# Properties

 Label 975.2.c.f Level $975$ Weight $2$ Character orbit 975.c Analytic conductor $7.785$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} + q^{4} - q^{6} - 4 i q^{7} + 3 i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 + i * q^3 + q^4 - q^6 - 4*i * q^7 + 3*i * q^8 - q^9 $$q + i q^{2} + i q^{3} + q^{4} - q^{6} - 4 i q^{7} + 3 i q^{8} - q^{9} + 4 q^{11} + i q^{12} - i q^{13} + 4 q^{14} - q^{16} + 2 i q^{17} - i q^{18} + 4 q^{21} + 4 i q^{22} - 3 q^{24} + q^{26} - i q^{27} - 4 i q^{28} + 10 q^{29} + 4 q^{31} + 5 i q^{32} + 4 i q^{33} - 2 q^{34} - q^{36} - 2 i q^{37} + q^{39} + 6 q^{41} + 4 i q^{42} + 12 i q^{43} + 4 q^{44} - i q^{48} - 9 q^{49} - 2 q^{51} - i q^{52} - 6 i q^{53} + q^{54} + 12 q^{56} + 10 i q^{58} - 12 q^{59} - 2 q^{61} + 4 i q^{62} + 4 i q^{63} - 7 q^{64} - 4 q^{66} - 8 i q^{67} + 2 i q^{68} - 3 i q^{72} - 2 i q^{73} + 2 q^{74} - 16 i q^{77} + i q^{78} - 8 q^{79} + q^{81} + 6 i q^{82} - 4 i q^{83} + 4 q^{84} - 12 q^{86} + 10 i q^{87} + 12 i q^{88} + 2 q^{89} - 4 q^{91} + 4 i q^{93} - 5 q^{96} + 10 i q^{97} - 9 i q^{98} - 4 q^{99} +O(q^{100})$$ q + i * q^2 + i * q^3 + q^4 - q^6 - 4*i * q^7 + 3*i * q^8 - q^9 + 4 * q^11 + i * q^12 - i * q^13 + 4 * q^14 - q^16 + 2*i * q^17 - i * q^18 + 4 * q^21 + 4*i * q^22 - 3 * q^24 + q^26 - i * q^27 - 4*i * q^28 + 10 * q^29 + 4 * q^31 + 5*i * q^32 + 4*i * q^33 - 2 * q^34 - q^36 - 2*i * q^37 + q^39 + 6 * q^41 + 4*i * q^42 + 12*i * q^43 + 4 * q^44 - i * q^48 - 9 * q^49 - 2 * q^51 - i * q^52 - 6*i * q^53 + q^54 + 12 * q^56 + 10*i * q^58 - 12 * q^59 - 2 * q^61 + 4*i * q^62 + 4*i * q^63 - 7 * q^64 - 4 * q^66 - 8*i * q^67 + 2*i * q^68 - 3*i * q^72 - 2*i * q^73 + 2 * q^74 - 16*i * q^77 + i * q^78 - 8 * q^79 + q^81 + 6*i * q^82 - 4*i * q^83 + 4 * q^84 - 12 * q^86 + 10*i * q^87 + 12*i * q^88 + 2 * q^89 - 4 * q^91 + 4*i * q^93 - 5 * q^96 + 10*i * q^97 - 9*i * q^98 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 8 q^{14} - 2 q^{16} + 8 q^{21} - 6 q^{24} + 2 q^{26} + 20 q^{29} + 8 q^{31} - 4 q^{34} - 2 q^{36} + 2 q^{39} + 12 q^{41} + 8 q^{44} - 18 q^{49} - 4 q^{51} + 2 q^{54} + 24 q^{56} - 24 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{66} + 4 q^{74} - 16 q^{79} + 2 q^{81} + 8 q^{84} - 24 q^{86} + 4 q^{89} - 8 q^{91} - 10 q^{96} - 8 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 8 * q^14 - 2 * q^16 + 8 * q^21 - 6 * q^24 + 2 * q^26 + 20 * q^29 + 8 * q^31 - 4 * q^34 - 2 * q^36 + 2 * q^39 + 12 * q^41 + 8 * q^44 - 18 * q^49 - 4 * q^51 + 2 * q^54 + 24 * q^56 - 24 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^66 + 4 * q^74 - 16 * q^79 + 2 * q^81 + 8 * q^84 - 24 * q^86 + 4 * q^89 - 8 * q^91 - 10 * q^96 - 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 − 1.00000i 1.00000i
1.00000i 1.00000i 1.00000 0 −1.00000 4.00000i 3.00000i −1.00000 0
274.2 1.00000i 1.00000i 1.00000 0 −1.00000 4.00000i 3.00000i −1.00000 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 975.2.c.f 2
3.b odd 2 1 2925.2.c.e 2
5.b even 2 1 inner 975.2.c.f 2
5.c odd 4 1 39.2.a.a 1
5.c odd 4 1 975.2.a.f 1
15.d odd 2 1 2925.2.c.e 2
15.e even 4 1 117.2.a.a 1
15.e even 4 1 2925.2.a.p 1
20.e even 4 1 624.2.a.i 1
35.f even 4 1 1911.2.a.f 1
40.i odd 4 1 2496.2.a.q 1
40.k even 4 1 2496.2.a.e 1
45.k odd 12 2 1053.2.e.b 2
45.l even 12 2 1053.2.e.d 2
55.e even 4 1 4719.2.a.c 1
60.l odd 4 1 1872.2.a.h 1
65.f even 4 1 507.2.b.a 2
65.h odd 4 1 507.2.a.a 1
65.k even 4 1 507.2.b.a 2
65.o even 12 2 507.2.j.e 4
65.q odd 12 2 507.2.e.a 2
65.r odd 12 2 507.2.e.b 2
65.t even 12 2 507.2.j.e 4
105.k odd 4 1 5733.2.a.e 1
120.q odd 4 1 7488.2.a.by 1
120.w even 4 1 7488.2.a.bl 1
195.j odd 4 1 1521.2.b.b 2
195.s even 4 1 1521.2.a.e 1
195.u odd 4 1 1521.2.b.b 2
260.p even 4 1 8112.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 5.c odd 4 1
117.2.a.a 1 15.e even 4 1
507.2.a.a 1 65.h odd 4 1
507.2.b.a 2 65.f even 4 1
507.2.b.a 2 65.k even 4 1
507.2.e.a 2 65.q odd 12 2
507.2.e.b 2 65.r odd 12 2
507.2.j.e 4 65.o even 12 2
507.2.j.e 4 65.t even 12 2
624.2.a.i 1 20.e even 4 1
975.2.a.f 1 5.c odd 4 1
975.2.c.f 2 1.a even 1 1 trivial
975.2.c.f 2 5.b even 2 1 inner
1053.2.e.b 2 45.k odd 12 2
1053.2.e.d 2 45.l even 12 2
1521.2.a.e 1 195.s even 4 1
1521.2.b.b 2 195.j odd 4 1
1521.2.b.b 2 195.u odd 4 1
1872.2.a.h 1 60.l odd 4 1
1911.2.a.f 1 35.f even 4 1
2496.2.a.e 1 40.k even 4 1
2496.2.a.q 1 40.i odd 4 1
2925.2.a.p 1 15.e even 4 1
2925.2.c.e 2 3.b odd 2 1
2925.2.c.e 2 15.d odd 2 1
4719.2.a.c 1 55.e even 4 1
5733.2.a.e 1 105.k odd 4 1
7488.2.a.bl 1 120.w even 4 1
7488.2.a.by 1 120.q odd 4 1
8112.2.a.s 1 260.p even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(975, [\chi])$$:

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} - 4$$ T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 10)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 144$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} + 100$$