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Properties

Label 1935.2.b.g.1549.33
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 1549.33
Character \(\chi\) \(=\) 1935.1549
Dual form 1935.2.b.g.1549.7

$q$-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)
 
\(f(q)\) \(=\) \(q+1.83296i q^{2} -1.35974 q^{4} +(-0.954173 - 2.02226i) q^{5} +3.64733i q^{7} +1.17356i q^{8} +(3.70673 - 1.74896i) q^{10} +0.124282 q^{11} +2.43266i q^{13} -6.68540 q^{14} -4.87058 q^{16} +2.46992i q^{17} +6.77389 q^{19} +(1.29743 + 2.74976i) q^{20} +0.227804i q^{22} -4.21466i q^{23} +(-3.17911 + 3.85918i) q^{25} -4.45897 q^{26} -4.95943i q^{28} +9.62369 q^{29} -10.6152 q^{31} -6.58046i q^{32} -4.52726 q^{34} +(7.37586 - 3.48018i) q^{35} +6.85378i q^{37} +12.4163i q^{38} +(2.37325 - 1.11978i) q^{40} -11.0350 q^{41} -1.00000i q^{43} -0.168992 q^{44} +7.72530 q^{46} -1.18498i q^{47} -6.30298 q^{49} +(-7.07373 - 5.82718i) q^{50} -3.30780i q^{52} -1.82984i q^{53} +(-0.118587 - 0.251331i) q^{55} -4.28036 q^{56} +17.6399i q^{58} +0.0841352 q^{59} +0.606586 q^{61} -19.4573i q^{62} +2.32056 q^{64} +(4.91948 - 2.32118i) q^{65} +8.00020i q^{67} -3.35845i q^{68} +(6.37903 + 13.5197i) q^{70} -11.2901 q^{71} +16.6712i q^{73} -12.5627 q^{74} -9.21076 q^{76} +0.453297i q^{77} -10.0899 q^{79} +(4.64738 + 9.84961i) q^{80} -20.2268i q^{82} +4.80853i q^{83} +(4.99482 - 2.35673i) q^{85} +1.83296 q^{86} +0.145853i q^{88} -4.53049 q^{89} -8.87270 q^{91} +5.73086i q^{92} +2.17202 q^{94} +(-6.46346 - 13.6986i) q^{95} -10.7693i q^{97} -11.5531i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 32 q^{4} - 12 q^{10} + 16 q^{16} + 20 q^{25} - 24 q^{31} - 48 q^{34} + 60 q^{40} + 16 q^{46} - 64 q^{49} + 4 q^{55} + 24 q^{61} + 16 q^{64} - 40 q^{70} - 40 q^{76} + 48 q^{79} - 20 q^{85} + 40 q^{91}+ \cdots + 80 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.83296i 1.29610i 0.761598 + 0.648049i \(0.224415\pi\)
−0.761598 + 0.648049i \(0.775585\pi\)
\(3\) 0 0
\(4\) −1.35974 −0.679872
\(5\) −0.954173 − 2.02226i −0.426719 − 0.904384i
\(6\) 0 0
\(7\) 3.64733i 1.37856i 0.724495 + 0.689280i \(0.242073\pi\)
−0.724495 + 0.689280i \(0.757927\pi\)
\(8\) 1.17356i 0.414917i
\(9\) 0 0
\(10\) 3.70673 − 1.74896i 1.17217 − 0.553070i
\(11\) 0.124282 0.0374725 0.0187362 − 0.999824i \(-0.494036\pi\)
0.0187362 + 0.999824i \(0.494036\pi\)
\(12\) 0 0
\(13\) 2.43266i 0.674698i 0.941380 + 0.337349i \(0.109530\pi\)
−0.941380 + 0.337349i \(0.890470\pi\)
\(14\) −6.68540 −1.78675
\(15\) 0 0
\(16\) −4.87058 −1.21765
\(17\) 2.46992i 0.599042i 0.954090 + 0.299521i \(0.0968269\pi\)
−0.954090 + 0.299521i \(0.903173\pi\)
\(18\) 0 0
\(19\) 6.77389 1.55404 0.777018 − 0.629478i \(-0.216731\pi\)
0.777018 + 0.629478i \(0.216731\pi\)
\(20\) 1.29743 + 2.74976i 0.290115 + 0.614866i
\(21\) 0 0
\(22\) 0.227804i 0.0485680i
\(23\) − 4.21466i − 0.878817i −0.898287 − 0.439408i \(-0.855188\pi\)
0.898287 − 0.439408i \(-0.144812\pi\)
\(24\) 0 0
\(25\) −3.17911 + 3.85918i −0.635821 + 0.771836i
\(26\) −4.45897 −0.874476
\(27\) 0 0
\(28\) − 4.95943i − 0.937245i
\(29\) 9.62369 1.78708 0.893538 − 0.448988i \(-0.148216\pi\)
0.893538 + 0.448988i \(0.148216\pi\)
\(30\) 0 0
\(31\) −10.6152 −1.90655 −0.953274 − 0.302106i \(-0.902310\pi\)
−0.953274 + 0.302106i \(0.902310\pi\)
\(32\) − 6.58046i − 1.16327i
\(33\) 0 0
\(34\) −4.52726 −0.776418
\(35\) 7.37586 − 3.48018i 1.24675 − 0.588258i
\(36\) 0 0
\(37\) 6.85378i 1.12675i 0.826200 + 0.563377i \(0.190498\pi\)
−0.826200 + 0.563377i \(0.809502\pi\)
\(38\) 12.4163i 2.01418i
\(39\) 0 0
\(40\) 2.37325 − 1.11978i 0.375244 − 0.177053i
\(41\) −11.0350 −1.72338 −0.861692 − 0.507432i \(-0.830595\pi\)
−0.861692 + 0.507432i \(0.830595\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i
\(44\) −0.168992 −0.0254765
\(45\) 0 0
\(46\) 7.72530 1.13903
\(47\) − 1.18498i − 0.172847i −0.996258 − 0.0864235i \(-0.972456\pi\)
0.996258 − 0.0864235i \(-0.0275438\pi\)
\(48\) 0 0
\(49\) −6.30298 −0.900426
\(50\) −7.07373 − 5.82718i −1.00038 − 0.824087i
\(51\) 0 0
\(52\) − 3.30780i − 0.458709i
\(53\) − 1.82984i − 0.251348i −0.992072 − 0.125674i \(-0.959891\pi\)
0.992072 − 0.125674i \(-0.0401094\pi\)
\(54\) 0 0
\(55\) −0.118587 − 0.251331i −0.0159902 − 0.0338895i
\(56\) −4.28036 −0.571988
\(57\) 0 0
\(58\) 17.6399i 2.31623i
\(59\) 0.0841352 0.0109535 0.00547674 − 0.999985i \(-0.498257\pi\)
0.00547674 + 0.999985i \(0.498257\pi\)
\(60\) 0 0
\(61\) 0.606586 0.0776654 0.0388327 − 0.999246i \(-0.487636\pi\)
0.0388327 + 0.999246i \(0.487636\pi\)
\(62\) − 19.4573i − 2.47108i
\(63\) 0 0
\(64\) 2.32056 0.290070
\(65\) 4.91948 − 2.32118i 0.610186 − 0.287907i
\(66\) 0 0
\(67\) 8.00020i 0.977380i 0.872458 + 0.488690i \(0.162525\pi\)
−0.872458 + 0.488690i \(0.837475\pi\)
\(68\) − 3.35845i − 0.407272i
\(69\) 0 0
\(70\) 6.37903 + 13.5197i 0.762440 + 1.61591i
\(71\) −11.2901 −1.33989 −0.669946 − 0.742410i \(-0.733683\pi\)
−0.669946 + 0.742410i \(0.733683\pi\)
\(72\) 0 0
\(73\) 16.6712i 1.95122i 0.219517 + 0.975609i \(0.429552\pi\)
−0.219517 + 0.975609i \(0.570448\pi\)
\(74\) −12.5627 −1.46039
\(75\) 0 0
\(76\) −9.21076 −1.05655
\(77\) 0.453297i 0.0516580i
\(78\) 0 0
\(79\) −10.0899 −1.13521 −0.567604 − 0.823302i \(-0.692129\pi\)
−0.567604 + 0.823302i \(0.692129\pi\)
\(80\) 4.64738 + 9.84961i 0.519593 + 1.10122i
\(81\) 0 0
\(82\) − 20.2268i − 2.23368i
\(83\) 4.80853i 0.527804i 0.964549 + 0.263902i \(0.0850096\pi\)
−0.964549 + 0.263902i \(0.914990\pi\)
\(84\) 0 0
\(85\) 4.99482 − 2.35673i 0.541764 − 0.255623i
\(86\) 1.83296 0.197653
\(87\) 0 0
\(88\) 0.145853i 0.0155480i
\(89\) −4.53049 −0.480231 −0.240116 − 0.970744i \(-0.577185\pi\)
−0.240116 + 0.970744i \(0.577185\pi\)
\(90\) 0 0
\(91\) −8.87270 −0.930112
\(92\) 5.73086i 0.597483i
\(93\) 0 0
\(94\) 2.17202 0.224027
\(95\) −6.46346 − 13.6986i −0.663137 − 1.40545i
\(96\) 0 0
\(97\) − 10.7693i − 1.09346i −0.837309 − 0.546730i \(-0.815872\pi\)
0.837309 − 0.546730i \(-0.184128\pi\)
\(98\) − 11.5531i − 1.16704i
\(99\) 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.1549.33 yes 40
3.2 odd 2 inner 1935.2.b.g.1549.8 yes 40
5.2 odd 4 9675.2.a.db.1.4 20
5.3 odd 4 9675.2.a.da.1.17 20
5.4 even 2 inner 1935.2.b.g.1549.7 ✓ 40
15.2 even 4 9675.2.a.db.1.17 20
15.8 even 4 9675.2.a.da.1.4 20
15.14 odd 2 inner 1935.2.b.g.1549.34 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1935.2.b.g.1549.7 ✓ 40 5.4 even 2 inner
1935.2.b.g.1549.8 yes 40 3.2 odd 2 inner
1935.2.b.g.1549.33 yes 40 1.1 even 1 trivial
1935.2.b.g.1549.34 yes 40 15.14 odd 2 inner
9675.2.a.da.1.4 20 15.8 even 4
9675.2.a.da.1.17 20 5.3 odd 4
9675.2.a.db.1.4 20 5.2 odd 4
9675.2.a.db.1.17 20 15.2 even 4