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Properties

Label 2415.4.a.l
Level $2415$
Weight $4$
Character orbit 2415.a
Self dual yes
Analytic conductor $142.490$
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] = [2415,4,Mod(1,2415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2415, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2415.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2415.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: \(142.489612664\)
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: \( x^{15} - 4 x^{14} - 80 x^{13} + 304 x^{12} + 2478 x^{11} - 8718 x^{10} - 37926 x^{9} + 118408 x^{8} + \cdots + 164864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{7} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 4) q^{4} - 5 q^{5} + 3 \beta_1 q^{6} - 7 q^{7} + (\beta_{3} + 3 \beta_1 + 2) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 4) q^{4} - 5 q^{5} + 3 \beta_1 q^{6} - 7 q^{7} + (\beta_{3} + 3 \beta_1 + 2) q^{8} + 9 q^{9} - 5 \beta_1 q^{10} + (\beta_{12} - \beta_{2} - \beta_1 + 3) q^{11} + (3 \beta_{2} + 12) q^{12} + ( - \beta_{14} - \beta_{2} + 2 \beta_1 - 2) q^{13} - 7 \beta_1 q^{14} - 15 q^{15} + (\beta_{14} + \beta_{13} - \beta_{10} + \cdots + 9) q^{16}+ \cdots + (9 \beta_{12} - 9 \beta_{2} + \cdots + 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 4 q^{2} + 45 q^{3} + 56 q^{4} - 75 q^{5} + 12 q^{6} - 105 q^{7} + 48 q^{8} + 135 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 4 q^{2} + 45 q^{3} + 56 q^{4} - 75 q^{5} + 12 q^{6} - 105 q^{7} + 48 q^{8} + 135 q^{9} - 20 q^{10} + 41 q^{11} + 168 q^{12} - 17 q^{13} - 28 q^{14} - 225 q^{15} + 136 q^{16} - 215 q^{17} + 36 q^{18} - 91 q^{19} - 280 q^{20} - 315 q^{21} - 222 q^{22} - 345 q^{23} + 144 q^{24} + 375 q^{25} + 346 q^{26} + 405 q^{27} - 392 q^{28} + 234 q^{29} - 60 q^{30} + 232 q^{31} - 82 q^{32} + 123 q^{33} - 544 q^{34} + 525 q^{35} + 504 q^{36} - 559 q^{37} - 544 q^{38} - 51 q^{39} - 240 q^{40} - 261 q^{41} - 84 q^{42} - 655 q^{43} - 1050 q^{44} - 675 q^{45} - 92 q^{46} - 274 q^{47} + 408 q^{48} + 735 q^{49} + 100 q^{50} - 645 q^{51} - 1512 q^{52} - 362 q^{53} + 108 q^{54} - 205 q^{55} - 336 q^{56} - 273 q^{57} - 912 q^{58} + 543 q^{59} - 840 q^{60} - 857 q^{61} - 1612 q^{62} - 945 q^{63} - 908 q^{64} + 85 q^{65} - 666 q^{66} - 2491 q^{67} - 1174 q^{68} - 1035 q^{69} + 140 q^{70} + 1052 q^{71} + 432 q^{72} - 1279 q^{73} - 944 q^{74} + 1125 q^{75} - 1910 q^{76} - 287 q^{77} + 1038 q^{78} - 450 q^{79} - 680 q^{80} + 1215 q^{81} - 1394 q^{82} - 2645 q^{83} - 1176 q^{84} + 1075 q^{85} - 3008 q^{86} + 702 q^{87} - 3908 q^{88} - 960 q^{89} - 180 q^{90} + 119 q^{91} - 1288 q^{92} + 696 q^{93} - 486 q^{94} + 455 q^{95} - 246 q^{96} - 1514 q^{97} + 196 q^{98} + 369 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 4 x^{14} - 80 x^{13} + 304 x^{12} + 2478 x^{11} - 8718 x^{10} - 37926 x^{9} + 118408 x^{8} + \cdots + 164864 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 19\nu - 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 353954239 \nu^{14} - 19974051 \nu^{13} + 33968584397 \nu^{12} + 21057840445 \nu^{11} + \cdots + 25\!\cdots\!52 ) / 46246925701120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 128028331 \nu^{14} + 1813576561 \nu^{13} + 1977258473 \nu^{12} - 120848284735 \nu^{11} + \cdots - 516683299802112 ) / 11561731425280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 763821383 \nu^{14} + 15330943893 \nu^{13} + 33760941509 \nu^{12} - 1166422417035 \nu^{11} + \cdots + 24\!\cdots\!44 ) / 46246925701120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1166195909 \nu^{14} - 7922701359 \nu^{13} - 77162891007 \nu^{12} + 549197925425 \nu^{11} + \cdots + 947229166537728 ) / 46246925701120 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3185244661 \nu^{14} - 11997034529 \nu^{13} + 277303962223 \nu^{12} + \cdots - 13\!\cdots\!52 ) / 46246925701120 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5011794437 \nu^{14} - 11607871473 \nu^{13} + 348833977471 \nu^{12} + \cdots - 37\!\cdots\!04 ) / 46246925701120 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 656225793 \nu^{14} + 1436680217 \nu^{13} - 53027998839 \nu^{12} - 114881034135 \nu^{11} + \cdots - 210638892606464 ) / 5780865712640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2853702387 \nu^{14} - 7728534217 \nu^{13} - 225751573161 \nu^{12} + 556134219895 \nu^{11} + \cdots - 462396436489216 ) / 23123462850560 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1335529867 \nu^{14} - 4380156225 \nu^{13} - 104672154097 \nu^{12} + 323047041375 \nu^{11} + \cdots - 403135061801984 ) / 9249385140224 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 8434294271 \nu^{14} + 5569636061 \nu^{13} + 677143392813 \nu^{12} + \cdots + 658644844715008 ) / 46246925701120 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5249427977 \nu^{14} - 3036614427 \nu^{13} - 412031710651 \nu^{12} + 199427782885 \nu^{11} + \cdots - 7336736183296 ) / 23123462850560 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 19\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + \beta_{13} - \beta_{10} - \beta_{8} + 26\beta_{2} + \beta _1 + 233 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{14} + \beta_{13} - \beta_{12} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} + \cdots + 57 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 40 \beta_{14} + 44 \beta_{13} - 6 \beta_{12} + 8 \beta_{11} - 30 \beta_{10} + 4 \beta_{9} - 38 \beta_{8} + \cdots + 5158 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 118 \beta_{14} + 46 \beta_{13} - 76 \beta_{12} - 8 \beta_{11} + 98 \beta_{10} + 112 \beta_{9} + \cdots + 1314 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1293 \beta_{14} + 1507 \beta_{13} - 302 \beta_{12} + 388 \beta_{11} - 741 \beta_{10} + 228 \beta_{9} + \cdots + 121369 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4730 \beta_{14} + 1557 \beta_{13} - 3493 \beta_{12} - 324 \beta_{11} + 3486 \beta_{10} + 4354 \beta_{9} + \cdots + 30175 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 39038 \beta_{14} + 46878 \beta_{13} - 11230 \beta_{12} + 13644 \beta_{11} - 17404 \beta_{10} + \cdots + 2965894 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 163152 \beta_{14} + 47798 \beta_{13} - 130746 \beta_{12} - 8868 \beta_{11} + 109896 \beta_{10} + \cdots + 761328 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1143625 \beta_{14} + 1389093 \beta_{13} - 374952 \beta_{12} + 426384 \beta_{11} - 402757 \beta_{10} + \cdots + 74476721 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5216034 \beta_{14} + 1418649 \beta_{13} - 4398953 \beta_{12} - 199776 \beta_{11} + 3269082 \beta_{10} + \cdots + 21675789 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 32982168 \beta_{14} + 40043364 \beta_{13} - 11890534 \beta_{12} + 12609616 \beta_{11} + \cdots + 1909488218 \) 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
−5.15604
−4.55742
−3.65248
−2.93493
−2.25789
−1.51566
−0.551239
0.104752
1.41672
1.92317
2.72057
4.20218
4.45486
4.50422
5.29918
−5.15604 3.00000 18.5848 −5.00000 −15.4681 −7.00000 −54.5754 9.00000 25.7802
1.2 −4.55742 3.00000 12.7701 −5.00000 −13.6723 −7.00000 −21.7391 9.00000 22.7871
1.3 −3.65248 3.00000 5.34064 −5.00000 −10.9575 −7.00000 9.71328 9.00000 18.2624
1.4 −2.93493 3.00000 0.613790 −5.00000 −8.80478 −7.00000 21.6780 9.00000 14.6746
1.5 −2.25789 3.00000 −2.90192 −5.00000 −6.77368 −7.00000 24.6154 9.00000 11.2895
1.6 −1.51566 3.00000 −5.70278 −5.00000 −4.54698 −7.00000 20.7687 9.00000 7.57830
1.7 −0.551239 3.00000 −7.69614 −5.00000 −1.65372 −7.00000 8.65233 9.00000 2.75620
1.8 0.104752 3.00000 −7.98903 −5.00000 0.314255 −7.00000 −1.67488 9.00000 −0.523758
1.9 1.41672 3.00000 −5.99290 −5.00000 4.25016 −7.00000 −19.8240 9.00000 −7.08361
1.10 1.92317 3.00000 −4.30141 −5.00000 5.76951 −7.00000 −23.6577 9.00000 −9.61586
1.11 2.72057 3.00000 −0.598500 −5.00000 8.16171 −7.00000 −23.3928 9.00000 −13.6028
1.12 4.20218 3.00000 9.65832 −5.00000 12.6065 −7.00000 6.96854 9.00000 −21.0109
1.13 4.45486 3.00000 11.8458 −5.00000 13.3646 −7.00000 17.1326 9.00000 −22.2743
1.14 4.50422 3.00000 12.2880 −5.00000 13.5127 −7.00000 19.3140 9.00000 −22.5211
1.15 5.29918 3.00000 20.0813 −5.00000 15.8975 −7.00000 64.0211 9.00000 −26.4959
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)
\(23\) \(1\)

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 2415.4.a.l ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.4.a.l ✓ 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2415))\):

\( T_{2}^{15} - 4 T_{2}^{14} - 80 T_{2}^{13} + 304 T_{2}^{12} + 2478 T_{2}^{11} - 8718 T_{2}^{10} + \cdots + 164864 \) Copy content Toggle raw display
\( T_{11}^{15} - 41 T_{11}^{14} - 9712 T_{11}^{13} + 344881 T_{11}^{12} + 35084393 T_{11}^{11} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 4 T^{14} + \cdots + 164864 \) Copy content Toggle raw display
$3$ \( (T - 3)^{15} \) Copy content Toggle raw display
$5$ \( (T + 5)^{15} \) Copy content Toggle raw display
$7$ \( (T + 7)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 17\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 63\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( (T + 23)^{15} \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 15\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 76\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 93\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 20\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 66\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 59\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 20\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 94\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 16\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 76\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 34\!\cdots\!64 \) Copy content Toggle raw display
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