Properties

Label 230.4.a.i
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: \(x^{4} - 84 x^{2} - 11 x + 1242\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( 1 - \beta_{1} ) q^{3} + 4 q^{4} -5 q^{5} + ( 2 - 2 \beta_{1} ) q^{6} + ( 6 - \beta_{1} - 2 \beta_{2} ) q^{7} + 8 q^{8} + ( 16 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + 2 q^{2} + ( 1 - \beta_{1} ) q^{3} + 4 q^{4} -5 q^{5} + ( 2 - 2 \beta_{1} ) q^{6} + ( 6 - \beta_{1} - 2 \beta_{2} ) q^{7} + 8 q^{8} + ( 16 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} -10 q^{10} + ( 24 + \beta_{1} + 3 \beta_{2} ) q^{11} + ( 4 - 4 \beta_{1} ) q^{12} + ( 9 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{13} + ( 12 - 2 \beta_{1} - 4 \beta_{2} ) q^{14} + ( -5 + 5 \beta_{1} ) q^{15} + 16 q^{16} + ( 26 + 4 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{17} + ( 32 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{18} + ( 48 + 6 \beta_{2} - \beta_{3} ) q^{19} -20 q^{20} + ( 76 - 5 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} ) q^{21} + ( 48 + 2 \beta_{1} + 6 \beta_{2} ) q^{22} + 23 q^{23} + ( 8 - 8 \beta_{1} ) q^{24} + 25 q^{25} + ( 18 + 4 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{26} + ( 61 + \beta_{1} - 10 \beta_{2} + 5 \beta_{3} ) q^{27} + ( 24 - 4 \beta_{1} - 8 \beta_{2} ) q^{28} + ( 69 + 23 \beta_{1} - 11 \beta_{2} + 7 \beta_{3} ) q^{29} + ( -10 + 10 \beta_{1} ) q^{30} + ( -55 + 13 \beta_{1} - 11 \beta_{2} - 2 \beta_{3} ) q^{31} + 32 q^{32} + ( -60 - 26 \beta_{1} - 10 \beta_{2} - 7 \beta_{3} ) q^{33} + ( 52 + 8 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} ) q^{34} + ( -30 + 5 \beta_{1} + 10 \beta_{2} ) q^{35} + ( 64 - 8 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{36} + ( 2 + 17 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} ) q^{37} + ( 96 + 12 \beta_{2} - 2 \beta_{3} ) q^{38} + ( -95 + 13 \beta_{1} + 22 \beta_{2} - 19 \beta_{3} ) q^{39} -40 q^{40} + ( -91 - 16 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} ) q^{41} + ( 152 - 10 \beta_{1} + 14 \beta_{2} + 10 \beta_{3} ) q^{42} + ( -32 - 4 \beta_{1} - 12 \beta_{2} + 16 \beta_{3} ) q^{43} + ( 96 + 4 \beta_{1} + 12 \beta_{2} ) q^{44} + ( -80 + 10 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{45} + 46 q^{46} + ( -89 + 50 \beta_{1} + 13 \beta_{2} + 6 \beta_{3} ) q^{47} + ( 16 - 16 \beta_{1} ) q^{48} + ( 143 + 32 \beta_{1} - 19 \beta_{2} + 9 \beta_{3} ) q^{49} + 50 q^{50} + ( -74 - 10 \beta_{1} + 20 \beta_{2} + \beta_{3} ) q^{51} + ( 36 + 8 \beta_{1} + 4 \beta_{2} - 12 \beta_{3} ) q^{52} + ( 14 - 41 \beta_{1} + 38 \beta_{2} + 3 \beta_{3} ) q^{53} + ( 122 + 2 \beta_{1} - 20 \beta_{2} + 10 \beta_{3} ) q^{54} + ( -120 - 5 \beta_{1} - 15 \beta_{2} ) q^{55} + ( 48 - 8 \beta_{1} - 16 \beta_{2} ) q^{56} + ( -38 - 47 \beta_{1} - 9 \beta_{2} - 17 \beta_{3} ) q^{57} + ( 138 + 46 \beta_{1} - 22 \beta_{2} + 14 \beta_{3} ) q^{58} + ( 53 \beta_{1} + 24 \beta_{2} - 9 \beta_{3} ) q^{59} + ( -20 + 20 \beta_{1} ) q^{60} + ( -80 - 51 \beta_{1} - 23 \beta_{2} - 10 \beta_{3} ) q^{61} + ( -110 + 26 \beta_{1} - 22 \beta_{2} - 4 \beta_{3} ) q^{62} + ( 36 - 96 \beta_{1} - 7 \beta_{2} + 16 \beta_{3} ) q^{63} + 64 q^{64} + ( -45 - 10 \beta_{1} - 5 \beta_{2} + 15 \beta_{3} ) q^{65} + ( -120 - 52 \beta_{1} - 20 \beta_{2} - 14 \beta_{3} ) q^{66} + ( 194 - 35 \beta_{1} + 52 \beta_{2} + \beta_{3} ) q^{67} + ( 104 + 16 \beta_{1} - 20 \beta_{2} - 4 \beta_{3} ) q^{68} + ( 23 - 23 \beta_{1} ) q^{69} + ( -60 + 10 \beta_{1} + 20 \beta_{2} ) q^{70} + ( -319 + 56 \beta_{1} - 18 \beta_{2} + \beta_{3} ) q^{71} + ( 128 - 16 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{72} + ( 129 - 102 \beta_{1} - 3 \beta_{2} - 10 \beta_{3} ) q^{73} + ( 4 + 34 \beta_{1} - 8 \beta_{2} - 14 \beta_{3} ) q^{74} + ( 25 - 25 \beta_{1} ) q^{75} + ( 192 + 24 \beta_{2} - 4 \beta_{3} ) q^{76} + ( -496 - 83 \beta_{1} - 33 \beta_{2} - 11 \beta_{3} ) q^{77} + ( -190 + 26 \beta_{1} + 44 \beta_{2} - 38 \beta_{3} ) q^{78} + ( 278 + 80 \beta_{1} - 62 \beta_{2} + 28 \beta_{3} ) q^{79} -80 q^{80} + ( -263 - 31 \beta_{1} - 43 \beta_{2} + 17 \beta_{3} ) q^{81} + ( -182 - 32 \beta_{1} + 16 \beta_{2} + 6 \beta_{3} ) q^{82} + ( -320 - 29 \beta_{1} + 60 \beta_{2} - 37 \beta_{3} ) q^{83} + ( 304 - 20 \beta_{1} + 28 \beta_{2} + 20 \beta_{3} ) q^{84} + ( -130 - 20 \beta_{1} + 25 \beta_{2} + 5 \beta_{3} ) q^{85} + ( -64 - 8 \beta_{1} - 24 \beta_{2} + 32 \beta_{3} ) q^{86} + ( -729 - 84 \beta_{1} - 53 \beta_{2} + 34 \beta_{3} ) q^{87} + ( 192 + 8 \beta_{1} + 24 \beta_{2} ) q^{88} + ( -228 + 60 \beta_{2} - 12 \beta_{3} ) q^{89} + ( -160 + 20 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} ) q^{90} + ( -246 + 131 \beta_{1} - 25 \beta_{2} - 36 \beta_{3} ) q^{91} + 92 q^{92} + ( -451 + 93 \beta_{1} + 38 \beta_{2} - \beta_{3} ) q^{93} + ( -178 + 100 \beta_{1} + 26 \beta_{2} + 12 \beta_{3} ) q^{94} + ( -240 - 30 \beta_{2} + 5 \beta_{3} ) q^{95} + ( 32 - 32 \beta_{1} ) q^{96} + ( 38 + 63 \beta_{1} + 59 \beta_{2} + 44 \beta_{3} ) q^{97} + ( 286 + 64 \beta_{1} - 38 \beta_{2} + 18 \beta_{3} ) q^{98} + ( 510 + 66 \beta_{1} + 38 \beta_{2} + 11 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} - 20 q^{5} + 8 q^{6} + 26 q^{7} + 32 q^{8} + 64 q^{9} + O(q^{10}) \) \( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} - 20 q^{5} + 8 q^{6} + 26 q^{7} + 32 q^{8} + 64 q^{9} - 40 q^{10} + 93 q^{11} + 16 q^{12} + 32 q^{13} + 52 q^{14} - 20 q^{15} + 64 q^{16} + 108 q^{17} + 128 q^{18} + 185 q^{19} - 80 q^{20} + 302 q^{21} + 186 q^{22} + 92 q^{23} + 32 q^{24} + 100 q^{25} + 64 q^{26} + 259 q^{27} + 104 q^{28} + 294 q^{29} - 40 q^{30} - 211 q^{31} + 128 q^{32} - 237 q^{33} + 216 q^{34} - 130 q^{35} + 256 q^{36} + 5 q^{37} + 370 q^{38} - 421 q^{39} - 160 q^{40} - 369 q^{41} + 604 q^{42} - 100 q^{43} + 372 q^{44} - 320 q^{45} + 184 q^{46} - 363 q^{47} + 64 q^{48} + 600 q^{49} + 200 q^{50} - 315 q^{51} + 128 q^{52} + 21 q^{53} + 518 q^{54} - 465 q^{55} + 208 q^{56} - 160 q^{57} + 588 q^{58} - 33 q^{59} - 80 q^{60} - 307 q^{61} - 422 q^{62} + 167 q^{63} + 256 q^{64} - 160 q^{65} - 474 q^{66} + 725 q^{67} + 432 q^{68} + 92 q^{69} - 260 q^{70} - 1257 q^{71} + 512 q^{72} + 509 q^{73} + 10 q^{74} + 100 q^{75} + 740 q^{76} - 1962 q^{77} - 842 q^{78} + 1202 q^{79} - 320 q^{80} - 992 q^{81} - 738 q^{82} - 1377 q^{83} + 1208 q^{84} - 540 q^{85} - 200 q^{86} - 2829 q^{87} + 744 q^{88} - 984 q^{89} - 640 q^{90} - 995 q^{91} + 368 q^{92} - 1843 q^{93} - 726 q^{94} - 925 q^{95} + 128 q^{96} + 137 q^{97} + 1200 q^{98} + 2013 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 84 x^{2} - 11 x + 1242\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 50 \nu - 96 \)\()/15\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 13 \nu^{2} + 50 \nu - 534 \)\()/15\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 42\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 13 \beta_{2} + 50 \beta_{1} + 12\)

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
8.16920
4.26018
−4.50148
−7.92791
2.00000 −7.16920 4.00000 −5.00000 −14.3384 −25.3945 8.00000 24.3974 −10.0000
1.2 2.00000 −3.26018 4.00000 −5.00000 −6.52037 27.7921 8.00000 −16.3712 −10.0000
1.3 2.00000 5.50148 4.00000 −5.00000 11.0030 0.0500526 8.00000 3.26627 −10.0000
1.4 2.00000 8.92791 4.00000 −5.00000 17.8558 23.5524 8.00000 52.7075 −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.i 4
3.b odd 2 1 2070.4.a.bi 4
4.b odd 2 1 1840.4.a.l 4
5.b even 2 1 1150.4.a.o 4
5.c odd 4 2 1150.4.b.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.i 4 1.a even 1 1 trivial
1150.4.a.o 4 5.b even 2 1
1150.4.b.m 8 5.c odd 4 2
1840.4.a.l 4 4.b odd 2 1
2070.4.a.bi 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4 T_{3}^{3} - 78 T_{3}^{2} + 175 T_{3} + 1148 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{4} \)
$3$ \( 1148 + 175 T - 78 T^{2} - 4 T^{3} + T^{4} \)
$5$ \( ( 5 + T )^{4} \)
$7$ \( -832 + 16655 T - 648 T^{2} - 26 T^{3} + T^{4} \)
$11$ \( -41700 + 23452 T + 1401 T^{2} - 93 T^{3} + T^{4} \)
$13$ \( 682562 + 263543 T - 7812 T^{2} - 32 T^{3} + T^{4} \)
$17$ \( 96120 + 9129 T - 1392 T^{2} - 108 T^{3} + T^{4} \)
$19$ \( 1404820 + 260908 T + 5787 T^{2} - 185 T^{3} + T^{4} \)
$23$ \( ( -23 + T )^{4} \)
$29$ \( -1735042500 + 23620150 T - 55221 T^{2} - 294 T^{3} + T^{4} \)
$31$ \( -311259365 - 5793467 T - 15780 T^{2} + 211 T^{3} + T^{4} \)
$37$ \( -14187104 + 5707840 T - 67782 T^{2} - 5 T^{3} + T^{4} \)
$41$ \( -114199911 - 2093721 T + 15870 T^{2} + 369 T^{3} + T^{4} \)
$43$ \( 93158400 - 14280960 T - 235824 T^{2} + 100 T^{3} + T^{4} \)
$47$ \( -9732540576 - 119094548 T - 251790 T^{2} + 363 T^{3} + T^{4} \)
$53$ \( -8104397736 + 99349844 T - 326490 T^{2} - 21 T^{3} + T^{4} \)
$59$ \( -623752800 + 63983540 T - 477930 T^{2} + 33 T^{3} + T^{4} \)
$61$ \( -5277943032 + 89074878 T - 420069 T^{2} + 307 T^{3} + T^{4} \)
$67$ \( -49731103520 + 302222200 T - 290226 T^{2} - 725 T^{3} + T^{4} \)
$71$ \( -3703975749 - 49677283 T + 326922 T^{2} + 1257 T^{3} + T^{4} \)
$73$ \( 80640087688 + 390630616 T - 858096 T^{2} - 509 T^{3} + T^{4} \)
$79$ \( -498954065920 + 1631351936 T - 968472 T^{2} - 1202 T^{3} + T^{4} \)
$83$ \( -424764340752 - 1684887452 T - 963084 T^{2} + 1377 T^{3} + T^{4} \)
$89$ \( 92383027200 - 305567424 T - 374976 T^{2} + 984 T^{3} + T^{4} \)
$97$ \( 734679597128 - 882068230 T - 2783397 T^{2} - 137 T^{3} + T^{4} \)
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