Properties

Label 154.2.m.a
Level $154$
Weight $2$
Character orbit 154.m
Analytic conductor $1.230$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [154,2,Mod(9,154)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(154, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([10, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("154.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 154.m (of order \(15\), degree \(8\), 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: \(1.22969619113\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{15}^{7} q^{2} + (2 \zeta_{15}^{6} + \cdots + 2 \zeta_{15}) q^{3} + \cdots + ( - \zeta_{15}^{7} + 3 \zeta_{15}^{5} + \cdots + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{15}^{7} q^{2} + (2 \zeta_{15}^{6} + \cdots + 2 \zeta_{15}) q^{3} + \cdots + ( - 6 \zeta_{15}^{7} + 14 \zeta_{15}^{6} + \cdots + 5) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + q^{3} + q^{4} - 5 q^{5} - 8 q^{6} - 4 q^{7} + 2 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} + q^{3} + q^{4} - 5 q^{5} - 8 q^{6} - 4 q^{7} + 2 q^{8} + 8 q^{9} - 11 q^{11} + 6 q^{12} - 18 q^{13} - 5 q^{14} + 10 q^{15} + q^{16} + 9 q^{17} + 7 q^{18} - 3 q^{19} + 10 q^{20} - 6 q^{21} - 2 q^{22} + 4 q^{23} + 4 q^{24} + 6 q^{26} + 10 q^{27} - q^{28} - 18 q^{29} + 5 q^{30} + q^{31} + 4 q^{32} + 14 q^{33} - 32 q^{34} - 25 q^{35} + 14 q^{36} + 2 q^{37} + 8 q^{38} - 21 q^{39} - 5 q^{40} - 28 q^{41} - 16 q^{42} + 68 q^{43} + 9 q^{44} + 30 q^{45} - 4 q^{46} + 9 q^{47} - 2 q^{48} - 2 q^{49} + 9 q^{51} - 6 q^{52} + 5 q^{53} + 40 q^{55} - 16 q^{56} - 54 q^{57} - 9 q^{58} - 3 q^{59} + 10 q^{60} - 29 q^{61} - 28 q^{62} - 35 q^{63} - 2 q^{64} + 30 q^{65} + 16 q^{66} - 4 q^{67} + 9 q^{68} - 8 q^{69} + 5 q^{70} + 10 q^{71} - 8 q^{72} + 12 q^{73} - 2 q^{74} - 44 q^{76} - 9 q^{77} + 48 q^{78} + 15 q^{79} + 5 q^{80} + 16 q^{81} + 11 q^{82} + 32 q^{83} + 5 q^{84} - 30 q^{85} - q^{86} - 6 q^{87} + q^{88} - 18 q^{89} - 20 q^{90} + 24 q^{91} + 12 q^{92} + 11 q^{93} + 6 q^{94} - 25 q^{95} - q^{96} + 46 q^{97} + 52 q^{98} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-1 - \zeta_{15}^{5}\) \(-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.104528 0.994522i
0.669131 0.743145i
0.669131 + 0.743145i
0.913545 + 0.406737i
−0.978148 + 0.207912i
0.913545 0.406737i
−0.978148 0.207912i
−0.104528 + 0.994522i
−0.669131 0.743145i −0.348943 0.155360i −0.104528 + 0.994522i −2.18720 0.464905i 0.118034 + 0.363271i −0.599960 2.57683i 0.809017 0.587785i −1.90977 2.12101i 1.11803 + 1.93649i
25.1 −0.913545 0.406737i 2.56082 + 0.544320i 0.669131 + 0.743145i 0.233733 2.22382i −2.11803 1.53884i −1.02924 2.43735i −0.309017 0.951057i 3.52090 + 1.56760i −1.11803 + 1.93649i
37.1 −0.913545 + 0.406737i 2.56082 0.544320i 0.669131 0.743145i 0.233733 + 2.22382i −2.11803 + 1.53884i −1.02924 + 2.43735i −0.309017 + 0.951057i 3.52090 1.56760i −1.11803 1.93649i
53.1 0.978148 0.207912i 0.0399263 + 0.379874i 0.913545 0.406737i 1.49622 1.66172i 0.118034 + 0.363271i −2.63611 + 0.225688i 0.809017 0.587785i 2.79173 0.593401i 1.11803 1.93649i
81.1 0.104528 0.994522i −1.75181 1.94558i −0.978148 0.207912i −2.04275 0.909491i −2.11803 + 1.53884i 2.26531 + 1.36688i −0.309017 + 0.951057i −0.402863 + 3.83299i −1.11803 + 1.93649i
93.1 0.978148 + 0.207912i 0.0399263 0.379874i 0.913545 + 0.406737i 1.49622 + 1.66172i 0.118034 0.363271i −2.63611 0.225688i 0.809017 + 0.587785i 2.79173 + 0.593401i 1.11803 + 1.93649i
135.1 0.104528 + 0.994522i −1.75181 + 1.94558i −0.978148 + 0.207912i −2.04275 + 0.909491i −2.11803 1.53884i 2.26531 1.36688i −0.309017 0.951057i −0.402863 3.83299i −1.11803 1.93649i
137.1 −0.669131 + 0.743145i −0.348943 + 0.155360i −0.104528 0.994522i −2.18720 + 0.464905i 0.118034 0.363271i −0.599960 + 2.57683i 0.809017 + 0.587785i −1.90977 + 2.12101i 1.11803 1.93649i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.2.m.a 8
7.c even 3 1 inner 154.2.m.a 8
11.c even 5 1 inner 154.2.m.a 8
77.m even 15 1 inner 154.2.m.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.m.a 8 1.a even 1 1 trivial
154.2.m.a 8 7.c even 3 1 inner
154.2.m.a 8 11.c even 5 1 inner
154.2.m.a 8 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - T_{3}^{7} - 5T_{3}^{6} - 14T_{3}^{5} + 39T_{3}^{4} + 26T_{3}^{3} + 10T_{3}^{2} + 4T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(154, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} - 5 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} + 5 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 11 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} + 9 T^{3} + 36 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 9 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$19$ \( T^{8} + 3 T^{7} + \cdots + 707281 \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{3} + 8 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 9 T^{3} + 36 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - T^{7} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( T^{8} - 2 T^{7} + \cdots + 3748096 \) Copy content Toggle raw display
$41$ \( (T^{4} + 14 T^{3} + \cdots + 841)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 17 T + 61)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} - 9 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$53$ \( T^{8} - 5 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$59$ \( T^{8} + 3 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 1026625681 \) Copy content Toggle raw display
$67$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 5 T^{3} + \cdots + 9025)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 12 T^{7} + \cdots + 1679616 \) Copy content Toggle raw display
$79$ \( T^{8} - 15 T^{7} + \cdots + 9150625 \) Copy content Toggle raw display
$83$ \( (T^{4} - 16 T^{3} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 9 T^{3} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 23 T^{3} + \cdots + 14641)^{2} \) Copy content Toggle raw display
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