# Properties

 Label 3450.2.d.ba Level $3450$ Weight $2$ Character orbit 3450.d Analytic conductor $27.548$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.5483886973$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.181494784.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 12 x^{3} + 49 x^{2} - 14 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + \beta_{4} q^{3} - q^{4} - q^{6} -\beta_{3} q^{7} -\beta_{4} q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} + \beta_{4} q^{3} - q^{4} - q^{6} -\beta_{3} q^{7} -\beta_{4} q^{8} - q^{9} + ( 1 + \beta_{1} + \beta_{2} ) q^{11} -\beta_{4} q^{12} + ( \beta_{3} - \beta_{4} + \beta_{5} ) q^{13} -\beta_{1} q^{14} + q^{16} + ( \beta_{4} - \beta_{5} ) q^{17} -\beta_{4} q^{18} + ( 1 + \beta_{1} - \beta_{2} ) q^{19} -\beta_{1} q^{21} + ( -\beta_{3} + \beta_{4} + \beta_{5} ) q^{22} -\beta_{4} q^{23} + q^{24} + ( 1 + \beta_{1} - \beta_{2} ) q^{26} -\beta_{4} q^{27} + \beta_{3} q^{28} + q^{29} + 6 q^{31} + \beta_{4} q^{32} + ( -\beta_{3} + \beta_{4} + \beta_{5} ) q^{33} + ( -1 + \beta_{2} ) q^{34} + q^{36} + ( -\beta_{4} + \beta_{5} ) q^{37} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{38} + ( 1 + \beta_{1} - \beta_{2} ) q^{39} + ( 7 + \beta_{1} - \beta_{2} ) q^{41} + \beta_{3} q^{42} + ( -\beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{43} + ( -1 - \beta_{1} - \beta_{2} ) q^{44} + q^{46} + ( \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{47} + \beta_{4} q^{48} + ( -4 + 2 \beta_{2} ) q^{49} + ( -1 + \beta_{2} ) q^{51} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{52} + ( 4 \beta_{4} - 2 \beta_{5} ) q^{53} + q^{54} + \beta_{1} q^{56} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{57} + \beta_{4} q^{58} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 6 - 2 \beta_{1} ) q^{61} + 6 \beta_{4} q^{62} + \beta_{3} q^{63} - q^{64} + ( -1 - \beta_{1} - \beta_{2} ) q^{66} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{67} + ( -\beta_{4} + \beta_{5} ) q^{68} + q^{69} + ( 4 - \beta_{1} - \beta_{2} ) q^{71} + \beta_{4} q^{72} + ( -2 \beta_{3} + \beta_{4} ) q^{73} + ( 1 - \beta_{2} ) q^{74} + ( -1 - \beta_{1} + \beta_{2} ) q^{76} + ( \beta_{3} + 11 \beta_{4} - 3 \beta_{5} ) q^{77} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{78} + ( 1 + \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( -\beta_{3} + 7 \beta_{4} - \beta_{5} ) q^{82} + ( 3 \beta_{3} + 4 \beta_{4} ) q^{83} + \beta_{1} q^{84} + ( -5 - \beta_{1} + \beta_{2} ) q^{86} + \beta_{4} q^{87} + ( \beta_{3} - \beta_{4} - \beta_{5} ) q^{88} + ( -3 + \beta_{2} ) q^{89} + ( 11 + 3 \beta_{1} - \beta_{2} ) q^{91} + \beta_{4} q^{92} + 6 \beta_{4} q^{93} + ( -2 + \beta_{1} - \beta_{2} ) q^{94} - q^{96} + ( 2 \beta_{3} - 4 \beta_{4} ) q^{97} + ( -4 \beta_{4} + 2 \beta_{5} ) q^{98} + ( -1 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{4} - 6q^{6} - 6q^{9} + O(q^{10})$$ $$6q - 6q^{4} - 6q^{6} - 6q^{9} + 6q^{11} + 2q^{14} + 6q^{16} + 2q^{19} + 2q^{21} + 6q^{24} + 2q^{26} + 6q^{29} + 36q^{31} - 4q^{34} + 6q^{36} + 2q^{39} + 38q^{41} - 6q^{44} + 6q^{46} - 20q^{49} - 4q^{51} + 6q^{54} - 2q^{56} + 40q^{61} - 6q^{64} - 6q^{66} + 6q^{69} + 24q^{71} + 4q^{74} - 2q^{76} + 6q^{79} + 6q^{81} - 2q^{84} - 26q^{86} - 16q^{89} + 58q^{91} - 16q^{94} - 6q^{96} - 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 12 x^{3} + 49 x^{2} - 14 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{5} + 8 \nu^{4} - 27 \nu^{3} - 18 \nu^{2} + 6 \nu - 103$$$$)/141$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} + 29 \nu^{4} - 45 \nu^{3} - 30 \nu^{2} + 10 \nu + 737$$$$)/141$$ $$\beta_{3}$$ $$=$$ $$($$$$-16 \nu^{5} + 27 \nu^{4} - 3 \nu^{3} - 237 \nu^{2} - 673 \nu + 93$$$$)/141$$ $$\beta_{4}$$ $$=$$ $$($$$$-19 \nu^{5} + 35 \nu^{4} - 30 \nu^{3} - 255 \nu^{2} - 949 \nu + 131$$$$)/141$$ $$\beta_{5}$$ $$=$$ $$($$$$-98 \nu^{5} + 183 \nu^{4} - 177 \nu^{3} - 1152 \nu^{2} - 5021 \nu + 693$$$$)/141$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 6 \beta_{4} + \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{5} - 17 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} - 9 \beta_{1} - 17$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{2} - 15 \beta_{1} - 58$$ $$\nu^{5}$$ $$=$$ $$($$$$-30 \beta_{5} + 223 \beta_{4} - 91 \beta_{3} + 30 \beta_{2} - 91 \beta_{1} - 223$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2899.1
 2.38150 + 2.38150i 0.138157 + 0.138157i −1.51966 − 1.51966i −1.51966 + 1.51966i 0.138157 − 0.138157i 2.38150 − 2.38150i
1.00000i 1.00000i −1.00000 0 −1.00000 3.76300i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 −1.00000 0.723686i 1.00000i −1.00000 0
2899.3 1.00000i 1.00000i −1.00000 0 −1.00000 4.03932i 1.00000i −1.00000 0
2899.4 1.00000i 1.00000i −1.00000 0 −1.00000 4.03932i 1.00000i −1.00000 0
2899.5 1.00000i 1.00000i −1.00000 0 −1.00000 0.723686i 1.00000i −1.00000 0
2899.6 1.00000i 1.00000i −1.00000 0 −1.00000 3.76300i 1.00000i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2899.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3450.2.d.ba 6
5.b even 2 1 inner 3450.2.d.ba 6
5.c odd 4 1 3450.2.a.bp 3
5.c odd 4 1 3450.2.a.bs yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.bp 3 5.c odd 4 1
3450.2.a.bs yes 3 5.c odd 4 1
3450.2.d.ba 6 1.a even 1 1 trivial
3450.2.d.ba 6 5.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 $$S_{2}^{\mathrm{new}}(3450, [\chi])$$:

 $$T_{7}^{6} + 31 T_{7}^{4} + 247 T_{7}^{2} + 121$$ $$T_{11}^{3} - 3 T_{11}^{2} - 32 T_{11} + 100$$ $$T_{13}^{6} + 65 T_{13}^{4} + 1000 T_{13}^{2} + 144$$ $$T_{17}^{6} + 38 T_{17}^{4} + 449 T_{17}^{2} + 1600$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$( 1 + T^{2} )^{3}$$
$5$ $$T^{6}$$
$7$ $$121 + 247 T^{2} + 31 T^{4} + T^{6}$$
$11$ $$( 100 - 32 T - 3 T^{2} + T^{3} )^{2}$$
$13$ $$144 + 1000 T^{2} + 65 T^{4} + T^{6}$$
$17$ $$1600 + 449 T^{2} + 38 T^{4} + T^{6}$$
$19$ $$( -12 - 32 T - T^{2} + T^{3} )^{2}$$
$23$ $$( 1 + T^{2} )^{3}$$
$29$ $$( -1 + T )^{6}$$
$31$ $$( -6 + T )^{6}$$
$37$ $$1600 + 449 T^{2} + 38 T^{4} + T^{6}$$
$41$ $$( -72 + 88 T - 19 T^{2} + T^{3} )^{2}$$
$43$ $$1296 + 1512 T^{2} + 121 T^{4} + T^{6}$$
$47$ $$8100 + 1561 T^{2} + 86 T^{4} + T^{6}$$
$53$ $$186624 + 10240 T^{2} + 180 T^{4} + T^{6}$$
$59$ $$( 528 - 140 T + T^{3} )^{2}$$
$61$ $$( 160 + 72 T - 20 T^{2} + T^{3} )^{2}$$
$67$ $$123904 + 18192 T^{2} + 264 T^{4} + T^{6}$$
$71$ $$( 10 + 13 T - 12 T^{2} + T^{3} )^{2}$$
$73$ $$729 + 3667 T^{2} + 123 T^{4} + T^{6}$$
$79$ $$( 100 - 32 T - 3 T^{2} + T^{3} )^{2}$$
$83$ $$525625 + 25719 T^{2} + 351 T^{4} + T^{6}$$
$89$ $$( -58 + 3 T + 8 T^{2} + T^{3} )^{2}$$
$97$ $$14400 + 3184 T^{2} + 156 T^{4} + T^{6}$$