Properties

Label 325.2.o.c
Level $325$
Weight $2$
Character orbit 325.o
Analytic conductor $2.595$
Analytic rank $0$
Dimension $20$
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] = [325,2,Mod(74,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.74");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.o (of order \(6\), degree \(2\), 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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 16 x^{18} + 168 x^{16} - 1022 x^{14} + 4518 x^{12} - 12577 x^{10} + 24961 x^{8} - 25050 x^{6} + 17307 x^{4} - 1242 x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{11} + \beta_{7}) q^{4} + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{6} + (\beta_{17} - \beta_{14} + \beta_{13}) q^{7} + (\beta_{18} + \beta_{15} + \beta_{14} - \beta_{10} + \beta_1) q^{8} + ( - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{11} + \beta_{7}) q^{4} + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{6} + (\beta_{17} - \beta_{14} + \beta_{13}) q^{7} + (\beta_{18} + \beta_{15} + \beta_{14} - \beta_{10} + \beta_1) q^{8} + ( - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3}) q^{9} + (\beta_{7} - \beta_{6} - 1) q^{11} + \beta_{17} q^{12} + (\beta_{18} + \beta_{17} - \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{10} - \beta_1) q^{13} + (\beta_{5} - \beta_{3} + \beta_{2} + 2) q^{14} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} - 1) q^{16} + (\beta_{18} + \beta_{17} + \beta_{14} + \beta_{13} - \beta_{12}) q^{17} + ( - \beta_{18} + \beta_{17} - 2 \beta_{15} - \beta_{14} + 2 \beta_{10} - \beta_1) q^{18} + (\beta_{11} - \beta_{7}) q^{19} + ( - \beta_{4} - \beta_{2} - 1) q^{21} + ( - \beta_{17} - 2 \beta_{15} - \beta_{13}) q^{22} + ( - 2 \beta_{12} - \beta_{10} + \beta_1) q^{23} + ( - \beta_{9} + 3 \beta_{7} - \beta_{6} - 3) q^{24} + ( - \beta_{11} - 2 \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3}) q^{26} + (\beta_{19} - 2 \beta_{17} + 3 \beta_{15} - 3 \beta_{10}) q^{27} + (\beta_{13} + 2 \beta_{12} + \beta_{10} + 2 \beta_1) q^{28} + ( - \beta_{11} + \beta_{8} - \beta_{4} + \beta_{2}) q^{29} + (\beta_{5} + \beta_{2}) q^{31} + ( - \beta_{19} + 2 \beta_{18} + \beta_{16} - \beta_{15} - 2 \beta_{12}) q^{32} + (\beta_{18} - \beta_{17} + 2 \beta_{15} - 2 \beta_{14} - \beta_{13} - \beta_{12}) q^{33} + (\beta_{4} - \beta_{3} - 2 \beta_{2} - 5) q^{34} + ( - 2 \beta_{11} + \beta_{8} - 3 \beta_{7} + \beta_{6} - \beta_{4} + 2 \beta_{2} + 3) q^{36} + ( - \beta_{16} - \beta_{13} + \beta_{12} - \beta_{10}) q^{37} + (\beta_{18} + \beta_{15} + \beta_{14} - \beta_{10} + \beta_1) q^{38} + (\beta_{9} - 3 \beta_{7} - \beta_{4} - \beta_{2} + 3) q^{39} + (2 \beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{4} - 2 \beta_{2} - 1) q^{41} + ( - \beta_{16} - \beta_{13} + 2 \beta_{12} + 2 \beta_{10} - 4 \beta_1) q^{42} + ( - 2 \beta_{18} + 3 \beta_{14} + 2 \beta_{12}) q^{43} + ( - \beta_{5} + \beta_{3} + 1) q^{44} + ( - \beta_{11} - 2 \beta_{9} + 2 \beta_{8} + \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{46} + ( - \beta_{19} - \beta_{18} - \beta_{17} + 3 \beta_{14} + 3 \beta_1) q^{47} + ( - \beta_{18} - \beta_{17} - 2 \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12}) q^{48} + ( - 2 \beta_{11} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{2} + 1) q^{49} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{51} + ( - \beta_{18} - \beta_{17} + \beta_{16} - \beta_{15} - 3 \beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{10} + \cdots - 4 \beta_1) q^{52}+ \cdots + (2 \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 12 q^{4} - 6 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 12 q^{4} - 6 q^{6} + 8 q^{9} - 6 q^{11} + 32 q^{14} - 8 q^{16} - 8 q^{19} - 32 q^{21} - 28 q^{24} - 18 q^{26} - 2 q^{29} - 108 q^{34} + 26 q^{36} + 20 q^{39} - 12 q^{41} + 32 q^{44} + 14 q^{46} + 14 q^{49} + 38 q^{54} + 34 q^{56} - 24 q^{59} - 10 q^{61} + 20 q^{64} + 172 q^{66} + 40 q^{69} - 38 q^{71} + 4 q^{74} - 48 q^{76} + 56 q^{79} - 58 q^{81} - 68 q^{84} - 168 q^{86} - 20 q^{89} + 34 q^{91} - 62 q^{94} + 68 q^{96} - 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 16 x^{18} + 168 x^{16} - 1022 x^{14} + 4518 x^{12} - 12577 x^{10} + 24961 x^{8} - 25050 x^{6} + 17307 x^{4} - 1242 x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 26863568 \nu^{18} + 405681404 \nu^{16} - 4156515402 \nu^{14} + 23801296135 \nu^{12} - 100725879444 \nu^{10} + 249332478494 \nu^{8} + \cdots - 784835807949 ) / 268382163375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 184732631 \nu^{18} - 2787465974 \nu^{16} + 28559714937 \nu^{14} - 163780437865 \nu^{12} + 692094729714 \nu^{10} - 1713181312139 \nu^{8} + \cdots + 555085489302 ) / 483087894075 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1541220706 \nu^{18} - 24010628908 \nu^{16} + 246007206354 \nu^{14} - 1431129051095 \nu^{12} + 5961554285988 \nu^{10} + \cdots + 2517866659143 ) / 2415439470375 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 616989689 \nu^{18} - 9590339552 \nu^{16} + 98260343376 \nu^{14} - 570567788680 \nu^{12} + 2381167527072 \nu^{10} + \cdots + 1970521442442 ) / 805146490125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2247835573 \nu^{18} - 36643617454 \nu^{16} + 387026214582 \nu^{14} - 2393494010915 \nu^{12} + 10686512655534 \nu^{10} + \cdots - 3135575353986 ) / 2415439470375 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 752247488 \nu^{18} - 11955369104 \nu^{16} + 125160533772 \nu^{14} - 756327386530 \nu^{12} + 3327250262379 \nu^{10} + \cdots - 84588695091 ) / 805146490125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3034965602 \nu^{18} - 49691600891 \nu^{16} + 525170629968 \nu^{14} - 3246813163900 \nu^{12} + 14406712674306 \nu^{10} + \cdots - 370294114329 ) / 2415439470375 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 507278227 \nu^{18} - 8303208961 \nu^{16} + 88236686808 \nu^{14} - 549318509930 \nu^{12} + 2473855622691 \nu^{10} + \cdots - 740380583214 ) / 268382163375 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3940067489 \nu^{19} + 60296811032 \nu^{17} - 617786818116 \nu^{15} + 3563719409905 \nu^{13} - 14970982793352 \nu^{11} + \cdots - 16110502021512 \nu ) / 7246318411125 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 752247488 \nu^{18} + 11955369104 \nu^{16} - 125160533772 \nu^{14} + 756327386530 \nu^{12} - 3327250262379 \nu^{10} + \cdots + 84588695091 ) / 268382163375 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 933076765 \nu^{19} + 14250041788 \nu^{17} - 146002546794 \nu^{15} + 841270881110 \nu^{13} - 3538116307668 \nu^{11} + \cdots - 10358741131677 \nu ) / 1449263682225 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5109145513 \nu^{19} - 78781632454 \nu^{17} + 807177912177 \nu^{15} - 4670732251835 \nu^{13} + 19560544640994 \nu^{11} + \cdots + 31141040114619 \nu ) / 7246318411125 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 752247488 \nu^{19} - 11955369104 \nu^{17} + 125160533772 \nu^{15} - 756327386530 \nu^{13} + 3327250262379 \nu^{11} + \cdots - 889735185216 \nu ) / 805146490125 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2666577601 \nu^{19} - 43219439509 \nu^{17} + 456347434890 \nu^{15} - 2810921453033 \nu^{13} + 12538938914913 \nu^{11} + \cdots - 3618038185722 \nu ) / 1449263682225 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 6140567453 \nu^{19} + 93599061194 \nu^{17} - 958993771047 \nu^{15} + 5520042575035 \nu^{13} - 23239536397134 \nu^{11} + \cdots - 67441792514994 \nu ) / 2415439470375 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 31801642196 \nu^{19} + 508643497118 \nu^{17} - 5336018421279 \nu^{15} + 32406411228190 \nu^{13} - 142835648184933 \nu^{11} + \cdots + 3640232347587 \nu ) / 7246318411125 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 51124092454 \nu^{19} + 814385618257 \nu^{17} - 8531748012306 \nu^{15} + 51653059068920 \nu^{13} - 227391939174972 \nu^{11} + \cdots + 5786180186193 \nu ) / 7246318411125 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 66704068546 \nu^{19} + 1062239293243 \nu^{17} - 11127239788269 \nu^{15} + 67350140699030 \nu^{13} - 296467419138753 \nu^{11} + \cdots + 7542959242032 \nu ) / 2415439470375 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 3\beta_{7} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} + \beta_{15} + 5\beta_{14} - \beta_{10} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{11} + \beta_{9} - \beta_{8} + 15\beta_{7} + \beta_{6} + \beta_{4} - 6\beta_{2} - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{19} + 10\beta_{18} + \beta_{16} + 7\beta_{15} + 28\beta_{14} - 10\beta_{12} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{5} + 11\beta_{4} - 8\beta_{3} - 37\beta_{2} - 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{16} - \beta_{13} - 80\beta_{12} + 42\beta_{10} - 167\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -236\beta_{11} - 81\beta_{9} + 91\beta_{8} - 527\beta_{7} - 52\beta_{6} - 81\beta_{5} - 52\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 91\beta_{19} - 590\beta_{18} + 19\beta_{17} - 249\beta_{15} - 1041\beta_{14} + 249\beta_{10} - 1041\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1540\beta_{11} - 609\beta_{9} + 681\beta_{8} - 3354\beta_{7} - 321\beta_{6} - 681\beta_{4} + 1540\beta_{2} + 3354 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 681 \beta_{19} - 4192 \beta_{18} + 216 \beta_{17} - 681 \beta_{16} - 1501 \beta_{15} - 6683 \beta_{14} + 216 \beta_{13} + 4192 \beta_{12} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4408\beta_{5} - 4873\beta_{4} + 1966\beta_{3} + 10194\beta_{2} + 21843 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -4873\beta_{16} + 1977\beta_{13} + 29221\beta_{12} - 9253\beta_{10} + 43732\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 68080 \beta_{11} + 31198 \beta_{9} - 34094 \beta_{8} + 144314 \beta_{7} + 12149 \beta_{6} + 31198 \beta_{5} + 12149 \beta_{3} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 34094 \beta_{19} + 201560 \beta_{18} - 16153 \beta_{17} + 58284 \beta_{15} + 289727 \beta_{14} - 58284 \beta_{10} + 289727 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 457193 \beta_{11} + 217713 \beta_{9} - 235654 \beta_{8} + 962210 \beta_{7} + 76225 \beta_{6} + 235654 \beta_{4} - 457193 \beta_{2} - 962210 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 235654 \beta_{19} + 1381868 \beta_{18} - 123547 \beta_{17} + 235654 \beta_{16} + 373989 \beta_{15} + 1934880 \beta_{14} - 123547 \beta_{13} - 1381868 \beta_{12} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( -1505415\beta_{5} + 1617522\beta_{4} - 486096\beta_{3} - 3081094\beta_{2} - 6453318 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 1617522\beta_{16} - 907212\beta_{13} - 9439075\beta_{12} + 2435764\beta_{10} - 12989495\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
−2.25583 1.30241i
−1.87176 1.08066i
−1.56608 0.904178i
−0.949161 0.547998i
−0.232843 0.134432i
0.232843 + 0.134432i
0.949161 + 0.547998i
1.56608 + 0.904178i
1.87176 + 1.08066i
2.25583 + 1.30241i
−2.25583 + 1.30241i
−1.87176 + 1.08066i
−1.56608 + 0.904178i
−0.949161 + 0.547998i
−0.232843 + 0.134432i
0.232843 0.134432i
0.949161 0.547998i
1.56608 0.904178i
1.87176 1.08066i
2.25583 1.30241i
−2.25583 1.30241i 0.117191 + 0.0676602i 2.39253 + 4.14398i 0 −0.176242 0.305261i −2.81660 + 1.62616i 7.25455i −1.49084 2.58222i 0
74.2 −1.87176 1.08066i −1.69816 0.980433i 1.33565 + 2.31341i 0 2.11903 + 3.67026i 2.66453 1.53837i 1.45089i 0.422497 + 0.731787i 0
74.3 −1.56608 0.904178i 1.60910 + 0.929015i 0.635076 + 1.09998i 0 −1.67999 2.90983i −3.60988 + 2.08417i 1.31983i 0.226138 + 0.391682i 0
74.4 −0.949161 0.547998i 2.91389 + 1.68234i −0.399395 0.691773i 0 −1.84383 3.19362i 1.37843 0.795836i 3.06747i 4.16051 + 7.20621i 0
74.5 −0.232843 0.134432i −0.522064 0.301414i −0.963856 1.66945i 0 0.0810394 + 0.140364i −1.23923 + 0.715471i 1.05602i −1.31830 2.28336i 0
74.6 0.232843 + 0.134432i 0.522064 + 0.301414i −0.963856 1.66945i 0 0.0810394 + 0.140364i 1.23923 0.715471i 1.05602i −1.31830 2.28336i 0
74.7 0.949161 + 0.547998i −2.91389 1.68234i −0.399395 0.691773i 0 −1.84383 3.19362i −1.37843 + 0.795836i 3.06747i 4.16051 + 7.20621i 0
74.8 1.56608 + 0.904178i −1.60910 0.929015i 0.635076 + 1.09998i 0 −1.67999 2.90983i 3.60988 2.08417i 1.31983i 0.226138 + 0.391682i 0
74.9 1.87176 + 1.08066i 1.69816 + 0.980433i 1.33565 + 2.31341i 0 2.11903 + 3.67026i −2.66453 + 1.53837i 1.45089i 0.422497 + 0.731787i 0
74.10 2.25583 + 1.30241i −0.117191 0.0676602i 2.39253 + 4.14398i 0 −0.176242 0.305261i 2.81660 1.62616i 7.25455i −1.49084 2.58222i 0
224.1 −2.25583 + 1.30241i 0.117191 0.0676602i 2.39253 4.14398i 0 −0.176242 + 0.305261i −2.81660 1.62616i 7.25455i −1.49084 + 2.58222i 0
224.2 −1.87176 + 1.08066i −1.69816 + 0.980433i 1.33565 2.31341i 0 2.11903 3.67026i 2.66453 + 1.53837i 1.45089i 0.422497 0.731787i 0
224.3 −1.56608 + 0.904178i 1.60910 0.929015i 0.635076 1.09998i 0 −1.67999 + 2.90983i −3.60988 2.08417i 1.31983i 0.226138 0.391682i 0
224.4 −0.949161 + 0.547998i 2.91389 1.68234i −0.399395 + 0.691773i 0 −1.84383 + 3.19362i 1.37843 + 0.795836i 3.06747i 4.16051 7.20621i 0
224.5 −0.232843 + 0.134432i −0.522064 + 0.301414i −0.963856 + 1.66945i 0 0.0810394 0.140364i −1.23923 0.715471i 1.05602i −1.31830 + 2.28336i 0
224.6 0.232843 0.134432i 0.522064 0.301414i −0.963856 + 1.66945i 0 0.0810394 0.140364i 1.23923 + 0.715471i 1.05602i −1.31830 + 2.28336i 0
224.7 0.949161 0.547998i −2.91389 + 1.68234i −0.399395 + 0.691773i 0 −1.84383 + 3.19362i −1.37843 0.795836i 3.06747i 4.16051 7.20621i 0
224.8 1.56608 0.904178i −1.60910 + 0.929015i 0.635076 1.09998i 0 −1.67999 + 2.90983i 3.60988 + 2.08417i 1.31983i 0.226138 0.391682i 0
224.9 1.87176 1.08066i 1.69816 0.980433i 1.33565 2.31341i 0 2.11903 3.67026i −2.66453 1.53837i 1.45089i 0.422497 0.731787i 0
224.10 2.25583 1.30241i −0.117191 + 0.0676602i 2.39253 4.14398i 0 −0.176242 + 0.305261i 2.81660 + 1.62616i 7.25455i −1.49084 + 2.58222i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.o.c 20
5.b even 2 1 inner 325.2.o.c 20
5.c odd 4 1 325.2.e.c 10
5.c odd 4 1 325.2.e.d yes 10
13.c even 3 1 inner 325.2.o.c 20
65.n even 6 1 inner 325.2.o.c 20
65.q odd 12 1 325.2.e.c 10
65.q odd 12 1 325.2.e.d yes 10
65.q odd 12 1 4225.2.a.bn 5
65.q odd 12 1 4225.2.a.bp 5
65.r odd 12 1 4225.2.a.bm 5
65.r odd 12 1 4225.2.a.bo 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.e.c 10 5.c odd 4 1
325.2.e.c 10 65.q odd 12 1
325.2.e.d yes 10 5.c odd 4 1
325.2.e.d yes 10 65.q odd 12 1
325.2.o.c 20 1.a even 1 1 trivial
325.2.o.c 20 5.b even 2 1 inner
325.2.o.c 20 13.c even 3 1 inner
325.2.o.c 20 65.n even 6 1 inner
4225.2.a.bm 5 65.r odd 12 1
4225.2.a.bn 5 65.q odd 12 1
4225.2.a.bo 5 65.r odd 12 1
4225.2.a.bp 5 65.q odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 16 T_{2}^{18} + 168 T_{2}^{16} - 1022 T_{2}^{14} + 4518 T_{2}^{12} - 12577 T_{2}^{10} + 24961 T_{2}^{8} - 25050 T_{2}^{6} + 17307 T_{2}^{4} - 1242 T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 16 T^{18} + 168 T^{16} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{20} - 19 T^{18} + 258 T^{16} - 1583 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 42 T^{18} + 1139 T^{16} + \cdots + 81450625 \) Copy content Toggle raw display
$11$ \( (T^{10} + 3 T^{9} + 26 T^{8} + 61 T^{7} + \cdots + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} - 20 T^{18} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( T^{20} - 88 T^{18} + \cdots + 57178852641 \) Copy content Toggle raw display
$19$ \( (T^{10} + 4 T^{9} + 24 T^{8} + 34 T^{7} + \cdots + 225)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} - 169 T^{18} + \cdots + 79502005521 \) Copy content Toggle raw display
$29$ \( (T^{10} + T^{9} + 66 T^{8} + 233 T^{7} + \cdots + 881721)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 57 T^{3} - 29 T^{2} + 495 T + 225)^{4} \) Copy content Toggle raw display
$37$ \( T^{20} - 183 T^{18} + \cdots + 682740290961 \) Copy content Toggle raw display
$41$ \( (T^{10} + 6 T^{9} + 110 T^{8} + 354 T^{7} + \cdots + 50625)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 201364301903521 \) Copy content Toggle raw display
$47$ \( (T^{10} + 184 T^{8} + 8044 T^{6} + \cdots + 4100625)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 298 T^{8} + 25443 T^{6} + \cdots + 36905625)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 12 T^{9} + 231 T^{8} + \cdots + 4782969)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 5 T^{9} + 72 T^{8} + 145 T^{7} + \cdots + 403225)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} - 518 T^{18} + \cdots + 65\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{10} + 19 T^{9} + 368 T^{8} + \cdots + 101787921)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 302 T^{8} + 31801 T^{6} + \cdots + 102515625)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 14 T^{4} - 95 T^{3} + 1350 T^{2} + \cdots - 16875)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + 563 T^{8} + 110723 T^{6} + \cdots + 2891750625)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 10 T^{9} + 139 T^{8} + \cdots + 2537649)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} - 633 T^{18} + \cdots + 28\!\cdots\!81 \) Copy content Toggle raw display
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