# Properties

 Label 7.11.b.b Level $7$ Weight $11$ Character orbit 7.b Analytic conductor $4.448$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 7.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.44750076872$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.373770240.2 Defining polynomial: $$x^{4} + 368x^{2} + 2760$$ x^4 + 368*x^2 + 2760 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}\cdot 3\cdot 5\cdot 7$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 12) q^{2} - \beta_1 q^{3} + ( - 24 \beta_{2} - 144) q^{4} + ( - \beta_{3} + 2 \beta_1) q^{5} + ( - 3 \beta_{3} + 11 \beta_1) q^{6} + ( - 7 \beta_{3} + 147 \beta_{2} + 1225) q^{7} + ( - 880 \beta_{2} - 3648) q^{8} + ( - 90 \beta_{2} - 7191) q^{9}+O(q^{10})$$ q + (b2 - 12) * q^2 - b1 * q^3 + (-24*b2 - 144) * q^4 + (-b3 + 2*b1) * q^5 + (-3*b3 + 11*b1) * q^6 + (-7*b3 + 147*b2 + 1225) * q^7 + (-880*b2 - 3648) * q^8 + (-90*b2 - 7191) * q^9 $$q + (\beta_{2} - 12) q^{2} - \beta_1 q^{3} + ( - 24 \beta_{2} - 144) q^{4} + ( - \beta_{3} + 2 \beta_1) q^{5} + ( - 3 \beta_{3} + 11 \beta_1) q^{6} + ( - 7 \beta_{3} + 147 \beta_{2} + 1225) q^{7} + ( - 880 \beta_{2} - 3648) q^{8} + ( - 90 \beta_{2} - 7191) q^{9} + (19 \beta_{3} - 267 \beta_1) q^{10} + (4204 \beta_{2} + 21498) q^{11} + (72 \beta_{3} + 168 \beta_1) q^{12} + (23 \beta_{3} + 1848 \beta_1) q^{13} + (91 \beta_{3} - 539 \beta_{2} - 1715 \beta_1 + 93492) q^{14} + ( - 21870 \beta_{2} + 132480) q^{15} + (31488 \beta_{2} - 456448) q^{16} + ( - 342 \beta_{3} - 3606 \beta_1) q^{17} + ( - 6111 \beta_{2} + 20052) q^{18} + ( - 942 \beta_{3} + 3033 \beta_1) q^{19} + ( - 24 \beta_{3} + 5544 \beta_1) q^{20} + ( - 441 \beta_{3} - 154350 \beta_{2} - 1372 \beta_1) q^{21} + ( - 28950 \beta_{2} + 2836168) q^{22} + (263824 \beta_{2} - 4815438) q^{23} + (2640 \beta_{3} + 4528 \beta_1) q^{24} + (95190 \beta_{2} + 4091065) q^{25} + (5245 \beta_{3} - 14693 \beta_1) q^{26} + (270 \beta_{3} - 51768 \beta_1) q^{27} + (840 \beta_{3} - 50568 \beta_{2} + 41160 \beta_1 - 2773008) q^{28} + ( - 562604 \beta_{2} + 5761002) q^{29} + (394920 \beta_{2} - 17686080) q^{30} + ( - 10834 \beta_{3} + 102402 \beta_1) q^{31} + (66816 \beta_{2} + 32388096) q^{32} + ( - 12612 \beta_{3} - 25702 \beta_1) q^{33} + ( - 6372 \beta_{3} - 44124 \beta_1) q^{34} + ( - 196 \beta_{3} + 360150 \beta_{2} - 33271 \beta_1 - 37867200) q^{35} + (185544 \beta_{2} + 2625264) q^{36} + ( - 2341884 \beta_{2} - 43729222) q^{37} + (21345 \beta_{3} - 264153 \beta_1) q^{38} + (673470 \beta_{2} + 122411520) q^{39} + ( - 2512 \beta_{3} + 206544 \beta_1) q^{40} + (38870 \beta_{3} + 346976 \beta_1) q^{41} + (1617 \beta_{3} + 1852200 \beta_{2} - 92953 \beta_1 - 113601600) q^{42} + (5694024 \beta_{2} + 40789562) q^{43} + ( - 1121328 \beta_{2} - 77355168) q^{44} + (6561 \beta_{3} + 7488 \beta_1) q^{45} + ( - 7981326 \beta_{2} + 251959720) q^{46} + (83878 \beta_{3} - 361394 \beta_1) q^{47} + ( - 94464 \beta_{3} + 424960 \beta_1) q^{48} + ( - 15092 \beta_{3} + 720300 \beta_{2} + \cdots - 247665551) q^{49}+ \cdots + ( - 32165784 \beta_{2} - 433065078) q^{99}+O(q^{100})$$ q + (b2 - 12) * q^2 - b1 * q^3 + (-24*b2 - 144) * q^4 + (-b3 + 2*b1) * q^5 + (-3*b3 + 11*b1) * q^6 + (-7*b3 + 147*b2 + 1225) * q^7 + (-880*b2 - 3648) * q^8 + (-90*b2 - 7191) * q^9 + (19*b3 - 267*b1) * q^10 + (4204*b2 + 21498) * q^11 + (72*b3 + 168*b1) * q^12 + (23*b3 + 1848*b1) * q^13 + (91*b3 - 539*b2 - 1715*b1 + 93492) * q^14 + (-21870*b2 + 132480) * q^15 + (31488*b2 - 456448) * q^16 + (-342*b3 - 3606*b1) * q^17 + (-6111*b2 + 20052) * q^18 + (-942*b3 + 3033*b1) * q^19 + (-24*b3 + 5544*b1) * q^20 + (-441*b3 - 154350*b2 - 1372*b1) * q^21 + (-28950*b2 + 2836168) * q^22 + (263824*b2 - 4815438) * q^23 + (2640*b3 + 4528*b1) * q^24 + (95190*b2 + 4091065) * q^25 + (5245*b3 - 14693*b1) * q^26 + (270*b3 - 51768*b1) * q^27 + (840*b3 - 50568*b2 + 41160*b1 - 2773008) * q^28 + (-562604*b2 + 5761002) * q^29 + (394920*b2 - 17686080) * q^30 + (-10834*b3 + 102402*b1) * q^31 + (66816*b2 + 32388096) * q^32 + (-12612*b3 - 25702*b1) * q^33 + (-6372*b3 - 44124*b1) * q^34 + (-196*b3 + 360150*b2 - 33271*b1 - 37867200) * q^35 + (185544*b2 + 2625264) * q^36 + (-2341884*b2 - 43729222) * q^37 + (21345*b3 - 264153*b1) * q^38 + (673470*b2 + 122411520) * q^39 + (-2512*b3 + 206544*b1) * q^40 + (38870*b3 + 346976*b1) * q^41 + (1617*b3 + 1852200*b2 - 92953*b1 - 113601600) * q^42 + (5694024*b2 + 40789562) * q^43 + (-1121328*b2 - 77355168) * q^44 + (6561*b3 + 7488*b1) * q^45 + (-7981326*b2 + 251959720) * q^46 + (83878*b3 - 361394*b1) * q^47 + (-94464*b3 + 424960*b1) * q^48 + (-15092*b3 + 720300*b2 - 504210*b1 - 247665551) * q^49 + (2948785*b2 + 20967060) * q^50 + (-7865640*b2 - 238861440) * q^51 + (-135816*b3 - 445704*b1) * q^52 + (11888776*b2 + 423482970) * q^53 + (-158814*b3 + 635598*b1) * q^54 + (7930*b3 - 978576*b1) * q^55 + (19376*b3 - 1614256*b2 + 1509200*b1 - 99677760) * q^56 + (-20498130*b2 + 200905920) * q^57 + (12512250*b2 - 483208568) * q^58 + (253334*b3 + 2973599*b1) * q^59 + (-30240*b2 + 367234560) * q^60 + (-1763*b3 - 1695426*b1) * q^61 + (448048*b3 - 3780752*b1) * q^62 + (49707*b3 - 1167327*b2 + 154350*b1 - 18546255) * q^63 + (-657408*b2 + 127922176) * q^64 + (39232410*b2 - 120402240) * q^65 + (86850*b3 - 2807218*b1) * q^66 + (7033296*b2 - 516191542) * q^67 + (300672*b3 + 2616768*b1) * q^68 + (-791472*b3 + 4551614*b1) * q^69 + (-97265*b3 - 42189000*b2 + 317961*b1 + 719476800) * q^70 + (-41954066*b2 + 282264978) * q^71 + (6656400*b2 + 84523968) * q^72 + (-915672*b3 + 2212062*b1) * q^73 + (-15626614*b2 - 1198875960) * q^74 + (-285570*b3 - 4186255*b1) * q^75 + (-105336*b3 + 5029416*b1) * q^76 + (-121058*b3 + 8310106*b2 - 7209860*b1 + 481174218) * q^77 + (114329880*b2 - 973264320) * q^78 + (-43145934*b2 + 2709073922) * q^79 + (676864*b3 - 8564480*b1) * q^80 + (-4020030*b2 - 3853733679) * q^81 + (535618*b3 + 5706414*b1) * q^82 + (1306996*b3 + 1663711*b1) * q^83 + (151704*b3 + 22226400*b2 + 2823576*b1 + 2726438400) * q^84 + (-61267320*b2 - 1372360320) * q^85 + (-27538726*b2 + 3701326920) * q^86 + (1687812*b3 - 5198398*b1) * q^87 + (-34254432*b2 - 2801271424) * q^88 + (-1472028*b3 + 10006446*b1) * q^89 + (-62829*b3 + 1525077*b1) * q^90 + (839762*b3 + 284055450*b2 + 3363801*b1 + 870945600) * q^91 + (77579856*b2 - 3966764064) * q^92 + (-229673520*b2 + 6783108480) * q^93 + (-2174596*b3 + 24525444*b1) * q^94 + (114797610*b2 - 5497655040) * q^95 + (-200448*b3 - 32454912*b1) * q^96 + (-617078*b3 - 33913194*b1) * q^97 + (-1316434*b3 - 256309151*b2 + 1848770*b1 + 3502127412) * q^98 + (-32165784*b2 - 433065078) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 48 q^{2} - 576 q^{4} + 4900 q^{7} - 14592 q^{8} - 28764 q^{9}+O(q^{10})$$ 4 * q - 48 * q^2 - 576 * q^4 + 4900 * q^7 - 14592 * q^8 - 28764 * q^9 $$4 q - 48 q^{2} - 576 q^{4} + 4900 q^{7} - 14592 q^{8} - 28764 q^{9} + 85992 q^{11} + 373968 q^{14} + 529920 q^{15} - 1825792 q^{16} + 80208 q^{18} + 11344672 q^{22} - 19261752 q^{23} + 16364260 q^{25} - 11092032 q^{28} + 23044008 q^{29} - 70744320 q^{30} + 129552384 q^{32} - 151468800 q^{35} + 10501056 q^{36} - 174916888 q^{37} + 489646080 q^{39} - 454406400 q^{42} + 163158248 q^{43} - 309420672 q^{44} + 1007838880 q^{46} - 990662204 q^{49} + 83868240 q^{50} - 955445760 q^{51} + 1693931880 q^{53} - 398711040 q^{56} + 803623680 q^{57} - 1932834272 q^{58} + 1468938240 q^{60} - 74185020 q^{63} + 511688704 q^{64} - 481608960 q^{65} - 2064766168 q^{67} + 2877907200 q^{70} + 1129059912 q^{71} + 338095872 q^{72} - 4795503840 q^{74} + 1924696872 q^{77} - 3893057280 q^{78} + 10836295688 q^{79} - 15414934716 q^{81} + 10905753600 q^{84} - 5489441280 q^{85} + 14805307680 q^{86} - 11205085696 q^{88} + 3483782400 q^{91} - 15867056256 q^{92} + 27132433920 q^{93} - 21990620160 q^{95} + 14008509648 q^{98} - 1732260312 q^{99}+O(q^{100})$$ 4 * q - 48 * q^2 - 576 * q^4 + 4900 * q^7 - 14592 * q^8 - 28764 * q^9 + 85992 * q^11 + 373968 * q^14 + 529920 * q^15 - 1825792 * q^16 + 80208 * q^18 + 11344672 * q^22 - 19261752 * q^23 + 16364260 * q^25 - 11092032 * q^28 + 23044008 * q^29 - 70744320 * q^30 + 129552384 * q^32 - 151468800 * q^35 + 10501056 * q^36 - 174916888 * q^37 + 489646080 * q^39 - 454406400 * q^42 + 163158248 * q^43 - 309420672 * q^44 + 1007838880 * q^46 - 990662204 * q^49 + 83868240 * q^50 - 955445760 * q^51 + 1693931880 * q^53 - 398711040 * q^56 + 803623680 * q^57 - 1932834272 * q^58 + 1468938240 * q^60 - 74185020 * q^63 + 511688704 * q^64 - 481608960 * q^65 - 2064766168 * q^67 + 2877907200 * q^70 + 1129059912 * q^71 + 338095872 * q^72 - 4795503840 * q^74 + 1924696872 * q^77 - 3893057280 * q^78 + 10836295688 * q^79 - 15414934716 * q^81 + 10905753600 * q^84 - 5489441280 * q^85 + 14805307680 * q^86 - 11205085696 * q^88 + 3483782400 * q^91 - 15867056256 * q^92 + 27132433920 * q^93 - 21990620160 * q^95 + 14008509648 * q^98 - 1732260312 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 368x^{2} + 2760$$ :

 $$\beta_{1}$$ $$=$$ $$( -3\nu^{3} - 1254\nu ) / 13$$ (-3*v^3 - 1254*v) / 13 $$\beta_{2}$$ $$=$$ $$( 2\nu^{2} + 368 ) / 13$$ (2*v^2 + 368) / 13 $$\beta_{3}$$ $$=$$ $$( -35\nu^{3} - 10990\nu ) / 13$$ (-35*v^3 - 10990*v) / 13
 $$\nu$$ $$=$$ $$( 3\beta_{3} - 35\beta_1 ) / 840$$ (3*b3 - 35*b1) / 840 $$\nu^{2}$$ $$=$$ $$( 13\beta_{2} - 368 ) / 2$$ (13*b2 - 368) / 2 $$\nu^{3}$$ $$=$$ $$( -627\beta_{3} + 5495\beta_1 ) / 420$$ (-627*b3 + 5495*b1) / 420

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 − 18.9826i 18.9826i − 2.76757i 2.76757i
−39.1293 252.583i 507.104 2873.50i 9883.42i −2763.01 + 16578.3i 20225.8 −4749.36 112438.i
6.2 −39.1293 252.583i 507.104 2873.50i 9883.42i −2763.01 16578.3i 20225.8 −4749.36 112438.i
6.3 15.1293 262.072i −795.104 1758.44i 3964.97i 5213.01 15978.1i −27521.8 −9632.64 26604.0i
6.4 15.1293 262.072i −795.104 1758.44i 3964.97i 5213.01 + 15978.1i −27521.8 −9632.64 26604.0i
 $$n$$: 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.11.b.b 4
3.b odd 2 1 63.11.d.c 4
4.b odd 2 1 112.11.c.b 4
7.b odd 2 1 inner 7.11.b.b 4
7.c even 3 2 49.11.d.b 8
7.d odd 6 2 49.11.d.b 8
21.c even 2 1 63.11.d.c 4
28.d even 2 1 112.11.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.11.b.b 4 1.a even 1 1 trivial
7.11.b.b 4 7.b odd 2 1 inner
49.11.d.b 8 7.c even 3 2
49.11.d.b 8 7.d odd 6 2
63.11.d.c 4 3.b odd 2 1
63.11.d.c 4 21.c even 2 1
112.11.c.b 4 4.b odd 2 1
112.11.c.b 4 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 24T_{2} - 592$$ acting on $$S_{11}^{\mathrm{new}}(7, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 24 T - 592)^{2}$$
$3$ $$T^{4} + 132480 T^{2} + \cdots + 4381776000$$
$5$ $$T^{4} + 11349120 T^{2} + \cdots + 25531635024000$$
$7$ $$T^{4} - 4900 T^{3} + \cdots + 79\!\cdots\!01$$
$11$ $$(T^{2} - 42996 T - 12545617372)^{2}$$
$13$ $$T^{4} + 458156334720 T^{2} + \cdots + 48\!\cdots\!00$$
$17$ $$T^{4} + 2988125614080 T^{2} + \cdots + 30\!\cdots\!00$$
$19$ $$T^{4} + 10819263899520 T^{2} + \cdots + 16\!\cdots\!00$$
$23$ $$(T^{2} + 9630876 T - 28039440658492)^{2}$$
$29$ $$(T^{2} - 11522004 T - 199771975916572)^{2}$$
$31$ $$T^{4} + \cdots + 11\!\cdots\!00$$
$37$ $$(T^{2} + 87458444 T - 21\!\cdots\!32)^{2}$$
$41$ $$T^{4} + \cdots + 62\!\cdots\!00$$
$43$ $$(T^{2} - 81579124 T - 22\!\cdots\!92)^{2}$$
$47$ $$T^{4} + \cdots + 78\!\cdots\!00$$
$53$ $$(T^{2} - 846965940 T + 75\!\cdots\!64)^{2}$$
$59$ $$T^{4} + \cdots + 42\!\cdots\!00$$
$61$ $$T^{4} + \cdots + 36\!\cdots\!00$$
$67$ $$(T^{2} + 1032383084 T + 23\!\cdots\!88)^{2}$$
$71$ $$(T^{2} - 564529956 T - 12\!\cdots\!32)^{2}$$
$73$ $$T^{4} + \cdots + 16\!\cdots\!00$$
$79$ $$(T^{2} - 5418147844 T + 59\!\cdots\!68)^{2}$$
$83$ $$T^{4} + \cdots + 83\!\cdots\!00$$
$89$ $$T^{4} + \cdots + 19\!\cdots\!00$$
$97$ $$T^{4} + \cdots + 53\!\cdots\!00$$