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Properties

Label 560.2.bj.b.97.1
Level $560$
Weight $2$
Character 560.97
Analytic conductor $4.472$
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] = [560,2,Mod(97,560)] 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("560.97"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.bj (of order \(4\), degree \(2\), not 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: \(4.47162251319\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 97.1
Root \(-1.83051 - 1.83051i\) of defining polynomial
Character \(\chi\) \(=\) 560.97
Dual form 560.2.bj.b.433.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.83051 - 1.83051i) q^{3} +(1.83051 - 1.28422i) q^{5} +(2.12393 + 1.57763i) q^{7} +3.70156i q^{9} +4.70156 q^{11} +(1.83051 + 1.83051i) q^{13} +(-5.70156 - 1.00000i) q^{15} +(-0.737925 + 0.737925i) q^{17} +4.75362 q^{19} +(-1.00000 - 6.77576i) q^{21} +(-3.70156 + 3.70156i) q^{23} +(1.70156 - 4.70156i) q^{25} +(1.28422 - 1.28422i) q^{27} +0.701562i q^{29} -8.79790i q^{31} +(-8.60627 - 8.60627i) q^{33} +(5.91391 + 0.160291i) q^{35} +(-3.70156 - 3.70156i) q^{37} -6.70156i q^{39} -4.75362i q^{41} +(5.00000 - 5.00000i) q^{43} +(4.75362 + 6.77576i) q^{45} +(-8.05998 + 8.05998i) q^{47} +(2.02214 + 6.70156i) q^{49} +2.70156 q^{51} +(5.00000 - 5.00000i) q^{53} +(8.60627 - 6.03784i) q^{55} +(-8.70156 - 8.70156i) q^{57} -4.75362 q^{59} +9.50723i q^{61} +(-5.83971 + 7.86185i) q^{63} +(5.70156 + 1.00000i) q^{65} +(-5.00000 - 5.00000i) q^{67} +13.5515 q^{69} +5.40312 q^{71} +(3.11473 + 3.11473i) q^{73} +(-11.7210 + 5.49154i) q^{75} +(9.98578 + 7.41734i) q^{77} +6.70156i q^{79} +6.40312 q^{81} +(-7.86835 - 7.86835i) q^{83} +(-0.403124 + 2.29844i) q^{85} +(1.28422 - 1.28422i) q^{87} -13.5515 q^{89} +(1.00000 + 6.77576i) q^{91} +(-16.1047 + 16.1047i) q^{93} +(8.70156 - 6.10469i) q^{95} +(-5.49154 + 5.49154i) q^{97} +17.4031i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{7} + 12 q^{11} - 20 q^{15} - 8 q^{21} - 4 q^{23} - 12 q^{25} + 14 q^{35} - 4 q^{37} + 40 q^{43} - 4 q^{51} + 40 q^{53} - 44 q^{57} - 42 q^{63} + 20 q^{65} - 40 q^{67} - 8 q^{71} + 44 q^{77} + 48 q^{85}+ \cdots + 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

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
\(3\) −1.83051 − 1.83051i −1.05685 − 1.05685i −0.998284 − 0.0585640i \(-0.981348\pi\)
−0.0585640 − 0.998284i \(-0.518652\pi\)
\(4\) 0 0
\(5\) 1.83051 − 1.28422i 0.818631 − 0.574320i
\(6\) 0 0
\(7\) 2.12393 + 1.57763i 0.802769 + 0.596289i
\(8\) 0 0
\(9\) 3.70156i 1.23385i
\(10\) 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 0.913818 − 0.406125i \(-0.133120\pi\)
−0.406125 + 0.913818i \(0.633120\pi\)
\(14\) 0 0
\(15\) −5.70156 − 1.00000i −1.47214 − 0.258199i
\(16\) 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
\(21\) −1.00000 − 6.77576i −0.218218 − 1.47859i
\(22\) 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
\(27\) 1.28422 − 1.28422i 0.247148 − 0.247148i
\(28\) 0 0
\(29\) 0.701562i 0.130277i 0.997876 + 0.0651384i \(0.0207489\pi\)
−0.997876 + 0.0651384i \(0.979251\pi\)
\(30\) 0 0
\(31\) − 8.79790i − 1.58015i −0.613010 − 0.790075i \(-0.710041\pi\)
0.613010 − 0.790075i \(-0.289959\pi\)
\(32\) 0 0
\(33\) −8.60627 − 8.60627i −1.49816 − 1.49816i
\(34\) 0 0
\(35\) 5.91391 + 0.160291i 0.999633 + 0.0270941i
\(36\) 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
\(39\) − 6.70156i − 1.07311i
\(40\) 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 −0.214280 − 0.976772i \(-0.568740\pi\)
0.976772 + 0.214280i \(0.0687403\pi\)
\(44\) 0 0
\(45\) 4.75362 + 6.77576i 0.708627 + 1.01007i
\(46\) 0 0
\(47\) −8.05998 + 8.05998i −1.17567 + 1.17567i −0.194832 + 0.980837i \(0.562416\pi\)
−0.980837 + 0.194832i \(0.937584\pi\)
\(48\) 0 0
\(49\) 2.02214 + 6.70156i 0.288878 + 0.957366i
\(50\) 0 0
\(51\) 2.70156 0.378294
\(52\) 0 0
\(53\) 5.00000 − 5.00000i 0.686803 − 0.686803i −0.274721 − 0.961524i \(-0.588586\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 0 0
\(55\) 8.60627 − 6.03784i 1.16047 − 0.814142i
\(56\) 0 0
\(57\) −8.70156 − 8.70156i −1.15255 − 1.15255i
\(58\) 0 0
\(59\) −4.75362 −0.618868 −0.309434 − 0.950921i \(-0.600140\pi\)
−0.309434 + 0.950921i \(0.600140\pi\)
\(60\) 0 0
\(61\) 9.50723i 1.21728i 0.793448 + 0.608638i \(0.208284\pi\)
−0.793448 + 0.608638i \(0.791716\pi\)
\(62\) 0 0
\(63\) −5.83971 + 7.86185i −0.735734 + 0.990500i
\(64\) 0 0
\(65\) 5.70156 + 1.00000i 0.707192 + 0.124035i
\(66\) 0 0
\(67\) −5.00000 − 5.00000i −0.610847 − 0.610847i 0.332320 − 0.943167i \(-0.392169\pi\)
−0.943167 + 0.332320i \(0.892169\pi\)
\(68\) 0 0
\(69\) 13.5515 1.63141
\(70\) 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 0.865486 − 0.500934i \(-0.167010\pi\)
−0.500934 + 0.865486i \(0.667010\pi\)
\(74\) 0 0
\(75\) −11.7210 + 5.49154i −1.35343 + 0.634109i
\(76\) 0 0
\(77\) 9.98578 + 7.41734i 1.13799 + 0.845285i
\(78\) 0 0
\(79\) 6.70156i 0.753985i 0.926216 + 0.376992i \(0.123042\pi\)
−0.926216 + 0.376992i \(0.876958\pi\)
\(80\) 0 0
\(81\) 6.40312 0.711458
\(82\) 0 0
\(83\) −7.86835 − 7.86835i −0.863664 − 0.863664i 0.128098 − 0.991762i \(-0.459113\pi\)
−0.991762 + 0.128098i \(0.959113\pi\)
\(84\) 0 0
\(85\) −0.403124 + 2.29844i −0.0437250 + 0.249301i
\(86\) 0 0
\(87\) 1.28422 − 1.28422i 0.137683 − 0.137683i
\(88\) 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
\(93\) −16.1047 + 16.1047i −1.66998 + 1.66998i
\(94\) 0 0
\(95\) 8.70156 − 6.10469i 0.892761 − 0.626328i
\(96\) 0 0
\(97\) −5.49154 + 5.49154i −0.557582 + 0.557582i −0.928618 − 0.371037i \(-0.879002\pi\)
0.371037 + 0.928618i \(0.379002\pi\)
\(98\) 0 0
\(99\) 17.4031i 1.74908i
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 560.2.bj.b.97.1 8
4.3 odd 2 140.2.m.a.97.4 yes 8
5.3 odd 4 inner 560.2.bj.b.433.4 8
7.6 odd 2 inner 560.2.bj.b.97.4 8
12.11 even 2 1260.2.ba.a.937.1 8
20.3 even 4 140.2.m.a.13.1 ✓ 8
20.7 even 4 700.2.m.c.293.4 8
20.19 odd 2 700.2.m.c.657.1 8
28.3 even 6 980.2.v.b.117.1 16
28.11 odd 6 980.2.v.b.117.4 16
28.19 even 6 980.2.v.b.717.4 16
28.23 odd 6 980.2.v.b.717.1 16
28.27 even 2 140.2.m.a.97.1 yes 8
35.13 even 4 inner 560.2.bj.b.433.1 8
60.23 odd 4 1260.2.ba.a.433.4 8
84.83 odd 2 1260.2.ba.a.937.4 8
140.3 odd 12 980.2.v.b.313.1 16
140.23 even 12 980.2.v.b.913.1 16
140.27 odd 4 700.2.m.c.293.1 8
140.83 odd 4 140.2.m.a.13.4 yes 8
140.103 odd 12 980.2.v.b.913.4 16
140.123 even 12 980.2.v.b.313.4 16
140.139 even 2 700.2.m.c.657.4 8
420.83 even 4 1260.2.ba.a.433.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.m.a.13.1 ✓ 8 20.3 even 4
140.2.m.a.13.4 yes 8 140.83 odd 4
140.2.m.a.97.1 yes 8 28.27 even 2
140.2.m.a.97.4 yes 8 4.3 odd 2
560.2.bj.b.97.1 8 1.1 even 1 trivial
560.2.bj.b.97.4 8 7.6 odd 2 inner
560.2.bj.b.433.1 8 35.13 even 4 inner
560.2.bj.b.433.4 8 5.3 odd 4 inner
700.2.m.c.293.1 8 140.27 odd 4
700.2.m.c.293.4 8 20.7 even 4
700.2.m.c.657.1 8 20.19 odd 2
700.2.m.c.657.4 8 140.139 even 2
980.2.v.b.117.1 16 28.3 even 6
980.2.v.b.117.4 16 28.11 odd 6
980.2.v.b.313.1 16 140.3 odd 12
980.2.v.b.313.4 16 140.123 even 12
980.2.v.b.717.1 16 28.23 odd 6
980.2.v.b.717.4 16 28.19 even 6
980.2.v.b.913.1 16 140.23 even 12
980.2.v.b.913.4 16 140.103 odd 12
1260.2.ba.a.433.1 8 420.83 even 4
1260.2.ba.a.433.4 8 60.23 odd 4
1260.2.ba.a.937.1 8 12.11 even 2
1260.2.ba.a.937.4 8 84.83 odd 2