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 1.3
Root \(-1.29363\) 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)
 
\(f(q)\) \(=\) \(q+1.29363 q^{2} +1.00000 q^{3} -0.326531 q^{4} +1.00000 q^{5} +1.29363 q^{6} -0.504831 q^{7} -3.00966 q^{8} +1.00000 q^{9} +1.29363 q^{10} -3.12499 q^{11} -0.326531 q^{12} -4.12499 q^{13} -0.653062 q^{14} +1.00000 q^{15} -3.24031 q^{16} +0.706374 q^{17} +1.29363 q^{18} -2.41861 q^{19} -0.326531 q^{20} -0.504831 q^{21} -4.04257 q^{22} -3.00966 q^{24} +1.00000 q^{25} -5.33619 q^{26} +1.00000 q^{27} +0.164843 q^{28} +2.00000 q^{29} +1.29363 q^{30} +0.560979 q^{31} +1.82757 q^{32} -3.12499 q^{33} +0.913784 q^{34} -0.504831 q^{35} -0.326531 q^{36} +9.72190 q^{37} -3.12878 q^{38} -4.12499 q^{39} -3.00966 q^{40} +7.87393 q^{41} -0.653062 q^{42} +9.78771 q^{43} +1.02041 q^{44} +1.00000 q^{45} +7.66272 q^{47} -3.24031 q^{48} -6.74515 q^{49} +1.29363 q^{50} +0.706374 q^{51} +1.34694 q^{52} +9.88088 q^{53} +1.29363 q^{54} -3.12499 q^{55} +1.51937 q^{56} -2.41861 q^{57} +2.58725 q^{58} -8.44489 q^{59} -0.326531 q^{60} +4.85764 q^{61} +0.725697 q^{62} -0.504831 q^{63} +8.84482 q^{64} -4.12499 q^{65} -4.04257 q^{66} -2.50483 q^{67} -0.230653 q^{68} -0.653062 q^{70} -5.12499 q^{71} -3.00966 q^{72} +7.59691 q^{73} +12.5765 q^{74} +1.00000 q^{75} +0.789753 q^{76} +1.57759 q^{77} -5.33619 q^{78} +8.22370 q^{79} -3.24031 q^{80} +1.00000 q^{81} +10.1859 q^{82} -1.35944 q^{83} +0.164843 q^{84} +0.706374 q^{85} +12.6616 q^{86} +2.00000 q^{87} +9.40516 q^{88} -6.32653 q^{89} +1.29363 q^{90} +2.08242 q^{91} +0.560979 q^{93} +9.91270 q^{94} -2.41861 q^{95} +1.82757 q^{96} +16.6298 q^{97} -8.72570 q^{98} -3.12499 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 8 q^{4} + 4 q^{5} + q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} + 8 q^{12} + q^{13} + 16 q^{14} + 4 q^{15} + 16 q^{16} + 8 q^{17} + 13 q^{19} + 8 q^{20} + q^{21} - 6 q^{22} - 6 q^{24} + 4 q^{25}+ \cdots + 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.29363 0.914732 0.457366 − 0.889279i \(-0.348793\pi\)
0.457366 + 0.889279i \(0.348793\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.326531 −0.163266
\(5\) 1.00000 0.447214
\(6\) 1.29363 0.528121
\(7\) −0.504831 −0.190808 −0.0954041 − 0.995439i \(-0.530414\pi\)
−0.0954041 + 0.995439i \(0.530414\pi\)
\(8\) −3.00966 −1.06408
\(9\) 1.00000 0.333333
\(10\) 1.29363 0.409081
\(11\) −3.12499 −0.942219 −0.471110 − 0.882075i \(-0.656146\pi\)
−0.471110 + 0.882075i \(0.656146\pi\)
\(12\) −0.326531 −0.0942614
\(13\) −4.12499 −1.14407 −0.572033 − 0.820231i \(-0.693845\pi\)
−0.572033 + 0.820231i \(0.693845\pi\)
\(14\) −0.653062 −0.174538
\(15\) 1.00000 0.258199
\(16\) −3.24031 −0.810079
\(17\) 0.706374 0.171321 0.0856604 − 0.996324i \(-0.472700\pi\)
0.0856604 + 0.996324i \(0.472700\pi\)
\(18\) 1.29363 0.304911
\(19\) −2.41861 −0.554868 −0.277434 − 0.960745i \(-0.589484\pi\)
−0.277434 + 0.960745i \(0.589484\pi\)
\(20\) −0.326531 −0.0730146
\(21\) −0.504831 −0.110163
\(22\) −4.04257 −0.861878
\(23\) 0 0
\(24\) −3.00966 −0.614345
\(25\) 1.00000 0.200000
\(26\) −5.33619 −1.04651
\(27\) 1.00000 0.192450
\(28\) 0.164843 0.0311524
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 1.29363 0.236183
\(31\) 0.560979 0.100755 0.0503774 − 0.998730i \(-0.483958\pi\)
0.0503774 + 0.998730i \(0.483958\pi\)
\(32\) 1.82757 0.323071
\(33\) −3.12499 −0.543991
\(34\) 0.913784 0.156713
\(35\) −0.504831 −0.0853320
\(36\) −0.326531 −0.0544219
\(37\) 9.72190 1.59827 0.799135 − 0.601151i \(-0.205291\pi\)
0.799135 + 0.601151i \(0.205291\pi\)
\(38\) −3.12878 −0.507556
\(39\) −4.12499 −0.660527
\(40\) −3.00966 −0.475869
\(41\) 7.87393 1.22970 0.614851 − 0.788644i \(-0.289216\pi\)
0.614851 + 0.788644i \(0.289216\pi\)
\(42\) −0.653062 −0.100770
\(43\) 9.78771 1.49261 0.746306 − 0.665603i \(-0.231826\pi\)
0.746306 + 0.665603i \(0.231826\pi\)
\(44\) 1.02041 0.153832
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 7.66272 1.11772 0.558862 − 0.829261i \(-0.311238\pi\)
0.558862 + 0.829261i \(0.311238\pi\)
\(48\) −3.24031 −0.467699
\(49\) −6.74515 −0.963592
\(50\) 1.29363 0.182946
\(51\) 0.706374 0.0989121
\(52\) 1.34694 0.186787
\(53\) 9.88088 1.35724 0.678622 − 0.734488i \(-0.262578\pi\)
0.678622 + 0.734488i \(0.262578\pi\)
\(54\) 1.29363 0.176040
\(55\) −3.12499 −0.421373
\(56\) 1.51937 0.203034
\(57\) −2.41861 −0.320353
\(58\) 2.58725 0.339723
\(59\) −8.44489 −1.09943 −0.549715 − 0.835352i \(-0.685264\pi\)
−0.549715 + 0.835352i \(0.685264\pi\)
\(60\) −0.326531 −0.0421550
\(61\) 4.85764 0.621956 0.310978 − 0.950417i \(-0.399343\pi\)
0.310978 + 0.950417i \(0.399343\pi\)
\(62\) 0.725697 0.0921637
\(63\) −0.504831 −0.0636027
\(64\) 8.84482 1.10560
\(65\) −4.12499 −0.511642
\(66\) −4.04257 −0.497606
\(67\) −2.50483 −0.306014 −0.153007 − 0.988225i \(-0.548896\pi\)
−0.153007 + 0.988225i \(0.548896\pi\)
\(68\) −0.230653 −0.0279708
\(69\) 0 0
\(70\) −0.653062 −0.0780559
\(71\) −5.12499 −0.608224 −0.304112 − 0.952636i \(-0.598360\pi\)
−0.304112 + 0.952636i \(0.598360\pi\)
\(72\) −3.00966 −0.354692
\(73\) 7.59691 0.889152 0.444576 − 0.895741i \(-0.353354\pi\)
0.444576 + 0.895741i \(0.353354\pi\)
\(74\) 12.5765 1.46199
\(75\) 1.00000 0.115470
\(76\) 0.789753 0.0905909
\(77\) 1.57759 0.179783
\(78\) −5.33619 −0.604205
\(79\) 8.22370 0.925239 0.462619 − 0.886557i \(-0.346910\pi\)
0.462619 + 0.886557i \(0.346910\pi\)
\(80\) −3.24031 −0.362278
\(81\) 1.00000 0.111111
\(82\) 10.1859 1.12485
\(83\) −1.35944 −0.149218 −0.0746088 − 0.997213i \(-0.523771\pi\)
−0.0746088 + 0.997213i \(0.523771\pi\)
\(84\) 0.164843 0.0179859
\(85\) 0.706374 0.0766170
\(86\) 12.6616 1.36534
\(87\) 2.00000 0.214423
\(88\) 9.40516 1.00259
\(89\) −6.32653 −0.670611 −0.335305 − 0.942109i \(-0.608840\pi\)
−0.335305 + 0.942109i \(0.608840\pi\)
\(90\) 1.29363 0.136360
\(91\) 2.08242 0.218297
\(92\) 0 0
\(93\) 0.560979 0.0581708
\(94\) 9.91270 1.02242
\(95\) −2.41861 −0.248145
\(96\) 1.82757 0.186525
\(97\) 16.6298 1.68850 0.844251 − 0.535948i \(-0.180046\pi\)
0.844251 + 0.535948i \(0.180046\pi\)
\(98\) −8.72570 −0.881429
\(99\) −3.12499 −0.314073
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.1.3 yes 4
23.22 odd 2 7935.2.a.z.1.3 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7935.2.a.z.1.3 ✓ 4 23.22 odd 2
7935.2.a.ba.1.3 yes 4 1.1 even 1 trivial