Properties

Label 245.3.g.c
Level $245$
Weight $3$
Character orbit 245.g
Analytic conductor $6.676$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,3,Mod(148,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.148");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 245.g (of order \(4\), degree \(2\), 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: \(6.67576647683\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} + 10 x^{9} + 127 x^{8} - 160 x^{7} + 116 x^{6} + 288 x^{5} + 1471 x^{4} - 1438 x^{3} + 882 x^{2} + 1302 x + 961 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{10} q^{3} + (\beta_{9} - \beta_{4}) q^{4} + (\beta_{10} - \beta_{8} + \beta_{6} - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{6} + \beta_{5} + \beta_1) q^{6} + (\beta_{11} + 3 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 2 \beta_{2} + \cdots - 2) q^{8}+ \cdots + (\beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{10} q^{3} + (\beta_{9} - \beta_{4}) q^{4} + (\beta_{10} - \beta_{8} + \beta_{6} - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{6} + \beta_{5} + \beta_1) q^{6} + (\beta_{11} + 3 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 2 \beta_{2} + \cdots - 2) q^{8}+ \cdots + ( - 8 \beta_{11} - 14 \beta_{10} - 11 \beta_{9} - 8 \beta_{8} - 17 \beta_{7} + \cdots + 12 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 2 q^{3} + 4 q^{5} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} + 2 q^{3} + 4 q^{5} - 18 q^{8} - 14 q^{10} + 24 q^{11} + 46 q^{12} - 4 q^{13} + 26 q^{15} - 20 q^{16} + 48 q^{17} + 4 q^{18} - 36 q^{20} + 52 q^{22} + 86 q^{23} + 16 q^{25} - 140 q^{26} + 38 q^{27} - 64 q^{30} - 120 q^{31} - 130 q^{32} - 116 q^{33} - 248 q^{36} - 44 q^{37} - 16 q^{38} + 158 q^{40} + 8 q^{41} - 98 q^{43} + 104 q^{45} + 148 q^{46} + 208 q^{47} - 26 q^{48} + 290 q^{50} + 160 q^{51} + 288 q^{52} + 72 q^{53} + 104 q^{55} + 328 q^{57} + 2 q^{58} - 262 q^{60} - 308 q^{61} + 88 q^{62} - 132 q^{65} - 316 q^{66} - 198 q^{67} - 332 q^{68} - 396 q^{71} - 308 q^{72} - 380 q^{73} + 450 q^{75} - 200 q^{76} - 360 q^{78} + 324 q^{80} + 352 q^{81} + 818 q^{82} - 230 q^{83} + 72 q^{85} + 336 q^{86} + 214 q^{87} + 288 q^{88} + 60 q^{90} + 686 q^{92} + 68 q^{93} + 88 q^{95} - 816 q^{96} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 2 x^{10} + 10 x^{9} + 127 x^{8} - 160 x^{7} + 116 x^{6} + 288 x^{5} + 1471 x^{4} - 1438 x^{3} + 882 x^{2} + 1302 x + 961 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 207214373 \nu^{11} + 279378723 \nu^{10} + 114063165 \nu^{9} - 4004733585 \nu^{8} - 26173200100 \nu^{7} + 17928092994 \nu^{6} + \cdots - 5302990066201 ) / 1164340619252 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 93879285759 \nu^{11} + 369235883682 \nu^{10} - 1358053708882 \nu^{9} + 184250606282 \nu^{8} - 10592846805002 \nu^{7} + \cdots + 140052025311286 ) / 252661914377684 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15415712325 \nu^{11} - 47564103039 \nu^{10} + 57873135865 \nu^{9} + 129352506885 \nu^{8} + 1794004639500 \nu^{7} + \cdots + 9742730098325 ) / 36094559196812 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 539763819 \nu^{11} + 872313265 \nu^{10} - 800148915 \nu^{9} - 5283575025 \nu^{8} - 72554738598 \nu^{7} + 60189010940 \nu^{6} + \cdots - 477887082075 ) / 1164340619252 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5778495270 \nu^{11} - 7079783421 \nu^{10} - 5778035820 \nu^{9} + 40955406676 \nu^{8} + 763790965974 \nu^{7} + \cdots - 30615963709484 ) / 8150384334764 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9632453577 \nu^{11} + 43857648899 \nu^{10} - 41828250917 \nu^{9} - 79882524736 \nu^{8} - 1006934482154 \nu^{7} + \cdots - 6255467433054 ) / 8150384334764 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 434461978482 \nu^{11} - 1608806305797 \nu^{10} + 2599503442075 \nu^{9} + 1464550402835 \nu^{8} + 49154340379480 \nu^{7} + \cdots - 636714006900551 ) / 252661914377684 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 77078561625 \nu^{11} - 237820515195 \nu^{10} + 289365679325 \nu^{9} + 646762534425 \nu^{8} + 8970023197500 \nu^{7} + \cdots + 48713650491625 ) / 36094559196812 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 879701746289 \nu^{11} + 1463141390485 \nu^{10} - 1350638228022 \nu^{9} - 8791530667025 \nu^{8} + \cdots - 815549092870187 ) / 252661914377684 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 987809736035 \nu^{11} - 2370475867085 \nu^{10} + 2423053750490 \nu^{9} + 10633992315945 \nu^{8} + 119711268639884 \nu^{7} + \cdots + 15\!\cdots\!51 ) / 252661914377684 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 5\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 3\beta_{10} + 2\beta_{9} + 2\beta_{7} + 2\beta_{6} - 8\beta_{5} - 2\beta_{4} - 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{8} + 2\beta_{6} - 3\beta_{5} - \beta_{3} - 14\beta_{2} - 3\beta _1 - 45 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -33\beta_{9} - 14\beta_{8} - 30\beta_{7} + 30\beta_{6} + 40\beta_{4} - 40\beta_{3} - 33\beta_{2} - 83\beta _1 - 40 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{11} - 25 \beta_{10} - 187 \beta_{9} + 11 \beta_{8} - 50 \beta_{7} + 69 \beta_{5} + 501 \beta_{4} - 25 \beta_{3} - 69 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 159 \beta_{11} - 483 \beta_{10} - 465 \beta_{9} - 382 \beta_{7} - 382 \beta_{6} + 962 \beta_{5} + 658 \beta_{4} + 465 \beta_{2} + 658 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 76 \beta_{11} - 472 \beta_{10} - 76 \beta_{8} - 900 \beta_{6} + 1188 \beta_{5} + 472 \beta_{3} + 2504 \beta_{2} + 1188 \beta _1 + 6065 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6348 \beta_{9} + 1756 \beta_{8} + 4764 \beta_{7} - 4764 \beta_{6} - 9972 \beta_{4} + 5864 \beta_{3} + 6348 \beta_{2} + 11733 \beta _1 + 9972 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 172 \beta_{11} + 7760 \beta_{10} + 33613 \beta_{9} - 172 \beta_{8} + 14260 \beta_{7} - 18388 \beta_{5} - 76285 \beta_{4} + 7760 \beta_{3} + 18388 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 19697 \beta_{11} + 72663 \beta_{10} + 85958 \beta_{9} + 59982 \beta_{7} + 59982 \beta_{6} - 147344 \beta_{5} - 144734 \beta_{4} - 85958 \beta_{2} - 144734 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1
−2.16577 + 2.16577i
−1.28771 + 1.28771i
−0.428659 + 0.428659i
0.901678 0.901678i
1.38509 1.38509i
2.59538 2.59538i
−2.16577 2.16577i
−1.28771 1.28771i
−0.428659 0.428659i
0.901678 + 0.901678i
1.38509 + 1.38509i
2.59538 + 2.59538i
−2.16577 + 2.16577i 1.73090 + 1.73090i 5.38116i 4.47562 2.22908i −7.49746 0 2.99127 + 2.99127i 3.00800i −4.86551 + 14.5209i
148.2 −1.28771 + 1.28771i 0.143294 + 0.143294i 0.683608i −3.65165 + 3.41547i −0.369042 0 −6.03113 6.03113i 8.95893i 0.304133 9.10040i
148.3 −0.428659 + 0.428659i −2.92848 2.92848i 3.63250i 0.392189 4.98460i 2.51064 0 −3.27174 3.27174i 8.15197i 1.96858 + 2.30481i
148.4 0.901678 0.901678i 2.93984 + 2.93984i 2.37395i 4.93168 + 0.823729i 5.30157 0 5.74725 + 5.74725i 8.28526i 5.18953 3.70405i
148.5 1.38509 1.38509i −1.92141 1.92141i 0.163078i 0.431739 + 4.98133i −5.32264 0 5.76622 + 5.76622i 1.61634i 7.49756 + 6.30156i
148.6 2.59538 2.59538i 1.03587 + 1.03587i 9.47199i −4.57958 2.00685i 5.37694 0 −14.2019 14.2019i 6.85396i −17.0943 + 6.67721i
197.1 −2.16577 2.16577i 1.73090 1.73090i 5.38116i 4.47562 + 2.22908i −7.49746 0 2.99127 2.99127i 3.00800i −4.86551 14.5209i
197.2 −1.28771 1.28771i 0.143294 0.143294i 0.683608i −3.65165 3.41547i −0.369042 0 −6.03113 + 6.03113i 8.95893i 0.304133 + 9.10040i
197.3 −0.428659 0.428659i −2.92848 + 2.92848i 3.63250i 0.392189 + 4.98460i 2.51064 0 −3.27174 + 3.27174i 8.15197i 1.96858 2.30481i
197.4 0.901678 + 0.901678i 2.93984 2.93984i 2.37395i 4.93168 0.823729i 5.30157 0 5.74725 5.74725i 8.28526i 5.18953 + 3.70405i
197.5 1.38509 + 1.38509i −1.92141 + 1.92141i 0.163078i 0.431739 4.98133i −5.32264 0 5.76622 5.76622i 1.61634i 7.49756 6.30156i
197.6 2.59538 + 2.59538i 1.03587 1.03587i 9.47199i −4.57958 + 2.00685i 5.37694 0 −14.2019 + 14.2019i 6.85396i −17.0943 6.67721i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 148.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.3.g.c 12
5.c odd 4 1 inner 245.3.g.c 12
7.b odd 2 1 245.3.g.b 12
7.c even 3 2 35.3.l.a 24
7.d odd 6 2 245.3.m.b 24
21.h odd 6 2 315.3.ca.a 24
35.f even 4 1 245.3.g.b 12
35.j even 6 2 175.3.p.c 24
35.k even 12 2 245.3.m.b 24
35.l odd 12 2 35.3.l.a 24
35.l odd 12 2 175.3.p.c 24
105.x even 12 2 315.3.ca.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.l.a 24 7.c even 3 2
35.3.l.a 24 35.l odd 12 2
175.3.p.c 24 35.j even 6 2
175.3.p.c 24 35.l odd 12 2
245.3.g.b 12 7.b odd 2 1
245.3.g.b 12 35.f even 4 1
245.3.g.c 12 1.a even 1 1 trivial
245.3.g.c 12 5.c odd 4 1 inner
245.3.m.b 24 7.d odd 6 2
245.3.m.b 24 35.k even 12 2
315.3.ca.a 24 21.h odd 6 2
315.3.ca.a 24 105.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{12} - 2 T_{2}^{11} + 2 T_{2}^{10} + 10 T_{2}^{9} + 127 T_{2}^{8} - 160 T_{2}^{7} + 116 T_{2}^{6} + 288 T_{2}^{5} + 1471 T_{2}^{4} - 1438 T_{2}^{3} + 882 T_{2}^{2} + 1302 T_{2} + 961 \) Copy content Toggle raw display
\( T_{3}^{12} - 2 T_{3}^{11} + 2 T_{3}^{10} - 2 T_{3}^{9} + 346 T_{3}^{8} - 698 T_{3}^{7} + 706 T_{3}^{6} - 710 T_{3}^{5} + 14857 T_{3}^{4} - 32980 T_{3}^{3} + 36992 T_{3}^{2} - 9248 T_{3} + 1156 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + 2 T^{10} + 10 T^{9} + \cdots + 961 \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + 2 T^{10} - 2 T^{9} + \cdots + 1156 \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} - 52 T^{9} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 12 T^{5} - 251 T^{4} + \cdots + 52100)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} + \cdots + 12744603664 \) Copy content Toggle raw display
$17$ \( T^{12} - 48 T^{11} + \cdots + 109735837696 \) Copy content Toggle raw display
$19$ \( T^{12} + 1578 T^{10} + \cdots + 2665395820816 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 445115941168441 \) Copy content Toggle raw display
$29$ \( T^{12} + 3620 T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + 60 T^{5} + 144 T^{4} + \cdots + 129716824)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 202458749440000 \) Copy content Toggle raw display
$41$ \( (T^{6} - 4 T^{5} - 5709 T^{4} + \cdots - 1783280375)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 98 T^{11} + \cdots + 47\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{12} - 208 T^{11} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} - 72 T^{11} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{12} + 9840 T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + 154 T^{5} + 3482 T^{4} + \cdots + 247538)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 198 T^{11} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} + 198 T^{5} + \cdots + 35847684344)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 380 T^{11} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{12} + 29064 T^{10} + \cdots + 84\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{12} + 230 T^{11} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{12} + 43772 T^{10} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{12} + 36 T^{11} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
show more
show less