Properties

Label 230.4.a.g
Level $230$
Weight $4$
Character orbit 230.a
Self dual yes
Analytic conductor $13.570$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.318165.1
Defining polynomial: \(x^{3} - x^{2} - 45 x + 60\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{2} -\beta_{1} q^{3} + 4 q^{4} + 5 q^{5} + 2 \beta_{1} q^{6} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{7} -8 q^{8} + ( 4 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q -2 q^{2} -\beta_{1} q^{3} + 4 q^{4} + 5 q^{5} + 2 \beta_{1} q^{6} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{7} -8 q^{8} + ( 4 - \beta_{1} + \beta_{2} ) q^{9} -10 q^{10} + ( 10 - 5 \beta_{1} - 2 \beta_{2} ) q^{11} -4 \beta_{1} q^{12} + ( 27 - 3 \beta_{1} + 3 \beta_{2} ) q^{13} + ( -2 - 6 \beta_{1} + 2 \beta_{2} ) q^{14} -5 \beta_{1} q^{15} + 16 q^{16} + ( 46 - 11 \beta_{1} ) q^{17} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{18} + ( -59 - 9 \beta_{1} - \beta_{2} ) q^{19} + 20 q^{20} + ( -91 + 14 \beta_{1} - \beta_{2} ) q^{21} + ( -20 + 10 \beta_{1} + 4 \beta_{2} ) q^{22} + 23 q^{23} + 8 \beta_{1} q^{24} + 25 q^{25} + ( -54 + 6 \beta_{1} - 6 \beta_{2} ) q^{26} + ( 29 + 10 \beta_{1} - \beta_{2} ) q^{27} + ( 4 + 12 \beta_{1} - 4 \beta_{2} ) q^{28} + ( 122 - 18 \beta_{1} + 4 \beta_{2} ) q^{29} + 10 \beta_{1} q^{30} + ( 125 + 17 \beta_{1} - 5 \beta_{2} ) q^{31} -32 q^{32} + ( 159 + 9 \beta_{1} + 9 \beta_{2} ) q^{33} + ( -92 + 22 \beta_{1} ) q^{34} + ( 5 + 15 \beta_{1} - 5 \beta_{2} ) q^{35} + ( 16 - 4 \beta_{1} + 4 \beta_{2} ) q^{36} + ( 324 + 8 \beta_{1} + 2 \beta_{2} ) q^{37} + ( 118 + 18 \beta_{1} + 2 \beta_{2} ) q^{38} + ( 87 - 66 \beta_{1} - 3 \beta_{2} ) q^{39} -40 q^{40} + ( -204 + 47 \beta_{1} + 10 \beta_{2} ) q^{41} + ( 182 - 28 \beta_{1} + 2 \beta_{2} ) q^{42} + ( 256 + 40 \beta_{1} - 4 \beta_{2} ) q^{43} + ( 40 - 20 \beta_{1} - 8 \beta_{2} ) q^{44} + ( 20 - 5 \beta_{1} + 5 \beta_{2} ) q^{45} -46 q^{46} + ( -106 + 42 \beta_{1} - 6 \beta_{2} ) q^{47} -16 \beta_{1} q^{48} + ( 303 - 49 \beta_{1} - 18 \beta_{2} ) q^{49} -50 q^{50} + ( 341 - 57 \beta_{1} + 11 \beta_{2} ) q^{51} + ( 108 - 12 \beta_{1} + 12 \beta_{2} ) q^{52} + ( 186 - 60 \beta_{1} - 12 \beta_{2} ) q^{53} + ( -58 - 20 \beta_{1} + 2 \beta_{2} ) q^{54} + ( 50 - 25 \beta_{1} - 10 \beta_{2} ) q^{55} + ( -8 - 24 \beta_{1} + 8 \beta_{2} ) q^{56} + ( 281 + 62 \beta_{1} + 11 \beta_{2} ) q^{57} + ( -244 + 36 \beta_{1} - 8 \beta_{2} ) q^{58} + ( 40 + 34 \beta_{1} + 12 \beta_{2} ) q^{59} -20 \beta_{1} q^{60} + ( -30 + 49 \beta_{1} + 8 \beta_{2} ) q^{61} + ( -250 - 34 \beta_{1} + 10 \beta_{2} ) q^{62} + ( -459 + 36 \beta_{1} + 15 \beta_{2} ) q^{63} + 64 q^{64} + ( 135 - 15 \beta_{1} + 15 \beta_{2} ) q^{65} + ( -318 - 18 \beta_{1} - 18 \beta_{2} ) q^{66} + ( 528 + 20 \beta_{1} - 12 \beta_{2} ) q^{67} + ( 184 - 44 \beta_{1} ) q^{68} -23 \beta_{1} q^{69} + ( -10 - 30 \beta_{1} + 10 \beta_{2} ) q^{70} + ( -132 - 67 \beta_{1} + 8 \beta_{2} ) q^{71} + ( -32 + 8 \beta_{1} - 8 \beta_{2} ) q^{72} + ( -284 + 30 \beta_{1} - 42 \beta_{2} ) q^{73} + ( -648 - 16 \beta_{1} - 4 \beta_{2} ) q^{74} -25 \beta_{1} q^{75} + ( -236 - 36 \beta_{1} - 4 \beta_{2} ) q^{76} + ( 299 + 80 \beta_{1} - 55 \beta_{2} ) q^{77} + ( -174 + 132 \beta_{1} + 6 \beta_{2} ) q^{78} + ( -272 - 28 \beta_{1} + 16 \beta_{2} ) q^{79} + 80 q^{80} + ( -416 + 20 \beta_{1} - 35 \beta_{2} ) q^{81} + ( 408 - 94 \beta_{1} - 20 \beta_{2} ) q^{82} + ( -106 + 52 \beta_{1} + 22 \beta_{2} ) q^{83} + ( -364 + 56 \beta_{1} - 4 \beta_{2} ) q^{84} + ( 230 - 55 \beta_{1} ) q^{85} + ( -512 - 80 \beta_{1} + 8 \beta_{2} ) q^{86} + ( 550 - 188 \beta_{1} + 10 \beta_{2} ) q^{87} + ( -80 + 40 \beta_{1} + 16 \beta_{2} ) q^{88} + ( -88 + 200 \beta_{1} + 26 \beta_{2} ) q^{89} + ( -40 + 10 \beta_{1} - 10 \beta_{2} ) q^{90} + ( -1362 + 153 \beta_{1} + 30 \beta_{2} ) q^{91} + 92 q^{92} + ( -517 - 48 \beta_{1} - 7 \beta_{2} ) q^{93} + ( 212 - 84 \beta_{1} + 12 \beta_{2} ) q^{94} + ( -295 - 45 \beta_{1} - 5 \beta_{2} ) q^{95} + 32 \beta_{1} q^{96} + ( 462 - 25 \beta_{1} + 40 \beta_{2} ) q^{97} + ( -606 + 98 \beta_{1} + 36 \beta_{2} ) q^{98} + ( -567 - 123 \beta_{1} + 27 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 6q^{2} - q^{3} + 12q^{4} + 15q^{5} + 2q^{6} + 7q^{7} - 24q^{8} + 10q^{9} + O(q^{10}) \) \( 3q - 6q^{2} - q^{3} + 12q^{4} + 15q^{5} + 2q^{6} + 7q^{7} - 24q^{8} + 10q^{9} - 30q^{10} + 27q^{11} - 4q^{12} + 75q^{13} - 14q^{14} - 5q^{15} + 48q^{16} + 127q^{17} - 20q^{18} - 185q^{19} + 60q^{20} - 258q^{21} - 54q^{22} + 69q^{23} + 8q^{24} + 75q^{25} - 150q^{26} + 98q^{27} + 28q^{28} + 344q^{29} + 10q^{30} + 397q^{31} - 96q^{32} + 477q^{33} - 254q^{34} + 35q^{35} + 40q^{36} + 978q^{37} + 370q^{38} + 198q^{39} - 120q^{40} - 575q^{41} + 516q^{42} + 812q^{43} + 108q^{44} + 50q^{45} - 138q^{46} - 270q^{47} - 16q^{48} + 878q^{49} - 150q^{50} + 955q^{51} + 300q^{52} + 510q^{53} - 196q^{54} + 135q^{55} - 56q^{56} + 894q^{57} - 688q^{58} + 142q^{59} - 20q^{60} - 49q^{61} - 794q^{62} - 1356q^{63} + 192q^{64} + 375q^{65} - 954q^{66} + 1616q^{67} + 508q^{68} - 23q^{69} - 70q^{70} - 471q^{71} - 80q^{72} - 780q^{73} - 1956q^{74} - 25q^{75} - 740q^{76} + 1032q^{77} - 396q^{78} - 860q^{79} + 240q^{80} - 1193q^{81} + 1150q^{82} - 288q^{83} - 1032q^{84} + 635q^{85} - 1624q^{86} + 1452q^{87} - 216q^{88} - 90q^{89} - 100q^{90} - 3963q^{91} + 276q^{92} - 1592q^{93} + 540q^{94} - 925q^{95} + 32q^{96} + 1321q^{97} - 1756q^{98} - 1851q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 31 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 31\)

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
6.50182
1.34735
−6.84916
−2.00000 −6.50182 4.00000 5.00000 13.0036 2.73001 −8.00000 15.2736 −10.0000
1.2 −2.00000 −1.34735 4.00000 5.00000 2.69469 32.8794 −8.00000 −25.1847 −10.0000
1.3 −2.00000 6.84916 4.00000 5.00000 −13.6983 −28.6094 −8.00000 19.9110 −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.g 3
3.b odd 2 1 2070.4.a.ba 3
4.b odd 2 1 1840.4.a.j 3
5.b even 2 1 1150.4.a.m 3
5.c odd 4 2 1150.4.b.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.g 3 1.a even 1 1 trivial
1150.4.a.m 3 5.b even 2 1
1150.4.b.l 6 5.c odd 4 2
1840.4.a.j 3 4.b odd 2 1
2070.4.a.ba 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + T_{3}^{2} - 45 T_{3} - 60 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T )^{3} \)
$3$ \( -60 - 45 T + T^{2} + T^{3} \)
$5$ \( ( -5 + T )^{3} \)
$7$ \( 2568 - 929 T - 7 T^{2} + T^{3} \)
$11$ \( 89388 - 3399 T - 27 T^{2} + T^{3} \)
$13$ \( 275238 - 3663 T - 75 T^{2} + T^{3} \)
$17$ \( 96550 - 109 T - 127 T^{2} + T^{3} \)
$19$ \( 37596 + 7003 T + 185 T^{2} + T^{3} \)
$23$ \( ( -23 + T )^{3} \)
$29$ \( 291288 + 16552 T - 344 T^{2} + T^{3} \)
$31$ \( 1547080 + 26165 T - 397 T^{2} + T^{3} \)
$37$ \( -33005536 + 313320 T - 978 T^{2} + T^{3} \)
$41$ \( -50953878 - 56243 T + 575 T^{2} + T^{3} \)
$43$ \( 10161920 + 140480 T - 812 T^{2} + T^{3} \)
$47$ \( 3184000 - 72660 T + 270 T^{2} + T^{3} \)
$53$ \( 89503704 - 172692 T - 510 T^{2} + T^{3} \)
$59$ \( -9906704 - 136780 T - 142 T^{2} + T^{3} \)
$61$ \( -23572158 - 151973 T + 49 T^{2} + T^{3} \)
$67$ \( -111600960 + 771840 T - 1616 T^{2} + T^{3} \)
$71$ \( -75603760 - 158325 T + 471 T^{2} + T^{3} \)
$73$ \( -672863896 - 851712 T + 780 T^{2} + T^{3} \)
$79$ \( -8296704 + 68208 T + 860 T^{2} + T^{3} \)
$83$ \( -106176592 - 397404 T + 288 T^{2} + T^{3} \)
$89$ \( -1109897568 - 2291928 T + 90 T^{2} + T^{3} \)
$97$ \( 689377182 - 368453 T - 1321 T^{2} + T^{3} \)
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