Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3762,2,Mod(1,3762)] 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("3762.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3762, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3762.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,0,3,3,0,-6,3,0,3,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(30.0397212404\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.66908\) of defining polynomial
Character \(\chi\) \(=\) 3762.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.12398 q^{5} -4.21417 q^{7} +1.00000 q^{8} +4.12398 q^{10} +1.00000 q^{11} -2.21417 q^{13} -4.21417 q^{14} +1.00000 q^{16} +3.45490 q^{17} -1.00000 q^{19} +4.12398 q^{20} +1.00000 q^{22} +5.45490 q^{23} +12.0072 q^{25} -2.21417 q^{26} -4.21417 q^{28} -5.57889 q^{29} +7.00724 q^{31} +1.00000 q^{32} +3.45490 q^{34} -17.3792 q^{35} -2.90981 q^{37} -1.00000 q^{38} +4.12398 q^{40} +11.9170 q^{41} +1.46214 q^{43} +1.00000 q^{44} +5.45490 q^{46} -7.58612 q^{47} +10.7593 q^{49} +12.0072 q^{50} -2.21417 q^{52} +13.2214 q^{53} +4.12398 q^{55} -4.21417 q^{56} -5.57889 q^{58} -4.79306 q^{59} +8.90981 q^{61} +7.00724 q^{62} +1.00000 q^{64} -9.13122 q^{65} +1.30437 q^{67} +3.45490 q^{68} -17.3792 q^{70} +6.80030 q^{71} -1.45490 q^{73} -2.90981 q^{74} -1.00000 q^{76} -4.21417 q^{77} -9.15777 q^{79} +4.12398 q^{80} +11.9170 q^{82} +13.2142 q^{83} +14.2480 q^{85} +1.46214 q^{86} +1.00000 q^{88} -8.24797 q^{89} +9.33092 q^{91} +5.45490 q^{92} -7.58612 q^{94} -4.12398 q^{95} +2.18038 q^{97} +10.7593 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} - 6 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 15 q^{23} + 12 q^{25} - 6 q^{28} - 6 q^{29} - 3 q^{31} + 3 q^{32}+ \cdots + 27 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 4.12398 1.84430 0.922151 − 0.386831i \(-0.126430\pi\)
0.922151 + 0.386831i \(0.126430\pi\)
\(6\) 0 0
\(7\) −4.21417 −1.59281 −0.796404 − 0.604765i \(-0.793267\pi\)
−0.796404 + 0.604765i \(0.793267\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 4.12398 1.30412
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.21417 −0.614102 −0.307051 − 0.951693i \(-0.599342\pi\)
−0.307051 + 0.951693i \(0.599342\pi\)
\(14\) −4.21417 −1.12629
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.45490 0.837937 0.418969 − 0.908001i \(-0.362392\pi\)
0.418969 + 0.908001i \(0.362392\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 4.12398 0.922151
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 5.45490 1.13743 0.568713 − 0.822536i \(-0.307441\pi\)
0.568713 + 0.822536i \(0.307441\pi\)
\(24\) 0 0
\(25\) 12.0072 2.40145
\(26\) −2.21417 −0.434235
\(27\) 0 0
\(28\) −4.21417 −0.796404
\(29\) −5.57889 −1.03597 −0.517987 − 0.855389i \(-0.673318\pi\)
−0.517987 + 0.855389i \(0.673318\pi\)
\(30\) 0 0
\(31\) 7.00724 1.25854 0.629268 − 0.777188i \(-0.283355\pi\)
0.629268 + 0.777188i \(0.283355\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.45490 0.592511
\(35\) −17.3792 −2.93762
\(36\) 0 0
\(37\) −2.90981 −0.478370 −0.239185 − 0.970974i \(-0.576880\pi\)
−0.239185 + 0.970974i \(0.576880\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 4.12398 0.652059
\(41\) 11.9170 1.86113 0.930565 − 0.366127i \(-0.119316\pi\)
0.930565 + 0.366127i \(0.119316\pi\)
\(42\) 0 0
\(43\) 1.46214 0.222974 0.111487 − 0.993766i \(-0.464439\pi\)
0.111487 + 0.993766i \(0.464439\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 5.45490 0.804282
\(47\) −7.58612 −1.10655 −0.553275 − 0.832999i \(-0.686622\pi\)
−0.553275 + 0.832999i \(0.686622\pi\)
\(48\) 0 0
\(49\) 10.7593 1.53704
\(50\) 12.0072 1.69808
\(51\) 0 0
\(52\) −2.21417 −0.307051
\(53\) 13.2214 1.81610 0.908050 − 0.418861i \(-0.137571\pi\)
0.908050 + 0.418861i \(0.137571\pi\)
\(54\) 0 0
\(55\) 4.12398 0.556078
\(56\) −4.21417 −0.563143
\(57\) 0 0
\(58\) −5.57889 −0.732544
\(59\) −4.79306 −0.624004 −0.312002 − 0.950082i \(-0.600999\pi\)
−0.312002 + 0.950082i \(0.600999\pi\)
\(60\) 0 0
\(61\) 8.90981 1.14078 0.570392 − 0.821373i \(-0.306791\pi\)
0.570392 + 0.821373i \(0.306791\pi\)
\(62\) 7.00724 0.889920
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −9.13122 −1.13259
\(66\) 0 0
\(67\) 1.30437 0.159354 0.0796769 − 0.996821i \(-0.474611\pi\)
0.0796769 + 0.996821i \(0.474611\pi\)
\(68\) 3.45490 0.418969
\(69\) 0 0
\(70\) −17.3792 −2.07721
\(71\) 6.80030 0.807047 0.403524 − 0.914969i \(-0.367785\pi\)
0.403524 + 0.914969i \(0.367785\pi\)
\(72\) 0 0
\(73\) −1.45490 −0.170284 −0.0851418 − 0.996369i \(-0.527134\pi\)
−0.0851418 + 0.996369i \(0.527134\pi\)
\(74\) −2.90981 −0.338258
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −4.21417 −0.480250
\(78\) 0 0
\(79\) −9.15777 −1.03033 −0.515165 − 0.857091i \(-0.672269\pi\)
−0.515165 + 0.857091i \(0.672269\pi\)
\(80\) 4.12398 0.461075
\(81\) 0 0
\(82\) 11.9170 1.31602
\(83\) 13.2142 1.45044 0.725222 − 0.688515i \(-0.241737\pi\)
0.725222 + 0.688515i \(0.241737\pi\)
\(84\) 0 0
\(85\) 14.2480 1.54541
\(86\) 1.46214 0.157667
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −8.24797 −0.874283 −0.437141 − 0.899393i \(-0.644009\pi\)
−0.437141 + 0.899393i \(0.644009\pi\)
\(90\) 0 0
\(91\) 9.33092 0.978146
\(92\) 5.45490 0.568713
\(93\) 0 0
\(94\) −7.58612 −0.782449
\(95\) −4.12398 −0.423112
\(96\) 0 0
\(97\) 2.18038 0.221384 0.110692 − 0.993855i \(-0.464693\pi\)
0.110692 + 0.993855i \(0.464693\pi\)
\(98\) 10.7593 1.08685
\(99\) 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 3762.2.a.bg.1.3 3
3.2 odd 2 418.2.a.g.1.3 ✓ 3
12.11 even 2 3344.2.a.q.1.1 3
33.32 even 2 4598.2.a.bo.1.3 3
57.56 even 2 7942.2.a.bi.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.3 ✓ 3 3.2 odd 2
3344.2.a.q.1.1 3 12.11 even 2
3762.2.a.bg.1.3 3 1.1 even 1 trivial
4598.2.a.bo.1.3 3 33.32 even 2
7942.2.a.bi.1.1 3 57.56 even 2