Properties

Label 3174.2.a.z
Level $3174$
Weight $2$
Character orbit 3174.a
Self dual yes
Analytic conductor $25.345$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3174,2,Mod(1,3174)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3174, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3174.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3174.a (trivial)

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: yes
Analytic conductor: \(25.3445176016\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{5} - q^{6} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{5} - q^{6} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} - q^{8} + q^{9} + (\beta_{4} + \beta_{2} + \beta_1 - 2) q^{10} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{11} + q^{12} + (\beta_{4} - \beta_{2} + 2) q^{13} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{14} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{15} + q^{16} + (5 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 5) q^{17} - q^{18} + ( - 4 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{19} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{20} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{21} + (\beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{22} - q^{24} + ( - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 5) q^{25} + ( - \beta_{4} + \beta_{2} - 2) q^{26} + q^{27} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{28} + (\beta_{3} - 3 \beta_{2} + 4 \beta_1 - 1) q^{29} + (\beta_{4} + \beta_{2} + \beta_1 - 2) q^{30} + ( - \beta_{4} - 3 \beta_{3} + \beta_{2} + 1) q^{31} - q^{32} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{33} + ( - 5 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 5) q^{34} + ( - 2 \beta_{4} + 7 \beta_{3} - 7 \beta_{2} + 3 \beta_1 - 6) q^{35} + q^{36} + ( - \beta_{3} + 3 \beta_{2} + 4) q^{37} + (4 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{38} + (\beta_{4} - \beta_{2} + 2) q^{39} + (\beta_{4} + \beta_{2} + \beta_1 - 2) q^{40} + ( - \beta_{4} + \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 1) q^{41} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{42} + (3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{43} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{44} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{45} + (2 \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{47} + q^{48} + ( - 3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2) q^{49} + (3 \beta_{4} - 3 \beta_{3} + \beta_{2} + 2 \beta_1 - 5) q^{50} + (5 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 5) q^{51} + (\beta_{4} - \beta_{2} + 2) q^{52} + ( - 4 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 3) q^{53} - q^{54} + ( - 7 \beta_{4} + 2 \beta_{3} - 10 \beta_{2} + 6 \beta_1 - 5) q^{55} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{56} + ( - 4 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{57} + ( - \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 1) q^{58} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{59} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{60} + ( - 5 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{61} + (\beta_{4} + 3 \beta_{3} - \beta_{2} - 1) q^{62} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{63} + q^{64} + (2 \beta_{3} - 5 \beta_{2} + 3) q^{65} + (\beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{66} + ( - 4 \beta_{4} + \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{67} + (5 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 5) q^{68} + (2 \beta_{4} - 7 \beta_{3} + 7 \beta_{2} - 3 \beta_1 + 6) q^{70} + ( - 4 \beta_{3} + \beta_{2} - 6 \beta_1) q^{71} - q^{72} + (2 \beta_{4} - 4 \beta_{3} + \beta_{2} - 5 \beta_1) q^{73} + (\beta_{3} - 3 \beta_{2} - 4) q^{74} + ( - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 5) q^{75} + ( - 4 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{76} + (4 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 4) q^{77} + ( - \beta_{4} + \beta_{2} - 2) q^{78} + ( - \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - \beta_1 + 3) q^{79} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{80} + q^{81} + (\beta_{4} - \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{82} + ( - 2 \beta_{3} - 2 \beta_{2} + 7 \beta_1 + 3) q^{83} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{84} + (10 \beta_{4} - 11 \beta_{3} + 8 \beta_{2} - 17 \beta_1 + 7) q^{85} + ( - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{86} + (\beta_{3} - 3 \beta_{2} + 4 \beta_1 - 1) q^{87} + (\beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{88} + (\beta_{4} - 5 \beta_{3} + 5 \beta_{2} - \beta_1 + 10) q^{89} + (\beta_{4} + \beta_{2} + \beta_1 - 2) q^{90} + ( - 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + \beta_1) q^{91} + ( - \beta_{4} - 3 \beta_{3} + \beta_{2} + 1) q^{93} + ( - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{94} + (4 \beta_{4} - 13 \beta_{3} + 11 \beta_{2} - \beta_1 + 8) q^{95} - q^{96} + (2 \beta_{4} + 2 \beta_{3} - 9 \beta_{2} + \beta_1 - 1) q^{97} + (3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2) q^{98} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + 11 q^{5} - 5 q^{6} - q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} + 11 q^{5} - 5 q^{6} - q^{7} - 5 q^{8} + 5 q^{9} - 11 q^{10} + 11 q^{11} + 5 q^{12} + 10 q^{13} + q^{14} + 11 q^{15} + 5 q^{16} + 11 q^{17} - 5 q^{18} - q^{19} + 11 q^{20} - q^{21} - 11 q^{22} - 5 q^{24} + 30 q^{25} - 10 q^{26} + 5 q^{27} - q^{28} + 3 q^{29} - 11 q^{30} + 2 q^{31} - 5 q^{32} + 11 q^{33} - 11 q^{34} - 11 q^{35} + 5 q^{36} + 16 q^{37} + q^{38} + 10 q^{39} - 11 q^{40} - 8 q^{41} + q^{42} - 12 q^{43} + 11 q^{44} + 11 q^{45} + 5 q^{48} - 4 q^{49} - 30 q^{50} + 11 q^{51} + 10 q^{52} + 27 q^{53} - 5 q^{54} + q^{56} - q^{57} - 3 q^{58} - 6 q^{59} + 11 q^{60} + 5 q^{61} - 2 q^{62} - q^{63} + 5 q^{64} + 22 q^{65} - 11 q^{66} - 7 q^{67} + 11 q^{68} + 11 q^{70} - 11 q^{71} - 5 q^{72} - 12 q^{73} - 16 q^{74} + 30 q^{75} - q^{76} + 11 q^{77} - 10 q^{78} + 16 q^{79} + 11 q^{80} + 5 q^{81} + 8 q^{82} + 22 q^{83} - q^{84} - 11 q^{85} + 12 q^{86} + 3 q^{87} - 11 q^{88} + 38 q^{89} - 11 q^{90} + 9 q^{91} + 2 q^{93} + 11 q^{95} - 5 q^{96} + 5 q^{97} + 4 q^{98} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{22} + \zeta_{22}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} + 6 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.91899
0.284630
1.30972
−1.68251
−0.830830
−1.00000 1.00000 1.00000 −2.43232 −1.00000 1.85592 −1.00000 1.00000 2.43232
1.2 −1.00000 1.00000 1.00000 1.95185 −1.00000 −0.458044 −1.00000 1.00000 −1.95185
1.3 −1.00000 1.00000 1.00000 2.89389 −1.00000 −2.77066 −1.00000 1.00000 −2.89389
1.4 −1.00000 1.00000 1.00000 4.16140 −1.00000 −2.94408 −1.00000 1.00000 −4.16140
1.5 −1.00000 1.00000 1.00000 4.42518 −1.00000 3.31686 −1.00000 1.00000 −4.42518
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3174.2.a.z 5
3.b odd 2 1 9522.2.a.bv 5
23.b odd 2 1 3174.2.a.y 5
23.c even 11 2 138.2.e.c 10
69.c even 2 1 9522.2.a.ca 5
69.h odd 22 2 414.2.i.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.c 10 23.c even 11 2
414.2.i.b 10 69.h odd 22 2
3174.2.a.y 5 23.b odd 2 1
3174.2.a.z 5 1.a even 1 1 trivial
9522.2.a.bv 5 3.b odd 2 1
9522.2.a.ca 5 69.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3174))\):

\( T_{5}^{5} - 11T_{5}^{4} + 33T_{5}^{3} + 22T_{5}^{2} - 231T_{5} + 253 \) Copy content Toggle raw display
\( T_{7}^{5} + T_{7}^{4} - 15T_{7}^{3} - 14T_{7}^{2} + 47T_{7} + 23 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 11 T^{4} + 33 T^{3} + \cdots + 253 \) Copy content Toggle raw display
$7$ \( T^{5} + T^{4} - 15 T^{3} - 14 T^{2} + \cdots + 23 \) Copy content Toggle raw display
$11$ \( T^{5} - 11 T^{4} + 22 T^{3} + \cdots + 253 \) Copy content Toggle raw display
$13$ \( T^{5} - 10 T^{4} + 29 T^{3} - 25 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{5} - 11 T^{4} - 33 T^{3} + \cdots - 5819 \) Copy content Toggle raw display
$19$ \( T^{5} + T^{4} - 59 T^{3} + 8 T^{2} + \cdots - 989 \) Copy content Toggle raw display
$23$ \( T^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 3 T^{4} - 69 T^{3} + 477 T^{2} + \cdots + 769 \) Copy content Toggle raw display
$31$ \( T^{5} - 2 T^{4} - 49 T^{3} + 13 T^{2} + \cdots + 23 \) Copy content Toggle raw display
$37$ \( T^{5} - 16 T^{4} + 65 T^{3} + \cdots - 397 \) Copy content Toggle raw display
$41$ \( T^{5} + 8 T^{4} - 47 T^{3} - 293 T^{2} + \cdots + 197 \) Copy content Toggle raw display
$43$ \( T^{5} + 12 T^{4} - 15 T^{3} + \cdots - 989 \) Copy content Toggle raw display
$47$ \( T^{5} - 55 T^{3} - 22 T^{2} + \cdots + 253 \) Copy content Toggle raw display
$53$ \( T^{5} - 27 T^{4} + 230 T^{3} + \cdots + 5653 \) Copy content Toggle raw display
$59$ \( T^{5} + 6 T^{4} - T^{3} - 32 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$61$ \( T^{5} - 5 T^{4} - 67 T^{3} + 386 T^{2} + \cdots - 199 \) Copy content Toggle raw display
$67$ \( T^{5} + 7 T^{4} - 119 T^{3} + \cdots - 3169 \) Copy content Toggle raw display
$71$ \( T^{5} + 11 T^{4} - 110 T^{3} + \cdots - 253 \) Copy content Toggle raw display
$73$ \( T^{5} + 12 T^{4} - 37 T^{3} + \cdots + 4489 \) Copy content Toggle raw display
$79$ \( T^{5} - 16 T^{4} - 45 T^{3} + \cdots - 25499 \) Copy content Toggle raw display
$83$ \( T^{5} - 22 T^{4} - 66 T^{3} + \cdots - 78397 \) Copy content Toggle raw display
$89$ \( T^{5} - 38 T^{4} + 450 T^{3} + \cdots + 64789 \) Copy content Toggle raw display
$97$ \( T^{5} - 5 T^{4} - 375 T^{3} + \cdots + 126983 \) Copy content Toggle raw display
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