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Properties

Label 245.6.b.f
Level $245$
Weight $6$
Character orbit 245.b
Analytic conductor $39.294$
Analytic rank $0$
Dimension $18$
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] = [245,6,Mod(99,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.99");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.b (of order \(2\), degree \(1\), 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: \(39.2940358542\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 415 x^{16} + 70143 x^{14} + 6242821 x^{12} + 316316372 x^{10} + 9248605616 x^{8} + \cdots + 5063868092416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{18}\cdot 5^{7}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} - 14) q^{4} + ( - \beta_{7} + 1) q^{5} + ( - \beta_{3} + 2) q^{6} + (\beta_{5} + \beta_{4} - 13 \beta_1) q^{8} + ( - \beta_{9} + \beta_{7} - \beta_{2} - 47) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} - 14) q^{4} + ( - \beta_{7} + 1) q^{5} + ( - \beta_{3} + 2) q^{6} + (\beta_{5} + \beta_{4} - 13 \beta_1) q^{8} + ( - \beta_{9} + \beta_{7} - \beta_{2} - 47) q^{9} + ( - \beta_{12} + \beta_{5} - \beta_{2} + \cdots + 5) q^{10}+ \cdots + (21 \beta_{16} + 5 \beta_{14} + \cdots - 3491) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 254 q^{4} + 12 q^{5} + 36 q^{6} - 832 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 254 q^{4} + 12 q^{5} + 36 q^{6} - 832 q^{9} + 84 q^{10} + 360 q^{11} + 344 q^{15} + 2630 q^{16} - 3316 q^{19} - 548 q^{20} - 3300 q^{24} + 404 q^{25} + 13570 q^{26} + 8002 q^{29} - 6482 q^{30} + 14860 q^{31} - 19440 q^{34} - 19878 q^{36} + 1988 q^{39} - 17098 q^{40} + 55926 q^{41} - 7670 q^{44} - 30876 q^{45} + 4818 q^{46} + 55938 q^{50} + 52732 q^{51} - 20724 q^{54} - 90260 q^{55} - 89124 q^{59} - 6156 q^{60} + 23122 q^{61} - 36822 q^{64} - 58748 q^{65} + 286316 q^{66} - 55282 q^{69} - 62132 q^{71} + 102486 q^{74} + 68328 q^{75} + 208510 q^{76} - 113556 q^{79} - 263952 q^{80} - 225014 q^{81} + 110778 q^{85} + 427796 q^{86} - 302422 q^{89} - 46262 q^{90} - 305370 q^{94} + 216808 q^{95} + 169668 q^{96} - 60408 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 415 x^{16} + 70143 x^{14} + 6242821 x^{12} + 316316372 x^{10} + 9248605616 x^{8} + \cdots + 5063868092416 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2760143981 \nu^{16} + 1089188907039 \nu^{14} + 170715023238519 \nu^{12} + \cdots + 44\!\cdots\!76 ) / 27\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1761796082507 \nu^{17} - 675688561374153 \nu^{15} + \cdots + 47\!\cdots\!88 \nu ) / 54\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1761796082507 \nu^{17} + 675688561374153 \nu^{15} + \cdots - 49\!\cdots\!08 \nu ) / 54\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 18206385135391 \nu^{17} + \cdots + 20\!\cdots\!24 \nu ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 226866738112299 \nu^{17} + 501295523093638 \nu^{16} + \cdots + 38\!\cdots\!88 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 226866738112299 \nu^{17} - 633919152512912 \nu^{16} + \cdots + 29\!\cdots\!88 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 226866738112299 \nu^{17} + \cdots - 22\!\cdots\!52 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 75622246037433 \nu^{17} + \cdots - 43\!\cdots\!84 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 226866738112299 \nu^{17} + \cdots - 36\!\cdots\!32 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 527722464331243 \nu^{17} + \cdots - 11\!\cdots\!84 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 17762860678841 \nu^{17} + \cdots - 55\!\cdots\!36 \nu ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 168561692228764 \nu^{17} - 948113649420533 \nu^{16} + \cdots - 21\!\cdots\!08 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 552021680325709 \nu^{17} + \cdots + 59\!\cdots\!92 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 226866738112299 \nu^{17} + 501295523093638 \nu^{16} + \cdots + 38\!\cdots\!48 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 898806093765632 \nu^{17} + 250647761546819 \nu^{16} + \cdots + 19\!\cdots\!44 ) / 41\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 46 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 77\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + \beta_{10} - \beta_{9} + 5\beta_{7} - \beta_{3} - 109\beta_{2} - \beta _1 + 3535 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} - 5 \beta_{16} + 2 \beta_{15} - 8 \beta_{14} + 18 \beta_{13} + 10 \beta_{12} + 2 \beta_{7} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 254 \beta_{16} + 14 \beta_{14} - 14 \beta_{13} + 14 \beta_{12} - 28 \beta_{11} - 170 \beta_{10} + \cdots - 317674 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 254 \beta_{17} + 1070 \beta_{16} - 372 \beta_{15} + 1764 \beta_{14} - 3896 \beta_{13} - 2136 \beta_{12} + \cdots + 376 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 41043 \beta_{16} - 2132 \beta_{14} + 2132 \beta_{13} - 2132 \beta_{12} + 6168 \beta_{11} + \cdots + 30677749 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 37475 \beta_{17} - 160639 \beta_{16} + 56550 \beta_{15} - 274784 \beta_{14} + 597918 \beta_{13} + \cdots - 46494 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5648496 \beta_{16} + 197354 \beta_{14} - 197354 \beta_{13} + 197354 \beta_{12} - 947652 \beta_{11} + \cdots - 3098071976 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4586832 \beta_{17} + 21084984 \beta_{16} - 7926488 \beta_{15} + 37456284 \beta_{14} - 80207140 \beta_{13} + \cdots + 4713684 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 721159941 \beta_{16} - 11407536 \beta_{14} + 11407536 \beta_{13} - 11407536 \beta_{12} + \cdots + 323106090451 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 517933813 \beta_{17} - 2599573065 \beta_{16} + 1056244394 \beta_{15} - 4778526552 \beta_{14} + \cdots - 420619578 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 88423591810 \beta_{16} - 144254650 \beta_{14} + 144254650 \beta_{13} - 144254650 \beta_{12} + \cdots - 34520332955142 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 56411028930 \beta_{17} + 310380658018 \beta_{16} - 135807701564 \beta_{15} + 587221865940 \beta_{14} + \cdots + 33539450096 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 10587691285079 \beta_{16} + 173010554932 \beta_{14} - 173010554932 \beta_{13} + 173010554932 \beta_{12} + \cdots + 37\!\cdots\!01 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 6057885030183 \beta_{17} - 36420112513075 \beta_{16} + 17005207345262 \beta_{15} + \cdots - 2310578189590 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
− 10.6997i
− 9.35914i
− 8.57527i
− 8.00995i
− 5.38664i
− 4.54441i
− 4.27576i
− 2.34601i
− 1.33235i
1.33235i
2.34601i
4.27576i
4.54441i
5.38664i
8.00995i
8.57527i
9.35914i
10.6997i
− 10.6997i 7.65116i −82.4833 −55.8997 − 0.468620i 81.8650 0 540.155i 184.460 −5.01409 + 598.110i
99.2 − 9.35914i − 19.5327i −55.5936 51.5790 + 21.5546i −182.810 0 220.815i −138.528 201.733 − 482.736i
99.3 − 8.57527i 21.4006i −41.5353 47.3017 + 29.7918i 183.516 0 81.7681i −214.986 255.473 − 405.625i
99.4 − 8.00995i − 7.64020i −32.1593 4.64688 − 55.7082i −61.1976 0 1.27596i 184.627 −446.220 − 37.2213i
99.5 − 5.38664i − 17.7926i 2.98414 −19.0612 + 52.5516i −95.8422 0 − 188.447i −73.5760 283.076 + 102.676i
99.6 − 4.54441i 24.8039i 11.3483 6.36900 − 55.5377i 112.719 0 − 196.993i −372.235 −252.386 − 28.9433i
99.7 − 4.27576i 8.27994i 13.7179 −52.2165 + 19.9608i 35.4030 0 − 195.479i 174.443 85.3476 + 223.265i
99.8 − 2.34601i − 25.2735i 26.4962 −25.2230 − 49.8879i −59.2919 0 − 137.233i −395.749 −117.038 + 59.1734i
99.9 − 1.33235i 2.73065i 30.2248 48.5038 + 27.7917i 3.63818 0 − 82.9052i 235.544 37.0283 − 64.6240i
99.10 1.33235i − 2.73065i 30.2248 48.5038 − 27.7917i 3.63818 0 82.9052i 235.544 37.0283 + 64.6240i
99.11 2.34601i 25.2735i 26.4962 −25.2230 + 49.8879i −59.2919 0 137.233i −395.749 −117.038 − 59.1734i
99.12 4.27576i − 8.27994i 13.7179 −52.2165 − 19.9608i 35.4030 0 195.479i 174.443 85.3476 − 223.265i
99.13 4.54441i − 24.8039i 11.3483 6.36900 + 55.5377i 112.719 0 196.993i −372.235 −252.386 + 28.9433i
99.14 5.38664i 17.7926i 2.98414 −19.0612 − 52.5516i −95.8422 0 188.447i −73.5760 283.076 − 102.676i
99.15 8.00995i 7.64020i −32.1593 4.64688 + 55.7082i −61.1976 0 − 1.27596i 184.627 −446.220 + 37.2213i
99.16 8.57527i − 21.4006i −41.5353 47.3017 − 29.7918i 183.516 0 − 81.7681i −214.986 255.473 + 405.625i
99.17 9.35914i 19.5327i −55.5936 51.5790 − 21.5546i −182.810 0 − 220.815i −138.528 201.733 + 482.736i
99.18 10.6997i − 7.65116i −82.4833 −55.8997 + 0.468620i 81.8650 0 − 540.155i 184.460 −5.01409 − 598.110i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.6.b.f 18
5.b even 2 1 inner 245.6.b.f 18
7.b odd 2 1 245.6.b.e 18
7.c even 3 2 35.6.j.a ✓ 36
35.c odd 2 1 245.6.b.e 18
35.j even 6 2 35.6.j.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.j.a ✓ 36 7.c even 3 2
35.6.j.a ✓ 36 35.j even 6 2
245.6.b.e 18 7.b odd 2 1
245.6.b.e 18 35.c odd 2 1
245.6.b.f 18 1.a even 1 1 trivial
245.6.b.f 18 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{18} + 415 T_{2}^{16} + 70143 T_{2}^{14} + 6242821 T_{2}^{12} + 316316372 T_{2}^{10} + \cdots + 5063868092416 \) Copy content Toggle raw display
\( T_{19}^{9} + 1658 T_{19}^{8} - 9966714 T_{19}^{7} - 9217161600 T_{19}^{6} + 27061039578345 T_{19}^{5} + \cdots + 38\!\cdots\!48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 5063868092416 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{18} \) Copy content Toggle raw display
$11$ \( (T^{9} + \cdots - 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 45\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots + 38\!\cdots\!48)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 53\!\cdots\!69 \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots + 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots - 47\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 59\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 96\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 57\!\cdots\!92)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 59\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots + 19\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 79\!\cdots\!56 \) Copy content Toggle raw display
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