Properties

Label 14.14.a.d
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: \(\Q(\sqrt{78985}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 19746 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \sqrt{78985}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 64 q^{2} + ( - 5 \beta + 553) q^{3} + 4096 q^{4} + ( - 39 \beta + 37765) q^{5} + ( - 320 \beta + 35392) q^{6} - 117649 q^{7} + 262144 q^{8} + ( - 5530 \beta + 686111) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 64 q^{2} + ( - 5 \beta + 553) q^{3} + 4096 q^{4} + ( - 39 \beta + 37765) q^{5} + ( - 320 \beta + 35392) q^{6} - 117649 q^{7} + 262144 q^{8} + ( - 5530 \beta + 686111) q^{9} + ( - 2496 \beta + 2416960) q^{10} + (34650 \beta + 1667930) q^{11} + ( - 20480 \beta + 2265088) q^{12} + (62475 \beta + 3999051) q^{13} - 7529536 q^{14} + ( - 210392 \beta + 36286120) q^{15} + 16777216 q^{16} + (358290 \beta + 19512416) q^{17} + ( - 353920 \beta + 43911104) q^{18} + ( - 543855 \beta + 62046467) q^{19} + ( - 159744 \beta + 154685440) q^{20} + (588245 \beta - 65059897) q^{21} + (2217600 \beta + 106747520) q^{22} + (448560 \beta + 950408000) q^{23} + ( - 1310720 \beta + 144965632) q^{24} + ( - 2945670 \beta + 325628285) q^{25} + (3998400 \beta + 255939264) q^{26} + (1482970 \beta + 1681694014) q^{27} - 481890304 q^{28} + ( - 10680810 \beta - 122567576) q^{29} + ( - 13465088 \beta + 2322311680) q^{30} + (9271530 \beta - 5505280074) q^{31} + 1073741824 q^{32} + (10821800 \beta - 12761785960) q^{33} + (22930560 \beta + 1248794624) q^{34} + (4588311 \beta - 4443014485) q^{35} + ( - 22650880 \beta + 2810310656) q^{36} + (36470910 \beta - 19815245608) q^{37} + ( - 34806720 \beta + 3970973888) q^{38} + (14553420 \beta - 22461464172) q^{39} + ( - 10223616 \beta + 9899868160) q^{40} + ( - 8828610 \beta - 1215513684) q^{41} + (37647680 \beta - 4163833408) q^{42} + ( - 177554370 \beta + 8911569158) q^{43} + (141926400 \beta + 6831841280) q^{44} + ( - 235598779 \beta + 42945676865) q^{45} + (28707840 \beta + 60826112000) q^{46} + (349815630 \beta + 32244155538) q^{47} + ( - 83886080 \beta + 9277800448) q^{48} + 13841287201 q^{49} + ( - 188522880 \beta + 20840210240) q^{50} + (100572290 \beta - 130707312202) q^{51} + (255897600 \beta + 16380112896) q^{52} + ( - 317904720 \beta - 63252088314) q^{53} + (94910080 \beta + 107628416896) q^{54} + (1243507980 \beta - 43747003300) q^{55} - 30840979456 q^{56} + ( - 610984150 \beta + 249093632126) q^{57} + ( - 683571840 \beta - 7844324864) q^{58} + ( - 896167935 \beta + 170629980619) q^{59} + ( - 861765632 \beta + 148627947520) q^{60} + ( - 616179855 \beta + 223620350373) q^{61} + (593377920 \beta - 352337924736) q^{62} + (650598970 \beta - 80720273039) q^{63} + 68719476736 q^{64} + (2203405386 \beta - 41424766110) q^{65} + (692595200 \beta - 816754301440) q^{66} + (846793920 \beta - 1035661145444) q^{67} + (1467555840 \beta + 79922855936) q^{68} + ( - 4503986320 \beta + 348428066000) q^{69} + (293651904 \beta - 284352927040) q^{70} + (1552163340 \beta + 325217232860) q^{71} + ( - 1449656320 \beta + 179859881984) q^{72} + (2850319740 \beta - 724904665058) q^{73} + (2334138240 \beta - 1268175718912) q^{74} + ( - 3257096935 \beta + 1343391166355) q^{75} + ( - 2227630080 \beta + 254142328832) q^{76} + ( - 4076537850 \beta - 196230296570) q^{77} + (931418880 \beta - 1437533707008) q^{78} + ( - 5153135820 \beta + 762576198828) q^{79} + ( - 654311424 \beta + 633591562240) q^{80} + (1228218530 \beta - 749567685361) q^{81} + ( - 565031040 \beta - 77792875776) q^{82} + (10286107965 \beta + 2128758819719) q^{83} + (2409451520 \beta - 266485338112) q^{84} + (12769837626 \beta - 366795500110) q^{85} + ( - 11363479680 \beta + 570340426112) q^{86} + ( - 5293650050 \beta + 4150339019722) q^{87} + (9083289600 \beta + 437237841920) q^{88} + ( - 2018475960 \beta + 6296825611314) q^{89} + ( - 15078321856 \beta + 2748523319360) q^{90} + ( - 7350121275 \beta - 470484351099) q^{91} + (1837301760 \beta + 3892871168000) q^{92} + (32653556460 \beta - 6705978866172) q^{93} + (22388200320 \beta + 2063625954432) q^{94} + ( - 22958496288 \beta + 4018483926080) q^{95} + ( - 5368709120 \beta + 593779228672) q^{96} + ( - 22566578790 \beta - 8270785003880) q^{97} + 885842380864 q^{98} + (14550093250 \beta - 13990286162270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 128 q^{2} + 1106 q^{3} + 8192 q^{4} + 75530 q^{5} + 70784 q^{6} - 235298 q^{7} + 524288 q^{8} + 1372222 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 128 q^{2} + 1106 q^{3} + 8192 q^{4} + 75530 q^{5} + 70784 q^{6} - 235298 q^{7} + 524288 q^{8} + 1372222 q^{9} + 4833920 q^{10} + 3335860 q^{11} + 4530176 q^{12} + 7998102 q^{13} - 15059072 q^{14} + 72572240 q^{15} + 33554432 q^{16} + 39024832 q^{17} + 87822208 q^{18} + 124092934 q^{19} + 309370880 q^{20} - 130119794 q^{21} + 213495040 q^{22} + 1900816000 q^{23} + 289931264 q^{24} + 651256570 q^{25} + 511878528 q^{26} + 3363388028 q^{27} - 963780608 q^{28} - 245135152 q^{29} + 4644623360 q^{30} - 11010560148 q^{31} + 2147483648 q^{32} - 25523571920 q^{33} + 2497589248 q^{34} - 8886028970 q^{35} + 5620621312 q^{36} - 39630491216 q^{37} + 7941947776 q^{38} - 44922928344 q^{39} + 19799736320 q^{40} - 2431027368 q^{41} - 8327666816 q^{42} + 17823138316 q^{43} + 13663682560 q^{44} + 85891353730 q^{45} + 121652224000 q^{46} + 64488311076 q^{47} + 18555600896 q^{48} + 27682574402 q^{49} + 41680420480 q^{50} - 261414624404 q^{51} + 32760225792 q^{52} - 126504176628 q^{53} + 215256833792 q^{54} - 87494006600 q^{55} - 61681958912 q^{56} + 498187264252 q^{57} - 15688649728 q^{58} + 341259961238 q^{59} + 297255895040 q^{60} + 447240700746 q^{61} - 704675849472 q^{62} - 161440546078 q^{63} + 137438953472 q^{64} - 82849532220 q^{65} - 1633508602880 q^{66} - 2071322290888 q^{67} + 159845711872 q^{68} + 696856132000 q^{69} - 568705854080 q^{70} + 650434465720 q^{71} + 359719763968 q^{72} - 1449809330116 q^{73} - 2536351437824 q^{74} + 2686782332710 q^{75} + 508284657664 q^{76} - 392460593140 q^{77} - 2875067414016 q^{78} + 1525152397656 q^{79} + 1267183124480 q^{80} - 1499135370722 q^{81} - 155585751552 q^{82} + 4257517639438 q^{83} - 532970676224 q^{84} - 733591000220 q^{85} + 1140680852224 q^{86} + 8300678039444 q^{87} + 874475683840 q^{88} + 12593651222628 q^{89} + 5497046638720 q^{90} - 940968702198 q^{91} + 7785742336000 q^{92} - 13411957732344 q^{93} + 4127251908864 q^{94} + 8036967852160 q^{95} + 1187558457344 q^{96} - 16541570007760 q^{97} + 1771684761728 q^{98} - 27980572324540 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
141.021
−140.021
64.0000 −852.214 4096.00 26804.3 −54541.7 −117649. 262144. −868055. 1.71548e6
1.2 64.0000 1958.21 4096.00 48725.7 125326. −117649. 262144. 2.24028e6 3.11844e6
\(n\): 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.d 2
3.b odd 2 1 126.14.a.g 2
4.b odd 2 1 112.14.a.c 2
7.b odd 2 1 98.14.a.f 2
7.c even 3 2 98.14.c.i 4
7.d odd 6 2 98.14.c.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.a.d 2 1.a even 1 1 trivial
98.14.a.f 2 7.b odd 2 1
98.14.c.i 4 7.c even 3 2
98.14.c.k 4 7.d odd 6 2
112.14.a.c 2 4.b odd 2 1
126.14.a.g 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 1106T_{3} - 1668816 \) 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
$3$ \( T^{2} - 1106 T - 1668816 \) Copy content Toggle raw display
$5$ \( T^{2} - 75530 T + 1306059040 \) Copy content Toggle raw display
$7$ \( (T + 117649)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 3335860 T - 92049177677600 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 292295968590024 \) Copy content Toggle raw display
$17$ \( T^{2} - 39024832 T - 97\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{2} - 124092934 T - 19\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{2} - 1900816000 T + 88\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{2} + 245135152 T - 89\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{2} + 11010560148 T + 23\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{2} + 39630491216 T + 28\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{2} + 2431027368 T - 46\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{2} - 17823138316 T - 24\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{2} - 64488311076 T - 86\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{2} + 126504176628 T - 39\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{2} - 341259961238 T - 34\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{2} - 447240700746 T + 20\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{2} + 2071322290888 T + 10\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{2} - 650434465720 T - 84\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{2} + 1449809330116 T - 11\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{2} - 1525152397656 T - 15\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{2} - 4257517639438 T - 38\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{2} - 12593651222628 T + 39\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{2} + 16541570007760 T + 28\!\cdots\!00 \) Copy content Toggle raw display
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