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Properties

Label 354.5.b.a
Level $354$
Weight $5$
Character orbit 354.b
Analytic conductor $36.593$
Analytic rank $0$
Dimension $76$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [354,5,Mod(119,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("354.119");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 354.b (of order \(2\), degree \(1\), 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: \(36.5929669317\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 608 q^{4} - 64 q^{6} - 184 q^{7} + 168 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q - 608 q^{4} - 64 q^{6} - 184 q^{7} + 168 q^{9} + 256 q^{10} - 200 q^{13} - 26 q^{15} + 4864 q^{16} - 512 q^{18} + 616 q^{19} + 330 q^{21} + 640 q^{22} + 512 q^{24} - 10540 q^{25} - 354 q^{27} + 1472 q^{28} - 832 q^{30} - 3920 q^{31} - 188 q^{33} + 2560 q^{34} - 1344 q^{36} - 1440 q^{37} + 8204 q^{39} - 2048 q^{40} - 5760 q^{42} - 1944 q^{43} + 3886 q^{45} + 4864 q^{46} + 33636 q^{49} - 7544 q^{51} + 1600 q^{52} + 3392 q^{54} - 10536 q^{55} - 12182 q^{57} - 7168 q^{58} + 208 q^{60} + 6360 q^{61} + 10860 q^{63} - 38912 q^{64} + 19712 q^{66} + 30744 q^{67} - 34208 q^{69} - 23808 q^{70} + 4096 q^{72} + 4032 q^{73} + 22324 q^{75} - 4928 q^{76} + 12864 q^{78} - 29824 q^{79} - 22584 q^{81} + 13184 q^{82} - 2640 q^{84} + 9240 q^{85} + 32850 q^{87} - 5120 q^{88} - 16448 q^{90} - 31160 q^{91} - 1780 q^{93} + 5248 q^{94} - 4096 q^{96} + 77504 q^{97} - 15412 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
119.1 − 2.82843i −8.93212 + 1.10332i −8.00000 5.83517i 3.12065 + 25.2638i −18.4051 22.6274i 78.5654 − 19.7099i 16.5043
119.2 − 2.82843i −8.91789 + 1.21294i −8.00000 12.8667i 3.43072 + 25.2236i −25.0597 22.6274i 78.0575 − 21.6338i 36.3926
119.3 − 2.82843i −8.83171 + 1.73229i −8.00000 − 40.6238i 4.89965 + 24.9799i 62.0657 22.6274i 74.9983 − 30.5982i −114.901
119.4 − 2.82843i −8.68666 + 2.35412i −8.00000 − 2.07514i 6.65846 + 24.5696i −53.2290 22.6274i 69.9162 − 40.8989i −5.86940
119.5 − 2.82843i −8.09468 − 3.93398i −8.00000 − 40.1568i −11.1270 + 22.8952i −61.8003 22.6274i 50.0476 + 63.6886i −113.581
119.6 − 2.82843i −8.08417 + 3.95552i −8.00000 31.0925i 11.1879 + 22.8655i 88.3750 22.6274i 49.7077 − 63.9542i 87.9429
119.7 − 2.82843i −7.91044 − 4.29243i −8.00000 − 19.2338i −12.1408 + 22.3741i 28.5043 22.6274i 44.1501 + 67.9100i −54.4013
119.8 − 2.82843i −7.33533 − 5.21468i −8.00000 49.1667i −14.7493 + 20.7475i −32.7313 22.6274i 26.6142 + 76.5028i 139.065
119.9 − 2.82843i −7.27743 − 5.29519i −8.00000 30.6811i −14.9771 + 20.5837i 36.2554 22.6274i 24.9218 + 77.0708i 86.7794
119.10 − 2.82843i −6.71583 + 5.99146i −8.00000 − 40.1609i 16.9464 + 18.9952i 33.0388 22.6274i 9.20482 − 80.4753i −113.592
119.11 − 2.82843i −6.39595 + 6.33181i −8.00000 30.3892i 17.9091 + 18.0905i −23.0970 22.6274i 0.816430 − 80.9959i 85.9536
119.12 − 2.82843i −4.28434 − 7.91482i −8.00000 3.69126i −22.3865 + 12.1179i 38.4283 22.6274i −44.2889 + 67.8196i 10.4405
119.13 − 2.82843i −4.23958 − 7.93889i −8.00000 15.5384i −22.4546 + 11.9913i −95.5047 22.6274i −45.0520 + 67.3151i 43.9493
119.14 − 2.82843i −4.23649 + 7.94054i −8.00000 − 25.1711i 22.4592 + 11.9826i −73.8012 22.6274i −45.1043 − 67.2801i −71.1945
119.15 − 2.82843i −3.88189 + 8.11979i −8.00000 43.7610i 22.9662 + 10.9796i −67.6729 22.6274i −50.8619 − 63.0402i 123.775
119.16 − 2.82843i −3.50299 + 8.29030i −8.00000 4.07079i 23.4485 + 9.90795i 46.9846 22.6274i −56.4581 − 58.0817i 11.5139
119.17 − 2.82843i −3.47377 − 8.30259i −8.00000 − 12.2691i −23.4833 + 9.82529i −13.7545 22.6274i −56.8659 + 57.6825i −34.7024
119.18 − 2.82843i −1.31589 − 8.90328i −8.00000 − 31.9146i −25.1823 + 3.72191i 95.4804 22.6274i −77.5369 + 23.4315i −90.2682
119.19 − 2.82843i 0.730836 + 8.97028i −8.00000 13.2882i 25.3718 − 2.06712i 42.9270 22.6274i −79.9318 + 13.1116i 37.5847
119.20 − 2.82843i 0.769993 − 8.96700i −8.00000 26.9261i −25.3625 − 2.17787i 20.9184 22.6274i −79.8142 − 13.8091i 76.1586
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 119.76
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 354.5.b.a ✓ 76
3.b odd 2 1 inner 354.5.b.a ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.5.b.a ✓ 76 1.a even 1 1 trivial
354.5.b.a ✓ 76 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(354, [\chi])\).