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Properties

Label 37.8.a.a
Level $37$
Weight $8$
Character orbit 37.a
Self dual yes
Analytic conductor $11.558$
Analytic rank $1$
Dimension $10$
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] = [37,8,Mod(1,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 37.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: \(11.5582459429\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} - 905 x^{8} + 4018 x^{7} + 291290 x^{6} - 1367036 x^{5} - 39566544 x^{4} + 175732512 x^{3} + 2143908736 x^{2} - 5929608704 x - 45399525376 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{2} + ( - \beta_{7} + \beta_1 - 10) q^{3} + (\beta_{7} + \beta_{6} + 3 \beta_1 + 59) q^{4} + (\beta_{7} - 2 \beta_{6} - \beta_{4} - 5 \beta_1 - 60) q^{5} + (9 \beta_{7} - 2 \beta_{6} + 3 \beta_{4} - \beta_{3} + 13 \beta_1 - 82) q^{6} + (6 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 4 \beta_{3} - \beta_{2} + 32 \beta_1 - 62) q^{7} + ( - \beta_{9} - 4 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + \cdots - 380) q^{8}+ \cdots + (6 \beta_{9} - \beta_{8} + 9 \beta_{7} + 8 \beta_{6} - 4 \beta_{5} - 6 \beta_{4} + \cdots + 587) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{2} + ( - \beta_{7} + \beta_1 - 10) q^{3} + (\beta_{7} + \beta_{6} + 3 \beta_1 + 59) q^{4} + (\beta_{7} - 2 \beta_{6} - \beta_{4} - 5 \beta_1 - 60) q^{5} + (9 \beta_{7} - 2 \beta_{6} + 3 \beta_{4} - \beta_{3} + 13 \beta_1 - 82) q^{6} + (6 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 4 \beta_{3} - \beta_{2} + 32 \beta_1 - 62) q^{7} + ( - \beta_{9} - 4 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + \cdots - 380) q^{8}+ \cdots + ( - 21932 \beta_{9} - 15469 \beta_{8} + 53964 \beta_{7} + \cdots - 1723977) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 24 q^{2} - 95 q^{3} + 602 q^{4} - 624 q^{5} - 777 q^{6} - 501 q^{7} - 3810 q^{8} + 6181 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 24 q^{2} - 95 q^{3} + 602 q^{4} - 624 q^{5} - 777 q^{6} - 501 q^{7} - 3810 q^{8} + 6181 q^{9} + 8595 q^{10} - 8325 q^{11} - 19645 q^{12} - 17108 q^{13} - 65418 q^{14} - 55756 q^{15} - 56998 q^{16} - 72924 q^{17} - 156165 q^{18} - 47786 q^{19} - 226209 q^{20} - 65313 q^{21} - 138973 q^{22} - 148086 q^{23} - 68031 q^{24} + 108736 q^{25} - 60237 q^{26} - 87329 q^{27} + 219974 q^{28} - 164154 q^{29} + 78864 q^{30} - 189560 q^{31} - 30114 q^{32} - 179737 q^{33} + 532624 q^{34} - 705156 q^{35} + 1923693 q^{36} + 506530 q^{37} + 1256412 q^{38} + 1322800 q^{39} + 2936777 q^{40} + 814263 q^{41} + 3415826 q^{42} - 590572 q^{43} + 610311 q^{44} - 250574 q^{45} + 2903897 q^{46} - 1534185 q^{47} + 2082419 q^{48} - 214337 q^{49} - 2313525 q^{50} + 722138 q^{51} + 149159 q^{52} - 2518209 q^{53} + 1095990 q^{54} - 3482468 q^{55} - 3645834 q^{56} - 9225638 q^{57} + 5626023 q^{58} - 5894748 q^{59} - 1289832 q^{60} - 2569480 q^{61} - 863697 q^{62} - 2836574 q^{63} - 4093742 q^{64} - 6774600 q^{65} + 17251556 q^{66} - 6983232 q^{67} - 8114412 q^{68} - 11557564 q^{69} + 8982748 q^{70} - 5013963 q^{71} - 7567137 q^{72} - 11678449 q^{73} - 1215672 q^{74} - 6586901 q^{75} + 4912252 q^{76} + 1333113 q^{77} - 7352119 q^{78} - 3853378 q^{79} - 11975661 q^{80} - 7381718 q^{81} + 564093 q^{82} - 15677895 q^{83} + 4781738 q^{84} + 11909320 q^{85} + 34274010 q^{86} - 12611710 q^{87} + 14448317 q^{88} - 25836 q^{89} + 64591590 q^{90} + 12335744 q^{91} + 7579845 q^{92} + 4592632 q^{93} + 26251718 q^{94} + 11723664 q^{95} + 42299113 q^{96} + 4648834 q^{97} + 15230184 q^{98} - 16904018 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} - 905 x^{8} + 4018 x^{7} + 291290 x^{6} - 1367036 x^{5} - 39566544 x^{4} + 175732512 x^{3} + 2143908736 x^{2} - 5929608704 x - 45399525376 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6903850151 \nu^{9} - 44030216064 \nu^{8} + 5848661370847 \nu^{7} + 31740379054046 \nu^{6} + \cdots - 29\!\cdots\!24 ) / 52\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 62575071425 \nu^{9} - 287016653328 \nu^{8} + 51295267891465 \nu^{7} + 193693281740162 \nu^{6} + \cdots - 13\!\cdots\!64 ) / 31\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 824744993 \nu^{9} + 5598640392 \nu^{8} - 698144499529 \nu^{7} - 3961595322746 \nu^{6} + 204042096531554 \nu^{5} + \cdots + 34\!\cdots\!60 ) / 372548736817152 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23822773327 \nu^{9} + 160382537808 \nu^{8} - 20325864531719 \nu^{7} - 115632112359166 \nu^{6} + \cdots + 10\!\cdots\!64 ) / 10\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 169096231607 \nu^{9} - 1115891967696 \nu^{8} + 141291187862959 \nu^{7} + 819419713942478 \nu^{6} + \cdots - 74\!\cdots\!88 ) / 31\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 169096231607 \nu^{9} + 1115891967696 \nu^{8} - 141291187862959 \nu^{7} - 819419713942478 \nu^{6} + \cdots + 74\!\cdots\!44 ) / 31\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 81089504555 \nu^{9} + 473877845808 \nu^{8} - 67919217844483 \nu^{7} - 344607611020502 \nu^{6} + \cdots + 32\!\cdots\!72 ) / 78\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 396026017655 \nu^{9} + 2579512909920 \nu^{8} - 331845919248943 \nu^{7} + \cdots + 17\!\cdots\!04 ) / 15\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - \beta _1 + 183 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 6\beta_{7} - 2\beta_{6} - 5\beta_{5} + 2\beta_{4} + \beta_{3} - 4\beta_{2} + 258\beta _1 - 214 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 11 \beta_{9} + \beta_{8} + 374 \beta_{7} + 335 \beta_{6} - 14 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 77 \beta_{2} - 532 \beta _1 + 46927 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 329 \beta_{9} + 123 \beta_{8} - 1888 \beta_{7} - 469 \beta_{6} - 1956 \beta_{5} + 1055 \beta_{4} + 523 \beta_{3} - 939 \beta_{2} + 74374 \beta _1 - 111261 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 7185 \beta_{9} + 495 \beta_{8} + 128160 \beta_{7} + 102799 \beta_{6} - 4486 \beta_{5} - 785 \beta_{4} + 401 \beta_{3} - 39639 \beta_{2} - 201250 \beta _1 + 13564839 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 83577 \beta_{9} + 72597 \beta_{8} - 520180 \beta_{7} - 131975 \beta_{6} - 654390 \beta_{5} + 425293 \beta_{4} + 201703 \beta_{3} - 140629 \beta_{2} + 22416470 \beta _1 - 41517167 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3247261 \beta_{9} + 23355 \beta_{8} + 43190140 \beta_{7} + 31563511 \beta_{6} - 902246 \beta_{5} + 66803 \beta_{4} - 265667 \beta_{3} - 15489787 \beta_{2} + \cdots + 4109646639 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 17596401 \beta_{9} + 32119609 \beta_{8} - 138258172 \beta_{7} - 46361323 \beta_{6} - 211558658 \beta_{5} + 155327385 \beta_{4} + 70077991 \beta_{3} + \cdots - 14473144979 \) 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
17.8339
16.8983
10.5631
10.4614
9.55485
−5.27803
−5.89746
−14.2020
−17.7200
−18.2140
−19.8339 −69.2191 265.384 −437.776 1372.88 1289.75 −2724.87 2604.28 8682.81
1.2 −18.8983 75.2961 229.145 −439.542 −1422.97 380.249 −1911.48 3482.50 8306.59
1.3 −12.5631 −77.5936 29.8308 318.311 974.814 −75.8943 1233.31 3833.76 −3998.96
1.4 −12.4614 49.0521 27.2868 −16.1140 −611.258 −704.986 1255.03 219.106 200.803
1.5 −11.5549 −22.2601 5.51464 118.339 257.212 925.681 1415.30 −1691.49 −1367.39
1.6 3.27803 −7.54665 −117.254 243.887 −24.7382 767.318 −803.953 −2130.05 799.468
1.7 3.89746 51.1724 −112.810 −91.5007 199.442 −1347.68 −938.547 431.610 −356.620
1.8 12.2020 8.53392 20.8894 −415.456 104.131 362.670 −1306.97 −2114.17 −5069.41
1.9 15.7200 −69.5612 119.119 316.724 −1093.50 −1289.70 −139.607 2651.76 4978.91
1.10 16.2140 −32.8739 134.894 −220.871 −533.017 −808.413 111.778 −1106.31 −3581.21
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(37\) \(-1\)

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 37.8.a.a ✓ 10
3.b odd 2 1 333.8.a.c 10
4.b odd 2 1 592.8.a.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.8.a.a ✓ 10 1.a even 1 1 trivial
333.8.a.c 10 3.b odd 2 1
592.8.a.f 10 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 24 T_{2}^{9} - 653 T_{2}^{8} - 16962 T_{2}^{7} + 139726 T_{2}^{6} + 4135692 T_{2}^{5} - 10527856 T_{2}^{4} - 394812000 T_{2}^{3} + 314897184 T_{2}^{2} + 11292902208 T_{2} - 26941953024 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(37))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 24 T^{9} + \cdots - 26941953024 \) Copy content Toggle raw display
$3$ \( T^{10} + 95 T^{9} + \cdots + 33\!\cdots\!44 \) Copy content Toggle raw display
$5$ \( T^{10} + 624 T^{9} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + 501 T^{9} + \cdots - 94\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{10} + 8325 T^{9} + \cdots - 11\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( T^{10} + 17108 T^{9} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{10} + 72924 T^{9} + \cdots + 53\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( T^{10} + 47786 T^{9} + \cdots + 50\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{10} + 148086 T^{9} + \cdots - 83\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{10} + 164154 T^{9} + \cdots - 13\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{10} + 189560 T^{9} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T - 50653)^{10} \) Copy content Toggle raw display
$41$ \( T^{10} - 814263 T^{9} + \cdots - 28\!\cdots\!98 \) Copy content Toggle raw display
$43$ \( T^{10} + 590572 T^{9} + \cdots + 30\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{10} + 1534185 T^{9} + \cdots - 32\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{10} + 2518209 T^{9} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{10} + 5894748 T^{9} + \cdots - 91\!\cdots\!52 \) Copy content Toggle raw display
$61$ \( T^{10} + 2569480 T^{9} + \cdots - 19\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{10} + 6983232 T^{9} + \cdots + 31\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{10} + 5013963 T^{9} + \cdots + 38\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{10} + 11678449 T^{9} + \cdots + 32\!\cdots\!06 \) Copy content Toggle raw display
$79$ \( T^{10} + 3853378 T^{9} + \cdots - 64\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{10} + 15677895 T^{9} + \cdots - 30\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{10} + 25836 T^{9} + \cdots + 74\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{10} - 4648834 T^{9} + \cdots + 48\!\cdots\!24 \) Copy content Toggle raw display
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