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 $4$

Related objects

Downloads

Learn more

Newspace parameters

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

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 17 x^{14} + 215 x^{12} - 1176 x^{10} + 4775 x^{8} - 2898 x^{6} + 1385 x^{4} - 164 x^{2} + 16\)
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

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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_{14} q^{7} + ( \beta_{4} - \beta_{5} + \beta_{10} ) q^{9} +O(q^{10})\) \( q + \beta_{6} q^{3} + \beta_{14} q^{7} + ( \beta_{4} - \beta_{5} + \beta_{10} ) q^{9} + ( 1 - 2 \beta_{3} - \beta_{7} ) q^{11} + ( \beta_{1} + \beta_{8} + \beta_{9} ) q^{13} + ( \beta_{6} + \beta_{12} - \beta_{15} ) q^{17} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{19} + ( 2 + \beta_{2} + 2 \beta_{5} - \beta_{10} ) q^{21} + ( -\beta_{8} - \beta_{12} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{6} - \beta_{9} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{27} + ( -\beta_{4} + \beta_{5} - 2 \beta_{11} ) q^{29} + ( -3 - \beta_{2} + 2 \beta_{3} + \beta_{7} ) q^{31} + ( -\beta_{6} - \beta_{12} + 3 \beta_{13} + \beta_{15} ) q^{33} + ( 2 \beta_{1} - 2 \beta_{6} + 3 \beta_{9} + 3 \beta_{13} - \beta_{14} ) q^{37} + ( 6 + 2 \beta_{2} + 3 \beta_{3} ) q^{39} + ( -2 - 2 \beta_{5} + \beta_{7} - \beta_{11} ) q^{41} + ( \beta_{6} + \beta_{8} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{43} + ( -\beta_{8} - 2 \beta_{9} - 2 \beta_{12} ) q^{47} + ( -3 + \beta_{2} + 2 \beta_{3} + \beta_{7} ) q^{49} + ( 4 \beta_{4} - 2 \beta_{5} + \beta_{10} - \beta_{11} ) q^{51} + ( -3 \beta_{1} + 2 \beta_{8} + \beta_{9} + \beta_{12} ) q^{53} + ( 3 \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{57} + ( -3 + \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{7} + 2 \beta_{11} ) q^{59} + ( -3 \beta_{5} + 2 \beta_{11} ) q^{61} + ( 5 \beta_{1} - 2 \beta_{8} - \beta_{9} ) q^{63} -2 \beta_{12} q^{67} + ( -\beta_{2} - 3 \beta_{3} + \beta_{7} ) q^{69} + ( 3 - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{10} ) q^{71} + ( 4 \beta_{6} - \beta_{8} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{73} + ( -2 \beta_{1} + 2 \beta_{6} + 3 \beta_{9} + 3 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{77} + ( -3 + 2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{7} - 2 \beta_{10} - \beta_{11} ) q^{79} + ( -5 - 4 \beta_{3} + 4 \beta_{4} - 5 \beta_{5} + \beta_{7} - \beta_{11} ) q^{81} + ( -3 \beta_{1} + 3 \beta_{6} + \beta_{9} + \beta_{13} - \beta_{15} ) q^{83} + ( 2 \beta_{1} - 2 \beta_{6} - \beta_{15} ) q^{87} + ( -\beta_{4} + \beta_{5} + 2 \beta_{10} - \beta_{11} ) q^{89} + ( 2 \beta_{4} + 12 \beta_{5} + 3 \beta_{11} ) q^{91} + ( -4 \beta_{6} - \beta_{8} + \beta_{12} - 4 \beta_{13} + \beta_{14} - \beta_{15} ) q^{93} + ( -\beta_{6} - 2 \beta_{12} + 2 \beta_{15} ) q^{97} + ( -4 \beta_{4} + 5 \beta_{5} - \beta_{10} + 4 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 10q^{9} + O(q^{10}) \) \( 16q + 10q^{9} + 8q^{11} - 6q^{19} + 16q^{21} - 10q^{29} - 40q^{31} + 108q^{39} - 16q^{41} - 40q^{49} + 24q^{51} - 22q^{59} + 24q^{61} - 12q^{69} + 28q^{71} - 26q^{79} - 48q^{81} - 10q^{89} - 92q^{91} - 48q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(26971964 \nu^{14} - 444227322 \nu^{12} + 5546383462 \nu^{10} - 28530870622 \nu^{8} + 110759128493 \nu^{6} - 8522047166 \nu^{4} + 1010355192 \nu^{2} - 60023019862\)\()/ 17067637587 \)
\(\beta_{3}\)\(=\)\((\)\(-41610718 \nu^{14} + 684671526 \nu^{12} - 8556625619 \nu^{10} + 44015705039 \nu^{8} - 171758222896 \nu^{6} + 13147299967 \nu^{4} - 1558715004 \nu^{2} - 6161981722\)\()/ 17067637587 \)
\(\beta_{4}\)\(=\)\((\)\( 7345 \nu^{14} - 127233 \nu^{12} + 1618437 \nu^{10} - 9124664 \nu^{8} + 37577239 \nu^{6} - 30836000 \nu^{4} + 10921017 \nu^{2} - 1293284 \)\()/2504606\)
\(\beta_{5}\)\(=\)\((\)\(40106707 \nu^{14} - 677309787 \nu^{12} + 8548959173 \nu^{10} - 46239259076 \nu^{8} + 186744961489 \nu^{6} - 97460284762 \nu^{4} + 54124634487 \nu^{2} - 6408773852\)\()/ 5251580796 \)
\(\beta_{6}\)\(=\)\((\)\(-40106707 \nu^{15} + 677309787 \nu^{13} - 8548959173 \nu^{11} + 46239259076 \nu^{9} - 186744961489 \nu^{7} + 97460284762 \nu^{5} - 54124634487 \nu^{3} + 6408773852 \nu\)\()/ 5251580796 \)
\(\beta_{7}\)\(=\)\((\)\(-96592348 \nu^{14} + 1588531329 \nu^{12} - 19862780534 \nu^{10} + 102175124654 \nu^{8} - 399408961640 \nu^{6} + 30519266062 \nu^{4} - 3618297144 \nu^{2} - 24794252808\)\()/ 5689212529 \)
\(\beta_{8}\)\(=\)\((\)\(-304415798 \nu^{15} + 5006038191 \nu^{13} - 62598583759 \nu^{11} + 322010208379 \nu^{9} - 1259225979323 \nu^{7} + 96183050987 \nu^{5} - 11403251244 \nu^{3} - 208838310356 \nu\)\()/ 17067637587 \)
\(\beta_{9}\)\(=\)\((\)\(-331387762 \nu^{15} + 5450265513 \nu^{13} - 68144967221 \nu^{11} + 350541079001 \nu^{9} - 1369985107816 \nu^{7} + 104705098153 \nu^{5} - 12413606436 \nu^{3} - 97612377733 \nu\)\()/ 17067637587 \)
\(\beta_{10}\)\(=\)\((\)\(913391869 \nu^{14} - 15395107497 \nu^{12} + 194194776311 \nu^{10} - 1046891021486 \nu^{8} + 4221230619577 \nu^{6} - 2104453054210 \nu^{4} + 1223165961171 \nu^{2} - 144830854988\)\()/ 34135275174 \)
\(\beta_{11}\)\(=\)\((\)\(589848415 \nu^{14} - 10109146509 \nu^{12} + 128159645582 \nu^{10} - 710466949547 \nu^{8} + 2902973030392 \nu^{6} - 2048816477110 \nu^{4} + 842761353126 \nu^{2} - 99796456472\)\()/ 17067637587 \)
\(\beta_{12}\)\(=\)\((\)\(621164806 \nu^{15} - 10215859500 \nu^{13} + 127733308823 \nu^{11} - 657066452963 \nu^{9} + 2568211992736 \nu^{7} - 196262896339 \nu^{5} + 23268497868 \nu^{3} + 171995136157 \nu\)\()/ 17067637587 \)
\(\beta_{13}\)\(=\)\((\)\(-3080990861 \nu^{15} + 52709730381 \nu^{13} - 667890407323 \nu^{11} + 3691698257488 \nu^{9} - 15063857001587 \nu^{7} + 10302777298346 \nu^{5} - 4372350742221 \nu^{3} + 517752221236 \nu\)\()/ 68270550348 \)
\(\beta_{14}\)\(=\)\((\)\(-640799947 \nu^{15} + 10828264986 \nu^{13} - 136673641894 \nu^{11} + 739737686495 \nu^{9} - 2985522971342 \nu^{7} + 1558113893036 \nu^{5} - 747929701211 \nu^{3} + 18500239168 \nu\)\()/ 5689212529 \)
\(\beta_{15}\)\(=\)\((\)\(7925043745 \nu^{15} - 134009754417 \nu^{13} + 1691462224943 \nu^{11} - 9161831867528 \nu^{9} + 36948597094099 \nu^{7} - 19283094792142 \nu^{5} + 7836470146197 \nu^{3} - 228957502496 \nu\)\()/ 68270550348 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{10} + 4 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{15} + \beta_{14} + \beta_{13} + \beta_{9} - 8 \beta_{6} + 8 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{11} - 9 \beta_{10} + 32 \beta_{5} - 13 \beta_{4}\)
\(\nu^{5}\)\(=\)\(14 \beta_{15} + 9 \beta_{14} + 18 \beta_{13} - 14 \beta_{12} - 9 \beta_{8} - 72 \beta_{6}\)
\(\nu^{6}\)\(=\)\(14 \beta_{7} - 150 \beta_{3} - 81 \beta_{2} - 278\)
\(\nu^{7}\)\(=\)\(-164 \beta_{12} - 233 \beta_{9} - 81 \beta_{8} - 671 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-164 \beta_{11} + 752 \beta_{10} + 164 \beta_{7} - 2518 \beta_{5} + 1629 \beta_{4} - 1629 \beta_{3} - 752 \beta_{2} - 2518\)
\(\nu^{9}\)\(=\)\(-1793 \beta_{15} - 752 \beta_{14} - 2670 \beta_{13} - 2670 \beta_{9} + 6403 \beta_{6} - 6403 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-1793 \beta_{11} + 7155 \beta_{10} - 23530 \beta_{5} + 17122 \beta_{4}\)
\(\nu^{11}\)\(=\)\(-18915 \beta_{15} - 7155 \beta_{14} - 28882 \beta_{13} + 18915 \beta_{12} + 7155 \beta_{8} + 62117 \beta_{6}\)
\(\nu^{12}\)\(=\)\(-18915 \beta_{7} + 176626 \beta_{3} + 69272 \beta_{2} + 224948\)
\(\nu^{13}\)\(=\)\(195541 \beta_{12} + 302895 \beta_{9} + 69272 \beta_{8} + 609390 \beta_{1}\)
\(\nu^{14}\)\(=\)\(195541 \beta_{11} - 678662 \beta_{10} - 195541 \beta_{7} + 2185022 \beta_{5} - 1801803 \beta_{4} + 1801803 \beta_{3} + 678662 \beta_{2} + 2185022\)
\(\nu^{15}\)\(=\)\(1997344 \beta_{15} + 678662 \beta_{14} + 3120485 \beta_{13} + 3120485 \beta_{9} - 6022811 \beta_{6} + 6022811 \beta_{1}\)

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.

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 349.8
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 \(T_{3}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 16 - 164 T^{2} + 1385 T^{4} - 2898 T^{6} + 4775 T^{8} - 1176 T^{10} + 215 T^{12} - 17 T^{14} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( ( 3136 + 2249 T^{2} + 473 T^{4} + 38 T^{6} + T^{8} )^{2} \)
$11$ \( ( 267 + 21 T - 35 T^{2} - 2 T^{3} + T^{4} )^{4} \)
$13$ \( 7311616 - 46403344 T^{2} + 287872417 T^{4} - 41569483 T^{6} + 4443046 T^{8} - 188719 T^{10} + 5830 T^{12} - 91 T^{14} + T^{16} \)
$17$ \( 1358954496 - 484282368 T^{2} + 116473761 T^{4} - 14907282 T^{6} + 1373167 T^{8} - 78744 T^{10} + 3239 T^{12} - 69 T^{14} + T^{16} \)
$19$ \( ( 130321 + 20577 T - 15523 T^{2} - 399 T^{3} + 1227 T^{4} - 21 T^{5} - 43 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$23$ \( 8503056 - 16297524 T^{2} + 27116613 T^{4} - 7442361 T^{6} + 1557711 T^{8} - 99036 T^{10} + 4671 T^{12} - 78 T^{14} + T^{16} \)
$29$ \( ( 74529 + 86814 T + 85017 T^{2} + 21492 T^{3} + 5344 T^{4} + 341 T^{5} + 84 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$31$ \( ( -277 - 303 T - 29 T^{2} + 10 T^{3} + T^{4} )^{4} \)
$37$ \( ( 171396 + 191205 T^{2} + 16249 T^{4} + 242 T^{6} + T^{8} )^{2} \)
$41$ \( ( 324 - 594 T + 999 T^{2} - 453 T^{3} + 271 T^{4} + 106 T^{5} + 59 T^{6} + 8 T^{7} + T^{8} )^{2} \)
$43$ \( 1003875856 - 631683908 T^{2} + 297806105 T^{4} - 55561218 T^{6} + 7612751 T^{8} - 315624 T^{10} + 9623 T^{12} - 113 T^{14} + T^{16} \)
$47$ \( 1912622616576 - 1265933358144 T^{2} + 796608891729 T^{4} - 26483790921 T^{6} + 609954559 T^{8} - 7305504 T^{10} + 63779 T^{12} - 306 T^{14} + T^{16} \)
$53$ \( 2200843458576 - 965858684868 T^{2} + 390659114889 T^{4} - 13802766702 T^{6} + 329902699 T^{8} - 4541676 T^{10} + 45731 T^{12} - 261 T^{14} + T^{16} \)
$59$ \( ( 2518569 - 733194 T + 300729 T^{2} - 9504 T^{3} + 6520 T^{4} + 319 T^{5} + 176 T^{6} + 11 T^{7} + T^{8} )^{2} \)
$61$ \( ( 841 + 12644 T + 190734 T^{2} - 8896 T^{3} + 5687 T^{4} - 608 T^{5} + 166 T^{6} - 12 T^{7} + T^{8} )^{2} \)
$67$ \( 118018152595456 - 8914074927104 T^{2} + 434640539648 T^{4} - 12637357056 T^{6} + 268234496 T^{8} - 3806976 T^{10} + 39536 T^{12} - 248 T^{14} + T^{16} \)
$71$ \( ( 5049009 - 1503243 T + 445314 T^{2} - 63585 T^{3} + 11614 T^{4} - 1324 T^{5} + 197 T^{6} - 14 T^{7} + T^{8} )^{2} \)
$73$ \( 96947540496 - 49356487188 T^{2} + 19122984549 T^{4} - 2876409225 T^{6} + 325629931 T^{8} - 5275616 T^{10} + 64815 T^{12} - 290 T^{14} + T^{16} \)
$79$ \( ( 9096256 + 5464992 T + 2897296 T^{2} + 310352 T^{3} + 42956 T^{4} + 1960 T^{5} + 297 T^{6} + 13 T^{7} + T^{8} )^{2} \)
$83$ \( ( 1695204 + 234033 T^{2} + 10810 T^{4} + 189 T^{6} + T^{8} )^{2} \)
$89$ \( ( 1382976 - 98784 T + 150528 T^{2} - 1512 T^{3} + 14128 T^{4} - 442 T^{5} + 147 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$97$ \( 178268320144656 - 11938797357732 T^{2} + 501889350825 T^{4} - 13392441198 T^{6} + 264597355 T^{8} - 3673676 T^{10} + 37731 T^{12} - 245 T^{14} + T^{16} \)
show more
show less