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Properties

Label 1260.2.ba.a.937.4
Level $1260$
Weight $2$
Character 1260.937
Analytic conductor $10.061$
Analytic rank $0$
Dimension $8$
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] = [1260,2,Mod(433,1260)] 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("1260.433"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1260, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.ba (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,-2] 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: no
Analytic conductor: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.11574317056.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 45x^{4} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 937.4
Root \(-1.83051 - 1.83051i\) of defining polynomial
Character \(\chi\) \(=\) 1260.937
Dual form 1260.2.ba.a.433.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.83051 - 1.28422i) q^{5} +(-1.57763 - 2.12393i) q^{7} +4.70156 q^{11} +(-1.83051 - 1.83051i) q^{13} +(-0.737925 + 0.737925i) q^{17} +4.75362 q^{19} +(-3.70156 + 3.70156i) q^{23} +(1.70156 - 4.70156i) q^{25} -0.701562i q^{29} -8.79790i q^{31} +(-5.61547 - 1.86185i) q^{35} +(-3.70156 - 3.70156i) q^{37} -4.75362i q^{41} +(-5.00000 + 5.00000i) q^{43} +(8.05998 - 8.05998i) q^{47} +(-2.02214 + 6.70156i) q^{49} +(-5.00000 + 5.00000i) q^{53} +(8.60627 - 6.03784i) q^{55} +4.75362 q^{59} -9.50723i q^{61} +(-5.70156 - 1.00000i) q^{65} +(5.00000 + 5.00000i) q^{67} +5.40312 q^{71} +(-3.11473 - 3.11473i) q^{73} +(-7.41734 - 9.98578i) q^{77} -6.70156i q^{79} +(7.86835 + 7.86835i) q^{83} +(-0.403124 + 2.29844i) q^{85} -13.5515 q^{89} +(-1.00000 + 6.77576i) q^{91} +(8.70156 - 6.10469i) q^{95} +(5.49154 - 5.49154i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{7} + 12 q^{11} - 4 q^{23} - 12 q^{25} + 14 q^{35} - 4 q^{37} - 40 q^{43} - 40 q^{53} - 20 q^{65} + 40 q^{67} - 8 q^{71} - 44 q^{77} + 48 q^{85} - 8 q^{91} + 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.83051 − 1.28422i 0.818631 − 0.574320i
\(6\) 0 0
\(7\) −1.57763 − 2.12393i −0.596289 − 0.802769i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.70156 1.41757 0.708787 − 0.705422i \(-0.249243\pi\)
0.708787 + 0.705422i \(0.249243\pi\)
\(12\) 0 0
\(13\) −1.83051 − 1.83051i −0.507693 − 0.507693i 0.406125 − 0.913818i \(-0.366880\pi\)
−0.913818 + 0.406125i \(0.866880\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.737925 + 0.737925i −0.178973 + 0.178973i −0.790908 − 0.611935i \(-0.790391\pi\)
0.611935 + 0.790908i \(0.290391\pi\)
\(18\) 0 0
\(19\) 4.75362 1.09055 0.545277 − 0.838256i \(-0.316424\pi\)
0.545277 + 0.838256i \(0.316424\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.70156 + 3.70156i −0.771829 + 0.771829i −0.978426 − 0.206597i \(-0.933761\pi\)
0.206597 + 0.978426i \(0.433761\pi\)
\(24\) 0 0
\(25\) 1.70156 − 4.70156i 0.340312 − 0.940312i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 0.701562i − 0.130277i −0.997876 − 0.0651384i \(-0.979251\pi\)
0.997876 − 0.0651384i \(-0.0207489\pi\)
\(30\) 0 0
\(31\) − 8.79790i − 1.58015i −0.613010 − 0.790075i \(-0.710041\pi\)
0.613010 − 0.790075i \(-0.289959\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.61547 − 1.86185i −0.949188 − 0.314711i
\(36\) 0 0
\(37\) −3.70156 − 3.70156i −0.608533 − 0.608533i 0.334030 − 0.942563i \(-0.391591\pi\)
−0.942563 + 0.334030i \(0.891591\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 4.75362i − 0.742390i −0.928555 − 0.371195i \(-0.878948\pi\)
0.928555 − 0.371195i \(-0.121052\pi\)
\(42\) 0 0
\(43\) −5.00000 + 5.00000i −0.762493 + 0.762493i −0.976772 − 0.214280i \(-0.931260\pi\)
0.214280 + 0.976772i \(0.431260\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.05998 − 8.05998i 1.17567 − 1.17567i 0.194832 − 0.980837i \(-0.437584\pi\)
0.980837 − 0.194832i \(-0.0624163\pi\)
\(48\) 0 0
\(49\) −2.02214 + 6.70156i −0.288878 + 0.957366i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.00000 + 5.00000i −0.686803 + 0.686803i −0.961524 − 0.274721i \(-0.911414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 8.60627 − 6.03784i 1.16047 − 0.814142i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.75362 0.618868 0.309434 − 0.950921i \(-0.399860\pi\)
0.309434 + 0.950921i \(0.399860\pi\)
\(60\) 0 0
\(61\) − 9.50723i − 1.21728i −0.793448 − 0.608638i \(-0.791716\pi\)
0.793448 − 0.608638i \(-0.208284\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.70156 − 1.00000i −0.707192 − 0.124035i
\(66\) 0 0
\(67\) 5.00000 + 5.00000i 0.610847 + 0.610847i 0.943167 − 0.332320i \(-0.107831\pi\)
−0.332320 + 0.943167i \(0.607831\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.40312 0.641233 0.320616 − 0.947209i \(-0.396110\pi\)
0.320616 + 0.947209i \(0.396110\pi\)
\(72\) 0 0
\(73\) −3.11473 − 3.11473i −0.364552 − 0.364552i 0.500934 − 0.865486i \(-0.332990\pi\)
−0.865486 + 0.500934i \(0.832990\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.41734 − 9.98578i −0.845285 − 1.13799i
\(78\) 0 0
\(79\) − 6.70156i − 0.753985i −0.926216 − 0.376992i \(-0.876958\pi\)
0.926216 − 0.376992i \(-0.123042\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.86835 + 7.86835i 0.863664 + 0.863664i 0.991762 − 0.128098i \(-0.0408872\pi\)
−0.128098 + 0.991762i \(0.540887\pi\)
\(84\) 0 0
\(85\) −0.403124 + 2.29844i −0.0437250 + 0.249301i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13.5515 −1.43646 −0.718229 − 0.695807i \(-0.755047\pi\)
−0.718229 + 0.695807i \(0.755047\pi\)
\(90\) 0 0
\(91\) −1.00000 + 6.77576i −0.104828 + 0.710293i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.70156 − 6.10469i 0.892761 − 0.626328i
\(96\) 0 0
\(97\) 5.49154 − 5.49154i 0.557582 − 0.557582i −0.371037 − 0.928618i \(-0.620998\pi\)
0.928618 + 0.371037i \(0.120998\pi\)
\(98\) 0 0
\(99\) 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 1260.2.ba.a.937.4 8
3.2 odd 2 140.2.m.a.97.1 yes 8
5.3 odd 4 inner 1260.2.ba.a.433.1 8
7.6 odd 2 inner 1260.2.ba.a.937.1 8
12.11 even 2 560.2.bj.b.97.4 8
15.2 even 4 700.2.m.c.293.1 8
15.8 even 4 140.2.m.a.13.4 yes 8
15.14 odd 2 700.2.m.c.657.4 8
21.2 odd 6 980.2.v.b.717.4 16
21.5 even 6 980.2.v.b.717.1 16
21.11 odd 6 980.2.v.b.117.1 16
21.17 even 6 980.2.v.b.117.4 16
21.20 even 2 140.2.m.a.97.4 yes 8
35.13 even 4 inner 1260.2.ba.a.433.4 8
60.23 odd 4 560.2.bj.b.433.1 8
84.83 odd 2 560.2.bj.b.97.1 8
105.23 even 12 980.2.v.b.913.4 16
105.38 odd 12 980.2.v.b.313.4 16
105.53 even 12 980.2.v.b.313.1 16
105.62 odd 4 700.2.m.c.293.4 8
105.68 odd 12 980.2.v.b.913.1 16
105.83 odd 4 140.2.m.a.13.1 ✓ 8
105.104 even 2 700.2.m.c.657.1 8
420.83 even 4 560.2.bj.b.433.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.m.a.13.1 ✓ 8 105.83 odd 4
140.2.m.a.13.4 yes 8 15.8 even 4
140.2.m.a.97.1 yes 8 3.2 odd 2
140.2.m.a.97.4 yes 8 21.20 even 2
560.2.bj.b.97.1 8 84.83 odd 2
560.2.bj.b.97.4 8 12.11 even 2
560.2.bj.b.433.1 8 60.23 odd 4
560.2.bj.b.433.4 8 420.83 even 4
700.2.m.c.293.1 8 15.2 even 4
700.2.m.c.293.4 8 105.62 odd 4
700.2.m.c.657.1 8 105.104 even 2
700.2.m.c.657.4 8 15.14 odd 2
980.2.v.b.117.1 16 21.11 odd 6
980.2.v.b.117.4 16 21.17 even 6
980.2.v.b.313.1 16 105.53 even 12
980.2.v.b.313.4 16 105.38 odd 12
980.2.v.b.717.1 16 21.5 even 6
980.2.v.b.717.4 16 21.2 odd 6
980.2.v.b.913.1 16 105.68 odd 12
980.2.v.b.913.4 16 105.23 even 12
1260.2.ba.a.433.1 8 5.3 odd 4 inner
1260.2.ba.a.433.4 8 35.13 even 4 inner
1260.2.ba.a.937.1 8 7.6 odd 2 inner
1260.2.ba.a.937.4 8 1.1 even 1 trivial