Properties

Label 1950.2.y.k
Level $1950$
Weight $2$
Character orbit 1950.y
Analytic conductor $15.571$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.y (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.17284886784.1
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 30 x^{5} + 185 x^{4} + 36 x^{3} + 8 x^{2} + 208 x + 2704\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{6} ) q^{2} + ( -\beta_{2} - \beta_{5} ) q^{3} -\beta_{6} q^{4} -\beta_{5} q^{6} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{7} - q^{8} + \beta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{6} ) q^{2} + ( -\beta_{2} - \beta_{5} ) q^{3} -\beta_{6} q^{4} -\beta_{5} q^{6} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{7} - q^{8} + \beta_{6} q^{9} + ( 2 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} ) q^{11} + \beta_{2} q^{12} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{14} + ( -1 + \beta_{6} ) q^{16} + 4 \beta_{5} q^{17} + q^{18} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{19} + ( \beta_{2} - \beta_{3} + \beta_{7} ) q^{21} + ( 1 - \beta_{2} + \beta_{4} - \beta_{7} ) q^{22} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{23} + ( \beta_{2} + \beta_{5} ) q^{24} + ( -1 - 2 \beta_{2} - 2 \beta_{5} - \beta_{7} ) q^{26} -\beta_{2} q^{27} + ( \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{28} + ( 2 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{29} + ( 4 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - 6 \beta_{6} - \beta_{7} ) q^{31} + \beta_{6} q^{32} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{33} -4 \beta_{2} q^{34} + ( 1 - \beta_{6} ) q^{36} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{37} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{38} + ( -1 + 2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{39} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{41} + ( -\beta_{1} + \beta_{4} ) q^{42} + ( 8 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{43} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{44} + ( -1 + \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{46} + ( 1 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{47} + \beta_{5} q^{48} + ( -3 + 5 \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{49} -4 q^{51} + ( -2 + \beta_{1} - \beta_{5} + 2 \beta_{6} ) q^{52} + ( -1 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{53} + ( -\beta_{2} - \beta_{5} ) q^{54} + ( 1 - \beta_{1} - \beta_{3} - \beta_{5} ) q^{56} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{57} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{58} + ( -4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{59} + ( -4 \beta_{2} - 2 \beta_{5} - 4 \beta_{6} ) q^{61} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{62} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{63} + q^{64} + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{66} + ( 4 - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{67} + ( -4 \beta_{2} - 4 \beta_{5} ) q^{68} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{69} + ( -2 + 2 \beta_{1} - 2 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} ) q^{71} -\beta_{6} q^{72} + ( -4 + 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{4} + 10 \beta_{5} + 2 \beta_{7} ) q^{73} + ( -1 - 5 \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{74} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{76} + ( 5 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 12 \beta_{6} - \beta_{7} ) q^{77} + ( 1 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{78} + ( 4 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} ) q^{79} + ( -1 + \beta_{6} ) q^{81} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{82} + ( 2 + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} + 2 \beta_{7} ) q^{83} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{84} + ( 4 + 4 \beta_{2} - \beta_{3} - 8 \beta_{6} + \beta_{7} ) q^{86} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{87} + ( -2 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{7} ) q^{88} + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{89} + ( 2 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} ) q^{91} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{92} + ( -\beta_{1} + 3 \beta_{2} - \beta_{4} - 3 \beta_{5} ) q^{93} + ( 3 + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{94} -\beta_{2} q^{96} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} - 8 \beta_{6} ) q^{97} + ( 1 - \beta_{1} + 8 \beta_{2} - \beta_{3} + 3 \beta_{5} + 3 \beta_{6} ) q^{98} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{2} - 4q^{4} - 2q^{7} - 8q^{8} + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{2} - 4q^{4} - 2q^{7} - 8q^{8} + 4q^{9} + 6q^{11} - 6q^{13} - 4q^{14} - 4q^{16} + 8q^{18} + 6q^{19} + 6q^{22} - 6q^{23} - 12q^{26} - 2q^{28} + 8q^{29} + 4q^{32} - 2q^{33} + 4q^{36} + 10q^{37} - 6q^{39} + 48q^{43} - 6q^{46} + 16q^{47} - 14q^{49} - 32q^{51} - 6q^{52} + 2q^{56} - 8q^{58} - 24q^{59} - 16q^{61} - 30q^{62} + 2q^{63} + 8q^{64} - 4q^{66} + 12q^{67} - 4q^{69} - 12q^{71} - 4q^{72} - 24q^{73} - 10q^{74} - 6q^{76} + 6q^{78} + 20q^{79} - 4q^{81} + 32q^{83} - 6q^{87} - 6q^{88} - 42q^{89} - 10q^{91} - 4q^{93} + 8q^{94} - 36q^{97} + 14q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 2 x^{6} + 30 x^{5} + 185 x^{4} + 36 x^{3} + 8 x^{2} + 208 x + 2704\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -2225 \nu^{7} + 27304 \nu^{6} - 47714 \nu^{5} - 21770 \nu^{4} + 204731 \nu^{3} + 5488658 \nu^{2} + 258164 \nu - 320632 \)\()/20561424\)
\(\beta_{3}\)\(=\)\((\)\( -1777 \nu^{7} - 125666 \nu^{6} + 800522 \nu^{5} - 2370014 \nu^{4} - 784473 \nu^{3} - 5277648 \nu^{2} + 34618460 \nu - 62031216 \)\()/6853808\)
\(\beta_{4}\)\(=\)\((\)\( -299 \nu^{7} + 7801 \nu^{6} - 39200 \nu^{5} + 84154 \nu^{4} + 80915 \nu^{3} + 360425 \nu^{2} - 1640788 \nu + 2274116 \)\()/395412\)
\(\beta_{5}\)\(=\)\((\)\( 43733 \nu^{7} - 71918 \nu^{6} - 318186 \nu^{5} + 3350390 \nu^{4} + 3714597 \nu^{3} - 2633192 \nu^{2} - 18392236 \nu + 94417440 \)\()/20561424\)
\(\beta_{6}\)\(=\)\((\)\( -151 \nu^{7} + 822 \nu^{6} - 2018 \nu^{5} - 2554 \nu^{4} - 7135 \nu^{3} + 37828 \nu^{2} - 46500 \nu - 3328 \)\()/51792\)
\(\beta_{7}\)\(=\)\((\)\( -123974 \nu^{7} + 362933 \nu^{6} - 209286 \nu^{5} - 5793110 \nu^{4} - 10547568 \nu^{3} + 7562729 \nu^{2} + 3493156 \nu - 117816036 \)\()/10280712\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - 4 \beta_{6} + \beta_{5} + \beta_{4} + 9 \beta_{2} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(6 \beta_{7} - 10 \beta_{6} + 26 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} + 18 \beta_{2} + 3 \beta_{1} - 18\)
\(\nu^{4}\)\(=\)\(23 \beta_{7} + 139 \beta_{5} + 5 \beta_{4} + 28 \beta_{3} + 67 \beta_{2} - 5 \beta_{1} - 149\)
\(\nu^{5}\)\(=\)\(250 \beta_{6} + 322 \beta_{5} - 72 \beta_{4} + 72 \beta_{3} - 76 \beta_{2} - 149 \beta_{1} - 398\)
\(\nu^{6}\)\(=\)\(-471 \beta_{7} + 1748 \beta_{6} - 619 \beta_{5} - 619 \beta_{4} - 148 \beta_{3} - 2095 \beta_{2} - 619 \beta_{1} - 255\)
\(\nu^{7}\)\(=\)\(-2986 \beta_{7} + 5302 \beta_{6} - 11002 \beta_{5} - 2714 \beta_{4} - 2714 \beta_{3} - 10118 \beta_{2} - 1493 \beta_{1} + 7530\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(\beta_{6}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−1.70006 + 1.70006i
1.33404 1.33404i
3.17270 + 3.17270i
−1.80668 1.80668i
−1.70006 1.70006i
1.33404 + 1.33404i
3.17270 3.17270i
−1.80668 + 1.80668i
0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i −2.32233 + 4.02239i −1.00000 0.500000 0.866025i 0
49.2 0.500000 + 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i 1.82233 3.15637i −1.00000 0.500000 0.866025i 0
49.3 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i −1.16129 + 2.01141i −1.00000 0.500000 0.866025i 0
49.4 0.500000 + 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i 0.661290 1.14539i −1.00000 0.500000 0.866025i 0
199.1 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i −2.32233 4.02239i −1.00000 0.500000 + 0.866025i 0
199.2 0.500000 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i 1.82233 + 3.15637i −1.00000 0.500000 + 0.866025i 0
199.3 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i −1.16129 2.01141i −1.00000 0.500000 + 0.866025i 0
199.4 0.500000 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i 0.661290 + 1.14539i −1.00000 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.y.k 8
5.b even 2 1 1950.2.y.j 8
5.c odd 4 1 390.2.bb.c 8
5.c odd 4 1 1950.2.bc.g 8
13.e even 6 1 1950.2.y.j 8
15.e even 4 1 1170.2.bs.f 8
65.l even 6 1 inner 1950.2.y.k 8
65.o even 12 1 5070.2.a.bz 4
65.q odd 12 1 5070.2.b.ba 8
65.r odd 12 1 390.2.bb.c 8
65.r odd 12 1 1950.2.bc.g 8
65.r odd 12 1 5070.2.b.ba 8
65.t even 12 1 5070.2.a.ca 4
195.bf even 12 1 1170.2.bs.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.c 8 5.c odd 4 1
390.2.bb.c 8 65.r odd 12 1
1170.2.bs.f 8 15.e even 4 1
1170.2.bs.f 8 195.bf even 12 1
1950.2.y.j 8 5.b even 2 1
1950.2.y.j 8 13.e even 6 1
1950.2.y.k 8 1.a even 1 1 trivial
1950.2.y.k 8 65.l even 6 1 inner
1950.2.bc.g 8 5.c odd 4 1
1950.2.bc.g 8 65.r odd 12 1
5070.2.a.bz 4 65.o even 12 1
5070.2.a.ca 4 65.t even 12 1
5070.2.b.ba 8 65.q odd 12 1
5070.2.b.ba 8 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{4} \)
$3$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( 2704 - 1040 T + 1388 T^{2} + 172 T^{3} + 349 T^{4} + 2 T^{5} + 23 T^{6} + 2 T^{7} + T^{8} \)
$11$ \( 32761 + 22806 T - 138 T^{2} - 3780 T^{3} + 467 T^{4} + 180 T^{5} - 18 T^{6} - 6 T^{7} + T^{8} \)
$13$ \( 28561 + 13182 T + 7605 T^{2} + 2418 T^{3} + 848 T^{4} + 186 T^{5} + 45 T^{6} + 6 T^{7} + T^{8} \)
$17$ \( ( 256 - 16 T^{2} + T^{4} )^{2} \)
$19$ \( 219024 + 84240 T - 15876 T^{2} - 10260 T^{3} + 2421 T^{4} + 342 T^{5} - 45 T^{6} - 6 T^{7} + T^{8} \)
$23$ \( 2704 + 624 T - 2292 T^{2} - 540 T^{3} + 1997 T^{4} - 270 T^{5} - 33 T^{6} + 6 T^{7} + T^{8} \)
$29$ \( 141376 + 55648 T + 36568 T^{2} + 244 T^{3} + 2329 T^{4} + 16 T^{5} + 103 T^{6} - 8 T^{7} + T^{8} \)
$31$ \( 913936 + 158088 T^{2} + 8777 T^{4} + 174 T^{6} + T^{8} \)
$37$ \( 644809 - 345290 T + 157598 T^{2} - 30680 T^{3} + 6259 T^{4} - 520 T^{5} + 134 T^{6} - 10 T^{7} + T^{8} \)
$41$ \( 692224 - 159744 T - 57600 T^{2} + 16128 T^{3} + 6224 T^{4} - 84 T^{6} + T^{8} \)
$43$ \( 327184 + 604032 T + 232716 T^{2} - 256608 T^{3} + 76517 T^{4} - 11664 T^{5} + 1011 T^{6} - 48 T^{7} + T^{8} \)
$47$ \( ( -1103 + 776 T - 82 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$53$ \( 2704 + 7816 T^{2} + 5793 T^{4} + 214 T^{6} + T^{8} \)
$59$ \( 80656 + 23856 T - 10996 T^{2} - 3948 T^{3} + 1821 T^{4} + 1128 T^{5} + 239 T^{6} + 24 T^{7} + T^{8} \)
$61$ \( ( 16 + 32 T + 60 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$67$ \( 123904 + 50688 T + 39040 T^{2} + 960 T^{3} + 4080 T^{4} + 336 T^{5} + 196 T^{6} - 12 T^{7} + T^{8} \)
$71$ \( 25240576 + 19051008 T + 5817984 T^{2} + 773568 T^{3} + 31472 T^{4} - 2448 T^{5} - 156 T^{6} + 12 T^{7} + T^{8} \)
$73$ \( ( -4544 - 1728 T - 124 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$79$ \( ( 3508 + 860 T - 147 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$83$ \( ( -5024 + 1936 T - 100 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$89$ \( 1008016 - 349392 T - 131316 T^{2} + 59508 T^{3} + 35117 T^{4} + 7182 T^{5} + 759 T^{6} + 42 T^{7} + T^{8} \)
$97$ \( 29246464 - 3374592 T + 2444416 T^{2} + 626496 T^{3} + 127344 T^{4} + 12432 T^{5} + 916 T^{6} + 36 T^{7} + T^{8} \)
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