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Properties

Label 1400.2.g.b.449.2
Level $1400$
Weight $2$
Character 1400.449
Analytic conductor $11.179$
Analytic rank $0$
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] = [1400,2,Mod(449,1400)] 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("1400.449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-2,0,0] 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: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
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 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1400.449
Dual form 1400.2.g.b.449.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +2.00000i q^{17} +2.00000 q^{19} +2.00000 q^{21} +8.00000i q^{23} +4.00000i q^{27} -2.00000 q^{29} +4.00000 q^{31} +6.00000i q^{37} -2.00000 q^{41} +8.00000i q^{43} +4.00000i q^{47} -1.00000 q^{49} -4.00000 q^{51} -10.0000i q^{53} +4.00000i q^{57} -6.00000 q^{59} +4.00000 q^{61} +1.00000i q^{63} +12.0000i q^{67} -16.0000 q^{69} -14.0000i q^{73} +8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} -4.00000i q^{87} -10.0000 q^{89} +8.00000i q^{93} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} + 4 q^{19} + 4 q^{21} - 4 q^{29} + 8 q^{31} - 4 q^{41} - 2 q^{49} - 8 q^{51} - 12 q^{59} + 8 q^{61} - 32 q^{69} + 16 q^{79} - 22 q^{81} - 20 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 8.00000i 1.66812i 0.551677 + 0.834058i \(0.313988\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) − 10.0000i − 1.37361i −0.726844 − 0.686803i \(-0.759014\pi\)
0.726844 − 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 0 0
\(69\) −16.0000 −1.92617
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 14.0000i − 1.63858i −0.573382 − 0.819288i \(-0.694369\pi\)
0.573382 − 0.819288i \(-0.305631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 4.00000i − 0.428845i
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 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 1400.2.g.b.449.2 2
4.3 odd 2 2800.2.g.g.449.1 2
5.2 odd 4 56.2.a.b.1.1 ✓ 1
5.3 odd 4 1400.2.a.a.1.1 1
5.4 even 2 inner 1400.2.g.b.449.1 2
15.2 even 4 504.2.a.h.1.1 1
20.3 even 4 2800.2.a.bd.1.1 1
20.7 even 4 112.2.a.a.1.1 1
20.19 odd 2 2800.2.g.g.449.2 2
35.2 odd 12 392.2.i.a.361.1 2
35.12 even 12 392.2.i.e.361.1 2
35.13 even 4 9800.2.a.bj.1.1 1
35.17 even 12 392.2.i.e.177.1 2
35.27 even 4 392.2.a.b.1.1 1
35.32 odd 12 392.2.i.a.177.1 2
40.27 even 4 448.2.a.h.1.1 1
40.37 odd 4 448.2.a.c.1.1 1
55.32 even 4 6776.2.a.h.1.1 1
60.47 odd 4 1008.2.a.m.1.1 1
65.12 odd 4 9464.2.a.h.1.1 1
80.27 even 4 1792.2.b.h.897.2 2
80.37 odd 4 1792.2.b.a.897.1 2
80.67 even 4 1792.2.b.h.897.1 2
80.77 odd 4 1792.2.b.a.897.2 2
105.2 even 12 3528.2.s.a.361.1 2
105.17 odd 12 3528.2.s.ba.3313.1 2
105.32 even 12 3528.2.s.a.3313.1 2
105.47 odd 12 3528.2.s.ba.361.1 2
105.62 odd 4 3528.2.a.b.1.1 1
120.77 even 4 4032.2.a.d.1.1 1
120.107 odd 4 4032.2.a.a.1.1 1
140.27 odd 4 784.2.a.i.1.1 1
140.47 odd 12 784.2.i.b.753.1 2
140.67 even 12 784.2.i.j.177.1 2
140.87 odd 12 784.2.i.b.177.1 2
140.107 even 12 784.2.i.j.753.1 2
280.27 odd 4 3136.2.a.c.1.1 1
280.237 even 4 3136.2.a.w.1.1 1
420.167 even 4 7056.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.a.b.1.1 ✓ 1 5.2 odd 4
112.2.a.a.1.1 1 20.7 even 4
392.2.a.b.1.1 1 35.27 even 4
392.2.i.a.177.1 2 35.32 odd 12
392.2.i.a.361.1 2 35.2 odd 12
392.2.i.e.177.1 2 35.17 even 12
392.2.i.e.361.1 2 35.12 even 12
448.2.a.c.1.1 1 40.37 odd 4
448.2.a.h.1.1 1 40.27 even 4
504.2.a.h.1.1 1 15.2 even 4
784.2.a.i.1.1 1 140.27 odd 4
784.2.i.b.177.1 2 140.87 odd 12
784.2.i.b.753.1 2 140.47 odd 12
784.2.i.j.177.1 2 140.67 even 12
784.2.i.j.753.1 2 140.107 even 12
1008.2.a.m.1.1 1 60.47 odd 4
1400.2.a.a.1.1 1 5.3 odd 4
1400.2.g.b.449.1 2 5.4 even 2 inner
1400.2.g.b.449.2 2 1.1 even 1 trivial
1792.2.b.a.897.1 2 80.37 odd 4
1792.2.b.a.897.2 2 80.77 odd 4
1792.2.b.h.897.1 2 80.67 even 4
1792.2.b.h.897.2 2 80.27 even 4
2800.2.a.bd.1.1 1 20.3 even 4
2800.2.g.g.449.1 2 4.3 odd 2
2800.2.g.g.449.2 2 20.19 odd 2
3136.2.a.c.1.1 1 280.27 odd 4
3136.2.a.w.1.1 1 280.237 even 4
3528.2.a.b.1.1 1 105.62 odd 4
3528.2.s.a.361.1 2 105.2 even 12
3528.2.s.a.3313.1 2 105.32 even 12
3528.2.s.ba.361.1 2 105.47 odd 12
3528.2.s.ba.3313.1 2 105.17 odd 12
4032.2.a.a.1.1 1 120.107 odd 4
4032.2.a.d.1.1 1 120.77 even 4
6776.2.a.h.1.1 1 55.32 even 4
7056.2.a.c.1.1 1 420.167 even 4
9464.2.a.h.1.1 1 65.12 odd 4
9800.2.a.bj.1.1 1 35.13 even 4