# Properties

 Label 230.6.a.d 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: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + ( -11 - \beta_{2} ) q^{3} + 16 q^{4} + 25 q^{5} + ( -44 - 4 \beta_{2} ) q^{6} + ( -39 + 3 \beta_{1} - 4 \beta_{2} ) q^{7} + 64 q^{8} + ( 22 - 6 \beta_{1} + 23 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + 4 q^{2} + ( -11 - \beta_{2} ) q^{3} + 16 q^{4} + 25 q^{5} + ( -44 - 4 \beta_{2} ) q^{6} + ( -39 + 3 \beta_{1} - 4 \beta_{2} ) q^{7} + 64 q^{8} + ( 22 - 6 \beta_{1} + 23 \beta_{2} ) q^{9} + 100 q^{10} + ( -173 - 19 \beta_{1} + 17 \beta_{2} ) q^{11} + ( -176 - 16 \beta_{2} ) q^{12} + ( 10 + 37 \beta_{1} + 46 \beta_{2} ) q^{13} + ( -156 + 12 \beta_{1} - 16 \beta_{2} ) q^{14} + ( -275 - 25 \beta_{2} ) q^{15} + 256 q^{16} + ( -402 + 32 \beta_{1} + 39 \beta_{2} ) q^{17} + ( 88 - 24 \beta_{1} + 92 \beta_{2} ) q^{18} + ( -563 - 77 \beta_{1} - 5 \beta_{2} ) q^{19} + 400 q^{20} + ( 1167 - 57 \beta_{1} + 123 \beta_{2} ) q^{21} + ( -692 - 76 \beta_{1} + 68 \beta_{2} ) q^{22} -529 q^{23} + ( -704 - 64 \beta_{2} ) q^{24} + 625 q^{25} + ( 40 + 148 \beta_{1} + 184 \beta_{2} ) q^{26} + ( -1205 + 204 \beta_{1} - 127 \beta_{2} ) q^{27} + ( -624 + 48 \beta_{1} - 64 \beta_{2} ) q^{28} + ( -1589 - 158 \beta_{1} - 232 \beta_{2} ) q^{29} + ( -1100 - 100 \beta_{2} ) q^{30} + ( -3180 - 63 \beta_{1} - 151 \beta_{2} ) q^{31} + 1024 q^{32} + ( -1571 + 311 \beta_{1} - 259 \beta_{2} ) q^{33} + ( -1608 + 128 \beta_{1} + 156 \beta_{2} ) q^{34} + ( -975 + 75 \beta_{1} - 100 \beta_{2} ) q^{35} + ( 352 - 96 \beta_{1} + 368 \beta_{2} ) q^{36} + ( 241 + 441 \beta_{1} + 202 \beta_{2} ) q^{37} + ( -2252 - 308 \beta_{1} - 20 \beta_{2} ) q^{38} + ( -4736 - 131 \beta_{1} - 118 \beta_{2} ) q^{39} + 1600 q^{40} + ( -713 - 200 \beta_{1} + 626 \beta_{2} ) q^{41} + ( 4668 - 228 \beta_{1} + 492 \beta_{2} ) q^{42} + ( -9829 - 473 \beta_{1} - 1065 \beta_{2} ) q^{43} + ( -2768 - 304 \beta_{1} + 272 \beta_{2} ) q^{44} + ( 550 - 150 \beta_{1} + 575 \beta_{2} ) q^{45} -2116 q^{46} + ( -429 - 602 \beta_{1} - 783 \beta_{2} ) q^{47} + ( -2816 - 256 \beta_{2} ) q^{48} + ( -9175 - 222 \beta_{1} + 697 \beta_{2} ) q^{49} + 2500 q^{50} + ( 534 - 118 \beta_{1} + 318 \beta_{2} ) q^{51} + ( 160 + 592 \beta_{1} + 736 \beta_{2} ) q^{52} + ( -11682 + 1438 \beta_{1} + 123 \beta_{2} ) q^{53} + ( -4820 + 816 \beta_{1} - 508 \beta_{2} ) q^{54} + ( -4325 - 475 \beta_{1} + 425 \beta_{2} ) q^{55} + ( -2496 + 192 \beta_{1} - 256 \beta_{2} ) q^{56} + ( 2755 + 817 \beta_{1} - 301 \beta_{2} ) q^{57} + ( -6356 - 632 \beta_{1} - 928 \beta_{2} ) q^{58} + ( -12046 + 368 \beta_{1} - 527 \beta_{2} ) q^{59} + ( -4400 - 400 \beta_{2} ) q^{60} + ( -14605 - 1651 \beta_{1} - 549 \beta_{2} ) q^{61} + ( -12720 - 252 \beta_{1} - 604 \beta_{2} ) q^{62} + ( -24150 + 636 \beta_{1} - 2355 \beta_{2} ) q^{63} + 4096 q^{64} + ( 250 + 925 \beta_{1} + 1150 \beta_{2} ) q^{65} + ( -6284 + 1244 \beta_{1} - 1036 \beta_{2} ) q^{66} + ( -28455 - 1847 \beta_{1} - 604 \beta_{2} ) q^{67} + ( -6432 + 512 \beta_{1} + 624 \beta_{2} ) q^{68} + ( 5819 + 529 \beta_{2} ) q^{69} + ( -3900 + 300 \beta_{1} - 400 \beta_{2} ) q^{70} + ( -34862 + 1563 \beta_{1} - 1231 \beta_{2} ) q^{71} + ( 1408 - 384 \beta_{1} + 1472 \beta_{2} ) q^{72} + ( -23156 - 2879 \beta_{1} - 1880 \beta_{2} ) q^{73} + ( 964 + 1764 \beta_{1} + 808 \beta_{2} ) q^{74} + ( -6875 - 625 \beta_{2} ) q^{75} + ( -9008 - 1232 \beta_{1} - 80 \beta_{2} ) q^{76} + ( -25806 - 54 \beta_{1} - 2076 \beta_{2} ) q^{77} + ( -18944 - 524 \beta_{1} - 472 \beta_{2} ) q^{78} + ( 3523 + 3201 \beta_{1} + 1627 \beta_{2} ) q^{79} + 6400 q^{80} + ( 37213 - 1548 \beta_{1} - 412 \beta_{2} ) q^{81} + ( -2852 - 800 \beta_{1} + 2504 \beta_{2} ) q^{82} + ( -22438 + 1436 \beta_{1} + 625 \beta_{2} ) q^{83} + ( 18672 - 912 \beta_{1} + 1968 \beta_{2} ) q^{84} + ( -10050 + 800 \beta_{1} + 975 \beta_{2} ) q^{85} + ( -39316 - 1892 \beta_{1} - 4260 \beta_{2} ) q^{86} + ( 42355 + 346 \beta_{1} + 2477 \beta_{2} ) q^{87} + ( -11072 - 1216 \beta_{1} + 1088 \beta_{2} ) q^{88} + ( 50523 - 2405 \beta_{1} - 1701 \beta_{2} ) q^{89} + ( 2200 - 600 \beta_{1} + 2300 \beta_{2} ) q^{90} + ( 4623 + 1023 \beta_{1} - 899 \beta_{2} ) q^{91} -8464 q^{92} + ( 53322 - 213 \beta_{1} + 4236 \beta_{2} ) q^{93} + ( -1716 - 2408 \beta_{1} - 3132 \beta_{2} ) q^{94} + ( -14075 - 1925 \beta_{1} - 125 \beta_{2} ) q^{95} + ( -11264 - 1024 \beta_{2} ) q^{96} + ( 70229 + 1809 \beta_{1} + 4551 \beta_{2} ) q^{97} + ( -36700 - 888 \beta_{1} + 2788 \beta_{2} ) q^{98} + ( 113410 - 358 \beta_{1} + 4280 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 12q^{2} - 34q^{3} + 48q^{4} + 75q^{5} - 136q^{6} - 121q^{7} + 192q^{8} + 89q^{9} + O(q^{10})$$ $$3q + 12q^{2} - 34q^{3} + 48q^{4} + 75q^{5} - 136q^{6} - 121q^{7} + 192q^{8} + 89q^{9} + 300q^{10} - 502q^{11} - 544q^{12} + 76q^{13} - 484q^{14} - 850q^{15} + 768q^{16} - 1167q^{17} + 356q^{18} - 1694q^{19} + 1200q^{20} + 3624q^{21} - 2008q^{22} - 1587q^{23} - 2176q^{24} + 1875q^{25} + 304q^{26} - 3742q^{27} - 1936q^{28} - 4999q^{29} - 3400q^{30} - 9691q^{31} + 3072q^{32} - 4972q^{33} - 4668q^{34} - 3025q^{35} + 1424q^{36} + 925q^{37} - 6776q^{38} - 14326q^{39} + 4800q^{40} - 1513q^{41} + 14496q^{42} - 30552q^{43} - 8032q^{44} + 2225q^{45} - 6348q^{46} - 2070q^{47} - 8704q^{48} - 26828q^{49} + 7500q^{50} + 1920q^{51} + 1216q^{52} - 34923q^{53} - 14968q^{54} - 12550q^{55} - 7744q^{56} + 7964q^{57} - 19996q^{58} - 36665q^{59} - 13600q^{60} - 44364q^{61} - 38764q^{62} - 74805q^{63} + 12288q^{64} + 1900q^{65} - 19888q^{66} - 85969q^{67} - 18672q^{68} + 17986q^{69} - 12100q^{70} - 105817q^{71} + 5696q^{72} - 71348q^{73} + 3700q^{74} - 21250q^{75} - 27104q^{76} - 79494q^{77} - 57304q^{78} + 12196q^{79} + 19200q^{80} + 111227q^{81} - 6052q^{82} - 66689q^{83} + 57984q^{84} - 29175q^{85} - 122208q^{86} + 129542q^{87} - 32128q^{88} + 149868q^{89} + 8900q^{90} + 12970q^{91} - 25392q^{92} + 164202q^{93} - 8280q^{94} - 42350q^{95} - 34816q^{96} + 215238q^{97} - 107312q^{98} + 344510q^{99} + O(q^{100})$$

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
 −5.13326 7.78556 −2.65230
4.00000 −26.8834 16.0000 25.0000 −107.534 −148.733 64.0000 479.717 100.000
1.2 4.00000 −9.47271 16.0000 25.0000 −37.8908 37.1792 64.0000 −153.268 100.000
1.3 4.00000 2.35610 16.0000 25.0000 9.42439 −9.44632 64.0000 −237.449 100.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.d 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.6.a.d 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 34 T_{3}^{2} + 169 T_{3} - 600$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(230))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -4 + T )^{3}$$
$3$ $$-600 + 169 T + 34 T^{2} + T^{3}$$
$5$ $$( -25 + T )^{3}$$
$7$ $$-52236 - 4476 T + 121 T^{2} + T^{3}$$
$11$ $$-62322768 - 187424 T + 502 T^{2} + T^{3}$$
$13$ $$123447082 - 738775 T - 76 T^{2} + T^{3}$$
$17$ $$-92676960 - 91080 T + 1167 T^{2} + T^{3}$$
$19$ $$148953712 - 1489840 T + 1694 T^{2} + T^{3}$$
$23$ $$( 529 + T )^{3}$$
$29$ $$-38710135623 - 7495553 T + 4999 T^{2} + T^{3}$$
$31$ $$13485256605 + 26348919 T + 9691 T^{2} + T^{3}$$
$37$ $$-204013791224 - 75307252 T - 925 T^{2} + T^{3}$$
$41$ $$-584014306329 - 122717585 T + 1513 T^{2} + T^{3}$$
$43$ $$-3056483949440 + 58785156 T + 30552 T^{2} + T^{3}$$
$47$ $$-708220332900 - 202482495 T + 2070 T^{2} + T^{3}$$
$53$ $$-17143408525284 - 440656200 T + 34923 T^{2} + T^{3}$$
$59$ $$547656475488 + 296996152 T + 36665 T^{2} + T^{3}$$
$61$ $$644446767232 - 404448648 T + 44364 T^{2} + T^{3}$$
$67$ $$3863516553720 + 1135720764 T + 85969 T^{2} + T^{3}$$
$71$ $$-8535894304605 + 2036428291 T + 105817 T^{2} + T^{3}$$
$73$ $$-18890151779714 - 1632079495 T + 71348 T^{2} + T^{3}$$
$79$ $$-65983735462208 - 3951112360 T - 12196 T^{2} + T^{3}$$
$83$ $$-14831556891504 + 681867424 T + 66689 T^{2} + T^{3}$$
$89$ $$17068429826064 + 5126041620 T - 149868 T^{2} + T^{3}$$
$97$ $$59877067847944 + 11010096468 T - 215238 T^{2} + T^{3}$$
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