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Properties

Label 103.14.a.a
Level $103$
Weight $14$
Character orbit 103.a
Self dual yes
Analytic conductor $110.448$
Analytic rank $1$
Dimension $53$
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] = [103,14,Mod(1,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, 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("103.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 103.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: \(110.447856820\)
Analytic rank: \(1\)
Dimension: \(53\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 53 q - 321 q^{2} - 2318 q^{3} + 200703 q^{4} - 88497 q^{5} - 264833 q^{6} - 350426 q^{7} - 3401916 q^{8} + 22892547 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 53 q - 321 q^{2} - 2318 q^{3} + 200703 q^{4} - 88497 q^{5} - 264833 q^{6} - 350426 q^{7} - 3401916 q^{8} + 22892547 q^{9} - 3227122 q^{10} - 16678503 q^{11} - 34851657 q^{12} - 17999289 q^{13} - 71140889 q^{14} - 95075573 q^{15} + 756395899 q^{16} - 561559595 q^{17} - 450456648 q^{18} - 26823607 q^{19} - 948357411 q^{20} - 192180398 q^{21} - 1119874847 q^{22} - 2000759560 q^{23} - 3229175502 q^{24} + 9307943754 q^{25} - 9943699890 q^{26} - 7958498246 q^{27} + 4170944787 q^{28} - 7527275068 q^{29} + 28888950878 q^{30} - 2856027504 q^{31} - 31664451103 q^{32} - 30745866021 q^{33} - 17858429618 q^{34} - 63960624357 q^{35} - 19922196044 q^{36} - 71135747310 q^{37} - 105795449475 q^{38} - 121690258597 q^{39} - 240488695117 q^{40} - 105160091282 q^{41} - 373894271377 q^{42} - 116931565384 q^{43} - 267365145170 q^{44} - 147064333536 q^{45} - 43731100858 q^{46} - 189013497571 q^{47} - 393889599621 q^{48} + 763458246039 q^{49} - 94909139174 q^{50} + 99473859537 q^{51} + 983993822919 q^{52} - 857201119575 q^{53} + 1728463920457 q^{54} + 789423851330 q^{55} + 632328778762 q^{56} - 785061637965 q^{57} + 868941795890 q^{58} + 257600097691 q^{59} + 2464400330567 q^{60} - 29066976327 q^{61} - 353172126115 q^{62} - 345337563532 q^{63} + 7821711348238 q^{64} - 2231140184791 q^{65} + 4988814967767 q^{66} + 308819744774 q^{67} - 575435238289 q^{68} + 462369648072 q^{69} + 1835621544436 q^{70} - 755617715837 q^{71} + 1464796084386 q^{72} - 4820173701945 q^{73} - 566353238439 q^{74} + 3980536435815 q^{75} + 5402860060260 q^{76} - 9532058413709 q^{77} - 1269921117316 q^{78} + 2498727057607 q^{79} - 8308027600258 q^{80} + 3062386299141 q^{81} - 4534052089028 q^{82} - 6614317794937 q^{83} - 10962728969337 q^{84} - 15359510487649 q^{85} - 19416717040457 q^{86} - 21012946484858 q^{87} - 32857588795775 q^{88} - 14627690279236 q^{89} - 34910566252334 q^{90} - 21245054069925 q^{91} - 35056167069072 q^{92} - 17783414024208 q^{93} - 40916611134680 q^{94} - 47164907479144 q^{95} - 100371077543243 q^{96} - 27010762673216 q^{97} - 42934174492250 q^{98} - 45249710368086 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −177.482 −406.181 23307.8 10690.8 72089.8 351896. −2.68279e6 −1.42934e6 −1.89743e6
1.2 −175.366 1176.95 22561.3 −43556.0 −206397. 443815. −2.51988e6 −209114. 7.63825e6
1.3 −170.006 −447.421 20710.0 −16914.3 76064.2 −488845. −2.12813e6 −1.39414e6 2.87553e6
1.4 −168.975 1182.87 20360.5 53368.0 −199875. −210653. −2.05616e6 −195139. −9.01785e6
1.5 −161.252 844.322 17810.3 37522.9 −136149. −224029. −1.55098e6 −881443. −6.05065e6
1.6 −159.873 −2380.81 17367.4 19284.3 380627. −355481. −1.46689e6 4.07393e6 −3.08304e6
1.7 −139.082 −433.200 11151.9 −43435.0 60250.5 396339. −411671. −1.40666e6 6.04104e6
1.8 −133.214 −534.857 9553.95 1660.05 71250.4 −373065. −181430. −1.30825e6 −221141.
1.9 −133.144 −2082.29 9535.21 −29159.7 277243. 322443. −178839. 2.74159e6 3.88242e6
1.10 −132.634 1971.07 9399.65 7808.70 −261430. 125295. −160175. 2.29078e6 −1.03569e6
1.11 −115.546 −1999.16 5158.84 9720.95 230995. 359868. 350469. 2.40233e6 −1.12322e6
1.12 −108.890 1087.30 3665.05 56335.6 −118396. −89295.9 492940. −412109. −6.13438e6
1.13 −98.0505 −2112.61 1421.90 −12582.0 207142. −478415. 663812. 2.86880e6 1.23367e6
1.14 −97.7541 2237.05 1363.86 −66695.5 −218680. 442226. 667479. 3.41005e6 6.51975e6
1.15 −97.2873 584.731 1272.82 −54026.7 −56886.9 −152492. 673148. −1.25241e6 5.25612e6
1.16 −93.4934 1430.34 549.025 2493.54 −133728. 493788. 714568. 451553. −233130.
1.17 −89.8965 449.666 −110.623 −67466.7 −40423.4 −352108. 746377. −1.39212e6 6.06502e6
1.18 −88.0551 2015.65 −438.308 −19106.1 −177488. −101168. 759942. 2.46852e6 1.68238e6
1.19 −81.0075 −778.764 −1629.79 33080.5 63085.7 164934. 795638. −987850. −2.67977e6
1.20 −69.3148 −2180.30 −3387.46 54393.0 151127. 318626. 802628. 3.15937e6 −3.77024e6
See all 53 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.53
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(103\) \(1\)

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 103.14.a.a ✓ 53
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.14.a.a ✓ 53 1.a even 1 1 trivial