Properties

Label 318.2.c.a
Level $318$
Weight $2$
Character orbit 318.c
Analytic conductor $2.539$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 318 = 2 \cdot 3 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 318.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.53924278428\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} -4 i q^{5} - q^{6} + q^{7} -i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} + i q^{3} - q^{4} -4 i q^{5} - q^{6} + q^{7} -i q^{8} - q^{9} + 4 q^{10} + 5 q^{11} -i q^{12} + 2 q^{13} + i q^{14} + 4 q^{15} + q^{16} -i q^{18} -i q^{19} + 4 i q^{20} + i q^{21} + 5 i q^{22} -3 i q^{23} + q^{24} -11 q^{25} + 2 i q^{26} -i q^{27} - q^{28} + 5 q^{29} + 4 i q^{30} + 6 i q^{31} + i q^{32} + 5 i q^{33} -4 i q^{35} + q^{36} -10 q^{37} + q^{38} + 2 i q^{39} -4 q^{40} -3 i q^{41} - q^{42} + 6 q^{43} -5 q^{44} + 4 i q^{45} + 3 q^{46} + 6 q^{47} + i q^{48} -6 q^{49} -11 i q^{50} -2 q^{52} + ( 7 - 2 i ) q^{53} + q^{54} -20 i q^{55} -i q^{56} + q^{57} + 5 i q^{58} -12 q^{59} -4 q^{60} + 13 i q^{61} -6 q^{62} - q^{63} - q^{64} -8 i q^{65} -5 q^{66} -13 i q^{67} + 3 q^{69} + 4 q^{70} -7 i q^{71} + i q^{72} + 4 i q^{73} -10 i q^{74} -11 i q^{75} + i q^{76} + 5 q^{77} -2 q^{78} + 16 i q^{79} -4 i q^{80} + q^{81} + 3 q^{82} + 12 i q^{83} -i q^{84} + 6 i q^{86} + 5 i q^{87} -5 i q^{88} -14 q^{89} -4 q^{90} + 2 q^{91} + 3 i q^{92} -6 q^{93} + 6 i q^{94} -4 q^{95} - q^{96} + 5 q^{97} -6 i q^{98} -5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} + 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} + 2q^{7} - 2q^{9} + 8q^{10} + 10q^{11} + 4q^{13} + 8q^{15} + 2q^{16} + 2q^{24} - 22q^{25} - 2q^{28} + 10q^{29} + 2q^{36} - 20q^{37} + 2q^{38} - 8q^{40} - 2q^{42} + 12q^{43} - 10q^{44} + 6q^{46} + 12q^{47} - 12q^{49} - 4q^{52} + 14q^{53} + 2q^{54} + 2q^{57} - 24q^{59} - 8q^{60} - 12q^{62} - 2q^{63} - 2q^{64} - 10q^{66} + 6q^{69} + 8q^{70} + 10q^{77} - 4q^{78} + 2q^{81} + 6q^{82} - 28q^{89} - 8q^{90} + 4q^{91} - 12q^{93} - 8q^{95} - 2q^{96} + 10q^{97} - 10q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/318\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(107\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 4.00000i −1.00000 1.00000 1.00000i −1.00000 4.00000
211.2 1.00000i 1.00000i −1.00000 4.00000i −1.00000 1.00000 1.00000i −1.00000 4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
53.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 318.2.c.a 2
3.b odd 2 1 954.2.c.d 2
4.b odd 2 1 2544.2.i.a 2
53.b even 2 1 inner 318.2.c.a 2
159.d odd 2 1 954.2.c.d 2
212.d odd 2 1 2544.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
318.2.c.a 2 1.a even 1 1 trivial
318.2.c.a 2 53.b even 2 1 inner
954.2.c.d 2 3.b odd 2 1
954.2.c.d 2 159.d odd 2 1
2544.2.i.a 2 4.b odd 2 1
2544.2.i.a 2 212.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(318, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 16 + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -5 + T )^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 1 + T^{2} \)
$23$ \( 9 + T^{2} \)
$29$ \( ( -5 + T )^{2} \)
$31$ \( 36 + T^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( 9 + T^{2} \)
$43$ \( ( -6 + T )^{2} \)
$47$ \( ( -6 + T )^{2} \)
$53$ \( 53 - 14 T + T^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( 169 + T^{2} \)
$67$ \( 169 + T^{2} \)
$71$ \( 49 + T^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( 256 + T^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( 14 + T )^{2} \)
$97$ \( ( -5 + T )^{2} \)
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