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Properties

Label 170.3.e.b
Level $170$
Weight $3$
Character orbit 170.e
Analytic conductor $4.632$
Analytic rank $0$
Dimension $16$
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] = [170,3,Mod(13,170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(170, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("170.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 170.e (of order \(4\), degree \(2\), 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: \(4.63216449413\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 80 x^{14} + 2532 x^{12} + 40532 x^{10} + 346464 x^{8} + 1518752 x^{6} + 2895224 x^{4} + \cdots + 148996 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + 1) q^{2} - \beta_1 q^{3} - 2 \beta_{5} q^{4} - \beta_{12} q^{5} + ( - \beta_{3} - \beta_1) q^{6} + (\beta_{13} - \beta_{5}) q^{7} + ( - 2 \beta_{5} - 2) q^{8} + ( - \beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + 1) q^{2} - \beta_1 q^{3} - 2 \beta_{5} q^{4} - \beta_{12} q^{5} + ( - \beta_{3} - \beta_1) q^{6} + (\beta_{13} - \beta_{5}) q^{7} + ( - 2 \beta_{5} - 2) q^{8} + ( - \beta_{3} + \beta_{2} - 1) q^{9} + ( - \beta_{12} - \beta_{11}) q^{10} + (\beta_{15} - \beta_{12} - \beta_{8} + \cdots + 1) q^{11}+ \cdots + (5 \beta_{15} + 2 \beta_{14} + \cdots - 23) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{2} - 2 q^{5} - 32 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{2} - 2 q^{5} - 32 q^{8} - 16 q^{9} - 4 q^{10} + 20 q^{11} + 4 q^{13} - 12 q^{14} + 12 q^{15} - 64 q^{16} + 36 q^{17} - 16 q^{18} - 16 q^{19} - 4 q^{20} + 40 q^{22} + 16 q^{23} + 44 q^{25} - 24 q^{28} - 20 q^{29} + 12 q^{30} + 92 q^{31} - 64 q^{32} - 60 q^{33} + 24 q^{34} - 124 q^{35} + 32 q^{37} - 16 q^{38} - 140 q^{39} - 60 q^{41} + 52 q^{43} + 40 q^{44} + 198 q^{45} + 16 q^{46} + 112 q^{47} + 136 q^{49} - 4 q^{50} - 140 q^{51} - 8 q^{52} + 48 q^{53} + 108 q^{54} + 40 q^{55} - 24 q^{56} - 40 q^{58} + 76 q^{61} - 40 q^{65} + 116 q^{67} - 24 q^{68} - 124 q^{70} - 268 q^{71} + 32 q^{72} + 32 q^{74} + 136 q^{75} - 116 q^{77} - 280 q^{78} - 88 q^{79} + 8 q^{80} - 352 q^{81} - 120 q^{82} - 160 q^{83} + 310 q^{85} + 104 q^{86} + 236 q^{87} + 260 q^{90} - 168 q^{91} + 48 q^{93} + 224 q^{94} + 264 q^{95} - 256 q^{97} + 136 q^{98} - 348 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 80 x^{14} + 2532 x^{12} + 40532 x^{10} + 346464 x^{8} + 1518752 x^{6} + 2895224 x^{4} + \cdots + 148996 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 52488 \nu^{14} - 3721365 \nu^{12} - 99736812 \nu^{10} - 1262016176 \nu^{8} + \cdots + 16798844292 ) / 1824877532 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 52488 \nu^{14} - 3721365 \nu^{12} - 99736812 \nu^{10} - 1262016176 \nu^{8} + \cdots - 1449931028 ) / 1824877532 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38771169 \nu^{14} - 3120350251 \nu^{12} - 98395469092 \nu^{10} - 1538231293646 \nu^{8} + \cdots - 16735205365204 ) / 501841321300 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1878149 \nu^{15} - 140121736 \nu^{13} - 4037249823 \nu^{11} - 56875930552 \nu^{9} + \cdots - 1122503633000 \nu ) / 352201363676 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2602409126 \nu^{15} - 4697035017 \nu^{14} - 197010242144 \nu^{13} + \cdots - 30\!\cdots\!72 ) / 193710750021800 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8651306 \nu^{15} + 682993915 \nu^{13} + 21123293514 \nu^{11} + 325190183552 \nu^{9} + \cdots + 10592998277920 \nu ) / 352201363676 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8001013426 \nu^{15} - 20198230299 \nu^{14} - 646446829364 \nu^{13} + \cdots - 88\!\cdots\!04 ) / 193710750021800 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9480497978 \nu^{15} - 10047017019 \nu^{14} + 763694289102 \nu^{13} + \cdots - 198748215235204 ) / 193710750021800 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9775375811 \nu^{15} + 14531097959 \nu^{14} + 798726604504 \nu^{13} + 1187307739396 \nu^{12} + \cdots + 11\!\cdots\!84 ) / 193710750021800 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10148364291 \nu^{15} - 3324084741 \nu^{14} - 791615026164 \nu^{13} + \cdots - 39\!\cdots\!96 ) / 193710750021800 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13034502089 \nu^{15} - 4536951939 \nu^{14} + 1046737198846 \nu^{13} + \cdots - 19\!\cdots\!44 ) / 193710750021800 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 9996988953 \nu^{15} - 802227766502 \nu^{13} - 25449566649579 \nu^{11} + \cdots - 64\!\cdots\!88 \nu ) / 96855375010900 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 31183879806 \nu^{15} + 12337193619 \nu^{14} - 2484799054374 \nu^{13} + \cdots + 28\!\cdots\!64 ) / 193710750021800 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 32958242191 \nu^{15} + 6670061279 \nu^{14} - 2637078829514 \nu^{13} + \cdots + 51\!\cdots\!44 ) / 193710750021800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} + 2\beta_{11} + \beta_{9} - \beta_{8} - 3\beta_{6} - 6\beta_{5} + 2\beta_{4} - 15\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7 \beta_{14} - 3 \beta_{13} + 11 \beta_{12} - 3 \beta_{11} - 2 \beta_{9} + \beta_{8} - 4 \beta_{6} + \cdots + 165 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 21 \beta_{14} + 14 \beta_{13} - 8 \beta_{12} - 66 \beta_{11} - 29 \beta_{9} + 37 \beta_{8} + \cdots - 29 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15 \beta_{15} - 256 \beta_{14} + 116 \beta_{13} - 379 \beta_{12} + 103 \beta_{11} + 15 \beta_{10} + \cdots - 3152 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10 \beta_{15} + 382 \beta_{14} - 77 \beta_{13} + 360 \beta_{12} + 1864 \beta_{11} - 10 \beta_{10} + \cdots + 677 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 600 \beta_{15} + 7284 \beta_{14} - 3566 \beta_{13} + 10502 \beta_{12} - 2866 \beta_{11} + \cdots + 65710 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 670 \beta_{15} - 6958 \beta_{14} - 2196 \beta_{13} - 11446 \beta_{12} - 49594 \beta_{11} + \cdots - 14994 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17190 \beta_{15} - 190732 \beta_{14} + 100906 \beta_{13} - 272450 \beta_{12} + 74634 \beta_{11} + \cdots - 1449982 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 27540 \beta_{15} + 132118 \beta_{14} + 109460 \beta_{13} + 320932 \beta_{12} + 1282112 \beta_{11} + \cdots + 328960 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 438420 \beta_{15} + 4827050 \beta_{14} - 2733802 \beta_{13} + 6886726 \beta_{12} - 1890534 \beta_{11} + \cdots + 33231586 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 920600 \beta_{15} - 2641318 \beta_{14} - 3414176 \beta_{13} - 8504256 \beta_{12} - 32636604 \beta_{11} + \cdots - 7266234 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 10680410 \beta_{15} - 120390564 \beta_{14} + 72109080 \beta_{13} - 172130754 \beta_{12} + \cdots - 781212184 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 27604660 \beta_{15} + 55457500 \beta_{14} + 93300442 \beta_{13} + 219098308 \beta_{12} + \cdots + 162700878 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/170\mathbb{Z}\right)^\times\).

\(n\) \(71\) \(137\)
\(\chi(n)\) \(-\beta_{5}\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
4.26845i
3.50651i
2.23421i
0.574749i
0.458184i
− 2.41271i
− 3.64449i
− 4.98490i
4.98490i
3.64449i
2.41271i
− 0.458184i
− 0.574749i
− 2.23421i
− 3.50651i
− 4.26845i
1.00000 + 1.00000i − 4.26845i 2.00000i −4.54937 − 2.07442i 4.26845 − 4.26845i − 3.42439i −2.00000 + 2.00000i −9.21970 −2.47495 − 6.62379i
13.2 1.00000 + 1.00000i − 3.50651i 2.00000i 1.39979 + 4.80006i 3.50651 − 3.50651i 11.2754i −2.00000 + 2.00000i −3.29560 −3.40027 + 6.19985i
13.3 1.00000 + 1.00000i − 2.23421i 2.00000i 4.93002 + 0.833579i 2.23421 − 2.23421i − 6.45533i −2.00000 + 2.00000i 4.00831 4.09645 + 5.76360i
13.4 1.00000 + 1.00000i − 0.574749i 2.00000i −4.36413 + 2.44015i 0.574749 − 0.574749i 5.24832i −2.00000 + 2.00000i 8.66966 −6.80428 − 1.92398i
13.5 1.00000 + 1.00000i − 0.458184i 2.00000i 0.150408 − 4.99774i 0.458184 − 0.458184i − 6.37601i −2.00000 + 2.00000i 8.79007 5.14815 − 4.84733i
13.6 1.00000 + 1.00000i 2.41271i 2.00000i 3.70308 − 3.35964i −2.41271 + 2.41271i 8.30225i −2.00000 + 2.00000i 3.17881 7.06272 + 0.343440i
13.7 1.00000 + 1.00000i 3.64449i 2.00000i 2.65202 + 4.23872i −3.64449 + 3.64449i − 2.51762i −2.00000 + 2.00000i −4.28231 −1.58670 + 6.89075i
13.8 1.00000 + 1.00000i 4.98490i 2.00000i −4.92182 − 0.880714i −4.98490 + 4.98490i − 0.0526451i −2.00000 + 2.00000i −15.8492 −4.04111 − 5.80254i
157.1 1.00000 − 1.00000i − 4.98490i − 2.00000i −4.92182 + 0.880714i −4.98490 − 4.98490i 0.0526451i −2.00000 − 2.00000i −15.8492 −4.04111 + 5.80254i
157.2 1.00000 − 1.00000i − 3.64449i − 2.00000i 2.65202 − 4.23872i −3.64449 − 3.64449i 2.51762i −2.00000 − 2.00000i −4.28231 −1.58670 − 6.89075i
157.3 1.00000 − 1.00000i − 2.41271i − 2.00000i 3.70308 + 3.35964i −2.41271 − 2.41271i − 8.30225i −2.00000 − 2.00000i 3.17881 7.06272 − 0.343440i
157.4 1.00000 − 1.00000i 0.458184i − 2.00000i 0.150408 + 4.99774i 0.458184 + 0.458184i 6.37601i −2.00000 − 2.00000i 8.79007 5.14815 + 4.84733i
157.5 1.00000 − 1.00000i 0.574749i − 2.00000i −4.36413 − 2.44015i 0.574749 + 0.574749i − 5.24832i −2.00000 − 2.00000i 8.66966 −6.80428 + 1.92398i
157.6 1.00000 − 1.00000i 2.23421i − 2.00000i 4.93002 − 0.833579i 2.23421 + 2.23421i 6.45533i −2.00000 − 2.00000i 4.00831 4.09645 − 5.76360i
157.7 1.00000 − 1.00000i 3.50651i − 2.00000i 1.39979 − 4.80006i 3.50651 + 3.50651i − 11.2754i −2.00000 − 2.00000i −3.29560 −3.40027 − 6.19985i
157.8 1.00000 − 1.00000i 4.26845i − 2.00000i −4.54937 + 2.07442i 4.26845 + 4.26845i 3.42439i −2.00000 − 2.00000i −9.21970 −2.47495 + 6.62379i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.3.e.b ✓ 16
5.c odd 4 1 170.3.j.b yes 16
17.c even 4 1 170.3.j.b yes 16
85.f odd 4 1 inner 170.3.e.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.3.e.b ✓ 16 1.a even 1 1 trivial
170.3.e.b ✓ 16 85.f odd 4 1 inner
170.3.j.b yes 16 5.c odd 4 1
170.3.j.b yes 16 17.c even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 80 T_{3}^{14} + 2532 T_{3}^{12} + 40532 T_{3}^{10} + 346464 T_{3}^{8} + 1518752 T_{3}^{6} + \cdots + 148996 \) acting on \(S_{3}^{\mathrm{new}}(170, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 80 T^{14} + \cdots + 148996 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{16} + 324 T^{14} + \cdots + 84235684 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 33057785124 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 179720728752016 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 48\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( (T^{8} + 8 T^{7} + \cdots - 3702760816)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 8 T^{7} + \cdots - 35274650)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{8} - 16 T^{7} + \cdots + 573578476400)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 49\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 27492639236080)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 63\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 31\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 146667506161200)^{2} \) Copy content Toggle raw display
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