Properties

 Label 75.9.c.d Level $75$ Weight $9$ Character orbit 75.c Analytic conductor $30.553$ Analytic rank $0$ Dimension $2$ CM discriminant -15 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 75.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$30.5533957546$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q + 17 i q^{2} - 81 i q^{3} - 33 q^{4} + 1377 q^{6} + 3791 i q^{8} - 6561 q^{9} +O(q^{10})$$ q + 17*i * q^2 - 81*i * q^3 - 33 * q^4 + 1377 * q^6 + 3791*i * q^8 - 6561 * q^9 $$q + 17 i q^{2} - 81 i q^{3} - 33 q^{4} + 1377 q^{6} + 3791 i q^{8} - 6561 q^{9} + 2673 i q^{12} - 72895 q^{16} - 21118 i q^{17} - 111537 i q^{18} + 203998 q^{19} + 550078 i q^{23} + 307071 q^{24} + 531441 i q^{27} + 1831682 q^{31} - 268719 i q^{32} + 359006 q^{34} + 216513 q^{36} + 3467966 i q^{38} - 9351326 q^{46} + 8065922 i q^{47} + 5904495 i q^{48} - 5764801 q^{49} - 1710558 q^{51} + 12619678 i q^{53} - 9034497 q^{54} - 16523838 i q^{57} + 14324642 q^{61} + 31138594 i q^{62} - 14092897 q^{64} + 696894 i q^{68} + 44556318 q^{69} - 24872751 i q^{72} - 6731934 q^{76} + 69617278 q^{79} + 43046721 q^{81} - 3847202 i q^{83} - 18152574 i q^{92} - 148366242 i q^{93} - 137120674 q^{94} - 21766239 q^{96} - 98001617 i q^{98} +O(q^{100})$$ q + 17*i * q^2 - 81*i * q^3 - 33 * q^4 + 1377 * q^6 + 3791*i * q^8 - 6561 * q^9 + 2673*i * q^12 - 72895 * q^16 - 21118*i * q^17 - 111537*i * q^18 + 203998 * q^19 + 550078*i * q^23 + 307071 * q^24 + 531441*i * q^27 + 1831682 * q^31 - 268719*i * q^32 + 359006 * q^34 + 216513 * q^36 + 3467966*i * q^38 - 9351326 * q^46 + 8065922*i * q^47 + 5904495*i * q^48 - 5764801 * q^49 - 1710558 * q^51 + 12619678*i * q^53 - 9034497 * q^54 - 16523838*i * q^57 + 14324642 * q^61 + 31138594*i * q^62 - 14092897 * q^64 + 696894*i * q^68 + 44556318 * q^69 - 24872751*i * q^72 - 6731934 * q^76 + 69617278 * q^79 + 43046721 * q^81 - 3847202*i * q^83 - 18152574*i * q^92 - 148366242*i * q^93 - 137120674 * q^94 - 21766239 * q^96 - 98001617*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 66 q^{4} + 2754 q^{6} - 13122 q^{9}+O(q^{10})$$ 2 * q - 66 * q^4 + 2754 * q^6 - 13122 * q^9 $$2 q - 66 q^{4} + 2754 q^{6} - 13122 q^{9} - 145790 q^{16} + 407996 q^{19} + 614142 q^{24} + 3663364 q^{31} + 718012 q^{34} + 433026 q^{36} - 18702652 q^{46} - 11529602 q^{49} - 3421116 q^{51} - 18068994 q^{54} + 28649284 q^{61} - 28185794 q^{64} + 89112636 q^{69} - 13463868 q^{76} + 139234556 q^{79} + 86093442 q^{81} - 274241348 q^{94} - 43532478 q^{96}+O(q^{100})$$ 2 * q - 66 * q^4 + 2754 * q^6 - 13122 * q^9 - 145790 * q^16 + 407996 * q^19 + 614142 * q^24 + 3663364 * q^31 + 718012 * q^34 + 433026 * q^36 - 18702652 * q^46 - 11529602 * q^49 - 3421116 * q^51 - 18068994 * q^54 + 28649284 * q^61 - 28185794 * q^64 + 89112636 * q^69 - 13463868 * q^76 + 139234556 * q^79 + 86093442 * q^81 - 274241348 * q^94 - 43532478 * q^96

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 − 1.00000i 1.00000i
17.0000i 81.0000i −33.0000 0 1377.00 0 3791.00i −6561.00 0
26.2 17.0000i 81.0000i −33.0000 0 1377.00 0 3791.00i −6561.00 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
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 75.9.c.d 2
3.b odd 2 1 inner 75.9.c.d 2
5.b even 2 1 inner 75.9.c.d 2
5.c odd 4 1 15.9.d.a 1
5.c odd 4 1 15.9.d.b yes 1
15.d odd 2 1 CM 75.9.c.d 2
15.e even 4 1 15.9.d.a 1
15.e even 4 1 15.9.d.b yes 1
20.e even 4 1 240.9.c.a 1
20.e even 4 1 240.9.c.b 1
60.l odd 4 1 240.9.c.a 1
60.l odd 4 1 240.9.c.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.d.a 1 5.c odd 4 1
15.9.d.a 1 15.e even 4 1
15.9.d.b yes 1 5.c odd 4 1
15.9.d.b yes 1 15.e even 4 1
75.9.c.d 2 1.a even 1 1 trivial
75.9.c.d 2 3.b odd 2 1 inner
75.9.c.d 2 5.b even 2 1 inner
75.9.c.d 2 15.d odd 2 1 CM
240.9.c.a 1 20.e even 4 1
240.9.c.a 1 60.l odd 4 1
240.9.c.b 1 20.e even 4 1
240.9.c.b 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(75, [\chi])$$:

 $$T_{2}^{2} + 289$$ T2^2 + 289 $$T_{7}$$ T7

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 289$$
$3$ $$T^{2} + 6561$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 445969924$$
$19$ $$(T - 203998)^{2}$$
$23$ $$T^{2} + 302585806084$$
$29$ $$T^{2}$$
$31$ $$(T - 1831682)^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 65059097710084$$
$53$ $$T^{2} + \cdots + 159256272823684$$
$59$ $$T^{2}$$
$61$ $$(T - 14324642)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$(T - 69617278)^{2}$$
$83$ $$T^{2} + 14800963228804$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$