Properties

Label 900.3.f.f
Level $900$
Weight $3$
Character orbit 900.f
Analytic conductor $24.523$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(199,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.199");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.f (of order \(2\), degree \(1\), 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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{14} + 12x^{12} + 25x^{10} + 53x^{8} + 100x^{6} + 192x^{4} + 320x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + ( - \beta_{8} - 1) q^{4} + ( - \beta_{13} - \beta_{11}) q^{7} + (\beta_{15} + \beta_{13} + 2 \beta_{9} + 2 \beta_{6}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + ( - \beta_{8} - 1) q^{4} + ( - \beta_{13} - \beta_{11}) q^{7} + (\beta_{15} + \beta_{13} + 2 \beta_{9} + 2 \beta_{6}) q^{8} + (\beta_{8} + \beta_{5} + \beta_{4} + 2 \beta_1 - 1) q^{11} + ( - 2 \beta_{15} - 2 \beta_{12} - \beta_{11} + 2 \beta_{9} + 2 \beta_{7} + \beta_{6}) q^{13} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_1 - 2) q^{14} + ( - \beta_{10} + 2 \beta_{8} + 3 \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} + 1) q^{16} + (3 \beta_{15} + 3 \beta_{14} + \beta_{13} + \beta_{12} + 2 \beta_{9} + \beta_{7} + 3 \beta_{6}) q^{17} + (\beta_{10} + \beta_{8} + 2 \beta_{5} - 3 \beta_{4} - 2 \beta_1 - 1) q^{19} + ( - \beta_{15} - \beta_{14} + \beta_{13} + 3 \beta_{12} + 2 \beta_{11} - 4 \beta_{9} + 2 \beta_{7} + \beta_{6}) q^{22} + ( - 3 \beta_{15} + 3 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - \beta_{11} - 2 \beta_{7} - 4 \beta_{6}) q^{23} + ( - 2 \beta_{10} - \beta_{5} - 3 \beta_{3} - 5 \beta_{2} + 4 \beta_1 + 1) q^{26} + ( - 3 \beta_{15} - \beta_{14} - \beta_{12} + 4 \beta_{11} + \beta_{9} - 2 \beta_{7} + 4 \beta_{6}) q^{28} + (2 \beta_{10} + 3 \beta_{8} - 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + \cdots + 10) q^{29}+ \cdots + (4 \beta_{15} + 4 \beta_{14} - 4 \beta_{13} - 12 \beta_{12} - 48 \beta_{9} - 27 \beta_{6}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 20 q^{4} - 40 q^{14} + 68 q^{16} + 72 q^{26} + 128 q^{29} + 184 q^{34} + 32 q^{41} + 344 q^{44} + 304 q^{46} + 112 q^{49} - 232 q^{56} - 352 q^{61} + 220 q^{64} + 264 q^{74} - 48 q^{76} + 400 q^{86} + 160 q^{89} + 192 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 5x^{14} + 12x^{12} + 25x^{10} + 53x^{8} + 100x^{6} + 192x^{4} + 320x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\nu^{14} + 115\nu^{12} + 96\nu^{10} + 467\nu^{8} + 35\nu^{6} + 1512\nu^{4} + 1184\nu^{2} + 2240 ) / 2432 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -47\nu^{14} - 139\nu^{12} - 196\nu^{10} - 807\nu^{8} - 1179\nu^{6} - 2156\nu^{4} - 5584\nu^{2} - 6144 ) / 1216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -127\nu^{14} - 471\nu^{12} - 1136\nu^{10} - 2407\nu^{8} - 5959\nu^{6} - 11736\nu^{4} - 20192\nu^{2} - 28736 ) / 2432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -35\nu^{14} - 93\nu^{12} - 74\nu^{10} - 339\nu^{8} - 861\nu^{6} - 1650\nu^{4} - 2040\nu^{2} - 2816 ) / 608 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -179\nu^{14} - 835\nu^{12} - 1576\nu^{10} - 3675\nu^{8} - 7603\nu^{6} - 12016\nu^{4} - 24960\nu^{2} - 36672 ) / 2432 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 165 \nu^{15} - 661 \nu^{13} - 984 \nu^{11} - 2749 \nu^{9} - 5541 \nu^{7} - 10064 \nu^{5} - 17152 \nu^{3} - 19008 \nu ) / 9728 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -101\nu^{15} - 213\nu^{13} - 536\nu^{11} - 1469\nu^{9} - 2021\nu^{7} - 7568\nu^{5} - 10816\nu^{3} - 17472\nu ) / 4864 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -59\nu^{14} - 185\nu^{12} - 318\nu^{10} - 667\nu^{8} - 1497\nu^{6} - 2662\nu^{4} - 6088\nu^{2} - 7040 ) / 608 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 71\nu^{15} + 231\nu^{13} + 440\nu^{11} + 1135\nu^{9} + 1815\nu^{7} + 3168\nu^{5} + 7200\nu^{3} + 9152\nu ) / 2432 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 211\nu^{14} + 603\nu^{12} + 1344\nu^{10} + 3099\nu^{8} + 5259\nu^{6} + 10376\nu^{4} + 20832\nu^{2} + 24064 ) / 1216 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 431\nu^{15} + 927\nu^{13} + 680\nu^{11} + 3319\nu^{9} + 7631\nu^{7} + 6416\nu^{5} + 17152\nu^{3} + 36032\nu ) / 9728 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -495\nu^{15} - 767\nu^{13} - 1736\nu^{11} - 3383\nu^{9} - 5679\nu^{7} - 9520\nu^{5} - 17408\nu^{3} + 1344\nu ) / 9728 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 621 \nu^{15} - 2333 \nu^{13} - 3416 \nu^{11} - 6853 \nu^{9} - 14509 \nu^{7} - 23440 \nu^{5} - 63360 \nu^{3} - 87104 \nu ) / 9728 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 977 \nu^{15} - 2849 \nu^{13} - 5528 \nu^{11} - 12681 \nu^{9} - 24817 \nu^{7} - 44848 \nu^{5} - 100352 \nu^{3} - 104768 \nu ) / 9728 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 525 \nu^{15} + 1661 \nu^{13} + 3352 \nu^{11} + 6757 \nu^{9} + 14093 \nu^{7} + 25168 \nu^{5} + 50816 \nu^{3} + 65344 \nu ) / 4864 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{7} + 3\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - 2\beta_{8} + \beta_{5} + 2\beta_{4} - \beta_{3} - 3\beta_{2} - 6 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{15} - 5\beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} - 2\beta_{9} + \beta_{7} + 4\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{5} - \beta_{4} - 4\beta_{3} + 2\beta_{2} + 7\beta _1 - 3 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{14} - \beta_{13} - 2\beta_{12} - 2\beta_{11} - 9\beta_{9} - 8\beta_{7} - 13\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{10} + 4\beta_{8} - 6\beta_{5} - 5\beta_{4} + \beta_{3} + \beta_{2} - 17\beta _1 - 15 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 9\beta_{15} - 7\beta_{14} + 12\beta_{13} + 9\beta_{12} + 9\beta_{11} - 19\beta_{9} + 15\beta_{7} - \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5\beta_{10} + 18\beta_{8} - 5\beta_{5} - 2\beta_{4} + 5\beta_{3} - 17\beta_{2} - 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -13\beta_{15} + 5\beta_{14} + 9\beta_{13} - \beta_{12} + \beta_{11} + 74\beta_{9} - \beta_{7} - 12\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 16\beta_{10} - 3\beta_{5} + 17\beta_{4} + 4\beta_{3} + 46\beta_{2} - 7\beta _1 + 3 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 48\beta_{15} + 30\beta_{14} + 9\beta_{13} - 14\beta_{12} - 30\beta_{11} + 33\beta_{9} + 24\beta_{7} + 85\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -31\beta_{10} - 52\beta_{8} - 34\beta_{5} - 3\beta_{4} + 31\beta_{3} + 31\beta_{2} + 105\beta _1 - 89 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -41\beta_{15} + 71\beta_{14} - 108\beta_{13} - 41\beta_{12} - 41\beta_{11} + 3\beta_{9} + \beta_{7} - 255\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 83\beta_{10} - 18\beta_{8} + 93\beta_{5} - 94\beta_{4} + 83\beta_{3} + 41\beta_{2} - 176\beta _1 + 466 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 11 \beta_{15} - 29 \beta_{14} - 9 \beta_{13} - 135 \beta_{12} + 135 \beta_{11} - 162 \beta_{9} - 135 \beta_{7} + 308 \beta_{6} ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.28061 + 0.600040i
1.28061 0.600040i
−0.120653 1.40906i
−0.120653 + 1.40906i
0.422403 + 1.34966i
0.422403 1.34966i
−0.957636 + 1.04064i
−0.957636 1.04064i
0.957636 + 1.04064i
0.957636 1.04064i
−0.422403 + 1.34966i
−0.422403 1.34966i
0.120653 1.40906i
0.120653 + 1.40906i
−1.28061 + 0.600040i
−1.28061 0.600040i
−1.95141 0.438172i 0 3.61601 + 1.71011i 0 0 6.33166 −6.30701 4.92155i 0 0
199.2 −1.95141 + 0.438172i 0 3.61601 1.71011i 0 0 6.33166 −6.30701 + 4.92155i 0 0
199.3 −1.08539 1.67986i 0 −1.64388 + 3.64660i 0 0 −0.596540 7.91002 1.19648i 0 0
199.4 −1.08539 + 1.67986i 0 −1.64388 3.64660i 0 0 −0.596540 7.91002 + 1.19648i 0 0
199.5 −0.696577 1.87477i 0 −3.02956 + 2.61185i 0 0 −5.46770 7.00695 + 3.86039i 0 0
199.6 −0.696577 + 1.87477i 0 −3.02956 2.61185i 0 0 −5.46770 7.00695 3.86039i 0 0
199.7 −0.169449 1.99281i 0 −3.94257 + 0.675358i 0 0 12.3959 2.01392 + 7.74236i 0 0
199.8 −0.169449 + 1.99281i 0 −3.94257 0.675358i 0 0 12.3959 2.01392 7.74236i 0 0
199.9 0.169449 1.99281i 0 −3.94257 0.675358i 0 0 −12.3959 −2.01392 + 7.74236i 0 0
199.10 0.169449 + 1.99281i 0 −3.94257 + 0.675358i 0 0 −12.3959 −2.01392 7.74236i 0 0
199.11 0.696577 1.87477i 0 −3.02956 2.61185i 0 0 5.46770 −7.00695 + 3.86039i 0 0
199.12 0.696577 + 1.87477i 0 −3.02956 + 2.61185i 0 0 5.46770 −7.00695 3.86039i 0 0
199.13 1.08539 1.67986i 0 −1.64388 3.64660i 0 0 0.596540 −7.91002 1.19648i 0 0
199.14 1.08539 + 1.67986i 0 −1.64388 + 3.64660i 0 0 0.596540 −7.91002 + 1.19648i 0 0
199.15 1.95141 0.438172i 0 3.61601 1.71011i 0 0 −6.33166 6.30701 4.92155i 0 0
199.16 1.95141 + 0.438172i 0 3.61601 + 1.71011i 0 0 −6.33166 6.30701 + 4.92155i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.f.f 16
3.b odd 2 1 300.3.f.b 16
4.b odd 2 1 inner 900.3.f.f 16
5.b even 2 1 inner 900.3.f.f 16
5.c odd 4 1 180.3.c.b 8
5.c odd 4 1 900.3.c.u 8
12.b even 2 1 300.3.f.b 16
15.d odd 2 1 300.3.f.b 16
15.e even 4 1 60.3.c.a 8
15.e even 4 1 300.3.c.d 8
20.d odd 2 1 inner 900.3.f.f 16
20.e even 4 1 180.3.c.b 8
20.e even 4 1 900.3.c.u 8
40.i odd 4 1 2880.3.e.j 8
40.k even 4 1 2880.3.e.j 8
60.h even 2 1 300.3.f.b 16
60.l odd 4 1 60.3.c.a 8
60.l odd 4 1 300.3.c.d 8
120.q odd 4 1 960.3.e.c 8
120.w even 4 1 960.3.e.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.c.a 8 15.e even 4 1
60.3.c.a 8 60.l odd 4 1
180.3.c.b 8 5.c odd 4 1
180.3.c.b 8 20.e even 4 1
300.3.c.d 8 15.e even 4 1
300.3.c.d 8 60.l odd 4 1
300.3.f.b 16 3.b odd 2 1
300.3.f.b 16 12.b even 2 1
300.3.f.b 16 15.d odd 2 1
300.3.f.b 16 60.h even 2 1
900.3.c.u 8 5.c odd 4 1
900.3.c.u 8 20.e even 4 1
900.3.f.f 16 1.a even 1 1 trivial
900.3.f.f 16 4.b odd 2 1 inner
900.3.f.f 16 5.b even 2 1 inner
900.3.f.f 16 20.d odd 2 1 inner
960.3.e.c 8 120.q odd 4 1
960.3.e.c 8 120.w even 4 1
2880.3.e.j 8 40.i odd 4 1
2880.3.e.j 8 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{8} - 224T_{7}^{6} + 12032T_{7}^{4} - 188416T_{7}^{2} + 65536 \) Copy content Toggle raw display
\( T_{29}^{4} - 32T_{29}^{3} - 2152T_{29}^{2} + 34688T_{29} + 1334416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 10 T^{14} + 33 T^{12} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 224 T^{6} + 12032 T^{4} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 208 T^{2} + 10496)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 1008 T^{6} + 290528 T^{4} + \cdots + 155351296)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 848 T^{6} + 162144 T^{4} + \cdots + 77721856)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1696 T^{6} + \cdots + 6544162816)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 3616 T^{6} + \cdots + 101419319296)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 32 T^{3} - 2152 T^{2} + \cdots + 1334416)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 5408 T^{6} + \cdots + 59895709696)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 6192 T^{6} + \cdots + 59919206656)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} - 1800 T^{2} + \cdots + 87184)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 10816 T^{6} + \cdots + 33624411406336)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 8032 T^{6} + \cdots + 1056981385216)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 11472 T^{6} + \cdots + 228545188096)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 4896 T^{6} + \cdots + 173909016576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 88 T^{3} - 2536 T^{2} + \cdots - 2142704)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 16064 T^{6} + \cdots + 281086590976)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 13952 T^{6} + \cdots + 16079971680256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 17552 T^{6} + \cdots + 24622079140096)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 41888 T^{6} + \cdots + 31\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 36928 T^{6} + \cdots + 42\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 40 T^{3} - 20584 T^{2} + \cdots + 70652944)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 34896 T^{6} + \cdots + 35\!\cdots\!76)^{2} \) Copy content Toggle raw display
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