# Properties

 Label 175.7.c.c Level $175$ Weight $7$ Character orbit 175.c Analytic conductor $40.259$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,7,Mod(174,175)]

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, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("175.174");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 175.c (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: $$40.2594646335$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{510})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 65025$$ x^4 + 65025 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 7) 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.

 $$f(q)$$ $$=$$ $$q - 8 \beta_1 q^{2} - \beta_{3} q^{3} + 8 \beta_{2} q^{6} + (7 \beta_{3} + 133 \beta_1) q^{7} - 512 \beta_1 q^{8} + 1311 q^{9}+O(q^{10})$$ q - 8*b1 * q^2 - b3 * q^3 + 8*b2 * q^6 + (7*b3 + 133*b1) * q^7 - 512*b1 * q^8 + 1311 * q^9 $$q - 8 \beta_1 q^{2} - \beta_{3} q^{3} + 8 \beta_{2} q^{6} + (7 \beta_{3} + 133 \beta_1) q^{7} - 512 \beta_1 q^{8} + 1311 q^{9} + 874 q^{11} - 49 \beta_{3} q^{13} + ( - 56 \beta_{2} + 1064) q^{14} - 4096 q^{16} - 132 \beta_{3} q^{17} - 10488 \beta_1 q^{18} + 69 \beta_{2} q^{19} + ( - 133 \beta_{2} - 14280) q^{21} - 6992 \beta_1 q^{22} - 4738 \beta_1 q^{23} + 512 \beta_{2} q^{24} + 392 \beta_{2} q^{26} - 582 \beta_{3} q^{27} - 11146 q^{29} + 608 \beta_{2} q^{31} - 874 \beta_{3} q^{33} + 1056 \beta_{2} q^{34} + 3002 \beta_1 q^{37} + 552 \beta_{3} q^{38} + 99960 q^{39} - 1274 \beta_{2} q^{41} + ( - 1064 \beta_{3} + 114240 \beta_1) q^{42} - 31418 \beta_1 q^{43} - 37904 q^{46} - 1604 \beta_{3} q^{47} + 4096 \beta_{3} q^{48} + (1862 \beta_{2} + 82271) q^{49} + 269280 q^{51} + 76406 \beta_1 q^{53} + 4656 \beta_{2} q^{54} + ( - 3584 \beta_{2} + 68096) q^{56} - 140760 \beta_1 q^{57} + 89168 \beta_1 q^{58} + 2507 \beta_{2} q^{59} + 6091 \beta_{2} q^{61} + 4864 \beta_{3} q^{62} + (9177 \beta_{3} + 174363 \beta_1) q^{63} - 262144 q^{64} + 6992 \beta_{2} q^{66} + 495242 \beta_1 q^{67} + 4738 \beta_{2} q^{69} - 184406 q^{71} - 671232 \beta_1 q^{72} + 1350 \beta_{3} q^{73} + 24016 q^{74} + (6118 \beta_{3} + 116242 \beta_1) q^{77} - 799680 \beta_1 q^{78} + 534934 q^{79} + 231561 q^{81} - 10192 \beta_{3} q^{82} + 15827 \beta_{3} q^{83} - 251344 q^{86} + 11146 \beta_{3} q^{87} - 447488 \beta_1 q^{88} - 13938 \beta_{2} q^{89} + ( - 6517 \beta_{2} - 699720) q^{91} - 1240320 \beta_1 q^{93} + 12832 \beta_{2} q^{94} + 18032 \beta_{3} q^{97} + (14896 \beta_{3} - 658168 \beta_1) q^{98} + 1145814 q^{99}+O(q^{100})$$ q - 8*b1 * q^2 - b3 * q^3 + 8*b2 * q^6 + (7*b3 + 133*b1) * q^7 - 512*b1 * q^8 + 1311 * q^9 + 874 * q^11 - 49*b3 * q^13 + (-56*b2 + 1064) * q^14 - 4096 * q^16 - 132*b3 * q^17 - 10488*b1 * q^18 + 69*b2 * q^19 + (-133*b2 - 14280) * q^21 - 6992*b1 * q^22 - 4738*b1 * q^23 + 512*b2 * q^24 + 392*b2 * q^26 - 582*b3 * q^27 - 11146 * q^29 + 608*b2 * q^31 - 874*b3 * q^33 + 1056*b2 * q^34 + 3002*b1 * q^37 + 552*b3 * q^38 + 99960 * q^39 - 1274*b2 * q^41 + (-1064*b3 + 114240*b1) * q^42 - 31418*b1 * q^43 - 37904 * q^46 - 1604*b3 * q^47 + 4096*b3 * q^48 + (1862*b2 + 82271) * q^49 + 269280 * q^51 + 76406*b1 * q^53 + 4656*b2 * q^54 + (-3584*b2 + 68096) * q^56 - 140760*b1 * q^57 + 89168*b1 * q^58 + 2507*b2 * q^59 + 6091*b2 * q^61 + 4864*b3 * q^62 + (9177*b3 + 174363*b1) * q^63 - 262144 * q^64 + 6992*b2 * q^66 + 495242*b1 * q^67 + 4738*b2 * q^69 - 184406 * q^71 - 671232*b1 * q^72 + 1350*b3 * q^73 + 24016 * q^74 + (6118*b3 + 116242*b1) * q^77 - 799680*b1 * q^78 + 534934 * q^79 + 231561 * q^81 - 10192*b3 * q^82 + 15827*b3 * q^83 - 251344 * q^86 + 11146*b3 * q^87 - 447488*b1 * q^88 - 13938*b2 * q^89 + (-6517*b2 - 699720) * q^91 - 1240320*b1 * q^93 + 12832*b2 * q^94 + 18032*b3 * q^97 + (14896*b3 - 658168*b1) * q^98 + 1145814 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 5244 q^{9}+O(q^{10})$$ 4 * q + 5244 * q^9 $$4 q + 5244 q^{9} + 3496 q^{11} + 4256 q^{14} - 16384 q^{16} - 57120 q^{21} - 44584 q^{29} + 399840 q^{39} - 151616 q^{46} + 329084 q^{49} + 1077120 q^{51} + 272384 q^{56} - 1048576 q^{64} - 737624 q^{71} + 96064 q^{74} + 2139736 q^{79} + 926244 q^{81} - 1005376 q^{86} - 2798880 q^{91} + 4583256 q^{99}+O(q^{100})$$ 4 * q + 5244 * q^9 + 3496 * q^11 + 4256 * q^14 - 16384 * q^16 - 57120 * q^21 - 44584 * q^29 + 399840 * q^39 - 151616 * q^46 + 329084 * q^49 + 1077120 * q^51 + 272384 * q^56 - 1048576 * q^64 - 737624 * q^71 + 96064 * q^74 + 2139736 * q^79 + 926244 * q^81 - 1005376 * q^86 - 2798880 * q^91 + 4583256 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 65025$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 255$$ (v^2) / 255 $$\beta_{2}$$ $$=$$ $$( 2\nu^{3} + 510\nu ) / 255$$ (2*v^3 + 510*v) / 255 $$\beta_{3}$$ $$=$$ $$( -2\nu^{3} + 510\nu ) / 255$$ (-2*v^3 + 510*v) / 255
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 4$$ (b3 + b2) / 4 $$\nu^{2}$$ $$=$$ $$255\beta_1$$ 255*b1 $$\nu^{3}$$ $$=$$ $$( -255\beta_{3} + 255\beta_{2} ) / 4$$ (-255*b3 + 255*b2) / 4

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
174.1
 11.2916 + 11.2916i −11.2916 − 11.2916i 11.2916 − 11.2916i −11.2916 + 11.2916i
8.00000i −45.1664 0 0 361.331i 316.165 + 133.000i 512.000i 1311.00 0
174.2 8.00000i 45.1664 0 0 361.331i −316.165 + 133.000i 512.000i 1311.00 0
174.3 8.00000i −45.1664 0 0 361.331i 316.165 133.000i 512.000i 1311.00 0
174.4 8.00000i 45.1664 0 0 361.331i −316.165 133.000i 512.000i 1311.00 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
7.b odd 2 1 inner
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.7.c.c 4
5.b even 2 1 inner 175.7.c.c 4
5.c odd 4 1 7.7.b.b 2
5.c odd 4 1 175.7.d.e 2
7.b odd 2 1 inner 175.7.c.c 4
15.e even 4 1 63.7.d.d 2
20.e even 4 1 112.7.c.b 2
35.c odd 2 1 inner 175.7.c.c 4
35.f even 4 1 7.7.b.b 2
35.f even 4 1 175.7.d.e 2
35.k even 12 2 49.7.d.d 4
35.l odd 12 2 49.7.d.d 4
40.i odd 4 1 448.7.c.d 2
40.k even 4 1 448.7.c.c 2
105.k odd 4 1 63.7.d.d 2
140.j odd 4 1 112.7.c.b 2
280.s even 4 1 448.7.c.d 2
280.y odd 4 1 448.7.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.b.b 2 5.c odd 4 1
7.7.b.b 2 35.f even 4 1
49.7.d.d 4 35.k even 12 2
49.7.d.d 4 35.l odd 12 2
63.7.d.d 2 15.e even 4 1
63.7.d.d 2 105.k odd 4 1
112.7.c.b 2 20.e even 4 1
112.7.c.b 2 140.j odd 4 1
175.7.c.c 4 1.a even 1 1 trivial
175.7.c.c 4 5.b even 2 1 inner
175.7.c.c 4 7.b odd 2 1 inner
175.7.c.c 4 35.c odd 2 1 inner
175.7.d.e 2 5.c odd 4 1
175.7.d.e 2 35.f even 4 1
448.7.c.c 2 40.k even 4 1
448.7.c.c 2 280.y odd 4 1
448.7.c.d 2 40.i odd 4 1
448.7.c.d 2 280.s even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 64$$ acting on $$S_{7}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 64)^{2}$$
$3$ $$(T^{2} - 2040)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + \cdots + 13841287201$$
$11$ $$(T - 874)^{4}$$
$13$ $$(T^{2} - 4898040)^{2}$$
$17$ $$(T^{2} - 35544960)^{2}$$
$19$ $$(T^{2} + 9712440)^{2}$$
$23$ $$(T^{2} + 22448644)^{2}$$
$29$ $$(T + 11146)^{4}$$
$31$ $$(T^{2} + 754114560)^{2}$$
$37$ $$(T^{2} + 9012004)^{2}$$
$41$ $$(T^{2} + 3311075040)^{2}$$
$43$ $$(T^{2} + 987090724)^{2}$$
$47$ $$(T^{2} - 5248544640)^{2}$$
$53$ $$(T^{2} + 5837876836)^{2}$$
$59$ $$(T^{2} + 12821499960)^{2}$$
$61$ $$(T^{2} + 75684573240)^{2}$$
$67$ $$(T^{2} + 245264638564)^{2}$$
$71$ $$(T + 184406)^{4}$$
$73$ $$(T^{2} - 3717900000)^{2}$$
$79$ $$(T - 534934)^{4}$$
$83$ $$(T^{2} - 511007615160)^{2}$$
$89$ $$(T^{2} + 396306401760)^{2}$$
$97$ $$(T^{2} - 663312168960)^{2}$$