# Properties

 Label 175.4.b.c Level $175$ Weight $4$ Character orbit 175.b Analytic conductor $10.325$ 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] = [175,4,Mod(99,175)]

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([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("175.99");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 175.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: $$10.3253342510$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 35) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 4 \beta_1) q^{2} + ( - 4 \beta_{2} - \beta_1) q^{3} + (8 \beta_{3} - 10) q^{4} + (15 \beta_{3} - 4) q^{6} - 7 \beta_1 q^{7} + (34 \beta_{2} - 24 \beta_1) q^{8} + ( - 8 \beta_{3} - 6) q^{9}+O(q^{10})$$ q + (-b2 + 4*b1) * q^2 + (-4*b2 - b1) * q^3 + (8*b3 - 10) * q^4 + (15*b3 - 4) * q^6 - 7*b1 * q^7 + (34*b2 - 24*b1) * q^8 + (-8*b3 - 6) * q^9 $$q + ( - \beta_{2} + 4 \beta_1) q^{2} + ( - 4 \beta_{2} - \beta_1) q^{3} + (8 \beta_{3} - 10) q^{4} + (15 \beta_{3} - 4) q^{6} - 7 \beta_1 q^{7} + (34 \beta_{2} - 24 \beta_1) q^{8} + ( - 8 \beta_{3} - 6) q^{9} + (32 \beta_{3} - 7) q^{11} + (32 \beta_{2} - 54 \beta_1) q^{12} + (4 \beta_{2} - 25 \beta_1) q^{13} + ( - 7 \beta_{3} + 28) q^{14} + ( - 96 \beta_{3} + 84) q^{16} + ( - 44 \beta_{2} - 25 \beta_1) q^{17} + ( - 26 \beta_{2} - 8 \beta_1) q^{18} + (44 \beta_{3} - 18) q^{19} + ( - 28 \beta_{3} - 7) q^{21} + (135 \beta_{2} - 92 \beta_1) q^{22} + ( - 68 \beta_{2} - 122 \beta_1) q^{23} + ( - 62 \beta_{3} + 248) q^{24} + ( - 41 \beta_{3} + 108) q^{26} + ( - 76 \beta_{2} + 43 \beta_1) q^{27} + ( - 56 \beta_{2} + 70 \beta_1) q^{28} + (24 \beta_{3} + 13) q^{29} + (180 \beta_{3} - 60) q^{31} + ( - 196 \beta_{2} + 336 \beta_1) q^{32} + ( - 4 \beta_{2} - 249 \beta_1) q^{33} + (151 \beta_{3} + 12) q^{34} + (32 \beta_{3} - 68) q^{36} + (60 \beta_{2} + 282 \beta_1) q^{37} + (194 \beta_{2} - 160 \beta_1) q^{38} + ( - 96 \beta_{3} + 7) q^{39} + ( - 124 \beta_{3} - 164) q^{41} + ( - 105 \beta_{2} + 28 \beta_1) q^{42} + (68 \beta_{2} + 130 \beta_1) q^{43} + ( - 376 \beta_{3} + 582) q^{44} + (150 \beta_{3} + 352) q^{46} + (132 \beta_{2} - 175 \beta_1) q^{47} + ( - 240 \beta_{2} + 684 \beta_1) q^{48} - 49 q^{49} + ( - 144 \beta_{3} - 377) q^{51} + ( - 240 \beta_{2} + 314 \beta_1) q^{52} + (128 \beta_{2} + 28 \beta_1) q^{53} + (347 \beta_{3} - 324) q^{54} + (238 \beta_{3} - 168) q^{56} + (28 \beta_{2} - 334 \beta_1) q^{57} + (83 \beta_{2} + 4 \beta_1) q^{58} + 616 q^{59} + (108 \beta_{3} + 168) q^{61} + (780 \beta_{2} - 600 \beta_1) q^{62} + (56 \beta_{2} + 42 \beta_1) q^{63} + (352 \beta_{3} - 1064) q^{64} + ( - 233 \beta_{3} + 988) q^{66} + (64 \beta_{2} - 76 \beta_1) q^{67} + (240 \beta_{2} - 454 \beta_1) q^{68} + ( - 556 \beta_{3} - 666) q^{69} - 952 q^{71} + ( - 12 \beta_{2} - 400 \beta_1) q^{72} + ( - 344 \beta_{2} - 338 \beta_1) q^{73} + (42 \beta_{3} - 1008) q^{74} + ( - 584 \beta_{3} + 884) q^{76} + ( - 224 \beta_{2} + 49 \beta_1) q^{77} + ( - 391 \beta_{2} + 220 \beta_1) q^{78} + (248 \beta_{3} - 507) q^{79} + ( - 120 \beta_{3} - 727) q^{81} + ( - 332 \beta_{2} - 408 \beta_1) q^{82} + (600 \beta_{2} + 188 \beta_1) q^{83} + (224 \beta_{3} - 378) q^{84} + ( - 142 \beta_{3} - 384) q^{86} + ( - 76 \beta_{2} - 205 \beta_1) q^{87} + ( - 1006 \beta_{2} + 2344 \beta_1) q^{88} + (44 \beta_{3} + 108) q^{89} + (28 \beta_{3} - 175) q^{91} + ( - 296 \beta_{2} + 132 \beta_1) q^{92} + (60 \beta_{2} - 1380 \beta_1) q^{93} + ( - 703 \beta_{3} + 964) q^{94} + (1148 \beta_{3} - 1232) q^{96} + ( - 220 \beta_{2} + 1371 \beta_1) q^{97} + (49 \beta_{2} - 196 \beta_1) q^{98} + ( - 136 \beta_{3} - 470) q^{99}+O(q^{100})$$ q + (-b2 + 4*b1) * q^2 + (-4*b2 - b1) * q^3 + (8*b3 - 10) * q^4 + (15*b3 - 4) * q^6 - 7*b1 * q^7 + (34*b2 - 24*b1) * q^8 + (-8*b3 - 6) * q^9 + (32*b3 - 7) * q^11 + (32*b2 - 54*b1) * q^12 + (4*b2 - 25*b1) * q^13 + (-7*b3 + 28) * q^14 + (-96*b3 + 84) * q^16 + (-44*b2 - 25*b1) * q^17 + (-26*b2 - 8*b1) * q^18 + (44*b3 - 18) * q^19 + (-28*b3 - 7) * q^21 + (135*b2 - 92*b1) * q^22 + (-68*b2 - 122*b1) * q^23 + (-62*b3 + 248) * q^24 + (-41*b3 + 108) * q^26 + (-76*b2 + 43*b1) * q^27 + (-56*b2 + 70*b1) * q^28 + (24*b3 + 13) * q^29 + (180*b3 - 60) * q^31 + (-196*b2 + 336*b1) * q^32 + (-4*b2 - 249*b1) * q^33 + (151*b3 + 12) * q^34 + (32*b3 - 68) * q^36 + (60*b2 + 282*b1) * q^37 + (194*b2 - 160*b1) * q^38 + (-96*b3 + 7) * q^39 + (-124*b3 - 164) * q^41 + (-105*b2 + 28*b1) * q^42 + (68*b2 + 130*b1) * q^43 + (-376*b3 + 582) * q^44 + (150*b3 + 352) * q^46 + (132*b2 - 175*b1) * q^47 + (-240*b2 + 684*b1) * q^48 - 49 * q^49 + (-144*b3 - 377) * q^51 + (-240*b2 + 314*b1) * q^52 + (128*b2 + 28*b1) * q^53 + (347*b3 - 324) * q^54 + (238*b3 - 168) * q^56 + (28*b2 - 334*b1) * q^57 + (83*b2 + 4*b1) * q^58 + 616 * q^59 + (108*b3 + 168) * q^61 + (780*b2 - 600*b1) * q^62 + (56*b2 + 42*b1) * q^63 + (352*b3 - 1064) * q^64 + (-233*b3 + 988) * q^66 + (64*b2 - 76*b1) * q^67 + (240*b2 - 454*b1) * q^68 + (-556*b3 - 666) * q^69 - 952 * q^71 + (-12*b2 - 400*b1) * q^72 + (-344*b2 - 338*b1) * q^73 + (42*b3 - 1008) * q^74 + (-584*b3 + 884) * q^76 + (-224*b2 + 49*b1) * q^77 + (-391*b2 + 220*b1) * q^78 + (248*b3 - 507) * q^79 + (-120*b3 - 727) * q^81 + (-332*b2 - 408*b1) * q^82 + (600*b2 + 188*b1) * q^83 + (224*b3 - 378) * q^84 + (-142*b3 - 384) * q^86 + (-76*b2 - 205*b1) * q^87 + (-1006*b2 + 2344*b1) * q^88 + (44*b3 + 108) * q^89 + (28*b3 - 175) * q^91 + (-296*b2 + 132*b1) * q^92 + (60*b2 - 1380*b1) * q^93 + (-703*b3 + 964) * q^94 + (1148*b3 - 1232) * q^96 + (-220*b2 + 1371*b1) * q^97 + (49*b2 - 196*b1) * q^98 + (-136*b3 - 470) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 40 q^{4} - 16 q^{6} - 24 q^{9}+O(q^{10})$$ 4 * q - 40 * q^4 - 16 * q^6 - 24 * q^9 $$4 q - 40 q^{4} - 16 q^{6} - 24 q^{9} - 28 q^{11} + 112 q^{14} + 336 q^{16} - 72 q^{19} - 28 q^{21} + 992 q^{24} + 432 q^{26} + 52 q^{29} - 240 q^{31} + 48 q^{34} - 272 q^{36} + 28 q^{39} - 656 q^{41} + 2328 q^{44} + 1408 q^{46} - 196 q^{49} - 1508 q^{51} - 1296 q^{54} - 672 q^{56} + 2464 q^{59} + 672 q^{61} - 4256 q^{64} + 3952 q^{66} - 2664 q^{69} - 3808 q^{71} - 4032 q^{74} + 3536 q^{76} - 2028 q^{79} - 2908 q^{81} - 1512 q^{84} - 1536 q^{86} + 432 q^{89} - 700 q^{91} + 3856 q^{94} - 4928 q^{96} - 1880 q^{99}+O(q^{100})$$ 4 * q - 40 * q^4 - 16 * q^6 - 24 * q^9 - 28 * q^11 + 112 * q^14 + 336 * q^16 - 72 * q^19 - 28 * q^21 + 992 * q^24 + 432 * q^26 + 52 * q^29 - 240 * q^31 + 48 * q^34 - 272 * q^36 + 28 * q^39 - 656 * q^41 + 2328 * q^44 + 1408 * q^46 - 196 * q^49 - 1508 * q^51 - 1296 * q^54 - 672 * q^56 + 2464 * q^59 + 672 * q^61 - 4256 * q^64 + 3952 * q^66 - 2664 * q^69 - 3808 * q^71 - 4032 * q^74 + 3536 * q^76 - 2028 * q^79 - 2908 * q^81 - 1512 * q^84 - 1536 * q^86 + 432 * q^89 - 700 * q^91 + 3856 * q^94 - 4928 * q^96 - 1880 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
5.41421i 4.65685i −21.3137 0 −25.2132 7.00000i 72.0833i 5.31371 0
99.2 2.58579i 6.65685i 1.31371 0 17.2132 7.00000i 24.0833i −17.3137 0
99.3 2.58579i 6.65685i 1.31371 0 17.2132 7.00000i 24.0833i −17.3137 0
99.4 5.41421i 4.65685i −21.3137 0 −25.2132 7.00000i 72.0833i 5.31371 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 175.4.b.c 4
5.b even 2 1 inner 175.4.b.c 4
5.c odd 4 1 35.4.a.b 2
5.c odd 4 1 175.4.a.c 2
15.e even 4 1 315.4.a.f 2
15.e even 4 1 1575.4.a.z 2
20.e even 4 1 560.4.a.r 2
35.f even 4 1 245.4.a.k 2
35.f even 4 1 1225.4.a.m 2
35.k even 12 2 245.4.e.i 4
35.l odd 12 2 245.4.e.h 4
40.i odd 4 1 2240.4.a.bn 2
40.k even 4 1 2240.4.a.bo 2
105.k odd 4 1 2205.4.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 5.c odd 4 1
175.4.a.c 2 5.c odd 4 1
175.4.b.c 4 1.a even 1 1 trivial
175.4.b.c 4 5.b even 2 1 inner
245.4.a.k 2 35.f even 4 1
245.4.e.h 4 35.l odd 12 2
245.4.e.i 4 35.k even 12 2
315.4.a.f 2 15.e even 4 1
560.4.a.r 2 20.e even 4 1
1225.4.a.m 2 35.f even 4 1
1575.4.a.z 2 15.e even 4 1
2205.4.a.u 2 105.k odd 4 1
2240.4.a.bn 2 40.i odd 4 1
2240.4.a.bo 2 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(175, [\chi])$$:

 $$T_{2}^{4} + 36T_{2}^{2} + 196$$ T2^4 + 36*T2^2 + 196 $$T_{3}^{4} + 66T_{3}^{2} + 961$$ T3^4 + 66*T3^2 + 961

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 36T^{2} + 196$$
$3$ $$T^{4} + 66T^{2} + 961$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 49)^{2}$$
$11$ $$(T^{2} + 14 T - 1999)^{2}$$
$13$ $$T^{4} + 1314 T^{2} + 351649$$
$17$ $$T^{4} + 8994 T^{2} + 10543009$$
$19$ $$(T^{2} + 36 T - 3548)^{2}$$
$23$ $$T^{4} + 48264 T^{2} + 31764496$$
$29$ $$(T^{2} - 26 T - 983)^{2}$$
$31$ $$(T^{2} + 120 T - 61200)^{2}$$
$37$ $$T^{4} + \cdots + 5230760976$$
$41$ $$(T^{2} + 328 T - 3856)^{2}$$
$43$ $$T^{4} + 52296 T^{2} + 58553104$$
$47$ $$T^{4} + 130946 T^{2} + 17833729$$
$53$ $$T^{4} + \cdots + 1022976256$$
$59$ $$(T - 616)^{4}$$
$61$ $$(T^{2} - 336 T + 4896)^{2}$$
$67$ $$T^{4} + 27936 T^{2} + 5837056$$
$71$ $$(T + 952)^{4}$$
$73$ $$T^{4} + \cdots + 14988615184$$
$79$ $$(T^{2} + 1014 T + 134041)^{2}$$
$83$ $$T^{4} + \cdots + 468753838336$$
$89$ $$(T^{2} - 216 T + 7792)^{2}$$
$97$ $$T^{4} + \cdots + 3178522031281$$