# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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