Properties

Label 210.6.i.b
Level $210$
Weight $6$
Character orbit 210.i
Analytic conductor $33.681$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 210.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.6806021607\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{151})\)
Defining polynomial: \(x^{4} + 151 x^{2} + 22801\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -4 - 4 \beta_{2} ) q^{2} -9 \beta_{2} q^{3} + 16 \beta_{2} q^{4} + ( 25 + 25 \beta_{2} ) q^{5} -36 q^{6} + ( -56 + 7 \beta_{1} + 56 \beta_{2} ) q^{7} + 64 q^{8} + ( -81 - 81 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -4 - 4 \beta_{2} ) q^{2} -9 \beta_{2} q^{3} + 16 \beta_{2} q^{4} + ( 25 + 25 \beta_{2} ) q^{5} -36 q^{6} + ( -56 + 7 \beta_{1} + 56 \beta_{2} ) q^{7} + 64 q^{8} + ( -81 - 81 \beta_{2} ) q^{9} -100 \beta_{2} q^{10} + ( 11 \beta_{1} - 241 \beta_{2} + 11 \beta_{3} ) q^{11} + ( 144 + 144 \beta_{2} ) q^{12} + ( -14 - 35 \beta_{3} ) q^{13} + ( 448 - 28 \beta_{1} + 224 \beta_{2} - 28 \beta_{3} ) q^{14} + 225 q^{15} + ( -256 - 256 \beta_{2} ) q^{16} + ( 109 \beta_{1} + 469 \beta_{2} + 109 \beta_{3} ) q^{17} + 324 \beta_{2} q^{18} + ( -1099 - 68 \beta_{1} - 1099 \beta_{2} ) q^{19} -400 q^{20} + ( 504 + 1008 \beta_{2} - 63 \beta_{3} ) q^{21} + ( -964 - 44 \beta_{3} ) q^{22} + ( -309 - 369 \beta_{1} - 309 \beta_{2} ) q^{23} -576 \beta_{2} q^{24} + 625 \beta_{2} q^{25} + ( 56 - 140 \beta_{1} + 56 \beta_{2} ) q^{26} -729 q^{27} + ( -896 - 1792 \beta_{2} + 112 \beta_{3} ) q^{28} + ( -5043 - 183 \beta_{3} ) q^{29} + ( -900 - 900 \beta_{2} ) q^{30} + ( -540 \beta_{1} + 2069 \beta_{2} - 540 \beta_{3} ) q^{31} + 1024 \beta_{2} q^{32} + ( -2169 + 99 \beta_{1} - 2169 \beta_{2} ) q^{33} + ( 1876 - 436 \beta_{3} ) q^{34} + ( -2800 + 175 \beta_{1} - 1400 \beta_{2} + 175 \beta_{3} ) q^{35} + 1296 q^{36} + ( 1036 - 141 \beta_{1} + 1036 \beta_{2} ) q^{37} + ( 272 \beta_{1} + 4396 \beta_{2} + 272 \beta_{3} ) q^{38} + ( -315 \beta_{1} + 126 \beta_{2} - 315 \beta_{3} ) q^{39} + ( 1600 + 1600 \beta_{2} ) q^{40} + ( 1273 + 767 \beta_{3} ) q^{41} + ( 2016 - 252 \beta_{1} - 2016 \beta_{2} ) q^{42} + ( -13350 - 205 \beta_{3} ) q^{43} + ( 3856 - 176 \beta_{1} + 3856 \beta_{2} ) q^{44} -2025 \beta_{2} q^{45} + ( 1476 \beta_{1} + 1236 \beta_{2} + 1476 \beta_{3} ) q^{46} + ( 1742 + 236 \beta_{1} + 1742 \beta_{2} ) q^{47} -2304 q^{48} + ( -784 \beta_{1} - 2009 \beta_{2} + 784 \beta_{3} ) q^{49} + 2500 q^{50} + ( 4221 + 981 \beta_{1} + 4221 \beta_{2} ) q^{51} + ( 560 \beta_{1} - 224 \beta_{2} + 560 \beta_{3} ) q^{52} + ( -2614 \beta_{1} + 158 \beta_{2} - 2614 \beta_{3} ) q^{53} + ( 2916 + 2916 \beta_{2} ) q^{54} + ( 6025 + 275 \beta_{3} ) q^{55} + ( -3584 + 448 \beta_{1} + 3584 \beta_{2} ) q^{56} + ( -9891 + 612 \beta_{3} ) q^{57} + ( 20172 - 732 \beta_{1} + 20172 \beta_{2} ) q^{58} + ( -277 \beta_{1} + 33101 \beta_{2} - 277 \beta_{3} ) q^{59} + 3600 \beta_{2} q^{60} + ( 22236 - 1564 \beta_{1} + 22236 \beta_{2} ) q^{61} + ( 8276 + 2160 \beta_{3} ) q^{62} + ( 9072 - 567 \beta_{1} + 4536 \beta_{2} - 567 \beta_{3} ) q^{63} + 4096 q^{64} + ( -350 + 875 \beta_{1} - 350 \beta_{2} ) q^{65} + ( -396 \beta_{1} + 8676 \beta_{2} - 396 \beta_{3} ) q^{66} + ( 1673 \beta_{1} - 9482 \beta_{2} + 1673 \beta_{3} ) q^{67} + ( -7504 - 1744 \beta_{1} - 7504 \beta_{2} ) q^{68} + ( -2781 + 3321 \beta_{3} ) q^{69} + ( 5600 + 11200 \beta_{2} - 700 \beta_{3} ) q^{70} + ( -17219 + 4463 \beta_{3} ) q^{71} + ( -5184 - 5184 \beta_{2} ) q^{72} + ( 4041 \beta_{1} + 17258 \beta_{2} + 4041 \beta_{3} ) q^{73} + ( 564 \beta_{1} - 4144 \beta_{2} + 564 \beta_{3} ) q^{74} + ( 5625 + 5625 \beta_{2} ) q^{75} + ( 17584 - 1088 \beta_{3} ) q^{76} + ( 1869 - 1232 \beta_{1} + 26992 \beta_{2} - 2303 \beta_{3} ) q^{77} + ( 504 + 1260 \beta_{3} ) q^{78} + ( 60795 - 316 \beta_{1} + 60795 \beta_{2} ) q^{79} -6400 \beta_{2} q^{80} + 6561 \beta_{2} q^{81} + ( -5092 + 3068 \beta_{1} - 5092 \beta_{2} ) q^{82} + ( -55397 - 4057 \beta_{3} ) q^{83} + ( -16128 + 1008 \beta_{1} - 8064 \beta_{2} + 1008 \beta_{3} ) q^{84} + ( -11725 + 2725 \beta_{3} ) q^{85} + ( 53400 - 820 \beta_{1} + 53400 \beta_{2} ) q^{86} + ( -1647 \beta_{1} + 45387 \beta_{2} - 1647 \beta_{3} ) q^{87} + ( 704 \beta_{1} - 15424 \beta_{2} + 704 \beta_{3} ) q^{88} + ( -47123 - 5657 \beta_{1} - 47123 \beta_{2} ) q^{89} -8100 q^{90} + ( 37779 + 1862 \beta_{1} + 36211 \beta_{2} + 3920 \beta_{3} ) q^{91} + ( 4944 - 5904 \beta_{3} ) q^{92} + ( 18621 - 4860 \beta_{1} + 18621 \beta_{2} ) q^{93} + ( -944 \beta_{1} - 6968 \beta_{2} - 944 \beta_{3} ) q^{94} + ( -1700 \beta_{1} - 27475 \beta_{2} - 1700 \beta_{3} ) q^{95} + ( 9216 + 9216 \beta_{2} ) q^{96} + ( 35876 + 108 \beta_{3} ) q^{97} + ( -8036 + 6272 \beta_{1} + 3136 \beta_{3} ) q^{98} + ( -19521 - 891 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{2} + 18q^{3} - 32q^{4} + 50q^{5} - 144q^{6} - 336q^{7} + 256q^{8} - 162q^{9} + O(q^{10}) \) \( 4q - 8q^{2} + 18q^{3} - 32q^{4} + 50q^{5} - 144q^{6} - 336q^{7} + 256q^{8} - 162q^{9} + 200q^{10} + 482q^{11} + 288q^{12} - 56q^{13} + 1344q^{14} + 900q^{15} - 512q^{16} - 938q^{17} - 648q^{18} - 2198q^{19} - 1600q^{20} - 3856q^{22} - 618q^{23} + 1152q^{24} - 1250q^{25} + 112q^{26} - 2916q^{27} - 20172q^{29} - 1800q^{30} - 4138q^{31} - 2048q^{32} - 4338q^{33} + 7504q^{34} - 8400q^{35} + 5184q^{36} + 2072q^{37} - 8792q^{38} - 252q^{39} + 3200q^{40} + 5092q^{41} + 12096q^{42} - 53400q^{43} + 7712q^{44} + 4050q^{45} - 2472q^{46} + 3484q^{47} - 9216q^{48} + 4018q^{49} + 10000q^{50} + 8442q^{51} + 448q^{52} - 316q^{53} + 5832q^{54} + 24100q^{55} - 21504q^{56} - 39564q^{57} + 40344q^{58} - 66202q^{59} - 7200q^{60} + 44472q^{61} + 33104q^{62} + 27216q^{63} + 16384q^{64} - 700q^{65} - 17352q^{66} + 18964q^{67} - 15008q^{68} - 11124q^{69} - 68876q^{71} - 10368q^{72} - 34516q^{73} + 8288q^{74} + 11250q^{75} + 70336q^{76} - 46508q^{77} + 2016q^{78} + 121590q^{79} + 12800q^{80} - 13122q^{81} - 10184q^{82} - 221588q^{83} - 48384q^{84} - 46900q^{85} + 106800q^{86} - 90774q^{87} + 30848q^{88} - 94246q^{89} - 32400q^{90} + 78694q^{91} + 19776q^{92} + 37242q^{93} + 13936q^{94} + 54950q^{95} + 18432q^{96} + 143504q^{97} - 32144q^{98} - 78084q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 151 x^{2} + 22801\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/151\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/151\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(151 \beta_{2}\)
\(\nu^{3}\)\(=\)\(151 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−6.14410 + 10.6419i
6.14410 10.6419i
−6.14410 10.6419i
6.14410 + 10.6419i
−2.00000 + 3.46410i 4.50000 + 7.79423i −8.00000 13.8564i 12.5000 21.6506i −36.0000 −127.009 + 25.9959i 64.0000 −40.5000 + 70.1481i 50.0000 + 86.6025i
121.2 −2.00000 + 3.46410i 4.50000 + 7.79423i −8.00000 13.8564i 12.5000 21.6506i −36.0000 −40.9913 122.991i 64.0000 −40.5000 + 70.1481i 50.0000 + 86.6025i
151.1 −2.00000 3.46410i 4.50000 7.79423i −8.00000 + 13.8564i 12.5000 + 21.6506i −36.0000 −127.009 25.9959i 64.0000 −40.5000 70.1481i 50.0000 86.6025i
151.2 −2.00000 3.46410i 4.50000 7.79423i −8.00000 + 13.8564i 12.5000 + 21.6506i −36.0000 −40.9913 + 122.991i 64.0000 −40.5000 70.1481i 50.0000 86.6025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.6.i.b 4
7.c even 3 1 inner 210.6.i.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.6.i.b 4 1.a even 1 1 trivial
210.6.i.b 4 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} - 482 T_{11}^{3} + 192514 T_{11}^{2} - 19188420 T_{11} + 1584836100 \) acting on \(S_{6}^{\mathrm{new}}(210, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 + 4 T + T^{2} )^{2} \)
$3$ \( ( 81 - 9 T + T^{2} )^{2} \)
$5$ \( ( 625 - 25 T + T^{2} )^{2} \)
$7$ \( 282475249 + 5647152 T + 54439 T^{2} + 336 T^{3} + T^{4} \)
$11$ \( 1584836100 - 19188420 T + 192514 T^{2} - 482 T^{3} + T^{4} \)
$13$ \( ( -184779 + 28 T + T^{2} )^{2} \)
$17$ \( 2477696364900 - 1476477660 T + 2453914 T^{2} + 938 T^{3} + T^{4} \)
$19$ \( 259668718929 + 1120050246 T + 4321627 T^{2} + 2198 T^{3} + T^{4} \)
$23$ \( 418809266928900 - 12647264940 T + 20846754 T^{2} + 618 T^{3} + T^{4} \)
$29$ \( ( 20375010 + 10086 T + T^{2} )^{2} \)
$31$ \( 1580129201203921 - 164488971782 T + 56873883 T^{2} + 4138 T^{3} + T^{4} \)
$37$ \( 3720018700225 + 3996338920 T + 6221919 T^{2} - 2072 T^{3} + T^{4} \)
$41$ \( ( -87211110 - 2546 T + T^{2} )^{2} \)
$43$ \( ( 171876725 + 26700 T + T^{2} )^{2} \)
$47$ \( 28896344283024 + 18728353488 T + 17513788 T^{2} - 3484 T^{3} + T^{4} \)
$53$ \( 1064523398487234624 - 326035348512 T + 1031857288 T^{2} + 316 T^{3} + T^{4} \)
$59$ \( 1175251392617974884 + 71768934256644 T + 3298614682 T^{2} + 66202 T^{3} + T^{4} \)
$61$ \( 15644806272640000 - 5562522182400 T + 1852679584 T^{2} - 44472 T^{3} + T^{4} \)
$67$ \( 110709222954302025 + 6309890866620 T + 692363251 T^{2} - 18964 T^{3} + T^{4} \)
$71$ \( ( -2711179758 + 34438 T + T^{2} )^{2} \)
$73$ \( 4699978008930633289 - 74828729803772 T + 3359297523 T^{2} + 34516 T^{3} + T^{4} \)
$79$ \( 13549420649515305361 - 447567168772710 T + 11103174331 T^{2} - 121590 T^{3} + T^{4} \)
$83$ \( ( 583481010 + 110794 T + T^{2} )^{2} \)
$89$ \( 6820829956549296900 - 246139627060020 T + 11493980386 T^{2} + 94246 T^{3} + T^{4} \)
$97$ \( ( 1285326112 - 71752 T + T^{2} )^{2} \)
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