Properties

Label 40.6.d.a
Level $40$
Weight $6$
Character orbit 40.d
Analytic conductor $6.415$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [40,6,Mod(21,40)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(40, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.21");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 40.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.41535279252\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} - 109104 x^{12} - 96128 x^{11} + 3580672 x^{10} - 1538048 x^{9} + \cdots + 1099511627776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{4}\cdot 5^{12} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} - 2) q^{4} + \beta_{5} q^{5} + (\beta_{9} - \beta_{2} + 10) q^{6} + ( - \beta_{8} - 4 \beta_1 - 10) q^{7} + ( - \beta_{6} - \beta_{5} + 2 \beta_1 + 13) q^{8} + ( - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 8 \beta_1 - 80) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} - 2) q^{4} + \beta_{5} q^{5} + (\beta_{9} - \beta_{2} + 10) q^{6} + ( - \beta_{8} - 4 \beta_1 - 10) q^{7} + ( - \beta_{6} - \beta_{5} + 2 \beta_1 + 13) q^{8} + ( - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 8 \beta_1 - 80) q^{9} + ( - \beta_{11} + \beta_{2} - 3) q^{10} + (\beta_{17} - \beta_{11} + \beta_{9} - 2 \beta_{7} + 3 \beta_{5} - 4 \beta_{3} + \beta_{2} + 22 \beta_1 + 4) q^{11} + ( - \beta_{18} + \beta_{13} + \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{2} + \cdots - 95) q^{12}+ \cdots + ( - 276 \beta_{18} - 43 \beta_{17} - 154 \beta_{16} + 10 \beta_{15} + \cdots - 1324) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} - 32 q^{4} + 204 q^{6} - 196 q^{7} + 248 q^{8} - 1620 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} - 32 q^{4} + 204 q^{6} - 196 q^{7} + 248 q^{8} - 1620 q^{9} - 50 q^{10} - 1876 q^{12} + 2708 q^{14} + 900 q^{15} + 3080 q^{16} - 5294 q^{18} - 1900 q^{20} + 13836 q^{22} - 4676 q^{23} + 1032 q^{24} - 12500 q^{25} - 8084 q^{26} + 2108 q^{28} + 5800 q^{30} + 7160 q^{31} + 6792 q^{32} + 5672 q^{33} + 21132 q^{34} + 18344 q^{36} - 19580 q^{38} - 44904 q^{39} + 6200 q^{40} + 11608 q^{41} - 17116 q^{42} + 72296 q^{44} - 28516 q^{46} + 44180 q^{47} - 88856 q^{48} + 18756 q^{49} - 1250 q^{50} - 39680 q^{52} - 100584 q^{54} - 24200 q^{55} - 53624 q^{56} + 5032 q^{57} + 59496 q^{58} - 31300 q^{60} + 59824 q^{62} + 240620 q^{63} - 11264 q^{64} - 56688 q^{66} + 11576 q^{68} + 29800 q^{70} - 200312 q^{71} + 235912 q^{72} - 105136 q^{73} + 78876 q^{74} - 153872 q^{76} + 95864 q^{78} + 282080 q^{79} + 16000 q^{80} + 65172 q^{81} - 223032 q^{82} - 297128 q^{84} + 27452 q^{86} - 332592 q^{87} + 86896 q^{88} - 3160 q^{89} + 51750 q^{90} + 107916 q^{92} + 148820 q^{94} + 144400 q^{95} + 395168 q^{96} + 147376 q^{97} + 216942 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} - 109104 x^{12} - 96128 x^{11} + 3580672 x^{10} - 1538048 x^{9} + \cdots + 1099511627776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 67985 \nu^{19} + 62912 \nu^{18} + 1230021 \nu^{17} - 3396972 \nu^{16} - 12508553 \nu^{15} + 17579546 \nu^{14} - 167940308 \nu^{13} + \cdots + 11\!\cdots\!16 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 79348087 \nu^{19} + 164467438 \nu^{18} + 2750967783 \nu^{17} + 2347892670 \nu^{16} - 17200470427 \nu^{15} + \cdots + 38\!\cdots\!76 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 36529 \nu^{19} - 37138 \nu^{18} - 952929 \nu^{17} - 2345826 \nu^{16} + 21259597 \nu^{15} + 2524124 \nu^{14} - 129135524 \nu^{13} + \cdots - 36\!\cdots\!00 ) / 321582991933440 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 97985891 \nu^{19} - 2347107250 \nu^{18} + 8955951363 \nu^{17} + 45014733006 \nu^{16} - 72698446823 \nu^{15} + \cdots + 17\!\cdots\!80 ) / 53\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6673115 \nu^{19} + 30164170 \nu^{18} - 153706635 \nu^{17} - 266710470 \nu^{16} + 3862360175 \nu^{15} + 2106440260 \nu^{14} + \cdots - 15\!\cdots\!20 ) / 32\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 69160775 \nu^{19} + 35235854 \nu^{18} + 3743303607 \nu^{17} - 5689165602 \nu^{16} - 40339659563 \nu^{15} + 104897574572 \nu^{14} + \cdots + 33\!\cdots\!84 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2121323 \nu^{19} + 2680522 \nu^{18} - 3689307 \nu^{17} + 6591738 \nu^{16} + 264714623 \nu^{15} + 2131216804 \nu^{14} + \cdots + 15\!\cdots\!28 ) / 34\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1652193653 \nu^{19} - 1541606518 \nu^{18} + 22547436165 \nu^{17} - 28897641030 \nu^{16} - 311157762785 \nu^{15} + \cdots + 18\!\cdots\!72 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 191297309 \nu^{19} - 782315974 \nu^{18} + 2082932205 \nu^{17} + 2536434474 \nu^{16} - 58787611817 \nu^{15} + \cdots + 46\!\cdots\!24 ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 242971859 \nu^{19} + 2811987922 \nu^{18} + 3207553677 \nu^{17} - 20184358830 \nu^{16} + 39588349847 \nu^{15} + \cdots + 77\!\cdots\!32 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3066220261 \nu^{19} + 3110541514 \nu^{18} + 59421776949 \nu^{17} - 134270357190 \nu^{16} - 571957216081 \nu^{15} + \cdots + 58\!\cdots\!08 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1100290003 \nu^{19} - 721259066 \nu^{18} + 12218919363 \nu^{17} - 23374232682 \nu^{16} - 231772749511 \nu^{15} + \cdots + 93\!\cdots\!80 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1149574177 \nu^{19} + 757890370 \nu^{18} + 47847494961 \nu^{17} - 17357641998 \nu^{16} - 524496762781 \nu^{15} + \cdots + 62\!\cdots\!00 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1378084499 \nu^{19} + 1748073274 \nu^{18} - 38128546179 \nu^{17} - 35922205590 \nu^{16} + 102019001735 \nu^{15} + \cdots - 60\!\cdots\!32 ) / 71\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2613608813 \nu^{19} + 397290362 \nu^{18} + 34407860733 \nu^{17} - 7553046486 \nu^{16} - 442322344697 \nu^{15} + \cdots + 24\!\cdots\!32 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 2626005709 \nu^{19} - 1743424454 \nu^{18} + 57278130525 \nu^{17} - 5901549846 \nu^{16} - 828211435225 \nu^{15} + \cdots + 79\!\cdots\!28 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 2753749541 \nu^{19} - 750933770 \nu^{18} - 67876189173 \nu^{17} - 119712276474 \nu^{16} + 704230722833 \nu^{15} + \cdots - 12\!\cdots\!24 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 6092522125 \nu^{19} - 6606216634 \nu^{18} - 167461819677 \nu^{17} - 161753870442 \nu^{16} + 2039605194649 \nu^{15} + \cdots - 22\!\cdots\!84 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 4725901625 \nu^{19} + 2761897870 \nu^{18} - 58020083145 \nu^{17} + 63571279710 \nu^{16} + 970185399893 \nu^{15} + \cdots - 53\!\cdots\!24 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} - \beta_{2} - 25\beta _1 + 3 ) / 50 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{19} + 4\beta_{12} - \beta_{11} + \beta_{7} + \beta_{5} + \beta_{3} + 6\beta_{2} - \beta _1 + 92 ) / 50 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{19} + 7 \beta_{18} - 14 \beta_{17} - 2 \beta_{16} - 10 \beta_{15} + 2 \beta_{14} - 3 \beta_{13} + 16 \beta_{12} + 5 \beta_{11} - 6 \beta_{10} - 13 \beta_{9} - 15 \beta_{8} + 8 \beta_{7} + 18 \beta_{6} + 3 \beta_{5} - 10 \beta_{4} - 2 \beta_{3} + \cdots - 655 ) / 100 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{19} - \beta_{17} - \beta_{16} - 4 \beta_{15} + \beta_{14} - \beta_{13} + 4 \beta_{12} + 2 \beta_{11} - 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 16 \beta_{2} + 18 \beta _1 - 154 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 98 \beta_{19} - 59 \beta_{18} + 23 \beta_{17} - 41 \beta_{16} - 50 \beta_{15} - 159 \beta_{14} + 96 \beta_{13} + 248 \beta_{12} - 99 \beta_{11} - 18 \beta_{10} - 209 \beta_{9} - 400 \beta_{8} + 459 \beta_{7} - 241 \beta_{6} + \cdots - 13307 ) / 100 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 203 \beta_{19} - 321 \beta_{18} + 502 \beta_{17} + 446 \beta_{16} + 350 \beta_{15} - 246 \beta_{14} - 431 \beta_{13} - 1748 \beta_{12} - 683 \beta_{11} + 298 \beta_{10} + 2679 \beta_{9} + 585 \beta_{8} - 384 \beta_{7} + \cdots - 154599 ) / 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 815 \beta_{19} + 188 \beta_{18} - 181 \beta_{17} + 947 \beta_{16} - 1820 \beta_{15} - 947 \beta_{14} + 1463 \beta_{13} - 660 \beta_{12} - 3122 \beta_{11} - 264 \beta_{10} - 5302 \beta_{9} + 5695 \beta_{8} - 7219 \beta_{7} + \cdots - 605926 ) / 100 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 58 \beta_{19} - 11 \beta_{18} + 263 \beta_{17} - 281 \beta_{16} + 334 \beta_{15} - 159 \beta_{14} - 96 \beta_{13} - 1032 \beta_{12} - 459 \beta_{11} + 222 \beta_{10} + 607 \beta_{9} - 144 \beta_{8} - 1141 \beta_{7} + \cdots + 54821 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 6317 \beta_{19} - 8969 \beta_{18} - 13322 \beta_{17} - 3986 \beta_{16} - 8290 \beta_{15} + 11386 \beta_{14} - 25319 \beta_{13} - 86148 \beta_{12} + 97357 \beta_{11} - 8918 \beta_{10} - 87249 \beta_{9} + \cdots + 7594441 ) / 100 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 58639 \beta_{19} + 17716 \beta_{18} + 148603 \beta_{17} - 72781 \beta_{16} + 284420 \beta_{15} - 59939 \beta_{14} + 46031 \beta_{13} - 54084 \beta_{12} + 143502 \beta_{11} + 50312 \beta_{10} + \cdots - 22761814 ) / 100 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 383022 \beta_{19} + 19549 \beta_{18} - 168673 \beta_{17} + 201311 \beta_{16} + 834430 \beta_{15} + 376889 \beta_{14} - 429696 \beta_{13} - 509192 \beta_{12} - 806123 \beta_{11} + \cdots + 103852861 ) / 100 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 57389 \beta_{19} + 90535 \beta_{18} - 86266 \beta_{17} - 34098 \beta_{16} - 40242 \beta_{15} + 77370 \beta_{14} - 22071 \beta_{13} + 69772 \beta_{12} + 121389 \beta_{11} + 8378 \beta_{10} + \cdots - 19775167 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 5174193 \beta_{19} - 5852340 \beta_{18} - 4240725 \beta_{17} - 9784845 \beta_{16} - 4087580 \beta_{15} + 6880045 \beta_{14} + 2672775 \beta_{13} + 23306188 \beta_{12} + \cdots + 2125002202 ) / 100 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 22835066 \beta_{19} - 872211 \beta_{18} - 820753 \beta_{17} + 14290831 \beta_{16} + 18184990 \beta_{15} - 7229671 \beta_{14} + 23437664 \beta_{13} - 21828424 \beta_{12} + \cdots - 2914135811 ) / 100 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 79597763 \beta_{19} - 24695161 \beta_{18} + 59342582 \beta_{17} + 113252366 \beta_{16} - 71557410 \beta_{15} - 894566 \beta_{14} + 91308489 \beta_{13} + 129913052 \beta_{12} + \cdots - 25887410871 ) / 100 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 3041913 \beta_{19} + 257348 \beta_{18} - 3990861 \beta_{17} + 5918283 \beta_{16} + 4133092 \beta_{15} - 15273371 \beta_{14} + 22439799 \beta_{13} + 10740028 \beta_{12} + \cdots + 2454452570 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 134431070 \beta_{19} - 52573091 \beta_{18} - 65313953 \beta_{17} + 658056031 \beta_{16} - 1776630530 \beta_{15} + 266743769 \beta_{14} - 985457856 \beta_{13} + \cdots + 419029465245 ) / 100 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 1589925323 \beta_{19} - 2663673185 \beta_{18} - 2473747690 \beta_{17} - 772959170 \beta_{16} - 5842034210 \beta_{15} - 2664210870 \beta_{14} + 5739559825 \beta_{13} + \cdots + 510369921177 ) / 100 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 40320191407 \beta_{19} + 6137817148 \beta_{18} + 53603331819 \beta_{17} + 31875087827 \beta_{16} + 53419435300 \beta_{15} - 18528567827 \beta_{14} + \cdots + 2088153966970 ) / 100 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
−2.63430 3.01006i
−2.63430 + 3.01006i
−3.90102 + 0.884346i
−3.90102 0.884346i
0.593959 3.95566i
0.593959 + 3.95566i
−3.80026 1.24819i
−3.80026 + 1.24819i
−2.80358 2.85306i
−2.80358 + 2.85306i
3.18502 2.41984i
3.18502 + 2.41984i
3.46430 1.99965i
3.46430 + 1.99965i
0.236693 3.99299i
0.236693 + 3.99299i
3.72553 + 1.45618i
3.72553 1.45618i
2.93366 + 2.71913i
2.93366 2.71913i
−5.64436 0.375761i 6.67450i 31.7176 + 4.24186i 25.0000i 2.50801 37.6733i −38.2812 −177.432 35.8608i 198.451 −9.39401 + 141.109i
21.2 −5.64436 + 0.375761i 6.67450i 31.7176 4.24186i 25.0000i 2.50801 + 37.6733i −38.2812 −177.432 + 35.8608i 198.451 −9.39401 141.109i
21.3 −4.78536 3.01667i 25.4343i 13.7994 + 28.8717i 25.0000i 76.7270 121.712i −56.4938 21.0614 179.790i −403.904 75.4168 119.634i
21.4 −4.78536 + 3.01667i 25.4343i 13.7994 28.8717i 25.0000i 76.7270 + 121.712i −56.4938 21.0614 + 179.790i −403.904 75.4168 + 119.634i
21.5 −3.36170 4.54962i 6.93089i −9.39799 + 30.5888i 25.0000i 31.5329 23.2996i 47.1406 170.761 60.0732i 194.963 −113.740 + 84.0424i
21.6 −3.36170 + 4.54962i 6.93089i −9.39799 30.5888i 25.0000i 31.5329 + 23.2996i 47.1406 170.761 + 60.0732i 194.963 −113.740 84.0424i
21.7 −2.55207 5.04846i 11.5927i −18.9739 + 25.7680i 25.0000i −58.5253 + 29.5854i −231.529 178.512 + 30.0270i 108.609 126.211 63.8017i
21.8 −2.55207 + 5.04846i 11.5927i −18.9739 25.7680i 25.0000i −58.5253 29.5854i −231.529 178.512 30.0270i 108.609 126.211 + 63.8017i
21.9 0.0494789 5.65664i 10.7455i −31.9951 0.559768i 25.0000i 60.7833 + 0.531674i 198.733 −4.74949 + 180.957i 127.535 141.416 + 1.23697i
21.10 0.0494789 + 5.65664i 10.7455i −31.9951 + 0.559768i 25.0000i 60.7833 0.531674i 198.733 −4.74949 180.957i 127.535 141.416 1.23697i
21.11 0.765181 5.60486i 17.3148i −30.8290 8.57748i 25.0000i −97.0471 13.2490i −9.19080 −71.6654 + 166.229i −56.8021 −140.122 19.1295i
21.12 0.765181 + 5.60486i 17.3148i −30.8290 + 8.57748i 25.0000i −97.0471 + 13.2490i −9.19080 −71.6654 166.229i −56.8021 −140.122 + 19.1295i
21.13 1.46465 5.46395i 29.2080i −27.7096 16.0056i 25.0000i 159.591 + 42.7797i −168.173 −128.039 + 127.961i −610.110 −136.599 36.6164i
21.14 1.46465 + 5.46395i 29.2080i −27.7096 + 16.0056i 25.0000i 159.591 42.7797i −168.173 −128.039 127.961i −610.110 −136.599 + 36.6164i
21.15 4.22968 3.75630i 25.0521i 3.78045 31.7759i 25.0000i −94.1031 105.962i 103.624 −103.370 148.603i −384.607 93.9075 + 105.742i
21.16 4.22968 + 3.75630i 25.0521i 3.78045 + 31.7759i 25.0000i −94.1031 + 105.962i 103.624 −103.370 + 148.603i −384.607 93.9075 105.742i
21.17 5.18171 2.26935i 10.8240i 21.7001 23.5182i 25.0000i 24.5634 + 56.0868i 163.706 59.0729 171.109i 125.841 −56.7336 129.543i
21.18 5.18171 + 2.26935i 10.8240i 21.7001 + 23.5182i 25.0000i 24.5634 56.0868i 163.706 59.0729 + 171.109i 125.841 −56.7336 + 129.543i
21.19 5.65278 0.214529i 18.7876i 31.9080 2.42537i 25.0000i −4.03048 106.202i −107.536 179.848 20.5552i −109.975 −5.36321 141.320i
21.20 5.65278 + 0.214529i 18.7876i 31.9080 + 2.42537i 25.0000i −4.03048 + 106.202i −107.536 179.848 + 20.5552i −109.975 −5.36321 + 141.320i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.6.d.a 20
3.b odd 2 1 360.6.k.b 20
4.b odd 2 1 160.6.d.a 20
5.b even 2 1 200.6.d.b 20
5.c odd 4 1 200.6.f.b 20
5.c odd 4 1 200.6.f.c 20
8.b even 2 1 inner 40.6.d.a 20
8.d odd 2 1 160.6.d.a 20
20.d odd 2 1 800.6.d.c 20
20.e even 4 1 800.6.f.b 20
20.e even 4 1 800.6.f.c 20
24.h odd 2 1 360.6.k.b 20
40.e odd 2 1 800.6.d.c 20
40.f even 2 1 200.6.d.b 20
40.i odd 4 1 200.6.f.b 20
40.i odd 4 1 200.6.f.c 20
40.k even 4 1 800.6.f.b 20
40.k even 4 1 800.6.f.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.d.a 20 1.a even 1 1 trivial
40.6.d.a 20 8.b even 2 1 inner
160.6.d.a 20 4.b odd 2 1
160.6.d.a 20 8.d odd 2 1
200.6.d.b 20 5.b even 2 1
200.6.d.b 20 40.f even 2 1
200.6.f.b 20 5.c odd 4 1
200.6.f.b 20 40.i odd 4 1
200.6.f.c 20 5.c odd 4 1
200.6.f.c 20 40.i odd 4 1
360.6.k.b 20 3.b odd 2 1
360.6.k.b 20 24.h odd 2 1
800.6.d.c 20 20.d odd 2 1
800.6.d.c 20 40.e odd 2 1
800.6.f.b 20 20.e even 4 1
800.6.f.b 20 40.k even 4 1
800.6.f.c 20 20.e even 4 1
800.6.f.c 20 40.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(40, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 2 T^{19} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{20} + 3240 T^{18} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( (T^{2} + 625)^{10} \) Copy content Toggle raw display
$7$ \( (T^{10} + 98 T^{9} + \cdots + 13\!\cdots\!08)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + 1711016 T^{18} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{20} + 3905528 T^{18} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{10} - 8750908 T^{8} + \cdots + 29\!\cdots\!68)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + 25883288 T^{18} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{10} + 2338 T^{9} + \cdots - 88\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + 195130080 T^{18} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} - 3580 T^{9} + \cdots - 42\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + 821506472 T^{18} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{10} - 5804 T^{9} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + 1732042120 T^{18} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( (T^{10} - 22090 T^{9} + \cdots + 16\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + 4360240504 T^{18} + \cdots + 82\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{20} + 4301536152 T^{18} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{20} + 9739073672 T^{18} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + 15173983176 T^{18} + \cdots + 50\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{10} + 100156 T^{9} + \cdots + 19\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 52568 T^{9} + \cdots - 24\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} - 141040 T^{9} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + 44012159480 T^{18} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{10} + 1580 T^{9} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} - 73688 T^{9} + \cdots + 22\!\cdots\!48)^{2} \) Copy content Toggle raw display
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