Properties

Label 1425.2.c.l
Level $1425$
Weight $2$
Character orbit 1425.c
Analytic conductor $11.379$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{2} -\zeta_{8}^{2} q^{3} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{4} + ( 1 + \zeta_{8} - \zeta_{8}^{3} ) q^{6} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{7} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{8} - q^{9} +O(q^{10})\) \( q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{2} -\zeta_{8}^{2} q^{3} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{4} + ( 1 + \zeta_{8} - \zeta_{8}^{3} ) q^{6} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{7} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{8} - q^{9} + ( 2 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{11} + ( 2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{12} + ( \zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{13} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{14} + 3 q^{16} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{17} + ( -\zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{18} + q^{19} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{21} + ( -\zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{22} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{23} + ( -3 - \zeta_{8} + \zeta_{8}^{3} ) q^{24} + ( -4 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{26} + \zeta_{8}^{2} q^{27} + ( \zeta_{8} + 4 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{28} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{29} + ( -6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{31} + ( \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{32} + ( 3 \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{33} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{34} + ( 1 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{36} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{37} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{38} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{39} + ( 4 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{41} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{42} + ( -3 \zeta_{8} - 8 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{43} + ( 10 - \zeta_{8} + \zeta_{8}^{3} ) q^{44} + ( 10 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{46} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{47} -3 \zeta_{8}^{2} q^{48} + 5 q^{49} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{51} + ( -5 \zeta_{8} - 6 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{52} -8 \zeta_{8}^{2} q^{53} + ( -1 - \zeta_{8} + \zeta_{8}^{3} ) q^{54} + ( -2 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{56} -\zeta_{8}^{2} q^{57} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{58} + ( -4 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{59} + ( -4 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{61} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{62} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{63} + ( 7 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{64} + ( -4 - \zeta_{8} + \zeta_{8}^{3} ) q^{66} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{67} + ( -6 \zeta_{8} + 4 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{68} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{69} + ( -8 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{71} + ( \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{72} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{73} + ( 4 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{74} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{76} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{77} + ( 3 \zeta_{8} + 4 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{78} + q^{81} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{82} + ( -2 \zeta_{8} - 10 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{83} + ( 4 + \zeta_{8} - \zeta_{8}^{3} ) q^{84} + ( 14 + 11 \zeta_{8} - 11 \zeta_{8}^{3} ) q^{86} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{87} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{88} + ( 4 - 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{89} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{91} + ( 8 \zeta_{8} + 18 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{92} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{93} + ( 10 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{94} + ( -3 + \zeta_{8} - \zeta_{8}^{3} ) q^{96} + ( 3 \zeta_{8} - 14 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{97} + ( 5 \zeta_{8} + 5 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{98} + ( -2 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + 8q^{11} + 8q^{14} + 12q^{16} + 4q^{19} - 12q^{24} - 16q^{26} - 24q^{31} + 4q^{36} + 8q^{39} + 16q^{41} + 40q^{44} + 40q^{46} + 20q^{49} + 16q^{51} - 4q^{54} - 8q^{56} - 16q^{59} - 16q^{61} + 28q^{64} - 16q^{66} - 8q^{69} - 32q^{71} + 16q^{74} - 4q^{76} + 4q^{81} + 16q^{84} + 56q^{86} + 16q^{89} + 8q^{91} + 40q^{94} - 12q^{96} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i 1.00000i −3.82843 0 2.41421 1.41421i 4.41421i −1.00000 0
799.2 0.414214i 1.00000i 1.82843 0 −0.414214 1.41421i 1.58579i −1.00000 0
799.3 0.414214i 1.00000i 1.82843 0 −0.414214 1.41421i 1.58579i −1.00000 0
799.4 2.41421i 1.00000i −3.82843 0 2.41421 1.41421i 4.41421i −1.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1425.2.c.l 4
5.b even 2 1 inner 1425.2.c.l 4
5.c odd 4 1 285.2.a.g 2
5.c odd 4 1 1425.2.a.k 2
15.e even 4 1 855.2.a.d 2
15.e even 4 1 4275.2.a.y 2
20.e even 4 1 4560.2.a.bf 2
95.g even 4 1 5415.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.g 2 5.c odd 4 1
855.2.a.d 2 15.e even 4 1
1425.2.a.k 2 5.c odd 4 1
1425.2.c.l 4 1.a even 1 1 trivial
1425.2.c.l 4 5.b even 2 1 inner
4275.2.a.y 2 15.e even 4 1
4560.2.a.bf 2 20.e even 4 1
5415.2.a.n 2 95.g even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1425, [\chi])\):

\( T_{2}^{4} + 6 T_{2}^{2} + 1 \)
\( T_{7}^{2} + 2 \)
\( T_{11}^{2} - 4 T_{11} - 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T^{2} + T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 2 + T^{2} )^{2} \)
$11$ \( ( -14 - 4 T + T^{2} )^{2} \)
$13$ \( 4 + 12 T^{2} + T^{4} \)
$17$ \( 64 + 48 T^{2} + T^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 784 + 72 T^{2} + T^{4} \)
$29$ \( ( -2 + T^{2} )^{2} \)
$31$ \( ( 28 + 12 T + T^{2} )^{2} \)
$37$ \( 4 + 12 T^{2} + T^{4} \)
$41$ \( ( -2 - 8 T + T^{2} )^{2} \)
$43$ \( 2116 + 164 T^{2} + T^{4} \)
$47$ \( 784 + 72 T^{2} + T^{4} \)
$53$ \( ( 64 + T^{2} )^{2} \)
$59$ \( ( -56 + 8 T + T^{2} )^{2} \)
$61$ \( ( -112 + 8 T + T^{2} )^{2} \)
$67$ \( 256 + 96 T^{2} + T^{4} \)
$71$ \( ( 56 + 16 T + T^{2} )^{2} \)
$73$ \( 784 + 72 T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( 8464 + 216 T^{2} + T^{4} \)
$89$ \( ( -82 - 8 T + T^{2} )^{2} \)
$97$ \( 31684 + 428 T^{2} + T^{4} \)
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