Properties

Label 25.9.c.c
Level $25$
Weight $9$
Character orbit 25.c
Analytic conductor $10.184$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,9,Mod(7,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.7");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 25.c (of order \(4\), degree \(2\), 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: \(10.1844652515\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} + 124 x^{9} + 1665 x^{8} - 2456 x^{7} + 4192 x^{6} + 50576 x^{5} + 221184 x^{4} + 133760 x^{3} + 3200 x^{2} - 80000 x + 1000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{6}\cdot 5^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{6} - \beta_{5}) q^{3} + (\beta_{11} + 281 \beta_1) q^{4} + ( - \beta_{9} + 7 \beta_{8} - 533) q^{6} + (4 \beta_{4} + 18 \beta_{3} - 42 \beta_{2}) q^{7} + (60 \beta_{7} + 27 \beta_{6} - 316 \beta_{5}) q^{8} + ( - 4 \beta_{11} - 22 \beta_{10} - 1586 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{6} - \beta_{5}) q^{3} + (\beta_{11} + 281 \beta_1) q^{4} + ( - \beta_{9} + 7 \beta_{8} - 533) q^{6} + (4 \beta_{4} + 18 \beta_{3} - 42 \beta_{2}) q^{7} + (60 \beta_{7} + 27 \beta_{6} - 316 \beta_{5}) q^{8} + ( - 4 \beta_{11} - 22 \beta_{10} - 1586 \beta_1) q^{9} + ( - 22 \beta_{9} + 21 \beta_{8} - 9843) q^{11} + (812 \beta_{4} - 233 \beta_{3} + 140 \beta_{2}) q^{12} + (258 \beta_{7} - 270 \beta_{6} + 906 \beta_{5}) q^{13} + (28 \beta_{11} + 60 \beta_{10} + 24132 \beta_1) q^{14} + (147 \beta_{9} - 288 \beta_{8} - 100339) q^{16} + (1149 \beta_{4} + 1134 \beta_{3} - 134 \beta_{2}) q^{17} + (2928 \beta_{7} + 1344 \beta_{6} + 3234 \beta_{5}) q^{18} + ( - 214 \beta_{11} + 369 \beta_{10} - 14875 \beta_1) q^{19} + ( - 84 \beta_{9} + 410 \beta_{8} - 113788) q^{21} + (4344 \beta_{4} - 1980 \beta_{3} + 3903 \beta_{2}) q^{22} + ( - 2154 \beta_{7} - 3294 \beta_{6} - 7822 \beta_{5}) q^{23} + (649 \beta_{11} - 1576 \beta_{10} + 105625 \beta_1) q^{24} + ( - 918 \beta_{9} + 306 \beta_{8} + 448002) q^{26} + ( - 6345 \beta_{4} - 2769 \beta_{3} - 23127 \beta_{2}) q^{27} + ( - 7984 \beta_{7} + 1404 \beta_{6} - 22848 \beta_{5}) q^{28} + ( - 740 \beta_{11} + 978 \beta_{10} - 401550 \beta_1) q^{29} + (1244 \beta_{9} + 2358 \beta_{8} + 268642) q^{31} + ( - 34932 \beta_{4} + 16065 \beta_{3} + 55884 \beta_{2}) q^{32} + (6939 \beta_{7} + 13188 \beta_{6} + 40308 \beta_{5}) q^{33} + (149 \beta_{11} + 1089 \beta_{10} + 238647 \beta_1) q^{34} + (2062 \beta_{9} - 8528 \beta_{8} + 1206466) q^{36} + ( - 20464 \beta_{4} - 14094 \beta_{3} - 85098 \beta_{2}) q^{37} + ( - 65976 \beta_{7} - 30132 \beta_{6} + 69031 \beta_{5}) q^{38} + (912 \beta_{11} + 5988 \beta_{10} - 477282 \beta_1) q^{39} + ( - 4016 \beta_{9} - 2712 \beta_{8} + 911697) q^{41} + (64080 \beta_{4} - 29328 \beta_{3} + 98364 \beta_{2}) q^{42} + (64228 \beta_{7} + 4698 \beta_{6} - 7998 \beta_{5}) q^{43} + ( - 3211 \beta_{11} - 15576 \beta_{10} + 609117 \beta_1) q^{44} + (2374 \beta_{9} + 19638 \beta_{8} - 4272738) q^{46} + ( - 2382 \beta_{4} + 50112 \beta_{3} + 58680 \beta_{2}) q^{47} + (58012 \beta_{7} + 61891 \beta_{6} - 295652 \beta_{5}) q^{48} + ( - 2688 \beta_{11} + 4464 \beta_{10} - 3012049 \beta_1) q^{49} + ( - 8162 \beta_{9} + 1967 \beta_{8} - 3968473) q^{51} + (165192 \beta_{4} + 24138 \beta_{3} - 476472 \beta_{2}) q^{52} + (125742 \beta_{7} - 45036 \beta_{6} - 139196 \beta_{5}) q^{53} + (19551 \beta_{11} + 7959 \beta_{10} + 11746725 \beta_1) q^{54} + (9100 \beta_{9} + 2976 \beta_{8} - 5393100) q^{56} + (25733 \beta_{4} - 35618 \beta_{3} + 496106 \beta_{2}) q^{57} + ( - 185232 \beta_{7} - 84528 \beta_{6} + 595374 \beta_{5}) q^{58} + (9312 \beta_{11} - 28980 \beta_{10} - 4117950 \beta_1) q^{59} + (5500 \beta_{9} - 24750 \beta_{8} - 1966468) q^{61} + (264912 \beta_{4} - 122040 \beta_{3} + 145238 \beta_{2}) q^{62} + (266808 \beta_{7} + 44412 \beta_{6} + 384300 \beta_{5}) q^{63} + ( - 69249 \beta_{11} + 95328 \beta_{10} - 5802209 \beta_1) q^{64} + ( - 20181 \beta_{9} - 73569 \beta_{8} + 22061319) q^{66} + ( - 614361 \beta_{4} + 78939 \beta_{3} - 286419 \beta_{2}) q^{67} + ( - 442020 \beta_{7} + 222453 \beta_{6} - 271660 \beta_{5}) q^{68} + ( - 1808 \beta_{11} - 96184 \beta_{10} - 26884142 \beta_1) q^{69} + (77000 \beta_{9} + 3000 \beta_{8} - 3989238) q^{71} + ( - 602184 \beta_{4} + 274458 \beta_{3} + 33864 \beta_{2}) q^{72} + (80197 \beta_{7} - 125694 \beta_{6} - 603222 \beta_{5}) q^{73} + (78728 \beta_{11} + 5016 \beta_{10} + 43162152 \beta_1) q^{74} + ( - 110355 \beta_{9} + 223992 \beta_{8} + \cdots + 36070475) q^{76}+ \cdots + (132480 \beta_{11} + 298452 \beta_{10} + 44350998 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6396 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6396 q^{6} - 118116 q^{11} - 1204068 q^{16} - 1365456 q^{21} + 5376024 q^{26} + 3223704 q^{31} + 14477592 q^{36} + 10940364 q^{41} - 51272856 q^{46} - 47621676 q^{51} - 64717200 q^{56} - 23597616 q^{61} + 264735828 q^{66} - 47870856 q^{71} + 432845700 q^{76} + 3008052 q^{81} - 105341616 q^{86} - 64237536 q^{91} - 1580181156 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} + 124 x^{9} + 1665 x^{8} - 2456 x^{7} + 4192 x^{6} + 50576 x^{5} + 221184 x^{4} + 133760 x^{3} + 3200 x^{2} - 80000 x + 1000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 449334647 \nu^{11} + 12061467443 \nu^{10} - 45928215186 \nu^{9} + 33350532992 \nu^{8} + 485787845405 \nu^{7} + \cdots - 93665468020000 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 381080397548 \nu^{11} + 6650619527307 \nu^{10} - 37023789923414 \nu^{9} + 65801651136208 \nu^{8} + \cdots + 89\!\cdots\!00 ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 435927172721 \nu^{11} + 7533818317764 \nu^{10} - 43325832090128 \nu^{9} + 78139416210016 \nu^{8} + \cdots + 92\!\cdots\!00 ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 207349351901 \nu^{11} + 3608923885464 \nu^{10} - 20095877149928 \nu^{9} + 35763821961616 \nu^{8} + \cdots + 47\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2727925523508 \nu^{11} + 14877294336467 \nu^{10} - 35907038981134 \nu^{9} - 337439888728952 \nu^{8} + \cdots - 11\!\cdots\!00 ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4175127571641 \nu^{11} - 21153383834384 \nu^{10} + 49390928549668 \nu^{9} + 514100031045104 \nu^{8} + \cdots + 12\!\cdots\!00 ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1550015596341 \nu^{11} + 8356009540724 \nu^{10} - 20060042759548 \nu^{9} - 191658965219744 \nu^{8} + \cdots - 61\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 572380746383 \nu^{11} - 2036265285124 \nu^{10} + 181744291018 \nu^{9} + 95492643662604 \nu^{8} + 909244071727231 \nu^{7} + \cdots + 38\!\cdots\!00 ) / 111630409428400 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39828320474 \nu^{11} - 141775272712 \nu^{10} + 13469724484 \nu^{9} + 6653307104727 \nu^{8} + 63244284483538 \nu^{7} + \cdots + 29\!\cdots\!75 ) / 6976900589275 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3672375956607 \nu^{11} - 24100275366453 \nu^{10} + 80120961677856 \nu^{9} + 333658706564268 \nu^{8} + \cdots + 27\!\cdots\!00 ) / 558152047142000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8224243606507 \nu^{11} - 54768226676503 \nu^{10} + 182306064041706 \nu^{9} + 741572009869568 \nu^{8} + \cdots + 61\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{11} - \beta_{10} - 2\beta_{9} + \beta_{8} + 12\beta_{4} + 100\beta _1 + 100 ) / 300 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{11} - 6\beta_{10} + 15\beta_{6} - 45\beta_{5} - 15\beta_{3} - 45\beta_{2} + 2850\beta_1 ) / 150 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 17\beta_{11} - 11\beta_{10} + 17\beta_{9} - 11\beta_{8} + 144\beta_{7} - 225\beta_{5} + 2525\beta _1 - 2525 ) / 75 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 73 \beta_{9} - 144 \beta_{8} + 264 \beta_{7} + 390 \beta_{6} - 1470 \beta_{5} - 264 \beta_{4} + 390 \beta_{3} + 1470 \beta_{2} - 54825 ) / 75 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 747 \beta_{11} + 586 \beta_{10} + 747 \beta_{9} - 586 \beta_{8} - 7212 \beta_{4} + 1125 \beta_{3} + 15000 \beta_{2} - 163975 \beta _1 - 163975 ) / 75 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1583 \beta_{11} + 2424 \beta_{10} - 7320 \beta_{7} - 6210 \beta_{6} + 28380 \beta_{5} - 7320 \beta_{4} + 6210 \beta_{3} + 28380 \beta_{2} - 865775 \beta_1 ) / 25 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 36613 \beta_{11} + 33104 \beta_{10} - 36613 \beta_{9} + 33104 \beta_{8} - 367476 \beta_{7} - 99225 \beta_{6} + 875700 \beta_{5} - 9996725 \beta _1 + 9996725 ) / 75 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 290297 \beta_{9} + 386016 \beta_{8} - 1424736 \beta_{7} - 920760 \beta_{6} + 4776480 \beta_{5} + 1424736 \beta_{4} - 920760 \beta_{3} - 4776480 \beta_{2} + 133078425 ) / 75 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1898923 \beta_{11} - 1876624 \beta_{10} - 1898923 \beta_{9} + 1876624 \beta_{8} + 19240308 \beta_{4} - 6627825 \beta_{3} - 49553100 \beta_{2} + 588239275 \beta _1 + 588239275 ) / 75 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 17056301 \beta_{11} - 20989728 \beta_{10} + 85237080 \beta_{7} + 47378670 \beta_{6} - 265898760 \beta_{5} + 85237080 \beta_{4} - 47378670 \beta_{3} - 265898760 \beta_{2} + 7082438925 \beta_1 ) / 75 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 101596857 \beta_{11} - 105815456 \beta_{10} + 101596857 \beta_{9} - 105815456 \beta_{8} + 1030566804 \beta_{7} + 402640425 \beta_{6} - 2776867500 \beta_{5} + \cdots - 33828064025 ) / 75 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
2.82685 + 2.82685i
0.908990 + 0.908990i
−1.51110 1.51110i
−3.96059 3.96059i
−1.54050 1.54050i
5.27634 + 5.27634i
2.82685 2.82685i
0.908990 0.908990i
−1.51110 + 1.51110i
−3.96059 + 3.96059i
−1.54050 + 1.54050i
5.27634 5.27634i
−21.8383 21.8383i −42.5089 + 42.5089i 697.824i 0 1856.64 −432.570 432.570i 9648.70 9648.70i 2946.99i 0
7.2 −18.1270 18.1270i 95.3921 95.3921i 401.173i 0 −3458.34 −1478.49 1478.49i 2631.55 2631.55i 11638.3i 0
7.3 −0.0371264 0.0371264i −36.2471 + 36.2471i 255.997i 0 2.69145 1325.17 + 1325.17i −19.0086 + 19.0086i 3933.30i 0
7.4 0.0371264 + 0.0371264i 36.2471 36.2471i 255.997i 0 2.69145 −1325.17 1325.17i 19.0086 19.0086i 3933.30i 0
7.5 18.1270 + 18.1270i −95.3921 + 95.3921i 401.173i 0 −3458.34 1478.49 + 1478.49i −2631.55 + 2631.55i 11638.3i 0
7.6 21.8383 + 21.8383i 42.5089 42.5089i 697.824i 0 1856.64 432.570 + 432.570i −9648.70 + 9648.70i 2946.99i 0
18.1 −21.8383 + 21.8383i −42.5089 42.5089i 697.824i 0 1856.64 −432.570 + 432.570i 9648.70 + 9648.70i 2946.99i 0
18.2 −18.1270 + 18.1270i 95.3921 + 95.3921i 401.173i 0 −3458.34 −1478.49 + 1478.49i 2631.55 + 2631.55i 11638.3i 0
18.3 −0.0371264 + 0.0371264i −36.2471 36.2471i 255.997i 0 2.69145 1325.17 1325.17i −19.0086 19.0086i 3933.30i 0
18.4 0.0371264 0.0371264i 36.2471 + 36.2471i 255.997i 0 2.69145 −1325.17 + 1325.17i 19.0086 + 19.0086i 3933.30i 0
18.5 18.1270 18.1270i −95.3921 95.3921i 401.173i 0 −3458.34 1478.49 1478.49i −2631.55 2631.55i 11638.3i 0
18.6 21.8383 21.8383i 42.5089 + 42.5089i 697.824i 0 1856.64 432.570 432.570i −9648.70 9648.70i 2946.99i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.9.c.c 12
5.b even 2 1 inner 25.9.c.c 12
5.c odd 4 2 inner 25.9.c.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.9.c.c 12 1.a even 1 1 trivial
25.9.c.c 12 5.b even 2 1 inner
25.9.c.c 12 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 1341657T_{2}^{8} + 392912788608T_{2}^{4} + 2985984 \) acting on \(S_{9}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 1341657 T^{8} + \cdots + 2985984 \) Copy content Toggle raw display
$3$ \( T^{12} + 351180027 T^{8} + \cdots + 29\!\cdots\!29 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 31588550560512 T^{8} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( (T^{3} + 29529 T^{2} + \cdots - 1580036164893)^{4} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 63\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( (T^{6} + 59573487675 T^{4} + \cdots + 11\!\cdots\!25)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 93\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{6} + 999419388300 T^{4} + \cdots + 97\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 805926 T^{2} + \cdots + 48\!\cdots\!12)^{4} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{3} - 2735091 T^{2} + \cdots + 77\!\cdots\!27)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 42\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T^{6} + 328367873378700 T^{4} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 5899404 T^{2} + \cdots - 25\!\cdots\!68)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 67\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{3} + 11967714 T^{2} + \cdots - 19\!\cdots\!28)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 62\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 13\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 43\!\cdots\!25)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
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