# Properties

 Label 1400.2.bh.i Level $1400$ Weight $2$ Character orbit 1400.bh Analytic conductor $11.179$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.bh (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.32905425960566784.37 Defining polynomial: $$x^{12} + 36 x^{10} + 432 x^{8} + 2040 x^{6} + 3780 x^{4} + 2592 x^{2} + 576$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 280) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{3} + ( \beta_{1} + \beta_{3} + \beta_{5} - \beta_{8} ) q^{7} + ( 3 \beta_{4} - \beta_{6} - 2 \beta_{10} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{3} + ( \beta_{1} + \beta_{3} + \beta_{5} - \beta_{8} ) q^{7} + ( 3 \beta_{4} - \beta_{6} - 2 \beta_{10} ) q^{9} + ( -1 + \beta_{4} - 2 \beta_{6} - \beta_{10} ) q^{11} + ( -2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} + \beta_{7} + 2 \beta_{8} ) q^{13} + ( -\beta_{1} - \beta_{3} ) q^{17} + ( \beta_{4} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{19} + ( 2 - 4 \beta_{4} + 2 \beta_{6} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{21} + ( -\beta_{2} - \beta_{7} ) q^{23} + ( -6 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} + 6 \beta_{8} ) q^{27} + ( -4 + \beta_{6} - \beta_{10} ) q^{29} + ( -4 + 4 \beta_{4} + \beta_{6} + \beta_{10} + \beta_{11} ) q^{31} + ( -4 \beta_{1} + 4 \beta_{2} - \beta_{3} + 4 \beta_{8} ) q^{33} + ( -\beta_{1} + 2 \beta_{2} + \beta_{7} + \beta_{8} ) q^{37} + ( -6 + 6 \beta_{4} - 3 \beta_{6} - \beta_{10} + \beta_{11} ) q^{39} + ( 3 + \beta_{6} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{41} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{7} - 2 \beta_{8} ) q^{43} + ( \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{7} - \beta_{8} ) q^{47} + ( 1 + 2 \beta_{4} - \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{49} + ( -\beta_{9} - \beta_{10} ) q^{51} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{5} + 4 \beta_{7} + 3 \beta_{8} ) q^{53} + ( 4 \beta_{2} - 6 \beta_{3} - 4 \beta_{5} + 2 \beta_{7} + 4 \beta_{8} ) q^{57} + ( 8 - 8 \beta_{4} ) q^{59} + ( 2 \beta_{4} + \beta_{6} + \beta_{9} + 3 \beta_{10} ) q^{61} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{7} - 7 \beta_{8} ) q^{63} + ( -\beta_{1} - 4 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} ) q^{67} + ( \beta_{9} - \beta_{10} - \beta_{11} ) q^{69} + ( \beta_{1} + 3 \beta_{3} + 4 \beta_{5} - 2 \beta_{8} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} + 4 \beta_{7} + \beta_{8} ) q^{77} + ( -6 \beta_{4} - 2 \beta_{6} + 2 \beta_{9} - 2 \beta_{10} ) q^{79} + ( -9 + 9 \beta_{4} - 3 \beta_{6} - 3 \beta_{10} - 3 \beta_{11} ) q^{81} + ( 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{7} + 2 \beta_{8} ) q^{83} + ( -3 \beta_{1} - 2 \beta_{3} + \beta_{5} - \beta_{7} - \beta_{8} ) q^{87} + ( \beta_{6} - \beta_{9} + \beta_{10} ) q^{89} + ( 9 - 10 \beta_{4} - \beta_{6} - \beta_{9} - 2 \beta_{10} ) q^{91} + ( -2 \beta_{1} + 4 \beta_{2} - 9 \beta_{3} + 2 \beta_{7} + 2 \beta_{8} ) q^{93} -\beta_{1} q^{97} + ( -21 - 4 \beta_{6} + 3 \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 18q^{9} + O(q^{10})$$ $$12q + 18q^{9} - 6q^{11} + 6q^{19} - 48q^{29} - 24q^{31} - 36q^{39} + 36q^{41} + 24q^{49} + 48q^{59} + 12q^{61} - 36q^{79} - 54q^{81} + 48q^{91} - 252q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 36 x^{10} + 432 x^{8} + 2040 x^{6} + 3780 x^{4} + 2592 x^{2} + 576$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{11} + 104 \nu^{9} + 1176 \nu^{7} + 5112 \nu^{5} + 9564 \nu^{3} + 9360 \nu$$$$)/2016$$ $$\beta_{2}$$ $$=$$ $$($$$$-65 \nu^{11} + 5 \nu^{10} - 2272 \nu^{9} + 192 \nu^{8} - 25704 \nu^{7} + 2688 \nu^{6} - 105720 \nu^{5} + 17592 \nu^{4} - 134868 \nu^{3} + 53460 \nu^{2} - 28080 \nu + 40464$$$$)/8064$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{11} + 69 \nu^{10} - 104 \nu^{9} + 2448 \nu^{8} - 1176 \nu^{7} + 28560 \nu^{6} - 5112 \nu^{5} + 126648 \nu^{4} - 9564 \nu^{3} + 199476 \nu^{2} - 9360 \nu + 76176$$$$)/4032$$ $$\beta_{4}$$ $$=$$ $$($$$$9 \nu^{11} + 320 \nu^{9} + 3744 \nu^{7} + 16632 \nu^{5} + 25956 \nu^{3} + 9936 \nu + 384$$$$)/768$$ $$\beta_{5}$$ $$=$$ $$($$$$71 \nu^{11} - 133 \nu^{10} + 2480 \nu^{9} - 4704 \nu^{8} + 28056 \nu^{7} - 54432 \nu^{6} + 115944 \nu^{5} - 235704 \nu^{4} + 153996 \nu^{3} - 345492 \nu^{2} + 46800 \nu - 111888$$$$)/8064$$ $$\beta_{6}$$ $$=$$ $$($$$$-101 \nu^{11} + 48 \nu^{10} - 3576 \nu^{9} + 1776 \nu^{8} - 41496 \nu^{7} + 22176 \nu^{6} - 181176 \nu^{5} + 110016 \nu^{4} - 274500 \nu^{3} + 202752 \nu^{2} - 110160 \nu + 93312$$$$)/8064$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{11} - 82 \nu^{10} - 104 \nu^{9} - 2880 \nu^{8} - 1176 \nu^{7} - 32928 \nu^{6} - 5112 \nu^{5} - 139728 \nu^{4} - 9564 \nu^{3} - 203400 \nu^{2} - 9360 \nu - 79776$$$$)/4032$$ $$\beta_{8}$$ $$=$$ $$($$$$-99 \nu^{11} + 151 \nu^{10} - 3544 \nu^{9} + 5328 \nu^{8} - 42000 \nu^{7} + 61488 \nu^{6} - 191880 \nu^{5} + 266376 \nu^{4} - 319980 \nu^{3} + 402876 \nu^{2} - 135504 \nu + 155952$$$$)/8064$$ $$\beta_{9}$$ $$=$$ $$($$$$35 \nu^{11} - 192 \nu^{10} + 1176 \nu^{9} - 6768 \nu^{8} + 12264 \nu^{7} - 77952 \nu^{6} + 40488 \nu^{5} - 335232 \nu^{4} + 14364 \nu^{3} - 488448 \nu^{2} - 51408 \nu - 171648$$$$)/8064$$ $$\beta_{10}$$ $$=$$ $$($$$$-101 \nu^{11} - 48 \nu^{10} - 3576 \nu^{9} - 1776 \nu^{8} - 41496 \nu^{7} - 22176 \nu^{6} - 181176 \nu^{5} - 110016 \nu^{4} - 274500 \nu^{3} - 202752 \nu^{2} - 110160 \nu - 93312$$$$)/8064$$ $$\beta_{11}$$ $$=$$ $$($$$$17 \nu^{11} + 30 \nu^{10} + 594 \nu^{9} + 1068 \nu^{8} + 6720 \nu^{7} + 12516 \nu^{6} + 27708 \nu^{5} + 55656 \nu^{4} + 36108 \nu^{3} + 86400 \nu^{2} + 7344 \nu + 33120$$$$)/1008$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{11} - \beta_{9} + \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} - \beta_{9} - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_{1} - 12$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} + 3 \beta_{7} - 6 \beta_{6} - 9 \beta_{5} + 4 \beta_{4} - 12 \beta_{3} + 9 \beta_{2} - 3 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-12 \beta_{11} - 4 \beta_{10} + 12 \beta_{9} + 10 \beta_{7} - 8 \beta_{6} - 16 \beta_{5} + 36 \beta_{3} - 16 \beta_{2} + 31 \beta_{1} + 72$$ $$\nu^{5}$$ $$=$$ $$($$$$-89 \beta_{11} + 92 \beta_{10} - 89 \beta_{9} - 264 \beta_{8} - 132 \beta_{7} + 181 \beta_{6} + 294 \beta_{5} - 240 \beta_{4} + 426 \beta_{3} - 294 \beta_{2} + 93 \beta_{1} + 120$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$225 \beta_{11} + 108 \beta_{10} - 225 \beta_{9} - 90 \beta_{7} + 117 \beta_{6} + 270 \beta_{5} - 564 \beta_{3} + 270 \beta_{2} - 462 \beta_{1} - 1020$$ $$\nu^{7}$$ $$=$$ $$693 \beta_{11} - 618 \beta_{10} + 693 \beta_{9} + 2388 \beta_{8} + 1194 \beta_{7} - 1311 \beta_{6} - 2364 \beta_{5} + 2520 \beta_{4} - 3558 \beta_{3} + 2364 \beta_{2} - 645 \beta_{1} - 1260$$ $$\nu^{8}$$ $$=$$ $$-3948 \beta_{11} - 2268 \beta_{10} + 3948 \beta_{9} + 720 \beta_{7} - 1680 \beta_{6} - 4608 \beta_{5} + 8736 \beta_{3} - 4608 \beta_{2} + 7032 \beta_{1} + 15336$$ $$\nu^{9}$$ $$=$$ $$-11043 \beta_{11} + 8208 \beta_{10} - 11043 \beta_{9} - 41040 \beta_{8} - 20520 \beta_{7} + 19251 \beta_{6} + 38070 \beta_{5} - 47136 \beta_{4} + 58590 \beta_{3} - 38070 \beta_{2} + 8667 \beta_{1} + 23568$$ $$\nu^{10}$$ $$=$$ $$67518 \beta_{11} + 43104 \beta_{10} - 67518 \beta_{9} - 3756 \beta_{7} + 24414 \beta_{6} + 78204 \beta_{5} - 136728 \beta_{3} + 78204 \beta_{2} - 109344 \beta_{1} - 237816$$ $$\nu^{11}$$ $$=$$ $$178212 \beta_{11} - 111108 \beta_{10} + 178212 \beta_{9} + 692424 \beta_{8} + 346212 \beta_{7} - 289320 \beta_{6} - 616428 \beta_{5} + 837936 \beta_{4} - 962640 \beta_{3} + 616428 \beta_{2} - 117396 \beta_{1} - 418968$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
249.1
 0.689786i − 0.802567i − 1.42101i 3.37523i − 4.06501i 2.22358i − 0.689786i 0.802567i 1.42101i − 3.37523i 4.06501i − 2.22358i
0 −2.84918 1.64497i 0 0 0 0.0641892 + 2.64497i 0 3.91187 + 6.77556i 0
249.2 0 −2.23800 1.29211i 0 0 0 2.62958 + 0.292113i 0 1.83911 + 3.18543i 0
249.3 0 −0.611171 0.352860i 0 0 0 −2.56539 0.647140i 0 −1.25098 2.16676i 0
249.4 0 0.611171 + 0.352860i 0 0 0 2.56539 + 0.647140i 0 −1.25098 2.16676i 0
249.5 0 2.23800 + 1.29211i 0 0 0 −2.62958 0.292113i 0 1.83911 + 3.18543i 0
249.6 0 2.84918 + 1.64497i 0 0 0 −0.0641892 2.64497i 0 3.91187 + 6.77556i 0
849.1 0 −2.84918 + 1.64497i 0 0 0 0.0641892 2.64497i 0 3.91187 6.77556i 0
849.2 0 −2.23800 + 1.29211i 0 0 0 2.62958 0.292113i 0 1.83911 3.18543i 0
849.3 0 −0.611171 + 0.352860i 0 0 0 −2.56539 + 0.647140i 0 −1.25098 + 2.16676i 0
849.4 0 0.611171 0.352860i 0 0 0 2.56539 0.647140i 0 −1.25098 + 2.16676i 0
849.5 0 2.23800 1.29211i 0 0 0 −2.62958 + 0.292113i 0 1.83911 3.18543i 0
849.6 0 2.84918 1.64497i 0 0 0 −0.0641892 + 2.64497i 0 3.91187 6.77556i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 849.6 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 1400.2.bh.i 12
5.b even 2 1 inner 1400.2.bh.i 12
5.c odd 4 1 280.2.q.e 6
5.c odd 4 1 1400.2.q.j 6
7.c even 3 1 inner 1400.2.bh.i 12
15.e even 4 1 2520.2.bi.q 6
20.e even 4 1 560.2.q.l 6
35.f even 4 1 1960.2.q.w 6
35.j even 6 1 inner 1400.2.bh.i 12
35.k even 12 1 1960.2.a.v 3
35.k even 12 1 1960.2.q.w 6
35.k even 12 1 9800.2.a.cf 3
35.l odd 12 1 280.2.q.e 6
35.l odd 12 1 1400.2.q.j 6
35.l odd 12 1 1960.2.a.w 3
35.l odd 12 1 9800.2.a.ce 3
105.x even 12 1 2520.2.bi.q 6
140.w even 12 1 560.2.q.l 6
140.w even 12 1 3920.2.a.cc 3
140.x odd 12 1 3920.2.a.cb 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.e 6 5.c odd 4 1
280.2.q.e 6 35.l odd 12 1
560.2.q.l 6 20.e even 4 1
560.2.q.l 6 140.w even 12 1
1400.2.q.j 6 5.c odd 4 1
1400.2.q.j 6 35.l odd 12 1
1400.2.bh.i 12 1.a even 1 1 trivial
1400.2.bh.i 12 5.b even 2 1 inner
1400.2.bh.i 12 7.c even 3 1 inner
1400.2.bh.i 12 35.j even 6 1 inner
1960.2.a.v 3 35.k even 12 1
1960.2.a.w 3 35.l odd 12 1
1960.2.q.w 6 35.f even 4 1
1960.2.q.w 6 35.k even 12 1
2520.2.bi.q 6 15.e even 4 1
2520.2.bi.q 6 105.x even 12 1
3920.2.a.cb 3 140.x odd 12 1
3920.2.a.cc 3 140.w even 12 1
9800.2.a.ce 3 35.l odd 12 1
9800.2.a.cf 3 35.k even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1400, [\chi])$$:

 $$T_{3}^{12} - 18 T_{3}^{10} + 243 T_{3}^{8} - 1386 T_{3}^{6} + 5913 T_{3}^{4} - 2916 T_{3}^{2} + 1296$$ $$T_{11}^{6} + 3 T_{11}^{5} + 33 T_{11}^{4} + 16 T_{11}^{3} + 708 T_{11}^{2} + 1056 T_{11} + 1936$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$1296 - 2916 T^{2} + 5913 T^{4} - 1386 T^{6} + 243 T^{8} - 18 T^{10} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$117649 - 28812 T^{2} - 2352 T^{4} + 1178 T^{6} - 48 T^{8} - 12 T^{10} + T^{12}$$
$11$ $$( 1936 + 1056 T + 708 T^{2} + 16 T^{3} + 33 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$13$ $$( 4624 + 984 T^{2} + 57 T^{4} + T^{6} )^{2}$$
$17$ $$( 16 - 4 T^{2} + T^{4} )^{3}$$
$19$ $$( 256 + 384 T + 528 T^{2} + 104 T^{3} + 33 T^{4} - 3 T^{5} + T^{6} )^{2}$$
$23$ $$2401 - 8967 T^{2} + 31578 T^{4} - 7039 T^{6} + 1338 T^{8} - 39 T^{10} + T^{12}$$
$29$ $$( -26 + 21 T + 12 T^{2} + T^{3} )^{4}$$
$31$ $$( 16384 - 1536 T + 1680 T^{2} + 400 T^{3} + 132 T^{4} + 12 T^{5} + T^{6} )^{2}$$
$37$ $$84934656 - 15925248 T^{2} + 2239488 T^{4} - 121536 T^{6} + 4833 T^{8} - 81 T^{10} + T^{12}$$
$41$ $$( 381 - 45 T - 9 T^{2} + T^{3} )^{4}$$
$43$ $$( 484 + 993 T^{2} + 66 T^{4} + T^{6} )^{2}$$
$47$ $$6359192801536 - 143376276864 T^{2} + 2181037488 T^{4} - 18665464 T^{6} + 117033 T^{8} - 417 T^{10} + T^{12}$$
$53$ $$151613669376 - 6391996416 T^{2} + 181875456 T^{4} - 2914848 T^{6} + 34209 T^{8} - 225 T^{10} + T^{12}$$
$59$ $$( 64 - 8 T + T^{2} )^{6}$$
$61$ $$( 295936 - 47328 T + 10833 T^{2} - 566 T^{3} + 123 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$67$ $$4096 - 298560 T^{2} + 21751089 T^{4} - 811582 T^{6} + 25611 T^{8} - 174 T^{10} + T^{12}$$
$71$ $$T^{12}$$
$73$ $$12745506816 - 1316818944 T^{2} + 111663360 T^{4} - 2293632 T^{6} + 34992 T^{8} - 216 T^{10} + T^{12}$$
$79$ $$( 589824 + 13824 T^{2} + 1536 T^{3} + 324 T^{4} + 18 T^{5} + T^{6} )^{2}$$
$83$ $$( 817216 + 30441 T^{2} + 318 T^{4} + T^{6} )^{2}$$
$89$ $$( 1764 - 1134 T + 729 T^{2} - 84 T^{3} + 27 T^{4} + T^{6} )^{2}$$
$97$ $$( 4 + T^{2} )^{6}$$