Properties

Label 10.11.c.a
Level $10$
Weight $11$
Character orbit 10.c
Analytic conductor $6.354$
Analytic rank $0$
Dimension $2$
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] = [10,11,Mod(3,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.3");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\), degree \(2\), 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: \(6.35357252674\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (16 i + 16) q^{2} + (183 i - 183) q^{3} + 512 i q^{4} + ( - 2500 i - 1875) q^{5} - 5856 q^{6} + ( - 8407 i - 8407) q^{7} + (8192 i - 8192) q^{8} - 7929 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (16 i + 16) q^{2} + (183 i - 183) q^{3} + 512 i q^{4} + ( - 2500 i - 1875) q^{5} - 5856 q^{6} + ( - 8407 i - 8407) q^{7} + (8192 i - 8192) q^{8} - 7929 i q^{9} + ( - 70000 i + 10000) q^{10} - 173398 q^{11} + ( - 93696 i - 93696) q^{12} + (232623 i - 232623) q^{13} - 269024 i q^{14} + (114375 i + 800625) q^{15} - 262144 q^{16} + (1880033 i + 1880033) q^{17} + ( - 126864 i + 126864) q^{18} - 1101560 i q^{19} + ( - 960000 i + 1280000) q^{20} + 3076962 q^{21} + ( - 2774368 i - 2774368) q^{22} + (5228263 i - 5228263) q^{23} - 2998272 i q^{24} + (9375000 i - 2734375) q^{25} - 7443936 q^{26} + ( - 9354960 i - 9354960) q^{27} + ( - 4304384 i + 4304384) q^{28} - 24790840 i q^{29} + (14640000 i + 10980000) q^{30} - 10065998 q^{31} + ( - 4194304 i - 4194304) q^{32} + ( - 31731834 i + 31731834) q^{33} + 60161056 i q^{34} + (36780625 i - 5254375) q^{35} + 4059648 q^{36} + (56387913 i + 56387913) q^{37} + ( - 17624960 i + 17624960) q^{38} - 85140018 i q^{39} + (5120000 i + 35840000) q^{40} - 153003598 q^{41} + (49231392 i + 49231392) q^{42} + ( - 59372457 i + 59372457) q^{43} - 88779776 i q^{44} + (14866875 i - 19822500) q^{45} - 167304416 q^{46} + (172339353 i + 172339353) q^{47} + ( - 47972352 i + 47972352) q^{48} - 141119951 i q^{49} + (106250000 i - 193750000) q^{50} - 688092078 q^{51} + ( - 119102976 i - 119102976) q^{52} + ( - 196385617 i + 196385617) q^{53} - 299358720 i q^{54} + (433495000 i + 325121250) q^{55} + 137740288 q^{56} + (201585480 i + 201585480) q^{57} + ( - 396653440 i + 396653440) q^{58} + 694068920 i q^{59} + (409920000 i - 58560000) q^{60} + 906185802 q^{61} + ( - 161055968 i - 161055968) q^{62} + (66659103 i - 66659103) q^{63} - 134217728 i q^{64} + (145389375 i + 1017725625) q^{65} + 1015418688 q^{66} + ( - 962073967 i - 962073967) q^{67} + (962576896 i - 962576896) q^{68} - 1913544258 i q^{69} + (504420000 i - 672560000) q^{70} - 3120877598 q^{71} + (64954368 i + 64954368) q^{72} + (636339263 i - 636339263) q^{73} + 1804413216 i q^{74} + ( - 2216015625 i - 1215234375) q^{75} + 563998720 q^{76} + (1457756986 i + 1457756986) q^{77} + ( - 1362240288 i + 1362240288) q^{78} - 1967996640 i q^{79} + (655360000 i + 491520000) q^{80} + 3892114881 q^{81} + ( - 2448057568 i - 2448057568) q^{82} + (5183822103 i - 5183822103) q^{83} + 1575404544 i q^{84} + ( - 8225144375 i + 1175020625) q^{85} + 1899918624 q^{86} + (4536723720 i + 4536723720) q^{87} + ( - 1420476416 i + 1420476416) q^{88} + 7771383280 i q^{89} + ( - 79290000 i - 555030000) q^{90} + 3911323122 q^{91} + ( - 2676870656 i - 2676870656) q^{92} + ( - 1842077634 i + 1842077634) q^{93} + 5514859296 i q^{94} + (2065425000 i - 2753900000) q^{95} + 1535115264 q^{96} + ( - 640361247 i - 640361247) q^{97} + ( - 2257919216 i + 2257919216) q^{98} + 1374872742 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32 q^{2} - 366 q^{3} - 3750 q^{5} - 11712 q^{6} - 16814 q^{7} - 16384 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 32 q^{2} - 366 q^{3} - 3750 q^{5} - 11712 q^{6} - 16814 q^{7} - 16384 q^{8} + 20000 q^{10} - 346796 q^{11} - 187392 q^{12} - 465246 q^{13} + 1601250 q^{15} - 524288 q^{16} + 3760066 q^{17} + 253728 q^{18} + 2560000 q^{20} + 6153924 q^{21} - 5548736 q^{22} - 10456526 q^{23} - 5468750 q^{25} - 14887872 q^{26} - 18709920 q^{27} + 8608768 q^{28} + 21960000 q^{30} - 20131996 q^{31} - 8388608 q^{32} + 63463668 q^{33} - 10508750 q^{35} + 8119296 q^{36} + 112775826 q^{37} + 35249920 q^{38} + 71680000 q^{40} - 306007196 q^{41} + 98462784 q^{42} + 118744914 q^{43} - 39645000 q^{45} - 334608832 q^{46} + 344678706 q^{47} + 95944704 q^{48} - 387500000 q^{50} - 1376184156 q^{51} - 238205952 q^{52} + 392771234 q^{53} + 650242500 q^{55} + 275480576 q^{56} + 403170960 q^{57} + 793306880 q^{58} - 117120000 q^{60} + 1812371604 q^{61} - 322111936 q^{62} - 133318206 q^{63} + 2035451250 q^{65} + 2030837376 q^{66} - 1924147934 q^{67} - 1925153792 q^{68} - 1345120000 q^{70} - 6241755196 q^{71} + 129908736 q^{72} - 1272678526 q^{73} - 2430468750 q^{75} + 1127997440 q^{76} + 2915513972 q^{77} + 2724480576 q^{78} + 983040000 q^{80} + 7784229762 q^{81} - 4896115136 q^{82} - 10367644206 q^{83} + 2350041250 q^{85} + 3799837248 q^{86} + 9073447440 q^{87} + 2840952832 q^{88} - 1110060000 q^{90} + 7822646244 q^{91} - 5353741312 q^{92} + 3684155268 q^{93} - 5507800000 q^{95} + 3070230528 q^{96} - 1280722494 q^{97} + 4515838432 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.00000i
1.00000i
16.0000 16.0000i −183.000 183.000i 512.000i −1875.00 + 2500.00i −5856.00 −8407.00 + 8407.00i −8192.00 8192.00i 7929.00i 10000.0 + 70000.0i
7.1 16.0000 + 16.0000i −183.000 + 183.000i 512.000i −1875.00 2500.00i −5856.00 −8407.00 8407.00i −8192.00 + 8192.00i 7929.00i 10000.0 70000.0i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.11.c.a 2
3.b odd 2 1 90.11.g.b 2
4.b odd 2 1 80.11.p.b 2
5.b even 2 1 50.11.c.c 2
5.c odd 4 1 inner 10.11.c.a 2
5.c odd 4 1 50.11.c.c 2
15.e even 4 1 90.11.g.b 2
20.e even 4 1 80.11.p.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.11.c.a 2 1.a even 1 1 trivial
10.11.c.a 2 5.c odd 4 1 inner
50.11.c.c 2 5.b even 2 1
50.11.c.c 2 5.c odd 4 1
80.11.p.b 2 4.b odd 2 1
80.11.p.b 2 20.e even 4 1
90.11.g.b 2 3.b odd 2 1
90.11.g.b 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 366T_{3} + 66978 \) acting on \(S_{11}^{\mathrm{new}}(10, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 32T + 512 \) Copy content Toggle raw display
$3$ \( T^{2} + 366T + 66978 \) Copy content Toggle raw display
$5$ \( T^{2} + 3750 T + 9765625 \) Copy content Toggle raw display
$7$ \( T^{2} + 16814 T + 141355298 \) Copy content Toggle raw display
$11$ \( (T + 173398)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 465246 T + 108226920258 \) Copy content Toggle raw display
$17$ \( T^{2} - 3760066 T + 7069048162178 \) Copy content Toggle raw display
$19$ \( T^{2} + 1213434433600 \) Copy content Toggle raw display
$23$ \( T^{2} + 10456526 T + 54669467994338 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 614585747905600 \) Copy content Toggle raw display
$31$ \( (T + 10065998)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 112775826 T + 63\!\cdots\!38 \) Copy content Toggle raw display
$41$ \( (T + 153003598)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 118744914 T + 70\!\cdots\!98 \) Copy content Toggle raw display
$47$ \( T^{2} - 344678706 T + 59\!\cdots\!18 \) Copy content Toggle raw display
$53$ \( T^{2} - 392771234 T + 77\!\cdots\!78 \) Copy content Toggle raw display
$59$ \( T^{2} + 48\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T - 906185802)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1924147934 T + 18\!\cdots\!78 \) Copy content Toggle raw display
$71$ \( (T + 3120877598)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1272678526 T + 80\!\cdots\!38 \) Copy content Toggle raw display
$79$ \( T^{2} + 38\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + 10367644206 T + 53\!\cdots\!18 \) Copy content Toggle raw display
$89$ \( T^{2} + 60\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + 1280722494 T + 82\!\cdots\!18 \) Copy content Toggle raw display
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