Properties

 Label 136.4.a.a Level $136$ Weight $4$ Character orbit 136.a Self dual yes Analytic conductor $8.024$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 136.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$8.02425976078$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + \beta ) q^{3} + ( -6 - 2 \beta ) q^{5} + ( -18 - 3 \beta ) q^{7} + ( -11 + 4 \beta ) q^{9} +O(q^{10})$$ $$q + ( 2 + \beta ) q^{3} + ( -6 - 2 \beta ) q^{5} + ( -18 - 3 \beta ) q^{7} + ( -11 + 4 \beta ) q^{9} + ( -10 - 13 \beta ) q^{11} + ( -42 + 12 \beta ) q^{13} + ( -36 - 10 \beta ) q^{15} + 17 q^{17} + ( 16 + 38 \beta ) q^{19} + ( -72 - 24 \beta ) q^{21} + ( 22 + 25 \beta ) q^{23} + ( -41 + 24 \beta ) q^{25} + ( -28 - 30 \beta ) q^{27} + ( -198 + 26 \beta ) q^{29} + ( 58 - 37 \beta ) q^{31} + ( -176 - 36 \beta ) q^{33} + ( 180 + 54 \beta ) q^{35} + ( -70 - 90 \beta ) q^{37} + ( 60 - 18 \beta ) q^{39} + ( -30 + 4 \beta ) q^{41} + ( 320 + 18 \beta ) q^{43} + ( -30 - 2 \beta ) q^{45} + ( -248 - 60 \beta ) q^{47} + ( 89 + 108 \beta ) q^{49} + ( 34 + 17 \beta ) q^{51} + ( 118 - 28 \beta ) q^{53} + ( 372 + 98 \beta ) q^{55} + ( 488 + 92 \beta ) q^{57} + ( 288 - 86 \beta ) q^{59} + ( -174 - 146 \beta ) q^{61} + ( 54 - 39 \beta ) q^{63} + ( -36 + 12 \beta ) q^{65} + ( 764 - 16 \beta ) q^{67} + ( 344 + 72 \beta ) q^{69} + ( -438 + 151 \beta ) q^{71} + ( -190 + 44 \beta ) q^{73} + ( 206 + 7 \beta ) q^{75} + ( 648 + 264 \beta ) q^{77} + ( 86 + 173 \beta ) q^{79} + ( -119 - 196 \beta ) q^{81} + ( -512 - 226 \beta ) q^{83} + ( -102 - 34 \beta ) q^{85} + ( -84 - 146 \beta ) q^{87} + ( -422 - 332 \beta ) q^{89} + ( 324 - 90 \beta ) q^{91} + ( -328 - 16 \beta ) q^{93} + ( -1008 - 260 \beta ) q^{95} + ( 18 - 56 \beta ) q^{97} + ( -514 + 103 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} - 12 q^{5} - 36 q^{7} - 22 q^{9} + O(q^{10})$$ $$2 q + 4 q^{3} - 12 q^{5} - 36 q^{7} - 22 q^{9} - 20 q^{11} - 84 q^{13} - 72 q^{15} + 34 q^{17} + 32 q^{19} - 144 q^{21} + 44 q^{23} - 82 q^{25} - 56 q^{27} - 396 q^{29} + 116 q^{31} - 352 q^{33} + 360 q^{35} - 140 q^{37} + 120 q^{39} - 60 q^{41} + 640 q^{43} - 60 q^{45} - 496 q^{47} + 178 q^{49} + 68 q^{51} + 236 q^{53} + 744 q^{55} + 976 q^{57} + 576 q^{59} - 348 q^{61} + 108 q^{63} - 72 q^{65} + 1528 q^{67} + 688 q^{69} - 876 q^{71} - 380 q^{73} + 412 q^{75} + 1296 q^{77} + 172 q^{79} - 238 q^{81} - 1024 q^{83} - 204 q^{85} - 168 q^{87} - 844 q^{89} + 648 q^{91} - 656 q^{93} - 2016 q^{95} + 36 q^{97} - 1028 q^{99} + O(q^{100})$$

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
 −1.73205 1.73205
0 −1.46410 0 0.928203 0 −7.60770 0 −24.8564 0
1.2 0 5.46410 0 −12.9282 0 −28.3923 0 2.85641 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
$$2$$ $$1$$
$$17$$ $$-1$$

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 136.4.a.a 2
3.b odd 2 1 1224.4.a.d 2
4.b odd 2 1 272.4.a.f 2
8.b even 2 1 1088.4.a.n 2
8.d odd 2 1 1088.4.a.p 2
12.b even 2 1 2448.4.a.z 2
17.b even 2 1 2312.4.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.a.a 2 1.a even 1 1 trivial
272.4.a.f 2 4.b odd 2 1
1088.4.a.n 2 8.b even 2 1
1088.4.a.p 2 8.d odd 2 1
1224.4.a.d 2 3.b odd 2 1
2312.4.a.b 2 17.b even 2 1
2448.4.a.z 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 4 T_{3} - 8$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(136))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-8 - 4 T + T^{2}$$
$5$ $$-12 + 12 T + T^{2}$$
$7$ $$216 + 36 T + T^{2}$$
$11$ $$-1928 + 20 T + T^{2}$$
$13$ $$36 + 84 T + T^{2}$$
$17$ $$( -17 + T )^{2}$$
$19$ $$-17072 - 32 T + T^{2}$$
$23$ $$-7016 - 44 T + T^{2}$$
$29$ $$31092 + 396 T + T^{2}$$
$31$ $$-13064 - 116 T + T^{2}$$
$37$ $$-92300 + 140 T + T^{2}$$
$41$ $$708 + 60 T + T^{2}$$
$43$ $$98512 - 640 T + T^{2}$$
$47$ $$18304 + 496 T + T^{2}$$
$53$ $$4516 - 236 T + T^{2}$$
$59$ $$-5808 - 576 T + T^{2}$$
$61$ $$-225516 + 348 T + T^{2}$$
$67$ $$580624 - 1528 T + T^{2}$$
$71$ $$-81768 + 876 T + T^{2}$$
$73$ $$12868 + 380 T + T^{2}$$
$79$ $$-351752 - 172 T + T^{2}$$
$83$ $$-350768 + 1024 T + T^{2}$$
$89$ $$-1144604 + 844 T + T^{2}$$
$97$ $$-37308 - 36 T + T^{2}$$