# Properties

 Label 1216.2.b.f Level $1216$ Weight $2$ Character orbit 1216.b Analytic conductor $9.710$ Analytic rank $0$ Dimension $12$ CM discriminant -152 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 18 x^{10} + 207 x^{8} + 1014 x^{6} + 1065 x^{4} - 5508 x^{2} + 8464$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + \beta_{7} q^{7} + ( -3 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} + \beta_{7} q^{7} + ( -3 + \beta_{2} ) q^{9} + \beta_{5} q^{13} -\beta_{8} q^{17} -\beta_{10} q^{19} + ( -\beta_{5} - \beta_{6} - \beta_{11} ) q^{21} -\beta_{1} q^{23} + 5 q^{25} + ( -3 \beta_{4} - 2 \beta_{10} ) q^{27} + ( -\beta_{5} + \beta_{6} ) q^{29} -\beta_{11} q^{37} + ( \beta_{1} + \beta_{3} + 2 \beta_{7} ) q^{39} -3 \beta_{3} q^{47} + ( -7 + 2 \beta_{2} + \beta_{8} ) q^{49} + ( \beta_{9} - 2 \beta_{10} ) q^{51} + ( \beta_{5} + 2 \beta_{6} ) q^{53} + ( -2 \beta_{2} + \beta_{8} ) q^{57} + ( -2 \beta_{4} - \beta_{9} ) q^{59} + ( \beta_{1} + 5 \beta_{3} - 5 \beta_{7} ) q^{63} + ( 2 \beta_{4} - \beta_{9} ) q^{67} + ( \beta_{5} - 2 \beta_{6} - \beta_{11} ) q^{69} + ( -\beta_{2} - 2 \beta_{8} ) q^{73} + 5 \beta_{4} q^{75} + ( 9 - 4 \beta_{2} + 2 \beta_{8} ) q^{81} + ( -2 \beta_{1} - 7 \beta_{3} - \beta_{7} ) q^{87} + ( 4 \beta_{4} + \beta_{9} - 2 \beta_{10} ) q^{91} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 36q^{9} + O(q^{10})$$ $$12q - 36q^{9} + 60q^{25} - 84q^{49} + 108q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 18 x^{10} + 207 x^{8} + 1014 x^{6} + 1065 x^{4} - 5508 x^{2} + 8464$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-857 \nu^{11} - 108754 \nu^{9} - 1882257 \nu^{7} - 19921624 \nu^{5} - 91847935 \nu^{3} + 56141760 \nu$$$$)/60769588$$ $$\beta_{2}$$ $$=$$ $$($$$$4848 \nu^{10} + 67487 \nu^{8} + 612280 \nu^{6} + 1941799 \nu^{4} - 6158396 \nu^{2} + 20243266$$$$)/15192397$$ $$\beta_{3}$$ $$=$$ $$($$$$-171 \nu^{11} - 3142 \nu^{9} - 37173 \nu^{7} - 203958 \nu^{5} - 397511 \nu^{3} + 101088 \nu$$$$)/589996$$ $$\beta_{4}$$ $$=$$ $$($$$$6496 \nu^{10} + 113219 \nu^{8} + 1399303 \nu^{6} + 7488246 \nu^{4} + 15491689 \nu^{2} - 17025842$$$$)/15192397$$ $$\beta_{5}$$ $$=$$ $$($$$$26777 \nu^{11} + 555190 \nu^{9} + 6273577 \nu^{7} + 34391608 \nu^{5} + 16535587 \nu^{3} - 306111640 \nu$$$$)/60769588$$ $$\beta_{6}$$ $$=$$ $$($$$$-14647 \nu^{11} - 250356 \nu^{9} - 2790517 \nu^{7} - 12120654 \nu^{5} - 6158189 \nu^{3} + 112303816 \nu$$$$)/30384794$$ $$\beta_{7}$$ $$=$$ $$($$$$-2741 \nu^{11} - 55406 \nu^{9} - 678481 \nu^{7} - 4014812 \nu^{5} - 9469483 \nu^{3} + 3424640 \nu$$$$)/5524508$$ $$\beta_{8}$$ $$=$$ $$($$$$11493 \nu^{10} + 188193 \nu^{8} + 1977982 \nu^{6} + 6662232 \nu^{4} - 20729120 \nu^{2} - 100079394$$$$)/15192397$$ $$\beta_{9}$$ $$=$$ $$($$$$11527 \nu^{10} + 356749 \nu^{8} + 4352527 \nu^{6} + 33217533 \nu^{4} + 72677113 \nu^{2} - 83718298$$$$)/15192397$$ $$\beta_{10}$$ $$=$$ $$($$$$-1347 \nu^{10} - 22950 \nu^{8} - 268517 \nu^{6} - 1266390 \nu^{4} - 2440467 \nu^{2} + 2608806$$$$)/1321078$$ $$\beta_{11}$$ $$=$$ $$($$$$9 \nu^{11} + 146 \nu^{9} + 1695 \nu^{7} + 7074 \nu^{5} + 4013 \nu^{3} - 67464 \nu$$$$)/9476$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{10} - \beta_{8} - 3 \beta_{4} - 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{11} - 3 \beta_{7} - 15 \beta_{6} - 6 \beta_{5} + 8 \beta_{3} - 3 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$18 \beta_{10} + \beta_{9} - 2 \beta_{8} + 32 \beta_{4} + 17 \beta_{2} - 30$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{11} + 66 \beta_{7} + 35 \beta_{6} + 19 \beta_{5} - 125 \beta_{3} - 19 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-4 \beta_{10} - 21 \beta_{9} + 69 \beta_{8} + 87 \beta_{4} - 243 \beta_{2} + 768$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$462 \beta_{11} - 1141 \beta_{7} + 889 \beta_{6} + 4 \beta_{5} + 1974 \beta_{3} + 425 \beta_{1}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-1674 \beta_{10} + 319 \beta_{9} - 282 \beta_{8} - 4612 \beta_{4} + 741 \beta_{2} - 2950$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-6592 \beta_{11} - 3717 \beta_{7} - 15345 \beta_{6} - 1998 \beta_{5} + 7204 \beta_{3} + 855 \beta_{1}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$14058 \beta_{10} - 2189 \beta_{9} - 5258 \beta_{8} + 36586 \beta_{4} + 19833 \beta_{2} - 59886$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$9031 \beta_{11} + 82638 \beta_{7} + 20333 \beta_{6} + 2233 \beta_{5} - 147851 \beta_{3} - 27605 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
607.1
 1.70027 + 2.78183i −1.70027 − 2.78183i −1.13617 + 0.418247i 1.13617 − 0.418247i −0.564101 + 2.36358i 0.564101 − 2.36358i 0.564101 + 2.36358i −0.564101 − 2.36358i 1.13617 + 0.418247i −1.13617 − 0.418247i −1.70027 + 2.78183i 1.70027 − 2.78183i
0 3.40054i 0 0 0 4.62947i 0 −8.56365 0
607.2 0 3.40054i 0 0 0 4.62947i 0 −8.56365 0
607.3 0 2.27234i 0 0 0 4.53419i 0 −2.16351 0
607.4 0 2.27234i 0 0 0 4.53419i 0 −2.16351 0
607.5 0 1.12820i 0 0 0 0.0952793i 0 1.72716 0
607.6 0 1.12820i 0 0 0 0.0952793i 0 1.72716 0
607.7 0 1.12820i 0 0 0 0.0952793i 0 1.72716 0
607.8 0 1.12820i 0 0 0 0.0952793i 0 1.72716 0
607.9 0 2.27234i 0 0 0 4.53419i 0 −2.16351 0
607.10 0 2.27234i 0 0 0 4.53419i 0 −2.16351 0
607.11 0 3.40054i 0 0 0 4.62947i 0 −8.56365 0
607.12 0 3.40054i 0 0 0 4.62947i 0 −8.56365 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 607.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
152.g odd 2 1 CM by $$\Q(\sqrt{-38})$$
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner
152.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.b.f 12
4.b odd 2 1 inner 1216.2.b.f 12
8.b even 2 1 inner 1216.2.b.f 12
8.d odd 2 1 inner 1216.2.b.f 12
19.b odd 2 1 inner 1216.2.b.f 12
76.d even 2 1 inner 1216.2.b.f 12
152.b even 2 1 inner 1216.2.b.f 12
152.g odd 2 1 CM 1216.2.b.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.b.f 12 1.a even 1 1 trivial
1216.2.b.f 12 4.b odd 2 1 inner
1216.2.b.f 12 8.b even 2 1 inner
1216.2.b.f 12 8.d odd 2 1 inner
1216.2.b.f 12 19.b odd 2 1 inner
1216.2.b.f 12 76.d even 2 1 inner
1216.2.b.f 12 152.b even 2 1 inner
1216.2.b.f 12 152.g odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1216, [\chi])$$:

 $$T_{3}^{6} + 18 T_{3}^{4} + 81 T_{3}^{2} + 76$$ $$T_{5}$$ $$T_{13}^{6} - 78 T_{13}^{4} + 1521 T_{13}^{2} - 76$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$( 76 + 81 T^{2} + 18 T^{4} + T^{6} )^{2}$$
$5$ $$T^{12}$$
$7$ $$( 4 + 441 T^{2} + 42 T^{4} + T^{6} )^{2}$$
$11$ $$T^{12}$$
$13$ $$( -76 + 1521 T^{2} - 78 T^{4} + T^{6} )^{2}$$
$17$ $$( -138 - 51 T + T^{3} )^{4}$$
$19$ $$( 19 + T^{2} )^{6}$$
$23$ $$( 30276 + 4761 T^{2} + 138 T^{4} + T^{6} )^{2}$$
$29$ $$( -82764 + 7569 T^{2} - 174 T^{4} + T^{6} )^{2}$$
$31$ $$T^{12}$$
$37$ $$( -76 + T^{2} )^{6}$$
$41$ $$T^{12}$$
$43$ $$T^{12}$$
$47$ $$( 36 + T^{2} )^{6}$$
$53$ $$( -197676 + 25281 T^{2} - 318 T^{4} + T^{6} )^{2}$$
$59$ $$( 246924 + 31329 T^{2} + 354 T^{4} + T^{6} )^{2}$$
$61$ $$T^{12}$$
$67$ $$( 870124 + 40401 T^{2} + 402 T^{4} + T^{6} )^{2}$$
$71$ $$T^{12}$$
$73$ $$( -394 - 219 T + T^{3} )^{4}$$
$79$ $$T^{12}$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$T^{12}$$