# Properties

 Label 2151.1.f.b Level $2151$ Weight $1$ Character orbit 2151.f Analytic conductor $1.073$ Analytic rank $0$ Dimension $24$ Projective image $D_{45}$ CM discriminant -239 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2151 = 3^{2} \cdot 239$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2151.f (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.07348884217$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{45})$$ Defining polynomial: $$x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{45}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{45} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{90}^{7} - \zeta_{90}^{23} ) q^{2} -\zeta_{90}^{3} q^{3} + ( -\zeta_{90} + \zeta_{90}^{14} + \zeta_{90}^{30} ) q^{4} + ( -\zeta_{90}^{17} - \zeta_{90}^{43} ) q^{5} + ( \zeta_{90}^{10} + \zeta_{90}^{26} ) q^{6} + ( \zeta_{90}^{8} - \zeta_{90}^{21} + \zeta_{90}^{24} - \zeta_{90}^{37} ) q^{8} + \zeta_{90}^{6} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{90}^{7} - \zeta_{90}^{23} ) q^{2} -\zeta_{90}^{3} q^{3} + ( -\zeta_{90} + \zeta_{90}^{14} + \zeta_{90}^{30} ) q^{4} + ( -\zeta_{90}^{17} - \zeta_{90}^{43} ) q^{5} + ( \zeta_{90}^{10} + \zeta_{90}^{26} ) q^{6} + ( \zeta_{90}^{8} - \zeta_{90}^{21} + \zeta_{90}^{24} - \zeta_{90}^{37} ) q^{8} + \zeta_{90}^{6} q^{9} + ( -\zeta_{90}^{5} - \zeta_{90}^{21} + \zeta_{90}^{24} + \zeta_{90}^{40} ) q^{10} + ( -\zeta_{90}^{11} - \zeta_{90}^{19} ) q^{11} + ( \zeta_{90}^{4} - \zeta_{90}^{17} - \zeta_{90}^{33} ) q^{12} + ( -\zeta_{90} + \zeta_{90}^{20} ) q^{15} + ( \zeta_{90}^{2} - \zeta_{90}^{15} + \zeta_{90}^{28} - \zeta_{90}^{31} + \zeta_{90}^{44} ) q^{16} + ( \zeta_{90}^{10} - \zeta_{90}^{35} ) q^{17} + ( -\zeta_{90}^{13} - \zeta_{90}^{29} ) q^{18} + ( \zeta_{90}^{2} + \zeta_{90}^{12} + \zeta_{90}^{18} + \zeta_{90}^{28} - \zeta_{90}^{31} + \zeta_{90}^{44} ) q^{20} + ( \zeta_{90}^{18} + \zeta_{90}^{26} + \zeta_{90}^{34} + \zeta_{90}^{42} ) q^{22} + ( -\zeta_{90}^{11} + \zeta_{90}^{24} - \zeta_{90}^{27} + \zeta_{90}^{40} ) q^{24} + ( -\zeta_{90}^{15} + \zeta_{90}^{34} - \zeta_{90}^{41} ) q^{25} -\zeta_{90}^{9} q^{27} + ( -\zeta_{90} - \zeta_{90}^{29} ) q^{29} + ( \zeta_{90}^{8} + \zeta_{90}^{24} - \zeta_{90}^{27} - \zeta_{90}^{43} ) q^{30} + ( -\zeta_{90}^{23} - \zeta_{90}^{37} ) q^{31} + ( \zeta_{90}^{6} - \zeta_{90}^{9} + \zeta_{90}^{22} - \zeta_{90}^{25} - \zeta_{90}^{35} + \zeta_{90}^{38} ) q^{32} + ( \zeta_{90}^{14} + \zeta_{90}^{22} ) q^{33} + ( -\zeta_{90}^{13} - \zeta_{90}^{17} - \zeta_{90}^{33} + \zeta_{90}^{42} ) q^{34} + ( -\zeta_{90}^{7} + \zeta_{90}^{20} + \zeta_{90}^{36} ) q^{36} + ( \zeta_{90}^{6} - \zeta_{90}^{9} - \zeta_{90}^{19} + \zeta_{90}^{22} - \zeta_{90}^{25} - \zeta_{90}^{35} + \zeta_{90}^{38} - \zeta_{90}^{41} ) q^{40} + ( \zeta_{90}^{4} + \zeta_{90}^{12} + \zeta_{90}^{20} - \zeta_{90}^{25} - \zeta_{90}^{33} - \zeta_{90}^{41} ) q^{44} + ( \zeta_{90}^{4} - \zeta_{90}^{23} ) q^{45} + ( \zeta_{90}^{2} - \zeta_{90}^{5} + \zeta_{90}^{18} - \zeta_{90}^{31} + \zeta_{90}^{34} ) q^{48} + \zeta_{90}^{30} q^{49} + ( -\zeta_{90}^{3} + \zeta_{90}^{12} - \zeta_{90}^{19} + \zeta_{90}^{22} + \zeta_{90}^{38} - \zeta_{90}^{41} ) q^{50} + ( -\zeta_{90}^{13} + \zeta_{90}^{38} ) q^{51} + ( \zeta_{90}^{16} + \zeta_{90}^{32} ) q^{54} + ( -\zeta_{90}^{9} - \zeta_{90}^{17} + \zeta_{90}^{28} + \zeta_{90}^{36} ) q^{55} + ( -\zeta_{90}^{7} + \zeta_{90}^{8} + \zeta_{90}^{24} + \zeta_{90}^{36} ) q^{58} + ( \zeta_{90}^{2} - \zeta_{90}^{5} - \zeta_{90}^{15} - \zeta_{90}^{21} - \zeta_{90}^{31} + \zeta_{90}^{34} ) q^{60} + ( -\zeta_{90}^{5} - \zeta_{90}^{25} ) q^{61} + ( -\zeta_{90} - \zeta_{90}^{15} + \zeta_{90}^{30} + \zeta_{90}^{44} ) q^{62} + ( 1 - \zeta_{90}^{3} - \zeta_{90}^{13} + \zeta_{90}^{16} - \zeta_{90}^{29} + \zeta_{90}^{32} + \zeta_{90}^{42} ) q^{64} + ( 1 - \zeta_{90}^{21} - \zeta_{90}^{29} - \zeta_{90}^{37} ) q^{66} + ( -\zeta_{90}^{5} + \zeta_{90}^{10} ) q^{67} + ( \zeta_{90}^{4} - \zeta_{90}^{11} + \zeta_{90}^{20} + \zeta_{90}^{24} + \zeta_{90}^{36} + \zeta_{90}^{40} ) q^{68} + ( -\zeta_{90}^{9} + \zeta_{90}^{36} ) q^{71} + ( \zeta_{90}^{14} - \zeta_{90}^{27} + \zeta_{90}^{30} - \zeta_{90}^{43} ) q^{72} + ( \zeta_{90}^{18} - \zeta_{90}^{37} + \zeta_{90}^{44} ) q^{75} + ( 2 - \zeta_{90}^{3} - \zeta_{90}^{13} + \zeta_{90}^{16} - \zeta_{90}^{19} + \zeta_{90}^{26} - \zeta_{90}^{29} + \zeta_{90}^{32} + \zeta_{90}^{42} ) q^{80} + \zeta_{90}^{12} q^{81} + ( \zeta_{90}^{14} + \zeta_{90}^{16} ) q^{83} + ( -\zeta_{90}^{7} + \zeta_{90}^{8} - \zeta_{90}^{27} - \zeta_{90}^{33} ) q^{85} + ( \zeta_{90}^{4} + \zeta_{90}^{32} ) q^{87} + ( -\zeta_{90}^{3} - \zeta_{90}^{11} - \zeta_{90}^{19} - \zeta_{90}^{27} + \zeta_{90}^{32} - \zeta_{90}^{35} + \zeta_{90}^{40} - \zeta_{90}^{43} ) q^{88} + ( -\zeta_{90} - \zeta_{90}^{11} - \zeta_{90}^{27} + \zeta_{90}^{30} ) q^{90} + ( \zeta_{90}^{26} + \zeta_{90}^{40} ) q^{93} + ( -\zeta_{90}^{9} + \zeta_{90}^{12} - \zeta_{90}^{25} + \zeta_{90}^{28} + \zeta_{90}^{38} - \zeta_{90}^{41} ) q^{96} + ( \zeta_{90}^{8} - \zeta_{90}^{37} ) q^{98} + ( -\zeta_{90}^{17} - \zeta_{90}^{25} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$24q + 3q^{3} - 12q^{4} + 6q^{8} + 3q^{9} + O(q^{10})$$ $$24q + 3q^{3} - 12q^{4} + 6q^{8} + 3q^{9} + 6q^{10} + 3q^{12} - 12q^{16} - 3q^{20} - 3q^{22} - 3q^{24} - 12q^{25} - 6q^{27} - 3q^{30} - 3q^{32} + 6q^{34} - 6q^{36} - 3q^{40} + 6q^{44} - 6q^{48} - 12q^{49} + 6q^{50} - 12q^{55} - 3q^{58} - 9q^{60} - 24q^{62} + 30q^{64} + 27q^{66} - 3q^{68} - 12q^{71} - 18q^{72} - 6q^{75} + 54q^{80} + 3q^{81} - 3q^{85} - 3q^{88} - 18q^{90} - 3q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times$$.

 $$n$$ $$479$$ $$1441$$ $$\chi(n)$$ $$\zeta_{90}^{30}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
238.1
 −0.719340 + 0.694658i 0.0348995 + 0.999391i −0.997564 − 0.0697565i −0.615661 − 0.788011i 0.990268 − 0.139173i 0.559193 − 0.829038i 0.848048 − 0.529919i −0.241922 − 0.970296i 0.961262 + 0.275637i −0.882948 − 0.469472i 0.438371 + 0.898794i −0.374607 + 0.927184i −0.719340 − 0.694658i 0.0348995 − 0.999391i −0.997564 + 0.0697565i −0.615661 + 0.788011i 0.990268 + 0.139173i 0.559193 + 0.829038i 0.848048 + 0.529919i −0.241922 + 0.970296i
−0.990268 1.71519i 0.669131 + 0.743145i −1.46126 + 2.53098i −0.848048 + 1.46886i 0.612019 1.88360i 0 3.80763 −0.104528 + 0.994522i 3.35918
238.2 −0.961262 1.66495i −0.104528 0.994522i −1.34805 + 2.33489i −0.438371 + 0.759281i −1.55535 + 1.13003i 0 3.26078 −0.978148 + 0.207912i 1.68556
238.3 −0.848048 1.46886i −0.978148 0.207912i −0.938371 + 1.62531i 0.615661 1.06636i 0.524123 + 1.61308i 0 1.48704 0.913545 + 0.406737i −2.08844
238.4 −0.559193 0.968551i 0.913545 0.406737i −0.125393 + 0.217188i 0.719340 1.24593i −0.904793 0.657371i 0 −0.837909 0.669131 0.743145i −1.60900
238.5 −0.438371 0.759281i 0.913545 0.406737i 0.115661 0.200332i 0.241922 0.419021i −0.709299 0.515336i 0 −1.07955 0.669131 0.743145i −0.424206
238.6 −0.0348995 0.0604477i −0.978148 0.207912i 0.497564 0.861806i −0.990268 + 1.71519i 0.0215691 + 0.0663828i 0 −0.139258 0.913545 + 0.406737i 0.138239
238.7 0.241922 + 0.419021i −0.104528 0.994522i 0.382948 0.663285i −0.559193 + 0.968551i 0.391438 0.284396i 0 0.854417 −0.978148 + 0.207912i −0.541124
238.8 0.374607 + 0.648838i 0.669131 + 0.743145i 0.219340 0.379908i −0.0348995 + 0.0604477i −0.231520 + 0.712544i 0 1.07788 −0.104528 + 0.994522i −0.0522943
238.9 0.615661 + 1.06636i 0.669131 + 0.743145i −0.258078 + 0.447004i 0.882948 1.52931i −0.380500 + 1.17106i 0 0.595768 −0.104528 + 0.994522i 2.17439
238.10 0.719340 + 1.24593i −0.104528 0.994522i −0.534899 + 0.926473i 0.997564 1.72783i 1.16392 0.845635i 0 −0.100418 −0.978148 + 0.207912i 2.87035
238.11 0.882948 + 1.52931i −0.978148 0.207912i −1.05919 + 1.83458i 0.374607 0.648838i −0.545692 1.67947i 0 −1.97495 0.913545 + 0.406737i 1.32303
238.12 0.997564 + 1.72783i 0.913545 0.406737i −1.49027 + 2.58122i −0.961262 + 1.66495i 1.61409 + 1.17271i 0 −3.95142 0.669131 0.743145i −3.83568
1672.1 −0.990268 + 1.71519i 0.669131 0.743145i −1.46126 2.53098i −0.848048 1.46886i 0.612019 + 1.88360i 0 3.80763 −0.104528 0.994522i 3.35918
1672.2 −0.961262 + 1.66495i −0.104528 + 0.994522i −1.34805 2.33489i −0.438371 0.759281i −1.55535 1.13003i 0 3.26078 −0.978148 0.207912i 1.68556
1672.3 −0.848048 + 1.46886i −0.978148 + 0.207912i −0.938371 1.62531i 0.615661 + 1.06636i 0.524123 1.61308i 0 1.48704 0.913545 0.406737i −2.08844
1672.4 −0.559193 + 0.968551i 0.913545 + 0.406737i −0.125393 0.217188i 0.719340 + 1.24593i −0.904793 + 0.657371i 0 −0.837909 0.669131 + 0.743145i −1.60900
1672.5 −0.438371 + 0.759281i 0.913545 + 0.406737i 0.115661 + 0.200332i 0.241922 + 0.419021i −0.709299 + 0.515336i 0 −1.07955 0.669131 + 0.743145i −0.424206
1672.6 −0.0348995 + 0.0604477i −0.978148 + 0.207912i 0.497564 + 0.861806i −0.990268 1.71519i 0.0215691 0.0663828i 0 −0.139258 0.913545 0.406737i 0.138239
1672.7 0.241922 0.419021i −0.104528 + 0.994522i 0.382948 + 0.663285i −0.559193 0.968551i 0.391438 + 0.284396i 0 0.854417 −0.978148 0.207912i −0.541124
1672.8 0.374607 0.648838i 0.669131 0.743145i 0.219340 + 0.379908i −0.0348995 0.0604477i −0.231520 0.712544i 0 1.07788 −0.104528 0.994522i −0.0522943
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1672.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
239.b odd 2 1 CM by $$\Q(\sqrt{-239})$$
9.c even 3 1 inner
2151.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2151.1.f.b 24
9.c even 3 1 inner 2151.1.f.b 24
239.b odd 2 1 CM 2151.1.f.b 24
2151.f odd 6 1 inner 2151.1.f.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2151.1.f.b 24 1.a even 1 1 trivial
2151.1.f.b 24 9.c even 3 1 inner
2151.1.f.b 24 239.b odd 2 1 CM
2151.1.f.b 24 2151.f odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} + \cdots$$ acting on $$S_{1}^{\mathrm{new}}(2151, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 12 T + 180 T^{2} - 370 T^{3} + 1563 T^{4} - 1377 T^{5} + 4193 T^{6} - 2664 T^{7} + 7560 T^{8} - 3139 T^{9} + 8322 T^{10} - 2556 T^{11} + 6625 T^{12} - 1419 T^{13} + 3618 T^{14} - 524 T^{15} + 1476 T^{16} - 135 T^{17} + 425 T^{18} - 21 T^{19} + 90 T^{20} - 2 T^{21} + 12 T^{22} + T^{24}$$
$3$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{3}$$
$5$ $$1 + 12 T + 180 T^{2} - 370 T^{3} + 1563 T^{4} - 1377 T^{5} + 4193 T^{6} - 2664 T^{7} + 7560 T^{8} - 3139 T^{9} + 8322 T^{10} - 2556 T^{11} + 6625 T^{12} - 1419 T^{13} + 3618 T^{14} - 524 T^{15} + 1476 T^{16} - 135 T^{17} + 425 T^{18} - 21 T^{19} + 90 T^{20} - 2 T^{21} + 12 T^{22} + T^{24}$$
$7$ $$T^{24}$$
$11$ $$1 + 12 T + 180 T^{2} - 370 T^{3} + 1563 T^{4} - 1377 T^{5} + 4193 T^{6} - 2664 T^{7} + 7560 T^{8} - 3139 T^{9} + 8322 T^{10} - 2556 T^{11} + 6625 T^{12} - 1419 T^{13} + 3618 T^{14} - 524 T^{15} + 1476 T^{16} - 135 T^{17} + 425 T^{18} - 21 T^{19} + 90 T^{20} - 2 T^{21} + 12 T^{22} + T^{24}$$
$13$ $$T^{24}$$
$17$ $$( 1 - 3 T + T^{3} )^{8}$$
$19$ $$T^{24}$$
$23$ $$T^{24}$$
$29$ $$1 + 12 T + 180 T^{2} - 370 T^{3} + 1563 T^{4} - 1377 T^{5} + 4193 T^{6} - 2664 T^{7} + 7560 T^{8} - 3139 T^{9} + 8322 T^{10} - 2556 T^{11} + 6625 T^{12} - 1419 T^{13} + 3618 T^{14} - 524 T^{15} + 1476 T^{16} - 135 T^{17} + 425 T^{18} - 21 T^{19} + 90 T^{20} - 2 T^{21} + 12 T^{22} + T^{24}$$
$31$ $$1 + 12 T + 180 T^{2} - 370 T^{3} + 1563 T^{4} - 1377 T^{5} + 4193 T^{6} - 2664 T^{7} + 7560 T^{8} - 3139 T^{9} + 8322 T^{10} - 2556 T^{11} + 6625 T^{12} - 1419 T^{13} + 3618 T^{14} - 524 T^{15} + 1476 T^{16} - 135 T^{17} + 425 T^{18} - 21 T^{19} + 90 T^{20} - 2 T^{21} + 12 T^{22} + T^{24}$$
$37$ $$T^{24}$$
$41$ $$T^{24}$$
$43$ $$T^{24}$$
$47$ $$T^{24}$$
$53$ $$T^{24}$$
$59$ $$T^{24}$$
$61$ $$( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} )^{4}$$
$67$ $$( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} )^{4}$$
$71$ $$( -1 + T + T^{2} )^{12}$$
$73$ $$T^{24}$$
$79$ $$T^{24}$$
$83$ $$1 + 12 T + 180 T^{2} - 370 T^{3} + 1563 T^{4} - 1377 T^{5} + 4193 T^{6} - 2664 T^{7} + 7560 T^{8} - 3139 T^{9} + 8322 T^{10} - 2556 T^{11} + 6625 T^{12} - 1419 T^{13} + 3618 T^{14} - 524 T^{15} + 1476 T^{16} - 135 T^{17} + 425 T^{18} - 21 T^{19} + 90 T^{20} - 2 T^{21} + 12 T^{22} + T^{24}$$
$89$ $$T^{24}$$
$97$ $$T^{24}$$