Properties

Label 14.7.b.a
Level $14$
Weight $7$
Character orbit 14.b
Analytic conductor $3.221$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [14,7,Mod(13,14)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(14, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("14.13");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 14.b (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: \(3.22075717068\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.211968.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 30x^{2} + 207 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_1 q^{3} + 32 q^{4} - 5 \beta_{2} q^{5} + ( - \beta_{2} - 5 \beta_1) q^{6} + ( - 21 \beta_{3} - 7 \beta_{2} + 14 \beta_1 + 77) q^{7} - 32 \beta_{3} q^{8} + (78 \beta_{3} + 273) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_1 q^{3} + 32 q^{4} - 5 \beta_{2} q^{5} + ( - \beta_{2} - 5 \beta_1) q^{6} + ( - 21 \beta_{3} - 7 \beta_{2} + 14 \beta_1 + 77) q^{7} - 32 \beta_{3} q^{8} + (78 \beta_{3} + 273) q^{9} + (25 \beta_{2} - 35 \beta_1) q^{10} + ( - 12 \beta_{3} - 1110) q^{11} - 32 \beta_1 q^{12} + (43 \beta_{2} + 94 \beta_1) q^{13} + ( - 77 \beta_{3} + 49 \beta_{2} + 21 \beta_1 + 672) q^{14} + (330 \beta_{3} - 1080) q^{15} + 1024 q^{16} + ( - 134 \beta_{2} + 150 \beta_1) q^{17} + ( - 273 \beta_{3} - 2496) q^{18} + (50 \beta_{2} - 283 \beta_1) q^{19} - 160 \beta_{2} q^{20} + ( - 630 \beta_{3} - 21 \beta_{2} - 182 \beta_1 + 4872) q^{21} + (1110 \beta_{3} + 384) q^{22} + ( - 264 \beta_{3} + 10146) q^{23} + ( - 32 \beta_{2} - 160 \beta_1) q^{24} + ( - 2850 \beta_{3} - 10175) q^{25} + ( - 121 \beta_{2} + 771 \beta_1) q^{26} + (78 \beta_{2} - 612 \beta_1) q^{27} + ( - 672 \beta_{3} - 224 \beta_{2} + 448 \beta_1 + 2464) q^{28} + (6564 \beta_{3} - 4566) q^{29} + (1080 \beta_{3} - 10560) q^{30} + (1038 \beta_{2} + 54 \beta_1) q^{31} - 1024 \beta_{3} q^{32} + ( - 12 \beta_{2} + 1050 \beta_1) q^{33} + (820 \beta_{2} - 188 \beta_1) q^{34} + ( - 8610 \beta_{3} + 140 \beta_{2} - 735 \beta_1 - 21000) q^{35} + (2496 \beta_{3} + 8736) q^{36} + (7668 \beta_{3} - 5798) q^{37} + ( - 533 \beta_{2} - 1065 \beta_1) q^{38} + ( - 10170 \beta_{3} + 52152) q^{39} + (800 \beta_{2} - 1120 \beta_1) q^{40} + ( - 2098 \beta_{2} + 1980 \beta_1) q^{41} + ( - 4872 \beta_{3} - 77 \beta_{2} - 1057 \beta_1 + 20160) q^{42} + (18240 \beta_{3} - 11174) q^{43} + ( - 384 \beta_{3} - 35520) q^{44} + ( - 3315 \beta_{2} + 2730 \beta_1) q^{45} + ( - 10146 \beta_{3} + 8448) q^{46} + ( - 986 \beta_{2} - 4014 \beta_1) q^{47} - 1024 \beta_1 q^{48} + ( - 6468 \beta_{3} + 980 \beta_{2} + 3038 \beta_1 - 77567) q^{49} + (10175 \beta_{3} + 91200) q^{50} + ( - 2856 \beta_{3} + 39456) q^{51} + (1376 \beta_{2} + 3008 \beta_1) q^{52} + ( - 10968 \beta_{3} + 62154) q^{53} + ( - 1002 \beta_{2} - 2514 \beta_1) q^{54} + (5850 \beta_{2} - 420 \beta_1) q^{55} + ( - 2464 \beta_{3} + 1568 \beta_{2} + 672 \beta_1 + 21504) q^{56} + (18774 \beta_{3} - 118248) q^{57} + (4566 \beta_{3} - 210048) q^{58} + (6710 \beta_{2} - 13485 \beta_1) q^{59} + (10560 \beta_{3} - 34560) q^{60} + (2945 \beta_{2} + 9368 \beta_1) q^{61} + ( - 5136 \beta_{2} + 7536 \beta_1) q^{62} + (273 \beta_{3} - 5733 \beta_{2} + 2184 \beta_1 - 31395) q^{63} + 32768 q^{64} + ( - 6510 \beta_{3} + 323400) q^{65} + (1110 \beta_{2} + 5166 \beta_1) q^{66} + ( - 408 \beta_{3} - 108694) q^{67} + ( - 4288 \beta_{2} + 4800 \beta_1) q^{68} + ( - 264 \beta_{2} - 11466 \beta_1) q^{69} + (21000 \beta_{3} - 1435 \beta_{2} - 2695 \beta_1 + 275520) q^{70} + ( - 58146 \beta_{3} - 112902) q^{71} + ( - 8736 \beta_{3} - 79872) q^{72} + ( - 11008 \beta_{2} + 3494 \beta_1) q^{73} + (5798 \beta_{3} - 245376) q^{74} + ( - 2850 \beta_{2} - 4075 \beta_1) q^{75} + (1600 \beta_{2} - 9056 \beta_1) q^{76} + (22386 \beta_{3} + 8358 \beta_{2} - 15288 \beta_1 - 77406) q^{77} + ( - 52152 \beta_{3} + 325440) q^{78} + (12690 \beta_{3} + 523226) q^{79} - 5120 \beta_{2} q^{80} + (99450 \beta_{3} - 63207) q^{81} + (12470 \beta_{2} - 4786 \beta_1) q^{82} + ( - 14380 \beta_{2} + 16983 \beta_1) q^{83} + ( - 20160 \beta_{3} - 672 \beta_{2} - 5824 \beta_1 + 155904) q^{84} + ( - 125880 \beta_{3} - 529440) q^{85} + (11174 \beta_{3} - 583680) q^{86} + (6564 \beta_{2} + 37386 \beta_1) q^{87} + (35520 \beta_{3} + 12288) q^{88} + (8076 \beta_{2} + 8430 \beta_1) q^{89} + (19305 \beta_{2} - 9555 \beta_1) q^{90} + (133266 \beta_{3} + 770 \beta_{2} + 23429 \beta_1 - 277368) q^{91} + ( - 8448 \beta_{3} + 324672) q^{92} + ( - 72720 \beta_{3} + 248832) q^{93} + (916 \beta_{2} - 26972 \beta_1) q^{94} + (121890 \beta_{3} - 47640) q^{95} + ( - 1024 \beta_{2} - 5120 \beta_1) q^{96} + (3562 \beta_{2} - 6254 \beta_1) q^{97} + (77567 \beta_{3} - 1862 \beta_{2} + 22050 \beta_1 + 206976) q^{98} + ( - 89856 \beta_{3} - 332982) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 128 q^{4} + 308 q^{7} + 1092 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 128 q^{4} + 308 q^{7} + 1092 q^{9} - 4440 q^{11} + 2688 q^{14} - 4320 q^{15} + 4096 q^{16} - 9984 q^{18} + 19488 q^{21} + 1536 q^{22} + 40584 q^{23} - 40700 q^{25} + 9856 q^{28} - 18264 q^{29} - 42240 q^{30} - 84000 q^{35} + 34944 q^{36} - 23192 q^{37} + 208608 q^{39} + 80640 q^{42} - 44696 q^{43} - 142080 q^{44} + 33792 q^{46} - 310268 q^{49} + 364800 q^{50} + 157824 q^{51} + 248616 q^{53} + 86016 q^{56} - 472992 q^{57} - 840192 q^{58} - 138240 q^{60} - 125580 q^{63} + 131072 q^{64} + 1293600 q^{65} - 434776 q^{67} + 1102080 q^{70} - 451608 q^{71} - 319488 q^{72} - 981504 q^{74} - 309624 q^{77} + 1301760 q^{78} + 2092904 q^{79} - 252828 q^{81} + 623616 q^{84} - 2117760 q^{85} - 2334720 q^{86} + 49152 q^{88} - 1109472 q^{91} + 1298688 q^{92} + 995328 q^{93} - 190560 q^{95} + 827904 q^{98} - 1331928 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 30x^{2} + 207 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 18\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 34\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{2} + 60 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 3\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} - 60 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{2} + 51\beta_1 ) / 16 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
3.27984i
3.27984i
4.38664i
4.38664i
−5.65685 3.84257i 32.0000 204.749i 21.7369i −41.7939 + 340.444i −181.019 714.235 1158.23i
13.2 −5.65685 3.84257i 32.0000 204.749i 21.7369i −41.7939 340.444i −181.019 714.235 1158.23i
13.3 5.65685 29.9539i 32.0000 98.3767i 169.445i 195.794 + 281.627i 181.019 −168.235 556.502i
13.4 5.65685 29.9539i 32.0000 98.3767i 169.445i 195.794 281.627i 181.019 −168.235 556.502i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.7.b.a 4
3.b odd 2 1 126.7.c.a 4
4.b odd 2 1 112.7.c.c 4
5.b even 2 1 350.7.b.a 4
5.c odd 4 2 350.7.d.a 8
7.b odd 2 1 inner 14.7.b.a 4
7.c even 3 2 98.7.d.b 8
7.d odd 6 2 98.7.d.b 8
8.b even 2 1 448.7.c.h 4
8.d odd 2 1 448.7.c.e 4
21.c even 2 1 126.7.c.a 4
28.d even 2 1 112.7.c.c 4
35.c odd 2 1 350.7.b.a 4
35.f even 4 2 350.7.d.a 8
56.e even 2 1 448.7.c.e 4
56.h odd 2 1 448.7.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.7.b.a 4 1.a even 1 1 trivial
14.7.b.a 4 7.b odd 2 1 inner
98.7.d.b 8 7.c even 3 2
98.7.d.b 8 7.d odd 6 2
112.7.c.c 4 4.b odd 2 1
112.7.c.c 4 28.d even 2 1
126.7.c.a 4 3.b odd 2 1
126.7.c.a 4 21.c even 2 1
350.7.b.a 4 5.b even 2 1
350.7.b.a 4 35.c odd 2 1
350.7.d.a 8 5.c odd 4 2
350.7.d.a 8 35.f even 4 2
448.7.c.e 4 8.d odd 2 1
448.7.c.e 4 56.e even 2 1
448.7.c.h 4 8.b even 2 1
448.7.c.h 4 56.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 912 T^{2} + 13248 \) Copy content Toggle raw display
$5$ \( T^{4} + 51600 T^{2} + \cdots + 405720000 \) Copy content Toggle raw display
$7$ \( T^{4} - 308 T^{3} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2220 T + 1227492)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 15367056 T^{2} + \cdots + 26266196601792 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 126735731638272 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 551809934313408 \) Copy content Toggle raw display
$23$ \( (T^{2} - 20292 T + 100711044)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 9132 T - 1357906716)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2274932736 T^{2} + \cdots + 86\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( (T^{2} + 11596 T - 1847926364)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 9071224896 T^{2} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( (T^{2} + 22348 T - 10521464924)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 20120477952 T^{2} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( (T^{2} - 124308 T + 13614948)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 180594109200 T^{2} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + 121774406928 T^{2} + \cdots + 82\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( (T^{2} + 217388 T + 11809058788)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 225804 T - 95443772508)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 228009998400 T^{2} + \cdots + 26\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( (T^{2} - 1046452 T + 268612291876)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 478841902608 T^{2} + \cdots + 21\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{4} + 258250641984 T^{2} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
$97$ \( T^{4} + 42611214336 T^{2} + \cdots + 39\!\cdots\!12 \) Copy content Toggle raw display
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