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Properties

Label 475.2.l.c
Level $475$
Weight $2$
Character orbit 475.l
Analytic conductor $3.793$
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] = [475,2,Mod(101,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.l (of order \(9\), degree \(6\), 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 15 x^{16} - 14 x^{15} + 72 x^{14} - 51 x^{13} + 231 x^{12} - 93 x^{11} + 438 x^{10} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{15} + \beta_{4}) q^{2} + (\beta_{15} + \beta_{11} + \cdots - \beta_{5}) q^{3}+ \cdots + (\beta_{16} + \beta_{15} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{15} + \beta_{4}) q^{2} + (\beta_{15} + \beta_{11} + \cdots - \beta_{5}) q^{3}+ \cdots + (2 \beta_{17} + 4 \beta_{16} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - 6 q^{6} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - 6 q^{6} + 6 q^{8} + 3 q^{9} + 18 q^{12} + 3 q^{13} - 12 q^{14} - 3 q^{16} - 24 q^{17} - 48 q^{18} - 21 q^{21} - 9 q^{22} + 9 q^{23} - 15 q^{24} + 3 q^{26} + 24 q^{27} + 12 q^{28} + 15 q^{29} - 18 q^{31} - 15 q^{32} + 33 q^{33} - 12 q^{34} + 75 q^{36} - 36 q^{37} + 33 q^{38} + 36 q^{39} - 30 q^{41} + 9 q^{42} + 36 q^{43} + 42 q^{44} + 9 q^{46} - 21 q^{47} - 33 q^{48} + 9 q^{49} - 45 q^{51} + 39 q^{52} + 12 q^{53} - 66 q^{54} - 72 q^{57} - 12 q^{58} + 18 q^{59} - 30 q^{61} + 24 q^{62} - 54 q^{63} + 36 q^{64} + 39 q^{66} - 51 q^{68} + 15 q^{69} - 12 q^{71} + 66 q^{72} - 24 q^{73} - 15 q^{74} - 33 q^{76} + 60 q^{77} + 48 q^{78} - 51 q^{79} + 27 q^{81} + 15 q^{82} + 48 q^{84} + 63 q^{86} + 15 q^{87} + 27 q^{88} - 54 q^{89} + 30 q^{91} + 42 q^{92} - 72 q^{93} + 30 q^{94} - 66 q^{96} - 27 q^{97} + 3 q^{98} - 93 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 15 x^{16} - 14 x^{15} + 72 x^{14} - 51 x^{13} + 231 x^{12} - 93 x^{11} + 438 x^{10} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2362215094 \nu^{17} - 176902925324 \nu^{16} + 523927730528 \nu^{15} + \cdots - 2601214274036 ) / 2269131592089 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 17485732004 \nu^{17} - 88297854546 \nu^{16} + 51093388053 \nu^{15} - 1446829381050 \nu^{14} + \cdots + 1249494973861 ) / 6807394776267 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31170269891 \nu^{17} + 67213359096 \nu^{16} + 77458832787 \nu^{15} + 1646328025947 \nu^{14} + \cdots + 2036474776337 ) / 6807394776267 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 150661038729 \nu^{17} + 22583410048 \nu^{16} - 1207986331145 \nu^{15} + \cdots + 9924998536452 ) / 6807394776267 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 271722740134 \nu^{17} - 484705722592 \nu^{16} + 3183507161538 \nu^{15} + \cdots - 2451247768033 ) / 6807394776267 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14393251623 \nu^{17} - 18779546977 \nu^{16} + 163974948404 \nu^{15} + 84439865154 \nu^{14} + \cdots - 139873473393 ) / 358283935593 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 99295558576 \nu^{17} + 144550753822 \nu^{16} - 988488698886 \nu^{15} + \cdots - 2743148781339 ) / 2269131592089 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 330462497810 \nu^{17} + 892333940472 \nu^{16} - 4644440262270 \nu^{15} + \cdots + 271722740134 ) / 6807394776267 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 335323421585 \nu^{17} - 1203461029533 \nu^{16} + 5639724390405 \nu^{15} + \cdots - 759862203949 ) / 6807394776267 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 169369527342 \nu^{17} - 512091105045 \nu^{16} + 2467349014493 \nu^{15} - 2179007028305 \nu^{14} + \cdots + 671726483761 ) / 2269131592089 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 183989570606 \nu^{17} + 559360956938 \nu^{16} - 2765267144950 \nu^{15} + \cdots + 2362215094 ) / 2269131592089 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 759862203949 \nu^{17} + 1944263190262 \nu^{16} - 10194472029702 \nu^{15} + \cdots + 422383778863 ) / 6807394776267 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1201752794172 \nu^{17} + 3351021374386 \nu^{16} - 17453288335442 \nu^{15} + \cdots - 12113076116529 ) / 6807394776267 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1764752036203 \nu^{17} + 5655888876310 \nu^{16} - 27296401124421 \nu^{15} + \cdots + 1723605145714 ) / 6807394776267 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 2009357177810 \nu^{17} + 5675262379841 \nu^{16} - 29025194492163 \nu^{15} + \cdots - 3751600650469 ) / 6807394776267 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 2169011193168 \nu^{17} + 7176985755886 \nu^{16} - 34139016228998 \nu^{15} + \cdots + 2187209095662 ) / 6807394776267 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - 5\beta_{12} - 3\beta_{10} + 3\beta_{9} + 3\beta_{7} + 2\beta_{5} - 2\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{17} + 3 \beta_{14} - 2 \beta_{11} - 3 \beta_{10} + 9 \beta_{9} - 5 \beta_{8} + \cdots - 11 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{17} + \beta_{15} + 3 \beta_{14} - \beta_{13} + 39 \beta_{12} - 12 \beta_{11} + 25 \beta_{10} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13 \beta_{17} + 2 \beta_{16} - \beta_{15} - 25 \beta_{14} - 3 \beta_{13} + 114 \beta_{12} - 13 \beta_{11} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 128 \beta_{17} + 13 \beta_{16} - 13 \beta_{15} - 128 \beta_{14} - 3 \beta_{13} + 87 \beta_{11} + \cdots + 375 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 269 \beta_{17} + 14 \beta_{16} - 27 \beta_{15} - 142 \beta_{14} + 27 \beta_{13} - 1181 \beta_{12} + \cdots - 46 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 455 \beta_{17} - 97 \beta_{16} + 44 \beta_{15} + 883 \beta_{14} + 141 \beta_{13} - 3823 \beta_{12} + \cdots - 126 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4309 \beta_{17} - 452 \beta_{16} + 452 \beta_{15} + 4309 \beta_{14} + 156 \beta_{13} + \cdots - 12254 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 9112 \beta_{17} - 499 \beta_{16} + 980 \beta_{15} + 4808 \beta_{14} - 980 \beta_{13} + 39530 \beta_{12} + \cdots + 1307 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 15570 \beta_{17} + 3111 \beta_{16} - 1650 \beta_{15} - 29285 \beta_{14} - 4761 \beta_{13} + 127231 \beta_{12} + \cdots + 4244 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 144639 \beta_{17} + 15399 \beta_{16} - 15399 \beta_{15} - 144639 \beta_{14} - 5307 \beta_{13} + \cdots + 410121 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 304266 \beta_{17} + 17220 \beta_{16} - 32396 \beta_{15} - 161859 \beta_{14} + 32396 \beta_{13} + \cdots - 43774 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 521594 \beta_{17} - 104569 \beta_{16} + 55469 \beta_{15} + 980815 \beta_{14} + 160038 \beta_{13} + \cdots - 140792 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 4841863 \beta_{17} - 515741 \beta_{16} + 515741 \beta_{15} + 4841863 \beta_{14} + \cdots - 13722757 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 10185670 \beta_{17} - 577063 \beta_{16} + 1085384 \beta_{15} + 5418926 \beta_{14} - 1085384 \beta_{13} + \cdots + 1462096 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(\beta_{13} - \beta_{16}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−0.644984 + 1.11715i
0.154946 − 0.268374i
0.816390 − 1.41403i
0.653994 + 1.13275i
−0.128481 − 0.222535i
−0.791558 − 1.37102i
−0.566185 − 0.980662i
0.394508 + 0.683308i
1.61137 + 2.79097i
0.653994 − 1.13275i
−0.128481 + 0.222535i
−0.791558 + 1.37102i
−0.644984 − 1.11715i
0.154946 + 0.268374i
0.816390 + 1.41403i
−0.566185 + 0.980662i
0.394508 − 0.683308i
1.61137 − 2.79097i
−1.75422 + 1.47196i 3.10785 + 1.13116i 0.563307 − 3.19467i 0 −7.11687 + 2.59033i −1.46955 + 2.54534i 1.42431 + 2.46697i 6.08105 + 5.10261i 0
101.2 −0.528654 + 0.443593i −0.652945 − 0.237653i −0.264596 + 1.50060i 0 0.450603 − 0.164006i −1.16732 + 2.02186i −1.21589 − 2.10598i −1.92827 − 1.61801i 0
101.3 0.484738 − 0.406743i 1.80387 + 0.656554i −0.277766 + 1.57529i 0 1.14145 − 0.415455i 2.04448 − 3.54114i 1.13887 + 1.97259i 0.524744 + 0.440313i 0
176.1 −0.289414 + 0.105338i 0.285463 − 1.61894i −1.45942 + 1.22460i 0 0.0879194 + 0.498616i 0.0445979 + 0.0772459i 0.601369 − 1.04160i 0.279590 + 0.101762i 0
176.2 1.18116 − 0.429906i −0.523072 + 2.96649i −0.321776 + 0.270002i 0 0.657481 + 3.72876i 1.86196 + 3.22501i −1.52095 + 2.63437i −5.70738 − 2.07732i 0
176.3 2.42733 − 0.883478i 0.0430161 − 0.243956i 3.57933 − 3.00342i 0 −0.111115 − 0.630167i −0.200820 − 0.347830i 3.45167 − 5.97847i 2.76141 + 1.00507i 0
226.1 −0.370282 + 2.09998i −1.70859 − 1.43367i −2.39340 − 0.871127i 0 3.64334 − 3.05712i 0.742812 + 1.28659i 0.583208 − 1.01015i 0.342900 + 1.94468i 0
226.2 −0.0366369 + 0.207778i −0.0612035 − 0.0513559i 1.83756 + 0.668816i 0 0.0129129 − 0.0108352i −0.843614 − 1.46118i −0.417271 + 0.722735i −0.519836 − 2.94814i 0
226.3 0.385975 − 2.18897i −0.794389 − 0.666572i −2.76323 − 1.00573i 0 −1.76572 + 1.48162i −1.01254 − 1.75377i −1.04532 + 1.81055i −0.334208 − 1.89539i 0
251.1 −0.289414 − 0.105338i 0.285463 + 1.61894i −1.45942 − 1.22460i 0 0.0879194 − 0.498616i 0.0445979 − 0.0772459i 0.601369 + 1.04160i 0.279590 − 0.101762i 0
251.2 1.18116 + 0.429906i −0.523072 − 2.96649i −0.321776 − 0.270002i 0 0.657481 − 3.72876i 1.86196 − 3.22501i −1.52095 − 2.63437i −5.70738 + 2.07732i 0
251.3 2.42733 + 0.883478i 0.0430161 + 0.243956i 3.57933 + 3.00342i 0 −0.111115 + 0.630167i −0.200820 + 0.347830i 3.45167 + 5.97847i 2.76141 − 1.00507i 0
301.1 −1.75422 − 1.47196i 3.10785 − 1.13116i 0.563307 + 3.19467i 0 −7.11687 − 2.59033i −1.46955 − 2.54534i 1.42431 − 2.46697i 6.08105 − 5.10261i 0
301.2 −0.528654 − 0.443593i −0.652945 + 0.237653i −0.264596 − 1.50060i 0 0.450603 + 0.164006i −1.16732 − 2.02186i −1.21589 + 2.10598i −1.92827 + 1.61801i 0
301.3 0.484738 + 0.406743i 1.80387 − 0.656554i −0.277766 − 1.57529i 0 1.14145 + 0.415455i 2.04448 + 3.54114i 1.13887 − 1.97259i 0.524744 − 0.440313i 0
351.1 −0.370282 − 2.09998i −1.70859 + 1.43367i −2.39340 + 0.871127i 0 3.64334 + 3.05712i 0.742812 − 1.28659i 0.583208 + 1.01015i 0.342900 − 1.94468i 0
351.2 −0.0366369 − 0.207778i −0.0612035 + 0.0513559i 1.83756 − 0.668816i 0 0.0129129 + 0.0108352i −0.843614 + 1.46118i −0.417271 − 0.722735i −0.519836 + 2.94814i 0
351.3 0.385975 + 2.18897i −0.794389 + 0.666572i −2.76323 + 1.00573i 0 −1.76572 − 1.48162i −1.01254 + 1.75377i −1.04532 − 1.81055i −0.334208 + 1.89539i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.l.c 18
5.b even 2 1 95.2.k.a ✓ 18
5.c odd 4 2 475.2.u.b 36
15.d odd 2 1 855.2.bs.c 18
19.e even 9 1 inner 475.2.l.c 18
19.e even 9 1 9025.2.a.cc 9
19.f odd 18 1 9025.2.a.cf 9
95.o odd 18 1 1805.2.a.s 9
95.p even 18 1 95.2.k.a ✓ 18
95.p even 18 1 1805.2.a.v 9
95.q odd 36 2 475.2.u.b 36
285.bd odd 18 1 855.2.bs.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.a ✓ 18 5.b even 2 1
95.2.k.a ✓ 18 95.p even 18 1
475.2.l.c 18 1.a even 1 1 trivial
475.2.l.c 18 19.e even 9 1 inner
475.2.u.b 36 5.c odd 4 2
475.2.u.b 36 95.q odd 36 2
855.2.bs.c 18 15.d odd 2 1
855.2.bs.c 18 285.bd odd 18 1
1805.2.a.s 9 95.o odd 18 1
1805.2.a.v 9 95.p even 18 1
9025.2.a.cc 9 19.e even 9 1
9025.2.a.cf 9 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 3 T_{2}^{17} + 6 T_{2}^{16} - 19 T_{2}^{15} + 33 T_{2}^{14} - 66 T_{2}^{13} + 268 T_{2}^{12} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 3 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{18} - 3 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{18} \) Copy content Toggle raw display
$7$ \( T^{18} + 27 T^{16} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( T^{18} + 60 T^{16} + \cdots + 597529 \) Copy content Toggle raw display
$13$ \( T^{18} - 3 T^{17} + \cdots + 2809 \) Copy content Toggle raw display
$17$ \( T^{18} + 24 T^{17} + \cdots + 2859481 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 2085474889 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 44269422409 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 47085094081 \) Copy content Toggle raw display
$37$ \( (T^{9} + 18 T^{8} + \cdots + 11125)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + 30 T^{17} + \cdots + 130321 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 2531341458361 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 2033313439249 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 44884235881 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 538419880441 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 3789756721 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 42166698100921 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 28\!\cdots\!49 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 585504633023209 \) Copy content Toggle raw display
$79$ \( T^{18} + 51 T^{17} + \cdots + 14691889 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 36\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 916651141561 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 3665788890625 \) Copy content Toggle raw display
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