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Properties

Label 1935.2.b.g.1549.6
Level $1935$
Weight $2$
Character 1935.1549
Analytic conductor $15.451$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1935,2,Mod(1549,1935)] 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("1935.1549"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1935, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1935 = 3^{2} \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1935.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0,0,-32,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.4510527911\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1549.6
Character \(\chi\) \(=\) 1935.1549
Dual form 1935.2.b.g.1549.36

$q$-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)
 
\(f(q)\) \(=\) \(q-2.19006i q^{2} -2.79638 q^{4} +(2.22627 + 0.209145i) q^{5} -4.99426i q^{7} +1.74413i q^{8} +(0.458041 - 4.87567i) q^{10} -5.42090 q^{11} +5.08752i q^{13} -10.9377 q^{14} -1.77300 q^{16} -2.09728i q^{17} -7.09005 q^{19} +(-6.22549 - 0.584849i) q^{20} +11.8721i q^{22} +1.86354i q^{23} +(4.91252 + 0.931223i) q^{25} +11.1420 q^{26} +13.9659i q^{28} +1.21100 q^{29} -3.60902 q^{31} +7.37126i q^{32} -4.59317 q^{34} +(1.04452 - 11.1185i) q^{35} +0.533626i q^{37} +15.5277i q^{38} +(-0.364776 + 3.88290i) q^{40} +8.69797 q^{41} -1.00000i q^{43} +15.1589 q^{44} +4.08128 q^{46} +1.27299i q^{47} -17.9426 q^{49} +(2.03944 - 10.7587i) q^{50} -14.2267i q^{52} -6.46214i q^{53} +(-12.0684 - 1.13375i) q^{55} +8.71064 q^{56} -2.65217i q^{58} -11.2530 q^{59} -11.8390 q^{61} +7.90398i q^{62} +12.5975 q^{64} +(-1.06403 + 11.3262i) q^{65} -12.1852i q^{67} +5.86479i q^{68} +(-24.3503 - 2.28757i) q^{70} +6.49765 q^{71} -0.220835i q^{73} +1.16868 q^{74} +19.8265 q^{76} +27.0733i q^{77} -4.51633 q^{79} +(-3.94718 - 0.370814i) q^{80} -19.0491i q^{82} -9.92115i q^{83} +(0.438634 - 4.66909i) q^{85} -2.19006 q^{86} -9.45477i q^{88} +11.1048 q^{89} +25.4084 q^{91} -5.21118i q^{92} +2.78794 q^{94} +(-15.7843 - 1.48285i) q^{95} +9.19491i q^{97} +39.2954i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 32 q^{4} - 12 q^{10} + 16 q^{16} + 20 q^{25} - 24 q^{31} - 48 q^{34} + 60 q^{40} + 16 q^{46} - 64 q^{49} + 4 q^{55} + 24 q^{61} + 16 q^{64} - 40 q^{70} - 40 q^{76} + 48 q^{79} - 20 q^{85} + 40 q^{91}+ \cdots + 80 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1935\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(1162\) \(1721\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.19006i − 1.54861i −0.632813 − 0.774305i \(-0.718100\pi\)
0.632813 − 0.774305i \(-0.281900\pi\)
\(3\) 0 0
\(4\) −2.79638 −1.39819
\(5\) 2.22627 + 0.209145i 0.995616 + 0.0935324i
\(6\) 0 0
\(7\) − 4.99426i − 1.88765i −0.330444 − 0.943826i \(-0.607199\pi\)
0.330444 − 0.943826i \(-0.392801\pi\)
\(8\) 1.74413i 0.616644i
\(9\) 0 0
\(10\) 0.458041 − 4.87567i 0.144845 − 1.54182i
\(11\) −5.42090 −1.63446 −0.817231 − 0.576310i \(-0.804492\pi\)
−0.817231 + 0.576310i \(0.804492\pi\)
\(12\) 0 0
\(13\) 5.08752i 1.41102i 0.708698 + 0.705512i \(0.249283\pi\)
−0.708698 + 0.705512i \(0.750717\pi\)
\(14\) −10.9377 −2.92323
\(15\) 0 0
\(16\) −1.77300 −0.443251
\(17\) − 2.09728i − 0.508664i −0.967117 − 0.254332i \(-0.918144\pi\)
0.967117 − 0.254332i \(-0.0818556\pi\)
\(18\) 0 0
\(19\) −7.09005 −1.62657 −0.813284 − 0.581866i \(-0.802323\pi\)
−0.813284 + 0.581866i \(0.802323\pi\)
\(20\) −6.22549 − 0.584849i −1.39206 − 0.130776i
\(21\) 0 0
\(22\) 11.8721i 2.53114i
\(23\) 1.86354i 0.388575i 0.980945 + 0.194288i \(0.0622395\pi\)
−0.980945 + 0.194288i \(0.937760\pi\)
\(24\) 0 0
\(25\) 4.91252 + 0.931223i 0.982503 + 0.186245i
\(26\) 11.1420 2.18513
\(27\) 0 0
\(28\) 13.9659i 2.63930i
\(29\) 1.21100 0.224878 0.112439 − 0.993659i \(-0.464134\pi\)
0.112439 + 0.993659i \(0.464134\pi\)
\(30\) 0 0
\(31\) −3.60902 −0.648199 −0.324099 − 0.946023i \(-0.605061\pi\)
−0.324099 + 0.946023i \(0.605061\pi\)
\(32\) 7.37126i 1.30307i
\(33\) 0 0
\(34\) −4.59317 −0.787722
\(35\) 1.04452 − 11.1185i 0.176556 − 1.87938i
\(36\) 0 0
\(37\) 0.533626i 0.0877276i 0.999038 + 0.0438638i \(0.0139668\pi\)
−0.999038 + 0.0438638i \(0.986033\pi\)
\(38\) 15.5277i 2.51892i
\(39\) 0 0
\(40\) −0.364776 + 3.88290i −0.0576762 + 0.613941i
\(41\) 8.69797 1.35839 0.679197 − 0.733956i \(-0.262328\pi\)
0.679197 + 0.733956i \(0.262328\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i
\(44\) 15.1589 2.28529
\(45\) 0 0
\(46\) 4.08128 0.601752
\(47\) 1.27299i 0.185685i 0.995681 + 0.0928425i \(0.0295953\pi\)
−0.995681 + 0.0928425i \(0.970405\pi\)
\(48\) 0 0
\(49\) −17.9426 −2.56323
\(50\) 2.03944 − 10.7587i 0.288420 − 1.52151i
\(51\) 0 0
\(52\) − 14.2267i − 1.97288i
\(53\) − 6.46214i − 0.887643i −0.896115 − 0.443822i \(-0.853622\pi\)
0.896115 − 0.443822i \(-0.146378\pi\)
\(54\) 0 0
\(55\) −12.0684 − 1.13375i −1.62730 − 0.152875i
\(56\) 8.71064 1.16401
\(57\) 0 0
\(58\) − 2.65217i − 0.348248i
\(59\) −11.2530 −1.46501 −0.732507 − 0.680759i \(-0.761650\pi\)
−0.732507 + 0.680759i \(0.761650\pi\)
\(60\) 0 0
\(61\) −11.8390 −1.51583 −0.757916 − 0.652352i \(-0.773782\pi\)
−0.757916 + 0.652352i \(0.773782\pi\)
\(62\) 7.90398i 1.00381i
\(63\) 0 0
\(64\) 12.5975 1.57469
\(65\) −1.06403 + 11.3262i −0.131976 + 1.40484i
\(66\) 0 0
\(67\) − 12.1852i − 1.48865i −0.667815 − 0.744327i \(-0.732770\pi\)
0.667815 − 0.744327i \(-0.267230\pi\)
\(68\) 5.86479i 0.711210i
\(69\) 0 0
\(70\) −24.3503 − 2.28757i −2.91042 − 0.273417i
\(71\) 6.49765 0.771129 0.385564 − 0.922681i \(-0.374007\pi\)
0.385564 + 0.922681i \(0.374007\pi\)
\(72\) 0 0
\(73\) − 0.220835i − 0.0258468i −0.999916 − 0.0129234i \(-0.995886\pi\)
0.999916 − 0.0129234i \(-0.00411376\pi\)
\(74\) 1.16868 0.135856
\(75\) 0 0
\(76\) 19.8265 2.27426
\(77\) 27.0733i 3.08529i
\(78\) 0 0
\(79\) −4.51633 −0.508127 −0.254063 − 0.967188i \(-0.581767\pi\)
−0.254063 + 0.967188i \(0.581767\pi\)
\(80\) −3.94718 − 0.370814i −0.441308 − 0.0414583i
\(81\) 0 0
\(82\) − 19.0491i − 2.10362i
\(83\) − 9.92115i − 1.08899i −0.838765 − 0.544494i \(-0.816722\pi\)
0.838765 − 0.544494i \(-0.183278\pi\)
\(84\) 0 0
\(85\) 0.438634 − 4.66909i 0.0475766 − 0.506434i
\(86\) −2.19006 −0.236161
\(87\) 0 0
\(88\) − 9.45477i − 1.00788i
\(89\) 11.1048 1.17711 0.588554 − 0.808458i \(-0.299697\pi\)
0.588554 + 0.808458i \(0.299697\pi\)
\(90\) 0 0
\(91\) 25.4084 2.66352
\(92\) − 5.21118i − 0.543303i
\(93\) 0 0
\(94\) 2.78794 0.287554
\(95\) −15.7843 − 1.48285i −1.61944 − 0.152137i
\(96\) 0 0
\(97\) 9.19491i 0.933602i 0.884362 + 0.466801i \(0.154593\pi\)
−0.884362 + 0.466801i \(0.845407\pi\)
\(98\) 39.2954i 3.96944i
\(99\) 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 1935.2.b.g.1549.6 yes 40
3.2 odd 2 inner 1935.2.b.g.1549.35 yes 40
5.2 odd 4 9675.2.a.db.1.18 20
5.3 odd 4 9675.2.a.da.1.3 20
5.4 even 2 inner 1935.2.b.g.1549.36 yes 40
15.2 even 4 9675.2.a.db.1.3 20
15.8 even 4 9675.2.a.da.1.18 20
15.14 odd 2 inner 1935.2.b.g.1549.5 ✓ 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1935.2.b.g.1549.5 ✓ 40 15.14 odd 2 inner
1935.2.b.g.1549.6 yes 40 1.1 even 1 trivial
1935.2.b.g.1549.35 yes 40 3.2 odd 2 inner
1935.2.b.g.1549.36 yes 40 5.4 even 2 inner
9675.2.a.da.1.3 20 5.3 odd 4
9675.2.a.da.1.18 20 15.8 even 4
9675.2.a.db.1.3 20 15.2 even 4
9675.2.a.db.1.18 20 5.2 odd 4