Properties

Label 5.10.a.b
Level $5$
Weight $10$
Character orbit 5.a
Self dual yes
Analytic conductor $2.575$
Analytic rank $0$
Dimension $2$
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] = [5,10,Mod(1,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 5.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: \(2.57517918082\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1009}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 252 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{1009}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 5) q^{2} + (2 \beta + 130) q^{3} + (10 \beta + 522) q^{4} + 625 q^{5} + ( - 140 \beta - 2668) q^{6} + (214 \beta + 850) q^{7} + ( - 60 \beta - 10140) q^{8} + (520 \beta + 1253) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 5) q^{2} + (2 \beta + 130) q^{3} + (10 \beta + 522) q^{4} + 625 q^{5} + ( - 140 \beta - 2668) q^{6} + (214 \beta + 850) q^{7} + ( - 60 \beta - 10140) q^{8} + (520 \beta + 1253) q^{9} + ( - 625 \beta - 3125) q^{10} + ( - 1900 \beta + 11992) q^{11} + (2344 \beta + 88040) q^{12} + (1352 \beta + 57510) q^{13} + ( - 1920 \beta - 220176) q^{14} + (1250 \beta + 81250) q^{15} + (5320 \beta - 156024) q^{16} + ( - 12856 \beta + 206410) q^{17} + ( - 3853 \beta - 530945) q^{18} + (2840 \beta - 148260) q^{19} + (6250 \beta + 326250) q^{20} + (29520 \beta + 542352) q^{21} + ( - 2492 \beta + 1857140) q^{22} + ( - 19398 \beta - 524610) q^{23} + ( - 28080 \beta - 1439280) q^{24} + 390625 q^{25} + ( - 64270 \beta - 1651718) q^{26} + (30740 \beta - 1346540) q^{27} + (120208 \beta + 2602960) q^{28} + (106960 \beta - 1833490) q^{29} + ( - 87500 \beta - 1667500) q^{30} + ( - 154700 \beta + 806572) q^{31} + (160144 \beta + 603920) q^{32} + ( - 223016 \beta - 2275240) q^{33} + ( - 142130 \beta + 11939654) q^{34} + (133750 \beta + 531250) q^{35} + (283970 \beta + 5900866) q^{36} + ( - 205296 \beta - 10560970) q^{37} + (134060 \beta - 2124260) q^{38} + (290780 \beta + 10204636) q^{39} + ( - 37500 \beta - 6337500) q^{40} + (155800 \beta - 13478638) q^{41} + ( - 689952 \beta - 32497440) q^{42} + ( - 25798 \beta + 26444850) q^{43} + ( - 871880 \beta - 12911176) q^{44} + (325000 \beta + 783125) q^{45} + (621600 \beta + 22195632) q^{46} + (523334 \beta + 29206090) q^{47} + (379552 \beta - 9547360) q^{48} + (363800 \beta + 6577057) q^{49} + ( - 390625 \beta - 1953125) q^{50} + ( - 1258460 \beta + 889892) q^{51} + (1280844 \beta + 43661900) q^{52} + ( - 1137448 \beta - 19517570) q^{53} + (1192840 \beta - 24283960) q^{54} + ( - 1187500 \beta + 7495000) q^{55} + ( - 2220960 \beta - 21574560) q^{56} + (72680 \beta - 13542680) q^{57} + (1298690 \beta - 98755190) q^{58} + (1544720 \beta - 27497780) q^{59} + (1465000 \beta + 55025000) q^{60} + (692000 \beta - 137289858) q^{61} + ( - 33072 \beta + 152059440) q^{62} + (710142 \beta + 113346570) q^{63} + ( - 4128480 \beta - 84720608) q^{64} + (845000 \beta + 35943750) q^{65} + (3390320 \beta + 236399344) q^{66} + ( - 2416706 \beta - 159290) q^{67} + ( - 4646732 \beta - 21971020) q^{68} + ( - 3570960 \beta - 107344464) q^{69} + ( - 1200000 \beta - 137610000) q^{70} + (6278500 \beta - 3565468) q^{71} + ( - 5347980 \beta - 44186220) q^{72} + (8830952 \beta + 60429090) q^{73} + (11587450 \beta + 259948514) q^{74} + (781250 \beta + 50781250) q^{75} + ( - 120 \beta - 48736120) q^{76} + (951288 \beta - 400066200) q^{77} + ( - 11658536 \beta - 344420200) q^{78} + ( - 18775640 \beta + 3438760) q^{79} + (3325000 \beta - 97515000) q^{80} + ( - 8932040 \beta - 137679679) q^{81} + (12699638 \beta - 89809010) q^{82} + (2748402 \beta + 701174370) q^{83} + (20832960 \beta + 580964544) q^{84} + ( - 8035000 \beta + 129006250) q^{85} + ( - 26315860 \beta - 106194068) q^{86} + (10237820 \beta - 22508420) q^{87} + (18546480 \beta - 6572880) q^{88} + (13381680 \beta + 415044330) q^{89} + ( - 2408125 \beta - 331840625) q^{90} + (13456340 \beta + 340815452) q^{91} + ( - 15371856 \beta - 469572240) q^{92} + ( - 18497856 \beta - 207330240) q^{93} + ( - 31822760 \beta - 674074456) q^{94} + (1775000 \beta - 92662500) q^{95} + (22026560 \beta + 401680192) q^{96} + ( - 2622216 \beta + 319197290) q^{97} + ( - 8396057 \beta - 399959485) q^{98} + (3855140 \beta - 981866024) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{2} + 260 q^{3} + 1044 q^{4} + 1250 q^{5} - 5336 q^{6} + 1700 q^{7} - 20280 q^{8} + 2506 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{2} + 260 q^{3} + 1044 q^{4} + 1250 q^{5} - 5336 q^{6} + 1700 q^{7} - 20280 q^{8} + 2506 q^{9} - 6250 q^{10} + 23984 q^{11} + 176080 q^{12} + 115020 q^{13} - 440352 q^{14} + 162500 q^{15} - 312048 q^{16} + 412820 q^{17} - 1061890 q^{18} - 296520 q^{19} + 652500 q^{20} + 1084704 q^{21} + 3714280 q^{22} - 1049220 q^{23} - 2878560 q^{24} + 781250 q^{25} - 3303436 q^{26} - 2693080 q^{27} + 5205920 q^{28} - 3666980 q^{29} - 3335000 q^{30} + 1613144 q^{31} + 1207840 q^{32} - 4550480 q^{33} + 23879308 q^{34} + 1062500 q^{35} + 11801732 q^{36} - 21121940 q^{37} - 4248520 q^{38} + 20409272 q^{39} - 12675000 q^{40} - 26957276 q^{41} - 64994880 q^{42} + 52889700 q^{43} - 25822352 q^{44} + 1566250 q^{45} + 44391264 q^{46} + 58412180 q^{47} - 19094720 q^{48} + 13154114 q^{49} - 3906250 q^{50} + 1779784 q^{51} + 87323800 q^{52} - 39035140 q^{53} - 48567920 q^{54} + 14990000 q^{55} - 43149120 q^{56} - 27085360 q^{57} - 197510380 q^{58} - 54995560 q^{59} + 110050000 q^{60} - 274579716 q^{61} + 304118880 q^{62} + 226693140 q^{63} - 169441216 q^{64} + 71887500 q^{65} + 472798688 q^{66} - 318580 q^{67} - 43942040 q^{68} - 214688928 q^{69} - 275220000 q^{70} - 7130936 q^{71} - 88372440 q^{72} + 120858180 q^{73} + 519897028 q^{74} + 101562500 q^{75} - 97472240 q^{76} - 800132400 q^{77} - 688840400 q^{78} + 6877520 q^{79} - 195030000 q^{80} - 275359358 q^{81} - 179618020 q^{82} + 1402348740 q^{83} + 1161929088 q^{84} + 258012500 q^{85} - 212388136 q^{86} - 45016840 q^{87} - 13145760 q^{88} + 830088660 q^{89} - 663681250 q^{90} + 681630904 q^{91} - 939144480 q^{92} - 414660480 q^{93} - 1348148912 q^{94} - 185325000 q^{95} + 803360384 q^{96} + 638394580 q^{97} - 799918970 q^{98} - 1963732048 q^{99}+O(q^{100}) \) 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
16.3824
−15.3824
−36.7648 193.530 839.648 625.000 −7115.07 7647.66 −12045.9 17770.7 −22978.0
1.2 26.7648 66.4705 204.352 625.000 1779.07 −5947.66 −8234.11 −15264.7 16728.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)

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 5.10.a.b 2
3.b odd 2 1 45.10.a.f 2
4.b odd 2 1 80.10.a.f 2
5.b even 2 1 25.10.a.b 2
5.c odd 4 2 25.10.b.b 4
7.b odd 2 1 245.10.a.d 2
8.b even 2 1 320.10.a.k 2
8.d odd 2 1 320.10.a.s 2
15.d odd 2 1 225.10.a.h 2
15.e even 4 2 225.10.b.h 4
20.d odd 2 1 400.10.a.t 2
20.e even 4 2 400.10.c.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.a.b 2 1.a even 1 1 trivial
25.10.a.b 2 5.b even 2 1
25.10.b.b 4 5.c odd 4 2
45.10.a.f 2 3.b odd 2 1
80.10.a.f 2 4.b odd 2 1
225.10.a.h 2 15.d odd 2 1
225.10.b.h 4 15.e even 4 2
245.10.a.d 2 7.b odd 2 1
320.10.a.k 2 8.b even 2 1
320.10.a.s 2 8.d odd 2 1
400.10.a.t 2 20.d odd 2 1
400.10.c.p 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 10T_{2} - 984 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 10T - 984 \) Copy content Toggle raw display
$3$ \( T^{2} - 260T + 12864 \) Copy content Toggle raw display
$5$ \( (T - 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 1700 T - 45485664 \) Copy content Toggle raw display
$11$ \( T^{2} - 23984 T - 3498681936 \) Copy content Toggle raw display
$13$ \( T^{2} - 115020 T + 1463044964 \) Copy content Toggle raw display
$17$ \( T^{2} - 412820 T - 124159138524 \) Copy content Toggle raw display
$19$ \( T^{2} + 296520 T + 13842837200 \) Copy content Toggle raw display
$23$ \( T^{2} + 1049220 T - 104453293536 \) Copy content Toggle raw display
$29$ \( T^{2} + 3666980 T - 8181719994300 \) Copy content Toggle raw display
$31$ \( T^{2} - 1613144 T - 23496920418816 \) Copy content Toggle raw display
$37$ \( T^{2} + 21121940 T + 69008321696356 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 157181579575044 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 698658564887264 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 576652311252096 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 924496505573436 \) Copy content Toggle raw display
$59$ \( T^{2} + 54995560 T - 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + 274579716 T + 18\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{2} + 318580 T - 58\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{2} + 7130936 T - 39\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{2} - 120858180 T - 75\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{2} - 6877520 T - 35\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} - 1402348740 T + 48\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{2} - 830088660 T - 84\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} - 638394580 T + 94\!\cdots\!96 \) Copy content Toggle raw display
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