# 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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