# Properties

 Label 819.2.y.h Level $819$ Weight $2$ Character orbit 819.y Analytic conductor $6.540$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.y (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 35 x^{8} + 295 x^{4} + 169$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{7} q^{2} + ( \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{4} -\beta_{11} q^{5} + ( -\beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} ) q^{7} + ( -\beta_{3} - \beta_{6} ) q^{8} +O(q^{10})$$ $$q -\beta_{7} q^{2} + ( \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{4} -\beta_{11} q^{5} + ( -\beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} ) q^{7} + ( -\beta_{3} - \beta_{6} ) q^{8} + ( -\beta_{2} + \beta_{9} - \beta_{11} ) q^{10} + ( 1 + \beta_{4} - \beta_{8} ) q^{11} + ( -\beta_{2} + \beta_{5} - \beta_{10} - \beta_{11} ) q^{13} + ( 1 - \beta_{3} + 2 \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{14} + ( 2 \beta_{3} - \beta_{7} + \beta_{8} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{5} + \beta_{9} ) q^{17} + ( -\beta_{5} + 2 \beta_{11} ) q^{19} + ( -\beta_{1} - \beta_{2} + \beta_{9} + \beta_{10} ) q^{20} + ( -1 - \beta_{3} ) q^{22} -3 \beta_{6} q^{23} + ( -3 \beta_{4} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{25} + ( -2 \beta_{2} - \beta_{5} + \beta_{10} - \beta_{11} ) q^{26} + ( -1 + \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{11} ) q^{28} + ( 3 - 2 \beta_{3} + 3 \beta_{7} - 3 \beta_{8} ) q^{29} + ( -2 \beta_{5} - \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{31} + ( -1 + 3 \beta_{3} + \beta_{4} - 3 \beta_{6} - 4 \beta_{7} ) q^{32} + ( -\beta_{1} - \beta_{2} + 2 \beta_{9} + 2 \beta_{10} ) q^{34} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{35} + ( 1 - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{8} ) q^{37} + ( 2 \beta_{2} - \beta_{9} + 2 \beta_{11} ) q^{38} + ( -\beta_{1} + \beta_{5} ) q^{40} + ( -\beta_{9} + \beta_{10} ) q^{41} + ( 3 \beta_{4} - 6 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} ) q^{43} + ( -1 + \beta_{4} ) q^{44} + ( 3 + 3 \beta_{4} - 3 \beta_{8} ) q^{46} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 2 \beta_{2} - \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{49} + ( 1 + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{8} ) q^{50} + ( -2 \beta_{1} - \beta_{2} + \beta_{5} + 2 \beta_{10} ) q^{52} + ( 4 - 5 \beta_{3} + 2 \beta_{7} - 2 \beta_{8} ) q^{53} + \beta_{10} q^{55} + ( \beta_{2} + 4 \beta_{4} - \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{56} + ( -1 - 3 \beta_{3} + \beta_{4} + 3 \beta_{6} + 5 \beta_{7} ) q^{58} + ( -3 \beta_{1} - \beta_{2} - 2 \beta_{9} - 2 \beta_{10} ) q^{59} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{5} - \beta_{11} ) q^{61} + ( \beta_{1} - 2 \beta_{2} + \beta_{5} + 5 \beta_{9} - 2 \beta_{11} ) q^{62} + ( 2 \beta_{4} - 4 \beta_{7} - 4 \beta_{8} ) q^{64} + ( 3 + 2 \beta_{4} + 2 \beta_{7} + 3 \beta_{8} ) q^{65} + ( 5 - 2 \beta_{3} - 5 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{67} + ( 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{5} + 3 \beta_{11} ) q^{68} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{70} + ( 2 - \beta_{3} - 2 \beta_{4} + \beta_{6} + 5 \beta_{7} ) q^{71} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{9} - 3 \beta_{10} ) q^{73} + ( 3 + \beta_{3} - \beta_{7} + \beta_{8} ) q^{74} + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{9} - 2 \beta_{10} ) q^{76} + ( -\beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{77} + ( 3 + 2 \beta_{3} + 6 \beta_{7} - 6 \beta_{8} ) q^{79} + ( -2 \beta_{5} + \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{80} + ( \beta_{1} + \beta_{2} + \beta_{5} + \beta_{11} ) q^{82} + ( 2 \beta_{5} - 5 \beta_{9} + 5 \beta_{10} - 3 \beta_{11} ) q^{83} + ( -3 + 5 \beta_{3} - 3 \beta_{4} + 5 \beta_{6} + 7 \beta_{8} ) q^{85} + ( 2 - 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{6} - 11 \beta_{8} ) q^{86} + ( -2 \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{88} + ( -\beta_{1} + 4 \beta_{2} + \beta_{9} + \beta_{10} ) q^{89} + ( -5 + \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{91} + ( -3 + 3 \beta_{3} ) q^{92} + ( 2 \beta_{2} - 5 \beta_{10} - 2 \beta_{11} ) q^{94} + ( -5 \beta_{4} - 3 \beta_{7} - 3 \beta_{8} ) q^{95} + ( \beta_{5} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{97} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{2} - 8 q^{7} - 4 q^{8} + O(q^{10})$$ $$12 q + 4 q^{2} - 8 q^{7} - 4 q^{8} + 8 q^{11} - 8 q^{14} + 16 q^{16} - 16 q^{22} - 20 q^{28} + 4 q^{29} + 16 q^{32} + 4 q^{35} + 12 q^{37} - 12 q^{44} + 24 q^{46} + 28 q^{50} + 12 q^{53} - 44 q^{58} + 40 q^{65} + 60 q^{67} + 4 q^{70} + 48 q^{74} - 4 q^{79} + 12 q^{85} - 36 q^{86} - 32 q^{91} - 24 q^{92} + 28 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 35 x^{8} + 295 x^{4} + 169$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{9} + 58 \nu^{5} + 659 \nu$$$$)/194$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{8} - 96 \nu^{4} + 3$$$$)/194$$ $$\beta_{4}$$ $$=$$ $$($$$$-9 \nu^{10} - 328 \nu^{6} - 3409 \nu^{2}$$$$)/2522$$ $$\beta_{5}$$ $$=$$ $$($$$$-9 \nu^{11} - 328 \nu^{7} - 3409 \nu^{3}$$$$)/2522$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{10} + 115 \nu^{6} - 431 \nu^{2}$$$$)/1261$$ $$\beta_{7}$$ $$=$$ $$($$$$-25 \nu^{10} - 39 \nu^{8} - 771 \nu^{6} - 1001 \nu^{4} - 5126 \nu^{2} - 4264$$$$)/2522$$ $$\beta_{8}$$ $$=$$ $$($$$$-25 \nu^{10} + 39 \nu^{8} - 771 \nu^{6} + 1001 \nu^{4} - 5126 \nu^{2} + 4264$$$$)/2522$$ $$\beta_{9}$$ $$=$$ $$($$$$25 \nu^{11} - 39 \nu^{9} + 771 \nu^{7} - 1001 \nu^{5} + 5126 \nu^{3} - 4264 \nu$$$$)/2522$$ $$\beta_{10}$$ $$=$$ $$($$$$-25 \nu^{11} - 39 \nu^{9} - 771 \nu^{7} - 1001 \nu^{5} - 5126 \nu^{3} - 4264 \nu$$$$)/2522$$ $$\beta_{11}$$ $$=$$ $$($$$$-18 \nu^{11} - 656 \nu^{7} - 5557 \nu^{3}$$$$)/1261$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} + \beta_{7} + \beta_{6} - 4 \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} - 4 \beta_{5}$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{8} - 5 \beta_{7} + 6 \beta_{3} - 17$$ $$\nu^{5}$$ $$=$$ $$\beta_{10} + \beta_{9} + 6 \beta_{2} - 17 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-22 \beta_{8} - 22 \beta_{7} - 31 \beta_{6} + 74 \beta_{4}$$ $$\nu^{7}$$ $$=$$ $$-31 \beta_{11} + 9 \beta_{10} - 9 \beta_{9} + 74 \beta_{5}$$ $$\nu^{8}$$ $$=$$ $$-96 \beta_{8} + 96 \beta_{7} - 154 \beta_{3} + 327$$ $$\nu^{9}$$ $$=$$ $$-58 \beta_{10} - 58 \beta_{9} - 154 \beta_{2} + 327 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$423 \beta_{8} + 423 \beta_{7} + 751 \beta_{6} - 1462 \beta_{4}$$ $$\nu^{11}$$ $$=$$ $$751 \beta_{11} - 328 \beta_{10} + 328 \beta_{9} - 1462 \beta_{5}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$379$$ $$703$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 1.33026 + 1.33026i −1.33026 − 1.33026i −1.52891 − 1.52891i 1.52891 + 1.52891i 0.626770 + 0.626770i −0.626770 − 0.626770i 1.33026 − 1.33026i −1.33026 + 1.33026i −1.52891 + 1.52891i 1.52891 − 1.52891i 0.626770 − 0.626770i −0.626770 + 0.626770i
−0.854638 0.854638i 0 0.539189i −0.612999 + 0.612999i 0 −2.64571 0.0148122i −2.17009 + 2.17009i 0 1.04778
307.2 −0.854638 0.854638i 0 0.539189i 0.612999 0.612999i 0 0.0148122 + 2.64571i −2.17009 + 2.17009i 0 −1.04778
307.3 0.403032 + 0.403032i 0 1.67513i −1.03221 + 1.03221i 0 2.60707 + 0.450747i 1.48119 1.48119i 0 −0.832030
307.4 0.403032 + 0.403032i 0 1.67513i 1.03221 1.03221i 0 −0.450747 2.60707i 1.48119 1.48119i 0 0.832030
307.5 1.45161 + 1.45161i 0 2.21432i −2.01464 + 2.01464i 0 −2.38948 + 1.13594i −0.311108 + 0.311108i 0 −5.84892
307.6 1.45161 + 1.45161i 0 2.21432i 2.01464 2.01464i 0 −1.13594 + 2.38948i −0.311108 + 0.311108i 0 5.84892
811.1 −0.854638 + 0.854638i 0 0.539189i −0.612999 0.612999i 0 −2.64571 + 0.0148122i −2.17009 2.17009i 0 1.04778
811.2 −0.854638 + 0.854638i 0 0.539189i 0.612999 + 0.612999i 0 0.0148122 2.64571i −2.17009 2.17009i 0 −1.04778
811.3 0.403032 0.403032i 0 1.67513i −1.03221 1.03221i 0 2.60707 0.450747i 1.48119 + 1.48119i 0 −0.832030
811.4 0.403032 0.403032i 0 1.67513i 1.03221 + 1.03221i 0 −0.450747 + 2.60707i 1.48119 + 1.48119i 0 0.832030
811.5 1.45161 1.45161i 0 2.21432i −2.01464 2.01464i 0 −2.38948 1.13594i −0.311108 0.311108i 0 −5.84892
811.6 1.45161 1.45161i 0 2.21432i 2.01464 + 2.01464i 0 −1.13594 2.38948i −0.311108 0.311108i 0 5.84892
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 811.6 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.b odd 2 1 inner
13.d odd 4 1 inner
91.i even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.y.h 12
3.b odd 2 1 91.2.i.a 12
7.b odd 2 1 inner 819.2.y.h 12
13.d odd 4 1 inner 819.2.y.h 12
21.c even 2 1 91.2.i.a 12
21.g even 6 2 637.2.bc.a 24
21.h odd 6 2 637.2.bc.a 24
39.f even 4 1 91.2.i.a 12
91.i even 4 1 inner 819.2.y.h 12
273.o odd 4 1 91.2.i.a 12
273.cb odd 12 2 637.2.bc.a 24
273.cd even 12 2 637.2.bc.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.i.a 12 3.b odd 2 1
91.2.i.a 12 21.c even 2 1
91.2.i.a 12 39.f even 4 1
91.2.i.a 12 273.o odd 4 1
637.2.bc.a 24 21.g even 6 2
637.2.bc.a 24 21.h odd 6 2
637.2.bc.a 24 273.cb odd 12 2
637.2.bc.a 24 273.cd even 12 2
819.2.y.h 12 1.a even 1 1 trivial
819.2.y.h 12 7.b odd 2 1 inner
819.2.y.h 12 13.d odd 4 1 inner
819.2.y.h 12 91.i even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$:

 $$T_{2}^{6} - 2 T_{2}^{5} + 2 T_{2}^{4} + 2 T_{2}^{3} + 4 T_{2}^{2} - 4 T_{2} + 2$$ $$T_{5}^{12} + 71 T_{5}^{8} + 339 T_{5}^{4} + 169$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 - 4 T + 4 T^{2} + 2 T^{3} + 2 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$3$ $$T^{12}$$
$5$ $$169 + 339 T^{4} + 71 T^{8} + T^{12}$$
$7$ $$117649 + 134456 T + 76832 T^{2} + 24696 T^{3} - 637 T^{4} - 4928 T^{5} - 2624 T^{6} - 704 T^{7} - 13 T^{8} + 72 T^{9} + 32 T^{10} + 8 T^{11} + T^{12}$$
$11$ $$( 2 - 2 T^{3} + 8 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$13$ $$4826809 - 571220 T^{2} - 27885 T^{4} + 6760 T^{6} - 165 T^{8} - 20 T^{10} + T^{12}$$
$17$ $$( -9386 + 1686 T^{2} - 82 T^{4} + T^{6} )^{2}$$
$19$ $$169 + 357527 T^{4} + 1203 T^{8} + T^{12}$$
$23$ $$( 729 + 891 T^{2} + 63 T^{4} + T^{6} )^{2}$$
$29$ $$( 85 - 33 T - T^{2} + T^{3} )^{4}$$
$31$ $$316733209 + 30446939 T^{4} + 12031 T^{8} + T^{12}$$
$37$ $$( 1250 - 400 T + 64 T^{2} - 2 T^{3} + 18 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$41$ $$43264 + 6080 T^{4} + 160 T^{8} + T^{12}$$
$43$ $$( 83521 + 10115 T^{2} + 195 T^{4} + T^{6} )^{2}$$
$47$ $$2339947129 + 15722487 T^{4} + 8003 T^{8} + T^{12}$$
$53$ $$( -113 - 55 T - 3 T^{2} + T^{3} )^{4}$$
$59$ $$2704 + 158720 T^{4} + 28060 T^{8} + T^{12}$$
$61$ $$( 260000 + 12496 T^{2} + 196 T^{4} + T^{6} )^{2}$$
$67$ $$( 35912 + 25192 T + 8836 T^{2} - 3088 T^{3} + 450 T^{4} - 30 T^{5} + T^{6} )^{2}$$
$71$ $$( 8978 - 6700 T + 2500 T^{2} - 134 T^{3} + T^{6} )^{2}$$
$73$ $$105625 + 226970591 T^{4} + 33867 T^{8} + T^{12}$$
$79$ $$( -1165 - 261 T + T^{2} + T^{3} )^{4}$$
$83$ $$6276893869129 + 4751604443 T^{4} + 142927 T^{8} + T^{12}$$
$89$ $$19021098889 + 42676175 T^{4} + 13515 T^{8} + T^{12}$$
$97$ $$105625 + 8400471 T^{4} + 6947 T^{8} + T^{12}$$