# Properties

 Label 336.2.h.b Level 336 Weight 2 Character orbit 336.h Analytic conductor 2.683 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.56070144.2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} + ( \beta_{3} + \beta_{5} ) q^{5} + \beta_{4} q^{7} + ( 2 + \beta_{3} - \beta_{5} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{6} q^{3} + ( \beta_{3} + \beta_{5} ) q^{5} + \beta_{4} q^{7} + ( 2 + \beta_{3} - \beta_{5} + \beta_{7} ) q^{9} + 2 \beta_{1} q^{11} + ( -4 - 3 \beta_{3} + 3 \beta_{5} ) q^{13} + ( \beta_{1} + \beta_{2} + 5 \beta_{4} - \beta_{6} ) q^{15} -2 \beta_{7} q^{17} + ( -\beta_{2} + \beta_{6} ) q^{19} + \beta_{3} q^{21} + ( 2 \beta_{2} + 2 \beta_{6} ) q^{23} + ( -5 - 2 \beta_{3} + 2 \beta_{5} ) q^{25} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{27} + 2 \beta_{7} q^{29} + ( -2 \beta_{2} - 8 \beta_{4} + 2 \beta_{6} ) q^{31} + ( 2 + 2 \beta_{3} - 4 \beta_{5} - 2 \beta_{7} ) q^{33} + ( -\beta_{2} - \beta_{6} ) q^{35} + ( 6 - 2 \beta_{3} + 2 \beta_{5} ) q^{37} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{39} + ( -2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{41} -2 \beta_{4} q^{43} + ( 2 + 5 \beta_{3} - \beta_{5} - 2 \beta_{7} ) q^{45} - q^{49} + ( -2 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{51} + ( 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{53} + ( 6 \beta_{2} + 4 \beta_{4} - 6 \beta_{6} ) q^{55} + ( -1 + \beta_{3} - \beta_{5} + \beta_{7} ) q^{57} + ( -4 \beta_{1} - \beta_{2} - \beta_{6} ) q^{59} + ( -4 - \beta_{3} + \beta_{5} ) q^{61} + ( \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{63} + ( -4 \beta_{3} - 4 \beta_{5} + 6 \beta_{7} ) q^{65} + ( 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{67} + ( 10 + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{69} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{6} ) q^{71} + ( -6 + 2 \beta_{3} - 2 \beta_{5} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{6} ) q^{75} -2 \beta_{7} q^{77} -12 \beta_{4} q^{79} + ( 3 + 2 \beta_{3} - 6 \beta_{5} ) q^{81} + ( -4 \beta_{1} - \beta_{2} - \beta_{6} ) q^{83} + ( -4 - 6 \beta_{3} + 6 \beta_{5} ) q^{85} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{87} + ( 2 \beta_{3} + 2 \beta_{5} - 6 \beta_{7} ) q^{89} + ( -3 \beta_{2} - 4 \beta_{4} + 3 \beta_{6} ) q^{91} + ( -2 - 6 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{93} + 2 \beta_{1} q^{95} + ( 6 + 4 \beta_{3} - 4 \beta_{5} ) q^{97} + ( 6 \beta_{2} - 6 \beta_{4} - 2 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{9} + O(q^{10})$$ $$8q + 8q^{9} - 8q^{13} - 4q^{21} - 24q^{25} - 8q^{33} + 64q^{37} - 8q^{45} - 8q^{49} - 16q^{57} - 24q^{61} + 64q^{69} - 64q^{73} - 8q^{81} + 16q^{85} + 16q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{4} + 2 \nu^{3} - 7 \nu^{2} + 6 \nu - 6$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{7} - \nu^{6} - 25 \nu^{5} - 46 \nu^{4} - 5 \nu^{3} - 95 \nu^{2} - 9 \nu + 19$$$$)/37$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} - 8 \nu^{6} + 22 \nu^{5} - 146 \nu^{4} + 256 \nu^{3} - 427 \nu^{2} + 372 \nu - 218$$$$)/37$$ $$\beta_{4}$$ $$=$$ $$($$$$-6 \nu^{7} + 21 \nu^{6} - 67 \nu^{5} + 115 \nu^{4} - 117 \nu^{3} + 71 \nu^{2} + 41 \nu - 29$$$$)/37$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 29 \nu^{6} - 89 \nu^{5} + 261 \nu^{4} - 373 \nu^{3} + 498 \nu^{2} - 257 \nu + 152$$$$)/37$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{7} - 36 \nu^{6} + 136 \nu^{5} - 361 \nu^{4} + 634 \nu^{3} - 756 \nu^{2} + 564 \nu - 167$$$$)/37$$ $$\beta_{7}$$ $$=$$ $$($$$$42 \nu^{7} - 147 \nu^{6} + 543 \nu^{5} - 990 \nu^{4} + 1559 \nu^{3} - 1422 \nu^{2} + 971 \nu - 278$$$$)/37$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{3} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{2} - 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} - \beta_{5} + 7 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 8$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{7} - 7 \beta_{6} - 10 \beta_{5} + 15 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_{1} + 13$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-4 \beta_{7} - 5 \beta_{6} - 7 \beta_{5} - 21 \beta_{4} + 18 \beta_{3} - 20 \beta_{2} - 5 \beta_{1} + 46$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-17 \beta_{7} + 37 \beta_{6} + 41 \beta_{5} - 101 \beta_{4} + 23 \beta_{3} - 23 \beta_{2} + 7 \beta_{1} - 19$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$4 \beta_{7} + 63 \beta_{6} + 80 \beta_{5} + \beta_{4} - 74 \beta_{3} + 77 \beta_{2} + 42 \beta_{1} - 237$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 0.5 − 2.19293i 0.5 + 2.19293i 0.5 + 1.56488i 0.5 − 1.56488i 0.5 − 0.564882i 0.5 + 0.564882i 0.5 + 1.19293i 0.5 − 1.19293i
0 −1.69293 0.366025i 0 3.38587i 0 1.00000i 0 2.73205 + 1.23931i 0
239.2 0 −1.69293 + 0.366025i 0 3.38587i 0 1.00000i 0 2.73205 1.23931i 0
239.3 0 −1.06488 1.36603i 0 2.12976i 0 1.00000i 0 −0.732051 + 2.90931i 0
239.4 0 −1.06488 + 1.36603i 0 2.12976i 0 1.00000i 0 −0.732051 2.90931i 0
239.5 0 1.06488 1.36603i 0 2.12976i 0 1.00000i 0 −0.732051 2.90931i 0
239.6 0 1.06488 + 1.36603i 0 2.12976i 0 1.00000i 0 −0.732051 + 2.90931i 0
239.7 0 1.69293 0.366025i 0 3.38587i 0 1.00000i 0 2.73205 1.23931i 0
239.8 0 1.69293 + 0.366025i 0 3.38587i 0 1.00000i 0 2.73205 + 1.23931i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.h.b 8
3.b odd 2 1 inner 336.2.h.b 8
4.b odd 2 1 inner 336.2.h.b 8
7.b odd 2 1 2352.2.h.o 8
8.b even 2 1 1344.2.h.g 8
8.d odd 2 1 1344.2.h.g 8
12.b even 2 1 inner 336.2.h.b 8
21.c even 2 1 2352.2.h.o 8
24.f even 2 1 1344.2.h.g 8
24.h odd 2 1 1344.2.h.g 8
28.d even 2 1 2352.2.h.o 8
84.h odd 2 1 2352.2.h.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.h.b 8 1.a even 1 1 trivial
336.2.h.b 8 3.b odd 2 1 inner
336.2.h.b 8 4.b odd 2 1 inner
336.2.h.b 8 12.b even 2 1 inner
1344.2.h.g 8 8.b even 2 1
1344.2.h.g 8 8.d odd 2 1
1344.2.h.g 8 24.f even 2 1
1344.2.h.g 8 24.h odd 2 1
2352.2.h.o 8 7.b odd 2 1
2352.2.h.o 8 21.c even 2 1
2352.2.h.o 8 28.d even 2 1
2352.2.h.o 8 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 16 T_{5}^{2} + 52$$ acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 4 T^{2} + 10 T^{4} - 36 T^{6} + 81 T^{8}$$
$5$ $$( 1 - 4 T^{2} + 42 T^{4} - 100 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$( 1 + 4 T^{2} + 54 T^{4} + 484 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + 2 T + 26 T^{3} + 169 T^{4} )^{4}$$
$17$ $$( 1 - 28 T^{2} + 582 T^{4} - 8092 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 68 T^{2} + 1866 T^{4} - 24548 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 28 T^{2} + 1062 T^{4} + 14812 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 76 T^{2} + 2934 T^{4} - 63916 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 28 T^{2} + 390 T^{4} - 26908 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 16 T + 126 T^{2} - 592 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 76 T^{2} + 3078 T^{4} - 127756 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 82 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 47 T^{2} )^{8}$$
$53$ $$( 1 - 124 T^{2} + 7734 T^{4} - 348316 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 76 T^{2} + 8298 T^{4} + 264556 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 6 T + 128 T^{2} + 366 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$( 1 - 16 T + 67 T^{2} )^{4}( 1 + 16 T + 67 T^{2} )^{4}$$
$71$ $$( 1 + 196 T^{2} + 17958 T^{4} + 988036 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 + 16 T + 198 T^{2} + 1168 T^{3} + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 14 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 172 T^{2} + 21066 T^{4} + 1184908 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 20 T^{2} + 15750 T^{4} + 158420 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 4 T + 150 T^{2} - 388 T^{3} + 9409 T^{4} )^{4}$$