# Properties

 Label 1925.4.a.q Level $1925$ Weight $4$ Character orbit 1925.a Self dual yes Analytic conductor $113.579$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,4,Mod(1,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1925.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1925 = 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1925.a (trivial)

## 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: yes Analytic conductor: $$113.578676761$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.509800.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 10x^{2} + 5x + 15$$ x^4 - x^3 - 10*x^2 + 5*x + 15 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 1) q^{2} + ( - \beta_{3} + 3) q^{3} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{4} + ( - 3 \beta_{3} - 5 \beta_{2} + \cdots - 1) q^{6}+ \cdots + ( - 5 \beta_{3} - 6 \beta_{2} + \cdots + 18) q^{9}+O(q^{10})$$ q + (-b2 + 1) * q^2 + (-b3 + 3) * q^3 + (-b3 - 2*b2 - b1 + 6) * q^4 + (-3*b3 - 5*b2 + 3*b1 - 1) * q^6 + 7 * q^7 + (-3*b3 - 2*b2 + 2*b1 + 14) * q^8 + (-5*b3 - 6*b2 - 3*b1 + 18) * q^9 $$q + ( - \beta_{2} + 1) q^{2} + ( - \beta_{3} + 3) q^{3} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{4} + ( - 3 \beta_{3} - 5 \beta_{2} + \cdots - 1) q^{6}+ \cdots + ( - 55 \beta_{3} - 66 \beta_{2} + \cdots + 198) q^{99}+O(q^{100})$$ q + (-b2 + 1) * q^2 + (-b3 + 3) * q^3 + (-b3 - 2*b2 - b1 + 6) * q^4 + (-3*b3 - 5*b2 + 3*b1 - 1) * q^6 + 7 * q^7 + (-3*b3 - 2*b2 + 2*b1 + 14) * q^8 + (-5*b3 - 6*b2 - 3*b1 + 18) * q^9 + 11 * q^11 + (-12*b3 - 10*b2 + b1 + 46) * q^12 + (4*b3 - 10*b2 + 3*b1 + 32) * q^13 + (-7*b2 + 7) * q^14 + (-7*b3 - 6*b2 + 13*b1 - 8) * q^16 + (6*b3 + 10*b2 + b1 + 18) * q^17 + (-15*b3 - 34*b2 + 12*b1 + 58) * q^18 + (2*b3 + 12*b2 + 26*b1 - 54) * q^19 + (-7*b3 + 21) * q^21 + (-11*b2 + 11) * q^22 + (-b3 + 10*b2 + 21*b1 - 59) * q^23 + (-24*b3 - 40*b2 + b1 + 142) * q^24 + (-4*b3 - 34*b2 - 25*b1 + 196) * q^26 + (-13*b3 - 42*b2 - 3*b1 + 129) * q^27 + (-7*b3 - 14*b2 - 7*b1 + 42) * q^28 + (12*b3 + 40*b2 + 30*b1 - 42) * q^29 + (33*b3 - 32*b2 - 32*b1 - 87) * q^31 + (-29*b3 + 4*b2 - 14*b1 + 8) * q^32 + (-11*b3 + 33) * q^33 + (26*b3 + 4*b2 - 9*b1 - 82) * q^34 + (-63*b3 - 74*b2 + 23*b1 + 368) * q^36 + (b3 - 2*b2 - 49*b1 - 47) * q^37 + (-34*b3 + 70*b2 - 20*b1 - 46) * q^38 + (-44*b3 - 44*b2 + 48*b1 - 88) * q^39 + (-14*b3 + 18*b2 - 53*b1 + 22) * q^41 + (-21*b3 - 35*b2 + 21*b1 - 7) * q^42 + (-36*b3 - 12*b2 + 16*b1 + 116) * q^43 + (-11*b3 - 22*b2 - 11*b1 + 66) * q^44 + (-35*b3 + 67*b2 - 8*b1 - 67) * q^46 + (-30*b3 + 14*b2 + b1 - 16) * q^47 + (-18*b3 - 150*b2 + 23*b1 + 204) * q^48 + 49 * q^49 + (14*b3 + 80*b2 - 10*b1 - 122) * q^51 + (-28*b3 - 158*b2 - 21*b1 + 216) * q^52 + (-4*b3 + 68*b2 - 14*b1 + 214) * q^53 + (-75*b3 - 197*b2 + 605) * q^54 + (-21*b3 - 14*b2 + 14*b1 + 98) * q^56 + (82*b3 - 84*b2 + 22*b1 - 186) * q^57 + (16*b3 + 106*b2 - 26*b1 - 334) * q^58 + (19*b3 + 12*b2 - 42*b1 - 285) * q^59 + (-42*b3 - 6*b2 - 99*b1 - 38) * q^61 + (131*b3 + 121*b2 - 99*b1 + 269) * q^62 + (-35*b3 - 42*b2 - 21*b1 + 126) * q^63 + (b3 - 14*b2 + b1 - 180) * q^64 + (-33*b3 - 55*b2 + 33*b1 - 11) * q^66 + (61*b3 + 70*b2 + 7*b1 + 371) * q^67 + (52*b3 + 58*b2 - 73*b1 - 228) * q^68 + (77*b3 - 82*b2 + 9*b1 - 101) * q^69 + (11*b3 - 14*b2 - 7*b1 + 585) * q^71 + (-189*b3 - 296*b2 - 4*b1 + 752) * q^72 + (-2*b3 - 62*b2 + 51*b1 + 382) * q^73 + (99*b3 + 47*b2 + 44*b1 - 311) * q^74 + (-8*b3 - 48*b2 - 16*b1 - 780) * q^76 + 77 * q^77 + (-272*b3 - 44*b2 + 40*b1 + 596) * q^78 + (54*b3 - 76*b2 - 26*b1 - 242) * q^79 + (-104*b3 - 108*b2 + 162*b1 + 201) * q^81 + (82*b3 - 32*b2 + 113*b1 - 586) * q^82 + (-20*b3 + 204*b2 + 48*b1 + 456) * q^83 + (-84*b3 - 70*b2 + 7*b1 + 322) * q^84 + (-152*b3 - 200*b2 + 80*b1 + 224) * q^86 + (146*b3 + 92*b2 - 24*b1 - 398) * q^87 + (-33*b3 - 22*b2 + 22*b1 + 154) * q^88 + (-99*b3 + 166*b2 + 141*b1 + 317) * q^89 + (28*b3 - 70*b2 + 21*b1 + 224) * q^91 + (-14*b3 - 16*b2 + 12*b1 - 654) * q^92 + (89*b3 + 230*b2 + 131*b1 - 1577) * q^93 + (-78*b3 - 30*b2 + 103*b1 - 312) * q^94 + (-58*b3 - 70*b2 - 127*b1 + 1084) * q^96 + (57*b3 - 30*b2 + 59*b1 + 521) * q^97 + (-49*b2 + 49) * q^98 + (-55*b3 - 66*b2 - 33*b1 + 198) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 12 q^{3} + 22 q^{4} + 2 q^{6} + 28 q^{7} + 60 q^{8} + 66 q^{9}+O(q^{10})$$ 4 * q + 4 * q^2 + 12 * q^3 + 22 * q^4 + 2 * q^6 + 28 * q^7 + 60 * q^8 + 66 * q^9 $$4 q + 4 q^{2} + 12 q^{3} + 22 q^{4} + 2 q^{6} + 28 q^{7} + 60 q^{8} + 66 q^{9} + 44 q^{11} + 186 q^{12} + 134 q^{13} + 28 q^{14} - 6 q^{16} + 74 q^{17} + 256 q^{18} - 164 q^{19} + 84 q^{21} + 44 q^{22} - 194 q^{23} + 570 q^{24} + 734 q^{26} + 510 q^{27} + 154 q^{28} - 108 q^{29} - 412 q^{31} + 4 q^{32} + 132 q^{33} - 346 q^{34} + 1518 q^{36} - 286 q^{37} - 224 q^{38} - 256 q^{39} - 18 q^{41} + 14 q^{42} + 496 q^{43} + 242 q^{44} - 284 q^{46} - 62 q^{47} + 862 q^{48} + 196 q^{49} - 508 q^{51} + 822 q^{52} + 828 q^{53} + 2420 q^{54} + 420 q^{56} - 700 q^{57} - 1388 q^{58} - 1224 q^{59} - 350 q^{61} + 878 q^{62} + 462 q^{63} - 718 q^{64} + 22 q^{66} + 1498 q^{67} - 1058 q^{68} - 386 q^{69} + 2326 q^{71} + 3000 q^{72} + 1630 q^{73} - 1156 q^{74} - 3152 q^{76} + 308 q^{77} + 2464 q^{78} - 1020 q^{79} + 1128 q^{81} - 2118 q^{82} + 1920 q^{83} + 1302 q^{84} + 1056 q^{86} - 1640 q^{87} + 660 q^{88} + 1550 q^{89} + 938 q^{91} - 2592 q^{92} - 6046 q^{93} - 1042 q^{94} + 4082 q^{96} + 2202 q^{97} + 196 q^{98} + 726 q^{99}+O(q^{100})$$ 4 * q + 4 * q^2 + 12 * q^3 + 22 * q^4 + 2 * q^6 + 28 * q^7 + 60 * q^8 + 66 * q^9 + 44 * q^11 + 186 * q^12 + 134 * q^13 + 28 * q^14 - 6 * q^16 + 74 * q^17 + 256 * q^18 - 164 * q^19 + 84 * q^21 + 44 * q^22 - 194 * q^23 + 570 * q^24 + 734 * q^26 + 510 * q^27 + 154 * q^28 - 108 * q^29 - 412 * q^31 + 4 * q^32 + 132 * q^33 - 346 * q^34 + 1518 * q^36 - 286 * q^37 - 224 * q^38 - 256 * q^39 - 18 * q^41 + 14 * q^42 + 496 * q^43 + 242 * q^44 - 284 * q^46 - 62 * q^47 + 862 * q^48 + 196 * q^49 - 508 * q^51 + 822 * q^52 + 828 * q^53 + 2420 * q^54 + 420 * q^56 - 700 * q^57 - 1388 * q^58 - 1224 * q^59 - 350 * q^61 + 878 * q^62 + 462 * q^63 - 718 * q^64 + 22 * q^66 + 1498 * q^67 - 1058 * q^68 - 386 * q^69 + 2326 * q^71 + 3000 * q^72 + 1630 * q^73 - 1156 * q^74 - 3152 * q^76 + 308 * q^77 + 2464 * q^78 - 1020 * q^79 + 1128 * q^81 - 2118 * q^82 + 1920 * q^83 + 1302 * q^84 + 1056 * q^86 - 1640 * q^87 + 660 * q^88 + 1550 * q^89 + 938 * q^91 - 2592 * q^92 - 6046 * q^93 - 1042 * q^94 + 4082 * q^96 + 2202 * q^97 + 196 * q^98 + 726 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 10x^{2} + 5x + 15$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 5$$ v^2 - v - 5 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 7\nu + 3$$ v^3 - v^2 - 7*v + 3
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} + \beta _1 + 10 ) / 2$$ (2*b2 + b1 + 10) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 2$$ b3 + b2 + 4*b1 + 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.66444 3.18303 −1.11082 1.59222
−3.76366 7.36360 6.16515 0 −27.7141 7.00000 6.90574 27.2227 0
1.2 −0.948670 0.163384 −7.10002 0 −0.154997 7.00000 14.3249 −26.9733 0
1.3 3.65527 −5.17115 5.36103 0 −18.9020 7.00000 −9.64616 −0.259212 0
1.4 5.05706 9.64416 17.5738 0 48.7711 7.00000 48.4155 66.0098 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1925.4.a.q 4
5.b even 2 1 77.4.a.c 4
15.d odd 2 1 693.4.a.m 4
20.d odd 2 1 1232.4.a.w 4
35.c odd 2 1 539.4.a.f 4
55.d odd 2 1 847.4.a.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.a.c 4 5.b even 2 1
539.4.a.f 4 35.c odd 2 1
693.4.a.m 4 15.d odd 2 1
847.4.a.e 4 55.d odd 2 1
1232.4.a.w 4 20.d odd 2 1
1925.4.a.q 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1925))$$:

 $$T_{2}^{4} - 4T_{2}^{3} - 19T_{2}^{2} + 56T_{2} + 66$$ T2^4 - 4*T2^3 - 19*T2^2 + 56*T2 + 66 $$T_{3}^{4} - 12T_{3}^{3} - 15T_{3}^{2} + 370T_{3} - 60$$ T3^4 - 12*T3^3 - 15*T3^2 + 370*T3 - 60 $$T_{13}^{4} - 134T_{13}^{3} + 1732T_{13}^{2} + 324160T_{13} - 9904192$$ T13^4 - 134*T13^3 + 1732*T13^2 + 324160*T13 - 9904192

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4 T^{3} + \cdots + 66$$
$3$ $$T^{4} - 12 T^{3} + \cdots - 60$$
$5$ $$T^{4}$$
$7$ $$(T - 7)^{4}$$
$11$ $$(T - 11)^{4}$$
$13$ $$T^{4} - 134 T^{3} + \cdots - 9904192$$
$17$ $$T^{4} - 74 T^{3} + \cdots - 4708304$$
$19$ $$T^{4} + 164 T^{3} + \cdots - 85552320$$
$23$ $$T^{4} + 194 T^{3} + \cdots - 39720496$$
$29$ $$T^{4} + 108 T^{3} + \cdots - 365881040$$
$31$ $$T^{4} + \cdots - 5207968724$$
$37$ $$T^{4} + \cdots + 1094639996$$
$41$ $$T^{4} + 18 T^{3} + \cdots - 410971280$$
$43$ $$T^{4} - 496 T^{3} + \cdots - 998066176$$
$47$ $$T^{4} + 62 T^{3} + \cdots + 463480064$$
$53$ $$T^{4} - 828 T^{3} + \cdots - 394495824$$
$59$ $$T^{4} + \cdots + 1674727140$$
$61$ $$T^{4} + \cdots - 3730099088$$
$67$ $$T^{4} + \cdots - 57482107536$$
$71$ $$T^{4} + \cdots + 109860635344$$
$73$ $$T^{4} + \cdots - 34532794928$$
$79$ $$T^{4} + \cdots - 48577598400$$
$83$ $$T^{4} + \cdots + 42421669632$$
$89$ $$T^{4} + \cdots - 926653158300$$
$97$ $$T^{4} + \cdots - 78194289572$$