Properties

Label 4334.2.a.c
Level $4334$
Weight $2$
Character orbit 4334.a
Self dual yes
Analytic conductor $34.607$
Analytic rank $1$
Dimension $17$
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] = [4334,2,Mod(1,4334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4334, 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("4334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4334 = 2 \cdot 11 \cdot 197 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4334.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: \(34.6071642360\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 5 x^{16} - 19 x^{15} + 121 x^{14} + 112 x^{13} - 1172 x^{12} - 25 x^{11} + 5845 x^{10} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{3} q^{5} - \beta_1 q^{6} + (\beta_{10} - 1) q^{7} - q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{3} q^{5} - \beta_1 q^{6} + (\beta_{10} - 1) q^{7} - q^{8} + (\beta_{2} + 1) q^{9} - \beta_{3} q^{10} - q^{11} + \beta_1 q^{12} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1) q^{13}+ \cdots + ( - \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 17 q^{2} + 5 q^{3} + 17 q^{4} + 6 q^{5} - 5 q^{6} - 9 q^{7} - 17 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - 17 q^{2} + 5 q^{3} + 17 q^{4} + 6 q^{5} - 5 q^{6} - 9 q^{7} - 17 q^{8} + 12 q^{9} - 6 q^{10} - 17 q^{11} + 5 q^{12} - 16 q^{13} + 9 q^{14} + 17 q^{16} - 8 q^{17} - 12 q^{18} - 23 q^{19} + 6 q^{20} - 15 q^{21} + 17 q^{22} + 12 q^{23} - 5 q^{24} + 11 q^{25} + 16 q^{26} + 17 q^{27} - 9 q^{28} - 8 q^{31} - 17 q^{32} - 5 q^{33} + 8 q^{34} + 6 q^{35} + 12 q^{36} - 7 q^{37} + 23 q^{38} - 9 q^{39} - 6 q^{40} - 27 q^{41} + 15 q^{42} - 13 q^{43} - 17 q^{44} - 11 q^{45} - 12 q^{46} + 23 q^{47} + 5 q^{48} - 8 q^{49} - 11 q^{50} - 40 q^{51} - 16 q^{52} + 14 q^{53} - 17 q^{54} - 6 q^{55} + 9 q^{56} - 18 q^{57} + 2 q^{59} - 49 q^{61} + 8 q^{62} - 42 q^{63} + 17 q^{64} - 57 q^{65} + 5 q^{66} - 5 q^{67} - 8 q^{68} - 9 q^{69} - 6 q^{70} - 5 q^{71} - 12 q^{72} - 54 q^{73} + 7 q^{74} + 7 q^{75} - 23 q^{76} + 9 q^{77} + 9 q^{78} - 11 q^{79} + 6 q^{80} - 35 q^{81} + 27 q^{82} - 8 q^{83} - 15 q^{84} - 65 q^{85} + 13 q^{86} - 20 q^{87} + 17 q^{88} - 9 q^{89} + 11 q^{90} - 9 q^{91} + 12 q^{92} - 50 q^{93} - 23 q^{94} - 27 q^{95} - 5 q^{96} - 42 q^{97} + 8 q^{98} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 5 x^{16} - 19 x^{15} + 121 x^{14} + 112 x^{13} - 1172 x^{12} - 25 x^{11} + 5845 x^{10} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 91723570911 \nu^{16} + 786165390578 \nu^{15} - 276196522876 \nu^{14} + \cdots - 4964148712500 ) / 20274514939304 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 509076733869 \nu^{16} + 1184304247799 \nu^{15} + 13788245789985 \nu^{14} + \cdots - 67533919204240 ) / 40549029878608 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 308784775837 \nu^{16} + 1021132038096 \nu^{15} + 7484292506002 \nu^{14} + \cdots + 162671423964068 ) / 20274514939304 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 170336316612 \nu^{16} + 101241374385 \nu^{15} + 6139477315997 \nu^{14} + \cdots + 7745938632420 ) / 10137257469652 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 792005346607 \nu^{16} - 3155392683109 \nu^{15} - 17190975335075 \nu^{14} + \cdots - 71489484909776 ) / 40549029878608 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 551710330124 \nu^{16} - 2209375896877 \nu^{15} - 11895288323205 \nu^{14} + \cdots + 55249696318436 ) / 20274514939304 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1123737079895 \nu^{16} + 4428902062779 \nu^{15} + 24984965598945 \nu^{14} + \cdots - 98451203132072 ) / 40549029878608 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1187737792121 \nu^{16} + 4528746272453 \nu^{15} + 26996550686535 \nu^{14} + \cdots + 178861298752392 ) / 40549029878608 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1445357887021 \nu^{16} - 4190428658205 \nu^{15} - 38465640240887 \nu^{14} + \cdots + 23820479731352 ) / 40549029878608 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 383736935734 \nu^{16} - 1475838823975 \nu^{15} - 8493223265449 \nu^{14} + \cdots - 12473615943512 ) / 10137257469652 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1590141174737 \nu^{16} + 4717981222477 \nu^{15} + 41714738148287 \nu^{14} + \cdots + 16731542670136 ) / 40549029878608 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1962307634045 \nu^{16} + 6784781527537 \nu^{15} + 46859464594843 \nu^{14} + \cdots - 125227015713032 ) / 40549029878608 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2185052574659 \nu^{16} - 7277989925923 \nu^{15} - 53559488228625 \nu^{14} + \cdots + 119091380419960 ) / 40549029878608 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1162170575998 \nu^{16} - 4066576751287 \nu^{15} - 27394453721783 \nu^{14} + \cdots + 3463324569132 ) / 20274514939304 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{14} - \beta_{11} - \beta_{9} - \beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{16} - 2 \beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} - 2 \beta_{9} + 3 \beta_{7} + \cdots + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{16} - 11 \beta_{15} - 11 \beta_{14} + 3 \beta_{13} + 5 \beta_{12} - 13 \beta_{11} + 2 \beta_{10} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 13 \beta_{16} - 27 \beta_{15} - 14 \beta_{14} + 5 \beta_{13} + 19 \beta_{12} - 21 \beta_{11} + \cdots + 185 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 20 \beta_{16} - 107 \beta_{15} - 110 \beta_{14} + 55 \beta_{13} + 87 \beta_{12} - 145 \beta_{11} + \cdots + 186 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 142 \beta_{16} - 299 \beta_{15} - 177 \beta_{14} + 116 \beta_{13} + 266 \beta_{12} - 296 \beta_{11} + \cdots + 1509 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 290 \beta_{16} - 1057 \beta_{15} - 1114 \beta_{14} + 750 \beta_{13} + 1123 \beta_{12} - 1552 \beta_{11} + \cdots + 2149 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1522 \beta_{16} - 3181 \beta_{15} - 2208 \beta_{14} + 1846 \beta_{13} + 3320 \beta_{12} + \cdots + 13192 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3730 \beta_{16} - 10748 \beta_{15} - 11573 \beta_{14} + 9202 \beta_{13} + 13093 \beta_{12} + \cdots + 24013 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 16465 \beta_{16} - 33664 \beta_{15} - 26941 \beta_{14} + 25029 \beta_{13} + 39099 \beta_{12} + \cdots + 121681 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 45210 \beta_{16} - 111677 \beta_{15} - 122645 \beta_{14} + 107366 \beta_{13} + 146245 \beta_{12} + \cdots + 263867 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 179749 \beta_{16} - 357177 \beta_{15} - 319955 \beta_{14} + 311498 \beta_{13} + 445290 \beta_{12} + \cdots + 1169988 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 528961 \beta_{16} - 1176068 \beta_{15} - 1315430 \beta_{14} + 1218013 \beta_{13} + 1601403 \beta_{12} + \cdots + 2873549 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1969972 \beta_{16} - 3802503 \beta_{15} - 3707046 \beta_{14} + 3684622 \beta_{13} + 4967488 \beta_{12} + \cdots + 11607813 \) Copy content Toggle raw display

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
−2.70875
−2.21770
−2.12974
−1.86328
−1.29060
−1.22229
−0.704982
0.00685268
0.390704
0.983229
1.49756
1.67869
1.82096
2.18116
2.50775
2.79763
3.27281
−1.00000 −2.70875 1.00000 1.80981 2.70875 −1.02803 −1.00000 4.33730 −1.80981
1.2 −1.00000 −2.21770 1.00000 −0.358958 2.21770 1.80298 −1.00000 1.91818 0.358958
1.3 −1.00000 −2.12974 1.00000 −1.71658 2.12974 1.55260 −1.00000 1.53579 1.71658
1.4 −1.00000 −1.86328 1.00000 −0.189598 1.86328 −4.05138 −1.00000 0.471817 0.189598
1.5 −1.00000 −1.29060 1.00000 −3.65188 1.29060 −4.10704 −1.00000 −1.33435 3.65188
1.6 −1.00000 −1.22229 1.00000 4.12347 1.22229 −0.673376 −1.00000 −1.50600 −4.12347
1.7 −1.00000 −0.704982 1.00000 3.10328 0.704982 3.48081 −1.00000 −2.50300 −3.10328
1.8 −1.00000 0.00685268 1.00000 −0.0502054 −0.00685268 3.10390 −1.00000 −2.99995 0.0502054
1.9 −1.00000 0.390704 1.00000 0.981263 −0.390704 −0.959354 −1.00000 −2.84735 −0.981263
1.10 −1.00000 0.983229 1.00000 1.92930 −0.983229 2.08175 −1.00000 −2.03326 −1.92930
1.11 −1.00000 1.49756 1.00000 2.18758 −1.49756 −3.55528 −1.00000 −0.757323 −2.18758
1.12 −1.00000 1.67869 1.00000 −2.31468 −1.67869 0.643391 −1.00000 −0.181998 2.31468
1.13 −1.00000 1.82096 1.00000 −3.88691 −1.82096 −1.09055 −1.00000 0.315898 3.88691
1.14 −1.00000 2.18116 1.00000 4.06589 −2.18116 −1.87368 −1.00000 1.75745 −4.06589
1.15 −1.00000 2.50775 1.00000 1.44362 −2.50775 −2.73850 −1.00000 3.28879 −1.44362
1.16 −1.00000 2.79763 1.00000 −0.756604 −2.79763 2.12377 −1.00000 4.82674 0.756604
1.17 −1.00000 3.27281 1.00000 −0.718794 −3.27281 −3.71200 −1.00000 7.71127 0.718794
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(11\) \( +1 \)
\(197\) \( +1 \)

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 4334.2.a.c 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4334.2.a.c 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{17} - 5 T_{3}^{16} - 19 T_{3}^{15} + 121 T_{3}^{14} + 112 T_{3}^{13} - 1172 T_{3}^{12} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4334))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{17} \) Copy content Toggle raw display
$3$ \( T^{17} - 5 T^{16} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{17} - 6 T^{16} + \cdots + 59 \) Copy content Toggle raw display
$7$ \( T^{17} + 9 T^{16} + \cdots - 70204 \) Copy content Toggle raw display
$11$ \( (T + 1)^{17} \) Copy content Toggle raw display
$13$ \( T^{17} + 16 T^{16} + \cdots + 4772 \) Copy content Toggle raw display
$17$ \( T^{17} + 8 T^{16} + \cdots - 12481868 \) Copy content Toggle raw display
$19$ \( T^{17} + 23 T^{16} + \cdots - 520852 \) Copy content Toggle raw display
$23$ \( T^{17} - 12 T^{16} + \cdots - 11579504 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots + 151663444 \) Copy content Toggle raw display
$31$ \( T^{17} + 8 T^{16} + \cdots - 2634679 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots + 1082618944 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots + 504538797980 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 2209597888 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 3671094604336 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 53757717392 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 15161840124931 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 101563158656816 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 564209728960 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 44058229429 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 4111126564172 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 10646788395196 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 130397400052 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 78583924480 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 97\!\cdots\!65 \) Copy content Toggle raw display
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