# Properties

 Label 230.2.b.b Level $230$ Weight $2$ Character orbit 230.b Analytic conductor $1.837$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 230.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.83655924649$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.11574317056.3 Defining polynomial: $$x^{8} + 45 x^{4} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( \beta_{2} - \beta_{5} ) q^{3} - q^{4} + ( \beta_{1} - \beta_{6} ) q^{5} + ( 1 - \beta_{4} ) q^{6} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{7} + \beta_{2} q^{8} + ( -2 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( \beta_{2} - \beta_{5} ) q^{3} - q^{4} + ( \beta_{1} - \beta_{6} ) q^{5} + ( 1 - \beta_{4} ) q^{6} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{7} + \beta_{2} q^{8} + ( -2 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{9} + ( -\beta_{3} + \beta_{7} ) q^{10} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{11} + ( -\beta_{2} + \beta_{5} ) q^{12} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{13} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} ) q^{14} + ( -2 - \beta_{2} - \beta_{5} - 2 \beta_{7} ) q^{15} + q^{16} + ( \beta_{2} + \beta_{5} - 2 \beta_{6} ) q^{17} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{18} + ( 1 + 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - \beta_{7} ) q^{19} + ( -\beta_{1} + \beta_{6} ) q^{20} + ( 2 + \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{21} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{22} + \beta_{2} q^{23} + ( -1 + \beta_{4} ) q^{24} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{25} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} ) q^{26} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + 3 \beta_{6} ) q^{27} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{28} + ( -2 \beta_{4} + 2 \beta_{7} ) q^{29} + ( -1 + 2 \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{30} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{31} -\beta_{2} q^{32} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{33} + ( 1 + \beta_{4} + 2 \beta_{7} ) q^{34} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{35} + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{7} ) q^{36} + ( -3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{37} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} - \beta_{6} ) q^{38} + ( \beta_{1} + \beta_{3} - \beta_{7} ) q^{39} + ( \beta_{3} - \beta_{7} ) q^{40} + ( 5 - 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + 4 \beta_{7} ) q^{41} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{42} + ( -\beta_{1} + \beta_{3} - 2 \beta_{5} + 4 \beta_{6} ) q^{43} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{44} + ( -3 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{45} + q^{46} + ( -2 \beta_{5} + 2 \beta_{6} ) q^{47} + ( \beta_{2} - \beta_{5} ) q^{48} + ( 2 + \beta_{4} + 2 \beta_{7} ) q^{49} + ( -1 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{50} + ( 1 + \beta_{1} + \beta_{3} + 3 \beta_{4} - 3 \beta_{7} ) q^{51} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{52} + ( -\beta_{1} + \beta_{3} - 4 \beta_{6} ) q^{53} + ( -2 - \beta_{1} - \beta_{3} + 2 \beta_{4} - 3 \beta_{7} ) q^{54} + ( -2 + 4 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{55} + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{56} + ( -3 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} - 4 \beta_{5} - \beta_{6} ) q^{57} + ( 2 \beta_{5} + 2 \beta_{6} ) q^{58} + ( -4 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{7} ) q^{59} + ( 2 + \beta_{2} + \beta_{5} + 2 \beta_{7} ) q^{60} + ( 3 + 3 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 4 \beta_{7} ) q^{61} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{62} + ( \beta_{1} + 4 \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{63} - q^{64} + ( -2 - \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{5} + \beta_{6} ) q^{65} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{66} + ( 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{67} + ( -\beta_{2} - \beta_{5} + 2 \beta_{6} ) q^{68} + ( -1 + \beta_{4} ) q^{69} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{70} + ( 5 - \beta_{4} + 2 \beta_{7} ) q^{71} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{72} + ( -4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{6} ) q^{73} + ( 2 + 3 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{74} + ( -4 - 7 \beta_{2} - 4 \beta_{3} + 3 \beta_{5} ) q^{75} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + \beta_{7} ) q^{76} + ( 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} ) q^{77} + ( \beta_{1} - \beta_{3} - \beta_{6} ) q^{78} + ( 2 + 4 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} - 6 \beta_{7} ) q^{79} + ( \beta_{1} - \beta_{6} ) q^{80} + ( 5 - \beta_{1} - \beta_{3} - 4 \beta_{4} + 5 \beta_{7} ) q^{81} + ( -2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + 4 \beta_{6} ) q^{82} + ( -3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{83} + ( -2 - \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{7} ) q^{84} + ( -4 - 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{85} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} - 4 \beta_{7} ) q^{86} + ( -2 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 6 \beta_{6} ) q^{87} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{88} + ( -8 - 4 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{7} ) q^{89} + ( -4 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{90} + ( -3 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{91} -\beta_{2} q^{92} + ( \beta_{1} - 6 \beta_{2} - \beta_{3} - 2 \beta_{5} + 3 \beta_{6} ) q^{93} + ( -2 \beta_{4} - 2 \beta_{7} ) q^{94} + ( 2 - 5 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{95} + ( 1 - \beta_{4} ) q^{96} + ( 3 \beta_{2} - \beta_{5} + 2 \beta_{6} ) q^{97} + ( -2 \beta_{2} - \beta_{5} + 2 \beta_{6} ) q^{98} + ( 7 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} + 6q^{6} - 14q^{9} + O(q^{10})$$ $$8q - 8q^{4} + 6q^{6} - 14q^{9} + 10q^{11} - 6q^{14} - 16q^{15} + 8q^{16} + 2q^{19} + 12q^{21} - 6q^{24} - 12q^{25} + 10q^{26} - 4q^{29} - 10q^{30} + 10q^{31} + 10q^{34} - 16q^{35} + 14q^{36} + 46q^{41} - 10q^{44} - 26q^{45} + 8q^{46} + 18q^{49} - 12q^{50} + 14q^{51} - 12q^{54} - 18q^{55} + 6q^{56} - 32q^{59} + 16q^{60} + 18q^{61} - 8q^{64} - 16q^{65} - 6q^{66} - 6q^{69} - 10q^{70} + 38q^{71} + 12q^{74} - 32q^{75} - 2q^{76} + 12q^{79} + 32q^{81} - 12q^{84} - 24q^{85} - 4q^{86} - 60q^{89} - 24q^{90} - 26q^{91} - 4q^{94} + 18q^{95} + 6q^{96} + 54q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 45 x^{4} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} - 47 \nu^{2}$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 47 \nu^{3}$$$$)/14$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{7} - 2 \nu^{5} - 4 \nu^{4} + 221 \nu^{3} - 66 \nu - 76$$$$)/56$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{7} + 6 \nu^{6} + 2 \nu^{5} + 221 \nu^{3} + 254 \nu^{2} + 66 \nu$$$$)/56$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{7} + 2 \nu^{5} + 315 \nu^{3} + 94 \nu$$$$)/28$$ $$\beta_{7}$$ $$=$$ $$($$$$-7 \nu^{7} + 2 \nu^{5} - 315 \nu^{3} + 94 \nu$$$$)/28$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2 \beta_{5} + \beta_{3} - 3 \beta_{2} - \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} - 7 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{7} - 14 \beta_{4} + 7 \beta_{3} + 7 \beta_{1} - 19$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{7} + 7 \beta_{6} - 47 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-47 \beta_{6} + 94 \beta_{5} - 47 \beta_{3} + 127 \beta_{2} + 47 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-47 \beta_{7} + 47 \beta_{6} + 315 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/230\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$51$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 1.83051 − 1.83051i −1.83051 + 1.83051i −0.386289 + 0.386289i 0.386289 − 0.386289i 0.386289 + 0.386289i −0.386289 − 0.386289i −1.83051 − 1.83051i 1.83051 + 1.83051i
1.00000i 2.40815i −1.00000 1.83051 1.28422i −2.40815 0.706585i 1.00000i −2.79917 −1.28422 1.83051i
139.2 1.00000i 0.706585i −1.00000 −1.83051 + 1.28422i 0.706585 2.40815i 1.00000i 2.50074 1.28422 + 1.83051i
139.3 1.00000i 1.44270i −1.00000 −0.386289 2.20245i 1.44270 3.25886i 1.00000i 0.918614 −2.20245 + 0.386289i
139.4 1.00000i 3.25886i −1.00000 0.386289 + 2.20245i 3.25886 1.44270i 1.00000i −7.62018 2.20245 0.386289i
139.5 1.00000i 3.25886i −1.00000 0.386289 2.20245i 3.25886 1.44270i 1.00000i −7.62018 2.20245 + 0.386289i
139.6 1.00000i 1.44270i −1.00000 −0.386289 + 2.20245i 1.44270 3.25886i 1.00000i 0.918614 −2.20245 0.386289i
139.7 1.00000i 0.706585i −1.00000 −1.83051 1.28422i 0.706585 2.40815i 1.00000i 2.50074 1.28422 1.83051i
139.8 1.00000i 2.40815i −1.00000 1.83051 + 1.28422i −2.40815 0.706585i 1.00000i −2.79917 −1.28422 + 1.83051i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 139.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.2.b.b 8
3.b odd 2 1 2070.2.d.f 8
4.b odd 2 1 1840.2.e.e 8
5.b even 2 1 inner 230.2.b.b 8
5.c odd 4 1 1150.2.a.r 4
5.c odd 4 1 1150.2.a.s 4
15.d odd 2 1 2070.2.d.f 8
20.d odd 2 1 1840.2.e.e 8
20.e even 4 1 9200.2.a.cj 4
20.e even 4 1 9200.2.a.cr 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.b.b 8 1.a even 1 1 trivial
230.2.b.b 8 5.b even 2 1 inner
1150.2.a.r 4 5.c odd 4 1
1150.2.a.s 4 5.c odd 4 1
1840.2.e.e 8 4.b odd 2 1
1840.2.e.e 8 20.d odd 2 1
2070.2.d.f 8 3.b odd 2 1
2070.2.d.f 8 15.d odd 2 1
9200.2.a.cj 4 20.e even 4 1
9200.2.a.cr 4 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 19 T_{3}^{6} + 105 T_{3}^{4} + 176 T_{3}^{2} + 64$$ acting on $$S_{2}^{\mathrm{new}}(230, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ $$64 + 176 T^{2} + 105 T^{4} + 19 T^{6} + T^{8}$$
$5$ $$625 + 150 T^{2} + 18 T^{4} + 6 T^{6} + T^{8}$$
$7$ $$64 + 176 T^{2} + 105 T^{4} + 19 T^{6} + T^{8}$$
$11$ $$( -10 + 18 T - 3 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$13$ $$16 + 72 T^{2} + 73 T^{4} + 23 T^{6} + T^{8}$$
$17$ $$400 + 9976 T^{2} + 1441 T^{4} + 67 T^{6} + T^{8}$$
$19$ $$( 370 + 42 T - 53 T^{2} - T^{3} + T^{4} )^{2}$$
$23$ $$( 1 + T^{2} )^{4}$$
$29$ $$( -400 - 360 T - 72 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$31$ $$( 100 + 32 T - 25 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$37$ $$817216 + 128944 T^{2} + 6820 T^{4} + 144 T^{6} + T^{8}$$
$41$ $$( -4016 + 552 T + 115 T^{2} - 23 T^{3} + T^{4} )^{2}$$
$43$ $$341056 + 108528 T^{2} + 7364 T^{4} + 168 T^{6} + T^{8}$$
$47$ $$256 + 1600 T^{2} + 2368 T^{4} + 100 T^{6} + T^{8}$$
$53$ $$( 4232 + 158 T^{2} + T^{4} )^{2}$$
$59$ $$( 1600 - 672 T - 20 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$61$ $$( -226 + 258 T - 61 T^{2} - 9 T^{3} + T^{4} )^{2}$$
$67$ $$27962944 + 2019120 T^{2} + 46084 T^{4} + 376 T^{6} + T^{8}$$
$71$ $$( 32 - 144 T + 93 T^{2} - 19 T^{3} + T^{4} )^{2}$$
$73$ $$54405376 + 3354368 T^{2} + 61408 T^{4} + 432 T^{6} + T^{8}$$
$79$ $$( 5920 + 304 T - 188 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$83$ $$15178816 + 3996208 T^{2} + 74308 T^{4} + 472 T^{6} + T^{8}$$
$89$ $$( -5840 - 744 T + 200 T^{2} + 30 T^{3} + T^{4} )^{2}$$
$97$ $$16 + 2424 T^{2} + 1273 T^{4} + 91 T^{6} + T^{8}$$