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Properties

Label 103.12.a.b
Level $103$
Weight $12$
Character orbit 103.a
Self dual yes
Analytic conductor $79.139$
Analytic rank $0$
Dimension $49$
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] = [103,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(79.1393475976\)
Analytic rank: \(0\)
Dimension: \(49\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 49 q + 183 q^{2} + 466 q^{3} + 55743 q^{4} + 16065 q^{5} + 33631 q^{6} + 31238 q^{7} + 367572 q^{8} + 3580275 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 49 q + 183 q^{2} + 466 q^{3} + 55743 q^{4} + 16065 q^{5} + 33631 q^{6} + 31238 q^{7} + 367572 q^{8} + 3580275 q^{9} + 750478 q^{10} + 727533 q^{11} + 2347431 q^{12} + 1686993 q^{13} + 1917175 q^{14} + 5732779 q^{15} + 63104603 q^{16} + 42447415 q^{17} + 38328816 q^{18} + 5625397 q^{19} + 23363733 q^{20} + 9056758 q^{21} + 35431841 q^{22} + 99653000 q^{23} + 60107634 q^{24} + 612692400 q^{25} - 287465619 q^{26} + 33237874 q^{27} + 94647576 q^{28} + 582553712 q^{29} + 2028243566 q^{30} + 574615536 q^{31} + 2461710524 q^{32} + 1398060597 q^{33} + 1095877670 q^{34} + 1224276663 q^{35} + 5178232888 q^{36} + 1054414698 q^{37} + 168008610 q^{38} - 682764349 q^{39} - 3058491845 q^{40} + 1814310718 q^{41} - 7019276677 q^{42} + 322591648 q^{43} - 1892602784 q^{44} - 4067379516 q^{45} - 11163159263 q^{46} + 15760013 q^{47} - 3956976183 q^{48} + 10317761631 q^{49} + 4788432202 q^{50} + 4167009909 q^{51} - 16558761537 q^{52} + 16561990335 q^{53} - 22053068789 q^{54} - 2528406158 q^{55} - 2175101045 q^{56} + 10578500421 q^{57} + 10570052770 q^{58} + 12504156727 q^{59} + 27219988055 q^{60} + 12858649383 q^{61} + 29953793573 q^{62} + 21928998164 q^{63} + 107776677998 q^{64} + 75670521263 q^{65} + 70097909751 q^{66} + 34656499990 q^{67} + 134555459108 q^{68} + 60441944808 q^{69} + 144324418256 q^{70} - 4976323973 q^{71} + 184674525846 q^{72} + 80843930205 q^{73} + 147160169907 q^{74} + 106406736723 q^{75} + 89376434340 q^{76} + 187650162769 q^{77} + 217624188512 q^{78} + 33073221431 q^{79} + 220351536446 q^{80} + 373198656177 q^{81} + 417339951569 q^{82} + 180678060299 q^{83} + 622749162897 q^{84} + 295945248373 q^{85} + 223228441021 q^{86} + 351128502358 q^{87} + 327529458125 q^{88} + 229672691360 q^{89} + 878270116936 q^{90} + 269714836479 q^{91} + 470788684476 q^{92} + 255181811376 q^{93} + 556696972040 q^{94} + 455878930472 q^{95} + 1296231412687 q^{96} + 248534514476 q^{97} + 747851098165 q^{98} + 465588303306 q^{99}+O(q^{100}) \) 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 −85.4977 −643.557 5261.86 3664.70 55022.7 −52149.3 −274778. 237019. −313324.
1.2 −83.5140 −305.450 4926.59 13563.0 25509.3 7547.99 −240403. −83847.4 −1.13270e6
1.3 −82.8620 49.7733 4818.11 4959.23 −4124.32 39697.9 −229537. −174670. −410932.
1.4 −79.9842 628.826 4349.46 −9125.87 −50296.1 72830.4 −184081. 218276. 729925.
1.5 −74.2508 −125.019 3465.19 −2224.90 9282.79 36712.6 −105227. −161517. 165201.
1.6 −72.6433 −417.329 3229.05 −5596.60 30316.1 −25505.9 −85795.5 −2983.71 406555.
1.7 −70.1667 450.509 2875.37 −10111.9 −31610.7 −28611.5 −58053.8 25811.4 709516.
1.8 −68.3542 785.372 2624.30 −5865.58 −53683.5 −6241.75 −39392.2 439662. 400937.
1.9 −63.2626 −32.9435 1954.15 9892.65 2084.09 −66793.0 5937.09 −176062. −625834.
1.10 −62.5574 644.007 1865.43 10838.6 −40287.4 21084.0 11421.3 237598. −678034.
1.11 −55.8511 −801.343 1071.35 −10888.4 44755.9 −30294.9 54547.1 465003. 608129.
1.12 −49.9289 −86.2321 444.890 −10809.1 4305.47 34378.8 80041.4 −169711. 539687.
1.13 −46.8129 118.150 143.448 10866.5 −5530.94 27751.2 89157.6 −163188. −508691.
1.14 −46.6984 747.639 132.740 −5397.56 −34913.5 −39035.3 89439.6 381817. 252057.
1.15 −38.8879 282.791 −535.729 −4153.29 −10997.2 −32659.9 100476. −97176.1 161513.
1.16 −37.9026 −238.341 −611.390 3192.36 9033.76 −12227.8 100798. −120340. −120999.
1.17 −34.5051 −538.024 −857.400 9278.33 18564.6 −65879.2 100251. 112323. −320149.
1.18 −27.3198 −618.068 −1301.63 −4904.34 16885.5 15960.0 91511.2 204862. 133986.
1.19 −20.0865 −608.225 −1644.53 −1762.68 12217.1 81502.2 74170.1 192791. 35406.1
1.20 −11.3914 −688.935 −1918.24 −2260.31 7847.93 −5266.15 45180.9 297485. 25748.1
See all 49 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.49
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 103.12.a.b ✓ 49
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.12.a.b ✓ 49 1.a even 1 1 trivial