Properties

Label 1350.4.c.a
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: no (minimal twist has level 54)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} - 4 q^{4} + 7 i q^{7} - 8 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} - 4 q^{4} + 7 i q^{7} - 8 i q^{8} - 60 q^{11} - 79 i q^{13} - 14 q^{14} + 16 q^{16} - 108 i q^{17} - 11 q^{19} - 120 i q^{22} + 132 i q^{23} + 158 q^{26} - 28 i q^{28} + 96 q^{29} + 20 q^{31} + 32 i q^{32} + 216 q^{34} + 169 i q^{37} - 22 i q^{38} - 192 q^{41} + 488 i q^{43} + 240 q^{44} - 264 q^{46} + 204 i q^{47} + 294 q^{49} + 316 i q^{52} - 360 i q^{53} + 56 q^{56} + 192 i q^{58} + 156 q^{59} + 83 q^{61} + 40 i q^{62} - 64 q^{64} - 47 i q^{67} + 432 i q^{68} - 216 q^{71} - 511 i q^{73} - 338 q^{74} + 44 q^{76} - 420 i q^{77} + 529 q^{79} - 384 i q^{82} + 1128 i q^{83} - 976 q^{86} + 480 i q^{88} + 36 q^{89} + 553 q^{91} - 528 i q^{92} - 408 q^{94} - 605 i q^{97} + 588 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 120 q^{11} - 28 q^{14} + 32 q^{16} - 22 q^{19} + 316 q^{26} + 192 q^{29} + 40 q^{31} + 432 q^{34} - 384 q^{41} + 480 q^{44} - 528 q^{46} + 588 q^{49} + 112 q^{56} + 312 q^{59} + 166 q^{61} - 128 q^{64} - 432 q^{71} - 676 q^{74} + 88 q^{76} + 1058 q^{79} - 1952 q^{86} + 72 q^{89} + 1106 q^{91} - 816 q^{94}+O(q^{100}) \) 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 7.00000i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 7.00000i 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.a 2
3.b odd 2 1 1350.4.c.t 2
5.b even 2 1 inner 1350.4.c.a 2
5.c odd 4 1 54.4.a.a 1
5.c odd 4 1 1350.4.a.v 1
15.d odd 2 1 1350.4.c.t 2
15.e even 4 1 54.4.a.d yes 1
15.e even 4 1 1350.4.a.h 1
20.e even 4 1 432.4.a.b 1
40.i odd 4 1 1728.4.a.ba 1
40.k even 4 1 1728.4.a.bb 1
45.k odd 12 2 162.4.c.h 2
45.l even 12 2 162.4.c.a 2
60.l odd 4 1 432.4.a.m 1
120.q odd 4 1 1728.4.a.f 1
120.w even 4 1 1728.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.a.a 1 5.c odd 4 1
54.4.a.d yes 1 15.e even 4 1
162.4.c.a 2 45.l even 12 2
162.4.c.h 2 45.k odd 12 2
432.4.a.b 1 20.e even 4 1
432.4.a.m 1 60.l odd 4 1
1350.4.a.h 1 15.e even 4 1
1350.4.a.v 1 5.c odd 4 1
1350.4.c.a 2 1.a even 1 1 trivial
1350.4.c.a 2 5.b even 2 1 inner
1350.4.c.t 2 3.b odd 2 1
1350.4.c.t 2 15.d odd 2 1
1728.4.a.e 1 120.w even 4 1
1728.4.a.f 1 120.q odd 4 1
1728.4.a.ba 1 40.i odd 4 1
1728.4.a.bb 1 40.k even 4 1

Hecke kernels

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

\( T_{7}^{2} + 49 \) Copy content Toggle raw display
\( T_{11} + 60 \) 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
$7$ \( T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T + 60)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 6241 \) Copy content Toggle raw display
$17$ \( T^{2} + 11664 \) Copy content Toggle raw display
$19$ \( (T + 11)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 17424 \) Copy content Toggle raw display
$29$ \( (T - 96)^{2} \) Copy content Toggle raw display
$31$ \( (T - 20)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 28561 \) Copy content Toggle raw display
$41$ \( (T + 192)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 238144 \) Copy content Toggle raw display
$47$ \( T^{2} + 41616 \) Copy content Toggle raw display
$53$ \( T^{2} + 129600 \) Copy content Toggle raw display
$59$ \( (T - 156)^{2} \) Copy content Toggle raw display
$61$ \( (T - 83)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 2209 \) Copy content Toggle raw display
$71$ \( (T + 216)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 261121 \) Copy content Toggle raw display
$79$ \( (T - 529)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1272384 \) Copy content Toggle raw display
$89$ \( (T - 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 366025 \) Copy content Toggle raw display
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