Properties

Label 175.8.b.c
Level $175$
Weight $8$
Character orbit 175.b
Analytic conductor $54.667$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.99");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(54.6673794597\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 8 \beta_1) q^{2} + (6 \beta_{3} + 15 \beta_1) q^{3} + ( - 16 \beta_{2} + 20) q^{4} + ( - 63 \beta_{2} - 384) q^{6} - 343 \beta_1 q^{7} + (20 \beta_{3} + 480 \beta_1) q^{8} + ( - 180 \beta_{2} + 378) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 8 \beta_1) q^{2} + (6 \beta_{3} + 15 \beta_1) q^{3} + ( - 16 \beta_{2} + 20) q^{4} + ( - 63 \beta_{2} - 384) q^{6} - 343 \beta_1 q^{7} + (20 \beta_{3} + 480 \beta_1) q^{8} + ( - 180 \beta_{2} + 378) q^{9} + ( - 380 \beta_{2} - 3953) q^{11} + ( - 120 \beta_{3} - 3924 \beta_1) q^{12} + (418 \beta_{3} + 8909 \beta_1) q^{13} + (343 \beta_{2} + 2744) q^{14} + ( - 2688 \beta_{2} - 2160) q^{16} + (2210 \beta_{3} - 1199 \beta_1) q^{17} + ( - 1062 \beta_{3} - 4896 \beta_1) q^{18} + ( - 5542 \beta_{2} + 1806) q^{19} + (2058 \beta_{2} + 5145) q^{21} + ( - 6993 \beta_{3} - 48344 \beta_1) q^{22} + ( - 10690 \beta_{3} - 6922 \beta_1) q^{23} + ( - 3180 \beta_{2} - 12480) q^{24} + ( - 12253 \beta_{2} - 89664) q^{26} + (12690 \beta_{3} - 9045 \beta_1) q^{27} + (5488 \beta_{3} - 6860 \beta_1) q^{28} + (23772 \beta_{2} + 63449) q^{29} + ( - 22554 \beta_{2} + 126384) q^{31} + ( - 21104 \beta_{3} - 74112 \beta_1) q^{32} + ( - 29418 \beta_{3} - 159615 \beta_1) q^{33} + ( - 16481 \beta_{2} - 87648) q^{34} + ( - 9648 \beta_{2} + 134280) q^{36} + ( - 43638 \beta_{3} - 132930 \beta_1) q^{37} + ( - 42530 \beta_{3} - 229400 \beta_1) q^{38} + ( - 59724 \beta_{2} - 243987) q^{39} + (37354 \beta_{2} - 55960) q^{41} + (21609 \beta_{3} + 131712 \beta_1) q^{42} + (24578 \beta_{3} - 473786 \beta_1) q^{43} + (55648 \beta_{2} + 188460) q^{44} + (92442 \beta_{2} + 525736) q^{46} + ( - 56742 \beta_{3} + 135637 \beta_1) q^{47} + ( - 53280 \beta_{3} - 742032 \beta_1) q^{48} - 117649 q^{49} + ( - 25956 \beta_{2} - 565455) q^{51} + ( - 134184 \beta_{3} - 116092 \beta_1) q^{52} + (65224 \beta_{3} + 633896 \beta_1) q^{53} + ( - 92475 \beta_{2} - 486000) q^{54} + (6860 \beta_{2} + 164640) q^{56} + ( - 72294 \beta_{3} - 1435998 \beta_1) q^{57} + (253625 \beta_{3} + 1553560 \beta_1) q^{58} + ( - 170640 \beta_{2} + 680060) q^{59} + (11334 \beta_{2} - 906840) q^{61} + ( - 54048 \beta_{3} + 18696 \beta_1) q^{62} + (61740 \beta_{3} - 129654 \beta_1) q^{63} + ( - 101120 \beta_{2} + 1244992) q^{64} + (394959 \beta_{2} + 2571312) q^{66} + (506344 \beta_{3} - 1094656 \beta_1) q^{67} + (63384 \beta_{3} - 1579820 \beta_1) q^{68} + (201882 \beta_{2} + 2925990) q^{69} + ( - 222048 \beta_{2} - 747464) q^{71} + ( - 78840 \beta_{3} + 23040 \beta_1) q^{72} + ( - 212396 \beta_{3} - 3584894 \beta_1) q^{73} + (482034 \beta_{2} + 2983512) q^{74} + ( - 139736 \beta_{2} + 3937688) q^{76} + (130340 \beta_{3} + 1355879 \beta_1) q^{77} + ( - 721779 \beta_{3} - 4579752 \beta_1) q^{78} + ( - 187504 \beta_{2} + 3971487) q^{79} + ( - 529740 \beta_{2} - 2387799) q^{81} + (242872 \beta_{3} + 1195896 \beta_1) q^{82} + (939444 \beta_{3} + 152356 \beta_1) q^{83} + ( - 41160 \beta_{2} - 1345932) q^{84} + (277162 \beta_{2} + 2708856) q^{86} + (737274 \beta_{3} + 7227543 \beta_1) q^{87} + ( - 261460 \beta_{3} - 2231840 \beta_1) q^{88} + ( - 247538 \beta_{2} + 8971764) q^{89} + (143374 \beta_{2} + 3055787) q^{91} + ( - 103048 \beta_{3} + 7387320 \beta_1) q^{92} + (419994 \beta_{3} - 4058496 \beta_1) q^{93} + (318299 \beta_{2} + 1411552) q^{94} + (761232 \beta_{2} + 6683136) q^{96} + ( - 680782 \beta_{3} + 2129037 \beta_1) q^{97} + ( - 117649 \beta_{3} - 941192 \beta_1) q^{98} + (567900 \beta_{2} + 1515366) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 80 q^{4} - 1536 q^{6} + 1512 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 80 q^{4} - 1536 q^{6} + 1512 q^{9} - 15812 q^{11} + 10976 q^{14} - 8640 q^{16} + 7224 q^{19} + 20580 q^{21} - 49920 q^{24} - 358656 q^{26} + 253796 q^{29} + 505536 q^{31} - 350592 q^{34} + 537120 q^{36} - 975948 q^{39} - 223840 q^{41} + 753840 q^{44} + 2102944 q^{46} - 470596 q^{49} - 2261820 q^{51} - 1944000 q^{54} + 658560 q^{56} + 2720240 q^{59} - 3627360 q^{61} + 4979968 q^{64} + 10285248 q^{66} + 11703960 q^{69} - 2989856 q^{71} + 11934048 q^{74} + 15750752 q^{76} + 15885948 q^{79} - 9551196 q^{81} - 5383728 q^{84} + 10835424 q^{86} + 35887056 q^{89} + 12223148 q^{91} + 5646208 q^{94} + 26732544 q^{96} + 6061464 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 16\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} - 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 8\beta_1 ) / 2 \) 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
1.65831 0.500000i
−1.65831 0.500000i
−1.65831 + 0.500000i
1.65831 + 0.500000i
14.6332i 54.7995i −86.1320 0 −801.895 343.000i 612.665i −815.985 0
99.2 1.36675i 24.7995i 126.132 0 33.8947 343.000i 347.335i 1571.98 0
99.3 1.36675i 24.7995i 126.132 0 33.8947 343.000i 347.335i 1571.98 0
99.4 14.6332i 54.7995i −86.1320 0 −801.895 343.000i 612.665i −815.985 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 175.8.b.c 4
5.b even 2 1 inner 175.8.b.c 4
5.c odd 4 1 35.8.a.a 2
5.c odd 4 1 175.8.a.b 2
15.e even 4 1 315.8.a.c 2
20.e even 4 1 560.8.a.i 2
35.f even 4 1 245.8.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.8.a.a 2 5.c odd 4 1
175.8.a.b 2 5.c odd 4 1
175.8.b.c 4 1.a even 1 1 trivial
175.8.b.c 4 5.b even 2 1 inner
245.8.a.b 2 35.f even 4 1
315.8.a.c 2 15.e even 4 1
560.8.a.i 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 216T_{2}^{2} + 400 \) acting on \(S_{8}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 216T^{2} + 400 \) Copy content Toggle raw display
$3$ \( T^{4} + 3618 T^{2} + 1846881 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 117649)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 7906 T + 9272609)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 51\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 45\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( (T^{2} - 3612 T - 1348143980)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{2} - 126898 T - 20838975695)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 252768 T - 6409132848)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{2} + 111920 T - 58262616304)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 15\!\cdots\!09 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{2} - 1360120 T - 818710818800)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 1813680 T + 816706565136)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots - 1610731398080)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 14225767990465)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots + 77796446568160)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 25\!\cdots\!69 \) Copy content Toggle raw display
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