Properties

Label 1900.2.s.d
Level $1900$
Weight $2$
Character orbit 1900.s
Analytic conductor $15.172$
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] = [1900,2,Mod(49,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1900.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.s (of order \(6\), degree \(2\), 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: \(15.1715763840\)
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: \( x^{16} - 17x^{14} + 215x^{12} - 1176x^{10} + 4775x^{8} - 2898x^{6} + 1385x^{4} - 164x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 380)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_{14} q^{7} + (\beta_{10} - \beta_{5} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + \beta_{14} q^{7} + (\beta_{10} - \beta_{5} + \beta_{4}) q^{9} + ( - \beta_{7} - 2 \beta_{3} + 1) q^{11} + (\beta_{9} + \beta_{8} + \beta_1) q^{13} + ( - \beta_{15} + \beta_{12} + \beta_{6}) q^{17} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{19} + ( - \beta_{10} + 2 \beta_{5} + \beta_{2} + 2) q^{21} + ( - \beta_{12} - \beta_{8}) q^{23} + ( - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{9} + 2 \beta_{6} - 2 \beta_1) q^{27} + ( - 2 \beta_{11} + \beta_{5} - \beta_{4}) q^{29} + (\beta_{7} + 2 \beta_{3} - \beta_{2} - 3) q^{31} + (\beta_{15} + 3 \beta_{13} - \beta_{12} - \beta_{6}) q^{33} + ( - \beta_{14} + 3 \beta_{13} + 3 \beta_{9} - 2 \beta_{6} + 2 \beta_1) q^{37} + (3 \beta_{3} + 2 \beta_{2} + 6) q^{39} + ( - \beta_{11} + \beta_{7} - 2 \beta_{5} - 2) q^{41} + ( - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{8} + \beta_{6}) q^{43} + ( - 2 \beta_{12} - 2 \beta_{9} - \beta_{8}) q^{47} + (\beta_{7} + 2 \beta_{3} + \beta_{2} - 3) q^{49} + ( - \beta_{11} + \beta_{10} - 2 \beta_{5} + 4 \beta_{4}) q^{51} + (\beta_{12} + \beta_{9} + 2 \beta_{8} - 3 \beta_1) q^{53} + ( - \beta_{15} - \beta_{14} - \beta_{13} - 2 \beta_{9} + \beta_{8} + 3 \beta_{6}) q^{57} + (2 \beta_{11} - 2 \beta_{7} - 3 \beta_{5} - \beta_{4} + \beta_{3} - 3) q^{59} + (2 \beta_{11} - 3 \beta_{5}) q^{61} + ( - \beta_{9} - 2 \beta_{8} + 5 \beta_1) q^{63} - 2 \beta_{12} q^{67} + (\beta_{7} - 3 \beta_{3} - \beta_{2}) q^{69} + (\beta_{10} + 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 3) q^{71} + ( - \beta_{15} + \beta_{14} + \beta_{12} - \beta_{8} + 4 \beta_{6}) q^{73} + ( - 3 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} + 3 \beta_{9} + 2 \beta_{6} - 2 \beta_1) q^{77} + ( - \beta_{11} - 2 \beta_{10} + \beta_{7} - 3 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \cdots - 3) q^{79}+ \cdots + (4 \beta_{11} - \beta_{10} + 5 \beta_{5} - 4 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 10 q^{9} + 8 q^{11} - 6 q^{19} + 16 q^{21} - 10 q^{29} - 40 q^{31} + 108 q^{39} - 16 q^{41} - 40 q^{49} + 24 q^{51} - 22 q^{59} + 24 q^{61} - 12 q^{69} + 28 q^{71} - 26 q^{79} - 48 q^{81} - 10 q^{89} - 92 q^{91} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 17x^{14} + 215x^{12} - 1176x^{10} + 4775x^{8} - 2898x^{6} + 1385x^{4} - 164x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 26971964 \nu^{14} - 444227322 \nu^{12} + 5546383462 \nu^{10} - 28530870622 \nu^{8} + 110759128493 \nu^{6} - 8522047166 \nu^{4} + \cdots - 60023019862 ) / 17067637587 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 41610718 \nu^{14} + 684671526 \nu^{12} - 8556625619 \nu^{10} + 44015705039 \nu^{8} - 171758222896 \nu^{6} + 13147299967 \nu^{4} + \cdots - 6161981722 ) / 17067637587 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7345 \nu^{14} - 127233 \nu^{12} + 1618437 \nu^{10} - 9124664 \nu^{8} + 37577239 \nu^{6} - 30836000 \nu^{4} + 10921017 \nu^{2} - 1293284 ) / 2504606 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40106707 \nu^{14} - 677309787 \nu^{12} + 8548959173 \nu^{10} - 46239259076 \nu^{8} + 186744961489 \nu^{6} - 97460284762 \nu^{4} + \cdots - 6408773852 ) / 5251580796 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 40106707 \nu^{15} + 677309787 \nu^{13} - 8548959173 \nu^{11} + 46239259076 \nu^{9} - 186744961489 \nu^{7} + 97460284762 \nu^{5} + \cdots + 6408773852 \nu ) / 5251580796 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 96592348 \nu^{14} + 1588531329 \nu^{12} - 19862780534 \nu^{10} + 102175124654 \nu^{8} - 399408961640 \nu^{6} + 30519266062 \nu^{4} + \cdots - 24794252808 ) / 5689212529 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 304415798 \nu^{15} + 5006038191 \nu^{13} - 62598583759 \nu^{11} + 322010208379 \nu^{9} - 1259225979323 \nu^{7} + \cdots - 208838310356 \nu ) / 17067637587 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 331387762 \nu^{15} + 5450265513 \nu^{13} - 68144967221 \nu^{11} + 350541079001 \nu^{9} - 1369985107816 \nu^{7} + \cdots - 97612377733 \nu ) / 17067637587 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 913391869 \nu^{14} - 15395107497 \nu^{12} + 194194776311 \nu^{10} - 1046891021486 \nu^{8} + 4221230619577 \nu^{6} + \cdots - 144830854988 ) / 34135275174 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 589848415 \nu^{14} - 10109146509 \nu^{12} + 128159645582 \nu^{10} - 710466949547 \nu^{8} + 2902973030392 \nu^{6} + \cdots - 99796456472 ) / 17067637587 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 621164806 \nu^{15} - 10215859500 \nu^{13} + 127733308823 \nu^{11} - 657066452963 \nu^{9} + 2568211992736 \nu^{7} + \cdots + 171995136157 \nu ) / 17067637587 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3080990861 \nu^{15} + 52709730381 \nu^{13} - 667890407323 \nu^{11} + 3691698257488 \nu^{9} - 15063857001587 \nu^{7} + \cdots + 517752221236 \nu ) / 68270550348 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 640799947 \nu^{15} + 10828264986 \nu^{13} - 136673641894 \nu^{11} + 739737686495 \nu^{9} - 2985522971342 \nu^{7} + \cdots + 18500239168 \nu ) / 5689212529 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 7925043745 \nu^{15} - 134009754417 \nu^{13} + 1691462224943 \nu^{11} - 9161831867528 \nu^{9} + 36948597094099 \nu^{7} + \cdots - 228957502496 \nu ) / 68270550348 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + 4\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{9} - 8\beta_{6} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 9\beta_{10} + 32\beta_{5} - 13\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{15} + 9\beta_{14} + 18\beta_{13} - 14\beta_{12} - 9\beta_{8} - 72\beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{7} - 150\beta_{3} - 81\beta_{2} - 278 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -164\beta_{12} - 233\beta_{9} - 81\beta_{8} - 671\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 164 \beta_{11} + 752 \beta_{10} + 164 \beta_{7} - 2518 \beta_{5} + 1629 \beta_{4} - 1629 \beta_{3} - 752 \beta_{2} - 2518 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -1793\beta_{15} - 752\beta_{14} - 2670\beta_{13} - 2670\beta_{9} + 6403\beta_{6} - 6403\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1793\beta_{11} + 7155\beta_{10} - 23530\beta_{5} + 17122\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -18915\beta_{15} - 7155\beta_{14} - 28882\beta_{13} + 18915\beta_{12} + 7155\beta_{8} + 62117\beta_{6} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -18915\beta_{7} + 176626\beta_{3} + 69272\beta_{2} + 224948 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 195541\beta_{12} + 302895\beta_{9} + 69272\beta_{8} + 609390\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 195541 \beta_{11} - 678662 \beta_{10} - 195541 \beta_{7} + 2185022 \beta_{5} - 1801803 \beta_{4} + 1801803 \beta_{3} + 678662 \beta_{2} + 2185022 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1997344\beta_{15} + 678662\beta_{14} + 3120485\beta_{13} + 3120485\beta_{9} - 6022811\beta_{6} + 6022811\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−2.74101 + 1.58253i
−2.18309 + 1.26041i
−0.614201 + 0.354609i
−0.306096 + 0.176725i
0.306096 0.176725i
0.614201 0.354609i
2.18309 1.26041i
2.74101 1.58253i
−2.74101 1.58253i
−2.18309 1.26041i
−0.614201 0.354609i
−0.306096 0.176725i
0.306096 + 0.176725i
0.614201 + 0.354609i
2.18309 + 1.26041i
2.74101 + 1.58253i
0 −2.74101 1.58253i 0 0 0 1.53315i 0 3.50877 + 6.07738i 0
49.2 0 −2.18309 1.26041i 0 0 0 2.72743i 0 1.67727 + 2.90511i 0
49.3 0 −0.614201 0.354609i 0 0 0 3.11079i 0 −1.24850 2.16247i 0
49.4 0 −0.306096 0.176725i 0 0 0 4.30507i 0 −1.43754 2.48989i 0
49.5 0 0.306096 + 0.176725i 0 0 0 4.30507i 0 −1.43754 2.48989i 0
49.6 0 0.614201 + 0.354609i 0 0 0 3.11079i 0 −1.24850 2.16247i 0
49.7 0 2.18309 + 1.26041i 0 0 0 2.72743i 0 1.67727 + 2.90511i 0
49.8 0 2.74101 + 1.58253i 0 0 0 1.53315i 0 3.50877 + 6.07738i 0
349.1 0 −2.74101 + 1.58253i 0 0 0 1.53315i 0 3.50877 6.07738i 0
349.2 0 −2.18309 + 1.26041i 0 0 0 2.72743i 0 1.67727 2.90511i 0
349.3 0 −0.614201 + 0.354609i 0 0 0 3.11079i 0 −1.24850 + 2.16247i 0
349.4 0 −0.306096 + 0.176725i 0 0 0 4.30507i 0 −1.43754 + 2.48989i 0
349.5 0 0.306096 0.176725i 0 0 0 4.30507i 0 −1.43754 + 2.48989i 0
349.6 0 0.614201 0.354609i 0 0 0 3.11079i 0 −1.24850 + 2.16247i 0
349.7 0 2.18309 1.26041i 0 0 0 2.72743i 0 1.67727 2.90511i 0
349.8 0 2.74101 1.58253i 0 0 0 1.53315i 0 3.50877 6.07738i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.s.d 16
5.b even 2 1 inner 1900.2.s.d 16
5.c odd 4 1 380.2.i.c 8
5.c odd 4 1 1900.2.i.d 8
15.e even 4 1 3420.2.t.w 8
19.c even 3 1 inner 1900.2.s.d 16
20.e even 4 1 1520.2.q.m 8
95.i even 6 1 inner 1900.2.s.d 16
95.l even 12 1 7220.2.a.p 4
95.m odd 12 1 380.2.i.c 8
95.m odd 12 1 1900.2.i.d 8
95.m odd 12 1 7220.2.a.r 4
285.v even 12 1 3420.2.t.w 8
380.v even 12 1 1520.2.q.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.c 8 5.c odd 4 1
380.2.i.c 8 95.m odd 12 1
1520.2.q.m 8 20.e even 4 1
1520.2.q.m 8 380.v even 12 1
1900.2.i.d 8 5.c odd 4 1
1900.2.i.d 8 95.m odd 12 1
1900.2.s.d 16 1.a even 1 1 trivial
1900.2.s.d 16 5.b even 2 1 inner
1900.2.s.d 16 19.c even 3 1 inner
1900.2.s.d 16 95.i even 6 1 inner
3420.2.t.w 8 15.e even 4 1
3420.2.t.w 8 285.v even 12 1
7220.2.a.p 4 95.l even 12 1
7220.2.a.r 4 95.m odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 17T_{3}^{14} + 215T_{3}^{12} - 1176T_{3}^{10} + 4775T_{3}^{8} - 2898T_{3}^{6} + 1385T_{3}^{4} - 164T_{3}^{2} + 16 \) 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
$3$ \( T^{16} - 17 T^{14} + 215 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 38 T^{6} + 473 T^{4} + 2249 T^{2} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 35 T^{2} + 21 T + 267)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} - 91 T^{14} + 5830 T^{12} + \cdots + 7311616 \) Copy content Toggle raw display
$17$ \( T^{16} - 69 T^{14} + \cdots + 1358954496 \) Copy content Toggle raw display
$19$ \( (T^{8} + 3 T^{7} - 43 T^{6} - 21 T^{5} + \cdots + 130321)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 78 T^{14} + 4671 T^{12} + \cdots + 8503056 \) Copy content Toggle raw display
$29$ \( (T^{8} + 5 T^{7} + 84 T^{6} + 341 T^{5} + \cdots + 74529)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 10 T^{3} - 29 T^{2} - 303 T - 277)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 242 T^{6} + 16249 T^{4} + \cdots + 171396)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 8 T^{7} + 59 T^{6} + 106 T^{5} + \cdots + 324)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 113 T^{14} + \cdots + 1003875856 \) Copy content Toggle raw display
$47$ \( T^{16} - 306 T^{14} + \cdots + 1912622616576 \) Copy content Toggle raw display
$53$ \( T^{16} - 261 T^{14} + \cdots + 2200843458576 \) Copy content Toggle raw display
$59$ \( (T^{8} + 11 T^{7} + 176 T^{6} + \cdots + 2518569)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 12 T^{7} + 166 T^{6} - 608 T^{5} + \cdots + 841)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 118018152595456 \) Copy content Toggle raw display
$71$ \( (T^{8} - 14 T^{7} + 197 T^{6} + \cdots + 5049009)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} - 290 T^{14} + \cdots + 96947540496 \) Copy content Toggle raw display
$79$ \( (T^{8} + 13 T^{7} + 297 T^{6} + \cdots + 9096256)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 189 T^{6} + 10810 T^{4} + \cdots + 1695204)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 5 T^{7} + 147 T^{6} + \cdots + 1382976)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 178268320144656 \) Copy content Toggle raw display
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