Properties

Label 175.10.b.c
Level $175$
Weight $10$
Character orbit 175.b
Analytic conductor $90.131$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.99");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(90.1312713287\)
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: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 12 \beta_1) q^{2} + (54 \beta_{2} - 87 \beta_1) q^{3} + (24 \beta_{3} + 360) q^{4} + ( - 735 \beta_{3} + 1476) q^{6} - 2401 \beta_1 q^{7} + ( - 584 \beta_{2} + 10272 \beta_1) q^{8} + (9396 \beta_{3} - 11214) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 12 \beta_1) q^{2} + (54 \beta_{2} - 87 \beta_1) q^{3} + (24 \beta_{3} + 360) q^{4} + ( - 735 \beta_{3} + 1476) q^{6} - 2401 \beta_1 q^{7} + ( - 584 \beta_{2} + 10272 \beta_1) q^{8} + (9396 \beta_{3} - 11214) q^{9} + (9124 \beta_{3} + 9283) q^{11} + (17352 \beta_{2} - 20952 \beta_1) q^{12} + ( - 18334 \beta_{2} - 25545 \beta_1) q^{13} + ( - 2401 \beta_{3} + 28812) q^{14} + (29568 \beta_{3} + 56384) q^{16} + (15186 \beta_{2} + 186955 \beta_1) q^{17} + (123966 \beta_{2} - 209736 \beta_1) q^{18} + ( - 31138 \beta_{3} + 71638) q^{19} + (129654 \beta_{3} - 208887) q^{21} + (100205 \beta_{2} + 38404 \beta_1) q^{22} + ( - 821442 \beta_{2} - 249454 \beta_1) q^{23} + ( - 605496 \beta_{3} + 1145952) q^{24} + (194463 \beta_{3} + 159868) q^{26} + ( - 360126 \beta_{2} + 3322269 \beta_1) q^{27} + ( - 57624 \beta_{2} - 864360 \beta_1) q^{28} + (296772 \beta_{3} + 5788777) q^{29} + ( - 2191070 \beta_{3} - 1976880) q^{31} + ( - 576 \beta_{2} + 5699328 \beta_1) q^{32} + ( - 292506 \beta_{2} + 3133947 \beta_1) q^{33} + (4723 \beta_{3} - 2121972) q^{34} + (3113424 \beta_{3} - 2233008) q^{36} + ( - 3997070 \beta_{2} + 1602706 \beta_1) q^{37} + ( - 445294 \beta_{2} + 1108760 \beta_1) q^{38} + ( - 215628 \beta_{3} + 5697873) q^{39} + (10352502 \beta_{3} + 529496) q^{41} + (1764735 \beta_{2} - 3543876 \beta_1) q^{42} + ( - 9725678 \beta_{2} + 7974090 \beta_1) q^{43} + (3507432 \beta_{3} + 5093688) q^{44} + (9607850 \beta_{3} - 3578088) q^{46} + (9479722 \beta_{2} - 32750645 \beta_1) q^{47} + (472320 \beta_{2} + 7867968 \beta_1) q^{48} - 5764801 q^{49} + ( - 8774388 \beta_{3} + 9704733) q^{51} + ( - 7213320 \beta_{2} - 12716328 \beta_1) q^{52} + ( - 5256968 \beta_{2} - 12557344 \beta_1) q^{53} + (7643781 \beta_{3} - 42748236) q^{54} + ( - 1402184 \beta_{3} + 24663072) q^{56} + (6577458 \beta_{2} - 19684122 \beta_1) q^{57} + ( - 2227513 \beta_{2} + 67091148 \beta_1) q^{58} + ( - 39092800 \beta_{3} + 58079604) q^{59} + (15794106 \beta_{3} - 22344272) q^{61} + ( - 24315960 \beta_{2} - 6194000 \beta_1) q^{62} + ( - 22559796 \beta_{2} + 26924814 \beta_1) q^{63} + (20845056 \beta_{3} - 39527936) q^{64} + (6644019 \beta_{3} - 39947412) q^{66} + ( - 76121736 \beta_{2} - 59046248 \beta_1) q^{67} + (9953880 \beta_{2} + 70219512 \beta_1) q^{68} + ( - 57994938 \beta_{3} + 333160446) q^{69} + ( - 76576000 \beta_{3} - 147082912) q^{71} + (103064688 \beta_{2} - 159088320 \beta_1) q^{72} + (17548324 \beta_{2} - 28709666 \beta_1) q^{73} + (49567546 \beta_{3} - 51209032) q^{74} + ( - 9490368 \beta_{3} + 19811184) q^{76} + ( - 21906724 \beta_{2} - 22288483 \beta_1) q^{77} + ( - 8285409 \beta_{2} + 70099500 \beta_1) q^{78} + ( - 28471816 \beta_{3} + 346426427) q^{79} + ( - 25792020 \beta_{3} + 223886673) q^{81} + (123700528 \beta_{2} - 76466064 \beta_1) q^{82} + (92242900 \beta_{2} - 270339964 \beta_1) q^{83} + (41662152 \beta_{3} - 50305752) q^{84} + (124682226 \beta_{3} - 173494504) q^{86} + (286774794 \beta_{2} - 375418095 \beta_1) q^{87} + (88300456 \beta_{2} + 52727648 \beta_1) q^{88} + (97352322 \beta_{3} + 389521852) q^{89} + ( - 44019934 \beta_{3} - 61333545) q^{91} + ( - 301706016 \beta_{2} - 247520304 \beta_1) q^{92} + (83871570 \beta_{2} - 774553680 \beta_1) q^{93} + ( - 146507309 \beta_{3} + 468845516) q^{94} + ( - 307813824 \beta_{3} + 496090368) q^{96} + ( - 45681470 \beta_{2} + 1336519703 \beta_1) q^{97} + (5764801 \beta_{2} - 69177612 \beta_1) q^{98} + ( - 15093468 \beta_{3} + 581733270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 1440 q^{4} + 5904 q^{6} - 44856 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 1440 q^{4} + 5904 q^{6} - 44856 q^{9} + 37132 q^{11} + 115248 q^{14} + 225536 q^{16} + 286552 q^{19} - 835548 q^{21} + 4583808 q^{24} + 639472 q^{26} + 23155108 q^{29} - 7907520 q^{31} - 8487888 q^{34} - 8932032 q^{36} + 22791492 q^{39} + 2117984 q^{41} + 20374752 q^{44} - 14312352 q^{46} - 23059204 q^{49} + 38818932 q^{51} - 170992944 q^{54} + 98652288 q^{56} + 232318416 q^{59} - 89377088 q^{61} - 158111744 q^{64} - 159789648 q^{66} + 1332641784 q^{69} - 588331648 q^{71} - 204836128 q^{74} + 79244736 q^{76} + 1385705708 q^{79} + 895546692 q^{81} - 201223008 q^{84} - 693978016 q^{86} + 1558087408 q^{89} - 245334180 q^{91} + 1875382064 q^{94} + 1984361472 q^{96} + 2326933080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display

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
14.8284i 239.735i 292.118 0 3554.89 2401.00i 11923.8i −37789.9 0
99.2 9.17157i 65.7351i 427.882 0 −602.894 2401.00i 8620.20i 15361.9 0
99.3 9.17157i 65.7351i 427.882 0 −602.894 2401.00i 8620.20i 15361.9 0
99.4 14.8284i 239.735i 292.118 0 3554.89 2401.00i 11923.8i −37789.9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 175.10.b.c 4
5.b even 2 1 inner 175.10.b.c 4
5.c odd 4 1 35.10.a.b 2
5.c odd 4 1 175.10.a.c 2
15.e even 4 1 315.10.a.b 2
35.f even 4 1 245.10.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.a.b 2 5.c odd 4 1
175.10.a.c 2 5.c odd 4 1
175.10.b.c 4 1.a even 1 1 trivial
175.10.b.c 4 5.b even 2 1 inner
245.10.a.c 2 35.f even 4 1
315.10.a.b 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 304T_{2}^{2} + 18496 \) acting on \(S_{10}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 304 T^{2} + 18496 \) Copy content Toggle raw display
$3$ \( T^{4} + 61794 T^{2} + 248346081 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 5764801)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 18566 T - 579804919)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 41\!\cdots\!29 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 10\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( (T^{2} - 143276 T - 2624597308)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{2} + \cdots + 32805350195857)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots - 34498247424800)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots - 857114015266016)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 48\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 12\!\cdots\!09 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 40\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{2} + \cdots - 88\!\cdots\!84)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots - 25\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 26\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 11\!\cdots\!81)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots + 75\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 31\!\cdots\!81 \) Copy content Toggle raw display
show more
show less