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Properties

Label 17.8.b.a
Level $17$
Weight $8$
Character orbit 17.b
Analytic conductor $5.311$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [17,8,Mod(16,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.16");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 17.b (of order \(2\), degree \(1\), 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: \(5.31054543323\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 16832x^{8} + 93191572x^{6} + 192821327856x^{4} + 116860780245888x^{2} + 9421474370420736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 2) q^{2} + \beta_1 q^{3} + (\beta_{4} - 2 \beta_{2} + 69) q^{4} - \beta_{5} q^{5} + (\beta_{5} - \beta_{3} - 4 \beta_1) q^{6} + ( - \beta_{7} - \beta_{5} + 4 \beta_1) q^{7} + (\beta_{8} - 4 \beta_{4} + 89 \beta_{2} - 349) q^{8} + (\beta_{6} + 38 \beta_{2} - 1187) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 2) q^{2} + \beta_1 q^{3} + (\beta_{4} - 2 \beta_{2} + 69) q^{4} - \beta_{5} q^{5} + (\beta_{5} - \beta_{3} - 4 \beta_1) q^{6} + ( - \beta_{7} - \beta_{5} + 4 \beta_1) q^{7} + (\beta_{8} - 4 \beta_{4} + 89 \beta_{2} - 349) q^{8} + (\beta_{6} + 38 \beta_{2} - 1187) q^{9} + (\beta_{9} + \beta_{7} + \cdots - 23 \beta_1) q^{10}+ \cdots + ( - 1080 \beta_{9} + \cdots - 71071 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 18 q^{2} + 690 q^{4} - 3330 q^{8} - 11794 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 18 q^{2} + 690 q^{4} - 3330 q^{8} - 11794 q^{9} - 316 q^{13} + 13824 q^{15} + 92226 q^{16} - 29454 q^{17} + 94106 q^{18} - 90184 q^{19} - 137648 q^{21} + 78370 q^{25} - 180420 q^{26} + 781392 q^{30} - 1019778 q^{32} + 418160 q^{33} + 1123478 q^{34} - 460704 q^{35} - 1026218 q^{36} - 1583880 q^{38} + 1939504 q^{42} - 112936 q^{43} + 3214512 q^{47} - 230842 q^{49} - 359514 q^{50} - 102928 q^{51} + 4611732 q^{52} - 4396356 q^{53} + 2187488 q^{55} - 6096552 q^{59} - 12859152 q^{60} + 4541698 q^{64} + 4831808 q^{66} - 10368488 q^{67} - 712902 q^{68} + 7389760 q^{69} + 11368032 q^{70} + 15426330 q^{72} - 14210712 q^{76} + 7163520 q^{77} + 20826346 q^{81} + 7060104 q^{83} - 85725280 q^{84} + 8300608 q^{85} + 30033048 q^{86} + 855264 q^{87} + 24760956 q^{89} - 39868160 q^{93} - 54883072 q^{94} + 81770994 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 16832x^{8} + 93191572x^{6} + 192821327856x^{4} + 116860780245888x^{2} + 9421474370420736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1676135 \nu^{8} + 13102253446 \nu^{6} - 34069440596992 \nu^{4} + \cdots - 11\!\cdots\!72 ) / 94\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1914131 \nu^{9} + 2906370896 \nu^{7} + 245089108547044 \nu^{5} + \cdots - 29\!\cdots\!88 \nu ) / 32\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2139880 \nu^{8} + 35581864841 \nu^{6} + 167408025377638 \nu^{4} + \cdots - 55\!\cdots\!92 ) / 79\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26828323 \nu^{9} + 334798678976 \nu^{7} + 897957185501500 \nu^{5} + \cdots - 43\!\cdots\!84 \nu ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31846565 \nu^{8} - 248942815474 \nu^{6} + 647319371342848 \nu^{4} + \cdots + 18\!\cdots\!08 ) / 47\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 62946217 \nu^{9} + 1058591379584 \nu^{7} + \cdots + 72\!\cdots\!44 \nu ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 335971693 \nu^{8} + 5189833909118 \nu^{6} + \cdots + 10\!\cdots\!32 ) / 31\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 504283189 \nu^{9} - 8236425527504 \nu^{7} + \cdots - 31\!\cdots\!56 \nu ) / 75\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 38\beta_{2} - 3374 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -18\beta_{9} - 51\beta_{7} - 354\beta_{5} - 114\beta_{3} - 5599\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 1035\beta_{8} - 7840\beta_{6} - 35181\beta_{4} - 381323\beta_{2} + 19477160 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 237348\beta_{9} + 731994\beta_{7} + 3570492\beta_{5} + 983364\beta_{3} + 36676090\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11577330\beta_{8} + 59935672\beta_{6} + 435504798\beta_{4} + 2567406530\beta_{2} - 128978207936 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2281160520\beta_{9} - 7149922404\beta_{7} - 31594041624\beta_{5} - 7233857160\beta_{3} - 258055749316\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 111536948340 \beta_{8} - 462191037904 \beta_{6} - 4119412356396 \beta_{4} - 15858073800020 \beta_{2} + 913144698198080 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 19673843216400 \beta_{9} + 62319508365768 \beta_{7} + 266558804455728 \beta_{5} + \cdots + 18\!\cdots\!80 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
− 71.2067i
71.2067i
− 9.73524i
9.73524i
− 88.6129i
88.6129i
− 28.9681i
28.9681i
− 54.5475i
54.5475i
−21.1595 − 71.2067i 319.725 − 244.729i 1506.70i − 1289.23i −4056.79 −2883.39 5178.33i
16.2 −21.1595 71.2067i 319.725 244.729i − 1506.70i 1289.23i −4056.79 −2883.39 − 5178.33i
16.3 −11.2698 − 9.73524i −0.992706 − 181.480i 109.714i 1340.88i 1453.72 2092.23 2045.24i
16.4 −11.2698 9.73524i −0.992706 181.480i − 109.714i − 1340.88i 1453.72 2092.23 − 2045.24i
16.5 −2.57355 − 88.6129i −121.377 321.949i 228.050i 569.789i 641.783 −5665.25 − 828.552i
16.6 −2.57355 88.6129i −121.377 − 321.949i − 228.050i − 569.789i 641.783 −5665.25 828.552i
16.7 7.27548 − 28.9681i −75.0673 − 364.342i − 210.757i − 192.987i −1477.41 1347.85 − 2650.76i
16.8 7.27548 28.9681i −75.0673 364.342i 210.757i 192.987i −1477.41 1347.85 2650.76i
16.9 18.7273 − 54.5475i 222.712 149.052i − 1021.53i − 641.202i 1773.71 −788.433 2791.35i
16.10 18.7273 54.5475i 222.712 − 149.052i 1021.53i 641.202i 1773.71 −788.433 − 2791.35i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.8.b.a ✓ 10
3.b odd 2 1 153.8.d.b 10
4.b odd 2 1 272.8.b.c 10
17.b even 2 1 inner 17.8.b.a ✓ 10
17.c even 4 2 289.8.a.f 10
51.c odd 2 1 153.8.d.b 10
68.d odd 2 1 272.8.b.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.8.b.a ✓ 10 1.a even 1 1 trivial
17.8.b.a ✓ 10 17.b even 2 1 inner
153.8.d.b 10 3.b odd 2 1
153.8.d.b 10 51.c odd 2 1
272.8.b.c 10 4.b odd 2 1
272.8.b.c 10 68.d odd 2 1
289.8.a.f 10 17.c even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(17, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} + 9 T^{4} + \cdots + 83616)^{2} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 94\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots - 23\!\cdots\!48)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 11\!\cdots\!93 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 69\!\cdots\!80)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 74\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 16\!\cdots\!08)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 13\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 60\!\cdots\!20)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 28\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 58\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 75\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 16\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
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