Properties

Label 145.2.j.a
Level $145$
Weight $2$
Character orbit 145.j
Analytic conductor $1.158$
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] = [145,2,Mod(17,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("145.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.j (of order \(4\), degree \(2\), 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: \(1.15783082931\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 6 q^{2} + 22 q^{4} - 4 q^{7} - 18 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 6 q^{2} + 22 q^{4} - 4 q^{7} - 18 q^{8} - 10 q^{9} - 6 q^{10} - 8 q^{11} + 14 q^{13} - 4 q^{14} + 10 q^{15} + 6 q^{16} + 20 q^{17} - 18 q^{18} - 20 q^{20} + 16 q^{21} - 8 q^{22} - 4 q^{23} + 10 q^{25} + 6 q^{26} - 8 q^{28} + 16 q^{30} + 8 q^{31} - 42 q^{32} - 32 q^{34} - 16 q^{35} - 22 q^{36} - 8 q^{38} - 16 q^{39} - 22 q^{40} - 6 q^{41} - 4 q^{42} - 44 q^{45} - 32 q^{46} + 8 q^{50} + 26 q^{52} + 14 q^{53} + 6 q^{55} - 32 q^{56} - 12 q^{57} + 28 q^{58} + 110 q^{60} + 18 q^{61} + 28 q^{62} + 60 q^{63} + 30 q^{64} - 18 q^{65} + 20 q^{66} + 32 q^{67} + 72 q^{68} + 12 q^{69} - 12 q^{70} + 10 q^{72} - 4 q^{73} + 6 q^{75} + 20 q^{76} - 12 q^{77} + 56 q^{78} + 4 q^{79} - 12 q^{80} - 86 q^{81} - 58 q^{82} - 60 q^{83} + 76 q^{84} + 60 q^{87} - 68 q^{88} - 46 q^{89} + 44 q^{90} - 28 q^{92} - 8 q^{93} + 60 q^{95} + 36 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −2.73482 1.63186i 5.47925 −0.903087 + 2.04559i 4.46285i 1.05887 + 1.05887i −9.51511 0.337028 2.46978 5.59432i
17.2 −2.41122 1.85973i 3.81399 1.59267 1.56952i 4.48422i −0.291676 0.291676i −4.37394 −0.458586 −3.84028 + 3.78447i
17.3 −1.82099 2.59340i 1.31599 1.75450 1.38626i 4.72256i −0.820621 0.820621i 1.24557 −3.72575 −3.19492 + 2.52437i
17.4 −1.77873 1.38965i 1.16389 −2.14366 + 0.636182i 2.47181i −3.41296 3.41296i 1.48721 1.06888 3.81300 1.13160i
17.5 −1.36192 0.228160i −0.145179 −1.87852 1.21291i 0.310736i 3.45046 + 3.45046i 2.92156 2.94794 2.55840 + 1.65188i
17.6 −0.839004 0.711801i −1.29607 1.24503 + 1.85739i 0.597203i 1.13987 + 1.13987i 2.76542 2.49334 −1.04459 1.55836i
17.7 −0.342532 2.64611i −1.88267 1.82632 + 1.29018i 0.906377i −1.55474 1.55474i 1.32994 −4.00189 −0.625573 0.441927i
17.8 0.222351 1.02589i −1.95056 −0.621451 2.14798i 0.228107i −2.35964 2.35964i −0.878412 1.94756 −0.138180 0.477605i
17.9 0.895351 2.11245i −1.19835 −1.71183 + 1.43862i 1.89138i 1.38465 + 1.38465i −2.86364 −1.46244 −1.53269 + 1.28807i
17.10 1.26373 0.913274i −0.402981 1.78577 1.34575i 1.15413i 2.03055 + 2.03055i −3.03672 2.16593 2.25673 1.70067i
17.11 1.41066 2.58872i −0.0100302 1.93694 + 1.11726i 3.65181i −0.510628 0.510628i −2.83547 −3.70148 2.73237 + 1.57607i
17.12 2.23019 1.25170i 2.97373 −2.19887 + 0.406147i 2.79153i 0.483409 + 0.483409i 2.17160 1.43324 −4.90390 + 0.905783i
17.13 2.26693 2.65401i 3.13899 −0.683805 2.12895i 6.01647i −2.59753 2.59753i 2.58201 −4.04378 −1.55014 4.82618i
128.1 −2.73482 1.63186i 5.47925 −0.903087 2.04559i 4.46285i 1.05887 1.05887i −9.51511 0.337028 2.46978 + 5.59432i
128.2 −2.41122 1.85973i 3.81399 1.59267 + 1.56952i 4.48422i −0.291676 + 0.291676i −4.37394 −0.458586 −3.84028 3.78447i
128.3 −1.82099 2.59340i 1.31599 1.75450 + 1.38626i 4.72256i −0.820621 + 0.820621i 1.24557 −3.72575 −3.19492 2.52437i
128.4 −1.77873 1.38965i 1.16389 −2.14366 0.636182i 2.47181i −3.41296 + 3.41296i 1.48721 1.06888 3.81300 + 1.13160i
128.5 −1.36192 0.228160i −0.145179 −1.87852 + 1.21291i 0.310736i 3.45046 3.45046i 2.92156 2.94794 2.55840 1.65188i
128.6 −0.839004 0.711801i −1.29607 1.24503 1.85739i 0.597203i 1.13987 1.13987i 2.76542 2.49334 −1.04459 + 1.55836i
128.7 −0.342532 2.64611i −1.88267 1.82632 1.29018i 0.906377i −1.55474 + 1.55474i 1.32994 −4.00189 −0.625573 + 0.441927i
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.2.j.a yes 26
5.b even 2 1 725.2.j.c 26
5.c odd 4 1 145.2.e.a 26
5.c odd 4 1 725.2.e.c 26
29.c odd 4 1 145.2.e.a 26
145.e even 4 1 725.2.j.c 26
145.f odd 4 1 725.2.e.c 26
145.j even 4 1 inner 145.2.j.a yes 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.e.a 26 5.c odd 4 1
145.2.e.a 26 29.c odd 4 1
145.2.j.a yes 26 1.a even 1 1 trivial
145.2.j.a yes 26 145.j even 4 1 inner
725.2.e.c 26 5.c odd 4 1
725.2.e.c 26 145.f odd 4 1
725.2.j.c 26 5.b even 2 1
725.2.j.c 26 145.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(145, [\chi])\).