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Properties

Label 700.1.j.a.643.2
Level $700$
Weight $1$
Character 700.643
Analytic conductor $0.349$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -7, -20, 140
Inner twists $16$

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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] = [700,1,Mod(307,700)] 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("700.307"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 700.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.349345508843\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-5}, \sqrt{-7})\)
Artin image: $\OD_{16}:C_2$
Artin field: Galois closure of 16.0.960400000000000000.1

Embedding invariants

Embedding label 643.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 700.643
Dual form 700.1.j.a.307.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.707107 + 0.707107i) q^{2} +1.00000i q^{4} +(0.707107 + 0.707107i) q^{7} +(-0.707107 + 0.707107i) q^{8} -1.00000i q^{9} +1.00000i q^{14} -1.00000 q^{16} +(0.707107 - 0.707107i) q^{18} +(-1.41421 + 1.41421i) q^{23} +(-0.707107 + 0.707107i) q^{28} -2.00000i q^{29} +(-0.707107 - 0.707107i) q^{32} +1.00000 q^{36} +(1.41421 - 1.41421i) q^{43} -2.00000 q^{46} +1.00000i q^{49} -1.00000 q^{56} +(1.41421 - 1.41421i) q^{58} +(0.707107 - 0.707107i) q^{63} -1.00000i q^{64} +(-1.41421 - 1.41421i) q^{67} +(0.707107 + 0.707107i) q^{72} -1.00000 q^{81} +2.00000 q^{86} +(-1.41421 - 1.41421i) q^{92} +(-0.707107 + 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{16} + 4 q^{36} - 8 q^{46} - 4 q^{56} - 4 q^{81} + 8 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(3\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 1.00000i 1.00000i
\(5\) 0 0
\(6\) 0 0
\(7\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(8\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(9\) − 1.00000i − 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 1.00000i 1.00000i
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.41421 + 1.41421i −1.41421 + 1.41421i −0.707107 + 0.707107i \(0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(29\) − 2.00000i − 2.00000i − 1.00000i \(-0.5\pi\)
− 1.00000i \(-0.5\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 1.41421 − 1.41421i 1.41421 − 1.41421i 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.00000 −2.00000
\(47\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 0 0
\(49\) 1.00000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −1.00000
\(57\) 0 0
\(58\) 1.41421 − 1.41421i 1.41421 − 1.41421i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(64\) − 1.00000i − 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −1.41421 − 1.41421i −1.41421 − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 − 0.707107i \(-0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(73\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 2.00000
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.41421 − 1.41421i −1.41421 − 1.41421i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(99\) 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 700.1.j.a.643.2 yes 4
4.3 odd 2 inner 700.1.j.a.643.1 yes 4
5.2 odd 4 inner 700.1.j.a.307.1 ✓ 4
5.3 odd 4 inner 700.1.j.a.307.2 yes 4
5.4 even 2 inner 700.1.j.a.643.1 yes 4
7.6 odd 2 CM 700.1.j.a.643.2 yes 4
20.3 even 4 inner 700.1.j.a.307.1 ✓ 4
20.7 even 4 inner 700.1.j.a.307.2 yes 4
20.19 odd 2 CM 700.1.j.a.643.2 yes 4
28.27 even 2 inner 700.1.j.a.643.1 yes 4
35.13 even 4 inner 700.1.j.a.307.2 yes 4
35.27 even 4 inner 700.1.j.a.307.1 ✓ 4
35.34 odd 2 inner 700.1.j.a.643.1 yes 4
140.27 odd 4 inner 700.1.j.a.307.2 yes 4
140.83 odd 4 inner 700.1.j.a.307.1 ✓ 4
140.139 even 2 RM 700.1.j.a.643.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.1.j.a.307.1 ✓ 4 5.2 odd 4 inner
700.1.j.a.307.1 ✓ 4 20.3 even 4 inner
700.1.j.a.307.1 ✓ 4 35.27 even 4 inner
700.1.j.a.307.1 ✓ 4 140.83 odd 4 inner
700.1.j.a.307.2 yes 4 5.3 odd 4 inner
700.1.j.a.307.2 yes 4 20.7 even 4 inner
700.1.j.a.307.2 yes 4 35.13 even 4 inner
700.1.j.a.307.2 yes 4 140.27 odd 4 inner
700.1.j.a.643.1 yes 4 4.3 odd 2 inner
700.1.j.a.643.1 yes 4 5.4 even 2 inner
700.1.j.a.643.1 yes 4 28.27 even 2 inner
700.1.j.a.643.1 yes 4 35.34 odd 2 inner
700.1.j.a.643.2 yes 4 1.1 even 1 trivial
700.1.j.a.643.2 yes 4 7.6 odd 2 CM
700.1.j.a.643.2 yes 4 20.19 odd 2 CM
700.1.j.a.643.2 yes 4 140.139 even 2 RM