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Properties

Label 361.3.f.i
Level $361$
Weight $3$
Character orbit 361.f
Analytic conductor $9.837$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $6$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [361,3,Mod(116,361)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(361, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("361.116");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 361.f (of order \(18\), degree \(6\), 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: \(9.83653754341\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 19x^{15} + 722x^{12} + 6857x^{9} + 130340x^{6} - 361x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{17} + \beta_{5}) q^{2} + ( - \beta_{17} - \beta_{16} + \cdots + \beta_{4}) q^{3}+ \cdots + ( - \beta_{14} + 2 \beta_{13} + \cdots + 6 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{17} + \beta_{5}) q^{2} + ( - \beta_{17} - \beta_{16} + \cdots + \beta_{4}) q^{3}+ \cdots + (5 \beta_{14} + 5 \beta_{13} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 63 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 63 q^{8} - 39 q^{11} - 225 q^{12} + 324 q^{20} + 222 q^{26} + 81 q^{27} - 456 q^{30} - 594 q^{31} - 132 q^{39} + 42 q^{45} - 306 q^{46} - 27 q^{49} + 243 q^{50} + 516 q^{58} - 249 q^{64} + 684 q^{65} + 126 q^{68} - 18 q^{69} - 1128 q^{77} - 219 q^{83} - 342 q^{84} + 852 q^{87} + 621 q^{88} + 378 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 19x^{15} + 722x^{12} + 6857x^{9} + 130340x^{6} - 361x^{3} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -88\nu^{15} + 3344\nu^{12} - 113353\nu^{9} + 603680\nu^{6} - 1672\nu^{3} + 14386172 ) / 102439773 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 311\nu^{16} - 11818\nu^{13} + 339332\nu^{10} - 2133460\nu^{7} + 5909\nu^{4} - 729444268\nu ) / 102439773 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 311\nu^{17} - 11818\nu^{14} + 339332\nu^{11} - 2133460\nu^{8} + 5909\nu^{5} - 729444268\nu^{2} ) / 102439773 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2089\nu^{17} + 79382\nu^{14} - 2261971\nu^{11} + 14330540\nu^{8} - 39691\nu^{5} + 5194163477\nu^{2} ) / 102439773 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -39691\nu^{15} + 753713\nu^{12} - 28654813\nu^{9} - 272240569\nu^{6} - 5193409348\nu^{3} - 2089 ) / 102439773 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -722\nu^{15} + 13717\nu^{12} - 521246\nu^{9} - 4952198\nu^{6} - 94098620\nu^{3} + 260623 ) / 260661 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -722\nu^{16} + 13717\nu^{13} - 521246\nu^{10} - 4952198\nu^{7} - 94098620\nu^{4} - 38\nu ) / 260661 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 376284 \nu^{15} + 7146620 \nu^{12} - 271635582 \nu^{9} - 2581951735 \nu^{6} + \cdots - 135932734 ) / 102439773 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 376507 \nu^{15} - 7155094 \nu^{12} + 271861561 \nu^{9} + 2580421955 \nu^{6} + 49064569039 \nu^{3} - 374245816 ) / 102439773 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2020004 \nu^{17} - 38374390 \nu^{14} + 1458336572 \nu^{11} + 13855207436 \nu^{8} + \cdots + 106316 \nu^{2} ) / 102439773 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2020315 \nu^{16} + 38386208 \nu^{13} - 1458675904 \nu^{10} - 13853073976 \nu^{7} + \cdots + 729337952 \nu ) / 102439773 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2679723 \nu^{16} + 50899973 \nu^{13} - 1934482500 \nu^{10} - 18385506445 \nu^{7} + \cdots - 967938025 \nu ) / 102439773 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2681190 \nu^{16} - 50955719 \nu^{13} + 1936065807 \nu^{10} + 18375442825 \nu^{7} + \cdots - 2664897143 \nu ) / 102439773 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14386172 \nu^{17} + 273337180 \nu^{14} - 10386812840 \nu^{11} - 98646094757 \nu^{8} + \cdots + 5193406420 \nu^{2} ) / 102439773 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 19081410 \nu^{17} + 362440961 \nu^{14} - 13774768638 \nu^{11} - 130917603178 \nu^{8} + \cdots - 6892409143 \nu^{2} ) / 102439773 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 19091855 \nu^{17} + 362837871 \nu^{14} - 13786078493 \nu^{11} - 130845950478 \nu^{8} + \cdots + 18975968469 \nu^{2} ) / 102439773 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} + \beta_{16} + 5\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + 2\beta_{9} - 3\beta_{7} - 7\beta_{6} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{14} + 16\beta_{13} - 31\beta_{12} - 6\beta_{8} - 23\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{17} + 28\beta_{16} - 48\beta_{15} + 55\beta_{11} + 34\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -138\beta_{10} + 69\beta_{9} - 262\beta_{7} - 90\beta_{6} - 90\beta_{2} - 138 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -318\beta_{14} + 159\beta_{13} - 567\beta_{12} - 469\beta_{8} + 318\beta_{3} - 469\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1256\beta_{17} + 628\beta_{16} - 2353\beta_{15} + 1044\beta_{11} - 1256\beta_{5} + 1044\beta_{4} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -1672\beta_{10} - 1672\beta_{9} - 4237\beta_{2} - 7732 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -5909\beta_{14} - 5909\beta_{13} + 27873\beta_{3} - 11076\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16985\beta_{17} - 16985\beta_{16} - 79016\beta_{5} + 39691\beta_{4} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 56676\beta_{10} - 113352\beta_{9} + 209719\beta_{7} + 112986\beta_{6} - 153043 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 169662\beta_{14} - 339324\beta_{13} + 621972\beta_{12} + 379747\beta_{8} + 452310\beta_{3} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -549409\beta_{17} - 1098818\beta_{16} + 2027974\beta_{15} - 1130958\beta_{11} - 1478565\beta_{5} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3360734\beta_{10} - 1680367\beta_{9} + 6172059\beta_{7} + 3676201\beta_{6} + 3676201\beta_{2} + 3360734 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 10713136 \beta_{14} - 5356568 \beta_{13} + 19745905 \beta_{12} + 11213160 \beta_{8} + \cdots + 11213160 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 33139456 \beta_{17} - 16569728 \beta_{16} + 60922344 \beta_{15} - 35815609 \beta_{11} + \cdots - 35815609 \beta_{4} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
0.131966 + 0.0480317i
−2.13599 − 0.777437i
2.94372 + 1.07143i
−0.543977 + 3.08505i
0.394715 − 2.23854i
−0.0243863 + 0.138302i
−2.39974 − 2.01362i
1.74127 + 1.46110i
−0.107580 − 0.0902701i
−2.39974 + 2.01362i
1.74127 − 1.46110i
−0.107580 + 0.0902701i
−0.543977 − 3.08505i
0.394715 + 2.23854i
−0.0243863 − 0.138302i
0.131966 − 0.0480317i
−2.13599 + 0.777437i
2.94372 − 1.07143i
−2.26310 − 0.399046i −2.74683 + 3.27354i 1.20362 + 0.438084i −5.48766 + 1.99734i 7.52265 − 6.31225i −4.69093 − 8.12493i 5.41145 + 3.12430i −1.60818 − 9.12045i 13.2162 − 2.33037i
116.2 0.663452 + 0.116985i 1.85040 − 2.20522i −3.33229 − 1.21285i −2.91889 + 1.06239i 1.48563 − 1.24659i 4.07647 + 7.06065i −4.40265 − 2.54187i 0.123812 + 0.702174i −2.06082 + 0.363379i
116.3 3.30539 + 0.582829i −2.44360 + 2.91216i 6.82713 + 2.48487i 6.52716 − 2.37569i −9.77432 + 8.20163i 0.614461 + 1.06428i 9.49120 + 5.47975i −0.946705 − 5.36903i 22.9594 − 4.04836i
127.1 −2.15744 − 2.57113i −1.30021 + 3.57230i −1.26160 + 7.15490i −1.20617 − 6.84053i 11.9900 − 4.36400i 0.614461 − 1.06428i 9.49120 − 5.47975i −4.17637 − 3.50439i −14.9857 + 17.8592i
127.2 −0.433038 − 0.516074i 0.984578 − 2.70511i 0.615782 − 3.49227i 0.539388 + 3.05902i −1.82240 + 0.663298i 4.07647 − 7.06065i −4.40265 + 2.54187i 0.546194 + 0.458311i 1.34511 − 1.60304i
127.3 1.47714 + 1.76038i −1.46156 + 4.01559i −0.222421 + 1.26141i 1.01408 + 5.75112i −9.22789 + 3.35868i −4.69093 + 8.12493i 5.41145 − 3.12430i −7.09445 − 5.95295i −8.62624 + 10.2803i
262.1 −1.14795 + 3.15396i 3.74381 − 0.660134i −5.56552 − 4.67003i −5.32099 + 4.46484i −2.21566 + 12.5656i 0.614461 − 1.06428i 9.49120 − 5.47975i 5.12307 − 1.86465i −7.97372 − 21.9076i
262.2 −0.230415 + 0.633059i −2.83498 + 0.499883i 2.71651 + 2.27942i 2.37950 − 1.99663i 0.336765 − 1.90989i 4.07647 − 7.06065i −4.40265 + 2.54187i −0.670006 + 0.243862i 0.715717 + 1.96642i
262.3 0.785967 − 2.15943i 4.20839 − 0.742052i −0.981204 − 0.823328i 4.47358 − 3.75378i 1.70525 − 9.67093i −4.69093 + 8.12493i 5.41145 − 3.12430i 8.70263 − 3.16750i −4.58993 − 12.6107i
299.1 −1.14795 − 3.15396i 3.74381 + 0.660134i −5.56552 + 4.67003i −5.32099 − 4.46484i −2.21566 − 12.5656i 0.614461 + 1.06428i 9.49120 + 5.47975i 5.12307 + 1.86465i −7.97372 + 21.9076i
299.2 −0.230415 − 0.633059i −2.83498 − 0.499883i 2.71651 − 2.27942i 2.37950 + 1.99663i 0.336765 + 1.90989i 4.07647 + 7.06065i −4.40265 − 2.54187i −0.670006 − 0.243862i 0.715717 − 1.96642i
299.3 0.785967 + 2.15943i 4.20839 + 0.742052i −0.981204 + 0.823328i 4.47358 + 3.75378i 1.70525 + 9.67093i −4.69093 − 8.12493i 5.41145 + 3.12430i 8.70263 + 3.16750i −4.58993 + 12.6107i
307.1 −2.15744 + 2.57113i −1.30021 − 3.57230i −1.26160 − 7.15490i −1.20617 + 6.84053i 11.9900 + 4.36400i 0.614461 + 1.06428i 9.49120 + 5.47975i −4.17637 + 3.50439i −14.9857 − 17.8592i
307.2 −0.433038 + 0.516074i 0.984578 + 2.70511i 0.615782 + 3.49227i 0.539388 − 3.05902i −1.82240 − 0.663298i 4.07647 + 7.06065i −4.40265 − 2.54187i 0.546194 − 0.458311i 1.34511 + 1.60304i
307.3 1.47714 − 1.76038i −1.46156 − 4.01559i −0.222421 − 1.26141i 1.01408 − 5.75112i −9.22789 − 3.35868i −4.69093 − 8.12493i 5.41145 + 3.12430i −7.09445 + 5.95295i −8.62624 − 10.2803i
333.1 −2.26310 + 0.399046i −2.74683 − 3.27354i 1.20362 − 0.438084i −5.48766 − 1.99734i 7.52265 + 6.31225i −4.69093 + 8.12493i 5.41145 − 3.12430i −1.60818 + 9.12045i 13.2162 + 2.33037i
333.2 0.663452 − 0.116985i 1.85040 + 2.20522i −3.33229 + 1.21285i −2.91889 − 1.06239i 1.48563 + 1.24659i 4.07647 − 7.06065i −4.40265 + 2.54187i 0.123812 − 0.702174i −2.06082 − 0.363379i
333.3 3.30539 − 0.582829i −2.44360 − 2.91216i 6.82713 − 2.48487i 6.52716 + 2.37569i −9.77432 − 8.20163i 0.614461 − 1.06428i 9.49120 − 5.47975i −0.946705 + 5.36903i 22.9594 + 4.04836i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 2 inner
19.f odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.3.f.i 18
19.b odd 2 1 361.3.f.h 18
19.c even 3 2 inner 361.3.f.i 18
19.d odd 6 2 361.3.f.h 18
19.e even 9 1 19.3.d.a ✓ 6
19.e even 9 1 361.3.b.b 6
19.e even 9 1 361.3.d.c 6
19.e even 9 3 361.3.f.h 18
19.f odd 18 1 19.3.d.a ✓ 6
19.f odd 18 1 361.3.b.b 6
19.f odd 18 1 361.3.d.c 6
19.f odd 18 3 inner 361.3.f.i 18
57.j even 18 1 171.3.p.d 6
57.l odd 18 1 171.3.p.d 6
76.k even 18 1 304.3.r.b 6
76.l odd 18 1 304.3.r.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.d.a ✓ 6 19.e even 9 1
19.3.d.a ✓ 6 19.f odd 18 1
171.3.p.d 6 57.j even 18 1
171.3.p.d 6 57.l odd 18 1
304.3.r.b 6 76.k even 18 1
304.3.r.b 6 76.l odd 18 1
361.3.b.b 6 19.e even 9 1
361.3.b.b 6 19.f odd 18 1
361.3.d.c 6 19.e even 9 1
361.3.d.c 6 19.f odd 18 1
361.3.f.h 18 19.b odd 2 1
361.3.f.h 18 19.d odd 6 2
361.3.f.h 18 19.e even 9 3
361.3.f.i 18 1.a even 1 1 trivial
361.3.f.i 18 19.c even 3 2 inner
361.3.f.i 18 19.f odd 18 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 45T_{2}^{15} + 224T_{2}^{12} + 20295T_{2}^{9} + 199756T_{2}^{6} - 109593T_{2}^{3} + 19683 \) acting on \(S_{3}^{\mathrm{new}}(361, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 45 T^{15} + \cdots + 19683 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 10460353203 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 4001504141376 \) Copy content Toggle raw display
$7$ \( (T^{6} + 78 T^{4} + \cdots + 8836)^{3} \) Copy content Toggle raw display
$11$ \( (T^{6} + 13 T^{5} + 252 T^{4} + \cdots + 9)^{3} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 44\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{18} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 16\!\cdots\!92 \) Copy content Toggle raw display
$31$ \( (T^{6} + 198 T^{5} + \cdots + 2642351052)^{3} \) Copy content Toggle raw display
$37$ \( (T^{6} + 3024 T^{4} + \cdots + 38988)^{3} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 12\!\cdots\!07 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 94\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 84\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 57\!\cdots\!47 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 51\!\cdots\!27 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 19\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 39\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 86\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( (T^{6} + 73 T^{5} + \cdots + 158094507321)^{3} \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 61\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 95\!\cdots\!07 \) Copy content Toggle raw display
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