# Properties

 Label 100.7.d.a Level $100$ Weight $7$ Character orbit 100.d Analytic conductor $23.005$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 100.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.0054083620$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{15})$$ Defining polynomial: $$x^{4} - 7 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 4) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} -4 \beta_{2} q^{3} + ( 56 - 2 \beta_{3} ) q^{4} + ( 240 - 4 \beta_{3} ) q^{6} -40 \beta_{2} q^{7} + ( 176 \beta_{1} - 48 \beta_{2} ) q^{8} + 231 q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} -4 \beta_{2} q^{3} + ( 56 - 2 \beta_{3} ) q^{4} + ( 240 - 4 \beta_{3} ) q^{6} -40 \beta_{2} q^{7} + ( 176 \beta_{1} - 48 \beta_{2} ) q^{8} + 231 q^{9} -62 \beta_{3} q^{11} + ( 480 \beta_{1} - 224 \beta_{2} ) q^{12} -733 \beta_{1} q^{13} + ( 2400 - 40 \beta_{3} ) q^{14} + ( 2176 - 224 \beta_{3} ) q^{16} -2383 \beta_{1} q^{17} + ( 231 \beta_{1} - 231 \beta_{2} ) q^{18} -486 \beta_{3} q^{19} + 9600 q^{21} + ( 3720 \beta_{1} + 248 \beta_{2} ) q^{22} -1352 \beta_{2} q^{23} + ( 11520 - 704 \beta_{3} ) q^{24} + ( 2932 + 733 \beta_{3} ) q^{26} + 1992 \beta_{2} q^{27} + ( 4800 \beta_{1} - 2240 \beta_{2} ) q^{28} -25498 q^{29} + 2704 \beta_{3} q^{31} + ( 15616 \beta_{1} - 1280 \beta_{2} ) q^{32} + 14880 \beta_{1} q^{33} + ( 9532 + 2383 \beta_{3} ) q^{34} + ( 12936 - 462 \beta_{3} ) q^{36} + 997 \beta_{1} q^{37} + ( 29160 \beta_{1} + 1944 \beta_{2} ) q^{38} + 2932 \beta_{3} q^{39} + 29362 q^{41} + ( 9600 \beta_{1} - 9600 \beta_{2} ) q^{42} -2780 \beta_{2} q^{43} + ( -29760 - 3472 \beta_{3} ) q^{44} + ( 81120 - 1352 \beta_{3} ) q^{46} + 976 \beta_{2} q^{47} + ( 53760 \beta_{1} - 8704 \beta_{2} ) q^{48} -21649 q^{49} + 9532 \beta_{3} q^{51} + ( -41048 \beta_{1} - 5864 \beta_{2} ) q^{52} + 96427 \beta_{1} q^{53} + ( -119520 + 1992 \beta_{3} ) q^{54} + ( 115200 - 7040 \beta_{3} ) q^{56} + 116640 \beta_{1} q^{57} + ( -25498 \beta_{1} + 25498 \beta_{2} ) q^{58} + 5062 \beta_{3} q^{59} -10918 q^{61} + ( -162240 \beta_{1} - 10816 \beta_{2} ) q^{62} -9240 \beta_{2} q^{63} + ( 14336 - 16896 \beta_{3} ) q^{64} + ( -59520 - 14880 \beta_{3} ) q^{66} + 50884 \beta_{2} q^{67} + ( -133448 \beta_{1} - 19064 \beta_{2} ) q^{68} + 324480 q^{69} + 34356 \beta_{3} q^{71} + ( 40656 \beta_{1} - 11088 \beta_{2} ) q^{72} -144313 \beta_{1} q^{73} + ( -3988 - 997 \beta_{3} ) q^{74} + ( -233280 - 27216 \beta_{3} ) q^{76} + 148800 \beta_{1} q^{77} + ( -175920 \beta_{1} - 11728 \beta_{2} ) q^{78} + 20056 \beta_{3} q^{79} -646479 q^{81} + ( 29362 \beta_{1} - 29362 \beta_{2} ) q^{82} -26356 \beta_{2} q^{83} + ( 537600 - 19200 \beta_{3} ) q^{84} + ( 166800 - 2780 \beta_{3} ) q^{86} + 101992 \beta_{2} q^{87} + ( 178560 \beta_{1} + 43648 \beta_{2} ) q^{88} -310738 q^{89} + 29320 \beta_{3} q^{91} + ( 162240 \beta_{1} - 75712 \beta_{2} ) q^{92} -648960 \beta_{1} q^{93} + ( -58560 + 976 \beta_{3} ) q^{94} + ( 307200 - 62464 \beta_{3} ) q^{96} -728543 \beta_{1} q^{97} + ( -21649 \beta_{1} + 21649 \beta_{2} ) q^{98} -14322 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 224 q^{4} + 960 q^{6} + 924 q^{9} + O(q^{10})$$ $$4 q + 224 q^{4} + 960 q^{6} + 924 q^{9} + 9600 q^{14} + 8704 q^{16} + 38400 q^{21} + 46080 q^{24} + 11728 q^{26} - 101992 q^{29} + 38128 q^{34} + 51744 q^{36} + 117448 q^{41} - 119040 q^{44} + 324480 q^{46} - 86596 q^{49} - 478080 q^{54} + 460800 q^{56} - 43672 q^{61} + 57344 q^{64} - 238080 q^{66} + 1297920 q^{69} - 15952 q^{74} - 933120 q^{76} - 2585916 q^{81} + 2150400 q^{84} + 667200 q^{86} - 1242952 q^{89} - 234240 q^{94} + 1228800 q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 7 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 3 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 11 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$8 \nu^{2} - 28$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 28$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{2} + 11 \beta_{1}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$77$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 1.93649 − 0.500000i 1.93649 + 0.500000i −1.93649 − 0.500000i −1.93649 + 0.500000i
−7.74597 2.00000i −30.9839 56.0000 + 30.9839i 0 240.000 + 61.9677i −309.839 −371.806 352.000i 231.000 0
99.2 −7.74597 + 2.00000i −30.9839 56.0000 30.9839i 0 240.000 61.9677i −309.839 −371.806 + 352.000i 231.000 0
99.3 7.74597 2.00000i 30.9839 56.0000 30.9839i 0 240.000 61.9677i 309.839 371.806 352.000i 231.000 0
99.4 7.74597 + 2.00000i 30.9839 56.0000 + 30.9839i 0 240.000 + 61.9677i 309.839 371.806 + 352.000i 231.000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.7.d.a 4
4.b odd 2 1 inner 100.7.d.a 4
5.b even 2 1 inner 100.7.d.a 4
5.c odd 4 1 4.7.b.a 2
5.c odd 4 1 100.7.b.c 2
15.e even 4 1 36.7.d.c 2
20.d odd 2 1 inner 100.7.d.a 4
20.e even 4 1 4.7.b.a 2
20.e even 4 1 100.7.b.c 2
40.i odd 4 1 64.7.c.c 2
40.k even 4 1 64.7.c.c 2
60.l odd 4 1 36.7.d.c 2
80.i odd 4 1 256.7.d.f 4
80.j even 4 1 256.7.d.f 4
80.s even 4 1 256.7.d.f 4
80.t odd 4 1 256.7.d.f 4
120.q odd 4 1 576.7.g.h 2
120.w even 4 1 576.7.g.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.7.b.a 2 5.c odd 4 1
4.7.b.a 2 20.e even 4 1
36.7.d.c 2 15.e even 4 1
36.7.d.c 2 60.l odd 4 1
64.7.c.c 2 40.i odd 4 1
64.7.c.c 2 40.k even 4 1
100.7.b.c 2 5.c odd 4 1
100.7.b.c 2 20.e even 4 1
100.7.d.a 4 1.a even 1 1 trivial
100.7.d.a 4 4.b odd 2 1 inner
100.7.d.a 4 5.b even 2 1 inner
100.7.d.a 4 20.d odd 2 1 inner
256.7.d.f 4 80.i odd 4 1
256.7.d.f 4 80.j even 4 1
256.7.d.f 4 80.s even 4 1
256.7.d.f 4 80.t odd 4 1
576.7.g.h 2 120.q odd 4 1
576.7.g.h 2 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 960$$ acting on $$S_{7}^{\mathrm{new}}(100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4096 - 112 T^{2} + T^{4}$$
$3$ $$( -960 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( -96000 + T^{2} )^{2}$$
$11$ $$( 922560 + T^{2} )^{2}$$
$13$ $$( 2149156 + T^{2} )^{2}$$
$17$ $$( 22714756 + T^{2} )^{2}$$
$19$ $$( 56687040 + T^{2} )^{2}$$
$23$ $$( -109674240 + T^{2} )^{2}$$
$29$ $$( 25498 + T )^{4}$$
$31$ $$( 1754787840 + T^{2} )^{2}$$
$37$ $$( 3976036 + T^{2} )^{2}$$
$41$ $$( -29362 + T )^{4}$$
$43$ $$( -463704000 + T^{2} )^{2}$$
$47$ $$( -57154560 + T^{2} )^{2}$$
$53$ $$( 37192665316 + T^{2} )^{2}$$
$59$ $$( 6149722560 + T^{2} )^{2}$$
$61$ $$( 10918 + T )^{4}$$
$67$ $$( -155350887360 + T^{2} )^{2}$$
$71$ $$( 283280336640 + T^{2} )^{2}$$
$73$ $$( 83304967876 + T^{2} )^{2}$$
$79$ $$( 96538352640 + T^{2} )^{2}$$
$83$ $$( -41678324160 + T^{2} )^{2}$$
$89$ $$( 310738 + T )^{4}$$
$97$ $$( 2123099611396 + T^{2} )^{2}$$