# Properties

 Label 230.4.a.h Level $230$ Weight $4$ Character orbit 230.a Self dual yes Analytic conductor $13.570$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## 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: $$x^{4} - 68 x^{2} - 111 x + 342$$ 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.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + ( -1 + \beta_{1} ) q^{3} + 4 q^{4} -5 q^{5} + ( 2 - 2 \beta_{1} ) q^{6} + ( 2 \beta_{1} + \beta_{2} ) q^{7} -8 q^{8} + ( 8 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -2 q^{2} + ( -1 + \beta_{1} ) q^{3} + 4 q^{4} -5 q^{5} + ( 2 - 2 \beta_{1} ) q^{6} + ( 2 \beta_{1} + \beta_{2} ) q^{7} -8 q^{8} + ( 8 + \beta_{3} ) q^{9} + 10 q^{10} + ( -10 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + ( -4 + 4 \beta_{1} ) q^{12} + ( -5 + 8 \beta_{1} - \beta_{3} ) q^{13} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{14} + ( 5 - 5 \beta_{1} ) q^{15} + 16 q^{16} + ( -6 + 13 \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( -16 - 2 \beta_{3} ) q^{18} + ( 14 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{19} -20 q^{20} + ( 74 + \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{21} + ( 20 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{22} -23 q^{23} + ( 8 - 8 \beta_{1} ) q^{24} + 25 q^{25} + ( 10 - 16 \beta_{1} + 2 \beta_{3} ) q^{26} + ( 35 - \beta_{1} + 3 \beta_{2} ) q^{27} + ( 8 \beta_{1} + 4 \beta_{2} ) q^{28} + ( 41 - 14 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{29} + ( -10 + 10 \beta_{1} ) q^{30} + ( 97 + 16 \beta_{1} - 5 \beta_{3} ) q^{31} -32 q^{32} + ( 22 - 26 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{33} + ( 12 - 26 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{34} + ( -10 \beta_{1} - 5 \beta_{2} ) q^{35} + ( 32 + 4 \beta_{3} ) q^{36} + ( 116 - 18 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{37} + ( -28 - 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{38} + ( 261 - 15 \beta_{1} - 3 \beta_{2} + 8 \beta_{3} ) q^{39} + 40 q^{40} + ( 121 - \beta_{1} - 10 \beta_{3} ) q^{41} + ( -148 - 2 \beta_{1} + 8 \beta_{2} - 10 \beta_{3} ) q^{42} + ( 222 + 24 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{43} + ( -40 + 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{44} + ( -40 - 5 \beta_{3} ) q^{45} + 46 q^{46} + ( -67 + 44 \beta_{1} - 3 \beta_{2} - 11 \beta_{3} ) q^{47} + ( -16 + 16 \beta_{1} ) q^{48} + ( 411 - 49 \beta_{1} + \beta_{2} + 9 \beta_{3} ) q^{49} -50 q^{50} + ( 458 + 26 \beta_{1} + 7 \beta_{2} + 10 \beta_{3} ) q^{51} + ( -20 + 32 \beta_{1} - 4 \beta_{3} ) q^{52} + ( 146 - 40 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{53} + ( -70 + 2 \beta_{1} - 6 \beta_{2} ) q^{54} + ( 50 - 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{55} + ( -16 \beta_{1} - 8 \beta_{2} ) q^{56} + ( 40 - 23 \beta_{1} - 18 \beta_{2} + 11 \beta_{3} ) q^{57} + ( -82 + 28 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{58} + ( -20 + 10 \beta_{1} + 14 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 20 - 20 \beta_{1} ) q^{60} + ( 290 + 33 \beta_{1} + 7 \beta_{2} - 19 \beta_{3} ) q^{61} + ( -194 - 32 \beta_{1} + 10 \beta_{3} ) q^{62} + ( 16 + 115 \beta_{1} + 4 \beta_{2} - 11 \beta_{3} ) q^{63} + 64 q^{64} + ( 25 - 40 \beta_{1} + 5 \beta_{3} ) q^{65} + ( -44 + 52 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{66} + ( -368 + 28 \beta_{1} - 12 \beta_{3} ) q^{67} + ( -24 + 52 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{68} + ( 23 - 23 \beta_{1} ) q^{69} + ( 20 \beta_{1} + 10 \beta_{2} ) q^{70} + ( 57 + 39 \beta_{1} + 28 \beta_{2} + 4 \beta_{3} ) q^{71} + ( -64 - 8 \beta_{3} ) q^{72} + ( 287 + 44 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{73} + ( -232 + 36 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{74} + ( -25 + 25 \beta_{1} ) q^{75} + ( 56 + 8 \beta_{1} + 12 \beta_{2} - 8 \beta_{3} ) q^{76} + ( -548 - 61 \beta_{1} - 16 \beta_{2} + 17 \beta_{3} ) q^{77} + ( -522 + 30 \beta_{1} + 6 \beta_{2} - 16 \beta_{3} ) q^{78} + ( -232 - 72 \beta_{1} - 20 \beta_{2} - 16 \beta_{3} ) q^{79} -80 q^{80} + ( -267 + 31 \beta_{1} - 12 \beta_{2} - 19 \beta_{3} ) q^{81} + ( -242 + 2 \beta_{1} + 20 \beta_{3} ) q^{82} + ( -256 - 26 \beta_{1} + 24 \beta_{2} - 6 \beta_{3} ) q^{83} + ( 296 + 4 \beta_{1} - 16 \beta_{2} + 20 \beta_{3} ) q^{84} + ( 30 - 65 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{85} + ( -444 - 48 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{86} + ( -547 - 30 \beta_{1} - 21 \beta_{2} - 5 \beta_{3} ) q^{87} + ( 80 - 8 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{88} + ( -450 + 30 \beta_{1} - 16 \beta_{2} - 34 \beta_{3} ) q^{89} + ( 80 + 10 \beta_{3} ) q^{90} + ( 576 - 85 \beta_{1} - 25 \beta_{2} + 51 \beta_{3} ) q^{91} -92 q^{92} + ( 367 + 23 \beta_{1} - 15 \beta_{2} + 16 \beta_{3} ) q^{93} + ( 134 - 88 \beta_{1} + 6 \beta_{2} + 22 \beta_{3} ) q^{94} + ( -70 - 10 \beta_{1} - 15 \beta_{2} + 10 \beta_{3} ) q^{95} + ( 32 - 32 \beta_{1} ) q^{96} + ( -504 - 137 \beta_{1} + 31 \beta_{2} + \beta_{3} ) q^{97} + ( -822 + 98 \beta_{1} - 2 \beta_{2} - 18 \beta_{3} ) q^{98} + ( -662 - 68 \beta_{1} + 17 \beta_{2} + 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} - 20 q^{5} + 8 q^{6} - q^{7} - 32 q^{8} + 32 q^{9} + O(q^{10})$$ $$4 q - 8 q^{2} - 4 q^{3} + 16 q^{4} - 20 q^{5} + 8 q^{6} - q^{7} - 32 q^{8} + 32 q^{9} + 40 q^{10} - 39 q^{11} - 16 q^{12} - 20 q^{13} + 2 q^{14} + 20 q^{15} + 64 q^{16} - 23 q^{17} - 64 q^{18} + 53 q^{19} - 80 q^{20} + 300 q^{21} + 78 q^{22} - 92 q^{23} + 32 q^{24} + 100 q^{25} + 40 q^{26} + 137 q^{27} - 4 q^{28} + 161 q^{29} - 40 q^{30} + 388 q^{31} - 128 q^{32} + 87 q^{33} + 46 q^{34} + 5 q^{35} + 128 q^{36} + 466 q^{37} - 106 q^{38} + 1047 q^{39} + 160 q^{40} + 484 q^{41} - 600 q^{42} + 894 q^{43} - 156 q^{44} - 160 q^{45} + 184 q^{46} - 265 q^{47} - 64 q^{48} + 1643 q^{49} - 200 q^{50} + 1825 q^{51} - 80 q^{52} + 576 q^{53} - 274 q^{54} + 195 q^{55} + 8 q^{56} + 178 q^{57} - 322 q^{58} - 94 q^{59} + 80 q^{60} + 1153 q^{61} - 776 q^{62} + 60 q^{63} + 256 q^{64} + 100 q^{65} - 174 q^{66} - 1472 q^{67} - 92 q^{68} + 92 q^{69} - 10 q^{70} + 200 q^{71} - 256 q^{72} + 1147 q^{73} - 932 q^{74} - 100 q^{75} + 212 q^{76} - 2176 q^{77} - 2094 q^{78} - 908 q^{79} - 320 q^{80} - 1056 q^{81} - 968 q^{82} - 1048 q^{83} + 1200 q^{84} + 115 q^{85} - 1788 q^{86} - 2167 q^{87} + 312 q^{88} - 1784 q^{89} + 320 q^{90} + 2329 q^{91} - 368 q^{92} + 1483 q^{93} + 530 q^{94} - 265 q^{95} + 128 q^{96} - 2047 q^{97} - 3286 q^{98} - 2665 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 68 x^{2} - 111 x + 342$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 3 \nu^{2} - 50 \nu + 18$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 2 \nu - 34$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2 \beta_{1} + 34$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3} + 3 \beta_{2} + 56 \beta_{1} + 84$$

## 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
 −6.57209 −3.74869 1.58997 8.73081
−2.00000 −7.57209 4.00000 −5.00000 15.1442 −35.4229 −8.00000 30.3365 10.0000
1.2 −2.00000 −4.74869 4.00000 −5.00000 9.49738 29.3684 −8.00000 −4.44993 10.0000
1.3 −2.00000 0.589969 4.00000 −5.00000 −1.17994 −18.5077 −8.00000 −26.6519 10.0000
1.4 −2.00000 7.73081 4.00000 −5.00000 −15.4616 23.5622 −8.00000 32.7654 10.0000
 $$n$$: 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 230.4.a.h 4
3.b odd 2 1 2070.4.a.bj 4
4.b odd 2 1 1840.4.a.m 4
5.b even 2 1 1150.4.a.p 4
5.c odd 4 2 1150.4.b.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.h 4 1.a even 1 1 trivial
1150.4.a.p 4 5.b even 2 1
1150.4.b.n 8 5.c odd 4 2
1840.4.a.m 4 4.b odd 2 1
2070.4.a.bj 4 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 4 T_{3}^{3} - 62 T_{3}^{2} - 243 T_{3} + 164$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(230))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{4}$$
$3$ $$164 - 243 T - 62 T^{2} + 4 T^{3} + T^{4}$$
$5$ $$( 5 + T )^{4}$$
$7$ $$453664 + 2618 T - 1507 T^{2} + T^{3} + T^{4}$$
$11$ $$-977536 - 94420 T - 1771 T^{2} + 39 T^{3} + T^{4}$$
$13$ $$3059002 + 25481 T - 4948 T^{2} + 20 T^{3} + T^{4}$$
$17$ $$1048184 - 590494 T - 14205 T^{2} + 23 T^{3} + T^{4}$$
$19$ $$-78336 + 340080 T - 15029 T^{2} - 53 T^{3} + T^{4}$$
$23$ $$( 23 + T )^{4}$$
$29$ $$1597064 + 1886712 T - 27260 T^{2} - 161 T^{3} + T^{4}$$
$31$ $$-397027152 + 5546709 T + 13076 T^{2} - 388 T^{3} + T^{4}$$
$37$ $$-7921024 - 681856 T + 37920 T^{2} - 466 T^{3} + T^{4}$$
$41$ $$1506099394 + 17572043 T - 36232 T^{2} - 484 T^{3} + T^{4}$$
$43$ $$-5392023552 + 22823424 T + 166752 T^{2} - 894 T^{3} + T^{4}$$
$47$ $$1306137888 - 43523988 T - 230390 T^{2} + 265 T^{3} + T^{4}$$
$53$ $$-3236491568 + 79775808 T - 107352 T^{2} - 576 T^{3} + T^{4}$$
$59$ $$11673423616 - 20596976 T - 243460 T^{2} + 94 T^{3} + T^{4}$$
$61$ $$-42772329400 + 315388730 T - 54057 T^{2} - 1153 T^{3} + T^{4}$$
$67$ $$-10308616192 + 42702336 T + 602032 T^{2} + 1472 T^{3} + T^{4}$$
$71$ $$274201266224 + 93278335 T - 1067322 T^{2} - 200 T^{3} + T^{4}$$
$73$ $$-4754107544 - 9049904 T + 356120 T^{2} - 1147 T^{3} + T^{4}$$
$79$ $$145785325568 - 272612224 T - 923664 T^{2} + 908 T^{3} + T^{4}$$
$83$ $$-178673654272 - 675145440 T - 359444 T^{2} + 1048 T^{3} + T^{4}$$
$89$ $$-969417760000 - 2249185600 T - 492084 T^{2} + 1784 T^{3} + T^{4}$$
$97$ $$-658266694276 - 2746911284 T - 716799 T^{2} + 2047 T^{3} + T^{4}$$