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Properties

Label 300.5.c.b
Level $300$
Weight $5$
Character orbit 300.c
Analytic conductor $31.011$
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] = [300,5,Mod(151,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.151");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 300.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: \(31.0109889252\)
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: \( x^{16} - 6 x^{15} + 20 x^{14} - 108 x^{13} + 492 x^{12} - 2160 x^{11} + 7360 x^{10} - 39552 x^{9} + \cdots + 4294967296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_1 q^{3} + \beta_{7} q^{4} + (\beta_{5} + 1) q^{6} + ( - \beta_{12} + \beta_{7}) q^{7} + (\beta_{3} - \beta_{2} + 12) q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_1 q^{3} + \beta_{7} q^{4} + (\beta_{5} + 1) q^{6} + ( - \beta_{12} + \beta_{7}) q^{7} + (\beta_{3} - \beta_{2} + 12) q^{8} - 27 q^{9} + ( - \beta_{14} - \beta_{11} + \cdots + 3 \beta_{2}) q^{11}+ \cdots + (27 \beta_{14} + 27 \beta_{11} + \cdots - 81 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{2} + 8 q^{4} + 18 q^{6} + 180 q^{8} - 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{2} + 8 q^{4} + 18 q^{6} + 180 q^{8} - 432 q^{9} - 176 q^{13} + 78 q^{14} - 376 q^{16} + 162 q^{18} - 144 q^{21} + 788 q^{22} + 108 q^{24} + 678 q^{26} - 3368 q^{28} + 1728 q^{29} - 2016 q^{32} - 2932 q^{34} - 216 q^{36} + 1568 q^{37} + 6990 q^{38} + 1248 q^{41} - 162 q^{42} + 8088 q^{44} + 5956 q^{46} - 2088 q^{48} - 10720 q^{49} - 3128 q^{52} + 288 q^{53} - 486 q^{54} - 10236 q^{56} - 5616 q^{57} + 16164 q^{58} - 3760 q^{61} + 12714 q^{62} + 10544 q^{64} + 8100 q^{66} - 26136 q^{68} + 9792 q^{69} - 4860 q^{72} - 11040 q^{73} - 17004 q^{74} - 28344 q^{76} - 768 q^{77} + 16830 q^{78} + 11664 q^{81} + 21280 q^{82} + 15120 q^{84} + 24414 q^{86} - 52840 q^{88} - 768 q^{89} - 23736 q^{92} + 9936 q^{93} - 45156 q^{94} - 11088 q^{96} - 7248 q^{97} + 58140 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 20 x^{14} - 108 x^{13} + 492 x^{12} - 2160 x^{11} + 7360 x^{10} - 39552 x^{9} + \cdots + 4294967296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 22761 \nu^{15} + 1177062 \nu^{14} - 419412 \nu^{13} - 17236980 \nu^{12} + \cdots - 91883040669696 ) / 21382226247680 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 21677 \nu^{15} + 746 \nu^{14} - 410316 \nu^{13} - 410780 \nu^{12} + 7032284 \nu^{11} + \cdots + 2036619804672 ) / 2672778280960 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21677 \nu^{15} + 746 \nu^{14} - 410316 \nu^{13} - 410780 \nu^{12} + 7032284 \nu^{11} + \cdots - 30036719566848 ) / 2672778280960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 109161 \nu^{15} - 2497938 \nu^{14} + 17073788 \nu^{13} - 139010900 \nu^{12} + \cdots - 16\!\cdots\!36 ) / 10691113123840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 65031 \nu^{15} - 2238 \nu^{14} + 1230948 \nu^{13} + 1232340 \nu^{12} + \cdots - 8782637694976 ) / 2672778280960 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 699729 \nu^{15} - 4691478 \nu^{14} + 62915828 \nu^{13} - 142283500 \nu^{12} + \cdots + 597877969649664 ) / 21382226247680 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16351 \nu^{15} - 105482 \nu^{14} + 241292 \nu^{13} - 454100 \nu^{12} + 6009172 \nu^{11} + \cdots - 11637750759424 ) / 334097285120 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 528347 \nu^{15} - 4346834 \nu^{14} + 5696444 \nu^{13} - 156454180 \nu^{12} + \cdots - 22\!\cdots\!48 ) / 10691113123840 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1660583 \nu^{15} - 11482214 \nu^{14} + 34112084 \nu^{13} + 59858740 \nu^{12} + \cdots + 19\!\cdots\!32 ) / 21382226247680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1668199 \nu^{15} + 15383658 \nu^{14} - 71945868 \nu^{13} + 66251380 \nu^{12} + \cdots + 10\!\cdots\!56 ) / 21382226247680 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1651199 \nu^{15} - 10261698 \nu^{14} + 29651228 \nu^{13} - 103582260 \nu^{12} + \cdots - 19\!\cdots\!36 ) / 10691113123840 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3400017 \nu^{15} + 7613094 \nu^{14} - 3207764 \nu^{13} + 50899500 \nu^{12} + \cdots - 80052016381952 ) / 21382226247680 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 362307 \nu^{15} + 2146026 \nu^{14} - 203980 \nu^{13} + 7231524 \nu^{12} + \cdots - 386897901780992 ) / 2138222624768 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1827337 \nu^{15} + 8732214 \nu^{14} - 63863604 \nu^{13} + 291078860 \nu^{12} + \cdots + 44\!\cdots\!08 ) / 10691113123840 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 244581 \nu^{15} - 1253922 \nu^{14} + 3714332 \nu^{13} - 16702380 \nu^{12} + \cdots + 1607357956096 ) / 1336389140480 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} + 3\beta_{2} + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} + \beta_{8} - 3\beta_{7} - \beta_{4} + \beta_{2} - \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 9 \beta_{15} + 3 \beta_{14} - 12 \beta_{12} - 5 \beta_{11} - 5 \beta_{10} + 6 \beta_{8} + 2 \beta_{7} + \cdots - 30 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 18 \beta_{15} + 6 \beta_{14} + 18 \beta_{13} - 18 \beta_{12} + 6 \beta_{11} + 6 \beta_{10} + 12 \beta_{9} + \cdots + 612 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{15} - 34 \beta_{14} - 10 \beta_{13} - 18 \beta_{12} - 46 \beta_{11} + 2 \beta_{10} + \cdots + 604 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 528 \beta_{15} - 144 \beta_{14} - 228 \beta_{13} - 516 \beta_{12} - 72 \beta_{11} + 136 \beta_{10} + \cdots + 24000 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 300 \beta_{15} + 108 \beta_{14} + 360 \beta_{13} - 216 \beta_{12} + 1516 \beta_{11} + 828 \beta_{10} + \cdots - 61208 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1528 \beta_{15} - 552 \beta_{14} - 640 \beta_{13} - 1168 \beta_{12} - 2200 \beta_{11} - 1080 \beta_{10} + \cdots - 126928 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 16728 \beta_{15} + 4200 \beta_{14} - 1992 \beta_{13} - 21000 \beta_{12} + 3096 \beta_{11} + \cdots + 472080 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 73152 \beta_{15} + 19584 \beta_{14} + 41664 \beta_{13} - 75264 \beta_{12} - 42816 \beta_{11} + \cdots - 1040192 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 41936 \beta_{15} - 26096 \beta_{14} + 33280 \beta_{13} + 17344 \beta_{12} - 37488 \beta_{11} + \cdots - 400800 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 25248 \beta_{15} - 1057824 \beta_{14} - 101856 \beta_{13} + 1459680 \beta_{12} + 158176 \beta_{11} + \cdots + 42598720 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 5928288 \beta_{15} + 997920 \beta_{14} - 800736 \beta_{13} + 1137312 \beta_{12} - 277024 \beta_{11} + \cdots + 29247552 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 4799232 \beta_{15} + 6359296 \beta_{14} + 4069184 \beta_{13} + 2290496 \beta_{12} + 8967296 \beta_{11} + \cdots - 532596736 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
1.95664 + 3.48878i
1.95664 − 3.48878i
3.79586 − 1.26152i
3.79586 + 1.26152i
−1.06635 + 3.85524i
−1.06635 − 3.85524i
3.97720 + 0.426493i
3.97720 − 0.426493i
−3.04390 + 2.59512i
−3.04390 − 2.59512i
2.11879 + 3.39275i
2.11879 − 3.39275i
−3.55818 − 1.82740i
−3.55818 + 1.82740i
−1.18006 + 3.82197i
−1.18006 − 3.82197i
−3.99969 − 0.0498899i − 5.19615i 15.9950 + 0.399088i 0 −0.259236 + 20.7830i − 35.2842i −63.9552 − 2.39422i −27.0000 0
151.2 −3.99969 + 0.0498899i 5.19615i 15.9950 − 0.399088i 0 −0.259236 − 20.7830i 35.2842i −63.9552 + 2.39422i −27.0000 0
151.3 −2.99044 − 2.65655i 5.19615i 1.88545 + 15.8885i 0 13.8039 − 15.5388i − 74.3539i 36.5704 − 52.5224i −27.0000 0
151.4 −2.99044 + 2.65655i − 5.19615i 1.88545 − 15.8885i 0 13.8039 + 15.5388i 74.3539i 36.5704 + 52.5224i −27.0000 0
151.5 −2.80556 − 2.85111i − 5.19615i −0.257663 + 15.9979i 0 −14.8148 + 14.5781i 63.6232i 46.3347 − 44.1485i −27.0000 0
151.6 −2.80556 + 2.85111i 5.19615i −0.257663 − 15.9979i 0 −14.8148 − 14.5781i − 63.6232i 46.3347 + 44.1485i −27.0000 0
151.7 −1.61924 − 3.65760i 5.19615i −10.7561 + 11.8451i 0 19.0055 − 8.41384i 95.1090i 60.7414 + 20.1614i −27.0000 0
151.8 −1.61924 + 3.65760i − 5.19615i −10.7561 − 11.8451i 0 19.0055 + 8.41384i − 95.1090i 60.7414 − 20.1614i −27.0000 0
151.9 −0.725488 − 3.93366i − 5.19615i −14.9473 + 5.70765i 0 −20.4399 + 3.76975i − 36.7329i 33.2960 + 54.6569i −27.0000 0
151.10 −0.725488 + 3.93366i 5.19615i −14.9473 − 5.70765i 0 −20.4399 − 3.76975i 36.7329i 33.2960 − 54.6569i −27.0000 0
151.11 1.87881 − 3.53130i 5.19615i −8.94016 − 13.2693i 0 18.3492 + 9.76257i − 7.86734i −63.6546 + 6.63999i −27.0000 0
151.12 1.87881 + 3.53130i − 5.19615i −8.94016 + 13.2693i 0 18.3492 − 9.76257i 7.86734i −63.6546 − 6.63999i −27.0000 0
151.13 3.36166 − 2.16777i − 5.19615i 6.60152 − 14.5746i 0 −11.2641 − 17.4677i − 36.6738i −9.40241 − 63.3056i −27.0000 0
151.14 3.36166 + 2.16777i 5.19615i 6.60152 + 14.5746i 0 −11.2641 + 17.4677i 36.6738i −9.40241 + 63.3056i −27.0000 0
151.15 3.89995 − 0.889027i 5.19615i 14.4193 − 6.93433i 0 4.61952 + 20.2647i − 44.0991i 50.0696 − 39.8627i −27.0000 0
151.16 3.89995 + 0.889027i − 5.19615i 14.4193 + 6.93433i 0 4.61952 − 20.2647i 44.0991i 50.0696 + 39.8627i −27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.5.c.b ✓ 16
4.b odd 2 1 inner 300.5.c.b ✓ 16
5.b even 2 1 300.5.c.c yes 16
5.c odd 4 2 300.5.f.c 32
20.d odd 2 1 300.5.c.c yes 16
20.e even 4 2 300.5.f.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.5.c.b ✓ 16 1.a even 1 1 trivial
300.5.c.b ✓ 16 4.b odd 2 1 inner
300.5.c.c yes 16 5.b even 2 1
300.5.c.c yes 16 20.d odd 2 1
300.5.f.c 32 5.c odd 4 2
300.5.f.c 32 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{16} + 24568 T_{7}^{14} + 232923164 T_{7}^{12} + 1109365466056 T_{7}^{10} + \cdots + 55\!\cdots\!61 \) Copy content Toggle raw display
\( T_{13}^{8} + 88 T_{13}^{7} - 85588 T_{13}^{6} - 3887096 T_{13}^{5} + 2076207574 T_{13}^{4} + \cdots + 62\!\cdots\!21 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
$3$ \( (T^{2} + 27)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 55\!\cdots\!61 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 62\!\cdots\!21)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 97\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 19\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 87\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 10\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 75\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 86\!\cdots\!76)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 51\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 44\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 44\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 62\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 22\!\cdots\!11)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 16\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 23\!\cdots\!48)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 56\!\cdots\!12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 64\!\cdots\!77)^{2} \) Copy content Toggle raw display
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