# Properties

 Label 676.6.d.a Level $676$ Weight $6$ Character orbit 676.d Analytic conductor $108.419$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$676 = 2^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 676.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$108.419462194$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 12 q^{3} + 27 \beta q^{5} + 44 \beta q^{7} - 99 q^{9}+O(q^{10})$$ q - 12 * q^3 + 27*b * q^5 + 44*b * q^7 - 99 * q^9 $$q - 12 q^{3} + 27 \beta q^{5} + 44 \beta q^{7} - 99 q^{9} - 270 \beta q^{11} - 324 \beta q^{15} - 594 q^{17} + 418 \beta q^{19} - 528 \beta q^{21} + 4104 q^{23} + 209 q^{25} + 4104 q^{27} - 594 q^{29} + 2128 \beta q^{31} + 3240 \beta q^{33} - 4752 q^{35} + 149 \beta q^{37} + 8613 \beta q^{41} + 12100 q^{43} - 2673 \beta q^{45} + 648 \beta q^{47} + 9063 q^{49} + 7128 q^{51} + 19494 q^{53} + 29160 q^{55} - 5016 \beta q^{57} + 3834 \beta q^{59} - 34738 q^{61} - 4356 \beta q^{63} + 10906 \beta q^{67} - 49248 q^{69} - 23436 \beta q^{71} - 33781 \beta q^{73} - 2508 q^{75} + 47520 q^{77} - 76912 q^{79} - 25191 q^{81} + 33858 \beta q^{83} - 16038 \beta q^{85} + 7128 q^{87} - 14877 \beta q^{89} - 25536 \beta q^{93} - 45144 q^{95} - 61199 \beta q^{97} + 26730 \beta q^{99} +O(q^{100})$$ q - 12 * q^3 + 27*b * q^5 + 44*b * q^7 - 99 * q^9 - 270*b * q^11 - 324*b * q^15 - 594 * q^17 + 418*b * q^19 - 528*b * q^21 + 4104 * q^23 + 209 * q^25 + 4104 * q^27 - 594 * q^29 + 2128*b * q^31 + 3240*b * q^33 - 4752 * q^35 + 149*b * q^37 + 8613*b * q^41 + 12100 * q^43 - 2673*b * q^45 + 648*b * q^47 + 9063 * q^49 + 7128 * q^51 + 19494 * q^53 + 29160 * q^55 - 5016*b * q^57 + 3834*b * q^59 - 34738 * q^61 - 4356*b * q^63 + 10906*b * q^67 - 49248 * q^69 - 23436*b * q^71 - 33781*b * q^73 - 2508 * q^75 + 47520 * q^77 - 76912 * q^79 - 25191 * q^81 + 33858*b * q^83 - 16038*b * q^85 + 7128 * q^87 - 14877*b * q^89 - 25536*b * q^93 - 45144 * q^95 - 61199*b * q^97 + 26730*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 24 q^{3} - 198 q^{9}+O(q^{10})$$ 2 * q - 24 * q^3 - 198 * q^9 $$2 q - 24 q^{3} - 198 q^{9} - 1188 q^{17} + 8208 q^{23} + 418 q^{25} + 8208 q^{27} - 1188 q^{29} - 9504 q^{35} + 24200 q^{43} + 18126 q^{49} + 14256 q^{51} + 38988 q^{53} + 58320 q^{55} - 69476 q^{61} - 98496 q^{69} - 5016 q^{75} + 95040 q^{77} - 153824 q^{79} - 50382 q^{81} + 14256 q^{87} - 90288 q^{95}+O(q^{100})$$ 2 * q - 24 * q^3 - 198 * q^9 - 1188 * q^17 + 8208 * q^23 + 418 * q^25 + 8208 * q^27 - 1188 * q^29 - 9504 * q^35 + 24200 * q^43 + 18126 * q^49 + 14256 * q^51 + 38988 * q^53 + 58320 * q^55 - 69476 * q^61 - 98496 * q^69 - 5016 * q^75 + 95040 * q^77 - 153824 * q^79 - 50382 * q^81 + 14256 * q^87 - 90288 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/676\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$509$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
0 −12.0000 0 54.0000i 0 88.0000i 0 −99.0000 0
337.2 0 −12.0000 0 54.0000i 0 88.0000i 0 −99.0000 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.6.d.a 2
13.b even 2 1 inner 676.6.d.a 2
13.d odd 4 1 4.6.a.a 1
13.d odd 4 1 676.6.a.a 1
39.f even 4 1 36.6.a.a 1
52.f even 4 1 16.6.a.b 1
65.f even 4 1 100.6.c.b 2
65.g odd 4 1 100.6.a.b 1
65.k even 4 1 100.6.c.b 2
91.i even 4 1 196.6.a.e 1
91.z odd 12 2 196.6.e.g 2
91.bb even 12 2 196.6.e.d 2
104.j odd 4 1 64.6.a.f 1
104.m even 4 1 64.6.a.b 1
117.y odd 12 2 324.6.e.a 2
117.z even 12 2 324.6.e.d 2
143.g even 4 1 484.6.a.a 1
156.l odd 4 1 144.6.a.c 1
195.j odd 4 1 900.6.d.a 2
195.n even 4 1 900.6.a.h 1
195.u odd 4 1 900.6.d.a 2
208.l even 4 1 256.6.b.c 2
208.m odd 4 1 256.6.b.g 2
208.r odd 4 1 256.6.b.g 2
208.s even 4 1 256.6.b.c 2
260.l odd 4 1 400.6.c.f 2
260.s odd 4 1 400.6.c.f 2
260.u even 4 1 400.6.a.d 1
312.w odd 4 1 576.6.a.bd 1
312.y even 4 1 576.6.a.bc 1
364.p odd 4 1 784.6.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 13.d odd 4 1
16.6.a.b 1 52.f even 4 1
36.6.a.a 1 39.f even 4 1
64.6.a.b 1 104.m even 4 1
64.6.a.f 1 104.j odd 4 1
100.6.a.b 1 65.g odd 4 1
100.6.c.b 2 65.f even 4 1
100.6.c.b 2 65.k even 4 1
144.6.a.c 1 156.l odd 4 1
196.6.a.e 1 91.i even 4 1
196.6.e.d 2 91.bb even 12 2
196.6.e.g 2 91.z odd 12 2
256.6.b.c 2 208.l even 4 1
256.6.b.c 2 208.s even 4 1
256.6.b.g 2 208.m odd 4 1
256.6.b.g 2 208.r odd 4 1
324.6.e.a 2 117.y odd 12 2
324.6.e.d 2 117.z even 12 2
400.6.a.d 1 260.u even 4 1
400.6.c.f 2 260.l odd 4 1
400.6.c.f 2 260.s odd 4 1
484.6.a.a 1 143.g even 4 1
576.6.a.bc 1 312.y even 4 1
576.6.a.bd 1 312.w odd 4 1
676.6.a.a 1 13.d odd 4 1
676.6.d.a 2 1.a even 1 1 trivial
676.6.d.a 2 13.b even 2 1 inner
784.6.a.d 1 364.p odd 4 1
900.6.a.h 1 195.n even 4 1
900.6.d.a 2 195.j odd 4 1
900.6.d.a 2 195.u odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 12$$ acting on $$S_{6}^{\mathrm{new}}(676, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 12)^{2}$$
$5$ $$T^{2} + 2916$$
$7$ $$T^{2} + 7744$$
$11$ $$T^{2} + 291600$$
$13$ $$T^{2}$$
$17$ $$(T + 594)^{2}$$
$19$ $$T^{2} + 698896$$
$23$ $$(T - 4104)^{2}$$
$29$ $$(T + 594)^{2}$$
$31$ $$T^{2} + 18113536$$
$37$ $$T^{2} + 88804$$
$41$ $$T^{2} + 296735076$$
$43$ $$(T - 12100)^{2}$$
$47$ $$T^{2} + 1679616$$
$53$ $$(T - 19494)^{2}$$
$59$ $$T^{2} + 58798224$$
$61$ $$(T + 34738)^{2}$$
$67$ $$T^{2} + 475763344$$
$71$ $$T^{2} + 2196984384$$
$73$ $$T^{2} + 4564623844$$
$79$ $$(T + 76912)^{2}$$
$83$ $$T^{2} + 4585456656$$
$89$ $$T^{2} + 885300516$$
$97$ $$T^{2} + 14981270404$$