# Properties

 Label 1900.3.e.f Level $1900$ Weight $3$ Character orbit 1900.e Analytic conductor $51.771$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# 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: $$x^{12} + 62 x^{10} + 1445 x^{8} + 15924 x^{6} + 83244 x^{4} + 170640 x^{2} + 55600$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{12}\cdot 5$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 1 + \beta_{9} ) q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 1 + \beta_{9} ) q^{7} + ( -1 + \beta_{2} ) q^{9} + ( -2 + \beta_{2} + \beta_{4} - \beta_{9} ) q^{11} + ( -\beta_{1} + \beta_{3} ) q^{13} + ( -2 \beta_{2} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{17} + ( 2 + \beta_{1} - \beta_{4} - \beta_{5} ) q^{19} + ( 2 \beta_{1} - \beta_{8} - \beta_{11} ) q^{21} + ( -1 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{23} + ( 2 \beta_{1} + \beta_{10} + \beta_{11} ) q^{27} + ( -2 \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{29} + ( -\beta_{1} - \beta_{3} - \beta_{8} ) q^{31} + ( -6 \beta_{1} - \beta_{3} + 2 \beta_{11} ) q^{33} + ( \beta_{3} - 2 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{37} + ( 12 + \beta_{2} + 4 \beta_{4} + \beta_{7} ) q^{39} + ( 3 \beta_{1} + \beta_{3} - \beta_{8} - 2 \beta_{11} ) q^{41} + ( 15 - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} - 2 \beta_{9} ) q^{43} + ( 7 - 2 \beta_{2} + 5 \beta_{4} ) q^{47} + ( 1 + 2 \beta_{2} + 5 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{49} + ( 7 \beta_{1} + \beta_{3} + 2 \beta_{8} - \beta_{11} ) q^{51} + ( -2 \beta_{1} + 4 \beta_{8} + 3 \beta_{10} + 2 \beta_{11} ) q^{53} + ( -13 - 4 \beta_{1} + \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{57} + ( -3 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{59} + ( 12 + \beta_{2} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + \beta_{9} ) q^{61} + ( -3 + 2 \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{63} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{8} + 4 \beta_{10} - \beta_{11} ) q^{69} + ( -11 \beta_{1} - 5 \beta_{3} + \beta_{8} + 4 \beta_{11} ) q^{71} + ( 14 + 3 \beta_{4} - \beta_{5} - \beta_{6} + 5 \beta_{7} + 5 \beta_{9} ) q^{73} + ( -32 - 2 \beta_{4} - 5 \beta_{7} + 4 \beta_{9} ) q^{77} + ( 4 \beta_{1} + \beta_{3} - 3 \beta_{8} + \beta_{10} + 4 \beta_{11} ) q^{79} + ( -39 + 5 \beta_{2} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 9 \beta_{7} + 3 \beta_{9} ) q^{81} + ( 19 + 6 \beta_{2} - 3 \beta_{4} - \beta_{5} - \beta_{6} + 8 \beta_{9} ) q^{83} + ( 10 + 8 \beta_{2} + 9 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 6 \beta_{7} - \beta_{9} ) q^{87} + ( -3 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} - 5 \beta_{10} ) q^{89} + ( -9 \beta_{1} + 4 \beta_{3} + 3 \beta_{8} - 3 \beta_{10} + 3 \beta_{11} ) q^{91} + ( 10 - 8 \beta_{2} - 8 \beta_{4} + \beta_{7} - 4 \beta_{9} ) q^{93} + ( 20 \beta_{1} - \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{10} - 3 \beta_{11} ) q^{97} + ( 28 - 9 \beta_{2} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 12q^{7} - 16q^{9} + O(q^{10})$$ $$12q + 12q^{7} - 16q^{9} - 32q^{11} + 12q^{17} + 24q^{19} - 4q^{23} + 124q^{39} + 176q^{43} + 72q^{47} - 24q^{49} - 140q^{57} + 152q^{61} - 48q^{63} + 148q^{73} - 376q^{77} - 468q^{81} + 208q^{83} + 84q^{87} + 184q^{93} + 392q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 62 x^{10} + 1445 x^{8} + 15924 x^{6} + 83244 x^{4} + 170640 x^{2} + 55600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 10$$ $$\beta_{3}$$ $$=$$ $$($$$$401 \nu^{11} + 20007 \nu^{9} + 344590 \nu^{7} + 2594164 \nu^{5} + 8665084 \nu^{3} + 10740520 \nu$$$$)/203940$$ $$\beta_{4}$$ $$=$$ $$($$$$-301 \nu^{10} - 14382 \nu^{8} - 228185 \nu^{6} - 1462904 \nu^{4} - 3463424 \nu^{2} - 1849460$$$$)/67980$$ $$\beta_{5}$$ $$=$$ $$($$$$-59 \nu^{11} + 2166 \nu^{10} - 5868 \nu^{9} + 106542 \nu^{8} - 184075 \nu^{7} + 1747380 \nu^{6} - 2290126 \nu^{5} + 11340804 \nu^{4} - 10898236 \nu^{3} + 24741264 \nu^{2} - 13285720 \nu + 8353200$$$$)/407880$$ $$\beta_{6}$$ $$=$$ $$($$$$59 \nu^{11} + 2166 \nu^{10} + 5868 \nu^{9} + 106542 \nu^{8} + 184075 \nu^{7} + 1747380 \nu^{6} + 2290126 \nu^{5} + 11340804 \nu^{4} + 10898236 \nu^{3} + 24741264 \nu^{2} + 13285720 \nu + 8353200$$$$)/407880$$ $$\beta_{7}$$ $$=$$ $$($$$$-113 \nu^{10} - 5661 \nu^{8} - 95740 \nu^{6} - 650992 \nu^{4} - 1502992 \nu^{2} - 417160$$$$)/18540$$ $$\beta_{8}$$ $$=$$ $$($$$$-247 \nu^{11} - 13044 \nu^{9} - 237725 \nu^{7} - 1788788 \nu^{5} - 4935908 \nu^{3} - 3124340 \nu$$$$)/67980$$ $$\beta_{9}$$ $$=$$ $$($$$$1984 \nu^{10} + 101403 \nu^{8} + 1766315 \nu^{6} + 12527276 \nu^{4} + 31136696 \nu^{2} + 11106620$$$$)/203940$$ $$\beta_{10}$$ $$=$$ $$($$$$113 \nu^{11} + 5661 \nu^{9} + 95740 \nu^{7} + 650992 \nu^{5} + 1502992 \nu^{3} + 435700 \nu$$$$)/18540$$ $$\beta_{11}$$ $$=$$ $$($$$$-113 \nu^{11} - 5661 \nu^{9} - 95740 \nu^{7} - 650992 \nu^{5} - 1484452 \nu^{3} - 139060 \nu$$$$)/18540$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 10$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{10} - 16 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{9} + 9 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - 22 \beta_{2} + 150$$ $$\nu^{5}$$ $$=$$ $$-29 \beta_{11} - 34 \beta_{10} - 6 \beta_{8} - 4 \beta_{6} + 4 \beta_{5} + 5 \beta_{3} + 303 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-93 \beta_{9} - 297 \beta_{7} - 79 \beta_{6} - 79 \beta_{5} + 15 \beta_{4} + 470 \beta_{2} - 2674$$ $$\nu^{7}$$ $$=$$ $$721 \beta_{11} + 910 \beta_{10} + 236 \beta_{8} + 158 \beta_{6} - 158 \beta_{5} - 173 \beta_{3} - 6313 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$2413 \beta_{9} + 7861 \beta_{7} + 2263 \beta_{6} + 2263 \beta_{5} - 91 \beta_{4} - 10310 \beta_{2} + 53398$$ $$\nu^{9}$$ $$=$$ $$-17249 \beta_{11} - 22606 \beta_{10} - 6848 \beta_{8} - 4526 \beta_{6} + 4526 \beta_{5} + 4617 \beta_{3} + 138837 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-59373 \beta_{9} - 194193 \beta_{7} - 57959 \beta_{6} - 57959 \beta_{5} - 2389 \beta_{4} + 231734 \beta_{2} - 1144366$$ $$\nu^{11}$$ $$=$$ $$407025 \beta_{11} + 544234 \beta_{10} + 177680 \beta_{8} + 115918 \beta_{6} - 115918 \beta_{5} - 113529 \beta_{3} - 3143257 \beta_{1}$$

## Character values

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

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

## 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
 − 4.83157i − 3.53755i − 3.42504i − 3.14288i − 2.03369i − 0.630185i 0.630185i 2.03369i 3.14288i 3.42504i 3.53755i 4.83157i
0 4.83157i 0 0 0 3.72856 0 −14.3441 0
1101.2 0 3.53755i 0 0 0 −0.468611 0 −3.51429 0
1101.3 0 3.42504i 0 0 0 −9.18599 0 −2.73093 0
1101.4 0 3.14288i 0 0 0 12.2192 0 −0.877687 0
1101.5 0 2.03369i 0 0 0 −4.27843 0 4.86411 0
1101.6 0 0.630185i 0 0 0 3.98530 0 8.60287 0
1101.7 0 0.630185i 0 0 0 3.98530 0 8.60287 0
1101.8 0 2.03369i 0 0 0 −4.27843 0 4.86411 0
1101.9 0 3.14288i 0 0 0 12.2192 0 −0.877687 0
1101.10 0 3.42504i 0 0 0 −9.18599 0 −2.73093 0
1101.11 0 3.53755i 0 0 0 −0.468611 0 −3.51429 0
1101.12 0 4.83157i 0 0 0 3.72856 0 −14.3441 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
19.b 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.f 12
5.b even 2 1 380.3.e.a 12
5.c odd 4 2 1900.3.g.c 24
15.d odd 2 1 3420.3.o.a 12
19.b odd 2 1 inner 1900.3.e.f 12
20.d odd 2 1 1520.3.h.b 12
95.d odd 2 1 380.3.e.a 12
95.g even 4 2 1900.3.g.c 24
285.b even 2 1 3420.3.o.a 12
380.d even 2 1 1520.3.h.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.e.a 12 5.b even 2 1
380.3.e.a 12 95.d odd 2 1
1520.3.h.b 12 20.d odd 2 1
1520.3.h.b 12 380.d even 2 1
1900.3.e.f 12 1.a even 1 1 trivial
1900.3.e.f 12 19.b odd 2 1 inner
1900.3.g.c 24 5.c odd 4 2
1900.3.g.c 24 95.g even 4 2
3420.3.o.a 12 15.d odd 2 1
3420.3.o.a 12 285.b even 2 1

## Hecke kernels

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

 $$T_{3}^{12} + 62 T_{3}^{10} + 1445 T_{3}^{8} + 15924 T_{3}^{6} + 83244 T_{3}^{4} + 170640 T_{3}^{2} + 55600$$ $$T_{7}^{6} - 6 T_{7}^{5} - 123 T_{7}^{4} + 448 T_{7}^{3} + 2080 T_{7}^{2} - 6272 T_{7} - 3344$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$55600 + 170640 T^{2} + 83244 T^{4} + 15924 T^{6} + 1445 T^{8} + 62 T^{10} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$( -3344 - 6272 T + 2080 T^{2} + 448 T^{3} - 123 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$11$ $$( -43264 + 65728 T + 5360 T^{2} - 2656 T^{3} - 164 T^{4} + 16 T^{5} + T^{6} )^{2}$$
$13$ $$18127879600 + 46566284240 T^{2} + 3704449004 T^{4} + 67065412 T^{6} + 454605 T^{8} + 1174 T^{10} + T^{12}$$
$17$ $$( 818576 - 680352 T + 111560 T^{2} + 3968 T^{3} - 703 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$19$ $$2213314919066161 - 147145590187224 T + 12194198263438 T^{2} - 538581245688 T^{3} + 40699639263 T^{4} - 1182272112 T^{5} + 69221028 T^{6} - 3274992 T^{7} + 312303 T^{8} - 11448 T^{9} + 718 T^{10} - 24 T^{11} + T^{12}$$
$23$ $$( -752400 + 748800 T + 62080 T^{2} - 18400 T^{3} - 1379 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$29$ $$24578837040640000 + 577751990604800 T^{2} + 4466410158784 T^{4} + 13344932784 T^{6} + 14527705 T^{8} + 6442 T^{10} + T^{12}$$
$31$ $$119374584217600 + 9209673482240 T^{2} + 189626036224 T^{4} + 1469386752 T^{6} + 4308160 T^{8} + 3904 T^{10} + T^{12}$$
$37$ $$2556834339961600 + 179296520786560 T^{2} + 3535386239344 T^{4} + 16197016912 T^{6} + 24619340 T^{8} + 9204 T^{10} + T^{12}$$
$41$ $$122746616519065600 + 1540001874575360 T^{2} + 7218693013504 T^{4} + 15782277120 T^{6} + 16359424 T^{8} + 7120 T^{10} + T^{12}$$
$43$ $$( -2451136 - 4996608 T - 2018960 T^{2} + 110560 T^{3} + 460 T^{4} - 88 T^{5} + T^{6} )^{2}$$
$47$ $$( -9167899456 - 205846592 T + 15902576 T^{2} + 188128 T^{3} - 7500 T^{4} - 36 T^{5} + T^{6} )^{2}$$
$53$ $$31\!\cdots\!00$$$$+ 6145069971405169040 T^{2} + 4516584793096844 T^{4} + 1637397541300 T^{6} + 308301741 T^{8} + 28390 T^{10} + T^{12}$$
$59$ $$492787956121600 + 7814344462704640 T^{2} + 56540303646464 T^{4} + 123820877984 T^{6} + 81450305 T^{8} + 17122 T^{10} + T^{12}$$
$61$ $$( -1301500864 - 116799296 T + 3171968 T^{2} + 189280 T^{3} - 2952 T^{4} - 76 T^{5} + T^{6} )^{2}$$
$67$ $$392055999889143600 + 13255845003660240 T^{2} + 76653074089804 T^{4} + 153163111300 T^{6} + 112421269 T^{8} + 19710 T^{10} + T^{12}$$
$71$ $$33\!\cdots\!00$$$$+ 20557086414940733440 T^{2} + 30856628712812544 T^{4} + 8826848235520 T^{6} + 1014731456 T^{8} + 52160 T^{10} + T^{12}$$
$73$ $$( 5637612816 - 490085088 T + 5561000 T^{2} + 404672 T^{3} - 7103 T^{4} - 74 T^{5} + T^{6} )^{2}$$
$79$ $$36\!\cdots\!00$$$$+ 72155660704890880000 T^{2} + 37408890436358144 T^{4} + 8400762836480 T^{6} + 924364304 T^{8} + 49000 T^{10} + T^{12}$$
$83$ $$( -2227337984 - 1670714112 T + 37448240 T^{2} + 941184 T^{3} - 12064 T^{4} - 104 T^{5} + T^{6} )^{2}$$
$89$ $$10\!\cdots\!00$$$$+ 8824799886414397440 T^{2} + 10690415477484544 T^{4} + 4018121756928 T^{6} + 630894800 T^{8} + 42776 T^{10} + T^{12}$$
$97$ $$23\!\cdots\!00$$$$+ 13144169535713183360 T^{2} + 20739756167840624 T^{4} + 7602023198288 T^{6} + 1027748300 T^{8} + 55476 T^{10} + T^{12}$$