Properties

Label 230.6.a.g
Level $230$
Weight $6$
Character orbit 230.a
Self dual yes
Analytic conductor $36.888$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 772 x^{3} - 255 x^{2} + 13416 x + 10080\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 q^{2} + ( 1 + \beta_{1} ) q^{3} + 16 q^{4} -25 q^{5} + ( -4 - 4 \beta_{1} ) q^{6} + ( 26 - 2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{7} -64 q^{8} + ( 66 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q -4 q^{2} + ( 1 + \beta_{1} ) q^{3} + 16 q^{4} -25 q^{5} + ( -4 - 4 \beta_{1} ) q^{6} + ( 26 - 2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{7} -64 q^{8} + ( 66 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{9} + 100 q^{10} + ( 15 - 16 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{11} + ( 16 + 16 \beta_{1} ) q^{12} + ( -149 + 10 \beta_{1} + \beta_{2} - 4 \beta_{3} - 6 \beta_{4} ) q^{13} + ( -104 + 8 \beta_{1} + 4 \beta_{2} + 4 \beta_{4} ) q^{14} + ( -25 - 25 \beta_{1} ) q^{15} + 256 q^{16} + ( -352 + 27 \beta_{1} + 2 \beta_{2} - 10 \beta_{3} + 5 \beta_{4} ) q^{17} + ( -264 - 4 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} - 4 \beta_{4} ) q^{18} + ( 383 + 56 \beta_{1} - 2 \beta_{2} + 9 \beta_{3} + \beta_{4} ) q^{19} -400 q^{20} + ( -506 + 99 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} ) q^{21} + ( -60 + 64 \beta_{1} + 8 \beta_{2} - 12 \beta_{3} ) q^{22} + 529 q^{23} + ( -64 - 64 \beta_{1} ) q^{24} + 625 q^{25} + ( 596 - 40 \beta_{1} - 4 \beta_{2} + 16 \beta_{3} + 24 \beta_{4} ) q^{26} + ( 586 + 244 \beta_{1} - 9 \beta_{2} + 21 \beta_{3} - 4 \beta_{4} ) q^{27} + ( 416 - 32 \beta_{1} - 16 \beta_{2} - 16 \beta_{4} ) q^{28} + ( 707 + 12 \beta_{1} - 34 \beta_{2} - 4 \beta_{3} + 25 \beta_{4} ) q^{29} + ( 100 + 100 \beta_{1} ) q^{30} + ( 362 + 128 \beta_{1} + 16 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} ) q^{31} -1024 q^{32} + ( -4385 + 272 \beta_{1} + 44 \beta_{2} - 11 \beta_{3} - 21 \beta_{4} ) q^{33} + ( 1408 - 108 \beta_{1} - 8 \beta_{2} + 40 \beta_{3} - 20 \beta_{4} ) q^{34} + ( -650 + 50 \beta_{1} + 25 \beta_{2} + 25 \beta_{4} ) q^{35} + ( 1056 + 16 \beta_{1} - 64 \beta_{2} + 32 \beta_{3} + 16 \beta_{4} ) q^{36} + ( 2345 + 301 \beta_{1} - 52 \beta_{2} - 21 \beta_{3} - 5 \beta_{4} ) q^{37} + ( -1532 - 224 \beta_{1} + 8 \beta_{2} - 36 \beta_{3} - 4 \beta_{4} ) q^{38} + ( 2510 - 435 \beta_{1} - 28 \beta_{2} + 15 \beta_{3} + 9 \beta_{4} ) q^{39} + 1600 q^{40} + ( -1904 + 121 \beta_{1} + 9 \beta_{2} - 41 \beta_{3} + 38 \beta_{4} ) q^{41} + ( 2024 - 396 \beta_{1} - 28 \beta_{2} - 8 \beta_{3} + 16 \beta_{4} ) q^{42} + ( -428 - 319 \beta_{1} - 3 \beta_{2} - 46 \beta_{3} - 23 \beta_{4} ) q^{43} + ( 240 - 256 \beta_{1} - 32 \beta_{2} + 48 \beta_{3} ) q^{44} + ( -1650 - 25 \beta_{1} + 100 \beta_{2} - 50 \beta_{3} - 25 \beta_{4} ) q^{45} -2116 q^{46} + ( -2930 + 52 \beta_{1} + 33 \beta_{2} + 49 \beta_{3} - 103 \beta_{4} ) q^{47} + ( 256 + 256 \beta_{1} ) q^{48} + ( 5201 - 337 \beta_{1} + 60 \beta_{2} - 58 \beta_{3} - 87 \beta_{4} ) q^{49} -2500 q^{50} + ( 6277 - 723 \beta_{1} - 17 \beta_{2} - 17 \beta_{3} + 44 \beta_{4} ) q^{51} + ( -2384 + 160 \beta_{1} + 16 \beta_{2} - 64 \beta_{3} - 96 \beta_{4} ) q^{52} + ( -4667 + 106 \beta_{1} + 129 \beta_{2} + 133 \beta_{3} - 2 \beta_{4} ) q^{53} + ( -2344 - 976 \beta_{1} + 36 \beta_{2} - 84 \beta_{3} + 16 \beta_{4} ) q^{54} + ( -375 + 400 \beta_{1} + 50 \beta_{2} - 75 \beta_{3} ) q^{55} + ( -1664 + 128 \beta_{1} + 64 \beta_{2} + 64 \beta_{4} ) q^{56} + ( 18978 + 817 \beta_{1} - 289 \beta_{2} + 160 \beta_{3} + 46 \beta_{4} ) q^{57} + ( -2828 - 48 \beta_{1} + 136 \beta_{2} + 16 \beta_{3} - 100 \beta_{4} ) q^{58} + ( 8799 - 846 \beta_{1} + 39 \beta_{2} - 39 \beta_{3} - 94 \beta_{4} ) q^{59} + ( -400 - 400 \beta_{1} ) q^{60} + ( 21525 - 166 \beta_{1} - 34 \beta_{2} - 71 \beta_{3} + 122 \beta_{4} ) q^{61} + ( -1448 - 512 \beta_{1} - 64 \beta_{2} - 12 \beta_{3} + 16 \beta_{4} ) q^{62} + ( 23727 - 642 \beta_{1} - 195 \beta_{2} + 199 \beta_{3} + 343 \beta_{4} ) q^{63} + 4096 q^{64} + ( 3725 - 250 \beta_{1} - 25 \beta_{2} + 100 \beta_{3} + 150 \beta_{4} ) q^{65} + ( 17540 - 1088 \beta_{1} - 176 \beta_{2} + 44 \beta_{3} + 84 \beta_{4} ) q^{66} + ( 26575 - 817 \beta_{1} + 60 \beta_{2} + 125 \beta_{3} - 107 \beta_{4} ) q^{67} + ( -5632 + 432 \beta_{1} + 32 \beta_{2} - 160 \beta_{3} + 80 \beta_{4} ) q^{68} + ( 529 + 529 \beta_{1} ) q^{69} + ( 2600 - 200 \beta_{1} - 100 \beta_{2} - 100 \beta_{4} ) q^{70} + ( 20640 + 2180 \beta_{1} + 346 \beta_{2} + 63 \beta_{3} - 85 \beta_{4} ) q^{71} + ( -4224 - 64 \beta_{1} + 256 \beta_{2} - 128 \beta_{3} - 64 \beta_{4} ) q^{72} + ( 25188 - 87 \beta_{1} - 176 \beta_{2} - 35 \beta_{3} + 132 \beta_{4} ) q^{73} + ( -9380 - 1204 \beta_{1} + 208 \beta_{2} + 84 \beta_{3} + 20 \beta_{4} ) q^{74} + ( 625 + 625 \beta_{1} ) q^{75} + ( 6128 + 896 \beta_{1} - 32 \beta_{2} + 144 \beta_{3} + 16 \beta_{4} ) q^{76} + ( 22150 - 1356 \beta_{1} - 278 \beta_{2} - 164 \beta_{3} + 347 \beta_{4} ) q^{77} + ( -10040 + 1740 \beta_{1} + 112 \beta_{2} - 60 \beta_{3} - 36 \beta_{4} ) q^{78} + ( 55140 - 401 \beta_{1} - 447 \beta_{2} - 404 \beta_{3} - 131 \beta_{4} ) q^{79} -6400 q^{80} + ( 63273 + 1642 \beta_{1} - 166 \beta_{2} + 146 \beta_{3} - 33 \beta_{4} ) q^{81} + ( 7616 - 484 \beta_{1} - 36 \beta_{2} + 164 \beta_{3} - 152 \beta_{4} ) q^{82} + ( 21355 + 1328 \beta_{1} - 49 \beta_{2} - 287 \beta_{3} - 226 \beta_{4} ) q^{83} + ( -8096 + 1584 \beta_{1} + 112 \beta_{2} + 32 \beta_{3} - 64 \beta_{4} ) q^{84} + ( 8800 - 675 \beta_{1} - 50 \beta_{2} + 250 \beta_{3} - 125 \beta_{4} ) q^{85} + ( 1712 + 1276 \beta_{1} + 12 \beta_{2} + 184 \beta_{3} + 92 \beta_{4} ) q^{86} + ( 4744 + 3966 \beta_{1} + 127 \beta_{2} + 31 \beta_{3} + 7 \beta_{4} ) q^{87} + ( -960 + 1024 \beta_{1} + 128 \beta_{2} - 192 \beta_{3} ) q^{88} + ( 11492 - 581 \beta_{1} + 283 \beta_{2} + 170 \beta_{3} - 573 \beta_{4} ) q^{89} + ( 6600 + 100 \beta_{1} - 400 \beta_{2} + 200 \beta_{3} + 100 \beta_{4} ) q^{90} + ( 54593 + 1472 \beta_{1} + 956 \beta_{2} - 401 \beta_{3} - 862 \beta_{4} ) q^{91} + 8464 q^{92} + ( 39528 - 1025 \beta_{1} - 580 \beta_{2} + 235 \beta_{3} + 137 \beta_{4} ) q^{93} + ( 11720 - 208 \beta_{1} - 132 \beta_{2} - 196 \beta_{3} + 412 \beta_{4} ) q^{94} + ( -9575 - 1400 \beta_{1} + 50 \beta_{2} - 225 \beta_{3} - 25 \beta_{4} ) q^{95} + ( -1024 - 1024 \beta_{1} ) q^{96} + ( -4957 - 2074 \beta_{1} + 754 \beta_{2} - 75 \beta_{3} + 220 \beta_{4} ) q^{97} + ( -20804 + 1348 \beta_{1} - 240 \beta_{2} + 232 \beta_{3} + 348 \beta_{4} ) q^{98} + ( 72633 - 4981 \beta_{1} - 665 \beta_{2} - 309 \beta_{3} + 306 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 20q^{2} + 5q^{3} + 80q^{4} - 125q^{5} - 20q^{6} + 130q^{7} - 320q^{8} + 334q^{9} + O(q^{10}) \) \( 5q - 20q^{2} + 5q^{3} + 80q^{4} - 125q^{5} - 20q^{6} + 130q^{7} - 320q^{8} + 334q^{9} + 500q^{10} + 81q^{11} + 80q^{12} - 753q^{13} - 520q^{14} - 125q^{15} + 1280q^{16} - 1780q^{17} - 1336q^{18} + 1933q^{19} - 2000q^{20} - 2526q^{21} - 324q^{22} + 2645q^{23} - 320q^{24} + 3125q^{25} + 3012q^{26} + 2972q^{27} + 2080q^{28} + 3527q^{29} + 500q^{30} + 1816q^{31} - 5120q^{32} - 21947q^{33} + 7120q^{34} - 3250q^{35} + 5344q^{36} + 11683q^{37} - 7732q^{38} + 12580q^{39} + 8000q^{40} - 9602q^{41} + 10104q^{42} - 2232q^{43} + 1296q^{44} - 8350q^{45} - 10580q^{46} - 14552q^{47} + 1280q^{48} + 25889q^{49} - 12500q^{50} + 31351q^{51} - 12048q^{52} - 23069q^{53} - 11888q^{54} - 2025q^{55} - 8320q^{56} + 95210q^{57} - 14108q^{58} + 43917q^{59} - 2000q^{60} + 107483q^{61} - 7264q^{62} + 119033q^{63} + 20480q^{64} + 18825q^{65} + 87788q^{66} + 133125q^{67} - 28480q^{68} + 2645q^{69} + 13000q^{70} + 103326q^{71} - 21376q^{72} + 125870q^{73} - 46732q^{74} + 3125q^{75} + 30928q^{76} + 110422q^{77} - 50320q^{78} + 274892q^{79} - 32000q^{80} + 316657q^{81} + 38408q^{82} + 106201q^{83} - 40416q^{84} + 44500q^{85} + 8928q^{86} + 23782q^{87} - 5184q^{88} + 57800q^{89} + 33400q^{90} + 272163q^{91} + 42320q^{92} + 198110q^{93} + 58208q^{94} - 48325q^{95} - 5120q^{96} - 24935q^{97} - 103556q^{98} + 362547q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 772 x^{3} - 255 x^{2} + 13416 x + 10080\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -29 \nu^{4} + 96 \nu^{3} + 21494 \nu^{2} - 64449 \nu - 49872 \)\()/3342\)
\(\beta_{3}\)\(=\)\((\)\( -25 \nu^{4} + 198 \nu^{3} + 19336 \nu^{2} - 138879 \nu - 308394 \)\()/3342\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{4} - 2 \nu^{3} + 8441 \nu^{2} + 3884 \nu - 102006 \)\()/557\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + 2 \beta_{3} - 4 \beta_{2} - \beta_{1} + 308\)
\(\nu^{3}\)\(=\)\(-7 \beta_{4} + 15 \beta_{3} + 3 \beta_{2} + 730 \beta_{1} + 147\)
\(\nu^{4}\)\(=\)\(718 \beta_{4} + 1532 \beta_{3} - 3070 \beta_{2} - 547 \beta_{1} + 227048\)

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
−27.2974
−3.96404
−0.766035
4.40191
27.6256
−4.00000 −26.2974 16.0000 −25.0000 105.190 152.657 −64.0000 448.555 100.000
1.2 −4.00000 −2.96404 16.0000 −25.0000 11.8561 −147.425 −64.0000 −234.214 100.000
1.3 −4.00000 0.233965 16.0000 −25.0000 −0.935860 203.513 −64.0000 −242.945 100.000
1.4 −4.00000 5.40191 16.0000 −25.0000 −21.6076 −140.289 −64.0000 −213.819 100.000
1.5 −4.00000 28.6256 16.0000 −25.0000 −114.502 61.5435 −64.0000 576.424 100.000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.6.a.g 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 5 T_{3}^{4} - 762 T_{3}^{3} + 2051 T_{3}^{2} + 11615 T_{3} - 2820 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + T )^{5} \)
$3$ \( -2820 + 11615 T + 2051 T^{2} - 762 T^{3} - 5 T^{4} + T^{5} \)
$5$ \( ( 25 + T )^{5} \)
$7$ \( -39544118760 + 545776316 T + 4694065 T^{2} - 46512 T^{3} - 130 T^{4} + T^{5} \)
$11$ \( 647661944664 + 14758006236 T + 12776538 T^{2} - 408793 T^{3} - 81 T^{4} + T^{5} \)
$13$ \( -30567624422454 - 330695667939 T - 1108905785 T^{2} - 966276 T^{3} + 753 T^{4} + T^{5} \)
$17$ \( -1206093341515680 - 5571433845000 T - 8024261559 T^{2} - 2848272 T^{3} + 1780 T^{4} + T^{5} \)
$19$ \( 893102873667144 + 3593957389796 T + 2156685670 T^{2} - 3433791 T^{3} - 1933 T^{4} + T^{5} \)
$23$ \( ( -529 + T )^{5} \)
$29$ \( -1125680364375995484 + 308280440091294 T + 130771194627 T^{2} - 36274239 T^{3} - 3527 T^{4} + T^{5} \)
$31$ \( 9243598456543185 - 47017058945500 T + 58186641967 T^{2} - 17956413 T^{3} - 1816 T^{4} + T^{5} \)
$37$ \( -2311395120314256576 - 2031900856251424 T + 1191211064020 T^{2} - 104243316 T^{3} - 11683 T^{4} + T^{5} \)
$41$ \( -4834760538656941695 - 4186030443699456 T - 1088530924899 T^{2} - 65832033 T^{3} + 9602 T^{4} + T^{5} \)
$43$ \( -11644455411079060480 + 5456362496745600 T + 79578966304 T^{2} - 155340232 T^{3} + 2232 T^{4} + T^{5} \)
$47$ \( 18519575406484168800 + 22621584761394300 T - 2052152886630 T^{2} - 338141535 T^{3} + 14552 T^{4} + T^{5} \)
$53$ \( -\)\(81\!\cdots\!76\)\( - 323578554763091472 T - 34385918322744 T^{2} - 948506880 T^{3} + 23069 T^{4} + T^{5} \)
$59$ \( -39793826244866359680 - 146234565199418976 T + 18734516788284 T^{2} - 36807738 T^{3} - 43917 T^{4} + T^{5} \)
$61$ \( -\)\(71\!\cdots\!00\)\( + 364278092779022116 T - 59421574588232 T^{2} + 3943979149 T^{3} - 107483 T^{4} + T^{5} \)
$67$ \( \)\(65\!\cdots\!20\)\( - 669229239431272560 T - 63225723444884 T^{2} + 5620729448 T^{3} - 133125 T^{4} + T^{5} \)
$71$ \( \)\(16\!\cdots\!65\)\( - 18332502243315099300 T + 560327804651817 T^{2} - 2487172225 T^{3} - 103326 T^{4} + T^{5} \)
$73$ \( \)\(10\!\cdots\!48\)\( - 228857654563967168 T - 62842031911472 T^{2} + 5135870493 T^{3} - 125870 T^{4} + T^{5} \)
$79$ \( \)\(44\!\cdots\!28\)\( - \)\(16\!\cdots\!72\)\( T + 1111644429221488 T^{2} + 17890963140 T^{3} - 274892 T^{4} + T^{5} \)
$83$ \( -\)\(86\!\cdots\!16\)\( - 2853328475410608384 T + 237137830259772 T^{2} - 164009672 T^{3} - 106201 T^{4} + T^{5} \)
$89$ \( \)\(18\!\cdots\!28\)\( + 20068200128786063232 T + 390315941840928 T^{2} - 10743366632 T^{3} - 57800 T^{4} + T^{5} \)
$97$ \( \)\(41\!\cdots\!76\)\( + 27335336390892144560 T - 165520786963550 T^{2} - 17229883725 T^{3} + 24935 T^{4} + T^{5} \)
show more
show less