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 $2$

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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:

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

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