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 1.2
Root \(0.927719\) 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)
 
\(f(q)\) \(=\) \(q-0.927719 q^{2} +1.00000 q^{3} -1.13934 q^{4} +1.00000 q^{5} -0.927719 q^{6} +2.45621 q^{7} +2.91242 q^{8} +1.00000 q^{9} -0.927719 q^{10} +1.24459 q^{11} -1.13934 q^{12} +0.244593 q^{13} -2.27867 q^{14} +1.00000 q^{15} -0.423236 q^{16} +2.92772 q^{17} -0.927719 q^{18} +4.17231 q^{19} -1.13934 q^{20} +2.45621 q^{21} -1.15463 q^{22} +2.91242 q^{24} +1.00000 q^{25} -0.226914 q^{26} +1.00000 q^{27} -2.79845 q^{28} +2.00000 q^{29} -0.927719 q^{30} +9.59032 q^{31} -5.43220 q^{32} +1.24459 q^{33} -2.71610 q^{34} +2.45621 q^{35} -1.13934 q^{36} -5.01245 q^{37} -3.87073 q^{38} +0.244593 q^{39} +2.91242 q^{40} +2.83776 q^{41} -2.27867 q^{42} +1.12166 q^{43} -1.41801 q^{44} +1.00000 q^{45} +3.36625 q^{47} -0.423236 q^{48} -0.967025 q^{49} -0.927719 q^{50} +2.92772 q^{51} -0.278674 q^{52} +3.21684 q^{53} -0.927719 q^{54} +1.24459 q^{55} +7.15353 q^{56} +4.17231 q^{57} -1.85544 q^{58} +11.6181 q^{59} -1.13934 q^{60} -10.7626 q^{61} -8.89713 q^{62} +2.45621 q^{63} +5.88603 q^{64} +0.244593 q^{65} -1.15463 q^{66} +0.456212 q^{67} -3.33566 q^{68} -2.27867 q^{70} -0.755407 q^{71} +2.91242 q^{72} -2.76786 q^{73} +4.65015 q^{74} +1.00000 q^{75} -4.75367 q^{76} +3.05698 q^{77} -0.226914 q^{78} +12.9566 q^{79} -0.423236 q^{80} +1.00000 q^{81} -2.63264 q^{82} -5.20639 q^{83} -2.79845 q^{84} +2.92772 q^{85} -1.04058 q^{86} +2.00000 q^{87} +3.62478 q^{88} -7.13934 q^{89} -0.927719 q^{90} +0.600773 q^{91} +9.59032 q^{93} -3.12294 q^{94} +4.17231 q^{95} -5.43220 q^{96} +9.29919 q^{97} +0.897127 q^{98} +1.24459 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 8 q^{4} + 4 q^{5} + q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} + 8 q^{12} + q^{13} + 16 q^{14} + 4 q^{15} + 16 q^{16} + 8 q^{17} + 13 q^{19} + 8 q^{20} + q^{21} - 6 q^{22} - 6 q^{24} + 4 q^{25}+ \cdots + 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.927719 −0.655996 −0.327998 − 0.944678i \(-0.606374\pi\)
−0.327998 + 0.944678i \(0.606374\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.13934 −0.569669
\(5\) 1.00000 0.447214
\(6\) −0.927719 −0.378740
\(7\) 2.45621 0.928361 0.464180 − 0.885741i \(-0.346349\pi\)
0.464180 + 0.885741i \(0.346349\pi\)
\(8\) 2.91242 1.02970
\(9\) 1.00000 0.333333
\(10\) −0.927719 −0.293371
\(11\) 1.24459 0.375259 0.187630 − 0.982240i \(-0.439920\pi\)
0.187630 + 0.982240i \(0.439920\pi\)
\(12\) −1.13934 −0.328898
\(13\) 0.244593 0.0678380 0.0339190 − 0.999425i \(-0.489201\pi\)
0.0339190 + 0.999425i \(0.489201\pi\)
\(14\) −2.27867 −0.609001
\(15\) 1.00000 0.258199
\(16\) −0.423236 −0.105809
\(17\) 2.92772 0.710076 0.355038 − 0.934852i \(-0.384468\pi\)
0.355038 + 0.934852i \(0.384468\pi\)
\(18\) −0.927719 −0.218665
\(19\) 4.17231 0.957194 0.478597 − 0.878035i \(-0.341145\pi\)
0.478597 + 0.878035i \(0.341145\pi\)
\(20\) −1.13934 −0.254764
\(21\) 2.45621 0.535989
\(22\) −1.15463 −0.246169
\(23\) 0 0
\(24\) 2.91242 0.594496
\(25\) 1.00000 0.200000
\(26\) −0.226914 −0.0445015
\(27\) 1.00000 0.192450
\(28\) −2.79845 −0.528858
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −0.927719 −0.169378
\(31\) 9.59032 1.72247 0.861237 − 0.508204i \(-0.169691\pi\)
0.861237 + 0.508204i \(0.169691\pi\)
\(32\) −5.43220 −0.960287
\(33\) 1.24459 0.216656
\(34\) −2.71610 −0.465807
\(35\) 2.45621 0.415176
\(36\) −1.13934 −0.189890
\(37\) −5.01245 −0.824043 −0.412021 − 0.911174i \(-0.635177\pi\)
−0.412021 + 0.911174i \(0.635177\pi\)
\(38\) −3.87073 −0.627916
\(39\) 0.244593 0.0391663
\(40\) 2.91242 0.460495
\(41\) 2.83776 0.443183 0.221592 − 0.975140i \(-0.428875\pi\)
0.221592 + 0.975140i \(0.428875\pi\)
\(42\) −2.27867 −0.351607
\(43\) 1.12166 0.171051 0.0855256 − 0.996336i \(-0.472743\pi\)
0.0855256 + 0.996336i \(0.472743\pi\)
\(44\) −1.41801 −0.213773
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 3.36625 0.491018 0.245509 − 0.969394i \(-0.421045\pi\)
0.245509 + 0.969394i \(0.421045\pi\)
\(48\) −0.423236 −0.0610889
\(49\) −0.967025 −0.138146
\(50\) −0.927719 −0.131199
\(51\) 2.92772 0.409963
\(52\) −0.278674 −0.0386452
\(53\) 3.21684 0.441867 0.220934 − 0.975289i \(-0.429090\pi\)
0.220934 + 0.975289i \(0.429090\pi\)
\(54\) −0.927719 −0.126247
\(55\) 1.24459 0.167821
\(56\) 7.15353 0.955930
\(57\) 4.17231 0.552636
\(58\) −1.85544 −0.243631
\(59\) 11.6181 1.51254 0.756272 − 0.654257i \(-0.227019\pi\)
0.756272 + 0.654257i \(0.227019\pi\)
\(60\) −1.13934 −0.147088
\(61\) −10.7626 −1.37801 −0.689007 − 0.724754i \(-0.741953\pi\)
−0.689007 + 0.724754i \(0.741953\pi\)
\(62\) −8.89713 −1.12994
\(63\) 2.45621 0.309454
\(64\) 5.88603 0.735754
\(65\) 0.244593 0.0303381
\(66\) −1.15463 −0.142126
\(67\) 0.456212 0.0557351 0.0278676 − 0.999612i \(-0.491128\pi\)
0.0278676 + 0.999612i \(0.491128\pi\)
\(68\) −3.33566 −0.404508
\(69\) 0 0
\(70\) −2.27867 −0.272354
\(71\) −0.755407 −0.0896503 −0.0448251 − 0.998995i \(-0.514273\pi\)
−0.0448251 + 0.998995i \(0.514273\pi\)
\(72\) 2.91242 0.343232
\(73\) −2.76786 −0.323954 −0.161977 − 0.986795i \(-0.551787\pi\)
−0.161977 + 0.986795i \(0.551787\pi\)
\(74\) 4.65015 0.540569
\(75\) 1.00000 0.115470
\(76\) −4.75367 −0.545283
\(77\) 3.05698 0.348376
\(78\) −0.226914 −0.0256930
\(79\) 12.9566 1.45773 0.728864 − 0.684658i \(-0.240048\pi\)
0.728864 + 0.684658i \(0.240048\pi\)
\(80\) −0.423236 −0.0473192
\(81\) 1.00000 0.111111
\(82\) −2.63264 −0.290727
\(83\) −5.20639 −0.571476 −0.285738 − 0.958308i \(-0.592239\pi\)
−0.285738 + 0.958308i \(0.592239\pi\)
\(84\) −2.79845 −0.305336
\(85\) 2.92772 0.317556
\(86\) −1.04058 −0.112209
\(87\) 2.00000 0.214423
\(88\) 3.62478 0.386403
\(89\) −7.13934 −0.756768 −0.378384 − 0.925649i \(-0.623520\pi\)
−0.378384 + 0.925649i \(0.623520\pi\)
\(90\) −0.927719 −0.0977902
\(91\) 0.600773 0.0629782
\(92\) 0 0
\(93\) 9.59032 0.994470
\(94\) −3.12294 −0.322106
\(95\) 4.17231 0.428070
\(96\) −5.43220 −0.554422
\(97\) 9.29919 0.944190 0.472095 − 0.881548i \(-0.343498\pi\)
0.472095 + 0.881548i \(0.343498\pi\)
\(98\) 0.897127 0.0906235
\(99\) 1.24459 0.125086
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.1.2 yes 4
23.22 odd 2 7935.2.a.z.1.2 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7935.2.a.z.1.2 ✓ 4 23.22 odd 2
7935.2.a.ba.1.2 yes 4 1.1 even 1 trivial