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Properties

Label 245.4.e.q
Level $245$
Weight $4$
Character orbit 245.e
Analytic conductor $14.455$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 27 x^{10} + 22 x^{9} + 399 x^{8} + 492 x^{7} + 4046 x^{6} + 8784 x^{5} + 22536 x^{4} + \cdots + 784 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{9} - \beta_{7} + 3 \beta_{3} + 3) q^{3} + ( - \beta_{11} - 2 \beta_{10} + \cdots - 2) q^{4}+ \cdots + ( - 3 \beta_{11} - \beta_{10} + \cdots + 16 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{9} - \beta_{7} + 3 \beta_{3} + 3) q^{3} + ( - \beta_{11} - 2 \beta_{10} + \cdots - 2) q^{4}+ \cdots + (26 \beta_{9} + 244 \beta_{8} + \cdots - 536) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 16 q^{3} - 14 q^{4} - 30 q^{5} - 48 q^{6} - 132 q^{8} - 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} + 16 q^{3} - 14 q^{4} - 30 q^{5} - 48 q^{6} - 132 q^{8} - 70 q^{9} + 10 q^{10} + 16 q^{11} + 160 q^{12} - 336 q^{13} - 160 q^{15} - 298 q^{16} - 4 q^{17} - 354 q^{18} + 308 q^{19} + 140 q^{20} - 472 q^{22} + 336 q^{23} - 92 q^{24} - 150 q^{25} + 56 q^{26} - 1928 q^{27} + 352 q^{29} + 120 q^{30} + 392 q^{31} + 770 q^{32} + 188 q^{33} - 1624 q^{34} + 460 q^{36} + 140 q^{37} + 20 q^{38} - 140 q^{39} + 330 q^{40} - 1312 q^{41} - 776 q^{43} + 160 q^{44} - 350 q^{45} + 388 q^{46} + 628 q^{47} - 2792 q^{48} - 100 q^{50} - 744 q^{51} + 1520 q^{52} + 676 q^{53} + 2284 q^{54} - 160 q^{55} + 2936 q^{57} + 2012 q^{58} + 996 q^{59} + 800 q^{60} + 740 q^{61} + 728 q^{62} + 2852 q^{64} + 840 q^{65} - 3620 q^{66} - 1768 q^{67} - 2940 q^{68} + 2096 q^{69} - 448 q^{71} - 2858 q^{72} + 2640 q^{73} - 928 q^{74} + 400 q^{75} + 2680 q^{76} + 16 q^{78} - 1636 q^{79} - 1490 q^{80} - 4442 q^{81} - 1756 q^{82} - 280 q^{83} + 40 q^{85} - 1180 q^{86} + 1940 q^{87} + 5652 q^{88} - 1904 q^{89} + 3540 q^{90} - 3904 q^{92} + 1592 q^{93} - 3332 q^{94} + 1540 q^{95} - 6460 q^{96} - 1032 q^{97} - 5608 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 27 x^{10} + 22 x^{9} + 399 x^{8} + 492 x^{7} + 4046 x^{6} + 8784 x^{5} + 22536 x^{4} + \cdots + 784 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 28982856459 \nu^{11} - 100035188374 \nu^{10} + 862940207031 \nu^{9} + \cdots + 17\!\cdots\!20 ) / 294724713250240 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 64037594762 \nu^{11} - 188357432449 \nu^{10} + 1870129815820 \nu^{9} + \cdots - 492665609549624 ) / 294724713250240 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13582777089 \nu^{11} + 30056420270 \nu^{10} - 375018884157 \nu^{9} + \cdots - 56425021720928 ) / 45342263576960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 111467791035 \nu^{11} + 52968478226 \nu^{10} + 2081845750599 \nu^{9} + \cdots + 572806666180672 ) / 294724713250240 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 76723499462 \nu^{11} - 139795554401 \nu^{10} + 2025141954592 \nu^{9} + \cdots + 331284635996552 ) / 147362356625120 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 236742116523 \nu^{11} + 798546202123 \nu^{10} - 7023150156897 \nu^{9} + \cdots + 334278957216120 ) / 294724713250240 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 329010546210 \nu^{11} + 296347940193 \nu^{10} - 7628430332148 \nu^{9} + \cdots - 15\!\cdots\!24 ) / 294724713250240 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12353465669 \nu^{11} - 35572783963 \nu^{10} + 357053140915 \nu^{9} - 37375371979 \nu^{8} + \cdots - 127876575513208 ) / 10525882616080 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 188811098545 \nu^{11} + 604767616393 \nu^{10} - 5588134013203 \nu^{9} + \cdots + 503193043313576 ) / 147362356625120 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 534066780106 \nu^{11} - 1229497377337 \nu^{10} + 14908874191660 \nu^{9} + \cdots + 22\!\cdots\!08 ) / 294724713250240 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1105671611289 \nu^{11} + 2180810036423 \nu^{10} - 29871225937975 \nu^{9} + \cdots - 47\!\cdots\!72 ) / 294724713250240 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 6\beta_{5} + \beta_{4} + 6\beta_{2} - \beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 8\beta_{10} + \beta_{7} + 13\beta_{5} + \beta_{4} + 56\beta_{3} ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -15\beta_{9} + 8\beta_{8} + 15\beta_{6} - 83\beta_{2} + 15\beta _1 - 98 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 29 \beta_{11} - 127 \beta_{10} - 36 \beta_{9} + 29 \beta_{8} - 36 \beta_{7} + 43 \beta_{6} + \cdots - 784 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -106\beta_{11} - 323\beta_{10} - 267\beta_{7} - 1315\beta_{5} - 267\beta_{4} - 2254\beta_{3} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 876\beta_{9} - 477\beta_{8} - 1037\beta_{6} + 4815\beta_{2} - 2003\beta _1 + 12544 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1632 \beta_{11} + 6357 \beta_{10} + 4971 \beta_{9} - 1632 \beta_{8} + 4971 \beta_{7} - 5209 \beta_{6} + \cdots + 43918 ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7253\beta_{11} + 32845\beta_{10} + 19034\beta_{7} + 84713\beta_{5} + 22135\beta_{4} + 212100\beta_{3} ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -94879\beta_{9} + 25362\beta_{8} + 103559\beta_{6} - 367219\beta_{2} + 118399\beta _1 - 814758 ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 107577 \beta_{11} - 554835 \beta_{10} - 391392 \beta_{9} + 107577 \beta_{8} - 391392 \beta_{7} + \cdots - 3684464 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 384644 \beta_{11} - 2146229 \beta_{10} - 1830431 \beta_{7} - 6253953 \beta_{5} + \cdots - 14829990 \beta_{3} ) / 7 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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
−1.52662 + 2.64418i
−0.120924 + 0.209447i
−1.02943 + 1.78303i
−0.522449 + 0.904909i
2.14754 − 3.71965i
2.05188 − 3.55396i
−1.52662 − 2.64418i
−0.120924 − 0.209447i
−1.02943 − 1.78303i
−0.522449 − 0.904909i
2.14754 + 3.71965i
2.05188 + 3.55396i
−2.23372 − 3.86892i 4.90460 − 8.49501i −5.97903 + 10.3560i −2.50000 − 4.33013i −43.8221 0 17.6824 −34.6102 − 59.9466i −11.1686 + 19.3446i
116.2 −0.828031 − 1.43419i −0.166444 + 0.288289i 2.62873 − 4.55309i −2.50000 − 4.33013i 0.551283 0 −21.9552 13.4446 + 23.2867i −4.14016 + 7.17096i
116.3 −0.322324 − 0.558282i −2.09344 + 3.62594i 3.79221 − 6.56831i −2.50000 − 4.33013i 2.69906 0 −10.0465 4.73504 + 8.20133i −1.61162 + 2.79141i
116.4 0.184657 + 0.319836i 4.87035 − 8.43569i 3.93180 − 6.81008i −2.50000 − 4.33013i 3.59738 0 5.85867 −33.9406 − 58.7868i 0.923287 − 1.59918i
116.5 1.44043 + 2.49490i −1.44526 + 2.50327i −0.149696 + 0.259281i −2.50000 − 4.33013i −8.32721 0 22.1844 9.32244 + 16.1469i 7.20217 − 12.4745i
116.6 2.75899 + 4.77871i 1.93020 − 3.34320i −11.2240 + 19.4406i −2.50000 − 4.33013i 21.3015 0 −79.7239 6.04869 + 10.4766i 13.7949 − 23.8935i
226.1 −2.23372 + 3.86892i 4.90460 + 8.49501i −5.97903 − 10.3560i −2.50000 + 4.33013i −43.8221 0 17.6824 −34.6102 + 59.9466i −11.1686 − 19.3446i
226.2 −0.828031 + 1.43419i −0.166444 − 0.288289i 2.62873 + 4.55309i −2.50000 + 4.33013i 0.551283 0 −21.9552 13.4446 − 23.2867i −4.14016 − 7.17096i
226.3 −0.322324 + 0.558282i −2.09344 − 3.62594i 3.79221 + 6.56831i −2.50000 + 4.33013i 2.69906 0 −10.0465 4.73504 − 8.20133i −1.61162 − 2.79141i
226.4 0.184657 − 0.319836i 4.87035 + 8.43569i 3.93180 + 6.81008i −2.50000 + 4.33013i 3.59738 0 5.85867 −33.9406 + 58.7868i 0.923287 + 1.59918i
226.5 1.44043 − 2.49490i −1.44526 − 2.50327i −0.149696 − 0.259281i −2.50000 + 4.33013i −8.32721 0 22.1844 9.32244 − 16.1469i 7.20217 + 12.4745i
226.6 2.75899 − 4.77871i 1.93020 + 3.34320i −11.2240 − 19.4406i −2.50000 + 4.33013i 21.3015 0 −79.7239 6.04869 − 10.4766i 13.7949 + 23.8935i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.6
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.q 12
7.b odd 2 1 245.4.e.p 12
7.c even 3 1 245.4.a.o ✓ 6
7.c even 3 1 inner 245.4.e.q 12
7.d odd 6 1 245.4.a.p yes 6
7.d odd 6 1 245.4.e.p 12
21.g even 6 1 2205.4.a.ca 6
21.h odd 6 1 2205.4.a.bz 6
35.i odd 6 1 1225.4.a.bi 6
35.j even 6 1 1225.4.a.bj 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.o ✓ 6 7.c even 3 1
245.4.a.p yes 6 7.d odd 6 1
245.4.e.p 12 7.b odd 2 1
245.4.e.p 12 7.d odd 6 1
245.4.e.q 12 1.a even 1 1 trivial
245.4.e.q 12 7.c even 3 1 inner
1225.4.a.bi 6 35.i odd 6 1
1225.4.a.bj 6 35.j even 6 1
2205.4.a.bz 6 21.h odd 6 1
2205.4.a.ca 6 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 \(S_{4}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{12} - 2 T_{2}^{11} + 33 T_{2}^{10} + 2 T_{2}^{9} + 763 T_{2}^{8} - 252 T_{2}^{7} + 4662 T_{2}^{6} + \cdots + 784 \) Copy content Toggle raw display
\( T_{3}^{12} - 16 T_{3}^{11} + 244 T_{3}^{10} - 1320 T_{3}^{9} + 9523 T_{3}^{8} - 9236 T_{3}^{7} + \cdots + 2208196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 784 \) Copy content Toggle raw display
$3$ \( T^{12} - 16 T^{11} + \cdots + 2208196 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} + 168 T^{5} + \cdots + 12513937372)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 89\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{6} - 176 T^{5} + \cdots - 544215793700)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 73\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 144691772208184)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 388 T^{5} + \cdots + 440374360000)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 34\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 70\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 78\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 12\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 73\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 510868966648482)^{2} \) Copy content Toggle raw display
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