# Properties

 Label 3330.2.e.d Level $3330$ Weight $2$ Character orbit 3330.e Analytic conductor $26.590$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3330.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.5901838731$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 19 x^{8} + 103 x^{6} + 210 x^{4} + 140 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 370) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} -\beta_{5} q^{5} -\beta_{2} q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} -\beta_{5} q^{5} -\beta_{2} q^{7} + q^{8} -\beta_{5} q^{10} -\beta_{7} q^{11} + ( -\beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{13} -\beta_{2} q^{14} + q^{16} + ( -2 + \beta_{8} ) q^{17} + ( -\beta_{4} - \beta_{6} + \beta_{9} ) q^{19} -\beta_{5} q^{20} -\beta_{7} q^{22} + ( -\beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{23} + ( 1 - \beta_{3} - \beta_{4} + \beta_{9} ) q^{25} + ( -\beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{26} -\beta_{2} q^{28} + ( \beta_{2} + \beta_{3} + \beta_{5} + \beta_{9} ) q^{29} + ( -\beta_{1} - \beta_{2} + \beta_{9} ) q^{31} + q^{32} + ( -2 + \beta_{8} ) q^{34} + ( \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{35} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{37} + ( -\beta_{4} - \beta_{6} + \beta_{9} ) q^{38} -\beta_{5} q^{40} + ( \beta_{4} - \beta_{6} - \beta_{7} ) q^{41} + ( -2 + \beta_{3} - \beta_{5} - \beta_{7} + 2 \beta_{8} ) q^{43} -\beta_{7} q^{44} + ( -\beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{9} ) q^{47} + ( -1 + 2 \beta_{7} + \beta_{8} ) q^{49} + ( 1 - \beta_{3} - \beta_{4} + \beta_{9} ) q^{50} + ( -\beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{52} + ( -\beta_{2} + \beta_{4} + \beta_{6} + \beta_{9} ) q^{53} + ( -\beta_{2} + \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{55} -\beta_{2} q^{56} + ( \beta_{2} + \beta_{3} + \beta_{5} + \beta_{9} ) q^{58} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{59} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} ) q^{61} + ( -\beta_{1} - \beta_{2} + \beta_{9} ) q^{62} + q^{64} + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{65} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{9} ) q^{67} + ( -2 + \beta_{8} ) q^{68} + ( \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{70} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{71} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{73} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} ) q^{74} + ( -\beta_{4} - \beta_{6} + \beta_{9} ) q^{76} + ( 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{9} ) q^{77} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{79} -\beta_{5} q^{80} + ( \beta_{4} - \beta_{6} - \beta_{7} ) q^{82} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{83} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{85} + ( -2 + \beta_{3} - \beta_{5} - \beta_{7} + 2 \beta_{8} ) q^{86} -\beta_{7} q^{88} + ( -2 \beta_{2} + 2 \beta_{9} ) q^{89} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{9} ) q^{91} + ( -\beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{92} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{9} ) q^{94} + ( -\beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{95} + ( \beta_{3} - \beta_{5} + \beta_{7} - 2 \beta_{8} ) q^{97} + ( -1 + 2 \beta_{7} + \beta_{8} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 10q^{2} + 10q^{4} + 3q^{5} + 10q^{8} + O(q^{10})$$ $$10q + 10q^{2} + 10q^{4} + 3q^{5} + 10q^{8} + 3q^{10} - 2q^{13} + 10q^{16} - 18q^{17} + 3q^{20} - 10q^{23} + 5q^{25} - 2q^{26} + 10q^{32} - 18q^{34} - 8q^{37} + 3q^{40} + 4q^{41} - 10q^{43} - 10q^{46} - 8q^{49} + 5q^{50} - 2q^{52} + 5q^{55} + 10q^{64} - 2q^{65} - 18q^{68} + 20q^{71} - 8q^{74} + 3q^{80} + 4q^{82} - 28q^{85} - 10q^{86} - 10q^{92} - 2q^{95} + 2q^{97} - 8q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 19 x^{8} + 103 x^{6} + 210 x^{4} + 140 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{9} - 45 \nu^{7} - 121 \nu^{5} - 26 \nu^{3} + 12 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{9} + 6 \nu^{8} + 15 \nu^{7} + 94 \nu^{6} + 39 \nu^{5} + 302 \nu^{4} - 10 \nu^{3} + 212 \nu^{2} - 44 \nu + 16$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{9} + 2 \nu^{8} + 49 \nu^{7} + 34 \nu^{6} + 181 \nu^{5} + 142 \nu^{4} + 186 \nu^{3} + 196 \nu^{2} + 20 \nu + 64$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{9} - 6 \nu^{8} + 15 \nu^{7} - 94 \nu^{6} + 39 \nu^{5} - 302 \nu^{4} - 10 \nu^{3} - 212 \nu^{2} - 44 \nu - 16$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{9} - 2 \nu^{8} + 49 \nu^{7} - 34 \nu^{6} + 181 \nu^{5} - 142 \nu^{4} + 186 \nu^{3} - 196 \nu^{2} + 20 \nu - 64$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{8} + 49 \nu^{6} + 181 \nu^{4} + 186 \nu^{2} + 28$$$$)/4$$ $$\beta_{8}$$ $$=$$ $$($$$$3 \nu^{8} + 49 \nu^{6} + 181 \nu^{4} + 190 \nu^{2} + 44$$$$)/4$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{9} - 17 \nu^{7} - 71 \nu^{5} - 100 \nu^{3} - 46 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{7} - 4$$ $$\nu^{3}$$ $$=$$ $$-\beta_{9} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 7 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-14 \beta_{8} + 10 \beta_{7} - 3 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} + 32$$ $$\nu^{5}$$ $$=$$ $$14 \beta_{9} + 14 \beta_{6} - 17 \beta_{5} + 14 \beta_{4} - 17 \beta_{3} - 2 \beta_{2} + 68 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$170 \beta_{8} - 108 \beta_{7} + 45 \beta_{6} + 16 \beta_{5} - 45 \beta_{4} - 16 \beta_{3} - 330$$ $$\nu^{7}$$ $$=$$ $$-170 \beta_{9} - 169 \beta_{6} + 215 \beta_{5} - 169 \beta_{4} + 215 \beta_{3} + 32 \beta_{2} - 748 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-1994 \beta_{8} + 1224 \beta_{7} - 554 \beta_{6} - 201 \beta_{5} + 554 \beta_{4} + 201 \beta_{3} + 3698$$ $$\nu^{9}$$ $$=$$ $$1994 \beta_{9} + 1979 \beta_{6} - 2548 \beta_{5} + 1979 \beta_{4} - 2548 \beta_{3} - 402 \beta_{2} + 8542 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times$$.

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
739.1
 1.78647i − 1.78647i − 0.377861i 0.377861i − 3.40359i 3.40359i 1.76216i − 1.76216i − 0.987983i 0.987983i
1.00000 0 1.00000 −2.21736 0.288618i 0 3.14934i 1.00000 0 −2.21736 0.288618i
739.2 1.00000 0 1.00000 −2.21736 + 0.288618i 0 3.14934i 1.00000 0 −2.21736 + 0.288618i
739.3 1.00000 0 1.00000 −1.04797 1.97529i 0 0.631751i 1.00000 0 −1.04797 1.97529i
739.4 1.00000 0 1.00000 −1.04797 + 1.97529i 0 0.631751i 1.00000 0 −1.04797 + 1.97529i
739.5 1.00000 0 1.00000 1.28269 1.83159i 0 2.06225i 1.00000 0 1.28269 1.83159i
739.6 1.00000 0 1.00000 1.28269 + 1.83159i 0 2.06225i 1.00000 0 1.28269 + 1.83159i
739.7 1.00000 0 1.00000 1.62868 1.53213i 0 1.22131i 1.00000 0 1.62868 1.53213i
739.8 1.00000 0 1.00000 1.62868 + 1.53213i 0 1.22131i 1.00000 0 1.62868 + 1.53213i
739.9 1.00000 0 1.00000 1.85396 1.25013i 0 4.78937i 1.00000 0 1.85396 1.25013i
739.10 1.00000 0 1.00000 1.85396 + 1.25013i 0 4.78937i 1.00000 0 1.85396 + 1.25013i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 739.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3330.2.e.d 10
3.b odd 2 1 370.2.c.a 10
5.b even 2 1 3330.2.e.c 10
15.d odd 2 1 370.2.c.b yes 10
15.e even 4 2 1850.2.d.i 20
37.b even 2 1 3330.2.e.c 10
111.d odd 2 1 370.2.c.b yes 10
185.d even 2 1 inner 3330.2.e.d 10
555.b odd 2 1 370.2.c.a 10
555.n even 4 2 1850.2.d.i 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.c.a 10 3.b odd 2 1
370.2.c.a 10 555.b odd 2 1
370.2.c.b yes 10 15.d odd 2 1
370.2.c.b yes 10 111.d odd 2 1
1850.2.d.i 20 15.e even 4 2
1850.2.d.i 20 555.n even 4 2
3330.2.e.c 10 5.b even 2 1
3330.2.e.c 10 37.b even 2 1
3330.2.e.d 10 1.a even 1 1 trivial
3330.2.e.d 10 185.d 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 $$S_{2}^{\mathrm{new}}(3330, [\chi])$$:

 $$T_{7}^{10} + 39 T_{7}^{8} + 438 T_{7}^{6} + 1684 T_{7}^{4} + 2048 T_{7}^{2} + 576$$ $$T_{13}^{5} + T_{13}^{4} - 39 T_{13}^{3} - 100 T_{13}^{2} + 160 T_{13} + 488$$ $$T_{17}^{5} + 9 T_{17}^{4} + 4 T_{17}^{3} - 108 T_{17}^{2} - 112 T_{17} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{10}$$
$3$ $$T^{10}$$
$5$ $$3125 - 1875 T + 250 T^{2} + 400 T^{3} - 95 T^{4} + 22 T^{5} - 19 T^{6} + 16 T^{7} + 2 T^{8} - 3 T^{9} + T^{10}$$
$7$ $$576 + 2048 T^{2} + 1684 T^{4} + 438 T^{6} + 39 T^{8} + T^{10}$$
$11$ $$( 48 + 16 T - 51 T^{2} - 28 T^{3} + T^{5} )^{2}$$
$13$ $$( 488 + 160 T - 100 T^{2} - 39 T^{3} + T^{4} + T^{5} )^{2}$$
$17$ $$( 144 - 112 T - 108 T^{2} + 4 T^{3} + 9 T^{4} + T^{5} )^{2}$$
$19$ $$9216 + 374336 T^{2} + 72240 T^{4} + 4668 T^{6} + 118 T^{8} + T^{10}$$
$23$ $$( 768 + 256 T - 308 T^{2} - 63 T^{3} + 5 T^{4} + T^{5} )^{2}$$
$29$ $$2262016 + 1251872 T^{2} + 194729 T^{4} + 10258 T^{6} + 182 T^{8} + T^{10}$$
$31$ $$60516 + 346130 T^{2} + 68053 T^{4} + 4502 T^{6} + 116 T^{8} + T^{10}$$
$37$ $$69343957 + 14993288 T + 4913341 T^{2} + 788544 T^{3} + 195434 T^{4} + 25584 T^{5} + 5282 T^{6} + 576 T^{7} + 97 T^{8} + 8 T^{9} + T^{10}$$
$41$ $$( 36 - 8 T - 89 T^{2} - 44 T^{3} - 2 T^{4} + T^{5} )^{2}$$
$43$ $$( -2624 + 1856 T - 76 T^{2} - 88 T^{3} + 5 T^{4} + T^{5} )^{2}$$
$47$ $$4596736 + 10494016 T^{2} + 992816 T^{4} + 29148 T^{6} + 310 T^{8} + T^{10}$$
$53$ $$39337984 + 10468352 T^{2} + 826000 T^{4} + 22964 T^{6} + 257 T^{8} + T^{10}$$
$59$ $$21827584 + 7068992 T^{2} + 590448 T^{4} + 18844 T^{6} + 238 T^{8} + T^{10}$$
$61$ $$82944 + 89792 T^{2} + 28169 T^{4} + 3402 T^{6} + 150 T^{8} + T^{10}$$
$67$ $$559417104 + 70672748 T^{2} + 2839394 T^{4} + 48727 T^{6} + 367 T^{8} + T^{10}$$
$71$ $$( -4608 - 2816 T + 1896 T^{2} - 164 T^{3} - 10 T^{4} + T^{5} )^{2}$$
$73$ $$589824 + 785408 T^{2} + 189152 T^{4} + 10965 T^{6} + 189 T^{8} + T^{10}$$
$79$ $$186486336 + 29601332 T^{2} + 1472598 T^{4} + 31019 T^{6} + 291 T^{8} + T^{10}$$
$83$ $$1024 + 23736128 T^{2} + 2459520 T^{4} + 62212 T^{6} + 466 T^{8} + T^{10}$$
$89$ $$1230045184 + 117901312 T^{2} + 3949632 T^{4} + 59888 T^{6} + 412 T^{8} + T^{10}$$
$97$ $$( 1168 + 2824 T - 248 T^{2} - 178 T^{3} - T^{4} + T^{5} )^{2}$$