# Properties

 Label 240.10.a.m 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: $$\Q(\sqrt{4729})$$ Defining polynomial: $$x^{2} - x - 1182$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ 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 $$\beta = 4\sqrt{4729}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -81 q^{3} -625 q^{5} + ( 5936 - 7 \beta ) q^{7} + 6561 q^{9} +O(q^{10})$$ $$q -81 q^{3} -625 q^{5} + ( 5936 - 7 \beta ) q^{7} + 6561 q^{9} + ( -17744 - 244 \beta ) q^{11} + ( 71838 + 173 \beta ) q^{13} + 50625 q^{15} + ( 192578 - 275 \beta ) q^{17} + ( 201648 + 121 \beta ) q^{19} + ( -480816 + 567 \beta ) q^{21} + ( -111852 - 8121 \beta ) q^{23} + 390625 q^{25} -531441 q^{27} + ( -37286 - 1552 \beta ) q^{29} + ( 2513564 + 9467 \beta ) q^{31} + ( 1437264 + 19764 \beta ) q^{33} + ( -3710000 + 4375 \beta ) q^{35} + ( 2686814 + 21837 \beta ) q^{37} + ( -5818878 - 14013 \beta ) q^{39} + ( 7105666 - 58762 \beta ) q^{41} + ( -13874460 + 19048 \beta ) q^{43} -4100625 q^{45} + ( -47983220 + 53921 \beta ) q^{47} + ( -1409975 - 83104 \beta ) q^{49} + ( -15598818 + 22275 \beta ) q^{51} + ( -32152798 - 116234 \beta ) q^{53} + ( 11090000 + 152500 \beta ) q^{55} + ( -16333488 - 9801 \beta ) q^{57} + ( -93931568 - 201676 \beta ) q^{59} + ( 77040030 + 282086 \beta ) q^{61} + ( 38946096 - 45927 \beta ) q^{63} + ( -44898750 - 108125 \beta ) q^{65} + ( -16796188 - 930520 \beta ) q^{67} + ( 9060012 + 657801 \beta ) q^{69} + ( 114135488 + 882640 \beta ) q^{71} + ( -16561158 + 810026 \beta ) q^{73} -31640625 q^{75} + ( 23905728 - 1324176 \beta ) q^{77} + ( 466203380 - 224755 \beta ) q^{79} + 43046721 q^{81} + ( -103520076 + 395136 \beta ) q^{83} + ( -120361250 + 171875 \beta ) q^{85} + ( 3020166 + 125712 \beta ) q^{87} + ( 112259082 + 1163274 \beta ) q^{89} + ( 334801264 + 524062 \beta ) q^{91} + ( -203598684 - 766827 \beta ) q^{93} + ( -126030000 - 75625 \beta ) q^{95} + ( 193567298 - 5504880 \beta ) q^{97} + ( -116418384 - 1600884 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 162q^{3} - 1250q^{5} + 11872q^{7} + 13122q^{9} + O(q^{10})$$ $$2q - 162q^{3} - 1250q^{5} + 11872q^{7} + 13122q^{9} - 35488q^{11} + 143676q^{13} + 101250q^{15} + 385156q^{17} + 403296q^{19} - 961632q^{21} - 223704q^{23} + 781250q^{25} - 1062882q^{27} - 74572q^{29} + 5027128q^{31} + 2874528q^{33} - 7420000q^{35} + 5373628q^{37} - 11637756q^{39} + 14211332q^{41} - 27748920q^{43} - 8201250q^{45} - 95966440q^{47} - 2819950q^{49} - 31197636q^{51} - 64305596q^{53} + 22180000q^{55} - 32666976q^{57} - 187863136q^{59} + 154080060q^{61} + 77892192q^{63} - 89797500q^{65} - 33592376q^{67} + 18120024q^{69} + 228270976q^{71} - 33122316q^{73} - 63281250q^{75} + 47811456q^{77} + 932406760q^{79} + 86093442q^{81} - 207040152q^{83} - 240722500q^{85} + 6040332q^{87} + 224518164q^{89} + 669602528q^{91} - 407197368q^{93} - 252060000q^{95} + 387134596q^{97} - 232836768q^{99} + O(q^{100})$$

## 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
 34.8839 −33.8839
0 −81.0000 0 −625.000 0 4010.50 0 6561.00 0
1.2 0 −81.0000 0 −625.000 0 7861.50 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.m 2
4.b odd 2 1 15.10.a.c 2
12.b even 2 1 45.10.a.e 2
20.d odd 2 1 75.10.a.g 2
20.e even 4 2 75.10.b.e 4
60.h even 2 1 225.10.a.j 2
60.l odd 4 2 225.10.b.g 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 11872 T_{7} + 31528560$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(240))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 81 T )^{2}$$
$5$ $$( 1 + 625 T )^{2}$$
$7$ $$1 - 11872 T + 112235774 T^{2} - 479078022304 T^{3} + 1628413597910449 T^{4}$$
$11$ $$1 + 35488 T + 526013014 T^{2} + 83678847658208 T^{3} + 5559917313492231481 T^{4}$$
$13$ $$1 - 143676 T + 24105149134 T^{2} - 1523612051915148 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 - 385156 T + 268539949078 T^{2} - 45674832160078532 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$1 - 403296 T + 684929514838 T^{2} - 130138657763479584 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$
$23$ $$1 + 223704 T - 1375273107794 T^{2} + 402925054979918952 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$1 + 74572 T + 28833430018078 T^{2} + 1081826889712503068 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$31$ $$1 - 5027128 T + 52415931233342 T^{2} -$$$$13\!\cdots\!88$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$
$37$ $$1 - 5373628 T + 231061724951934 T^{2} -$$$$69\!\cdots\!56$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$1 - 14211332 T + 443988635955862 T^{2} -$$$$46\!\cdots\!52$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$
$43$ $$1 + 27748920 T + 1170232974699430 T^{2} +$$$$13\!\cdots\!60$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 95966440 T + 4320659216802910 T^{2} +$$$$10\!\cdots\!80$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 + 64305596 T + 6611083028543086 T^{2} +$$$$21\!\cdots\!68$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$1 + 187863136 T + 23071633420288438 T^{2} +$$$$16\!\cdots\!04$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$
$61$ $$1 - 154080060 T + 23302683905802238 T^{2} -$$$$18\!\cdots\!60$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$
$67$ $$1 + 33592376 T - 10819815556424362 T^{2} +$$$$91\!\cdots\!72$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$1 - 228270976 T + 45777616900481806 T^{2} -$$$$10\!\cdots\!56$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$
$73$ $$1 + 33122316 T + 68371107952007926 T^{2} +$$$$19\!\cdots\!08$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$1 - 932406760 T + 453226630902929438 T^{2} -$$$$11\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$
$83$ $$1 + 207040152 T + 372783310330485238 T^{2} +$$$$38\!\cdots\!56$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$1 - 224518164 T + 610925899926766678 T^{2} -$$$$78\!\cdots\!76$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$
$97$ $$1 - 387134596 T - 734969029248610362 T^{2} -$$$$29\!\cdots\!32$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$