# Properties

 Label 975.2.c.h Level $975$ Weight $2$ Character orbit 975.c Analytic conductor $7.785$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1) q^{2} + \beta_1 q^{3} + ( - 2 \beta_{3} - 1) q^{4} + ( - \beta_{3} - 1) q^{6} + 2 \beta_{2} q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8} - q^{9}+O(q^{10})$$ q + (b2 + b1) * q^2 + b1 * q^3 + (-2*b3 - 1) * q^4 + (-b3 - 1) * q^6 + 2*b2 * q^7 + (-b2 - 3*b1) * q^8 - q^9 $$q + (\beta_{2} + \beta_1) q^{2} + \beta_1 q^{3} + ( - 2 \beta_{3} - 1) q^{4} + ( - \beta_{3} - 1) q^{6} + 2 \beta_{2} q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8} - q^{9} - 2 q^{11} + ( - 2 \beta_{2} - \beta_1) q^{12} - \beta_1 q^{13} + ( - 2 \beta_{3} - 4) q^{14} + 3 q^{16} + (4 \beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{2} - \beta_1) q^{18} - 2 \beta_{3} q^{19} - 2 \beta_{3} q^{21} + ( - 2 \beta_{2} - 2 \beta_1) q^{22} - 4 \beta_1 q^{23} + (\beta_{3} + 3) q^{24} + (\beta_{3} + 1) q^{26} - \beta_1 q^{27} + ( - 2 \beta_{2} - 8 \beta_1) q^{28} - 2 q^{29} + ( - 2 \beta_{3} - 4) q^{31} + (\beta_{2} - 3 \beta_1) q^{32} - 2 \beta_1 q^{33} + ( - 2 \beta_{3} - 6) q^{34} + (2 \beta_{3} + 1) q^{36} + ( - 4 \beta_{2} + 2 \beta_1) q^{37} + ( - 2 \beta_{2} - 4 \beta_1) q^{38} + q^{39} + (2 \beta_{3} + 8) q^{41} + ( - 2 \beta_{2} - 4 \beta_1) q^{42} + (4 \beta_{2} + 4 \beta_1) q^{43} + (4 \beta_{3} + 2) q^{44} + (4 \beta_{3} + 4) q^{46} + ( - 4 \beta_{2} + 6 \beta_1) q^{47} + 3 \beta_1 q^{48} - q^{49} + ( - 4 \beta_{3} + 2) q^{51} + (2 \beta_{2} + \beta_1) q^{52} - 2 \beta_1 q^{53} + (\beta_{3} + 1) q^{54} + (6 \beta_{3} + 4) q^{56} - 2 \beta_{2} q^{57} + ( - 2 \beta_{2} - 2 \beta_1) q^{58} + (4 \beta_{3} - 2) q^{59} + ( - 8 \beta_{3} + 2) q^{61} + ( - 6 \beta_{2} - 8 \beta_1) q^{62} - 2 \beta_{2} q^{63} + (2 \beta_{3} + 7) q^{64} + (2 \beta_{3} + 2) q^{66} + (2 \beta_{2} - 4 \beta_1) q^{67} - 14 \beta_1 q^{68} + 4 q^{69} + 2 q^{71} + (\beta_{2} + 3 \beta_1) q^{72} + (4 \beta_{2} + 6 \beta_1) q^{73} + (2 \beta_{3} + 6) q^{74} + (2 \beta_{3} + 8) q^{76} - 4 \beta_{2} q^{77} + (\beta_{2} + \beta_1) q^{78} - 8 \beta_{3} q^{79} + q^{81} + (10 \beta_{2} + 12 \beta_1) q^{82} + ( - 4 \beta_{2} - 2 \beta_1) q^{83} + (2 \beta_{3} + 8) q^{84} + ( - 8 \beta_{3} - 12) q^{86} - 2 \beta_1 q^{87} + (2 \beta_{2} + 6 \beta_1) q^{88} + (2 \beta_{3} - 12) q^{89} + 2 \beta_{3} q^{91} + (8 \beta_{2} + 4 \beta_1) q^{92} + ( - 2 \beta_{2} - 4 \beta_1) q^{93} + ( - 2 \beta_{3} + 2) q^{94} + ( - \beta_{3} + 3) q^{96} + (4 \beta_{2} + 2 \beta_1) q^{97} + ( - \beta_{2} - \beta_1) q^{98} + 2 q^{99}+O(q^{100})$$ q + (b2 + b1) * q^2 + b1 * q^3 + (-2*b3 - 1) * q^4 + (-b3 - 1) * q^6 + 2*b2 * q^7 + (-b2 - 3*b1) * q^8 - q^9 - 2 * q^11 + (-2*b2 - b1) * q^12 - b1 * q^13 + (-2*b3 - 4) * q^14 + 3 * q^16 + (4*b2 - 2*b1) * q^17 + (-b2 - b1) * q^18 - 2*b3 * q^19 - 2*b3 * q^21 + (-2*b2 - 2*b1) * q^22 - 4*b1 * q^23 + (b3 + 3) * q^24 + (b3 + 1) * q^26 - b1 * q^27 + (-2*b2 - 8*b1) * q^28 - 2 * q^29 + (-2*b3 - 4) * q^31 + (b2 - 3*b1) * q^32 - 2*b1 * q^33 + (-2*b3 - 6) * q^34 + (2*b3 + 1) * q^36 + (-4*b2 + 2*b1) * q^37 + (-2*b2 - 4*b1) * q^38 + q^39 + (2*b3 + 8) * q^41 + (-2*b2 - 4*b1) * q^42 + (4*b2 + 4*b1) * q^43 + (4*b3 + 2) * q^44 + (4*b3 + 4) * q^46 + (-4*b2 + 6*b1) * q^47 + 3*b1 * q^48 - q^49 + (-4*b3 + 2) * q^51 + (2*b2 + b1) * q^52 - 2*b1 * q^53 + (b3 + 1) * q^54 + (6*b3 + 4) * q^56 - 2*b2 * q^57 + (-2*b2 - 2*b1) * q^58 + (4*b3 - 2) * q^59 + (-8*b3 + 2) * q^61 + (-6*b2 - 8*b1) * q^62 - 2*b2 * q^63 + (2*b3 + 7) * q^64 + (2*b3 + 2) * q^66 + (2*b2 - 4*b1) * q^67 - 14*b1 * q^68 + 4 * q^69 + 2 * q^71 + (b2 + 3*b1) * q^72 + (4*b2 + 6*b1) * q^73 + (2*b3 + 6) * q^74 + (2*b3 + 8) * q^76 - 4*b2 * q^77 + (b2 + b1) * q^78 - 8*b3 * q^79 + q^81 + (10*b2 + 12*b1) * q^82 + (-4*b2 - 2*b1) * q^83 + (2*b3 + 8) * q^84 + (-8*b3 - 12) * q^86 - 2*b1 * q^87 + (2*b2 + 6*b1) * q^88 + (2*b3 - 12) * q^89 + 2*b3 * q^91 + (8*b2 + 4*b1) * q^92 + (-2*b2 - 4*b1) * q^93 + (-2*b3 + 2) * q^94 + (-b3 + 3) * q^96 + (4*b2 + 2*b1) * q^97 + (-b2 - b1) * q^98 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 $$4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 8 q^{11} - 16 q^{14} + 12 q^{16} + 12 q^{24} + 4 q^{26} - 8 q^{29} - 16 q^{31} - 24 q^{34} + 4 q^{36} + 4 q^{39} + 32 q^{41} + 8 q^{44} + 16 q^{46} - 4 q^{49} + 8 q^{51} + 4 q^{54} + 16 q^{56} - 8 q^{59} + 8 q^{61} + 28 q^{64} + 8 q^{66} + 16 q^{69} + 8 q^{71} + 24 q^{74} + 32 q^{76} + 4 q^{81} + 32 q^{84} - 48 q^{86} - 48 q^{89} + 8 q^{94} + 12 q^{96} + 8 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 - 8 * q^11 - 16 * q^14 + 12 * q^16 + 12 * q^24 + 4 * q^26 - 8 * q^29 - 16 * q^31 - 24 * q^34 + 4 * q^36 + 4 * q^39 + 32 * q^41 + 8 * q^44 + 16 * q^46 - 4 * q^49 + 8 * q^51 + 4 * q^54 + 16 * q^56 - 8 * q^59 + 8 * q^61 + 28 * q^64 + 8 * q^66 + 16 * q^69 + 8 * q^71 + 24 * q^74 + 32 * q^76 + 4 * q^81 + 32 * q^84 - 48 * q^86 - 48 * q^89 + 8 * q^94 + 12 * q^96 + 8 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 1.00000i −3.82843 0 −2.41421 2.82843i 4.41421i −1.00000 0
274.2 0.414214i 1.00000i 1.82843 0 0.414214 2.82843i 1.58579i −1.00000 0
274.3 0.414214i 1.00000i 1.82843 0 0.414214 2.82843i 1.58579i −1.00000 0
274.4 2.41421i 1.00000i −3.82843 0 −2.41421 2.82843i 4.41421i −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.h 4
3.b odd 2 1 2925.2.c.u 4
5.b even 2 1 inner 975.2.c.h 4
5.c odd 4 1 39.2.a.b 2
5.c odd 4 1 975.2.a.l 2
15.d odd 2 1 2925.2.c.u 4
15.e even 4 1 117.2.a.c 2
15.e even 4 1 2925.2.a.v 2
20.e even 4 1 624.2.a.k 2
35.f even 4 1 1911.2.a.h 2
40.i odd 4 1 2496.2.a.bf 2
40.k even 4 1 2496.2.a.bi 2
45.k odd 12 2 1053.2.e.m 4
45.l even 12 2 1053.2.e.e 4
55.e even 4 1 4719.2.a.p 2
60.l odd 4 1 1872.2.a.w 2
65.f even 4 1 507.2.b.e 4
65.h odd 4 1 507.2.a.h 2
65.k even 4 1 507.2.b.e 4
65.o even 12 2 507.2.j.f 8
65.q odd 12 2 507.2.e.h 4
65.r odd 12 2 507.2.e.d 4
65.t even 12 2 507.2.j.f 8
105.k odd 4 1 5733.2.a.u 2
120.q odd 4 1 7488.2.a.co 2
120.w even 4 1 7488.2.a.cl 2
195.j odd 4 1 1521.2.b.j 4
195.s even 4 1 1521.2.a.f 2
195.u odd 4 1 1521.2.b.j 4
260.p even 4 1 8112.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 5.c odd 4 1
117.2.a.c 2 15.e even 4 1
507.2.a.h 2 65.h odd 4 1
507.2.b.e 4 65.f even 4 1
507.2.b.e 4 65.k even 4 1
507.2.e.d 4 65.r odd 12 2
507.2.e.h 4 65.q odd 12 2
507.2.j.f 8 65.o even 12 2
507.2.j.f 8 65.t even 12 2
624.2.a.k 2 20.e even 4 1
975.2.a.l 2 5.c odd 4 1
975.2.c.h 4 1.a even 1 1 trivial
975.2.c.h 4 5.b even 2 1 inner
1053.2.e.e 4 45.l even 12 2
1053.2.e.m 4 45.k odd 12 2
1521.2.a.f 2 195.s even 4 1
1521.2.b.j 4 195.j odd 4 1
1521.2.b.j 4 195.u odd 4 1
1872.2.a.w 2 60.l odd 4 1
1911.2.a.h 2 35.f even 4 1
2496.2.a.bf 2 40.i odd 4 1
2496.2.a.bi 2 40.k even 4 1
2925.2.a.v 2 15.e even 4 1
2925.2.c.u 4 3.b odd 2 1
2925.2.c.u 4 15.d odd 2 1
4719.2.a.p 2 55.e even 4 1
5733.2.a.u 2 105.k odd 4 1
7488.2.a.cl 2 120.w even 4 1
7488.2.a.co 2 120.q odd 4 1
8112.2.a.bm 2 260.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 $$S_{2}^{\mathrm{new}}(975, [\chi])$$:

 $$T_{2}^{4} + 6T_{2}^{2} + 1$$ T2^4 + 6*T2^2 + 1 $$T_{7}^{2} + 8$$ T7^2 + 8 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 8)^{2}$$
$11$ $$(T + 2)^{4}$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$T^{4} + 72T^{2} + 784$$
$19$ $$(T^{2} - 8)^{2}$$
$23$ $$(T^{2} + 16)^{2}$$
$29$ $$(T + 2)^{4}$$
$31$ $$(T^{2} + 8 T + 8)^{2}$$
$37$ $$T^{4} + 72T^{2} + 784$$
$41$ $$(T^{2} - 16 T + 56)^{2}$$
$43$ $$T^{4} + 96T^{2} + 256$$
$47$ $$T^{4} + 136T^{2} + 16$$
$53$ $$(T^{2} + 4)^{2}$$
$59$ $$(T^{2} + 4 T - 28)^{2}$$
$61$ $$(T^{2} - 4 T - 124)^{2}$$
$67$ $$T^{4} + 48T^{2} + 64$$
$71$ $$(T - 2)^{4}$$
$73$ $$T^{4} + 136T^{2} + 16$$
$79$ $$(T^{2} - 128)^{2}$$
$83$ $$T^{4} + 72T^{2} + 784$$
$89$ $$(T^{2} + 24 T + 136)^{2}$$
$97$ $$T^{4} + 72T^{2} + 784$$