# Properties

 Label 740.2.e.a Level $740$ Weight $2$ Character orbit 740.e Analytic conductor $5.909$ Analytic rank $0$ Dimension $20$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(369,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("740.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.e (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: $$5.90892974957$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} + 4 x^{18} - 31 x^{16} - 216 x^{14} + 470 x^{12} + 6696 x^{10} + 11750 x^{8} - 135000 x^{6} + \cdots + 9765625$$ x^20 + 4*x^18 - 31*x^16 - 216*x^14 + 470*x^12 + 6696*x^10 + 11750*x^8 - 135000*x^6 - 484375*x^4 + 1562500*x^2 + 9765625 Coefficient ring: $$\Z[a_1, \ldots, a_{37}]$$ Coefficient ring index: $$2^{13}$$ 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 $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + \beta_1 q^{5} - \beta_{4} q^{7} + ( - \beta_{14} - 1) q^{9}+O(q^{10})$$ q + b3 * q^3 + b1 * q^5 - b4 * q^7 + (-b14 - 1) * q^9 $$q + \beta_{3} q^{3} + \beta_1 q^{5} - \beta_{4} q^{7} + ( - \beta_{14} - 1) q^{9} + \beta_{8} q^{11} - \beta_{16} q^{13} + \beta_{11} q^{15} + \beta_{15} q^{17} - \beta_{19} q^{19} + (\beta_{18} - \beta_{2}) q^{21} + (\beta_{17} + \beta_{11} + \cdots + \beta_1) q^{23}+ \cdots + (2 \beta_{14} + \beta_{12} + \beta_{2} + 2) q^{99}+O(q^{100})$$ q + b3 * q^3 + b1 * q^5 - b4 * q^7 + (-b14 - 1) * q^9 + b8 * q^11 - b16 * q^13 + b11 * q^15 + b15 * q^17 - b19 * q^19 + (b18 - b2) * q^21 + (b17 + b11 - b10 + b1) * q^23 + b2 * q^25 + (-b5 - b3) * q^27 + (-b9 + b7) * q^29 + (-b19 + b9) * q^31 + (-b13 - b12 - b11 + b10 + 2*b6 - b5 + b4 + b2 + b1) * q^33 + (b16 + b15 + b13 - b9) * q^35 + (-b17 - b13 - b11 + b10 + b6 + b4 + b3) * q^37 + (2*b19 - b13 - b11 + b10 - b9 + b1) * q^39 + (b18 + b14 + b12 + b8) * q^41 + (b17 + b16 + b15 + b13 - b10 + b1) * q^43 + (b19 - b17 - b16 - b13 - b11 - b9 + b7 - b1) * q^45 + (b13 + b12 + b11 - b10 - 2*b6 - b4 - b2 - b1) * q^47 + (-b14 - 1) * q^49 + (-b13 - b11 - b10 + b7 - b1) * q^51 + (b12 - b5 + b4 - b2) * q^53 + (b19 - b17 - b16 - b15 + 2*b10 - b9 - b7) * q^55 + (-2*b16 - b15 - b13 + b11 - b10 + b1) * q^57 + (-b19 + 2*b9 + b7) * q^59 + (-b13 - b11 - b10 + b9 + b7 - b1) * q^61 + (-b13 - b12 - b11 + b10 + 2*b6 + b5 + 3*b4 + b3 + b2 + b1) * q^63 + (-b14 - b12 - b5 - b4 - 1) * q^65 + (-b12 + b5 + b4 - b3 + b2) * q^67 + (2*b19 + b13 + b11 - b10 - 3*b9 + b7 - b1) * q^69 + (-b12 - b8 - b2 + 2) * q^71 + (b12 + b5 - b4 + 2*b3 - b2) * q^73 + (b13 + 2*b12 + b11 - b10 - 2*b6 - 2*b4 - b3 - b2 - b1 - 1) * q^75 + (-b13 - 2*b12 - b11 + b10 + 2*b6 - b5 + b4 + 2*b2 + b1) * q^77 + (-b19 + 2*b13 + 2*b11 - b10 + b9 - 2*b7 - b1) * q^79 + (b18 - b8 - b2 + 1) * q^81 + (-b13 - b11 + b10 + 2*b6 + b3 + b1) * q^83 + (-b14 + b13 + b11 - b10 - b8 - 2*b6 - 2*b3 - b2 - b1 - 1) * q^85 + (-2*b17 - b16 - b15 - 2*b11 + 2*b10 - 2*b1) * q^87 + (-b13 - b11 + 2*b10 - b9 - b7 + 2*b1) * q^89 + (2*b10 + b9 - 2*b7 + 2*b1) * q^91 + (b17 - 2*b16 - b15 - b13 + 2*b11 - b10 + b1) * q^93 + (-b18 - b14 - b12 + b5 - b4 - b3 + b2 - 1) * q^95 + (b17 + 2*b16 + b11 - b10 + b1) * q^97 + (2*b14 + b12 + b2 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q - 28 q^{9}+O(q^{10})$$ 20 * q - 28 * q^9 $$20 q - 28 q^{9} - 8 q^{25} - 8 q^{41} - 28 q^{49} - 20 q^{65} + 56 q^{71} - 28 q^{75} + 20 q^{81} - 20 q^{85} - 20 q^{95} + 40 q^{99}+O(q^{100})$$ 20 * q - 28 * q^9 - 8 * q^25 - 8 * q^41 - 28 * q^49 - 20 * q^65 + 56 * q^71 - 28 * q^75 + 20 * q^81 - 20 * q^85 - 20 * q^95 + 40 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 4 x^{18} - 31 x^{16} - 216 x^{14} + 470 x^{12} + 6696 x^{10} + 11750 x^{8} - 135000 x^{6} + \cdots + 9765625$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$( \nu^{18} - 7 \nu^{16} - 50 \nu^{14} - 50 \nu^{12} + 1596 \nu^{10} + 276 \nu^{8} - 22006 \nu^{6} + \cdots + 2376875 ) / 1920000$$ (v^18 - 7*v^16 - 50*v^14 - 50*v^12 + 1596*v^10 + 276*v^8 - 22006*v^6 - 257350*v^4 + 450475*v^2 + 2376875) / 1920000 $$\beta_{4}$$ $$=$$ $$( - 301 \nu^{18} - 2029 \nu^{16} + 32906 \nu^{14} + 57466 \nu^{12} - 187020 \nu^{10} + \cdots - 1427734375 ) / 400000000$$ (-301*v^18 - 2029*v^16 + 32906*v^14 + 57466*v^12 - 187020*v^10 - 2676996*v^8 + 351550*v^6 + 54958750*v^4 + 251640625*v^2 - 1427734375) / 400000000 $$\beta_{5}$$ $$=$$ $$( - 59 \nu^{18} - 667 \nu^{16} + 5830 \nu^{14} + 4630 \nu^{12} - 86484 \nu^{10} - 755484 \nu^{8} + \cdots - 497640625 ) / 48000000$$ (-59*v^18 - 667*v^16 + 5830*v^14 + 4630*v^12 - 86484*v^10 - 755484*v^8 + 1540274*v^6 + 18472850*v^4 + 31084375*v^2 - 497640625) / 48000000 $$\beta_{6}$$ $$=$$ $$( - 375 \nu^{19} + 956 \nu^{18} - 3375 \nu^{17} + 1924 \nu^{16} - 5250 \nu^{15} + \cdots + 1192187500 ) / 1200000000$$ (-375*v^19 + 956*v^18 - 3375*v^17 + 1924*v^16 - 5250*v^15 - 39736*v^14 + 54750*v^13 + 154904*v^12 + 97500*v^11 + 624720*v^10 - 2023500*v^9 - 2076624*v^8 - 14523750*v^7 + 7960600*v^6 - 21993750*v^5 + 38125000*v^4 + 71671875*v^3 - 262187500*v^2 - 227578125*v + 1192187500) / 1200000000 $$\beta_{7}$$ $$=$$ $$( - 2 \nu^{19} - 14 \nu^{17} + 63 \nu^{15} + 93 \nu^{13} - 2919 \nu^{11} - 2237 \nu^{9} + \cdots - 2265625 \nu ) / 5000000$$ (-2*v^19 - 14*v^17 + 63*v^15 + 93*v^13 - 2919*v^11 - 2237*v^9 + 4949*v^7 - 161225*v^5 + 153125*v^3 - 2265625*v) / 5000000 $$\beta_{8}$$ $$=$$ $$( - 108 \nu^{18} + 643 \nu^{16} + 3898 \nu^{14} - 9372 \nu^{12} - 104210 \nu^{10} + \cdots - 208203125 ) / 50000000$$ (-108*v^18 + 643*v^16 + 3898*v^14 - 9372*v^12 - 104210*v^10 - 282918*v^8 + 1572950*v^6 + 14132500*v^4 - 45531250*v^2 - 208203125) / 50000000 $$\beta_{9}$$ $$=$$ $$( 481 \nu^{19} - 2551 \nu^{17} - 12186 \nu^{15} + 70454 \nu^{13} + 756420 \nu^{11} + \cdots + 1926171875 \nu ) / 1000000000$$ (481*v^19 - 2551*v^17 - 12186*v^15 + 70454*v^13 + 756420*v^11 + 1631276*v^9 - 14650350*v^7 - 105348750*v^5 + 151171875*v^3 + 1926171875*v) / 1000000000 $$\beta_{10}$$ $$=$$ $$( \nu^{19} + 4 \nu^{17} - 31 \nu^{15} - 216 \nu^{13} + 470 \nu^{11} + 6696 \nu^{9} + \cdots + 1562500 \nu ) / 1953125$$ (v^19 + 4*v^17 - 31*v^15 - 216*v^13 + 470*v^11 + 6696*v^9 + 11750*v^7 - 135000*v^5 - 484375*v^3 + 1562500*v) / 1953125 $$\beta_{11}$$ $$=$$ $$( \nu^{19} - 7 \nu^{17} - 50 \nu^{15} - 50 \nu^{13} + 1596 \nu^{11} + 276 \nu^{9} + \cdots + 2376875 \nu ) / 1920000$$ (v^19 - 7*v^17 - 50*v^15 - 50*v^13 + 1596*v^11 + 276*v^9 - 22006*v^7 - 257350*v^5 + 450475*v^3 + 2376875*v) / 1920000 $$\beta_{12}$$ $$=$$ $$( - \nu^{18} - 4 \nu^{16} + 31 \nu^{14} + 216 \nu^{12} - 470 \nu^{10} - 6696 \nu^{8} + \cdots - 1562500 ) / 390625$$ (-v^18 - 4*v^16 + 31*v^14 + 216*v^12 - 470*v^10 - 6696*v^8 - 11750*v^6 + 135000*v^4 + 484375*v^2 - 1562500) / 390625 $$\beta_{13}$$ $$=$$ $$( - 3803 \nu^{19} + 413 \nu^{17} + 8518 \nu^{15} + 40198 \nu^{13} - 2568660 \nu^{11} + \cdots + 1096484375 \nu ) / 6000000000$$ (-3803*v^19 + 413*v^17 + 8518*v^15 + 40198*v^13 - 2568660*v^11 - 527388*v^9 - 40372750*v^7 + 169561250*v^5 - 2179015625*v^3 + 1096484375*v) / 6000000000 $$\beta_{14}$$ $$=$$ $$( 12 \nu^{18} - 27 \nu^{16} - 522 \nu^{14} - 292 \nu^{12} + 14690 \nu^{10} + 47702 \nu^{8} + \cdots + 28828125 ) / 4000000$$ (12*v^18 - 27*v^16 - 522*v^14 - 292*v^12 + 14690*v^10 + 47702*v^8 - 186950*v^6 - 1478100*v^4 + 1421250*v^2 + 28828125) / 4000000 $$\beta_{15}$$ $$=$$ $$( 871 \nu^{19} + 9 \nu^{17} - 29026 \nu^{15} - 17286 \nu^{13} + 463720 \nu^{11} + \cdots + 883984375 \nu ) / 1000000000$$ (871*v^19 + 9*v^17 - 29026*v^15 - 17286*v^13 + 463720*v^11 + 1540216*v^9 + 7140650*v^7 - 92651250*v^5 - 130484375*v^3 + 883984375*v) / 1000000000 $$\beta_{16}$$ $$=$$ $$( 2989 \nu^{19} + 7331 \nu^{17} - 200534 \nu^{15} - 187874 \nu^{13} + 3565080 \nu^{11} + \cdots + 13286328125 \nu ) / 3000000000$$ (2989*v^19 + 7331*v^17 - 200534*v^15 - 187874*v^13 + 3565080*v^11 + 20614344*v^9 - 47823250*v^7 - 535063750*v^5 + 47171875*v^3 + 13286328125*v) / 3000000000 $$\beta_{17}$$ $$=$$ $$( - 7709 \nu^{19} - 23461 \nu^{17} + 325354 \nu^{15} + 648394 \nu^{13} - 3619980 \nu^{11} + \cdots - 21180859375 \nu ) / 6000000000$$ (-7709*v^19 - 23461*v^17 + 325354*v^15 + 648394*v^13 - 3619980*v^11 - 24656964*v^9 + 58814750*v^7 + 1072268750*v^5 + 1176390625*v^3 - 21180859375*v) / 6000000000 $$\beta_{18}$$ $$=$$ $$( 212 \nu^{18} + 43 \nu^{16} - 5542 \nu^{14} - 13212 \nu^{12} + 104270 \nu^{10} + 648202 \nu^{8} + \cdots + 247421875 ) / 20000000$$ (212*v^18 + 43*v^16 - 5542*v^14 - 13212*v^12 + 104270*v^10 + 648202*v^8 - 595530*v^6 - 19339500*v^4 + 12368750*v^2 + 247421875) / 20000000 $$\beta_{19}$$ $$=$$ $$( - 5041 \nu^{19} + 3911 \nu^{17} + 136946 \nu^{15} + 301906 \nu^{13} - 2688220 \nu^{11} + \cdots - 5016796875 \nu ) / 2000000000$$ (-5041*v^19 + 3911*v^17 + 136946*v^15 + 301906*v^13 - 2688220*v^11 - 14058036*v^9 + 13536950*v^7 + 382003750*v^5 - 303046875*v^3 - 5016796875*v) / 2000000000
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\nu^{3}$$ $$=$$ $$\beta_{16} - \beta_{15} - 2\beta_{13} - \beta_{10} - \beta_{9} + \beta_{7}$$ b16 - b15 - 2*b13 - b10 - b9 + b7 $$\nu^{4}$$ $$=$$ $$4\beta_{14} + \beta_{12} + 2\beta_{8} + 2\beta_{5} - 2\beta_{4} - 8\beta_{3} + \beta_{2} + 7$$ 4*b14 + b12 + 2*b8 + 2*b5 - 2*b4 - 8*b3 + b2 + 7 $$\nu^{5}$$ $$=$$ $$- 4 \beta_{19} + 4 \beta_{17} + \beta_{16} - \beta_{15} + 4 \beta_{13} - 2 \beta_{11} - 4 \beta_{10} + \cdots + 9 \beta_1$$ -4*b19 + 4*b17 + b16 - b15 + 4*b13 - 2*b11 - 4*b10 - 3*b9 - 5*b7 + 9*b1 $$\nu^{6}$$ $$=$$ $$- 8 \beta_{18} - 8 \beta_{14} - 12 \beta_{13} - 23 \beta_{12} - 12 \beta_{11} + 12 \beta_{10} + \cdots + 20$$ -8*b18 - 8*b14 - 12*b13 - 23*b12 - 12*b11 + 12*b10 - 16*b8 + 24*b6 + 4*b5 + 4*b4 + 8*b3 + 20*b2 + 12*b1 + 20 $$\nu^{7}$$ $$=$$ $$12 \beta_{19} + 28 \beta_{17} + 16 \beta_{16} + 48 \beta_{15} - 24 \beta_{13} + 15 \beta_{10} + \cdots + 52 \beta_1$$ 12*b19 + 28*b17 + 16*b16 + 48*b15 - 24*b13 + 15*b10 - 20*b9 + 12*b7 + 52*b1 $$\nu^{8}$$ $$=$$ $$- 16 \beta_{18} + 32 \beta_{14} + 32 \beta_{13} - 36 \beta_{12} + 32 \beta_{11} - 32 \beta_{10} + \cdots - 143$$ -16*b18 + 32*b14 + 32*b13 - 36*b12 + 32*b11 - 32*b10 - 72*b8 - 64*b6 - 4*b5 + 4*b4 - 240*b3 + 20*b2 - 32*b1 - 143 $$\nu^{9}$$ $$=$$ $$- 52 \beta_{19} + 132 \beta_{17} + 128 \beta_{16} - 64 \beta_{15} - 12 \beta_{13} - 164 \beta_{11} + \cdots - 147 \beta_1$$ -52*b19 + 132*b17 + 128*b16 - 64*b15 - 12*b13 - 164*b11 + 100*b10 + 76*b9 + 108*b7 - 147*b1 $$\nu^{10}$$ $$=$$ $$- 184 \beta_{18} + 488 \beta_{14} - 252 \beta_{13} - 332 \beta_{12} - 252 \beta_{11} + 252 \beta_{10} + \cdots - 644$$ -184*b18 + 488*b14 - 252*b13 - 332*b12 - 252*b11 + 252*b10 + 224*b8 + 504*b6 - 28*b5 + 580*b4 + 232*b3 + 265*b2 + 252*b1 - 644 $$\nu^{11}$$ $$=$$ $$316 \beta_{19} + 604 \beta_{17} + 5 \beta_{16} + 123 \beta_{15} - 2 \beta_{13} + 624 \beta_{11} + \cdots - 52 \beta_1$$ 316*b19 + 604*b17 + 5*b16 + 123*b15 - 2*b13 + 624*b11 + 931*b10 + 423*b9 - 1135*b7 - 52*b1 $$\nu^{12}$$ $$=$$ $$- 288 \beta_{18} - 1468 \beta_{14} - 992 \beta_{13} - 351 \beta_{12} - 992 \beta_{11} + 992 \beta_{10} + \cdots - 5577$$ -288*b18 - 1468*b14 - 992*b13 - 351*b12 - 992*b11 + 992*b10 - 2574*b8 + 1984*b6 - 1338*b5 + 1658*b4 - 1880*b3 - 511*b2 + 992*b1 - 5577 $$\nu^{13}$$ $$=$$ $$1096 \beta_{19} + 344 \beta_{17} + 1613 \beta_{16} + 5683 \beta_{15} - 536 \beta_{13} + \cdots - 5347 \beta_1$$ 1096*b19 + 344*b17 + 1613*b16 + 5683*b15 - 536*b13 - 5390*b11 - 4808*b10 + 4325*b9 + 1259*b7 - 5347*b1 $$\nu^{14}$$ $$=$$ $$752 \beta_{18} - 112 \beta_{14} + 1208 \beta_{13} - 3175 \beta_{12} + 1208 \beta_{11} - 1208 \beta_{10} + \cdots + 20184$$ 752*b18 - 112*b14 + 1208*b13 - 3175*b12 + 1208*b11 - 1208*b10 + 816*b8 - 2416*b6 - 4152*b5 + 21224*b4 - 2320*b3 - 6392*b2 - 1208*b1 + 20184 $$\nu^{15}$$ $$=$$ $$2824 \beta_{19} - 6584 \beta_{17} - 20240 \beta_{16} - 21232 \beta_{15} - 11264 \beta_{13} + \cdots + 13320 \beta_1$$ 2824*b19 - 6584*b17 - 20240*b16 - 21232*b15 - 11264*b13 - 6128*b11 + 32427*b10 + 31592*b9 - 3928*b7 + 13320*b1 $$\nu^{16}$$ $$=$$ $$9408 \beta_{18} - 10624 \beta_{14} - 24704 \beta_{13} + 11160 \beta_{12} - 24704 \beta_{11} + \cdots - 166687$$ 9408*b18 - 10624*b14 - 24704*b13 + 11160*b12 - 24704*b11 + 24704*b10 + 51552*b8 + 49408*b6 - 48072*b5 + 55368*b4 + 29344*b3 + 31272*b2 + 24704*b1 - 166687 $$\nu^{17}$$ $$=$$ $$82024 \beta_{19} - 129064 \beta_{17} + 9872 \beta_{16} - 48784 \beta_{15} - 122648 \beta_{13} + \cdots - 208871 \beta_1$$ 82024*b19 - 129064*b17 + 9872*b16 - 48784*b15 - 122648*b13 - 63464*b11 - 600*b10 + 19400*b9 - 49656*b7 - 208871*b1 $$\nu^{18}$$ $$=$$ $$211088 \beta_{18} - 87696 \beta_{14} - 32840 \beta_{13} + 192840 \beta_{12} - 32840 \beta_{11} + \cdots + 495528$$ 211088*b18 - 87696*b14 - 32840*b13 + 192840*b12 - 32840*b11 + 32840*b10 + 97936*b8 + 65680*b6 + 37512*b5 + 178216*b4 - 271376*b3 - 307711*b2 + 32840*b1 + 495528 $$\nu^{19}$$ $$=$$ $$- 485144 \beta_{19} - 570296 \beta_{17} - 746583 \beta_{16} - 48169 \beta_{15} - 39826 \beta_{13} + \cdots + 143704 \beta_1$$ -485144*b19 - 570296*b17 - 746583*b16 - 48169*b15 - 39826*b13 - 565488*b11 - 385561*b10 + 473871*b9 - 172543*b7 + 143704*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/740\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$371$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
369.1
 −0.419357 + 2.19639i 0.419357 − 2.19639i −2.21322 − 0.318824i 2.21322 + 0.318824i −0.609336 + 2.15144i 0.609336 − 2.15144i −2.11814 − 0.716581i 2.11814 + 0.716581i −1.25219 + 1.85257i 1.25219 − 1.85257i −1.25219 − 1.85257i 1.25219 + 1.85257i −2.11814 + 0.716581i 2.11814 − 0.716581i −0.609336 − 2.15144i 0.609336 + 2.15144i −2.21322 + 0.318824i 2.21322 − 0.318824i −0.419357 − 2.19639i 0.419357 + 2.19639i
0 3.06614i 0 −0.419357 + 2.19639i 0 3.66077i 0 −6.40124 0
369.2 0 3.06614i 0 0.419357 2.19639i 0 3.66077i 0 −6.40124 0
369.3 0 2.69030i 0 −2.21322 0.318824i 0 3.35227i 0 −4.23773 0
369.4 0 2.69030i 0 2.21322 + 0.318824i 0 3.35227i 0 −4.23773 0
369.5 0 2.00519i 0 −0.609336 + 2.15144i 0 2.83210i 0 −1.02079 0
369.6 0 2.00519i 0 0.609336 2.15144i 0 2.83210i 0 −1.02079 0
369.7 0 0.979375i 0 −2.11814 0.716581i 0 2.22692i 0 2.04083 0
369.8 0 0.979375i 0 2.11814 + 0.716581i 0 2.22692i 0 2.04083 0
369.9 0 0.617308i 0 −1.25219 + 1.85257i 0 2.09310i 0 2.61893 0
369.10 0 0.617308i 0 1.25219 1.85257i 0 2.09310i 0 2.61893 0
369.11 0 0.617308i 0 −1.25219 1.85257i 0 2.09310i 0 2.61893 0
369.12 0 0.617308i 0 1.25219 + 1.85257i 0 2.09310i 0 2.61893 0
369.13 0 0.979375i 0 −2.11814 + 0.716581i 0 2.22692i 0 2.04083 0
369.14 0 0.979375i 0 2.11814 0.716581i 0 2.22692i 0 2.04083 0
369.15 0 2.00519i 0 −0.609336 2.15144i 0 2.83210i 0 −1.02079 0
369.16 0 2.00519i 0 0.609336 + 2.15144i 0 2.83210i 0 −1.02079 0
369.17 0 2.69030i 0 −2.21322 + 0.318824i 0 3.35227i 0 −4.23773 0
369.18 0 2.69030i 0 2.21322 0.318824i 0 3.35227i 0 −4.23773 0
369.19 0 3.06614i 0 −0.419357 2.19639i 0 3.66077i 0 −6.40124 0
369.20 0 3.06614i 0 0.419357 + 2.19639i 0 3.66077i 0 −6.40124 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 369.20 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.b even 2 1 inner
37.b even 2 1 inner
185.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.e.a 20
3.b odd 2 1 6660.2.i.b 20
5.b even 2 1 inner 740.2.e.a 20
5.c odd 4 2 3700.2.h.h 20
15.d odd 2 1 6660.2.i.b 20
37.b even 2 1 inner 740.2.e.a 20
111.d odd 2 1 6660.2.i.b 20
185.d even 2 1 inner 740.2.e.a 20
185.h odd 4 2 3700.2.h.h 20
555.b odd 2 1 6660.2.i.b 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.e.a 20 1.a even 1 1 trivial
740.2.e.a 20 5.b even 2 1 inner
740.2.e.a 20 37.b even 2 1 inner
740.2.e.a 20 185.d even 2 1 inner
3700.2.h.h 20 5.c odd 4 2
3700.2.h.h 20 185.h odd 4 2
6660.2.i.b 20 3.b odd 2 1
6660.2.i.b 20 15.d odd 2 1
6660.2.i.b 20 111.d odd 2 1
6660.2.i.b 20 555.b odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(740, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$(T^{10} + 22 T^{8} + \cdots + 100)^{2}$$
$5$ $$T^{20} + 4 T^{18} + \cdots + 9765625$$
$7$ $$(T^{10} + 42 T^{8} + \cdots + 26244)^{2}$$
$11$ $$(T^{5} - 37 T^{3} + \cdots + 96)^{4}$$
$13$ $$(T^{10} - 76 T^{8} + \cdots - 87616)^{2}$$
$17$ $$(T^{10} - 82 T^{8} + \cdots - 129600)^{2}$$
$19$ $$(T^{10} + 88 T^{8} + \cdots + 144)^{2}$$
$23$ $$(T^{10} - 142 T^{8} + \cdots - 20736)^{2}$$
$29$ $$(T^{10} + 146 T^{8} + \cdots + 10445824)^{2}$$
$31$ $$(T^{10} + 104 T^{8} + \cdots + 11664)^{2}$$
$37$ $$T^{20} + \cdots + 48\!\cdots\!49$$
$41$ $$(T^{5} + 2 T^{4} + \cdots + 2040)^{4}$$
$43$ $$(T^{10} - 174 T^{8} + \cdots - 473344)^{2}$$
$47$ $$(T^{10} + 250 T^{8} + \cdots + 109202500)^{2}$$
$53$ $$(T^{10} + 272 T^{8} + \cdots + 640000)^{2}$$
$59$ $$(T^{10} + 454 T^{8} + \cdots + 678498304)^{2}$$
$61$ $$(T^{10} + 282 T^{8} + \cdots + 291999744)^{2}$$
$67$ $$(T^{10} + 394 T^{8} + \cdots + 1458017856)^{2}$$
$71$ $$(T^{5} - 14 T^{4} + \cdots - 11808)^{4}$$
$73$ $$(T^{10} + 408 T^{8} + \cdots + 278784)^{2}$$
$79$ $$(T^{10} + 702 T^{8} + \cdots + 4859205264)^{2}$$
$83$ $$(T^{10} + 334 T^{8} + \cdots + 51076)^{2}$$
$89$ $$(T^{10} + 688 T^{8} + \cdots + 8097120256)^{2}$$
$97$ $$(T^{10} - 390 T^{8} + \cdots - 92416)^{2}$$