Properties

Label 41.8.a.b
Level $41$
Weight $8$
Character orbit 41.a
Self dual yes
Analytic conductor $12.808$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [41,8,Mod(1,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 41.a (trivial)

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: yes
Analytic conductor: \(12.8077860448\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 1463 x^{12} + 2401 x^{11} + 805624 x^{10} - 1589098 x^{9} - 207304606 x^{8} + 405007522 x^{7} + 25187266209 x^{6} + \cdots - 113773372735872 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + (\beta_{4} + 3) q^{3} + (\beta_{2} + \beta_1 + 82) q^{4} + (\beta_{7} + 2 \beta_{4} + \beta_{2} + 13) q^{5} + (\beta_{13} - \beta_{8} + 6 \beta_{4} - 3 \beta_1 + 36) q^{6} + ( - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} + 3 \beta_{4} - \beta_{2} + \cdots + 171) q^{7}+ \cdots + ( - 3 \beta_{13} + 3 \beta_{12} - 2 \beta_{11} + \beta_{10} + 4 \beta_{9} - \beta_{8} + \cdots + 867) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + (\beta_{4} + 3) q^{3} + (\beta_{2} + \beta_1 + 82) q^{4} + (\beta_{7} + 2 \beta_{4} + \beta_{2} + 13) q^{5} + (\beta_{13} - \beta_{8} + 6 \beta_{4} - 3 \beta_1 + 36) q^{6} + ( - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} + 3 \beta_{4} - \beta_{2} + \cdots + 171) q^{7}+ \cdots + (6354 \beta_{13} - 10008 \beta_{12} + 1822 \beta_{11} + \cdots - 124565) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 15 q^{2} + 40 q^{3} + 1151 q^{4} + 180 q^{5} + 486 q^{6} + 2400 q^{7} + 2145 q^{8} + 12198 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 15 q^{2} + 40 q^{3} + 1151 q^{4} + 180 q^{5} + 486 q^{6} + 2400 q^{7} + 2145 q^{8} + 12198 q^{9} + 542 q^{10} + 2048 q^{11} - 10130 q^{12} + 10800 q^{13} + 52472 q^{14} + 56896 q^{15} + 160083 q^{16} + 31820 q^{17} + 235075 q^{18} + 88808 q^{19} + 223710 q^{20} + 139008 q^{21} + 216590 q^{22} + 182280 q^{23} + 142618 q^{24} + 335394 q^{25} + 227316 q^{26} - 55520 q^{27} + 89040 q^{28} - 175984 q^{29} - 27324 q^{30} + 677152 q^{31} - 2215 q^{32} + 184840 q^{33} + 77354 q^{34} - 366336 q^{35} + 265587 q^{36} + 346980 q^{37} - 1907530 q^{38} + 478856 q^{39} - 711782 q^{40} - 964894 q^{41} - 4692940 q^{42} + 295200 q^{43} - 4124210 q^{44} - 3675460 q^{45} - 754176 q^{46} - 1392840 q^{47} - 8878890 q^{48} + 626398 q^{49} - 4364047 q^{50} - 197192 q^{51} + 921460 q^{52} - 1850560 q^{53} - 5498260 q^{54} + 408448 q^{55} + 5198648 q^{56} + 7867960 q^{57} - 5263160 q^{58} + 3372168 q^{59} - 2850740 q^{60} + 5479380 q^{61} - 4354720 q^{62} + 8386960 q^{63} + 13017523 q^{64} + 8483688 q^{65} - 6250512 q^{66} + 10917760 q^{67} - 2821990 q^{68} + 11637288 q^{69} + 2034328 q^{70} + 7769000 q^{71} + 17924885 q^{72} + 8689860 q^{73} - 862286 q^{74} + 9280928 q^{75} - 982434 q^{76} + 20593720 q^{77} - 1483160 q^{78} + 16493240 q^{79} - 4376442 q^{80} + 37048782 q^{81} - 1033815 q^{82} - 6140920 q^{83} - 13969980 q^{84} + 32466920 q^{85} - 1867244 q^{86} + 24094000 q^{87} + 9705610 q^{88} + 4370788 q^{89} - 30810194 q^{90} + 27603968 q^{91} + 21810960 q^{92} - 13863960 q^{93} - 47442312 q^{94} + 18025048 q^{95} - 64816854 q^{96} - 16968300 q^{97} - 77506525 q^{98} - 1593296 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 1463 x^{12} + 2401 x^{11} + 805624 x^{10} - 1589098 x^{9} - 207304606 x^{8} + 405007522 x^{7} + 25187266209 x^{6} + \cdots - 113773372735872 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 209 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 88\!\cdots\!37 \nu^{13} + \cdots + 38\!\cdots\!72 ) / 10\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 88\!\cdots\!01 \nu^{13} + \cdots - 27\!\cdots\!04 ) / 10\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45\!\cdots\!59 \nu^{13} + \cdots + 69\!\cdots\!16 ) / 10\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 46\!\cdots\!53 \nu^{13} + \cdots - 47\!\cdots\!80 ) / 10\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 32\!\cdots\!41 \nu^{13} + \cdots + 10\!\cdots\!88 ) / 54\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 43\!\cdots\!77 \nu^{13} + \cdots + 99\!\cdots\!60 ) / 54\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 45\!\cdots\!33 \nu^{13} + \cdots + 88\!\cdots\!84 ) / 54\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13\!\cdots\!56 \nu^{13} + \cdots + 18\!\cdots\!72 ) / 13\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!85 \nu^{13} + \cdots + 47\!\cdots\!60 ) / 10\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 35\!\cdots\!71 \nu^{13} + \cdots + 70\!\cdots\!68 ) / 27\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 36\!\cdots\!55 \nu^{13} + \cdots + 10\!\cdots\!16 ) / 27\!\cdots\!64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 209 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{13} - 3 \beta_{12} + 3 \beta_{11} + 2 \beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{7} + 2 \beta_{4} - \beta_{3} + 362 \beta _1 - 228 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 8 \beta_{13} + 16 \beta_{12} - 28 \beta_{11} - 28 \beta_{10} + 34 \beta_{9} + 6 \beta_{8} - 20 \beta_{7} - 22 \beta_{6} - 2 \beta_{5} - 202 \beta_{4} + 26 \beta_{3} + 445 \beta_{2} - 995 \beta _1 + 75313 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 639 \beta_{13} - 1977 \beta_{12} + 1901 \beta_{11} + 1398 \beta_{10} - 753 \beta_{9} + 1204 \beta_{8} - 203 \beta_{7} + 270 \beta_{6} - 162 \beta_{5} + 408 \beta_{4} - 801 \beta_{3} - 428 \beta_{2} + 149394 \beta _1 - 222172 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2790 \beta_{13} + 12006 \beta_{12} - 23734 \beta_{11} - 19676 \beta_{10} + 23842 \beta_{9} + 140 \beta_{8} - 14118 \beta_{7} - 17832 \beta_{6} + 1856 \beta_{5} - 149548 \beta_{4} + \cdots + 31003497 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 376225 \beta_{13} - 1036507 \beta_{12} + 1027731 \beta_{11} + 767714 \beta_{10} - 425341 \beta_{9} + 674870 \beta_{8} + 22903 \beta_{7} + 248340 \beta_{6} - 168604 \beta_{5} + \cdots - 156688996 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 121828 \beta_{13} + 7317092 \beta_{12} - 15153592 \beta_{11} - 11204956 \beta_{10} + 13160650 \beta_{9} - 2545214 \beta_{8} - 8216080 \beta_{7} - 11392874 \beta_{6} + \cdots + 13628240993 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 213765619 \beta_{13} - 506782941 \beta_{12} + 535361721 \beta_{11} + 394931422 \beta_{10} - 225027433 \beta_{9} + 373187432 \beta_{8} + 54332969 \beta_{7} + \cdots - 98369535628 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 943195982 \beta_{13} + 4210838482 \beta_{12} - 8833590666 \beta_{11} - 6057037260 \beta_{10} + 6790245442 \beta_{9} - 2801343192 \beta_{8} - 4449934650 \beta_{7} + \cdots + 6255731237689 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 118393862373 \beta_{13} - 242465171215 \beta_{12} + 276720184943 \beta_{11} + 199381660682 \beta_{10} - 117634793605 \beta_{9} + 205134577930 \beta_{8} + \cdots - 58330762637588 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 918662557432 \beta_{13} + 2364272985936 \beta_{12} - 4957011752140 \beta_{11} - 3229146440236 \beta_{10} + 3434680174698 \beta_{9} + \cdots + 29\!\cdots\!17 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 64532183621623 \beta_{13} - 115769780979121 \beta_{12} + 143290254985973 \beta_{11} + 100444030952998 \beta_{10} - 61597405538385 \beta_{9} + \cdots - 33\!\cdots\!20 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−23.1882
−20.5582
−16.6759
−12.9953
−7.31453
−3.83006
−3.36176
2.21303
6.69987
7.58629
15.3972
16.2287
20.1035
20.6954
−22.1882 −47.7619 364.315 −20.2326 1059.75 −1255.51 −5243.40 94.1971 448.925
1.2 −19.5582 25.9017 254.524 198.852 −506.592 1681.52 −2474.59 −1516.10 −3889.19
1.3 −15.6759 2.81296 117.733 529.264 −44.0956 −1075.25 160.949 −2179.09 −8296.67
1.4 −11.9953 −7.45913 15.8880 −439.756 89.4748 −467.875 1344.82 −2131.36 5275.02
1.5 −6.31453 84.6190 −88.1267 178.532 −534.329 401.625 1364.74 4973.38 −1127.34
1.6 −2.83006 44.8732 −119.991 −496.659 −126.994 1471.76 701.829 −173.395 1405.58
1.7 −2.36176 −33.5892 −122.422 −22.5446 79.3297 −918.563 591.438 −1058.77 53.2451
1.8 3.21303 −85.2828 −117.676 −486.874 −274.016 −165.944 −789.366 5086.16 −1564.34
1.9 7.69987 −55.0219 −68.7120 88.1001 −423.662 791.814 −1514.66 840.415 678.359
1.10 8.58629 46.4126 −54.2756 541.053 398.512 297.785 −1565.07 −32.8677 4645.64
1.11 16.3972 92.0824 140.867 −201.442 1509.89 37.9012 210.982 6292.18 −3303.07
1.12 17.2287 28.2812 168.828 92.8129 487.249 939.532 703.410 −1387.17 1599.05
1.13 21.1035 27.7605 317.358 223.327 585.845 −494.040 3996.11 −1416.35 4712.99
1.14 21.6954 −83.6288 342.691 −4.43317 −1814.36 1155.23 4657.81 4806.78 −96.1795
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(41\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.8.a.b 14
3.b odd 2 1 369.8.a.f 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.8.a.b 14 1.a even 1 1 trivial
369.8.a.f 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 15 T_{2}^{13} - 1359 T_{2}^{12} + 19515 T_{2}^{11} + 683942 T_{2}^{10} - 9194140 T_{2}^{9} - 157865704 T_{2}^{8} + 1911507840 T_{2}^{7} + 16847893504 T_{2}^{6} + \cdots - 94627126738944 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(41))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 15 T^{13} + \cdots - 94627126738944 \) Copy content Toggle raw display
$3$ \( T^{14} - 40 T^{13} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{14} - 180 T^{13} + \cdots - 80\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} - 2400 T^{13} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{14} - 2048 T^{13} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{14} - 10800 T^{13} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{14} - 31820 T^{13} + \cdots + 64\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{14} - 88808 T^{13} + \cdots - 39\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{14} - 182280 T^{13} + \cdots - 35\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{14} + 175984 T^{13} + \cdots - 83\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{14} - 677152 T^{13} + \cdots - 81\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{14} - 346980 T^{13} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T + 68921)^{14} \) Copy content Toggle raw display
$43$ \( T^{14} - 295200 T^{13} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{14} + 1392840 T^{13} + \cdots - 44\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{14} + 1850560 T^{13} + \cdots - 13\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{14} - 3372168 T^{13} + \cdots + 86\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{14} - 5479380 T^{13} + \cdots - 33\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{14} - 10917760 T^{13} + \cdots - 16\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{14} - 7769000 T^{13} + \cdots + 88\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{14} - 8689860 T^{13} + \cdots + 67\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{14} - 16493240 T^{13} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{14} + 6140920 T^{13} + \cdots - 31\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{14} - 4370788 T^{13} + \cdots + 43\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{14} + 16968300 T^{13} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
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