967 0.364669
\(53\) −3.81681 −0.524279 −0.262140 − 0.965030i \(-0.584428\pi\)
−0.262140 + 0.965030i \(0.584428\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 5.10083 0.681627
\(57\) 0 0
\(58\) −1.11930 −0.146971
\(59\) 6.24482 0.813006 0.406503 − 0.913649i \(-0.366748\pi\)
0.406503 + 0.913649i \(0.366748\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0.571993 0.0726432
\(63\) 0 0
\(64\) −1.18319 −0.147899
\(65\) 0 0
\(66\) 0 0
\(67\) 0.917641 0.112108 0.0560538 − 0.998428i \(-0.482148\pi\)
0.0560538 + 0.998428i \(0.482148\pi\)
\(68\) −7.81681 −0.947927
\(69\) 0 0
\(70\) 0 0
\(71\) 6.44648 0.765056 0.382528 − 0.923944i \(-0.375054\pi\)
0.382528 + 0.923944i \(0.375054\pi\)
\(72\) 0 0
\(73\) −6.71598 −0.786046 −0.393023 − 0.919529i \(-0.628571\pi\)
−0.393023 + 0.919529i \(0.628571\pi\)
\(74\) −2.04316 −0.237512
\(75\) 0 0
\(76\) −8.94233 −1.02576
\(77\) 2.77761 0.316538
\(78\) 0 0
\(79\) 9.52884 1.07208 0.536039 − 0.844193i \(-0.319920\pi\)
0.536039 + 0.844193i \(0.319920\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.79834 0.198593
\(83\) −3.16246 −0.347125 −0.173562 − 0.984823i \(-0.555528\pi\)
−0.173562 + 0.984823i \(0.555528\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.63362 −0.176158
\(87\) 0 0
\(88\) 2.40332 0.256195
\(89\) 15.2241 1.61375 0.806875 − 0.590722i \(-0.201157\pi\)
0.806875 + 0.590722i \(0.201157\pi\)
\(90\) 0 0
\(91\) 3.81681 0.400110
\(92\) 3.03920 0.316859
\(93\) 0 0
\(94\) 4.09688 0.422561
\(95\) 0 0
\(96\) 0 0
\(97\) −10.7776 −1.09430 −0.547150 − 0.837035i \(-0.684287\pi\)
−0.547150 + 0.837035i \(0.684287\pi\)
\(98\) −0.631929 −0.0638344
\(99\) 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.1.2 3
3.2 odd 2 2325.2.a.r.1.2 3
5.4 even 2 1395.2.a.j.1.2 3
15.2 even 4 2325.2.c.k.1024.3 6
15.8 even 4 2325.2.c.k.1024.4 6
15.14 odd 2 465.2.a.e.1.2 ✓ 3
60.59 even 2 7440.2.a.bs.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.e.1.2 ✓ 3 15.14 odd 2
1395.2.a.j.1.2 3 5.4 even 2
2325.2.a.r.1.2 3 3.2 odd 2
2325.2.c.k.1024.3 6 15.2 even 4
2325.2.c.k.1024.4 6 15.8 even 4
6975.2.a.bf.1.2 3 1.1 even 1 trivial
7440.2.a.bs.1.2 3 60.59 even 2