Properties

Label 14.14.a.c
Level $14$
Weight $14$
Character orbit 14.a
Self dual yes
Analytic conductor $15.012$
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] = [14,14,Mod(1,14)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(14, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("14.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 14.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: \(15.0123300533\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{100129}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 25032 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{100129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 64 q^{2} + ( - \beta + 476) q^{3} + 4096 q^{4} + ( - 63 \beta + 16002) q^{5} + (64 \beta - 30464) q^{6} + 117649 q^{7} - 262144 q^{8} + ( - 952 \beta - 967231) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 64 q^{2} + ( - \beta + 476) q^{3} + 4096 q^{4} + ( - 63 \beta + 16002) q^{5} + (64 \beta - 30464) q^{6} + 117649 q^{7} - 262144 q^{8} + ( - 952 \beta - 967231) q^{9} + (4032 \beta - 1024128) q^{10} + ( - 11088 \beta - 676368) q^{11} + ( - 4096 \beta + 1949696) q^{12} + (50895 \beta + 1755194) q^{13} - 7529536 q^{14} + ( - 45990 \beta + 32849460) q^{15} + 16777216 q^{16} + ( - 128286 \beta + 108855978) q^{17} + (60928 \beta + 61902784) q^{18} + (114129 \beta + 295667876) q^{19} + ( - 258048 \beta + 65544192) q^{20} + ( - 117649 \beta + 56000924) q^{21} + (709632 \beta + 43287552) q^{22} + (1099350 \beta + 420367500) q^{23} + (262144 \beta - 124780544) q^{24} + ( - 2016252 \beta + 625008883) q^{25} + ( - 3257280 \beta - 112332416) q^{26} + (2108402 \beta - 838008472) q^{27} + 481890304 q^{28} + (2389086 \beta - 243811770) q^{29} + (2943360 \beta - 2102365440) q^{30} + ( - 3506274 \beta + 1096538072) q^{31} - 1073741824 q^{32} + ( - 4601520 \beta + 4118970240) q^{33} + (8210304 \beta - 6966782592) q^{34} + ( - 7411887 \beta + 1882619298) q^{35} + ( - 3899392 \beta - 3961778176) q^{36} + (9857106 \beta + 202530134) q^{37} + ( - 7304256 \beta - 18922744064) q^{38} + (22470826 \beta - 19548789476) q^{39} + (16515072 \beta - 4194828288) q^{40} + ( - 51860862 \beta + 4259086314) q^{41} + (7529536 \beta - 3584059136) q^{42} + (36481536 \beta + 13112522648) q^{43} + ( - 45416448 \beta - 2770403328) q^{44} + (45701649 \beta + 8543717154) q^{45} + ( - 70358400 \beta - 26903520000) q^{46} + ( - 23291442 \beta + 77524424880) q^{47} + ( - 16777216 \beta + 7985954816) q^{48} + 13841287201 q^{49} + (129040128 \beta - 40000568512) q^{50} + ( - 169920114 \beta + 103196041104) q^{51} + (208465920 \beta + 7189274624) q^{52} + (339839892 \beta + 33003525246) q^{53} + ( - 134937728 \beta + 53632542208) q^{54} + ( - 134818992 \beta + 268954807968) q^{55} - 30840979456 q^{56} + ( - 241342472 \beta + 95027418412) q^{57} + ( - 152901504 \beta + 15603953280) q^{58} + ( - 225693891 \beta - 238181148492) q^{59} + ( - 188375040 \beta + 134551388160) q^{60} + (401532849 \beta + 98689425002) q^{61} + (224401536 \beta - 70178436608) q^{62} + ( - 112001848 \beta - 113793759919) q^{63} + 68719476736 q^{64} + (703844568 \beta - 1256121880272) q^{65} + (294497280 \beta - 263614095360) q^{66} + ( - 840313278 \beta - 859366429744) q^{67} + ( - 525459456 \beta + 445874085888) q^{68} + (102923100 \beta - 240212334600) q^{69} + (474360768 \beta - 120487635072) q^{70} + (1823602032 \beta - 347771739168) q^{71} + (249561088 \beta + 253553803264) q^{72} + ( - 1429760268 \beta - 233042619670) q^{73} + ( - 630854784 \beta - 12961928576) q^{74} + ( - 1584744835 \beta + 1105045414340) q^{75} + (467472384 \beta + 1211055620096) q^{76} + ( - 1304492112 \beta - 79574018832) q^{77} + ( - 1438132864 \beta + 1251122526464) q^{78} + (2547097812 \beta - 1216008287920) q^{79} + ( - 1056964608 \beta + 268469010432) q^{80} + (3359403320 \beta + 298737861509) q^{81} + (3319095168 \beta - 272581524096) q^{82} + (1600755741 \beta - 871992247308) q^{83} + ( - 481890304 \beta + 229379784704) q^{84} + ( - 8910759186 \beta + 4978890881244) q^{85} + ( - 2334818304 \beta - 839201449472) q^{86} + (1381016706 \beta - 1072921570896) q^{87} + (2906652672 \beta + 177305812992) q^{88} + (2399150268 \beta + 1511290120242) q^{89} + ( - 2924905536 \beta - 546797897856) q^{90} + (5987745855 \beta + 206496818906) q^{91} + (4502937600 \beta + 1721825280000) q^{92} + ( - 2765524496 \beta + 1926270959656) q^{93} + (1490652288 \beta - 4961563192320) q^{94} + ( - 16800783930 \beta + 1851516446220) q^{95} + (1073741824 \beta - 511101108224) q^{96} + (4519008846 \beta + 3880031330546) q^{97} - 885842380864 q^{98} + (11368559664 \beta + 4881961277424) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 128 q^{2} + 952 q^{3} + 8192 q^{4} + 32004 q^{5} - 60928 q^{6} + 235298 q^{7} - 524288 q^{8} - 1934462 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 128 q^{2} + 952 q^{3} + 8192 q^{4} + 32004 q^{5} - 60928 q^{6} + 235298 q^{7} - 524288 q^{8} - 1934462 q^{9} - 2048256 q^{10} - 1352736 q^{11} + 3899392 q^{12} + 3510388 q^{13} - 15059072 q^{14} + 65698920 q^{15} + 33554432 q^{16} + 217711956 q^{17} + 123805568 q^{18} + 591335752 q^{19} + 131088384 q^{20} + 112001848 q^{21} + 86575104 q^{22} + 840735000 q^{23} - 249561088 q^{24} + 1250017766 q^{25} - 224664832 q^{26} - 1676016944 q^{27} + 963780608 q^{28} - 487623540 q^{29} - 4204730880 q^{30} + 2193076144 q^{31} - 2147483648 q^{32} + 8237940480 q^{33} - 13933565184 q^{34} + 3765238596 q^{35} - 7923556352 q^{36} + 405060268 q^{37} - 37845488128 q^{38} - 39097578952 q^{39} - 8389656576 q^{40} + 8518172628 q^{41} - 7168118272 q^{42} + 26225045296 q^{43} - 5540806656 q^{44} + 17087434308 q^{45} - 53807040000 q^{46} + 155048849760 q^{47} + 15971909632 q^{48} + 27682574402 q^{49} - 80001137024 q^{50} + 206392082208 q^{51} + 14378549248 q^{52} + 66007050492 q^{53} + 107265084416 q^{54} + 537909615936 q^{55} - 61681958912 q^{56} + 190054836824 q^{57} + 31207906560 q^{58} - 476362296984 q^{59} + 269102776320 q^{60} + 197378850004 q^{61} - 140356873216 q^{62} - 227587519838 q^{63} + 137438953472 q^{64} - 2512243760544 q^{65} - 527228190720 q^{66} - 1718732859488 q^{67} + 891748171776 q^{68} - 480424669200 q^{69} - 240975270144 q^{70} - 695543478336 q^{71} + 507107606528 q^{72} - 466085239340 q^{73} - 25923857152 q^{74} + 2210090828680 q^{75} + 2422111240192 q^{76} - 159148037664 q^{77} + 2502245052928 q^{78} - 2432016575840 q^{79} + 536938020864 q^{80} + 597475723018 q^{81} - 545163048192 q^{82} - 1743984494616 q^{83} + 458759569408 q^{84} + 9957781762488 q^{85} - 1678402898944 q^{86} - 2145843141792 q^{87} + 354611625984 q^{88} + 3022580240484 q^{89} - 1093595795712 q^{90} + 412993637812 q^{91} + 3443650560000 q^{92} + 3852541919312 q^{93} - 9923126384640 q^{94} + 3703032892440 q^{95} - 1022202216448 q^{96} + 7760062661092 q^{97} - 1771684761728 q^{98} + 9763922554848 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
158.716
−157.716
−64.0000 −156.863 4096.00 −23868.4 10039.3 117649. −262144. −1.56972e6 1.52758e6
1.2 −64.0000 1108.86 4096.00 55872.4 −70967.3 117649. −262144. −364745. −3.57583e6
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)

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 14.14.a.c 2
3.b odd 2 1 126.14.a.l 2
4.b odd 2 1 112.14.a.d 2
7.b odd 2 1 98.14.a.e 2
7.c even 3 2 98.14.c.l 4
7.d odd 6 2 98.14.c.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.a.c 2 1.a even 1 1 trivial
98.14.a.e 2 7.b odd 2 1
98.14.c.l 4 7.c even 3 2
98.14.c.m 4 7.d odd 6 2
112.14.a.d 2 4.b odd 2 1
126.14.a.l 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 952T_{3} - 173940 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(14))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 952T - 173940 \) Copy content Toggle raw display
$5$ \( T^{2} - 32004 T - 1333584000 \) Copy content Toggle raw display
$7$ \( (T - 117649)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1352736 T - 48783462900480 \) Copy content Toggle raw display
$13$ \( T^{2} - 3510388 T - 10\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{2} - 217711956 T + 52\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( T^{2} - 591335752 T + 82\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{2} - 840735000 T - 30\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{2} + 487623540 T - 22\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{2} - 2193076144 T - 37\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( T^{2} - 405060268 T - 38\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{2} - 8518172628 T - 10\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{2} - 26225045296 T - 36\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{2} - 155048849760 T + 57\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{2} - 66007050492 T - 45\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{2} + 476362296984 T + 36\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{2} - 197378850004 T - 54\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{2} + 1718732859488 T + 45\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{2} + 695543478336 T - 12\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{2} + 466085239340 T - 76\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{2} + 2432016575840 T - 11\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{2} + 1743984494616 T - 26\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{2} - 3022580240484 T - 21\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{2} - 7760062661092 T + 68\!\cdots\!60 \) Copy content Toggle raw display
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