Properties

Label 2300.2.i.f
Level $2300$
Weight $2$
Character orbit 2300.i
Analytic conductor $18.366$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3655924649\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 74 x^{12} + 1357 x^{8} + 3177 x^{4} + 1296\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{5} + \beta_{9} ) q^{7} + ( \beta_{6} - \beta_{7} + \beta_{11} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{5} + \beta_{9} ) q^{7} + ( \beta_{6} - \beta_{7} + \beta_{11} ) q^{9} + ( 2 \beta_{6} + \beta_{13} ) q^{11} + ( 2 \beta_{5} - \beta_{9} ) q^{13} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{17} + ( 4 - \beta_{4} ) q^{19} + ( -4 \beta_{6} - 2 \beta_{11} ) q^{21} + ( \beta_{1} - \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} - \beta_{15} ) q^{23} + ( \beta_{12} - \beta_{14} + \beta_{15} ) q^{27} + ( 2 \beta_{6} - \beta_{7} + 2 \beta_{11} ) q^{29} + ( \beta_{2} + \beta_{4} ) q^{31} + ( -2 \beta_{10} + 2 \beta_{14} ) q^{33} + ( 4 \beta_{8} + 2 \beta_{9} ) q^{37} + ( \beta_{6} + 2 \beta_{7} + \beta_{11} ) q^{39} + \beta_{3} q^{41} + ( \beta_{10} + 3 \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{43} + ( -\beta_{10} + 3 \beta_{12} ) q^{47} + ( 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{11} + \beta_{13} ) q^{49} + ( 6 \beta_{6} - 4 \beta_{7} + 2 \beta_{11} ) q^{51} + ( -2 \beta_{10} - 2 \beta_{14} ) q^{53} + ( 4 \beta_{1} - 2 \beta_{8} ) q^{57} + ( 3 \beta_{6} - 4 \beta_{7} + 2 \beta_{11} + \beta_{13} ) q^{59} + ( 2 \beta_{6} + 4 \beta_{11} ) q^{61} + ( 3 \beta_{10} - \beta_{12} + 4 \beta_{14} - \beta_{15} ) q^{63} + ( 2 \beta_{1} + 2 \beta_{9} ) q^{67} + ( -4 + 2 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} + 3 \beta_{11} - \beta_{13} ) q^{69} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{71} + ( -\beta_{1} + 2 \beta_{8} + 3 \beta_{9} ) q^{73} + ( \beta_{10} + 5 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{77} + ( -2 + 2 \beta_{3} - 3 \beta_{4} ) q^{79} + ( 5 - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{81} + ( -3 \beta_{10} + 3 \beta_{12} - 2 \beta_{14} + 3 \beta_{15} ) q^{83} + ( -5 \beta_{10} - 3 \beta_{14} + 3 \beta_{15} ) q^{87} + ( 6 + 4 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -10 \beta_{6} - 4 \beta_{7} - 3 \beta_{13} ) q^{91} + ( -\beta_{1} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{93} + ( 2 \beta_{1} - 4 \beta_{5} - 2 \beta_{8} ) q^{97} + ( -2 + 2 \beta_{2} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 68q^{19} - 4q^{41} - 56q^{69} - 8q^{71} - 28q^{79} + 64q^{81} + 120q^{89} - 28q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 74 x^{12} + 1357 x^{8} + 3177 x^{4} + 1296\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -347 \nu^{12} - 23572 \nu^{8} - 353096 \nu^{4} - 57267 \)\()/179019\)
\(\beta_{3}\)\(=\)\((\)\( -389 \nu^{12} - 26941 \nu^{8} - 443813 \nu^{4} - 690507 \)\()/179019\)
\(\beta_{4}\)\(=\)\((\)\( -683 \nu^{12} - 50524 \nu^{8} - 899813 \nu^{4} - 1184769 \)\()/179019\)
\(\beta_{5}\)\(=\)\((\)\( -305 \nu^{13} - 20203 \nu^{9} - 262379 \nu^{5} + 1113030 \nu \)\()/537057\)
\(\beta_{6}\)\(=\)\((\)\( 767 \nu^{14} + 57262 \nu^{10} + 1081247 \nu^{6} + 3525363 \nu^{2} \)\()/2148228\)
\(\beta_{7}\)\(=\)\((\)\( -1987 \nu^{14} - 138074 \nu^{10} - 2130763 \nu^{6} + 926757 \nu^{2} \)\()/2148228\)
\(\beta_{8}\)\(=\)\((\)\( -683 \nu^{13} - 50524 \nu^{9} - 899813 \nu^{5} - 1184769 \nu \)\()/358038\)
\(\beta_{9}\)\(=\)\((\)\( 2911 \nu^{13} + 212192 \nu^{9} + 3768499 \nu^{5} + 7275915 \nu \)\()/1074114\)
\(\beta_{10}\)\(=\)\((\)\( -767 \nu^{15} - 57262 \nu^{11} - 1081247 \nu^{7} - 3525363 \nu^{3} \)\()/2148228\)
\(\beta_{11}\)\(=\)\((\)\( -1685 \nu^{14} - 122374 \nu^{10} - 2151917 \nu^{6} - 3675489 \nu^{2} \)\()/716076\)
\(\beta_{12}\)\(=\)\((\)\( -1691 \nu^{15} - 131380 \nu^{11} - 2718983 \nu^{7} - 11728035 \nu^{3} \)\()/3222342\)
\(\beta_{13}\)\(=\)\((\)\( -1017 \nu^{14} - 74474 \nu^{10} - 1317181 \nu^{6} - 2015009 \nu^{2} \)\()/238692\)
\(\beta_{14}\)\(=\)\((\)\( -1017 \nu^{15} - 74474 \nu^{11} - 1317181 \nu^{7} - 2015009 \nu^{3} \)\()/477384\)
\(\beta_{15}\)\(=\)\((\)\( -48307 \nu^{15} - 3546710 \nu^{11} - 63612847 \nu^{7} - 121516803 \nu^{3} \)\()/12889368\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{11} - \beta_{7} + 4 \beta_{6}\)
\(\nu^{3}\)\(=\)\(\beta_{15} - \beta_{14} + \beta_{12} - 6 \beta_{10}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 8 \beta_{3} + 7 \beta_{2} - 22\)
\(\nu^{5}\)\(=\)\(9 \beta_{9} + 11 \beta_{8} + 6 \beta_{5} - 37 \beta_{1}\)
\(\nu^{6}\)\(=\)\(11 \beta_{13} - 57 \beta_{11} + 43 \beta_{7} - 133 \beta_{6}\)
\(\nu^{7}\)\(=\)\(-71 \beta_{15} + 93 \beta_{14} - 29 \beta_{12} + 233 \beta_{10}\)
\(\nu^{8}\)\(=\)\(-93 \beta_{4} + 397 \beta_{3} - 262 \beta_{2} + 832\)
\(\nu^{9}\)\(=\)\(-532 \beta_{9} - 718 \beta_{8} - 127 \beta_{5} + 1491 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-718 \beta_{13} + 2741 \beta_{11} - 1618 \beta_{7} + 5305 \beta_{6}\)
\(\nu^{11}\)\(=\)\(3864 \beta_{15} - 5300 \beta_{14} + 495 \beta_{12} - 9664 \beta_{10}\)
\(\nu^{12}\)\(=\)\(5300 \beta_{4} - 18828 \beta_{3} + 10159 \beta_{2} - 34297\)
\(\nu^{13}\)\(=\)\(27497 \beta_{9} + 38097 \beta_{8} + 1490 \beta_{5} - 63284 \beta_{1}\)
\(\nu^{14}\)\(=\)\(38097 \beta_{13} - 128878 \beta_{11} + 64774 \beta_{7} - 224149 \beta_{6}\)
\(\nu^{15}\)\(=\)\(-192982 \beta_{15} + 269176 \beta_{14} - 670 \beta_{12} + 417801 \beta_{10}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1057.1
−1.84123 + 1.84123i
−1.58699 + 1.58699i
−0.854787 + 0.854787i
−0.600551 + 0.600551i
0.600551 0.600551i
0.854787 0.854787i
1.58699 1.58699i
1.84123 1.84123i
−1.84123 1.84123i
−1.58699 1.58699i
−0.854787 0.854787i
−0.600551 0.600551i
0.600551 + 0.600551i
0.854787 + 0.854787i
1.58699 + 1.58699i
1.84123 + 1.84123i
0 −1.84123 + 1.84123i 0 0 0 2.81318 2.81318i 0 3.78026i 0
1057.2 0 −1.58699 + 1.58699i 0 0 0 −0.216666 + 0.216666i 0 2.03710i 0
1057.3 0 −0.854787 + 0.854787i 0 0 0 3.24270 3.24270i 0 1.53868i 0
1057.4 0 −0.600551 + 0.600551i 0 0 0 −1.01189 + 1.01189i 0 2.27868i 0
1057.5 0 0.600551 0.600551i 0 0 0 1.01189 1.01189i 0 2.27868i 0
1057.6 0 0.854787 0.854787i 0 0 0 −3.24270 + 3.24270i 0 1.53868i 0
1057.7 0 1.58699 1.58699i 0 0 0 0.216666 0.216666i 0 2.03710i 0
1057.8 0 1.84123 1.84123i 0 0 0 −2.81318 + 2.81318i 0 3.78026i 0
1793.1 0 −1.84123 1.84123i 0 0 0 2.81318 + 2.81318i 0 3.78026i 0
1793.2 0 −1.58699 1.58699i 0 0 0 −0.216666 0.216666i 0 2.03710i 0
1793.3 0 −0.854787 0.854787i 0 0 0 3.24270 + 3.24270i 0 1.53868i 0
1793.4 0 −0.600551 0.600551i 0 0 0 −1.01189 1.01189i 0 2.27868i 0
1793.5 0 0.600551 + 0.600551i 0 0 0 1.01189 + 1.01189i 0 2.27868i 0
1793.6 0 0.854787 + 0.854787i 0 0 0 −3.24270 3.24270i 0 1.53868i 0
1793.7 0 1.58699 + 1.58699i 0 0 0 0.216666 + 0.216666i 0 2.03710i 0
1793.8 0 1.84123 + 1.84123i 0 0 0 −2.81318 2.81318i 0 3.78026i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1793.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
115.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.2.i.f yes 16
5.b even 2 1 inner 2300.2.i.f yes 16
5.c odd 4 2 2300.2.i.e 16
23.b odd 2 1 2300.2.i.e 16
115.c odd 2 1 2300.2.i.e 16
115.e even 4 2 inner 2300.2.i.f yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.i.e 16 5.c odd 4 2
2300.2.i.e 16 23.b odd 2 1
2300.2.i.e 16 115.c odd 2 1
2300.2.i.f yes 16 1.a even 1 1 trivial
2300.2.i.f yes 16 5.b even 2 1 inner
2300.2.i.f yes 16 115.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2300, [\chi])\):

\( T_{3}^{16} + 74 T_{3}^{12} + 1357 T_{3}^{8} + 3177 T_{3}^{4} + 1296 \)
\( T_{19}^{4} - 17 T_{19}^{3} + 94 T_{19}^{2} - 192 T_{19} + 120 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 1296 + 3177 T^{4} + 1357 T^{8} + 74 T^{12} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( 4096 + 465664 T^{4} + 113712 T^{8} + 697 T^{12} + T^{16} \)
$11$ \( ( 576 + 1488 T^{2} + 472 T^{4} + 41 T^{6} + T^{8} )^{2} \)
$13$ \( 50625 + 5287374 T^{4} + 428671 T^{8} + 2507 T^{12} + T^{16} \)
$17$ \( 6879707136 + 316182528 T^{4} + 1711360 T^{8} + 2720 T^{12} + T^{16} \)
$19$ \( ( 120 - 192 T + 94 T^{2} - 17 T^{3} + T^{4} )^{4} \)
$23$ \( 78310985281 + 2368574224 T^{2} + 72758660 T^{4} - 8506320 T^{6} - 346042 T^{8} - 16080 T^{10} + 260 T^{12} + 16 T^{14} + T^{16} \)
$29$ \( ( 25 + 514 T^{2} + 999 T^{4} + 79 T^{6} + T^{8} )^{2} \)
$31$ \( ( -36 - 63 T - 27 T^{2} + T^{4} )^{4} \)
$37$ \( 907401035776 + 150918160384 T^{4} + 95936256 T^{8} + 17968 T^{12} + T^{16} \)
$41$ \( ( 15 - 12 T - 11 T^{2} + T^{3} + T^{4} )^{4} \)
$43$ \( 37341681094656 + 75558069504 T^{4} + 50449312 T^{8} + 12377 T^{12} + T^{16} \)
$47$ \( 4430766096 + 4596369561 T^{4} + 12890341 T^{8} + 9626 T^{12} + T^{16} \)
$53$ \( 5308416 + 9252864 T^{4} + 4032256 T^{8} + 6368 T^{12} + T^{16} \)
$59$ \( ( 20304036 + 1453029 T^{2} + 35407 T^{4} + 335 T^{6} + T^{8} )^{2} \)
$61$ \( ( 21678336 + 1641216 T^{2} + 39904 T^{4} + 368 T^{6} + T^{8} )^{2} \)
$67$ \( 319794774016 + 7422140416 T^{4} + 36628224 T^{8} + 14128 T^{12} + T^{16} \)
$71$ \( ( 240 - 171 T - 101 T^{2} + 2 T^{3} + T^{4} )^{4} \)
$73$ \( 1816891022241 + 31097085678 T^{4} + 39058111 T^{8} + 11579 T^{12} + T^{16} \)
$79$ \( ( 4296 - 516 T - 128 T^{2} + 7 T^{3} + T^{4} )^{4} \)
$83$ \( 131135844519936 + 8520250975488 T^{4} + 1281259296 T^{8} + 62649 T^{12} + T^{16} \)
$89$ \( ( -864 + 504 T + 192 T^{2} - 30 T^{3} + T^{4} )^{4} \)
$97$ \( 4111875506176 + 84244148224 T^{4} + 146194176 T^{8} + 57568 T^{12} + T^{16} \)
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