Properties

Label 98.10.c.m
Level $98$
Weight $10$
Character orbit 98.c
Analytic conductor $50.474$
Analytic rank $0$
Dimension $8$
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] = [98,10,Mod(67,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.67");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), 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: \(50.4735119441\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4 x^{7} - 4806 x^{6} + 14432 x^{5} + 8678459 x^{4} - 17380976 x^{3} - 6978412850 x^{2} + 6987105744 x + 2108388231292 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 16 \beta_1 + 16) q^{2} + (5 \beta_{3} - \beta_{2}) q^{3} - 256 \beta_1 q^{4} + (104 \beta_{6} + 7 \beta_{5} - 104 \beta_{3} - 7 \beta_{2}) q^{5} + (80 \beta_{6} - 16 \beta_{5}) q^{6} - 4096 q^{8} + (9 \beta_{7} + 3978 \beta_1 - 3978) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 16 \beta_1 + 16) q^{2} + (5 \beta_{3} - \beta_{2}) q^{3} - 256 \beta_1 q^{4} + (104 \beta_{6} + 7 \beta_{5} - 104 \beta_{3} - 7 \beta_{2}) q^{5} + (80 \beta_{6} - 16 \beta_{5}) q^{6} - 4096 q^{8} + (9 \beta_{7} + 3978 \beta_1 - 3978) q^{9} + ( - 1664 \beta_{3} - 112 \beta_{2}) q^{10} + (37 \beta_{7} - 37 \beta_{4} - 2387 \beta_1) q^{11} + (1280 \beta_{6} - 256 \beta_{5} - 1280 \beta_{3} + 256 \beta_{2}) q^{12} + (182 \beta_{6} + 809 \beta_{5}) q^{13} + ( - 76 \beta_{4} - 104328) q^{15} + (65536 \beta_1 - 65536) q^{16} + ( - 43071 \beta_{3} + 248 \beta_{2}) q^{17} + (144 \beta_{7} - 144 \beta_{4} + 63648 \beta_1) q^{18} + ( - 31639 \beta_{6} + 3679 \beta_{5} + 31639 \beta_{3} - 3679 \beta_{2}) q^{19} + ( - 26624 \beta_{6} - 1792 \beta_{5}) q^{20} + ( - 592 \beta_{4} - 38192) q^{22} + (692 \beta_{7} - 709184 \beta_1 + 709184) q^{23} + ( - 20480 \beta_{3} + 4096 \beta_{2}) q^{24} + (1505 \beta_{7} - 1505 \beta_{4} - 241536 \beta_1) q^{25} + (2912 \beta_{6} + 12944 \beta_{5} - 2912 \beta_{3} - 12944 \beta_{2}) q^{26} + ( - 118548 \beta_{6} - 11736 \beta_{5}) q^{27} + (1087 \beta_{4} + 4167493) q^{29} + ( - 1216 \beta_{7} + 1669248 \beta_1 - 1669248) q^{30} + ( - 100134 \beta_{3} + 41130 \beta_{2}) q^{31} + 1048576 \beta_1 q^{32} + ( - 798254 \beta_{6} + 13930 \beta_{5} + 798254 \beta_{3} - 13930 \beta_{2}) q^{33} + ( - 689136 \beta_{6} + 3968 \beta_{5}) q^{34} + ( - 2304 \beta_{4} + 1018368) q^{36} + (9021 \beta_{7} + 806593 \beta_1 - 806593) q^{37} + (506224 \beta_{3} - 58864 \beta_{2}) q^{38} + ( - 3054 \beta_{7} + 3054 \beta_{4} - 17277642 \beta_1) q^{39} + ( - 425984 \beta_{6} - 28672 \beta_{5} + 425984 \beta_{3} + 28672 \beta_{2}) q^{40} + ( - 1430791 \beta_{6} + 56652 \beta_{5}) q^{41} + (9147 \beta_{4} - 19120051) q^{43} + ( - 9472 \beta_{7} + 611072 \beta_1 - 611072) q^{44} + ( - 904500 \beta_{3} - 66969 \beta_{2}) q^{45} + (11072 \beta_{7} - 11072 \beta_{4} - 11346944 \beta_1) q^{46} + ( - 2664762 \beta_{6} - 163054 \beta_{5} + \cdots + 163054 \beta_{2}) q^{47}+ \cdots + (125703 \beta_{4} - 697895415) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 64 q^{2} - 1024 q^{4} - 32768 q^{8} - 15912 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 64 q^{2} - 1024 q^{4} - 32768 q^{8} - 15912 q^{9} - 9548 q^{11} - 834624 q^{15} - 262144 q^{16} + 254592 q^{18} - 305536 q^{22} + 2836736 q^{23} - 966144 q^{25} + 33339944 q^{29} - 6676992 q^{30} + 4194304 q^{32} + 8146944 q^{36} - 3226372 q^{37} - 69110568 q^{39} - 152960408 q^{43} - 2444288 q^{44} - 45387776 q^{46} - 30916608 q^{50} + 97261596 q^{51} - 13708112 q^{53} - 743115384 q^{57} + 266719552 q^{58} + 106831872 q^{60} + 134217728 q^{64} + 515731188 q^{65} + 814420776 q^{67} - 2649337488 q^{71} + 65175552 q^{72} + 51621952 q^{74} - 2211538176 q^{78} + 668411176 q^{79} + 1111307688 q^{81} - 3318529928 q^{85} - 1223683264 q^{86} + 39108608 q^{88} - 1452408832 q^{92} + 3706577496 q^{93} - 977188688 q^{95} - 5583163320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4 x^{7} - 4806 x^{6} + 14432 x^{5} + 8678459 x^{4} - 17380976 x^{3} - 6978412850 x^{2} + 6987105744 x + 2108388231292 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{6} + 6\nu^{5} + 12029\nu^{4} - 24068\nu^{3} - 20270573\nu^{2} + 20282608\nu + 10511155942 ) / 46445490 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 25942 \nu^{7} - 5272762 \nu^{6} + 147070813 \nu^{5} + 31931448273 \nu^{4} - 327481137590 \nu^{3} - 116244756773774 \nu^{2} + \cdots + 10\!\cdots\!28 ) / 41518854316485 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7412 \nu^{7} + 25942 \nu^{6} + 37422896 \nu^{5} - 93622095 \nu^{4} - 75124591024 \nu^{3} + 112780521602 \nu^{2} + \cdots - 24589480635776 ) / 5931264902355 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14824 \nu^{7} + 51884 \nu^{6} + 74845792 \nu^{5} - 187244190 \nu^{4} - 150249182048 \nu^{3} + 225561043204 \nu^{2} + \cdots - 90697815588037 ) / 1977088300785 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12468197 \nu^{7} + 12869129603 \nu^{6} - 83592352721 \nu^{5} - 46412459941545 \nu^{4} + 146945466630124 \nu^{3} + \cdots - 22\!\cdots\!44 ) / 83037708632970 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 67214 \nu^{7} - 235249 \nu^{6} - 241800797 \nu^{5} + 605090115 \nu^{4} + 290196735703 \nu^{3} - 435900311294 \nu^{2} + \cdots + 58163847633332 ) / 223821317070 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14276096 \nu^{7} - 49966336 \nu^{6} - 68632324838 \nu^{5} + 171705727935 \nu^{4} + 103199433438052 \nu^{3} + \cdots + 24\!\cdots\!48 ) / 3954176601570 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - 6\beta_{3} + 21 ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 8\beta_{3} - 28\beta_{2} + 84\beta _1 + 50505 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} + 12\beta_{6} + 1205\beta_{4} - 21660\beta_{3} - 42\beta_{2} + 126\beta _1 + 75747 ) / 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 12 \beta_{7} - 32 \beta_{6} + 112 \beta_{5} + 2409 \beta_{4} - 9616 \beta_{3} - 67480 \beta_{2} + 606900 \beta _1 + 60377961 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 24080 \beta_{7} + 144376 \beta_{6} + 280 \beta_{5} + 1453209 \beta_{4} - 43476396 \beta_{3} - 168630 \beta_{2} + 1517040 \beta _1 + 150818661 ) / 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 72210 \beta_{7} + 96256 \beta_{6} + 674464 \beta_{5} + 4353605 \beta_{4} - 69538296 \beta_{3} - 122070900 \beta_{2} + 1826515530 \beta _1 + 71748760503 ) / 42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 60866918 \beta_{7} + 608063400 \beta_{6} + 2359644 \beta_{5} + 1753872737 \beta_{4} - 73364375020 \beta_{3} - 426657994 \beta_{2} + 6387494862 \beta _1 + 250592884815 ) / 42 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−33.4952 1.22474i
34.4952 + 1.22474i
35.9094 1.22474i
−34.9094 + 1.22474i
−33.4952 + 1.22474i
34.4952 1.22474i
35.9094 + 1.22474i
−34.9094 1.22474i
8.00000 + 13.8564i −95.8886 + 166.084i −128.000 + 221.703i −16.7949 29.0896i −3068.43 0 −4096.00 −8547.74 14805.1i 268.719 465.434i
67.2 8.00000 + 13.8564i −51.3408 + 88.9249i −128.000 + 221.703i 1047.40 + 1814.15i −1642.91 0 −4096.00 4569.74 + 7915.01i −16758.4 + 29026.4i
67.3 8.00000 + 13.8564i 51.3408 88.9249i −128.000 + 221.703i −1047.40 1814.15i 1642.91 0 −4096.00 4569.74 + 7915.01i 16758.4 29026.4i
67.4 8.00000 + 13.8564i 95.8886 166.084i −128.000 + 221.703i 16.7949 + 29.0896i 3068.43 0 −4096.00 −8547.74 14805.1i −268.719 + 465.434i
79.1 8.00000 13.8564i −95.8886 166.084i −128.000 221.703i −16.7949 + 29.0896i −3068.43 0 −4096.00 −8547.74 + 14805.1i 268.719 + 465.434i
79.2 8.00000 13.8564i −51.3408 88.9249i −128.000 221.703i 1047.40 1814.15i −1642.91 0 −4096.00 4569.74 7915.01i −16758.4 29026.4i
79.3 8.00000 13.8564i 51.3408 + 88.9249i −128.000 221.703i −1047.40 + 1814.15i 1642.91 0 −4096.00 4569.74 7915.01i 16758.4 + 29026.4i
79.4 8.00000 13.8564i 95.8886 + 166.084i −128.000 221.703i 16.7949 29.0896i 3068.43 0 −4096.00 −8547.74 + 14805.1i −268.719 465.434i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.10.c.m 8
7.b odd 2 1 inner 98.10.c.m 8
7.c even 3 1 98.10.a.k 4
7.c even 3 1 inner 98.10.c.m 8
7.d odd 6 1 98.10.a.k 4
7.d odd 6 1 inner 98.10.c.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.10.a.k 4 7.c even 3 1
98.10.a.k 4 7.d odd 6 1
98.10.c.m 8 1.a even 1 1 trivial
98.10.c.m 8 7.b odd 2 1 inner
98.10.c.m 8 7.c even 3 1 inner
98.10.c.m 8 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 47322T_{3}^{6} + 1851596820T_{3}^{4} + 18350282114208T_{3}^{2} + 150369345150218496 \) acting on \(S_{10}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 16 T + 256)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 47322 T^{6} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{8} + 4389322 T^{6} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4774 T^{3} + \cdots + 84\!\cdots\!76)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 28441135314 T^{2} + \cdots + 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 364177565476 T^{6} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{8} + 760834631722 T^{6} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{4} - 1418368 T^{3} + \cdots + 26\!\cdots\!04)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8334986 T + 14857994423056)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + 74580188894856 T^{6} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{4} + 1613186 T^{3} + \cdots + 29\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 524653229945812 T^{2} + \cdots + 15\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 38240102 T + 187841499446728)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 70\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{4} + 6854056 T^{3} + \cdots + 51\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 61\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{4} - 407210388 T^{3} + \cdots + 96\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 662334372 T + 10\!\cdots\!64)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{4} - 334205588 T^{3} + \cdots + 59\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 89\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 62\!\cdots\!56)^{2} \) Copy content Toggle raw display
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