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Properties

Label 40.3.i.a
Level 4040
Weight 33
Character orbit 40.i
Analytic conductor 1.0901.090
Analytic rank 00
Dimension 2020
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [40,3,Mod(13,40)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(40, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 40=23⋅5 40 = 2^{3} \cdot 5
Weight: k k == 3 3
Character orbit: [χ][\chi] == 40.i (of order 44, degree 22, 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: 1.089921057441.08992105744
Analytic rank: 00
Dimension: 2020
Relative dimension: 1010 over Q(i)\Q(i)
Coefficient field: Q[x]/(x20−⋯ )\mathbb{Q}[x]/(x^{20} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x20−x18−3x16+11x14+x12−40x10+4x8+176x6−192x4−256x2+1024 x^{20} - x^{18} - 3x^{16} + 11x^{14} + x^{12} - 40x^{10} + 4x^{8} + 176x^{6} - 192x^{4} - 256x^{2} + 1024 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a9]\Z[a_1, \ldots, a_{9}]
Coefficient ring index: 220 2^{20}
Twist minimal: yes
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,…,β191,\beta_1,\ldots,\beta_{19} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β4q2+β16q3+β3q4+β8q5+(−β15−1)q6−β10q7+(−β18−β17+⋯−β7)q8+(−β19+2β18+⋯+β2)q9+⋯+(β19−7β17+⋯−2β3)q99+O(q100) q + \beta_{4} q^{2} + \beta_{16} q^{3} + \beta_{3} q^{4} + \beta_{8} q^{5} + ( - \beta_{15} - 1) q^{6} - \beta_{10} q^{7} + ( - \beta_{18} - \beta_{17} + \cdots - \beta_{7}) q^{8} + ( - \beta_{19} + 2 \beta_{18} + \cdots + \beta_{2}) q^{9}+ \cdots + (\beta_{19} - 7 \beta_{17} + \cdots - 2 \beta_{3}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 20q−2q2−16q6−4q7+4q8+6q10−44q12−4q15−56q16−12q17+10q18−24q20+92q22−4q23−28q25+100q26+68q28+100q30−136q31+⋯+546q98+O(q100) 20 q - 2 q^{2} - 16 q^{6} - 4 q^{7} + 4 q^{8} + 6 q^{10} - 44 q^{12} - 4 q^{15} - 56 q^{16} - 12 q^{17} + 10 q^{18} - 24 q^{20} + 92 q^{22} - 4 q^{23} - 28 q^{25} + 100 q^{26} + 68 q^{28} + 100 q^{30} - 136 q^{31}+ \cdots + 546 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x20−x18−3x16+11x14+x12−40x10+4x8+176x6−192x4−256x2+1024 x^{20} - x^{18} - 3x^{16} + 11x^{14} + x^{12} - 40x^{10} + 4x^{8} + 176x^{6} - 192x^{4} - 256x^{2} + 1024 : Copy content Toggle raw display

β1\beta_{1}== 4ν2 4\nu^{2} Copy content Toggle raw display
β2\beta_{2}== (−2ν19+19ν17−7ν15−93ν13−35ν11−35ν9+132ν7+⋯+1408ν)/9600 ( - 2 \nu^{19} + 19 \nu^{17} - 7 \nu^{15} - 93 \nu^{13} - 35 \nu^{11} - 35 \nu^{9} + 132 \nu^{7} + \cdots + 1408 \nu ) / 9600 Copy content Toggle raw display
β3\beta_{3}== (11ν19+5ν17−185ν15+177ν13+755ν11−160ν9+⋯−13312ν)/9600 ( 11 \nu^{19} + 5 \nu^{17} - 185 \nu^{15} + 177 \nu^{13} + 755 \nu^{11} - 160 \nu^{9} + \cdots - 13312 \nu ) / 9600 Copy content Toggle raw display
β4\beta_{4}== (75ν19−16ν18−75ν17+152ν16−225ν15−56ν14+825ν13+⋯+11264)/38400 ( 75 \nu^{19} - 16 \nu^{18} - 75 \nu^{17} + 152 \nu^{16} - 225 \nu^{15} - 56 \nu^{14} + 825 \nu^{13} + \cdots + 11264 ) / 38400 Copy content Toggle raw display
β5\beta_{5}== (75ν19+16ν18−75ν17−152ν16−225ν15+56ν14+825ν13+⋯−11264)/38400 ( 75 \nu^{19} + 16 \nu^{18} - 75 \nu^{17} - 152 \nu^{16} - 225 \nu^{15} + 56 \nu^{14} + 825 \nu^{13} + \cdots - 11264 ) / 38400 Copy content Toggle raw display
β6\beta_{6}== (−17ν18+13ν16+71ν14+33ν12+115ν10−140ν8+972ν6+⋯−2048)/4800 ( - 17 \nu^{18} + 13 \nu^{16} + 71 \nu^{14} + 33 \nu^{12} + 115 \nu^{10} - 140 \nu^{8} + 972 \nu^{6} + \cdots - 2048 ) / 4800 Copy content Toggle raw display
β7\beta_{7}== (−17ν18+13ν16+71ν14+33ν12+115ν10−140ν8+972ν6+⋯−2048)/4800 ( - 17 \nu^{18} + 13 \nu^{16} + 71 \nu^{14} + 33 \nu^{12} + 115 \nu^{10} - 140 \nu^{8} + 972 \nu^{6} + \cdots - 2048 ) / 4800 Copy content Toggle raw display
β8\beta_{8}== (105ν19−122ν18−489ν17+250ν16+357ν15+590ν14+⋯+54784)/38400 ( 105 \nu^{19} - 122 \nu^{18} - 489 \nu^{17} + 250 \nu^{16} + 357 \nu^{15} + 590 \nu^{14} + \cdots + 54784 ) / 38400 Copy content Toggle raw display
β9\beta_{9}== (−5ν19+234ν18−215ν17+246ν16−805ν15−1278ν14+⋯−118272)/38400 ( - 5 \nu^{19} + 234 \nu^{18} - 215 \nu^{17} + 246 \nu^{16} - 805 \nu^{15} - 1278 \nu^{14} + \cdots - 118272 ) / 38400 Copy content Toggle raw display
β10\beta_{10}== (−5ν19−234ν18−215ν17−246ν16−805ν15+1278ν14+⋯+118272)/38400 ( - 5 \nu^{19} - 234 \nu^{18} - 215 \nu^{17} - 246 \nu^{16} - 805 \nu^{15} + 1278 \nu^{14} + \cdots + 118272 ) / 38400 Copy content Toggle raw display
β11\beta_{11}== (−105ν19−122ν18+489ν17+250ν16−357ν15+590ν14+⋯+54784)/38400 ( - 105 \nu^{19} - 122 \nu^{18} + 489 \nu^{17} + 250 \nu^{16} - 357 \nu^{15} + 590 \nu^{14} + \cdots + 54784 ) / 38400 Copy content Toggle raw display
β12\beta_{12}== (−25ν19+82ν18+45ν17−50ν16−185ν15+10ν14−95ν13+⋯+16896)/12800 ( - 25 \nu^{19} + 82 \nu^{18} + 45 \nu^{17} - 50 \nu^{16} - 185 \nu^{15} + 10 \nu^{14} - 95 \nu^{13} + \cdots + 16896 ) / 12800 Copy content Toggle raw display
β13\beta_{13}== (−165ν19+166ν18−183ν17−950ν16+1179ν15+830ν14+⋯+130048)/38400 ( - 165 \nu^{19} + 166 \nu^{18} - 183 \nu^{17} - 950 \nu^{16} + 1179 \nu^{15} + 830 \nu^{14} + \cdots + 130048 ) / 38400 Copy content Toggle raw display
β14\beta_{14}== (ν18−ν16−3ν14+11ν12+ν10−40ν8+4ν6+176ν4−192ν2−256)/128 ( \nu^{18} - \nu^{16} - 3\nu^{14} + 11\nu^{12} + \nu^{10} - 40\nu^{8} + 4\nu^{6} + 176\nu^{4} - 192\nu^{2} - 256 ) / 128 Copy content Toggle raw display
β15\beta_{15}== (5ν18−9ν16+37ν14+19ν12−55ν10−60ν8+100ν6+⋯+3456)/640 ( 5 \nu^{18} - 9 \nu^{16} + 37 \nu^{14} + 19 \nu^{12} - 55 \nu^{10} - 60 \nu^{8} + 100 \nu^{6} + \cdots + 3456 ) / 640 Copy content Toggle raw display
β16\beta_{16}== (25ν19+82ν18−45ν17−50ν16+185ν15+10ν14+95ν13+⋯+16896)/12800 ( 25 \nu^{19} + 82 \nu^{18} - 45 \nu^{17} - 50 \nu^{16} + 185 \nu^{15} + 10 \nu^{14} + 95 \nu^{13} + \cdots + 16896 ) / 12800 Copy content Toggle raw display
β17\beta_{17}== (165ν19+166ν18+183ν17−950ν16−1179ν15+830ν14+⋯+130048)/38400 ( 165 \nu^{19} + 166 \nu^{18} + 183 \nu^{17} - 950 \nu^{16} - 1179 \nu^{15} + 830 \nu^{14} + \cdots + 130048 ) / 38400 Copy content Toggle raw display
β18\beta_{18}== (−95ν19+553ν17+11ν15−843ν13+775ν11+5290ν9+⋯−8192ν)/19200 ( - 95 \nu^{19} + 553 \nu^{17} + 11 \nu^{15} - 843 \nu^{13} + 775 \nu^{11} + 5290 \nu^{9} + \cdots - 8192 \nu ) / 19200 Copy content Toggle raw display
β19\beta_{19}== (−17ν19+13ν17+71ν15+33ν13+115ν11−140ν9+⋯−2048ν)/2400 ( - 17 \nu^{19} + 13 \nu^{17} + 71 \nu^{15} + 33 \nu^{13} + 115 \nu^{11} - 140 \nu^{9} + \cdots - 2048 \nu ) / 2400 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (−β7+β6)/4 ( -\beta_{7} + \beta_{6} ) / 4 Copy content Toggle raw display
ν2\nu^{2}== (β1)/4 ( \beta_1 ) / 4 Copy content Toggle raw display
ν3\nu^{3}== (−β19+2β18−β17+β16+β13−β12−β11+⋯+2β3)/4 ( - \beta_{19} + 2 \beta_{18} - \beta_{17} + \beta_{16} + \beta_{13} - \beta_{12} - \beta_{11} + \cdots + 2 \beta_{3} ) / 4 Copy content Toggle raw display
ν4\nu^{4}== (−β17+β16−β13+β12−β11+β10−β9−β8+⋯+2)/4 ( - \beta_{17} + \beta_{16} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \cdots + 2 ) / 4 Copy content Toggle raw display
ν5\nu^{5}== (β18−β16+β12−β11+β8+β5+β4−β3−3β2)/2 ( \beta_{18} - \beta_{16} + \beta_{12} - \beta_{11} + \beta_{8} + \beta_{5} + \beta_{4} - \beta_{3} - 3\beta_{2} ) / 2 Copy content Toggle raw display
ν6\nu^{6}== (2β17+4β16−4β15+2β13+4β12+2β11+2β8+⋯−8)/4 ( 2 \beta_{17} + 4 \beta_{16} - 4 \beta_{15} + 2 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} + 2 \beta_{8} + \cdots - 8 ) / 4 Copy content Toggle raw display
ν7\nu^{7}== (β19−2β17−2β16+2β13+2β12−2β11+⋯+20β2)/4 ( \beta_{19} - 2 \beta_{17} - 2 \beta_{16} + 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + \cdots + 20 \beta_{2} ) / 4 Copy content Toggle raw display
ν8\nu^{8}== (−β17+3β16−2β15−2β14−β13+3β12+5β11+⋯−12)/4 ( - \beta_{17} + 3 \beta_{16} - 2 \beta_{15} - 2 \beta_{14} - \beta_{13} + 3 \beta_{12} + 5 \beta_{11} + \cdots - 12 ) / 4 Copy content Toggle raw display
ν9\nu^{9}== (−2β19−4β18+β17−5β16−β13+5β12+⋯−50β2)/4 ( - 2 \beta_{19} - 4 \beta_{18} + \beta_{17} - 5 \beta_{16} - \beta_{13} + 5 \beta_{12} + \cdots - 50 \beta_{2} ) / 4 Copy content Toggle raw display
ν10\nu^{10}== (4β17+8β16−7β15−9β14+4β13+8β12+β11+⋯+11)/2 ( 4 \beta_{17} + 8 \beta_{16} - 7 \beta_{15} - 9 \beta_{14} + 4 \beta_{13} + 8 \beta_{12} + \beta_{11} + \cdots + 11 ) / 2 Copy content Toggle raw display
ν11\nu^{11}== (14β19−4β18+12β17+6β16−12β13−6β12+⋯−28β2)/4 ( 14 \beta_{19} - 4 \beta_{18} + 12 \beta_{17} + 6 \beta_{16} - 12 \beta_{13} - 6 \beta_{12} + \cdots - 28 \beta_{2} ) / 4 Copy content Toggle raw display
ν12\nu^{12}== (−8β17−28β16+16β15+48β14−8β13−28β12+⋯+96)/4 ( - 8 \beta_{17} - 28 \beta_{16} + 16 \beta_{15} + 48 \beta_{14} - 8 \beta_{13} - 28 \beta_{12} + \cdots + 96 ) / 4 Copy content Toggle raw display
ν13\nu^{13}== (5β19−2β18−11β17+15β16+11β13−15β12+⋯−184β2)/4 ( 5 \beta_{19} - 2 \beta_{18} - 11 \beta_{17} + 15 \beta_{16} + 11 \beta_{13} - 15 \beta_{12} + \cdots - 184 \beta_{2} ) / 4 Copy content Toggle raw display
ν14\nu^{14}== (−31β17+7β16+64β15−16β14−31β13+7β12+⋯−162)/4 ( - 31 \beta_{17} + 7 \beta_{16} + 64 \beta_{15} - 16 \beta_{14} - 31 \beta_{13} + 7 \beta_{12} + \cdots - 162 ) / 4 Copy content Toggle raw display
ν15\nu^{15}== (4β19+51β18−6β17+51β16+6β13−51β12+⋯−85β2)/2 ( 4 \beta_{19} + 51 \beta_{18} - 6 \beta_{17} + 51 \beta_{16} + 6 \beta_{13} - 51 \beta_{12} + \cdots - 85 \beta_{2} ) / 2 Copy content Toggle raw display
ν16\nu^{16}== (−122β17−36β16+108β15+200β14−122β13−36β12+⋯+208)/4 ( - 122 \beta_{17} - 36 \beta_{16} + 108 \beta_{15} + 200 \beta_{14} - 122 \beta_{13} - 36 \beta_{12} + \cdots + 208 ) / 4 Copy content Toggle raw display
ν17\nu^{17}== (87β19+40β18+46β17−34β16−46β13+34β12+⋯+460β2)/4 ( 87 \beta_{19} + 40 \beta_{18} + 46 \beta_{17} - 34 \beta_{16} - 46 \beta_{13} + 34 \beta_{12} + \cdots + 460 \beta_{2} ) / 4 Copy content Toggle raw display
ν18\nu^{18}== (−7β17+205β16+74β15+74β14−7β13+205β12+⋯−1132)/4 ( - 7 \beta_{17} + 205 \beta_{16} + 74 \beta_{15} + 74 \beta_{14} - 7 \beta_{13} + 205 \beta_{12} + \cdots - 1132 ) / 4 Copy content Toggle raw display
ν19\nu^{19}== (−234β19+244β18−25β17+453β16+25β13+⋯+978β2)/4 ( - 234 \beta_{19} + 244 \beta_{18} - 25 \beta_{17} + 453 \beta_{16} + 25 \beta_{13} + \cdots + 978 \beta_{2} ) / 4 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

Character values

We give the values of χ\chi on generators for (Z/40Z)×\left(\mathbb{Z}/40\mathbb{Z}\right)^\times.

nn 1717 2121 3131
χ(n)\chi(n) β2\beta_{2} −1-1 11

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
13.1
1.17039 + 0.793843i
1.39859 + 0.209644i
0.0552378 + 1.41313i
−0.0552378 + 1.41313i
1.27574 − 0.610320i
−1.17039 + 0.793843i
0.541828 − 1.30630i
−1.39859 + 0.209644i
−0.541828 − 1.30630i
−1.27574 − 0.610320i
1.17039 − 0.793843i
1.39859 − 0.209644i
0.0552378 − 1.41313i
−0.0552378 − 1.41313i
1.27574 + 0.610320i
−1.17039 − 0.793843i
0.541828 + 1.30630i
−1.39859 − 0.209644i
−0.541828 + 1.30630i
−1.27574 + 0.610320i
−1.96423 − 0.376547i 0.977390 − 0.977390i 3.71643 + 1.47925i −0.801246 − 4.93538i −2.28785 + 1.55179i 8.39950 − 8.39950i −6.74292 − 4.30500i 7.08942i −0.284568 + 9.99595i
13.2 −1.60823 − 1.18894i −2.52630 + 2.52630i 1.17282 + 3.82420i 3.09141 + 3.92978i 7.06650 − 1.05925i −5.20520 + 5.20520i 2.66059 − 7.54462i − 3.76437i −0.299408 − 9.99552i
13.3 −1.46837 + 1.35790i −2.57493 + 2.57493i 0.312234 − 3.98780i −4.90427 − 0.973739i 0.284467 − 7.27744i −4.07624 + 4.07624i 4.95654 + 6.27955i − 4.26050i 8.52353 − 5.22967i
13.4 −1.35790 + 1.46837i 2.57493 − 2.57493i −0.312234 − 3.98780i 4.90427 + 0.973739i 0.284467 + 7.27744i −4.07624 + 4.07624i 6.27955 + 4.95654i − 4.26050i −8.08930 + 5.87905i
13.5 −0.665418 − 1.88606i 3.60765 − 3.60765i −3.11444 + 2.51004i −2.34539 + 4.41578i −9.20485 − 4.40365i 1.47907 − 1.47907i 6.80648 + 4.20379i − 17.0303i 9.88909 + 1.48520i
13.6 0.376547 + 1.96423i −0.977390 + 0.977390i −3.71643 + 1.47925i 0.801246 + 4.93538i −2.28785 − 1.55179i 8.39950 − 8.39950i −4.30500 − 6.74292i 7.08942i −9.39254 + 3.43224i
13.7 0.764474 − 1.84813i −0.130791 + 0.130791i −2.83116 − 2.82569i 4.38731 − 2.39823i 0.141733 + 0.341706i −1.59713 + 1.59713i −7.38659 + 3.07218i 8.96579i −1.07825 − 9.94170i
13.8 1.18894 + 1.60823i 2.52630 − 2.52630i −1.17282 + 3.82420i −3.09141 − 3.92978i 7.06650 + 1.05925i −5.20520 + 5.20520i −7.54462 + 2.66059i − 3.76437i 2.64449 − 9.64400i
13.9 1.84813 − 0.764474i 0.130791 − 0.130791i 2.83116 − 2.82569i −4.38731 + 2.39823i 0.141733 − 0.341706i −1.59713 + 1.59713i 3.07218 − 7.38659i 8.96579i −6.27494 + 7.78621i
13.10 1.88606 + 0.665418i −3.60765 + 3.60765i 3.11444 + 2.51004i 2.34539 − 4.41578i −9.20485 + 4.40365i 1.47907 − 1.47907i 4.20379 + 6.80648i − 17.0303i 7.36189 − 6.76776i
37.1 −1.96423 + 0.376547i 0.977390 + 0.977390i 3.71643 − 1.47925i −0.801246 + 4.93538i −2.28785 − 1.55179i 8.39950 + 8.39950i −6.74292 + 4.30500i − 7.08942i −0.284568 − 9.99595i
37.2 −1.60823 + 1.18894i −2.52630 − 2.52630i 1.17282 − 3.82420i 3.09141 − 3.92978i 7.06650 + 1.05925i −5.20520 − 5.20520i 2.66059 + 7.54462i 3.76437i −0.299408 + 9.99552i
37.3 −1.46837 − 1.35790i −2.57493 − 2.57493i 0.312234 + 3.98780i −4.90427 + 0.973739i 0.284467 + 7.27744i −4.07624 − 4.07624i 4.95654 − 6.27955i 4.26050i 8.52353 + 5.22967i
37.4 −1.35790 − 1.46837i 2.57493 + 2.57493i −0.312234 + 3.98780i 4.90427 − 0.973739i 0.284467 − 7.27744i −4.07624 − 4.07624i 6.27955 − 4.95654i 4.26050i −8.08930 − 5.87905i
37.5 −0.665418 + 1.88606i 3.60765 + 3.60765i −3.11444 − 2.51004i −2.34539 − 4.41578i −9.20485 + 4.40365i 1.47907 + 1.47907i 6.80648 − 4.20379i 17.0303i 9.88909 − 1.48520i
37.6 0.376547 − 1.96423i −0.977390 − 0.977390i −3.71643 − 1.47925i 0.801246 − 4.93538i −2.28785 + 1.55179i 8.39950 + 8.39950i −4.30500 + 6.74292i − 7.08942i −9.39254 − 3.43224i
37.7 0.764474 + 1.84813i −0.130791 − 0.130791i −2.83116 + 2.82569i 4.38731 + 2.39823i 0.141733 − 0.341706i −1.59713 − 1.59713i −7.38659 − 3.07218i − 8.96579i −1.07825 + 9.94170i
37.8 1.18894 − 1.60823i 2.52630 + 2.52630i −1.17282 − 3.82420i −3.09141 + 3.92978i 7.06650 − 1.05925i −5.20520 − 5.20520i −7.54462 − 2.66059i 3.76437i 2.64449 + 9.64400i
37.9 1.84813 + 0.764474i 0.130791 + 0.130791i 2.83116 + 2.82569i −4.38731 − 2.39823i 0.141733 + 0.341706i −1.59713 − 1.59713i 3.07218 + 7.38659i − 8.96579i −6.27494 − 7.78621i
37.10 1.88606 − 0.665418i −3.60765 − 3.60765i 3.11444 − 2.51004i 2.34539 + 4.41578i −9.20485 − 4.40365i 1.47907 + 1.47907i 4.20379 − 6.80648i 17.0303i 7.36189 + 6.76776i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.b even 2 1 inner
40.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.3.i.a ✓ 20
3.b odd 2 1 360.3.u.b 20
4.b odd 2 1 160.3.m.a 20
5.b even 2 1 200.3.i.b 20
5.c odd 4 1 inner 40.3.i.a ✓ 20
5.c odd 4 1 200.3.i.b 20
8.b even 2 1 inner 40.3.i.a ✓ 20
8.d odd 2 1 160.3.m.a 20
15.e even 4 1 360.3.u.b 20
20.d odd 2 1 800.3.m.b 20
20.e even 4 1 160.3.m.a 20
20.e even 4 1 800.3.m.b 20
24.h odd 2 1 360.3.u.b 20
40.e odd 2 1 800.3.m.b 20
40.f even 2 1 200.3.i.b 20
40.i odd 4 1 inner 40.3.i.a ✓ 20
40.i odd 4 1 200.3.i.b 20
40.k even 4 1 160.3.m.a 20
40.k even 4 1 800.3.m.b 20
120.w even 4 1 360.3.u.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.3.i.a ✓ 20 1.a even 1 1 trivial
40.3.i.a ✓ 20 5.c odd 4 1 inner
40.3.i.a ✓ 20 8.b even 2 1 inner
40.3.i.a ✓ 20 40.i odd 4 1 inner
160.3.m.a 20 4.b odd 2 1
160.3.m.a 20 8.d odd 2 1
160.3.m.a 20 20.e even 4 1
160.3.m.a 20 40.k even 4 1
200.3.i.b 20 5.b even 2 1
200.3.i.b 20 5.c odd 4 1
200.3.i.b 20 40.f even 2 1
200.3.i.b 20 40.i odd 4 1
360.3.u.b 20 3.b odd 2 1
360.3.u.b 20 15.e even 4 1
360.3.u.b 20 24.h odd 2 1
360.3.u.b 20 120.w even 4 1
800.3.m.b 20 20.d odd 2 1
800.3.m.b 20 20.e even 4 1
800.3.m.b 20 40.e odd 2 1
800.3.m.b 20 40.k even 4 1

Hecke kernels

This newform subspace is the entire newspace S3new(40,[χ])S_{3}^{\mathrm{new}}(40, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T20+2T19+⋯+1048576 T^{20} + 2 T^{19} + \cdots + 1048576 Copy content Toggle raw display
33 T20+1020T16+⋯+82944 T^{20} + 1020 T^{16} + \cdots + 82944 Copy content Toggle raw display
55 T20+⋯+95367431640625 T^{20} + \cdots + 95367431640625 Copy content Toggle raw display
77 (T10+2T9+⋯+5671712)2 (T^{10} + 2 T^{9} + \cdots + 5671712)^{2} Copy content Toggle raw display
1111 (T10+552T8+⋯+473497600)2 (T^{10} + 552 T^{8} + \cdots + 473497600)^{2} Copy content Toggle raw display
1313 T20+⋯+23 ⁣⋯ ⁣00 T^{20} + \cdots + 23\!\cdots\!00 Copy content Toggle raw display
1717 (T10+6T9+⋯+2344207392)2 (T^{10} + 6 T^{9} + \cdots + 2344207392)^{2} Copy content Toggle raw display
1919 (T10+⋯−384717505536)2 (T^{10} + \cdots - 384717505536)^{2} Copy content Toggle raw display
2323 (T10+2T9+⋯+415411488)2 (T^{10} + 2 T^{9} + \cdots + 415411488)^{2} Copy content Toggle raw display
2929 (T10−2548T8+⋯−31360000)2 (T^{10} - 2548 T^{8} + \cdots - 31360000)^{2} Copy content Toggle raw display
3131 (T5+34T4+⋯−1252704)4 (T^{5} + 34 T^{4} + \cdots - 1252704)^{4} Copy content Toggle raw display
3737 T20+⋯+59 ⁣⋯ ⁣44 T^{20} + \cdots + 59\!\cdots\!44 Copy content Toggle raw display
4141 (T5+2T4+⋯+74680800)4 (T^{5} + 2 T^{4} + \cdots + 74680800)^{4} Copy content Toggle raw display
4343 T20+⋯+81 ⁣⋯ ⁣64 T^{20} + \cdots + 81\!\cdots\!64 Copy content Toggle raw display
4747 (T10+⋯+418650446283552)2 (T^{10} + \cdots + 418650446283552)^{2} Copy content Toggle raw display
5353 T20+⋯+30 ⁣⋯ ⁣84 T^{20} + \cdots + 30\!\cdots\!84 Copy content Toggle raw display
5959 (T10+⋯−26 ⁣⋯ ⁣16)2 (T^{10} + \cdots - 26\!\cdots\!16)^{2} Copy content Toggle raw display
6161 (T10+⋯+26 ⁣⋯ ⁣00)2 (T^{10} + \cdots + 26\!\cdots\!00)^{2} Copy content Toggle raw display
6767 T20+⋯+23 ⁣⋯ ⁣44 T^{20} + \cdots + 23\!\cdots\!44 Copy content Toggle raw display
7171 (T5−62T4+⋯+423110304)4 (T^{5} - 62 T^{4} + \cdots + 423110304)^{4} Copy content Toggle raw display
7373 (T10+⋯+37 ⁣⋯ ⁣88)2 (T^{10} + \cdots + 37\!\cdots\!88)^{2} Copy content Toggle raw display
7979 (T10+⋯+66 ⁣⋯ ⁣00)2 (T^{10} + \cdots + 66\!\cdots\!00)^{2} Copy content Toggle raw display
8383 T20+⋯+23 ⁣⋯ ⁣64 T^{20} + \cdots + 23\!\cdots\!64 Copy content Toggle raw display
8989 (T10+⋯+51 ⁣⋯ ⁣00)2 (T^{10} + \cdots + 51\!\cdots\!00)^{2} Copy content Toggle raw display
9797 (T10+⋯+90 ⁣⋯ ⁣32)2 (T^{10} + \cdots + 90\!\cdots\!32)^{2} Copy content Toggle raw display
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