# Properties

 Label 36.7.d.c Level $36$ Weight $7$ Character orbit 36.d Analytic conductor $8.282$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 36.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.28194701031$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-15})$$ Defining polynomial: $$x^{2} - x + 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 4) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 - \beta ) q^{2} + ( -56 + 4 \beta ) q^{4} -10 q^{5} + 40 \beta q^{7} + ( 352 + 48 \beta ) q^{8} +O(q^{10})$$ $$q + ( -2 - \beta ) q^{2} + ( -56 + 4 \beta ) q^{4} -10 q^{5} + 40 \beta q^{7} + ( 352 + 48 \beta ) q^{8} + ( 20 + 10 \beta ) q^{10} + 124 \beta q^{11} + 1466 q^{13} + ( 2400 - 80 \beta ) q^{14} + ( 2176 - 448 \beta ) q^{16} + 4766 q^{17} + 972 \beta q^{19} + ( 560 - 40 \beta ) q^{20} + ( 7440 - 248 \beta ) q^{22} + 1352 \beta q^{23} -15525 q^{25} + ( -2932 - 1466 \beta ) q^{26} + ( -9600 - 2240 \beta ) q^{28} -25498 q^{29} + 5408 \beta q^{31} + ( -31232 - 1280 \beta ) q^{32} + ( -9532 - 4766 \beta ) q^{34} -400 \beta q^{35} + 1994 q^{37} + ( 58320 - 1944 \beta ) q^{38} + ( -3520 - 480 \beta ) q^{40} -29362 q^{41} -2780 \beta q^{43} + ( -29760 - 6944 \beta ) q^{44} + ( 81120 - 2704 \beta ) q^{46} + 976 \beta q^{47} + 21649 q^{49} + ( 31050 + 15525 \beta ) q^{50} + ( -82096 + 5864 \beta ) q^{52} + 192854 q^{53} -1240 \beta q^{55} + ( -115200 + 14080 \beta ) q^{56} + ( 50996 + 25498 \beta ) q^{58} + 10124 \beta q^{59} -10918 q^{61} + ( 324480 - 10816 \beta ) q^{62} + ( -14336 + 33792 \beta ) q^{64} -14660 q^{65} -50884 \beta q^{67} + ( -266896 + 19064 \beta ) q^{68} + ( -24000 + 800 \beta ) q^{70} -68712 \beta q^{71} + 288626 q^{73} + ( -3988 - 1994 \beta ) q^{74} + ( -233280 - 54432 \beta ) q^{76} -297600 q^{77} -40112 \beta q^{79} + ( -21760 + 4480 \beta ) q^{80} + ( 58724 + 29362 \beta ) q^{82} + 26356 \beta q^{83} -47660 q^{85} + ( -166800 + 5560 \beta ) q^{86} + ( -357120 + 43648 \beta ) q^{88} -310738 q^{89} + 58640 \beta q^{91} + ( -324480 - 75712 \beta ) q^{92} + ( 58560 - 1952 \beta ) q^{94} -9720 \beta q^{95} -1457086 q^{97} + ( -43298 - 21649 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} - 112 q^{4} - 20 q^{5} + 704 q^{8} + O(q^{10})$$ $$2 q - 4 q^{2} - 112 q^{4} - 20 q^{5} + 704 q^{8} + 40 q^{10} + 2932 q^{13} + 4800 q^{14} + 4352 q^{16} + 9532 q^{17} + 1120 q^{20} + 14880 q^{22} - 31050 q^{25} - 5864 q^{26} - 19200 q^{28} - 50996 q^{29} - 62464 q^{32} - 19064 q^{34} + 3988 q^{37} + 116640 q^{38} - 7040 q^{40} - 58724 q^{41} - 59520 q^{44} + 162240 q^{46} + 43298 q^{49} + 62100 q^{50} - 164192 q^{52} + 385708 q^{53} - 230400 q^{56} + 101992 q^{58} - 21836 q^{61} + 648960 q^{62} - 28672 q^{64} - 29320 q^{65} - 533792 q^{68} - 48000 q^{70} + 577252 q^{73} - 7976 q^{74} - 466560 q^{76} - 595200 q^{77} - 43520 q^{80} + 117448 q^{82} - 95320 q^{85} - 333600 q^{86} - 714240 q^{88} - 621476 q^{89} - 648960 q^{92} + 117120 q^{94} - 2914172 q^{97} - 86596 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/36\mathbb{Z}\right)^\times$$.

 $$n$$ $$19$$ $$29$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.5 + 1.93649i 0.5 − 1.93649i
−2.00000 7.74597i 0 −56.0000 + 30.9839i −10.0000 0 309.839i 352.000 + 371.806i 0 20.0000 + 77.4597i
19.2 −2.00000 + 7.74597i 0 −56.0000 30.9839i −10.0000 0 309.839i 352.000 371.806i 0 20.0000 77.4597i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.7.d.c 2
3.b odd 2 1 4.7.b.a 2
4.b odd 2 1 inner 36.7.d.c 2
8.b even 2 1 576.7.g.h 2
8.d odd 2 1 576.7.g.h 2
12.b even 2 1 4.7.b.a 2
15.d odd 2 1 100.7.b.c 2
15.e even 4 2 100.7.d.a 4
24.f even 2 1 64.7.c.c 2
24.h odd 2 1 64.7.c.c 2
48.i odd 4 2 256.7.d.f 4
48.k even 4 2 256.7.d.f 4
60.h even 2 1 100.7.b.c 2
60.l odd 4 2 100.7.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.7.b.a 2 3.b odd 2 1
4.7.b.a 2 12.b even 2 1
36.7.d.c 2 1.a even 1 1 trivial
36.7.d.c 2 4.b odd 2 1 inner
64.7.c.c 2 24.f even 2 1
64.7.c.c 2 24.h odd 2 1
100.7.b.c 2 15.d odd 2 1
100.7.b.c 2 60.h even 2 1
100.7.d.a 4 15.e even 4 2
100.7.d.a 4 60.l odd 4 2
256.7.d.f 4 48.i odd 4 2
256.7.d.f 4 48.k even 4 2
576.7.g.h 2 8.b even 2 1
576.7.g.h 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 10$$ acting on $$S_{7}^{\mathrm{new}}(36, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 + 4 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 10 + T )^{2}$$
$7$ $$96000 + T^{2}$$
$11$ $$922560 + T^{2}$$
$13$ $$( -1466 + T )^{2}$$
$17$ $$( -4766 + T )^{2}$$
$19$ $$56687040 + T^{2}$$
$23$ $$109674240 + T^{2}$$
$29$ $$( 25498 + T )^{2}$$
$31$ $$1754787840 + T^{2}$$
$37$ $$( -1994 + T )^{2}$$
$41$ $$( 29362 + T )^{2}$$
$43$ $$463704000 + T^{2}$$
$47$ $$57154560 + T^{2}$$
$53$ $$( -192854 + T )^{2}$$
$59$ $$6149722560 + T^{2}$$
$61$ $$( 10918 + T )^{2}$$
$67$ $$155350887360 + T^{2}$$
$71$ $$283280336640 + T^{2}$$
$73$ $$( -288626 + T )^{2}$$
$79$ $$96538352640 + T^{2}$$
$83$ $$41678324160 + T^{2}$$
$89$ $$( 310738 + T )^{2}$$
$97$ $$( 1457086 + T )^{2}$$