Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7935,2,Mod(1,7935)] 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("7935.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7935, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7935.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,8,4,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.3612940039\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.25492.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 8x^{2} - 2x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.77129\) of defining polynomial
Character \(\chi\) \(=\) 7935.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.77129 q^{2} +1.00000 q^{3} +5.68002 q^{4} +1.00000 q^{5} -2.77129 q^{6} -4.09920 q^{7} -10.1984 q^{8} +1.00000 q^{9} -2.77129 q^{10} +3.35211 q^{11} +5.68002 q^{12} +2.35211 q^{13} +11.3600 q^{14} +1.00000 q^{15} +16.9026 q^{16} +4.77129 q^{17} -2.77129 q^{18} +8.12340 q^{19} +5.68002 q^{20} -4.09920 q^{21} -9.28966 q^{22} -10.1984 q^{24} +1.00000 q^{25} -6.51837 q^{26} +1.00000 q^{27} -23.2835 q^{28} +2.00000 q^{29} -2.77129 q^{30} -6.91667 q^{31} -26.4452 q^{32} +3.35211 q^{33} -13.2226 q^{34} -4.09920 q^{35} +5.68002 q^{36} +2.30371 q^{37} -22.5123 q^{38} +2.35211 q^{39} -10.1984 q^{40} +10.7088 q^{41} +11.3600 q^{42} -1.51376 q^{43} +19.0401 q^{44} +1.00000 q^{45} +2.83835 q^{47} +16.9026 q^{48} +9.80342 q^{49} -2.77129 q^{50} +4.77129 q^{51} +13.3600 q^{52} -2.31386 q^{53} -2.77129 q^{54} +3.35211 q^{55} +41.8052 q^{56} +8.12340 q^{57} -5.54257 q^{58} +2.74929 q^{59} +5.68002 q^{60} +1.79328 q^{61} +19.1681 q^{62} -4.09920 q^{63} +39.4819 q^{64} +2.35211 q^{65} -9.28966 q^{66} -6.09920 q^{67} +27.1010 q^{68} +11.3600 q^{70} +1.35211 q^{71} -10.1984 q^{72} +6.65582 q^{73} -6.38425 q^{74} +1.00000 q^{75} +46.1411 q^{76} -13.7410 q^{77} -6.51837 q^{78} -4.07832 q^{79} +16.9026 q^{80} +1.00000 q^{81} -29.6772 q^{82} +6.58876 q^{83} -23.2835 q^{84} +4.77129 q^{85} +4.19507 q^{86} +2.00000 q^{87} -34.1862 q^{88} -0.319977 q^{89} -2.77129 q^{90} -9.64177 q^{91} -6.91667 q^{93} -7.86587 q^{94} +8.12340 q^{95} -26.4452 q^{96} +13.7471 q^{97} -27.1681 q^{98} +3.35211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 8 q^{4} + 4 q^{5} + q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} + 8 q^{12} + q^{13} + 16 q^{14} + 4 q^{15} + 16 q^{16} + 8 q^{17} + 13 q^{19} + 8 q^{20} + q^{21} - 6 q^{22} - 6 q^{24} + 4 q^{25}+ \cdots + 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.77129 −1.95959 −0.979797 − 0.199993i \(-0.935908\pi\)
−0.979797 + 0.199993i \(0.935908\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.68002 2.84001
\(5\) 1.00000 0.447214
\(6\) −2.77129 −1.13137
\(7\) −4.09920 −1.54935 −0.774676 − 0.632359i \(-0.782087\pi\)
−0.774676 + 0.632359i \(0.782087\pi\)
\(8\) −10.1984 −3.60568
\(9\) 1.00000 0.333333
\(10\) −2.77129 −0.876357
\(11\) 3.35211 1.01070 0.505350 − 0.862915i \(-0.331363\pi\)
0.505350 + 0.862915i \(0.331363\pi\)
\(12\) 5.68002 1.63968
\(13\) 2.35211 0.652358 0.326179 − 0.945308i \(-0.394239\pi\)
0.326179 + 0.945308i \(0.394239\pi\)
\(14\) 11.3600 3.03610
\(15\) 1.00000 0.258199
\(16\) 16.9026 4.22565
\(17\) 4.77129 1.15721 0.578603 − 0.815609i \(-0.303598\pi\)
0.578603 + 0.815609i \(0.303598\pi\)
\(18\) −2.77129 −0.653198
\(19\) 8.12340 1.86364 0.931818 − 0.362927i \(-0.118223\pi\)
0.931818 + 0.362927i \(0.118223\pi\)
\(20\) 5.68002 1.27009
\(21\) −4.09920 −0.894518
\(22\) −9.28966 −1.98056
\(23\) 0 0
\(24\) −10.1984 −2.08174
\(25\) 1.00000 0.200000
\(26\) −6.51837 −1.27836
\(27\) 1.00000 0.192450
\(28\) −23.2835 −4.40017
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −2.77129 −0.505965
\(31\) −6.91667 −1.24227 −0.621136 − 0.783703i \(-0.713328\pi\)
−0.621136 + 0.783703i \(0.713328\pi\)
\(32\) −26.4452 −4.67489
\(33\) 3.35211 0.583528
\(34\) −13.2226 −2.26766
\(35\) −4.09920 −0.692891
\(36\) 5.68002 0.946671
\(37\) 2.30371 0.378728 0.189364 − 0.981907i \(-0.439357\pi\)
0.189364 + 0.981907i \(0.439357\pi\)
\(38\) −22.5123 −3.65197
\(39\) 2.35211 0.376639
\(40\) −10.1984 −1.61251
\(41\) 10.7088 1.67244 0.836219 − 0.548395i \(-0.184761\pi\)
0.836219 + 0.548395i \(0.184761\pi\)
\(42\) 11.3600 1.75289
\(43\) −1.51376 −0.230847 −0.115423 − 0.993316i \(-0.536822\pi\)
−0.115423 + 0.993316i \(0.536822\pi\)
\(44\) 19.0401 2.87040
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 2.83835 0.414016 0.207008 − 0.978339i \(-0.433627\pi\)
0.207008 + 0.978339i \(0.433627\pi\)
\(48\) 16.9026 2.43968
\(49\) 9.80342 1.40049
\(50\) −2.77129 −0.391919
\(51\) 4.77129 0.668114
\(52\) 13.3600 1.85271
\(53\) −2.31386 −0.317833 −0.158916 − 0.987292i \(-0.550800\pi\)
−0.158916 + 0.987292i \(0.550800\pi\)
\(54\) −2.77129 −0.377124
\(55\) 3.35211 0.451999
\(56\) 41.8052 5.58646
\(57\) 8.12340 1.07597
\(58\) −5.54257 −0.727775
\(59\) 2.74929 0.357928 0.178964 − 0.983856i \(-0.442725\pi\)
0.178964 + 0.983856i \(0.442725\pi\)
\(60\) 5.68002 0.733288
\(61\) 1.79328 0.229605 0.114803 − 0.993388i \(-0.463376\pi\)
0.114803 + 0.993388i \(0.463376\pi\)
\(62\) 19.1681 2.43435
\(63\) −4.09920 −0.516450
\(64\) 39.4819 4.93524
\(65\) 2.35211 0.291743
\(66\) −9.28966 −1.14348
\(67\) −6.09920 −0.745136 −0.372568 − 0.928005i \(-0.621523\pi\)
−0.372568 + 0.928005i \(0.621523\pi\)
\(68\) 27.1010 3.28648
\(69\) 0 0
\(70\) 11.3600 1.35779
\(71\) 1.35211 0.160466 0.0802330 − 0.996776i \(-0.474434\pi\)
0.0802330 + 0.996776i \(0.474434\pi\)
\(72\) −10.1984 −1.20189
\(73\) 6.65582 0.779005 0.389503 − 0.921025i \(-0.372647\pi\)
0.389503 + 0.921025i \(0.372647\pi\)
\(74\) −6.38425 −0.742153
\(75\) 1.00000 0.115470
\(76\) 46.1411 5.29275
\(77\) −13.7410 −1.56593
\(78\) −6.51837 −0.738060
\(79\) −4.07832 −0.458847 −0.229424 − 0.973327i \(-0.573684\pi\)
−0.229424 + 0.973327i \(0.573684\pi\)
\(80\) 16.9026 1.88977
\(81\) 1.00000 0.111111
\(82\) −29.6772 −3.27730
\(83\) 6.58876 0.723210 0.361605 − 0.932331i \(-0.382229\pi\)
0.361605 + 0.932331i \(0.382229\pi\)
\(84\) −23.2835 −2.54044
\(85\) 4.77129 0.517519
\(86\) 4.19507 0.452366
\(87\) 2.00000 0.214423
\(88\) −34.1862 −3.64426
\(89\) −0.319977 −0.0339175 −0.0169587 − 0.999856i \(-0.505398\pi\)
−0.0169587 + 0.999856i \(0.505398\pi\)
\(90\) −2.77129 −0.292119
\(91\) −9.64177 −1.01073
\(92\) 0 0
\(93\) −6.91667 −0.717226
\(94\) −7.86587 −0.811303
\(95\) 8.12340 0.833443
\(96\) −26.4452 −2.69905
\(97\) 13.7471 1.39581 0.697903 − 0.716193i \(-0.254117\pi\)
0.697903 + 0.716193i \(0.254117\pi\)
\(98\) −27.1681 −2.74439
\(99\) 3.35211 0.336900
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 7935.2.a.ba.1.1 yes 4
23.22 odd 2 7935.2.a.z.1.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7935.2.a.z.1.1 ✓ 4 23.22 odd 2
7935.2.a.ba.1.1 yes 4 1.1 even 1 trivial