# Properties

 Label 1575.4.a.z Level $1575$ Weight $4$ Character orbit 1575.a Self dual yes Analytic conductor $92.928$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1575.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: $$92.9280082590$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 4) q^{2} + (8 \beta + 10) q^{4} + 7 q^{7} + (34 \beta + 24) q^{8}+O(q^{10})$$ q + (b + 4) * q^2 + (8*b + 10) * q^4 + 7 * q^7 + (34*b + 24) * q^8 $$q + (\beta + 4) q^{2} + (8 \beta + 10) q^{4} + 7 q^{7} + (34 \beta + 24) q^{8} + (32 \beta + 7) q^{11} + ( - 4 \beta - 25) q^{13} + (7 \beta + 28) q^{14} + (96 \beta + 84) q^{16} + (44 \beta - 25) q^{17} + (44 \beta + 18) q^{19} + (135 \beta + 92) q^{22} + ( - 68 \beta + 122) q^{23} + ( - 41 \beta - 108) q^{26} + (56 \beta + 70) q^{28} + ( - 24 \beta + 13) q^{29} + ( - 180 \beta - 60) q^{31} + (196 \beta + 336) q^{32} + (151 \beta - 12) q^{34} + (60 \beta - 282) q^{37} + (194 \beta + 160) q^{38} + ( - 124 \beta + 164) q^{41} + ( - 68 \beta + 130) q^{43} + (376 \beta + 582) q^{44} + ( - 150 \beta + 352) q^{46} + ( - 132 \beta - 175) q^{47} + 49 q^{49} + ( - 240 \beta - 314) q^{52} + (128 \beta - 28) q^{53} + (238 \beta + 168) q^{56} + ( - 83 \beta + 4) q^{58} + 616 q^{59} + ( - 108 \beta + 168) q^{61} + ( - 780 \beta - 600) q^{62} + (352 \beta + 1064) q^{64} + (64 \beta + 76) q^{67} + (240 \beta + 454) q^{68} + 952 q^{71} + (344 \beta - 338) q^{73} + ( - 42 \beta - 1008) q^{74} + (584 \beta + 884) q^{76} + (224 \beta + 49) q^{77} + (248 \beta + 507) q^{79} + ( - 332 \beta + 408) q^{82} + (600 \beta - 188) q^{83} + ( - 142 \beta + 384) q^{86} + (1006 \beta + 2344) q^{88} + ( - 44 \beta + 108) q^{89} + ( - 28 \beta - 175) q^{91} + (296 \beta + 132) q^{92} + ( - 703 \beta - 964) q^{94} + ( - 220 \beta - 1371) q^{97} + (49 \beta + 196) q^{98}+O(q^{100})$$ q + (b + 4) * q^2 + (8*b + 10) * q^4 + 7 * q^7 + (34*b + 24) * q^8 + (32*b + 7) * q^11 + (-4*b - 25) * q^13 + (7*b + 28) * q^14 + (96*b + 84) * q^16 + (44*b - 25) * q^17 + (44*b + 18) * q^19 + (135*b + 92) * q^22 + (-68*b + 122) * q^23 + (-41*b - 108) * q^26 + (56*b + 70) * q^28 + (-24*b + 13) * q^29 + (-180*b - 60) * q^31 + (196*b + 336) * q^32 + (151*b - 12) * q^34 + (60*b - 282) * q^37 + (194*b + 160) * q^38 + (-124*b + 164) * q^41 + (-68*b + 130) * q^43 + (376*b + 582) * q^44 + (-150*b + 352) * q^46 + (-132*b - 175) * q^47 + 49 * q^49 + (-240*b - 314) * q^52 + (128*b - 28) * q^53 + (238*b + 168) * q^56 + (-83*b + 4) * q^58 + 616 * q^59 + (-108*b + 168) * q^61 + (-780*b - 600) * q^62 + (352*b + 1064) * q^64 + (64*b + 76) * q^67 + (240*b + 454) * q^68 + 952 * q^71 + (344*b - 338) * q^73 + (-42*b - 1008) * q^74 + (584*b + 884) * q^76 + (224*b + 49) * q^77 + (248*b + 507) * q^79 + (-332*b + 408) * q^82 + (600*b - 188) * q^83 + (-142*b + 384) * q^86 + (1006*b + 2344) * q^88 + (-44*b + 108) * q^89 + (-28*b - 175) * q^91 + (296*b + 132) * q^92 + (-703*b - 964) * q^94 + (-220*b - 1371) * q^97 + (49*b + 196) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{2} + 20 q^{4} + 14 q^{7} + 48 q^{8}+O(q^{10})$$ 2 * q + 8 * q^2 + 20 * q^4 + 14 * q^7 + 48 * q^8 $$2 q + 8 q^{2} + 20 q^{4} + 14 q^{7} + 48 q^{8} + 14 q^{11} - 50 q^{13} + 56 q^{14} + 168 q^{16} - 50 q^{17} + 36 q^{19} + 184 q^{22} + 244 q^{23} - 216 q^{26} + 140 q^{28} + 26 q^{29} - 120 q^{31} + 672 q^{32} - 24 q^{34} - 564 q^{37} + 320 q^{38} + 328 q^{41} + 260 q^{43} + 1164 q^{44} + 704 q^{46} - 350 q^{47} + 98 q^{49} - 628 q^{52} - 56 q^{53} + 336 q^{56} + 8 q^{58} + 1232 q^{59} + 336 q^{61} - 1200 q^{62} + 2128 q^{64} + 152 q^{67} + 908 q^{68} + 1904 q^{71} - 676 q^{73} - 2016 q^{74} + 1768 q^{76} + 98 q^{77} + 1014 q^{79} + 816 q^{82} - 376 q^{83} + 768 q^{86} + 4688 q^{88} + 216 q^{89} - 350 q^{91} + 264 q^{92} - 1928 q^{94} - 2742 q^{97} + 392 q^{98}+O(q^{100})$$ 2 * q + 8 * q^2 + 20 * q^4 + 14 * q^7 + 48 * q^8 + 14 * q^11 - 50 * q^13 + 56 * q^14 + 168 * q^16 - 50 * q^17 + 36 * q^19 + 184 * q^22 + 244 * q^23 - 216 * q^26 + 140 * q^28 + 26 * q^29 - 120 * q^31 + 672 * q^32 - 24 * q^34 - 564 * q^37 + 320 * q^38 + 328 * q^41 + 260 * q^43 + 1164 * q^44 + 704 * q^46 - 350 * q^47 + 98 * q^49 - 628 * q^52 - 56 * q^53 + 336 * q^56 + 8 * q^58 + 1232 * q^59 + 336 * q^61 - 1200 * q^62 + 2128 * q^64 + 152 * q^67 + 908 * q^68 + 1904 * q^71 - 676 * q^73 - 2016 * q^74 + 1768 * q^76 + 98 * q^77 + 1014 * q^79 + 816 * q^82 - 376 * q^83 + 768 * q^86 + 4688 * q^88 + 216 * q^89 - 350 * q^91 + 264 * q^92 - 1928 * q^94 - 2742 * q^97 + 392 * q^98

## 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
 −1.41421 1.41421
2.58579 0 −1.31371 0 0 7.00000 −24.0833 0 0
1.2 5.41421 0 21.3137 0 0 7.00000 72.0833 0 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
$$3$$ $$-1$$
$$5$$ $$+1$$
$$7$$ $$-1$$

## 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 1575.4.a.z 2
3.b odd 2 1 175.4.a.c 2
5.b even 2 1 315.4.a.f 2
15.d odd 2 1 35.4.a.b 2
15.e even 4 2 175.4.b.c 4
21.c even 2 1 1225.4.a.m 2
35.c odd 2 1 2205.4.a.u 2
60.h even 2 1 560.4.a.r 2
105.g even 2 1 245.4.a.k 2
105.o odd 6 2 245.4.e.h 4
105.p even 6 2 245.4.e.i 4
120.i odd 2 1 2240.4.a.bn 2
120.m even 2 1 2240.4.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 15.d odd 2 1
175.4.a.c 2 3.b odd 2 1
175.4.b.c 4 15.e even 4 2
245.4.a.k 2 105.g even 2 1
245.4.e.h 4 105.o odd 6 2
245.4.e.i 4 105.p even 6 2
315.4.a.f 2 5.b even 2 1
560.4.a.r 2 60.h even 2 1
1225.4.a.m 2 21.c even 2 1
1575.4.a.z 2 1.a even 1 1 trivial
2205.4.a.u 2 35.c odd 2 1
2240.4.a.bn 2 120.i odd 2 1
2240.4.a.bo 2 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}^{2} - 8T_{2} + 14$$ T2^2 - 8*T2 + 14 $$T_{11}^{2} - 14T_{11} - 1999$$ T11^2 - 14*T11 - 1999 $$T_{13}^{2} + 50T_{13} + 593$$ T13^2 + 50*T13 + 593

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 8T + 14$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 7)^{2}$$
$11$ $$T^{2} - 14T - 1999$$
$13$ $$T^{2} + 50T + 593$$
$17$ $$T^{2} + 50T - 3247$$
$19$ $$T^{2} - 36T - 3548$$
$23$ $$T^{2} - 244T + 5636$$
$29$ $$T^{2} - 26T - 983$$
$31$ $$T^{2} + 120T - 61200$$
$37$ $$T^{2} + 564T + 72324$$
$41$ $$T^{2} - 328T - 3856$$
$43$ $$T^{2} - 260T + 7652$$
$47$ $$T^{2} + 350T - 4223$$
$53$ $$T^{2} + 56T - 31984$$
$59$ $$(T - 616)^{2}$$
$61$ $$T^{2} - 336T + 4896$$
$67$ $$T^{2} - 152T - 2416$$
$71$ $$(T - 952)^{2}$$
$73$ $$T^{2} + 676T - 122428$$
$79$ $$T^{2} - 1014 T + 134041$$
$83$ $$T^{2} + 376T - 684656$$
$89$ $$T^{2} - 216T + 7792$$
$97$ $$T^{2} + 2742 T + 1782841$$