# Properties

 Label 3744.2.m.h Level $3744$ Weight $2$ Character orbit 3744.m Analytic conductor $29.896$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3744 = 2^{5} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3744.m (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.8959905168$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{20}$$ Twist minimal: no (minimal twist has level 936) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{14} q^{5} + \beta_{5} q^{7} +O(q^{10})$$ $$q + \beta_{14} q^{5} + \beta_{5} q^{7} + ( -\beta_{11} + \beta_{14} ) q^{11} + ( -\beta_{6} - \beta_{15} ) q^{13} -\beta_{3} q^{17} + ( -\beta_{6} + \beta_{12} - \beta_{15} ) q^{19} -\beta_{2} q^{23} + \beta_{1} q^{25} + \beta_{7} q^{29} + \beta_{10} q^{31} + ( -2 \beta_{7} + \beta_{9} ) q^{35} + ( -2 \beta_{13} + 2 \beta_{15} ) q^{37} + ( -3 \beta_{4} - 6 \beta_{8} ) q^{41} + ( \beta_{12} + \beta_{13} ) q^{43} + ( -4 \beta_{4} - \beta_{8} ) q^{47} + ( -8 - \beta_{1} ) q^{49} + ( -2 \beta_{7} - \beta_{9} ) q^{53} + ( 5 + 3 \beta_{1} ) q^{55} + ( -3 \beta_{11} - \beta_{14} ) q^{59} -3 \beta_{6} q^{61} + ( \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{8} ) q^{65} + ( 2 \beta_{6} - 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{67} + ( -4 \beta_{4} + \beta_{8} ) q^{71} + ( -\beta_{5} - \beta_{10} ) q^{73} + ( -\beta_{7} + 3 \beta_{9} ) q^{77} + ( -2 - 4 \beta_{1} ) q^{79} + ( \beta_{11} - 3 \beta_{14} ) q^{83} + ( -3 \beta_{6} + 3 \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{85} + ( \beta_{4} + 8 \beta_{8} ) q^{89} + ( 4 \beta_{6} - 3 \beta_{12} - \beta_{15} ) q^{91} + ( \beta_{2} - 2 \beta_{3} ) q^{95} + ( \beta_{5} - \beta_{10} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + O(q^{10})$$ $$16 q - 128 q^{49} + 80 q^{55} - 32 q^{79} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$396 \nu^{14} + 1812 \nu^{12} - 7496 \nu^{10} - 25482 \nu^{8} - 34944 \nu^{6} - 37770 \nu^{4} - 5400 \nu^{2} + 1991875$$$$)/881375$$ $$\beta_{2}$$ $$=$$ $$($$$$-7468 \nu^{14} - 15136 \nu^{12} + 146738 \nu^{10} + 531446 \nu^{8} - 573668 \nu^{6} - 3703230 \nu^{4} - 4157900 \nu^{2} - 4522500$$$$)/881375$$ $$\beta_{3}$$ $$=$$ $$($$$$2458 \nu^{14} + 5148 \nu^{12} - 42734 \nu^{10} - 183828 \nu^{8} + 123374 \nu^{6} + 1088952 \nu^{4} + 1198950 \nu^{2} + 1368050$$$$)/176275$$ $$\beta_{4}$$ $$=$$ $$($$$$-7797 \nu^{15} - 1879 \nu^{13} + 194307 \nu^{11} + 471569 \nu^{9} - 1255952 \nu^{7} - 4205580 \nu^{5} - 3942175 \nu^{3} - 4720750 \nu$$$$)/4406875$$ $$\beta_{5}$$ $$=$$ $$($$$$10661 \nu^{15} + 14207 \nu^{13} - 225956 \nu^{11} - 888177 \nu^{9} + 521116 \nu^{7} + 5727895 \nu^{5} + 13225300 \nu^{3} + 26244000 \nu$$$$)/4406875$$ $$\beta_{6}$$ $$=$$ $$($$$$23588 \nu^{14} + 17416 \nu^{12} - 373728 \nu^{10} - 1083526 \nu^{8} + 2038808 \nu^{6} + 4471770 \nu^{4} + 6997400 \nu^{2} + 8274750$$$$)/881375$$ $$\beta_{7}$$ $$=$$ $$($$$$28806 \nu^{14} + 29832 \nu^{12} - 448156 \nu^{10} - 1496702 \nu^{8} + 2047066 \nu^{6} + 6175530 \nu^{4} + 12354350 \nu^{2} + 13465750$$$$)/881375$$ $$\beta_{8}$$ $$=$$ $$($$$$-1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} - 478015 \nu^{5} - 412275 \nu^{3} - 598500 \nu$$$$)/400625$$ $$\beta_{9}$$ $$=$$ $$($$$$-44196 \nu^{14} - 27412 \nu^{12} + 754046 \nu^{10} + 2232082 \nu^{8} - 3966856 \nu^{6} - 11081230 \nu^{4} - 21249600 \nu^{2} - 22476000$$$$)/881375$$ $$\beta_{10}$$ $$=$$ $$($$$$8289 \nu^{15} + 23943 \nu^{13} - 103294 \nu^{11} - 572573 \nu^{9} - 243116 \nu^{7} + 1825705 \nu^{5} + 5160650 \nu^{3} + 11379000 \nu$$$$)/881375$$ $$\beta_{11}$$ $$=$$ $$($$$$-68641 \nu^{15} - 53217 \nu^{13} + 1141536 \nu^{11} + 3533287 \nu^{9} - 5705496 \nu^{7} - 17001745 \nu^{5} - 30732300 \nu^{3} - 30934000 \nu$$$$)/4406875$$ $$\beta_{12}$$ $$=$$ $$($$$$-78923 \nu^{15} - 26760 \nu^{14} - 34611 \nu^{13} + 11580 \nu^{12} + 1339113 \nu^{11} + 580360 \nu^{10} + 3553821 \nu^{9} + 1252870 \nu^{8} - 8128668 \nu^{7} - 4374960 \nu^{6} - 15701920 \nu^{5} - 8822500 \nu^{4} - 26400825 \nu^{3} - 12863000 \nu^{2} - 32586750 \nu - 13844375$$$$)/4406875$$ $$\beta_{13}$$ $$=$$ $$($$$$78923 \nu^{15} - 144700 \nu^{14} + 34611 \nu^{13} - 75500 \nu^{12} - 1339113 \nu^{11} + 2449000 \nu^{10} - 3553821 \nu^{9} + 6670500 \nu^{8} + 8128668 \nu^{7} - 14569000 \nu^{6} + 15701920 \nu^{5} - 31181350 \nu^{4} + 26400825 \nu^{3} - 47850000 \nu^{2} + 32586750 \nu - 55218125$$$$)/4406875$$ $$\beta_{14}$$ $$=$$ $$($$$$108813 \nu^{15} + 85331 \nu^{13} - 1829673 \nu^{11} - 5646091 \nu^{9} + 9386828 \nu^{7} + 27713410 \nu^{5} + 49911275 \nu^{3} + 50144500 \nu$$$$)/4406875$$ $$\beta_{15}$$ $$=$$ $$($$$$128244 \nu^{15} - 144700 \nu^{14} + 47208 \nu^{13} - 75500 \nu^{12} - 2180589 \nu^{11} + 2449000 \nu^{10} - 5658588 \nu^{9} + 6670500 \nu^{8} + 13459254 \nu^{7} - 14569000 \nu^{6} + 25923185 \nu^{5} - 31181350 \nu^{4} + 43314975 \nu^{3} - 47850000 \nu^{2} + 53041500 \nu - 55218125$$$$)/4406875$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - \beta_{13} + \beta_{11} - \beta_{8} + \beta_{5} + \beta_{4}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} - \beta_{1} - 1$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$10 \beta_{14} - \beta_{13} + \beta_{12} + 12 \beta_{11} + \beta_{10} + 8 \beta_{8} - \beta_{6} - 3 \beta_{5} - 2 \beta_{4}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{13} + 3 \beta_{12} - 2 \beta_{9} - 4 \beta_{7} + 5 \beta_{6} - 2 \beta_{2} - 8 \beta_{1} + 18$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$14 \beta_{15} - 19 \beta_{13} + 5 \beta_{12} - 6 \beta_{8} - 5 \beta_{6}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-11 \beta_{13} - 11 \beta_{12} + 11 \beta_{9} + 17 \beta_{7} - 15 \beta_{6} - 5 \beta_{3} - 6 \beta_{2} - 26 \beta_{1} + 56$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$10 \beta_{15} + 110 \beta_{14} - 9 \beta_{13} - \beta_{12} + 182 \beta_{11} - 9 \beta_{10} - 38 \beta_{8} + \beta_{6} + 17 \beta_{5} + 72 \beta_{4}$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$35 \beta_{13} + 35 \beta_{12} - 20 \beta_{9} - 32 \beta_{7} + 53 \beta_{6} - 32 \beta_{3} - 52 \beta_{2} - 18 \beta_{1} + 52$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$274 \beta_{15} + 114 \beta_{14} - 361 \beta_{13} + 87 \beta_{12} + 190 \beta_{11} + 87 \beta_{10} + 190 \beta_{8} - 87 \beta_{6} - 187 \beta_{5} - 304 \beta_{4}$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$21 \beta_{9} + 36 \beta_{7} - 275 \beta_{1} + 607$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$768 \beta_{15} + 682 \beta_{14} - 1007 \beta_{13} + 239 \beta_{12} + 1080 \beta_{11} - 239 \beta_{10} - 1080 \beta_{8} - 239 \beta_{6} + 529 \beta_{5} + 1762 \beta_{4}$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$($$$$-110 \beta_{13} - 110 \beta_{12} + 290 \beta_{9} + 462 \beta_{7} - 177 \beta_{6} - 462 \beta_{3} - 752 \beta_{2} - 67 \beta_{1} + 153$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$790 \beta_{15} + 4630 \beta_{14} - 1037 \beta_{13} + 247 \beta_{12} + 7514 \beta_{11} + 1037 \beta_{10} + 1746 \beta_{8} - 247 \beta_{6} - 2321 \beta_{5} - 2884 \beta_{4}$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$2071 \beta_{13} + 2071 \beta_{12} - 1039 \beta_{9} - 1678 \beta_{7} + 3360 \beta_{6} - 400 \beta_{3} - 639 \beta_{2} - 5431 \beta_{1} + 12151$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$4399 \beta_{15} - 5754 \beta_{13} + 1355 \beta_{12} - 1521 \beta_{8} - 1355 \beta_{6} + 2475 \beta_{4}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$$.

 $$n$$ $$703$$ $$2017$$ $$2081$$ $$2341$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1585.1
 0.752864 − 0.902863i 0.0783900 − 1.17295i 0.752864 + 0.902863i 0.0783900 + 1.17295i 0.556839 − 1.81878i −1.90184 + 0.0324487i 0.556839 + 1.81878i −1.90184 − 0.0324487i 1.90184 + 0.0324487i −0.556839 − 1.81878i 1.90184 − 0.0324487i −0.556839 + 1.81878i −0.0783900 − 1.17295i −0.752864 − 0.902863i −0.0783900 + 1.17295i −0.752864 + 0.902863i
0 0 0 −2.68999 0 4.15163i 0 0 0
1585.2 0 0 0 −2.68999 0 4.15163i 0 0 0
1585.3 0 0 0 −2.68999 0 4.15163i 0 0 0
1585.4 0 0 0 −2.68999 0 4.15163i 0 0 0
1585.5 0 0 0 −1.66251 0 3.57266i 0 0 0
1585.6 0 0 0 −1.66251 0 3.57266i 0 0 0
1585.7 0 0 0 −1.66251 0 3.57266i 0 0 0
1585.8 0 0 0 −1.66251 0 3.57266i 0 0 0
1585.9 0 0 0 1.66251 0 3.57266i 0 0 0
1585.10 0 0 0 1.66251 0 3.57266i 0 0 0
1585.11 0 0 0 1.66251 0 3.57266i 0 0 0
1585.12 0 0 0 1.66251 0 3.57266i 0 0 0
1585.13 0 0 0 2.68999 0 4.15163i 0 0 0
1585.14 0 0 0 2.68999 0 4.15163i 0 0 0
1585.15 0 0 0 2.68999 0 4.15163i 0 0 0
1585.16 0 0 0 2.68999 0 4.15163i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1585.16 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
8.b even 2 1 inner
13.b even 2 1 inner
24.h odd 2 1 inner
39.d odd 2 1 inner
104.e even 2 1 inner
312.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.2.m.h 16
3.b odd 2 1 inner 3744.2.m.h 16
4.b odd 2 1 936.2.m.h 16
8.b even 2 1 inner 3744.2.m.h 16
8.d odd 2 1 936.2.m.h 16
12.b even 2 1 936.2.m.h 16
13.b even 2 1 inner 3744.2.m.h 16
24.f even 2 1 936.2.m.h 16
24.h odd 2 1 inner 3744.2.m.h 16
39.d odd 2 1 inner 3744.2.m.h 16
52.b odd 2 1 936.2.m.h 16
104.e even 2 1 inner 3744.2.m.h 16
104.h odd 2 1 936.2.m.h 16
156.h even 2 1 936.2.m.h 16
312.b odd 2 1 inner 3744.2.m.h 16
312.h even 2 1 936.2.m.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.m.h 16 4.b odd 2 1
936.2.m.h 16 8.d odd 2 1
936.2.m.h 16 12.b even 2 1
936.2.m.h 16 24.f even 2 1
936.2.m.h 16 52.b odd 2 1
936.2.m.h 16 104.h odd 2 1
936.2.m.h 16 156.h even 2 1
936.2.m.h 16 312.h even 2 1
3744.2.m.h 16 1.a even 1 1 trivial
3744.2.m.h 16 3.b odd 2 1 inner
3744.2.m.h 16 8.b even 2 1 inner
3744.2.m.h 16 13.b even 2 1 inner
3744.2.m.h 16 24.h odd 2 1 inner
3744.2.m.h 16 39.d odd 2 1 inner
3744.2.m.h 16 104.e even 2 1 inner
3744.2.m.h 16 312.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 10 T_{5}^{2} + 20$$ acting on $$S_{2}^{\mathrm{new}}(3744, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 20 - 10 T^{2} + T^{4} )^{4}$$
$7$ $$( 220 + 30 T^{2} + T^{4} )^{4}$$
$11$ $$( 20 - 20 T^{2} + T^{4} )^{4}$$
$13$ $$( 28561 - 2028 T^{2} + 54 T^{4} - 12 T^{6} + T^{8} )^{2}$$
$17$ $$( 880 - 60 T^{2} + T^{4} )^{4}$$
$19$ $$( 44 - 34 T^{2} + T^{4} )^{4}$$
$23$ $$( 880 - 80 T^{2} + T^{4} )^{4}$$
$29$ $$( 176 + 28 T^{2} + T^{4} )^{4}$$
$31$ $$( 5500 + 150 T^{2} + T^{4} )^{4}$$
$37$ $$( 704 - 104 T^{2} + T^{4} )^{4}$$
$41$ $$( 324 + 126 T^{2} + T^{4} )^{4}$$
$43$ $$( 80 + 20 T^{2} + T^{4} )^{4}$$
$47$ $$( 1444 + 84 T^{2} + T^{4} )^{4}$$
$53$ $$( 176 + 128 T^{2} + T^{4} )^{4}$$
$59$ $$( 500 - 100 T^{2} + T^{4} )^{4}$$
$61$ $$( 6480 + 180 T^{2} + T^{4} )^{4}$$
$67$ $$( 5324 - 154 T^{2} + T^{4} )^{4}$$
$71$ $$( 484 + 116 T^{2} + T^{4} )^{4}$$
$73$ $$( 3520 + 200 T^{2} + T^{4} )^{4}$$
$79$ $$( -76 + 4 T + T^{2} )^{8}$$
$83$ $$( 500 - 100 T^{2} + T^{4} )^{4}$$
$89$ $$( 12100 + 230 T^{2} + T^{4} )^{4}$$
$97$ $$( 3520 + 160 T^{2} + T^{4} )^{4}$$