Properties

Label 1900.3.e.g
Level $1900$
Weight $3$
Character orbit 1900.e
Analytic conductor $51.771$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1900.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.7712502285\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Defining polynomial: \(x^{14} + 89 x^{12} + 3026 x^{10} + 49092 x^{8} + 390297 x^{6} + 1440495 x^{4} + 1994425 x^{2} + 151875\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{7} q^{7} + ( -4 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{7} q^{7} + ( -4 + \beta_{2} ) q^{9} -\beta_{3} q^{11} -\beta_{12} q^{13} -\beta_{4} q^{17} + ( -2 + \beta_{5} ) q^{19} + ( \beta_{1} - \beta_{8} - \beta_{10} ) q^{21} + ( 2 - \beta_{9} ) q^{23} + ( -3 \beta_{1} + \beta_{10} - \beta_{11} ) q^{27} + ( -\beta_{11} + \beta_{12} ) q^{29} + ( \beta_{5} - \beta_{6} + \beta_{8} - \beta_{11} ) q^{31} + ( -\beta_{5} + \beta_{6} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{33} + ( -\beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{37} + ( -4 - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{39} + ( 4 \beta_{1} + \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{41} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} ) q^{43} + ( -6 + \beta_{2} + 2 \beta_{3} + \beta_{7} ) q^{47} + ( 10 - \beta_{2} - \beta_{4} - \beta_{7} - 2 \beta_{9} ) q^{49} + ( -2 \beta_{1} + \beta_{5} - \beta_{6} + \beta_{10} - 3 \beta_{11} - \beta_{13} ) q^{51} + ( 3 \beta_{1} - \beta_{5} + \beta_{6} - 2 \beta_{10} + \beta_{11} - \beta_{12} ) q^{53} + ( -4 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{57} + ( -\beta_{1} - \beta_{5} + \beta_{6} + 2 \beta_{8} - \beta_{12} - \beta_{13} ) q^{59} + ( -4 - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{7} + \beta_{9} ) q^{61} + ( -17 + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{9} ) q^{63} + ( 7 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{11} - \beta_{13} ) q^{67} + ( 3 \beta_{1} - \beta_{5} + \beta_{6} - 2 \beta_{8} - 2 \beta_{10} + 3 \beta_{11} - 4 \beta_{12} + \beta_{13} ) q^{69} + ( 4 \beta_{1} + \beta_{5} - \beta_{6} - \beta_{8} - 3 \beta_{12} - \beta_{13} ) q^{71} + ( 15 - 5 \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{73} + ( -3 - \beta_{2} + 4 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{9} ) q^{77} + ( 6 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} - \beta_{13} ) q^{79} + ( 6 + \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{81} + ( -21 + 5 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} - 2 \beta_{9} ) q^{83} + ( -\beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{7} - 2 \beta_{9} ) q^{87} + ( -4 \beta_{1} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{10} - \beta_{11} - 2 \beta_{13} ) q^{89} + ( 11 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - \beta_{8} - 5 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} + \beta_{13} ) q^{91} + ( -7 + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{9} ) q^{93} + ( -8 \beta_{1} - 3 \beta_{5} + 3 \beta_{6} - 5 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{97} + ( 8 + 4 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{7} - 52 q^{9} + O(q^{10}) \) \( 14 q - 4 q^{7} - 52 q^{9} + 4 q^{11} + 6 q^{17} - 29 q^{19} + 28 q^{23} - 56 q^{39} - 22 q^{43} - 84 q^{47} + 138 q^{49} + 5 q^{57} - 50 q^{61} - 234 q^{63} + 204 q^{73} - 68 q^{77} + 66 q^{81} - 256 q^{83} - 18 q^{87} - 118 q^{93} + 92 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} + 89 x^{12} + 3026 x^{10} + 49092 x^{8} + 390297 x^{6} + 1440495 x^{4} + 1994425 x^{2} + 151875\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 13 \)
\(\beta_{3}\)\(=\)\((\)\( -107 \nu^{12} + 4422 \nu^{10} + 631973 \nu^{8} + 17246726 \nu^{6} + 168176486 \nu^{4} + 507707525 \nu^{2} + 88760925 \)\()/22723050\)
\(\beta_{4}\)\(=\)\((\)\( -648 \nu^{12} - 34907 \nu^{10} - 499913 \nu^{8} + 407849 \nu^{6} + 49911024 \nu^{4} + 451211695 \nu^{2} + 843924750 \)\()/37871750\)
\(\beta_{5}\)\(=\)\((\)\(1462 \nu^{13} + 23670 \nu^{12} + 110483 \nu^{11} + 2161755 \nu^{10} + 2732597 \nu^{9} + 72077670 \nu^{8} + 14407194 \nu^{7} + 1050305265 \nu^{6} - 382439106 \nu^{5} + 6162209865 \nu^{4} - 5407506030 \nu^{3} + 10133859150 \nu^{2} - 17209386875 \nu - 2922880500\)\()/ 340845750 \)
\(\beta_{6}\)\(=\)\((\)\(-1462 \nu^{13} + 23670 \nu^{12} - 110483 \nu^{11} + 2161755 \nu^{10} - 2732597 \nu^{9} + 72077670 \nu^{8} - 14407194 \nu^{7} + 1050305265 \nu^{6} + 382439106 \nu^{5} + 6162209865 \nu^{4} + 5407506030 \nu^{3} + 10133859150 \nu^{2} + 17209386875 \nu - 2922880500\)\()/ 340845750 \)
\(\beta_{7}\)\(=\)\((\)\( -13561 \nu^{12} - 1135449 \nu^{10} - 35328191 \nu^{8} - 496542482 \nu^{6} - 3026541407 \nu^{4} - 6362924960 \nu^{2} - 743885250 \)\()/ 113615250 \)
\(\beta_{8}\)\(=\)\((\)\( -14264 \nu^{13} - 1102351 \nu^{11} - 29091859 \nu^{9} - 259998543 \nu^{7} + 619030032 \nu^{5} + 16213033785 \nu^{3} + 40788043750 \nu \)\()/ 340845750 \)
\(\beta_{9}\)\(=\)\((\)\( 1401 \nu^{12} + 117049 \nu^{10} + 3662221 \nu^{8} + 52791402 \nu^{6} + 343822347 \nu^{4} + 805742830 \nu^{2} + 146643050 \)\()/7574350\)
\(\beta_{10}\)\(=\)\((\)\( -26419 \nu^{13} - 2303996 \nu^{11} - 76892714 \nu^{9} - 1229628903 \nu^{7} - 9698654253 \nu^{5} - 35301808665 \nu^{3} - 42678853750 \nu \)\()/ 340845750 \)
\(\beta_{11}\)\(=\)\((\)\( -26419 \nu^{13} - 2303996 \nu^{11} - 76892714 \nu^{9} - 1229628903 \nu^{7} - 9698654253 \nu^{5} - 35642654415 \nu^{3} - 49836614500 \nu \)\()/ 340845750 \)
\(\beta_{12}\)\(=\)\((\)\( 5608 \nu^{13} + 473087 \nu^{11} + 14768858 \nu^{9} + 206445486 \nu^{7} + 1219089951 \nu^{5} + 2159753685 \nu^{3} - 1252190975 \nu \)\()/68169150\)
\(\beta_{13}\)\(=\)\((\)\( 9986 \nu^{13} + 902959 \nu^{11} + 30950281 \nu^{9} + 498348567 \nu^{7} + 3816325902 \nu^{5} + 12974409555 \nu^{3} + 15865169450 \nu \)\()/68169150\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 13\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{10} - 21 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{9} - 3 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} - 26 \beta_{2} + 276\)
\(\nu^{5}\)\(=\)\(\beta_{13} - 6 \beta_{12} + 33 \beta_{11} - 35 \beta_{10} - 5 \beta_{8} + 2 \beta_{6} - 2 \beta_{5} + 496 \beta_{1}\)
\(\nu^{6}\)\(=\)\(58 \beta_{9} + 144 \beta_{7} + 39 \beta_{6} + 39 \beta_{5} - 56 \beta_{4} - 18 \beta_{3} + 645 \beta_{2} - 6578\)
\(\nu^{7}\)\(=\)\(-54 \beta_{13} + 328 \beta_{12} - 927 \beta_{11} + 1039 \beta_{10} + 260 \beta_{8} - 57 \beta_{6} + 57 \beta_{5} - 12208 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-2269 \beta_{9} - 5267 \beta_{7} - 1314 \beta_{6} - 1314 \beta_{5} + 1228 \beta_{4} + 1160 \beta_{3} - 16017 \beta_{2} + 163233\)
\(\nu^{9}\)\(=\)\(2029 \beta_{13} - 12864 \beta_{12} + 25040 \beta_{11} - 29678 \beta_{10} - 9805 \beta_{8} + 1023 \beta_{6} - 1023 \beta_{5} + 306617 \beta_{1}\)
\(\nu^{10}\)\(=\)\(77045 \beta_{9} + 173705 \beta_{7} + 42099 \beta_{6} + 42099 \beta_{5} - 23710 \beta_{4} - 48782 \beta_{3} + 401423 \beta_{2} - 4131880\)
\(\nu^{11}\)\(=\)\(-65339 \beta_{13} + 441160 \beta_{12} - 668272 \beta_{11} + 837126 \beta_{10} + 327795 \beta_{8} - 9874 \beta_{6} + 9874 \beta_{5} - 7809830 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-2440375 \beta_{9} - 5434385 \beta_{7} - 1306588 \beta_{6} - 1306588 \beta_{5} + 390220 \beta_{4} + 1721594 \beta_{3} - 10167868 \beta_{2} + 106018551\)
\(\nu^{13}\)\(=\)\(1939013 \beta_{13} - 14096270 \beta_{12} + 17768071 \beta_{11} - 23486399 \beta_{10} - 10315135 \beta_{8} - 197587 \beta_{6} + 197587 \beta_{5} + 201243047 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1101.1
5.26531i
4.84257i
4.58743i
2.93221i
2.16913i
1.84332i
0.284180i
0.284180i
1.84332i
2.16913i
2.93221i
4.58743i
4.84257i
5.26531i
0 5.26531i 0 0 0 12.1735 0 −18.7235 0
1101.2 0 4.84257i 0 0 0 −6.85668 0 −14.4504 0
1101.3 0 4.58743i 0 0 0 −1.52935 0 −12.0445 0
1101.4 0 2.93221i 0 0 0 −5.62705 0 0.402160 0
1101.5 0 2.16913i 0 0 0 8.18099 0 4.29489 0
1101.6 0 1.84332i 0 0 0 −10.5375 0 5.60216 0
1101.7 0 0.284180i 0 0 0 2.19606 0 8.91924 0
1101.8 0 0.284180i 0 0 0 2.19606 0 8.91924 0
1101.9 0 1.84332i 0 0 0 −10.5375 0 5.60216 0
1101.10 0 2.16913i 0 0 0 8.18099 0 4.29489 0
1101.11 0 2.93221i 0 0 0 −5.62705 0 0.402160 0
1101.12 0 4.58743i 0 0 0 −1.52935 0 −12.0445 0
1101.13 0 4.84257i 0 0 0 −6.85668 0 −14.4504 0
1101.14 0 5.26531i 0 0 0 12.1735 0 −18.7235 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1101.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.3.e.g 14
5.b even 2 1 1900.3.e.h yes 14
5.c odd 4 2 1900.3.g.d 28
19.b odd 2 1 inner 1900.3.e.g 14
95.d odd 2 1 1900.3.e.h yes 14
95.g even 4 2 1900.3.g.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.3.e.g 14 1.a even 1 1 trivial
1900.3.e.g 14 19.b odd 2 1 inner
1900.3.e.h yes 14 5.b even 2 1
1900.3.e.h yes 14 95.d odd 2 1
1900.3.g.d 28 5.c odd 4 2
1900.3.g.d 28 95.g even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1900, [\chi])\):

\( T_{3}^{14} + 89 T_{3}^{12} + 3026 T_{3}^{10} + 49092 T_{3}^{8} + 390297 T_{3}^{6} + 1440495 T_{3}^{4} + 1994425 T_{3}^{2} + 151875 \)
\( T_{7}^{7} + 2 T_{7}^{6} - 204 T_{7}^{5} - 640 T_{7}^{4} + 9845 T_{7}^{3} + 37276 T_{7}^{2} - 56107 T_{7} - 135988 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \)
$3$ \( 151875 + 1994425 T^{2} + 1440495 T^{4} + 390297 T^{6} + 49092 T^{8} + 3026 T^{10} + 89 T^{12} + T^{14} \)
$5$ \( T^{14} \)
$7$ \( ( -135988 - 56107 T + 37276 T^{2} + 9845 T^{3} - 640 T^{4} - 204 T^{5} + 2 T^{6} + T^{7} )^{2} \)
$11$ \( ( -1375296 - 566944 T + 143260 T^{2} + 52709 T^{3} - 187 T^{4} - 491 T^{5} - 2 T^{6} + T^{7} )^{2} \)
$13$ \( 835543129687500 + 65048234903125 T^{2} + 1929559114925 T^{4} + 27930202763 T^{6} + 209455663 T^{8} + 800283 T^{10} + 1459 T^{12} + T^{14} \)
$17$ \( ( 48317883 - 4000903 T - 2774055 T^{2} + 306059 T^{3} + 6724 T^{4} - 1132 T^{5} - 3 T^{6} + T^{7} )^{2} \)
$19$ \( 799006685782884121 + 64186132652918669 T - 410781439272667 T^{2} - 219869207128786 T^{3} - 14600642122469 T^{4} - 362775740589 T^{5} + 33077424615 T^{6} + 3302937732 T^{7} + 91627215 T^{8} - 2783709 T^{9} - 310349 T^{10} - 12946 T^{11} - 67 T^{12} + 29 T^{13} + T^{14} \)
$23$ \( ( 60378750 - 66767625 T - 1618050 T^{2} + 1064485 T^{3} + 21124 T^{4} - 2126 T^{5} - 14 T^{6} + T^{7} )^{2} \)
$29$ \( 139157556567187500 + 8185037983368325 T^{2} + 115385205118835 T^{4} + 722146001707 T^{6} + 2330952117 T^{8} + 3945061 T^{10} + 3229 T^{12} + T^{14} \)
$31$ \( 62870637263155680000 + 4269589200370412800 T^{2} + 23216867887846640 T^{4} + 49130733549268 T^{6} + 52012954041 T^{8} + 29450980 T^{10} + 8545 T^{12} + T^{14} \)
$37$ \( \)\(20\!\cdots\!00\)\( + 6013875372966958825 T^{2} + 48994492235858545 T^{4} + 118380339024868 T^{6} + 117610092001 T^{8} + 55061870 T^{10} + 12090 T^{12} + T^{14} \)
$41$ \( \)\(10\!\cdots\!00\)\( + 13803332456922761200 T^{2} + 69413681562706040 T^{4} + 163769224031707 T^{6} + 179488225761 T^{8} + 78974173 T^{10} + 14818 T^{12} + T^{14} \)
$43$ \( ( -5955376816 - 949508824 T + 51012820 T^{2} + 3923436 T^{3} - 69183 T^{4} - 4429 T^{5} + 11 T^{6} + T^{7} )^{2} \)
$47$ \( ( -588156954 + 6357071 T + 11774525 T^{2} + 112028 T^{3} - 65413 T^{4} - 1740 T^{5} + 42 T^{6} + T^{7} )^{2} \)
$53$ \( \)\(55\!\cdots\!00\)\( + 96856552722325890625 T^{2} + 552354458214305000 T^{4} + 1184644176585200 T^{6} + 758527686355 T^{8} + 203092278 T^{10} + 23785 T^{12} + T^{14} \)
$59$ \( \)\(32\!\cdots\!00\)\( + \)\(16\!\cdots\!00\)\( T^{2} + 3983659526653723840 T^{4} + 3580384257345092 T^{6} + 1497578884805 T^{8} + 306407345 T^{10} + 28918 T^{12} + T^{14} \)
$61$ \( ( 1184496469600 - 75337068200 T - 443506444 T^{2} + 58011820 T^{3} - 88571 T^{4} - 13406 T^{5} + 25 T^{6} + T^{7} )^{2} \)
$67$ \( \)\(14\!\cdots\!75\)\( + \)\(43\!\cdots\!50\)\( T^{2} + 20418688324750649880 T^{4} + 16756036216816282 T^{6} + 5207164719664 T^{8} + 710936548 T^{10} + 43812 T^{12} + T^{14} \)
$71$ \( \)\(16\!\cdots\!00\)\( + \)\(25\!\cdots\!00\)\( T^{2} + 6872379900868940560 T^{4} + 8092668880751808 T^{6} + 3176214808129 T^{8} + 523289759 T^{10} + 37943 T^{12} + T^{14} \)
$73$ \( ( -322844910273 + 60569768502 T - 1921442800 T^{2} - 33813126 T^{3} + 1703216 T^{4} - 11892 T^{5} - 102 T^{6} + T^{7} )^{2} \)
$79$ \( \)\(13\!\cdots\!00\)\( + \)\(29\!\cdots\!00\)\( T^{2} + \)\(25\!\cdots\!60\)\( T^{4} + 85573794542590952 T^{6} + 14519018308409 T^{8} + 1302087023 T^{10} + 58132 T^{12} + T^{14} \)
$83$ \( ( -10116036761400 - 80185887260 T + 9908051484 T^{2} + 83062091 T^{3} - 2639073 T^{4} - 22183 T^{5} + 128 T^{6} + T^{7} )^{2} \)
$89$ \( \)\(75\!\cdots\!00\)\( + \)\(11\!\cdots\!00\)\( T^{2} + \)\(61\!\cdots\!40\)\( T^{4} + 164320081120969723 T^{6} + 23463339400932 T^{8} + 1796791733 T^{10} + 68471 T^{12} + T^{14} \)
$97$ \( \)\(64\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( T^{2} + \)\(66\!\cdots\!00\)\( T^{4} + 1156284576864875108 T^{6} + 101057584591649 T^{8} + 4664670080 T^{10} + 108465 T^{12} + T^{14} \)
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