Properties

Label 925.2.e.a
Level $925$
Weight $2$
Character orbit 925.e
Analytic conductor $7.386$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 37)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{4} + 2 \zeta_{6} q^{7} + 3 q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{4} + 2 \zeta_{6} q^{7} + 3 q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} -2 q^{11} -2 \zeta_{6} q^{13} + 2 q^{14} + ( 1 - \zeta_{6} ) q^{16} + ( 3 - 3 \zeta_{6} ) q^{17} -3 \zeta_{6} q^{18} + 6 \zeta_{6} q^{19} + ( -2 + 2 \zeta_{6} ) q^{22} + 4 q^{23} -2 q^{26} + ( -2 + 2 \zeta_{6} ) q^{28} + 9 q^{29} -10 q^{31} + 5 \zeta_{6} q^{32} -3 \zeta_{6} q^{34} + 3 q^{36} + ( 4 + 3 \zeta_{6} ) q^{37} + 6 q^{38} + 9 \zeta_{6} q^{41} -2 q^{43} -2 \zeta_{6} q^{44} + ( 4 - 4 \zeta_{6} ) q^{46} -6 q^{47} + ( 3 - 3 \zeta_{6} ) q^{49} + ( 2 - 2 \zeta_{6} ) q^{52} + ( -2 + 2 \zeta_{6} ) q^{53} + 6 \zeta_{6} q^{56} + ( 9 - 9 \zeta_{6} ) q^{58} + ( 4 - 4 \zeta_{6} ) q^{59} -\zeta_{6} q^{61} + ( -10 + 10 \zeta_{6} ) q^{62} + 6 q^{63} + 7 q^{64} -10 \zeta_{6} q^{67} + 3 q^{68} -6 \zeta_{6} q^{71} + ( 9 - 9 \zeta_{6} ) q^{72} + 10 q^{73} + ( 7 - 4 \zeta_{6} ) q^{74} + ( -6 + 6 \zeta_{6} ) q^{76} -4 \zeta_{6} q^{77} -10 \zeta_{6} q^{79} -9 \zeta_{6} q^{81} + 9 q^{82} + ( -12 + 12 \zeta_{6} ) q^{83} + ( -2 + 2 \zeta_{6} ) q^{86} -6 q^{88} + ( -7 + 7 \zeta_{6} ) q^{89} + ( 4 - 4 \zeta_{6} ) q^{91} + 4 \zeta_{6} q^{92} + ( -6 + 6 \zeta_{6} ) q^{94} -7 q^{97} -3 \zeta_{6} q^{98} + ( -6 + 6 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{4} + 2q^{7} + 6q^{8} + 3q^{9} + O(q^{10}) \) \( 2q + q^{2} + q^{4} + 2q^{7} + 6q^{8} + 3q^{9} - 4q^{11} - 2q^{13} + 4q^{14} + q^{16} + 3q^{17} - 3q^{18} + 6q^{19} - 2q^{22} + 8q^{23} - 4q^{26} - 2q^{28} + 18q^{29} - 20q^{31} + 5q^{32} - 3q^{34} + 6q^{36} + 11q^{37} + 12q^{38} + 9q^{41} - 4q^{43} - 2q^{44} + 4q^{46} - 12q^{47} + 3q^{49} + 2q^{52} - 2q^{53} + 6q^{56} + 9q^{58} + 4q^{59} - q^{61} - 10q^{62} + 12q^{63} + 14q^{64} - 10q^{67} + 6q^{68} - 6q^{71} + 9q^{72} + 20q^{73} + 10q^{74} - 6q^{76} - 4q^{77} - 10q^{79} - 9q^{81} + 18q^{82} - 12q^{83} - 2q^{86} - 12q^{88} - 7q^{89} + 4q^{91} + 4q^{92} - 6q^{94} - 14q^{97} - 3q^{98} - 6q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 1.00000 1.73205i 3.00000 1.50000 + 2.59808i 0
676.1 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 1.00000 + 1.73205i 3.00000 1.50000 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 925.2.e.a 2
5.b even 2 1 37.2.c.a 2
5.c odd 4 2 925.2.o.a 4
15.d odd 2 1 333.2.f.a 2
20.d odd 2 1 592.2.i.c 2
37.c even 3 1 inner 925.2.e.a 2
185.l even 6 1 1369.2.a.b 1
185.n even 6 1 37.2.c.a 2
185.n even 6 1 1369.2.a.d 1
185.q odd 12 2 1369.2.b.b 2
185.s odd 12 2 925.2.o.a 4
555.w odd 6 1 333.2.f.a 2
740.w odd 6 1 592.2.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.2.c.a 2 5.b even 2 1
37.2.c.a 2 185.n even 6 1
333.2.f.a 2 15.d odd 2 1
333.2.f.a 2 555.w odd 6 1
592.2.i.c 2 20.d odd 2 1
592.2.i.c 2 740.w odd 6 1
925.2.e.a 2 1.a even 1 1 trivial
925.2.e.a 2 37.c even 3 1 inner
925.2.o.a 4 5.c odd 4 2
925.2.o.a 4 185.s odd 12 2
1369.2.a.b 1 185.l even 6 1
1369.2.a.d 1 185.n even 6 1
1369.2.b.b 2 185.q odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 - 2 T + T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( 4 + 2 T + T^{2} \)
$17$ \( 9 - 3 T + T^{2} \)
$19$ \( 36 - 6 T + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( ( -9 + T )^{2} \)
$31$ \( ( 10 + T )^{2} \)
$37$ \( 37 - 11 T + T^{2} \)
$41$ \( 81 - 9 T + T^{2} \)
$43$ \( ( 2 + T )^{2} \)
$47$ \( ( 6 + T )^{2} \)
$53$ \( 4 + 2 T + T^{2} \)
$59$ \( 16 - 4 T + T^{2} \)
$61$ \( 1 + T + T^{2} \)
$67$ \( 100 + 10 T + T^{2} \)
$71$ \( 36 + 6 T + T^{2} \)
$73$ \( ( -10 + T )^{2} \)
$79$ \( 100 + 10 T + T^{2} \)
$83$ \( 144 + 12 T + T^{2} \)
$89$ \( 49 + 7 T + T^{2} \)
$97$ \( ( 7 + T )^{2} \)
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