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Properties

Label 400.3.h.a.399.1
Level $400$
Weight $3$
Character 400.399
Analytic conductor $10.899$
Analytic rank $0$
Dimension $2$
CM discriminant -4
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] = [400,3,Mod(399,400)] 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("400.399"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.h (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,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.8992105744\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 399.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.399
Dual form 400.3.h.a.399.2

$q$-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)
 
\(f(q)\) \(=\) \(q-9.00000 q^{9} -10.0000i q^{13} -30.0000i q^{17} -42.0000 q^{29} -70.0000i q^{37} +18.0000 q^{41} -49.0000 q^{49} -90.0000i q^{53} -22.0000 q^{61} +110.000i q^{73} +81.0000 q^{81} +78.0000 q^{89} +130.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} - 84 q^{29} + 36 q^{41} - 98 q^{49} - 44 q^{61} + 162 q^{81} + 156 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −9.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) − 10.0000i − 0.769231i −0.923077 − 0.384615i \(-0.874334\pi\)
0.923077 − 0.384615i \(-0.125666\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 30.0000i − 1.76471i −0.470588 − 0.882353i \(-0.655958\pi\)
0.470588 − 0.882353i \(-0.344042\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −42.0000 −1.44828 −0.724138 − 0.689655i \(-0.757762\pi\)
−0.724138 + 0.689655i \(0.757762\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 70.0000i − 1.89189i −0.324324 − 0.945946i \(-0.605137\pi\)
0.324324 − 0.945946i \(-0.394863\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 18.0000 0.439024 0.219512 − 0.975610i \(-0.429553\pi\)
0.219512 + 0.975610i \(0.429553\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −49.0000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 90.0000i − 1.69811i −0.528302 − 0.849057i \(-0.677171\pi\)
0.528302 − 0.849057i \(-0.322829\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −22.0000 −0.360656 −0.180328 − 0.983607i \(-0.557716\pi\)
−0.180328 + 0.983607i \(0.557716\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 110.000i 1.50685i 0.657534 + 0.753425i \(0.271599\pi\)
−0.657534 + 0.753425i \(0.728401\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 78.0000 0.876404 0.438202 − 0.898876i \(-0.355615\pi\)
0.438202 + 0.898876i \(0.355615\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 130.000i 1.34021i 0.742268 + 0.670103i \(0.233750\pi\)
−0.742268 + 0.670103i \(0.766250\pi\)
\(98\) 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 400.3.h.a.399.1 2
3.2 odd 2 3600.3.j.a.1999.1 2
4.3 odd 2 CM 400.3.h.a.399.1 2
5.2 odd 4 400.3.b.a.351.1 1
5.3 odd 4 16.3.c.a.15.1 ✓ 1
5.4 even 2 inner 400.3.h.a.399.2 2
8.3 odd 2 1600.3.h.b.1599.2 2
8.5 even 2 1600.3.h.b.1599.2 2
12.11 even 2 3600.3.j.a.1999.1 2
15.2 even 4 3600.3.e.c.3151.1 1
15.8 even 4 144.3.g.a.127.1 1
15.14 odd 2 3600.3.j.a.1999.2 2
20.3 even 4 16.3.c.a.15.1 ✓ 1
20.7 even 4 400.3.b.a.351.1 1
20.19 odd 2 inner 400.3.h.a.399.2 2
35.3 even 12 784.3.r.d.79.1 2
35.13 even 4 784.3.d.b.687.1 1
35.18 odd 12 784.3.r.e.79.1 2
35.23 odd 12 784.3.r.e.655.1 2
35.33 even 12 784.3.r.d.655.1 2
40.3 even 4 64.3.c.a.63.1 1
40.13 odd 4 64.3.c.a.63.1 1
40.19 odd 2 1600.3.h.b.1599.1 2
40.27 even 4 1600.3.b.b.1151.1 1
40.29 even 2 1600.3.h.b.1599.1 2
40.37 odd 4 1600.3.b.b.1151.1 1
45.13 odd 12 1296.3.o.o.703.1 2
45.23 even 12 1296.3.o.b.703.1 2
45.38 even 12 1296.3.o.b.271.1 2
45.43 odd 12 1296.3.o.o.271.1 2
60.23 odd 4 144.3.g.a.127.1 1
60.47 odd 4 3600.3.e.c.3151.1 1
60.59 even 2 3600.3.j.a.1999.2 2
80.3 even 4 256.3.d.b.127.2 2
80.13 odd 4 256.3.d.b.127.2 2
80.43 even 4 256.3.d.b.127.1 2
80.53 odd 4 256.3.d.b.127.1 2
120.53 even 4 576.3.g.b.127.1 1
120.83 odd 4 576.3.g.b.127.1 1
140.3 odd 12 784.3.r.d.79.1 2
140.23 even 12 784.3.r.e.655.1 2
140.83 odd 4 784.3.d.b.687.1 1
140.103 odd 12 784.3.r.d.655.1 2
140.123 even 12 784.3.r.e.79.1 2
180.23 odd 12 1296.3.o.b.703.1 2
180.43 even 12 1296.3.o.o.271.1 2
180.83 odd 12 1296.3.o.b.271.1 2
180.103 even 12 1296.3.o.o.703.1 2
240.53 even 4 2304.3.b.f.127.2 2
240.83 odd 4 2304.3.b.f.127.1 2
240.173 even 4 2304.3.b.f.127.1 2
240.203 odd 4 2304.3.b.f.127.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.3.c.a.15.1 ✓ 1 5.3 odd 4
16.3.c.a.15.1 ✓ 1 20.3 even 4
64.3.c.a.63.1 1 40.3 even 4
64.3.c.a.63.1 1 40.13 odd 4
144.3.g.a.127.1 1 15.8 even 4
144.3.g.a.127.1 1 60.23 odd 4
256.3.d.b.127.1 2 80.43 even 4
256.3.d.b.127.1 2 80.53 odd 4
256.3.d.b.127.2 2 80.3 even 4
256.3.d.b.127.2 2 80.13 odd 4
400.3.b.a.351.1 1 5.2 odd 4
400.3.b.a.351.1 1 20.7 even 4
400.3.h.a.399.1 2 1.1 even 1 trivial
400.3.h.a.399.1 2 4.3 odd 2 CM
400.3.h.a.399.2 2 5.4 even 2 inner
400.3.h.a.399.2 2 20.19 odd 2 inner
576.3.g.b.127.1 1 120.53 even 4
576.3.g.b.127.1 1 120.83 odd 4
784.3.d.b.687.1 1 35.13 even 4
784.3.d.b.687.1 1 140.83 odd 4
784.3.r.d.79.1 2 35.3 even 12
784.3.r.d.79.1 2 140.3 odd 12
784.3.r.d.655.1 2 35.33 even 12
784.3.r.d.655.1 2 140.103 odd 12
784.3.r.e.79.1 2 35.18 odd 12
784.3.r.e.79.1 2 140.123 even 12
784.3.r.e.655.1 2 35.23 odd 12
784.3.r.e.655.1 2 140.23 even 12
1296.3.o.b.271.1 2 45.38 even 12
1296.3.o.b.271.1 2 180.83 odd 12
1296.3.o.b.703.1 2 45.23 even 12
1296.3.o.b.703.1 2 180.23 odd 12
1296.3.o.o.271.1 2 45.43 odd 12
1296.3.o.o.271.1 2 180.43 even 12
1296.3.o.o.703.1 2 45.13 odd 12
1296.3.o.o.703.1 2 180.103 even 12
1600.3.b.b.1151.1 1 40.27 even 4
1600.3.b.b.1151.1 1 40.37 odd 4
1600.3.h.b.1599.1 2 40.19 odd 2
1600.3.h.b.1599.1 2 40.29 even 2
1600.3.h.b.1599.2 2 8.3 odd 2
1600.3.h.b.1599.2 2 8.5 even 2
2304.3.b.f.127.1 2 240.83 odd 4
2304.3.b.f.127.1 2 240.173 even 4
2304.3.b.f.127.2 2 240.53 even 4
2304.3.b.f.127.2 2 240.203 odd 4
3600.3.e.c.3151.1 1 15.2 even 4
3600.3.e.c.3151.1 1 60.47 odd 4
3600.3.j.a.1999.1 2 3.2 odd 2
3600.3.j.a.1999.1 2 12.11 even 2
3600.3.j.a.1999.2 2 15.14 odd 2
3600.3.j.a.1999.2 2 60.59 even 2