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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7942,2,Mod(1,7942)] 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("7942.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7942, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-12,3,12,-9,-3,0] 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: \(63.4171892853\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 15 x^{10} + 41 x^{9} + 90 x^{8} - 198 x^{7} - 285 x^{6} + 396 x^{5} + 486 x^{4} + \cdots - 3 \) 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 1.4
Root \(-1.33949\) of defining polynomial
Character \(\chi\) \(=\) 7942.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.33949 q^{3} +1.00000 q^{4} +0.688863 q^{5} +1.33949 q^{6} -0.213889 q^{7} -1.00000 q^{8} -1.20577 q^{9} -0.688863 q^{10} -1.00000 q^{11} -1.33949 q^{12} +3.34855 q^{13} +0.213889 q^{14} -0.922724 q^{15} +1.00000 q^{16} +0.666758 q^{17} +1.20577 q^{18} +0.688863 q^{20} +0.286502 q^{21} +1.00000 q^{22} -2.71889 q^{23} +1.33949 q^{24} -4.52547 q^{25} -3.34855 q^{26} +5.63358 q^{27} -0.213889 q^{28} +0.247332 q^{29} +0.922724 q^{30} +0.310630 q^{31} -1.00000 q^{32} +1.33949 q^{33} -0.666758 q^{34} -0.147340 q^{35} -1.20577 q^{36} +4.31489 q^{37} -4.48534 q^{39} -0.688863 q^{40} +1.50548 q^{41} -0.286502 q^{42} -8.26857 q^{43} -1.00000 q^{44} -0.830610 q^{45} +2.71889 q^{46} +10.9225 q^{47} -1.33949 q^{48} -6.95425 q^{49} +4.52547 q^{50} -0.893115 q^{51} +3.34855 q^{52} +7.44823 q^{53} -5.63358 q^{54} -0.688863 q^{55} +0.213889 q^{56} -0.247332 q^{58} -8.09079 q^{59} -0.922724 q^{60} -5.23294 q^{61} -0.310630 q^{62} +0.257901 q^{63} +1.00000 q^{64} +2.30669 q^{65} -1.33949 q^{66} +15.3275 q^{67} +0.666758 q^{68} +3.64193 q^{69} +0.147340 q^{70} -8.95419 q^{71} +1.20577 q^{72} -5.83233 q^{73} -4.31489 q^{74} +6.06181 q^{75} +0.213889 q^{77} +4.48534 q^{78} -1.22852 q^{79} +0.688863 q^{80} -3.92881 q^{81} -1.50548 q^{82} +8.00255 q^{83} +0.286502 q^{84} +0.459305 q^{85} +8.26857 q^{86} -0.331299 q^{87} +1.00000 q^{88} -1.94981 q^{89} +0.830610 q^{90} -0.716217 q^{91} -2.71889 q^{92} -0.416085 q^{93} -10.9225 q^{94} +1.33949 q^{96} -10.7104 q^{97} +6.95425 q^{98} +1.20577 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 3 q^{3} + 12 q^{4} - 9 q^{5} - 3 q^{6} - 12 q^{8} + 3 q^{9} + 9 q^{10} - 12 q^{11} + 3 q^{12} - 9 q^{15} + 12 q^{16} - 9 q^{17} - 3 q^{18} - 9 q^{20} + 9 q^{21} + 12 q^{22} - 18 q^{23}+ \cdots - 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.33949 −0.773354 −0.386677 − 0.922215i \(-0.626377\pi\)
−0.386677 + 0.922215i \(0.626377\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.688863 0.308069 0.154034 − 0.988065i \(-0.450773\pi\)
0.154034 + 0.988065i \(0.450773\pi\)
\(6\) 1.33949 0.546844
\(7\) −0.213889 −0.0808424 −0.0404212 − 0.999183i \(-0.512870\pi\)
−0.0404212 + 0.999183i \(0.512870\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.20577 −0.401923
\(10\) −0.688863 −0.217838
\(11\) −1.00000 −0.301511
\(12\) −1.33949 −0.386677
\(13\) 3.34855 0.928720 0.464360 − 0.885647i \(-0.346284\pi\)
0.464360 + 0.885647i \(0.346284\pi\)
\(14\) 0.213889 0.0571642
\(15\) −0.922724 −0.238246
\(16\) 1.00000 0.250000
\(17\) 0.666758 0.161713 0.0808563 − 0.996726i \(-0.474235\pi\)
0.0808563 + 0.996726i \(0.474235\pi\)
\(18\) 1.20577 0.284203
\(19\) 0 0
\(20\) 0.688863 0.154034
\(21\) 0.286502 0.0625198
\(22\) 1.00000 0.213201
\(23\) −2.71889 −0.566928 −0.283464 − 0.958983i \(-0.591484\pi\)
−0.283464 + 0.958983i \(0.591484\pi\)
\(24\) 1.33949 0.273422
\(25\) −4.52547 −0.905094
\(26\) −3.34855 −0.656704
\(27\) 5.63358 1.08418
\(28\) −0.213889 −0.0404212
\(29\) 0.247332 0.0459284 0.0229642 − 0.999736i \(-0.492690\pi\)
0.0229642 + 0.999736i \(0.492690\pi\)
\(30\) 0.922724 0.168466
\(31\) 0.310630 0.0557908 0.0278954 − 0.999611i \(-0.491119\pi\)
0.0278954 + 0.999611i \(0.491119\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.33949 0.233175
\(34\) −0.666758 −0.114348
\(35\) −0.147340 −0.0249050
\(36\) −1.20577 −0.200962
\(37\) 4.31489 0.709364 0.354682 − 0.934987i \(-0.384589\pi\)
0.354682 + 0.934987i \(0.384589\pi\)
\(38\) 0 0
\(39\) −4.48534 −0.718229
\(40\) −0.688863 −0.108919
\(41\) 1.50548 0.235116 0.117558 − 0.993066i \(-0.462493\pi\)
0.117558 + 0.993066i \(0.462493\pi\)
\(42\) −0.286502 −0.0442082
\(43\) −8.26857 −1.26094 −0.630472 − 0.776212i \(-0.717139\pi\)
−0.630472 + 0.776212i \(0.717139\pi\)
\(44\) −1.00000 −0.150756
\(45\) −0.830610 −0.123820
\(46\) 2.71889 0.400879
\(47\) 10.9225 1.59321 0.796606 − 0.604499i \(-0.206627\pi\)
0.796606 + 0.604499i \(0.206627\pi\)
\(48\) −1.33949 −0.193339
\(49\) −6.95425 −0.993465
\(50\) 4.52547 0.639998
\(51\) −0.893115 −0.125061
\(52\) 3.34855 0.464360
\(53\) 7.44823 1.02309 0.511546 − 0.859256i \(-0.329073\pi\)
0.511546 + 0.859256i \(0.329073\pi\)
\(54\) −5.63358 −0.766633
\(55\) −0.688863 −0.0928862
\(56\) 0.213889 0.0285821
\(57\) 0 0
\(58\) −0.247332 −0.0324763
\(59\) −8.09079 −1.05333 −0.526666 − 0.850072i \(-0.676558\pi\)
−0.526666 + 0.850072i \(0.676558\pi\)
\(60\) −0.922724 −0.119123
\(61\) −5.23294 −0.670010 −0.335005 − 0.942216i \(-0.608738\pi\)
−0.335005 + 0.942216i \(0.608738\pi\)
\(62\) −0.310630 −0.0394500
\(63\) 0.257901 0.0324924
\(64\) 1.00000 0.125000
\(65\) 2.30669 0.286110
\(66\) −1.33949 −0.164880
\(67\) 15.3275 1.87255 0.936274 − 0.351272i \(-0.114251\pi\)
0.936274 + 0.351272i \(0.114251\pi\)
\(68\) 0.666758 0.0808563
\(69\) 3.64193 0.438436
\(70\) 0.147340 0.0176105
\(71\) −8.95419 −1.06267 −0.531333 − 0.847163i \(-0.678309\pi\)
−0.531333 + 0.847163i \(0.678309\pi\)
\(72\) 1.20577 0.142101
\(73\) −5.83233 −0.682622 −0.341311 − 0.939950i \(-0.610871\pi\)
−0.341311 + 0.939950i \(0.610871\pi\)
\(74\) −4.31489 −0.501596
\(75\) 6.06181 0.699958
\(76\) 0 0
\(77\) 0.213889 0.0243749
\(78\) 4.48534 0.507865
\(79\) −1.22852 −0.138219 −0.0691096 − 0.997609i \(-0.522016\pi\)
−0.0691096 + 0.997609i \(0.522016\pi\)
\(80\) 0.688863 0.0770172
\(81\) −3.92881 −0.436535
\(82\) −1.50548 −0.166252
\(83\) 8.00255 0.878394 0.439197 − 0.898391i \(-0.355263\pi\)
0.439197 + 0.898391i \(0.355263\pi\)
\(84\) 0.286502 0.0312599
\(85\) 0.459305 0.0498186
\(86\) 8.26857 0.891622
\(87\) −0.331299 −0.0355189
\(88\) 1.00000 0.106600
\(89\) −1.94981 −0.206680 −0.103340 − 0.994646i \(-0.532953\pi\)
−0.103340 + 0.994646i \(0.532953\pi\)
\(90\) 0.830610 0.0875539
\(91\) −0.716217 −0.0750799
\(92\) −2.71889 −0.283464
\(93\) −0.416085 −0.0431460
\(94\) −10.9225 −1.12657
\(95\) 0 0
\(96\) 1.33949 0.136711
\(97\) −10.7104 −1.08748 −0.543739 − 0.839254i \(-0.682992\pi\)
−0.543739 + 0.839254i \(0.682992\pi\)
\(98\) 6.95425 0.702485
\(99\) 1.20577 0.121184
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 7942.2.a.bt.1.4 12
19.9 even 9 418.2.j.a.309.2 yes 24
19.17 even 9 418.2.j.a.23.2 ✓ 24
19.18 odd 2 7942.2.a.bx.1.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.j.a.23.2 ✓ 24 19.17 even 9
418.2.j.a.309.2 yes 24 19.9 even 9
7942.2.a.bt.1.4 12 1.1 even 1 trivial
7942.2.a.bx.1.9 12 19.18 odd 2