# Properties

 Label 4020.2.q.j Level 4020 Weight 2 Character orbit 4020.q Analytic conductor 32.100 Analytic rank 0 Dimension 12 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4020.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.0998616126$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + ( \beta_{1} - \beta_{6} - \beta_{9} ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} - q^{5} + ( \beta_{1} - \beta_{6} - \beta_{9} ) q^{7} + q^{9} + ( -1 + \beta_{5} + \beta_{10} ) q^{11} + \beta_{5} q^{13} - q^{15} + ( \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{10} ) q^{17} + ( -\beta_{1} + \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{19} + ( \beta_{1} - \beta_{6} - \beta_{9} ) q^{21} + ( \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{5} - \beta_{8} ) q^{23} + q^{25} + q^{27} + ( 1 + \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{29} + ( -1 + \beta_{5} + \beta_{10} + \beta_{11} ) q^{31} + ( -1 + \beta_{5} + \beta_{10} ) q^{33} + ( -\beta_{1} + \beta_{6} + \beta_{9} ) q^{35} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + 2 \beta_{8} + \beta_{9} ) q^{37} + \beta_{5} q^{39} + ( -1 + \beta_{1} + \beta_{5} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{41} + ( -1 - \beta_{3} + \beta_{6} - \beta_{7} ) q^{43} - q^{45} + ( -2 + 2 \beta_{5} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{47} + ( \beta_{1} - \beta_{2} - \beta_{3} - 8 \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{49} + ( \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{10} ) q^{51} + ( -2 + \beta_{4} - \beta_{7} ) q^{53} + ( 1 - \beta_{5} - \beta_{10} ) q^{55} + ( -\beta_{1} + \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{57} + ( -2 \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{59} + ( -\beta_{2} + \beta_{4} + 4 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{61} + ( \beta_{1} - \beta_{6} - \beta_{9} ) q^{63} -\beta_{5} q^{65} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{67} + ( \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{5} - \beta_{8} ) q^{69} + ( 4 + \beta_{1} - 4 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{71} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{73} + q^{75} + ( 4 \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{77} + ( \beta_{1} - 2 \beta_{10} - \beta_{11} ) q^{79} + q^{81} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{7} + \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{83} + ( -\beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{8} - \beta_{10} ) q^{85} + ( 1 + \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{87} + ( 1 - \beta_{2} - 2 \beta_{4} ) q^{89} + ( \beta_{3} - \beta_{6} ) q^{91} + ( -1 + \beta_{5} + \beta_{10} + \beta_{11} ) q^{93} + ( \beta_{1} - \beta_{3} + 2 \beta_{5} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{95} + ( -\beta_{2} - 3 \beta_{4} - \beta_{8} + 3 \beta_{10} ) q^{97} + ( -1 + \beta_{5} + \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 12q^{3} - 12q^{5} + 2q^{7} + 12q^{9} + O(q^{10})$$ $$12q + 12q^{3} - 12q^{5} + 2q^{7} + 12q^{9} - 5q^{11} + 6q^{13} - 12q^{15} + 9q^{17} - 11q^{19} + 2q^{21} + 19q^{23} + 12q^{25} + 12q^{27} + 8q^{29} - 4q^{31} - 5q^{33} - 2q^{35} + 5q^{37} + 6q^{39} - 9q^{41} - 14q^{43} - 12q^{45} - 6q^{47} - 46q^{49} + 9q^{51} - 20q^{53} + 5q^{55} - 11q^{57} + 6q^{59} + 27q^{61} + 2q^{63} - 6q^{65} + 3q^{67} + 19q^{69} + 25q^{71} - 8q^{73} + 12q^{75} + 10q^{77} - 2q^{79} + 12q^{81} + 20q^{83} - 9q^{85} + 8q^{87} + 12q^{89} + 4q^{91} - 4q^{93} + 11q^{95} - q^{97} - 5q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 30 x^{10} - 53 x^{9} + 798 x^{8} - 1096 x^{7} + 4060 x^{6} - 915 x^{5} + 10392 x^{4} - 7038 x^{3} + 4869 x^{2} - 675 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$3882223435 \nu^{11} + 10214206017 \nu^{10} + 117178408376 \nu^{9} + 172229907007 \nu^{8} + 2805124258771 \nu^{7} + 5330805726143 \nu^{6} + 13095529592739 \nu^{5} + 14682367453768 \nu^{4} + 104356504552668 \nu^{3} + 10811165099478 \nu^{2} - 1517742210897 \nu - 213693056523513$$$$)/ 41206785731685$$ $$\beta_{3}$$ $$=$$ $$($$$$-52241027 \nu^{11} - 8330484 \nu^{10} - 1527067746 \nu^{9} + 1015516512 \nu^{8} - 39224741894 \nu^{7} + 11206364358 \nu^{6} - 166492823853 \nu^{5} - 141488390232 \nu^{4} - 685298055296 \nu^{3} - 115195094163 \nu^{2} + 16205642379 \nu + 52425010656$$$$)/ 392445578397$$ $$\beta_{4}$$ $$=$$ $$($$$$-18208180335 \nu^{11} + 1146781944 \nu^{10} - 532350662473 \nu^{9} + 458507766964 \nu^{8} - 13717814681058 \nu^{7} + 6580973669456 \nu^{6} - 57724581657177 \nu^{5} - 47555282937464 \nu^{4} - 185567304670144 \nu^{3} - 39200281496859 \nu^{2} + 5516033458671 \nu - 55721026347906$$$$)/ 13735595243895$$ $$\beta_{5}$$ $$=$$ $$($$$$-5825001184 \nu^{11} + 5354831941 \nu^{10} - 174825009876 \nu^{9} + 294981453038 \nu^{8} - 4639211296224 \nu^{7} + 6031178620618 \nu^{6} - 23548647527818 \nu^{5} + 3831440668683 \nu^{4} - 61806807816216 \nu^{3} + 34828675835328 \nu^{2} - 29398686612363 \nu + 4077726580611$$$$)/ 3532010205573$$ $$\beta_{6}$$ $$=$$ $$($$$$-17210840888 \nu^{11} + 1079200536 \nu^{10} - 502794995191 \nu^{9} + 433270280680 \nu^{8} - 12966377092280 \nu^{7} + 6193265819117 \nu^{6} - 54563120045688 \nu^{5} - 44952548682560 \nu^{4} - 181135313239308 \nu^{3} - 37054233616944 \nu^{2} + 5214052586664 \nu - 37541561931450$$$$)/ 8241357146337$$ $$\beta_{7}$$ $$=$$ $$($$$$99006849920 \nu^{11} - 12877775961 \nu^{10} + 2892833484967 \nu^{9} - 2664598755331 \nu^{8} + 74666534698007 \nu^{7} - 40028044437344 \nu^{6} + 313376622706938 \nu^{5} + 255696353386256 \nu^{4} + 958346242038351 \nu^{3} + 211593294448401 \nu^{2} - 29776415210874 \nu + 136395667228734$$$$)/ 41206785731685$$ $$\beta_{8}$$ $$=$$ $$($$$$340088114552 \nu^{11} - 310530618669 \nu^{10} + 10196741983762 \nu^{9} - 17126290059559 \nu^{8} + 270520940621483 \nu^{7} - 349406047914461 \nu^{6} + 1366034458141293 \nu^{5} - 191204974749472 \nu^{4} + 3583136534671908 \nu^{3} - 2018885081453712 \nu^{2} + 1769719192585947 \nu - 31631837094135$$$$)/ 41206785731685$$ $$\beta_{9}$$ $$=$$ $$($$$$-208917470212 \nu^{11} + 225445934212 \nu^{10} - 6274826024079 \nu^{9} + 11552472669332 \nu^{8} - 167293976034885 \nu^{7} + 241497537648025 \nu^{6} - 858137457348736 \nu^{5} + 243087280642482 \nu^{4} - 2130621634155120 \nu^{3} + 1617132043932120 \nu^{2} - 983103667365429 \nu + 136216640185254$$$$)/ 24724071439011$$ $$\beta_{10}$$ $$=$$ $$($$$$-1061664495556 \nu^{11} + 934960473697 \nu^{10} - 31728339975486 \nu^{9} + 52500424408727 \nu^{8} - 840619144671909 \nu^{7} + 1063455151268233 \nu^{6} - 4175717123058529 \nu^{5} + 478464801171081 \nu^{4} - 10930594128742824 \nu^{3} + 6156279135530136 \nu^{2} - 4264311775658751 \nu + 96443745926355$$$$)/ 123620357195055$$ $$\beta_{11}$$ $$=$$ $$($$$$-361920964801 \nu^{11} + 304890086297 \nu^{10} - 10768625400391 \nu^{9} + 17453392390817 \nu^{8} - 284857156713719 \nu^{7} + 349950039715648 \nu^{6} - 1382628324291494 \nu^{5} + 78133074301936 \nu^{4} - 3608703960512079 \nu^{3} + 2031316101387126 \nu^{2} - 1184548981579686 \nu + 31816447355970$$$$)/ 41206785731685$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{11} + 2 \beta_{10} - \beta_{8} - \beta_{7} - 10 \beta_{5} - 2 \beta_{4} - \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{7} + 5 \beta_{6} - 25 \beta_{3} - \beta_{2} + 11$$ $$\nu^{4}$$ $$=$$ $$28 \beta_{11} - 57 \beta_{10} + 2 \beta_{9} + 17 \beta_{8} + 2 \beta_{6} + 220 \beta_{5} + 18 \beta_{1} - 220$$ $$\nu^{5}$$ $$=$$ $$-103 \beta_{11} + 22 \beta_{10} + 144 \beta_{9} + 8 \beta_{8} - 103 \beta_{7} - 446 \beta_{5} - 22 \beta_{4} + 623 \beta_{3} + 8 \beta_{2} - 623 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$748 \beta_{7} + 3 \beta_{6} + 1463 \beta_{4} - 849 \beta_{3} + 376 \beta_{2} + 5536$$ $$\nu^{7}$$ $$=$$ $$3060 \beta_{11} - 1362 \beta_{10} - 3671 \beta_{9} + 98 \beta_{8} - 3671 \beta_{6} + 14744 \beta_{5} + 15847 \beta_{1} - 14744$$ $$\nu^{8}$$ $$=$$ $$-20269 \beta_{11} + 37161 \beta_{10} + 2113 \beta_{9} - 9116 \beta_{8} - 20269 \beta_{7} - 144691 \beta_{5} - 37161 \beta_{4} + 30930 \beta_{3} - 9116 \beta_{2} - 30930 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$88360 \beta_{7} + 92478 \beta_{6} + 56197 \beta_{4} - 409535 \beta_{3} + 8548 \beta_{2} + 455798$$ $$\nu^{10}$$ $$=$$ $$554092 \beta_{11} - 952259 \beta_{10} - 108276 \beta_{9} + 228697 \beta_{8} - 108276 \beta_{6} + 3844291 \beta_{5} + 1018959 \beta_{1} - 3844291$$ $$\nu^{11}$$ $$=$$ $$-2525310 \beta_{11} + 1976724 \beta_{10} + 2350334 \beta_{9} - 356591 \beta_{8} - 2525310 \beta_{7} - 13648472 \beta_{5} - 1976724 \beta_{4} + 10725256 \beta_{3} - 356591 \beta_{2} - 10725256 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$$.

 $$n$$ $$1141$$ $$2011$$ $$2681$$ $$3217$$ $$\chi(n)$$ $$-1 + \beta_{5}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
841.1
 −2.64313 + 4.57803i 1.25129 − 2.16730i 0.307496 − 0.532598i −0.818190 + 1.41715i 0.0725333 − 0.125631i 2.33000 − 4.03568i −2.64313 − 4.57803i 1.25129 + 2.16730i 0.307496 + 0.532598i −0.818190 − 1.41715i 0.0725333 + 0.125631i 2.33000 + 4.03568i
0 1.00000 0 −1.00000 0 −2.33998 + 4.05297i 0 1.00000 0
841.2 0 1.00000 0 −1.00000 0 −1.97947 + 3.42854i 0 1.00000 0
841.3 0 1.00000 0 −1.00000 0 −0.550200 + 0.952975i 0 1.00000 0
841.4 0 1.00000 0 −1.00000 0 1.22030 2.11362i 0 1.00000 0
841.5 0 1.00000 0 −1.00000 0 2.26952 3.93092i 0 1.00000 0
841.6 0 1.00000 0 −1.00000 0 2.37984 4.12200i 0 1.00000 0
3781.1 0 1.00000 0 −1.00000 0 −2.33998 4.05297i 0 1.00000 0
3781.2 0 1.00000 0 −1.00000 0 −1.97947 3.42854i 0 1.00000 0
3781.3 0 1.00000 0 −1.00000 0 −0.550200 0.952975i 0 1.00000 0
3781.4 0 1.00000 0 −1.00000 0 1.22030 + 2.11362i 0 1.00000 0
3781.5 0 1.00000 0 −1.00000 0 2.26952 + 3.93092i 0 1.00000 0
3781.6 0 1.00000 0 −1.00000 0 2.37984 + 4.12200i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3781.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4020.2.q.j 12
67.c even 3 1 inner 4020.2.q.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.q.j 12 1.a even 1 1 trivial
4020.2.q.j 12 67.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4020, [\chi])$$:

 $$T_{7}^{12} - \cdots$$ $$T_{11}^{12} + \cdots$$ $$T_{17}^{12} - \cdots$$