LMFDB - Newform orbit 162.7.b.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,7,Mod(161,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.161"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$N$	$=$	$162 = 2 \cdot 3^{4}$
Weight:	$k$	$=$	$7$
Character orbit:	$[\chi]$	$=$	162.b (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$37.2687615464$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600$ x^12 + 370x^10 + 51793x^8 + 3491832x^6 + 117603792x^4 + 1832032512*x^2 + 10453017600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{16}\cdot 3^{42}$
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

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.

$f(q)$	$=$	$q + \beta_{2} q^{2} - 32 q^{4} + \beta_{8} q^{5} + ( - \beta_1 - 40) q^{7} - 32 \beta_{2} q^{8} + \beta_{3} q^{10} + ( - \beta_{10} - 3 \beta_{8} - 19 \beta_{2}) q^{11} + (\beta_{4} - 280) q^{13} + (\beta_{9} + 2 \beta_{8} - 40 \beta_{2}) q^{14}+ \cdots + ( - 84 \beta_{11} + \cdots + 22525 \beta_{2}) q^{98}+O(q^{100})$ q + b2 * q^2 - 32 * q^4 + b8 * q^5 + (-b1 - 40) * q^7 - 32b2 q^8 + b3 * q^10 + (-b10 - 3b8 - 19b2) * q^11 + (b4 - 280) * q^13 + (b9 + 2b8 - 40b2) * q^14 + 1024 * q^16 + (b11 - 2b10 - b9 + 15b8 - b7 - 143b2) q^17 + (-b5 - 4b4 - b3 + 4b1 - 235) * q^19 - 32b8 q^20 + (-b6 + 2b4 - 3b3 + 5b1 + 600) q^22 + (-6b11 + 10b10 + 6b9 + 9b8 - 3b7 + 352b2) * q^23 + (3b6 + b5 - 9b3 - 9b1 - 1349) q^25 + (4b11 + 4b10 - b9 + 2b8 - 278b2) * q^26 + (32b1 + 1280) q^28 + (7b11 + 7b10 - 2b9 - 102b8 + 833b2) q^29 + (-b6 + 5b5 + b4 + 5b3 - 3580) * q^31 + 1024b2 q^32 + (-5b6 + 4b5 - 2b4 + 18b3 - 23b1 + 4560) q^34 + (-7b11 + b10 + 6b9 - 122b8 - 7b7 + 809b2) q^35 + (-5b6 + 9b5 - 3b4 - 7b3 - 61b1 - 2128) q^37 + (-20b11 - 4b10 + 7b9 + 10b8 - 8b7 - 243b2) * q^38 - 32b3 q^40 + (19b11 + 8b10 - 3b9 - 274b8 - 5b7 - 2789b2) * q^41 + (-8b6 + 7b5 + 2b4 - 55b3 + 157b1 - 11905) q^43 + (32b10 + 96b8 + 608b2) q^44 + (10b6 + 12b5 + 16b4 + 9b3 + 22b1 - 11256) q^46 + (11b11 + 11b10 + 34b9 + 608b8 - 11b7 - 3835b2) * q^47 + (8b6 + 14b5 - 51b4 + 4b3 + 104b1 + 22659) q^49 + (28b11 - 84b10 - 19b9 + 294b8 + 8b7 - 1361b2) * q^50 + (-32b4 + 8960) q^52 + (-37b11 - 37b10 + 68b9 - 373b8 + 799b2) q^53 + (-17b6 + 21b5 + 35b4 - 207b3 + 116b1 + 48366) q^55 + (-32b9 - 64b8 + 1280b2) q^56 + (-56b4 - 105b3 - 8b1 - 26544) q^58 + (-27b11 - 44b10 - 70b9 - 128b8 - 8b7 - 2728b2) * q^59 + (22b5 + 11b4 + 22b3 + 200b1 - 22624) * q^61 + (16b11 - 32b10 - 29b9 - 122b8 + 40b7 - 3574b2) * q^62 - 32768 * q^64 + (21b11 + 27b10 - 38b9 - 32b8 + 26b7 + 563b2) * q^65 + (-41b6 + 44b5 + 39b4 + 122b3 + 18b1 + 48323) q^67 + (-32b11 + 64b10 + 32b9 - 480b8 + 32b7 + 4576b2) * q^68 + (-6b6 + 28b5 + 40b4 - 113b3 - 2b1 - 25992) q^70 + (59b11 + 21b10 - 158b9 + 1115b8 + 4b7 + 14139b2) * q^71 + (-8b6 + 40b5 - 129b4 + 242b3 - 60b1 - 81475) q^73 + (-16b11 + 36b9 + 424b8 + 72b7 - 2114b2) q^74 + (32b5 + 128b4 + 32b3 - 128b1 + 7520) * q^76 + (-10b11 + 230b10 + 174b9 + 851b8 + 20b7 + 18180b2) * q^77 + (-17b6 + 85b5 - 61b4 - 143b3 - 260b1 + 127466) q^79 + 1024b8 q^80 + (-21b6 + 20b5 - 130b4 - 277b3 + 41b1 + 89424) q^82 + (-89b11 - 113b10 - 120b9 - 2880b8 - 23b7 + 30117b2) * q^83 + (11b6 + 57b5 + 115b4 - 867b3 + 487b1 - 269928) q^85 + (-28b11 + 116b10 - 152b9 + 1540b8 + 56b7 - 11869b2) * q^86 + (32b6 - 64b4 + 96b3 - 160b1 - 19200) * q^88 + (53b11 + 133b10 - 16b9 - 739b8 - 60b7 - 61311b2) * q^89 + (39b6 + 29b5 - 67b4 + 111b3 + 1594b1 + 29632) q^91 + (192b11 - 320b10 - 192b9 - 288b8 + 96b7 - 11264b2) * q^92 + (-22b6 + 44b5 - 88b4 + 551b3 + 1022b1 + 122808) q^94 + (-84b11 - 298b10 + 222b9 + 1438b8 - 84b7 + 7448b2) * q^95 + (11b6 + 25b5 - 173b4 - 755b3 - 945b1 + 6479) q^97 + (-84b11 - 564b10 - 207b9 - 402b8 + 112b7 + 22525b2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 384 q^{4} - 480 q^{7} - 3360 q^{13} + 12288 q^{16} - 2820 q^{19} + 7200 q^{22} - 16188 q^{25} + 15360 q^{28} - 42960 q^{31} + 54720 q^{34} - 25536 q^{37} - 142860 q^{43} - 135072 q^{46} + 271908 q^{49}+ \cdots + 77748 q^{97}+O(q^{100})$ 12 * q - 384 * q^4 - 480 * q^7 - 3360 * q^13 + 12288 * q^16 - 2820 * q^19 + 7200 * q^22 - 16188 * q^25 + 15360 * q^28 - 42960 * q^31 + 54720 * q^34 - 25536 * q^37 - 142860 * q^43 - 135072 * q^46 + 271908 * q^49 + 107520 * q^52 + 580392 * q^55 - 318528 * q^58 - 271488 * q^61 - 393216 * q^64 + 579876 * q^67 - 311904 * q^70 - 977700 * q^73 + 90240 * q^76 + 1529592 * q^79 + 1073088 * q^82 - 3239136 * q^85 - 230400 * q^88 + 355584 * q^91 + 1473696 * q^94 + 77748 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600$ x^12 + 370*x^10 + 51793*x^8 + 3491832*x^6 + 117603792*x^4 + 1832032512*x^2 + 10453017600 :

$\beta_{1}$	$=$	$( - 13081 \nu^{10} - 5302927 \nu^{8} - 773457883 \nu^{6} - 48532556637 \nu^{4} + \cdots - 10233017042400 ) / 3086858160$ (-13081v^10 - 5302927v^8 - 773457883v^6 - 48532556637v^4 - 1247487287112*v^2 - 10233017042400) / 3086858160
$\beta_{2}$	$=$	$( 26659 \nu^{11} + 7658854 \nu^{9} + 735955987 \nu^{7} + 28995600888 \nu^{5} + \cdots - 806248446336 \nu ) / 1753335434880$ (26659v^11 + 7658854v^9 + 735955987v^7 + 28995600888v^5 + 410789456208v^3 - 806248446336v) / 1753335434880
$\beta_{3}$	$=$	$( 210097 \nu^{10} + 67295716 \nu^{8} + 7678096429 \nu^{6} + 391082105466 \nu^{4} + \cdots + 66081730824000 ) / 3086858160$ (210097v^10 + 67295716v^8 + 7678096429v^6 + 391082105466v^4 + 8756856645192*v^2 + 66081730824000) / 3086858160
$\beta_{4}$	$=$	$( 225662 \nu^{10} + 73086479 \nu^{8} + 8515647236 \nu^{6} + 451736622819 \nu^{4} + \cdots + 98574178616640 ) / 3086858160$ (225662v^10 + 73086479v^8 + 8515647236v^6 + 451736622819v^4 + 11025749885544*v^2 + 98574178616640) / 3086858160
$\beta_{5}$	$=$	$( 1009529 \nu^{10} + 306190910 \nu^{8} + 31261162145 \nu^{6} + 1265226360660 \nu^{4} + \cdots + 59476995947520 ) / 4115810880$ (1009529v^10 + 306190910v^8 + 31261162145v^6 + 1265226360660v^4 + 17476432457616*v^2 + 59476995947520) / 4115810880
$\beta_{6}$	$=$	$( 3267469 \nu^{10} + 1216138966 \nu^{8} + 166433693269 \nu^{6} + 10303228617156 \nu^{4} + \cdots + 24\!\cdots\!00 ) / 12347432640$ (3267469v^10 + 1216138966v^8 + 166433693269v^6 + 10303228617156v^4 + 279754337012400*v^2 + 2459838034713600) / 12347432640
$\beta_{7}$	$=$	$( - 1385069 \nu^{11} + 3076022942 \nu^{9} + 982609562195 \nu^{7} + 97465949305360 \nu^{5} + \cdots + 37\!\cdots\!68 \nu ) / 2337780579840$ (-1385069v^11 + 3076022942v^9 + 982609562195v^7 + 97465949305360v^5 + 3617974672064304v^3 + 37729052485757568v) / 2337780579840
$\beta_{8}$	$=$	$( 9191227 \nu^{11} + 3200148886 \nu^{9} + 405374895019 \nu^{7} + 23127582217656 \nu^{5} + \cdots + 46\!\cdots\!80 \nu ) / 7013341739520$ (9191227v^11 + 3200148886v^9 + 405374895019v^7 + 23127582217656v^5 + 578696810407632v^3 + 4695274771908480v) / 7013341739520
$\beta_{9}$	$=$	$( - 42893003 \nu^{11} - 14985851078 \nu^{9} - 1900173246107 \nu^{7} + \cdots - 21\!\cdots\!40 \nu ) / 3506670869760$ (-42893003v^11 - 14985851078v^9 - 1900173246107v^7 - 107709393330888v^5 - 2633148078796944v^3 - 21101455040250240v) / 3506670869760
$\beta_{10}$	$=$	$( 186692297 \nu^{11} + 65156440394 \nu^{9} + 8325664614761 \nu^{7} + \cdots + 10\!\cdots\!20 \nu ) / 14026683479040$ (186692297v^11 + 65156440394v^9 + 8325664614761v^7 + 483226339488384v^5 + 12321432128900304v^3 + 101244861147611520v) / 14026683479040
$\beta_{11}$	$=$	$( - 500693495 \nu^{11} - 172073646566 \nu^{9} - 21610924592375 \nu^{7} + \cdots - 28\!\cdots\!44 \nu ) / 14026683479040$ (-500693495v^11 - 172073646566v^9 - 21610924592375v^7 - 1239699467732400v^5 - 31933855280617200v^3 - 280984712354722944v) / 14026683479040

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( -\beta_{11} - \beta_{10} - 4\beta_{9} - 53\beta_{8} - 122\beta_{2} ) / 1458$ (-b11 - b10 - 4b9 - 53b8 - 122*b2) / 1458
$\nu^{2}$	$=$	$( -7\beta_{6} + \beta_{5} + 79\beta_{4} - 75\beta_{3} - 221\beta _1 - 269730 ) / 4374$ (-7b6 + b5 + 79b4 - 75b3 - 221b1 - 269730) / 4374
$\nu^{3}$	$=$	$( 47\beta_{11} + 263\beta_{10} + 476\beta_{9} + 3679\beta_{8} - 36\beta_{7} - 55457\beta_{2} ) / 1458$ (47b11 + 263b10 + 476b9 + 3679b8 - 36b7 - 55457b2) / 1458
$\nu^{4}$	$=$	$( 1531\beta_{6} + 275\beta_{5} - 10039\beta_{4} + 6819\beta_{3} + 47861\beta _1 + 24285906 ) / 4374$ (1531b6 + 275b5 - 10039b4 + 6819b3 + 47861*b1 + 24285906) / 4374
$\nu^{5}$	$=$	$( - 17841 \beta_{11} - 153073 \beta_{10} - 176680 \beta_{9} - 910113 \beta_{8} + \cdots + 29162487 \beta_{2} ) / 4374$ (-17841b11 - 153073b10 - 176680b9 - 910113b8 + 19024b7 + 29162487b2) / 4374
$\nu^{6}$	$=$	$( -84433\beta_{6} - 30377\beta_{5} + 412409\beta_{4} - 164953\beta_{3} - 2565667\beta _1 - 884098638 ) / 1458$ (-84433b6 - 30377b5 + 412409b4 - 164953b3 - 2565667*b1 - 884098638) / 1458
$\nu^{7}$	$=$	$( 2532885 \beta_{11} + 24684205 \beta_{10} + 23196280 \beta_{9} + 82540461 \beta_{8} + \cdots - 4223775003 \beta_{2} ) / 4374$ (2532885b11 + 24684205b10 + 23196280b9 + 82540461b8 - 2791348b7 - 4223775003b2) / 4374
$\nu^{8}$	$=$	$( 38381363 \beta_{6} + 17581675 \beta_{5} - 160101575 \beta_{4} + 29748507 \beta_{3} + \cdots + 322428746610 ) / 4374$ (38381363b6 + 17581675b5 - 160101575b4 + 29748507b3 + 1130312437*b1 + 322428746610) / 4374
$\nu^{9}$	$=$	$( - 117062251 \beta_{11} - 1228877051 \beta_{10} - 1049537384 \beta_{9} + \cdots + 196243285213 \beta_{2} ) / 1458$ (-117062251b11 - 1228877051b10 - 1049537384b9 - 2721206603b8 + 131450920b7 + 196243285213b2) / 1458
$\nu^{10}$	$=$	$( - 5595014987 \beta_{6} - 2854696819 \beta_{5} + 21460960859 \beta_{4} - 946680195 \beta_{3} + \cdots - 41687065381770 ) / 4374$ (-5595014987b6 - 2854696819b5 + 21460960859b4 - 946680195b3 - 160636202281*b1 - 41687065381770) / 4374
$\nu^{11}$	$=$	$( 48110080821 \beta_{11} + 531932678909 \beta_{10} + 433999027448 \beta_{9} + \cdots - 81411526600971 \beta_{2} ) / 4374$ (48110080821b11 + 531932678909b10 + 433999027448b9 + 881691825117b8 - 55261937468b7 - 81411526600971b2) / 4374

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$ .

$n$	$83$
$\chi(n)$	$-1$

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.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

161.1

		4.28281i
		8.15670i
		3.87527i
	−	11.8022i
		7.20150i
	−	8.88570i
		8.88570i
	−	7.20150i
		11.8022i
	−	3.87527i
	−	8.15670i
	−	4.28281i

−

5.65685i

−32.0000

−

181.232i

−208.612

181.019i

−1025.20

161.2

−

5.65685i

−32.0000

−

110.679i

326.338

181.019i

−626.092

161.3

−

5.65685i

−32.0000

1.84488i

−12.6882

181.019i

10.4362

161.4

−

5.65685i

−32.0000

10.8441i

−644.082

181.019i

61.3435

161.5

−

5.65685i

−32.0000

45.6802i

490.195

181.019i

258.406

161.6

−

5.65685i

−32.0000

233.541i

−191.150

181.019i

1321.11

161.7

5.65685i

−32.0000

−

233.541i

−191.150

−

181.019i

1321.11

161.8

5.65685i

−32.0000

−

45.6802i

490.195

−

181.019i

258.406

161.9

5.65685i

−32.0000

−

10.8441i

−644.082

−

181.019i

61.3435

161.10

5.65685i

−32.0000

−

1.84488i

−12.6882

−

181.019i

10.4362

161.11

5.65685i

−32.0000

110.679i

326.338

−

181.019i

−626.092

161.12

5.65685i

−32.0000

181.232i

−208.612

−

181.019i

−1025.20

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	162.7.b.c		12
3.b	odd	2	1	inner	162.7.b.c		12
9.c	even	3	1		18.7.d.a	✓	12
9.c	even	3	1		54.7.d.a		12
9.d	odd	6	1		18.7.d.a	✓	12
9.d	odd	6	1		54.7.d.a		12
36.f	odd	6	1		144.7.q.c		12
36.f	odd	6	1		432.7.q.b		12
36.h	even	6	1		144.7.q.c		12
36.h	even	6	1		432.7.q.b		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
18.7.d.a	✓	12	9.c	even	3	1
18.7.d.a	✓	12	9.d	odd	6	1
54.7.d.a		12	9.c	even	3	1
54.7.d.a		12	9.d	odd	6	1
144.7.q.c		12	36.f	odd	6	1
144.7.q.c		12	36.h	even	6	1
162.7.b.c		12	1.a	even	1	1	trivial
162.7.b.c		12	3.b	odd	2	1	inner
432.7.q.b		12	36.f	odd	6	1
432.7.q.b		12	36.h	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{12} + 101844 T_{5}^{10} + 3082099734 T_{5}^{8} + 28287732473100 T_{5}^{6} + \cdots + 18\!\cdots\!00$ T5^12 + 101844*T5^10 + 3082099734*T5^8 + 28287732473100*T5^6 + 49170069564710625*T5^4 + 5551794796611375000*T5^2 + 18327410507756250000 acting on $S_{7}^{\mathrm{new}}(162, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} + 32)^{6}$ (T^2 + 32)^6
$3$	$T^{12}$ T^12
$5$	$T^{12} + \cdots + 18\!\cdots\!00$ T^12 + 101844T^10 + 3082099734T^8 + 28287732473100T^6 + 49170069564710625T^4 + 5551794796611375000*T^2 + 18327410507756250000
$7$	$(T^{6} + \cdots + 52130539391500)^{2}$ (T^6 + 240T^5 - 392124T^4 - 55146328T^3 + 25960208811T^2 + 4446040761060*T + 52130539391500)^2
$11$	$T^{12} + \cdots + 83\!\cdots\!09$ T^12 + 10575414T^10 + 29066350433391T^8 + 32270582355839795988T^6 + 15347811718723259758268799T^4 + 2588767524926685496381340733174*T^2 + 8396331763261806334931312268678609
$13$	$(T^{6} + \cdots + 99\!\cdots\!04)^{2}$ (T^6 + 1680T^5 - 8159682T^4 - 15757013464T^3 + 4082544117537T^2 + 11403560244690744*T + 994262964821436304)^2
$17$	$T^{12} + \cdots + 64\!\cdots\!00$ T^12 + 215347950T^10 + 18049393789365897T^8 + 744245857097477811160200T^6 + 15758822831665576279553603396496T^4 + 163019786997880499892255044713099468800*T^2 + 647615845301140144271862466834882809692160000
$19$	$(T^{6} + \cdots - 71\!\cdots\!00)^{2}$ (T^6 + 1410T^5 - 191867007T^4 - 382630142176T^3 + 8146570540915536T^2 + 10520181713913585216*T - 71656106187277592075600)^2
$23$	$T^{12} + \cdots + 18\!\cdots\!84$ T^12 + 1572370272T^10 + 986814766259255862T^8 + 316978130829381946284296592T^6 + 55221840693370860046390796852576025T^4 + 4970366415762720821310254683382073545247360*T^2 + 181274508901672738342803717379344218269940306448384
$29$	$T^{12} + \cdots + 50\!\cdots\!00$ T^12 + 3053068740T^10 + 3176774649999521766T^8 + 1278686398779790229104936092T^6 + 155026724487719161371797580161350545T^4 + 1741362037654040738643734450561954258698392*T^2 + 5027955642030876021255666273173542092600730976400
$31$	$(T^{6} + \cdots - 15\!\cdots\!40)^{2}$ (T^6 + 21480T^5 - 1454594160T^4 - 23532175741012T^3 + 232995605666379171T^2 + 2292663668715229007904*T - 15085208683835164120221440)^2
$37$	$(T^{6} + \cdots + 54\!\cdots\!48)^{2}$ (T^6 + 12768T^5 - 8080025376T^4 - 290657217707008T^3 + 7707967525365861840T^2 + 469731024283389663754752*T + 5433204259070256480499243648)^2
$41$	$T^{12} + \cdots + 10\!\cdots\!25$ T^12 + 22899698514T^10 + 204388751561180881383T^8 + 908175034324898785502044932636T^6 + 2115890835830226181604439565401824247807T^4 + 2444822188907796198916690984976438850226229246322*T^2 + 1093108903660218461443878297074432990317363772002989236025
$43$	$(T^{6} + \cdots + 13\!\cdots\!55)^{2}$ (T^6 + 71430T^5 - 19451843031T^4 - 2101393973600548T^3 - 688128595459799925T^2 + 5540401975799043971812878*T + 139940106815041240108843786555)^2
$47$	$T^{12} + \cdots + 11\!\cdots\!84$ T^12 + 81383295024T^10 + 2296938542781412473606T^8 + 25781973207317087654480445921888T^6 + 95110951318350617343321792372008684134089T^4 + 69548638930485713387226249211504819402417927872864*T^2 + 11352325507954713356415354329196820130174509398775533129984
$53$	$T^{12} + \cdots + 27\!\cdots\!00$ T^12 + 154278905256T^10 + 8929906998468019106832T^8 + 242364011133635868206769530075136T^6 + 3115411623824000275138802196347949885751296T^4 + 16594198074713880237911795558339712988334889369600000*T^2 + 27589190625545302529950674339088056232779384635365785600000000
$59$	$T^{12} + \cdots + 10\!\cdots\!25$ T^12 + 208169231526T^10 + 11640269148823256101695T^8 + 157974059853798371268624740982708T^6 + 535270872961769052596761723357357892714079T^4 + 537761633636372109209374504450231548241131731683302*T^2 + 107845408856258274391484618313163672169674229202492971208225
$61$	$(T^{6} + \cdots - 10\!\cdots\!52)^{2}$ (T^6 + 135744T^5 - 43933928682T^4 - 1932063220934656T^3 + 129976021141079429625T^2 + 2749503284100061829621520*T - 100612235400443333905146287552)^2
$67$	$(T^{6} + \cdots - 22\!\cdots\!45)^{2}$ (T^6 - 289938T^5 - 218991722871T^4 + 62105624235201044T^3 + 10540183365224733764283T^2 - 2170547738162640439030115778*T - 224344684993932975881534855111045)^2
$71$	$T^{12} + \cdots + 58\!\cdots\!76$ T^12 + 745338015792T^10 + 211841158536200161770336T^8 + 28713696316493810302171965937294080T^6 + 1893914369983922634606747358098864471034941696T^4 + 55745672477261347012021180913449485413864611458529886208*T^2 + 582020661731963214582116915368004114657702490515906873653173682176
$73$	$(T^{6} + \cdots - 49\!\cdots\!60)^{2}$ (T^6 + 488850T^5 - 288767754939T^4 - 107502764253120628T^3 + 23592072114350415315708T^2 + 5039310210860382018228509568*T - 490992446252487733477028437974560)^2
$79$	$(T^{6} + \cdots - 18\!\cdots\!00)^{2}$ (T^6 - 764796T^5 - 376982580180T^4 + 196897165313692484T^3 + 46665826532094930953019T^2 - 12444035566937602730456425620*T - 1853044026966572310725283136114100)^2
$83$	$T^{12} + \cdots + 42\!\cdots\!36$ T^12 + 1934136240336T^10 + 1113173908435832197079718T^8 + 183585490901956048708389465614198592T^6 + 8299806760406099223930037188577351237182090153T^4 + 125467396346269293918229947342064473972018449411551369056*T^2 + 428234183684530679408589452662634798291924379086971779603014881536
$89$	$T^{12} + \cdots + 74\!\cdots\!00$ T^12 + 1318308583464T^10 + 565622194723694711012880T^8 + 99406449004429134652575271908317184T^6 + 6536931386011703438167308234917695080675999744T^4 + 58291273547747417478210468610446183294029635135104614400*T^2 + 74277694524736955403755792553253864629289012250507854479360000
$97$	$(T^{6} + \cdots - 12\!\cdots\!55)^{2}$ (T^6 - 38874T^5 - 1781930852337T^4 + 517722176935493876T^3 + 449902704158409366729375T^2 - 114777293257140786199157416986*T - 12763236632317836815428259595623855)^2
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 161.12
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	162.7.b.c

Level	$162$
Weight	$7$
Character orbit	162.b
Analytic conductor	$37.269$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$