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 = [6,0,6,4,6,0,-2] 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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4507648.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.20475\) 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)
 
\(f(q)\) \(=\) \(q-0.656184 q^{2} +1.00000 q^{3} -1.56942 q^{4} +1.00000 q^{5} -0.656184 q^{6} +3.56472 q^{7} +2.34220 q^{8} +1.00000 q^{9} -0.656184 q^{10} -0.568022 q^{11} -1.56942 q^{12} -5.22090 q^{13} -2.33911 q^{14} +1.00000 q^{15} +1.60193 q^{16} -6.16754 q^{17} -0.656184 q^{18} +0.843173 q^{19} -1.56942 q^{20} +3.56472 q^{21} +0.372727 q^{22} +2.34220 q^{24} +1.00000 q^{25} +3.42587 q^{26} +1.00000 q^{27} -5.59455 q^{28} -9.53754 q^{29} -0.656184 q^{30} +8.95484 q^{31} -5.73556 q^{32} -0.568022 q^{33} +4.04704 q^{34} +3.56472 q^{35} -1.56942 q^{36} +8.61566 q^{37} -0.553276 q^{38} -5.22090 q^{39} +2.34220 q^{40} -4.65964 q^{41} -2.33911 q^{42} +0.662292 q^{43} +0.891466 q^{44} +1.00000 q^{45} +1.81032 q^{47} +1.60193 q^{48} +5.70720 q^{49} -0.656184 q^{50} -6.16754 q^{51} +8.19380 q^{52} +3.54029 q^{53} -0.656184 q^{54} -0.568022 q^{55} +8.34927 q^{56} +0.843173 q^{57} +6.25838 q^{58} -8.82983 q^{59} -1.56942 q^{60} +1.27875 q^{61} -5.87602 q^{62} +3.56472 q^{63} +0.559715 q^{64} -5.22090 q^{65} +0.372727 q^{66} -15.5810 q^{67} +9.67947 q^{68} -2.33911 q^{70} -8.77371 q^{71} +2.34220 q^{72} -12.6389 q^{73} -5.65346 q^{74} +1.00000 q^{75} -1.32329 q^{76} -2.02484 q^{77} +3.42587 q^{78} -8.83550 q^{79} +1.60193 q^{80} +1.00000 q^{81} +3.05758 q^{82} +5.72967 q^{83} -5.59455 q^{84} -6.16754 q^{85} -0.434585 q^{86} -9.53754 q^{87} -1.33042 q^{88} -1.62654 q^{89} -0.656184 q^{90} -18.6110 q^{91} +8.95484 q^{93} -1.18791 q^{94} +0.843173 q^{95} -5.73556 q^{96} +2.78620 q^{97} -3.74497 q^{98} -0.568022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{7} + 6 q^{9} - 12 q^{11} + 4 q^{12} - 4 q^{13} - 8 q^{14} + 6 q^{15} - 14 q^{17} - 8 q^{19} + 4 q^{20} - 2 q^{21} - 12 q^{22} + 6 q^{25} + 24 q^{26} + 6 q^{27} - 4 q^{28}+ \cdots - 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.656184 −0.463992 −0.231996 − 0.972717i \(-0.574526\pi\)
−0.231996 + 0.972717i \(0.574526\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.56942 −0.784711
\(5\) 1.00000 0.447214
\(6\) −0.656184 −0.267886
\(7\) 3.56472 1.34734 0.673668 − 0.739034i \(-0.264718\pi\)
0.673668 + 0.739034i \(0.264718\pi\)
\(8\) 2.34220 0.828092
\(9\) 1.00000 0.333333
\(10\) −0.656184 −0.207504
\(11\) −0.568022 −0.171265 −0.0856325 − 0.996327i \(-0.527291\pi\)
−0.0856325 + 0.996327i \(0.527291\pi\)
\(12\) −1.56942 −0.453053
\(13\) −5.22090 −1.44802 −0.724009 − 0.689791i \(-0.757702\pi\)
−0.724009 + 0.689791i \(0.757702\pi\)
\(14\) −2.33911 −0.625153
\(15\) 1.00000 0.258199
\(16\) 1.60193 0.400483
\(17\) −6.16754 −1.49585 −0.747924 − 0.663785i \(-0.768949\pi\)
−0.747924 + 0.663785i \(0.768949\pi\)
\(18\) −0.656184 −0.154664
\(19\) 0.843173 0.193437 0.0967185 − 0.995312i \(-0.469165\pi\)
0.0967185 + 0.995312i \(0.469165\pi\)
\(20\) −1.56942 −0.350934
\(21\) 3.56472 0.777885
\(22\) 0.372727 0.0794656
\(23\) 0 0
\(24\) 2.34220 0.478099
\(25\) 1.00000 0.200000
\(26\) 3.42587 0.671868
\(27\) 1.00000 0.192450
\(28\) −5.59455 −1.05727
\(29\) −9.53754 −1.77108 −0.885538 − 0.464567i \(-0.846210\pi\)
−0.885538 + 0.464567i \(0.846210\pi\)
\(30\) −0.656184 −0.119802
\(31\) 8.95484 1.60834 0.804168 − 0.594402i \(-0.202611\pi\)
0.804168 + 0.594402i \(0.202611\pi\)
\(32\) −5.73556 −1.01391
\(33\) −0.568022 −0.0988799
\(34\) 4.04704 0.694061
\(35\) 3.56472 0.602547
\(36\) −1.56942 −0.261570
\(37\) 8.61566 1.41641 0.708203 − 0.706009i \(-0.249506\pi\)
0.708203 + 0.706009i \(0.249506\pi\)
\(38\) −0.553276 −0.0897533
\(39\) −5.22090 −0.836013
\(40\) 2.34220 0.370334
\(41\) −4.65964 −0.727713 −0.363856 − 0.931455i \(-0.618540\pi\)
−0.363856 + 0.931455i \(0.618540\pi\)
\(42\) −2.33911 −0.360932
\(43\) 0.662292 0.100999 0.0504993 − 0.998724i \(-0.483919\pi\)
0.0504993 + 0.998724i \(0.483919\pi\)
\(44\) 0.891466 0.134394
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 1.81032 0.264063 0.132031 − 0.991246i \(-0.457850\pi\)
0.132031 + 0.991246i \(0.457850\pi\)
\(48\) 1.60193 0.231219
\(49\) 5.70720 0.815314
\(50\) −0.656184 −0.0927984
\(51\) −6.16754 −0.863628
\(52\) 8.19380 1.13628
\(53\) 3.54029 0.486297 0.243148 − 0.969989i \(-0.421820\pi\)
0.243148 + 0.969989i \(0.421820\pi\)
\(54\) −0.656184 −0.0892953
\(55\) −0.568022 −0.0765920
\(56\) 8.34927 1.11572
\(57\) 0.843173 0.111681
\(58\) 6.25838 0.821765
\(59\) −8.82983 −1.14955 −0.574773 − 0.818313i \(-0.694910\pi\)
−0.574773 + 0.818313i \(0.694910\pi\)
\(60\) −1.56942 −0.202612
\(61\) 1.27875 0.163727 0.0818636 − 0.996644i \(-0.473913\pi\)
0.0818636 + 0.996644i \(0.473913\pi\)
\(62\) −5.87602 −0.746256
\(63\) 3.56472 0.449112
\(64\) 0.559715 0.0699644
\(65\) −5.22090 −0.647573
\(66\) 0.372727 0.0458795
\(67\) −15.5810 −1.90352 −0.951760 − 0.306843i \(-0.900727\pi\)
−0.951760 + 0.306843i \(0.900727\pi\)
\(68\) 9.67947 1.17381
\(69\) 0 0
\(70\) −2.33911 −0.279577
\(71\) −8.77371 −1.04125 −0.520624 − 0.853786i \(-0.674301\pi\)
−0.520624 + 0.853786i \(0.674301\pi\)
\(72\) 2.34220 0.276031
\(73\) −12.6389 −1.47927 −0.739636 − 0.673007i \(-0.765002\pi\)
−0.739636 + 0.673007i \(0.765002\pi\)
\(74\) −5.65346 −0.657201
\(75\) 1.00000 0.115470
\(76\) −1.32329 −0.151792
\(77\) −2.02484 −0.230751
\(78\) 3.42587 0.387903
\(79\) −8.83550 −0.994071 −0.497036 − 0.867730i \(-0.665578\pi\)
−0.497036 + 0.867730i \(0.665578\pi\)
\(80\) 1.60193 0.179102
\(81\) 1.00000 0.111111
\(82\) 3.05758 0.337653
\(83\) 5.72967 0.628913 0.314456 − 0.949272i \(-0.398178\pi\)
0.314456 + 0.949272i \(0.398178\pi\)
\(84\) −5.59455 −0.610415
\(85\) −6.16754 −0.668963
\(86\) −0.434585 −0.0468625
\(87\) −9.53754 −1.02253
\(88\) −1.33042 −0.141823
\(89\) −1.62654 −0.172413 −0.0862066 − 0.996277i \(-0.527475\pi\)
−0.0862066 + 0.996277i \(0.527475\pi\)
\(90\) −0.656184 −0.0691679
\(91\) −18.6110 −1.95097
\(92\) 0 0
\(93\) 8.95484 0.928574
\(94\) −1.18791 −0.122523
\(95\) 0.843173 0.0865077
\(96\) −5.73556 −0.585383
\(97\) 2.78620 0.282895 0.141448 − 0.989946i \(-0.454824\pi\)
0.141448 + 0.989946i \(0.454824\pi\)
\(98\) −3.74497 −0.378299
\(99\) −0.568022 −0.0570883
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.bg.1.3 yes 6
23.22 odd 2 7935.2.a.bf.1.3 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7935.2.a.bf.1.3 ✓ 6 23.22 odd 2
7935.2.a.bg.1.3 yes 6 1.1 even 1 trivial