Properties

Label 1900.2.l.c
Level $1900$
Weight $2$
Character orbit 1900.l
Analytic conductor $15.172$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1900,2,Mod(493,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1900.493");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.l (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: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 48 x^{14} - 184 x^{13} + 588 x^{12} - 1440 x^{11} + 3064 x^{10} - 5344 x^{9} + 8028 x^{8} - 9824 x^{7} + 9952 x^{6} - 5472 x^{5} + 1248 x^{4} + 6112 x^{3} + \cdots + 5800 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + \beta_{9} q^{7} + ( - \beta_{11} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + \beta_{9} q^{7} + ( - \beta_{11} - \beta_{2}) q^{9} + \beta_{6} q^{11} - \beta_{5} q^{13} - \beta_{14} q^{17} + (\beta_{12} + \beta_{11} + \beta_{2}) q^{19} + ( - \beta_{7} + \beta_1) q^{21} + (\beta_{13} + \beta_{3}) q^{23} + (\beta_{10} + \beta_{8}) q^{27} + 2 \beta_{12} q^{29} + (\beta_{7} + \beta_1) q^{31} + ( - \beta_{5} - \beta_{4}) q^{33} + ( - 2 \beta_{10} + \beta_{8}) q^{37} + ( - 3 \beta_{11} - 2 \beta_{2}) q^{39} + (\beta_{7} - \beta_1) q^{41} + ( - \beta_{13} + \beta_{3}) q^{43} - \beta_{9} q^{47} + ( - 4 \beta_{11} - 5 \beta_{2}) q^{49} + 2 \beta_1 q^{51} + (2 \beta_{5} - \beta_{4}) q^{53} + (\beta_{14} - \beta_{10} - \beta_{9} + 2 \beta_{8}) q^{57} + ( - \beta_{15} + \beta_{12}) q^{59} + (3 \beta_{6} - 2) q^{61} + (3 \beta_{13} + \beta_{3}) q^{63} + ( - \beta_{10} - 4 \beta_{8}) q^{67} + (\beta_{15} + 3 \beta_{12}) q^{69} + ( - \beta_{7} + \beta_1) q^{71} - 2 \beta_{3} q^{73} + ( - \beta_{14} + 2 \beta_{9}) q^{77} + (\beta_{15} + \beta_{12}) q^{79} + (\beta_{6} + 5) q^{81} + (3 \beta_{13} - 2 \beta_{3}) q^{83} + (2 \beta_{14} - 2 \beta_{9}) q^{87} + 2 \beta_{12} q^{89} + ( - \beta_{7} + 3 \beta_1) q^{91} + ( - 4 \beta_{13} + \beta_{3}) q^{93} + (2 \beta_{10} + 5 \beta_{8}) q^{97} + ( - \beta_{11} - 6 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{61} + 80 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 48 x^{14} - 184 x^{13} + 588 x^{12} - 1440 x^{11} + 3064 x^{10} - 5344 x^{9} + 8028 x^{8} - 9824 x^{7} + 9952 x^{6} - 5472 x^{5} + 1248 x^{4} + 6112 x^{3} + \cdots + 5800 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3404742364 \nu^{15} - 63834217534 \nu^{14} + 427804418714 \nu^{13} - 2161411700658 \nu^{12} + 7483730081226 \nu^{11} + \cdots + 9822248453700 ) / 45470877735650 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 603090818534 \nu^{15} - 5243849724144 \nu^{14} + 31506273291774 \nu^{13} - 118945960487813 \nu^{12} + 353600813370636 \nu^{11} + \cdots - 51\!\cdots\!50 ) / 74\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 129262203733548 \nu^{15} + \cdots + 65\!\cdots\!00 ) / 99\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 608903147273088 \nu^{15} + \cdots + 28\!\cdots\!00 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 631490110904546 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5747364 \nu^{15} + 51262207 \nu^{14} - 315413196 \nu^{13} + 1294440423 \nu^{12} - 4232991322 \nu^{11} + 10934282618 \nu^{10} + \cdots + 14137291952 ) / 27146792678 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2482704232 \nu^{15} - 6162325547 \nu^{14} + 33239984062 \nu^{13} + 3359208941 \nu^{12} + 99837424838 \nu^{11} + \cdots + 45411323499100 ) / 9094175547130 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!56 \nu^{15} + \cdots - 34\!\cdots\!00 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 628431571537036 \nu^{15} + \cdots + 57\!\cdots\!00 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\!\cdots\!98 \nu^{15} + \cdots + 93\!\cdots\!00 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4176 \nu^{15} - 37891 \nu^{14} + 232586 \nu^{13} - 934932 \nu^{12} + 2974454 \nu^{11} - 7309692 \nu^{10} + 15117340 \nu^{9} - 25921809 \nu^{8} + 38402400 \nu^{7} + \cdots - 805100 ) / 10374950 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2166898748 \nu^{15} - 15097277260 \nu^{14} + 80885054370 \nu^{13} - 256170024905 \nu^{12} + 677613701850 \nu^{11} + \cdots - 11035714715932 ) / 4424927206514 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!78 \nu^{15} + \cdots + 68\!\cdots\!00 ) / 19\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 705129212158676 \nu^{15} + \cdots - 36\!\cdots\!00 ) / 99\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10709177408 \nu^{15} - 79061811993 \nu^{14} + 439445786448 \nu^{13} - 1512144077948 \nu^{12} + 4335120362722 \nu^{11} + \cdots + 179749858580 ) / 4424927206514 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{14} - 2\beta_{8} - 2\beta_{4} - 2\beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} - 2\beta_{12} + 2\beta_{11} - 4\beta_{8} - 2\beta_{6} - \beta_{3} - 8\beta_{2} - 2\beta _1 - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{14} + \beta_{13} - 6 \beta_{12} + 6 \beta_{11} - 4 \beta_{10} + 3 \beta_{9} - 4 \beta_{8} - 2 \beta_{5} + 14 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - 3 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - \beta_{10} + \beta_{9} + 5 \beta_{8} + 5 \beta_{6} - 3 \beta_{5} + 7 \beta_{4} + 15 \beta_{2} + 5 \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10 \beta_{15} - 13 \beta_{14} + 10 \beta_{13} + 30 \beta_{12} - 50 \beta_{11} + 34 \beta_{10} - 28 \beta_{9} + 90 \beta_{8} + 10 \beta_{7} + 30 \beta_{6} - 34 \beta_{5} - 10 \beta_{4} + 35 \beta_{3} + 108 \beta_{2} + 50 \beta _1 + 112 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 13 \beta_{15} + 32 \beta_{14} - 37 \beta_{13} + 79 \beta_{12} - 72 \beta_{11} + 72 \beta_{10} - 37 \beta_{9} + 54 \beta_{8} + 23 \beta_{7} - 66 \beta_{6} + 10 \beta_{5} - 100 \beta_{4} + 43 \beta_{3} - 62 \beta_{2} - \beta _1 + 206 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 91 \beta_{15} + 161 \beta_{14} - 138 \beta_{13} + 112 \beta_{12} + 70 \beta_{11} + 79 \beta_{10} + 84 \beta_{9} - 139 \beta_{8} - 308 \beta_{6} + 215 \beta_{5} - 251 \beta_{4} - 14 \beta_{3} - 552 \beta_{2} - 175 \beta _1 + 176 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 56 \beta_{15} + 120 \beta_{14} + 46 \beta_{13} - 112 \beta_{12} + 490 \beta_{11} - 216 \beta_{10} + 262 \beta_{9} - 360 \beta_{8} - 132 \beta_{7} - 110 \beta_{6} + 280 \beta_{5} - 32 \beta_{4} - 202 \beta_{3} - 630 \beta_{2} + \cdots - 458 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 600 \beta_{15} - 573 \beta_{14} + 1590 \beta_{13} - 1500 \beta_{12} + 2124 \beta_{11} - 2144 \beta_{10} + 318 \beta_{9} - 1268 \beta_{8} - 684 \beta_{7} + 2376 \beta_{6} - 316 \beta_{5} + 1372 \beta_{4} - 1275 \beta_{3} + \cdots - 4088 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1334 \beta_{15} - 2032 \beta_{14} + 1398 \beta_{13} - 1728 \beta_{12} - 1662 \beta_{11} - 1258 \beta_{10} - 2052 \beta_{9} + 240 \beta_{8} + 376 \beta_{7} + 4592 \beta_{6} - 3090 \beta_{5} + 2400 \beta_{4} + \cdots - 3872 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1562 \beta_{15} - 9339 \beta_{14} - 7746 \beta_{13} - 88 \beta_{12} - 29370 \beta_{11} + 12374 \beta_{10} - 12496 \beta_{9} + 10686 \beta_{8} + 8250 \beta_{7} + 4642 \beta_{6} - 16070 \beta_{5} + 8574 \beta_{4} + \cdots + 1816 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 9826 \beta_{15} + 2116 \beta_{14} - 21346 \beta_{13} + 15958 \beta_{12} - 27390 \beta_{11} + 30464 \beta_{10} + 2734 \beta_{9} + 21260 \beta_{8} + 7658 \beta_{7} - 34998 \beta_{6} + 3652 \beta_{5} + \cdots + 38844 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 33280 \beta_{15} + 52204 \beta_{14} - 19175 \beta_{13} + 63180 \beta_{12} + 50310 \beta_{11} + 37688 \beta_{10} + 58885 \beta_{9} + 43046 \beta_{8} - 14950 \beta_{7} - 121420 \beta_{6} + \cdots + 161818 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 8154 \beta_{15} + 165389 \beta_{14} + 126260 \beta_{13} + 79252 \beta_{12} + 381238 \beta_{11} - 154322 \beta_{10} + 121710 \beta_{9} - 62482 \beta_{8} - 105076 \beta_{7} - 49738 \beta_{6} + \cdots + 255148 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 249607 \beta_{15} + 72423 \beta_{14} + 461244 \beta_{13} - 367639 \beta_{12} + 686394 \beta_{11} - 809454 \beta_{10} - 162528 \beta_{9} - 800040 \beta_{8} - 167375 \beta_{7} + 877890 \beta_{6} + \cdots - 843668 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
493.1
0.870982 + 0.924776i
0.129018 + 1.66674i
−0.667207 + 1.54769i
1.66721 0.786721i
−0.667207 0.213279i
1.66721 2.54769i
0.870982 2.66674i
0.129018 1.92478i
0.870982 0.924776i
0.129018 1.66674i
−0.667207 1.54769i
1.66721 + 0.786721i
−0.667207 + 0.213279i
1.66721 + 2.54769i
0.870982 + 2.66674i
0.129018 + 1.92478i
0 −1.79576 + 1.79576i 0 0 0 −3.30136 + 3.30136i 0 3.44949i 0
493.2 0 −1.79576 + 1.79576i 0 0 0 3.30136 3.30136i 0 3.44949i 0
493.3 0 −0.880486 + 0.880486i 0 0 0 −1.04930 + 1.04930i 0 1.44949i 0
493.4 0 −0.880486 + 0.880486i 0 0 0 1.04930 1.04930i 0 1.44949i 0
493.5 0 0.880486 0.880486i 0 0 0 −1.04930 + 1.04930i 0 1.44949i 0
493.6 0 0.880486 0.880486i 0 0 0 1.04930 1.04930i 0 1.44949i 0
493.7 0 1.79576 1.79576i 0 0 0 −3.30136 + 3.30136i 0 3.44949i 0
493.8 0 1.79576 1.79576i 0 0 0 3.30136 3.30136i 0 3.44949i 0
1557.1 0 −1.79576 1.79576i 0 0 0 −3.30136 3.30136i 0 3.44949i 0
1557.2 0 −1.79576 1.79576i 0 0 0 3.30136 + 3.30136i 0 3.44949i 0
1557.3 0 −0.880486 0.880486i 0 0 0 −1.04930 1.04930i 0 1.44949i 0
1557.4 0 −0.880486 0.880486i 0 0 0 1.04930 + 1.04930i 0 1.44949i 0
1557.5 0 0.880486 + 0.880486i 0 0 0 −1.04930 1.04930i 0 1.44949i 0
1557.6 0 0.880486 + 0.880486i 0 0 0 1.04930 + 1.04930i 0 1.44949i 0
1557.7 0 1.79576 + 1.79576i 0 0 0 −3.30136 3.30136i 0 3.44949i 0
1557.8 0 1.79576 + 1.79576i 0 0 0 3.30136 + 3.30136i 0 3.44949i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 493.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
19.b odd 2 1 inner
95.d odd 2 1 inner
95.g even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.l.c 16
5.b even 2 1 inner 1900.2.l.c 16
5.c odd 4 2 inner 1900.2.l.c 16
19.b odd 2 1 inner 1900.2.l.c 16
95.d odd 2 1 inner 1900.2.l.c 16
95.g even 4 2 inner 1900.2.l.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.l.c 16 1.a even 1 1 trivial
1900.2.l.c 16 5.b even 2 1 inner
1900.2.l.c 16 5.c odd 4 2 inner
1900.2.l.c 16 19.b odd 2 1 inner
1900.2.l.c 16 95.d odd 2 1 inner
1900.2.l.c 16 95.g even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 44T_{3}^{4} + 100 \) acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 44 T^{4} + 100)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 480 T^{4} + 2304)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6)^{8} \) Copy content Toggle raw display
$13$ \( (T^{8} + 524 T^{4} + 62500)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1920 T^{4} + 36864)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 20 T^{6} + 438 T^{4} + \cdots + 130321)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2784 T^{4} + 1440000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 96 T^{2} + 1920)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 144 T^{2} + 4320)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 12524 T^{4} + 27984100)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 144 T^{2} + 480)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 4320 T^{4} + 186624)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 480 T^{4} + 2304)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 9900 T^{4} + 5062500)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 144 T^{2} + 4320)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 50)^{8} \) Copy content Toggle raw display
$67$ \( (T^{8} + 9164 T^{4} + 13032100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 144 T^{2} + 480)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 30720 T^{4} + 9437184)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 144 T^{2} + 480)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 106464 T^{4} + \cdots + 900000000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 96 T^{2} + 1920)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 49004 T^{4} + 39062500)^{2} \) Copy content Toggle raw display
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