# Properties

 Label 93.2.p.a Level $93$ Weight $2$ Character orbit 93.p Analytic conductor $0.743$ Analytic rank $0$ Dimension $8$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$93 = 3 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 93.p (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.742608738798$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{15})$$ Defining polynomial: $$x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{30}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{15}^{2} + 2 \zeta_{15}^{7} ) q^{3} + 2 \zeta_{15}^{6} q^{4} + ( -\zeta_{15} + \zeta_{15}^{2} + 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} + \zeta_{15}^{5} - \zeta_{15}^{6} ) q^{7} -3 \zeta_{15}^{4} q^{9} +O(q^{10})$$ $$q + ( \zeta_{15}^{2} + 2 \zeta_{15}^{7} ) q^{3} + 2 \zeta_{15}^{6} q^{4} + ( -\zeta_{15} + \zeta_{15}^{2} + 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} + \zeta_{15}^{5} - \zeta_{15}^{6} ) q^{7} -3 \zeta_{15}^{4} q^{9} + ( 2 - 2 \zeta_{15} - 2 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{5} - 2 \zeta_{15}^{7} ) q^{12} + ( -4 + 5 \zeta_{15} - 3 \zeta_{15}^{3} + 4 \zeta_{15}^{4} - 4 \zeta_{15}^{5} - 3 \zeta_{15}^{6} + 4 \zeta_{15}^{7} ) q^{13} + ( -4 \zeta_{15}^{2} - 4 \zeta_{15}^{7} ) q^{16} + ( 5 - 3 \zeta_{15}^{2} + 5 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 5 \zeta_{15}^{6} - 8 \zeta_{15}^{7} ) q^{19} + ( -5 - 5 \zeta_{15} - \zeta_{15}^{5} - 4 \zeta_{15}^{6} ) q^{21} + 5 \zeta_{15}^{5} q^{25} + ( 6 \zeta_{15} + 3 \zeta_{15}^{6} ) q^{27} + ( -10 + 6 \zeta_{15}^{2} - 6 \zeta_{15}^{3} + 2 \zeta_{15}^{4} - 6 \zeta_{15}^{5} - 6 \zeta_{15}^{6} + 6 \zeta_{15}^{7} ) q^{28} + ( -5 \zeta_{15}^{2} + \zeta_{15}^{7} ) q^{31} + ( 6 + 6 \zeta_{15}^{5} ) q^{36} + ( 7 - 7 \zeta_{15} + 7 \zeta_{15}^{2} + 4 \zeta_{15}^{3} - 7 \zeta_{15}^{4} + 7 \zeta_{15}^{5} - 4 \zeta_{15}^{7} ) q^{37} + ( -3 + 5 \zeta_{15} + 2 \zeta_{15}^{3} + 5 \zeta_{15}^{4} - 10 \zeta_{15}^{5} + 5 \zeta_{15}^{7} ) q^{39} + ( 7 - 6 \zeta_{15} + \zeta_{15}^{5} + \zeta_{15}^{6} ) q^{43} + ( 4 - 4 \zeta_{15}^{2} + 4 \zeta_{15}^{3} + 4 \zeta_{15}^{4} + 4 \zeta_{15}^{6} - 4 \zeta_{15}^{7} ) q^{48} + ( -3 + 8 \zeta_{15} - 8 \zeta_{15}^{3} + 3 \zeta_{15}^{4} - 3 \zeta_{15}^{5} + 8 \zeta_{15}^{6} + 10 \zeta_{15}^{7} ) q^{49} + ( 6 + 6 \zeta_{15}^{3} - 8 \zeta_{15}^{4} + 6 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{52} + ( -7 - \zeta_{15} + 7 \zeta_{15}^{2} - 7 \zeta_{15}^{3} + 8 \zeta_{15}^{4} + 7 \zeta_{15}^{7} ) q^{57} + ( -5 - 4 \zeta_{15} + 5 \zeta_{15}^{2} - 5 \zeta_{15}^{3} - 4 \zeta_{15}^{4} - 14 \zeta_{15}^{6} + 5 \zeta_{15}^{7} ) q^{61} + ( 6 - 6 \zeta_{15} - 3 \zeta_{15}^{2} + 9 \zeta_{15}^{3} - 6 \zeta_{15}^{4} + 6 \zeta_{15}^{5} - 15 \zeta_{15}^{7} ) q^{63} + 8 \zeta_{15}^{3} q^{64} + ( -2 + 7 \zeta_{15} + 2 \zeta_{15}^{2} - 2 \zeta_{15}^{3} + 9 \zeta_{15}^{4} - 4 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{67} + ( 9 - 8 \zeta_{15} - \zeta_{15}^{2} - 17 \zeta_{15}^{4} + 8 \zeta_{15}^{5} + \zeta_{15}^{6} - 9 \zeta_{15}^{7} ) q^{73} + ( -10 \zeta_{15}^{2} - 5 \zeta_{15}^{7} ) q^{75} + ( -16 + 10 \zeta_{15} - 4 \zeta_{15}^{3} + 10 \zeta_{15}^{4} - 6 \zeta_{15}^{5} + 10 \zeta_{15}^{7} ) q^{76} + ( -7 - 3 \zeta_{15} - 7 \zeta_{15}^{3} - 3 \zeta_{15}^{4} - 4 \zeta_{15}^{5} - 3 \zeta_{15}^{7} ) q^{79} + ( -9 + 9 \zeta_{15} - 9 \zeta_{15}^{3} + 9 \zeta_{15}^{4} - 9 \zeta_{15}^{5} + 9 \zeta_{15}^{7} ) q^{81} + ( 2 \zeta_{15} + 8 \zeta_{15}^{2} - 8 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{84} + ( 15 - 11 \zeta_{15} - 19 \zeta_{15}^{2} + 10 \zeta_{15}^{3} - \zeta_{15}^{4} + 11 \zeta_{15}^{5} + 5 \zeta_{15}^{6} - 20 \zeta_{15}^{7} ) q^{91} + ( 11 - 11 \zeta_{15}^{2} + 11 \zeta_{15}^{3} - 7 \zeta_{15}^{4} + 11 \zeta_{15}^{6} - 11 \zeta_{15}^{7} ) q^{93} + ( -3 + 3 \zeta_{15}^{2} - 11 \zeta_{15}^{5} - 8 \zeta_{15}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 3q^{3} - 4q^{4} - 4q^{7} - 3q^{9} + O(q^{10})$$ $$8q + 3q^{3} - 4q^{4} - 4q^{7} - 3q^{9} + 6q^{12} + 9q^{13} - 8q^{16} + 7q^{19} - 33q^{21} - 20q^{25} - 18q^{28} - 4q^{31} + 24q^{36} + 9q^{37} + 27q^{39} + 44q^{43} + 12q^{48} + 9q^{49} + 18q^{52} - 21q^{57} - 24q^{63} - 16q^{64} + 16q^{67} + 3q^{73} - 15q^{75} - 66q^{76} - 35q^{79} + 9q^{81} + 24q^{84} - 5q^{91} + 15q^{93} + 15q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/93\mathbb{Z}\right)^\times$$.

 $$n$$ $$32$$ $$34$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{15}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.913545 + 0.406737i 0.913545 − 0.406737i −0.978148 + 0.207912i −0.104528 + 0.994522i 0.669131 + 0.743145i −0.978148 − 0.207912i 0.669131 − 0.743145i −0.104528 − 0.994522i
0 −1.28716 + 1.15897i −1.61803 + 1.17557i 0 0 0.473579 + 4.50581i 0 0.313585 2.98357i 0
17.1 0 −1.28716 1.15897i −1.61803 1.17557i 0 0 0.473579 4.50581i 0 0.313585 + 2.98357i 0
44.1 0 0.704489 + 1.58231i 0.618034 1.90211i 0 0 0.802903 + 0.891714i 0 −2.00739 + 2.22943i 0
53.1 0 0.360114 1.69420i −1.61803 1.17557i 0 0 1.88052 0.837263i 0 −2.74064 1.22021i 0
65.1 0 1.72256 + 0.181049i 0.618034 1.90211i 0 0 −5.15701 + 1.09616i 0 2.93444 + 0.623735i 0
74.1 0 0.704489 1.58231i 0.618034 + 1.90211i 0 0 0.802903 0.891714i 0 −2.00739 2.22943i 0
83.1 0 1.72256 0.181049i 0.618034 + 1.90211i 0 0 −5.15701 1.09616i 0 2.93444 0.623735i 0
86.1 0 0.360114 + 1.69420i −1.61803 + 1.17557i 0 0 1.88052 + 0.837263i 0 −2.74064 + 1.22021i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 86.1 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 CM by $$\Q(\sqrt{-3})$$
31.h odd 30 1 inner
93.p even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.2.p.a 8
3.b odd 2 1 CM 93.2.p.a 8
31.h odd 30 1 inner 93.2.p.a 8
93.p even 30 1 inner 93.2.p.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.2.p.a 8 1.a even 1 1 trivial
93.2.p.a 8 3.b odd 2 1 CM
93.2.p.a 8 31.h odd 30 1 inner
93.2.p.a 8 93.p even 30 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(93, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$81 - 81 T + 54 T^{2} - 27 T^{3} + 9 T^{4} - 9 T^{5} + 6 T^{6} - 3 T^{7} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$3481 - 5841 T + 4655 T^{2} - 1304 T^{3} - 81 T^{4} + 76 T^{5} + 4 T^{7} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$32761 + 20453 T + 22336 T^{2} + 5724 T^{3} - 701 T^{4} + 82 T^{5} + 54 T^{6} - 9 T^{7} + T^{8}$$
$17$ $$T^{8}$$
$19$ $$201601 + 196662 T + 83030 T^{2} + 14147 T^{3} - 216 T^{4} - 322 T^{5} - 7 T^{7} + T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$923521 + 119164 T - 14415 T^{2} - 5704 T^{3} - 271 T^{4} - 184 T^{5} - 15 T^{6} + 4 T^{7} + T^{8}$$
$37$ $$6651241 + 3667338 T + 266546 T^{2} - 224676 T^{3} + 18119 T^{4} + 1422 T^{5} - 131 T^{6} - 9 T^{7} + T^{8}$$
$41$ $$T^{8}$$
$43$ $$564001 - 485897 T + 719531 T^{2} - 283016 T^{3} + 64179 T^{4} - 9168 T^{5} + 834 T^{6} - 44 T^{7} + T^{8}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$39300361 + 6392618 T^{2} + 103259 T^{4} + 562 T^{6} + T^{8}$$
$67$ $$4932841 + 2807344 T + 1773155 T^{2} - 28784 T^{3} + 24244 T^{4} - 1264 T^{5} + 335 T^{6} - 16 T^{7} + T^{8}$$
$71$ $$T^{8}$$
$73$ $$653262481 - 65533276 T + 9798844 T^{2} + 10923 T^{3} - 35396 T^{4} + 6196 T^{5} + 6 T^{6} - 3 T^{7} + T^{8}$$
$79$ $$7317025 + 8304350 T + 4438700 T^{2} + 704900 T^{3} + 86445 T^{4} + 9960 T^{5} + 735 T^{6} + 35 T^{7} + T^{8}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$1263447025 + 159063875 T + 7067250 T^{2} - 648175 T^{3} + 106215 T^{4} - 4975 T^{5} + 560 T^{6} - 15 T^{7} + T^{8}$$