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Properties

Label 225.3.j.b
Level $225$
Weight $3$
Character orbit 225.j
Analytic conductor $6.131$
Analytic rank $0$
Dimension $16$
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] = [225,3,Mod(101,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.j (of order \(6\), 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: \(6.13080594811\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{11} - \beta_{8}) q^{3} + (\beta_{15} - 2 \beta_{8} - \beta_{3} + \cdots + 2) q^{4}+ \cdots + ( - \beta_{15} - \beta_{14} - \beta_{13} + \cdots + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + ( - \beta_{11} - \beta_{8}) q^{3} + (\beta_{15} - 2 \beta_{8} - \beta_{3} + \cdots + 2) q^{4}+ \cdots + ( - 19 \beta_{15} - 4 \beta_{14} + \cdots - 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} + 16 q^{4} - 22 q^{6} - 2 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} + 16 q^{4} - 22 q^{6} - 2 q^{7} + 8 q^{9} - 18 q^{11} + 22 q^{12} + 10 q^{13} - 54 q^{14} - 32 q^{16} + 8 q^{18} - 52 q^{19} + 72 q^{21} + 24 q^{22} + 54 q^{23} + 108 q^{24} - 34 q^{27} - 32 q^{28} - 54 q^{29} + 32 q^{31} - 216 q^{32} - 62 q^{33} + 54 q^{34} - 86 q^{36} - 44 q^{37} - 252 q^{38} + 160 q^{39} + 144 q^{41} + 270 q^{42} + 124 q^{43} - 108 q^{46} + 216 q^{47} + 172 q^{48} - 54 q^{49} - 106 q^{51} - 62 q^{52} - 316 q^{54} - 18 q^{56} + 236 q^{57} - 90 q^{58} - 486 q^{59} + 62 q^{61} + 132 q^{63} + 256 q^{64} + 208 q^{66} - 14 q^{67} + 288 q^{68} + 90 q^{69} - 804 q^{72} + 268 q^{73} + 540 q^{74} - 106 q^{76} - 702 q^{77} - 290 q^{78} - 40 q^{79} - 112 q^{81} + 204 q^{82} - 522 q^{83} + 714 q^{84} + 54 q^{86} - 106 q^{87} - 144 q^{88} + 136 q^{91} + 1332 q^{92} - 90 q^{93} - 150 q^{94} + 166 q^{96} + 142 q^{97} - 824 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} + 43 \nu^{12} + 715 \nu^{10} + 5777 \nu^{8} + 23051 \nu^{6} + 39821 \nu^{4} + 25605 \nu^{2} + \cdots + 28755 ) / 4752 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{14} - 43 \nu^{12} - 715 \nu^{10} - 5777 \nu^{8} - 23051 \nu^{6} - 39821 \nu^{4} - 20853 \nu^{2} + \cdots - 243 ) / 4752 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13 \nu^{15} + 13 \nu^{14} - 643 \nu^{13} + 643 \nu^{12} - 12847 \nu^{11} + 12847 \nu^{10} + \cdots + 1290303 ) / 175824 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7 \nu^{15} - 68 \nu^{14} - 369 \nu^{13} - 3056 \nu^{12} - 7299 \nu^{11} - 53504 \nu^{10} + \cdots - 532116 ) / 175824 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8 \nu^{14} - 353 \nu^{12} - 6275 \nu^{10} - 58567 \nu^{8} - 308113 \nu^{6} - 860038 \nu^{4} + \cdots - 99036 ) / 21978 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 81 \nu^{15} - 70 \nu^{14} - 3699 \nu^{13} - 3394 \nu^{12} - 66351 \nu^{11} - 63148 \nu^{10} + \cdots + 1210356 ) / 175824 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{15} - 143 \nu^{13} - 2693 \nu^{11} - 25397 \nu^{9} - 125029 \nu^{7} - 304045 \nu^{5} + \cdots + 2376 ) / 4752 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 97 \nu^{15} + 116 \nu^{14} + 4553 \nu^{13} + 4952 \nu^{12} + 85043 \nu^{11} + 80720 \nu^{10} + \cdots + 167292 ) / 175824 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 45 \nu^{15} - 56 \nu^{14} - 2129 \nu^{13} - 2693 \nu^{12} - 39797 \nu^{11} - 50696 \nu^{10} + \cdots - 58887 ) / 87912 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 67 \nu^{15} - 5 \nu^{14} - 3257 \nu^{13} - 290 \nu^{12} - 62816 \nu^{11} - 6572 \nu^{10} + \cdots - 202257 ) / 87912 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 157 \nu^{15} + \nu^{14} - 7737 \nu^{13} + 613 \nu^{12} - 151623 \nu^{11} + 22027 \nu^{10} + \cdots + 1145745 ) / 175824 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 157 \nu^{15} - 12 \nu^{14} - 7737 \nu^{13} - 918 \nu^{12} - 151623 \nu^{11} - 22788 \nu^{10} + \cdots + 1425870 ) / 175824 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 161 \nu^{15} + 16 \nu^{14} + 7747 \nu^{13} + 632 \nu^{12} + 147749 \nu^{11} + 9072 \nu^{10} + \cdots - 92304 ) / 58608 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 17 \nu^{15} - 815 \nu^{13} - 15443 \nu^{11} - 146605 \nu^{9} - 727123 \nu^{7} - 1784449 \nu^{5} + \cdots + 14256 ) / 4752 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{13} - \beta_{11} - \beta_{9} + 4\beta_{8} - \beta_{7} - 11\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + 2\beta_{12} + 2\beta_{7} + \beta_{6} + \beta_{5} - 19\beta_{3} - 13\beta_{2} + 4\beta _1 + 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} - 16 \beta_{14} - 18 \beta_{13} + 3 \beta_{12} + 17 \beta_{11} + 2 \beta_{10} + 17 \beta_{9} + \cdots + 59 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 29 \beta_{14} + 4 \beta_{13} - 44 \beta_{12} - 18 \beta_{11} + 6 \beta_{9} - 12 \beta_{8} + \cdots - 675 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 46 \beta_{15} + 219 \beta_{14} + 267 \beta_{13} - 79 \beta_{12} - 254 \beta_{11} - 34 \beta_{10} + \cdots - 932 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 588 \beta_{14} - 108 \beta_{13} + 774 \beta_{12} + 510 \beta_{11} - 150 \beta_{9} + 360 \beta_{8} + \cdots + 8037 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 720 \beta_{15} - 2984 \beta_{14} - 3806 \beta_{13} + 1512 \beta_{12} + 3686 \beta_{11} + 468 \beta_{10} + \cdots + 13764 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 10310 \beta_{14} + 2088 \beta_{13} - 12490 \beta_{12} - 10218 \beta_{11} + 2694 \beta_{9} + \cdots - 99348 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 9628 \beta_{15} + 41261 \beta_{14} + 53703 \beta_{13} - 25494 \beta_{12} - 53005 \beta_{11} + \cdots - 197521 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 167293 \beta_{14} - 35528 \beta_{13} + 192484 \beta_{12} + 177630 \beta_{11} - 42750 \beta_{9} + \cdots + 1263516 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 118250 \beta_{15} - 576912 \beta_{14} - 755544 \beta_{13} + 402527 \beta_{12} + 758659 \beta_{11} + \cdots + 2794303 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2590167 \beta_{14} + 566496 \beta_{13} - 2883258 \beta_{12} - 2863572 \beta_{11} + 640284 \beta_{9} + \cdots - 16432077 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1374714 \beta_{15} + 8112283 \beta_{14} + 10619317 \beta_{13} - 6113709 \beta_{12} - 10818934 \beta_{11} + \cdots - 39231234 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1 - \beta_{8}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
− 3.09125i
− 2.82877i
− 1.83391i
− 0.692902i
0.0476108i
1.39204i
3.27064i
3.73655i
3.09125i
2.82877i
1.83391i
0.692902i
− 0.0476108i
− 1.39204i
− 3.27064i
− 3.73655i
−2.67710 − 1.54563i 2.82138 − 1.01971i 2.77793 + 4.81151i 0 −9.12922 − 1.63094i −1.10296 + 1.91037i − 4.80953i 6.92040 − 5.75396i 0
101.2 −2.44978 − 1.41438i −0.110987 + 2.99795i 2.00096 + 3.46576i 0 4.51214 − 7.18734i 3.97472 − 6.88441i − 0.00541780i −8.97536 − 0.665467i 0
101.3 −1.58822 − 0.916957i −0.822398 − 2.88508i −0.318381 − 0.551452i 0 −1.33934 + 5.33623i −3.23398 + 5.60142i 8.50342i −7.64732 + 4.74536i 0
101.4 −0.600071 − 0.346451i −2.96601 − 0.450312i −1.75994 − 3.04831i 0 1.62381 + 1.29780i 6.14112 − 10.6367i 5.21055i 8.59444 + 2.67126i 0
101.5 0.0412321 + 0.0238054i 1.76580 + 2.42528i −1.99887 − 3.46214i 0 0.0150729 + 0.142035i −1.90756 + 3.30399i − 0.380778i −2.76393 + 8.56508i 0
101.6 1.20554 + 0.696021i −2.33148 + 1.88791i −1.03111 − 1.78593i 0 −4.12473 + 0.653207i −4.41004 + 7.63842i − 8.43886i 1.87156 − 8.80325i 0
101.7 2.83245 + 1.63532i 2.26486 + 1.96733i 3.34853 + 5.79983i 0 3.19790 + 9.27615i 3.16931 − 5.48940i 8.82112i 1.25919 + 8.91148i 0
101.8 3.23594 + 1.86827i −2.62117 − 1.45927i 4.98088 + 8.62715i 0 −5.75562 − 9.61919i −3.63061 + 6.28840i 22.2764i 4.74103 + 7.65001i 0
176.1 −2.67710 + 1.54563i 2.82138 + 1.01971i 2.77793 − 4.81151i 0 −9.12922 + 1.63094i −1.10296 − 1.91037i 4.80953i 6.92040 + 5.75396i 0
176.2 −2.44978 + 1.41438i −0.110987 − 2.99795i 2.00096 − 3.46576i 0 4.51214 + 7.18734i 3.97472 + 6.88441i 0.00541780i −8.97536 + 0.665467i 0
176.3 −1.58822 + 0.916957i −0.822398 + 2.88508i −0.318381 + 0.551452i 0 −1.33934 − 5.33623i −3.23398 − 5.60142i − 8.50342i −7.64732 − 4.74536i 0
176.4 −0.600071 + 0.346451i −2.96601 + 0.450312i −1.75994 + 3.04831i 0 1.62381 − 1.29780i 6.14112 + 10.6367i − 5.21055i 8.59444 − 2.67126i 0
176.5 0.0412321 − 0.0238054i 1.76580 − 2.42528i −1.99887 + 3.46214i 0 0.0150729 − 0.142035i −1.90756 − 3.30399i 0.380778i −2.76393 − 8.56508i 0
176.6 1.20554 − 0.696021i −2.33148 − 1.88791i −1.03111 + 1.78593i 0 −4.12473 − 0.653207i −4.41004 − 7.63842i 8.43886i 1.87156 + 8.80325i 0
176.7 2.83245 − 1.63532i 2.26486 − 1.96733i 3.34853 − 5.79983i 0 3.19790 − 9.27615i 3.16931 + 5.48940i − 8.82112i 1.25919 − 8.91148i 0
176.8 3.23594 − 1.86827i −2.62117 + 1.45927i 4.98088 − 8.62715i 0 −5.75562 + 9.61919i −3.63061 − 6.28840i − 22.2764i 4.74103 − 7.65001i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.3.j.b 16
3.b odd 2 1 675.3.j.b 16
5.b even 2 1 45.3.i.a ✓ 16
5.c odd 4 2 225.3.i.b 32
9.c even 3 1 675.3.j.b 16
9.d odd 6 1 inner 225.3.j.b 16
15.d odd 2 1 135.3.i.a 16
15.e even 4 2 675.3.i.c 32
20.d odd 2 1 720.3.bs.c 16
45.h odd 6 1 45.3.i.a ✓ 16
45.h odd 6 1 405.3.c.a 16
45.j even 6 1 135.3.i.a 16
45.j even 6 1 405.3.c.a 16
45.k odd 12 2 675.3.i.c 32
45.l even 12 2 225.3.i.b 32
60.h even 2 1 2160.3.bs.c 16
180.n even 6 1 720.3.bs.c 16
180.p odd 6 1 2160.3.bs.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.i.a ✓ 16 5.b even 2 1
45.3.i.a ✓ 16 45.h odd 6 1
135.3.i.a 16 15.d odd 2 1
135.3.i.a 16 45.j even 6 1
225.3.i.b 32 5.c odd 4 2
225.3.i.b 32 45.l even 12 2
225.3.j.b 16 1.a even 1 1 trivial
225.3.j.b 16 9.d odd 6 1 inner
405.3.c.a 16 45.h odd 6 1
405.3.c.a 16 45.j even 6 1
675.3.i.c 32 15.e even 4 2
675.3.i.c 32 45.k odd 12 2
675.3.j.b 16 3.b odd 2 1
675.3.j.b 16 9.c even 3 1
720.3.bs.c 16 20.d odd 2 1
720.3.bs.c 16 180.n even 6 1
2160.3.bs.c 16 60.h even 2 1
2160.3.bs.c 16 180.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 24 T_{2}^{14} + 408 T_{2}^{12} + 216 T_{2}^{11} - 3392 T_{2}^{10} - 2718 T_{2}^{9} + \cdots + 81 \) acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 24 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{15} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 4654875290256 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 112356358416 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 69380636886016 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + 26 T^{7} + \cdots - 3133657244)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 64\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 396718580736 \) Copy content Toggle raw display
$37$ \( (T^{8} + 22 T^{7} + \cdots - 143779124336)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 44\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 39\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 3873480104384)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
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