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Properties

Label 3520.2.f.g.2111.1
Level $3520$
Weight $2$
Character 3520.2111
Analytic conductor $28.107$
Analytic rank $0$
Dimension $4$
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] = [3520,2,Mod(2111,3520)] 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("3520.2111"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3520.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-4,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0, 0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(33)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.1073415115\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 880)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2111.1
Root \(0.517638i\) of defining polynomial
Character \(\chi\) \(=\) 3520.2111
Dual form 3520.2.f.g.2111.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421i q^{3} -1.00000 q^{5} -3.46410 q^{7} +1.00000 q^{9} +(-1.73205 - 2.82843i) q^{11} +2.44949i q^{13} +1.41421i q^{15} +7.34847i q^{17} +3.46410 q^{19} +4.89898i q^{21} +1.41421i q^{23} +1.00000 q^{25} -5.65685i q^{27} -4.89898i q^{29} +(-4.00000 + 2.44949i) q^{33} +3.46410 q^{35} +2.00000 q^{37} +3.46410 q^{39} +3.46410 q^{43} -1.00000 q^{45} +7.07107i q^{47} +5.00000 q^{49} +10.3923 q^{51} +6.00000 q^{53} +(1.73205 + 2.82843i) q^{55} -4.89898i q^{57} +2.82843i q^{59} +9.79796i q^{61} -3.46410 q^{63} -2.44949i q^{65} -12.7279i q^{67} +2.00000 q^{69} -5.65685i q^{71} -7.34847i q^{73} -1.41421i q^{75} +(6.00000 + 9.79796i) q^{77} +10.3923 q^{79} -5.00000 q^{81} -10.3923 q^{83} -7.34847i q^{85} -6.92820 q^{87} +12.0000 q^{89} -8.48528i q^{91} -3.46410 q^{95} -2.00000 q^{97} +(-1.73205 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 4 q^{9} + 4 q^{25} - 16 q^{33} + 8 q^{37} - 4 q^{45} + 20 q^{49} + 24 q^{53} + 8 q^{69} + 24 q^{77} - 20 q^{81} + 48 q^{89} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(1541\) \(2751\) \(2817\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.41421i − 0.816497i −0.912871 − 0.408248i \(-0.866140\pi\)
0.912871 − 0.408248i \(-0.133860\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 − 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.73205 − 2.82843i −0.522233 − 0.852803i
\(12\) 0 0
\(13\) 2.44949i 0.679366i 0.940540 + 0.339683i \(0.110320\pi\)
−0.940540 + 0.339683i \(0.889680\pi\)
\(14\) 0 0
\(15\) 1.41421i 0.365148i
\(16\) 0 0
\(17\) 7.34847i 1.78227i 0.453743 + 0.891133i \(0.350089\pi\)
−0.453743 + 0.891133i \(0.649911\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 − 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 4.89898i 1.06904i
\(22\) 0 0
\(23\) 1.41421i 0.294884i 0.989071 + 0.147442i \(0.0471040\pi\)
−0.989071 + 0.147442i \(0.952896\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) − 5.65685i − 1.08866i
\(28\) 0 0
\(29\) − 4.89898i − 0.909718i −0.890564 − 0.454859i \(-0.849690\pi\)
0.890564 − 0.454859i \(-0.150310\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) −4.00000 + 2.44949i −0.696311 + 0.426401i
\(34\) 0 0
\(35\) 3.46410 0.585540
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 3.46410 0.554700
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 3.46410 0.528271 0.264135 − 0.964486i \(-0.414913\pi\)
0.264135 + 0.964486i \(0.414913\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 7.07107i 1.03142i 0.856763 + 0.515711i \(0.172472\pi\)
−0.856763 + 0.515711i \(0.827528\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 10.3923 1.45521
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 1.73205 + 2.82843i 0.233550 + 0.381385i
\(56\) 0 0
\(57\) − 4.89898i − 0.648886i
\(58\) 0 0
\(59\) 2.82843i 0.368230i 0.982905 + 0.184115i \(0.0589419\pi\)
−0.982905 + 0.184115i \(0.941058\pi\)
\(60\) 0 0
\(61\) 9.79796i 1.25450i 0.778818 + 0.627250i \(0.215820\pi\)
−0.778818 + 0.627250i \(0.784180\pi\)
\(62\) 0 0
\(63\) −3.46410 −0.436436
\(64\) 0 0
\(65\) − 2.44949i − 0.303822i
\(66\) 0 0
\(67\) − 12.7279i − 1.55496i −0.628906 − 0.777482i \(-0.716497\pi\)
0.628906 − 0.777482i \(-0.283503\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) − 5.65685i − 0.671345i −0.941979 − 0.335673i \(-0.891036\pi\)
0.941979 − 0.335673i \(-0.108964\pi\)
\(72\) 0 0
\(73\) − 7.34847i − 0.860073i −0.902811 − 0.430037i \(-0.858501\pi\)
0.902811 − 0.430037i \(-0.141499\pi\)
\(74\) 0 0
\(75\) − 1.41421i − 0.163299i
\(76\) 0 0
\(77\) 6.00000 + 9.79796i 0.683763 + 1.11658i
\(78\) 0 0
\(79\) 10.3923 1.16923 0.584613 − 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) −10.3923 −1.14070 −0.570352 − 0.821401i \(-0.693193\pi\)
−0.570352 + 0.821401i \(0.693193\pi\)
\(84\) 0 0
\(85\) − 7.34847i − 0.797053i
\(86\) 0 0
\(87\) −6.92820 −0.742781
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) − 8.48528i − 0.889499i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.46410 −0.355409
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −1.73205 − 2.82843i −0.174078 − 0.284268i
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 3520.2.f.g.2111.1 4
4.3 odd 2 inner 3520.2.f.g.2111.4 4
8.3 odd 2 880.2.f.c.351.2 yes 4
8.5 even 2 880.2.f.c.351.3 yes 4
11.10 odd 2 inner 3520.2.f.g.2111.2 4
44.43 even 2 inner 3520.2.f.g.2111.3 4
88.21 odd 2 880.2.f.c.351.4 yes 4
88.43 even 2 880.2.f.c.351.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
880.2.f.c.351.1 ✓ 4 88.43 even 2
880.2.f.c.351.2 yes 4 8.3 odd 2
880.2.f.c.351.3 yes 4 8.5 even 2
880.2.f.c.351.4 yes 4 88.21 odd 2
3520.2.f.g.2111.1 4 1.1 even 1 trivial
3520.2.f.g.2111.2 4 11.10 odd 2 inner
3520.2.f.g.2111.3 4 44.43 even 2 inner
3520.2.f.g.2111.4 4 4.3 odd 2 inner