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Properties

Label 961.2.d.q
Level $961$
Weight $2$
Character orbit 961.d
Analytic conductor $7.674$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [961,2,Mod(374,961)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(961, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("961.374");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.d (of order \(5\), degree \(4\), 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: \(7.67362363425\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 31)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{14} + \beta_{12} + \beta_{11} + 1) q^{2} + (\beta_{13} - \beta_{11} - \beta_{8} + \cdots + 1) q^{3} + (\beta_{15} + 2 \beta_{11} + \cdots - \beta_{2}) q^{4} + (\beta_{15} - \beta_{14} + \cdots - 2 \beta_{3}) q^{5}+ \cdots + (\beta_{15} - \beta_{14} + 2 \beta_{13} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 6 q^{3} + 6 q^{4} + 6 q^{5} + 22 q^{6} - 9 q^{7} - 8 q^{8} - 10 q^{9} - 6 q^{10} - 4 q^{11} + 5 q^{12} - 9 q^{13} - 18 q^{14} - 4 q^{15} - 2 q^{16} - 17 q^{17} - 14 q^{18} - 7 q^{19} + 36 q^{20}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{14} - 65\nu^{12} - 358\nu^{10} - 641\nu^{8} + 691\nu^{6} + 3382\nu^{4} + 2839\nu^{2} + 255 ) / 93 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15 \nu^{15} - 4 \nu^{14} - 267 \nu^{13} - 65 \nu^{12} - 1792 \nu^{11} - 358 \nu^{10} - 5744 \nu^{9} + \cdots + 162 ) / 186 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{14} + 219\nu^{12} + 1334\nu^{10} + 3517\nu^{8} + 3497\nu^{6} - 33\nu^{4} - 1035\nu^{2} + 140 ) / 93 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -13\nu^{14} - 219\nu^{12} - 1334\nu^{10} - 3517\nu^{8} - 3497\nu^{6} + 33\nu^{4} + 1128\nu^{2} + 46 ) / 93 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8 \nu^{15} + 32 \nu^{14} - 192 \nu^{13} + 582 \nu^{12} - 1832 \nu^{11} + 4011 \nu^{10} - 8815 \nu^{9} + \cdots + 99 ) / 186 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 21 \nu^{15} + 43 \nu^{14} - 380 \nu^{13} + 753 \nu^{12} - 2608 \nu^{11} + 4887 \nu^{10} - 8581 \nu^{9} + \cdots - 52 ) / 186 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 42 \nu^{15} + 49 \nu^{14} - 791 \nu^{13} + 897 \nu^{12} - 5743 \nu^{11} + 6261 \nu^{10} + \cdots + 573 ) / 186 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 63 \nu^{15} + 34 \nu^{14} - 1140 \nu^{13} + 599 \nu^{12} - 7793 \nu^{11} + 3911 \nu^{10} + \cdots - 168 ) / 186 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 50 \nu^{15} - 25 \nu^{14} + 983 \nu^{13} - 445 \nu^{12} + 7575 \nu^{11} - 2966 \nu^{10} + 29263 \nu^{9} + \cdots + 36 ) / 186 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28 \nu^{15} + 38 \nu^{14} + 517 \nu^{13} + 664 \nu^{12} + 3653 \nu^{11} + 4300 \nu^{10} + 12516 \nu^{9} + \cdots + 11 ) / 186 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 50 \nu^{15} - 25 \nu^{14} - 983 \nu^{13} - 445 \nu^{12} - 7575 \nu^{11} - 2966 \nu^{10} - 29263 \nu^{9} + \cdots + 36 ) / 186 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 59 \nu^{15} + 25 \nu^{14} + 1106 \nu^{13} + 445 \nu^{12} + 7993 \nu^{11} + 2966 \nu^{10} + 28357 \nu^{9} + \cdots - 129 ) / 186 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 63 \nu^{15} - 34 \nu^{14} - 1140 \nu^{13} - 599 \nu^{12} - 7793 \nu^{11} - 3911 \nu^{10} + \cdots + 168 ) / 186 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 124 \nu^{15} - 23 \nu^{14} + 2325 \nu^{13} - 397 \nu^{12} + 16802 \nu^{11} - 2508 \nu^{10} + \cdots - 231 ) / 186 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 124 \nu^{15} + 23 \nu^{14} + 2325 \nu^{13} + 397 \nu^{12} + 16802 \nu^{11} + 2508 \nu^{10} + \cdots + 231 ) / 186 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} + \beta_{11} - 2 \beta_{10} - 3 \beta_{9} - \beta_{8} + \cdots + 2 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{15} - 9 \beta_{14} - 11 \beta_{13} + 12 \beta_{12} + 4 \beta_{10} + 13 \beta_{9} + 7 \beta_{8} + \cdots - 2 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + 2 \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 42 \beta_{15} + 42 \beta_{14} + 53 \beta_{13} - 82 \beta_{12} - 11 \beta_{11} - 12 \beta_{10} - 60 \beta_{9} + \cdots - 5 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 8 \beta_{15} + 8 \beta_{14} - 11 \beta_{13} - 6 \beta_{11} - 15 \beta_{9} + 11 \beta_{8} - 9 \beta_{7} + \cdots - 50 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 203 \beta_{15} - 203 \beta_{14} - 272 \beta_{13} + 500 \beta_{12} + 111 \beta_{11} + 68 \beta_{10} + \cdots + 97 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 56 \beta_{15} - 56 \beta_{14} + 92 \beta_{13} + 25 \beta_{11} + 89 \beta_{9} - 92 \beta_{8} + 64 \beta_{7} + \cdots + 296 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1016 \beta_{15} + 1016 \beta_{14} + 1439 \beta_{13} - 3038 \beta_{12} - 915 \beta_{11} - 516 \beta_{10} + \cdots - 937 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 385 \beta_{15} + 385 \beta_{14} - 688 \beta_{13} - 72 \beta_{11} - 495 \beta_{9} + 688 \beta_{8} + \cdots - 1792 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5253 \beta_{15} - 5253 \beta_{14} - 7797 \beta_{13} + 18688 \beta_{12} + 6944 \beta_{11} + \cdots + 7585 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2624 \beta_{15} - 2624 \beta_{14} + 4853 \beta_{13} - 9 \beta_{11} + 2712 \beta_{9} - 4853 \beta_{8} + \cdots + 10977 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 27997 \beta_{15} + 27997 \beta_{14} + 43223 \beta_{13} - 116270 \beta_{12} - 50099 \beta_{11} + \cdots - 56363 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 17693 \beta_{15} + 17693 \beta_{14} - 33083 \beta_{13} + 2393 \beta_{11} - 14930 \beta_{9} + \cdots - 67810 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 153509 \beta_{15} - 153509 \beta_{14} - 244981 \beta_{13} + 729212 \beta_{12} + 349095 \beta_{11} + \cdots + 398363 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
374.1
0.333129i
− 0.176392i
2.16544i
− 1.14660i
− 0.333129i
0.176392i
− 2.16544i
1.14660i
− 1.83925i
− 2.52368i
1.03739i
1.42343i
1.83925i
2.52368i
− 1.03739i
− 1.42343i
−1.67638 − 1.21796i −1.73080 + 1.25750i 0.708788 + 2.18143i 2.34791 4.43308 −1.13550 − 3.49470i 0.188054 − 0.578772i 0.487319 − 1.49981i −3.93600 − 2.85967i
374.2 −0.996848 − 0.724253i 1.67869 − 1.21964i −0.148869 − 0.458173i −1.54562 −2.55673 −1.17543 − 3.61759i −0.944957 + 2.90828i 0.403428 − 1.24162i 1.54075 + 1.11942i
374.3 1.49685 + 1.08752i 0.403986 − 0.293513i 0.439812 + 1.35360i 1.20736 0.923909 1.15357 + 3.55033i 0.329747 − 1.01486i −0.849996 + 2.61602i 1.80724 + 1.31303i
374.4 2.17638 + 1.58123i 1.14813 − 0.834164i 1.61830 + 4.98062i 0.608384 3.81777 −0.533635 − 1.64236i −2.69088 + 8.28167i −0.304682 + 0.937716i 1.32408 + 0.961997i
388.1 −1.67638 + 1.21796i −1.73080 − 1.25750i 0.708788 − 2.18143i 2.34791 4.43308 −1.13550 + 3.49470i 0.188054 + 0.578772i 0.487319 + 1.49981i −3.93600 + 2.85967i
388.2 −0.996848 + 0.724253i 1.67869 + 1.21964i −0.148869 + 0.458173i −1.54562 −2.55673 −1.17543 + 3.61759i −0.944957 − 2.90828i 0.403428 + 1.24162i 1.54075 − 1.11942i
388.3 1.49685 − 1.08752i 0.403986 + 0.293513i 0.439812 − 1.35360i 1.20736 0.923909 1.15357 − 3.55033i 0.329747 + 1.01486i −0.849996 − 2.61602i 1.80724 − 1.31303i
388.4 2.17638 − 1.58123i 1.14813 + 0.834164i 1.61830 − 4.98062i 0.608384 3.81777 −0.533635 + 1.64236i −2.69088 − 8.28167i −0.304682 − 0.937716i 1.32408 − 0.961997i
531.1 −0.213065 − 0.655747i 0.278857 − 0.858234i 1.23343 − 0.896137i 3.70752 −0.622199 −0.617599 + 0.448712i −1.96606 − 1.42843i 1.76825 + 1.28471i −0.789942 − 2.43119i
531.2 0.108599 + 0.334232i 0.894076 − 2.75168i 1.51812 − 1.10298i 2.97323 1.01680 0.875458 − 0.636058i 1.10215 + 0.800755i −4.34534 − 3.15708i 0.322889 + 0.993749i
531.3 0.391401 + 1.20461i −0.458679 + 1.41167i 0.320145 − 0.232599i −3.80032 −1.88004 −1.77093 + 1.28666i 2.45490 + 1.78359i 0.644632 + 0.468353i −1.48745 − 4.57790i
531.4 0.713065 + 2.19459i 0.785745 − 2.41827i −2.68972 + 1.95420i −2.49846 5.86740 −1.29595 + 0.941560i −2.47295 − 1.79670i −2.80360 − 2.03694i −1.78156 − 5.48309i
628.1 −0.213065 + 0.655747i 0.278857 + 0.858234i 1.23343 + 0.896137i 3.70752 −0.622199 −0.617599 − 0.448712i −1.96606 + 1.42843i 1.76825 − 1.28471i −0.789942 + 2.43119i
628.2 0.108599 − 0.334232i 0.894076 + 2.75168i 1.51812 + 1.10298i 2.97323 1.01680 0.875458 + 0.636058i 1.10215 − 0.800755i −4.34534 + 3.15708i 0.322889 − 0.993749i
628.3 0.391401 − 1.20461i −0.458679 − 1.41167i 0.320145 + 0.232599i −3.80032 −1.88004 −1.77093 − 1.28666i 2.45490 − 1.78359i 0.644632 − 0.468353i −1.48745 + 4.57790i
628.4 0.713065 − 2.19459i 0.785745 + 2.41827i −2.68972 − 1.95420i −2.49846 5.86740 −1.29595 − 0.941560i −2.47295 + 1.79670i −2.80360 + 2.03694i −1.78156 + 5.48309i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 374.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.d.q 16
31.b odd 2 1 961.2.d.p 16
31.c even 3 1 961.2.g.m 16
31.c even 3 1 961.2.g.n 16
31.d even 5 1 961.2.a.j 8
31.d even 5 2 961.2.d.n 16
31.d even 5 1 inner 961.2.d.q 16
31.e odd 6 1 961.2.g.s 16
31.e odd 6 1 961.2.g.t 16
31.f odd 10 1 961.2.a.i 8
31.f odd 10 2 961.2.d.o 16
31.f odd 10 1 961.2.d.p 16
31.g even 15 2 961.2.c.i 16
31.g even 15 2 961.2.g.j 16
31.g even 15 2 961.2.g.l 16
31.g even 15 1 961.2.g.m 16
31.g even 15 1 961.2.g.n 16
31.h odd 30 2 31.2.g.a ✓ 16
31.h odd 30 2 961.2.c.j 16
31.h odd 30 2 961.2.g.k 16
31.h odd 30 1 961.2.g.s 16
31.h odd 30 1 961.2.g.t 16
93.k even 10 1 8649.2.a.bf 8
93.l odd 10 1 8649.2.a.be 8
93.p even 30 2 279.2.y.c 16
124.p even 30 2 496.2.bg.c 16
155.v odd 30 2 775.2.bl.a 16
155.x even 60 4 775.2.ck.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.g.a ✓ 16 31.h odd 30 2
279.2.y.c 16 93.p even 30 2
496.2.bg.c 16 124.p even 30 2
775.2.bl.a 16 155.v odd 30 2
775.2.ck.a 32 155.x even 60 4
961.2.a.i 8 31.f odd 10 1
961.2.a.j 8 31.d even 5 1
961.2.c.i 16 31.g even 15 2
961.2.c.j 16 31.h odd 30 2
961.2.d.n 16 31.d even 5 2
961.2.d.o 16 31.f odd 10 2
961.2.d.p 16 31.b odd 2 1
961.2.d.p 16 31.f odd 10 1
961.2.d.q 16 1.a even 1 1 trivial
961.2.d.q 16 31.d even 5 1 inner
961.2.g.j 16 31.g even 15 2
961.2.g.k 16 31.h odd 30 2
961.2.g.l 16 31.g even 15 2
961.2.g.m 16 31.c even 3 1
961.2.g.m 16 31.g even 15 1
961.2.g.n 16 31.c even 3 1
961.2.g.n 16 31.g even 15 1
961.2.g.s 16 31.e odd 6 1
961.2.g.s 16 31.h odd 30 1
961.2.g.t 16 31.e odd 6 1
961.2.g.t 16 31.h odd 30 1
8649.2.a.be 8 93.l odd 10 1
8649.2.a.bf 8 93.k even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(961, [\chi])\):

\( T_{2}^{16} - 4 T_{2}^{15} + 9 T_{2}^{14} - 4 T_{2}^{13} + 6 T_{2}^{12} - 22 T_{2}^{11} + 221 T_{2}^{10} + \cdots + 81 \) Copy content Toggle raw display
\( T_{3}^{16} - 6 T_{3}^{15} + 29 T_{3}^{14} - 81 T_{3}^{13} + 186 T_{3}^{12} - 333 T_{3}^{11} + 721 T_{3}^{10} + \cdots + 961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 4 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} - 6 T^{15} + \cdots + 961 \) Copy content Toggle raw display
$5$ \( (T^{8} - 3 T^{7} + \cdots - 279)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 9 T^{15} + \cdots + 68121 \) Copy content Toggle raw display
$11$ \( T^{16} + 4 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$13$ \( T^{16} + 9 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$17$ \( T^{16} + 17 T^{15} + \cdots + 74805201 \) Copy content Toggle raw display
$19$ \( T^{16} + 7 T^{15} + \cdots + 361201 \) Copy content Toggle raw display
$23$ \( T^{16} + 21 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$29$ \( T^{16} + 26 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( (T^{8} + 8 T^{7} + \cdots - 18569)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 6 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{16} - 16 T^{15} + \cdots + 7612081 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 3306365001 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 366207732801 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 167728401 \) Copy content Toggle raw display
$61$ \( (T^{8} - 30 T^{7} + \cdots + 38161)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 13 T^{7} + \cdots + 86521)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 214944921 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 17441907675201 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 84609661119201 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1446653267361 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 117957215601 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 7131992195241 \) Copy content Toggle raw display
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