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Properties

Label 2075.2.a.n
Level $2075$
Weight $2$
Character orbit 2075.a
Self dual yes
Analytic conductor $16.569$
Analytic rank $1$
Dimension $20$
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] = [2075,2,Mod(1,2075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2075, 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("2075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2075 = 5^{2} \cdot 83 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2075.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: \(16.5689584194\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 5 x^{19} - 17 x^{18} + 115 x^{17} + 78 x^{16} - 1071 x^{15} + 196 x^{14} + 5232 x^{13} + \cdots - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 415)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{15} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{15} + \beta_{4} + \beta_1 - 1) q^{6} + (\beta_{10} - 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + (\beta_{12} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{15} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{15} + \beta_{4} + \beta_1 - 1) q^{6} + (\beta_{10} - 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + (\beta_{12} + 1) q^{9} + ( - \beta_{13} + \beta_{5}) q^{11} + (\beta_{17} + \beta_{14} + \beta_{13} + \cdots - 1) q^{12}+ \cdots + ( - \beta_{18} + \beta_{17} + \beta_{16} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 5 q^{2} - 6 q^{3} + 19 q^{4} - 14 q^{7} - 15 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 5 q^{2} - 6 q^{3} + 19 q^{4} - 14 q^{7} - 15 q^{8} + 16 q^{9} - 2 q^{11} - 18 q^{12} - 34 q^{13} - 4 q^{14} + 17 q^{16} - 8 q^{17} - 25 q^{18} + 2 q^{19} - 6 q^{21} - 23 q^{22} - 14 q^{23} + 8 q^{24} + 8 q^{26} - 6 q^{27} - 42 q^{28} - 4 q^{29} - 4 q^{31} - 10 q^{32} - 32 q^{33} - 12 q^{34} - 7 q^{36} - 70 q^{37} - 16 q^{38} - 2 q^{39} - 4 q^{41} + 22 q^{42} - 34 q^{43} + 2 q^{44} + 10 q^{46} - 12 q^{47} - 42 q^{48} + 30 q^{49} - 16 q^{51} - 13 q^{52} - 50 q^{53} - 16 q^{54} + 2 q^{56} - 24 q^{57} - 54 q^{58} - 10 q^{59} - 6 q^{61} + 26 q^{62} - 42 q^{63} - 5 q^{64} + 18 q^{66} - 56 q^{67} - 20 q^{68} - 10 q^{69} + 2 q^{71} - 7 q^{72} - 52 q^{73} + 36 q^{74} - 24 q^{76} + 6 q^{77} - 48 q^{78} + 30 q^{79} + 24 q^{81} + q^{82} + 20 q^{83} - 4 q^{84} - 28 q^{86} - 2 q^{87} - 74 q^{88} + 2 q^{89} + 28 q^{91} + 2 q^{92} - 86 q^{93} - 24 q^{94} + 18 q^{96} - 52 q^{97} - 19 q^{98} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} - 17 x^{18} + 115 x^{17} + 78 x^{16} - 1071 x^{15} + 196 x^{14} + 5232 x^{13} + \cdots - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 52966893 \nu^{19} + 97041342 \nu^{18} + 1522665259 \nu^{17} - 2545609093 \nu^{16} + \cdots + 1547199766 ) / 652834258 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 174395664 \nu^{19} + 739240825 \nu^{18} + 3412123885 \nu^{17} - 16922803767 \nu^{16} + \cdots - 4541363720 ) / 1305668516 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 174930923 \nu^{19} + 849331311 \nu^{18} + 3071221523 \nu^{17} - 19450205733 \nu^{16} + \cdots - 7200220352 ) / 1305668516 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 94049755 \nu^{19} - 254832657 \nu^{18} - 2349841130 \nu^{17} + 6035322467 \nu^{16} + \cdots + 512196476 ) / 652834258 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 195222161 \nu^{19} + 559727939 \nu^{18} + 4824252311 \nu^{17} - 13413535605 \nu^{16} + \cdots - 1911456328 ) / 1305668516 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 200767183 \nu^{19} - 634416806 \nu^{18} - 4774418578 \nu^{17} + 15102561242 \nu^{16} + \cdots + 5309784668 ) / 1305668516 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 224155478 \nu^{19} - 826874083 \nu^{18} - 4837946135 \nu^{17} + 19162418897 \nu^{16} + \cdots - 4381185604 ) / 1305668516 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 285553392 \nu^{19} - 1064662291 \nu^{18} - 6132985685 \nu^{17} + 24750632087 \nu^{16} + \cdots - 1716448396 ) / 1305668516 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 328002451 \nu^{19} - 1252007410 \nu^{18} - 7008249128 \nu^{17} + 29375858172 \nu^{16} + \cdots + 11601873008 ) / 1305668516 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 477557615 \nu^{19} - 1632162645 \nu^{18} - 10759443007 \nu^{17} + 38116704121 \nu^{16} + \cdots - 4320671256 ) / 1305668516 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 491829297 \nu^{19} + 1787553252 \nu^{18} + 10703286444 \nu^{17} - 41544698658 \nu^{16} + \cdots + 3965282632 ) / 1305668516 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 247655700 \nu^{19} - 937655907 \nu^{18} - 5244844149 \nu^{17} + 21712896092 \nu^{16} + \cdots + 2875611108 ) / 652834258 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 505218903 \nu^{19} + 1807880375 \nu^{18} + 11205209569 \nu^{17} - 42550667883 \nu^{16} + \cdots - 4571317800 ) / 1305668516 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 621196553 \nu^{19} - 2156937555 \nu^{18} - 13910181009 \nu^{17} + 50387852715 \nu^{16} + \cdots - 813731360 ) / 1305668516 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 738282768 \nu^{19} - 2607273497 \nu^{18} - 16340562245 \nu^{17} + 60607959631 \nu^{16} + \cdots - 1168424844 ) / 1305668516 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 375836187 \nu^{19} - 1313800310 \nu^{18} - 8421242685 \nu^{17} + 30806964428 \nu^{16} + \cdots + 3684087794 ) / 652834258 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{18} + \beta_{15} - \beta_{13} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{19} + \beta_{18} - \beta_{16} + \beta_{14} + 9\beta_{3} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} + 11 \beta_{18} - \beta_{17} - \beta_{16} + 10 \beta_{15} + 2 \beta_{14} - 9 \beta_{13} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 9 \beta_{19} + 14 \beta_{18} - 4 \beta_{17} - 12 \beta_{16} + \beta_{15} + 11 \beta_{14} - 2 \beta_{12} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{19} + 96 \beta_{18} - 17 \beta_{17} - 17 \beta_{16} + 81 \beta_{15} + 27 \beta_{14} + \cdots + 511 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 59 \beta_{19} + 143 \beta_{18} - 64 \beta_{17} - 111 \beta_{16} + 21 \beta_{15} + 92 \beta_{14} + \cdots + 78 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 159 \beta_{19} + 777 \beta_{18} - 192 \beta_{17} - 196 \beta_{16} + 613 \beta_{15} + 257 \beta_{14} + \cdots + 3167 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 329 \beta_{19} + 1292 \beta_{18} - 698 \beta_{17} - 938 \beta_{16} + 275 \beta_{15} + 699 \beta_{14} + \cdots + 834 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1459 \beta_{19} + 6070 \beta_{18} - 1837 \beta_{17} - 1909 \beta_{16} + 4503 \beta_{15} + 2138 \beta_{14} + \cdots + 19984 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1542 \beta_{19} + 10951 \beta_{18} - 6503 \beta_{17} - 7580 \beta_{16} + 2916 \beta_{15} + \cdots + 7647 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 12390 \beta_{19} + 46456 \beta_{18} - 16129 \beta_{17} - 16941 \beta_{16} + 32590 \beta_{15} + \cdots + 127696 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 4958 \beta_{19} + 89278 \beta_{18} - 55771 \beta_{17} - 59627 \beta_{16} + 27505 \beta_{15} + \cdots + 64754 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 100375 \beta_{19} + 350797 \beta_{18} - 134571 \beta_{17} - 141931 \beta_{16} + 234048 \beta_{15} + \cdots + 824284 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 7305 \beta_{19} + 709044 \beta_{18} - 455208 \beta_{17} - 460605 \beta_{16} + 241358 \beta_{15} + \cdots + 523335 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 788428 \beta_{19} + 2623934 \beta_{18} - 1086070 \beta_{17} - 1144596 \beta_{16} + 1674276 \beta_{15} + \cdots + 5368653 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 344581 \beta_{19} + 5526570 \beta_{18} - 3598084 \beta_{17} - 3511726 \beta_{16} + 2019301 \beta_{15} + \cdots + 4105856 \) 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
2.68559
2.52468
2.46656
2.37393
1.91276
1.57645
1.46659
1.44311
0.746031
0.220742
0.0837513
−0.196489
−0.514545
−0.704735
−1.25831
−1.36958
−1.54560
−2.01087
−2.38782
−2.51224
−2.68559 −1.30999 5.21239 0 3.51810 −1.01544 −8.62717 −1.28392 0
1.2 −2.52468 −1.35362 4.37400 0 3.41746 −3.72896 −5.99358 −1.16771 0
1.3 −2.46656 2.38524 4.08390 0 −5.88332 −1.64452 −5.14004 2.68936 0
1.4 −2.37393 −3.01949 3.63552 0 7.16805 3.13841 −3.88262 6.11733 0
1.5 −1.91276 1.13806 1.65864 0 −2.17684 −3.49585 0.652942 −1.70481 0
1.6 −1.57645 0.326178 0.485200 0 −0.514204 4.26977 2.38801 −2.89361 0
1.7 −1.46659 2.73457 0.150891 0 −4.01050 0.306682 2.71189 4.47787 0
1.8 −1.44311 −2.42358 0.0825627 0 3.49748 −3.81480 2.76707 2.87372 0
1.9 −0.746031 −2.46691 −1.44344 0 1.84039 2.76720 2.56891 3.08566 0
1.10 −0.220742 0.244497 −1.95127 0 −0.0539708 0.200882 0.872210 −2.94022 0
1.11 −0.0837513 3.21611 −1.99299 0 −0.269353 −4.22314 0.334418 7.34336 0
1.12 0.196489 −1.70081 −1.96139 0 −0.334192 −1.66637 −0.778372 −0.107242 0
1.13 0.514545 0.538603 −1.73524 0 0.277135 4.96313 −1.92195 −2.70991 0
1.14 0.704735 −3.21265 −1.50335 0 −2.26407 −3.52055 −2.46893 7.32114 0
1.15 1.25831 1.84534 −0.416663 0 2.32200 −0.928239 −3.04090 0.405269 0
1.16 1.36958 −1.38444 −0.124251 0 −1.89610 2.05056 −2.90933 −1.08333 0
1.17 1.54560 0.665788 0.388871 0 1.02904 −1.28556 −2.49016 −2.55673 0
1.18 2.01087 0.785813 2.04361 0 1.58017 −1.12749 0.0876973 −2.38250 0
1.19 2.38782 −2.50188 3.70168 0 −5.97403 −0.350694 4.06331 3.25939 0
1.20 2.51224 −0.506823 4.31133 0 −1.27326 −4.89502 5.80660 −2.74313 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(83\) \( -1 \)

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 2075.2.a.n 20
5.b even 2 1 2075.2.a.o 20
5.c odd 4 2 415.2.b.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
415.2.b.a ✓ 40 5.c odd 4 2
2075.2.a.n 20 1.a even 1 1 trivial
2075.2.a.o 20 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2075))\):

\( T_{2}^{20} + 5 T_{2}^{19} - 17 T_{2}^{18} - 115 T_{2}^{17} + 78 T_{2}^{16} + 1071 T_{2}^{15} + 196 T_{2}^{14} + \cdots - 8 \) Copy content Toggle raw display
\( T_{3}^{20} + 6 T_{3}^{19} - 20 T_{3}^{18} - 178 T_{3}^{17} + 44 T_{3}^{16} + 2016 T_{3}^{15} + \cdots + 304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 5 T^{19} + \cdots - 8 \) Copy content Toggle raw display
$3$ \( T^{20} + 6 T^{19} + \cdots + 304 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 14 T^{19} + \cdots - 110480 \) Copy content Toggle raw display
$11$ \( T^{20} + 2 T^{19} + \cdots - 4387024 \) Copy content Toggle raw display
$13$ \( T^{20} + 34 T^{19} + \cdots + 3419872 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 1114535888 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots - 419204608 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 27415936000 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 26637943211 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 2749910413 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 15712549072 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 18742209717248 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 335143208320 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 9298694070784 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 15585676288 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 55617096672095 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 693613188042109 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 19\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 15058061312 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 1614633125600 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( (T - 1)^{20} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 328599947548672 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 28\!\cdots\!28 \) Copy content Toggle raw display
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