# Properties

 Label 1900.3.e.e Level $1900$ Weight $3$ Character orbit 1900.e Analytic conductor $51.771$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1900.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$51.7712502285$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 20 x^{10} + 44 x^{8} + 270 x^{6} + 36676 x^{4} + 71664 x^{2} + 687241$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{22}$$ Twist minimal: no (minimal twist has level 380) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} + \beta_{8} q^{7} + ( -6 - \beta_{1} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{6} q^{3} + \beta_{8} q^{7} + ( -6 - \beta_{1} + \beta_{3} ) q^{9} + ( -2 + \beta_{1} ) q^{11} + ( -\beta_{6} + \beta_{9} ) q^{13} + ( \beta_{7} - \beta_{8} ) q^{17} + ( -6 + \beta_{2} - \beta_{3} ) q^{19} + ( \beta_{2} - \beta_{5} ) q^{21} -5 \beta_{8} q^{23} + ( -6 \beta_{6} - \beta_{11} ) q^{27} + ( -\beta_{2} - \beta_{5} ) q^{29} + ( -\beta_{2} - \beta_{4} + \beta_{5} ) q^{31} + \beta_{9} q^{33} + ( -\beta_{6} - \beta_{11} ) q^{37} + ( 21 - 4 \beta_{1} - 3 \beta_{3} ) q^{39} + ( -3 \beta_{2} - \beta_{4} - \beta_{5} ) q^{41} + ( -6 \beta_{7} + \beta_{8} + \beta_{10} ) q^{43} + ( 3 \beta_{7} - \beta_{8} + 3 \beta_{10} ) q^{47} + ( -18 + 3 \beta_{1} - \beta_{3} ) q^{49} + ( -2 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{51} + ( 7 \beta_{6} - \beta_{9} ) q^{53} + ( \beta_{6} + 3 \beta_{7} - 5 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{57} + ( 2 \beta_{2} + 3 \beta_{4} ) q^{59} + ( -50 + 7 \beta_{1} ) q^{61} + ( 7 \beta_{7} - 11 \beta_{8} + \beta_{10} ) q^{63} + ( -5 \beta_{6} - 2 \beta_{11} ) q^{67} + ( -5 \beta_{2} + 5 \beta_{5} ) q^{69} + ( -3 \beta_{2} + \beta_{4} - \beta_{5} ) q^{71} + ( 7 \beta_{7} - \beta_{8} - 2 \beta_{10} ) q^{73} + ( \beta_{8} + \beta_{10} ) q^{77} + ( -6 \beta_{2} + 2 \beta_{5} ) q^{79} + ( 37 - 12 \beta_{3} ) q^{81} + 4 \beta_{10} q^{83} + ( \beta_{7} - 10 \beta_{8} - 3 \beta_{10} ) q^{87} + ( \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{89} + ( -4 \beta_{2} + \beta_{4} ) q^{91} + ( 2 \beta_{7} + 25 \beta_{8} + \beta_{10} ) q^{93} + ( 15 \beta_{6} + 4 \beta_{9} + \beta_{11} ) q^{97} + ( -12 + 4 \beta_{1} - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 76 q^{9} + O(q^{10})$$ $$12 q - 76 q^{9} - 24 q^{11} - 68 q^{19} + 264 q^{39} - 212 q^{49} - 600 q^{61} + 492 q^{81} - 136 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 20 x^{10} + 44 x^{8} + 270 x^{6} + 36676 x^{4} + 71664 x^{2} + 687241$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-1092 \nu^{10} + 464639 \nu^{8} - 7852919 \nu^{6} - 2760266 \nu^{4} - 167525143 \nu^{2} + 11929187385$$$$)/ 1817684474$$ $$\beta_{2}$$ $$=$$ $$($$$$28618 \nu^{10} - 524952 \nu^{8} + 4391039 \nu^{6} - 54167333 \nu^{4} + 1825258945 \nu^{2} + 558303007$$$$)/ 908842237$$ $$\beta_{3}$$ $$=$$ $$($$$$-60512 \nu^{10} + 2443821 \nu^{8} - 32340835 \nu^{6} + 100053868 \nu^{4} - 517724371 \nu^{2} + 21947164823$$$$)/ 1817684474$$ $$\beta_{4}$$ $$=$$ $$($$$$-4088716 \nu^{10} + 85163504 \nu^{8} - 153849803 \nu^{6} - 1329918273 \nu^{4} - 157720051429 \nu^{2} - 184041084331$$$$)/ 17268002503$$ $$\beta_{5}$$ $$=$$ $$($$$$8648182 \nu^{10} - 172547115 \nu^{8} + 139206261 \nu^{6} + 10398083744 \nu^{4} + 260828374443 \nu^{2} + 473221869537$$$$)/ 34536005006$$ $$\beta_{6}$$ $$=$$ $$($$$$-9729001 \nu^{11} + 195485288 \nu^{9} - 813261775 \nu^{7} + 3883239581 \nu^{5} - 354532580162 \nu^{3} - 2065201213063 \nu$$$$)/ 1506860428946$$ $$\beta_{7}$$ $$=$$ $$($$$$-194997239 \nu^{11} + 5048505213 \nu^{9} - 28805936576 \nu^{7} + 162835339255 \nu^{5} - 12940529662267 \nu^{3} + 36331199656762 \nu$$$$)/ 28630348149974$$ $$\beta_{8}$$ $$=$$ $$($$$$-651724712 \nu^{11} + 15674641213 \nu^{9} - 65363306369 \nu^{7} - 565819505556 \nu^{5} - 17994962875673 \nu^{3} + 25942480991957 \nu$$$$)/ 28630348149974$$ $$\beta_{9}$$ $$=$$ $$($$$$-39207097 \nu^{11} + 503811445 \nu^{9} + 6471056538 \nu^{7} - 52572021371 \nu^{5} - 1532495976047 \nu^{3} - 10974976155720 \nu$$$$)/ 1506860428946$$ $$\beta_{10}$$ $$=$$ $$($$$$-1586126027 \nu^{11} + 35580028314 \nu^{9} - 155977137845 \nu^{7} - 107857958145 \nu^{5} - 57879968630252 \nu^{3} + 134361173655723 \nu$$$$)/ 28630348149974$$ $$\beta_{11}$$ $$=$$ $$($$$$108734495 \nu^{11} - 992670198 \nu^{9} - 24848763895 \nu^{7} + 240525327873 \nu^{5} + 4527798347604 \nu^{3} + 34021640801649 \nu$$$$)/ 1506860428946$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{10} - \beta_{8} - \beta_{7} - 4 \beta_{6}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 3 \beta_{1} + 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{11} + 7 \beta_{10} + 12 \beta_{9} - 3 \beta_{8} - 39 \beta_{7} - 12 \beta_{6}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$10 \beta_{5} + 11 \beta_{4} + 9 \beta_{3} + 12 \beta_{2} - 37 \beta_{1} + 107$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$104 \beta_{11} + 117 \beta_{10} + 348 \beta_{9} - 269 \beta_{8} - 193 \beta_{7} - 92 \beta_{6}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$138 \beta_{5} + 205 \beta_{4} - 38 \beta_{3} + 401 \beta_{2} - 152 \beta_{1} + 1504$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$1688 \beta_{11} - 61 \beta_{10} + 7144 \beta_{9} - 67 \beta_{8} + 629 \beta_{7} - 9828 \beta_{6}$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$2494 \beta_{5} + 3663 \beta_{4} - 339 \beta_{3} + 7438 \beta_{2} + 4487 \beta_{1} - 25057$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$31500 \beta_{11} - 42099 \beta_{10} + 114804 \beta_{9} + 72759 \beta_{8} + 91547 \beta_{7} - 102460 \beta_{6}$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$43502 \beta_{5} + 56558 \beta_{4} - 47133 \beta_{3} + 97359 \beta_{2} + 226937 \beta_{1} - 973351$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$387576 \beta_{11} - 1261457 \beta_{10} + 1411196 \beta_{9} + 1681777 \beta_{8} + 3343309 \beta_{7} - 1226604 \beta_{6}$$$$)/8$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1101.1
 1.46321 + 2.80718i −1.46321 + 2.80718i −1.34185 + 1.63363i 1.34185 + 1.63363i 4.19028 + 0.975196i −4.19028 + 0.975196i 4.19028 − 0.975196i −4.19028 − 0.975196i −1.34185 − 1.63363i 1.34185 − 1.63363i 1.46321 − 2.80718i −1.46321 − 2.80718i
0 5.61436i 0 0 0 −7.03410 0 −22.5210 0
1101.2 0 5.61436i 0 0 0 7.03410 0 −22.5210 0
1101.3 0 3.26725i 0 0 0 −6.30368 0 −1.67495 0
1101.4 0 3.26725i 0 0 0 6.30368 0 −1.67495 0
1101.5 0 1.95039i 0 0 0 −2.18749 0 5.19597 0
1101.6 0 1.95039i 0 0 0 2.18749 0 5.19597 0
1101.7 0 1.95039i 0 0 0 −2.18749 0 5.19597 0
1101.8 0 1.95039i 0 0 0 2.18749 0 5.19597 0
1101.9 0 3.26725i 0 0 0 −6.30368 0 −1.67495 0
1101.10 0 3.26725i 0 0 0 6.30368 0 −1.67495 0
1101.11 0 5.61436i 0 0 0 −7.03410 0 −22.5210 0
1101.12 0 5.61436i 0 0 0 7.03410 0 −22.5210 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1101.12 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.b odd 2 1 inner
95.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.3.e.e 12
5.b even 2 1 inner 1900.3.e.e 12
5.c odd 4 2 380.3.g.c 12
15.e even 4 2 3420.3.h.e 12
19.b odd 2 1 inner 1900.3.e.e 12
95.d odd 2 1 inner 1900.3.e.e 12
95.g even 4 2 380.3.g.c 12
285.j odd 4 2 3420.3.h.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.g.c 12 5.c odd 4 2
380.3.g.c 12 95.g even 4 2
1900.3.e.e 12 1.a even 1 1 trivial
1900.3.e.e 12 5.b even 2 1 inner
1900.3.e.e 12 19.b odd 2 1 inner
1900.3.e.e 12 95.d odd 2 1 inner
3420.3.h.e 12 15.e even 4 2
3420.3.h.e 12 285.j odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1900, [\chi])$$:

 $$T_{3}^{6} + 46 T_{3}^{4} + 497 T_{3}^{2} + 1280$$ $$T_{7}^{6} - 94 T_{7}^{4} + 2393 T_{7}^{2} - 9408$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$( 1280 + 497 T^{2} + 46 T^{4} + T^{6} )^{2}$$
$5$ $$T^{12}$$
$7$ $$( -9408 + 2393 T^{2} - 94 T^{4} + T^{6} )^{2}$$
$11$ $$( -44 - 38 T + 6 T^{2} + T^{3} )^{4}$$
$13$ $$( 7200000 + 127153 T^{2} + 682 T^{4} + T^{6} )^{2}$$
$17$ $$( -12288 + 10793 T^{2} - 314 T^{4} + T^{6} )^{2}$$
$19$ $$( 47045881 + 4430914 T + 261003 T^{2} + 12084 T^{3} + 723 T^{4} + 34 T^{5} + T^{6} )^{2}$$
$23$ $$( -147000000 + 1495625 T^{2} - 2350 T^{4} + T^{6} )^{2}$$
$29$ $$( 51671040 + 1274297 T^{2} + 2306 T^{4} + T^{6} )^{2}$$
$31$ $$( 1345781760 + 4670948 T^{2} + 4652 T^{4} + T^{6} )^{2}$$
$37$ $$( 5202247680 + 9192448 T^{2} + 5312 T^{4} + T^{6} )^{2}$$
$41$ $$( 318504960 + 3194532 T^{2} + 7140 T^{4} + T^{6} )^{2}$$
$43$ $$( -95158272 + 12661412 T^{2} - 7540 T^{4} + T^{6} )^{2}$$
$47$ $$( -75606592512 + 60919184 T^{2} - 14056 T^{4} + T^{6} )^{2}$$
$53$ $$( 136869120 + 2302273 T^{2} + 3130 T^{4} + T^{6} )^{2}$$
$59$ $$( 7228354560 + 82566857 T^{2} + 18086 T^{4} + T^{6} )^{2}$$
$61$ $$( 18964 + 5050 T + 150 T^{2} + T^{3} )^{4}$$
$67$ $$( 301871934720 + 147525697 T^{2} + 22430 T^{4} + T^{6} )^{2}$$
$71$ $$( 23313776640 + 31173668 T^{2} + 11980 T^{4} + T^{6} )^{2}$$
$73$ $$( -4401589248 + 24107193 T^{2} - 12714 T^{4} + T^{6} )^{2}$$
$79$ $$( 4282122240 + 115386512 T^{2} + 26440 T^{4} + T^{6} )^{2}$$
$83$ $$( -216760320000 + 120002816 T^{2} - 19616 T^{4} + T^{6} )^{2}$$
$89$ $$( 187730165760 + 107125028 T^{2} + 18508 T^{4} + T^{6} )^{2}$$
$97$ $$( 318730752000 + 150753808 T^{2} + 22008 T^{4} + T^{6} )^{2}$$