# Properties

 Label 276.2.g.a Level $276$ Weight $2$ Character orbit 276.g Analytic conductor $2.204$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$276 = 2^{2} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 276.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.20387109579$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.14453810176.7 Defining polynomial: $$x^{8} + 9 x^{6} + 15 x^{4} - 30 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + \beta_{5} q^{5} -\beta_{7} q^{7} + ( -1 - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} + \beta_{5} q^{5} -\beta_{7} q^{7} + ( -1 - \beta_{3} ) q^{9} + ( \beta_{5} + \beta_{6} ) q^{11} + ( -\beta_{2} + \beta_{4} ) q^{13} + ( -\beta_{1} + \beta_{6} ) q^{15} + ( -\beta_{5} - 2 \beta_{6} ) q^{17} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{19} + ( \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{21} + ( \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{23} + ( 1 + 2 \beta_{2} - 2 \beta_{4} ) q^{25} + ( 2 + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{27} + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{29} + ( -4 - 3 \beta_{2} + 3 \beta_{4} ) q^{31} + ( -\beta_{1} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{33} + ( 2 \beta_{3} + 2 \beta_{4} ) q^{35} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} ) q^{37} + ( 2 - \beta_{3} ) q^{39} + ( -3 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{41} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{43} + ( \beta_{1} - 2 \beta_{5} + \beta_{7} ) q^{45} + ( -\beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{47} + ( -3 + 2 \beta_{2} - 2 \beta_{4} ) q^{49} + ( \beta_{1} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{51} + ( 2 \beta_{5} - 3 \beta_{6} ) q^{53} + 4 q^{55} + ( -4 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{57} + ( -2 \beta_{2} - 2 \beta_{4} ) q^{59} + 2 \beta_{7} q^{61} + ( \beta_{1} + 3 \beta_{5} + 2 \beta_{6} ) q^{63} + ( -\beta_{5} + \beta_{6} ) q^{65} + 3 \beta_{7} q^{67} + ( -1 - \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{69} + ( \beta_{2} + \beta_{4} ) q^{71} + ( 2 + 3 \beta_{2} - 3 \beta_{4} ) q^{73} + ( -4 + 2 \beta_{3} + \beta_{4} ) q^{75} + ( -4 \beta_{2} + 4 \beta_{3} ) q^{77} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{79} + ( -5 + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{81} + ( -3 \beta_{5} + \beta_{6} ) q^{83} + ( -2 + 2 \beta_{2} - 2 \beta_{4} ) q^{85} + ( -2 - 6 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{87} + ( -2 \beta_{5} + \beta_{6} ) q^{89} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{91} + ( 6 - 3 \beta_{3} - 4 \beta_{4} ) q^{93} + ( 4 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} ) q^{95} + ( -2 \beta_{1} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{97} + ( -\beta_{1} + \beta_{6} + 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{3} - 6q^{9} + O(q^{10})$$ $$8q + 2q^{3} - 6q^{9} + 4q^{13} + 8q^{27} - 20q^{31} + 18q^{39} - 32q^{49} + 32q^{55} - 14q^{69} + 4q^{73} - 34q^{75} - 46q^{81} - 24q^{85} - 2q^{87} + 46q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 9 x^{6} + 15 x^{4} - 30 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{6} - 20 \nu^{4} - 67 \nu^{2} - 97$$$$)/103$$ $$\beta_{3}$$ $$=$$ $$($$$$4 \nu^{6} + 61 \nu^{4} + 158 \nu^{2} - 214$$$$)/103$$ $$\beta_{4}$$ $$=$$ $$($$$$-6 \nu^{6} - 40 \nu^{4} - 31 \nu^{2} + 115$$$$)/103$$ $$\beta_{5}$$ $$=$$ $$($$$$-23 \nu^{7} - 119 \nu^{5} - 33 \nu^{3} + 458 \nu$$$$)/824$$ $$\beta_{6}$$ $$=$$ $$($$$$-27 \nu^{7} - 283 \nu^{5} - 397 \nu^{3} + 2114 \nu$$$$)/824$$ $$\beta_{7}$$ $$=$$ $$($$$$-29 \nu^{7} - 365 \nu^{5} - 1403 \nu^{3} - 354 \nu$$$$)/824$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - 2 \beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} + 3 \beta_{6} - \beta_{5} - 4 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{4} + 3 \beta_{3} + 8 \beta_{2} + 16$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{7} - 19 \beta_{6} + 16 \beta_{5} + 21 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-9 \beta_{4} - 20 \beta_{3} - 43 \beta_{2} - 72$$ $$\nu^{7}$$ $$=$$ $$($$$$-23 \beta_{7} + 94 \beta_{6} - 153 \beta_{5} - 83 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/276\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$139$$ $$185$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
137.1
 −0.331077 − 2.33494i 0.331077 + 2.33494i −0.331077 + 2.33494i 0.331077 − 2.33494i −1.06789 − 0.545948i 1.06789 + 0.545948i −1.06789 + 0.545948i 1.06789 − 0.545948i
0 −0.780776 1.54609i 0 −3.02045 0 2.62238i 0 −1.78078 + 2.41430i 0
137.2 0 −0.780776 1.54609i 0 3.02045 0 2.62238i 0 −1.78078 + 2.41430i 0
137.3 0 −0.780776 + 1.54609i 0 −3.02045 0 2.62238i 0 −1.78078 2.41430i 0
137.4 0 −0.780776 + 1.54609i 0 3.02045 0 2.62238i 0 −1.78078 2.41430i 0
137.5 0 1.28078 1.16602i 0 −0.936426 0 3.88884i 0 0.280776 2.98683i 0
137.6 0 1.28078 1.16602i 0 0.936426 0 3.88884i 0 0.280776 2.98683i 0
137.7 0 1.28078 + 1.16602i 0 −0.936426 0 3.88884i 0 0.280776 + 2.98683i 0
137.8 0 1.28078 + 1.16602i 0 0.936426 0 3.88884i 0 0.280776 + 2.98683i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 137.8 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
23.b odd 2 1 inner
69.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.2.g.a 8
3.b odd 2 1 inner 276.2.g.a 8
4.b odd 2 1 1104.2.m.b 8
12.b even 2 1 1104.2.m.b 8
23.b odd 2 1 inner 276.2.g.a 8
69.c even 2 1 inner 276.2.g.a 8
92.b even 2 1 1104.2.m.b 8
276.h odd 2 1 1104.2.m.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.g.a 8 1.a even 1 1 trivial
276.2.g.a 8 3.b odd 2 1 inner
276.2.g.a 8 23.b odd 2 1 inner
276.2.g.a 8 69.c even 2 1 inner
1104.2.m.b 8 4.b odd 2 1
1104.2.m.b 8 12.b even 2 1
1104.2.m.b 8 92.b even 2 1
1104.2.m.b 8 276.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(276, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 9 - 3 T + 2 T^{2} - T^{3} + T^{4} )^{2}$$
$5$ $$( 8 - 10 T^{2} + T^{4} )^{2}$$
$7$ $$( 104 + 22 T^{2} + T^{4} )^{2}$$
$11$ $$( 32 - 20 T^{2} + T^{4} )^{2}$$
$13$ $$( -4 - T + T^{2} )^{4}$$
$17$ $$( 8 - 58 T^{2} + T^{4} )^{2}$$
$19$ $$( 1664 + 82 T^{2} + T^{4} )^{2}$$
$23$ $$279841 - 10580 T^{2} + 70 T^{4} - 20 T^{6} + T^{8}$$
$29$ $$( 832 + 59 T^{2} + T^{4} )^{2}$$
$31$ $$( -32 + 5 T + T^{2} )^{4}$$
$37$ $$( 416 + 92 T^{2} + T^{4} )^{2}$$
$41$ $$( 208 + 115 T^{2} + T^{4} )^{2}$$
$43$ $$( 1664 + 82 T^{2} + T^{4} )^{2}$$
$47$ $$( 208 + 123 T^{2} + T^{4} )^{2}$$
$53$ $$( 8192 - 190 T^{2} + T^{4} )^{2}$$
$59$ $$( 832 + 60 T^{2} + T^{4} )^{2}$$
$61$ $$( 1664 + 88 T^{2} + T^{4} )^{2}$$
$67$ $$( 8424 + 198 T^{2} + T^{4} )^{2}$$
$71$ $$( 52 + 15 T^{2} + T^{4} )^{2}$$
$73$ $$( -38 - T + T^{2} )^{4}$$
$79$ $$( 416 + 146 T^{2} + T^{4} )^{2}$$
$83$ $$( 32 - 116 T^{2} + T^{4} )^{2}$$
$89$ $$( 128 - 62 T^{2} + T^{4} )^{2}$$
$97$ $$( 1664 + 116 T^{2} + T^{4} )^{2}$$