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 = [3,0,0,2,0,0,4,-6,0,0,7,0,0,6,0,-4,7] 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: \(0\)
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 2325)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
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)
 
\(f(q)\) \(=\) \(q+0.539189 q^{2} -1.70928 q^{4} +3.70928 q^{7} -2.00000 q^{8} -1.87936 q^{11} -6.58864 q^{13} +2.00000 q^{14} +2.34017 q^{16} +3.09171 q^{17} -7.34017 q^{19} -1.01333 q^{22} -7.29791 q^{23} -3.55252 q^{26} -6.34017 q^{28} -0.170086 q^{29} -1.00000 q^{31} +5.26180 q^{32} +1.66701 q^{34} +6.82991 q^{37} -3.95774 q^{38} +7.95774 q^{41} +4.18342 q^{43} +3.21235 q^{44} -3.93495 q^{46} -11.7587 q^{47} +6.75872 q^{49} +11.2618 q^{52} +13.9288 q^{53} -7.41855 q^{56} -0.0917087 q^{58} +6.72261 q^{59} +1.84324 q^{61} -0.539189 q^{62} -1.84324 q^{64} -1.81432 q^{67} -5.28458 q^{68} +11.2618 q^{71} +2.92162 q^{73} +3.68261 q^{74} +12.5464 q^{76} -6.97107 q^{77} +6.24846 q^{79} +4.29072 q^{82} -6.21953 q^{83} +2.25565 q^{86} +3.75872 q^{88} +12.4030 q^{89} -24.4391 q^{91} +12.4741 q^{92} -6.34017 q^{94} +3.94441 q^{97} +3.64423 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{4} + 4 q^{7} - 6 q^{8} + 7 q^{11} + 6 q^{14} - 4 q^{16} + 7 q^{17} - 11 q^{19} - 4 q^{22} + 5 q^{23} - 10 q^{26} - 8 q^{28} + 5 q^{29} - 3 q^{31} + 8 q^{32} - 18 q^{34} + 26 q^{37} + 4 q^{38}+ \cdots + 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.539189 0.381264 0.190632 − 0.981662i \(-0.438946\pi\)
0.190632 + 0.981662i \(0.438946\pi\)
\(3\) 0 0
\(4\) −1.70928 −0.854638
\(5\) 0 0
\(6\) 0 0
\(7\) 3.70928 1.40197 0.700987 − 0.713174i \(-0.252743\pi\)
0.700987 + 0.713174i \(0.252743\pi\)
\(8\) −2.00000 −0.707107
\(9\) 0 0
\(10\) 0 0
\(11\) −1.87936 −0.566649 −0.283324 − 0.959024i \(-0.591437\pi\)
−0.283324 + 0.959024i \(0.591437\pi\)
\(12\) 0 0
\(13\) −6.58864 −1.82736 −0.913680 − 0.406435i \(-0.866772\pi\)
−0.913680 + 0.406435i \(0.866772\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 2.34017 0.585043
\(17\) 3.09171 0.749850 0.374925 − 0.927055i \(-0.377669\pi\)
0.374925 + 0.927055i \(0.377669\pi\)
\(18\) 0 0
\(19\) −7.34017 −1.68395 −0.841976 − 0.539516i \(-0.818607\pi\)
−0.841976 + 0.539516i \(0.818607\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.01333 −0.216043
\(23\) −7.29791 −1.52172 −0.760860 − 0.648916i \(-0.775223\pi\)
−0.760860 + 0.648916i \(0.775223\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.55252 −0.696706
\(27\) 0 0
\(28\) −6.34017 −1.19818
\(29\) −0.170086 −0.0315843 −0.0157921 − 0.999875i \(-0.505027\pi\)
−0.0157921 + 0.999875i \(0.505027\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 5.26180 0.930163
\(33\) 0 0
\(34\) 1.66701 0.285891
\(35\) 0 0
\(36\) 0 0
\(37\) 6.82991 1.12283 0.561415 − 0.827534i \(-0.310257\pi\)
0.561415 + 0.827534i \(0.310257\pi\)
\(38\) −3.95774 −0.642030
\(39\) 0 0
\(40\) 0 0
\(41\) 7.95774 1.24279 0.621395 − 0.783497i \(-0.286566\pi\)
0.621395 + 0.783497i \(0.286566\pi\)
\(42\) 0 0
\(43\) 4.18342 0.637965 0.318983 − 0.947761i \(-0.396659\pi\)
0.318983 + 0.947761i \(0.396659\pi\)
\(44\) 3.21235 0.484280
\(45\) 0 0
\(46\) −3.93495 −0.580177
\(47\) −11.7587 −1.71519 −0.857593 − 0.514329i \(-0.828041\pi\)
−0.857593 + 0.514329i \(0.828041\pi\)
\(48\) 0 0
\(49\) 6.75872 0.965532
\(50\) 0 0
\(51\) 0 0
\(52\) 11.2618 1.56173
\(53\) 13.9288 1.91327 0.956635 − 0.291291i \(-0.0940847\pi\)
0.956635 + 0.291291i \(0.0940847\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7.41855 −0.991346
\(57\) 0 0
\(58\) −0.0917087 −0.0120419
\(59\) 6.72261 0.875209 0.437604 − 0.899168i \(-0.355827\pi\)
0.437604 + 0.899168i \(0.355827\pi\)
\(60\) 0 0
\(61\) 1.84324 0.236003 0.118002 − 0.993013i \(-0.462351\pi\)
0.118002 + 0.993013i \(0.462351\pi\)
\(62\) −0.539189 −0.0684771
\(63\) 0 0
\(64\) −1.84324 −0.230406
\(65\) 0 0
\(66\) 0 0
\(67\) −1.81432 −0.221654 −0.110827 − 0.993840i \(-0.535350\pi\)
−0.110827 + 0.993840i \(0.535350\pi\)
\(68\) −5.28458 −0.640850
\(69\) 0 0
\(70\) 0 0
\(71\) 11.2618 1.33653 0.668265 − 0.743924i \(-0.267037\pi\)
0.668265 + 0.743924i \(0.267037\pi\)
\(72\) 0 0
\(73\) 2.92162 0.341950 0.170975 − 0.985275i \(-0.445308\pi\)
0.170975 + 0.985275i \(0.445308\pi\)
\(74\) 3.68261 0.428095
\(75\) 0 0
\(76\) 12.5464 1.43917
\(77\) −6.97107 −0.794427
\(78\) 0 0
\(79\) 6.24846 0.703007 0.351504 − 0.936187i \(-0.385671\pi\)
0.351504 + 0.936187i \(0.385671\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.29072 0.473831
\(83\) −6.21953 −0.682683 −0.341341 − 0.939939i \(-0.610881\pi\)
−0.341341 + 0.939939i \(0.610881\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.25565 0.243233
\(87\) 0 0
\(88\) 3.75872 0.400681
\(89\) 12.4030 1.31471 0.657355 − 0.753581i \(-0.271675\pi\)
0.657355 + 0.753581i \(0.271675\pi\)
\(90\) 0 0
\(91\) −24.4391 −2.56191
\(92\) 12.4741 1.30052
\(93\) 0 0
\(94\) −6.34017 −0.653939
\(95\) 0 0
\(96\) 0 0
\(97\) 3.94441 0.400494 0.200247 − 0.979745i \(-0.435826\pi\)
0.200247 + 0.979745i \(0.435826\pi\)
\(98\) 3.64423 0.368123
\(99\) 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.bd.1.2 3
3.2 odd 2 2325.2.a.t.1.2 ✓ 3
5.4 even 2 6975.2.a.bc.1.2 3
15.2 even 4 2325.2.c.m.1024.3 6
15.8 even 4 2325.2.c.m.1024.4 6
15.14 odd 2 2325.2.a.u.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2325.2.a.t.1.2 ✓ 3 3.2 odd 2
2325.2.a.u.1.2 yes 3 15.14 odd 2
2325.2.c.m.1024.3 6 15.2 even 4
2325.2.c.m.1024.4 6 15.8 even 4
6975.2.a.bc.1.2 3 5.4 even 2
6975.2.a.bd.1.2 3 1.1 even 1 trivial