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Properties

Label 175.5.c.c
Level $175$
Weight $5$
Character orbit 175.c
Analytic conductor $18.090$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.174");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(18.0897435397\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 8 x^{14} + 1152 x^{13} + 39936 x^{12} + 85792 x^{11} + 896 x^{10} + \cdots + 1691830895616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{6} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_{3} q^{3} - \beta_{4} q^{4} + \beta_{5} q^{6} + (\beta_{12} + 3 \beta_{2} - 3 \beta_1) q^{7} + ( - \beta_{13} - 5 \beta_{2} - 2 \beta_1) q^{8} + (2 \beta_{8} - 2 \beta_{6} + 3 \beta_{4} + 47) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - \beta_{3} q^{3} - \beta_{4} q^{4} + \beta_{5} q^{6} + (\beta_{12} + 3 \beta_{2} - 3 \beta_1) q^{7} + ( - \beta_{13} - 5 \beta_{2} - 2 \beta_1) q^{8} + (2 \beta_{8} - 2 \beta_{6} + 3 \beta_{4} + 47) q^{9} + (2 \beta_{8} - 3 \beta_{6} - \beta_{4} + 10) q^{11} + (\beta_{10} + 7 \beta_{3}) q^{12} + (\beta_{14} + \beta_{12} + \cdots - 5 \beta_{3}) q^{13}+ \cdots + (87 \beta_{8} - 317 \beta_{6} + \cdots + 7559) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} + 760 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} + 760 q^{9} + 128 q^{11} + 880 q^{14} - 1304 q^{16} + 528 q^{21} - 1832 q^{29} - 14496 q^{36} + 11856 q^{39} - 3816 q^{44} - 9936 q^{46} + 1592 q^{49} - 26424 q^{51} + 2464 q^{56} + 17608 q^{64} + 41768 q^{71} - 62944 q^{74} - 19112 q^{79} + 23072 q^{81} + 9648 q^{84} + 22384 q^{86} + 19848 q^{91} + 120584 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 8 x^{14} + 1152 x^{13} + 39936 x^{12} + 85792 x^{11} + 896 x^{10} + \cdots + 1691830895616 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 17\!\cdots\!97 \nu^{15} + \cdots - 86\!\cdots\!16 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 51\!\cdots\!01 \nu^{15} + \cdots + 15\!\cdots\!48 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12\!\cdots\!95 \nu^{15} + \cdots - 29\!\cdots\!20 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 55\!\cdots\!89 \nu^{15} + \cdots - 44\!\cdots\!32 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!61 \nu^{15} + \cdots + 37\!\cdots\!16 ) / 14\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41\!\cdots\!97 \nu^{15} + \cdots + 27\!\cdots\!84 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!74 \nu^{15} + \cdots - 49\!\cdots\!04 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 77\!\cdots\!41 \nu^{15} + \cdots - 19\!\cdots\!88 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53\!\cdots\!74 \nu^{15} + \cdots + 26\!\cdots\!52 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23\!\cdots\!89 \nu^{15} + \cdots + 93\!\cdots\!16 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\!\cdots\!15 \nu^{15} + \cdots + 57\!\cdots\!12 ) / 84\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13\!\cdots\!25 \nu^{15} + \cdots - 11\!\cdots\!64 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 75\!\cdots\!53 \nu^{15} + \cdots + 28\!\cdots\!04 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 16\!\cdots\!25 \nu^{15} + \cdots - 60\!\cdots\!40 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 39\!\cdots\!57 \nu^{15} + \cdots - 11\!\cdots\!76 ) / 30\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{6} + 5\beta_{3} - 5\beta_{2} + 3 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{13} - \beta_{12} + \beta_{7} - 5\beta_{5} - 35\beta_{2} + 68\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{15} - 5 \beta_{14} + 70 \beta_{13} - 10 \beta_{12} + 108 \beta_{11} - 20 \beta_{10} + \cdots - 2195 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 176 \beta_{14} + 266 \beta_{12} - 306 \beta_{10} - 800 \beta_{8} + 266 \beta_{7} + 1760 \beta_{6} + \cdots - 55150 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 167 \beta_{15} - 1110 \beta_{14} - 11135 \beta_{13} + 7460 \beta_{12} - 8991 \beta_{11} + \cdots - 397302 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2250 \beta_{15} - 284174 \beta_{13} + 117304 \beta_{12} - 131700 \beta_{11} + 41250 \beta_{9} + \cdots - 21750 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 26687 \beta_{15} + 337695 \beta_{14} - 3009580 \beta_{13} + 660550 \beta_{12} - 1932048 \beta_{11} + \cdots + 112702047 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 8865728 \beta_{14} - 15800648 \beta_{12} + 20025768 \beta_{10} + 40179800 \beta_{8} - 15800648 \beta_{7} + \cdots + 2794023540 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5951826 \beta_{15} + 92078560 \beta_{14} + 785034550 \beta_{13} - 516780880 \beta_{12} + \cdots + 30123201212 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 135701100 \beta_{15} + 18005129596 \beta_{13} - 7865389816 \beta_{12} + 10004518200 \beta_{11} + \cdots + 1525089300 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1492144442 \beta_{15} - 24179359130 \beta_{14} + 202701190440 \beta_{13} - 44132976820 \beta_{12} + \cdots - 7798367077906 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 559385215488 \beta_{14} + 1022087899008 \beta_{12} - 1312608396528 \beta_{10} + \cdots - 179034152546200 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 383423510596 \beta_{15} - 6263395806760 \beta_{14} - 52172089449220 \beta_{13} + \cdots - 20\!\cdots\!44 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 8793196335480 \beta_{15} + \cdots - 101383568919240 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 98788812978924 \beta_{15} + \cdots + 51\!\cdots\!56 ) / 5 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
11.3354 − 11.3354i
−5.43779 + 5.43779i
1.98778 + 1.98778i
−6.88539 − 6.88539i
3.21950 − 3.21950i
−0.884861 + 0.884861i
5.18677 + 5.18677i
−6.52141 − 6.52141i
5.18677 − 5.18677i
−6.52141 + 6.52141i
3.21950 + 3.21950i
−0.884861 − 0.884861i
1.98778 − 1.98778i
−6.88539 + 6.88539i
11.3354 + 11.3354i
−5.43779 − 5.43779i
− 5.89760i −16.7732 −18.7817 0 98.9215i −14.0035 + 46.9564i 16.4054i 200.340 0
174.2 − 5.89760i 16.7732 −18.7817 0 − 98.9215i 14.0035 + 46.9564i 16.4054i 200.340 0
174.3 − 4.89760i −8.87317 −7.98650 0 43.4572i 38.6355 − 30.1380i − 39.2469i −2.26685 0
174.4 − 4.89760i 8.87317 −7.98650 0 − 43.4572i −38.6355 − 30.1380i − 39.2469i −2.26685 0
174.5 − 2.33464i −4.10436 10.5495 0 9.58220i 44.7554 + 19.9488i − 61.9834i −64.1542 0
174.6 − 2.33464i 4.10436 10.5495 0 − 9.58220i −44.7554 + 19.9488i − 61.9834i −64.1542 0
174.7 − 1.33464i −11.7082 14.2187 0 15.6262i −36.1822 + 33.0431i − 40.3311i 56.0814 0
174.8 − 1.33464i 11.7082 14.2187 0 − 15.6262i 36.1822 + 33.0431i − 40.3311i 56.0814 0
174.9 1.33464i −11.7082 14.2187 0 − 15.6262i −36.1822 − 33.0431i 40.3311i 56.0814 0
174.10 1.33464i 11.7082 14.2187 0 15.6262i 36.1822 − 33.0431i 40.3311i 56.0814 0
174.11 2.33464i −4.10436 10.5495 0 − 9.58220i 44.7554 − 19.9488i 61.9834i −64.1542 0
174.12 2.33464i 4.10436 10.5495 0 9.58220i −44.7554 − 19.9488i 61.9834i −64.1542 0
174.13 4.89760i −8.87317 −7.98650 0 − 43.4572i 38.6355 + 30.1380i 39.2469i −2.26685 0
174.14 4.89760i 8.87317 −7.98650 0 43.4572i −38.6355 + 30.1380i 39.2469i −2.26685 0
174.15 5.89760i −16.7732 −18.7817 0 − 98.9215i −14.0035 − 46.9564i − 16.4054i 200.340 0
174.16 5.89760i 16.7732 −18.7817 0 98.9215i 14.0035 − 46.9564i − 16.4054i 200.340 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 174.16
Significant digits:
Format:

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 175.5.c.c 16
5.b even 2 1 inner 175.5.c.c 16
5.c odd 4 1 175.5.d.f ✓ 8
5.c odd 4 1 175.5.d.h yes 8
7.b odd 2 1 inner 175.5.c.c 16
35.c odd 2 1 inner 175.5.c.c 16
35.f even 4 1 175.5.d.f ✓ 8
35.f even 4 1 175.5.d.h yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.5.c.c 16 1.a even 1 1 trivial
175.5.c.c 16 5.b even 2 1 inner
175.5.c.c 16 7.b odd 2 1 inner
175.5.c.c 16 35.c odd 2 1 inner
175.5.d.f ✓ 8 5.c odd 4 1
175.5.d.f ✓ 8 35.f even 4 1
175.5.d.h yes 8 5.c odd 4 1
175.5.d.h yes 8 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 66T_{2}^{6} + 1269T_{2}^{4} + 6604T_{2}^{2} + 8100 \) acting on \(S_{5}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 66 T^{6} + \cdots + 8100)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - 514 T^{6} + \cdots + 51151500)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} - 32 T^{3} + \cdots + 37732329)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 458 T^{3} + \cdots - 64265126274)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 95\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 84\!\cdots\!25)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 394750647417186)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 246405347853736)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
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