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Properties

Label 705.2.c.c
Level $705$
Weight $2$
Character orbit 705.c
Analytic conductor $5.629$
Analytic rank $0$
Dimension $26$
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] = [705,2,Mod(424,705)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(705, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("705.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 705 = 3 \cdot 5 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 705.c (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: \(5.62945334250\)
Analytic rank: \(0\)
Dimension: \(26\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 34 q^{4} - 2 q^{5} + 6 q^{6} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 34 q^{4} - 2 q^{5} + 6 q^{6} - 26 q^{9} + 2 q^{10} + 32 q^{11} + 8 q^{14} - 2 q^{15} + 58 q^{16} - 56 q^{19} - 6 q^{20} - 8 q^{21} - 30 q^{24} - 8 q^{25} + 8 q^{26} - 24 q^{29} - 4 q^{30} + 52 q^{31} - 16 q^{34} + 16 q^{35} + 34 q^{36} + 16 q^{39} - 32 q^{40} + 48 q^{41} - 8 q^{44} + 2 q^{45} + 32 q^{46} - 42 q^{49} - 18 q^{50} + 24 q^{51} - 6 q^{54} - 18 q^{55} - 68 q^{59} - 24 q^{60} + 48 q^{61} - 26 q^{64} - 30 q^{65} - 12 q^{69} - 30 q^{70} + 40 q^{71} - 40 q^{74} - 16 q^{75} + 132 q^{76} - 12 q^{79} + 10 q^{80} + 26 q^{81} + 40 q^{84} + 6 q^{85} + 96 q^{86} - 72 q^{89} - 2 q^{90} + 8 q^{91} - 6 q^{94} - 12 q^{95} + 106 q^{96} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1 − 2.74237i 1.00000i −5.52059 1.68430 + 1.47076i 2.74237 − 1.18262i 9.65475i −1.00000 4.03337 − 4.61897i
424.2 − 2.59761i 1.00000i −4.74755 0.614635 − 2.14994i 2.59761 4.83808i 7.13706i −1.00000 −5.58468 − 1.59658i
424.3 − 2.57687i 1.00000i −4.64024 −2.23396 − 0.0970147i 2.57687 − 1.51741i 6.80354i −1.00000 −0.249994 + 5.75662i
424.4 − 2.32477i − 1.00000i −3.40453 −0.853202 + 2.06689i −2.32477 − 4.29166i 3.26521i −1.00000 4.80504 + 1.98349i
424.5 − 2.24873i − 1.00000i −3.05678 2.21793 + 0.284264i −2.24873 − 1.38491i 2.37640i −1.00000 0.639233 − 4.98751i
424.6 − 2.12231i − 1.00000i −2.50418 −2.20867 − 0.348946i −2.12231 4.63433i 1.07002i −1.00000 −0.740570 + 4.68748i
424.7 − 1.67335i 1.00000i −0.800100 −0.458681 + 2.18852i 1.67335 2.75180i − 2.00785i −1.00000 3.66216 + 0.767534i
424.8 − 1.32490i 1.00000i 0.244653 0.910113 − 2.04247i 1.32490 − 1.72082i − 2.97393i −1.00000 −2.70606 − 1.20580i
424.9 − 1.22883i − 1.00000i 0.489976 2.03984 − 0.915995i −1.22883 − 1.54483i − 3.05976i −1.00000 −1.12560 − 2.50662i
424.10 − 0.747203i 1.00000i 1.44169 −2.22629 + 0.208857i 0.747203 − 0.654660i − 2.57164i −1.00000 0.156058 + 1.66349i
424.11 − 0.697808i − 1.00000i 1.51306 0.442810 − 2.19178i −0.697808 3.46613i − 2.45144i −1.00000 −1.52944 − 0.308996i
424.12 − 0.105426i − 1.00000i 1.98889 0.237709 − 2.22340i −0.105426 − 3.91364i − 0.420531i −1.00000 −0.234403 − 0.0250606i
424.13 − 0.0655745i 1.00000i 1.99570 −1.16652 − 1.90768i 0.0655745 − 1.54896i − 0.262016i −1.00000 −0.125095 + 0.0764940i
424.14 0.0655745i − 1.00000i 1.99570 −1.16652 + 1.90768i 0.0655745 1.54896i 0.262016i −1.00000 −0.125095 − 0.0764940i
424.15 0.105426i 1.00000i 1.98889 0.237709 + 2.22340i −0.105426 3.91364i 0.420531i −1.00000 −0.234403 + 0.0250606i
424.16 0.697808i 1.00000i 1.51306 0.442810 + 2.19178i −0.697808 − 3.46613i 2.45144i −1.00000 −1.52944 + 0.308996i
424.17 0.747203i − 1.00000i 1.44169 −2.22629 − 0.208857i 0.747203 0.654660i 2.57164i −1.00000 0.156058 − 1.66349i
424.18 1.22883i 1.00000i 0.489976 2.03984 + 0.915995i −1.22883 1.54483i 3.05976i −1.00000 −1.12560 + 2.50662i
424.19 1.32490i − 1.00000i 0.244653 0.910113 + 2.04247i 1.32490 1.72082i 2.97393i −1.00000 −2.70606 + 1.20580i
424.20 1.67335i − 1.00000i −0.800100 −0.458681 − 2.18852i 1.67335 − 2.75180i 2.00785i −1.00000 3.66216 − 0.767534i
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.26
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 705.2.c.c ✓ 26
5.b even 2 1 inner 705.2.c.c ✓ 26
5.c odd 4 1 3525.2.a.bh 13
5.c odd 4 1 3525.2.a.bi 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.2.c.c ✓ 26 1.a even 1 1 trivial
705.2.c.c ✓ 26 5.b even 2 1 inner
3525.2.a.bh 13 5.c odd 4 1
3525.2.a.bi 13 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{26} + 43 T_{2}^{24} + 807 T_{2}^{22} + 8677 T_{2}^{20} + 58990 T_{2}^{18} + 264074 T_{2}^{16} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(705, [\chi])\). Copy content Toggle raw display