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Properties

Label 309.2.o.b
Level $309$
Weight $2$
Character orbit 309.o
Analytic conductor $2.467$
Analytic rank $0$
Dimension $1024$
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] = [309,2,Mod(5,309)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(102))
 
chi = DirichletCharacter(H, H._module([51, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 309.o (of order \(102\), degree \(32\), 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: \(2.46737742246\)
Analytic rank: \(0\)
Dimension: \(1024\)
Relative dimension: \(32\) over \(\Q(\zeta_{102})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{102}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1024 q - 34 q^{3} - 100 q^{4} - 31 q^{6} - 66 q^{7} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1024 q - 34 q^{3} - 100 q^{4} - 31 q^{6} - 66 q^{7} - 26 q^{9} - 136 q^{10} - 22 q^{12} - 64 q^{13} - 31 q^{15} - 52 q^{16} - 23 q^{18} - 56 q^{19} + 3 q^{21} - 68 q^{22} - 34 q^{24} - 108 q^{25} - 34 q^{27} - 22 q^{28} - 42 q^{30} - 68 q^{31} - 31 q^{33} + 38 q^{34} - 119 q^{36} - 170 q^{37} - 170 q^{39} - 80 q^{40} - 34 q^{42} - 26 q^{43} - 97 q^{45} - 48 q^{46} + 124 q^{48} - 62 q^{49} - 22 q^{51} - 62 q^{52} + 14 q^{54} + 14 q^{55} - 39 q^{57} - 148 q^{58} + 169 q^{60} - 16 q^{61} - 41 q^{63} - 122 q^{64} - 194 q^{66} + 76 q^{67} - 170 q^{69} - 74 q^{70} + 8 q^{72} + 34 q^{73} + 252 q^{75} - 12 q^{76} + 8 q^{78} - 10 q^{81} + 30 q^{82} + 221 q^{84} - 24 q^{85} - 67 q^{87} + 186 q^{88} - 34 q^{90} + 296 q^{91} - 134 q^{93} + 136 q^{94} + 29 q^{96} + 52 q^{97} - 94 q^{99}+O(q^{100}) \) 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 −2.72076 + 0.595855i 1.53394 + 0.804387i 5.23053 − 2.40643i 0.0101383 − 0.329063i −4.65276 − 1.27454i −0.926125 − 0.114679i −8.35179 + 6.30698i 1.70592 + 2.46776i 0.168490 + 0.901341i
5.2 −2.40993 + 0.527784i −0.622477 − 1.61633i 3.71229 − 1.70793i 0.0553439 − 1.79632i 2.35320 + 3.56671i 3.34582 + 0.414302i −4.10747 + 3.10182i −2.22505 + 2.01226i 0.814691 + 4.35821i
5.3 −2.34368 + 0.513273i 0.568568 − 1.63607i 3.41243 − 1.56997i −0.0886817 + 2.87837i −0.492788 + 4.12625i −2.68174 − 0.332071i −3.36257 + 2.53930i −2.35346 − 1.86044i −1.26955 − 6.79148i
5.4 −2.33510 + 0.511396i −1.67664 + 0.434610i 3.37425 − 1.55240i 0.0970698 − 3.15063i 3.69287 − 1.87228i −4.87217 − 0.603306i −3.27011 + 2.46947i 2.62223 − 1.45737i 1.38455 + 7.40668i
5.5 −1.96912 + 0.431243i 1.44782 − 0.950690i 1.87452 − 0.862415i 0.109878 − 3.56634i −2.44095 + 2.49638i 0.432924 + 0.0536076i −0.101972 + 0.0770058i 1.19238 − 2.75286i 1.32160 + 7.06992i
5.6 −1.96871 + 0.431154i −1.38688 − 1.03758i 1.87299 − 0.861711i −0.0275563 + 0.894404i 3.17772 + 1.44473i −0.0979314 − 0.0121265i −0.0992405 + 0.0749429i 0.846866 + 2.87799i −0.331375 − 1.77270i
5.7 −1.82198 + 0.399019i 0.766574 + 1.55318i 1.34346 − 0.618088i 0.0420806 − 1.36583i −2.01643 − 2.52398i 1.63994 + 0.203069i 0.775738 − 0.585810i −1.82473 + 2.38125i 0.468321 + 2.50530i
5.8 −1.81996 + 0.398578i 1.69424 + 0.359940i 1.33646 − 0.614870i −0.119447 + 3.87693i −3.22691 + 0.0202092i 3.97123 + 0.491745i 0.786328 − 0.593808i 2.74089 + 1.21965i −1.32787 − 7.10347i
5.9 −1.57401 + 0.344713i −1.64215 + 0.550756i 0.541736 − 0.249238i −0.0432380 + 1.40339i 2.39491 − 1.43296i 2.16635 + 0.268252i 1.80493 − 1.36302i 2.39334 − 1.80885i −0.415710 − 2.22385i
5.10 −1.05970 + 0.232077i 1.52762 − 0.816311i −0.747835 + 0.344059i −0.0388394 + 1.26062i −1.42937 + 1.21957i −1.36265 − 0.168732i 2.44403 − 1.84564i 1.66727 − 2.49403i −0.251404 − 1.34489i
5.11 −1.04781 + 0.229473i 1.36474 + 1.06653i −0.771691 + 0.355034i 0.0173193 − 0.562140i −1.67472 − 0.804342i −5.02334 − 0.622024i 2.43908 − 1.84191i 0.725042 + 2.91107i 0.110849 + 0.592988i
5.12 −0.771135 + 0.168881i −0.817071 + 1.52722i −1.25080 + 0.575461i 0.0670041 − 2.17478i 0.372154 − 1.31568i −0.275913 − 0.0341655i 2.12728 − 1.60645i −1.66479 − 2.49569i 0.315609 + 1.68836i
5.13 −0.744596 + 0.163069i −1.49998 − 0.866061i −1.28910 + 0.593080i −0.0633559 + 2.05637i 1.25811 + 0.400265i −3.43867 − 0.425799i 2.07971 − 1.57052i 1.49988 + 2.59815i −0.288155 − 1.54149i
5.14 −0.616561 + 0.135029i 0.553022 − 1.64139i −1.45502 + 0.669413i 0.00977411 − 0.317242i −0.119336 + 1.08669i 3.97609 + 0.492347i 1.81409 − 1.36994i −2.38833 − 1.81545i 0.0368105 + 0.196919i
5.15 −0.483298 + 0.105844i −1.07837 − 1.35540i −1.59456 + 0.733612i 0.113133 − 3.67200i 0.664635 + 0.540924i −1.09057 − 0.135042i 1.48264 − 1.11964i −0.674234 + 2.92325i 0.333982 + 1.78664i
5.16 −0.146893 + 0.0321700i −1.06683 + 1.36451i −1.79639 + 0.826470i −0.119985 + 3.89439i 0.112813 − 0.234756i 0.0376043 + 0.00465642i 0.477291 − 0.360434i −0.723765 − 2.91139i −0.107658 − 0.575918i
5.17 0.146893 − 0.0321700i 1.70766 + 0.289668i −1.79639 + 0.826470i 0.119985 − 3.89439i 0.260161 − 0.0123852i 0.0376043 + 0.00465642i −0.477291 + 0.360434i 2.83219 + 0.989307i −0.107658 − 0.575918i
5.18 0.483298 − 0.105844i −0.116204 − 1.72815i −1.59456 + 0.733612i −0.113133 + 3.67200i −0.239075 − 0.822911i −1.09057 − 0.135042i −1.48264 + 1.11964i −2.97299 + 0.401634i 0.333982 + 1.78664i
5.19 0.616561 − 0.135029i −1.51449 − 0.840435i −1.45502 + 0.669413i −0.00977411 + 0.317242i −1.04726 − 0.313680i 3.97609 + 0.492347i −1.81409 + 1.36994i 1.58734 + 2.54565i 0.0368105 + 0.196919i
5.20 0.744596 − 0.163069i 0.525036 − 1.65056i −1.28910 + 0.593080i 0.0633559 − 2.05637i 0.121785 − 1.31461i −3.43867 − 0.425799i −2.07971 + 1.57052i −2.44867 − 1.73320i −0.288155 − 1.54149i
See next 80 embeddings (of 1024 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
103.h odd 102 1 inner
309.o even 102 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 309.2.o.b ✓ 1024
3.b odd 2 1 inner 309.2.o.b ✓ 1024
103.h odd 102 1 inner 309.2.o.b ✓ 1024
309.o even 102 1 inner 309.2.o.b ✓ 1024
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.2.o.b ✓ 1024 1.a even 1 1 trivial
309.2.o.b ✓ 1024 3.b odd 2 1 inner
309.2.o.b ✓ 1024 103.h odd 102 1 inner
309.2.o.b ✓ 1024 309.o even 102 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{1024} + 82 T_{2}^{1022} + 3263 T_{2}^{1020} + 82933 T_{2}^{1018} + 1484648 T_{2}^{1016} + \cdots + 67\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(309, [\chi])\). Copy content Toggle raw display