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Properties

Label 4010.2.a.k
Level $4010$
Weight $2$
Character orbit 4010.a
Self dual yes
Analytic conductor $32.020$
Analytic rank $1$
Dimension $15$
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] = [4010,2,Mod(1,4010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4010.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: \(32.0200112105\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 7 x^{14} - 7 x^{13} + 133 x^{12} - 99 x^{11} - 941 x^{10} + 1290 x^{9} + 3031 x^{8} + \cdots - 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{9} q^{3} + q^{4} - q^{5} + \beta_{9} q^{6} + ( - \beta_{11} - \beta_1) q^{7} - q^{8} + (\beta_{12} - \beta_{10} + \cdots + \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{9} q^{3} + q^{4} - q^{5} + \beta_{9} q^{6} + ( - \beta_{11} - \beta_1) q^{7} - q^{8} + (\beta_{12} - \beta_{10} + \cdots + \beta_{6}) q^{9}+ \cdots + ( - \beta_{14} - \beta_{11} - \beta_{10} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 15 q^{2} - 6 q^{3} + 15 q^{4} - 15 q^{5} + 6 q^{6} - 5 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{10} - 2 q^{11} - 6 q^{12} - 13 q^{13} + 5 q^{14} + 6 q^{15} + 15 q^{16} + 11 q^{17} - 19 q^{18} - 15 q^{19} - 15 q^{20} - 2 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 15 q^{25} + 13 q^{26} - 12 q^{27} - 5 q^{28} + 28 q^{29} - 6 q^{30} - 12 q^{31} - 15 q^{32} - 22 q^{33} - 11 q^{34} + 5 q^{35} + 19 q^{36} - 23 q^{37} + 15 q^{38} - 2 q^{39} + 15 q^{40} + 24 q^{41} + 2 q^{42} - 24 q^{43} - 2 q^{44} - 19 q^{45} + 3 q^{46} - 3 q^{47} - 6 q^{48} + 20 q^{49} - 15 q^{50} - 5 q^{51} - 13 q^{52} + 10 q^{53} + 12 q^{54} + 2 q^{55} + 5 q^{56} - 11 q^{57} - 28 q^{58} + 2 q^{59} + 6 q^{60} + 15 q^{61} + 12 q^{62} - 2 q^{63} + 15 q^{64} + 13 q^{65} + 22 q^{66} - 48 q^{67} + 11 q^{68} + 21 q^{69} - 5 q^{70} + 15 q^{71} - 19 q^{72} - 47 q^{73} + 23 q^{74} - 6 q^{75} - 15 q^{76} + 7 q^{77} + 2 q^{78} - 34 q^{79} - 15 q^{80} + 43 q^{81} - 24 q^{82} - 32 q^{83} - 2 q^{84} - 11 q^{85} + 24 q^{86} + 14 q^{87} + 2 q^{88} + 25 q^{89} + 19 q^{90} - 32 q^{91} - 3 q^{92} - 42 q^{93} + 3 q^{94} + 15 q^{95} + 6 q^{96} - 34 q^{97} - 20 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 7 x^{14} - 7 x^{13} + 133 x^{12} - 99 x^{11} - 941 x^{10} + 1290 x^{9} + 3031 x^{8} + \cdots - 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58437627 \nu^{14} - 534122576 \nu^{13} + 115180567 \nu^{12} + 10329491164 \nu^{11} + \cdots + 117611134796 ) / 2559277300 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20542792 \nu^{14} - 146543981 \nu^{13} - 164096893 \nu^{12} + 2932664074 \nu^{11} + \cdots + 23702683016 ) / 255927730 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 129009841 \nu^{14} - 883037733 \nu^{13} - 1197107414 \nu^{12} + 17651323112 \nu^{11} + \cdots + 125216059318 ) / 1279638650 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 58921429 \nu^{14} + 416412812 \nu^{13} + 480260011 \nu^{12} - 8324831388 \nu^{11} + \cdots - 67211679492 ) / 511855460 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 355634499 \nu^{14} - 2493292912 \nu^{13} - 2946774221 \nu^{12} + 49230821268 \nu^{11} + \cdots + 330809703252 ) / 2559277300 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 357030859 \nu^{14} + 2078381242 \nu^{13} + 4813725111 \nu^{12} - 41262341138 \nu^{11} + \cdots - 192795588882 ) / 1279638650 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 91199013 \nu^{14} + 582879604 \nu^{13} + 1011640657 \nu^{12} - 11599830506 \nu^{11} + \cdots - 72900927894 ) / 255927730 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1161424079 \nu^{14} + 7214691402 \nu^{13} + 13595344791 \nu^{12} - 142730708728 \nu^{11} + \cdots - 835598044392 ) / 2559277300 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1173296833 \nu^{14} + 7231451704 \nu^{13} + 14132908607 \nu^{12} - 143965640456 \nu^{11} + \cdots - 845081247084 ) / 2559277300 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 371844021 \nu^{14} + 2450736548 \nu^{13} + 3766297784 \nu^{12} - 48621859147 \nu^{11} + \cdots - 316360212508 ) / 639819325 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2202353061 \nu^{14} + 13831059568 \nu^{13} + 25351582419 \nu^{12} - 274524650952 \nu^{11} + \cdots - 1574486438628 ) / 2559277300 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1275103963 \nu^{14} + 8052863369 \nu^{13} + 14434590052 \nu^{12} - 159745442766 \nu^{11} + \cdots - 960771925824 ) / 1279638650 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1426055511 \nu^{14} + 8868477043 \nu^{13} + 16772024394 \nu^{12} - 176015990502 \nu^{11} + \cdots - 1018945552978 ) / 1279638650 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} + \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - \beta_{7} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 17 \beta_{14} - 3 \beta_{13} + 5 \beta_{12} + 13 \beta_{11} + 5 \beta_{10} + 15 \beta_{9} + 16 \beta_{8} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 64 \beta_{14} - 8 \beta_{13} + 26 \beta_{12} + 38 \beta_{11} + 25 \beta_{10} + 47 \beta_{9} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 291 \beta_{14} - 63 \beta_{13} + 113 \beta_{12} + 190 \beta_{11} + 118 \beta_{10} + 239 \beta_{9} + \cdots + 165 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1180 \beta_{14} - 218 \beta_{13} + 503 \beta_{12} + 688 \beta_{11} + 503 \beta_{10} + 896 \beta_{9} + \cdots + 468 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5098 \beta_{14} - 1146 \beta_{13} + 2134 \beta_{12} + 3105 \beta_{11} + 2248 \beta_{10} + 4040 \beta_{9} + \cdots + 2333 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 21215 \beta_{14} - 4459 \beta_{13} + 9167 \beta_{12} + 12348 \beta_{11} + 9418 \beta_{10} + \cdots + 8640 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 90328 \beta_{14} - 20460 \beta_{13} + 38737 \beta_{12} + 53525 \beta_{11} + 40822 \beta_{10} + \cdots + 38650 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 379511 \beta_{14} - 83664 \beta_{13} + 164653 \beta_{12} + 220960 \beta_{11} + 171450 \beta_{10} + \cdots + 156303 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1608230 \beta_{14} - 364938 \beta_{13} + 695590 \beta_{12} + 942964 \beta_{11} + 732462 \beta_{10} + \cdots + 674056 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 6781632 \beta_{14} - 1522673 \beta_{13} + 2946121 \beta_{12} + 3949413 \beta_{11} + 3086141 \beta_{10} + \cdots + 2805548 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 28690546 \beta_{14} - 6514636 \beta_{13} + 12448701 \beta_{12} + 16753801 \beta_{11} + \cdots + 11945030 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−2.07374
0.679191
2.09866
4.22659
−2.62108
−1.71302
1.95650
1.81559
2.79590
−1.99921
0.698269
−0.560392
−1.05167
2.25248
0.495927
−1.00000 −3.34293 1.00000 −1.00000 3.34293 3.45446 −1.00000 8.17517 1.00000
1.2 −1.00000 −3.00420 1.00000 −1.00000 3.00420 −4.38135 −1.00000 6.02524 1.00000
1.3 −1.00000 −2.86623 1.00000 −1.00000 2.86623 0.981408 −1.00000 5.21525 1.00000
1.4 −1.00000 −2.24378 1.00000 −1.00000 2.24378 −2.96737 −1.00000 2.03457 1.00000
1.5 −1.00000 −1.88769 1.00000 −1.00000 1.88769 1.11871 −1.00000 0.563364 1.00000
1.6 −1.00000 −1.21181 1.00000 −1.00000 1.21181 0.441519 −1.00000 −1.53151 1.00000
1.7 −1.00000 −0.918908 1.00000 −1.00000 0.918908 −3.92602 −1.00000 −2.15561 1.00000
1.8 −1.00000 −0.876107 1.00000 −1.00000 0.876107 4.75824 −1.00000 −2.23244 1.00000
1.9 −1.00000 −0.154802 1.00000 −1.00000 0.154802 −4.41535 −1.00000 −2.97604 1.00000
1.10 −1.00000 0.271531 1.00000 −1.00000 −0.271531 1.75980 −1.00000 −2.92627 1.00000
1.11 −1.00000 1.23027 1.00000 −1.00000 −1.23027 3.50539 −1.00000 −1.48643 1.00000
1.12 −1.00000 1.40727 1.00000 −1.00000 −1.40727 −2.31473 −1.00000 −1.01959 1.00000
1.13 −1.00000 1.72412 1.00000 −1.00000 −1.72412 −1.94532 −1.00000 −0.0274146 1.00000
1.14 −1.00000 2.71977 1.00000 −1.00000 −2.71977 −0.594829 −1.00000 4.39715 1.00000
1.15 −1.00000 3.15350 1.00000 −1.00000 −3.15350 −0.474552 −1.00000 6.94456 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(401\) \( +1 \)

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 4010.2.a.k ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4010.2.a.k ✓ 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\( T_{3}^{15} + 6 T_{3}^{14} - 14 T_{3}^{13} - 140 T_{3}^{12} - 35 T_{3}^{11} + 1126 T_{3}^{10} + \cdots + 128 \) Copy content Toggle raw display
\( T_{7}^{15} + 5 T_{7}^{14} - 50 T_{7}^{13} - 266 T_{7}^{12} + 845 T_{7}^{11} + 5014 T_{7}^{10} + \cdots - 14080 \) Copy content Toggle raw display
\( T_{11}^{15} + 2 T_{11}^{14} - 72 T_{11}^{13} - 56 T_{11}^{12} + 1978 T_{11}^{11} - 320 T_{11}^{10} + \cdots - 2560 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + 6 T^{14} + \cdots + 128 \) Copy content Toggle raw display
$5$ \( (T + 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + 5 T^{14} + \cdots - 14080 \) Copy content Toggle raw display
$11$ \( T^{15} + 2 T^{14} + \cdots - 2560 \) Copy content Toggle raw display
$13$ \( T^{15} + 13 T^{14} + \cdots - 265268 \) Copy content Toggle raw display
$17$ \( T^{15} - 11 T^{14} + \cdots - 50096 \) Copy content Toggle raw display
$19$ \( T^{15} + 15 T^{14} + \cdots + 4323968 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 133743104 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 352010240 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 182380960 \) Copy content Toggle raw display
$37$ \( T^{15} + 23 T^{14} + \cdots + 4181512 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 119996288 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 195278336 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 185155072 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 125904188 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 49410713920 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 6149301808576 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 157282369024 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 31935740224 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 34713267712 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 7250505143072 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 9471377325856 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 30055051552 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 22082901745448 \) Copy content Toggle raw display
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