# Properties

 Label 3.11.b.a Level $3$ Weight $11$ Character orbit 3.b Analytic conductor $1.906$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 3.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.90607175802$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 12\sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (9 \beta - 27) q^{3} + 304 q^{4} - 106 \beta q^{5} + ( - 27 \beta - 6480) q^{6} + 17234 q^{7} + 1328 \beta q^{8} + ( - 486 \beta - 57591) q^{9} +O(q^{10})$$ q + b * q^2 + (9*b - 27) * q^3 + 304 * q^4 - 106*b * q^5 + (-27*b - 6480) * q^6 + 17234 * q^7 + 1328*b * q^8 + (-486*b - 57591) * q^9 $$q + \beta q^{2} + (9 \beta - 27) q^{3} + 304 q^{4} - 106 \beta q^{5} + ( - 27 \beta - 6480) q^{6} + 17234 q^{7} + 1328 \beta q^{8} + ( - 486 \beta - 57591) q^{9} + 76320 q^{10} - 6962 \beta q^{11} + (2736 \beta - 8208) q^{12} - 169654 q^{13} + 17234 \beta q^{14} + (2862 \beta + 686880) q^{15} - 644864 q^{16} - 12792 \beta q^{17} + ( - 57591 \beta + 349920) q^{18} - 949462 q^{19} - 32224 \beta q^{20} + (155106 \beta - 465318) q^{21} + 5012640 q^{22} + 99044 \beta q^{23} + ( - 35856 \beta - 8605440) q^{24} + 1675705 q^{25} - 169654 \beta q^{26} + ( - 505197 \beta + 4704237) q^{27} + 5239136 q^{28} + 118594 \beta q^{29} + (686880 \beta - 2060640) q^{30} - 29793118 q^{31} + 715008 \beta q^{32} + (187974 \beta + 45113760) q^{33} + 9210240 q^{34} - 1826804 \beta q^{35} + ( - 147744 \beta - 17507664) q^{36} - 60811846 q^{37} - 949462 \beta q^{38} + ( - 1526886 \beta + 4580658) q^{39} + 101352960 q^{40} + 6770372 \beta q^{41} + ( - 465318 \beta - 111676320) q^{42} + 107419706 q^{43} - 2116448 \beta q^{44} + (6104646 \beta - 37091520) q^{45} - 71311680 q^{46} - 9987608 \beta q^{47} + ( - 5803776 \beta + 17411328) q^{48} + 14535507 q^{49} + 1675705 \beta q^{50} + (345384 \beta + 82892160) q^{51} - 51574816 q^{52} + 7158798 \beta q^{53} + (4704237 \beta + 363741840) q^{54} - 531339840 q^{55} + 22886752 \beta q^{56} + ( - 8545158 \beta + 25635474) q^{57} - 85387680 q^{58} - 24192682 \beta q^{59} + (870048 \beta + 208811520) q^{60} + 1030793642 q^{61} - 29793118 \beta q^{62} + ( - 8375724 \beta - 992523294) q^{63} - 1175146496 q^{64} + 17983324 \beta q^{65} + (45113760 \beta - 135341280) q^{66} + 1876742474 q^{67} - 3888768 \beta q^{68} + ( - 2674188 \beta - 641805120) q^{69} + 1315298880 q^{70} + 100003596 \beta q^{71} + ( - 76480848 \beta + 464693760) q^{72} - 2846528494 q^{73} - 60811846 \beta q^{74} + (15081345 \beta - 45244035) q^{75} - 288636448 q^{76} - 119983108 \beta q^{77} + (4580658 \beta + 1099357920) q^{78} + 1488647618 q^{79} + 68355584 \beta q^{80} + (55978452 \beta + 3146662161) q^{81} - 4874667840 q^{82} + 47175562 \beta q^{83} + (47152224 \beta - 141456672) q^{84} - 976285440 q^{85} + 107419706 \beta q^{86} + ( - 3202038 \beta - 768489120) q^{87} + 6656785920 q^{88} - 224371428 \beta q^{89} + ( - 37091520 \beta - 4395345120) q^{90} - 2923817036 q^{91} + 30109376 \beta q^{92} + ( - 268138062 \beta + 804414186) q^{93} + 7191077760 q^{94} + 100642972 \beta q^{95} + ( - 19305216 \beta - 4633251840) q^{96} - 1592948926 q^{97} + 14535507 \beta q^{98} + (400948542 \beta - 2436143040) q^{99} +O(q^{100})$$ q + b * q^2 + (9*b - 27) * q^3 + 304 * q^4 - 106*b * q^5 + (-27*b - 6480) * q^6 + 17234 * q^7 + 1328*b * q^8 + (-486*b - 57591) * q^9 + 76320 * q^10 - 6962*b * q^11 + (2736*b - 8208) * q^12 - 169654 * q^13 + 17234*b * q^14 + (2862*b + 686880) * q^15 - 644864 * q^16 - 12792*b * q^17 + (-57591*b + 349920) * q^18 - 949462 * q^19 - 32224*b * q^20 + (155106*b - 465318) * q^21 + 5012640 * q^22 + 99044*b * q^23 + (-35856*b - 8605440) * q^24 + 1675705 * q^25 - 169654*b * q^26 + (-505197*b + 4704237) * q^27 + 5239136 * q^28 + 118594*b * q^29 + (686880*b - 2060640) * q^30 - 29793118 * q^31 + 715008*b * q^32 + (187974*b + 45113760) * q^33 + 9210240 * q^34 - 1826804*b * q^35 + (-147744*b - 17507664) * q^36 - 60811846 * q^37 - 949462*b * q^38 + (-1526886*b + 4580658) * q^39 + 101352960 * q^40 + 6770372*b * q^41 + (-465318*b - 111676320) * q^42 + 107419706 * q^43 - 2116448*b * q^44 + (6104646*b - 37091520) * q^45 - 71311680 * q^46 - 9987608*b * q^47 + (-5803776*b + 17411328) * q^48 + 14535507 * q^49 + 1675705*b * q^50 + (345384*b + 82892160) * q^51 - 51574816 * q^52 + 7158798*b * q^53 + (4704237*b + 363741840) * q^54 - 531339840 * q^55 + 22886752*b * q^56 + (-8545158*b + 25635474) * q^57 - 85387680 * q^58 - 24192682*b * q^59 + (870048*b + 208811520) * q^60 + 1030793642 * q^61 - 29793118*b * q^62 + (-8375724*b - 992523294) * q^63 - 1175146496 * q^64 + 17983324*b * q^65 + (45113760*b - 135341280) * q^66 + 1876742474 * q^67 - 3888768*b * q^68 + (-2674188*b - 641805120) * q^69 + 1315298880 * q^70 + 100003596*b * q^71 + (-76480848*b + 464693760) * q^72 - 2846528494 * q^73 - 60811846*b * q^74 + (15081345*b - 45244035) * q^75 - 288636448 * q^76 - 119983108*b * q^77 + (4580658*b + 1099357920) * q^78 + 1488647618 * q^79 + 68355584*b * q^80 + (55978452*b + 3146662161) * q^81 - 4874667840 * q^82 + 47175562*b * q^83 + (47152224*b - 141456672) * q^84 - 976285440 * q^85 + 107419706*b * q^86 + (-3202038*b - 768489120) * q^87 + 6656785920 * q^88 - 224371428*b * q^89 + (-37091520*b - 4395345120) * q^90 - 2923817036 * q^91 + 30109376*b * q^92 + (-268138062*b + 804414186) * q^93 + 7191077760 * q^94 + 100642972*b * q^95 + (-19305216*b - 4633251840) * q^96 - 1592948926 * q^97 + 14535507*b * q^98 + (400948542*b - 2436143040) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 54 q^{3} + 608 q^{4} - 12960 q^{6} + 34468 q^{7} - 115182 q^{9}+O(q^{10})$$ 2 * q - 54 * q^3 + 608 * q^4 - 12960 * q^6 + 34468 * q^7 - 115182 * q^9 $$2 q - 54 q^{3} + 608 q^{4} - 12960 q^{6} + 34468 q^{7} - 115182 q^{9} + 152640 q^{10} - 16416 q^{12} - 339308 q^{13} + 1373760 q^{15} - 1289728 q^{16} + 699840 q^{18} - 1898924 q^{19} - 930636 q^{21} + 10025280 q^{22} - 17210880 q^{24} + 3351410 q^{25} + 9408474 q^{27} + 10478272 q^{28} - 4121280 q^{30} - 59586236 q^{31} + 90227520 q^{33} + 18420480 q^{34} - 35015328 q^{36} - 121623692 q^{37} + 9161316 q^{39} + 202705920 q^{40} - 223352640 q^{42} + 214839412 q^{43} - 74183040 q^{45} - 142623360 q^{46} + 34822656 q^{48} + 29071014 q^{49} + 165784320 q^{51} - 103149632 q^{52} + 727483680 q^{54} - 1062679680 q^{55} + 51270948 q^{57} - 170775360 q^{58} + 417623040 q^{60} + 2061587284 q^{61} - 1985046588 q^{63} - 2350292992 q^{64} - 270682560 q^{66} + 3753484948 q^{67} - 1283610240 q^{69} + 2630597760 q^{70} + 929387520 q^{72} - 5693056988 q^{73} - 90488070 q^{75} - 577272896 q^{76} + 2198715840 q^{78} + 2977295236 q^{79} + 6293324322 q^{81} - 9749335680 q^{82} - 282913344 q^{84} - 1952570880 q^{85} - 1536978240 q^{87} + 13313571840 q^{88} - 8790690240 q^{90} - 5847634072 q^{91} + 1608828372 q^{93} + 14382155520 q^{94} - 9266503680 q^{96} - 3185897852 q^{97} - 4872286080 q^{99}+O(q^{100})$$ 2 * q - 54 * q^3 + 608 * q^4 - 12960 * q^6 + 34468 * q^7 - 115182 * q^9 + 152640 * q^10 - 16416 * q^12 - 339308 * q^13 + 1373760 * q^15 - 1289728 * q^16 + 699840 * q^18 - 1898924 * q^19 - 930636 * q^21 + 10025280 * q^22 - 17210880 * q^24 + 3351410 * q^25 + 9408474 * q^27 + 10478272 * q^28 - 4121280 * q^30 - 59586236 * q^31 + 90227520 * q^33 + 18420480 * q^34 - 35015328 * q^36 - 121623692 * q^37 + 9161316 * q^39 + 202705920 * q^40 - 223352640 * q^42 + 214839412 * q^43 - 74183040 * q^45 - 142623360 * q^46 + 34822656 * q^48 + 29071014 * q^49 + 165784320 * q^51 - 103149632 * q^52 + 727483680 * q^54 - 1062679680 * q^55 + 51270948 * q^57 - 170775360 * q^58 + 417623040 * q^60 + 2061587284 * q^61 - 1985046588 * q^63 - 2350292992 * q^64 - 270682560 * q^66 + 3753484948 * q^67 - 1283610240 * q^69 + 2630597760 * q^70 + 929387520 * q^72 - 5693056988 * q^73 - 90488070 * q^75 - 577272896 * q^76 + 2198715840 * q^78 + 2977295236 * q^79 + 6293324322 * q^81 - 9749335680 * q^82 - 282913344 * q^84 - 1952570880 * q^85 - 1536978240 * q^87 + 13313571840 * q^88 - 8790690240 * q^90 - 5847634072 * q^91 + 1608828372 * q^93 + 14382155520 * q^94 - 9266503680 * q^96 - 3185897852 * q^97 - 4872286080 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 − 2.23607i 2.23607i
26.8328i −27.0000 241.495i 304.000 2844.28i −6480.00 + 724.486i 17234.0 35634.0i −57591.0 + 13040.7i 76320.0
2.2 26.8328i −27.0000 + 241.495i 304.000 2844.28i −6480.00 724.486i 17234.0 35634.0i −57591.0 13040.7i 76320.0
 $$n$$: 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.11.b.a 2
3.b odd 2 1 inner 3.11.b.a 2
4.b odd 2 1 48.11.e.c 2
5.b even 2 1 75.11.c.d 2
5.c odd 4 2 75.11.d.b 4
8.b even 2 1 192.11.e.e 2
8.d odd 2 1 192.11.e.d 2
9.c even 3 2 81.11.d.d 4
9.d odd 6 2 81.11.d.d 4
12.b even 2 1 48.11.e.c 2
15.d odd 2 1 75.11.c.d 2
15.e even 4 2 75.11.d.b 4
24.f even 2 1 192.11.e.d 2
24.h odd 2 1 192.11.e.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.11.b.a 2 1.a even 1 1 trivial
3.11.b.a 2 3.b odd 2 1 inner
48.11.e.c 2 4.b odd 2 1
48.11.e.c 2 12.b even 2 1
75.11.c.d 2 5.b even 2 1
75.11.c.d 2 15.d odd 2 1
75.11.d.b 4 5.c odd 4 2
75.11.d.b 4 15.e even 4 2
81.11.d.d 4 9.c even 3 2
81.11.d.d 4 9.d odd 6 2
192.11.e.d 2 8.d odd 2 1
192.11.e.d 2 24.f even 2 1
192.11.e.e 2 8.b even 2 1
192.11.e.e 2 24.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{11}^{\mathrm{new}}(3, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 720$$
$3$ $$T^{2} + 54T + 59049$$
$5$ $$T^{2} + 8089920$$
$7$ $$(T - 17234)^{2}$$
$11$ $$T^{2} + 34897999680$$
$13$ $$(T + 169654)^{2}$$
$17$ $$T^{2} + 117817390080$$
$19$ $$(T + 949462)^{2}$$
$23$ $$T^{2} + 7062994033920$$
$29$ $$T^{2} + 10126466521920$$
$31$ $$(T + 29793118)^{2}$$
$37$ $$(T + 60811846)^{2}$$
$41$ $$T^{2} + 33\!\cdots\!80$$
$43$ $$(T - 107419706)^{2}$$
$47$ $$T^{2} + 71\!\cdots\!80$$
$53$ $$T^{2} + 36\!\cdots\!80$$
$59$ $$T^{2} + 42\!\cdots\!80$$
$61$ $$(T - 1030793642)^{2}$$
$67$ $$(T - 1876742474)^{2}$$
$71$ $$T^{2} + 72\!\cdots\!20$$
$73$ $$(T + 2846528494)^{2}$$
$79$ $$(T - 1488647618)^{2}$$
$83$ $$T^{2} + 16\!\cdots\!80$$
$89$ $$T^{2} + 36\!\cdots\!80$$
$97$ $$(T + 1592948926)^{2}$$