Properties

Label 230.4.a.j
Level $230$
Weight $4$
Character orbit 230.a
Self dual yes
Analytic conductor $13.570$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 2 x^{3} - 60 x^{2} - 45 x + 108\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( 4 - \beta_{1} ) q^{3} + 4 q^{4} + 5 q^{5} + ( 8 - 2 \beta_{1} ) q^{6} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{7} + 8 q^{8} + ( 18 - 4 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + 2 q^{2} + ( 4 - \beta_{1} ) q^{3} + 4 q^{4} + 5 q^{5} + ( 8 - 2 \beta_{1} ) q^{6} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{7} + 8 q^{8} + ( 18 - 4 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{9} + 10 q^{10} + ( 6 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{11} + ( 16 - 4 \beta_{1} ) q^{12} + ( 15 + 4 \beta_{1} + \beta_{3} ) q^{13} + ( 2 + 2 \beta_{1} + 4 \beta_{2} ) q^{14} + ( 20 - 5 \beta_{1} ) q^{15} + 16 q^{16} + ( 18 - 2 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{17} + ( 36 - 8 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{18} + ( 39 + 8 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{19} + 20 q^{20} + ( -37 + 7 \beta_{1} + 16 \beta_{2} - \beta_{3} ) q^{21} + ( 12 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{22} -23 q^{23} + ( 32 - 8 \beta_{1} ) q^{24} + 25 q^{25} + ( 30 + 8 \beta_{1} + 2 \beta_{3} ) q^{26} + ( 105 - 11 \beta_{1} - 23 \beta_{2} + 7 \beta_{3} ) q^{27} + ( 4 + 4 \beta_{1} + 8 \beta_{2} ) q^{28} + ( -34 + 9 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} ) q^{29} + ( 40 - 10 \beta_{1} ) q^{30} + ( -17 + 27 \beta_{1} + 15 \beta_{2} + 8 \beta_{3} ) q^{31} + 32 q^{32} + ( -25 + 14 \beta_{1} - 3 \beta_{2} - 7 \beta_{3} ) q^{33} + ( 36 - 4 \beta_{1} - 12 \beta_{2} - 6 \beta_{3} ) q^{34} + ( 5 + 5 \beta_{1} + 10 \beta_{2} ) q^{35} + ( 72 - 16 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{36} + ( -96 + 3 \beta_{1} + 33 \beta_{2} + \beta_{3} ) q^{37} + ( 78 + 16 \beta_{1} - 10 \beta_{2} + 6 \beta_{3} ) q^{38} + ( -43 - 27 \beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{39} + 40 q^{40} + ( 28 + 32 \beta_{1} - 15 \beta_{2} - 11 \beta_{3} ) q^{41} + ( -74 + 14 \beta_{1} + 32 \beta_{2} - 2 \beta_{3} ) q^{42} + ( -32 + 52 \beta_{1} - 18 \beta_{2} - 4 \beta_{3} ) q^{43} + ( 24 + 4 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} ) q^{44} + ( 90 - 20 \beta_{1} - 10 \beta_{2} + 5 \beta_{3} ) q^{45} -46 q^{46} + ( 106 - 16 \beta_{1} - 33 \beta_{2} + 4 \beta_{3} ) q^{47} + ( 64 - 16 \beta_{1} ) q^{48} + ( -17 - 42 \beta_{1} + 10 \beta_{2} + 9 \beta_{3} ) q^{49} + 50 q^{50} + ( 127 - 6 \beta_{1} - 43 \beta_{2} - 7 \beta_{3} ) q^{51} + ( 60 + 16 \beta_{1} + 4 \beta_{3} ) q^{52} + ( -114 + 37 \beta_{1} + 7 \beta_{2} + 11 \beta_{3} ) q^{53} + ( 210 - 22 \beta_{1} - 46 \beta_{2} + 14 \beta_{3} ) q^{54} + ( 30 + 5 \beta_{1} - 5 \beta_{2} - 10 \beta_{3} ) q^{55} + ( 8 + 8 \beta_{1} + 16 \beta_{2} ) q^{56} + ( -7 - 95 \beta_{1} - 22 \beta_{2} + \beta_{3} ) q^{57} + ( -68 + 18 \beta_{1} + 20 \beta_{2} - 10 \beta_{3} ) q^{58} + ( -76 - 79 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{59} + ( 80 - 20 \beta_{1} ) q^{60} + ( -66 - 47 \beta_{1} + 31 \beta_{2} ) q^{61} + ( -34 + 54 \beta_{1} + 30 \beta_{2} + 16 \beta_{3} ) q^{62} + ( -487 + 86 \beta_{1} + 73 \beta_{2} - 10 \beta_{3} ) q^{63} + 64 q^{64} + ( 75 + 20 \beta_{1} + 5 \beta_{3} ) q^{65} + ( -50 + 28 \beta_{1} - 6 \beta_{2} - 14 \beta_{3} ) q^{66} + ( -140 + 43 \beta_{1} + 33 \beta_{2} - 7 \beta_{3} ) q^{67} + ( 72 - 8 \beta_{1} - 24 \beta_{2} - 12 \beta_{3} ) q^{68} + ( -92 + 23 \beta_{1} ) q^{69} + ( 10 + 10 \beta_{1} + 20 \beta_{2} ) q^{70} + ( 68 - 72 \beta_{1} + 21 \beta_{2} + 23 \beta_{3} ) q^{71} + ( 144 - 32 \beta_{1} - 16 \beta_{2} + 8 \beta_{3} ) q^{72} + ( -140 - 44 \beta_{1} - 17 \beta_{2} + 16 \beta_{3} ) q^{73} + ( -192 + 6 \beta_{1} + 66 \beta_{2} + 2 \beta_{3} ) q^{74} + ( 100 - 25 \beta_{1} ) q^{75} + ( 156 + 32 \beta_{1} - 20 \beta_{2} + 12 \beta_{3} ) q^{76} + ( 19 - 9 \beta_{1} - 46 \beta_{2} - 7 \beta_{3} ) q^{77} + ( -86 - 54 \beta_{1} + 14 \beta_{2} - 2 \beta_{3} ) q^{78} + ( -220 - 92 \beta_{1} - 20 \beta_{2} - 28 \beta_{3} ) q^{79} + 80 q^{80} + ( 482 - 173 \beta_{1} - 136 \beta_{2} + 5 \beta_{3} ) q^{81} + ( 56 + 64 \beta_{1} - 30 \beta_{2} - 22 \beta_{3} ) q^{82} + ( -290 + 13 \beta_{1} - 47 \beta_{2} - 9 \beta_{3} ) q^{83} + ( -148 + 28 \beta_{1} + 64 \beta_{2} - 4 \beta_{3} ) q^{84} + ( 90 - 10 \beta_{1} - 30 \beta_{2} - 15 \beta_{3} ) q^{85} + ( -64 + 104 \beta_{1} - 36 \beta_{2} - 8 \beta_{3} ) q^{86} + ( -522 + 134 \beta_{1} + 93 \beta_{2} - 24 \beta_{3} ) q^{87} + ( 48 + 8 \beta_{1} - 8 \beta_{2} - 16 \beta_{3} ) q^{88} + ( -512 - 92 \beta_{1} + 30 \beta_{2} + 16 \beta_{3} ) q^{89} + ( 180 - 40 \beta_{1} - 20 \beta_{2} + 10 \beta_{3} ) q^{90} + ( 122 + 19 \beta_{1} + 19 \beta_{2} + 6 \beta_{3} ) q^{91} -92 q^{92} + ( -837 - 19 \beta_{1} + 151 \beta_{2} - 3 \beta_{3} ) q^{93} + ( 212 - 32 \beta_{1} - 66 \beta_{2} + 8 \beta_{3} ) q^{94} + ( 195 + 40 \beta_{1} - 25 \beta_{2} + 15 \beta_{3} ) q^{95} + ( 128 - 32 \beta_{1} ) q^{96} + ( -198 - 13 \beta_{1} + 55 \beta_{2} + 30 \beta_{3} ) q^{97} + ( -34 - 84 \beta_{1} + 20 \beta_{2} + 18 \beta_{3} ) q^{98} + ( -741 + 70 \beta_{1} + 41 \beta_{2} + 19 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{2} + 14q^{3} + 16q^{4} + 20q^{5} + 28q^{6} + 8q^{7} + 32q^{8} + 64q^{9} + O(q^{10}) \) \( 4q + 8q^{2} + 14q^{3} + 16q^{4} + 20q^{5} + 28q^{6} + 8q^{7} + 32q^{8} + 64q^{9} + 40q^{10} + 21q^{11} + 56q^{12} + 70q^{13} + 16q^{14} + 70q^{15} + 64q^{16} + 56q^{17} + 128q^{18} + 173q^{19} + 80q^{20} - 120q^{21} + 42q^{22} - 92q^{23} + 112q^{24} + 100q^{25} + 140q^{26} + 389q^{27} + 32q^{28} - 118q^{29} + 140q^{30} + 17q^{31} + 128q^{32} - 89q^{33} + 112q^{34} + 40q^{35} + 256q^{36} - 343q^{37} + 346q^{38} - 221q^{39} + 160q^{40} + 139q^{41} - 240q^{42} - 50q^{43} + 84q^{44} + 320q^{45} - 184q^{46} + 367q^{47} + 224q^{48} - 124q^{49} + 200q^{50} + 439q^{51} + 280q^{52} - 353q^{53} + 778q^{54} + 105q^{55} + 64q^{56} - 238q^{57} - 236q^{58} - 453q^{59} + 280q^{60} - 327q^{61} + 34q^{62} - 1723q^{63} + 256q^{64} + 350q^{65} - 178q^{66} - 455q^{67} + 224q^{68} - 322q^{69} + 80q^{70} + 195q^{71} + 512q^{72} - 633q^{73} - 686q^{74} + 350q^{75} + 692q^{76} - 2q^{77} - 442q^{78} - 1140q^{79} + 320q^{80} + 1456q^{81} + 278q^{82} - 1199q^{83} - 480q^{84} + 280q^{85} - 100q^{86} - 1775q^{87} + 168q^{88} - 2170q^{89} + 640q^{90} + 557q^{91} - 368q^{92} - 3241q^{93} + 734q^{94} + 865q^{95} + 448q^{96} - 703q^{97} - 248q^{98} - 2745q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 60 x^{2} - 45 x + 108\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 5 \nu^{2} - 45 \nu + 54 \)\()/9\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} - \nu^{2} - 126 \nu - 153 \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 2 \beta_{2} + 4 \beta_{1} + 29\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} - \beta_{2} + 65 \beta_{1} + 91\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
9.04090
1.01192
−1.92711
−6.12571
2.00000 −5.04090 4.00000 5.00000 −10.0818 5.03071 8.00000 −1.58932 10.0000
1.2 2.00000 2.98808 4.00000 5.00000 5.97615 2.98517 8.00000 −18.0714 10.0000
1.3 2.00000 5.92711 4.00000 5.00000 11.8542 24.6272 8.00000 8.13066 10.0000
1.4 2.00000 10.1257 4.00000 5.00000 20.2514 −24.6431 8.00000 75.5301 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.4.a.j 4
3.b odd 2 1 2070.4.a.bg 4
4.b odd 2 1 1840.4.a.k 4
5.b even 2 1 1150.4.a.n 4
5.c odd 4 2 1150.4.b.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.j 4 1.a even 1 1 trivial
1150.4.a.n 4 5.b even 2 1
1150.4.b.o 8 5.c odd 4 2
1840.4.a.k 4 4.b odd 2 1
2070.4.a.bg 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 14 T_{3}^{3} + 12 T_{3}^{2} + 365 T_{3} - 904 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{4} \)
$3$ \( -904 + 365 T + 12 T^{2} - 14 T^{3} + T^{4} \)
$5$ \( ( -5 + T )^{4} \)
$7$ \( -9114 + 4865 T - 592 T^{2} - 8 T^{3} + T^{4} \)
$11$ \( -164232 - 43134 T - 2605 T^{2} - 21 T^{3} + T^{4} \)
$13$ \( -46062 + 11409 T + 286 T^{2} - 70 T^{3} + T^{4} \)
$17$ \( 7970400 + 264771 T - 8532 T^{2} - 56 T^{3} + T^{4} \)
$19$ \( 9983784 + 626822 T - 2059 T^{2} - 173 T^{3} + T^{4} \)
$23$ \( ( 23 + T )^{4} \)
$29$ \( 44611452 - 118638 T - 35805 T^{2} + 118 T^{3} + T^{4} \)
$31$ \( 2678191911 - 120799 T - 107992 T^{2} - 17 T^{3} + T^{4} \)
$37$ \( -2095186944 - 32938816 T - 99964 T^{2} + 343 T^{3} + T^{4} \)
$41$ \( 4789051317 + 10257183 T - 157626 T^{2} - 139 T^{3} + T^{4} \)
$43$ \( 5363873792 - 14655968 T - 207336 T^{2} + 50 T^{3} + T^{4} \)
$47$ \( 668142000 + 7331796 T - 138234 T^{2} - 367 T^{3} + T^{4} \)
$53$ \( -289523592 - 36583380 T - 112122 T^{2} + 353 T^{3} + T^{4} \)
$59$ \( 8963853984 - 16917012 T - 308646 T^{2} + 453 T^{3} + T^{4} \)
$61$ \( -1898667392 - 41660040 T - 204755 T^{2} + 327 T^{3} + T^{4} \)
$67$ \( -1366509568 + 33923056 T - 255912 T^{2} + 455 T^{3} + T^{4} \)
$71$ \( 106065123651 + 103060557 T - 706330 T^{2} - 195 T^{3} + T^{4} \)
$73$ \( -15845099784 - 152518736 T - 226884 T^{2} + 633 T^{3} + T^{4} \)
$79$ \( 60635801088 - 326815808 T - 526896 T^{2} + 1140 T^{3} + T^{4} \)
$83$ \( 103460976 + 9039492 T + 218008 T^{2} + 1199 T^{3} + T^{4} \)
$89$ \( -5919819264 - 213089088 T + 986248 T^{2} + 2170 T^{3} + T^{4} \)
$97$ \( -22808262684 - 296475040 T - 708127 T^{2} + 703 T^{3} + T^{4} \)
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