# Properties

 Label 1900.2.l.b Level $1900$ Weight $2$ Character orbit 1900.l Analytic conductor $15.172$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(493,1900)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1900.493");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.l (of order $$4$$, degree $$2$$, 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: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 4 x^{11} + 28 x^{10} - 64 x^{9} + 236 x^{8} - 420 x^{7} + 946 x^{6} - 1216 x^{5} + 1896 x^{4} - 1564 x^{3} + 2284 x^{2} - 1088 x + 1370$$ x^12 - 4*x^11 + 28*x^10 - 64*x^9 + 236*x^8 - 420*x^7 + 946*x^6 - 1216*x^5 + 1896*x^4 - 1564*x^3 + 2284*x^2 - 1088*x + 1370 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 380) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} - \beta_{6} q^{7} + ( - \beta_{8} - \beta_{6} + 4 \beta_{5} + \beta_{3}) q^{9}+O(q^{10})$$ q - b2 * q^3 - b6 * q^7 + (-b8 - b6 + 4*b5 + b3) * q^9 $$q - \beta_{2} q^{3} - \beta_{6} q^{7} + ( - \beta_{8} - \beta_{6} + 4 \beta_{5} + \beta_{3}) q^{9} + ( - \beta_1 + 1) q^{11} + (\beta_{11} + \beta_{7} + \beta_{2}) q^{13} + (\beta_{8} + 2 \beta_{6} - \beta_1) q^{17} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3}) q^{19} + (\beta_{11} - \beta_{10}) q^{21} + ( - 2 \beta_{8} + \beta_{3} - 2 \beta_1) q^{23} + ( - \beta_{10} + \beta_{9} + 2 \beta_{4}) q^{27} + ( - \beta_{4} - \beta_{2}) q^{29} + (\beta_{11} - \beta_{10}) q^{31} + (\beta_{11} + \beta_{7}) q^{33} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{4}) q^{37} + (5 \beta_{8} - \beta_{5}) q^{39} + (\beta_{11} - \beta_{10} + \beta_{4} - \beta_{2}) q^{41} + (2 \beta_{8} - 2 \beta_{5} - \beta_{3} + 2 \beta_1 + 2) q^{43} + (2 \beta_{8} + 3 \beta_{6} - 2 \beta_{5} - 2 \beta_1 - 2) q^{47} + ( - 2 \beta_{8} - 3 \beta_{5}) q^{49} + (2 \beta_{10} - \beta_{4} + \beta_{2}) q^{51} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{2}) q^{53} + (\beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} + 2 \beta_1 - 3) q^{57} + (\beta_{9} + 3 \beta_{7} + 3 \beta_{4} + 3 \beta_{2}) q^{59} + (\beta_{6} + \beta_{3} - \beta_1 + 1) q^{61} + ( - \beta_{8} + 3 \beta_{5} + 5 \beta_{3} - \beta_1 - 3) q^{63} + (\beta_{11} - \beta_{7} - 3 \beta_{4}) q^{67} + (\beta_{9} + 3 \beta_{7} + 2 \beta_{4} + 2 \beta_{2}) q^{69} + (\beta_{11} + \beta_{10}) q^{71} + (\beta_{8} - 6 \beta_{5} + \beta_1 + 6) q^{73} + ( - \beta_{8} + \beta_{5} + \beta_1 + 1) q^{77} + (\beta_{9} + \beta_{7} - \beta_{4} - \beta_{2}) q^{79} + (6 \beta_{6} + 6 \beta_{3} - 5 \beta_1 - 2) q^{81} + (\beta_{8} + 7 \beta_{5} - \beta_{3} + \beta_1 - 7) q^{83} + ( - \beta_{8} - 2 \beta_{6} + 7 \beta_{5} + \beta_1 + 7) q^{87} + ( - 2 \beta_{9} - \beta_{4} - \beta_{2}) q^{89} + ( - 3 \beta_{11} + \beta_{10} + 2 \beta_{4} - 2 \beta_{2}) q^{91} + ( - \beta_{8} + 3 \beta_{5} + 8 \beta_{3} - \beta_1 - 3) q^{93} + (2 \beta_{10} - 2 \beta_{9} + \beta_{4}) q^{97} + (\beta_{8} - \beta_{6} + 3 \beta_{5} + \beta_{3}) q^{99}+O(q^{100})$$ q - b2 * q^3 - b6 * q^7 + (-b8 - b6 + 4*b5 + b3) * q^9 + (-b1 + 1) * q^11 + (b11 + b7 + b2) * q^13 + (b8 + 2*b6 - b1) * q^17 + (-b7 - b6 + b5 + b3) * q^19 + (b11 - b10) * q^21 + (-2*b8 + b3 - 2*b1) * q^23 + (-b10 + b9 + 2*b4) * q^27 + (-b4 - b2) * q^29 + (b11 - b10) * q^31 + (b11 + b7) * q^33 + (b11 + b10 - b9 - b7 - b4) * q^37 + (5*b8 - b5) * q^39 + (b11 - b10 + b4 - b2) * q^41 + (2*b8 - 2*b5 - b3 + 2*b1 + 2) * q^43 + (2*b8 + 3*b6 - 2*b5 - 2*b1 - 2) * q^47 + (-2*b8 - 3*b5) * q^49 + (2*b10 - b4 + b2) * q^51 + (b11 - b10 - b9 + b7 - b2) * q^53 + (b11 - b10 + b9 - 2*b8 - b7 + b6 - 3*b5 + b4 + 2*b1 - 3) * q^57 + (b9 + 3*b7 + 3*b4 + 3*b2) * q^59 + (b6 + b3 - b1 + 1) * q^61 + (-b8 + 3*b5 + 5*b3 - b1 - 3) * q^63 + (b11 - b7 - 3*b4) * q^67 + (b9 + 3*b7 + 2*b4 + 2*b2) * q^69 + (b11 + b10) * q^71 + (b8 - 6*b5 + b1 + 6) * q^73 + (-b8 + b5 + b1 + 1) * q^77 + (b9 + b7 - b4 - b2) * q^79 + (6*b6 + 6*b3 - 5*b1 - 2) * q^81 + (b8 + 7*b5 - b3 + b1 - 7) * q^83 + (-b8 - 2*b6 + 7*b5 + b1 + 7) * q^87 + (-2*b9 - b4 - b2) * q^89 + (-3*b11 + b10 + 2*b4 - 2*b2) * q^91 + (-b8 + 3*b5 + 8*b3 - b1 - 3) * q^93 + (2*b10 - 2*b9 + b4) * q^97 + (b8 - b6 + 3*b5 + b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 4 q^{7}+O(q^{10})$$ 12 * q - 4 * q^7 $$12 q - 4 q^{7} + 8 q^{11} + 4 q^{17} - 4 q^{23} + 28 q^{43} - 20 q^{47} - 24 q^{57} + 16 q^{61} - 20 q^{63} + 76 q^{73} + 16 q^{77} + 4 q^{81} - 84 q^{83} + 80 q^{87} - 8 q^{93}+O(q^{100})$$ 12 * q - 4 * q^7 + 8 * q^11 + 4 * q^17 - 4 * q^23 + 28 * q^43 - 20 * q^47 - 24 * q^57 + 16 * q^61 - 20 * q^63 + 76 * q^73 + 16 * q^77 + 4 * q^81 - 84 * q^83 + 80 * q^87 - 8 * q^93

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} + 28 x^{10} - 64 x^{9} + 236 x^{8} - 420 x^{7} + 946 x^{6} - 1216 x^{5} + 1896 x^{4} - 1564 x^{3} + 2284 x^{2} - 1088 x + 1370$$ :

 $$\beta_{1}$$ $$=$$ $$( 43066472536 \nu^{11} - 449979675808 \nu^{10} + 2364419690616 \nu^{9} - 10578246250319 \nu^{8} + 27902022591136 \nu^{7} + \cdots - 279575999635621 ) / 79197078563257$$ (43066472536*v^11 - 449979675808*v^10 + 2364419690616*v^9 - 10578246250319*v^8 + 27902022591136*v^7 - 79408221084824*v^6 + 141181287966024*v^5 - 265116944629568*v^4 + 312068843894400*v^3 - 358597198759640*v^2 + 218631797615712*v - 279575999635621) / 79197078563257 $$\beta_{2}$$ $$=$$ $$( - 20904914650254 \nu^{11} + 151211473768106 \nu^{10} - 794968673642910 \nu^{9} + \cdots + 61\!\cdots\!02 ) / 22\!\cdots\!17$$ (-20904914650254*v^11 + 151211473768106*v^10 - 794968673642910*v^9 + 2887911715476177*v^8 - 7417029052839577*v^7 + 19179607630692878*v^6 - 33837760885089870*v^5 + 55088809656985999*v^4 - 59614149648195498*v^3 + 73129468722892382*v^2 - 13260382630146966*v + 61336751231783802) / 22254379076275217 $$\beta_{3}$$ $$=$$ $$( - 1434891834382 \nu^{11} + 4349786383365 \nu^{10} - 30559554581954 \nu^{9} + 42878862093020 \nu^{8} + \cdots - 614674855329394 ) / 11\!\cdots\!43$$ (-1434891834382*v^11 + 4349786383365*v^10 - 30559554581954*v^9 + 42878862093020*v^8 - 159886688468036*v^7 + 171437976711065*v^6 - 135293109090940*v^5 - 195667629153178*v^4 + 1037776394961430*v^3 - 1640243274648676*v^2 + 1313836759023712*v - 614674855329394) / 1171283109277643 $$\beta_{4}$$ $$=$$ $$( - 36029778947630 \nu^{11} + 211750342123647 \nu^{10} + \cdots + 82\!\cdots\!06 ) / 22\!\cdots\!17$$ (-36029778947630*v^11 + 211750342123647*v^10 - 1238018225399786*v^9 + 3905072695392607*v^8 - 11494568432177163*v^7 + 26121941475654550*v^6 - 52601844594880850*v^5 + 76369650536598931*v^4 - 107621131739087582*v^3 + 103021520567154984*v^2 - 109128561548555516*v + 82760354423821606) / 22254379076275217 $$\beta_{5}$$ $$=$$ $$( 109895826 \nu^{11} - 371224964 \nu^{10} + 2654024378 \nu^{9} - 4749330090 \nu^{8} + 18447551030 \nu^{7} - 26443913423 \nu^{6} + \cdots + 32432735911 ) / 57697329013$$ (109895826*v^11 - 371224964*v^10 + 2654024378*v^9 - 4749330090*v^8 + 18447551030*v^7 - 26443913423*v^6 + 55312820760*v^5 - 45534731950*v^4 + 73343454746*v^3 + 3299301738*v^2 + 68436366132*v + 32432735911) / 57697329013 $$\beta_{6}$$ $$=$$ $$( 46589648822398 \nu^{11} - 179791490723729 \nu^{10} + \cdots + 68\!\cdots\!68 ) / 22\!\cdots\!17$$ (46589648822398*v^11 - 179791490723729*v^10 + 1294747135943166*v^9 - 2848570771913886*v^8 + 10820535063658264*v^7 - 17972775160857487*v^6 + 39813418259552772*v^5 - 44865504993333180*v^4 + 60390304202600358*v^3 - 34043951404704188*v^2 + 31530397540338688*v + 6809468628902868) / 22254379076275217 $$\beta_{7}$$ $$=$$ $$( 176200650116 \nu^{11} - 1329132544436 \nu^{10} + 6501003581929 \nu^{9} - 24026915922758 \nu^{8} + 54318387857268 \nu^{7} + \cdots - 625898181929096 ) / 79197078563257$$ (176200650116*v^11 - 1329132544436*v^10 + 6501003581929*v^9 - 24026915922758*v^8 + 54318387857268*v^7 - 145683835248313*v^6 + 220874607342796*v^5 - 357135080326986*v^4 + 349453006431700*v^3 - 471595750135346*v^2 + 272502725402962*v - 625898181929096) / 79197078563257 $$\beta_{8}$$ $$=$$ $$( 60614681294844 \nu^{11} - 176924960309206 \nu^{10} + \cdots - 21\!\cdots\!53 ) / 22\!\cdots\!17$$ (60614681294844*v^11 - 176924960309206*v^10 + 1343717589033390*v^9 - 1909836933490731*v^8 + 8609778550923730*v^7 - 10682537136506275*v^6 + 24968890071655488*v^5 - 24287455010154182*v^4 + 49085377800415758*v^3 - 35882940036857220*v^2 + 77615330419094180*v - 21606519557536053) / 22254379076275217 $$\beta_{9}$$ $$=$$ $$( 256791075958 \nu^{11} - 2213799473497 \nu^{10} + 10545830020631 \nu^{9} - 43417051770468 \nu^{8} + 98549420880028 \nu^{7} + \cdots - 965449441292380 ) / 79197078563257$$ (256791075958*v^11 - 2213799473497*v^10 + 10545830020631*v^9 - 43417051770468*v^8 + 98549420880028*v^7 - 293506877658653*v^6 + 456518283579298*v^5 - 854726461820000*v^4 + 874080236420692*v^3 - 1149195769701478*v^2 + 642334628102864*v - 965449441292380) / 79197078563257 $$\beta_{10}$$ $$=$$ $$( - 115790335988434 \nu^{11} + 395417420255441 \nu^{10} + \cdots + 88\!\cdots\!30 ) / 22\!\cdots\!17$$ (-115790335988434*v^11 + 395417420255441*v^10 - 2966086386470279*v^9 + 5628049459277240*v^8 - 23005474619523732*v^7 + 35317879974459683*v^6 - 80577050400795482*v^5 + 98786412752558234*v^4 - 135626919464003418*v^3 + 128200897738300130*v^2 - 191498153537718348*v + 88642056768645630) / 22254379076275217 $$\beta_{11}$$ $$=$$ $$( - 120425382922218 \nu^{11} + 468739736789324 \nu^{10} + \cdots - 68\!\cdots\!02 ) / 22\!\cdots\!17$$ (-120425382922218*v^11 + 468739736789324*v^10 - 3221660843066217*v^9 + 6816331515095650*v^8 - 24346391496367094*v^7 + 38423409192569915*v^6 - 81116872823766392*v^5 + 82309718088189014*v^4 - 110128017822072330*v^3 + 32522730416457886*v^2 - 114045130802806098*v - 6834518991404602) / 22254379076275217
 $$\nu$$ $$=$$ $$( -\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2$$ (-b5 - b4 - b3 + b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 2\beta_{6} - 3\beta_{5} + 2\beta_{2} - \beta _1 - 7 ) / 2$$ (-b11 + b10 + b9 + b8 - b7 + 2*b6 - 3*b5 + 2*b2 - b1 - 7) / 2 $$\nu^{3}$$ $$=$$ $$( - \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 3 \beta_{6} + 12 \beta_{5} + 10 \beta_{4} + 15 \beta_{3} + 2 \beta_{2} - 4 \beta _1 - 18 ) / 2$$ (-b10 + 3*b9 - 2*b8 + 3*b6 + 12*b5 + 10*b4 + 15*b3 + 2*b2 - 4*b1 - 18) / 2 $$\nu^{4}$$ $$=$$ $$( 8 \beta_{11} - 16 \beta_{10} - 4 \beta_{9} - 30 \beta_{8} + 4 \beta_{7} - 37 \beta_{6} + 84 \beta_{5} + 16 \beta_{4} + 29 \beta_{3} - 24 \beta_{2} + 4 \beta _1 + 20 ) / 2$$ (8*b11 - 16*b10 - 4*b9 - 30*b8 + 4*b7 - 37*b6 + 84*b5 + 16*b4 + 29*b3 - 24*b2 + 4*b1 + 20) / 2 $$\nu^{5}$$ $$=$$ $$( 2 \beta_{11} - 10 \beta_{10} - 55 \beta_{9} - 21 \beta_{8} + 5 \beta_{7} - 150 \beta_{6} + 37 \beta_{5} - 85 \beta_{4} - 116 \beta_{3} - 105 \beta_{2} + 89 \beta _1 + 273 ) / 2$$ (2*b11 - 10*b10 - 55*b9 - 21*b8 + 5*b7 - 150*b6 + 37*b5 - 85*b4 - 116*b3 - 105*b2 + 89*b1 + 273) / 2 $$\nu^{6}$$ $$=$$ $$- 32 \beta_{11} + 99 \beta_{10} - 51 \beta_{9} + 183 \beta_{8} + 8 \beta_{7} + 99 \beta_{6} - 498 \beta_{5} - 210 \beta_{4} - 325 \beta_{3} + 60 \beta_{2} + 93 \beta _1 + 238$$ -32*b11 + 99*b10 - 51*b9 + 183*b8 + 8*b7 + 99*b6 - 498*b5 - 210*b4 - 325*b3 + 60*b2 + 93*b1 + 238 $$\nu^{7}$$ $$=$$ $$( - 100 \beta_{11} + 586 \beta_{10} + 532 \beta_{9} + 1074 \beta_{8} - 70 \beta_{7} + 2615 \beta_{6} - 2858 \beta_{5} - 20 \beta_{4} - 77 \beta_{3} + 1742 \beta_{2} - 934 \beta _1 - 2648 ) / 2$$ (-100*b11 + 586*b10 + 532*b9 + 1074*b8 - 70*b7 + 2615*b6 - 2858*b5 - 20*b4 - 77*b3 + 1742*b2 - 934*b1 - 2648) / 2 $$\nu^{8}$$ $$=$$ $$176 \beta_{11} - 616 \beta_{10} + 1388 \beta_{9} - 1120 \beta_{8} - 292 \beta_{7} + 1768 \beta_{6} + 3024 \beta_{5} + 2876 \beta_{4} + 4400 \beta_{3} + 1164 \beta_{2} - 2527 \beta _1 - 6723$$ 176*b11 - 616*b10 + 1388*b9 - 1120*b8 - 292*b7 + 1768*b6 + 3024*b5 + 2876*b4 + 4400*b3 + 1164*b2 - 2527*b1 - 6723 $$\nu^{9}$$ $$=$$ $$993 \beta_{11} - 5524 \beta_{10} - 348 \beta_{9} - 10000 \beta_{8} - 13512 \beta_{6} + 26866 \beta_{5} + 7641 \beta_{4} + 11633 \beta_{3} - 8961 \beta_{2} + 584 \beta _1 + 1838$$ 993*b11 - 5524*b10 - 348*b9 - 10000*b8 - 13512*b6 + 26866*b5 + 7641*b4 + 11633*b3 - 8961*b2 + 584*b1 + 1838 $$\nu^{10}$$ $$=$$ $$877 \beta_{11} - 5469 \beta_{10} - 19989 \beta_{9} - 9929 \beta_{8} + 4117 \beta_{7} - 57259 \beta_{6} + 26452 \beta_{5} - 21000 \beta_{4} - 32160 \beta_{3} - 37564 \beta_{2} + 36399 \beta _1 + 97170$$ 877*b11 - 5469*b10 - 19989*b9 - 9929*b8 + 4117*b7 - 57259*b6 + 26452*b5 - 21000*b4 - 32160*b3 - 37564*b2 + 36399*b1 + 97170 $$\nu^{11}$$ $$=$$ $$( - 23101 \beta_{11} + 124939 \beta_{10} - 88121 \beta_{9} + 226350 \beta_{8} + 17347 \beta_{7} + 85400 \beta_{6} - 607538 \beta_{5} - 316704 \beta_{4} - 481316 \beta_{3} + 56994 \beta_{2} + \cdots + 427540 ) / 2$$ (-23101*b11 + 124939*b10 - 88121*b9 + 226350*b8 + 17347*b7 + 85400*b6 - 607538*b5 - 316704*b4 - 481316*b3 + 56994*b2 + 160388*b1 + 427540) / 2

## Character values

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

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$\beta_{5}$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
493.1
 1.24060 − 1.01288i −0.585043 − 2.22350i 0.344446 − 1.15131i 0.344446 + 1.84020i −0.585043 + 1.05342i 1.24060 + 3.49408i 1.24060 + 1.01288i −0.585043 + 2.22350i 0.344446 + 1.15131i 0.344446 − 1.84020i −0.585043 − 1.05342i 1.24060 − 3.49408i
0 −2.25348 + 2.25348i 0 0 0 1.48119 1.48119i 0 7.15633i 0
493.2 0 −1.63846 + 1.63846i 0 0 0 −2.17009 + 2.17009i 0 2.36910i 0
493.3 0 −1.49576 + 1.49576i 0 0 0 −0.311108 + 0.311108i 0 1.47457i 0
493.4 0 1.49576 1.49576i 0 0 0 −0.311108 + 0.311108i 0 1.47457i 0
493.5 0 1.63846 1.63846i 0 0 0 −2.17009 + 2.17009i 0 2.36910i 0
493.6 0 2.25348 2.25348i 0 0 0 1.48119 1.48119i 0 7.15633i 0
1557.1 0 −2.25348 2.25348i 0 0 0 1.48119 + 1.48119i 0 7.15633i 0
1557.2 0 −1.63846 1.63846i 0 0 0 −2.17009 2.17009i 0 2.36910i 0
1557.3 0 −1.49576 1.49576i 0 0 0 −0.311108 0.311108i 0 1.47457i 0
1557.4 0 1.49576 + 1.49576i 0 0 0 −0.311108 0.311108i 0 1.47457i 0
1557.5 0 1.63846 + 1.63846i 0 0 0 −2.17009 2.17009i 0 2.36910i 0
1557.6 0 2.25348 + 2.25348i 0 0 0 1.48119 + 1.48119i 0 7.15633i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 493.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.l.b 12
5.b even 2 1 380.2.l.b 12
5.c odd 4 1 380.2.l.b 12
5.c odd 4 1 inner 1900.2.l.b 12
15.d odd 2 1 3420.2.bb.d 12
15.e even 4 1 3420.2.bb.d 12
19.b odd 2 1 inner 1900.2.l.b 12
95.d odd 2 1 380.2.l.b 12
95.g even 4 1 380.2.l.b 12
95.g even 4 1 inner 1900.2.l.b 12
285.b even 2 1 3420.2.bb.d 12
285.j odd 4 1 3420.2.bb.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.l.b 12 5.b even 2 1
380.2.l.b 12 5.c odd 4 1
380.2.l.b 12 95.d odd 2 1
380.2.l.b 12 95.g even 4 1
1900.2.l.b 12 1.a even 1 1 trivial
1900.2.l.b 12 5.c odd 4 1 inner
1900.2.l.b 12 19.b odd 2 1 inner
1900.2.l.b 12 95.g even 4 1 inner
3420.2.bb.d 12 15.d odd 2 1
3420.2.bb.d 12 15.e even 4 1
3420.2.bb.d 12 285.b even 2 1
3420.2.bb.d 12 285.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 152T_{3}^{8} + 5616T_{3}^{4} + 59536$$ acting on $$S_{2}^{\mathrm{new}}(1900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 152 T^{8} + 5616 T^{4} + \cdots + 59536$$
$5$ $$T^{12}$$
$7$ $$(T^{6} + 2 T^{5} + 2 T^{4} - 8 T^{3} + 36 T^{2} + \cdots + 8)^{2}$$
$11$ $$(T^{3} - 2 T^{2} - 4 T + 4)^{4}$$
$13$ $$T^{12} + 1816 T^{8} + 408272 T^{4} + \cdots + 59536$$
$17$ $$(T^{6} - 2 T^{5} + 2 T^{4} - 16 T^{3} + \cdots + 1352)^{2}$$
$19$ $$T^{12} - 6 T^{10} + 71 T^{8} + \cdots + 47045881$$
$23$ $$(T^{6} + 2 T^{5} + 2 T^{4} + 40 T^{3} + \cdots + 5000)^{2}$$
$29$ $$(T^{6} - 40 T^{4} + 496 T^{2} - 1952)^{2}$$
$31$ $$(T^{6} + 96 T^{4} + 2336 T^{2} + \cdots + 1952)^{2}$$
$37$ $$T^{12} + 20536 T^{8} + 54727472 T^{4} + \cdots + 59536$$
$41$ $$(T^{6} + 144 T^{4} + 2432 T^{2} + \cdots + 7808)^{2}$$
$43$ $$(T^{6} - 14 T^{5} + 98 T^{4} - 200 T^{3} + \cdots + 14792)^{2}$$
$47$ $$(T^{6} + 10 T^{5} + 50 T^{4} - 232 T^{3} + \cdots + 2312)^{2}$$
$53$ $$T^{12} + 29016 T^{8} + \cdots + 37210000$$
$59$ $$(T^{6} - 448 T^{4} + 63808 T^{2} + \cdots - 2818688)^{2}$$
$61$ $$(T^{3} - 4 T^{2} - 4 T + 20)^{4}$$
$67$ $$T^{12} + 16568 T^{8} + 33883600 T^{4} + \cdots + 59536$$
$71$ $$(T^{6} + 104 T^{4} + 1088 T^{2} + \cdots + 1952)^{2}$$
$73$ $$(T^{6} - 38 T^{5} + 722 T^{4} + \cdots + 390728)^{2}$$
$79$ $$(T^{6} - 184 T^{4} + 4032 T^{2} + \cdots - 7808)^{2}$$
$83$ $$(T^{6} + 42 T^{5} + 882 T^{4} + \cdots + 803912)^{2}$$
$89$ $$(T^{6} - 288 T^{4} + 18960 T^{2} + \cdots - 329888)^{2}$$
$97$ $$T^{12} + 140856 T^{8} + \cdots + 46306881767056$$