# Properties

 Label 175.4.a.c Level $175$ Weight $4$ Character orbit 175.a Self dual yes Analytic conductor $10.325$ Analytic rank $1$ 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] = [175,4,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("175.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 175.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: $$10.3253342510$$ Analytic rank: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta - 4) q^{2} + ( - 4 \beta - 1) q^{3} + ( - 8 \beta + 10) q^{4} + (15 \beta - 4) q^{6} + 7 q^{7} + (34 \beta - 24) q^{8} + (8 \beta + 6) q^{9}+O(q^{10})$$ q + (b - 4) * q^2 + (-4*b - 1) * q^3 + (-8*b + 10) * q^4 + (15*b - 4) * q^6 + 7 * q^7 + (34*b - 24) * q^8 + (8*b + 6) * q^9 $$q + (\beta - 4) q^{2} + ( - 4 \beta - 1) q^{3} + ( - 8 \beta + 10) q^{4} + (15 \beta - 4) q^{6} + 7 q^{7} + (34 \beta - 24) q^{8} + (8 \beta + 6) q^{9} + (32 \beta - 7) q^{11} + ( - 32 \beta + 54) q^{12} + (4 \beta - 25) q^{13} + (7 \beta - 28) q^{14} + ( - 96 \beta + 84) q^{16} + (44 \beta + 25) q^{17} + ( - 26 \beta - 8) q^{18} + ( - 44 \beta + 18) q^{19} + ( - 28 \beta - 7) q^{21} + ( - 135 \beta + 92) q^{22} + ( - 68 \beta - 122) q^{23} + (62 \beta - 248) q^{24} + ( - 41 \beta + 108) q^{26} + (76 \beta - 43) q^{27} + ( - 56 \beta + 70) q^{28} + ( - 24 \beta - 13) q^{29} + (180 \beta - 60) q^{31} + (196 \beta - 336) q^{32} + ( - 4 \beta - 249) q^{33} + ( - 151 \beta - 12) q^{34} + (32 \beta - 68) q^{36} + ( - 60 \beta - 282) q^{37} + (194 \beta - 160) q^{38} + (96 \beta - 7) q^{39} + ( - 124 \beta - 164) q^{41} + (105 \beta - 28) q^{42} + (68 \beta + 130) q^{43} + (376 \beta - 582) q^{44} + (150 \beta + 352) q^{46} + ( - 132 \beta + 175) q^{47} + ( - 240 \beta + 684) q^{48} + 49 q^{49} + ( - 144 \beta - 377) q^{51} + (240 \beta - 314) q^{52} + (128 \beta + 28) q^{53} + ( - 347 \beta + 324) q^{54} + (238 \beta - 168) q^{56} + ( - 28 \beta + 334) q^{57} + (83 \beta + 4) q^{58} - 616 q^{59} + (108 \beta + 168) q^{61} + ( - 780 \beta + 600) q^{62} + (56 \beta + 42) q^{63} + ( - 352 \beta + 1064) q^{64} + ( - 233 \beta + 988) q^{66} + ( - 64 \beta + 76) q^{67} + (240 \beta - 454) q^{68} + (556 \beta + 666) q^{69} - 952 q^{71} + (12 \beta + 400) q^{72} + ( - 344 \beta - 338) q^{73} + ( - 42 \beta + 1008) q^{74} + ( - 584 \beta + 884) q^{76} + (224 \beta - 49) q^{77} + ( - 391 \beta + 220) q^{78} + ( - 248 \beta + 507) q^{79} + ( - 120 \beta - 727) q^{81} + (332 \beta + 408) q^{82} + (600 \beta + 188) q^{83} + ( - 224 \beta + 378) q^{84} + ( - 142 \beta - 384) q^{86} + (76 \beta + 205) q^{87} + ( - 1006 \beta + 2344) q^{88} + ( - 44 \beta - 108) q^{89} + (28 \beta - 175) q^{91} + (296 \beta - 132) q^{92} + (60 \beta - 1380) q^{93} + (703 \beta - 964) q^{94} + (1148 \beta - 1232) q^{96} + (220 \beta - 1371) q^{97} + (49 \beta - 196) q^{98} + (136 \beta + 470) q^{99}+O(q^{100})$$ q + (b - 4) * q^2 + (-4*b - 1) * q^3 + (-8*b + 10) * q^4 + (15*b - 4) * q^6 + 7 * q^7 + (34*b - 24) * q^8 + (8*b + 6) * q^9 + (32*b - 7) * q^11 + (-32*b + 54) * q^12 + (4*b - 25) * q^13 + (7*b - 28) * q^14 + (-96*b + 84) * q^16 + (44*b + 25) * q^17 + (-26*b - 8) * q^18 + (-44*b + 18) * q^19 + (-28*b - 7) * q^21 + (-135*b + 92) * q^22 + (-68*b - 122) * q^23 + (62*b - 248) * q^24 + (-41*b + 108) * q^26 + (76*b - 43) * q^27 + (-56*b + 70) * q^28 + (-24*b - 13) * q^29 + (180*b - 60) * q^31 + (196*b - 336) * q^32 + (-4*b - 249) * q^33 + (-151*b - 12) * q^34 + (32*b - 68) * q^36 + (-60*b - 282) * q^37 + (194*b - 160) * q^38 + (96*b - 7) * q^39 + (-124*b - 164) * q^41 + (105*b - 28) * q^42 + (68*b + 130) * q^43 + (376*b - 582) * q^44 + (150*b + 352) * q^46 + (-132*b + 175) * q^47 + (-240*b + 684) * q^48 + 49 * q^49 + (-144*b - 377) * q^51 + (240*b - 314) * q^52 + (128*b + 28) * q^53 + (-347*b + 324) * q^54 + (238*b - 168) * q^56 + (-28*b + 334) * q^57 + (83*b + 4) * q^58 - 616 * q^59 + (108*b + 168) * q^61 + (-780*b + 600) * q^62 + (56*b + 42) * q^63 + (-352*b + 1064) * q^64 + (-233*b + 988) * q^66 + (-64*b + 76) * q^67 + (240*b - 454) * q^68 + (556*b + 666) * q^69 - 952 * q^71 + (12*b + 400) * q^72 + (-344*b - 338) * q^73 + (-42*b + 1008) * q^74 + (-584*b + 884) * q^76 + (224*b - 49) * q^77 + (-391*b + 220) * q^78 + (-248*b + 507) * q^79 + (-120*b - 727) * q^81 + (332*b + 408) * q^82 + (600*b + 188) * q^83 + (-224*b + 378) * q^84 + (-142*b - 384) * q^86 + (76*b + 205) * q^87 + (-1006*b + 2344) * q^88 + (-44*b - 108) * q^89 + (28*b - 175) * q^91 + (296*b - 132) * q^92 + (60*b - 1380) * q^93 + (703*b - 964) * q^94 + (1148*b - 1232) * q^96 + (220*b - 1371) * q^97 + (49*b - 196) * q^98 + (136*b + 470) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{2} - 2 q^{3} + 20 q^{4} - 8 q^{6} + 14 q^{7} - 48 q^{8} + 12 q^{9}+O(q^{10})$$ 2 * q - 8 * q^2 - 2 * q^3 + 20 * q^4 - 8 * q^6 + 14 * q^7 - 48 * q^8 + 12 * q^9 $$2 q - 8 q^{2} - 2 q^{3} + 20 q^{4} - 8 q^{6} + 14 q^{7} - 48 q^{8} + 12 q^{9} - 14 q^{11} + 108 q^{12} - 50 q^{13} - 56 q^{14} + 168 q^{16} + 50 q^{17} - 16 q^{18} + 36 q^{19} - 14 q^{21} + 184 q^{22} - 244 q^{23} - 496 q^{24} + 216 q^{26} - 86 q^{27} + 140 q^{28} - 26 q^{29} - 120 q^{31} - 672 q^{32} - 498 q^{33} - 24 q^{34} - 136 q^{36} - 564 q^{37} - 320 q^{38} - 14 q^{39} - 328 q^{41} - 56 q^{42} + 260 q^{43} - 1164 q^{44} + 704 q^{46} + 350 q^{47} + 1368 q^{48} + 98 q^{49} - 754 q^{51} - 628 q^{52} + 56 q^{53} + 648 q^{54} - 336 q^{56} + 668 q^{57} + 8 q^{58} - 1232 q^{59} + 336 q^{61} + 1200 q^{62} + 84 q^{63} + 2128 q^{64} + 1976 q^{66} + 152 q^{67} - 908 q^{68} + 1332 q^{69} - 1904 q^{71} + 800 q^{72} - 676 q^{73} + 2016 q^{74} + 1768 q^{76} - 98 q^{77} + 440 q^{78} + 1014 q^{79} - 1454 q^{81} + 816 q^{82} + 376 q^{83} + 756 q^{84} - 768 q^{86} + 410 q^{87} + 4688 q^{88} - 216 q^{89} - 350 q^{91} - 264 q^{92} - 2760 q^{93} - 1928 q^{94} - 2464 q^{96} - 2742 q^{97} - 392 q^{98} + 940 q^{99}+O(q^{100})$$ 2 * q - 8 * q^2 - 2 * q^3 + 20 * q^4 - 8 * q^6 + 14 * q^7 - 48 * q^8 + 12 * q^9 - 14 * q^11 + 108 * q^12 - 50 * q^13 - 56 * q^14 + 168 * q^16 + 50 * q^17 - 16 * q^18 + 36 * q^19 - 14 * q^21 + 184 * q^22 - 244 * q^23 - 496 * q^24 + 216 * q^26 - 86 * q^27 + 140 * q^28 - 26 * q^29 - 120 * q^31 - 672 * q^32 - 498 * q^33 - 24 * q^34 - 136 * q^36 - 564 * q^37 - 320 * q^38 - 14 * q^39 - 328 * q^41 - 56 * q^42 + 260 * q^43 - 1164 * q^44 + 704 * q^46 + 350 * q^47 + 1368 * q^48 + 98 * q^49 - 754 * q^51 - 628 * q^52 + 56 * q^53 + 648 * q^54 - 336 * q^56 + 668 * q^57 + 8 * q^58 - 1232 * q^59 + 336 * q^61 + 1200 * q^62 + 84 * q^63 + 2128 * q^64 + 1976 * q^66 + 152 * q^67 - 908 * q^68 + 1332 * q^69 - 1904 * q^71 + 800 * q^72 - 676 * q^73 + 2016 * q^74 + 1768 * q^76 - 98 * q^77 + 440 * q^78 + 1014 * q^79 - 1454 * q^81 + 816 * q^82 + 376 * q^83 + 756 * q^84 - 768 * q^86 + 410 * q^87 + 4688 * q^88 - 216 * q^89 - 350 * q^91 - 264 * q^92 - 2760 * q^93 - 1928 * q^94 - 2464 * q^96 - 2742 * q^97 - 392 * q^98 + 940 * q^99

## 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
−5.41421 4.65685 21.3137 0 −25.2132 7.00000 −72.0833 −5.31371 0
1.2 −2.58579 −6.65685 −1.31371 0 17.2132 7.00000 24.0833 17.3137 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
$$5$$ $$+1$$
$$7$$ $$-1$$

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 5.b even 2 1
175.4.a.c 2 1.a even 1 1 trivial
175.4.b.c 4 5.c odd 4 2
245.4.a.k 2 35.c odd 2 1
245.4.e.h 4 35.j even 6 2
245.4.e.i 4 35.i odd 6 2
315.4.a.f 2 15.d odd 2 1
560.4.a.r 2 20.d odd 2 1
1225.4.a.m 2 7.b odd 2 1
1575.4.a.z 2 3.b odd 2 1
2205.4.a.u 2 105.g even 2 1
2240.4.a.bn 2 40.f even 2 1
2240.4.a.bo 2 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 8T_{2} + 14$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(175))$$.

## Hecke characteristic polynomials

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