# Properties

 Label 240.10.a.r Level $240$ Weight $10$ Character orbit 240.a Self dual yes Analytic conductor $123.609$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 240.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$123.608600679$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{241})$$ Defining polynomial: $$x^{2} - x - 60$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 48\sqrt{241}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 81 q^{3} + 625 q^{5} + ( -7056 - 7 \beta ) q^{7} + 6561 q^{9} +O(q^{10})$$ $$q + 81 q^{3} + 625 q^{5} + ( -7056 - 7 \beta ) q^{7} + 6561 q^{9} + ( 10756 + 74 \beta ) q^{11} + ( 12142 - 167 \beta ) q^{13} + 50625 q^{15} + ( -78478 + 235 \beta ) q^{17} + ( 47948 + 179 \beta ) q^{19} + ( -571536 - 567 \beta ) q^{21} + ( 367632 - 821 \beta ) q^{23} + 390625 q^{25} + 531441 q^{27} + ( -1339106 - 5268 \beta ) q^{29} + ( -5391216 + 4773 \beta ) q^{31} + ( 871236 + 5994 \beta ) q^{33} + ( -4410000 - 4375 \beta ) q^{35} + ( 10984166 - 6203 \beta ) q^{37} + ( 983502 - 13527 \beta ) q^{39} + ( 13030186 - 15378 \beta ) q^{41} + ( 3595580 + 21948 \beta ) q^{43} + 4100625 q^{45} + ( 15790120 - 61579 \beta ) q^{47} + ( 36641465 + 98784 \beta ) q^{49} + ( -6356718 + 19035 \beta ) q^{51} + ( 1565558 - 5554 \beta ) q^{53} + ( 6722500 + 46250 \beta ) q^{55} + ( 3883788 + 14499 \beta ) q^{57} + ( 17747332 + 135886 \beta ) q^{59} + ( 170748670 + 29694 \beta ) q^{61} + ( -46294416 - 45927 \beta ) q^{63} + ( 7588750 - 104375 \beta ) q^{65} + ( 144097908 - 32080 \beta ) q^{67} + ( 29778192 - 66501 \beta ) q^{69} + ( -105143032 + 142040 \beta ) q^{71} + ( -116331542 + 36506 \beta ) q^{73} + 31640625 q^{75} + ( -363521088 - 597436 \beta ) q^{77} + ( 12377520 - 595285 \beta ) q^{79} + 43046721 q^{81} + ( 186041076 - 226884 \beta ) q^{83} + ( -49048750 + 146875 \beta ) q^{85} + ( -108467586 - 426708 \beta ) q^{87} + ( -213819558 + 282786 \beta ) q^{89} + ( 563429664 + 1093358 \beta ) q^{91} + ( -436688496 + 386613 \beta ) q^{93} + ( 29967500 + 111875 \beta ) q^{95} + ( 885829442 - 420340 \beta ) q^{97} + ( 70570116 + 485514 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 162q^{3} + 1250q^{5} - 14112q^{7} + 13122q^{9} + O(q^{10})$$ $$2q + 162q^{3} + 1250q^{5} - 14112q^{7} + 13122q^{9} + 21512q^{11} + 24284q^{13} + 101250q^{15} - 156956q^{17} + 95896q^{19} - 1143072q^{21} + 735264q^{23} + 781250q^{25} + 1062882q^{27} - 2678212q^{29} - 10782432q^{31} + 1742472q^{33} - 8820000q^{35} + 21968332q^{37} + 1967004q^{39} + 26060372q^{41} + 7191160q^{43} + 8201250q^{45} + 31580240q^{47} + 73282930q^{49} - 12713436q^{51} + 3131116q^{53} + 13445000q^{55} + 7767576q^{57} + 35494664q^{59} + 341497340q^{61} - 92588832q^{63} + 15177500q^{65} + 288195816q^{67} + 59556384q^{69} - 210286064q^{71} - 232663084q^{73} + 63281250q^{75} - 727042176q^{77} + 24755040q^{79} + 86093442q^{81} + 372082152q^{83} - 98097500q^{85} - 216935172q^{87} - 427639116q^{89} + 1126859328q^{91} - 873376992q^{93} + 59935000q^{95} + 1771658884q^{97} + 141140232q^{99} + O(q^{100})$$

## 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
 8.26209 −7.26209
0 81.0000 0 625.000 0 −12272.1 0 6561.00 0
1.2 0 81.0000 0 625.000 0 −1839.88 0 6561.00 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## 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 240.10.a.r 2
4.b odd 2 1 15.10.a.d 2
12.b even 2 1 45.10.a.d 2
20.d odd 2 1 75.10.a.f 2
20.e even 4 2 75.10.b.f 4
60.h even 2 1 225.10.a.k 2
60.l odd 4 2 225.10.b.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.d 2 4.b odd 2 1
45.10.a.d 2 12.b even 2 1
75.10.a.f 2 20.d odd 2 1
75.10.b.f 4 20.e even 4 2
225.10.a.k 2 60.h even 2 1
225.10.b.i 4 60.l odd 4 2
240.10.a.r 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 14112 T_{7} + 22579200$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(240))$$.