Properties

Label 1350.4.c.u
Level $1350$
Weight $4$
Character orbit 1350.c
Analytic conductor $79.653$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{401})\)
Defining polynomial: \( x^{4} + 201x^{2} + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} - 4 q^{4} - \beta_{2} q^{7} + 8 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} - 4 q^{4} - \beta_{2} q^{7} + 8 \beta_1 q^{8} + ( - \beta_{3} - 16) q^{11} + ( - 2 \beta_{2} - 33 \beta_1) q^{13} + ( - 2 \beta_{3} + 2) q^{14} + 16 q^{16} + (\beta_{2} - 25 \beta_1) q^{17} + ( - \beta_{3} - 57) q^{19} + (2 \beta_{2} + 34 \beta_1) q^{22} + (5 \beta_{2} + 13 \beta_1) q^{23} + ( - 4 \beta_{3} - 62) q^{26} + 4 \beta_{2} q^{28} + ( - 7 \beta_{3} - 28) q^{29} + ( - 5 \beta_{3} + 150) q^{31} - 32 \beta_1 q^{32} + (2 \beta_{3} - 52) q^{34} + ( - 3 \beta_{2} + 68 \beta_1) q^{37} + (2 \beta_{2} + 116 \beta_1) q^{38} + (6 \beta_{3} + 318) q^{41} + ( - 5 \beta_{2} - 147 \beta_1) q^{43} + (4 \beta_{3} + 64) q^{44} + (10 \beta_{3} + 16) q^{46} + ( - 13 \beta_{2} - 95 \beta_1) q^{47} + (\beta_{3} - 560) q^{49} + (8 \beta_{2} + 132 \beta_1) q^{52} + ( - 2 \beta_{2} - 184 \beta_1) q^{53} + (8 \beta_{3} - 8) q^{56} + (14 \beta_{2} + 70 \beta_1) q^{58} + (8 \beta_{3} - 628) q^{59} + (3 \beta_{3} + 41) q^{61} + (10 \beta_{2} - 290 \beta_1) q^{62} - 64 q^{64} + (3 \beta_{2} + 494 \beta_1) q^{67} + ( - 4 \beta_{2} + 100 \beta_1) q^{68} + ( - 6 \beta_{3} + 750) q^{71} + ( - 3 \beta_{2} + 928 \beta_1) q^{73} + ( - 6 \beta_{3} + 142) q^{74} + (4 \beta_{3} + 228) q^{76} + (16 \beta_{2} + 902 \beta_1) q^{77} + (6 \beta_{3} - 635) q^{79} + ( - 12 \beta_{2} - 648 \beta_1) q^{82} + ( - 28 \beta_{2} + 526 \beta_1) q^{83} + ( - 10 \beta_{3} - 284) q^{86} + ( - 8 \beta_{2} - 136 \beta_1) q^{88} + (36 \beta_{3} - 360) q^{89} + ( - 31 \beta_{3} - 1773) q^{91} + ( - 20 \beta_{2} - 52 \beta_1) q^{92} + ( - 26 \beta_{3} - 164) q^{94} + ( - 17 \beta_{2} - 644 \beta_1) q^{97} + ( - 2 \beta_{2} + 1118 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{4} - 66 q^{11} + 4 q^{14} + 64 q^{16} - 230 q^{19} - 256 q^{26} - 126 q^{29} + 590 q^{31} - 204 q^{34} + 1284 q^{41} + 264 q^{44} + 84 q^{46} - 2238 q^{49} - 16 q^{56} - 2496 q^{59} + 170 q^{61} - 256 q^{64} + 2988 q^{71} + 556 q^{74} + 920 q^{76} - 2528 q^{79} - 1156 q^{86} - 1368 q^{89} - 7154 q^{91} - 708 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 201x^{2} + 10000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 101\nu ) / 100 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 401\nu ) / 100 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 302 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 302 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -101\beta_{2} + 401\beta_1 ) / 3 \) 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
9.51249i
10.5125i
10.5125i
9.51249i
2.00000i 0 −4.00000 0 0 29.5375i 8.00000i 0 0
649.2 2.00000i 0 −4.00000 0 0 30.5375i 8.00000i 0 0
649.3 2.00000i 0 −4.00000 0 0 30.5375i 8.00000i 0 0
649.4 2.00000i 0 −4.00000 0 0 29.5375i 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.u 4
3.b odd 2 1 1350.4.c.bb 4
5.b even 2 1 inner 1350.4.c.u 4
5.c odd 4 1 270.4.a.m 2
5.c odd 4 1 1350.4.a.bm 2
15.d odd 2 1 1350.4.c.bb 4
15.e even 4 1 270.4.a.n yes 2
15.e even 4 1 1350.4.a.bf 2
20.e even 4 1 2160.4.a.w 2
45.k odd 12 2 810.4.e.bd 4
45.l even 12 2 810.4.e.z 4
60.l odd 4 1 2160.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.m 2 5.c odd 4 1
270.4.a.n yes 2 15.e even 4 1
810.4.e.z 4 45.l even 12 2
810.4.e.bd 4 45.k odd 12 2
1350.4.a.bf 2 15.e even 4 1
1350.4.a.bm 2 5.c odd 4 1
1350.4.c.u 4 1.a even 1 1 trivial
1350.4.c.u 4 5.b even 2 1 inner
1350.4.c.bb 4 3.b odd 2 1
1350.4.c.bb 4 15.d odd 2 1
2160.4.a.w 2 20.e even 4 1
2160.4.a.bb 2 60.l odd 4 1

Hecke kernels

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

\( T_{7}^{4} + 1805T_{7}^{2} + 813604 \) Copy content Toggle raw display
\( T_{11}^{2} + 33T_{11} - 630 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 1805 T^{2} + 813604 \) Copy content Toggle raw display
$11$ \( (T^{2} + 33 T - 630)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 9266 T^{2} + \cdots + 6682225 \) Copy content Toggle raw display
$17$ \( T^{4} + 3105 T^{2} + 63504 \) Copy content Toggle raw display
$19$ \( (T^{2} + 115 T + 2404)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 45333 T^{2} + \cdots + 503822916 \) Copy content Toggle raw display
$29$ \( (T^{2} + 63 T - 43218)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 295 T - 800)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 25901 T^{2} + \cdots + 10824100 \) Copy content Toggle raw display
$41$ \( (T^{2} - 642 T + 70560)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 86873 T^{2} + \cdots + 2808976 \) Copy content Toggle raw display
$47$ \( T^{4} + 320625 T^{2} + \cdots + 20923043904 \) Copy content Toggle raw display
$53$ \( T^{4} + 74196 T^{2} + \cdots + 892814400 \) Copy content Toggle raw display
$59$ \( (T^{2} + 1248 T + 331632)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 85 T - 6314)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 501353 T^{2} + \cdots + 54960238096 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1494 T + 525528)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 1744181 T^{2} + \cdots + 732479222500 \) Copy content Toggle raw display
$79$ \( (T^{2} + 1264 T + 366943)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1997928 T^{2} + \cdots + 172859703696 \) Copy content Toggle raw display
$89$ \( (T^{2} + 684 T - 1052352)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 1329221 T^{2} + \cdots + 20480472100 \) Copy content Toggle raw display
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