# Properties

 Label 275.4.b.c Level $275$ Weight $4$ Character orbit 275.b Analytic conductor $16.226$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,4,Mod(199,275)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("275.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 275.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$16.2255252516$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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 11) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1) q^{2} + ( - 4 \beta_{2} + \beta_1) q^{3} + (2 \beta_{3} + 4) q^{4} + (5 \beta_{3} - 13) q^{6} + (4 \beta_{2} + 10 \beta_1) q^{7} + ( - 10 \beta_{2} + 6 \beta_1) q^{8} + (8 \beta_{3} - 22) q^{9}+O(q^{10})$$ q + (-b2 + b1) * q^2 + (-4*b2 + b1) * q^3 + (2*b3 + 4) * q^4 + (5*b3 - 13) * q^6 + (4*b2 + 10*b1) * q^7 + (-10*b2 + 6*b1) * q^8 + (8*b3 - 22) * q^9 $$q + ( - \beta_{2} + \beta_1) q^{2} + ( - 4 \beta_{2} + \beta_1) q^{3} + (2 \beta_{3} + 4) q^{4} + (5 \beta_{3} - 13) q^{6} + (4 \beta_{2} + 10 \beta_1) q^{7} + ( - 10 \beta_{2} + 6 \beta_1) q^{8} + (8 \beta_{3} - 22) q^{9} - 11 q^{11} + ( - 14 \beta_{2} - 20 \beta_1) q^{12} + ( - 20 \beta_{2} - 40 \beta_1) q^{13} + (6 \beta_{3} + 2) q^{14} + (32 \beta_{3} - 4) q^{16} + ( - 12 \beta_{2} - 62 \beta_1) q^{17} + (30 \beta_{2} - 46 \beta_1) q^{18} + (60 \beta_{3} - 36) q^{19} + (36 \beta_{3} + 38) q^{21} + (11 \beta_{2} - 11 \beta_1) q^{22} + ( - 36 \beta_{2} + 49 \beta_1) q^{23} + (34 \beta_{3} - 126) q^{24} + ( - 20 \beta_{3} - 20) q^{26} + ( - 12 \beta_{2} - 91 \beta_1) q^{27} + (36 \beta_{2} + 64 \beta_1) q^{28} + ( - 56 \beta_{3} - 72) q^{29} + ( - 28 \beta_{3} - 17) q^{31} + ( - 44 \beta_{2} - 52 \beta_1) q^{32} + (44 \beta_{2} - 11 \beta_1) q^{33} + ( - 50 \beta_{3} + 26) q^{34} + ( - 12 \beta_{3} - 40) q^{36} + (8 \beta_{2} + 27 \beta_1) q^{37} + (96 \beta_{2} - 216 \beta_1) q^{38} + ( - 140 \beta_{3} - 200) q^{39} + (4 \beta_{3} + 268) q^{41} + ( - 2 \beta_{2} - 70 \beta_1) q^{42} + ( - 16 \beta_{2} + 30 \beta_1) q^{43} + ( - 22 \beta_{3} - 44) q^{44} + (85 \beta_{3} - 157) q^{46} + (120 \beta_{2} - 136 \beta_1) q^{47} + (48 \beta_{2} - 388 \beta_1) q^{48} + ( - 80 \beta_{3} + 195) q^{49} + ( - 236 \beta_{3} - 82) q^{51} + ( - 160 \beta_{2} - 280 \beta_1) q^{52} + ( - 56 \beta_{2} + 246 \beta_1) q^{53} + ( - 79 \beta_{3} + 55) q^{54} + (76 \beta_{3} + 60) q^{56} + (204 \beta_{2} - 756 \beta_1) q^{57} + (16 \beta_{2} + 96 \beta_1) q^{58} + ( - 132 \beta_{3} - 317) q^{59} + ( - 184 \beta_{3} + 420) q^{61} + ( - 11 \beta_{2} + 67 \beta_1) q^{62} + ( - 8 \beta_{2} - 124 \beta_1) q^{63} + (248 \beta_{3} - 112) q^{64} + ( - 55 \beta_{3} + 143) q^{66} + (20 \beta_{2} + 377 \beta_1) q^{67} + ( - 172 \beta_{2} - 320 \beta_1) q^{68} + (232 \beta_{3} - 481) q^{69} + ( - 76 \beta_{3} - 339) q^{71} + (268 \beta_{2} - 372 \beta_1) q^{72} + ( - 468 \beta_{2} + 200 \beta_1) q^{73} + (19 \beta_{3} - 3) q^{74} + (168 \beta_{3} + 216) q^{76} + ( - 44 \beta_{2} - 110 \beta_1) q^{77} + (60 \beta_{2} + 220 \beta_1) q^{78} + (656 \beta_{3} - 158) q^{79} + ( - 136 \beta_{3} - 647) q^{81} + ( - 264 \beta_{2} + 256 \beta_1) q^{82} + (120 \beta_{2} - 234 \beta_1) q^{83} + (220 \beta_{3} + 368) q^{84} + (46 \beta_{3} - 78) q^{86} + (232 \beta_{2} + 600 \beta_1) q^{87} + (110 \beta_{2} - 66 \beta_1) q^{88} + ( - 328 \beta_{3} + 921) q^{89} + (360 \beta_{3} + 640) q^{91} + ( - 46 \beta_{2} - 20 \beta_1) q^{92} + (40 \beta_{2} + 319 \beta_1) q^{93} + ( - 256 \beta_{3} + 496) q^{94} + ( - 164 \beta_{3} - 476) q^{96} + ( - 144 \beta_{2} + 1097 \beta_1) q^{97} + ( - 275 \beta_{2} + 435 \beta_1) q^{98} + ( - 88 \beta_{3} + 242) q^{99}+O(q^{100})$$ q + (-b2 + b1) * q^2 + (-4*b2 + b1) * q^3 + (2*b3 + 4) * q^4 + (5*b3 - 13) * q^6 + (4*b2 + 10*b1) * q^7 + (-10*b2 + 6*b1) * q^8 + (8*b3 - 22) * q^9 - 11 * q^11 + (-14*b2 - 20*b1) * q^12 + (-20*b2 - 40*b1) * q^13 + (6*b3 + 2) * q^14 + (32*b3 - 4) * q^16 + (-12*b2 - 62*b1) * q^17 + (30*b2 - 46*b1) * q^18 + (60*b3 - 36) * q^19 + (36*b3 + 38) * q^21 + (11*b2 - 11*b1) * q^22 + (-36*b2 + 49*b1) * q^23 + (34*b3 - 126) * q^24 + (-20*b3 - 20) * q^26 + (-12*b2 - 91*b1) * q^27 + (36*b2 + 64*b1) * q^28 + (-56*b3 - 72) * q^29 + (-28*b3 - 17) * q^31 + (-44*b2 - 52*b1) * q^32 + (44*b2 - 11*b1) * q^33 + (-50*b3 + 26) * q^34 + (-12*b3 - 40) * q^36 + (8*b2 + 27*b1) * q^37 + (96*b2 - 216*b1) * q^38 + (-140*b3 - 200) * q^39 + (4*b3 + 268) * q^41 + (-2*b2 - 70*b1) * q^42 + (-16*b2 + 30*b1) * q^43 + (-22*b3 - 44) * q^44 + (85*b3 - 157) * q^46 + (120*b2 - 136*b1) * q^47 + (48*b2 - 388*b1) * q^48 + (-80*b3 + 195) * q^49 + (-236*b3 - 82) * q^51 + (-160*b2 - 280*b1) * q^52 + (-56*b2 + 246*b1) * q^53 + (-79*b3 + 55) * q^54 + (76*b3 + 60) * q^56 + (204*b2 - 756*b1) * q^57 + (16*b2 + 96*b1) * q^58 + (-132*b3 - 317) * q^59 + (-184*b3 + 420) * q^61 + (-11*b2 + 67*b1) * q^62 + (-8*b2 - 124*b1) * q^63 + (248*b3 - 112) * q^64 + (-55*b3 + 143) * q^66 + (20*b2 + 377*b1) * q^67 + (-172*b2 - 320*b1) * q^68 + (232*b3 - 481) * q^69 + (-76*b3 - 339) * q^71 + (268*b2 - 372*b1) * q^72 + (-468*b2 + 200*b1) * q^73 + (19*b3 - 3) * q^74 + (168*b3 + 216) * q^76 + (-44*b2 - 110*b1) * q^77 + (60*b2 + 220*b1) * q^78 + (656*b3 - 158) * q^79 + (-136*b3 - 647) * q^81 + (-264*b2 + 256*b1) * q^82 + (120*b2 - 234*b1) * q^83 + (220*b3 + 368) * q^84 + (46*b3 - 78) * q^86 + (232*b2 + 600*b1) * q^87 + (110*b2 - 66*b1) * q^88 + (-328*b3 + 921) * q^89 + (360*b3 + 640) * q^91 + (-46*b2 - 20*b1) * q^92 + (40*b2 + 319*b1) * q^93 + (-256*b3 + 496) * q^94 + (-164*b3 - 476) * q^96 + (-144*b2 + 1097*b1) * q^97 + (-275*b2 + 435*b1) * q^98 + (-88*b3 + 242) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 16 q^{4} - 52 q^{6} - 88 q^{9}+O(q^{10})$$ 4 * q + 16 * q^4 - 52 * q^6 - 88 * q^9 $$4 q + 16 q^{4} - 52 q^{6} - 88 q^{9} - 44 q^{11} + 8 q^{14} - 16 q^{16} - 144 q^{19} + 152 q^{21} - 504 q^{24} - 80 q^{26} - 288 q^{29} - 68 q^{31} + 104 q^{34} - 160 q^{36} - 800 q^{39} + 1072 q^{41} - 176 q^{44} - 628 q^{46} + 780 q^{49} - 328 q^{51} + 220 q^{54} + 240 q^{56} - 1268 q^{59} + 1680 q^{61} - 448 q^{64} + 572 q^{66} - 1924 q^{69} - 1356 q^{71} - 12 q^{74} + 864 q^{76} - 632 q^{79} - 2588 q^{81} + 1472 q^{84} - 312 q^{86} + 3684 q^{89} + 2560 q^{91} + 1984 q^{94} - 1904 q^{96} + 968 q^{99}+O(q^{100})$$ 4 * q + 16 * q^4 - 52 * q^6 - 88 * q^9 - 44 * q^11 + 8 * q^14 - 16 * q^16 - 144 * q^19 + 152 * q^21 - 504 * q^24 - 80 * q^26 - 288 * q^29 - 68 * q^31 + 104 * q^34 - 160 * q^36 - 800 * q^39 + 1072 * q^41 - 176 * q^44 - 628 * q^46 + 780 * q^49 - 328 * q^51 + 220 * q^54 + 240 * q^56 - 1268 * q^59 + 1680 * q^61 - 448 * q^64 + 572 * q^66 - 1924 * q^69 - 1356 * q^71 - 12 * q^74 + 864 * q^76 - 632 * q^79 - 2588 * q^81 + 1472 * q^84 - 312 * q^86 + 3684 * q^89 + 2560 * q^91 + 1984 * q^94 - 1904 * q^96 + 968 * q^99

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 $$\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$$.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
2.73205i 7.92820i 0.535898 0 −21.6603 3.07180i 23.3205i −35.8564 0
199.2 0.732051i 5.92820i 7.46410 0 −4.33975 16.9282i 11.3205i −8.14359 0
199.3 0.732051i 5.92820i 7.46410 0 −4.33975 16.9282i 11.3205i −8.14359 0
199.4 2.73205i 7.92820i 0.535898 0 −21.6603 3.07180i 23.3205i −35.8564 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 275.4.b.c 4
5.b even 2 1 inner 275.4.b.c 4
5.c odd 4 1 11.4.a.a 2
5.c odd 4 1 275.4.a.b 2
15.e even 4 1 99.4.a.c 2
15.e even 4 1 2475.4.a.q 2
20.e even 4 1 176.4.a.i 2
35.f even 4 1 539.4.a.e 2
40.i odd 4 1 704.4.a.p 2
40.k even 4 1 704.4.a.n 2
55.e even 4 1 121.4.a.c 2
55.k odd 20 4 121.4.c.c 8
55.l even 20 4 121.4.c.f 8
60.l odd 4 1 1584.4.a.bc 2
65.h odd 4 1 1859.4.a.a 2
165.l odd 4 1 1089.4.a.v 2
220.i odd 4 1 1936.4.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 5.c odd 4 1
99.4.a.c 2 15.e even 4 1
121.4.a.c 2 55.e even 4 1
121.4.c.c 8 55.k odd 20 4
121.4.c.f 8 55.l even 20 4
176.4.a.i 2 20.e even 4 1
275.4.a.b 2 5.c odd 4 1
275.4.b.c 4 1.a even 1 1 trivial
275.4.b.c 4 5.b even 2 1 inner
539.4.a.e 2 35.f even 4 1
704.4.a.n 2 40.k even 4 1
704.4.a.p 2 40.i odd 4 1
1089.4.a.v 2 165.l odd 4 1
1584.4.a.bc 2 60.l odd 4 1
1859.4.a.a 2 65.h odd 4 1
1936.4.a.w 2 220.i odd 4 1
2475.4.a.q 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 8T_{2}^{2} + 4$$ acting on $$S_{4}^{\mathrm{new}}(275, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 8T^{2} + 4$$
$3$ $$T^{4} + 98T^{2} + 2209$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 296T^{2} + 2704$$
$11$ $$(T + 11)^{4}$$
$13$ $$T^{4} + 5600 T^{2} + 160000$$
$17$ $$T^{4} + 8552 T^{2} + 11641744$$
$19$ $$(T^{2} + 72 T - 9504)^{2}$$
$23$ $$T^{4} + 12578 T^{2} + 2211169$$
$29$ $$(T^{2} + 144 T - 4224)^{2}$$
$31$ $$(T^{2} + 34 T - 2063)^{2}$$
$37$ $$T^{4} + 1842 T^{2} + 288369$$
$41$ $$(T^{2} - 536 T + 71776)^{2}$$
$43$ $$T^{4} + 3336 T^{2} + 17424$$
$47$ $$T^{4} + 123392 T^{2} + 610287616$$
$53$ $$T^{4} + \cdots + 2612027664$$
$59$ $$(T^{2} + 634 T + 48217)^{2}$$
$61$ $$(T^{2} - 840 T + 74832)^{2}$$
$67$ $$T^{4} + \cdots + 19860983041$$
$71$ $$(T^{2} + 678 T + 97593)^{2}$$
$73$ $$T^{4} + \cdots + 380777853184$$
$79$ $$(T^{2} + 316 T - 1266044)^{2}$$
$83$ $$T^{4} + 195912 T^{2} + 133541136$$
$89$ $$(T^{2} - 1842 T + 525489)^{2}$$
$97$ $$T^{4} + \cdots + 1302339722401$$