Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3721,2,Mod(1,3721)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3721, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3721.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3721 = 61^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3721.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: | \(29.7123345921\) |
| Analytic rank: | \(1\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 5 x^{15} - 11 x^{14} + 86 x^{13} + 5 x^{12} - 562 x^{11} + 362 x^{10} + 1761 x^{9} - 1799 x^{8} + \cdots - 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 61) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{16} - 5 x^{15} - 11 x^{14} + 86 x^{13} + 5 x^{12} - 562 x^{11} + 362 x^{10} + 1761 x^{9} - 1799 x^{8} + \cdots - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 243793 \nu^{15} + 1128643 \nu^{14} + 3378901 \nu^{13} - 20667477 \nu^{12} - 13129174 \nu^{11} + \cdots - 9695934 ) / 2107977 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 321733 \nu^{15} + 1118670 \nu^{14} + 5344938 \nu^{13} - 20059342 \nu^{12} - 32970081 \nu^{11} + \cdots - 9414651 ) / 2107977 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 321733 \nu^{15} - 1118670 \nu^{14} - 5344938 \nu^{13} + 20059342 \nu^{12} + 32970081 \nu^{11} + \cdots + 3090720 ) / 2107977 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 489995 \nu^{15} + 1805875 \nu^{14} + 7609696 \nu^{13} - 31361416 \nu^{12} - 42516742 \nu^{11} + \cdots - 2895597 ) / 2107977 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2507016 \nu^{15} - 9558194 \nu^{14} - 38842352 \nu^{13} + 169169491 \nu^{12} + 212112815 \nu^{11} + \cdots + 19336761 ) / 2107977 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 5770954 \nu^{15} + 23758687 \nu^{14} + 84675040 \nu^{13} - 422577930 \nu^{12} - 404078407 \nu^{11} + \cdots - 63277305 ) / 2107977 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 6298885 \nu^{15} + 26194499 \nu^{14} + 91389299 \nu^{13} - 464963603 \nu^{12} - 423960647 \nu^{11} + \cdots - 65792685 ) / 2107977 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 8350130 \nu^{15} - 34515709 \nu^{14} - 121836955 \nu^{13} + 612761083 \nu^{12} + 574120882 \nu^{11} + \cdots + 87896481 ) / 2107977 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 8806535 \nu^{15} - 35735104 \nu^{14} - 130319929 \nu^{13} + 634081177 \nu^{12} + 636977887 \nu^{11} + \cdots + 83357961 ) / 2107977 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 9853789 \nu^{15} - 40428820 \nu^{14} - 144713263 \nu^{13} + 717979203 \nu^{12} + 693391444 \nu^{11} + \cdots + 98821953 ) / 2107977 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 197646 \nu^{15} - 805955 \nu^{14} - 2917085 \nu^{13} + 14310061 \nu^{12} + 14162693 \nu^{11} + \cdots + 2028567 ) / 34557 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 4194779 \nu^{15} + 17199689 \nu^{14} + 61552148 \nu^{13} - 305099798 \nu^{12} - 294602171 \nu^{11} + \cdots - 40680671 ) / 702659 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 15912006 \nu^{15} + 65304908 \nu^{14} + 233424776 \nu^{13} - 1158870763 \nu^{12} + \cdots - 151855944 ) / 2107977 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 19542249 \nu^{15} - 79686754 \nu^{14} - 288268384 \nu^{13} + 1414438175 \nu^{12} + \cdots + 190691154 ) / 2107977 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} + \beta_{12} - \beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} + \beta_{14} + \beta_{11} - \beta_{10} - \beta_{8} + 2\beta_{7} + 7\beta_{4} + 6\beta_{3} - \beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{14} + 9 \beta_{13} + 9 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + \beta_{9} - 8 \beta_{8} + \cdots + 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{15} + 11 \beta_{14} + 3 \beta_{13} + 2 \beta_{12} + 13 \beta_{11} - 8 \beta_{10} + 2 \beta_{9} + \cdots + 83 \)
|
| \(\nu^{7}\) | \(=\) |
\( 26 \beta_{14} + 68 \beta_{13} + 68 \beta_{12} + 31 \beta_{11} + 22 \beta_{10} + 13 \beta_{9} - 56 \beta_{8} + \cdots + 40 \)
|
| \(\nu^{8}\) | \(=\) |
\( 78 \beta_{15} + 98 \beta_{14} + 42 \beta_{13} + 32 \beta_{12} + 124 \beta_{11} - 51 \beta_{10} + \cdots + 480 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2 \beta_{15} + 248 \beta_{14} + 489 \beta_{13} + 495 \beta_{12} + 324 \beta_{11} + 184 \beta_{10} + \cdots + 215 \)
|
| \(\nu^{10}\) | \(=\) |
\( 553 \beta_{15} + 806 \beta_{14} + 417 \beta_{13} + 357 \beta_{12} + 1062 \beta_{11} - 287 \beta_{10} + \cdots + 2842 \)
|
| \(\nu^{11}\) | \(=\) |
\( 34 \beta_{15} + 2103 \beta_{14} + 3453 \beta_{13} + 3569 \beta_{12} + 2898 \beta_{11} + 1413 \beta_{10} + \cdots + 1126 \)
|
| \(\nu^{12}\) | \(=\) |
\( 3741 \beta_{15} + 6359 \beta_{14} + 3631 \beta_{13} + 3416 \beta_{12} + 8642 \beta_{11} - 1413 \beta_{10} + \cdots + 17063 \)
|
| \(\nu^{13}\) | \(=\) |
\( 374 \beta_{15} + 16832 \beta_{14} + 24219 \beta_{13} + 25665 \beta_{12} + 24009 \beta_{11} + 10522 \beta_{10} + \cdots + 5814 \)
|
| \(\nu^{14}\) | \(=\) |
\( 24674 \beta_{15} + 48950 \beta_{14} + 29666 \beta_{13} + 30054 \beta_{12} + 68283 \beta_{11} + \cdots + 103361 \)
|
| \(\nu^{15}\) | \(=\) |
\( 3419 \beta_{15} + 130479 \beta_{14} + 169622 \beta_{13} + 184433 \beta_{12} + 190709 \beta_{11} + \cdots + 29677 \)
|
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 |
|
−2.50334 | −1.35778 | 4.26669 | 3.01729 | 3.39897 | 2.65496 | −5.67428 | −1.15645 | −7.55329 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.97162 | 1.22817 | 1.88727 | −1.66661 | −2.42149 | −2.98016 | 0.222255 | −1.49159 | 3.28592 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.64884 | 2.09020 | 0.718662 | −1.43417 | −3.44639 | 1.19792 | 2.11272 | 1.36893 | 2.36472 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.57355 | 0.310580 | 0.476071 | −1.11491 | −0.488714 | −4.94554 | 2.39798 | −2.90354 | 1.75437 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.21251 | −1.64090 | −0.529816 | 0.514718 | 1.98962 | 4.32543 | 3.06743 | −0.307433 | −0.624101 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.806392 | 0.304575 | −1.34973 | −0.775876 | −0.245607 | 0.0561731 | 2.70120 | −2.90723 | 0.625660 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.0292067 | −1.34632 | −1.99915 | −0.578559 | 0.0393215 | −3.17413 | 0.116802 | −1.18743 | 0.0168978 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.480002 | 1.68317 | −1.76960 | −3.59267 | 0.807924 | 3.90642 | −1.80942 | −0.166950 | −1.72449 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.885892 | −0.936447 | −1.21520 | 2.84875 | −0.829591 | 0.0947841 | −2.84832 | −2.12307 | 2.52368 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.892823 | 1.64555 | −1.20287 | 0.610081 | 1.46918 | 0.681125 | −2.85959 | −0.292175 | 0.544695 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.33501 | −2.85085 | −0.217749 | −2.94702 | −3.80591 | −3.02640 | −2.96072 | 5.12735 | −3.93430 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.39168 | 2.62064 | −0.0632328 | −0.263251 | 3.64709 | 0.409268 | −2.87136 | 3.86775 | −0.366360 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.12973 | −2.31072 | 2.53577 | 1.23124 | −4.92121 | 3.86982 | 1.14104 | 2.33940 | 2.62222 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.37540 | 2.18487 | 3.64253 | −3.76583 | 5.18994 | −2.89361 | 3.90166 | 1.77365 | −8.94534 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.54592 | −2.67533 | 4.48171 | −2.88840 | −6.81119 | −0.0556586 | 6.31824 | 4.15741 | −7.35365 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.70899 | −0.949406 | 5.33864 | −1.19478 | −2.57193 | −4.12040 | 9.04435 | −2.09863 | −3.23664 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(61\) | \( +1 \) |
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 | 3721.2.a.l | 16 | |
| 61.b | even | 2 | 1 | 3721.2.a.j | 16 | ||
| 61.k | even | 30 | 2 | 61.2.i.a | ✓ | 32 | |
| 183.v | odd | 30 | 2 | 549.2.bl.b | 32 | ||
| 244.v | odd | 30 | 2 | 976.2.bw.c | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 61.2.i.a | ✓ | 32 | 61.k | even | 30 | 2 | |
| 549.2.bl.b | 32 | 183.v | odd | 30 | 2 | ||
| 976.2.bw.c | 32 | 244.v | odd | 30 | 2 | ||
| 3721.2.a.j | 16 | 61.b | even | 2 | 1 | ||
| 3721.2.a.l | 16 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} - 5 T_{2}^{15} - 11 T_{2}^{14} + 86 T_{2}^{13} + 5 T_{2}^{12} - 562 T_{2}^{11} + 362 T_{2}^{10} + \cdots - 9 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3721))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 5 T^{15} + \cdots - 9 \)
|
| $3$ |
\( T^{16} + 2 T^{15} + \cdots + 181 \)
|
| $5$ |
\( T^{16} + 12 T^{15} + \cdots - 144 \)
|
| $7$ |
\( T^{16} + 4 T^{15} + \cdots - 29 \)
|
| $11$ |
\( T^{16} - 9 T^{15} + \cdots - 279 \)
|
| $13$ |
\( T^{16} - 109 T^{14} + \cdots - 310259 \)
|
| $17$ |
\( T^{16} + 4 T^{15} + \cdots + 399591 \)
|
| $19$ |
\( T^{16} + 19 T^{15} + \cdots + 74955571 \)
|
| $23$ |
\( T^{16} + 4 T^{15} + \cdots + 12640176 \)
|
| $29$ |
\( T^{16} + 4 T^{15} + \cdots + 48464181 \)
|
| $31$ |
\( T^{16} + \cdots + 21375239311 \)
|
| $37$ |
\( T^{16} + \cdots - 2683457969 \)
|
| $41$ |
\( T^{16} + \cdots + 687136041 \)
|
| $43$ |
\( T^{16} + \cdots - 38912496269 \)
|
| $47$ |
\( T^{16} + \cdots + 96685425621 \)
|
| $53$ |
\( T^{16} + \cdots + 66709118691 \)
|
| $59$ |
\( T^{16} + \cdots + 950361967251 \)
|
| $61$ |
\( T^{16} \)
|
| $67$ |
\( T^{16} + \cdots - 7098991769 \)
|
| $71$ |
\( T^{16} + \cdots + 1797540336 \)
|
| $73$ |
\( T^{16} - 6 T^{15} + \cdots + 63165691 \)
|
| $79$ |
\( T^{16} + \cdots - 65979493529 \)
|
| $83$ |
\( T^{16} + \cdots + 83550563361 \)
|
| $89$ |
\( T^{16} + \cdots - 11114295249069 \)
|
| $97$ |
\( T^{16} - 13 T^{15} + \cdots + 33725911 \)
|
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