Properties

Label 4410.2.b.a
Level $4410$
Weight $2$
Character orbit 4410.b
Analytic conductor $35.214$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{16}^{4} q^{2} - q^{4} - q^{5} + \zeta_{16}^{4} q^{8} +O(q^{10})\) \( q -\zeta_{16}^{4} q^{2} - q^{4} - q^{5} + \zeta_{16}^{4} q^{8} + \zeta_{16}^{4} q^{10} + ( \zeta_{16}^{3} - 2 \zeta_{16}^{4} + \zeta_{16}^{5} ) q^{11} + ( \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} + 4 \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{13} + q^{16} + ( -\zeta_{16}^{2} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{17} + ( -2 \zeta_{16} + \zeta_{16}^{2} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{19} + q^{20} + ( -2 + \zeta_{16} - \zeta_{16}^{7} ) q^{22} + ( \zeta_{16} - \zeta_{16}^{2} - 5 \zeta_{16}^{3} + 2 \zeta_{16}^{4} - 5 \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{23} + q^{25} + ( 4 + \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{26} + ( \zeta_{16} - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{4} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{29} + ( -2 \zeta_{16} + 2 \zeta_{16}^{2} - \zeta_{16}^{3} - 4 \zeta_{16}^{4} - \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{31} -\zeta_{16}^{4} q^{32} + ( -3 \zeta_{16} + \zeta_{16}^{2} + \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{34} + ( -2 + 3 \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{5} - 3 \zeta_{16}^{6} ) q^{37} + ( \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{38} -\zeta_{16}^{4} q^{40} + ( -6 + \zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{41} + ( -2 + 2 \zeta_{16} - \zeta_{16}^{2} - 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{43} + ( -\zeta_{16}^{3} + 2 \zeta_{16}^{4} - \zeta_{16}^{5} ) q^{44} + ( 2 - 5 \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{46} + ( 6 - \zeta_{16} + \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{47} -\zeta_{16}^{4} q^{50} + ( -\zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} - 4 \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{52} + ( 5 \zeta_{16} + 2 \zeta_{16}^{2} - \zeta_{16}^{3} + 4 \zeta_{16}^{4} - \zeta_{16}^{5} + 2 \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{53} + ( -\zeta_{16}^{3} + 2 \zeta_{16}^{4} - \zeta_{16}^{5} ) q^{55} + ( -2 + 2 \zeta_{16} - 2 \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{58} + ( -2 + 2 \zeta_{16} - \zeta_{16}^{2} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{59} + ( 2 \zeta_{16} + \zeta_{16}^{2} + 6 \zeta_{16}^{4} + \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{61} + ( -4 - \zeta_{16} + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{62} - q^{64} + ( -\zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} - 4 \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{65} + ( -5 \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{67} + ( \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{68} + ( 3 \zeta_{16}^{2} + 4 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 4 \zeta_{16}^{5} + 3 \zeta_{16}^{6} ) q^{71} + ( 5 \zeta_{16} + 4 \zeta_{16}^{2} - \zeta_{16}^{3} + 4 \zeta_{16}^{4} - \zeta_{16}^{5} + 4 \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{73} + ( \zeta_{16} - 3 \zeta_{16}^{2} + 2 \zeta_{16}^{4} - 3 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{74} + ( 2 \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{76} + ( -4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{79} - q^{80} + ( -\zeta_{16} - \zeta_{16}^{3} + 6 \zeta_{16}^{4} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{82} + ( -2 - 2 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{83} + ( \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{85} + ( 5 \zeta_{16} + \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{4} - 2 \zeta_{16}^{5} + \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{86} + ( 2 - \zeta_{16} + \zeta_{16}^{7} ) q^{88} + ( 6 + 4 \zeta_{16} - 2 \zeta_{16}^{2} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{89} + ( -\zeta_{16} + \zeta_{16}^{2} + 5 \zeta_{16}^{3} - 2 \zeta_{16}^{4} + 5 \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{92} + ( 2 \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - 6 \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{94} + ( 2 \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{95} + ( 4 \zeta_{16}^{3} + 6 \zeta_{16}^{4} + 4 \zeta_{16}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} - 8q^{5} + O(q^{10}) \) \( 8q - 8q^{4} - 8q^{5} + 8q^{16} + 8q^{20} - 16q^{22} + 8q^{25} + 32q^{26} - 16q^{37} - 48q^{41} - 16q^{43} + 16q^{46} + 48q^{47} - 16q^{58} - 16q^{59} - 32q^{62} - 8q^{64} - 8q^{80} - 16q^{83} + 16q^{88} + 48q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(2647\) \(3431\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−0.923880 0.382683i
−0.382683 + 0.923880i
0.382683 0.923880i
0.923880 + 0.382683i
0.923880 0.382683i
0.382683 + 0.923880i
−0.382683 0.923880i
−0.923880 + 0.382683i
1.00000i 0 −1.00000 −1.00000 0 0 1.00000i 0 1.00000i
881.2 1.00000i 0 −1.00000 −1.00000 0 0 1.00000i 0 1.00000i
881.3 1.00000i 0 −1.00000 −1.00000 0 0 1.00000i 0 1.00000i
881.4 1.00000i 0 −1.00000 −1.00000 0 0 1.00000i 0 1.00000i
881.5 1.00000i 0 −1.00000 −1.00000 0 0 1.00000i 0 1.00000i
881.6 1.00000i 0 −1.00000 −1.00000 0 0 1.00000i 0 1.00000i
881.7 1.00000i 0 −1.00000 −1.00000 0 0 1.00000i 0 1.00000i
881.8 1.00000i 0 −1.00000 −1.00000 0 0 1.00000i 0 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4410.2.b.a 8
3.b odd 2 1 4410.2.b.d yes 8
7.b odd 2 1 4410.2.b.d yes 8
21.c even 2 1 inner 4410.2.b.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4410.2.b.a 8 1.a even 1 1 trivial
4410.2.b.a 8 21.c even 2 1 inner
4410.2.b.d yes 8 3.b odd 2 1
4410.2.b.d yes 8 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4410, [\chi])\):

\( T_{11}^{8} + 24 T_{11}^{6} + 148 T_{11}^{4} + 176 T_{11}^{2} + 4 \)
\( T_{17}^{4} - 40 T_{17}^{2} - 72 T_{17} + 94 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ 1
$5$ \( ( 1 + T )^{8} \)
$7$ 1
$11$ \( 1 - 64 T^{2} + 1952 T^{4} - 37312 T^{6} + 489570 T^{8} - 4514752 T^{10} + 28579232 T^{12} - 113379904 T^{14} + 214358881 T^{16} \)
$13$ \( 1 - 16 T^{2} + 308 T^{4} - 5424 T^{6} + 50118 T^{8} - 916656 T^{10} + 8796788 T^{12} - 77228944 T^{14} + 815730721 T^{16} \)
$17$ \( ( 1 + 28 T^{2} - 72 T^{3} + 468 T^{4} - 1224 T^{5} + 8092 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( 1 - 112 T^{2} + 5956 T^{4} - 197328 T^{6} + 4474022 T^{8} - 71235408 T^{10} + 776191876 T^{12} - 5269138672 T^{14} + 16983563041 T^{16} \)
$23$ \( 1 + 48 T^{2} + 980 T^{4} - 4592 T^{6} - 398298 T^{8} - 2429168 T^{10} + 274244180 T^{12} + 7105722672 T^{14} + 78310985281 T^{16} \)
$29$ \( 1 - 144 T^{2} + 8384 T^{4} - 272784 T^{6} + 7329058 T^{8} - 229411344 T^{10} + 5929843904 T^{12} - 85654558224 T^{14} + 500246412961 T^{16} \)
$31$ \( 1 - 112 T^{2} + 5984 T^{4} - 201840 T^{6} + 5984898 T^{8} - 193968240 T^{10} + 5526349664 T^{12} - 99400412272 T^{14} + 852891037441 T^{16} \)
$37$ \( ( 1 + 8 T + 132 T^{2} + 784 T^{3} + 7188 T^{4} + 29008 T^{5} + 180708 T^{6} + 405224 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 + 24 T + 372 T^{2} + 3720 T^{3} + 28158 T^{4} + 152520 T^{5} + 625332 T^{6} + 1654104 T^{7} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 + 8 T + 76 T^{2} + 256 T^{3} + 1492 T^{4} + 11008 T^{5} + 140524 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 24 T + 380 T^{2} - 3968 T^{3} + 31844 T^{4} - 186496 T^{5} + 839420 T^{6} - 2491752 T^{7} + 4879681 T^{8} )^{2} \)
$53$ \( 1 - 120 T^{2} + 14060 T^{4} - 973896 T^{6} + 61968070 T^{8} - 2735673864 T^{10} + 110940162860 T^{12} - 2659723335480 T^{14} + 62259690411361 T^{16} \)
$59$ \( ( 1 + 8 T + 176 T^{2} + 888 T^{3} + 12914 T^{4} + 52392 T^{5} + 612656 T^{6} + 1643032 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( 1 - 304 T^{2} + 47620 T^{4} - 4871952 T^{6} + 350453414 T^{8} - 18128533392 T^{10} + 659338948420 T^{12} - 15662193805744 T^{14} + 191707312997281 T^{16} \)
$67$ \( ( 1 + 164 T^{2} - 200 T^{3} + 14052 T^{4} - 13400 T^{5} + 736196 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( 1 - 352 T^{2} + 54788 T^{4} - 5305888 T^{6} + 402214086 T^{8} - 26746981408 T^{10} + 1392255178628 T^{12} - 45091299940192 T^{14} + 645753531245761 T^{16} \)
$73$ \( 1 - 184 T^{2} + 31436 T^{4} - 3096008 T^{6} + 278954470 T^{8} - 16498626632 T^{10} + 892727104076 T^{12} - 27845497637176 T^{14} + 806460091894081 T^{16} \)
$79$ \( ( 1 + 126 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 8 T + 324 T^{2} + 1896 T^{3} + 40022 T^{4} + 157368 T^{5} + 2232036 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 24 T + 428 T^{2} - 5032 T^{3} + 54470 T^{4} - 447848 T^{5} + 3390188 T^{6} - 16919256 T^{7} + 62742241 T^{8} )^{2} \)
$97$ \( 1 - 504 T^{2} + 122652 T^{4} - 19352520 T^{6} + 2189485382 T^{8} - 182087860680 T^{10} + 10858293373212 T^{12} - 419817890484216 T^{14} + 7837433594376961 T^{16} \)
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