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Properties

Label 418.2.u.a
Level $418$
Weight $2$
Character orbit 418.u
Analytic conductor $3.338$
Analytic rank $0$
Dimension $216$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [418,2,Mod(5,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, base_ring=CyclotomicField(90))
 
chi = DirichletCharacter(H, H._module([36, 80]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("418.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.u (of order \(45\), degree \(24\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(9\) over \(\Q(\zeta_{45})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{45}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 3 q^{7} - 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 3 q^{7} - 27 q^{8} + 3 q^{11} + 6 q^{13} + 18 q^{14} - 9 q^{15} - 15 q^{17} + 42 q^{18} + 45 q^{19} - 48 q^{21} - 6 q^{23} + 6 q^{25} - 30 q^{26} + 18 q^{27} - 6 q^{28} - 18 q^{29} - 3 q^{31} + 45 q^{33} - 12 q^{34} - 30 q^{35} - 36 q^{37} + 15 q^{38} - 36 q^{39} - 51 q^{41} + 6 q^{42} - 36 q^{43} - 30 q^{44} + 132 q^{45} + 12 q^{46} - 36 q^{47} - 48 q^{49} - 24 q^{50} - 36 q^{51} - 3 q^{52} + 75 q^{53} + 120 q^{55} + 24 q^{56} - 36 q^{57} + 12 q^{58} - 42 q^{59} - 72 q^{60} - 108 q^{61} - 27 q^{62} + 81 q^{63} + 27 q^{64} - 102 q^{65} + 165 q^{66} - 84 q^{67} - 3 q^{68} - 30 q^{69} - 15 q^{70} + 24 q^{71} - 36 q^{73} - 6 q^{74} + 6 q^{76} + 6 q^{77} - 18 q^{78} - 36 q^{79} - 96 q^{81} - 39 q^{82} - 24 q^{84} + 6 q^{85} + 27 q^{86} - 132 q^{87} - 3 q^{88} - 42 q^{89} + 24 q^{90} + 108 q^{91} - 18 q^{92} - 75 q^{93} - 156 q^{94} + 66 q^{95} + 63 q^{97} - 12 q^{98} - 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0.615661 + 0.788011i −1.77121 + 1.71044i −0.241922 + 0.970296i −2.96594 + 1.85333i −2.43831 − 0.342682i 4.46328 + 1.98718i −0.913545 + 0.406737i 0.106889 − 3.06091i −3.28646 − 1.19617i
5.2 0.615661 + 0.788011i −1.53524 + 1.48257i −0.241922 + 0.970296i −1.71626 + 1.07244i −2.11347 − 0.297029i −2.79496 − 1.24439i −0.913545 + 0.406737i 0.0542680 − 1.55403i −1.90173 − 0.692174i
5.3 0.615661 + 0.788011i −1.28208 + 1.23809i −0.241922 + 0.970296i 1.84514 − 1.15297i −1.76495 − 0.248048i 1.49870 + 0.667264i −0.913545 + 0.406737i 0.00616292 − 0.176483i 2.04454 + 0.744151i
5.4 0.615661 + 0.788011i −1.06998 + 1.03327i −0.241922 + 0.970296i 1.17323 − 0.733116i −1.47297 − 0.207013i −4.20324 − 1.87140i −0.913545 + 0.406737i −0.0274833 + 0.787018i 1.30002 + 0.473167i
5.5 0.615661 + 0.788011i 0.343250 − 0.331472i −0.241922 + 0.970296i −1.37828 + 0.861248i 0.472530 + 0.0664097i 1.46222 + 0.651023i −0.913545 + 0.406737i −0.0967521 + 2.77062i −1.52723 − 0.555866i
5.6 0.615661 + 0.788011i 0.516073 − 0.498366i −0.241922 + 0.970296i 1.38205 − 0.863602i 0.710443 + 0.0998463i 2.58409 + 1.15051i −0.913545 + 0.406737i −0.0867357 + 2.48379i 1.53140 + 0.557385i
5.7 0.615661 + 0.788011i 0.736190 − 0.710930i −0.241922 + 0.970296i −2.96978 + 1.85573i 1.01346 + 0.142433i −1.76367 − 0.785235i −0.913545 + 0.406737i −0.0681449 + 1.95141i −3.29072 − 1.19772i
5.8 0.615661 + 0.788011i 1.77612 − 1.71518i −0.241922 + 0.970296i 2.38937 − 1.49305i 2.44506 + 0.343631i 1.14234 + 0.508603i −0.913545 + 0.406737i 0.108063 − 3.09453i 2.64758 + 0.963641i
5.9 0.615661 + 0.788011i 2.28688 − 2.20842i −0.241922 + 0.970296i −0.827792 + 0.517262i 3.14820 + 0.442451i −1.03766 − 0.461997i −0.913545 + 0.406737i 0.248029 − 7.10262i −0.917247 − 0.333851i
9.1 −0.990268 − 0.139173i −0.724333 + 2.90514i 0.961262 + 0.275637i −0.0381555 + 1.09263i 1.12160 − 2.77606i −3.43887 + 1.53108i −0.913545 − 0.406737i −5.26635 − 2.80017i 0.189849 − 1.07669i
9.2 −0.990268 − 0.139173i −0.551416 + 2.21161i 0.961262 + 0.275637i 0.0341180 − 0.977012i 0.853847 − 2.11334i 1.90848 − 0.849710i −0.913545 − 0.406737i −1.93832 − 1.03062i −0.169760 + 0.962755i
9.3 −0.990268 − 0.139173i −0.367573 + 1.47426i 0.961262 + 0.275637i −0.126277 + 3.61609i 0.569173 − 1.40875i −0.999064 + 0.444812i −0.913545 − 0.406737i 0.610521 + 0.324620i 0.628311 − 3.56333i
9.4 −0.990268 − 0.139173i −0.232391 + 0.932069i 0.961262 + 0.275637i 0.0720784 − 2.06405i 0.359848 − 0.890656i −2.22827 + 0.992090i −0.913545 − 0.406737i 1.83410 + 0.975206i −0.358638 + 2.03394i
9.5 −0.990268 − 0.139173i −0.0582954 + 0.233810i 0.961262 + 0.275637i 0.00205953 − 0.0589772i 0.0902681 − 0.223421i 3.37543 − 1.50284i −0.913545 − 0.406737i 2.59757 + 1.38115i −0.0102475 + 0.0581166i
9.6 −0.990268 − 0.139173i 0.352008 − 1.41183i 0.961262 + 0.275637i −0.0725922 + 2.07877i −0.545070 + 1.34910i −1.68057 + 0.748238i −0.913545 − 0.406737i 0.779498 + 0.414467i 0.361195 − 2.04844i
9.7 −0.990268 − 0.139173i 0.460826 − 1.84827i 0.961262 + 0.275637i 0.00305043 − 0.0873529i −0.713570 + 1.76615i 3.88390 − 1.72922i −0.913545 − 0.406737i −0.554900 − 0.295046i −0.0151779 + 0.0860782i
9.8 −0.990268 − 0.139173i 0.516596 − 2.07195i 0.961262 + 0.275637i −0.0850036 + 2.43418i −0.799928 + 1.97989i −1.17126 + 0.521480i −0.913545 − 0.406737i −1.37727 − 0.732308i 0.422949 − 2.39866i
9.9 −0.990268 − 0.139173i 0.604580 − 2.42484i 0.961262 + 0.275637i 0.0844542 − 2.41845i −0.936169 + 2.31710i −3.34132 + 1.48765i −0.913545 − 0.406737i −2.86548 − 1.52360i −0.420216 + 2.38316i
25.1 0.241922 − 0.970296i −0.0940796 + 2.69409i −0.882948 − 0.469472i 1.14013 − 2.33762i 2.59130 + 0.743043i 1.02329 + 1.13648i −0.669131 + 0.743145i −4.25656 − 0.297647i −1.99236 − 1.67179i
25.2 0.241922 − 0.970296i −0.0808654 + 2.31568i −0.882948 − 0.469472i −0.432270 + 0.886285i 2.22733 + 0.638678i 1.27217 + 1.41289i −0.669131 + 0.743145i −2.36315 − 0.165248i 0.755383 + 0.633841i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
19.e even 9 1 inner
209.u even 45 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.u.a ✓ 216
11.c even 5 1 inner 418.2.u.a ✓ 216
19.e even 9 1 inner 418.2.u.a ✓ 216
209.u even 45 1 inner 418.2.u.a ✓ 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.u.a ✓ 216 1.a even 1 1 trivial
418.2.u.a ✓ 216 11.c even 5 1 inner
418.2.u.a ✓ 216 19.e even 9 1 inner
418.2.u.a ✓ 216 209.u even 45 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{216} - 6 T_{3}^{213} + 24 T_{3}^{212} + 12 T_{3}^{211} - 953 T_{3}^{210} - 1410 T_{3}^{209} + \cdots + 23\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display