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 $4$

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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: \(x^{12} - 20 x^{10} + 44 x^{8} + 270 x^{6} + 36676 x^{4} + 71664 x^{2} + 687241\)
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:

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

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