Properties

Label 1350.4.c.h
Level $1350$
Weight $4$
Character orbit 1350.c
Analytic conductor $79.653$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.6525785077\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} - 4 q^{4} + 19 i q^{7} - 8 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} - 4 q^{4} + 19 i q^{7} - 8 i q^{8} - 12 q^{11} + 50 i q^{13} - 38 q^{14} + 16 q^{16} - 126 i q^{17} - 29 q^{19} - 24 i q^{22} - 18 i q^{23} - 100 q^{26} - 76 i q^{28} + 102 q^{29} - 265 q^{31} + 32 i q^{32} + 252 q^{34} - 65 i q^{37} - 58 i q^{38} - 240 q^{41} - 367 i q^{43} + 48 q^{44} + 36 q^{46} - 72 i q^{47} - 18 q^{49} - 200 i q^{52} + 636 i q^{53} + 152 q^{56} + 204 i q^{58} + 102 q^{59} - 103 q^{61} - 530 i q^{62} - 64 q^{64} + 52 i q^{67} + 504 i q^{68} + 582 q^{71} + 65 i q^{73} + 130 q^{74} + 116 q^{76} - 228 i q^{77} - 173 q^{79} - 480 i q^{82} - 498 i q^{83} + 734 q^{86} + 96 i q^{88} - 822 q^{89} - 950 q^{91} + 72 i q^{92} + 144 q^{94} - 821 i q^{97} - 36 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 24 q^{11} - 76 q^{14} + 32 q^{16} - 58 q^{19} - 200 q^{26} + 204 q^{29} - 530 q^{31} + 504 q^{34} - 480 q^{41} + 96 q^{44} + 72 q^{46} - 36 q^{49} + 304 q^{56} + 204 q^{59} - 206 q^{61} - 128 q^{64} + 1164 q^{71} + 260 q^{74} + 232 q^{76} - 346 q^{79} + 1468 q^{86} - 1644 q^{89} - 1900 q^{91} + 288 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
2.00000i 0 −4.00000 0 0 19.0000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 19.0000i 8.00000i 0 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 1350.4.c.h 2
3.b odd 2 1 1350.4.c.m 2
5.b even 2 1 inner 1350.4.c.h 2
5.c odd 4 1 1350.4.a.c 1
5.c odd 4 1 1350.4.a.y yes 1
15.d odd 2 1 1350.4.c.m 2
15.e even 4 1 1350.4.a.k yes 1
15.e even 4 1 1350.4.a.q yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.4.a.c 1 5.c odd 4 1
1350.4.a.k yes 1 15.e even 4 1
1350.4.a.q yes 1 15.e even 4 1
1350.4.a.y yes 1 5.c odd 4 1
1350.4.c.h 2 1.a even 1 1 trivial
1350.4.c.h 2 5.b even 2 1 inner
1350.4.c.m 2 3.b odd 2 1
1350.4.c.m 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{2} + 361 \) Copy content Toggle raw display
\( T_{11} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 361 \) Copy content Toggle raw display
$11$ \( (T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2500 \) Copy content Toggle raw display
$17$ \( T^{2} + 15876 \) Copy content Toggle raw display
$19$ \( (T + 29)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 324 \) Copy content Toggle raw display
$29$ \( (T - 102)^{2} \) Copy content Toggle raw display
$31$ \( (T + 265)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4225 \) Copy content Toggle raw display
$41$ \( (T + 240)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 134689 \) Copy content Toggle raw display
$47$ \( T^{2} + 5184 \) Copy content Toggle raw display
$53$ \( T^{2} + 404496 \) Copy content Toggle raw display
$59$ \( (T - 102)^{2} \) Copy content Toggle raw display
$61$ \( (T + 103)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 2704 \) Copy content Toggle raw display
$71$ \( (T - 582)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4225 \) Copy content Toggle raw display
$79$ \( (T + 173)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 248004 \) Copy content Toggle raw display
$89$ \( (T + 822)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 674041 \) Copy content Toggle raw display
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