Properties

Label 145.4.c.b
Level $145$
Weight $4$
Character orbit 145.c
Analytic conductor $8.555$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [145,4,Mod(86,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("145.86");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 145.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(8.55527695083\)
Analytic rank: \(0\)
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: \( x^{16} + 95 x^{14} + 3576 x^{12} + 68256 x^{10} + 700479 x^{8} + 3754089 x^{6} + 9373424 x^{4} + 8978880 x^{2} + 2560000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} - 4) q^{4} - 5 q^{5} + ( - \beta_{4} + 3) q^{6} + ( - \beta_{5} + 2) q^{7} + (\beta_{3} - 4 \beta_1) q^{8} + ( - \beta_{9} - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} - 4) q^{4} - 5 q^{5} + ( - \beta_{4} + 3) q^{6} + ( - \beta_{5} + 2) q^{7} + (\beta_{3} - 4 \beta_1) q^{8} + ( - \beta_{9} - 8) q^{9} - 5 \beta_1 q^{10} + (\beta_{11} + \beta_{8} + \beta_{3}) q^{11} + (\beta_{14} - 3 \beta_{8} + \beta_{7} + \beta_1) q^{12} + (\beta_{9} - \beta_{6} - \beta_{4} - 1) q^{13} + ( - \beta_{13} - \beta_{12} - 2 \beta_{8} - \beta_{7} - \beta_{3} + 3 \beta_1) q^{14} - 5 \beta_{8} q^{15} + (\beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} - 4 \beta_{2} + 13) q^{16} + (\beta_{14} - \beta_{13} + 2 \beta_{11} + \beta_{7} + 4 \beta_1) q^{17} + ( - \beta_{12} - \beta_{11} - \beta_{8} + 2 \beta_{7} - 7 \beta_1) q^{18} + (2 \beta_{13} + \beta_{12} + \beta_{11} + 3 \beta_{8} - \beta_{7} + 4 \beta_1) q^{19} + ( - 5 \beta_{2} + 20) q^{20} + ( - \beta_{14} + \beta_{13} + 2 \beta_{11} + 5 \beta_{8} + 3 \beta_{7} - 2 \beta_1) q^{21} + (\beta_{15} + 2 \beta_{10} + \beta_{9} - 2 \beta_{4} - 7 \beta_{2} + 6) q^{22} + ( - \beta_{15} - 2 \beta_{9} - \beta_{5} + \beta_{4} - \beta_{2} + 2) q^{23} + (\beta_{15} - \beta_{10} + 2 \beta_{9} - \beta_{6} - \beta_{5} + 3 \beta_{4} + 4 \beta_{2} + 4) q^{24} + 25 q^{25} + (\beta_{14} + \beta_{13} + 2 \beta_{12} + 3 \beta_{11} - 10 \beta_{8} - 2 \beta_{7} - \beta_{3} + \cdots + 2 \beta_1) q^{26}+ \cdots + ( - 2 \beta_{14} - 8 \beta_{13} - 14 \beta_{12} - 7 \beta_{11} + 5 \beta_{8} + \cdots - 60 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 62 q^{4} - 80 q^{5} + 50 q^{6} + 38 q^{7} - 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 62 q^{4} - 80 q^{5} + 50 q^{6} + 38 q^{7} - 126 q^{9} - 14 q^{13} + 210 q^{16} + 310 q^{20} + 88 q^{22} + 42 q^{23} + 62 q^{24} + 400 q^{25} - 346 q^{28} + 28 q^{29} - 250 q^{30} - 460 q^{33} - 626 q^{34} - 190 q^{35} - 12 q^{36} - 292 q^{38} + 584 q^{42} + 630 q^{45} + 1894 q^{49} - 320 q^{51} - 294 q^{52} + 614 q^{53} + 1840 q^{54} - 1360 q^{57} - 644 q^{58} - 2086 q^{59} - 30 q^{62} - 1456 q^{63} - 894 q^{64} + 70 q^{65} - 1604 q^{67} + 792 q^{71} + 1720 q^{74} + 4894 q^{78} - 1050 q^{80} - 192 q^{81} + 1276 q^{82} + 4400 q^{83} - 2042 q^{86} - 2046 q^{87} - 9264 q^{88} + 212 q^{91} - 2030 q^{92} - 1816 q^{93} + 4304 q^{94} + 2234 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 95 x^{14} + 3576 x^{12} + 68256 x^{10} + 700479 x^{8} + 3754089 x^{6} + 9373424 x^{4} + 8978880 x^{2} + 2560000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 20\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2921 \nu^{14} - 424754 \nu^{12} - 43504918 \nu^{10} - 1309446602 \nu^{8} - 16740400863 \nu^{6} - 90877248464 \nu^{4} - 164377632320 \nu^{2} + \cdots - 49537244160 ) / 3578977280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4989 \nu^{14} - 1514454 \nu^{12} - 101633938 \nu^{10} - 2824000782 \nu^{8} - 36563420213 \nu^{6} - 209267664544 \nu^{4} + \cdots - 147240898560 ) / 3578977280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35659 \nu^{14} + 366634 \nu^{12} - 114129122 \nu^{10} - 4715627198 \nu^{8} - 71915285837 \nu^{6} - 472643097616 \nu^{4} + \cdots - 730450017280 ) / 3578977280 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 87491 \nu^{15} - 10479950 \nu^{13} - 482308646 \nu^{11} - 10777422106 \nu^{9} - 121259364579 \nu^{7} - 644125587684 \nu^{5} + \cdots - 1261719578880 \nu ) / 17894886400 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 117723 \nu^{15} - 11198290 \nu^{13} - 418853678 \nu^{11} - 7817776498 \nu^{9} - 75915256307 \nu^{7} - 358240615032 \nu^{5} + \cdots - 235132528640 \nu ) / 17894886400 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 206911 \nu^{14} - 17850466 \nu^{12} - 595757222 \nu^{10} - 9869743418 \nu^{8} - 87348320967 \nu^{6} - 410236605776 \nu^{4} + \cdots - 474531251200 ) / 3578977280 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 237581 \nu^{14} + 16702646 \nu^{12} + 379994162 \nu^{10} + 2330115438 \nu^{8} - 21130385083 \nu^{6} - 268095179104 \nu^{4} + \cdots - 48840913920 ) / 3578977280 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 281889 \nu^{15} - 18808690 \nu^{13} - 361754674 \nu^{11} + 253871346 \nu^{9} + 74022144239 \nu^{7} + 702410702684 \nu^{5} + \cdots + 992640807680 \nu ) / 17894886400 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 32397 \nu^{15} - 3208210 \nu^{13} - 126511802 \nu^{11} - 2533586246 \nu^{9} - 27094688877 \nu^{7} - 147344171676 \nu^{5} + \cdots - 225428352256 \nu ) / 715795456 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1107917 \nu^{15} + 105509510 \nu^{13} + 3974641362 \nu^{11} + 75632627342 \nu^{9} + 767639998053 \nu^{7} + 3997872786928 \nu^{5} + \cdots + 6291486108160 \nu ) / 17894886400 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1222067 \nu^{15} - 110577470 \nu^{13} - 3907557222 \nu^{11} - 68670886362 \nu^{9} - 630106450483 \nu^{7} - 2842134935348 \nu^{5} + \cdots - 1041262242560 \nu ) / 17894886400 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1328823 \nu^{14} + 113276498 \nu^{12} + 3691161366 \nu^{10} + 58195959754 \nu^{8} + 464998109631 \nu^{6} + 1788892394928 \nu^{4} + \cdots + 956900162560 ) / 3578977280 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 20\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} - 28\beta_{2} + 237 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{14} - \beta_{13} - 2\beta_{12} - 3\beta_{11} - 7\beta_{8} - 3\beta_{7} - 38\beta_{3} + 461\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} - 43\beta_{10} - 38\beta_{9} + 35\beta_{6} + 53\beta_{5} + 14\beta_{4} + 755\beta_{2} - 5435 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 104 \beta_{14} + 63 \beta_{13} + 110 \beta_{12} + 145 \beta_{11} + 489 \beta_{8} + 107 \beta_{7} + 1189 \beta_{3} - 11412 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 69 \beta_{15} + 1438 \beta_{10} + 1073 \beta_{9} - 978 \beta_{6} - 1928 \beta_{5} - 918 \beta_{4} - 20512 \beta_{2} + 134854 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4070 \beta_{14} - 2526 \beta_{13} - 4260 \beta_{12} - 5254 \beta_{11} - 22130 \beta_{8} - 2696 \beta_{7} - 35060 \beta_{3} + 295071 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3076 \beta_{15} - 44384 \beta_{10} - 27672 \beta_{9} + 25716 \beta_{6} + 61360 \beta_{5} + 41360 \beta_{4} + 562981 \beta_{2} - 3507524 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 142432 \beta_{14} + 86180 \beta_{13} + 144200 \beta_{12} + 171820 \beta_{11} + 843996 \beta_{8} + 56676 \beta_{7} + 1011281 \beta_{3} - 7844180 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 114812 \beta_{15} + 1325533 \beta_{10} + 692033 \beta_{9} - 665757 \beta_{6} - 1844405 \beta_{5} - 1584696 \beta_{4} - 15589468 \beta_{2} + 93923993 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4695010 \beta_{14} - 2733805 \beta_{13} - 4578026 \beta_{12} - 5355591 \beta_{11} - 29414939 \beta_{8} - 983999 \beta_{7} - 28933518 \beta_{3} + 212434437 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3917445 \beta_{15} - 38984119 \beta_{10} - 17167438 \beta_{9} + 17250095 \beta_{6} + 54020649 \beta_{5} + 55504982 \beta_{4} + 434851755 \beta_{2} - 2562480439 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 149143000 \beta_{14} + 83589563 \beta_{13} + 140472918 \beta_{12} + 162649861 \beta_{11} + 970590381 \beta_{8} + 11116519 \beta_{7} + 825830341 \beta_{3} + \cdots - 5828898692 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/145\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(117\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1
5.36866i
4.81564i
3.81405i
3.70640i
3.07662i
1.89892i
1.06661i
0.702546i
0.702546i
1.06661i
1.89892i
3.07662i
3.70640i
3.81405i
4.81564i
5.36866i
5.36866i 4.69099i −20.8225 −5.00000 25.1844 14.8930 68.8399i 4.99460 26.8433i
86.2 4.81564i 7.51293i −15.1904 −5.00000 −36.1796 5.34977 34.6262i −29.4442 24.0782i
86.3 3.81405i 6.79246i −6.54695 −5.00000 25.9068 −13.4961 5.54200i −19.1376 19.0702i
86.4 3.70640i 2.07839i −5.73740 −5.00000 7.70333 −27.6525 8.38609i 22.6803 18.5320i
86.5 3.07662i 1.74979i −1.46560 −5.00000 −5.38343 25.6180 20.1039i 23.9382 15.3831i
86.6 1.89892i 8.84442i 4.39409 −5.00000 16.7949 36.3095 23.5354i −51.2237 9.49461i
86.7 1.06661i 8.29094i 6.86234 −5.00000 −8.84320 −22.7716 15.8523i −41.7397 5.33305i
86.8 0.702546i 0.260656i 7.50643 −5.00000 −0.183122 0.749935 10.8940i 26.9321 3.51273i
86.9 0.702546i 0.260656i 7.50643 −5.00000 −0.183122 0.749935 10.8940i 26.9321 3.51273i
86.10 1.06661i 8.29094i 6.86234 −5.00000 −8.84320 −22.7716 15.8523i −41.7397 5.33305i
86.11 1.89892i 8.84442i 4.39409 −5.00000 16.7949 36.3095 23.5354i −51.2237 9.49461i
86.12 3.07662i 1.74979i −1.46560 −5.00000 −5.38343 25.6180 20.1039i 23.9382 15.3831i
86.13 3.70640i 2.07839i −5.73740 −5.00000 7.70333 −27.6525 8.38609i 22.6803 18.5320i
86.14 3.81405i 6.79246i −6.54695 −5.00000 25.9068 −13.4961 5.54200i −19.1376 19.0702i
86.15 4.81564i 7.51293i −15.1904 −5.00000 −36.1796 5.34977 34.6262i −29.4442 24.0782i
86.16 5.36866i 4.69099i −20.8225 −5.00000 25.1844 14.8930 68.8399i 4.99460 26.8433i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.4.c.b 16
29.b even 2 1 inner 145.4.c.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.4.c.b 16 1.a even 1 1 trivial
145.4.c.b 16 29.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 95 T_{2}^{14} + 3576 T_{2}^{12} + 68256 T_{2}^{10} + 700479 T_{2}^{8} + 3754089 T_{2}^{6} + 9373424 T_{2}^{4} + 8978880 T_{2}^{2} + 2560000 \) acting on \(S_{4}^{\mathrm{new}}(145, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 95 T^{14} + 3576 T^{12} + \cdots + 2560000 \) Copy content Toggle raw display
$3$ \( T^{16} + 279 T^{14} + \cdots + 276889600 \) Copy content Toggle raw display
$5$ \( (T + 5)^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 19 T^{7} - 1665 T^{6} + \cdots - 472329184)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 11680 T^{14} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + 7 T^{7} - 8177 T^{6} + \cdots - 283267515392)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 52631 T^{14} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + 69156 T^{14} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{8} - 21 T^{7} + \cdots - 69040506420288)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} - 28 T^{15} + \cdots + 12\!\cdots\!81 \) Copy content Toggle raw display
$31$ \( T^{16} + 132075 T^{14} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + 356564 T^{14} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{16} + 327492 T^{14} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{16} + 889827 T^{14} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + 1325828 T^{14} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} - 307 T^{7} + \cdots + 17\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 1043 T^{7} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 656315 T^{14} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + 802 T^{7} + \cdots + 19\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 396 T^{7} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + 2549615 T^{14} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + 5069151 T^{14} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{8} - 2200 T^{7} + \cdots - 54\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + 5882764 T^{14} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + 10761167 T^{14} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
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