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Properties

Label 400.3.r.d
Level $400$
Weight $3$
Character orbit 400.r
Analytic conductor $10.899$
Analytic rank $0$
Dimension $28$
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] = [400,3,Mod(51,400)] 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("400.51"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.8992105744\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{3} + 10 q^{4} - 2 q^{6} - 28 q^{7} - 26 q^{11} - 52 q^{12} + 36 q^{13} + 6 q^{14} - 46 q^{16} + 16 q^{17} + 158 q^{18} - 46 q^{19} - 92 q^{21} - 14 q^{22} + 76 q^{23} + 74 q^{24} + 24 q^{26}+ \cdots - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −1.99847 + 0.0782547i 0.600049 − 0.600049i 3.98775 − 0.312779i 0 −1.15222 + 1.24614i 0.116286 −7.94492 + 0.937139i 8.27988i 0
51.2 −1.94649 − 0.459548i −3.49919 + 3.49919i 3.57763 + 1.78901i 0 8.41917 − 5.20308i −3.47820 −6.14168 − 5.12638i − 15.4886i 0
51.3 −1.53982 + 1.27630i 2.78870 − 2.78870i 0.742099 − 3.93056i 0 −0.734871 + 7.85332i 1.99582 3.87389 + 6.99950i − 6.55367i 0
51.4 −1.45774 − 1.36931i −0.258661 + 0.258661i 0.249991 + 3.99218i 0 0.731245 − 0.0228729i 10.8029 5.10210 − 6.16186i 8.86619i 0
51.5 −1.31780 + 1.50446i −2.26876 + 2.26876i −0.526785 − 3.96516i 0 −0.423473 − 6.40304i −6.63497 6.65962 + 4.43278i − 1.29458i 0
51.6 −1.14089 − 1.64267i 1.62818 − 1.62818i −1.39676 + 3.74821i 0 −4.53213 − 0.817001i −11.3290 7.75063 − 1.98186i 3.69807i 0
51.7 −0.166739 + 1.99304i −0.532090 + 0.532090i −3.94440 − 0.664633i 0 −0.971755 − 1.14919i 3.37769 1.98232 − 7.75051i 8.43376i 0
51.8 0.448476 + 1.94907i 3.45893 − 3.45893i −3.59774 + 1.74822i 0 8.29293 + 5.19044i −8.30977 −5.02090 − 6.22821i − 14.9283i 0
51.9 0.838693 − 1.81565i 0.825268 − 0.825268i −2.59319 − 3.04555i 0 −0.806253 − 2.19055i −8.39588 −7.70455 + 2.15405i 7.63787i 0
51.10 1.04893 + 1.70286i −4.12293 + 4.12293i −1.79949 + 3.57237i 0 −11.3454 − 2.69612i 7.21305 −7.97080 + 0.682875i − 24.9970i 0
51.11 1.64869 + 1.13217i 1.18548 − 1.18548i 1.43637 + 3.73321i 0 3.29666 − 0.612320i 6.06481 −1.85852 + 7.78113i 6.18928i 0
51.12 1.69904 − 1.05512i −1.83498 + 1.83498i 1.77346 − 3.58537i 0 −1.18158 + 5.05382i 7.32730 −0.769814 − 7.96288i 2.26569i 0
51.13 1.90775 − 0.600411i 3.01738 − 3.01738i 3.27901 − 2.29087i 0 3.94474 − 7.56808i −1.89720 4.88007 − 6.33916i − 9.20918i 0
51.14 1.97637 + 0.306552i −1.98737 + 1.98737i 3.81205 + 1.21172i 0 −4.53700 + 3.31854i −10.8528 7.16256 + 3.56339i 1.10072i 0
251.1 −1.99847 − 0.0782547i 0.600049 + 0.600049i 3.98775 + 0.312779i 0 −1.15222 − 1.24614i 0.116286 −7.94492 − 0.937139i − 8.27988i 0
251.2 −1.94649 + 0.459548i −3.49919 − 3.49919i 3.57763 − 1.78901i 0 8.41917 + 5.20308i −3.47820 −6.14168 + 5.12638i 15.4886i 0
251.3 −1.53982 − 1.27630i 2.78870 + 2.78870i 0.742099 + 3.93056i 0 −0.734871 − 7.85332i 1.99582 3.87389 − 6.99950i 6.55367i 0
251.4 −1.45774 + 1.36931i −0.258661 − 0.258661i 0.249991 − 3.99218i 0 0.731245 + 0.0228729i 10.8029 5.10210 + 6.16186i − 8.86619i 0
251.5 −1.31780 − 1.50446i −2.26876 − 2.26876i −0.526785 + 3.96516i 0 −0.423473 + 6.40304i −6.63497 6.65962 − 4.43278i 1.29458i 0
251.6 −1.14089 + 1.64267i 1.62818 + 1.62818i −1.39676 − 3.74821i 0 −4.53213 + 0.817001i −11.3290 7.75063 + 1.98186i − 3.69807i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.r.d ✓ 28
5.b even 2 1 400.3.r.e yes 28
5.c odd 4 1 400.3.k.e 28
5.c odd 4 1 400.3.k.f 28
16.f odd 4 1 inner 400.3.r.d ✓ 28
80.j even 4 1 400.3.k.e 28
80.k odd 4 1 400.3.r.e yes 28
80.s even 4 1 400.3.k.f 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.3.k.e 28 5.c odd 4 1
400.3.k.e 28 80.j even 4 1
400.3.k.f 28 5.c odd 4 1
400.3.k.f 28 80.s even 4 1
400.3.r.d ✓ 28 1.a even 1 1 trivial
400.3.r.d ✓ 28 16.f odd 4 1 inner
400.3.r.e yes 28 5.b even 2 1
400.3.r.e yes 28 80.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} + 2 T_{3}^{27} + 2 T_{3}^{26} - 54 T_{3}^{25} + 1467 T_{3}^{24} + 1604 T_{3}^{23} + \cdots + 3422133001 \) acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display