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Properties

Label 245.4.e.o
Level $245$
Weight $4$
Character orbit 245.e
Analytic conductor $14.455$
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] = [245,4,Mod(116,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.116");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.e (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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 38 x^{8} - 5 x^{7} + 1102 x^{6} - 137 x^{5} + 11161 x^{4} + 10784 x^{3} + 81600 x^{2} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 35)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3}) q^{3} + ( - \beta_{8} - 7 \beta_{7} - \beta_{4}) q^{4} + (5 \beta_{7} - 5) q^{5} + (\beta_{6} - 2 \beta_{4} + \beta_{3} + \cdots - 3) q^{6}+ \cdots + ( - \beta_{9} + \beta_{8} + 18 \beta_{7} + \cdots - 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3}) q^{3} + ( - \beta_{8} - 7 \beta_{7} - \beta_{4}) q^{4} + (5 \beta_{7} - 5) q^{5} + (\beta_{6} - 2 \beta_{4} + \beta_{3} + \cdots - 3) q^{6}+ \cdots + (15 \beta_{6} + \beta_{4} + 83 \beta_{3} + \cdots + 28) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} - 8 q^{3} - 35 q^{4} - 25 q^{5} - 32 q^{6} + 66 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} - 8 q^{3} - 35 q^{4} - 25 q^{5} - 32 q^{6} + 66 q^{8} - 81 q^{9} - 5 q^{10} - 47 q^{11} - 98 q^{12} - 2 q^{13} + 80 q^{15} - 171 q^{16} - 2 q^{17} + 51 q^{18} - 21 q^{19} + 350 q^{20} + 1046 q^{22} - 201 q^{23} + 848 q^{24} - 125 q^{25} - 47 q^{26} + 1036 q^{27} + 380 q^{29} + 80 q^{30} + 388 q^{31} + 95 q^{32} - 262 q^{33} - 260 q^{34} + 2458 q^{36} + 145 q^{37} + 835 q^{38} - 14 q^{39} - 165 q^{40} + 562 q^{41} + 1136 q^{43} - 1091 q^{44} - 405 q^{45} - 337 q^{46} - 473 q^{47} - 140 q^{48} + 50 q^{50} - 732 q^{51} - 379 q^{52} - 351 q^{53} - 774 q^{54} + 470 q^{55} + 1908 q^{57} - 1818 q^{58} + 708 q^{59} - 490 q^{60} + 1944 q^{61} - 896 q^{62} - 250 q^{64} + 5 q^{65} + 1482 q^{66} - 1118 q^{67} - 3118 q^{68} - 748 q^{69} + 1728 q^{71} + 2219 q^{72} - 1652 q^{73} + 3285 q^{74} - 200 q^{75} - 1382 q^{76} - 11148 q^{78} - 218 q^{79} - 855 q^{80} + 455 q^{81} + 1027 q^{82} + 3004 q^{83} + 20 q^{85} + 4264 q^{86} + 390 q^{87} - 2131 q^{88} + 2322 q^{89} - 510 q^{90} - 5914 q^{92} + 2288 q^{93} - 2677 q^{94} - 105 q^{95} + 4592 q^{96} - 1196 q^{97} + 70 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 38 x^{8} - 5 x^{7} + 1102 x^{6} - 137 x^{5} + 11161 x^{4} + 10784 x^{3} + 81600 x^{2} + \cdots + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 67990957 \nu^{9} + 304117491 \nu^{8} - 1131828850 \nu^{7} + 13487902815 \nu^{6} + \cdots - 6897867839488 ) / 31151769983040 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 67990957 \nu^{9} + 304117491 \nu^{8} - 1131828850 \nu^{7} + 13487902815 \nu^{6} + \cdots - 964197366528 ) / 5933670472960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11628389 \nu^{9} - 116108913 \nu^{8} + 432120550 \nu^{7} - 3172287075 \nu^{6} + \cdots - 14611095022046 ) / 973492811970 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4820632391 \nu^{9} + 390031347 \nu^{8} - 133770984310 \nu^{7} - 159796525845 \nu^{6} + \cdots + 19549052954624 ) / 124607079932160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1902564901 \nu^{9} - 28913288517 \nu^{8} + 107606089950 \nu^{7} - 1115463289785 \nu^{6} + \cdots - 11\!\cdots\!64 ) / 41535693310720 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3368099531 \nu^{9} + 3232117617 \nu^{8} - 128596017160 \nu^{7} + 19104155355 \nu^{6} + \cdots - 1111623972736 ) / 62303539966080 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3368099531 \nu^{9} - 3232117617 \nu^{8} + 128596017160 \nu^{7} - 19104155355 \nu^{6} + \cdots + 63415163938816 ) / 4153569331072 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 190353647257 \nu^{9} + 202320055689 \nu^{8} - 7179913917350 \nu^{7} + \cdots - 33\!\cdots\!12 ) / 124607079932160 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 15\beta_{7} - 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 21\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{9} - 29\beta_{8} - 327\beta_{7} + 4\beta_{6} + 4\beta_{5} - 29\beta_{4} + 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{9} - 8\beta_{8} - 96\beta_{7} + 152\beta_{5} + 489\beta_{2} - 489\beta _1 + 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -152\beta_{6} + 793\beta_{4} - 216\beta_{3} - 16\beta_{2} + 7791 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 216 \beta_{9} + 416 \beta_{8} + 2364 \beta_{7} - 216 \beta_{6} - 4604 \beta_{5} + 416 \beta_{4} + \cdots + 12029 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4604\beta_{9} + 21237\beta_{8} + 194183\beta_{7} - 7996\beta_{5} + 176\beta_{2} - 176\beta _1 - 194183 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 7996\beta_{6} - 15816\beta_{4} + 129776\beta_{3} - 304401\beta_{2} - 69464 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-1 + \beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
2.60181 − 4.50647i
1.92311 − 3.33093i
−0.113749 + 0.197018i
−1.39853 + 2.42233i
−2.51265 + 4.35203i
2.60181 + 4.50647i
1.92311 + 3.33093i
−0.113749 − 0.197018i
−1.39853 − 2.42233i
−2.51265 − 4.35203i
−2.60181 − 4.50647i 2.45326 − 4.24917i −9.53885 + 16.5218i −2.50000 − 4.33013i −25.5317 0 57.6442 1.46305 + 2.53407i −13.0091 + 22.5324i
116.2 −1.92311 − 3.33093i −4.48397 + 7.76647i −3.39672 + 5.88330i −2.50000 − 4.33013i 34.4927 0 −4.64067 −26.7120 − 46.2666i −9.61556 + 16.6546i
116.3 0.113749 + 0.197018i −0.904291 + 1.56628i 3.97412 − 6.88338i −2.50000 − 4.33013i −0.411448 0 3.62818 11.8645 + 20.5499i 0.568743 − 0.985092i
116.4 1.39853 + 2.42233i 3.10692 − 5.38134i 0.0882227 − 0.152806i −2.50000 − 4.33013i 17.3805 0 22.8700 −5.80586 − 10.0560i 6.99265 − 12.1116i
116.5 2.51265 + 4.35203i −4.17191 + 7.22596i −8.62677 + 14.9420i −2.50000 − 4.33013i −41.9301 0 −46.5017 −21.3097 − 36.9094i 12.5632 − 21.7601i
226.1 −2.60181 + 4.50647i 2.45326 + 4.24917i −9.53885 − 16.5218i −2.50000 + 4.33013i −25.5317 0 57.6442 1.46305 − 2.53407i −13.0091 − 22.5324i
226.2 −1.92311 + 3.33093i −4.48397 − 7.76647i −3.39672 − 5.88330i −2.50000 + 4.33013i 34.4927 0 −4.64067 −26.7120 + 46.2666i −9.61556 − 16.6546i
226.3 0.113749 − 0.197018i −0.904291 − 1.56628i 3.97412 + 6.88338i −2.50000 + 4.33013i −0.411448 0 3.62818 11.8645 − 20.5499i 0.568743 + 0.985092i
226.4 1.39853 − 2.42233i 3.10692 + 5.38134i 0.0882227 + 0.152806i −2.50000 + 4.33013i 17.3805 0 22.8700 −5.80586 + 10.0560i 6.99265 + 12.1116i
226.5 2.51265 − 4.35203i −4.17191 − 7.22596i −8.62677 − 14.9420i −2.50000 + 4.33013i −41.9301 0 −46.5017 −21.3097 + 36.9094i 12.5632 + 21.7601i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.e.o 10
7.b odd 2 1 35.4.e.c ✓ 10
7.c even 3 1 245.4.a.n 5
7.c even 3 1 inner 245.4.e.o 10
7.d odd 6 1 35.4.e.c ✓ 10
7.d odd 6 1 245.4.a.m 5
21.c even 2 1 315.4.j.g 10
21.g even 6 1 315.4.j.g 10
21.g even 6 1 2205.4.a.bu 5
21.h odd 6 1 2205.4.a.bt 5
28.d even 2 1 560.4.q.n 10
28.f even 6 1 560.4.q.n 10
35.c odd 2 1 175.4.e.d 10
35.f even 4 2 175.4.k.d 20
35.i odd 6 1 175.4.e.d 10
35.i odd 6 1 1225.4.a.bg 5
35.j even 6 1 1225.4.a.bf 5
35.k even 12 2 175.4.k.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.c ✓ 10 7.b odd 2 1
35.4.e.c ✓ 10 7.d odd 6 1
175.4.e.d 10 35.c odd 2 1
175.4.e.d 10 35.i odd 6 1
175.4.k.d 20 35.f even 4 2
175.4.k.d 20 35.k even 12 2
245.4.a.m 5 7.d odd 6 1
245.4.a.n 5 7.c even 3 1
245.4.e.o 10 1.a even 1 1 trivial
245.4.e.o 10 7.c even 3 1 inner
315.4.j.g 10 21.c even 2 1
315.4.j.g 10 21.g even 6 1
560.4.q.n 10 28.d even 2 1
560.4.q.n 10 28.f even 6 1
1225.4.a.bf 5 35.j even 6 1
1225.4.a.bg 5 35.i odd 6 1
2205.4.a.bt 5 21.h odd 6 1
2205.4.a.bu 5 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{10} + T_{2}^{9} + 38 T_{2}^{8} + 5 T_{2}^{7} + 1102 T_{2}^{6} + 137 T_{2}^{5} + 11161 T_{2}^{4} + \cdots + 4096 \) Copy content Toggle raw display
\( T_{3}^{10} + 8 T_{3}^{9} + 140 T_{3}^{8} + 316 T_{3}^{7} + 7741 T_{3}^{6} + 11542 T_{3}^{5} + \cdots + 17023876 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( T^{10} + 8 T^{9} + \cdots + 17023876 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 1044811065600 \) Copy content Toggle raw display
$13$ \( (T^{5} + T^{4} + \cdots - 307317696)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 113580345278464 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 12\!\cdots\!89 \) Copy content Toggle raw display
$29$ \( (T^{5} - 190 T^{4} + \cdots - 13190815450)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 63\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} - 281 T^{4} + \cdots - 77000100765)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 568 T^{4} + \cdots - 72928266842)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 6439908260352)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 17832616128012)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 85\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{5} + 598 T^{4} + \cdots - 863391264288)^{2} \) Copy content Toggle raw display
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