Properties

Label 1150.4.b.f
Level $1150$
Weight $4$
Character orbit 1150.b
Analytic conductor $67.852$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
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.

\(f(q)\) \(=\) \( q -2 i q^{2} + i q^{3} -4 q^{4} + 2 q^{6} + 18 i q^{7} + 8 i q^{8} + 26 q^{9} +O(q^{10})\) \( q -2 i q^{2} + i q^{3} -4 q^{4} + 2 q^{6} + 18 i q^{7} + 8 i q^{8} + 26 q^{9} -32 q^{11} -4 i q^{12} -47 i q^{13} + 36 q^{14} + 16 q^{16} -20 i q^{17} -52 i q^{18} -36 q^{19} -18 q^{21} + 64 i q^{22} -23 i q^{23} -8 q^{24} -94 q^{26} + 53 i q^{27} -72 i q^{28} + 27 q^{29} -33 q^{31} -32 i q^{32} -32 i q^{33} -40 q^{34} -104 q^{36} -56 i q^{37} + 72 i q^{38} + 47 q^{39} -157 q^{41} + 36 i q^{42} + 18 i q^{43} + 128 q^{44} -46 q^{46} -65 i q^{47} + 16 i q^{48} + 19 q^{49} + 20 q^{51} + 188 i q^{52} -14 i q^{53} + 106 q^{54} -144 q^{56} -36 i q^{57} -54 i q^{58} + 744 q^{59} + 552 q^{61} + 66 i q^{62} + 468 i q^{63} -64 q^{64} -64 q^{66} + 156 i q^{67} + 80 i q^{68} + 23 q^{69} + 699 q^{71} + 208 i q^{72} -609 i q^{73} -112 q^{74} + 144 q^{76} -576 i q^{77} -94 i q^{78} + 644 q^{79} + 649 q^{81} + 314 i q^{82} + 512 i q^{83} + 72 q^{84} + 36 q^{86} + 27 i q^{87} -256 i q^{88} + 102 q^{89} + 846 q^{91} + 92 i q^{92} -33 i q^{93} -130 q^{94} + 32 q^{96} -578 i q^{97} -38 i q^{98} -832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9} + O(q^{10}) \) \( 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9} - 64 q^{11} + 72 q^{14} + 32 q^{16} - 72 q^{19} - 36 q^{21} - 16 q^{24} - 188 q^{26} + 54 q^{29} - 66 q^{31} - 80 q^{34} - 208 q^{36} + 94 q^{39} - 314 q^{41} + 256 q^{44} - 92 q^{46} + 38 q^{49} + 40 q^{51} + 212 q^{54} - 288 q^{56} + 1488 q^{59} + 1104 q^{61} - 128 q^{64} - 128 q^{66} + 46 q^{69} + 1398 q^{71} - 224 q^{74} + 288 q^{76} + 1288 q^{79} + 1298 q^{81} + 144 q^{84} + 72 q^{86} + 204 q^{89} + 1692 q^{91} - 260 q^{94} + 64 q^{96} - 1664 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1
1.00000i
1.00000i
2.00000i 1.00000i −4.00000 0 2.00000 18.0000i 8.00000i 26.0000 0
599.2 2.00000i 1.00000i −4.00000 0 2.00000 18.0000i 8.00000i 26.0000 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 1150.4.b.f 2
5.b even 2 1 inner 1150.4.b.f 2
5.c odd 4 1 230.4.a.e 1
5.c odd 4 1 1150.4.a.b 1
15.e even 4 1 2070.4.a.e 1
20.e even 4 1 1840.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.e 1 5.c odd 4 1
1150.4.a.b 1 5.c odd 4 1
1150.4.b.f 2 1.a even 1 1 trivial
1150.4.b.f 2 5.b even 2 1 inner
1840.4.a.d 1 20.e even 4 1
2070.4.a.e 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1150, [\chi])\):

\( T_{3}^{2} + 1 \)
\( T_{7}^{2} + 324 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 324 + T^{2} \)
$11$ \( ( 32 + T )^{2} \)
$13$ \( 2209 + T^{2} \)
$17$ \( 400 + T^{2} \)
$19$ \( ( 36 + T )^{2} \)
$23$ \( 529 + T^{2} \)
$29$ \( ( -27 + T )^{2} \)
$31$ \( ( 33 + T )^{2} \)
$37$ \( 3136 + T^{2} \)
$41$ \( ( 157 + T )^{2} \)
$43$ \( 324 + T^{2} \)
$47$ \( 4225 + T^{2} \)
$53$ \( 196 + T^{2} \)
$59$ \( ( -744 + T )^{2} \)
$61$ \( ( -552 + T )^{2} \)
$67$ \( 24336 + T^{2} \)
$71$ \( ( -699 + T )^{2} \)
$73$ \( 370881 + T^{2} \)
$79$ \( ( -644 + T )^{2} \)
$83$ \( 262144 + T^{2} \)
$89$ \( ( -102 + T )^{2} \)
$97$ \( 334084 + T^{2} \)
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