Newform orbit 230.2.g.b

• Label: 230.2.g.b
• Level: $230$
• Weight: $2$
• Character orbit: 230.g
• Analytic conductor: $1.837$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.83655924649\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(2\) over \(\Q(\zeta_{11})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	\(x^{20} - 5 x^{19} + 12 x^{18} + 16 x^{17} - 49 x^{16} + 59 x^{15} - 197 x^{14} + 42 x^{13} + 1625 x^{12} - 5910 x^{11} + 14651 x^{10} - 22501 x^{9} + 26003 x^{8} - 19607 x^{7} + 13040 x^{6} + 348 x^{5} + 7193 x^{4} + 10771 x^{3} + 8781 x^{2} + 3105 x + 529\)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{2} + ( \beta_{1} - \beta_{2} + \beta_{5} - \beta_{13} + \beta_{18} ) q^{3} + \beta_{3} q^{4} + \beta_{4} q^{5} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{17} + \beta_{19} ) q^{6} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{14} - \beta_{15} - \beta_{19} ) q^{7} -\beta_{8} q^{8} + ( -1 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{9} - 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{17} ) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20q + 2q^{2} - 3q^{3} - 2q^{4} - 2q^{5} + 3q^{6} + 19q^{7} + 2q^{8} - 27q^{9} + O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 5 x^{19} + 12 x^{18} + 16 x^{17} - 49 x^{16} + 59 x^{15} - 197 x^{14} + 42 x^{13} + 1625 x^{12} - 5910 x^{11} + 14651 x^{10} - 22501 x^{9} + 26003 x^{8} - 19607 x^{7} + 13040 x^{6} + 348 x^{5} + 7193 x^{4} + 10771 x^{3} + 8781 x^{2} + 3105 x + 529\):

\(1\)	\(=\)	\(\beta_0\)
\(\nu\)	\(=\)	\(\beta_{1}\)
\(\nu^{2}\)	\(=\)	\(\beta_{18} - \beta_{17} + \beta_{16} + \beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)	\(=\)	\(3 \beta_{19} + \beta_{18} - 3 \beta_{17} + 6 \beta_{16} + 3 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} + 5 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - 2\)
\(\nu^{4}\)	\(=\)	\(18 \beta_{19} - 2 \beta_{18} + 25 \beta_{16} + 12 \beta_{15} - 27 \beta_{14} + 18 \beta_{13} - 15 \beta_{12} - 4 \beta_{11} + 15 \beta_{9} - 4 \beta_{8} - 15 \beta_{7} - 4 \beta_{6} + 13 \beta_{5} - 4 \beta_{3} - 5 \beta_{1} - 19\)
\(\nu^{5}\)	\(=\)	\(29 \beta_{19} - 79 \beta_{18} + 52 \beta_{17} - 70 \beta_{16} - 27 \beta_{15} - 32 \beta_{14} + 15 \beta_{13} - 62 \beta_{12} + 20 \beta_{11} - 80 \beta_{10} + 79 \beta_{9} + 80 \beta_{8} + 42 \beta_{6} + 39 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 52 \beta_{2} - 79 \beta_{1} - 62\)
\(\nu^{6}\)	\(=\)	\(-56 \beta_{19} - 239 \beta_{18} + 282 \beta_{17} - 369 \beta_{16} - 239 \beta_{15} + 37 \beta_{14} - 210 \beta_{13} - 166 \beta_{12} - 19 \beta_{11} - 356 \beta_{10} + 56 \beta_{9} + 359 \beta_{8} + 221 \beta_{7} + 123 \beta_{6} + 226 \beta_{5} + 60 \beta_{3} + 222 \beta_{2} - 178 \beta_{1} - 89\)
\(\nu^{7}\)	\(=\)	\(-1081 \beta_{19} - 361 \beta_{18} + 535 \beta_{17} - 1933 \beta_{16} - 1081 \beta_{15} + 1400 \beta_{14} - 1420 \beta_{13} + 361 \beta_{12} + 14 \beta_{11} - 854 \beta_{10} - 535 \beta_{9} + 640 \beta_{8} + 1038 \beta_{7} - 45 \beta_{5} + 105 \beta_{4} + 339 \beta_{3} + 503 \beta_{2} - 45 \beta_{1} + 852\)
\(\nu^{8}\)	\(=\)	\(-3798 \beta_{19} + 2438 \beta_{18} - 1058 \beta_{17} - 1773 \beta_{16} - 1058 \beta_{15} + 4464 \beta_{14} - 3798 \beta_{13} + 3798 \beta_{12} - 810 \beta_{11} + 1868 \beta_{10} - 4325 \beta_{9} - 1922 \beta_{8} + 2438 \beta_{7} - 1876 \beta_{6} - 2797 \beta_{5} + 413 \beta_{4} + 1221 \beta_{3} - 1001 \beta_{2} + 3267 \beta_{1} + 4464\)
\(\nu^{9}\)	\(=\)	\(-5135 \beta_{19} + 16473 \beta_{18} - 15916 \beta_{17} + 15193 \beta_{16} + 8789 \beta_{15} + 6870 \beta_{14} + 420 \beta_{13} + 15916 \beta_{12} - 1210 \beta_{11} + 21605 \beta_{10} - 12724 \beta_{9} - 21471 \beta_{8} - 5135 \beta_{7} - 10069 \beta_{6} - 15916 \beta_{5} + 1735 \beta_{4} - 12724 \beta_{2} + 14481 \beta_{1} + 14706\)
\(\nu^{10}\)	\(=\)	\(36734 \beta_{19} + 50703 \beta_{18} - 59591 \beta_{17} + 118069 \beta_{16} + 66327 \beta_{15} - 47053 \beta_{14} + 67366 \beta_{13} + 17683 \beta_{12} - 4813 \beta_{11} + 87212 \beta_{10} - 77404 \beta_{8} - 59591 \beta_{7} - 19274 \beta_{6} - 33307 \beta_{5} - 4813 \beta_{4} - 17813 \beta_{3} - 50703 \beta_{2} + 33020 \beta_{1} - 9938\)
\(\nu^{11}\)	\(=\)	\(256857 \beta_{19} - 76135 \beta_{17} + 344581 \beta_{16} + 196934 \beta_{15} - 318507 \beta_{14} + 318507 \beta_{13} - 138918 \beta_{12} + 22073 \beta_{11} + 101338 \beta_{10} + 196934 \beta_{9} - 76135 \beta_{8} - 240976 \beta_{7} + 37850 \beta_{6} + 59923 \beta_{5} - 25203 \beta_{4} - 87724 \beta_{3} - 59923 \beta_{2} - 76135 \beta_{1} - 232313\)
\(\nu^{12}\)	\(=\)	\(778490 \beta_{19} - 778490 \beta_{18} + 560354 \beta_{17} - 972564 \beta_{14} + 654831 \beta_{13} - 1019617 \beta_{12} + 123659 \beta_{11} - 777936 \beta_{10} + 1019617 \beta_{9} + 830113 \beta_{8} - 282060 \beta_{7} + 560908 \beta_{6} + 842414 \beta_{5} - 133529 \beta_{4} - 194074 \beta_{3} + 459263 \beta_{2} - 863156 \beta_{1} - 1153146\)
\(\nu^{13}\)	\(=\)	\(-3979257 \beta_{18} + 3979257 \beta_{17} - 5344858 \beta_{16} - 3042920 \beta_{15} - 1672508 \beta_{13} - 3042920 \beta_{12} + 397588 \beta_{11} - 5651765 \beta_{10} + 2164279 \beta_{9} + 5344858 \beta_{8} + 2164279 \beta_{7} + 2111735 \beta_{6} + 3414324 \beta_{5} - 117728 \beta_{4} + 397588 \beta_{3} + 3317425 \beta_{2} - 3317425 \beta_{1} - 2111735\)
\(\nu^{14}\)	\(=\)	\(-11985926 \beta_{19} - 8599271 \beta_{18} + 11985926 \beta_{17} - 27688350 \beta_{16} - 15700939 \beta_{15} + 14954050 \beta_{14} - 18419890 \beta_{13} - 16918620 \beta_{10} - 4413673 \beta_{9} + 14954050 \beta_{8} + 15700939 \beta_{7} + 2687957 \beta_{6} + 4413673 \beta_{5} + 1027056 \beta_{4} + 4932694 \beta_{3} + 9944941 \beta_{2} - 3715013 \beta_{1} + 6433964\)
\(\nu^{15}\)	\(=\)	\(-61492387 \beta_{19} + 17642007 \beta_{18} - 58907654 \beta_{16} - 33525393 \beta_{15} + 76549661 \beta_{14} - 69535225 \beta_{13} + 46978173 \beta_{12} - 6084356 \beta_{11} - 56720847 \beta_{9} - 6084356 \beta_{8} + 46978173 \beta_{7} - 19834963 \beta_{6} - 29336166 \beta_{5} + 8042838 \beta_{4} + 19558179 \beta_{3} + 32156221 \beta_{1} + 66536352\)
\(\nu^{16}\)	\(=\)	\(-132340428 \beta_{19} + 221906105 \beta_{18} - 184749586 \beta_{17} + 120271015 \beta_{16} + 68506823 \beta_{15} + 165278556 \beta_{14} - 72491788 \beta_{13} + 242001342 \beta_{12} - 30705338 \beta_{11} + 261063925 \beta_{10} - 221906105 \beta_{9} - 261063925 \beta_{8} - 140998611 \beta_{6} - 220942488 \beta_{5} + 23474981 \beta_{4} + 23474981 \beta_{3} - 153399282 \beta_{2} + 221906105 \beta_{1} + 242001342\)
\(\nu^{17}\)	\(=\)	\(271144644 \beta_{19} + 873784229 \beta_{18} - 949293604 \beta_{17} + 1539365880 \beta_{16} + 873784229 \beta_{15} - 338975821 \beta_{14} + 698859189 \beta_{13} + 518087766 \beta_{12} - 67831177 \beta_{11} + 1343730611 \beta_{10} - 271144644 \beta_{9} - 1250285953 \beta_{8} - 724971372 \beta_{7} - 424471848 \beta_{6} - 678148960 \beta_{5} - 178167737 \beta_{3} - 789232410 \beta_{2} + 649461997 \beta_{1} + 217095417\)
\(\nu^{18}\)	\(=\)	\(3426020339 \beta_{19} + 1059630164 \beta_{18} - 2039347103 \beta_{17} + 6028541248 \beta_{16} + 3426020339 \beta_{15} - 4266560602 \beta_{14} + 4710719625 \beta_{13} - 1059630164 \beta_{12} + 140416675 \beta_{11} + 2879887366 \beta_{10} + 2039347103 \beta_{9} - 2403439748 \beta_{8} - 3731718622 \beta_{7} - 97039871 \beta_{5} - 364092645 \beta_{4} - 1284699286 \beta_{3} - 1692371519 \beta_{2} - 97039871 \beta_{1} - 2602520909\)
\(\nu^{19}\)	\(=\)	\(13471243579 \beta_{19} - 7998279224 \beta_{18} + 4175877645 \beta_{17} + 7359415616 \beta_{16} + 4175877645 \beta_{15} - 16796005904 \beta_{14} + 13471243579 \beta_{13} - 13471243579 \beta_{12} + 1716522174 \beta_{11} - 5892399819 \beta_{10} + 14648648527 \beta_{9} + 6927879328 \beta_{8} - 7998279224 \beta_{7} + 6543364251 \beta_{6} + 10008173667 \beta_{5} - 1886451261 \beta_{4} - 3850103282 \beta_{3} + 3463069912 \beta_{2} - 10472770882 \beta_{1} - 16796005904\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(-\beta_{6}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

31.1

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Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	230.2.g.b	✓	20
23.c	even	11	1	inner	230.2.g.b	✓	20
23.c	even	11	1		5290.2.a.bj		10
23.d	odd	22	1		5290.2.a.bi		10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
230.2.g.b	✓	20	1.a	even	1	1	trivial
230.2.g.b	✓	20	23.c	even	11	1	inner
5290.2.a.bi		10	23.d	odd	22	1
5290.2.a.bj		10	23.c	even	11	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(230, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
$3$	\( 1024 + 8192 T + 32768 T^{2} + 126720 T^{3} + 366464 T^{4} + 431808 T^{5} + 252880 T^{6} + 67672 T^{7} + 31164 T^{8} + 46692 T^{9} + 34605 T^{10} + 15634 T^{11} + 7173 T^{12} + 3432 T^{13} + 1695 T^{14} + 701 T^{15} + 252 T^{16} + 66 T^{17} + 21 T^{18} + 3 T^{19} + T^{20} \)
$5$	\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
$7$	\( 38809 + 126277 T + 536876 T^{2} + 148808 T^{3} + 209230 T^{4} - 785936 T^{5} + 1709082 T^{6} - 1623479 T^{7} + 183019 T^{8} + 188999 T^{9} + 477641 T^{10} - 734599 T^{11} + 526637 T^{12} - 263428 T^{13} + 102312 T^{14} - 31386 T^{15} + 7556 T^{16} - 1408 T^{17} + 197 T^{18} - 19 T^{19} + T^{20} \)
$11$	\( 2374681 - 11682321 T + 20709116 T^{2} - 306162 T^{3} - 784055 T^{4} + 11498674 T^{5} + 1481982 T^{6} + 2003537 T^{7} + 4030360 T^{8} + 453856 T^{9} + 361008 T^{10} + 129010 T^{11} + 29129 T^{12} + 20470 T^{13} + 1922 T^{14} + 198 T^{15} + 334 T^{16} - 96 T^{17} + 40 T^{18} - 7 T^{19} + T^{20} \)