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Properties

Label 1210.2.a.l.1.1
Level $1210$
Weight $2$
Character 1210.1
Self dual yes
Analytic conductor $9.662$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [1210,2,Mod(1,1210)] 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("1210.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1210, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1210 = 2 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1210.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,1,1,1,1,1,1,-2,1,0,1,4,1,1,1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.66189864457\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1210.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} +1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} -5.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} +1.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} +1.00000 q^{35} -2.00000 q^{36} +8.00000 q^{37} +4.00000 q^{38} +4.00000 q^{39} +1.00000 q^{40} +3.00000 q^{41} +1.00000 q^{42} +1.00000 q^{43} -2.00000 q^{45} +9.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{50} +4.00000 q^{52} -12.0000 q^{53} -5.00000 q^{54} +1.00000 q^{56} +4.00000 q^{57} +6.00000 q^{58} +6.00000 q^{59} +1.00000 q^{60} -11.0000 q^{61} -10.0000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -1.00000 q^{67} +1.00000 q^{70} -6.00000 q^{71} -2.00000 q^{72} -8.00000 q^{73} +8.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +4.00000 q^{78} -14.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.00000 q^{82} +12.0000 q^{83} +1.00000 q^{84} +1.00000 q^{86} +6.00000 q^{87} -15.0000 q^{89} -2.00000 q^{90} +4.00000 q^{91} -10.0000 q^{93} +9.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} +8.00000 q^{97} -6.00000 q^{98} +O(q^{100})\)

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
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −2.00000 −0.471405
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) −5.00000 −0.962250
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 1.00000 0.182574
\(31\) −10.0000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) −2.00000 −0.333333
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 4.00000 0.648886
\(39\) 4.00000 0.640513
\(40\) 1.00000 0.158114
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 1.00000 0.154303
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 4.00000 0.529813
\(58\) 6.00000 0.787839
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 1.00000 0.129099
\(61\) −11.0000 −1.40841 −0.704203 − 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) −10.0000 −1.27000
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −1.00000 −0.122169 −0.0610847 − 0.998133i \(-0.519456\pi\)
−0.0610847 + 0.998133i \(0.519456\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −2.00000 −0.235702
\(73\) −8.00000 −0.936329 −0.468165 − 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 8.00000 0.929981
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) −14.0000 −1.57512 −0.787562 − 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 3.00000 0.331295
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 − 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) −2.00000 −0.210819
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) −10.0000 −1.03695
\(94\) 9.00000 0.928279
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −6.00000 −0.606092
\(99\) 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 1210.2.a.l.1.1 yes 1
4.3 odd 2 9680.2.a.k.1.1 1
5.4 even 2 6050.2.a.g.1.1 1
11.10 odd 2 1210.2.a.f.1.1 ✓ 1
44.43 even 2 9680.2.a.l.1.1 1
55.54 odd 2 6050.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1210.2.a.f.1.1 ✓ 1 11.10 odd 2
1210.2.a.l.1.1 yes 1 1.1 even 1 trivial
6050.2.a.g.1.1 1 5.4 even 2
6050.2.a.bd.1.1 1 55.54 odd 2
9680.2.a.k.1.1 1 4.3 odd 2
9680.2.a.l.1.1 1 44.43 even 2