Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3174.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.3445176016\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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.

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.830830
−1.68251
1.91899
1.30972
0.284630
1.00000 1.00000 1.00000 −3.20362 1.00000 1.51334 1.00000 1.00000 −3.20362
1.2 1.00000 1.00000 1.00000 −2.59435 1.00000 −1.23648 1.00000 1.00000 −2.59435
1.3 1.00000 1.00000 1.00000 −1.47889 1.00000 −3.20362 1.00000 1.00000 −1.47889
1.4 1.00000 1.00000 1.00000 −1.23648 1.00000 −1.47889 1.00000 1.00000 −1.23648
1.5 1.00000 1.00000 1.00000 1.51334 1.00000 −2.59435 1.00000 1.00000 1.51334
\(n\): 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.bc 5
3.b odd 2 1 9522.2.a.bt 5
23.b odd 2 1 3174.2.a.bd 5
23.c even 11 2 138.2.e.a 10
69.c even 2 1 9522.2.a.bq 5
69.h odd 22 2 414.2.i.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.a 10 23.c even 11 2
414.2.i.d 10 69.h odd 22 2
3174.2.a.bc 5 1.a even 1 1 trivial
3174.2.a.bd 5 23.b odd 2 1
9522.2.a.bq 5 69.c even 2 1
9522.2.a.bt 5 3.b odd 2 1

Hecke kernels

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

\( T_{5}^{5} + 7T_{5}^{4} + 13T_{5}^{3} - 6T_{5}^{2} - 35T_{5} - 23 \) Copy content Toggle raw display
\( T_{7}^{5} + 7T_{7}^{4} + 13T_{7}^{3} - 6T_{7}^{2} - 35T_{7} - 23 \) 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
$5$ \( T^{5} + 7 T^{4} + 13 T^{3} - 6 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$7$ \( T^{5} + 7 T^{4} + 13 T^{3} - 6 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$11$ \( T^{5} + 13 T^{4} + 50 T^{3} + 53 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$13$ \( T^{5} + 4 T^{4} - 9 T^{3} - 27 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{5} + 9 T^{4} + 17 T^{3} - 31 T^{2} + \cdots - 43 \) Copy content Toggle raw display
$19$ \( T^{5} + 11 T^{4} + 11 T^{3} + \cdots + 253 \) Copy content Toggle raw display
$23$ \( T^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 7 T^{4} - 75 T^{3} + \cdots + 1693 \) Copy content Toggle raw display
$31$ \( T^{5} + 8 T^{4} - 69 T^{3} - 733 T^{2} + \cdots - 947 \) Copy content Toggle raw display
$37$ \( T^{5} + 12 T^{4} + 51 T^{3} + 96 T^{2} + \cdots + 23 \) Copy content Toggle raw display
$41$ \( T^{5} + 10 T^{4} - 37 T^{3} - 195 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$43$ \( T^{5} + 4 T^{4} - 97 T^{3} + 149 T^{2} + \cdots - 439 \) Copy content Toggle raw display
$47$ \( T^{5} + 24 T^{4} + 83 T^{3} + \cdots - 10649 \) Copy content Toggle raw display
$53$ \( T^{5} + 9 T^{4} - 170 T^{3} + \cdots + 19009 \) Copy content Toggle raw display
$59$ \( T^{5} + 14 T^{4} - 3 T^{3} + \cdots + 5633 \) Copy content Toggle raw display
$61$ \( T^{5} + 5 T^{4} - 243 T^{3} + \cdots + 52933 \) Copy content Toggle raw display
$67$ \( T^{5} + 13 T^{4} - 181 T^{3} + \cdots + 42481 \) Copy content Toggle raw display
$71$ \( T^{5} + 19 T^{4} - 36 T^{3} + \cdots + 38609 \) Copy content Toggle raw display
$73$ \( T^{5} - 4 T^{4} - 317 T^{3} + \cdots + 15377 \) Copy content Toggle raw display
$79$ \( T^{5} + 4 T^{4} - 251 T^{3} + \cdots - 49169 \) Copy content Toggle raw display
$83$ \( T^{5} + 24 T^{4} + 50 T^{3} + \cdots + 25673 \) Copy content Toggle raw display
$89$ \( T^{5} + 4 T^{4} - 130 T^{3} + \cdots + 9637 \) Copy content Toggle raw display
$97$ \( T^{5} - 9 T^{4} - 335 T^{3} + \cdots - 149381 \) Copy content Toggle raw display
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