# Properties

 Label 75.4.b.b Level $75$ Weight $4$ Character orbit 75.b Analytic conductor $4.425$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.42514325043$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + i q^{2} -3 i q^{3} + 7 q^{4} + 3 q^{6} -24 i q^{7} + 15 i q^{8} -9 q^{9} +O(q^{10})$$ $$q + i q^{2} -3 i q^{3} + 7 q^{4} + 3 q^{6} -24 i q^{7} + 15 i q^{8} -9 q^{9} + 52 q^{11} -21 i q^{12} -22 i q^{13} + 24 q^{14} + 41 q^{16} -14 i q^{17} -9 i q^{18} + 20 q^{19} -72 q^{21} + 52 i q^{22} + 168 i q^{23} + 45 q^{24} + 22 q^{26} + 27 i q^{27} -168 i q^{28} -230 q^{29} -288 q^{31} + 161 i q^{32} -156 i q^{33} + 14 q^{34} -63 q^{36} -34 i q^{37} + 20 i q^{38} -66 q^{39} + 122 q^{41} -72 i q^{42} + 188 i q^{43} + 364 q^{44} -168 q^{46} + 256 i q^{47} -123 i q^{48} -233 q^{49} -42 q^{51} -154 i q^{52} + 338 i q^{53} -27 q^{54} + 360 q^{56} -60 i q^{57} -230 i q^{58} -100 q^{59} + 742 q^{61} -288 i q^{62} + 216 i q^{63} + 167 q^{64} + 156 q^{66} -84 i q^{67} -98 i q^{68} + 504 q^{69} -328 q^{71} -135 i q^{72} + 38 i q^{73} + 34 q^{74} + 140 q^{76} -1248 i q^{77} -66 i q^{78} + 240 q^{79} + 81 q^{81} + 122 i q^{82} -1212 i q^{83} -504 q^{84} -188 q^{86} + 690 i q^{87} + 780 i q^{88} -330 q^{89} -528 q^{91} + 1176 i q^{92} + 864 i q^{93} -256 q^{94} + 483 q^{96} + 866 i q^{97} -233 i q^{98} -468 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{4} + 6q^{6} - 18q^{9} + O(q^{10})$$ $$2q + 14q^{4} + 6q^{6} - 18q^{9} + 104q^{11} + 48q^{14} + 82q^{16} + 40q^{19} - 144q^{21} + 90q^{24} + 44q^{26} - 460q^{29} - 576q^{31} + 28q^{34} - 126q^{36} - 132q^{39} + 244q^{41} + 728q^{44} - 336q^{46} - 466q^{49} - 84q^{51} - 54q^{54} + 720q^{56} - 200q^{59} + 1484q^{61} + 334q^{64} + 312q^{66} + 1008q^{69} - 656q^{71} + 68q^{74} + 280q^{76} + 480q^{79} + 162q^{81} - 1008q^{84} - 376q^{86} - 660q^{89} - 1056q^{91} - 512q^{94} + 966q^{96} - 936q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
1.00000i 3.00000i 7.00000 0 3.00000 24.0000i 15.0000i −9.00000 0
49.2 1.00000i 3.00000i 7.00000 0 3.00000 24.0000i 15.0000i −9.00000 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
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 75.4.b.b 2
3.b odd 2 1 225.4.b.e 2
4.b odd 2 1 1200.4.f.b 2
5.b even 2 1 inner 75.4.b.b 2
5.c odd 4 1 15.4.a.a 1
5.c odd 4 1 75.4.a.b 1
15.d odd 2 1 225.4.b.e 2
15.e even 4 1 45.4.a.c 1
15.e even 4 1 225.4.a.f 1
20.d odd 2 1 1200.4.f.b 2
20.e even 4 1 240.4.a.e 1
20.e even 4 1 1200.4.a.t 1
35.f even 4 1 735.4.a.e 1
40.i odd 4 1 960.4.a.b 1
40.k even 4 1 960.4.a.ba 1
45.k odd 12 2 405.4.e.g 2
45.l even 12 2 405.4.e.i 2
55.e even 4 1 1815.4.a.e 1
60.l odd 4 1 720.4.a.n 1
105.k odd 4 1 2205.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 5.c odd 4 1
45.4.a.c 1 15.e even 4 1
75.4.a.b 1 5.c odd 4 1
75.4.b.b 2 1.a even 1 1 trivial
75.4.b.b 2 5.b even 2 1 inner
225.4.a.f 1 15.e even 4 1
225.4.b.e 2 3.b odd 2 1
225.4.b.e 2 15.d odd 2 1
240.4.a.e 1 20.e even 4 1
405.4.e.g 2 45.k odd 12 2
405.4.e.i 2 45.l even 12 2
720.4.a.n 1 60.l odd 4 1
735.4.a.e 1 35.f even 4 1
960.4.a.b 1 40.i odd 4 1
960.4.a.ba 1 40.k even 4 1
1200.4.a.t 1 20.e even 4 1
1200.4.f.b 2 4.b odd 2 1
1200.4.f.b 2 20.d odd 2 1
1815.4.a.e 1 55.e even 4 1
2205.4.a.l 1 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$576 + T^{2}$$
$11$ $$( -52 + T )^{2}$$
$13$ $$484 + T^{2}$$
$17$ $$196 + T^{2}$$
$19$ $$( -20 + T )^{2}$$
$23$ $$28224 + T^{2}$$
$29$ $$( 230 + T )^{2}$$
$31$ $$( 288 + T )^{2}$$
$37$ $$1156 + T^{2}$$
$41$ $$( -122 + T )^{2}$$
$43$ $$35344 + T^{2}$$
$47$ $$65536 + T^{2}$$
$53$ $$114244 + T^{2}$$
$59$ $$( 100 + T )^{2}$$
$61$ $$( -742 + T )^{2}$$
$67$ $$7056 + T^{2}$$
$71$ $$( 328 + T )^{2}$$
$73$ $$1444 + T^{2}$$
$79$ $$( -240 + T )^{2}$$
$83$ $$1468944 + T^{2}$$
$89$ $$( 330 + T )^{2}$$
$97$ $$749956 + T^{2}$$