# Properties

 Label 275.2.b.e Level $275$ Weight $2$ Character orbit 275.b Analytic conductor $2.196$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 275.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.19588605559$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + ( - 2 \beta_{2} + 1) q^{6} + ( - 2 \beta_{3} - 3 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + (3 \beta_{2} + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b3 - b1) * q^3 + (b2 + 1) * q^4 + (-2*b2 + 1) * q^6 + (-2*b3 - 3*b1) * q^7 + (b3 + 2*b1) * q^8 + (3*b2 + 1) * q^9 $$q + \beta_1 q^{2} + (\beta_{3} - \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + ( - 2 \beta_{2} + 1) q^{6} + ( - 2 \beta_{3} - 3 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + (3 \beta_{2} + 1) q^{9} + q^{11} + \beta_1 q^{12} + (3 \beta_{3} - 2 \beta_1) q^{13} + ( - \beta_{2} + 3) q^{14} + 3 \beta_{2} q^{16} + ( - \beta_{3} - \beta_1) q^{17} + (3 \beta_{3} - 2 \beta_1) q^{18} + ( - 6 \beta_{2} - 3) q^{19} + (4 \beta_{2} - 1) q^{21} + \beta_1 q^{22} + (4 \beta_{3} + 5 \beta_1) q^{23} + ( - 3 \beta_{2} + 1) q^{24} + ( - 5 \beta_{2} + 2) q^{26} + (\beta_{3} + 2 \beta_1) q^{27} + ( - 5 \beta_{3} - 2 \beta_1) q^{28} + (\beta_{2} + 3) q^{29} - 3 q^{31} + (5 \beta_{3} + \beta_1) q^{32} + (\beta_{3} - \beta_1) q^{33} + q^{34} + (\beta_{2} + 4) q^{36} + ( - 7 \beta_{3} + 2 \beta_1) q^{37} + ( - 6 \beta_{3} + 3 \beta_1) q^{38} + (7 \beta_{2} - 5) q^{39} - 3 q^{41} + (4 \beta_{3} - 5 \beta_1) q^{42} - 6 \beta_{3} q^{43} + (\beta_{2} + 1) q^{44} + (\beta_{2} - 5) q^{46} + (\beta_{3} + 8 \beta_1) q^{47} + ( - 3 \beta_{3} + 6 \beta_1) q^{48} + ( - 3 \beta_{2} - 6) q^{49} + \beta_{2} q^{51} + (\beta_{3} + 3 \beta_1) q^{52} + ( - 2 \beta_{3} - 7 \beta_1) q^{53} + (\beta_{2} - 2) q^{54} + (\beta_{2} + 8) q^{56} + (3 \beta_{3} - 9 \beta_1) q^{57} + (\beta_{3} + 2 \beta_1) q^{58} + ( - 4 \beta_{2} - 7) q^{59} + ( - 5 \beta_{2} - 8) q^{61} - 3 \beta_1 q^{62} - 11 \beta_{3} q^{63} + (2 \beta_{2} - 1) q^{64} + ( - 2 \beta_{2} + 1) q^{66} - 8 \beta_{3} q^{67} + ( - 2 \beta_{3} - \beta_1) q^{68} + ( - 6 \beta_{2} + 1) q^{69} + ( - 10 \beta_{2} - 8) q^{71} + (7 \beta_{3} - \beta_1) q^{72} + (12 \beta_{3} + \beta_1) q^{73} + (9 \beta_{2} - 2) q^{74} + ( - 3 \beta_{2} - 9) q^{76} + ( - 2 \beta_{3} - 3 \beta_1) q^{77} + (7 \beta_{3} - 12 \beta_1) q^{78} + (3 \beta_{2} - 1) q^{79} + (6 \beta_{2} + 4) q^{81} - 3 \beta_1 q^{82} + ( - 15 \beta_{3} - 3 \beta_1) q^{83} + ( - \beta_{2} + 3) q^{84} + 6 \beta_{2} q^{86} + (2 \beta_{3} - \beta_1) q^{87} + (\beta_{3} + 2 \beta_1) q^{88} + (5 \beta_{2} + 15) q^{89} + 11 \beta_{2} q^{91} + (9 \beta_{3} + 4 \beta_1) q^{92} + ( - 3 \beta_{3} + 3 \beta_1) q^{93} + (7 \beta_{2} - 8) q^{94} + (3 \beta_{2} - 4) q^{96} + \beta_1 q^{97} + ( - 3 \beta_{3} - 3 \beta_1) q^{98} + (3 \beta_{2} + 1) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b3 - b1) * q^3 + (b2 + 1) * q^4 + (-2*b2 + 1) * q^6 + (-2*b3 - 3*b1) * q^7 + (b3 + 2*b1) * q^8 + (3*b2 + 1) * q^9 + q^11 + b1 * q^12 + (3*b3 - 2*b1) * q^13 + (-b2 + 3) * q^14 + 3*b2 * q^16 + (-b3 - b1) * q^17 + (3*b3 - 2*b1) * q^18 + (-6*b2 - 3) * q^19 + (4*b2 - 1) * q^21 + b1 * q^22 + (4*b3 + 5*b1) * q^23 + (-3*b2 + 1) * q^24 + (-5*b2 + 2) * q^26 + (b3 + 2*b1) * q^27 + (-5*b3 - 2*b1) * q^28 + (b2 + 3) * q^29 - 3 * q^31 + (5*b3 + b1) * q^32 + (b3 - b1) * q^33 + q^34 + (b2 + 4) * q^36 + (-7*b3 + 2*b1) * q^37 + (-6*b3 + 3*b1) * q^38 + (7*b2 - 5) * q^39 - 3 * q^41 + (4*b3 - 5*b1) * q^42 - 6*b3 * q^43 + (b2 + 1) * q^44 + (b2 - 5) * q^46 + (b3 + 8*b1) * q^47 + (-3*b3 + 6*b1) * q^48 + (-3*b2 - 6) * q^49 + b2 * q^51 + (b3 + 3*b1) * q^52 + (-2*b3 - 7*b1) * q^53 + (b2 - 2) * q^54 + (b2 + 8) * q^56 + (3*b3 - 9*b1) * q^57 + (b3 + 2*b1) * q^58 + (-4*b2 - 7) * q^59 + (-5*b2 - 8) * q^61 - 3*b1 * q^62 - 11*b3 * q^63 + (2*b2 - 1) * q^64 + (-2*b2 + 1) * q^66 - 8*b3 * q^67 + (-2*b3 - b1) * q^68 + (-6*b2 + 1) * q^69 + (-10*b2 - 8) * q^71 + (7*b3 - b1) * q^72 + (12*b3 + b1) * q^73 + (9*b2 - 2) * q^74 + (-3*b2 - 9) * q^76 + (-2*b3 - 3*b1) * q^77 + (7*b3 - 12*b1) * q^78 + (3*b2 - 1) * q^79 + (6*b2 + 4) * q^81 - 3*b1 * q^82 + (-15*b3 - 3*b1) * q^83 + (-b2 + 3) * q^84 + 6*b2 * q^86 + (2*b3 - b1) * q^87 + (b3 + 2*b1) * q^88 + (5*b2 + 15) * q^89 + 11*b2 * q^91 + (9*b3 + 4*b1) * q^92 + (-3*b3 + 3*b1) * q^93 + (7*b2 - 8) * q^94 + (3*b2 - 4) * q^96 + b1 * q^97 + (-3*b3 - 3*b1) * q^98 + (3*b2 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 8 q^{6} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 + 8 * q^6 - 2 * q^9 $$4 q + 2 q^{4} + 8 q^{6} - 2 q^{9} + 4 q^{11} + 14 q^{14} - 6 q^{16} - 12 q^{21} + 10 q^{24} + 18 q^{26} + 10 q^{29} - 12 q^{31} + 4 q^{34} + 14 q^{36} - 34 q^{39} - 12 q^{41} + 2 q^{44} - 22 q^{46} - 18 q^{49} - 2 q^{51} - 10 q^{54} + 30 q^{56} - 20 q^{59} - 22 q^{61} - 8 q^{64} + 8 q^{66} + 16 q^{69} - 12 q^{71} - 26 q^{74} - 30 q^{76} - 10 q^{79} + 4 q^{81} + 14 q^{84} - 12 q^{86} + 50 q^{89} - 22 q^{91} - 46 q^{94} - 22 q^{96} - 2 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 + 8 * q^6 - 2 * q^9 + 4 * q^11 + 14 * q^14 - 6 * q^16 - 12 * q^21 + 10 * q^24 + 18 * q^26 + 10 * q^29 - 12 * q^31 + 4 * q^34 + 14 * q^36 - 34 * q^39 - 12 * q^41 + 2 * q^44 - 22 * q^46 - 18 * q^49 - 2 * q^51 - 10 * q^54 + 30 * q^56 - 20 * q^59 - 22 * q^61 - 8 * q^64 + 8 * q^66 + 16 * q^69 - 12 * q^71 - 26 * q^74 - 30 * q^76 - 10 * q^79 + 4 * q^81 + 14 * q^84 - 12 * q^86 + 50 * q^89 - 22 * q^91 - 46 * q^94 - 22 * q^96 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2\beta_1$$ b3 - 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 1.61803i − 0.618034i 0.618034i 1.61803i
1.61803i 2.61803i −0.618034 0 4.23607 2.85410i 2.23607i −3.85410 0
199.2 0.618034i 0.381966i 1.61803 0 −0.236068 3.85410i 2.23607i 2.85410 0
199.3 0.618034i 0.381966i 1.61803 0 −0.236068 3.85410i 2.23607i 2.85410 0
199.4 1.61803i 2.61803i −0.618034 0 4.23607 2.85410i 2.23607i −3.85410 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 275.2.b.e 4
3.b odd 2 1 2475.2.c.p 4
4.b odd 2 1 4400.2.b.x 4
5.b even 2 1 inner 275.2.b.e 4
5.c odd 4 1 275.2.a.d 2
5.c odd 4 1 275.2.a.g yes 2
15.d odd 2 1 2475.2.c.p 4
15.e even 4 1 2475.2.a.n 2
15.e even 4 1 2475.2.a.s 2
20.d odd 2 1 4400.2.b.x 4
20.e even 4 1 4400.2.a.bg 2
20.e even 4 1 4400.2.a.bv 2
55.e even 4 1 3025.2.a.i 2
55.e even 4 1 3025.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.d 2 5.c odd 4 1
275.2.a.g yes 2 5.c odd 4 1
275.2.b.e 4 1.a even 1 1 trivial
275.2.b.e 4 5.b even 2 1 inner
2475.2.a.n 2 15.e even 4 1
2475.2.a.s 2 15.e even 4 1
2475.2.c.p 4 3.b odd 2 1
2475.2.c.p 4 15.d odd 2 1
3025.2.a.i 2 55.e even 4 1
3025.2.a.m 2 55.e even 4 1
4400.2.a.bg 2 20.e even 4 1
4400.2.a.bv 2 20.e even 4 1
4400.2.b.x 4 4.b odd 2 1
4400.2.b.x 4 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(275, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 1$$
$3$ $$T^{4} + 7T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 23T^{2} + 121$$
$11$ $$(T - 1)^{4}$$
$13$ $$T^{4} + 42T^{2} + 121$$
$17$ $$T^{4} + 3T^{2} + 1$$
$19$ $$(T^{2} - 45)^{2}$$
$23$ $$T^{4} + 67T^{2} + 841$$
$29$ $$(T^{2} - 5 T + 5)^{2}$$
$31$ $$(T + 3)^{4}$$
$37$ $$T^{4} + 138T^{2} + 3481$$
$41$ $$(T + 3)^{4}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$T^{4} + 178T^{2} + 5041$$
$53$ $$T^{4} + 127T^{2} + 3481$$
$59$ $$(T^{2} + 10 T + 5)^{2}$$
$61$ $$(T^{2} + 11 T - 1)^{2}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} + 6 T - 116)^{2}$$
$73$ $$T^{4} + 267 T^{2} + 17161$$
$79$ $$(T^{2} + 5 T - 5)^{2}$$
$83$ $$T^{4} + 387 T^{2} + 29241$$
$89$ $$(T^{2} - 25 T + 125)^{2}$$
$97$ $$T^{4} + 3T^{2} + 1$$