# Properties

 Label 7200.2.h.m Level $7200$ Weight $2$ Character orbit 7200.h Analytic conductor $57.492$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7200 = 2^{5} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7200.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$57.4922894553$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.426337261060096.1 Defining polynomial: $$x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{41}]$$ Coefficient ring index: $$2^{15}$$ Twist minimal: no (minimal twist has level 1440) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{7} +O(q^{10})$$ $$q + \beta_{6} q^{7} -\beta_{4} q^{11} + ( 2 - \beta_{9} ) q^{13} -\beta_{5} q^{17} -2 \beta_{6} q^{19} + ( -\beta_{3} - \beta_{4} ) q^{23} + ( \beta_{5} + 2 \beta_{10} ) q^{29} + ( \beta_{1} - \beta_{6} - \beta_{11} ) q^{31} + ( 1 - \beta_{8} ) q^{37} + ( \beta_{2} - 2 \beta_{10} ) q^{41} + ( 2 \beta_{1} - 2 \beta_{11} ) q^{43} + ( 2 \beta_{3} - \beta_{4} - \beta_{7} ) q^{47} + ( -\beta_{8} + \beta_{9} ) q^{49} + ( 2 \beta_{2} + 3 \beta_{5} + 2 \beta_{10} ) q^{53} + ( 2 \beta_{3} - \beta_{7} ) q^{59} + ( 2 \beta_{8} + 2 \beta_{9} ) q^{61} + ( -2 \beta_{1} - 2 \beta_{6} ) q^{67} + ( 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{71} + ( 9 + \beta_{8} + \beta_{9} ) q^{73} + ( 2 \beta_{2} - \beta_{5} + \beta_{10} ) q^{77} + ( -3 \beta_{1} - \beta_{6} + \beta_{11} ) q^{79} + ( -2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{83} + ( \beta_{2} + 2 \beta_{10} ) q^{89} + ( -\beta_{1} + 3 \beta_{6} + 2 \beta_{11} ) q^{91} + ( 3 - \beta_{8} + 3 \beta_{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + O(q^{10})$$ $$12q + 24q^{13} + 16q^{37} + 4q^{49} - 8q^{61} + 104q^{73} + 40q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{11} - 8 \nu^{10} + 20 \nu^{9} + 12 \nu^{8} + 45 \nu^{7} + 12 \nu^{6} - 116 \nu^{5} - 24 \nu^{4} + 4 \nu^{3} - 32 \nu^{2} + 208 \nu - 448$$$$)/160$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{10} + \nu^{9} - 2 \nu^{7} - 3 \nu^{6} + 5 \nu^{5} + 4 \nu^{4} + 2 \nu^{3} - 8 \nu^{2} - 24 \nu$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{11} - 4 \nu^{9} - 4 \nu^{8} - 3 \nu^{7} + 20 \nu^{6} + 20 \nu^{5} - 8 \nu^{4} - 12 \nu^{3} - 64 \nu^{2} + 16 \nu + 64$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{11} - 12 \nu^{10} + 12 \nu^{9} + 4 \nu^{8} - 9 \nu^{7} - 28 \nu^{5} + 56 \nu^{4} + 76 \nu^{3} - 112 \nu^{2} - 144 \nu - 320$$$$)/160$$ $$\beta_{5}$$ $$=$$ $$($$$$7 \nu^{11} - 6 \nu^{10} - 30 \nu^{9} + 4 \nu^{8} + 15 \nu^{7} + 94 \nu^{6} + 18 \nu^{5} - 128 \nu^{4} - 232 \nu^{3} - 64 \nu^{2} + 256 \nu + 384$$$$)/160$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{11} + 12 \nu^{10} + 20 \nu^{9} + 12 \nu^{8} - 35 \nu^{7} - 48 \nu^{6} - 36 \nu^{5} + 136 \nu^{4} + 164 \nu^{3} + 48 \nu^{2} - 432 \nu - 448$$$$)/160$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{11} - 4 \nu^{10} - 4 \nu^{9} - 4 \nu^{8} + 13 \nu^{7} + 32 \nu^{6} + 4 \nu^{5} - 40 \nu^{4} - 44 \nu^{3} + 48 \nu^{2} + 144 \nu + 64$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$-11 \nu^{11} + 4 \nu^{10} - 4 \nu^{9} + 12 \nu^{8} + 33 \nu^{7} - 40 \nu^{6} + 36 \nu^{5} + 88 \nu^{4} - 12 \nu^{3} + 144 \nu^{2} - 272 \nu + 160$$$$)/160$$ $$\beta_{9}$$ $$=$$ $$($$$$-7 \nu^{11} - 12 \nu^{10} + 12 \nu^{9} + 44 \nu^{8} + 21 \nu^{7} - 40 \nu^{6} - 108 \nu^{5} - 24 \nu^{4} + 196 \nu^{3} + 208 \nu^{2} + 176 \nu - 320$$$$)/160$$ $$\beta_{10}$$ $$=$$ $$($$$$11 \nu^{11} + 22 \nu^{10} + 30 \nu^{9} - 28 \nu^{8} - 85 \nu^{7} - 38 \nu^{6} + 94 \nu^{5} + 256 \nu^{4} + 64 \nu^{3} - 352 \nu^{2} - 352 \nu - 448$$$$)/160$$ $$\beta_{11}$$ $$=$$ $$($$$$-3 \nu^{11} - \nu^{10} + 4 \nu^{8} + 5 \nu^{7} - \nu^{6} - 12 \nu^{5} - 8 \nu^{4} - 12 \nu^{3} + 36 \nu^{2} + 16 \nu + 64$$$$)/20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{10} + \beta_{9} - \beta_{6} - \beta_{4} - \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - \beta_{3} - \beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{11} - \beta_{10} + \beta_{9} + \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 4$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{10} + \beta_{9} + 3 \beta_{8} + \beta_{5} - 6 \beta_{2} + 5$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{11} + \beta_{10} + 3 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} + \beta_{4} + 7 \beta_{2} - 2 \beta_{1} - 6$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{11} + 3 \beta_{7} + 7 \beta_{6} + 5 \beta_{3} + \beta_{1}$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-5 \beta_{11} - 3 \beta_{10} - \beta_{9} + 8 \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 8 \beta_{1} + 12$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$5 \beta_{10} + 15 \beta_{9} + 5 \beta_{8} + 15 \beta_{5} + 14 \beta_{2} + 3$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$5 \beta_{11} + 3 \beta_{10} + \beta_{9} - 6 \beta_{8} + 10 \beta_{7} - \beta_{6} - 4 \beta_{5} + 7 \beta_{4} - 4 \beta_{3} + 33 \beta_{2} + 6 \beta_{1} + 26$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-2 \beta_{11} - 3 \beta_{7} + 17 \beta_{6} - 40 \beta_{4} + 11 \beta_{3} + 15 \beta_{1}$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-27 \beta_{11} - \beta_{10} + 5 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} + 5 \beta_{6} + 22 \beta_{5} + 3 \beta_{4} - 22 \beta_{3} + 7 \beta_{2} + 4 \beta_{1} + 104$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$6401$$ $$6751$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 −1.35818 + 0.394157i −0.394157 + 1.35818i −0.0912546 + 1.41127i 1.41127 − 0.0912546i −0.760198 + 1.19252i 1.19252 − 0.760198i −0.760198 − 1.19252i 1.19252 + 0.760198i −0.0912546 − 1.41127i 1.41127 + 0.0912546i −1.35818 − 0.394157i −0.394157 − 1.35818i
0 0 0 0 0 3.50466i 0 0 0
1151.2 0 0 0 0 0 3.50466i 0 0 0
1151.3 0 0 0 0 0 2.64002i 0 0 0
1151.4 0 0 0 0 0 2.64002i 0 0 0
1151.5 0 0 0 0 0 0.864641i 0 0 0
1151.6 0 0 0 0 0 0.864641i 0 0 0
1151.7 0 0 0 0 0 0.864641i 0 0 0
1151.8 0 0 0 0 0 0.864641i 0 0 0
1151.9 0 0 0 0 0 2.64002i 0 0 0
1151.10 0 0 0 0 0 2.64002i 0 0 0
1151.11 0 0 0 0 0 3.50466i 0 0 0
1151.12 0 0 0 0 0 3.50466i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1151.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 7200.2.h.m 12
3.b odd 2 1 inner 7200.2.h.m 12
4.b odd 2 1 inner 7200.2.h.m 12
5.b even 2 1 7200.2.h.l 12
5.c odd 4 1 1440.2.o.a 12
5.c odd 4 1 1440.2.o.b yes 12
12.b even 2 1 inner 7200.2.h.m 12
15.d odd 2 1 7200.2.h.l 12
15.e even 4 1 1440.2.o.a 12
15.e even 4 1 1440.2.o.b yes 12
20.d odd 2 1 7200.2.h.l 12
20.e even 4 1 1440.2.o.a 12
20.e even 4 1 1440.2.o.b yes 12
40.i odd 4 1 2880.2.o.e 12
40.i odd 4 1 2880.2.o.f 12
40.k even 4 1 2880.2.o.e 12
40.k even 4 1 2880.2.o.f 12
60.h even 2 1 7200.2.h.l 12
60.l odd 4 1 1440.2.o.a 12
60.l odd 4 1 1440.2.o.b yes 12
120.q odd 4 1 2880.2.o.e 12
120.q odd 4 1 2880.2.o.f 12
120.w even 4 1 2880.2.o.e 12
120.w even 4 1 2880.2.o.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.o.a 12 5.c odd 4 1
1440.2.o.a 12 15.e even 4 1
1440.2.o.a 12 20.e even 4 1
1440.2.o.a 12 60.l odd 4 1
1440.2.o.b yes 12 5.c odd 4 1
1440.2.o.b yes 12 15.e even 4 1
1440.2.o.b yes 12 20.e even 4 1
1440.2.o.b yes 12 60.l odd 4 1
2880.2.o.e 12 40.i odd 4 1
2880.2.o.e 12 40.k even 4 1
2880.2.o.e 12 120.q odd 4 1
2880.2.o.e 12 120.w even 4 1
2880.2.o.f 12 40.i odd 4 1
2880.2.o.f 12 40.k even 4 1
2880.2.o.f 12 120.q odd 4 1
2880.2.o.f 12 120.w even 4 1
7200.2.h.l 12 5.b even 2 1
7200.2.h.l 12 15.d odd 2 1
7200.2.h.l 12 20.d odd 2 1
7200.2.h.l 12 60.h even 2 1
7200.2.h.m 12 1.a even 1 1 trivial
7200.2.h.m 12 3.b odd 2 1 inner
7200.2.h.m 12 4.b odd 2 1 inner
7200.2.h.m 12 12.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(7200, [\chi])$$:

 $$T_{7}^{6} + 20 T_{7}^{4} + 100 T_{7}^{2} + 64$$ $$T_{11}^{6} - 28 T_{11}^{4} + 228 T_{11}^{2} - 512$$ $$T_{13}^{3} - 6 T_{13}^{2} + 2 T_{13} + 4$$ $$T_{23}^{6} - 88 T_{23}^{4} + 528 T_{23}^{2} - 512$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$( 64 + 100 T^{2} + 20 T^{4} + T^{6} )^{2}$$
$11$ $$( -512 + 228 T^{2} - 28 T^{4} + T^{6} )^{2}$$
$13$ $$( 4 + 2 T - 6 T^{2} + T^{3} )^{4}$$
$17$ $$( 128 + 288 T^{2} + 34 T^{4} + T^{6} )^{2}$$
$19$ $$( 4096 + 1600 T^{2} + 80 T^{4} + T^{6} )^{2}$$
$23$ $$( -512 + 528 T^{2} - 88 T^{4} + T^{6} )^{2}$$
$29$ $$( 3200 + 2144 T^{2} + 98 T^{4} + T^{6} )^{2}$$
$31$ $$( 6400 + 1424 T^{2} + 88 T^{4} + T^{6} )^{2}$$
$37$ $$( -8 - 14 T - 4 T^{2} + T^{3} )^{4}$$
$41$ $$( 13448 + 2828 T^{2} + 102 T^{4} + T^{6} )^{2}$$
$43$ $$( 262144 + 16448 T^{2} + 240 T^{4} + T^{6} )^{2}$$
$47$ $$( -51200 + 5696 T^{2} - 176 T^{4} + T^{6} )^{2}$$
$53$ $$( 991232 + 30464 T^{2} + 306 T^{4} + T^{6} )^{2}$$
$59$ $$( -359552 + 15652 T^{2} - 220 T^{4} + T^{6} )^{2}$$
$61$ $$( -328 - 100 T + 2 T^{2} + T^{3} )^{4}$$
$67$ $$( 262144 + 14592 T^{2} + 224 T^{4} + T^{6} )^{2}$$
$71$ $$( -204800 + 15616 T^{2} - 256 T^{4} + T^{6} )^{2}$$
$73$ $$( -464 + 200 T - 26 T^{2} + T^{3} )^{4}$$
$79$ $$( 215296 + 29584 T^{2} + 344 T^{4} + T^{6} )^{2}$$
$83$ $$( -204800 + 15616 T^{2} - 256 T^{4} + T^{6} )^{2}$$
$89$ $$( 200 + 1356 T^{2} + 118 T^{4} + T^{6} )^{2}$$
$97$ $$( 848 - 88 T - 10 T^{2} + T^{3} )^{4}$$