# Properties

 Label 975.2.h.f Level $975$ Weight $2$ Character orbit 975.h Analytic conductor $7.785$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + \beta_1 q^{3} + q^{4} - \beta_{2} q^{6} + 2 \beta_{3} q^{7} + \beta_{3} q^{8} - q^{9}+O(q^{10})$$ q - b3 * q^2 + b1 * q^3 + q^4 - b2 * q^6 + 2*b3 * q^7 + b3 * q^8 - q^9 $$q - \beta_{3} q^{2} + \beta_1 q^{3} + q^{4} - \beta_{2} q^{6} + 2 \beta_{3} q^{7} + \beta_{3} q^{8} - q^{9} - 2 \beta_{2} q^{11} + \beta_1 q^{12} + (2 \beta_{3} + \beta_1) q^{13} - 6 q^{14} - 5 q^{16} - 6 \beta_1 q^{17} + \beta_{3} q^{18} - 2 \beta_{2} q^{19} + 2 \beta_{2} q^{21} + 6 \beta_1 q^{22} + \beta_{2} q^{24} + ( - \beta_{2} - 6) q^{26} - \beta_1 q^{27} + 2 \beta_{3} q^{28} - 6 q^{29} - 2 \beta_{2} q^{31} + 3 \beta_{3} q^{32} + 2 \beta_{3} q^{33} + 6 \beta_{2} q^{34} - q^{36} - 4 \beta_{3} q^{37} + 6 \beta_1 q^{38} + (2 \beta_{2} - 1) q^{39} - 4 \beta_{2} q^{41} - 6 \beta_1 q^{42} + 4 \beta_1 q^{43} - 2 \beta_{2} q^{44} - 2 \beta_{3} q^{47} - 5 \beta_1 q^{48} + 5 q^{49} + 6 q^{51} + (2 \beta_{3} + \beta_1) q^{52} - 6 \beta_1 q^{53} + \beta_{2} q^{54} + 6 q^{56} + 2 \beta_{3} q^{57} + 6 \beta_{3} q^{58} - 6 \beta_{2} q^{59} - 2 q^{61} + 6 \beta_1 q^{62} - 2 \beta_{3} q^{63} + q^{64} - 6 q^{66} + 6 \beta_{3} q^{67} - 6 \beta_1 q^{68} + 2 \beta_{2} q^{71} - \beta_{3} q^{72} + 12 q^{74} - 2 \beta_{2} q^{76} - 12 \beta_1 q^{77} + (\beta_{3} - 6 \beta_1) q^{78} + 8 q^{79} + q^{81} + 12 \beta_1 q^{82} - 2 \beta_{3} q^{83} + 2 \beta_{2} q^{84} - 4 \beta_{2} q^{86} - 6 \beta_1 q^{87} - 6 \beta_1 q^{88} + 4 \beta_{2} q^{89} + (2 \beta_{2} + 12) q^{91} + 2 \beta_{3} q^{93} + 6 q^{94} + 3 \beta_{2} q^{96} + 8 \beta_{3} q^{97} - 5 \beta_{3} q^{98} + 2 \beta_{2} q^{99}+O(q^{100})$$ q - b3 * q^2 + b1 * q^3 + q^4 - b2 * q^6 + 2*b3 * q^7 + b3 * q^8 - q^9 - 2*b2 * q^11 + b1 * q^12 + (2*b3 + b1) * q^13 - 6 * q^14 - 5 * q^16 - 6*b1 * q^17 + b3 * q^18 - 2*b2 * q^19 + 2*b2 * q^21 + 6*b1 * q^22 + b2 * q^24 + (-b2 - 6) * q^26 - b1 * q^27 + 2*b3 * q^28 - 6 * q^29 - 2*b2 * q^31 + 3*b3 * q^32 + 2*b3 * q^33 + 6*b2 * q^34 - q^36 - 4*b3 * q^37 + 6*b1 * q^38 + (2*b2 - 1) * q^39 - 4*b2 * q^41 - 6*b1 * q^42 + 4*b1 * q^43 - 2*b2 * q^44 - 2*b3 * q^47 - 5*b1 * q^48 + 5 * q^49 + 6 * q^51 + (2*b3 + b1) * q^52 - 6*b1 * q^53 + b2 * q^54 + 6 * q^56 + 2*b3 * q^57 + 6*b3 * q^58 - 6*b2 * q^59 - 2 * q^61 + 6*b1 * q^62 - 2*b3 * q^63 + q^64 - 6 * q^66 + 6*b3 * q^67 - 6*b1 * q^68 + 2*b2 * q^71 - b3 * q^72 + 12 * q^74 - 2*b2 * q^76 - 12*b1 * q^77 + (b3 - 6*b1) * q^78 + 8 * q^79 + q^81 + 12*b1 * q^82 - 2*b3 * q^83 + 2*b2 * q^84 - 4*b2 * q^86 - 6*b1 * q^87 - 6*b1 * q^88 + 4*b2 * q^89 + (2*b2 + 12) * q^91 + 2*b3 * q^93 + 6 * q^94 + 3*b2 * q^96 + 8*b3 * q^97 - 5*b3 * q^98 + 2*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} - 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^4 - 4 * q^9 $$4 q + 4 q^{4} - 4 q^{9} - 24 q^{14} - 20 q^{16} - 24 q^{26} - 24 q^{29} - 4 q^{36} - 4 q^{39} + 20 q^{49} + 24 q^{51} + 24 q^{56} - 8 q^{61} + 4 q^{64} - 24 q^{66} + 48 q^{74} + 32 q^{79} + 4 q^{81} + 48 q^{91} + 24 q^{94}+O(q^{100})$$ 4 * q + 4 * q^4 - 4 * q^9 - 24 * q^14 - 20 * q^16 - 24 * q^26 - 24 * q^29 - 4 * q^36 - 4 * q^39 + 20 * q^49 + 24 * q^51 + 24 * q^56 - 8 * q^61 + 4 * q^64 - 24 * q^66 + 48 * q^74 + 32 * q^79 + 4 * q^81 + 48 * q^91 + 24 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i
−1.73205 1.00000i 1.00000 0 1.73205i 3.46410 1.73205 −1.00000 0
649.2 −1.73205 1.00000i 1.00000 0 1.73205i 3.46410 1.73205 −1.00000 0
649.3 1.73205 1.00000i 1.00000 0 1.73205i −3.46410 −1.73205 −1.00000 0
649.4 1.73205 1.00000i 1.00000 0 1.73205i −3.46410 −1.73205 −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
13.b even 2 1 inner
65.d 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.h.f 4
5.b even 2 1 inner 975.2.h.f 4
5.c odd 4 1 39.2.b.a 2
5.c odd 4 1 975.2.b.d 2
13.b even 2 1 inner 975.2.h.f 4
15.e even 4 1 117.2.b.a 2
20.e even 4 1 624.2.c.e 2
35.f even 4 1 1911.2.c.d 2
40.i odd 4 1 2496.2.c.k 2
40.k even 4 1 2496.2.c.d 2
60.l odd 4 1 1872.2.c.e 2
65.d even 2 1 inner 975.2.h.f 4
65.f even 4 1 507.2.a.f 2
65.h odd 4 1 39.2.b.a 2
65.h odd 4 1 975.2.b.d 2
65.k even 4 1 507.2.a.f 2
65.o even 12 2 507.2.e.e 4
65.q odd 12 1 507.2.j.a 2
65.q odd 12 1 507.2.j.c 2
65.r odd 12 1 507.2.j.a 2
65.r odd 12 1 507.2.j.c 2
65.t even 12 2 507.2.e.e 4
195.j odd 4 1 1521.2.a.l 2
195.s even 4 1 117.2.b.a 2
195.u odd 4 1 1521.2.a.l 2
260.l odd 4 1 8112.2.a.bv 2
260.p even 4 1 624.2.c.e 2
260.s odd 4 1 8112.2.a.bv 2
455.s even 4 1 1911.2.c.d 2
520.bc even 4 1 2496.2.c.d 2
520.bg odd 4 1 2496.2.c.k 2
780.w odd 4 1 1872.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 5.c odd 4 1
39.2.b.a 2 65.h odd 4 1
117.2.b.a 2 15.e even 4 1
117.2.b.a 2 195.s even 4 1
507.2.a.f 2 65.f even 4 1
507.2.a.f 2 65.k even 4 1
507.2.e.e 4 65.o even 12 2
507.2.e.e 4 65.t even 12 2
507.2.j.a 2 65.q odd 12 1
507.2.j.a 2 65.r odd 12 1
507.2.j.c 2 65.q odd 12 1
507.2.j.c 2 65.r odd 12 1
624.2.c.e 2 20.e even 4 1
624.2.c.e 2 260.p even 4 1
975.2.b.d 2 5.c odd 4 1
975.2.b.d 2 65.h odd 4 1
975.2.h.f 4 1.a even 1 1 trivial
975.2.h.f 4 5.b even 2 1 inner
975.2.h.f 4 13.b even 2 1 inner
975.2.h.f 4 65.d even 2 1 inner
1521.2.a.l 2 195.j odd 4 1
1521.2.a.l 2 195.u odd 4 1
1872.2.c.e 2 60.l odd 4 1
1872.2.c.e 2 780.w odd 4 1
1911.2.c.d 2 35.f even 4 1
1911.2.c.d 2 455.s even 4 1
2496.2.c.d 2 40.k even 4 1
2496.2.c.d 2 520.bc even 4 1
2496.2.c.k 2 40.i odd 4 1
2496.2.c.k 2 520.bg odd 4 1
8112.2.a.bv 2 260.l odd 4 1
8112.2.a.bv 2 260.s odd 4 1

## Hecke kernels

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

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{7}^{2} - 12$$ T7^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 3)^{2}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 12)^{2}$$
$11$ $$(T^{2} + 12)^{2}$$
$13$ $$T^{4} - 22T^{2} + 169$$
$17$ $$(T^{2} + 36)^{2}$$
$19$ $$(T^{2} + 12)^{2}$$
$23$ $$T^{4}$$
$29$ $$(T + 6)^{4}$$
$31$ $$(T^{2} + 12)^{2}$$
$37$ $$(T^{2} - 48)^{2}$$
$41$ $$(T^{2} + 48)^{2}$$
$43$ $$(T^{2} + 16)^{2}$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T^{2} + 108)^{2}$$
$61$ $$(T + 2)^{4}$$
$67$ $$(T^{2} - 108)^{2}$$
$71$ $$(T^{2} + 12)^{2}$$
$73$ $$T^{4}$$
$79$ $$(T - 8)^{4}$$
$83$ $$(T^{2} - 12)^{2}$$
$89$ $$(T^{2} + 48)^{2}$$
$97$ $$(T^{2} - 192)^{2}$$