# Properties

 Label 1232.2.q.n Level $1232$ Weight $2$ Character orbit 1232.q Analytic conductor $9.838$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1232.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.83756952902$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 9 x^{6} - 2 x^{5} + 66 x^{4} - 9 x^{3} + 136 x^{2} + 15 x + 225$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 616) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} - \beta_{4} ) q^{3} + ( 1 - \beta_{1} + \beta_{2} - \beta_{6} ) q^{5} + \beta_{7} q^{7} + ( -2 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} - \beta_{4} ) q^{3} + ( 1 - \beta_{1} + \beta_{2} - \beta_{6} ) q^{5} + \beta_{7} q^{7} + ( -2 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{9} + \beta_{2} q^{11} + \beta_{3} q^{13} + ( -1 - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{15} + ( -2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{7} ) q^{17} + ( -5 - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{19} + ( 3 + 3 \beta_{1} + \beta_{2} + 2 \beta_{6} ) q^{21} + ( 2 - \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{23} + ( -2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{25} + ( 4 - 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{27} + ( -2 \beta_{3} + \beta_{6} ) q^{29} + ( \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{31} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{33} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{35} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{6} ) q^{37} + ( -\beta_{1} - 6 \beta_{2} + \beta_{4} ) q^{39} + ( 4 + 2 \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{41} + ( 3 - \beta_{3} ) q^{43} + ( 2 \beta_{1} + 3 \beta_{2} + \beta_{5} - \beta_{7} ) q^{45} + ( -3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{47} + ( -2 - \beta_{1} - 5 \beta_{2} - \beta_{3} + \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{49} + ( -6 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + 4 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{51} + ( \beta_{1} + \beta_{5} - \beta_{7} ) q^{53} + ( -1 + \beta_{6} ) q^{55} + ( -4 - 3 \beta_{3} - 5 \beta_{6} ) q^{57} + ( \beta_{1} + 9 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{59} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{61} + ( -\beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{63} + ( -1 + \beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{65} + ( 3 \beta_{1} - 7 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{67} + ( -6 + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{69} + ( 1 + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{71} + ( -2 \beta_{1} - \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{73} + ( -7 + \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{75} + ( -1 - \beta_{3} + \beta_{5} ) q^{77} + ( 8 + 3 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} ) q^{79} + ( 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{81} + ( -3 - 2 \beta_{3} - \beta_{4} + \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{83} + ( -6 + \beta_{4} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{85} + ( 3 \beta_{1} + 13 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{87} + ( -2 - 4 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{89} + ( 2 + 2 \beta_{1} + 3 \beta_{2} - \beta_{6} ) q^{91} + ( -9 - 6 \beta_{1} - 10 \beta_{2} + \beta_{3} + \beta_{4} - 6 \beta_{6} ) q^{93} + ( 3 \beta_{1} - 8 \beta_{2} + 4 \beta_{4} + \beta_{5} - \beta_{7} ) q^{95} + ( 2 - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{7} ) q^{97} + ( 2 - \beta_{3} + \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{3} + 4 q^{5} + q^{7} - 10 q^{9} + O(q^{10})$$ $$8 q - 2 q^{3} + 4 q^{5} + q^{7} - 10 q^{9} - 4 q^{11} - 4 q^{13} + 2 q^{15} - 3 q^{17} - 13 q^{19} + 20 q^{21} + 10 q^{23} - 2 q^{25} + 46 q^{27} + 8 q^{29} - q^{31} - 2 q^{33} - 13 q^{35} + 8 q^{37} + 22 q^{39} + 18 q^{41} + 28 q^{43} - 11 q^{45} - 11 q^{47} + 11 q^{49} - 12 q^{51} + q^{53} - 8 q^{55} - 20 q^{57} - 33 q^{59} + 9 q^{61} - 26 q^{63} + q^{65} + 25 q^{67} - 46 q^{69} - 6 q^{71} - 16 q^{75} - 2 q^{77} + 28 q^{79} - 4 q^{81} - 10 q^{83} - 54 q^{85} - 47 q^{87} - 3 q^{89} + 4 q^{91} - 38 q^{93} + 25 q^{95} + 40 q^{97} + 20 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 9 x^{6} - 2 x^{5} + 66 x^{4} - 9 x^{3} + 136 x^{2} + 15 x + 225$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$27 \nu^{7} - 330 \nu^{6} + 198 \nu^{5} - 2474 \nu^{4} + 2112 \nu^{3} - 21978 \nu^{2} + 3122 \nu - 44550$$$$)/36705$$ $$\beta_{3}$$ $$=$$ $$($$$$9 \nu^{7} - 110 \nu^{6} + 66 \nu^{5} - 9 \nu^{4} + 704 \nu^{3} + 15 \nu^{2} + 225 \nu + 14514$$$$)/7341$$ $$\beta_{4}$$ $$=$$ $$($$$$18 \nu^{7} - 220 \nu^{6} + 132 \nu^{5} - 2465 \nu^{4} + 1408 \nu^{3} - 14652 \nu^{2} + 2897 \nu - 29700$$$$)/7341$$ $$\beta_{5}$$ $$=$$ $$($$$$-424 \nu^{7} - 1615 \nu^{6} - 11266 \nu^{5} - 6917 \nu^{4} - 63074 \nu^{3} - 21914 \nu^{2} - 247959 \nu + 112320$$$$)/110115$$ $$\beta_{6}$$ $$=$$ $$($$$$66 \nu^{7} + 9 \nu^{6} + 484 \nu^{5} - 66 \nu^{4} + 4347 \nu^{3} + 110 \nu^{2} + 1650 \nu + 1215$$$$)/7341$$ $$\beta_{7}$$ $$=$$ $$($$$$-1993 \nu^{7} - 2830 \nu^{6} - 10537 \nu^{5} - 12689 \nu^{4} - 55298 \nu^{3} - 102833 \nu^{2} + 87207 \nu - 51705$$$$)/110115$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} - 5 \beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} + 5 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 1$$ $$\nu^{4}$$ $$=$$ $$-9 \beta_{4} + 30 \beta_{2} + \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-9 \beta_{7} - 30 \beta_{6} - 18 \beta_{5} + 10 \beta_{4} + 19 \beta_{3} - 14 \beta_{2} - 30 \beta_{1} + 5$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{7} + 14 \beta_{6} + \beta_{5} - \beta_{4} - 68 \beta_{3} + 128$$ $$\nu^{7}$$ $$=$$ $$-66 \beta_{7} + 66 \beta_{5} - 18 \beta_{4} + 141 \beta_{2} + 196 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times$$.

 $$n$$ $$309$$ $$353$$ $$463$$ $$673$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 −1.35283 − 2.34317i 1.24132 + 2.15004i 0.805004 + 1.39431i −0.693499 − 1.20118i −1.35283 + 2.34317i 1.24132 − 2.15004i 0.805004 − 1.39431i −0.693499 + 1.20118i
0 −1.66029 2.87570i 0 −0.852828 + 1.47714i 0 1.47905 + 2.19372i 0 −4.01312 + 6.95092i 0
177.2 0 −1.08177 1.87368i 0 1.74132 3.01606i 0 −2.52609 0.786673i 0 −0.840445 + 1.45569i 0
177.3 0 0.703938 + 1.21926i 0 1.30500 2.26033i 0 2.64229 + 0.135337i 0 0.508942 0.881513i 0
177.4 0 1.03812 + 1.79807i 0 −0.193499 + 0.335150i 0 −1.09525 2.40841i 0 −0.655380 + 1.13515i 0
529.1 0 −1.66029 + 2.87570i 0 −0.852828 1.47714i 0 1.47905 2.19372i 0 −4.01312 6.95092i 0
529.2 0 −1.08177 + 1.87368i 0 1.74132 + 3.01606i 0 −2.52609 + 0.786673i 0 −0.840445 1.45569i 0
529.3 0 0.703938 1.21926i 0 1.30500 + 2.26033i 0 2.64229 0.135337i 0 0.508942 + 0.881513i 0
529.4 0 1.03812 1.79807i 0 −0.193499 0.335150i 0 −1.09525 + 2.40841i 0 −0.655380 1.13515i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.q.n 8
4.b odd 2 1 616.2.q.d 8
7.c even 3 1 inner 1232.2.q.n 8
7.c even 3 1 8624.2.a.cz 4
7.d odd 6 1 8624.2.a.cr 4
28.f even 6 1 4312.2.a.bd 4
28.g odd 6 1 616.2.q.d 8
28.g odd 6 1 4312.2.a.y 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.q.d 8 4.b odd 2 1
616.2.q.d 8 28.g odd 6 1
1232.2.q.n 8 1.a even 1 1 trivial
1232.2.q.n 8 7.c even 3 1 inner
4312.2.a.y 4 28.g odd 6 1
4312.2.a.bd 4 28.f even 6 1
8624.2.a.cr 4 7.d odd 6 1
8624.2.a.cz 4 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1232, [\chi])$$:

 $$T_{3}^{8} + 2 T_{3}^{7} + 13 T_{3}^{6} + 78 T_{3}^{4} - 3 T_{3}^{3} + 270 T_{3}^{2} - 189 T_{3} + 441$$ $$T_{13}^{4} + 2 T_{13}^{3} - 9 T_{13}^{2} - 9 T_{13} + 21$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$441 - 189 T + 270 T^{2} - 3 T^{3} + 78 T^{4} + 13 T^{6} + 2 T^{7} + T^{8}$$
$5$ $$36 + 90 T + 243 T^{2} + 3 T^{3} + 63 T^{4} - 18 T^{5} + 19 T^{6} - 4 T^{7} + T^{8}$$
$7$ $$2401 - 343 T - 245 T^{2} + 7 T^{3} + 5 T^{4} + T^{5} - 5 T^{6} - T^{7} + T^{8}$$
$11$ $$( 1 + T + T^{2} )^{4}$$
$13$ $$( 21 - 9 T - 9 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$17$ $$8100 + 13230 T + 17289 T^{2} + 7596 T^{3} + 2835 T^{4} + 150 T^{5} + 57 T^{6} + 3 T^{7} + T^{8}$$
$19$ $$256 + 2032 T + 16513 T^{2} - 2632 T^{3} + 2243 T^{4} + 566 T^{5} + 145 T^{6} + 13 T^{7} + T^{8}$$
$23$ $$3136 - 952 T + 1465 T^{2} - 763 T^{3} + 667 T^{4} - 244 T^{5} + 79 T^{6} - 10 T^{7} + T^{8}$$
$29$ $$( 211 - 13 T - 51 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$31$ $$25600 + 2080 T + 10729 T^{2} - 1178 T^{3} + 4183 T^{4} - 92 T^{5} + 67 T^{6} + T^{7} + T^{8}$$
$37$ $$16384 + 13312 T + 12352 T^{2} + 800 T^{3} + 848 T^{4} - 112 T^{5} + 76 T^{6} - 8 T^{7} + T^{8}$$
$41$ $$( 492 + 257 T - 54 T^{2} - 9 T^{3} + T^{4} )^{2}$$
$43$ $$( 48 - 99 T + 63 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$47$ $$11222500 + 5892650 T + 2651881 T^{2} + 305888 T^{3} + 40123 T^{4} + 2066 T^{5} + 253 T^{6} + 11 T^{7} + T^{8}$$
$53$ $$62500 + 31250 T + 30625 T^{2} - 7000 T^{3} + 3475 T^{4} - 190 T^{5} + 61 T^{6} - T^{7} + T^{8}$$
$59$ $$164025 - 497340 T + 1652569 T^{2} + 465126 T^{3} + 87330 T^{4} + 9325 T^{5} + 732 T^{6} + 33 T^{7} + T^{8}$$
$61$ $$34574400 - 11454240 T + 2947984 T^{2} - 386352 T^{3} + 44148 T^{4} - 2600 T^{5} + 225 T^{6} - 9 T^{7} + T^{8}$$
$67$ $$26460736 - 3683104 T + 1191664 T^{2} - 162688 T^{3} + 40468 T^{4} - 4732 T^{5} + 493 T^{6} - 25 T^{7} + T^{8}$$
$71$ $$( -90 - 147 T - 48 T^{2} + 3 T^{3} + T^{4} )^{2}$$
$73$ $$19289664 + 188856 T + 937345 T^{2} - 9159 T^{3} + 40977 T^{4} - 86 T^{5} + 213 T^{6} + T^{8}$$
$79$ $$12201049 - 841813 T + 718258 T^{2} - 150059 T^{3} + 45962 T^{4} - 5774 T^{5} + 595 T^{6} - 28 T^{7} + T^{8}$$
$83$ $$( 3242 - 263 T - 126 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$89$ $$10381284 - 1285578 T + 603837 T^{2} + 35730 T^{3} + 17019 T^{4} + 384 T^{5} + 147 T^{6} + 3 T^{7} + T^{8}$$
$97$ $$( 1687 + 859 T + 15 T^{2} - 20 T^{3} + T^{4} )^{2}$$