Properties

Label 105.4.i.e
Level $105$
Weight $4$
Character orbit 105.i
Analytic conductor $6.195$
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] = [105,4,Mod(16,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.16");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.i (of order \(3\), 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: \(6.19520055060\)
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: \( x^{10} - x^{9} + 31x^{8} + 26x^{7} + 738x^{6} + 352x^{5} + 5008x^{4} + 5368x^{3} + 26728x^{2} + 13776x + 7056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 \beta_{4} q^{3} + (\beta_{8} - 4 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - 5 \beta_{4} + 5) q^{5} + 3 \beta_{3} q^{6} + (\beta_{9} + \beta_{6} - 3 \beta_{4} + \beta_{2} + \beta_1 + 7) q^{7} + (\beta_{9} - \beta_{7} + 2 \beta_{5} + 5 \beta_{3} - \beta_{2} + \beta_1 - 12) q^{8} + (9 \beta_{4} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 \beta_{4} q^{3} + (\beta_{8} - 4 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - 5 \beta_{4} + 5) q^{5} + 3 \beta_{3} q^{6} + (\beta_{9} + \beta_{6} - 3 \beta_{4} + \beta_{2} + \beta_1 + 7) q^{7} + (\beta_{9} - \beta_{7} + 2 \beta_{5} + 5 \beta_{3} - \beta_{2} + \beta_1 - 12) q^{8} + (9 \beta_{4} - 9) q^{9} + (5 \beta_{3} + 5 \beta_1) q^{10} + (\beta_{9} - 4 \beta_{8} + 2 \beta_{6} - \beta_{5} + 6 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_1) q^{11} + ( - 3 \beta_{8} + 12 \beta_{4} - 3 \beta_1 - 12) q^{12} + ( - 2 \beta_{9} + 2 \beta_{7} + \beta_{6} - 3 \beta_{5} - 4 \beta_{3} - 3 \beta_{2} + \cdots - 7) q^{13}+ \cdots + (9 \beta_{9} - 9 \beta_{7} - 9 \beta_{6} + 9 \beta_{5} - 9 \beta_{3} - 36 \beta_{2} + \cdots - 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - 15 q^{3} - 21 q^{4} + 25 q^{5} - 6 q^{6} + 56 q^{7} - 138 q^{8} - 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} - 15 q^{3} - 21 q^{4} + 25 q^{5} - 6 q^{6} + 56 q^{7} - 138 q^{8} - 45 q^{9} - 5 q^{10} + 33 q^{11} - 63 q^{12} - 46 q^{13} - 73 q^{14} - 150 q^{15} - 113 q^{16} + 136 q^{17} + 9 q^{18} + 39 q^{19} - 210 q^{20} - 147 q^{21} - 174 q^{22} + 133 q^{23} + 207 q^{24} - 125 q^{25} + 73 q^{26} + 270 q^{27} - 809 q^{28} + 544 q^{29} - 15 q^{30} + 430 q^{31} + 573 q^{32} + 99 q^{33} - 744 q^{34} + 35 q^{35} + 378 q^{36} - 3 q^{37} + 837 q^{38} + 69 q^{39} - 345 q^{40} - 1254 q^{41} - 372 q^{42} + 216 q^{43} + 1809 q^{44} + 225 q^{45} - 1637 q^{46} + 553 q^{47} + 678 q^{48} - 386 q^{49} - 50 q^{50} + 408 q^{51} + 1047 q^{52} + 1135 q^{53} + 27 q^{54} + 330 q^{55} - 1356 q^{56} - 234 q^{57} + 2564 q^{58} + 332 q^{59} + 315 q^{60} + 584 q^{61} - 3124 q^{62} - 63 q^{63} + 2274 q^{64} - 115 q^{65} + 261 q^{66} - 412 q^{67} - 1712 q^{68} - 798 q^{69} - 985 q^{70} - 284 q^{71} + 621 q^{72} + 2074 q^{73} - 605 q^{74} - 375 q^{75} + 18 q^{76} - 751 q^{77} - 438 q^{78} - 28 q^{79} + 565 q^{80} - 405 q^{81} - 1515 q^{82} - 1680 q^{83} - 1077 q^{84} + 1360 q^{85} - 40 q^{86} - 816 q^{87} - 4181 q^{88} + 2978 q^{89} + 90 q^{90} - 2736 q^{91} + 1062 q^{92} + 1290 q^{93} + 843 q^{94} - 195 q^{95} + 1719 q^{96} - 4336 q^{97} - 5183 q^{98} - 594 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 31x^{8} + 26x^{7} + 738x^{6} + 352x^{5} + 5008x^{4} + 5368x^{3} + 26728x^{2} + 13776x + 7056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 547053 \nu^{9} - 39677237 \nu^{8} + 74096045 \nu^{7} - 1011660144 \nu^{6} - 43979588 \nu^{5} - 28238719808 \nu^{4} + \cdots - 1760861059296 ) / 152584070032 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1229309 \nu^{9} - 547053 \nu^{8} + 1021605 \nu^{7} - 105036474 \nu^{6} - 606372 \nu^{5} - 389343552 \nu^{4} + 21692996784 \nu^{3} + \cdots + 78821785392 ) / 152584070032 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 234588647 \nu^{9} - 208773158 \nu^{8} + 7283736170 \nu^{7} + 6077851117 \nu^{6} + 175332187440 \nu^{5} + 82587937556 \nu^{4} + \cdots + 3259806136104 ) / 3204265470672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 407749371 \nu^{9} + 1645652265 \nu^{8} - 17320513781 \nu^{7} + 30310144474 \nu^{6} - 363550408560 \nu^{5} + \cdots + 4857556590096 ) / 1068088490224 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 253478250 \nu^{9} + 501821041 \nu^{8} + 6186518193 \nu^{7} + 32231489069 \nu^{6} + 183243487894 \nu^{5} + 632905819744 \nu^{4} + \cdots + 14110741960056 ) / 534044245112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 408363607 \nu^{9} - 221600804 \nu^{8} + 11444006564 \nu^{7} + 19321193051 \nu^{6} + 265397721032 \nu^{5} + 303713239568 \nu^{4} + \cdots + 6396445900488 ) / 534044245112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 475394561 \nu^{9} - 554501960 \nu^{8} + 14823232880 \nu^{7} + 8982519389 \nu^{6} + 350512568624 \nu^{5} + 67703058216 \nu^{4} + \cdots + 80722315800 ) / 534044245112 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 908381875 \nu^{9} - 1965915003 \nu^{8} + 28948768535 \nu^{7} - 6809581088 \nu^{6} + 628838769376 \nu^{5} - 386225403248 \nu^{4} + \cdots - 4008994682112 ) / 534044245112 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 12\beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{7} + 2\beta_{5} + 21\beta_{3} - \beta_{2} + \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{9} - 27\beta_{8} - 4\beta_{7} + 3\beta_{6} + \beta_{5} + 240\beta_{4} + 3\beta_{3} - 42\beta _1 - 240 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{9} - 55 \beta_{8} - 31 \beta_{7} + 35 \beta_{6} - 33 \beta_{5} + 492 \beta_{4} - 476 \beta_{3} + 55 \beta_{2} - 509 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -121\beta_{9} + 121\beta_{7} + 62\beta_{6} - 180\beta_{5} - 1569\beta_{3} + 699\beta_{2} - 121\beta _1 + 5700 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1059 \beta_{9} + 2047 \beta_{8} + 1764 \beta_{7} - 941 \beta_{6} - 823 \beta_{5} - 17004 \beta_{4} - 941 \beta_{3} + 12280 \beta _1 + 17004 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1646 \beta_{9} + 18395 \beta_{8} + 2283 \beta_{7} - 5575 \beta_{6} + 3929 \beta_{5} - 146652 \beta_{4} + 46200 \beta_{3} - 18395 \beta_{2} + 50129 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 26253 \beta_{9} - 26253 \beta_{7} - 4566 \beta_{6} + 47940 \beta_{5} + 356365 \beta_{3} - 67119 \beta_{2} + 26253 \beta _1 - 544524 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
−2.13900 + 3.70486i
−1.18645 + 2.05499i
−0.272258 + 0.471565i
1.41747 2.45514i
2.68023 4.64230i
−2.13900 3.70486i
−1.18645 2.05499i
−0.272258 0.471565i
1.41747 + 2.45514i
2.68023 + 4.64230i
−2.13900 + 3.70486i −1.50000 2.59808i −5.15064 8.92117i 2.50000 4.33013i 12.8340 17.2464 + 6.74999i 9.84488 −4.50000 + 7.79423i 10.6950 + 18.5243i
16.2 −1.18645 + 2.05499i −1.50000 2.59808i 1.18467 + 2.05191i 2.50000 4.33013i 7.11870 −10.8596 15.0023i −24.6054 −4.50000 + 7.79423i 5.93225 + 10.2750i
16.3 −0.272258 + 0.471565i −1.50000 2.59808i 3.85175 + 6.67143i 2.50000 4.33013i 1.63355 −0.983423 + 18.4941i −8.55081 −4.50000 + 7.79423i 1.36129 + 2.35782i
16.4 1.41747 2.45514i −1.50000 2.59808i −0.0184684 0.0319883i 2.50000 4.33013i −8.50485 4.61394 17.9363i 22.5749 −4.50000 + 7.79423i −7.08737 12.2757i
16.5 2.68023 4.64230i −1.50000 2.59808i −10.3673 17.9567i 2.50000 4.33013i −16.0814 17.9827 4.42983i −68.2635 −4.50000 + 7.79423i −13.4012 23.2115i
46.1 −2.13900 3.70486i −1.50000 + 2.59808i −5.15064 + 8.92117i 2.50000 + 4.33013i 12.8340 17.2464 6.74999i 9.84488 −4.50000 7.79423i 10.6950 18.5243i
46.2 −1.18645 2.05499i −1.50000 + 2.59808i 1.18467 2.05191i 2.50000 + 4.33013i 7.11870 −10.8596 + 15.0023i −24.6054 −4.50000 7.79423i 5.93225 10.2750i
46.3 −0.272258 0.471565i −1.50000 + 2.59808i 3.85175 6.67143i 2.50000 + 4.33013i 1.63355 −0.983423 18.4941i −8.55081 −4.50000 7.79423i 1.36129 2.35782i
46.4 1.41747 + 2.45514i −1.50000 + 2.59808i −0.0184684 + 0.0319883i 2.50000 + 4.33013i −8.50485 4.61394 + 17.9363i 22.5749 −4.50000 7.79423i −7.08737 + 12.2757i
46.5 2.68023 + 4.64230i −1.50000 + 2.59808i −10.3673 + 17.9567i 2.50000 + 4.33013i −16.0814 17.9827 + 4.42983i −68.2635 −4.50000 7.79423i −13.4012 + 23.2115i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.5
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 105.4.i.e 10
3.b odd 2 1 315.4.j.f 10
7.c even 3 1 inner 105.4.i.e 10
7.c even 3 1 735.4.a.y 5
7.d odd 6 1 735.4.a.x 5
21.g even 6 1 2205.4.a.bv 5
21.h odd 6 1 315.4.j.f 10
21.h odd 6 1 2205.4.a.bw 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.i.e 10 1.a even 1 1 trivial
105.4.i.e 10 7.c even 3 1 inner
315.4.j.f 10 3.b odd 2 1
315.4.j.f 10 21.h odd 6 1
735.4.a.x 5 7.d odd 6 1
735.4.a.y 5 7.c even 3 1
2205.4.a.bv 5 21.g even 6 1
2205.4.a.bw 5 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - T_{2}^{9} + 31 T_{2}^{8} + 26 T_{2}^{7} + 738 T_{2}^{6} + 352 T_{2}^{5} + 5008 T_{2}^{4} + 5368 T_{2}^{3} + 26728 T_{2}^{2} + 13776 T_{2} + 7056 \) acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - T^{9} + 31 T^{8} + 26 T^{7} + \cdots + 7056 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{5} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} - 56 T^{9} + \cdots + 4747561509943 \) Copy content Toggle raw display
$11$ \( T^{10} - 33 T^{9} + \cdots + 1986036295824 \) Copy content Toggle raw display
$13$ \( (T^{5} + 23 T^{4} - 5870 T^{3} + \cdots - 25478281)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} - 136 T^{9} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} - 39 T^{9} + \cdots + 22\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{10} - 133 T^{9} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} - 272 T^{4} + \cdots - 12806983008)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 430 T^{9} + \cdots + 40\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{10} + 3 T^{9} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( (T^{5} + 627 T^{4} + \cdots + 45007298820)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 108 T^{4} + \cdots + 1217258521160)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} - 553 T^{9} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{10} - 1135 T^{9} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{10} - 332 T^{9} + \cdots + 84\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{10} - 584 T^{9} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + 412 T^{9} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + 142 T^{4} + \cdots - 27239555642904)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} - 2074 T^{9} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{10} + 28 T^{9} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{5} + 840 T^{4} + \cdots + 59475285699552)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 2978 T^{9} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{5} + 2168 T^{4} + \cdots + 701198534208512)^{2} \) Copy content Toggle raw display
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