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Properties

Label 896.2.m.b.225.1
Level $896$
Weight $2$
Character 896.225
Analytic conductor $7.155$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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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] = [896,2,Mod(225,896)] 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("896.225"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(896, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.m (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-4,0,0,0,0,0,-2,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.15459602111\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 225.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 896.225
Dual form 896.2.m.b.673.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.00000 - 2.00000i) q^{5} -1.00000i q^{7} +3.00000i q^{9} +(-1.00000 - 1.00000i) q^{11} -2.00000 q^{17} +(-2.00000 + 2.00000i) q^{19} -6.00000i q^{23} +3.00000i q^{25} +(-7.00000 + 7.00000i) q^{29} -8.00000 q^{31} +(-2.00000 + 2.00000i) q^{35} +(5.00000 + 5.00000i) q^{37} +10.0000i q^{41} +(1.00000 + 1.00000i) q^{43} +(6.00000 - 6.00000i) q^{45} -12.0000 q^{47} -1.00000 q^{49} +(1.00000 + 1.00000i) q^{53} +4.00000i q^{55} +(-8.00000 - 8.00000i) q^{59} +(-6.00000 + 6.00000i) q^{61} +3.00000 q^{63} +(-3.00000 + 3.00000i) q^{67} -6.00000i q^{73} +(-1.00000 + 1.00000i) q^{77} +10.0000 q^{79} -9.00000 q^{81} +(10.0000 - 10.0000i) q^{83} +(4.00000 + 4.00000i) q^{85} -14.0000i q^{89} +8.00000 q^{95} -2.00000 q^{97} +(3.00000 - 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 2 q^{11} - 4 q^{17} - 4 q^{19} - 14 q^{29} - 16 q^{31} - 4 q^{35} + 10 q^{37} + 2 q^{43} + 12 q^{45} - 24 q^{47} - 2 q^{49} + 2 q^{53} - 16 q^{59} - 12 q^{61} + 6 q^{63} - 6 q^{67} - 2 q^{77}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 0 0
\(5\) −2.00000 − 2.00000i −0.894427 − 0.894427i 0.100509 − 0.994936i \(-0.467953\pi\)
−0.994936 + 0.100509i \(0.967953\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) −1.00000 − 1.00000i −0.301511 − 0.301511i 0.540094 − 0.841605i \(-0.318389\pi\)
−0.841605 + 0.540094i \(0.818389\pi\)
\(12\) 0 0
\(13\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −2.00000 + 2.00000i −0.458831 + 0.458831i −0.898272 − 0.439440i \(-0.855177\pi\)
0.439440 + 0.898272i \(0.355177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 6.00000i − 1.25109i −0.780189 − 0.625543i \(-0.784877\pi\)
0.780189 − 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 3.00000i 0.600000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.00000 + 7.00000i −1.29987 + 1.29987i −0.371391 + 0.928477i \(0.621119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 + 2.00000i −0.338062 + 0.338062i
\(36\) 0 0
\(37\) 5.00000 + 5.00000i 0.821995 + 0.821995i 0.986394 − 0.164399i \(-0.0525685\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 1.00000 + 1.00000i 0.152499 + 0.152499i 0.779233 − 0.626734i \(-0.215609\pi\)
−0.626734 + 0.779233i \(0.715609\pi\)
\(44\) 0 0
\(45\) 6.00000 − 6.00000i 0.894427 − 0.894427i
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.00000 + 1.00000i 0.137361 + 0.137361i 0.772444 − 0.635083i \(-0.219034\pi\)
−0.635083 + 0.772444i \(0.719034\pi\)
\(54\) 0 0
\(55\) 4.00000i 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 − 8.00000i −1.04151 − 1.04151i −0.999100 − 0.0424110i \(-0.986496\pi\)
−0.0424110 − 0.999100i \(-0.513504\pi\)
\(60\) 0 0
\(61\) −6.00000 + 6.00000i −0.768221 + 0.768221i −0.977793 − 0.209572i \(-0.932793\pi\)
0.209572 + 0.977793i \(0.432793\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.00000 + 3.00000i −0.366508 + 0.366508i −0.866202 − 0.499694i \(-0.833446\pi\)
0.499694 + 0.866202i \(0.333446\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) − 6.00000i − 0.702247i −0.936329 − 0.351123i \(-0.885800\pi\)
0.936329 − 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 + 1.00000i −0.113961 + 0.113961i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 10.0000 − 10.0000i 1.09764 − 1.09764i 0.102957 − 0.994686i \(-0.467170\pi\)
0.994686 − 0.102957i \(-0.0328303\pi\)
\(84\) 0 0
\(85\) 4.00000 + 4.00000i 0.433861 + 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 14.0000i − 1.48400i −0.670402 − 0.741999i \(-0.733878\pi\)
0.670402 − 0.741999i \(-0.266122\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 3.00000 − 3.00000i 0.301511 − 0.301511i
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 896.2.m.b.225.1 2
4.3 odd 2 896.2.m.c.225.1 2
8.3 odd 2 448.2.m.a.113.1 2
8.5 even 2 112.2.m.b.85.1 yes 2
16.3 odd 4 896.2.m.c.673.1 2
16.5 even 4 112.2.m.b.29.1 ✓ 2
16.11 odd 4 448.2.m.a.337.1 2
16.13 even 4 inner 896.2.m.b.673.1 2
32.3 odd 8 7168.2.a.k.1.2 2
32.13 even 8 7168.2.a.b.1.1 2
32.19 odd 8 7168.2.a.k.1.1 2
32.29 even 8 7168.2.a.b.1.2 2
56.5 odd 6 784.2.x.e.165.1 4
56.13 odd 2 784.2.m.a.197.1 2
56.37 even 6 784.2.x.d.165.1 4
56.45 odd 6 784.2.x.e.373.1 4
56.53 even 6 784.2.x.d.373.1 4
112.5 odd 12 784.2.x.e.557.1 4
112.37 even 12 784.2.x.d.557.1 4
112.53 even 12 784.2.x.d.765.1 4
112.69 odd 4 784.2.m.a.589.1 2
112.101 odd 12 784.2.x.e.765.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.b.29.1 ✓ 2 16.5 even 4
112.2.m.b.85.1 yes 2 8.5 even 2
448.2.m.a.113.1 2 8.3 odd 2
448.2.m.a.337.1 2 16.11 odd 4
784.2.m.a.197.1 2 56.13 odd 2
784.2.m.a.589.1 2 112.69 odd 4
784.2.x.d.165.1 4 56.37 even 6
784.2.x.d.373.1 4 56.53 even 6
784.2.x.d.557.1 4 112.37 even 12
784.2.x.d.765.1 4 112.53 even 12
784.2.x.e.165.1 4 56.5 odd 6
784.2.x.e.373.1 4 56.45 odd 6
784.2.x.e.557.1 4 112.5 odd 12
784.2.x.e.765.1 4 112.101 odd 12
896.2.m.b.225.1 2 1.1 even 1 trivial
896.2.m.b.673.1 2 16.13 even 4 inner
896.2.m.c.225.1 2 4.3 odd 2
896.2.m.c.673.1 2 16.3 odd 4
7168.2.a.b.1.1 2 32.13 even 8
7168.2.a.b.1.2 2 32.29 even 8
7168.2.a.k.1.1 2 32.19 odd 8
7168.2.a.k.1.2 2 32.3 odd 8