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Properties

Label 2240.4.a.bo.1.2
Level $2240$
Weight $4$
Character 2240.1
Self dual yes
Analytic conductor $132.164$
Analytic rank $1$
Dimension $2$
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] = [2240,4,Mod(1,2240)] 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("2240.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2240.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,10,0,14,0,12,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(132.164278413\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2240.1

$q$-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)
 
\(f(q)\) \(=\) \(q+6.65685 q^{3} +5.00000 q^{5} +7.00000 q^{7} +17.3137 q^{9} +38.2548 q^{11} -19.3431 q^{13} +33.2843 q^{15} -87.2254 q^{17} -44.2254 q^{19} +46.5980 q^{21} -218.167 q^{23} +25.0000 q^{25} -64.4802 q^{27} +46.9411 q^{29} -194.558 q^{31} +254.657 q^{33} +35.0000 q^{35} -366.853 q^{37} -128.765 q^{39} -339.362 q^{41} -226.167 q^{43} +86.5685 q^{45} -11.6762 q^{47} +49.0000 q^{49} -580.647 q^{51} +209.019 q^{53} +191.274 q^{55} -294.402 q^{57} -616.000 q^{59} -320.735 q^{61} +121.196 q^{63} -96.7157 q^{65} +14.5097 q^{67} -1452.30 q^{69} +952.000 q^{71} +824.489 q^{73} +166.421 q^{75} +267.784 q^{77} -156.275 q^{79} -896.706 q^{81} -1036.53 q^{83} -436.127 q^{85} +312.480 q^{87} -170.225 q^{89} -135.402 q^{91} -1295.15 q^{93} -221.127 q^{95} +1059.87 q^{97} +662.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 10 q^{5} + 14 q^{7} + 12 q^{9} - 14 q^{11} - 50 q^{13} + 10 q^{15} - 50 q^{17} + 36 q^{19} + 14 q^{21} - 244 q^{23} + 50 q^{25} + 86 q^{27} + 26 q^{29} + 120 q^{31} + 498 q^{33} + 70 q^{35}+ \cdots + 940 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 6.65685 1.28111 0.640556 − 0.767911i \(-0.278704\pi\)
0.640556 + 0.767911i \(0.278704\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 17.3137 0.641248
\(10\) 0 0
\(11\) 38.2548 1.04857 0.524285 − 0.851543i \(-0.324333\pi\)
0.524285 + 0.851543i \(0.324333\pi\)
\(12\) 0 0
\(13\) −19.3431 −0.412679 −0.206339 − 0.978480i \(-0.566155\pi\)
−0.206339 + 0.978480i \(0.566155\pi\)
\(14\) 0 0
\(15\) 33.2843 0.572931
\(16\) 0 0
\(17\) −87.2254 −1.24443 −0.622214 − 0.782847i \(-0.713767\pi\)
−0.622214 + 0.782847i \(0.713767\pi\)
\(18\) 0 0
\(19\) −44.2254 −0.534000 −0.267000 − 0.963697i \(-0.586032\pi\)
−0.267000 + 0.963697i \(0.586032\pi\)
\(20\) 0 0
\(21\) 46.5980 0.484215
\(22\) 0 0
\(23\) −218.167 −1.97786 −0.988932 − 0.148371i \(-0.952597\pi\)
−0.988932 + 0.148371i \(0.952597\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −64.4802 −0.459601
\(28\) 0 0
\(29\) 46.9411 0.300578 0.150289 − 0.988642i \(-0.451980\pi\)
0.150289 + 0.988642i \(0.451980\pi\)
\(30\) 0 0
\(31\) −194.558 −1.12722 −0.563609 − 0.826042i \(-0.690587\pi\)
−0.563609 + 0.826042i \(0.690587\pi\)
\(32\) 0 0
\(33\) 254.657 1.34334
\(34\) 0 0
\(35\) 35.0000 0.169031
\(36\) 0 0
\(37\) −366.853 −1.63001 −0.815003 − 0.579457i \(-0.803265\pi\)
−0.815003 + 0.579457i \(0.803265\pi\)
\(38\) 0 0
\(39\) −128.765 −0.528688
\(40\) 0 0
\(41\) −339.362 −1.29267 −0.646336 − 0.763053i \(-0.723699\pi\)
−0.646336 + 0.763053i \(0.723699\pi\)
\(42\) 0 0
\(43\) −226.167 −0.802095 −0.401047 − 0.916057i \(-0.631354\pi\)
−0.401047 + 0.916057i \(0.631354\pi\)
\(44\) 0 0
\(45\) 86.5685 0.286775
\(46\) 0 0
\(47\) −11.6762 −0.0362372 −0.0181186 − 0.999836i \(-0.505768\pi\)
−0.0181186 + 0.999836i \(0.505768\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −580.647 −1.59425
\(52\) 0 0
\(53\) 209.019 0.541717 0.270859 − 0.962619i \(-0.412692\pi\)
0.270859 + 0.962619i \(0.412692\pi\)
\(54\) 0 0
\(55\) 191.274 0.468935
\(56\) 0 0
\(57\) −294.402 −0.684114
\(58\) 0 0
\(59\) −616.000 −1.35926 −0.679630 − 0.733555i \(-0.737860\pi\)
−0.679630 + 0.733555i \(0.737860\pi\)
\(60\) 0 0
\(61\) −320.735 −0.673212 −0.336606 − 0.941646i \(-0.609279\pi\)
−0.336606 + 0.941646i \(0.609279\pi\)
\(62\) 0 0
\(63\) 121.196 0.242369
\(64\) 0 0
\(65\) −96.7157 −0.184556
\(66\) 0 0
\(67\) 14.5097 0.0264573 0.0132286 − 0.999912i \(-0.495789\pi\)
0.0132286 + 0.999912i \(0.495789\pi\)
\(68\) 0 0
\(69\) −1452.30 −2.53387
\(70\) 0 0
\(71\) 952.000 1.59129 0.795645 − 0.605763i \(-0.207132\pi\)
0.795645 + 0.605763i \(0.207132\pi\)
\(72\) 0 0
\(73\) 824.489 1.32191 0.660953 − 0.750427i \(-0.270152\pi\)
0.660953 + 0.750427i \(0.270152\pi\)
\(74\) 0 0
\(75\) 166.421 0.256222
\(76\) 0 0
\(77\) 267.784 0.396322
\(78\) 0 0
\(79\) −156.275 −0.222561 −0.111280 − 0.993789i \(-0.535495\pi\)
−0.111280 + 0.993789i \(0.535495\pi\)
\(80\) 0 0
\(81\) −896.706 −1.23005
\(82\) 0 0
\(83\) −1036.53 −1.37077 −0.685384 − 0.728182i \(-0.740366\pi\)
−0.685384 + 0.728182i \(0.740366\pi\)
\(84\) 0 0
\(85\) −436.127 −0.556525
\(86\) 0 0
\(87\) 312.480 0.385074
\(88\) 0 0
\(89\) −170.225 −0.202740 −0.101370 − 0.994849i \(-0.532323\pi\)
−0.101370 + 0.994849i \(0.532323\pi\)
\(90\) 0 0
\(91\) −135.402 −0.155978
\(92\) 0 0
\(93\) −1295.15 −1.44409
\(94\) 0 0
\(95\) −221.127 −0.238812
\(96\) 0 0
\(97\) 1059.87 1.10942 0.554710 − 0.832044i \(-0.312829\pi\)
0.554710 + 0.832044i \(0.312829\pi\)
\(98\) 0 0
\(99\) 662.333 0.672394
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 2240.4.a.bo.1.2 2
4.3 odd 2 2240.4.a.bn.1.1 2
8.3 odd 2 35.4.a.b.1.1 ✓ 2
8.5 even 2 560.4.a.r.1.1 2
24.11 even 2 315.4.a.f.1.2 2
40.3 even 4 175.4.b.c.99.2 4
40.19 odd 2 175.4.a.c.1.2 2
40.27 even 4 175.4.b.c.99.3 4
56.3 even 6 245.4.e.i.226.2 4
56.11 odd 6 245.4.e.h.226.2 4
56.19 even 6 245.4.e.i.116.2 4
56.27 even 2 245.4.a.k.1.1 2
56.51 odd 6 245.4.e.h.116.2 4
120.59 even 2 1575.4.a.z.1.1 2
168.83 odd 2 2205.4.a.u.1.2 2
280.139 even 2 1225.4.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.1 ✓ 2 8.3 odd 2
175.4.a.c.1.2 2 40.19 odd 2
175.4.b.c.99.2 4 40.3 even 4
175.4.b.c.99.3 4 40.27 even 4
245.4.a.k.1.1 2 56.27 even 2
245.4.e.h.116.2 4 56.51 odd 6
245.4.e.h.226.2 4 56.11 odd 6
245.4.e.i.116.2 4 56.19 even 6
245.4.e.i.226.2 4 56.3 even 6
315.4.a.f.1.2 2 24.11 even 2
560.4.a.r.1.1 2 8.5 even 2
1225.4.a.m.1.2 2 280.139 even 2
1575.4.a.z.1.1 2 120.59 even 2
2205.4.a.u.1.2 2 168.83 odd 2
2240.4.a.bn.1.1 2 4.3 odd 2
2240.4.a.bo.1.2 2 1.1 even 1 trivial