Properties

Label 350.7.d.a
Level $350$
Weight $7$
Character orbit 350.d
Analytic conductor $80.519$
Analytic rank $0$
Dimension $8$
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] = [350,7,Mod(349,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.349");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 350.d (of order \(2\), degree \(1\), not 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: \(80.5189292669\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.179721732096.20
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 28x^{6} - 40x^{5} + 258x^{4} - 32x^{3} - 620x^{2} + 7480x + 11698 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: no (minimal twist has level 14)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{5} q^{3} - 32 q^{4} + (5 \beta_{7} - \beta_{6}) q^{6} + (14 \beta_{5} - 7 \beta_{4} + 21 \beta_{3} + 77 \beta_{2}) q^{7} - 32 \beta_{3} q^{8} + (78 \beta_1 - 273) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{5} q^{3} - 32 q^{4} + (5 \beta_{7} - \beta_{6}) q^{6} + (14 \beta_{5} - 7 \beta_{4} + 21 \beta_{3} + 77 \beta_{2}) q^{7} - 32 \beta_{3} q^{8} + (78 \beta_1 - 273) q^{9} + (12 \beta_1 - 1110) q^{11} - 32 \beta_{5} q^{12} + ( - 94 \beta_{5} - 43 \beta_{4}) q^{13} + (21 \beta_{7} - 49 \beta_{6} - 77 \beta_1 - 672) q^{14} + 1024 q^{16} + (150 \beta_{5} - 134 \beta_{4}) q^{17} + ( - 273 \beta_{3} + 2496 \beta_{2}) q^{18} + ( - 283 \beta_{7} - 50 \beta_{6}) q^{19} + (182 \beta_{7} - 21 \beta_{6} + 630 \beta_1 + 4872) q^{21} + ( - 1110 \beta_{3} + 384 \beta_{2}) q^{22} + ( - 264 \beta_{3} - 10146 \beta_{2}) q^{23} + ( - 160 \beta_{7} + 32 \beta_{6}) q^{24} + ( - 771 \beta_{7} - 121 \beta_{6}) q^{26} + ( - 612 \beta_{5} + 78 \beta_{4}) q^{27} + ( - 448 \beta_{5} + 224 \beta_{4} - 672 \beta_{3} - 2464 \beta_{2}) q^{28} + (6564 \beta_1 + 4566) q^{29} + ( - 54 \beta_{7} + 1038 \beta_{6}) q^{31} + 1024 \beta_{3} q^{32} + ( - 1050 \beta_{5} + 12 \beta_{4}) q^{33} + ( - 188 \beta_{7} - 820 \beta_{6}) q^{34} + ( - 2496 \beta_1 + 8736) q^{36} + ( - 7668 \beta_{3} - 5798 \beta_{2}) q^{37} + (1065 \beta_{5} + 533 \beta_{4}) q^{38} + ( - 10170 \beta_1 - 52152) q^{39} + ( - 1980 \beta_{7} - 2098 \beta_{6}) q^{41} + ( - 1057 \beta_{5} - 77 \beta_{4} + 4872 \beta_{3} + 20160 \beta_{2}) q^{42} + (18240 \beta_{3} + 11174 \beta_{2}) q^{43} + ( - 384 \beta_1 + 35520) q^{44} + (10146 \beta_1 + 8448) q^{46} + ( - 4014 \beta_{5} - 986 \beta_{4}) q^{47} + 1024 \beta_{5} q^{48} + (3038 \beta_{7} - 980 \beta_{6} - 6468 \beta_1 + 77567) q^{49} + (2856 \beta_1 + 39456) q^{51} + (3008 \beta_{5} + 1376 \beta_{4}) q^{52} + ( - 10968 \beta_{3} - 62154 \beta_{2}) q^{53} + ( - 2514 \beta_{7} + 1002 \beta_{6}) q^{54} + ( - 672 \beta_{7} + 1568 \beta_{6} + 2464 \beta_1 + 21504) q^{56} + ( - 18774 \beta_{3} - 118248 \beta_{2}) q^{57} + (4566 \beta_{3} + 210048 \beta_{2}) q^{58} + ( - 13485 \beta_{7} - 6710 \beta_{6}) q^{59} + ( - 9368 \beta_{7} + 2945 \beta_{6}) q^{61} + (7536 \beta_{5} - 5136 \beta_{4}) q^{62} + ( - 2184 \beta_{5} + 5733 \beta_{4} + 273 \beta_{3} + 31395 \beta_{2}) q^{63} - 32768 q^{64} + ( - 5166 \beta_{7} + 1110 \beta_{6}) q^{66} + (408 \beta_{3} - 108694 \beta_{2}) q^{67} + ( - 4800 \beta_{5} + 4288 \beta_{4}) q^{68} + ( - 11466 \beta_{7} + 264 \beta_{6}) q^{69} + (58146 \beta_1 - 112902) q^{71} + (8736 \beta_{3} - 79872 \beta_{2}) q^{72} + ( - 3494 \beta_{5} + 11008 \beta_{4}) q^{73} + (5798 \beta_1 + 245376) q^{74} + (9056 \beta_{7} + 1600 \beta_{6}) q^{76} + ( - 15288 \beta_{5} + 8358 \beta_{4} - 22386 \beta_{3} - 77406 \beta_{2}) q^{77} + ( - 52152 \beta_{3} - 325440 \beta_{2}) q^{78} + (12690 \beta_1 - 523226) q^{79} + ( - 99450 \beta_1 - 63207) q^{81} + ( - 4786 \beta_{5} + 12470 \beta_{4}) q^{82} + ( - 16983 \beta_{5} + 14380 \beta_{4}) q^{83} + ( - 5824 \beta_{7} + 672 \beta_{6} - 20160 \beta_1 - 155904) q^{84} + ( - 11174 \beta_1 - 583680) q^{86} + (37386 \beta_{5} + 6564 \beta_{4}) q^{87} + (35520 \beta_{3} - 12288 \beta_{2}) q^{88} + (8430 \beta_{7} - 8076 \beta_{6}) q^{89} + ( - 23429 \beta_{7} + 770 \beta_{6} - 133266 \beta_1 - 277368) q^{91} + (8448 \beta_{3} + 324672 \beta_{2}) q^{92} + ( - 72720 \beta_{3} - 248832 \beta_{2}) q^{93} + ( - 26972 \beta_{7} - 916 \beta_{6}) q^{94} + (5120 \beta_{7} - 1024 \beta_{6}) q^{96} + ( - 6254 \beta_{5} + 3562 \beta_{4}) q^{97} + ( - 22050 \beta_{5} + 1862 \beta_{4} + 77567 \beta_{3} - 206976 \beta_{2}) q^{98} + ( - 89856 \beta_1 + 332982) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 256 q^{4} - 2184 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 256 q^{4} - 2184 q^{9} - 8880 q^{11} - 5376 q^{14} + 8192 q^{16} + 38976 q^{21} + 36528 q^{29} + 69888 q^{36} - 417216 q^{39} + 284160 q^{44} + 67584 q^{46} + 620536 q^{49} + 315648 q^{51} + 172032 q^{56} - 262144 q^{64} - 903216 q^{71} + 1963008 q^{74} - 4185808 q^{79} - 505656 q^{81} - 1247232 q^{84} - 4669440 q^{86} - 2218944 q^{91} + 2663856 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 28x^{6} - 40x^{5} + 258x^{4} - 32x^{3} - 620x^{2} + 7480x + 11698 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 68088 \nu^{7} - 1192852 \nu^{6} + 3754608 \nu^{5} - 14542140 \nu^{4} + 2822728 \nu^{3} + 65626512 \nu^{2} - 319126368 \nu + 2280456368 ) / 502161073 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 88482 \nu^{7} - 496428 \nu^{6} + 3438200 \nu^{5} - 8169695 \nu^{4} + 40056132 \nu^{3} - 61256098 \nu^{2} + 110856524 \nu + 629909935 ) / 297167111 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 330528 \nu^{7} - 1756044 \nu^{6} + 12796256 \nu^{5} - 35079836 \nu^{4} + 160737304 \nu^{3} - 212573944 \nu^{2} + 330272096 \nu + 2243651672 ) / 197941847 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 168752282 \nu^{7} - 537771504 \nu^{6} + 5539642408 \nu^{5} - 5130738670 \nu^{4} + 19457898344 \nu^{3} + 81862631972 \nu^{2} + \cdots + 864763629972 ) / 68796067001 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 231831374 \nu^{7} - 656249376 \nu^{6} + 5414235240 \nu^{5} - 2422416074 \nu^{4} + 40448137544 \nu^{3} + 57693609052 \nu^{2} + \cdots + 1337919721068 ) / 68796067001 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 173141866 \nu^{7} + 1283513888 \nu^{6} - 8336224680 \nu^{5} + 28368672042 \nu^{4} - 101446760380 \nu^{3} + \cdots - 1378152307096 ) / 39670724767 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 369202382 \nu^{7} - 2207604480 \nu^{6} + 13578418280 \nu^{5} - 40765776686 \nu^{4} + 148677452196 \nu^{3} - 325911446316 \nu^{2} + \cdots + 2171463328472 ) / 39670724767 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{6} - 3\beta_{5} + \beta_{4} + 4\beta_{3} + 16\beta_{2} + 16 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 13\beta_{3} - 32\beta_{2} + 11\beta _1 - 40 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -15\beta_{7} - 9\beta_{6} + 21\beta_{5} - 27\beta_{4} + 44\beta_{3} + 64\beta_{2} + 24\beta _1 - 304 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -23\beta_{7} - 21\beta_{6} + 31\beta_{5} - 21\beta_{4} - 218\beta_{3} + 1464\beta_{2} - 18\beta _1 - 360 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -160\beta_{7} - 284\beta_{6} - 187\beta_{5} + 469\beta_{4} - 1262\beta_{3} + 6528\beta_{2} - 790\beta _1 + 688 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 60\beta_{7} - 30\beta_{6} - 216\beta_{5} + 438\beta_{4} + 953\beta_{3} - 7868\beta_{2} - 2098\beta _1 + 9772 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2024 \beta_{7} + 2516 \beta_{6} + 129 \beta_{5} - 169 \beta_{4} + 6436 \beta_{3} - 47364 \beta_{2} - 3836 \beta _1 + 21100 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
2.69332 2.11561i
2.13992 4.16623i
−1.13992 + 3.75201i
−1.69332 0.298603i
2.69332 + 2.11561i
2.13992 + 4.16623i
−1.13992 3.75201i
−1.69332 + 0.298603i
5.65685i −29.9539 −32.0000 0 169.445i −281.627 195.794i 181.019i 168.235 0
349.2 5.65685i −3.84257 −32.0000 0 21.7369i −340.444 41.7939i 181.019i −714.235 0
349.3 5.65685i 3.84257 −32.0000 0 21.7369i 340.444 41.7939i 181.019i −714.235 0
349.4 5.65685i 29.9539 −32.0000 0 169.445i 281.627 195.794i 181.019i 168.235 0
349.5 5.65685i −29.9539 −32.0000 0 169.445i −281.627 + 195.794i 181.019i 168.235 0
349.6 5.65685i −3.84257 −32.0000 0 21.7369i −340.444 + 41.7939i 181.019i −714.235 0
349.7 5.65685i 3.84257 −32.0000 0 21.7369i 340.444 + 41.7939i 181.019i −714.235 0
349.8 5.65685i 29.9539 −32.0000 0 169.445i 281.627 + 195.794i 181.019i 168.235 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.7.d.a 8
5.b even 2 1 inner 350.7.d.a 8
5.c odd 4 1 14.7.b.a 4
5.c odd 4 1 350.7.b.a 4
7.b odd 2 1 inner 350.7.d.a 8
15.e even 4 1 126.7.c.a 4
20.e even 4 1 112.7.c.c 4
35.c odd 2 1 inner 350.7.d.a 8
35.f even 4 1 14.7.b.a 4
35.f even 4 1 350.7.b.a 4
35.k even 12 2 98.7.d.b 8
35.l odd 12 2 98.7.d.b 8
40.i odd 4 1 448.7.c.h 4
40.k even 4 1 448.7.c.e 4
105.k odd 4 1 126.7.c.a 4
140.j odd 4 1 112.7.c.c 4
280.s even 4 1 448.7.c.h 4
280.y odd 4 1 448.7.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.7.b.a 4 5.c odd 4 1
14.7.b.a 4 35.f even 4 1
98.7.d.b 8 35.k even 12 2
98.7.d.b 8 35.l odd 12 2
112.7.c.c 4 20.e even 4 1
112.7.c.c 4 140.j odd 4 1
126.7.c.a 4 15.e even 4 1
126.7.c.a 4 105.k odd 4 1
350.7.b.a 4 5.c odd 4 1
350.7.b.a 4 35.f even 4 1
350.7.d.a 8 1.a even 1 1 trivial
350.7.d.a 8 5.b even 2 1 inner
350.7.d.a 8 7.b odd 2 1 inner
350.7.d.a 8 35.c odd 2 1 inner
448.7.c.e 4 40.k even 4 1
448.7.c.e 4 280.y odd 4 1
448.7.c.h 4 40.i odd 4 1
448.7.c.h 4 280.s even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 912T_{3}^{2} + 13248 \) acting on \(S_{7}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - 912 T^{2} + 13248)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 310268 T^{6} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2220 T + 1227492)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 15367056 T^{2} + \cdots + 26266196601792)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 40214784 T^{2} + \cdots + 126735731638272)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 65975568 T^{2} + \cdots + 551809934313408)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 210343176 T^{2} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 9132 T - 1357906716)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2274932736 T^{2} + \cdots + 86\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3830319944 T^{2} + \cdots + 34\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 9071224896 T^{2} + \cdots + 28\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 21542362952 T^{2} + \cdots + 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 20120477952 T^{2} + \cdots + 12\!\cdots\!88)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 15425248968 T^{2} + \cdots + 185366809042704)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 180594109200 T^{2} + \cdots + 78\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 121774406928 T^{2} + \cdots + 82\!\cdots\!48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 23639424968 T^{2} + \cdots + 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 225804 T - 95443772508)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 228009998400 T^{2} + \cdots + 26\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1046452 T + 268612291876)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 478841902608 T^{2} + \cdots + 21\!\cdots\!08)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 258250641984 T^{2} + \cdots + 15\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 42611214336 T^{2} + \cdots + 39\!\cdots\!12)^{2} \) Copy content Toggle raw display
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