# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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