# Properties

 Label 700.2.r.d Level $700$ Weight $2$ Character orbit 700.r Analytic conductor $5.590$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{6} + \beta_{11} ) q^{3} + ( \beta_{4} - \beta_{10} ) q^{7} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{6} + \beta_{11} ) q^{3} + ( \beta_{4} - \beta_{10} ) q^{7} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{9} + ( -1 + \beta_{2} + \beta_{5} - \beta_{8} ) q^{11} + ( 3 \beta_{10} + \beta_{11} ) q^{13} + ( -\beta_{4} - 4 \beta_{7} - 2 \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{17} + ( -2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{8} ) q^{19} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{8} ) q^{21} + ( \beta_{4} + 2 \beta_{6} - \beta_{9} ) q^{23} + ( 2 \beta_{4} + \beta_{6} + \beta_{9} - 3 \beta_{10} + 4 \beta_{11} ) q^{27} + ( -\beta_{1} - 2 \beta_{3} - \beta_{5} + \beta_{8} ) q^{29} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} + 2 \beta_{8} ) q^{31} + ( \beta_{4} - \beta_{6} - 6 \beta_{7} - \beta_{9} ) q^{33} + ( 3 \beta_{6} + \beta_{7} ) q^{37} + ( -6 - \beta_{1} + 6 \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{8} ) q^{39} + ( 2 + \beta_{1} + 2 \beta_{3} - 4 \beta_{5} - \beta_{8} ) q^{41} + ( 2 \beta_{4} + \beta_{6} + \beta_{9} - 4 \beta_{10} + 2 \beta_{11} ) q^{43} + ( \beta_{4} + \beta_{7} - \beta_{9} ) q^{47} + ( -1 - \beta_{1} - 3 \beta_{2} - 2 \beta_{5} + 2 \beta_{8} ) q^{49} + ( 3 \beta_{2} + 6 \beta_{8} ) q^{51} + ( -\beta_{4} - 3 \beta_{6} + 5 \beta_{7} - 2 \beta_{9} - 5 \beta_{10} - 4 \beta_{11} ) q^{53} + ( 3 \beta_{10} + 2 \beta_{11} ) q^{57} + ( -2 + 2 \beta_{2} - \beta_{5} + \beta_{8} ) q^{59} + ( 2 \beta_{1} - 7 \beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{8} ) q^{61} + ( -2 \beta_{6} - 12 \beta_{7} - \beta_{9} + 6 \beta_{10} - 4 \beta_{11} ) q^{63} + ( -2 \beta_{6} - 6 \beta_{7} + 6 \beta_{10} - 2 \beta_{11} ) q^{67} + ( 9 - \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + \beta_{8} ) q^{69} + ( -7 - 2 \beta_{5} ) q^{71} + ( 4 \beta_{7} - 4 \beta_{10} ) q^{73} + ( -2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} + \beta_{10} - 4 \beta_{11} ) q^{77} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} - 4 \beta_{8} ) q^{79} + ( -6 + 6 \beta_{2} + 5 \beta_{5} - 5 \beta_{8} ) q^{81} + ( 2 \beta_{4} + \beta_{6} + \beta_{9} + 10 \beta_{10} + 5 \beta_{11} ) q^{83} + ( 2 \beta_{4} + 4 \beta_{6} + 9 \beta_{7} + 4 \beta_{9} - 9 \beta_{10} + 6 \beta_{11} ) q^{87} + ( 2 \beta_{1} - 5 \beta_{2} + \beta_{3} - \beta_{5} ) q^{89} + ( 6 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{5} + \beta_{8} ) q^{91} + ( -2 \beta_{4} + 3 \beta_{6} + 9 \beta_{7} + 2 \beta_{9} ) q^{93} + ( -4 \beta_{4} - 2 \beta_{6} - 2 \beta_{9} - 4 \beta_{10} - 3 \beta_{11} ) q^{97} + ( -6 + 2 \beta_{1} + 4 \beta_{3} + 5 \beta_{5} - 2 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 16 q^{9} + O(q^{10})$$ $$12 q + 16 q^{9} - 8 q^{11} - 4 q^{19} - 22 q^{21} + 6 q^{31} - 28 q^{39} + 44 q^{41} - 24 q^{49} + 6 q^{51} - 10 q^{59} - 34 q^{61} + 96 q^{69} - 76 q^{71} - 2 q^{79} - 46 q^{81} - 28 q^{89} + 34 q^{91} - 84 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$117 \nu^{10} - 767 \nu^{8} + 7472 \nu^{6} - 27234 \nu^{4} + 90554 \nu^{2} - 60864$$$$)/21995$$ $$\beta_{2}$$ $$=$$ $$($$$$-1298 \nu^{10} + 10953 \nu^{8} - 71803 \nu^{6} + 202311 \nu^{4} - 490451 \nu^{2} + 240966$$$$)/197955$$ $$\beta_{3}$$ $$=$$ $$($$$$461 \nu^{10} - 5466 \nu^{8} + 28501 \nu^{6} - 98847 \nu^{4} + 142112 \nu^{2} - 186237$$$$)/65985$$ $$\beta_{4}$$ $$=$$ $$($$$$569 \nu^{11} - 6174 \nu^{9} + 40474 \nu^{7} - 169668 \nu^{5} + 474413 \nu^{3} - 729693 \nu$$$$)/197955$$ $$\beta_{5}$$ $$=$$ $$($$$$-288 \nu^{10} + 1888 \nu^{8} - 9933 \nu^{6} + 12896 \nu^{4} - 6336 \nu^{2} - 68439$$$$)/21995$$ $$\beta_{6}$$ $$=$$ $$($$$$81 \nu^{11} - 531 \nu^{9} + 3481 \nu^{7} - 3627 \nu^{5} + 1782 \nu^{3} + 98293 \nu$$$$)/21995$$ $$\beta_{7}$$ $$=$$ $$($$$$288 \nu^{11} - 1888 \nu^{9} + 9933 \nu^{7} - 12896 \nu^{5} + 6336 \nu^{3} + 90434 \nu$$$$)/65985$$ $$\beta_{8}$$ $$=$$ $$($$$$1138 \nu^{10} - 12348 \nu^{8} + 80948 \nu^{6} - 273351 \nu^{4} + 552916 \nu^{2} - 271656$$$$)/65985$$ $$\beta_{9}$$ $$=$$ $$($$$$2027 \nu^{11} - 15732 \nu^{9} + 103132 \nu^{7} - 234954 \nu^{5} + 506489 \nu^{3} + 445716 \nu$$$$)/197955$$ $$\beta_{10}$$ $$=$$ $$($$$$88 \nu^{11} - 783 \nu^{9} + 4868 \nu^{7} - 13716 \nu^{5} + 26581 \nu^{3} - 2916 \nu$$$$)/7155$$ $$\beta_{11}$$ $$=$$ $$($$$$-1298 \nu^{11} + 10953 \nu^{9} - 71803 \nu^{7} + 202311 \nu^{5} - 450860 \nu^{3} + 43011 \nu$$$$)/39591$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{9} + 2 \beta_{6} + \beta_{4}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{8} + 2 \beta_{5} + \beta_{3} - 9 \beta_{2} - \beta_{1} + 9$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{11} + 5 \beta_{9} + 5 \beta_{6} + 10 \beta_{4}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-\beta_{8} + 2 \beta_{5} - 2 \beta_{3} - 12 \beta_{2} - 4 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$18 \beta_{11} + 9 \beta_{10} + 46 \beta_{9} - 9 \beta_{7} - 5 \beta_{6} + 23 \beta_{4}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$32 \beta_{8} - 5 \beta_{5} - 64 \beta_{3} - 32 \beta_{1} - 153$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$106 \beta_{9} - 81 \beta_{7} - 116 \beta_{6} - 106 \beta_{4}$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$51 \beta_{8} - 51 \beta_{5} - 55 \beta_{3} + 222 \beta_{2} + 55 \beta_{1} - 222$$ $$\nu^{9}$$ $$=$$ $$($$$$-495 \beta_{11} - 531 \beta_{10} - 493 \beta_{9} - 493 \beta_{6} - 986 \beta_{4}$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$-191 \beta_{8} - 835 \beta_{5} + 835 \beta_{3} + 2952 \beta_{2} + 1670 \beta_{1}$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-2505 \beta_{11} - 3078 \beta_{10} - 4624 \beta_{9} + 3078 \beta_{7} - 193 \beta_{6} - 2312 \beta_{4}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −0.617942 − 0.356769i −1.65604 − 0.956115i −1.90412 − 1.09935i 1.90412 + 1.09935i 1.65604 + 0.956115i 0.617942 + 0.356769i −0.617942 + 0.356769i −1.65604 + 0.956115i −1.90412 + 1.09935i 1.90412 − 1.09935i 1.65604 − 0.956115i 0.617942 − 0.356769i
0 −2.77509 + 1.60220i 0 0 0 2.15715 + 1.53189i 0 3.63409 6.29444i 0
149.2 0 −2.22469 + 1.28442i 0 0 0 0.568650 2.58392i 0 1.79949 3.11682i 0
149.3 0 −0.315621 + 0.182224i 0 0 0 −1.58850 2.11581i 0 −1.43359 + 2.48305i 0
149.4 0 0.315621 0.182224i 0 0 0 1.58850 + 2.11581i 0 −1.43359 + 2.48305i 0
149.5 0 2.22469 1.28442i 0 0 0 −0.568650 + 2.58392i 0 1.79949 3.11682i 0
149.6 0 2.77509 1.60220i 0 0 0 −2.15715 1.53189i 0 3.63409 6.29444i 0
249.1 0 −2.77509 1.60220i 0 0 0 2.15715 1.53189i 0 3.63409 + 6.29444i 0
249.2 0 −2.22469 1.28442i 0 0 0 0.568650 + 2.58392i 0 1.79949 + 3.11682i 0
249.3 0 −0.315621 0.182224i 0 0 0 −1.58850 + 2.11581i 0 −1.43359 2.48305i 0
249.4 0 0.315621 + 0.182224i 0 0 0 1.58850 2.11581i 0 −1.43359 2.48305i 0
249.5 0 2.22469 + 1.28442i 0 0 0 −0.568650 2.58392i 0 1.79949 + 3.11682i 0
249.6 0 2.77509 + 1.60220i 0 0 0 −2.15715 + 1.53189i 0 3.63409 + 6.29444i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 249.6 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
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.r.d 12
5.b even 2 1 inner 700.2.r.d 12
5.c odd 4 1 700.2.i.d 6
5.c odd 4 1 700.2.i.e yes 6
7.c even 3 1 inner 700.2.r.d 12
7.c even 3 1 4900.2.e.t 6
7.d odd 6 1 4900.2.e.s 6
35.i odd 6 1 4900.2.e.s 6
35.j even 6 1 inner 700.2.r.d 12
35.j even 6 1 4900.2.e.t 6
35.k even 12 1 4900.2.a.bb 3
35.k even 12 1 4900.2.a.bd 3
35.l odd 12 1 700.2.i.d 6
35.l odd 12 1 700.2.i.e yes 6
35.l odd 12 1 4900.2.a.ba 3
35.l odd 12 1 4900.2.a.bc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.i.d 6 5.c odd 4 1
700.2.i.d 6 35.l odd 12 1
700.2.i.e yes 6 5.c odd 4 1
700.2.i.e yes 6 35.l odd 12 1
700.2.r.d 12 1.a even 1 1 trivial
700.2.r.d 12 5.b even 2 1 inner
700.2.r.d 12 7.c even 3 1 inner
700.2.r.d 12 35.j even 6 1 inner
4900.2.a.ba 3 35.l odd 12 1
4900.2.a.bb 3 35.k even 12 1
4900.2.a.bc 3 35.l odd 12 1
4900.2.a.bd 3 35.k even 12 1
4900.2.e.s 6 7.d odd 6 1
4900.2.e.s 6 35.i odd 6 1
4900.2.e.t 6 7.c even 3 1
4900.2.e.t 6 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(700, [\chi])$$:

 $$T_{3}^{12} - 17 T_{3}^{10} + 219 T_{3}^{8} - 1172 T_{3}^{6} + 4747 T_{3}^{4} - 630 T_{3}^{2} + 81$$ $$T_{11}^{6} + 4 T_{11}^{5} + 19 T_{11}^{4} + 6 T_{11}^{3} + 45 T_{11}^{2} + 27 T_{11} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$81 - 630 T^{2} + 4747 T^{4} - 1172 T^{6} + 219 T^{8} - 17 T^{10} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$117649 + 28812 T^{2} + 5880 T^{4} + 947 T^{6} + 120 T^{8} + 12 T^{10} + T^{12}$$
$11$ $$( 81 + 27 T + 45 T^{2} + 6 T^{3} + 19 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$13$ $$( 9 + 217 T^{2} + 38 T^{4} + T^{6} )^{2}$$
$17$ $$43046721 - 11160261 T^{2} + 2355399 T^{4} - 126360 T^{6} + 5023 T^{8} - 82 T^{10} + T^{12}$$
$19$ $$( 441 - 483 T + 487 T^{2} - 88 T^{3} + 27 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$23$ $$43046721 - 11691702 T^{2} + 2644083 T^{4} - 131220 T^{6} + 4779 T^{8} - 81 T^{10} + T^{12}$$
$29$ $$( -81 - 45 T + T^{3} )^{4}$$
$31$ $$( 1369 + 1554 T + 1653 T^{2} + 200 T^{3} + 51 T^{4} - 3 T^{5} + T^{6} )^{2}$$
$37$ $$492884401 - 127811157 T^{2} + 29546487 T^{4} - 888232 T^{6} + 20487 T^{8} - 162 T^{10} + T^{12}$$
$41$ $$( 873 - 78 T - 11 T^{2} + T^{3} )^{4}$$
$43$ $$( 5041 + 1758 T^{2} + 93 T^{4} + T^{6} )^{2}$$
$47$ $$6561 - 41553 T^{2} + 258471 T^{4} - 29592 T^{6} + 2851 T^{8} - 58 T^{10} + T^{12}$$
$53$ $$90842562801 - 4700951397 T^{2} + 175149783 T^{4} - 2922120 T^{6} + 35479 T^{8} - 226 T^{10} + T^{12}$$
$59$ $$( 81 + 45 T^{2} + 18 T^{3} + 25 T^{4} + 5 T^{5} + T^{6} )^{2}$$
$61$ $$( 1 + 26 T + 659 T^{2} + 440 T^{3} + 263 T^{4} + 17 T^{5} + T^{6} )^{2}$$
$67$ $$26873856 - 36910080 T^{2} + 49657600 T^{4} - 1413632 T^{6} + 32880 T^{8} - 200 T^{10} + T^{12}$$
$71$ $$( 45 + 87 T + 19 T^{2} + T^{3} )^{4}$$
$73$ $$( 256 - 16 T^{2} + T^{4} )^{3}$$
$79$ $$( 201601 - 52982 T + 13475 T^{2} - 1016 T^{3} + 119 T^{4} + T^{5} + T^{6} )^{2}$$
$83$ $$( 962361 + 71577 T^{2} + 526 T^{4} + T^{6} )^{2}$$
$89$ $$( 2025 - 1755 T + 2151 T^{2} + 636 T^{3} + 157 T^{4} + 14 T^{5} + T^{6} )^{2}$$
$97$ $$( 25 + 474 T^{2} + 261 T^{4} + T^{6} )^{2}$$