# Properties

 Label 75.7.d.a Level 75 Weight 7 Character orbit 75.d Analytic conductor 17.254 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 75.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.2540562715$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -27 i q^{3} -64 q^{4} + 286 i q^{7} -729 q^{9} +O(q^{10})$$ $$q -27 i q^{3} -64 q^{4} + 286 i q^{7} -729 q^{9} + 1728 i q^{12} + 506 i q^{13} + 4096 q^{16} + 10582 q^{19} + 7722 q^{21} + 19683 i q^{27} -18304 i q^{28} + 35282 q^{31} + 46656 q^{36} + 89206 i q^{37} + 13662 q^{39} + 111386 i q^{43} -110592 i q^{48} + 35853 q^{49} -32384 i q^{52} -285714 i q^{57} -420838 q^{61} -208494 i q^{63} -262144 q^{64} -172874 i q^{67} + 638066 i q^{73} -677248 q^{76} + 204622 q^{79} + 531441 q^{81} -494208 q^{84} -144716 q^{91} -952614 i q^{93} + 56446 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 128q^{4} - 1458q^{9} + O(q^{10})$$ $$2q - 128q^{4} - 1458q^{9} + 8192q^{16} + 21164q^{19} + 15444q^{21} + 70564q^{31} + 93312q^{36} + 27324q^{39} + 71706q^{49} - 841676q^{61} - 524288q^{64} - 1354496q^{76} + 409244q^{79} + 1062882q^{81} - 988416q^{84} - 289432q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
74.1
 1.00000i − 1.00000i
0 27.0000i −64.0000 0 0 286.000i 0 −729.000 0
74.2 0 27.0000i −64.0000 0 0 286.000i 0 −729.000 0
 $$n$$: e.g. 2-40 or 990-1000 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 $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.7.d.a 2
3.b odd 2 1 CM 75.7.d.a 2
5.b even 2 1 inner 75.7.d.a 2
5.c odd 4 1 3.7.b.a 1
5.c odd 4 1 75.7.c.a 1
15.d odd 2 1 inner 75.7.d.a 2
15.e even 4 1 3.7.b.a 1
15.e even 4 1 75.7.c.a 1
20.e even 4 1 48.7.e.a 1
35.f even 4 1 147.7.b.a 1
40.i odd 4 1 192.7.e.b 1
40.k even 4 1 192.7.e.a 1
45.k odd 12 2 81.7.d.a 2
45.l even 12 2 81.7.d.a 2
60.l odd 4 1 48.7.e.a 1
105.k odd 4 1 147.7.b.a 1
120.q odd 4 1 192.7.e.a 1
120.w even 4 1 192.7.e.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.7.b.a 1 5.c odd 4 1
3.7.b.a 1 15.e even 4 1
48.7.e.a 1 20.e even 4 1
48.7.e.a 1 60.l odd 4 1
75.7.c.a 1 5.c odd 4 1
75.7.c.a 1 15.e even 4 1
75.7.d.a 2 1.a even 1 1 trivial
75.7.d.a 2 3.b odd 2 1 CM
75.7.d.a 2 5.b even 2 1 inner
75.7.d.a 2 15.d odd 2 1 inner
81.7.d.a 2 45.k odd 12 2
81.7.d.a 2 45.l even 12 2
147.7.b.a 1 35.f even 4 1
147.7.b.a 1 105.k odd 4 1
192.7.e.a 1 40.k even 4 1
192.7.e.a 1 120.q odd 4 1
192.7.e.b 1 40.i odd 4 1
192.7.e.b 1 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{7}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 64 T^{2} )^{2}$$
$3$ $$1 + 729 T^{2}$$
$5$ 1
$7$ $$1 - 153502 T^{2} + 13841287201 T^{4}$$
$11$ $$( 1 - 1331 T )^{2}( 1 + 1331 T )^{2}$$
$13$ $$1 - 9397582 T^{2} + 23298085122481 T^{4}$$
$17$ $$( 1 + 24137569 T^{2} )^{2}$$
$19$ $$( 1 - 10582 T + 47045881 T^{2} )^{2}$$
$23$ $$( 1 + 148035889 T^{2} )^{2}$$
$29$ $$( 1 - 24389 T )^{2}( 1 + 24389 T )^{2}$$
$31$ $$( 1 - 35282 T + 887503681 T^{2} )^{2}$$
$37$ $$1 + 2826257618 T^{2} + 6582952005840035281 T^{4}$$
$41$ $$( 1 - 68921 T )^{2}( 1 + 68921 T )^{2}$$
$43$ $$1 - 235885102 T^{2} + 39959630797262576401 T^{4}$$
$47$ $$( 1 + 10779215329 T^{2} )^{2}$$
$53$ $$( 1 + 22164361129 T^{2} )^{2}$$
$59$ $$( 1 - 205379 T )^{2}( 1 + 205379 T )^{2}$$
$61$ $$( 1 + 420838 T + 51520374361 T^{2} )^{2}$$
$67$ $$1 - 151031344462 T^{2} +$$$$81\!\cdots\!61$$$$T^{4}$$
$71$ $$( 1 - 357911 T )^{2}( 1 + 357911 T )^{2}$$
$73$ $$1 + 104459767778 T^{2} +$$$$22\!\cdots\!21$$$$T^{4}$$
$79$ $$( 1 - 204622 T + 243087455521 T^{2} )^{2}$$
$83$ $$( 1 + 326940373369 T^{2} )^{2}$$
$89$ $$( 1 - 704969 T )^{2}( 1 + 704969 T )^{2}$$
$97$ $$1 - 1662757858942 T^{2} +$$$$69\!\cdots\!41$$$$T^{4}$$
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