Properties

Label 230.6.a.c
Level $230$
Weight $6$
Character orbit 230.a
Self dual yes
Analytic conductor $36.888$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.8882785570\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.27980.1
Defining polynomial: \(x^{3} - 47 x - 106\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 q^{2} + ( -9 + \beta_{2} ) q^{3} + 16 q^{4} + 25 q^{5} + ( 36 - 4 \beta_{2} ) q^{6} + ( -1 + 7 \beta_{1} + 4 \beta_{2} ) q^{7} -64 q^{8} + ( -18 - 6 \beta_{1} - 17 \beta_{2} ) q^{9} +O(q^{10})\) \( q -4 q^{2} + ( -9 + \beta_{2} ) q^{3} + 16 q^{4} + 25 q^{5} + ( 36 - 4 \beta_{2} ) q^{6} + ( -1 + 7 \beta_{1} + 4 \beta_{2} ) q^{7} -64 q^{8} + ( -18 - 6 \beta_{1} - 17 \beta_{2} ) q^{9} -100 q^{10} + ( 57 - 19 \beta_{1} + 7 \beta_{2} ) q^{11} + ( -144 + 16 \beta_{2} ) q^{12} + ( -90 + 3 \beta_{1} + 34 \beta_{2} ) q^{13} + ( 4 - 28 \beta_{1} - 16 \beta_{2} ) q^{14} + ( -225 + 25 \beta_{2} ) q^{15} + 256 q^{16} + ( -538 + 28 \beta_{1} - 69 \beta_{2} ) q^{17} + ( 72 + 24 \beta_{1} + 68 \beta_{2} ) q^{18} + ( -13 - 17 \beta_{1} - 155 \beta_{2} ) q^{19} + 400 q^{20} + ( 207 - 87 \beta_{1} - 117 \beta_{2} ) q^{21} + ( -228 + 76 \beta_{1} - 28 \beta_{2} ) q^{22} + 529 q^{23} + ( 576 - 64 \beta_{2} ) q^{24} + 625 q^{25} + ( 360 - 12 \beta_{1} - 136 \beta_{2} ) q^{26} + ( 225 + 156 \beta_{1} - 53 \beta_{2} ) q^{27} + ( -16 + 112 \beta_{1} + 64 \beta_{2} ) q^{28} + ( 2291 + 202 \beta_{1} + 28 \beta_{2} ) q^{29} + ( 900 - 100 \beta_{2} ) q^{30} + ( 2080 - 183 \beta_{1} - 121 \beta_{2} ) q^{31} -1024 q^{32} + ( 1521 + 129 \beta_{1} + 229 \beta_{2} ) q^{33} + ( 2152 - 112 \beta_{1} + 276 \beta_{2} ) q^{34} + ( -25 + 175 \beta_{1} + 100 \beta_{2} ) q^{35} + ( -288 - 96 \beta_{1} - 272 \beta_{2} ) q^{36} + ( 1729 - 601 \beta_{1} - 2 \beta_{2} ) q^{37} + ( 52 + 68 \beta_{1} + 620 \beta_{2} ) q^{38} + ( 5544 - 231 \beta_{1} - 398 \beta_{2} ) q^{39} -1600 q^{40} + ( 2167 - 60 \beta_{1} + 226 \beta_{2} ) q^{41} + ( -828 + 348 \beta_{1} + 468 \beta_{2} ) q^{42} + ( -7351 - 357 \beta_{1} - 935 \beta_{2} ) q^{43} + ( 912 - 304 \beta_{1} + 112 \beta_{2} ) q^{44} + ( -450 - 150 \beta_{1} - 425 \beta_{2} ) q^{45} -2116 q^{46} + ( -4631 + 282 \beta_{1} + 163 \beta_{2} ) q^{47} + ( -2304 + 256 \beta_{2} ) q^{48} + ( -3855 + 478 \beta_{1} - 223 \beta_{2} ) q^{49} -2500 q^{50} + ( -6606 + 162 \beta_{1} - 322 \beta_{2} ) q^{51} + ( -1440 + 48 \beta_{1} + 544 \beta_{2} ) q^{52} + ( -12678 + 282 \beta_{1} + 7 \beta_{2} ) q^{53} + ( -900 - 624 \beta_{1} + 212 \beta_{2} ) q^{54} + ( 1425 - 475 \beta_{1} + 175 \beta_{2} ) q^{55} + ( 64 - 448 \beta_{1} - 256 \beta_{2} ) q^{56} + ( -21285 + 1083 \beta_{1} + 1431 \beta_{2} ) q^{57} + ( -9164 - 808 \beta_{1} - 112 \beta_{2} ) q^{58} + ( -14446 + 1588 \beta_{1} - 377 \beta_{2} ) q^{59} + ( -3600 + 400 \beta_{2} ) q^{60} + ( -17465 - 441 \beta_{1} + 1031 \beta_{2} ) q^{61} + ( -8320 + 732 \beta_{1} + 484 \beta_{2} ) q^{62} + ( -13770 - 216 \beta_{1} + 1215 \beta_{2} ) q^{63} + 4096 q^{64} + ( -2250 + 75 \beta_{1} + 850 \beta_{2} ) q^{65} + ( -6084 - 516 \beta_{1} - 916 \beta_{2} ) q^{66} + ( -17325 - 383 \beta_{1} + 84 \beta_{2} ) q^{67} + ( -8608 + 448 \beta_{1} - 1104 \beta_{2} ) q^{68} + ( -4761 + 529 \beta_{2} ) q^{69} + ( 100 - 700 \beta_{1} - 400 \beta_{2} ) q^{70} + ( -8902 + 43 \beta_{1} + 1039 \beta_{2} ) q^{71} + ( 1152 + 384 \beta_{1} + 1088 \beta_{2} ) q^{72} + ( -36704 + 1599 \beta_{1} + 1580 \beta_{2} ) q^{73} + ( -6916 + 2404 \beta_{1} + 8 \beta_{2} ) q^{74} + ( -5625 + 625 \beta_{2} ) q^{75} + ( -208 - 272 \beta_{1} - 2480 \beta_{2} ) q^{76} + ( -31674 - 1346 \beta_{1} - 624 \beta_{2} ) q^{77} + ( -22176 + 924 \beta_{1} + 1592 \beta_{2} ) q^{78} + ( 813 - 579 \beta_{1} - 2663 \beta_{2} ) q^{79} + 6400 q^{80} + ( -13707 + 372 \beta_{1} + 2908 \beta_{2} ) q^{81} + ( -8668 + 240 \beta_{1} - 904 \beta_{2} ) q^{82} + ( -19222 - 4036 \beta_{1} - 3405 \beta_{2} ) q^{83} + ( 3312 - 1392 \beta_{1} - 1872 \beta_{2} ) q^{84} + ( -13450 + 700 \beta_{1} - 1725 \beta_{2} ) q^{85} + ( 29404 + 1428 \beta_{1} + 3740 \beta_{2} ) q^{86} + ( -27495 - 1986 \beta_{1} - 357 \beta_{2} ) q^{87} + ( -3648 + 1216 \beta_{1} - 448 \beta_{2} ) q^{88} + ( -28557 - 1835 \beta_{1} - 7701 \beta_{2} ) q^{89} + ( 1800 + 600 \beta_{1} + 1700 \beta_{2} ) q^{90} + ( 12033 - 1197 \beta_{1} - 3069 \beta_{2} ) q^{91} + 8464 q^{92} + ( -26262 + 2373 \beta_{1} + 5244 \beta_{2} ) q^{93} + ( 18524 - 1128 \beta_{1} - 652 \beta_{2} ) q^{94} + ( -325 - 425 \beta_{1} - 3875 \beta_{2} ) q^{95} + ( 9216 - 1024 \beta_{2} ) q^{96} + ( -85099 - 2709 \beta_{1} - 4941 \beta_{2} ) q^{97} + ( 15420 - 1912 \beta_{1} + 892 \beta_{2} ) q^{98} + ( -1530 + 2082 \beta_{1} - 3560 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 12 q^{2} - 26 q^{3} + 48 q^{4} + 75 q^{5} + 104 q^{6} + q^{7} - 192 q^{8} - 71 q^{9} + O(q^{10}) \) \( 3 q - 12 q^{2} - 26 q^{3} + 48 q^{4} + 75 q^{5} + 104 q^{6} + q^{7} - 192 q^{8} - 71 q^{9} - 300 q^{10} + 178 q^{11} - 416 q^{12} - 236 q^{13} - 4 q^{14} - 650 q^{15} + 768 q^{16} - 1683 q^{17} + 284 q^{18} - 194 q^{19} + 1200 q^{20} + 504 q^{21} - 712 q^{22} + 1587 q^{23} + 1664 q^{24} + 1875 q^{25} + 944 q^{26} + 622 q^{27} + 16 q^{28} + 6901 q^{29} + 2600 q^{30} + 6119 q^{31} - 3072 q^{32} + 4792 q^{33} + 6732 q^{34} + 25 q^{35} - 1136 q^{36} + 5185 q^{37} + 776 q^{38} + 16234 q^{39} - 4800 q^{40} + 6727 q^{41} - 2016 q^{42} - 22988 q^{43} + 2848 q^{44} - 1775 q^{45} - 6348 q^{46} - 13730 q^{47} - 6656 q^{48} - 11788 q^{49} - 7500 q^{50} - 20140 q^{51} - 3776 q^{52} - 38027 q^{53} - 2488 q^{54} + 4450 q^{55} - 64 q^{56} - 62424 q^{57} - 27604 q^{58} - 43715 q^{59} - 10400 q^{60} - 51364 q^{61} - 24476 q^{62} - 40095 q^{63} + 12288 q^{64} - 5900 q^{65} - 19168 q^{66} - 51891 q^{67} - 26928 q^{68} - 13754 q^{69} - 100 q^{70} - 25667 q^{71} + 4544 q^{72} - 108532 q^{73} - 20740 q^{74} - 16250 q^{75} - 3104 q^{76} - 95646 q^{77} - 64936 q^{78} - 224 q^{79} + 19200 q^{80} - 38213 q^{81} - 26908 q^{82} - 61071 q^{83} + 8064 q^{84} - 42075 q^{85} + 91952 q^{86} - 82842 q^{87} - 11392 q^{88} - 93372 q^{89} + 7100 q^{90} + 33030 q^{91} + 25392 q^{92} - 73542 q^{93} + 54920 q^{94} - 4850 q^{95} + 26624 q^{96} - 260238 q^{97} + 47152 q^{98} - 8150 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 47 x - 106\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \nu - 31 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/3\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{2} + 4 \beta_{1} + 93\)\()/3\)

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
−2.65230
7.78556
−5.13326
−4.00000 −22.3561 16.0000 25.0000 89.4244 −110.123 −64.0000 256.795 −100.000
1.2 −4.00000 −10.5273 16.0000 25.0000 42.1092 156.388 −64.0000 −132.176 −100.000
1.3 −4.00000 6.88339 16.0000 25.0000 −27.5336 −45.2649 −64.0000 −195.619 −100.000
\(n\): 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 230.6.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.6.a.c 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 26 T_{3}^{2} + 9 T_{3} - 1620 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T )^{3} \)
$3$ \( -1620 + 9 T + 26 T^{2} + T^{3} \)
$5$ \( ( -25 + T )^{3} \)
$7$ \( -779544 - 19316 T - T^{2} + T^{3} \)
$11$ \( 21004632 - 175884 T - 178 T^{2} + T^{3} \)
$13$ \( -16482042 - 217575 T + 236 T^{2} + T^{3} \)
$17$ \( 73501360 - 753600 T + 1683 T^{2} + T^{3} \)
$19$ \( -841047768 - 4848620 T + 194 T^{2} + T^{3} \)
$23$ \( ( -529 + T )^{3} \)
$29$ \( 809182077 - 570953 T - 6901 T^{2} + T^{3} \)
$31$ \( 30813892995 - 999581 T - 6119 T^{2} + T^{3} \)
$37$ \( 881126317804 - 143618332 T - 5185 T^{2} + T^{3} \)
$41$ \( 1050649591 + 152495 T - 6727 T^{2} + T^{3} \)
$43$ \( -1900451896320 - 8805224 T + 22988 T^{2} + T^{3} \)
$47$ \( -98633296700 + 31449305 T + 13730 T^{2} + T^{3} \)
$53$ \( 1549341371484 + 448711800 T + 38027 T^{2} + T^{3} \)
$59$ \( -22979507348832 - 564614168 T + 43715 T^{2} + T^{3} \)
$61$ \( -4638543486768 + 488088052 T + 51364 T^{2} + T^{3} \)
$67$ \( 4102954370460 + 828384804 T + 51891 T^{2} + T^{3} \)
$71$ \( -1521596424555 - 6946409 T + 25667 T^{2} + T^{3} \)
$73$ \( 4522640531874 + 2744414065 T + 108532 T^{2} + T^{3} \)
$79$ \( -11537777158848 - 1407646380 T + 224 T^{2} + T^{3} \)
$83$ \( -70485885180896 - 5764115016 T + 61071 T^{2} + T^{3} \)
$89$ \( -650267509666176 - 8889108840 T + 93372 T^{2} + T^{3} \)
$97$ \( -42022925072144 + 16517913168 T + 260238 T^{2} + T^{3} \)
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